Theory of Games and Economic Behavior: 60th Anniversary Commemorative Edition [60th Anniversary Commemorative ed.] 9781400829460

Table of contents :
Contents
Introduction
CONTENTS
Preface to First Edition. Preface to Second Edition
Preface to Third Edition
Technical Note
Acknowledgment
Chapter I. Formulation of the Economic Problem
Chapter II. General Formal Description of Games of Strategy
Chapter III. Zero-Sum Two-Person Games: Theory
Chapter IV. Zero-Sum Two-Person Games: Examples
Chapter V. Zero-Sum Three-Person Games
Chapter VI. Formulation of the General Theory, Zero-Sum n-Person Games
Chapter VII. Zero-Sum Four-Person Games
Chapter VIII. Some Remarks Concerning n≧ 5 Participants
Chapter IX. Composition and Decomposition of Games
Chapter X. Simple Games
Chapter XI. General Non-Zero-Sum Games
Chapter XII. Extension of the Concepts of Domination and Solution
Appendix: The Axiomatic Treatment of Utility
Afterword
Reviews
Heads, I Win, and Tails, You Lose
Big D
Mathematics Of Games And Economics
Theory Of Games
Mathematical Theory of Poker is Applied to Business Problems
A Theory of Strategy
The Collaboration between Oskar Morgenstern and John von Neumann on the Theory of Games
Index
CREDITS

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Theory of Games and Economic Behavior

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Theory of Games and Economic Behavior John von Neumann and Oskar Morgenstern With an introduction by H a rold W. Kuhn a nd a n afte rword by Ariel Rubin stein

S I X TI ET H -ANN I V ER SA RY EDI T I ON

PR I N CET O N UN I VE R S IT Y P RESS

Princeton and Oxford

Copyrig ht © 1944 by Pri ncelOn Un i\'crsi ty Press C..op)Tight © re newed 1972 Pri nccton Univcrsity Press Sixtieth-Anniversary Edition copyright © 2004 Prince ton Uni ve rsity Press Publi$ hed by Pri ncelOn University i'rC$s, 41 William Street, Pri ncelOn , New J crsey 08540 In the Uni ted Kingdom: PrincelOn Uni versity Press, 3 Market Place, Woodstock, Oxfordshirc OX20 IS\' All Ri ghts Rcscl"\'cd Second Edition. 1947 T hird Editio n. 1953 Fourth printing. and fi rst paperback printing. Or SiXlicth An n iversary Edit io n. 2007 Papcrback ISBN-1 3: 978-0-69 1- 1306 1-3 Papcrback ISBN-IO: 0-69 1-1306 1-2 Librr.ts that von Neumann 's theOl), was 100 math ematical for the econom ists. To illustrate the attitude of a lypical economics d epartment of the period and later, more than fifteen years after the publication of T GE B the economists at Pri nceton voted against instituting a mathematics requiremcnt fo r undergraduate majors, choosing to rim t\\'o tracks fo r stude nts, one which used the calcul us and o ne which avoidcd it. Ri chard Lester, who ahernated with Lester Cha ndl er as chairman of the department, had carried on a running debate with Fritz Machlup o\'er the valid ity of marginal product (a calcu lus notion) as a determinant of wages. Cou rses that lIsed mathematical terms and which cm'ered mathe matical topics Stich as linear programming were concealed by titles such as "Managerial theory of the fi nn. " Given such prevai ling views, there was no incentive or opportunity for graduate slllde n ts andjunior facul ty to study the theory of games. As a consequence, the theory o f games was d evelo ped almost excl usively by mathemalicians in Ihis period. To describe the spirit of the ti me as see n by another outside obser.'er, we shall paraphrase a sec tion of Robe rtJ. Aumann's magnifice nt a rticle on game them), from The New Palgrave Dictionary oj t'conomics 114]. The pe riod of the late '40s a nd early '50s was a period of excite ment in game theory. The discipline had broke n out of its cocoon and was testing iL v (read : u is preferable to v), a nd the "natu ral" operation aU (1 - a )v, (0 < a < 1), (read: center of gravity of u, v with the respective weights a, 1 - a; or : combina· t ion of u, v with the alternative probabilities a, 1 - a). If t he existenceand reproducible observability --of these concepts is conceded, then our way is clear: We must find a correspondence between utilities and numbers which carries the relation u > v and the operation aU + (1 - a)v for utilities into the sy nonymous concepts for numbers. Denote the correspondence by

+

U_p = v(u),

u being the utility and v(u) t he number which the correspondence attaches to it. Our requirements are then: (3, L a) (3, U )

U

v(au

>v

+ (1 -

implies v(u) > v(v), a)v) = av{u) + (i - a)v(v).!

If two such correspondences

(3, ",") (3,2,b)

u_p=v(u), p' = v'(u),

U _

should exist, then t hey set up a correspondence oetween numbers (3,3)

p!:::> p',

for wh ich we may also wri te ~~

(-.W

Since (3:2:a), (3: 2:b) fulfill (3 :l:a), (3:l:b), t he correspondence (3:3), i. e. the function .p(p) in (3:4) must leave t he relation p > u 'and the operation 'Observe that in in each case the leftrhand side has the "natural" concepts fo r utilities, and the right-hand side the conventional ones for numbers. t Now these are applied to numbers p, ",[

T H E NOT ION OF UTI LITY ap

+ (I

(H,.) (3,5,b)

- a)cr unaffected (d footnote 1 on p. 2-1). p ,p (a p

> cr

+ (1 -

25

I.e.

implies ,pep) > ,p{cr), a)cr) = a~(p) + (1 - a),p(cr).

Hencc ,pep) must be a linear functio n, Le. where 1010, WI are fixed numbers (constants) with 1010 > O. So we see: If such a numerical valuation of utilities' exists a t all, t hen it is determined up t o a. linear tra.nsformati on.2.~ I.e. then utility is a num ber up to a linear transformation. I n order t hat a nu merical valuation in the above sense should exist it is necessary to postulate certain propcrties of t he relation u > v and t he operation au + (1 - a)v for utilities. T he selection of these postulates or axioms and their subsequent analysis leads to problems of a certain mathematical interest. In what follows we gi\'e a general outline of the situation for the orientation of the reader; a complete discussion is found in the J\ ppend ix. 3.6.2. A choice of axioms is not a purcly objective task. It is usually expected to achieve some definite aim- some specific t heorem or theorems are to be derivable from the axioms- and to this extcnt t he problem is exact and objective. But beyond t his t here are always other impo rta nt desiderata of a less exact nature: T he axioms shou ld not be too numerous, their system is to be as simple a nd transpa rent as possible, and each axiom should ha\'e an immediate intuitive meaning by which its appropriatenes.'l may be judged directly.' In a situation like ours th is last requirement is par ticularl), vital , in spite of its mgueness: we want to make an intuitive concept amenable to mathematical t reatment and 1.0 see as clearly as possible what hypotheses this requ ires. T he objective part of our problem is clear: t he postulates muat imply the ex istence of a correspondence (3:2:a) with the properties (3: 1:a), (3: I :b) as described in 3.5. 1. T he further heuristic, and even esthetic desiderata, ind icated above, do not determine a unique way of fi nding th is ax iomatic treatment. I n what fo llows we shall formulate a set of axioms which seems to be essent ially satisfactory. , I.e. a correspondence (3:2:a) which fulfills (3:1 :a), (3:1 :b). • I.e. one of the form (3:6). • Remember the physical examplee of the lAme situation given in 3.4.4. (Our preteot discussio n is somewhat more de tailed.) We do not undertake to fix an absolute tero and a n absolute unit of utility. • T he first lIond the last principle mlly represent-at least to a certain extellt- "pposite influences: If we reduce the numbe r of axioms by mergi ng them as fa r as te v, and fo r any number a, (0 < a < I), an operation au

+ (I

- a)v = w.

These concepts satisfy the following axioms: (3:A) u > v i8 a complete ordering of U.2 This means: Write u < v when v> u. Then:

(3,A,.)

For any two u, v one and only one of the t hree following relations holds:

> v, imply u > w. J

u = v,

(3,A,b) (3,B) (3,B,,) (3,B,b) (3,B,c)

u

>

w

>

(3,C,,) (3,C,b)

< v.

+ (1

- a}v

< w.

v implies t he existence of an a with au

(3,C)

u

u > v, v > w Ordering and combining. ~ u < v implies that u < au + (1 - a)v. n > v implies that u > au + (1 - a)v. u < w < v implies the existence of an a with au

(3,B,d)

u

+ (I

- a)v

> w.

Algebra of combining. a(pu

where

a u + ( I - a)v = (l - a)1.I + au. + (1 - 13)1.1) + ( I - a)v = 1'u + (1 l' =

-y)v

ap.

One can show that t hese axioms imply the exi;;tence of a correspondence (3:2:a) with the properties (3 :l:a), (3: I: b) as described in 3.5. 1. Hence the conclusions of 3.5. 1. hold good: The system U- i. e. in our present in terpretation, t he system of (abstract) utilities-is one of numbers up to a linear t ransformation. The construction of (3:2:&) (with (3: I:a), (3: I:b) by means of the axioms (3:A)-(3:C» is a purely mathematical task which is somewhat lengthy, although it TUns along conventional lines and presents no par, This is, of course, meant to be the system of (abstract) utilities, to be characterized by our axiom8. Concerning the general nature of the axiomatic method, cf. the remarks and references in the last part of 10. 1.1. 'For a more systematic mathematical discllSl:!ion of th is notion, d. 65.3. 1. Th('. equivalent coneept of the completeness of the system of preferenees W8.8 previously considered at the beginning of 3.3.2. and of 3.4.6. J These cond itions (3:A :a ), (3:A :b) correspond to (65:A :a), (65:A :b) in 65.3. 1. • Remember that the a, (J, "'( occurring he re are a lways> 0, < I.

THE NOTION OF UTILITY

27

ticuJar difficulties. (Cf. Appendix.) It seems equally unnecessary to carry out the usual logistic discussion of these axioms' on this occasion. We shall however say a few more words about the intuitive meaningi.e. the justification-of each one of our axioms (3:A)-(3:C). 3.6.2. The analysis of our postulates follows: (3:A:a*)

(3:A:b*) (3:B:a *)

(3:B :b*)

This is the statement of the completeness of the system of individual preferences. It is customary to assume this when discussing utilities or preferences, e.g. in the" indifference curve analysis method." These questions were already considered in 3.3.4. and 3.4.6. This is the "transitivity" of preference, a plausible and generally accepted property. We state here: If v is preferable to u, then even a chance 1 - a of v-alternatively to u- is preferable. This is legitimate since any kind of com plementarity (or the opposite) has been excluded, d. the beginning of 3.3.2. This is the dual of (3:B:a *), with" less preferable" in place of preferable." We state here: If w is preferable to u, and an even more preferable v is also given, then the combination of u with a chance 1 - a of v will not affect w's preferability to it if this chance is small enough. I.e.: However desirable v may be in itself, one can make its influence as weak as desired by giving it a sufficiently small chance. This is a plausible "continuity" assu mption. This is the dual of (3:B:c *) , with " less preferable" in place,of " preferable. " This is the statement that it is irrelevant in which order the constituents u, v of a combination are named. It is legitimate, partiCUlarly since the constituents are alternative events, cf. (3:B:a*) above. This is the statement that it is irrelevant whether a combination of two constituents is obtained in two successive steps,-first the probabilities a, I - a, then the probabilities fj, 1 - fj; or in one operation,-the probabilities "1', 1 - "I' where "I' = afj.2 The same t hings can be said for this as for (3:C :a*) above. It may be, however, that this postulate has a deeper significance, to wh ich one allusion is made in 3.7.1. below. II

(3:B:c*)

(3:B:d*) (3:C :a*)

(3 :C :b*)

L A similar situation is dealt with mote exhaustively in 10.; those axioms describe a subject which is more vital for our maill objective. The logistic discussion is indicated there in 10.2. Some of the genetal remarks of 10.3. a.pply to the prescnt case also. I This is of course the correr.t arithmetic of accounting for two successive admixtures of v with u.

28

FORMULATI ON OF THE ECOKOMI C PROBL EM 3.7. General Rema.rks Concerning the Axioms

3.7.1. At this point it may be well to stop and to reconsider the situa· tion. H ave we not shown too much? We can deri·ve from the postulates (3:AH3 :C) the numerical character of utility in the sense of (3:2 :a) and (3: 1:a), (3:1:b) in 3.5.1.; and (3:I:b) states that the numerical values of utili ty combine (w ith probabilities) like mathematical expectations ! And yet the concept of mathematical expectation has been often questioned, and its legitimateness is certainly dependent upon some hypothesis con· cern ing the nature of an "expectation."L Have we not then begged the question? Do not our postulates introduce, in some oblique way , the hypotheses which bring in the mathematical expectation? More specifi cally: May there not exist in an individual a (positive or negative) utility of the mcre act of "taking a chance," of gambling, wl'tich the use of the mathematical expectation obliteratcs? How did our ax ioms (3:AH3:C) get around this possibility? As far as we can see, our postulate!! (3:A)·(3:C) do not attem pt to avoid it. Even that one which gets closest to excluding a "utility of gambling" (3:C:b) (cf. its discussion in 3.6.2.), seems to be plausible and legitimate,unless a much more refin ed system of psychology is used than the one now available for the purposes of economics. The fact that a numerical utilitywith a formula amounting to the use of mathematical expectations- can be buil t upon (3:A)·(3:C), seems to indicate this: We have practically defined nu merical utility as being that thing for wh ich the calculus of mathematical expectations is legitimate.' Since (3:A)- (3:C) secure that the necessary construction can be carried out, concepts like a "specific utility of gambling" cannot be formulatcd free of contrad iction on t his levcl.' 3.7.2. As we have stated, the last time in 3.6.1. , our axioms are based on the relation u > v and on the operation alt + (1 - a}v for utilities. It seems noteworthy that the latter may be regarded as more immediately given than the former: One can hardly doubt that anybody who could imagine two alternative situations with the respective utilitics u, v cou ld not also conceive the pros pect of having both with the given res pective probabilities a, 1 - a. On the other hand one may question the postulate of axiom (3 :A:a) for u > v, i.e. t he completeness of this ordering. Let us consider this point for a moment. We have conceded that one may doubt whether a person can always decide which of two alternativesL cr. Karl Menger: Das UnsicherhcitSlfioment in der Wertlehre, Zeitschrift fUr Kationalo konomie, vol. 5, (1934) pp. 459ff. LInd Gerh(lrd Tin/ner: A contribution to the non·static Theory of Choice, Quarterly Journal of Economics, vol. LVI , (1942) pp. 274ff . • Thus Daniel Bernoulli', well known suggestion to "solve" the "St. Peter6burg Paradox" by the use of the so-called "moral expectation" (i nstead of the mathematical expectation) means defining the utility numerically as the logarithm of one's monetary posscS6ions. • This may scem to be a paradoxical assertion. But anybody who has seriously tried to axiomatize that elusive concept, wit! probably concur with it.

THE NOTIO N OF UT ILITY

29

with the utilities u, v-he prefers. I But, whatever the merits of this doubt are, this possibility- Le. the completeness of the system of (individ ual) preferences- must be assumed even for the purposes of the" indifference curve met hod " (d. our remarks on (3:A :a.) in 3.6.2.). But if t his property of u > v 1 is assumed, then our use of the much less questionable au + (I - a )v ' y ields the numerical utilities toO!4 If the general comparability assumption is not made,5 a mathematical theory- based on au + (I - a)v together with what remains of u > vis still possible. ~ It leads to what may be described as a many-dimensional vector concept of utility, Thi5 is a more complicated and less satisfactory set-up, but we do not proposc to treat it systcmatically at t his time. 3.7.3, This brief exposition does not claim to exhaust the subject, but we hope to have conveyed the essential points. T o avoid misunderstandings, the following further remarks may be useful. (I) We re-emphasize that we are consideri ng only utilitic5 experienced by one person. These considerations do not imply anything concerning the comparisons of the utilities belonging to different individuals. (2) It cannot be denied t hat the analysis of t he methods which make use of mathematical expectation (ef. footnote I on p. 28 for the literature) i:,; far from concluded at pres:ent. Our remarks in 3.7.1. lie in this direction, but much more should be said in this respect. There are many interesting questions involved, which however lic beyond the scope of th is work. For ou r purposes it suffices to observe that the validity of the simpl e and plausible axioms (3:A)-{3 :C) in 3.6.1. for t he relation u > " and the operation au + (l - a)t' makcs t he utilities numbers up to a linear transformation in the sense discussed in these sections. 3.8. The Role of the Concept of Marginal Utility

3.8.1. The preceding analysis made it clear that \\·e feci free to make use of a numerical conception of utility. On the other hand, subsequent Or that he can assert that they are precisely equally desirab le. ' I.e. the completeness postulate (3:A:a). , I .e. the postulates (3 :B), (3 :C) together with the obvious postulate (3:A :b). I At this point the rellder may recall the familiar ar)l:ument acco rding to whieh the un numerical (" indifferellee curve" ) treatmclIl of ut.ilities is preferable to any nume ri cal one, because it is simpler and based on fewer hypotheses. This objection might be legitimate if the numerical treatment were based on Pareto's equality relation for utility differences (cf. the end of 3.4.6.). This reilltion is, indeed, a stronger and more com p licated hypothesis, added to the original ones cOlleerning the genera l com parability of utilities (co mpleteness of prefercnces). However, we used the operation au + (I - a)1> instead, and we hope that the reader will agree with us that it represents an even safcr assumption than that of the completeness of preferences. We think therefore that our procedure, as distinguished from Pareto's, ianot open to the objections bllSed on the necessity of artificial assumptions and a loss of simplicity . • Th is amounts to weakening (3: A:a) to an (3 :A:a') by replacing in it "one and only one" by "at most one." The conditions (3: A:a'), (3: A:b) then correspond to (65:U:n), (65 :B:b). ' In t his cuse BOme modifications in the groups of postulates (3 :B), (3:C) are abo necessary. 1

30

FORMULATION OF THE ECONOMIC PROBLEM

discussions will show that we cannot avoid the assumption that all subjects of the economy under consideration arc completely informed about the physical characteristics of the situation in which they operate and are able to perform all statistical, mathematical, etc., operat.ions which this knowledge makes possible. The nature and importance of this assumption has been given extensive attention in the literature and the subject is probably very far from being exhausted. We propose not. to enter upon it. The question is too vast and too difficult and we believe that it is best. to divide difficulties." I.e. we wish to avoid this complication which, while interest-ing in its own right, should be considered separately from our present problem. Actually we think that our investigations- although they assume "complete information " without any further discussion---do make a con~ tribution to the study of thii' subject. It will be seen that many economic and social phenomena which are usually ascribed to the individual's state of "incomplete information " make their appearance in our theory and can be satisfactorily interpreted with its help. Since our theory assumes" complete information," we conclude from this that those phenomena have nothing to do with the individual's "incomplete information." Some particularly striking examples of this will be found in the concepts of "discrimination" in 33.1. , of " incomplete exploitation" in 38.3., and of the "transfer" or "tribute" in 46.11., 46.12. On the basis of the above we would even venture to question the importance usually ascribed to incomplete information in its conventional sensei in economic and social theory. It will appear that some phenomena which would prima facie have to be attributed to this factor, have nothing to do with it. 2 3.8.2. Let us now consider an isolated individual with definite physical characteristics and with definite quantities of goods at his disposal. In view of what was said above, he is in a position to determine the maximum utility which can be obtained in this situation. Since the maximum is a well-defined quantity, the same is true for the increase which occurs when a unit of any definite good is added to the stock of all goods in the possession of the individual. This is, of course, t he classical notion of the marginal utility of a unit of the commodity in question.~ These quantitiea are clearly of decisive importance in the "Robinson Crusoe" economy. The above marginal utility obviously corresponds to II

I We shall aee that the rules of the games considered may explicitly prescribe that certain participante should not po88eS8 certain pieces of information. cr. 6.3., 6.4. (Games in which this does not happen are referred to in 14.8. and in (15:8) of 15.3.2., and are called games with "perfect information .") We shall reeognile and utilize this kind of "incomplete information" (according to the above, rather to be called "imperfellt information " ). But we reject all other types, vaguely defined by the use of concepte like complication, intelligence. etc. t Our theory attributes these phenomena to t he possibility of multiple "stable standards of behavior" cf. 4.6. and the end of 4.7. t More precisely; the so-called" illdirelltly dependent expected utility."

SOLUTIONS AND STANDARDS OF BEHAVIOR

31

the maximum effort which he will be willing to make- if he behaves according to the customary criteria of rationality- in order to obtain a further unit of that commodity. It is not clear at all, however, what significance it has in determining the behavior of a participant in a social exchange economy_ We saw that the principles of rational behavior in this case still await formulation, and that they are certainly not expressed by a maximum requirement of the Crusoe type. Thus it must be uncertain whether marginal utility has any meaning at all in this casc. 1 Positive statements on this subject will be possible only after we have succeeded in developing a theory of rational behavior in a social exchange economy,- that is, as was statcd before, with the help of the theory of "games of strategy." It will be seen that marginal utility does, indeed, play an important role in this case too, but in a more subtle way than is usually assumed.

4. Structure of the Theory: Solutions and Standards of Behavior '.1. The Simplest Concept of a Solution for One Participant

4.1.1. We have now reached the point where it becomes possible to give a positive description of our proposed procedure. This means primarily an outline and an account of the main technical concepts and devices. As we stated before, we wish to find the mathematically complete principles which define "rational behavior" for the participants in a social economy, and to derive from them the gencral characteristics of that behavior. And while the principles ought to be perfectly general- i.e., valid in all situations-we may be satisfied if we can find solutions, for the moment, only in some characteristic special cases. First of all we must obtain a clear notion of what can bc accepted as a solution of this problem; Le., what the amount of information is which a solution must convey, and what we should expect regarding its formal structure. A precise analysis becomes possible only after these matters have been clarified. 4.1.2. The immediate concept of a solution is plausibly a set of rules for each participant which tell him how to behave in every situation which may conceivably arise. One may object at this point that this view is unnecessarily inclusive. Since we want to theorize about" rational behavior," there seems to be no need tl' give the individual advice as to his behavior in situations other than those which arise in a rational community. This would justify assuming rational behavior on the part of the others as well,in whatever way we are going to characterize that. Such a procedure would probably lead to a unique sequence of situations to which alone our theory need refer. I All this is underat.ood within the domain of our severaisimplifying assumptions. they are relaxed, then variou8 further difficulties ensue.

If

32

FO R I\:I ULATIOK OF T H E ECOKOMIC PROBLEM

This objection seems to be invalid for two reasons: First, (,he "rules of the game,"- i.e. the physical laws which give the factua l background of the economic activities under consideration may be explicitly statistical. The actions of the participants oi the economy may determine the outcome only in conjunction with events which depend on chance (with known probabilities), cf. footnote 2 on p. 10 and 6.2. 1. If this is taken into consideration, then the rules of behavior even in a pcrfectly rational community must provide for a great variety of situations-some of which will be very far from optimum. I Sccond, and this is even more fundamental, t he rules of rational behavior must provide dcfinitely for t he possibility of irrational conduct on the part of others. In other words: I magine that we have discovered a set of rules for all participants-to be termed as "optimal " or "rational "-cach of which is indeed optimal provided that the other participants conform. Then the question remains as t.o what will happen if some of t he participants do not conform. If that. should turn out to be advantageous for them- and, quite particu larly, disadvantageous to the conformists- then the abo\·c "solution" would seem very questionable. We are in no position to give a positive discussion of these t.hings as yet- but we want to make it clear that u nder such conditions the "solution," or at least its motivation, must be considered as imperfect and incomplete. I n whatever way we formulate the guiding principles and the objectivc justificat.ion of "rational behavior," provisos will have to be made fo r every possible conduct of "the others." Only in this way can a satisfactory Rnd exhaustive t heory be developed. But if t he superiority of "rational behavior " over a ny other kind is to be established, then its description must include rules of conduct for all conceivable situations- including those wherc "the others" behaved irrat.ionally, in the sense of the standards which the theory will set for them. 4.1.3. At this stage the reader will observe a great similarity with t.he e\·eryday concept of games. We think t hat. this similarity is very essential ; indeed , that it is more t han that. For economic and social problems t.he games fulfill- or should fulfi ll-the same function which various geomctricomathematical models have successfully performed in the physical sciences. Such models arc theoret.ical constructs with a precise, exhaustive and not too complicated definition; and they must be similar to reality in those respects which arc essential in t.he invcstigation at hand . To recapitu late in detail: The definition must be precise and exhaustive in order to make a mathematical treatment possible. T he construct must not· be unduly complicat.cd, so that the mathematical treatment can be brought beyond the mere formalism to the point where it y ields complete numerical result-s. Similarity to rcality is nceded to make the operation significant. And this similarity must usually be restricted to a few traits , Thllt a uni move his king into a position of "check." This is a prohibition in the same amolute .wnse in which he rna}' not movea pawn sideways. But to move the king into a position whe re the opponent ean "che('kmate" him at the next move is ID.. of all th~se (I~, on which 0'" k" A, explicitly depend. In the first place great care must be exercised in oreler to avoid circularity i"n this I"cfj uirement, as far!1S A, is concerned. I But even if this difficulty does !lOt arise, becalls(' .\, de pends only on K and not on (II, . . . ,(I'_I- i.e. if the information available to e\'(~ ry player at e\'ery moment is independent of the previolls CO\lr;;e of the play- t he II.ho'·e pro('edme fitly still be inlldmissihle. "\SfiUIll(', e.g., that 0' , depends on a. cCl"tll.in combin:ttion uf so me :TA from among the A = I, , K - I, and that the rules of the game do indeed provide t!ltlt the pbSN k, at ~lt, >lhOllld k:loW t he value of this combinatio!l, bu ~ that it dc('~ not [dIal\" him 10 know more (i.e. the values of t he indivi[h:aI O"I, ··,0".__ 1). Kg.: He may kno\\" the value of 0",. + O"~ where p., >.. are both anterior 10..; (p., A < ..;), hut he is not allowed to know the separate values of 0",. and t]~. One could try various tricks to bril1g back the above s ituation to our earlier, simpll'r, sc heme, II"hieh d('scribes k.'s state of information by means of the set j\,. ~ liut it. becomes completely impossible to disentangle the various components of k.'s information at, ~l1" if they themselves ariginate from perso nal mo.'('s of difTerent pl:lyers, or of th~ same player but in c.ard to th~ n('xi tri,;k, :.e. k. at ;W:" is the one who took the last trick, i.e. again dependent upon the previous ('ourse of the play. In some fo~ms of Poker, and some other related /l:ame~ , the amount of i"forrl1alion available to a player at a given moment, i.e. A, at $" ucpcllds on what he and the others did previously. L The (l'A Ol! which, :WlOng ut ileriS, A. df'pcnd are only dr-fincd if the tctality of alt .\" for alt S('qW·tL l·" S(I',,· ,(1'. _10 is ('oltsi,If'T.7 • In the aho,· ,' ('X' lIllplt: ()n ~· Inightlry Iv n·piace the mu,·,· :)1"(,. hy a new onc in wl,i..,h not (1',. is choscn, out ",. + "A. :m. would rE'main un changed. Then k,:lt ~,would be informed about thc outcome of the choice connected w:th the new ~u only.

57

COMPLETE CONCE PT OF A GAME

different stages uf information. In our IIbo\'e example thi~ hapJl('n>' if k~ "#- kA, or if k~ = k~ but the sta te of information of this player is not ;.he same at mt~ and at mt~.1 7.2. The General Description

7.2. 1. There arc sti ll various, more or less artificial, tricks by which Gne could try to circumvent these difficuit ies. Dut the most natuml procedure seems to be to admit them, and to modify our definitions a.ceordingly. This is done by sacrificing t he A. as a means of describing the slate of information. Instead, we describe the st.ate of information of the player k.· at the time of his personal move m. explicitly: By ellumerating those functions of the variable u~ anterior to this movc--i.e. of the U I , ' " u._l- the numerical values of which he is supposcd to know at this moment. This is a system of fun ctions, to be denoted by 4>,. So 4>. is a set of functions

Since the elcments of.fl, describe the dependence on Ut, . is fixed, i.e. depend ing on ~ only.' a" k, may depend on (l"t, since their values are known to k. at m" these functions

,(1"._1,

so .... itseif and

• . • ,U'_i)

must belong to . happens to consist of all (unctions of certain variables "A--say of those for whi~h ). belongs to 8. given set ~f.-and of no others, then the 4>, description specializes back to the A, one: A, being the above set M.. But we have seen that we cannot, in general, count upon the existence of such a set.

58

DESC RIPTIO N Of' GAMES OF ST RATEGY

of the or iginal examples, which a re act.ual games. (E.g. Chess and Bridge can be described wit h t he help of the 11. •. ) There exist games which requi re discussion by means of the .> But in most of them olle could reve rt to t he A. by means of various extraneous tricks~and t he entire subject requires a rathe r delicate analysis upon which it does not seem worth while to enter here.' There exist unquestionably economic models where the 4>. a rc nece:ssary.z The most important point, however, is this. In pursuit of the objectives which we have set ourselves we must achieve the certainty of having exhausted a U combinat orial possibilities in connee· tion with the e ntire interplay of t he various decisions of the players, t heir changing states of information , etc. These a re problems, which have been dwelt upon extensively in economic literature. We hope to show that they can be d isposed of completely. But for this reason we want to be safe against a ny possible accusation of having overlooked some essential possibility by undue specialization. Besides, it will be seen that all the formal elements which we are int roducing now into the discussion do not com plicat e it ultima analysi. I. e. t hey complicate only t he present, prelimina ry stage of formal descrip· tion. The final form of the proble m turns out to be unaffected by t hem. (Cf. 11.2.) 7.2.3. There remains only one more point to discuss: The s pecializing assumption formulated at the very start of this discussion (at the beginning of 6.2. 1.) t hat both the number a nd the arrangement of t he moves are given (i.e. fixed) ab initio. We shall now see t hat this restriction is not essential. Consider first t he" arrangement" of the moves. T he possible variability of t he nature of eaeh move-i. e. of its k.- has already received full consideration (especially in 7.2.1.). The ordering of the moves m., k = I, . . . , ~, was from the start simply the chronological one. Thus there is nothing left to discuss on this score. Consider next t he number of moves~. This quantity t oo could be variable, i.e. dependent upon the course of thc play.' In describing this variability of ~ a certain amount of care must be exercised. I We mean card games where playcrs may discard !lOme cards without uncovering them, and arc allowed to take up or otherwise use openly a part of their discards later. There exists also a ga me of double-blind ChCll8--80metimes called " Kriegsspic1 It-which belongs in this class. (For ita description cr. 9.2.3. With referencc to that description: Each playcr knows about the "possibility" of the other'!! anterior choiccs, without knowing those choices themselves-and this "poSllibility" is a fun ction of all anterior choices.) 1 Let a participant be ignorant of the fuli details of the previous actions of the others, but let him be informed concerning certain statistical resultants of those actions . • It is, too, in moat games: Chees, Backgammon , Poker, Bridge. In the ca.seof Bridge this variability is due first to the variable length of the "bidding" phase, and second to the changing numbe r of contracta needed to make a " rubber" (i.e. a play). Examples of ga.mes with a fixed .. are harder to find: we shall see tha t we can make ~ fixed in every game by a.n artifice, but gamCl! in which .. is ab inifio fixed a re apt to be monotonous.

COMPLETE CONCEPT OF A GAME

59

The course of the play is characterized by the sequence (of choices) " , 11, (cf. 6.2.2.). Now one cannot state simply that JI may be a function of the variables 111, • • • ,11., because the full sequence CTI , • • • ,11 , cannot be visualized at all, without knowing beforehand what its length JI is going to be.' The correct formulation is this: Imagine that the variables 11.,111,113, • . • are chosen one after the other.' If this succession of choices is carried on indefinitely, then the rules of t he game must at some place JI slop the procedure. Then JI for which the stop occurs will, of course, depend on all the choices up to that moment. It is the number of moves in that particular play. Now this slop rule must be such as to give a certainty that every conceivable play will be stopped sometime. I.e. it must be impossible to in such a manner (subject arrange t he successive choices of 111,11.,113, to the restrictions of footnote 2 above) that the stop never comes. The obvious way to guarantee this is to devise a stop rule for which it is certain that the stop will come before a fixed moment, say JI.. I.e. that while JI may depend on 111, 11., 111, • • • , it is sure to be JI ;;;;; JI. where JI. does not depend on 111, 11:, 111, • • ' . If this is thc case we say that the stop rule is bounded by JI.. We shall assume for the games which we consider that they have stop rules bounded by (su itable, but fixed) numbers

111,

)1 • •

1.4

, I.e. one cannot say that the length of the game depends on all choices made in connection with all moves, since it will depend on the length of the game whether certain movea will occur at all. The argument is clearly circular. I The domain of variability of cr, is 1, . ,a,. The domain of variability of cr. is 1, . , a" and may depend on cr,: a. _ a.(cr,). The domain of variability of cr, is 1," ,a., and may depend on cr"cr.: ... _ ... (cr .. cr,). Etc.,etc . • This stop rule is indeed an Cll3Cntial part of e\'ery game. In most games it ia easy to find ~'s Ii)(cd upper bound ~.. Sometimes, however, the conventional form of the rules of the game does not exclude that the play might-under eX(lcptional conditions-go on od infinitum. In all these eases practical safcguards have heen subsequently incorporated into the rules of the game with the purpose of securing the existence of the hound ~.. It mUAt be said, however, that these safeguards are not always absolutely effective-although the intention is clear in every inatance, and even where exceptional infinite plays exist they are of little practical importance. It is nevertheless quite instructive, at least from a mathematicsl point of view, to discuss II. few typics.l examples. We give four examples, arranged according to dCllreasiDg effectiveness. t:eart~: A play is a "rubber," a "rubber" consists of winning two "games" out of three (ct. footnote I on p. 49), a "game" consists of winning five "points," and each "deal" I!;ivesone player or the other one or two points. Hence a " rubber " is complete after at most three "games," a "game" after at mOllt nine "deals," snd it is elUlY to verify that a "deal" consists of 13, 14 or 18 movea. Hence~ · - 3 ",9 . 18 - 486. Poker: A pn'ori two players could kecp "overbidding" each other ad infinitum. It is thcrefore customalY to add to the rulcs a proviso limiting the permissible number of "overbids." (The amounts of the bids are also limited, SO as to make the number of alternatives a. at these personal moves linite.) This of course secures a linite ~ •. Bridge: The play is a "rubber" and this could go on forever if both sides (players) invsriably failed to make their contract. It ia not inconceivable that the aide which is in danger of losing the "Iuhher," should in this \\.· ay permanently prevent a completion of the play by absurdly high bids. This is not done in practice, but there is nothing explir.it in the rules of the gsme to prevent it. In thL'Ory, at flny rale, some stop rule shou ld be introduced ill Bridge. Chess: It is easy to construct sequences of choices (i n the USUIII terminology:

60

DESCR I PTIO~

OF GA lVIES OF STRATEGY

Now lYe can make use of this bound v* t.o get entirely rid of the variability of ". This is done simply by extending the scheme of the game so that there are always 11* moves ~l, ,:nt,.. For every sequence O'lJ O't, 0'3, • . • everything is unchanged up to the move :m., and all moves beyond :nt. are "dummy moves." I.e. if we consider s. move ~m " I( = I, " , v·, for a sequence 0"1, 112, O"h' • for which II < K, then we make mI, a chance move with one alternative onlyL- i.e. onc at which nothing happens. Thus the assum:)tions made at the beginning of 6.2. I.- particularly that II is given ab initio-arc justified ex post.

8. Sets and P artitions 8.1. Des:rabillty of a Set-theoretical Descrip tion of a Game

8.1. We have obtained a satisfactory and gcneral description of the concept of a game, which could now be restated wit.h axiomatic precision and rigidity to serve as a basis for the subsequent mathematical discussion. It is worth while, however, before doing that, to pass to a different formu lation . This formulation is exactly equivalp.nt to the one which we reached in the preceding sections, but it is more unified, s impler when stated in a general form, and it leads to more elegant and transparent notations. I n order to arrive at this formulatioll we must use the symbolism of the theo;'Y of sets- and morc particularly of partitions- more extensively than we h{we done so far. This necessitates a certain amOU!lt of explanation and illustration , which we now proceed to give. "mO\'cs")-pa~ticu!ar!y in the "end J;nme"- whi('h can go 011 ad infinitum without ever cnding the play (i.e. pToduciug n "dlC~kmat e") . The simplest ones aTe periodical, i.e. indefinite rCJletitions of the snme (,yl'lc of choices, but there e"ist non-periodical ones as w ~11. AI! of them oITcr a vcry real Jlossihility for th e player who is in dl'.nge r of losing to secure somctj mes a .. tie." For this rcaS{Jn various" tic :'Iles "-i.e. sto p rules- are in use just to prevent that plll'tlomenon. One well known "tie rule" is this: Any cycle of choices (i.e. "mo\'es"), when three times repeated, terminates the play by a "tie." This rule excludes most but not all infinite sequences, anu hence is really not cffeeth·c. Another "tic rule" is this: If no pawn has been moved and no officer taken (these Ilre "irreversible" operations, which rannot be undone subsequently) fo r 40 moves, then the play is terminated by a "tie." It is easy to sce that this rule is effective, although the ~' is enormous . • From n purely rnalhemntical point of view, the following question could be asked: Let the stop rule he effective in this senae only, that it is impossible so to arrange the suc"eu ive choiccs a" a" a" . that the stop never comes. I.e. let there a lways be a finite ~ dependent upon a" 17t, ~ distinguished . since (a ) i" a 0""_(·]'·"11"11' >!L"t w]'ile a ;" nut. IJ~(~

62

(B,A,,)

(B,A,d)

(B,A,.)

(B,A of) (B,A,,)

DESCRIPTWN OF GAMES OF STRATEGY The sum of two sets A, B is the set of all elements of A together with all elements of B,- to be denoted by A u B. Similarly the su ms of more than two sets are formed. I The product, or intersection, of two sets A, B is t he set of all common elements of A and of B,-to be denoted by A n B. Similarly the products of more t han two sets are formed. I The difference of two sets A, B (A the minuerul, B the subtrahend) is the set of all those elements of A which do not belong to B,-to be denoted by A - B.L When B is a subset of A, we shall also call A - B the complement of B in A. Occasionally it win be so obvious which set A is meant that we shall simply write - B and talk about t he complement of B without any further specifications. Two sets A, B are disjunct if they have no elements in com· mon,~i.e. if A n B = e. A system (set) a of sets is said to be a system of pairwise disjunct sets if all pairs of different elements of a are disjunct sets,~ i.e. if for A, B belonging to a , A ~ B implies A n B = e.

8.2.3. At this point some graphical illustrations may be helpful. We denote the objects which are elements of sets in these considerations by dots (Figure I). We denote sets by encircling the dots (elements)

•

•

•

•

,

•

••

•

•

• • • •

• •

, •

• •

•

•

•

Figure I.

which belong to them, writing the symbol which denotes t he set across the encircling line in one or more places (Figure 1). The sets A, C in t his figure are, by the way, disjunct, while A, B are not. L This nomenclature of sums, products, differences, is traditional. It is hased on certain algebraic ana logies which we shall not use here. In fact, the algebra of these operations U, n, also known ill! Boolean algebra, has a considerable intereat of its own. Cf. e.g. A. Torah': Introduction w Logic, New York, 1941. Cf. fu rther Garreft BirkhoJ!: Lattice Theory, New York 1940. This book is of wider interest for t he understanding of the modern abstract method. Chapt. VI. deals with Boolean Algebras. Further literature is given th ere.

SET S AND PART ITIONS

63

Wit h this device we can also represent sums, products and differences of sets (Figure 2). In this figure neither A is a subset of B nor B one of A , hence neither t he difference A - B nor the difference B - A is a comple· ment. In the next figure, however, B is a subset of A, and so A - B is t he complement of B in A (Figure 3) .

•

• •

.._.-. Figure 2.

•

•

•

• • •

•

•

•

Figure 3,

8.3. Partitions, Their Properties, and Their Graphical Representation

8.3.1. Let a set fI a nd a system of sets a be given. We say t hat is a partition in fI if it fulfill s the two following requirements: (8 Jl,.) (8, B ,b)

a

Every element A of a is a. Subset of fI, and not empty. (t is a system of pai rwise disjunct sets.

This concept too has been the subject of an extensive li terature. I We say for two partitions a, ffi that a is a subparliliQn of ~

/, 1:'p . ,

~.,

-

-D'

',

•

"

,

Figllre 10.

ately from (10:1 :b) in 10.1.1. Now (lO:l:h) in to. 1.1. permits the inference from this (for details cI. 9.2.2. ) that a.+1 coincides with e. (k) in B. (k)- for all k = 0, I, . . , n. (We could equally have used the corresponding points in 10.1.2., i.e. t he meaning of these concepts. We leave the verbal expression of the argumen t to t he render.) Bu t e. (k) is a partition in B. (k); hence the above sta tement mcnns that e.(k) simply is that part of a.+1 which lies in B. (k ). We restate this : JJ

( IO,C)

JJ

If the condition of (lO:B) is fulfilled , then e .(k) is that part of a'+ l which lies in B. (k).

Thus when prelimina rity and anteriority coincide, then in our present formalism the sequence a I, . .. , a.,Ci.+ 1 and the sets B.(k), k = 0, 1, .. , n, for each J( = 1, . . . , 1' , describe the game fully . I.e. the picture

STRATEGIES AI\D FIKAL S IMPLIFTCATIQ!\'

79

of Figure 9 in 8.3.2. must be amplified on ly by bracketing together those elements of each d" which belong to the same set , '" are identical in all respects. I 13.2. The Operations Mu: and Min

13.2.1. Consider a function 4> which has real numbers for vslues

.(x, y,

).

Assume first that 4> is a one-variable function. If its variable can be chosen, say as x = Xa 80 that 4>(xo) ;;;; 4>(x') for all other choices x', then we say that 4> has the maximum 4>(xo) and assumes it at x = Xo. Observe that this maximum 4>(xo) is uniquely determined; i.e., the maximum may be assumed at x = Xo for several Xo, but they must all furnish the same value 4>(XO).2 We denote this value by Max 4>(x), the maximum value of 4>(x). If we replace ~ by ~, then the concept of 4>'s minimum, 4>(xo), obtains, and of Xo where 4> assumes it. Again there may be several such X9, but they must all furnish the same value ¢(X9)' We denote this value by Min 4>(x), the minimum value of .p. I The concept of a function is closely allied to that of a set. and the above should be viewed in parallel with the expoeition of 8.2. t Proof: Consider two such ::to, eay ::t~ and ::t~'. Then .p(~) iii:; .p{::t~) and .p l::t~ ) iii:; ~(::t~). Hence .p(::t;) _ .p(::t~).

ZERO-SUM TWO-PERSON GAMES, THEORY

90

Observe that there is no a priori guarantee that either Max 4>(x) or Min 4>(x} exist.' If, however, the domain of .;----over which the variable x may runconsists only of a finite number of elements, then the existence of both Max 4>(x) and Min tII(x) is obvious. This will actually be the case for most functions which we shall discuss. 2 For the remaining ones it will be a consequence of their continuity together with the geometrical limitations of their domains.' At any rate we are restricting our considerations to such functions , for which Max and Min exist. 13.2.2. Let now tP have any number of variables x, y, z, .. ' . By singling out one of the variables, say x, and t reating the others, y, Z, . . . , as constants, we can view 4>(x, y, z, . . . ) as a. one-va.ria.ble function, of the va.ria.ble x. Hence we may form Max $(x, y, Z, . . • ), Min $ (x, y, z,' .. ) as in 13.2.1., of course with respect to this x. But since we could have done this equally well for anyone of the other variables y, z, . . it becomes necessary to indicate that the operations Max, Min were performed with respect to the variable x. We do this by writing Max s $(x, y, Z,' . ) , Minx $(x, y, z, . . . ) instead of the incomplete expressions Max q" Min q,. Thus we can now apply to the function $(x, y, z, . . . ) anyone of the operators Max:., Min%, Max v, Min v, Ma.x., Min" . . '. They are all distinct and our notation is una.mbiguous. This notation is even adva.ntageous for one va.riable functions, and we will use it accordingly; i.e. we write Max:. q,(x), Min:. $(x) instead of the Max $(x), Min $(x) of 13.2.1. Sometimes it will be convenient-or even necessary- to indicate the domain S for a maximum or a minimum explicitly. E.g. when the function $(x) is defined also for (some) x outside of S, but it is desired to form the maximum or minimum within S only. In such a case we write

Max..... .! $(x), instead of Max:. $(x), Min:. $ (x). In certain other cases it may be simpler to enumerate the values of q,(x)say a J b, . . . - than to express $(x) as a function. We may then write I E.g. if 4>C z ) • :z: with all real numbers8./! domain , then neither Max 4>C:z:) nor Min 4>tz) exist. I Typical examples: The functions X.~"" . , "') of 11.2.3. (or of (e) in 13.1.2.), the function X C"" .. ,) of 14.1.1.

~

Typical examples: The functions K( fl.

17.4., the functiona Min"

-- , -- -~,

,, ), Max.... K (

~ XC"" ...)fT.' Max••

I, _I

~

~ ,

,,), Min_ K( f, ,,) in

,

fl.

L: XC", " )'1"

in 17.5.2.

The vari-

.~_I ~

abies of all these functions 1I.re t or " or both, with respect to which aub8equent maxima and minima are fo rmed . Another instance is diacU89Cd in 46.2.1. espec. footnote 1 on p. 384, where the mathematical background of this subject and it6 literature are considered. It seems unnecessary to enter upon these here, since the above examples are entirely elementary.

FUNCTIONAL CALCULUS

Max (a, b, ... ), [Min (a, b, ... )] instead of

91 Max~

~ (x),

[Min ..

• (x)I'

13.2.3. Observe that while t/t(x, y, Z, • . • ) is a function of the variables y, 21, • • • , Max.. .,,(x, y, z, . . . ), Min .. 4>(z, y, 21, • • • ) are st ill functions, but of t he variables y, Zt . • • only. Purely typographically. :z; is still present in Max.. 4>(x, y, Zt • . . ), Min" ¢ (x, y, z, . . . ), but it is no X,

longer a variable of these functions. We say that the operations Max.. , Min.. kill the variable x whieh appears as their index .! Since Max .. q,(x, y, z, . . . ), Min. t/J (x, y, Zt • . • ) are still functions of the variables y, Zt . • • " we can go on a nd form the expressions Max" Max" tIt{x, y, Z, Min" Max.. I/I(x, Y. Zt

. • . ),

. •• ),

Max" Min .. .,,(x, y, Zt Min" Min .. q,(x, y, %,

. , • ), • • • ).

We could equally form Max.. Max.. q,(x, y, z, . . . ),

Max. Min. ofJ;(3:, y,

%, • • • )

etc. ;· or use two other variables than 3:, y (if there are any); or use more variables than two (if there are any). [nfine, after having applied as many operations Max or Min as there are variables of ¢ (3:, y, %, • • • )-in any order and combination, but precisely one for each variable 3:, y, Z, -we obtain a fun ction of no variables at all, i.e. a constant. 13.S. Commuuti.-Ity Questions

13.3.1. The discussions of 13.2.3. provide the basis for viewing the Max., Min., Max., Min., Max" Min., . . . entirely as JUndional operationa, each one of which carries a function into a nother function.' We have seen that we can apply several of them successively. In this latter case it is prima Jacie relevant, in which order the succes.sive operatioIl8 are applied. But is it really relevant? Precisely : Two operations are said to commute if, in case of their successive application (to the same object), the order in which this is done does not matter. Now we ask: Do t he operations Max", Min., Max., Min., Max., Min" . . . all commute with each other or not ? We shall aIl8wer this question. For this purpose we need use only two variables, say 3:, y and t hen it is not necessary that q, be a function of further variables besides x, y.IOf course Max (a, h" .) [Min (a, h, . . . )I is simply the greatest IsmalleetJone among the numbers a, h, .. • A well known operation in a na lysis which kills a variable t is the defirlite integral :

10

1

,H%) is a function of %, but .,,(%)11% is a constant. I We treated 1/, Z, . as constant parameters in 13.2.2. But now that % ha.e been killed we relell80 the variables y, %, • • • Observe that if two or more operations are applied, the innermost applies first and kills ita variable; then the next one follows suit, etc , With one variable less, since these operationa kill one variable each. 'Further variable! of """ if a ny, may be treated as COI1lllanl.3 for the purpose of this an&lyeis.

92

ZERO-SUM TWO-PERSON GAMES, THEORY

So we consider a. two-varia.ble function ",(x, y). The significant questiona of commutativity arc then clearly these: Which of the three equations whi.ch follow are generally true: Max~ Max~ t/J(x , y) = Max il Max" t/1(x, y), Min% Minv ", (x, y) = Minll Min" q,(x, y), MR.>:" Minll q,(x, y) = M inll Max" ",(x, y), 1

(13, 1) (13 ,2) (13,3)

We shall see t hat (13:1 ), (13 :2) are true, while (13:3) is not; i.e., any two Max or any two Min commute, while a Max and a Min do not commute in general. We shall also obtain a criterion which determines in which special ca.sea Max and Min commute. This question of commutativity of Max and Min will turn out to be decisive for the zero-sum two-person game (ef. 14.4.2. and 17.6.). 13.3.2. Let us first consider (13 :1). It ought to be intuitively clear that Max~ Max~ ¢(x, y) is the maximum of ¢(x, y) if we treat x, y together as one variable .: i.e. that for ...,orne suitable x o, Yo, 4>(xo, Yo) = Max. Max v cP(z, y) and that for all x', y', ¢(xo, Yo) ~ ¢(x', y'). If a mathematical proof is nevertheless wanted, we give it here: Choose Xo so that Max~ ¢(:s;, y) assumes its x-maximum at x = Xo, and then choose Yo so that ¢ (xo, y) assumes its y-maximum at y = Yo. Then 4> (Xo, Yo) = Max" ¢(xo, y) = Max. Maxil ¢(x, y),

and for all z', y' .p(Xo, Yo) = Max" cf>(xo, y)

e:

Max" .p(x', y) !i;;; cf>(x', y').

This completes the proof. Now by interchanging x, y we see that Max" Max • .p (x, y) is equally the maximum of cf>(x, y) if we treat x, y as one variable. Thus both sides of ( 13:1) have the same characteristic property, and therefore they are equal to each other. This proves (I3: 1). Literally the same arguments apply to Min in place of Max: we need only use ~ consistently in place of ~ . This proves (1 3:2). This device of treating two variables x, y as one, is occasionally quite convenient in itself. When we use it (as, e.g., in 18.2.1., with TL, TI, X("!"I, TI) in place of our present x, y, .p(x, y», we shall write Max •. ~ ¢ (x, y) and Mins.~

.p(x, y).

13.3.3. At this point a graphical illustration may be useful. Assume that the domain of ¢ for x, y is a finite set. Denote, for the sake of simplicity, the possible values of x (in this domain) by 1, " , t and those of y by 1, . . . ,8. Then the values of .p(x, y) corresponding to all x, y in this domain- Leo to all combinations of x = 1, . . . ,t, Y = 1, . . . , 8-can be arranged in a rectangular scheme: We use a rectangle of t rows and 8 I The combination Min. Maxv requires no treatment of ita own, since it obtains from the above-Max~ Min,,-by interchanging z, y.

FU NCTIO NAL CALCULUS

93

columns, using the number x = I , " , t to enumerate the former and the number y = 1, . .. ,8 to enumerate the latter. Into the field of intersection of row x a nd column y-to bc known briefly as the field x , y- we write the value (x, y/)

~

Min" Max z ",(x, y) = q,(xo, Yo) = Max z ¢ (x , yo),

i.e. Max., ¢(x, Yo) ,;;;;; Max., ",(x, y')-SO Max z ¢(x, y) assumes its minimum at y = yo. Hence y belongs to B.... This completes the proof. The theorems (13:C·), (13:D*) indicate, by the way. the limitations of t he analogy described at the end of 13.4.2.; i.e. they show that our concept of a saddle point is narrower than the everyday (oreographical) idea of a saddle or a pass. Indeed , (13:C*) states that all saddles-provided that they exist- are at the same altitude. And (13:0*) states-if we depict the sets A*, B* as two intervals of numbers2- that all saddles together are an area which has the form of a rectangular plateau. ' 13.5.3. We conclude this section by proving the existence of a saddle point for a special kind of x, y and 41(X, y). This special case will be seen to be of a not inconsiderable generality. Let a function oJt(x, u) of two variables x, u be given. We consider all functions f(x) of the variable which have values in the domain of u. Now we keep the variable x but in place of the variable u we use the function 1 itself. 4 The expression "'(x,f{x » is determined by x,j; hence we may treat ",(x,f(x » fl.!; a funct,ion of the variables x , I and let it take th"! place of 41(x, y). We wish to prove that for these x, 1 and "' (x, I(x»-in place cf x, y and 41(X, y)- a saddle point exists; i.e. that

Max z Min, "'(x, I (x» = Min, Max z oJt(x, f (x».

Proof: For every x choose a Uo with oJt(x, uo) = Min .. "'(x, u). This Uo depends on x, hence we can define a function fo by Uo = fo (x). Thus ", (x, lo(x» = Min.. oJt (x, u). Consequently Max z oJt(x, fo (x»

=

Max z Min .. oJt(x, 1').

1 Only under our hypothesis at the beginning of this section! Otherwise there exi~t no saddle points at all. I If the z, yare positive integers, then this can certainly be brought about by two appropriate permutations of their domain. 'The general mathematical concepts alluded to in footnote 1 on p. 95 are free from these IimitatiomJ. They correspond precisely to the everyday idea of a pass. • The reader is asked to vi~ua.li~e this: Although itself a function, f may perfectly well be the varia.ble of another function.

98

ZERO-SUM TWO-PERSON GAMES, THEORY

A fortiori,

(13 ,F)

Min, Max z f(x, f(x» ;i!.

Max~

Min .. ",(x, u).

Now Min, ",(x, f(x» is the same thing as Min .. ", (x, u) since f enters into this expression only via its value at the one place x, i. e. f(x), for which we may write u. So Min, Hz, f (x» = l\·lin .. 1/;(x, u) and consequently, Max z Min, 1/t{x, !(;t» = Max z Min .. f (x, u).

(13,G)

(13 :F), (13:0) together establish the validity of a;;;; in (I3:E). The S in (l3:E) holds owing to (l3:A *). Hence we have = in (l3:E ), Le. the proof is completed.

14. Strictly Determined Games a.1. Formulation of the Problen: 14.1.1. We now proceed to the consideration of the zero-sum two-person game. Again we begin by using the normalized form. Accordi ng to this the game consists of two moves: Player 1 chooses a number T) = 1, " , fh player 2 chooses a number T~ = 1, , {j" each choice being made in complete igaorance of the other, and then players 1 and 2 get the amounts X \(T\, Tt} and Xt(TI, Tl }, respectively.\ Since the game is zero-sum, we have, by l1A. X\(T\, Tt}

+ Xt(TJ, T,}

=:

O.

We prefer to express this by writing X\(T\, TZ)

-= Je(T\, Tt},

X,(T1, T,)

5"

-Je(T!, Tt).

We shall now attempt to understand how the obvious desires of the players 1, 2 will determine the events, i.e. the choices TI, Tt. lt must again be remembered, of course, that Tl, TZ stand ultima analysi not for a choice (in a move) but for the players' strategies; i.e. their entire 'I theory" or "plan " concerning the game. For the moment we leave it at that. Subsequently we shall also go "behind" the T" T, and analyze the course of a play. 14.1.2. The desires of the players 1, '1, are ~imple enough. 1 wishes to make XI(TI, Tt) :;:;; Je(Tt, Tt) a maximum, 2 wishes to make Je,(TI, Tt) == -X(T\, T,) a maximum; Le. 1 wants to maximize and 2 wants to minimize X(T!, Tl)'

So the interests of the two players concentrate on the same object: the one function X(T\, Tt). But their intentions are-as is to be expected in a zero-sum two-person game-exactly opposite: 1 wants to maximize, 2 wants to mIDlmlZC. Now the peculiar difficulty in all this is that neither player has full control of the object of his endeavor-of X(T!, TI)-Le. of both its variables Tit Tt. 1 wants to maximize, but he controls only T\; 2 wants to minimize, but he controls only Tt: What is going to happen? I

C{.

til :D)

in 11.2.3.

99

STRICTLY DETERMINED GAMES

The difficulty is t hat no par ticular choice of, say 'TI, need in itself make Je(r!, 'T2) either great or small . The influence of T) on X(Tl, 'TI) is, in general, no definite thing; it becomes t hat. only in conjunction with t he choice of the other variable, in this case Tt . (ef. the cOlTesponding diffic ulty in economics as discussed in 2.2.3.) Observe that from the point of view of the player 1 who chooses a variable, say T I, the other variable can certainly not be considered as a chance event. The other variable, in this case 'TI, is dependent upon the will of the other player, which must be regarded in t he same light of " ration· slity" as one's own. (eL also the end of 2.2.3. and 2.2.4.) 14.1.3. At this point it is convenient to make use of the graphical representation developed in 13.3.3. We represent X(TI, T,) by a rectangular

matrix: We form a rectangle of ~l rows and (j, columns, using the number = 1, " , ~ I to enumerate t he former, and the number T t = 1, . . . ,~2 to enumerate the latter; and into the field TI, T2 we write the matrix element JC(T! , T2) ' (Cf. with Figure 11 in 13.3.3. The 4>, x, y, t, 8 there correspond to our X, TI, T2, ~I, (j, (Figure 15).)

Tl

2

o.

"

I

:Je (l , I )

:Je( 1, 2)

... ...

2

X{2, I )

X(2, 2)

...... ...

"

:Je{T" I )

O.

X (fl" I )

I :Je(I, fl.)

I :Je(I, "') X {2, "')

..... ..

3C(2, fl.)

X( .. " 2)

Xt T., "')

.. ........

X(.", fl.)

X(/J,,2)

X {fl ., T.)

X(fl" fl.)

It ought to be understood that the function X(71, T,) is subject to no restricLions whatsoever; i.e., we are free to choose it absolutely at will. I Indeed , any given function X(TI, T2) defines a zero-sum two-person game in t he sense of ( lI:D) of 11.2.3. by simply defining

(cf. 14.1.1.). The desires of the players 1, 2, as described above in the last section, can now be visualized as follows: Both players a re solely I

(1,;'"

The domain, of course, ia prescribed: It conaillta of all pain TI, .... with." _ 1, . . . , - 1, . . . ,fl.. Thill ill a finite set, 150 all Mu: and Min exist, ct. the end of 13.2.1 .

100

ZERO-SUM TWO-PERSON GAMES, THEORY

interested in the value of the matrix element JC(TI, TI). Player 1 tries to maximize it, but he controls only the fow, -i.e. the number T\. Player 2 tries to minimize it, but he controls only the coiumn,- i.e. the number r ,.

We must now attempt to find a satisfactory interpretation (or the outcome of this peculiar tug-oi-war,' 14..2. The Minor..at and the Majorant GaIllta

1'.2. Instead of attempting a direct attack on the game r itself- for which we are not yet prepared- let us consider two other games, which arc

closely connected with r and the discussion of which is immediately feasible. The difficulty in analyzing r is clearly that the player 1, in choosing TI does not know what choice TI of the player 2 he is going to face and vice verso. Let us therefore compare r with other games where this difficulty does not arise. We define first a game r l , which agrees with r in every detail except that player 1 has to make his choice of '1'1 before player 2 makes his choice of TI, and that player 2 makes his choice in full knowledge of the value given by player 1 to Tl (i.e. l's move is preliminary to 2's move). ! In this game r l player 1 is obviously at a disadvantage as compared to his position in the original game r. We shall therefore call r 1 the miMmnt game of r. We define similarly a second game r l which again agrees with r in every detail except that now player 2 has to make his choice of TI before player 1 makes his choice of Tl and that I makes his choice in full knowledge of the value given by 2 to (i.e. 2'8 move is preliminary to 1's move).J In this game r, the player I is obviously at an advantage as compared to his position in the game r. We shall therefore caB r , the majorant game of r. The introduction of these two games r l , r t achieves this: It ought to be evident by common sense-and we shall also establish it by an exact discussion- that for rio r l the "best way of playing " - Le. the concept of rational behavior-has a clear meaning. On the other hand, the game r lies clearly "between" the two games r l , r l ; e.g. from l's point of view r l is always less and r, is always more advantageous than r.( Thus r l , r l may be expected to provide lower and upper bounds for the significant quantities conceming r. We shall , of course, discuss all this in an entirely precise form. A priori, these "bounds" could differ widely and leave a considerable uncertainty as to the understanding of r. Indeed, prima facie this will seem to be the case for many games. But we shall succeed in manipulating this technique in such a way- by the introduction of certain

T,

1 The point is, of course, that this is Dot a tug-of-war. The two playere have opposite intereata, but the means by which they have to promote them are not in opposition to each other. On the contrary, theBe "mu\ls"-i.e. the choices of '1'" 'I',--are apparently independent. This discrepancy characterites the entire problem. , Thus r,-while extremely simple--is no longer in the normalized form. I Thus r,--while extremely simple-is no longer ill the normaliled form . • Of courae, to be precise we ahould aay " less than or equal to" instead of " less," and "more than or equal to" i08tead of "more."

STRICTLY DETERM l :\,ED G Al\'IES

101

further devices- as to obtain in the end a preci se theory of r , which gives complete answers to all questions. 14..S. Discussion of tbe Auziliary Games

14.3.1. Let us first consider the minorant game rl. After player I has made his choice TI the player 2 makes his choice T2 in full knowledge of the value of TI. Since 2'8 desire is to minimize JC(TI' Tt), it is certain that

he will choose T2 so as to make the value of JC (TI, T 2) a minimum for this TI · I n other words: When I chooses a particular value of TI he can already foresee with certainty what tbe value of le(TI' TI) will be. This will be l\'lin" X(Tj, T,).I This is a function of TI alone. Now 1 wishes to maximize X(TI, 12) and since bis choice of TI is conducive to the value Min, le(TI, T,)-which depends on TI only, and not at all on Tz-SO he will cho'ose TI so as to maximize Min,. le(TI, "1"1). Thus the value of thi.~ quantify will finally be

Summing up: (14:A:a)

T he good way (strategy) for I to play the minorant game

r l is to choose TI, belonging to the set A,- A being the set of (14:A:b)

those "1"1 for which Min, Je(TI, T2) assumes its maximum value Max, Min, le(TI, TI)' • The go~d way (strategy) for 2 to play is this: If I has chosen a definite value of TI,I then 12 should be chosen belonging to the set B, ,- B, being the set of those T1 for which :JC(T" TI) assumes i'ts minimum value M in,. X(T" T1).·

On the basis of this we can state furt her: (14,A:c)

If both players 1 and 2 play the minorant game r l well , i.c. if 11 belongs to A and 12 belollgs to B then the value of X(TI, Tt) will be equal to T ,

VI = Max" l'din" X(1J, 1,). I Observe that ~1 may not be uniquely determined : For a given n the ~ 1-function JC(n, H) may M8ume itL! ~.-minim um fo r several values of Tt. The value of JC(~" TI ) will, howeve r, be the same fo r all these Tt, nal:Jeiy the uniquely defined minimum value Min',JC(n, ~I)' (el. 13.2.1.) , For the same rea&On as in footnote I above, the value of~, may not be unique, but the value of Min" X(n, ~1) is the same fo r all ~1 in question, namely the uniquely-defined maximum value Max r , Min,.JC{n Tt).

'2 i.9 informed of the value of n whe n called upon to make his choice of ~ I,-this is the rule of r,. It follows from our concept of a at ratcg y (d. 4.1.2. and end of 11.1.1 .) that at t his point a rule must be provided fo r 2's choice of" for every value of n,-irrespective of whether I has played well or not, i.e. whether or not the value chosen belongs to A . • In all, this n is treated as a known parameter on which everything depende,- including the set Hr, from whieh T, ought to be chosen.

102

ZERO-SUM TWO-PERSON GAMES, THEORY

The truth of the above assertion is immediately established in the mathematical sense by remembering the definitions of the sets A and B." and by substituting accordingly in the assertion . We leave t his exercise-which is nothing but the classical operation of "substituting the defining for the defined"- to the reader. Moreover, the statement ought to be clear by common sense. The entire discussion should make it clear that every play of the game r l has a definite value for each player. This value is the above VI for the pl~ser 1 and therefore -VI for the player 2. An even more detailed idea of the significance of VI is obtained in this way:

Player 1 can, by playing appropriately, secure for himself a gain ;:;;; VI irrespective of what player 2 does. Player 2 can, by playing appropriately, secure for himself a gain ~ -v" irrespective of what player 1 does.

(Proof: The former obtains by any choice of 71 in A. The latter obtains by any choice of 7, in BT,,' Again we leave the details to the reader; they are altogether trivial.) The above can be stated equivalently thus: Player 2 can, by playing appropriately, make it sure that the gain of player 1 is ~ VI, Le. prevent him from gaining > VI irrespective of what player 1 does. Player 1 can, by playing appropriately, make it sure that the gain of player 2 is ;::; -v" i.e. prevent him from gaining> -VI irrespective of what player 2 does. 14.3.2. We have carried out the discussion of r l in rather profuse detail although the "solution" is a rather obv ious one. That is, it is very likely that anybody with a clear vision of the situation will easily reach the same conclusions "unmathematically," just by exercise oC common sense. Nevertheless we Celt it necessary to discuss this case so minutely because it is a prototype of several others to follow where the situation will be much less open to "unmathematical " vision. Also, all essential elements of complication as well as the bases for overcoming them are really present in this very simplest case. By seeing their respective positions clearly in this case, it will be possible to visualize them in the subsequent, more complicated, ones. And it will be possible, in this way only, to judge precisely how much can be achieved by every particular measure. 14.3.3. Let us now consider the majorant game r t . r t differs from fl only in that the roles of players 1 and 2 are interchanged: Now player 2 must make his choice 72 first and then the player 1 makes his choice of 71 in full knowledge of the value of 1'2. I Recall that n must be chosen without knowledge of T" while T, is chosen in full knowledge of n.

STRICTLY DETERMINED GAMES

!O3

But in saying that r l arises from f l by interchanging t he players I and 2, it must be remembered that these players conserve in the process their respective functions :lC1(r!, "1"2), leS(Tl, T2) , i. e. Je(r!, '1"2) , -Je(rl,1'2). That 18, 1 still desires to maximize and 2 still desires to minimize JC(Tl' 'Ts) . These being understood, we can leave tbe practically literal repetition of the considerations of 14.3. 1. to the reader. We confine ourselves to restating the significant definitions, in the form in which they apply to f,. (14,Boa)

The good way (strategy) for 2 to pla.y the majorant ga.me f, is to choose '1"2 belonging to the set B,-B being the set of t hose T! for which Max, .3C(Tl, T,) assumes its minimum value Min, Max. , X(TI, 'Ts). •

.

(l4,B,b)

The good way (stra.tegy) for 1 to play is this: If 2 bas chosen a definite value of '11,1 then T L should be chosen belonging to the set A~, ,-A~. being the Set of those TL for which X(TI, Tt) assumes its maximum value Max_, X(TI, Tt).%

On the basis of this we can state further: (!4 ,Boc)

If both players 1 and 2 play the majorant game r 2 well , Le. if '1, belongs to Band '11 belongs to A., then the value of X (Tl, Tt) will be equal to Vt

= Min., Max~,

JC(TL, '12).

The entire discussion should make it clear that every play of the game has a definite value for each player. This value is the above Vt for the player 1 and therefore -Vt for the player 2. In order to stress the symmetry of the entire arrangement, we repeat, mutatis mutandis, the considerations which concluded 14.3.1. They now serve to give a more detailed idea of the significance of VI.

rt

(!4 ,B,d)

Player 1 can, by playing appropriately, secure for himself a gain ~ VI, irrespective of what player 2 does. Player 2 can, by playing appropriately, secure for himself a gain ~ -VI, irrespective of what player I does.

(Proof: The latter obtains by any choice of '12 in B. The for mer obtains by any choice of '1 1 in A.,.' Cf. with the proof, loco cit.) The above can again be stated equivalently thus:

(!4,B,.)

Player 2 can, by playing appropriately, make it sure that the gain of player I is ;;:;; v" i.e. prevent him from gaining

1 I is informed of the value of .. , when called upon to make his choice of T,-this is the rule of r, (Cf. footnote 3 on p. 101 ). I In all this ... is treated 8.8 a known parameter on which everything depend!!, ineluding the Bet A., from which n ought to be chosen. I Remember that ... mUllt be chosen without any knowledge of "" while n is chosen with full knowledge of ....

104

ZERO-SUM TWO-PERSON GAMES, T H EORY

> V1 , irrespective of what player I does. Player I can, by playing appropriately, make it su re that the gain of player 2 is ~ -V1, i.e. prevent him from gaining> -VI, irrespective of what player 2 does. 14.3.4. T he d iscussions of r l and rt , as given in 14.3.1. and 14.3.3., respectively, are in a relationship of symmetry or duality to each other; they obtain from ench other, as was pointed out previollsly (at t he beg i n~ ning of 14.3.3.) by interchanging the roles of the players 1 and 2. I n itself neither game r 1 nor r t is symmetric with respect to this interchange; indeed,

this is nothing but a restatement of the fact that the inte rchange of the players 1 and 2 also interchanges the two games 1'1 and r l , and so modifies both. It is in harmony with t his that the various statements which we made in 14.3.1. and 14.3.3. concerning the good strategies of r l and r l , respectively-i.e. (14:A:a), ( 14:A:b), (14 :B:a), (14:B:b), loco cit.-were not symmetric with respect to the players 1 and 2 either. Again we see: An interchange of the players I and 2 interchanges the pertinent definitions for r l and r l , and so modifies both. I It is therefore very remarkable that the characterization of the value of a play (VI for r l , VI for r 2), as given at t he end of 14.3. 1. a nd 14.3.3.- i.e. (l4:A:c), (14:A:d), (14:A:e), (14:B:c), (14:B:d), (I4:B:e), loco cit. (except for the formulae at the end of (14:A:c) and of (14:B:c))- are fully symmetric with respect to the players 1 and 2. According to what was said above, this is the same thing as asserting that these characteri7.ations are stated exactly the same way fo r r l and r t .' All this is, of course, equally clear by immediate inspection of the relevant passages. Thus we ha.ve succeeded in defining t he value of a play in the sa.me way for the games r 1 and r l , and symmetrically for t he players I and 2: in (14:A:c), (14:A:d), (14 :A:e), (14:B:c), (14:B:d) and (I4 :B:e) in 14.3. 1. and in 14.3.3.,- this in spite of the fundamental difference of t he individual role of each player in these two games. From this we derive the hope that the definition of the value of a play may be used in the same form for other games as well-in particular for t he game r - which, as we know, occupies a midd le position between r, and rt. T his hope applies, of course, only to the concept of value itself, b ut not to the reasonings which lead to it; those were specific to r] and r t • indeed different for r l and for r l , and altogether impracticable for r itself; i.e., we expect for the future more from (l4:A:d), (I4:A:e), (14:B:d), (l4:B:e) than from (14:A:a), (!4,A,b), (!4,Boa), (14,B,b). , Observe thllt the original gllme r W!l8 symmetric with respect to the two players 1 llnd 2, if we let each player tllke his function X,(... , " ,), X ,(... , .,,) with him in an interchange; i.e. the personal moves of 1 and 2 had both the 8llme character in r. For a narrowe r concept of symmetry, where the funct ions X ,{." , ",), X , ("" .,,) are held fixed, d. 14.6. I This point deserves careful consideration: Naturally these two characte:izationB must obtain from each other by int.erchanging the roles of the players 1 and 2. But in tbis C!l8e the statements coincide also di rcctly whe n no interchangp, of t he players is made at all. This is d ue to thei r individual symmetry.

STRICTLY DETERMINED GAMES

105

These are clearly only heuristic indications. Thus far we have not even attempted the proof that a numerical value of a play can be defined in this manner for r. We shall now begin the detailed discussion by which this gap will be filled. It will be seen that at first definite and serious difficulties seem to limit the applicability of t his procedure, but that it will be possible to remove them by the introauction of a new device (eL 14.7.1. and 17.1.-17.3., respectively). 14..4. Conclusiolls

14.4.1. We have seen that a perfectly plausible interpretation of the value of a play rletermines this quantity as VI = Max, Min, JC(TL, TI), v, = Min, I Max: X{TL, TI),

•

•

for the games r l , I" respectively, as far as the player 1 is concerned,l Since the game r, is less advantageous for 1 than the game f r-in r, he must make his move prior to, and in full view of, his adversary, while in f: the Situation is reversed-it is a reasonable conclusion that the value of r l is less than, or equal to (i.e. certainly not greater than) the value of f,. One may argue whether this is a rigorous" proof." The question whether it is, is hard to decide, but at any rate a close analysis of the verbal argument involved shows that it is entirely parallel to the mathematical proof of the saffle proposition which we already possess. Indeed, t he proposition in question , coincides with (13:A·) in 13.4.3. (The.p, x, y there correspond to our X, T" Ti.) Instead of ascribing vI, V2 as values to two games 1' , and f: different from r we may alternatively correlate them with r itself, under suitable assumptions concerning the "intellect" of the players 1 and 2. Indeed , the rules of the game r prescribe that each player must make his choice (his personal move) in ignorance of the outcome of the choice of his adversary. It is nevertheless conceivable that one of the players, say 2, "finds out" his adversary; i.e., that he has somehow acquired the knowledge as to what his adversary's strategy is.' The basis for t his knowledge does not concern us; it may (but need not) be experience from previous plays. At any rate we assume that the player 2 possesses this knowledge. It is possible, of course, that in this situation 1 will change his strategy; but again let us assume that, for any reason whatever, he does not do it. 2 Under these assumptions we may then say that player 2 has "found out" his adversary. 'For player 2 the values are consequently -V" -v•. In the game r -which is in the normalized form-the strategy is just the actual choice at the unique personal move of the player. Remember how t his normalized form was derived from the original extensive fOfm of the game; consequently it appears that this choice corresponds equally to the strategy in the original game. I For an interpretation of all these assumption9, d. 17.3.1. I

106

ZERO-SUM TWO-PERSON GAMES, THEORY In this case, conditions in r become exactly the same as if the game were

r l • and hence all discussions of 14.3.1. apply literally.

Similarly, we may visualize the opposite possibility, that player I has "found out" his adversary. Then conditions in r become exactly the same as if the game were r,; and hence all discussions of 14.3.3. apply literally. In the light of the above we can say: The value of a play of the game r is a well-defined quantity if one of the following two extreme assumptions is made: Either that player 2 "finds out" his adversary, or that player I "finds out" his adversary. In the first case the value of a play is VI for I, and -VI for 2; in the second case the value of a play is v ~ for 1 and -v, for 2

14.4.2. This discussion shows that if the value of a play of r itselfwithout any further qualifications or modifications- can be defined at all, then it must lie between the values of VI and Vt. (We mean the values for the player 1.) I.e. if we write v for the hoped-for value of a play of r itself (for player 1), then there must be VI ~ V ~ Vt.

The length of this interval , which is still available for v, is

a

=

Vt -

VI ~

O.

At the same time a expresses the advantage which is gained (in the game r) by "finding out" one's adversary instead of being "found out" by him. I Now the game may be such that it does not matter which player" finds out" his opponent; i.e., that the advantage involved is zero. According to the above, this is the case if and only if

a=O or equivalently Or, if we replace

by their definitions; Max~, Min~ X(rl, T!) = Min, • Max, I X('I, . VI, Vt

12)'

r possesses these properties, then we call it strictly determined. The last form of this condition calls for comparison with (13:3) in 13.3.1. and with the discussions of 13.4.1.-13.5.2. (The t/J, x, y there again correspond to our X, 11, 1""t). Indeed , the statement of (13:8·) in 13.4.3. says that the game r is strictly determined if and only if a saddle point of JC(TI, "Tl) exists. If the game

1'.6. ADalr.i. of Strict Determin.tenea.

14.5.1. Let us assume the game saddle point of X(1""I, 1""t) exists.

r to

be strictly determined; i.e. tbat a

1 ObBerve that thiB expre88ion for the advantage in question applie8 for both pl.yen: The advantage for the player I is v, - V,; for the player 2 it is (-v,) - (-v,) and theae two expreaaioDB are equal to each other, i.e. to 0 i.e. in such a game it involves a positive advantage to "find out" onc's adversary. Hence there is an essential difference between the results, i.e. the values in r L snd in r t , and therefore also between the good ways of playing these games. The considerations of 14.3.1., 14.3.3. provide therefore no guidance for the tl'eatment of r. Those of 14.5.1., 14.5.2. do not apply either, since they make use of the existence of saddle points of JC(TI, T2) and of the validity of

Max., Min., JCtTh TI)

=

Min., Max" JC(TI, T1),

i.e. of r being strictly determined. There is, of course, some plausibility in the inequality at the beginning of 14.4.2. According to this, the value v of a play of r (for the player 1)-if such a concept can be formed at all in this generality, for which we have no evidence as yeP-is restricted by VI

& v :;;;

VI.

But this still leaves an interval of length .6. = V2 - VI > 0 open to Vi and, besides, the entire situation is conceptually most unsatisfactory. One might be inclined to give up a ltogether: Since thcre is a positive advantage in "finding out" one's opponent in such a game r, it seems plausible to say that there is no chance to find a solution unless one makes some definite assumption as to " who finds out whom," and to what extent.~ We shall see in 17. that this is not so, and that in spite of.6. > 0 a solution can be found along the same lines as before. But we propose first , without attacking that difficulty, to enumerate certain games r with.6. > 0, and others with .0. = O. The first- which are not strictly determinedwill be dealt with briefly now; their detailed investigation will be undertaken in 17.1. The second-which are strictly determined- will be analyzed in considerable detail. 14.7.2. Since there exist functions JC(TI, Tt) without saddle points (cf. 13.4.1., 13.4.2.; the tj,(x, y) there, is our JC(TI, T2» there exist not strictly determined games r. It is worth while to re-examine those examples-i.e. Thi~ is in harmony with footnote I on p. 106, &II it should be. 'Cf. however, 11.8.L I In plainer language: 0. > 0 means that it is not pol58ible in this game for each player &imultaneously to be cleverer than his opponent. Consequently it seem~ desirable to know just how clever each particular player is. I

STRICTLY DETERMINED GAMES

111

the functions described by the matrices of Figs. 12, 13 on p. 94-in the light of our present application. Tha.t is, to describe explicitly the games to which they belong. (In each case, replace tP(x, y) by our Je(rl' 1t), 12 being the column number and 11 the row number in every matrix. cr. also Fig. 15 on p. 99).

Fig. 12: This is the game of "Matching Pennies."

Let---for 1\ and for

12-1 be "heads" and 2 be "tails," then the matrix element has the value 1

if TI, Tt "match"-Le. arc equal to each other--snd -1, if they do not. So player 1 "matches" player 2: He wins (one unit) if they "match" and he loses (one unit), if they do not. Fig. 13: This is the game of "Stone, Paper, Scissors," Let-for 11 and for 11-1 be "stone," 2 be "paper," and 3 be "scissors," The distribution of elements 1 and -lover the matrix expresses that" paper" defeats .. stone," "scissors" defeat "paper," .. stone" defeats "scissors." I Thus player 1 wms lone unit) if he defeats player 2, and he loses (one unit) if he is defeated. Otherwise (if both players make the same choice) the game is tied. 14.7.3. These two examples show the difficulties which we encounter in a not strictly determined game, in a particularly clear form; just because of their extreme simplicity the difficulty is perfectly isolated here, in vitro. The point is that in "Matching Pennies" and in "Stone, Paper, Scissors," any way of playing- i.e. any 'Tt or any - is just 8.'l good as any other: There is no intrinsic advantage or disadvantage in "heads" or in "tails" per St, nor in "stone," "paper" or "scissors" per se. The only thing which matters is to guess correctly what the adversary is going to do; but how are we going to describe that without further assumptions about the players' "intellects" ?2 There are, of course, more complicated games which are not strictly determined and which are important from various more subtle, technical viewpoints (cf. IS., 19.). But as far as the main difficulty is concerned, the simple games of "Matching Pennies" and of "Stone, Paper, Scissors" are perfectly characteristic.

T,

14.8. Program of a Detailed Analysia of Strict Determinateneu

14.8. While the strictly determined games r - for which our solution is valid-are thus a special case only, one should not underestimate the size of the territory they cover. The fact that we are using the normalized form for the game r may tempt to s uch an underestimation: It makes things look more elementary than they really are. One must remember that the TI, Tt represent strategies in the extensive form of the game, which may be of a very complicated structu re, as mentioned in 14.1.1. In order to understand the meaning of strict determinateness, it is therefore necessary to investigate it in relation to the extensive form of the game. This brings up questions concerning the detailed nature of the moves, I" I

Paper covers the "wne, !Uli~rs cut the paper, "wne grinds the !Uli~rs."

AIJ mentioned before, we shall ahow in 17.1. that it can be done.

112

ZERO-SUM TWO-PERSON GAMES, THEORY

-chance or personal- the state of information of the players, etc .; Le. we come to the structural analysis based on the extensive form , as mentioned in 12.1.1. We are particularly interested in those games in which each player who makes a personal move is perfectly informed about the outcome of the choices of all anterior moves. These games were already mentioned in 6.4.1. and it was stated there that they are generally considered to be of a particular rational character. We shall now establish this in a precise sense, by proving that all such games are strictly determined. And this will be true not only when all moves are personal, but also when chance

moves too are present. 15. Games with Perfect Information 16.1. StatemeDt of Purpose.

Induction

16.1.1. We wish to investigate the zero·sum two· person games somewhat further, with the purpose of finding as wide a subclass among them as possible in which only strictly determined games occur; i.e. where the quantities VI = Max., Min., X(TI, Tt), Vt = Min, Max. , 3C(TI, Tt)

.

of 14.4.1.- which turned out to be so important for the appraisal of the game-fulfill V]=Vt=v.

We shall show that when perfect information prevails in f - i.e. when preliminarity is equivalent to anteriority (cf. 6.4.1. and the end of 14.8.)then f is strictly determined. We shall al~o discuss the conceptual sig· nificance of this rtlSult (cL 15.8.). Indeed , we shall obtain this as a special case of a more general rule concerning vI, Vt , (cf. 15.5.3.). We begin our discussions in even greater generality, by considering a perfectly unrestricted general n·person game f. The greater generality will be useful in a subsequent instance. 16.1.2. Let r be a general n·person game, given in it-s extensive form. We shall consider certain aspects of r , first in our original pre·set·theoretical terminology of 6. , 7., (d. 15.1.), and then translate everything into the par· tition and set terminoiogy of 9., 10. (cf. 15.2., et sequ.). The reader will probably obtain a full understanding with the help of the first discussion alone; and the second, with its rather formalistic machinery, is only under· taken for the sake of absolute rigor, in order to show that we are really proceeding strictly on the basis of our axioms of 10.1.1. We consider the sequence of all moves in r; ~I' ~t, . . . ,~.. Let us fix our attention on the first move, ~1, and the situation which exists at the moment of this move. Since nothing is anterior to this move, nothing is preliminary to it either; i.e. the characteristics of this move depend on nothing,- they are constants. This applies in the first place to t.he fact, whether ~I is a chance

GAMES WITH PERFECT INFORMATION

113

move or a personal move;and in the latter case, to which player~JRl belongs,i.e. to the value of k , = 0, I, " , n respectively, in the sense of 6.2. 1 . And it applies also to the number of a lternatives 0'1 at mIL and for a chance move (i.e. when kl = 0) to the values of the probabilities Pl(I). . . . , PI(aJ). The result of the choice at m\ - chance or personal- is a u[ = 1, " , aL. Now a plausible step suggests itself for the mathematical analysis of the game 1', which is entirely in the spiri t of the method of "complete induction" widely used in all branches of mathematics. It replaces, if successful, the analysis of r by the analysis of other games which contain onc move less than r. I This step consists in choosing a (J I = 1> , aI and denoting by r, a game which agrees with r in every detail except that the move fit, is omitted, and instead the choice UI is dictated (by the rules of the new game) the value u, = 61. Z f" has, indeed, one move less than f: Its moves are fit t , ' " m•.' And our "inductive" step will have been successful if we can dcrive the essentiai characteristics of r from those of all r" , 61 = I, . . . , al. 15.1.3. It must be noted, however, that the possibilities of forming r, are dependent upon a certain restriction on f. Indeed, every player who makes a certain personal move in the game f" must be fully informed about the rules of this game. Now this knowledge consists of the knowledge of the rules of the original game f plus the value of the dictated choice at fit l , i.e. 61. Hence r , can be formed out of r-without modifying the rules which govern the pla'yer's state of information in r-only if the outcome of the choice at fitl, by virtue of the original rules r, is known to every player at any personal move of his :m!,' " mt.; Le. "m l muet be preliminary to aU personal moves :m" ... ,mt.. We restate this:

r" can be formed-without essentially modifying the structure of r for that purpose-only if r possess the following property: mtl is preliminary to all personAl moves "mt,' " "m,.4 , I.e. have ~ - 1 instead of~. Repeated application of this "inductive" step--if feasible at all-will reduce the game r to one with 0 steps; i.e. to one of fixed, unalterable outcome. And this means, of course, a complete solution for r. (Cf. (15:C:aJ in 15.6. 1.) • E.g. r is the game of Chesa, and i, a particular opening move--i.e. choice at ml.,-of "white," i.e. player I. Then r" i8 again Chess, but beginning with a move of the character of the second move in ordinary Chess-a "black," player 2- and in the position created by the "open:ng move" i .. This dictated "opening move" may, but need not, be a conventional one Oike E2-E4J. Tbe same operation is exemplified by forms of Tournament Bridge where the "umpires" assign the players definite-known and previously seleeted-"hands." (This is done, e.g., in Duplicate Bridge.) In the firllt example, the dictated move gn, was originally pel1lOnal (of "white," player I ); in the second example it was originally chance (the" deal "). In &orne garnC!! occlUlionally "handicaps" are used which amount to one or more luch operations. I We should really use the mdices 1,' " ~ - I and indicate the dependence OD iIi e.g. by writing gn:" ... ,gn:~I' But we prefer the simpler notation gn., ", ~ •. 'This il the terminology of 6.3.; i.e. we use the special form of dependence in the sense of 7.2.1. Using the general description of 7.2.1. we must state (IS:A:a) like this: For every pel"8Onai move '31t., ~ _ 2, . . . ,~, tbe eet 4>. containa the fun ction v,.

114

ZERO-SUM TWO-PERSON GAM ES, THEORY 16.2 . Tbe Eu t t Conditio n (Fiul Step)

16.2.1. We now translate 15.1.2., 15.1.3. into the partition and sct terminology of 9.,10., (eL also the beginning of 15.1.2.), The notations of 10. 1. will therefore be used. fr. consists of t he one set fl ((1 0 :1 :f) in 1O.1.I .) , and it is a subpartition of (BI « 10 : I :9.) in 10.1.1.); hence CR. too consists of the onc set n (the others bei ng empty) ,t.t Thill. is: 8, (k)

~ 1~

for precisely one k, say k = k l' for all k ¢. k •.

Thisk. = 0, 1," ,n determines thecharacterofmI .;it is thek.of6 .2. 1. H k. = 1, ...• n-Le. if the move is personal- then a. is also a subpartitian of :DI(k.), « 10:1 :d ) in 10.1. 1. This was only postulated within BI(k.), but BI (k l ) E2 U). Hence :S:h (k .) too consists of the one set n.1 And for

e

k ~ k" t he :odk) which is a partition in B\ (k) = « lO :A:g) in to.1.I. ) must he empty. So we have precisely one AI of (iJ, which is 11, and for k\ = 1, . . . ,n precisely one D\ in aU :D\(k), which is also 11 ; while for k\ = 0 there are no DI in all :D,(k). The move mtl consists of the choice of a C\ from e ,(k l ); by chance if k\ = 0; by the player k\ if k\ - I , . . . ,n. C I is automatically a subset of the unique A I( = 11) in the former case, and of the unique D, ( - 11) in the latter. The number of these C, is a, (cf. 9.1.5., particularly footnot e 2 on p. 70); and since the A, or DI in question is fi xed, t his a, is a definite constant. a, is the number of alternatives at $t h the a\ of 6.2. 1. and 15. 1.2. These C I correspond to the 0'\ = I , . . . , a\ of 15. 1.2., and we denote them accordingly by C,(I), . . . ,C I(al) .4 Now (10 :1 :h) in 10.1. 1. showsas is readily verified-that (11 is also the set of the C ,( I ),' " Cda,), i.e. equal to el. So far our analysis has been perfectly general,-valid for $t l (and to a ce rtain extent for $t,) of a ny game 1'. The reader should translate t hese properties into everyday term inology in the sense of 8.4.2. and 10.4.2. We pass now to r ". This s hould obtain from r by dictating the move $t1-1lS described in 15.1.2.- by putting 0'1 = '1. At the same time the moves of the $tame are restricted to mI1, . . . • mI.. This means t hat t he 'This (S, is Rn exception from (8:B:a ) in 8.3. L;d. the remark concerning this (8:8:a ) in footnote 1 on p. 63, and also footnote 4 on p. 69. 'Proof : n belonga to (1 " which ia a l ubpartition of (i, ; hence n ill a 8ubaet of an ele ment of (B , . Thia element i8 neceaurily equal to n. All other elementl of ell, Ire therefore d isjunct from n (d . 8.3. 1.), i.e. empty. a a" 5Mk ,) unlike (B, (d. above) must fulfill both (8:8:a), (8:B:b) in 8.3.1.; hence both have no further elements beaides tl. • They represent the alternatives (1 ,(1),' " a, (",, ) of 6.2. Rnd 9.104., IU .5.

GAMES WITH PERFECT INFORMATION

115

element Jr-which represents the actual play-can no longer vary over all 0, but is restricted to C,(c1',). And the partitions enumerated in 9.2.1. are restricted to those with /C = 2, .. ,50, I (and I( = ~ + 1 for a.). 15.2.2. We now come to the equivalent of the rest.riction of 15.1.3. The possibility of carrying out the changes formulated at the end of 15.2.1. is dependent upon a. certain restriction on r. As indicated, we wish to restrict the play-Leo T-within C,(I1I). Therefore all those sets which figured in the description of r and which were subsets of fl, must be made over into subsets of C,(c1'I)-and the partitions into partitions within C 1(6,) (or within subsets of C,(I1,». How is this to be done? The partitions which make up the descriptions of r (cf. 9.2.1.) fall into two classes: those which represent objective facts-the a., the ffi, = (B.eO), B,(I),' " B.(n» and the e,(k), k = 0, 1, . . . , n-and those which represent only the player's state of information,: the :D.(k), k = I, . . . ,n. We assume, of course K ;;:; 2 (cf. the end of 15.2.1.). In the first class of partitions we need only replace each element by its intersection with CL(6"L). Thus ffi. is modified by replacing its elements B.(O), B.O), , B,(n) by CI("I) n B.(O), C L(6"L) n B.(I), . . . , C L(6"I) n B.(n). In a, even this is not necessary: It is a subpartition of a: (since K ;;:; 2, cf. 10.4.1.), i.e. of the system of pairwise disjunct sets (CI(I),' " CL(aL» (cf. 15.2.1.); hence we keep only those elements of a. which are subsets of CL("L), i.e. that part of a. which lies in CL("L). The e,(k) should be treated like ffi. but we prefer to postpone this discussion. In the second class of partitions- i.e. for the :D.(k)-we cannot do anything like it. Replacing the elements of :D.(k) by their intersections with CL("L) would involve a modification of a player's state of information' and should therefore be avoided. The only permissible procedure would be that which was feasible in the case of a.: replacement-of :D.(k)- by that part of itself which lies in CL("L). But this is applicable only if :D.(k)-like a. before-is a subpartition of a: (for K ~ 2). So we must postulate this. Now e.(k) takes care of itself: It is a 8ubpartition of :D.(k) «10:1 :c) in ID.I.1.), hence of a: (by the above assumption); and so we can replace it by that part of itself which lies in CL("L). So we see: The necessary restriction of r is that every :D,(k) (with K ;;; 2) must be a subpartition of a z. Recall now the interpretation of 8.4.2. and of (ID:A:d·), (lO:A:g·) in ID.1.2. They give to this restriction the meaning that every player at a personal move :ml, ... , :m. is fully I We do not wi6h to change the enumeration to K _ I, . . . , ~ - I, cf. footnote 3 on p. 113. 1 a. represents the umpire's state of information, but this is an objective fact: the events up to that momcnt have determined the course of the play precisely to that extent ld. 9.1.2.). ~ Namely, giving him additional information.

116

ZERO-SUM TWO-PERSON GAMES, THEORY

informed about the state of things after the move

;ntL

(i.e. before the move

mt 2) expressed by (i2. (Cr. also the discussion before (lO:B) in 10.4.2.) That is, ;m:L must be preliminary to all moves gyr2,' " mt •.

Thus we have again obtained the condition (15:A:a) of 15. 1.3. We leave to the reader tne simple verification that the game r, fulfills the requirements of ID.1.1. L

16.3. The Exact CODdition (Entire IDduction)

16.3.1. As indicated at the end of 15.1.2., we wish to obtain the characteristics of r from those of all r", 61 = 1, . . . , al, since this-if successful- would be a typical step of a "complete induction." For the moment, however, the only class of games fo r which we possess any kind of (mathematical) characteristiC{'! consists of the zero-sum twoperson games: for these we have the quantities v" v, (cL 15.1.1.). Let us therefore assume that r is a zero-sum two-person game. Now we shall see that the v" v. of r can indeed be expressed with the help of those of the r", Irl = I, " , elL (cf. 15.1.2.). This circumstance makes it desirable to push the "induction" further, to its conclusion: i.e., to form in the same way r".", r".".", . . . , r".", ... ",.' The point is that the number of steps in these games decreases successively . to 0 (for r".", ... " ,); from (for r), )I - 1 (for r ,,). over v - 2, )I - 3, i.e. r "., •..... '. is a "vacuous" game (like the one mentioned in the footnote 2 on p. 76). There are no moves; the player k gets the fixed amount 5'.(Ir" . . . , Ir,). This is the terminology of 15.1.2., 15.1.3.,-i.e. of 6., 7. In that of 15.2.1., 15.2.2.- i.e. of 9. , 1O. -we would say that n (for r) is gradually restricted to a C,(Ir,) of fi. (for r,,), a C.(lrlt Ir:) of Ci1 (for r".,,), a C.{lrl, Ir., Ir.) of fij (for r"., •. ,), etc., etc., and finally to a C.(lrl, 1r2,' " Ir.) of fi.+ 1 (for r".r, ..... ,). And this last set has a unique element «lO:l:g) in 10.1.1.), say... Hence the outcome of the game r"." ..... '. is fixed: The player k gets the fixed amount 5'.(.. ). Consequently, the nature of the game r".,., ....'. is- trivially-clear; it is clear what this game's value is for every player. Thcrefore the process which leads from the r" to r - if established-can be used to work backwards from r "., •.....'. to r"." ... .. '._1 to r"., •.....'._, etc., etc., to r"." to r and finally to r. " is feasible only if we are able to form all games of the sequence But this r", r".", r"., •. ", . . . , r".,., ... ".' i.e. if the final condition of 15.1.3. or 15.2.2. is fulfilled for all these games. This requil ement may again be formulated for any general n-person game r; so we return now to those r. 15.3.2. The requirement then is, in the terminology of 15. 1.2., 15. 1.3. (Le. of 6., 7.) that m, must be preliminary to all mi, m" ... , m.; that )I

1 i, _

a, -

1, . . . ,

alj i, _

.,,,(i,, i.); ew., etc.

I, . . . ,

0,

where

o. _ o.(i ,); i. _

1, ..

, a,

where

GAMES WITH PERFECT INFORMATION

117

m, must be preliminary to all m l , m l , ' , m,; etc., etc.; i.e. that preliminarity must coincide with anteriority. In the terminology of 15.2.1., 15.2.2.-i.e. of 9., 10.-of course the same is obtained: All :D.(k), Ie ~ 2 must be subpartitions of el2; all :D.{k), Ie ~ 3 must be subpartitions of ell, etc., etc.; i.e. all :D.{k) must be subpartitions of el~ if Ie E; },..' Since el, is a subpartition of elA in any case (d. 10.4.1.), it suffices to require that all :D.(k) be subpartitions of a.. However a, is a subpartition of :D,{k) within .'iax,..:" JC.:(r.:I1, :h(O'~)).

Now Min3, Max.: Max'.:/, X.:(r .:/ I, :h(O'Y)) = Max.: Mi nJ, Max,.: /! 3C.:(r.:II, ~2(O'n) =

Max.: Min'.:/. Max,.: It X.;(r.:II , r.; /t)

owing to (I3:E) and (13:G) in 13.5.3. ; there we need only replace the x, U , ((x) , .r(x, 14) by our O'~, T.:/2, ~2 (O'n , Max 0, then put t = Z tj, so 1 -

i

~

"p-i

Z tj - L

t=

t; =

{p.

HeneeO

< t;;;. 1.

I

Put

8' =

"

t-jt for " = 1 . . .

So

p-1.

"

Hence, by our

,,- 1 _ assumption for p - I,

L

D is convex, hence

is in D.

81 X i

i-I

,,-I

tZ

--+

_

8jX;+(l-t)x"

i-I

is also in Dj but this vector is equal to p-l

_

k

tjX

J~

which thus belongs to D.

I

_ j

+ lp x

p_ J>

=

k j~

ti I

X'

LINEARITY AND CONVEXITY

133

The proof is therefore completed. The h, . . ,II' of (16:2 :d) may themselves bc viewed as the com~

panents of a vector t = It ... , tp] in Lp. It is therefore appropriate to give a name to the set to "which they are restricted, defined by

and

, ~ Ii = 1. j-I

.,-a.,i,

" ",

"

,

"

a.·.xit

a,-., is

Figu re 19.

Figure IIi

"

"

"

j

,

'1

(i.e. for which the vectors z -

-->

-->

y and u -

-->

z areort.hogonal ). This means,

,

Hypo.pl . ... of ( 18 3)

Figure 25. _

Hypt.p) .... of (16 .4 )

Figure 26.

indeed , that. ~

( Zi -

Yi)(Ui -

%i) =

O.

Clearly all of C must. lie on this

i_ I ~

hyperplane, or on that side of it which is away from y. ~

~

If any point u ~

~

of C did lie on the y side, t.hen some points of the interval I z , u J would be ~

~

nearer to y than z is.

(Cf. Figure 25.

The computation on pp. 135- 136~

properly in terpreted-shows precisely this.) ~

and so alJ of [ Z

~

,

~

Si nce C contains z and u, ~

u 1. this would contradict the statement t hat z is as near

~

to y as possible in C. Now our passage from (16:3) to (I6:4) amounts to a parallel shift of ~

~

this hyperplane from z to y (parallel, because the coefficients

ai

=

Zi -

Yi

138

ZERO-SUM TWO-PERSON GAMES, THEORY

-

of Ui, i = 1, . . n al'e unaltered). Now y lies on the hyperplane, and all of C in one half-space produced by it (Figure 26). The case n = 3 (space) could be visualized in a similar way. It is even possible to account for a general n in this geometrical manner. If the reader can persuade himself that he possesses n-dimensional " geometrical intuition " he may accept the above as a proof which is equally valid in n dimensions. It is even possible to avoid this by arguing as follows: Whatever n, the entire proof deals with only three points at once, J

- --

namely Y Z u. Now it is always possible to lay a (2-dirnensional) plane through three given points. If we consider only the situation in this plane, then Figures 24- 26 and the associated argument can be used without any re-interpretation. Be this as it may , the purely algebraic proof given above is absolutely rigorous at any rate. We gave the geometrical analogies mainly in the hope that they may facilitate the understanding of the algebraic operations performed in that proof. J

J

16.(. The Theorem of the Aiternuive lor Matrices

16.4.1. The theorem (16:"8) of 16.3. permits an inference which will be fundamental for our subsequent work. We start by considering a rectangular matrix in the sense of 13.3.3. with n rows and m columns, and the matrix element a(i, j). (Cf. Figure 11 in 13.3.3. The 1/>, x, y, t, s there correspond to our a, i, j, n, m.) I.e. a(i, j) is a perfectly a rbitrary function of the two variables i = I, . . . ,"i j = I, . 1m. Next we form certain vectors inL.. : Foreachj = 1, . . . , ~

m the vector

Xi

=

I:r>;, ... ,

x~1

with x{ = a(i , j) and for each l

-

=

1,

.. , n the coordinate vector {; I = lliill. (Cf. for t he latter t he end of 16.1.3.; we have replaced the j there by our l.) Let us now apply the

- -- -

theorem (16:B) of 16.3. for p = n

x"', Ii

I, ... ,

Ii~.

--

+ m to these n + m vectors

(They replace the x

I, ... ,

y = O.

x', . . . ,

x p loco cit.)

- -- -

~

-

We put

The convex Cspanned by x I, ,x"', Ii', , {; "may contain If this is the case, then we can conclude from (l6:2:d) in 16.2.2. that

- "- -

... tj x~i ~ J -

1

+

~

81

Ii

I

= 0,

/ .. 1

with

(16,5) (16,6)

.

t ..

~

0,

81 ~

"

0, . . . ,

ZI;+ Z"~1. i-I

1-1

8~ ~

O.

o.

LINEARITY AND CONVEXITY tl,

,

, 8~

t .. , 81,'

replace the II>

,lp

139

(Ioc. cit.).

In terms of the

components this mcans

•

•

L l jaU, j) + ,., L 818,1 =

0.

The second term on the left-hand side is equal to

8;,

so we write

•

L a(i, j)lj =

- 8,.

i_I

If we:had

8"

•

L I, =

0, then II

=

.

=

= 0, thus contradicting (16:6).

t..

0, hence by (16:7) 81

=

•

L ii > O.

Hence

=

.

We replace (16:7)

i_I

by its corollary

•

~ ali, i)';

; _1

NowPutXi=li/ ~

t, forj= I , ' "

" o. ,m. Then we have

i_I

(16:5) gives

XI ~

0, . . .

•

L Xi =

1 and

i- I ,X .. ~

O.

Hence belongs to SOl

(16,9)

and (16 :8) gives (16010)

•

~ ali, j)x; " 0

i = I , ..

fo'

,n.

i .. 1 ~

Consider, on the other hand, the possibility that C does not contain O . Then the theorem (16 :B) of 16.3. permits us to infer the existence of a ~

hyperplane which contains y (cL 06 :2 :a) in 16.2.1. ), such that all of C is contained in one half-space produced by that hyperplane (cf. (16 :2 :b) in 16.2. 1. ). Denote this hyperplane by

•

~ a,x, ~ b. ~

Since 0 belongs to it, therefore b = O.

•

~ o,x, , .. I

So the half space in question is

> o.

140

-x

"

6

I,

ZERO-SUM TWO-PERSON GAMES: THEORY ,x .. ,6', . . . , ---> 6

"

(I6:!1) becomes ~

Stating it for

Xi,

R

belong to this half space.

> 0, i.e.

0;6 11

Stating this for

> O. So we have

aj

(16: 11 ) becomes

"

~ a(i, j)a;

(16"3)

> O.

;- 1

NOW Put Wi =

a;/ ±

a;

fori = I ,

..

;-1

(l6:12) gi ves

WI

>

0, . . . ,

WR

>

Then we have

L."

Wi

= 1 and

;-1

Hence

O.

{WI, . • . , VJR

,no

belongs to SR'

}

And (16:13) gives

•

~ a(i, J)W;

.;>

0

j = 1, '

fo'

"

m.

i-I

Summing up (16:9), (16:10), (16:14), (16:15), we may state : Let a rectangular mat rix with n rows and m columns be given. Denote its matrix element by a(i, j), i = 1, . . . , n; Then there exists either . ,x.. } in S. with

. . . , m.

r• a(i, j)Xj ~ 0 i-I

or a vector

W

•

=

~ a(i, j)w;

{WI,

>0

i

fo'

",

1,

=

LL

-

vector x =

" , n,

w.. l in S. with

fo,

j

1, . . .

=

,ffl.

;- 1

We observe furth er: The two alternatives ( 16:16 :a), (16 :16:b) exclude each other. Proof: Assume both (16 :16:a) a nd (16 :16 :b) . Multiply each (16:16 :8.) by Wi

and sum over i = I, .

, n; t.his gives

r•

•

L. a(i,j)w;Xj ~ O.

;- l i - I

Multiply

LINEARITY AND CONVEXITY eaeh (16:lfi:b) by

X,

141

and sum over i = I, . . . , lit; this gives

•

I I

a(i, i)w,x, > 0. 1

;_1 j_l

Thus we have a contradiction . 16.4.2. We replace t he matrix a(i, i) by its negative transposed matrix; i.e. we denote the columns (and not, as berore, the rows) by i = I , . . . ,n and the rows (and not, as before, the columns) by i = I , . . . ,m. And we let the matrix element be -a(i, i) (and not, as before a(i, i». (Thus n, m too are interchanged.) We restate now t he final results of 16.4.1. as applied to this new matrix.

-

--

But in formulating them, we let

w.J

W = /WI, . • • , /Xlo . • • ,x .. ! had .

x

I

-

", x:"1 play the role whic h " w'.! the role which x ~

= !x~,

had, and w' = Iw:,' And we announce the result in terms of the original

matrix. Then we have: Let a rectangular matrix with n rows alld m columns be given. Denote its matrix element by a(i, j), i = 1, . , n; j = I, ..

Ix;, ...

-

, m. Then t here exists either a vector x' = , x:"J in S... wit h

•

Z o(i, j)z', < 0 j-I

-

ora vector w'

=

i

Iwl ,

Z• o(i, j)w; ~ 0

,w~1

fo'

=

,n,

I , ..

in S. with

j = 1, . . . ,m.

i _I

And the two alternatives exclude each other. 16.4.3. We now combine the result.s of 16.4.1. and 16.4.2. They imply that we must have (16:17:a), or (I6:16:b), or ( 16: 16:a) and (16:17:b) simultaneously; and also that these three possibilities exclude each other.

--

--

Using the same matrix a(i, j) but writing x, w, ............ x', W I for the vectors

............ x I, W, I

x,

W I

in 16.4. 1., 16.4 .2. we obtain this:

> 0 and not only i:: O. Indeed, - 0 would nece8l!itate %,

•

is impossible since

Zz, i -I

1.

-

•• -

%.. -

Owhich

142 (I6oE)

ZERO-SUM TWO-PERSON GAMESo THEORY

-

T here exists either a vector x

•

(16 01Soa)

L. j-I

a(i, j)x;

-

or a vector w

•

(l6o ISob)

[WI,

=

~ a(i, j)Wl i_I

0

foc

•

( 16 oISoo)

, n,

fo,

,m,

j = 1, ..

-

,x~linS .. andw'=

0

fo,

i = I , ...

;;;;; 0

fo,

j = I,

•

L. a(i, j)w:

= 1, .

. ,w .. 1 in 8 .. with

or two vectors x I = [X'" in 8 .. with

L a(i, j)xj :;;

i

. , x ... 1 in S,.. with

{XL,

=

{w'.,··· ,w~1

,n,

. 1m.

i _I

The three alternatives (16: 18:a), (l6:8:b), (16:8:c) exclude each other. By combining (16:18:a) and (16: 18 :c) on one hand and (l6:l8:b) and (16 :18:c) on the other, we get this simpler but weaker statement.l,~ ( l .oF)

-

T here exists either a vector x

•

(16019oa)

L a(i, j)Xi ~ j -)

-

or a vector w

=

0

{WI,

•

(160 190b)

L o(i, j)w, "

0

fo'

=

S..

•

J

x .. ] in S", with

, n,

i = 1,

" , w .. 1 in

fo,

[Xl,

with

j = I , . . . ,m.

16.4.4. Consider now a skew symmetric matrix a(i, j), i.e. one which coincides with its negative transposed in the sense of 16.4.2.; i.e. n = m and

a(i, j) = -aU, i)

fo,

i,j=l,···,n.

I The two alternatives (16:19:a), (16:19:b) do not exclude each other : Their conjunction is precisely (16:18:c). 'This result could al90 have been obtained directly from the final result of 16.4.1.: (16:19:&) is precisely (16 :16:a) there, and (l6: 19 :b) is a weakened form of (16:16 :b) there. We gave the above more detailed diSCUSIlio n because it gives a better inaight into the entire situation.

MIXED STRAT EGI ES. T H E SOLUTION

143

T hen the condit ions (l6: 19:a) and (16:19:b) in 16.4.3. express t he same t hing : Indeed, (16:19:b) is

•

L

a(i, j)Wi ~ 0;

this may be written

•

•

- Z aU, i)Wi ~ 0, We need only write j, i for i, j ~

t hen

\\'C

L.

a(j,i)w; ;;;; O.

•

1

so that t his becomes

~

x fo r w,! so that

0'

have

.

L. a(i, j)x;;;;;; O.

L a(i, j)w; ;;;; 0, and And this is precisely

( 16 ,'h). Therefore we can replace the disjunction of (16: 19:a) a nd (16: 19:b) by either one of them,-say by ( 16: 19:b) . So we obtain : If t he matrix a(i, j) is skew-symmetric (and therefore n = m ~

cf. above), then t here exists a vector w = [WI, ... , w .. } in S .. with

•

L1ali, i)w, 1: 0 ._ 17. Mixed Strategies.

fo'

j = I,

",

n.

The Solution for All Games

17.1. Discussion of Two Elementary Examples

17.1.1. In order to overcome t he difficulties in the non-strictly determined case--which we observed particularly in 14. 7.-it is best to reconsider the simplest examples of this phenomenon. T hese are the games of Matching Pennies and of Stone, Paper, Scissors (cf. 14.7.2., 14.7 .3.). Since an empirical, common-sense attitude with respect to the" problems" of t hp.se games exist.s, we may hope to get a clue for the solution of non-strictly determined (zero-sum two-person) games by observi ng a nd analyzing these attitudes. I t was pointed out that, e.g. in Matchi ng Pennies, no particular way of playing- i.e. neither playing "heads" nor playing "tails"- is allY better t han the other, and all that matters is to fi nd out the o pponent's intent ions. This seems to block the way to a soilltion, since the rules of the game in question explicit ly bar each player from the knowled ge about the o pponent's actions , at the moment when he has to make his choice. But , Observe that now, wit.h

110 -

m this is only a change in notation!

144

ZERO SUM TWO·PERSON GAMES: THEORY

the above observation does not correspond full y to t he realities of the case; In playing Matchi ng Pennies against an at least moderately intelligent opponent, the player will not at tempt to find out the opponent's intentions but will concentrate on avoidi ng hav ing his own intentions found out, by playing irregularly" heads" and "tails" in successive games. Since we wish to describe the strategy in one play- indeed we must discuss the course in one play and not that of a sequencc of successive plays-it is prefcrable to express this as follows: The player's strategy consists neither of playing "tails" nor of playing "heads," but of playing "tails" with t he probability of t and "heads" with the probability of t. 17.1.2. One might imagine that in order to play Matching Pennies in a rational way the player will- bcfore his choice in each play-decide by some 50:50 chance device whether to play "heads" or "tails. " l The point is t hat t his procedure protects him from loss. Indeed, whatever strategy t he opponent follows, the player's expectation for the outcome of t he play will be zero.2 This is true in particular if with certainty the opponent plays "tails," a nd also if with certainty he plays "heads"; a nd also, finally, if he- like the player himself-may play both "heads" and "tails," with certain probabilit ies. 3 Thus, if we permit a player in Matching Pennies to use a "statistical" strategy, i. e. to " mix " the possible ways of playing with certain proba· bilities (chosen by him), then he can protect himself against loss. Indeed, we specified a bove such a statistical strategy with which he cannot lose, irrespective of what his opponent docs. The sa me is true for t he opponent, i.e. t he opponent can use a statistical strategy which prevents t he player from winning, irrespective of what t he player does. ~ The reader will observe the great similarity of t his with the discussions of 1 4 . 5.~ In t he spirit of those discussions it seems legitimate to consider zero as the value of a play of Matching Pennies and t he 50:50 statistical mixture of "heads" a nd "tails" as a good strategy. The situation in Paper, Stone, Scissors is entirely similar. Common sense will tell t hat the good way of playing is to play all three alternatives with t he probabilities of teach .· The value of a playas well as the inter· L E.g. he could throw a dic-Qf course without letting the opponent see the resulland then play "tails" if the number of spots showing is even, and '"heads " if that num· ber is odd. , I.e. hia probability of winning equals hia probability of losing, because unde r these conditions the probability of matching as well as that of not matching will be \, whatever the opponent's conduct. 'Say p, I - p. For the player himself we used the probabilities I. i . • All this, of CQUTlle, in the statistical sense: that the player cannot lose. melLna that his probability of losing is ;;:;: his probability of winning. That he cannot win , means that the former is ~ to the latter. Actually each play will be won or lost, since Matching Penniea knows no t iea. f We mea n specifically {l 4 :C:d ), (14 :C:c) in 14 .5.1. I A challce device could be introduced as befor('. The die Ill('nlioned in (oolnol(' I, abo\'c, 1I"0uid be a possible one. F..II:. the player could deeide "stone" if lor 2 ~P()ts show, " paper" is 3 or 4 spots show, ··scis~ors" if 5 or tl show.

MIXED STRAT EGIES.

TH E SOLUTION

145

pretation of the above strategy as a good one can be motivated as before. again in the sense of the quotation t here. ' 17.2. Ge neralization of This Viewpoin t

17.2.1. It is plausible to try to extend the results fou nd for Matching P ennies and Stone, P aper, Scissors to all zero-sum two· person games. We use the normalized form, the possible choices of t he two players being t, = 1, .. ,131 and t2 = I, . . . ,fJ2, and t he outcome for player 1 JC(rl' 72), as formerly. We make no assum ption of strict determinateness. Let us now try to repeat the procedure which was successfu l in 17. 1.; i.e. let us again visualize players whose "theory" of the game consists not in t he choice of definite strategies but rat her in t he choice of several strategies with defi nite probabilities. 2 T hus player 1 will not choose a number TI = 1, , PI- i.e. the corresponding strategy 2:~,-but PI nu mbers ~I, . , ~II" - the probabilities of t hese strategies 1:;, ... , 2:1" respectively. Eq~ally player 2 will not choose a number T 2 = 1, . . . , tJ:-i.e. the correspond ing strategy 2:;>- but tJt numbers 1)1, • , 1)II",- t he probabilities of these strategies 2:~ , . . . , 2:{" respectively. Since these probabilities belong to disjoint but exhaustive alternatives, t he nu mbers ~." 1)., arc subject to the conditions all ~" ~ 0, a ll 1).,

~

~

=

~

0,

L ""

{~"

. . . , ~~,I

and to no others. We form the vectors ~

.,

., - 1

-

I.

~

a nd"

=

{"I,'

" ~

"",I . to S",

T hen the above conditions state t hat ! must belong to SII", and 1) in t he sense of 16.2.2. I n t his setup a player docs not, as previously, choose his strategy, but he plays all possible strategies and chooses only t he probabilities with which he is going to play them respectively. This generalization meets t he ma jor difficulty of t he not strictly determined case to a certain point: We have seen that t he characteristic of t hat case was t hat it constituted a defin ite disadvantage 3 for each player to have his intentions fou nd out by his

I I n Stone, Paper, &:iSl!ors there exists 8. tie, but no loss still means that the probability of 10sinJl: is ;;; t he prohability of winning, and no gain means the reve rse. cr. footnote 4 on p. 144. t That these probabilities were the same for all strategies (~, i or I, I. t in the examples of the last paragraph ) was, of course accidental. It is to be expected tha t thi" equality wa" due to the symmetric way in which the var ious alternatives appea red in those games. We proceed now on the assumpt ion that the a ppearance of probabilitiel:l in formulating a strategy was the essential thing, while the pa rticular values' we re accidental. 'The j, > 0 of 14.7.1.

146

ZERO-SUM TWO-PERSO N GAMES, THEORY

opponent. Thus one important consideration] for a player in such a game is to protcct himself against having. his intentions found out by hi s opponent. Playing several different strategies at random, so that on ly their probabilities are determined, is a very effective way to achieve a degree of such protection: By this device t he opponent cannot possibly find out what the player's strategy is going to be, since the player does not know it himself.2 Ignorance is obviously a very good safeguard against disclosing information

directly or indirectly. 17.2.2. It may now seem that we have incidentally restricted the player's freedom of action. It may happen, after all, that he wishes to play one definite strategy to the exclusion of all others; or that, while desir· ing to use certain strategies with certain probabilities, he wants to exclude absolutely the remaining ones. 3 We emphasize that these possibilities are perfectly within t he scope of our scheme. A player who does not wish to play certain strategies at all will simply choose for them the probabilities zero. A player who wishes to play one strategy to the exclusion of all others will choose for this strategy the probability 1 and for all Qther strategies the probability zero.

-

Thus if player 1 wishes to play the strategy

-

the coordinate vector

2;;, and the vectors

-

1)

~',

-

(cL 16.1.3.) .

Similarly for player 2, the strategy

and 3 '0.

In view of all these considerations we call a vector

--

-

2:;, only, he will choose for (

~

of S8, or a vector

" of S(I, a statistical or mixed strategy of player 1 or 2, respectively.

The

coordinate vectors 3', or 3 " correspond, as we saw, to the original strategies T] or Tt-i.e. I'i' or Itt-of player 1 or 2, respectively. We call them strict or pure strategies. 17.3. Justification of the Procedure As Applied to an Individual Play

17.3.1. At t his stage the reader may have become uneasy and perceive a contradiction between two viewpoints which we have stressed as equally vital throughout our discussions. On the one hand we have always insisted that our theory is a static one (d. 4.8.2.), and that we analyze the course I But not necessarily the only one . • If the opponent has enough statistical experience about the player's "style," or if he is very shrewd in rationalizing his expected behavior, he may discover the probabilities - frequeneieS-Qf the various strategies. (We need not discuss whether and how this may happen. Cf. the argument of 17.3.1.) But by the very conccpt of probability and randomness nobody under any conditions can foresee what will actually happen in any particular case. (Exception must be made for such probabilities as may vanish; ee. below.) l In this case he clearly incrcases the danger of having his strategy found out by the opponent. But it may be that the strategy or stratcgies in question have such intrinsic advantagea over the othcrs as to make this worth while. This happenB,-e.g. in an extreme form for the " good II strategies of the stri('tly determined case (d. 14.5. , particu. larly (14:C:a), (14:C;b) in 14.5.2.).

MIXED STRATEGIES.

THE SOLUTION

147

of one play and not that of a sequence of successive plays (cf. 17.1.). But on the other hand we have placed considerations concerning the danger of one's strategy being found out by the opponent into an absolutely central position (cL 14.4., 14.7.1. and again the last part of 17.2.). How can the strategy of a player-particularly one who plays a random mixture of several different strategies-be found out if not by continued observation! We have wled out that this observation should extend over many plays. Thus it would seem necessary to carry it out in a single play. Now even if the rules of the game should be such as to make this possible--i.e. if they lead to long and repetitious plays-the observation would be effected only gradually and successively in the course of the play. It would not be available at the beginning. And the whole thing would be tied up with various dynamical considerations,-while we insisted on a static theory! Besides, the rules of the game may not even give such opportunities for observation; I they certainly do not in our original examples of Matching Pennies, and Stone, Paper, Scissors. These conflicts and contradictions occur both in the discussions of 14.-where we used no probabilities in connection with the choice of a strategy-and in our present discussions of 17. where probabilities will be used. How are they to be solved? 17.3.2. Our answer is this: To begin with, the ultimate proof of the results obtained in 14. and l7.i.e. the discussions of 14.5. and of 17.8.- do not contain any of these conflicting elements. So we could answer that our final proofs are correct even t hough the heuristic procedures wh ich lead to them are questionable. But even these procedures can be justified. We make no concessions: Our viewpoint is static and we are analyzing only a single play. We are trying to find a satisfactory thcory,- at this stage for the zero-sum twoperson game. Consequently we are not arguing deductively from the firm basis of an existing theory- which has already stood all reasonable testsbut we are search ing for such a theory.2 Now in doing this, it is perfectly legitimate for us to use the conventional tools of logics, and in particular that of the indirect proof. This consists in imagining that we have a satisfactory theory of a certain desired type,3 trying to picture the consequences of this imaginnry intellectual situation , and then in drawing conclusions from this as to what the hypothetical theory must be like in detail. If this process is applied successfully, it may narrow t he possibilities for the hypothetical theory of the type in question to such an extent that only one 'I.e. "gradual," "successive" observations of the behavior of the opponen t within one play. 'Our method is, of course, the empirical one: We are t rying to understand, formalize and generalize those features of the sim plest !l:amcs which im(Jrcss us 88 typi cal. This is, after all, the standard method of all scienc('S with an cmpiriraJ basis. I This is full cognizance of the fact that we do not (y ... t ) po>iSess one, and that we cannot imagine (,vet) what it woulJ be lik ... , if we hud one. All this is- in i~ own domain- no worse than any other indircet proof in uny part of science (e.g. the pu ab8urdum proofs in mathematics and in physics).

148

ZERO-SUM TWO-PERSON GAMES , THEORY

possibility is ieft,- i.e. that the theory is determined , discovered by this device. I Of course, it can happen that the application is even more" successful ," and that it narrows the possibilities down to nothing- i.e. that it demonstrates that a consistent theory of the desired kind is inconceivable. 2 17.3.3. Let us now imagine that there exists a complete theory of the zero-sum two-person game which tells a player what to do, and which is absolutely convincing. If the players knew such a theory then each player would have to assume that his strategy has been "found out" by his opponent. The opponent knows the theory, and he knows that a player would be unwise not to follow it.I Thus the hypothesis of the existence of a satisfactory theory legitimatizes our investigation of the situation when a play· er's strategy is "found out" by his opponent. And a satisfactory theory ' can exist only if we are able to harmonize the two extremes r 1 and r t , strategies of player 1 "found out" or of player 2 "found out." For the original treatment- free from probability (i.e. with pure strategies)- the extent to which this can be done was determined in 14.5. We saw that the strictly determined case is the one where there exists a theory satisfactory on that ba8is. We are now trying to push further, by using probabilities (i.e. with mixed strategies) . The same device which we used in 14.5. when there were no probabilities will do again,- the analysis of "finding out " the strategy of the other player. It will turn out that this time the hypothetieal theory can be determined completely and in all cases (not merely for the strictly determined case--cf. 17.5.1. , 17.6.). After t he theory is found we must justify it independently by a direct argument.~ This was done for the strictly determined case in 14.5., and we shall do it for the present complete theory in 17.8. 1 There are several important examples of this ~ rformance in physics. The successive approachcs to S~cial and to General lli!lativity or to Wave Mechanics may be viewed R8 such. Cf. A. D'Abro: The Decl ine of l\.lechanism in Modern Physics, New York 1939. • This too OCCUrfl in physics. The N. Bohr-Heisenberg analysis of "quantities which are not simultaneously observable" in Quantum l\lechanics permits this interpretation. Cf. N . Bohr: Atomic Theory and the Deacrlption of Nature, Cambridge 1934 and P. A . M . Dirac: The Principles of Quantum Mechanics, London 1931 , Chap. I. I Why it would be unwise not to follow it is none of our concern at present ; we have assumed that the theory is absolutely convincing. That this is not impossible will appear from our final rcsult. We shall find a theory which is satisfactory; nevertheless it implies that the player's strategy is found out by his opponent. But the theory gives him the directions which permit him to adjust himself so that this causes no loss. (CL the theorem of 17.6. and the discussion of our complete solution in 17.8.) • I.e. a theory using our present devices on ly. Of course we do not pretend to be able to make" absolute" statements. If our presen t requirements should turn out to be unfulfi!lahle we should have to look for another basis for a theo ry. We have actually done this once by passing from 14. (with pure strategics) to 17. (with mixed stra.tegies). • The indirect argument, as ou tlined above, gives only ncccssary conditions. Hence it may establish absurdity (per absurdum proof), or narrow down the possibilities to onC i but in the latter c&!le it is st ill necessary to show that the one remaining possibility is satisfactory.

T H E SOLUT IO\'

l\H XED STRATEGIES.

17.' . The M iooraot and th e M a joraot Gamu

149

(For M i%ed Strat egies )

17.4.1. Our present picture is then that player I chooses a n arbitrary ~

element

S'"

~

~

from S, and t hat player 2 chooses an arbitrary element

~

the coordinate vector 6 " (cL 16. 1.3.); similarly for player 2, the strategy ~

~ ;,

from

T hus if player 1 wishes to play t he strategy :r'i' on ly, he will choose for

~

~

'I

'

and the vectors

'1

~

and 6 ',. ~

We imagi ne again t hat player 1 ma kes his choice of

~

in ignorance of

~

player 2'8 choice of '1 and vice versa. T he meani ng is, of course, that when these choices have been made player I will actually use (every) TI .. I, . . . , ~I with t he probabilities ~" and the player 2 will use (every) T2 "'" I, . . . '~2 with the probabilities '1',' Since their choices a re independent, the mathematical expectation of the outcome is ~

( 17,2)

~

K( ( • , ) -

" xh, ")!'."" L" .,_1 L

" - I

In other words, we have replaced t he original game r by a new one of essentially the same structure, but with the following formal diffcrences: T he numbers TI, Tr-the choices of the players- are replaced by the vectors ~

t' ,

~

If .

T he functio n Je(TI, T2)- t he outcome, or rather the" mathematical ~~

expectation" of the outcome of a play- is replaced by 1( ~ , 'I)' All these considerations demonstrate t he identity of structure of our present view of f with that of 14.1. 2.,-the sole difference being t he replacement of

........................

Je(TI, TI) by t', '1, 1( t', '1), described above. T his isomorphism suggests the a pplication of t he sa me devices which we used on t he original r, t he com parison with the rnajorant and minorant games f l and f , as described in 14.2., 14.3. 1., 14.3.3.

T l, T2,

~

~

17.4.2. Thus in f l player I chooses his

~

first and player 2 chooses his

'1

~

afterwards in full knowledge of the ~ chosen by his opponent. In f, the order of their choices is reversed. So the discussion of 14.3.1. appliC!!I ~

literally.

Player I, choosing a certain

......

choose his II , so as to minimize K( ~

the value Min.... K (

•

............ ~

,

~,may

'1 );

expect t hat player 2 will

i.e. player 1's choice of

~

~, J)

...... ~

leads to

~

.

This is a fu nction of t' alone ; hence player 1

~

~~

should choose his t' so as to maximize Min- K( a play of fl is (for pl ayer 1)

•

~

,

'1).

Thus t he value of

150

ZERO-SUM TWO-PERSON GAMES, THEORY v;

=

.

,

Max- Min-

1(

-~,

I).

--

Similarly the value of a play of r t (for player I) turns out to be v~

.

,

= Min- Max- K( ~,

I).

(The apparent assumption of rational behavior of the opponent does not really matter, since the justifications (14 :A:a)-(14:A :e), (14 :B:a)-(14:B:e) of 14.3.1. and 14.3.3. again apply literally .) As in 14.4.1. we caD argue that the obvious fact that r 1 is less favorable for player I than r t constitutes a proof of

--

and that if this is questioned, a rigorous proof is contained in (13:A·) in 13.4.3. that

The x, y, q, there correspond to our

~

,

I)

,K.I

If it should happen

v~ = v~,

then the considerations of 14.5. apply literally.

-

-

The arguments (l4:C:a)-

(14:C :f), ( 14 :D:a), ( 14:D :b) loc. cit., determine the concept of a "good"

~

and,! and fix the "value " of a play of (for the player I) at Vi =

v;

=

v;.z

--

All this happens by (13 :B*) in 13.4.3. if and only if a saddle point of K exists.

(The x, y, ¢ there correspond to our

~, '1,

K.)

17.6. GeDeral Strict DetenniDl,teDeSl

17.6.1. We have replaced the vI, V z of (14:A:c) and (14:B:c) by our present v~, v~, and t he above discussion s hows that the latter can perform the function s of the former . But we are jus t as much dependent upon v~ = v~ as we were then upon VI = v 1 . It is natural to ask, therefore, whether there is any gain in this substitution. Evidently this is the case if, as and when there is a better prospect of having v; = v~ (for any given r) than of having VI = Vi. We called r 8trictly determined when VI = Vz; it now seems preferable to make a distinction and to designate r for VI = V2 as 8pect'ally 8trictly determined, and for v~ = v~ as generally 8triCtly determined. This nomenclature is justified only provided we can show that the former implies the latter.

--

Although E, " are vectors, i.e. sequences of real numbers (E,,' ,E~ and , "" . . . ,"~,) it is perfectly admi!lllible to view each as a single va riable in the maxima and minima which we are now forming . Their domains are, of course, the &et8 8(3" 8~. which we introduced in 17.2 . • For an exhaustive repetition of the arguments in question cr. 17.8. 1

MIXED STRATEGIES.

THE SOLUTION

151

This implication is plausible by common sense: Our introduetion of mixed strategies has increased the player's abi lity to defend himself against having his strategy found out; so it may be expected that v~ , v~ actually lie between VL, V2. For this reason one may even assert that (This inequality secures, of cou rse, the impl ication just mentioned.) To exclude all possibility of doubt we shall give a rigorous proof of (17:3). It is convenient to prove this as a corollary of another lemma. 17.6.2. First we prove t his lemma: ~

For every ~ in S~, ~

II,

~

~,

2:

Min, K ( E, '1) = Min-,

.,

L

= Min.,

~ X('TL, 'T2H.,I1~, X('T!, 'T1H~,·

', _ L ~

For every

,

11

in S/I, ~

~

/I,

,

II,

L

Max---< K( E, '1) = Max---< ~

" L

= Max.,

X('TL,'TzH.,'1.,

X(1'1, 1'2)'1, •.

• ,_1

Proof: We prove t he first formula only; t he proof of the second is exactly the same, only interchanging Max and Min as well as ~ and ~. ~

~,

Consideration of the special vector '1 = 05 ', (cf. 16. 1. 3. and the end of 17.2.) gives

Since this is true for all 1';, so J\'Iin.;

" L

X('T!, 'T ;H. ,.

" _ L

On the other band, for all

1' 1

/I,

II,

L X(1'\, 'T2H., ~ Min., L X(1'L,1't)E.,.

,, - I ~

Given any '1 in Sit,. multiply this by '1' , and sum over 1'Z = 1, . . . , Ph Since

L•

', - I

IJ~. = 1, therefore

152

ZERO-SUM TWO-PERSON GAMES: THEORY ~,

~

~,

~

~ Je(rL, rZ)~ •.'/T, ~ Min"

~ le(rL' rtH., " -I

~

results.

Since this is t rue for all

7J,

so

(17:4:a), (17:4:b) yield together the desired relation. If we combine the above formulae with the definition of v~ , v~ in 17.4., then we obtain

.,

v; = Max!' Min"

v~ = l\:lin - l\"lax,

"

~ Je(rL' ., - 1

.,

7tH."

'" Je{rL , rth •.

' ','-'_ L

,

These formul ae have a simple verbal interpretation: In computing v~ we need only to give player 1 the protection against having his strategy found ~

~

out which lies in the usc of

(instead of rL); player 2 might as well proceed ~

in the old way and use rt (and not 11). In computing v~ the roles are interchanged. This is plausible by common-sense: v; belongs to the game r L (cL 17.4. and 14.2.); there player 2 chooses after player 1 and is fully informed about the choice of player I ,- hence he needs no protection against having his strategy found out by player 1. For v; which belongs to the game rz (cf. id.) the roles are interchanged. Now the value of

v;

~

becomes

~

~

if we restrict the variability of ~

,

the Max- of the above formula.

Let us restrict. it to the vect.ors

(r'l = 1, . . . ,Pio cf. 16.1.3. and the end of 17.2.).

~

in

~,

=

6',

Since

,, _I this replaces our expression by l\'Iax,', Min. , Je(r;, rt)

= VL.

So we have shown that

- --,

Similarly (cL the remark at the beginning of the proof of om lemma above) restriction of

1'/

to the

7J

=

6', establishes

MIXED STRATEGIES. Together with

v~

;:;;

v~ (cf.

THE SOLUTION

153

17.4.), these inequalities prove

as desired. 17.6. Proof of the Main Theor em

17.6. We have established t hat general strict determinateness (v~ = v~) holds in all cases of special strict determinateness (VI = V2) 11.8 is to be expected . That it holds in some further cases as well- i.e. that we can have v; = v~ but not v\ = V2 - is clear from our discussions of Matching ' Pennies and Stone, Paper, Scissors.! Thus we may say, in the sense of 17.5.1. that the passage from special to general strict determinateness does constitute an advance. But for all \\Ie know at this moment this advance may not cover the entire ground which should be controlled; it could happen that certain games r are not even generally strictly deterrriined,- i.e. we have not yet excluded t he possibility

v; < v;. If this possibility should occur, t hen all that was said in 14.7.1. would apply again and to an increased extent: finding out one's opponent's strategy would constitute a definite advantage v~

11' =

- v; > 0,

and it would be difficult to see how a theory of t he game should be constructed without some additional hypotheses as to "who finds out whose strategy. " The decisive fact is, therefore, that it can be shown that this never happens. For all games r I.e.

,

.

Max- Min- K(

.

~~

~,

,/)

,

~

Min- Max- K(

~

~,

,/),

or equivalently (again use (13:B*) in 13.4.3. the x, y, ¢ there corresponding to our

--+

~,

--+ 7),

K): A saddle point of K{

--+--+

~,

,/) exists. ~

~

This is a general theorem valid for all functions K{ ~, ,/) of the form (IU)

',_1 ••_ 1

The coefficients :le(T!, Tt) are absolutely unrestricted; they form, as described ~

in 14.1.3. a perfectly arbitrary matrix.

The variables

~,

~

7)

are really

, In both games V, _ -1, v. _ 1 (eL 14.7.2., 14.7.3.), while the discussion of 17.1. ean be interpreted Ra establishing v; _ v; .. O.

154

ZERO-SUM TWO-PERSOJ\ G AM ES, THEORY

sequences of real numbers

, ~'I

~"

llnd 'II,

.. . , 1/.,: their domains ~

being the sets S" S~ (cf. footnote I on p. 150) . The function s K ( ~, II ) of the form (17:2) ar~ called bilinear form s. With the help of the results of 16.4.3. the proof is easy. ' This is it : We apply (16:19:8.), (16:19 :b) in 16.4.3. replacing the i , j, n, m, a (i, j)

there by our

1'" Tt,

--

--

fJ" /11. JC (TI , Tt) and the vecton> w , x there by our ~ , ~

If (l6: 19:b) holds, then we have a.

" JC(TJ. T!)~., ~ 0 L

~

in S, , with

T, =

fo,

.,_1

I).

1,

i.e. with Min.,

k"

•,-1

JC(Th TtH., ~ O.

Therefore the formula (17:5:8.) of 17.5.2. gives v~

e; O. ~

If (16: 19:a) holds, then we have an

k" X CTI' Tlh. , ;;§;; 0

.,_1

fo,

11

in S, , with

1', =

I , .. .

, t3 "

I Th~ theorem occurred and " ' lUI proved fi,.t in the original publication of one of the authora on the theory of gamel: J . I'On Neumann : " Zur Theorie der Gesellschdwpieie," Math . Annalen, Vol. 100 (1928), pp. 29S-320. A slightly mOrfl general form of this r.lin-Ma lC problem arises in another qUeltion of mathematical e v', irrespective of what player 1 does. Player 1 can, by playing appropriately, make it sure that the gain of player 2 is ;:;;: -VI , Le. prevent him from gaining > -Vi, irres pective of what player 2 does.

17.8.2. T hird, we may now assert-on the basis of (17:C:d) and (17:C:e) and of the considerations in the proof of (I7:C:d)- that:

-

(I7,C,,)

The good way (combination of strategies) fo r I to play

(17,C ,b)

the game r is to choose any ~ belonging to A- , -A- being the set of (l7:B:a) above. The good way (combination of strategies) for 2 to play the game r is to choose any of ( li: B:b) above.

I)

belonging to B,- B being the set

Fourth , combination of the assertions of (17:C:d)- or equally well of those of (17 :C :e}- gives: (17,C,,)

-

-

If both players I and 2 play the game r well- i.e. if

belongs to A and I) belongs to B- then the value of J{ ( will be equal to the value of a play (for I),-i.e. to v'.

~

~, ~ I)

We add the observation t hat (13:D*) in 13.5.2. and the remark concerning the sets A, f3 before (l7:B:a), (I7:B:b) above give together this; (17,CJ)

--

-

Both players 1 and 2 play the game r well-Le. ~ belongs to A and I) belongs to fJ-if and only if ~~, J/ is a sadd le point

of K( L

").!

160

ZERO-SUM TWO-PERSON GAMES , THEORY

All this s hould make it amply clear that v' may indeed be interpreted as the value of a play of r (for I ), and t hat A, jj contain t he good ways of playing r for 1, 2, respectively. There is nothing heuristic or u ncertain about the enti re argumentation (17:C:a)-( 17:C :f) . We have made no extra hypotheses about t he "intelligence" of t he players, about "who has found out whose strategy" etc. Nor are our results for one player based upon any belief in t he rational conduct of the other,- a point the im portance of which we have repeatedly stressed. (eL t he end of 4. 1.2.; also 15.8.3.) 17.9. Further Chautteristics of Good Strategies

17.9.1. The last results- ( 17:C:c) and (17 :C:f) in 17.8. 2.- givc also a simple explicit characterizat ion of the elements of our present soiut ion,i.e. of t he number v' and of the vector sets A and B. By (17:C:c) loc. cit. , A, tJ d etermine v'; hence we need only study A, B, and we shall do this by means of (1 7: C:f) id. ~

--

~

According to that criterion, ~ belongs to A, and 7J belongs to tJ if and only if

~,

--

7J is a saddle poin t of K ( ~, 7J).

~ ~

K(

~ ,

This means that

{M.,-" K((', ,) ~

7J) =

~

__

Min-;: K ( ~,

11 ')

We make t his explicit by using the expression (17:2) of 17.4.1. and

-~

~,

17.6. for K( Max--- K (

e

~

~', 7J )

and Min- K(

.'

Il,

Considering that

~ [Max.;

., -1

~" =

.,-1

,').

Then our eqnations become:

L

11.,

= 1, we can also write for t hese

.,_1 Il,

Il,

1.,~- 1Je(,;, ,.h.,} - .,~-1 Jeh ,.)",] (', ~ 0,

~

~

~,

Il,

L " - I

Il,

--

7J), and the expressions of the lemma ( 17:A) of 17.5. 2. for

Il,

[ -M ;n,;

Il,

1.,~_1xl"~ , ' ;)(" } + .,_1~

Je(", ")("] ",

Now on the left·hand side of these equations the ~. , which are all ~ 0.' The ~." 'I., themselves are als~ I

Obse rve how the Max and Min occur there!

'I.

~

0,

have coefficien ts

~ O. Hence these

MIXED STRATEGIES.

THE SOLUTION

16 1

equations hold only when all terms of their left hand sides vanish separately. I.e. when for each TI = 1, " , (31 for which the coefficient is not zero, we have~. , = 0; and for each T~ = 1,' " i3~ for which the coefficient is not zero, we have 'I., = O. Summing up: ~

~

~

belongs to A and '1 belongs to {j if and only if these are true: For each

TI

=

not assume its maximum (in TI) we For each

T!

L" JC(TI,

" , (31, fol" which

1,

= 1,'

have~. ,

i3!, for which

"

T~h., does

= O.

L"

JC(TI'

TtH.,

does

" - I

not assume its minimum (in

T~)

we have

It is easy to formulate these principles verbally. ~

~

~, 'I

'I., =

O.

They express this: If

~

are good mixed strategies, then

~

excludes all strategies TI which are

~

~

not optimal (for player 1) against '1, and

'I

~

excludes all strategies Tt which ~

~

are not optimal (for player 2) against ~; i.e. ~, 'I are-as was to be expected- optimal against each other. 17.9.2. Another remark which may be made at this point is this: The game is speeially strictly determined if and only if there exists for each player a good strategy which is a pure strategy. In view of our past discussions, and particularly of the process of generalization by which we passed from pure strategies to mixed strategies, this assertion may be intuitively convincing. But we shall also supply a mathematical proof, which is equally simple. This is it: We saw in the last. part of 17.5.2. that both VI and v~ obtain by applyi ng ~,

Max..... to Min, I

'

~ JC (Tt, .,£...J_1

T2H.,,

-+

only with different. domains for ~: The set

~

of all 6"(TI = 1,'" ,(3I) for vl,andallof S~,forv~;i.e.thepurestrategies in the first case, and t.he mixed ones in the second. Hence VI = v~, i.e. the two maxima are equal if and only if the maximum of the second domain is assumed (at least once) within the first domain. This means by (17:D) above that (at least) one pure strategy must belong to A, i.e. be a good one. I.e. VI = v~ if and only if there exists for player 1 a good strategy which is a pure strategy.

162

ZERO-SUM TWO-PERSON GAMES, THEORY

Similarly: VI = v; if and only if there exists for player 2 a good strategy which is a pure strategy.

Now i.e.

VI

=

v~ = Vii

v; = v' and strict determinateness means VI = VI = v', and v, = v;. So (17:F:a), (17:F:b) give together (17:E).

17.10. Mistakes Ind Their Consequences.

Permanent Optimality

17.10.1. Our past discussions have made clear what a good mixed strategy is. Let us now say a few words about the other mixed strategies. We want to express the distance from "goodness" for those strategies (i.e. ~

~

vectors ~, '/I) which are not good; and to obtain some picture of the consequences of a mistake- Leo of the use of a strategy which is not good. However, we shall not attempt to exhaust this subject, which has many intriguing ramifications. ~

For any

~

~

in

S~ , and

any " in ~

(17Jh)

a(

0

S~ , we

form the numerical functions ~~

=

v' - Min- K( L ,,),

•

~

~

(j( ,,) = Max7 K( L

By the lemma (17:A) of 17.5.2. equally ~

a( ~) = v' - Min., ~

f3( ,,) = Max"

"L

~

,,) - v'.

.,

L X(TJ, T,HT"

.,-1

X(TI, T2)"., -

Vi.

,, _ I

The definition v' = Max- Min- K( f.

-~,

,,) = Min- Max ..... K( •

f

-~

I

,,)

guarantees that always

~

And now (17:B:a), (17:B:b) and (l7:C:a), (17:C:b) in 17.8. imply that

--

is good if and only if a(

~

~) =

~

~

~

--

0, and " is good if and only if (j( ,,) =

o.

Thus a( ~), (j( " ) are convenient numerical measures for the general ~, " expressing their distance from goodness. The explicit verbal formulation ~

~

of what a( ~), (j(" ) are, makes this interpretation even more plausible: The formulae (17 :13:0.), (17:13:b) or (17: 13:11.·), (17:13:b*) above make clear

MIXED STRATEGIES. THE SOLUTION

163

how much of a loss the player risks- relative to the value of a play for him I - by using this particular strategy . We mean here " risk" in the sense of the worst that can happen under the given conditions.: ~

~

It must be understood, however, that a( ~) , fj( '1) do not disclose which strategy of the opponent will inflict this (maxim um) loss upon the player who ~

~

is using

~

It is, in particular, not at all certain that if the opponent

or " .

~

~

uses some particular good strategy, i.e. an ,,~in fJ or a ! 0 in A, this in itself ~

~

If a (not good) ! or '1 is used by the

implies the maximum loss in question.

~

~

player, then the maximum loss will occur for those,,' or !' of the opponent, for which ~

--

(17, 1"-)

•

KU', ,,)

(l7'IH)

-

=

,

Max- K(

~

-t,

Min- 1«

--

I),

t, ,,),

i.e. if ,, ' is optimal against the given t, or t' optimal against the given ~

And we have never ascertained whether any fixed ~

'10

~

1).

or to can be optimal

~

against all t or

'1.

--

~

~

--

11.10.2. Let us therefore call an 7J' or a t ' which is optimal against all

t or 'If - i.e. which fulfill s (17: 14:a). or (l7:14:b) in 17.10.1. for all ~

~

t

7J-

permanently optimal. Any permanently optimal 7J' or €' is necessarily good; this should be clear conceptually and an exact proof is easy.- But L

- - --

~~

I.e. we mean by loss the value or the play minU8 theaetual outcome : v' - K(

~ I

,,)

for player 1 and (- v' ) - (- K ( ~, '1 )) - K ( ~ , ,,) - Vi for player 2. "Indeed, using the previous footnote and (17:13:&), (17:13:b)

-- ----

a(

~)

~l in_

- v' -

,

•

---

K( f, ,, ) - Max_ lv' - K(

•

,

i3(,,) - Ma)[- K( E, ,,) - v' - Ma)[-lK(

~,

~,

,,)1,

,,) - v'J.

I.e. each is a ma)[imum loss. ~

~

• Proof: It suffiees to show this for" 'i the proof for f' ia analogous. ~

~

Let ,,' be permanently optimal. Choose a ~

~

~

~

• which is optimal against" " i.e. with

,

~

~

K( ~ " ,,') - Ma)[ _ K( f, ,,')

By definitibn

--

--

--

K( E " ,,') - Min_ K(

•

--

-

~.,

,,) .

Thus E " ,,' is a saddle point of K ( E, ,,) and therefore ,,' belongs to 8-i.e. it isgooclby (17:C:f) in 17.8.2.

164

ZERO-SUM TIVO-PERSO" GAMES, THEORY

t.he question remains: Are all good strategies also permanently optimal? And even; Do any permanently optimal strategies exist? In general the answer is no . Thus in Matching Pennies or in Stone, Paper, Scissors, the only good strategy (for player I as well as for player 2) is ~

~

~

= "1 = It,-!Ior It, t.t] , respectiveiy.1

Ifplayer lplayeddifferentlye.g. always" heads"2 or always" stone "!- then he would lose if the opponent countered by playing" tails " l or" paper. "I But then the opponent's strategy is not good- i.e. I-t, il or Ii, t, i-l. respectively-either. If the opponent played the good strategy, then the player's mistake would not

matter. ' We shall get another example of this-in a morc subtle and complicated way- in connection with Poker and the necessity of "bluffing," in 19.2 and 19.10.3. All this may be summed up by saying that while our good strategies are perfect from the defensive point of vicw, they will (in general) not. get the maximum out of the opponent's (possible) mistakes,- i.e. they are not calculated for the offensive. It should be remembered, however, that our deductions of 17.8. are nevertheless cogent; i.e. a theory of the offensive, in this sense, is not possible without essentially new ideas. The reader who is reluctant to accept t his, ought to visualize the situation in Matching Pennies or in Stone , Paper, Scissors once more; the extreme simpli city of these two games makes the decisive points particularly clear. Another caveat against overemphasizing this point is : A great deal goes, in common parlance, undcr the name of "offensive," which is not at all "offensive" in the above sense,- i.e. which is fully covered by our pres· ent t heory. This holds for all games in which perfect information prevails, as will be seen in 17.10.3. 5 Also such typically " aggressive" operations (and which are necessitated by imperfect information) as "bluffing" in Poker. 8 17.10.3. We conclude by remarking that there is an important class of (ze ro~su m t\\"o~person) games in which permanently optimal strategies exist. These are the games in which perfect information prevails, which we analyzed in 15. and particularly in 15.3.2., 15.6., 15.7. I ndeed, a small modification of the proof of special strict determinateness of these games, as given loc. cit., would suffice to establish this assertion too. It would give permanently optimal pure strategies. But we do not enter upon these con~ siderations here. l

Cf. 17.1.

Any other probabilities would lead to losses when" found out."

~

t Thisis

Cf. below.

~

t - 6'_11,010rI 1,0,0I,respectively. ~

~

• This is 'I - 6" - 10, I] or 10, 1,0], respeetively. • I.e. the bad strategy of "heads" (or "stone ") "an be defeated only by "tails" (o r " paper"), "·hieh is just as bad in itseif. • Thus Chess and Backgammon are included. ' The pre than 0, -50); hence the good strategies are again unique and mixed. The formulae used before give the vahle (for Moriarty) v' = 40

178

ZERO-SUM TWO-PERSON GAMES: EXAMPLES ~

and the good strategies

(~

for Moriarty,

1)

for Sherlock Holmes):

Thus Moriarty should go to Dover with a probabil ity of 60 %, while Sherlock Holmes should stop at the intermediate station with a probability o{60 % ,~the remaining 40% being left in each case for the other alternative. l 18.6. Discussion of Some Slightly More Complicat ed Games

18.6.1. The general solu tion of the zero-sum two-person game which we obtained in 17.8. brings certain alternatives and concepts particularly into the foreground: The presence or absence of strict determinateness, the value v' of a play, and t he sets A, B of good strategies. For all these we obtained very simple explicit characterizations and dcterminations in IS.2. These became even more striking in the reformulation of those results in 18.3. This simpl icity may even lead to some misunderstandings. I ndeed, the results of IS.2., IS.3. were obtained by explicit computations of the most elementary sort. T he combinatorial criteria of ( tS:A), (IS:C) in 18.3. for strict determinateness were-at least in their final fo rm- also considerably more straightforward t hlin anything we have experienced before. This may give occasion to doubts whether the somewhat involved considerations of 17.8. (and the corresponding considerations of \4.5. in the case of strict determinateness) were n~cessa ry,- particularly since they are based on the mathematical theorem of 17.6. which necessitates our analysis of linearity and convexity in 16. If all this could bc replaced by discussions in the style of 18.2., 18.3. then our mode of discussion of 16. and 17. would be entirely unjustified. 1 This is not· so. As pointed out at the end of 18.3. , the great simplicity of the procedures and results of 18.2. and 18.3. is due to the fact that they apply only to the simplest type of zero-sum two-person games: the Matching Pennies class of games, characterized by {JI = {J2 = 2. For the general case the more abstract machinery of 16. and 17. seems so far indispensable. I The narrative of Conan Doyle--excusabl y-disregards mixed strategies and states instead the actual developments. According to these Sherlock Holmes gets out at the intermediate station and triumphantly watches Moriarty's special train going on to Dover. CQ1Ian Doylt " !IOlution is the best possible under his limitations (to pure strategies), in!lOfnr 8.8 he attributes to each opponcnt the course which we found to be the more probable one (i.e. he replaces 60% probability by certainty). It is, however, !IOmewhat misleading that this procedure leads to Sherlock Holmes's complete victory, whereas, 8.8 we saw above, the odds (i.e. the vlLlue of a play) arc definitely in favor of ~~

Moriarty. (Our result for E, I'J yields that Sherlock Holmes is as good as 48% dead when his train pulls out from Victoria Station. Compare in this connection the suggest ion in Marl/tnf/ern, loc. cit., p. 98, that the whole trip is unnecessary because the loser could be dew rmined before the star t.) • Of course it would not lack rigor, but it would be an unnecessary use of heavy mathematical machinery on an elementary problem.

SOME ELEMENTARY GAMES

179

It may help to see these things in their right proportions if we show by some examples how the assertions of 18.2., 18.3. fail for greater values of

p,

18.6.2. It will actually suffice to consider games with 81 = 13, = 3. In fact t hey will be somewhat related to Matching P ennies,- more general on ly by introduction of a third alternative. Thus both players will have the alternative choices 1, 2, 3 (Le. the values for TI , 12)' T he reader will best. think of the choice I in terms of choosing "heads," the choice 2 of choosing" tails" a nd t he choice 3 as something like "calling oil." Player I again tries to match. If either player "calls off," then it will not matter whether t he other player chooses" heads" or "tails," - the only thing of importance is whether he chooses one of these two at all or whether he "calls off" too. ConsequenUy the matrix has now the appearance of Figure 30:

1

I

1

- I

.,.

: :1: :-

'---__ L_-'----'_ _ ' Figure 30.

The four first elements-i.e. the first two elements of the first two rowsare the familiar pattern of Matching Pennies (cL Fig. 12). T he two fields with 0: are operative when player 1 "calls off" and player 2 does not. The two elements with "I are operative in the opposite case. The element with {J refers to the case where both players "call off." By assigning appropriate values (positive, negative or zero) we can put a premium or a penalty on anyone of t hese occurrences, or make it indifferent. We shall obtain all t he examples we need at this juncture by specializing t his scheme,- i.e. by choosing t he above 0:, {J, 'Yappropriately. 18.5.3. Our purpose is to show that none of the results (I8 :A), (I8:B), (I8:C) of 18.3. is generally true. Ad (I8 :A): This criterion of strict determinateness is clearly tied to the special case {JI = Pt = 2: For greater values of PI. Pt the two diagonals do not even exhaust the matrix rcctangle, and therefore the occurrence on the diagonal alone can not be characteristic as before. Ad (18 :B): We shall give an example of a game wh ich is not strictly determined, but where nevertheless t here exists a good strategy which is pure for one player (but of course not for the other). This example has the further peculiarity that one of the players has several good strategies, while the other has only one.

180

ZERO-SUM TWO-PERSON GAMES: EXAMPLES We choose in the game of Figure 30 a, p,

'Y

s.s follows:

Figure 31.

> 0, IS > O. The reader will determine for himself which combinations of "calling off" are at a premium or are penalized in the previously indicated sense. This is a complete discussion of the game, using the criteria of 17.8. a

~

For

~

=

It, t , 01 always K(

~~

-

~ ,

I) )

-

--

= 0, i.e. with this strategy player 1

cannot lose. Hence v' :equenccs ii, .. ,is with a T[ = I , denote t he (pure) strategies of the players 1, 2 by 2:'i', 2:;., But we prefer to continue with our present notations. We must now express the payment which player 1 receives if the strategies l:](iIJ' . , is), ! 2(j] , , is) are used by the two players. This is the matrix eiementJC(i), .. ,isli), , i s) ,' If the players have actually the "hands" SI, 82 then the payment received by player 1 can be expressed in t his way (using the rules stated above): It is .c.u~ ' . "_. ,(i. ,, j. ,) where sgn(sl - 82) is the sign of 5, - 82,2 and w~ere the three function s ,c"f" (i, j), £n(i, j) , .c-(i, j) 1, } = 1,2, 3.

can be represented by the following matrix schcmcs: 1

I I I I ~ I I I ~I I I 2

I

I

3

2

3

0

b

---

b

2

b

b

-b

b

b

3

-b

0

~~

I

- - - - - ---2 0 0 0 --- ------

" -----

3

0

I

" --" ----

2

I

~a

2

0

~b

3

I

-b

~b

-b

£.(1,1)

.c.(" Jl

.c _(I, J )

Figure 37.

Figure 38.

originate from chance moves, as described above.

Sl, S2

,

" , j~} =

JC(i!, . . . ,islil'

~2

Z

b

~b

Figure 36.

Now

3

Hence

.e,gn{.,_.,)(i", j.,)..

1, .1, . 1

19.6.2. We now pass to the (mixed) strategies in the sense of 17.2. ~

These are the vectors 'The enlire sequence i"

i s is

~,

~

'/ belonging to S8. Considering the notations , is is the row index, and the:entire sequence i" ,

the column index. In our original notations the strategiCli were matrix element X(T" n)

+

E,',

>

xi' and the

I l.e. 0 for s, _ h respectively. It cxpresses in an arithmetical form whicb hand is < strongcr. • The readcr will do well to compare these matrix schemes with our verbal statements of the rules, and to verify their appropriateness. Another circumstance which is worth observing is that the aymmetry of the game corresponds w the identities .co(i, i ) •

~.co(j,

i)

• The reader may verify

XU"

. ,isli,,'

,is) -

~X(h ,

,isl!!' ... ,is)

as a consequence of the relations at the end of footnote 3 above. X{i"

,

!slh, .

I.e.

,is)

is skew·symmetric, e)(prc.'lSing once more the symmetry of the game.

POKER AND BL UFFI NG

193

which we arc now using, wc must index t he components of these vectors also in the new way: We must write €i, ..... i, ' 7/i, .•.. ,i. instead of €." 7/.,' We express (17:2) of 17.4 .1., which evaluates the expectation value of player 1's gain XCi i, ' . , i.llllj .. ' .. ,j.lll)!;,... , .;,'Ii, ..... i, i, .

. . ,•. i, .. .. J,

;, . .. . .i,i, .. .. ,;, ', .•,

Thel'e is an advantage in interchanging the two 1: and writing

.,."

i, •.. . .• .;,• ..•

J.

If we now put (19:1) i.

_ i

•

L

i, . . .. J, ududinl

7/i, . . ...i. j"

1.,-;

then the above equation becomes

It is worth while to expound t he meaning of (19: 1}-{l9:3) verbally. (19 :1) shows that P:' is the probability that player 1, using the mixed ~

strategy €, will choose i when his " hand" is 8.; (19:2) shows t.hat qi' is the ~

probabili ty that player 2, using the mixed strategy 7J, will choosej when his "hand " is .h. 1 Now it is intuitively clear that the expectation value ~~

K{ €, 7J) depends on these probabilities P~" 0',' only, and not. on the underly109 probabilities k .... i,' 'Ii, .. .. .i. thcmselves.t The formula ( 19 :3) can • We know from 19.4. that i or i - 1 mellnSI\ "high " bid, i - 2,311. " low" bid with (the in tention of) n subsequent "Seeing" or" l'nssing" respectively. t T his mCRns that two different mixtu res of (pure) 8tmle~ics mny in Ilctual effeet he the l\lIme thing. Let 'IS illustrate this by n simple example. Put S - 2, i.e. \('t thl'rf' be only a "high" and n " low" hand. Consider i - 2, 3 u one thing, i.e. let there be only a " high" and a

194

ZERO-SUM TWO-PERSON GAMESo EXAMPLES

easily be seen to be correct in this direct way : It suffices to remember the meaning of t he .e'Q ~ (.,_ •• )(i, Jl and the interpretation of the p~l, ""it, 19.5.3. It is clear, both from the meaning of the P~I, 111' and from t heir form al definition (19 :1), (19 :2), that they ful fill the conditions

, all

(1904)

p~, ;;;:

0,

~

p~1

~

,

i _I

all

(19 05)

0';., ;;;:

0,

r ui'

1

i- I

On the other hand, a ny ~

p~l,

ui' which fulfill these conditions can be obtained

~

from suitable ~, 1) by (19:1), (19:2). This is clear mathematically,1 and intuitively as wcll. Any such system of pi', ""if is one of probabilities which define a possible modus procedendi,- so they must correspond to some mixed strategy. (19:4), ( 19 :5) make it opportu ne to form the 3-dimensional vectors ~

~

p',

=

[p~"

P;', p;'I,

11" =

111"1, 11',2, l1't} . ~

~~

Then (19:4), (19:5) state precisely that all p", 11" belong to Sa. This shows how much of a simplification the introduction of these ~

vectors is:

~

~

(or '1) was a vector in SIf, i.e. depend ing on {3 - I = 3 s - 1 ~

~

numerical const.ants; the p " (or the 11 't) are S vectors in 8~, i.e. each one depends on 2 numerical constants; hence they amollnt together to 28 numerical constants. And 3$ - I is much greater than 2S, even for moderate 8. 2 "low" bid. Then there are four possible (pure) strategies, to which "'e ~hall give names: "Bold" : Did "high" on every hand. "Cautious"; Bid " low" on every hand. "Normal" : Bid "high" on a "high" hand, "low" on a "low" hand. " Bluff": Bid "high" on a " Jow" hand, "low " on a "high" hand. Then a 50-50 mixture of "Bold" and "Cautious" is in effect the same thing as a 50-50 mixtu re of "Norm al" and "Bluff" : both mtan that the player will-according to chance-bid 50-50 "hip:h" or "low" on any hand. Neve rtheless tbese a re. in our present notations, two different" mixed" strategies,_ ~

i.e. vectoJ1l t. This mcsns, of course, that our notations, which were perfectly suited to the general case, are redundant for many particular games. This is a freQuent occurrence in mathe_ matical discussions with general aims. There was no r!'ason to take accou nt of this redundance fill long fill we were working out the general theory. Hut we shall remove it now fo r thl! particular game under consideration. 'Put e.g. E;, . ... . ;. - p;,' . p'l., 'I " . ... . '. - ";,' ''''6 alld verify the (l 7:1:a) , (17:1 :h) of 17.2.1. 11.8 consequences of the II.bove (19:4), (19:5). 1 Actually 8 is ahout 2! millio ns (cf. footnote 4 on p. IS7); so 38 I a nd 28 are both great, but the forme r is Quite exorhitantly the greater.

POKER AND BLUFFING

195

19.6. Statement of the Problem

19.6. Since we are dealing with a symmetric game, we can use the char~

acterization of the good (mixed) strategies-i.e. of the ~ in A-given in ~

(17:H) of 17.11.2. --+ --+ K( ~, '1)

Min ....,

It stated this:

must be optimal against itseif,-i.e. --+

must be assumed for 7J =

Now we saw in 19.5. that K( -

--+ --+

~, 7J)

1;

--+ ~.

depends actually on the

--+--+ p I, . . . , P sll1 I,

we may write for it, K( states (we rearrange the

. ..

--+

11 S).

--+

--+ p ",

u ".

So

Then (19:3) in 19.5. 2.

somewhat)

K(;\ ' " -;81-;;\' "

(19:6)

~

;S) =

.

L:,,,, L:.,J

.c,v,,(.,_.,)(i, j)p:'uj'.

~

~

And the characteristic of the

~!

P8

pi,

of a good strategy is that

~

Min- ,.....•-, K(p\ --+

-+

-+--+

is assumed at 11 I = pI, IT S = P s. The explicit conditions for this can be found in essentially the same way as in the similar problem of 17.9. 1.; we will give a brief alternative discussion. The Min-.1 .....•• - of ( 19:6) amounts to a minimum with respect to ~

~

~

each (1 I , • 11 S separately. Consider therefore such a 11',. restricted only by the requirement to belong to S a,-i.e. by

It is

,

"a·,. ~ k

1.

(19:6) is a lineiu expression in these three components

I1~I, 11~',

~

IT;'.

Hence

it assumes its minimum with respect to IT ' , there where all those componenl s 11;,' which do not have the smallest possible coefficient (with respect to j, cf. below), vanish. The coefficient of 11;' is

~ '" .c.

Sl~

Q.. (. _.

I.

)(i, j)p!.

to be denoted by

1

S "1': "

"." Thus (19:6) becomes

(19,7)

Ke

~

p I, •

, ;8)

=

~

L: "1';111;', " ';

196

ZERO-SUM TIVO-PERSO:-< GAMES, EXAMPLES

And the

condi~ion

for the minimum (with respect to

ior each pair 81, j, for which (in j '), we have ui' = O.

--

q

I,) is t his:

1';' does not assume its minimum

Hence the characteristic of a good strategy-minimization at q" =

P "-is

-q

I

= pi,

this: -

--10

__

describe a good strategy, i.e. a ~ in A, if and only if this is true: For each pair 8t, j for which 1'i' does not assume its minimum (in j I), we have pi' = O. p','

,p

S

We finally state the explicit expressions for the "ti', of course by using the matrix schemes of Figures 36- 38. They are .,- I

(19,9,,)

'Y~, =

1

S

(2:

(- ap;' - ap;' - bpi') - bp~'

', _ I

S

+

2:

(ap~' + ap;'

- bp;,) } ,

,,-_, +1

.,-1

(19,9,b)

-ri' -

~ {L

(-apI' - bp;' - bp;)

.,_1

s

2:

+ _, _ _,+1

(apj'

+ bpi' + bp;~) },

.,- 1 (19,9,c)

I

'Yi' = S

( 2: (bpj. -

bpi' - bp;')

+ bp~'

I, " 1

S

+

2:

(bp~' + bp;l + bp~,)}

" · ,.+1 19.7. Passage from the Discrete to the Continuous Problem

19.7.1. The criterion (J9:A) of 19.6., iogether with the formulae (19:7), (19:9:a), (19:9 :b), (19 :9:c), can be used to det.ermine all good strategies.! This discussion is of a rather tiresome combinatorial character, involving the analysis of a number of alternatives. Thc results which are obtained I We mean in j and not in 3o, j! • Thi~ determination has been carr ied out by one of us and will be published elsewhere.

POKER AND BLUFFING

197

are qualitatively similar to those"which we shall derive below under some· what modified assumptions, except for certain differences in very delicate detail which may be called t he "fine structure" of the strategy. We shall say more about this in 19. 12. For the moment we a re chiefly interested in t he main features of the solution and not in the question of "fine structure." We begin by turning our attention to the "granular" structure of the sequence of possible hands s = 1, . . . ,S. If we t ry to picture t he strength of all possible" hands" on a scale from

0 % to 100%, or rather of fractions from 0 to 1, then the weakest possible hand, I, will correspond to 0, and the strongest possible hand, S, to 1. Hence the" hand" s( = 1, . . . , S) should be placed at z scale.

I.e. we have this

.

I Old seale:

PQMible "hands" (

I r\ew seale:

=

8- 1 S _ 1 on this

corre~pondence:

1_-1 1 1 2 1 3 1I S-I I~ '

I----I-'- I I S~l l s~ ·····I~= : I 0

I

Figure 39.

Thus t he values of z fill the interval

very densely,! but they form nevertheless a discrete sequence. This is the "granular" st ructure referred to above. , Ve will now replace it by a continuous one. I.e. we assu me that the chanee move which chooses t he hand 8- i.e. zmay produce any z of the interval (19:10). We assume that the probability of any part of (19:10) is the length of that part, i.e. that z is equidistributed over (19:10).2 We denote the "hands" of the two players 1, 2 by z) , Zl respecti vely. 19.7.2. This change entails that we replace the vectors

--

--

p ", 11" (81. 82, = I , , S) by vectors p ", 11" (0 ;;;; ZL, Z2 ;;;; 1); but they are, of course, still probabi lity vectors of the same nature as before, i.e. belonging to 81. In consequence, t he components (probabilities) P~" 11:' (810 82 = I , " , S; i, j = I, 2, 3) give way to the components P:', oj' (0 ;;;; Z), Z2 ;;;; 1; i,j = 1, 2, 3). Similarly the 'YHin (19:9:a), (19:9 :b), (19:9:c) of 19.6.) become 1;'. We now rewrite the expressions for K and the 'Yi in t.he formulae (19:7), (19:9:a), (19:9:b), (19:9:c) in 19.6. Clearly all su ms

1 I

It ·...·iIl be remembered (d. footnote 4 on p. 187) that S is about 2 j This is the so-called goometrical probability.

million~.

198

ZERO-SUM TWO-PERSON GAMES, EXAMPLES

.,_1

0,. L

must be replaced by integrals

10

1 •••

10

dz!,

1 .

sums

".1

_,_.,+1

by integrals

while isolated terms behind a factor l i S may be neglected. 1.2

These being

understood, the formulae for K and the 7i (i.e. -yj) become: (19:7*)

K

=

kf i

'Yi' = .... ;. =

(I9:9:c*)

'Y;'

=

'Yj>ui odz1

'

.

+ I I(api' + ap;. - bp;,)dz\, ' f.", (-api' - bpi' - bpi,)dz i + I (api' + bpi' + bp;,)dz" f.", (bpi' - bpi' - bp;,)dz\ + II'. (bp~1 + bp;' + bp;.)dz,.

10"' (-api'

- api' - bp;')dz i

. I

-

And the characterization (I9 :A) of 19.6. goes over into this :

The p. (0 ~ z ~ 1) (they all belong to 8 1) describe a good strategy if and only if this is true: For each z, j for which 'Yi does not assume its minimum (inj '), we have p~ = 0. 4 Specifically we mean t.he middle terms - bpi' and bp;. in (19:9:a) and (19:9:0). These termq correllpond to II, _ 4t, in our present set-up to z, - Zt, and since the Z" %t are continuou8 variablea, the probability of their (fortuitous) coincidence is indeed O. Mathematically one may describe these operations by flll.yinJ1: that we are now carrying out the limiting proce&! S ...... 00. I We mean in j and not in z, j! • The formulae (19:7*), (19:9 :a -), (19:9:6 -), (19:9:c -) lind this criterion could also have I

I

--

been derived directly by discU3/1ing this "continuous" arrangement, with the

-p ",

p',

in place of the E, " from the start. We preferred the lengthier and more explicit procedure followed in 19.4.-19.7. in order to make the rigor and the completene&! of our procedure &Pparent. The reader will find it a good exercise to carry out the qhorter direct disculISion, mentioned above. It would be tempting to build up a theory of gamel$, into which such continuous parametertl enter, systematically and directly; i.e. in sufficient generality for ap plications like the present one, and without the nece&!ity for a limiting process from discrete gamea. An interesting step in this direction was taken by J. Ville in the work referred to in footnote I on p. 154: pp. 110-1131oc. cit. The continuity 8!l8umptions made tht:reseem, however, to be too reIItrictive for many applications,-in partiCUlar for the present one.

POK ER AN D BLU FFI NG

199

19.8. Mathemalical Determination of the Solution ~

19.8.1. We now proceed to t he determination of the good strategies p', i.e. of t he solution of t he implicit condi tion 09:B ) of 19.7. Assu me fi rst t hat p~ > 0 ever happens. L For such a z necessarily Mini 1'j = 1'; hence 1' ~ ~ 1'; i.e.

Substituting (19 :9:a *), (19:9:b *) into this gives (19, 11 )

Now let zn be t he upper limit of t hese z with p; > 0. 2 Then ( 19 :11 ) holds by conti nuity for z = zn too. As p:i' > 0 does not occur for z! > z~by hypothesis-so the with

11 p~' "

dZ I term in (1 9 :11 ) is now O.

So we may write it

+ instead of -, and (19: 11 ) becomes: (a - b) [I pi,dz i )0

f 1 P~.dZL ;;;;;: O.

+ 2b

.'

But pi' is always ~ 0 and sometimes> 0, by hy pothesis; hencc t he first term is > O. U The second term is clearly ~ O. So we have derived a contradiction. I. e. we have shown (1 9 :12)

pi

EO

O .~

19.8.2. Having eliminated j = 2 we now analyze t he relationship of Since p~ = 0 so pi + p1 E 1 i.e.:

j = I and j = 3. ( 19: 13)

P; = 1 -

p~,

o ; ; ;:

1.

a nd consequently (19,14)

p ~ ;;;;;:

Now t here may exist in t he interval 0 ;;;;;: z ~ 1 subintervals in which always pi E" 0 or always pi is. 1. $ A z which is not inside any interval of I I.e. t hat the good strategy under consideration provides for j 2, i.e. " low" bidding with (the intention of) subsequent "Seeing." under certain conditions. I I.e. the greatest ztfor which > 0 occurs arbitra rily near to zo. (But we do not require p; > 0 for all z < zo.) This ZO exists certainly if the z with .ot > 0 exist. I Of course (I b > O. ' I t does not seem necessary to go into the detailed fine points of the theory of integration, measure, etc. We assume that our functions a re smooth enough 80 that a positive function has a positive integral etc. An exact t reatment could be give n with efl.8e if we made use of t he pertinent mathematical theories mentioned above. 'The reader should reformulate this verbally; We excluded " low" bids with (the intention of) subsequent" Seeing" by analysing conditions fo r the (hypothetical) upper limit of the hands for which this would be done; a nd showed t hat nea r the re, at least, an outright" high" bid would be p referable. T his is, of course, conditioned by our simplificstion which forbids "overbidding." ' I.e. whe re the strategy direct.s the player to bid al ways "hitth," or where it directs him to bid always " low" (with subsequent" Passing ").

p:

200

TWO-PF:RSO~

ZERO-SUM

CA!>.IER: r X:\:,\II' I. ES

('il h('r kind- i.e, arhitrarily Il ear 10 wh ic h bot.h pf "e 0 ~ 0 or II!' "e 1 Ilill he called i'l/rrmcdilllr, Si rH'e :,\lin, 'Yi' = yi' 0]' yf rCi'pcctivcly, thcrefore we see: 'Yf ~ )'j' occur arbitrarily near to an intermediate z, 'Y'i = 'Y1 by continuity,1 i.e,

IIr

pr pr

and ;t! 1 (H'('ur(i,c, ;t! 0) imply Both yf ~ yj' and Hence for suc h a z,

Substituting (19:!l:a."), (ItJ:9:c · ) and recalling (19 :12), (19: 13), gives (a

+ b) fo' 1I~ '(Iz!

(a - b)

-

f

p'i ,dz l

+ 2b

f

( I - p'i.)dz l = 0

i. e.

z =

Conside r next tll'O int e rm(>dinte z', Z" and subtract. Then 2(a

f" p'i",/z

+ b)

l

-

Apply (19 :15) to z = z' and

Z".

2b (z" - z') = 0

obtains, Le.

-,-'-, f'" p'i,dz

(19 :Hl)

Z

-

Z

b

i

a

"

+ b'

\'erbally : Det ween tll'O intermediate z' , z" the al'eragc of pf is

a~ b'

So neither pi :; 0 nor pi ., I can hold throughout the intcr\'al z' ;S z is z"

sinee that. .....ould yield the a\'erage 0 or 1. li e nee thi s interval must conta in (at least) a further intermediate t, i.e. hetwccn an .... tl\'O inlf'rmcdiate places there lies (at lells t) a third one. Iterati on of this re~mlt shows that bclwpen two intcrmediate i', i " the furth e r intermediate z lie el'e rywhere de nse. Hence the i', i" for which ( 19 :lG) holds lieevcrywhere dense bet\\'een z', Z". But then ( 19 :16) must hold for all i', i" between z' , z", by continuit.I,.1 This !eaves no alternative but that pi = a

~ b e\"~r)'where

bctwf!en

Zl,

z".'

1':

1 The am defined by integrals ( 19 :9:a ' ), (19 :9:b O), (19 :9:c 0), hence th ey nre certainly continuous, I The inlcgrnl ill ( 19:16) is certainl y continuous. J Clearly isobted cxcl'plion~ co \'('ring a Z 1Irt'1i o-i.e. of tOl31 probability Zl'ro (e.g, II finite numl>cr of fixed :" )-1:ollld be permitted. They alter no intl'grals. An eurt Inllthematical treatment would be easy but dOf"s 1I0t!;eem to he cfliled for in this contex t

(d. footnote 4 on p. 199).

So it sc('ma &ill1 pl(,,81 to assume p; - a

~b

in I' :iii z :ii 1"

without any exceptions. This ought to be kept in mind when Ilppmising rhe formulae of the next pages which deal "'itll the interval l' :ii z ~ i" on one hand, lind with the intervals 0 :iii z < l' Rnd i" < z' :iii I on the other; i.e, which COllnt the points !', I" to the first-Itlentioned interval. This is, of course, irrelevant: two fixed isolated points-f' and I" in thia cas~ould be disposed of in any way (d. above ), The rel\.dcr must observe, however, that while there is 110 significant diffe rence

POKER ,\XD BLUFFIXG

201

19.8.3. Now if intermediate z exist at all , then there exists a smallest one and a largest one; choose i', i" as t hese. We have ._ PI -

b

a

throughout

+b

f' ~

z

~

i".

°

If no intermediate z exist, t hen we must have P~ == (for all z) or P~ == (for all z). I t is easy to see that neither is a solution.' Thus intermediate z do exist and \\'ith them i', i" exist and (19: I 7) is valid. 19.8.4. The left hand side of (19: 15) is 1'; - 1'~ for all z; hence for z = 1'1 -

1'1

°

=

(a

+ b) 10

1

pi,dz l

> 0,

(since P ~ ' == is excluded). By continuity 1'~ - 1'~ > 0, i.e. 1'~ < 1'~ remains true even when z is merely near enough to I. Hence P~ = 0, i.e. pi = I for these z. Thus (19;17) neeessitatcs %" < I. Now no intermediate z exists in i" ~ z ~ 1; hence \\'c have p~ == or p~ == 1 throughout this interval. Our preceding result excludes the former. Hence

°

(19:18)

PI

==

I

throughout

i"

~

z

~ I.

19.8.6. Consider finally the lower end of (19;17), t. If %' > 0 then we have an interval 0 ~ z ~ i'. This interval contains no intermediate Zi hence we have pi -= Oorpl == 1 throughout 0 ~ Z ~ f'. The firstderivat.ive of 1'i - 1'1, i.e. of the left side of (19 :15), is clearly 2(a + b)pi - 2b. Hence in 0 ~ z < Z' this derivative is 2(a + b) . 0 - 2b = -2b < 0 if pi == 0 there, 2(a + b) . 1 - 2b = 2a > 0 if p~ == I there, i.e. 1'~ - 1'i is monotone decreasing or increasing respectively, throughout 0 ~ z < i'. Since its value is 0 at the upper end (the intermediate point z'), we have 1'3 - 1'; > 0 or < 0 respectively, i.e. 1'i < 1'; or 1'; < 1'i respectively, throughout o ~ z < i'. The former necessitates P3 == 0, pi -= 1 the latter pj = 0 in o ;;i z < :'; but the hypotheses with which we st·arted were p~ "" 0 or p~ == I respectively , there. So there is 10 contradiction in each case. Consequently f' =

O.

19.8.6. And now we determine z" by expressing the validily of (19;15) for l he intermediate z = i' = O. Then (19: 15) becomes

+ b) 10 p~>dZl + 2b 1

-(a

i1pj,dZ I =

=

0

a~b

hetween a < and a ::s: when the z'~ themselves are eompared, this is /lot so for the 1';. Thus we saw that 1': > 'Yi implies 0, while 1': ~ ,.i hall no such consequence. (CL also the diS\!ussion of Fig. 41 and of Figs. 47, 48.) 'I.e. bidding "Iow" (with subsequent "PM/l ") under all conditions is not a good strategy; nor is bidding" high" under all conditions. Mathemati.::al proof: For p; _ 0: Compute 'Yt - -b, ,..: - b hence ,..: < ,..: contradicting p; - I ;t O. In other words: (!9,D)

A mistake, i.e. a strategy IT; whieh deviates from the good strategy pi will causc no losses when the opponent sticks to t he good strategy if and only if the ": fulfill (I9;C) above.

Now one glance at Figure 41 suffices to make it clear that (i9;C) means IT:

=

IT~

= 0 for z

> a-b -a-

but merely

IT;

= 0 for z

~

a-b _ _ .2

I.e.: (I9:C)

a

prescribes "high" bidding, and nothing else, for strong hands

(Z > a ~ ~);

it forbids "low" bidding with subsequent "Seeing" for all hands, but it fails to prescribe the probability ratio of "high" bidding and of "low" bidding (with subsequent" Passing") for weak hands, i.e. in the zone of "Bluffing "

(z & a~ b}

19.10.3. Thus any deviation from the good strategy which involves morc than just inCOrl"c(;t "Ulllffillg," leads to immediate losses. It suffices for the opponent to stick to the good s trategy. Incorrect" Bluffing" causes no losses against an opponent playing the good strategy; but the

i, and not in z, i!

I

We mean in

I

Actually even at

;o!

a-'_ .

0 would be permitted at the one pla.ce z _ -

•

But this

isola.ted value of z has probability 0 and so it can be disregarded . Cf. footnote 3 p . 200.

011

206

ZERO-SUM TWO-PERSON GAMES, EXAMPLES

opponent could inflict losses by deviating appropriately from the good strategy. I.e. the importance of "Bluffing" lies not in the actual play, played against a good player, but. in the protection which it pt'ovides against the opponent's potential deviations from the good strategy_ This is in agreement with the remarks made at the end of 19.2., particulArly with the second interpretation which we propesed there for" Bluffing."1 Indeed, the

element of uncertainty created by "Bluffing" is just that type of constraint on the opponent's strategy to which we referred there, and which was analyzed at the end of 19.2. OUf results on "bluffing" 61. in also with the conclusions of 17.10.2. We see that the unique good strategy of this variant of Poker is not permanently optimal; hence no permanently optimal strategy exists there. (CL the first remarks in 17.10.2., particularly footnote 3 on p. 163.) And "Bluffing" is a defensive measure in the sense discussed in the second half of 17.10.2. 19.10.4. Third and last, let us take a look at the offensive steps indicated loco cit., i.e. the deviations from good strategy by which a player can profit from his opponent's failure to "Bluff" correctly. We reverse the roles : Let player 1 "Bluff " incorrectly, i.e. use p~. different from those of Figure 40. Since only incorrect "Bluffing" is involved, we still assume

for all

,

for all

b ,> a--- . a

So we are interested only in the consequences of for some

Z

=

Zn

< a-b - _., a

The left hand side of (19:15) in 19.8. is still a valid expression for"y; Consider now a .

Z

< zoo

Then ~ in (19:23) leaves

"y~.

10" piLliz l unaffected, but

fl.

increases decreases . . pi,dzl hence It . the left hand Side of (19:15), I.e. mcreases ecreases " "Y; - 'Yj . Since"Y~ - 'Yi would be 0 without the change (19:23) (cf. Figure 41), so it will now be ;; O. I.e."Yi "Yi. Consider next a Z in

,t d

s:

Zo

a - b < Z s: - -a-.

I All this holds for the form of Poker now under consideration. For further view. points cf. 19.16. I We need this really for more than one ~, d. footnote 3 on p. 200. Thp simplest B8Sumption is that these inequalitips hold in a amaH neighborhood of thp z. in question. It would be easy to ll"f.!at thi3 mattpr rigorously in the sense of footnote 4 on p. 199 and of footnote 3 on p. 200. We refrain from doing it for the reason atated there.

POKER AND BLUFFING Then

~

in ( 19 :23) dincreases

ecrea.ses

k' 0

p~tdZI

..

while It leaves

207

f' z

pj a + b for S --- 00 whatever the variability of s. I

1

• Actually 2- (,'I

+ ,'''J I

b

- a-+-b for most 3 < I"•

POKER AKD BLUFFIKC

209

given in Figur~ 40, and depend on arithmetical peculiariti~s of 8 and S (with respect to nib).' 19.12.2. Thus the strategy which corresponds more precisely to Figure 40 - i.e. where

p~ == a ~_

b for a.l l 8

< 8°-is not good, and it differs quite

considerably (rom the good one. Nevertheless it can be shown that the m!l.ximal loss which ('.an be incurred by playing this "average" strategy is not great. More precisely, it tends to 0 for S _ CJl . 2 So we see: In the discrete game the correct way of "Bluffing" has a very complicated" fine structure," which however secures only an extremely small advantage to the player \\'ho uses it. This phenomenon is possibly typical, and recurs in much more complicated real games. It shows how extremely careful one must be in asserting or expecting continuity in this theory . ~ But the practical importance - i.e. the gains and losses cau~ed-see ms to be small, and t he whole thing is probal,lly terra 1:ncognila to even the most experienced players. 19.13.

1>1

possible Bids

19.13.1. Consider, second, (B): I. e. let us keep the hands continuous, but permit bidding in more than two ways. I.e. we replace the two bids a>b(>O) by a greater number, say m, ordered :

a, > at > ... > a.._ L > a .. (> 0). In this case too the solution bears a certain similarity to that of Figure 40. 4 There exists a certain zG 10 such that for z > zG the player should make the highest bid, and nothing else, while for z < ZO he should make irregularly various bids (always including the highest bid a, but also others), with specified probabilities. Which bids he must make and with what proba'Thus in the equivalent of Figure 40, the left plI:rt of the figure wm not be a line

(p _a ~ b in

0 :;: z

~

a

~

b),

~traight

but one which oscillates violently around this

average . • It is actually of the order 1/8. Rcmember that in real Poker 8 isabout2J millions. (Cr. footnote 4 on p. 187.) I Recall in this connection the remarks made in the second part of footnote 4 on p. 198. • It has a.ctually been determined only under the further restriction of the rules which forbids" Seeing" a higher bid. l.e. each player is expected to make his final, highest bid a.t once, and to "Pass" (and accept the consequences) if the opponent's bid should turn out higher than his. h a-b ·,n F'gure · I Analogue of t e z _ - ,40.

210

ZERO-8U .M TWO- PER80X GAMES: EXA]'vfPLE8

bilitics, if< d('t('rmined by the value of z.' So we have a zone of "Bluffi ng" and abo\'c it. a zone of" high ,. bids- actually of the hi.!!:her,t bid and noth ing else- just as in Figure 40. But the "Bluffing"- in its own zon~ z ~ zo_ has a much more complicated and varying structu re t.han in Figure 40. We shall not go into a detailed analysi5 of this structure, alt hough it offers some quite interesting aspects. We shall , however, mention one of it.s peculiarities. 19.13.2. Let two values

a>b>O be given, and usc them as highest a nd lowest bids:

a,

=

a,

a ..

=

b.

Now let m _ 00 and choose t.he remaining bids az, fill the interval

, a .._ l so t hat t hey

b ~ x ~.a

(19,24)

with unlim ited increasing density. (Cf. the two ('xamples to be given in footnote 2 below.) If the good strategy def'lc ribcd above now tends to a limit-i. e. to an asymptotic strategy for m _ 00 - thf'n one could in terpret this a.'l a good strategy for the game in which on ly upper and lower bounds are set for t he bids (a and b), and the bids can be a nything in between (i.e. in (19:24)). I. e. the requirement of a minimum interval between bids mentioned at the begin ning of 19.3. is removed. Now t his is not the case. E .g. we can interpolat.e t he az, . . . , a.._1 between al = a and a .. = b both in arithmetic and in geometric sequence.: I n both cases an asymptotic strategy obtains for m -> oc but the two strategies differ in many essential details. If we consider the game in which all bids (19:24) are permitted , as one in its own right, then a direct det.ermination of its good strategies is possible. IIf the bids which he must ma.ke are a" ~

then it

caLL

a", a.,

. . , (1 " (1
O. A detailed discussion of this game, with particular respect to the considerations of 21.2., would be of a certain interest, but we do not propose to continue this subject c - b - (l - I,

further at present.

23G

ZERO-SUM THRE E-P ERSON GAMES

players, say 1,2 respec tively. III this situation we have assumed that player 2 will certainly choose rI. so M to maxim ize 52((1"1, • . • , 0'. _ 1, 0'.). This gives a rI . = 0'.(0' 1, • • • I 0' ._1). Now we have also assumed t hat player 1, in choosing 00 . _1 can rely on this; i.e. t hat he may safely replace t.he !F1(erl, . . . ,cr._I, IT.), (w hich is what he will really obtain), by 5".(0'1, . . . • 11._1t (J.("'" .. ,0"._1» and maximize this latter quantity,l But can he rely on this assumption? To begin with, 0'.(0'1, . . . , 0'._1) may not even be uniquely determined : !f,(ITI, . . . ) IT._I, 0".) may assume its maximum (for given 0"1 , . . . ,CT._I) at several places 17,_ In the zero-sum two-person game this was irrelevant: there IT 1 .. -!ft, hence two (/. which give the same value to U'2, also give the same value to 5'1. 2 But even in the zero·su m three-person game, 5'2 does 110t determine 5'1) due to the existence of the third player and his !I.! So it happens here for the first time that a difference which is unimportant for one player may be significant for anothcr player. This was impossible in t he zerO-Bum two-person game, where each player won (precisely) what the other lost. What then must player 1 expect if two (f. are of t he same importance for player 2, but not fo r player I ? One must expect t hat he will try to induce player 2 to choose the fl. which is more favorable to him . He could offer to pay to player 2 any amount up to t he diffe rence t his makes for him. This being conceded, onc must envisage t hat player 1 may even try to ind uce player 2 to choose afl. which does not maximizeff,(fI!, . . . ,(f._I, fl.). As long as this change causes player 2 less of a loss than it causes player 1 a gain,' player 1 can compensate player 2 for his loss. and possibly even give up to him some part of his profit. 24.2.3. But if player I can offer this to player 2, then he must also count on simil ar offers coming from player 3 to player 2. I.e. there is no certainty at all that player 2 will, by his choice of (f., maximize ~h{(fl' . . . ,fl._ I, fl.). In comparing two fl. one must consider whether player 2's loss is over· compensated by player 1's or player 3's gain, since this could lead to understandings and compensations. I. e. one must analyze whether a coalition 1,2 or 2,3 would gain by any modification of 11 •• 24.2.4. This brings the coalitions bac k into the picture. A closer a nalysis would lead us to t he considerations and results of 22.2. , 22.3., 23. in every detail. But it does not seem necessary to carry this out here in complete detail ; after all, this is just a special case, a nd the discussion of 22.2., 22.3 ., 23. was of absolutely general valid ity (for the zero-sum three·person game) I Since this is a fun ction of "" , " •. 1, ". _ , ody, of which"" . . . ," ._1 are known at :m._" and ".~L is controlled by player i, he is able to maximize it. .. , "._1, "') since that also depends on ". He cannot in any sense mnximize ~'("" which he neithe r knows nor controls . • Indeed, we refrained ill i5.8.2. from mentioning ~I at all : instead of maximizing ~" we talked of minimizing :1,. There was no need even to introduce ",(,,1, . , "._,) and everything was described by Max and Min operlltkms on ~,. ~ I.e. when it happens at the expense of pillyer 3.

DISCUSSION OF AN OBJECTION

237

provided that the consideration of understandings and com pensations, i.e. of coalitions, is permitted. We wanted to show that the weakness of the argument of 15.8.2. , already recognized in 15.8.3. , becomes destructive exactly when we go beyond the zero-sum two-person games, and that it leads precisely to the mechanism of coalitions etc. foreseen in the earlier paragraphs of this chapter. This should be clear from the above analysis, and so we can return to our original method in dealing with zero-sum three-person games,-i.e. claim full validity for the rceults of 22.2. , 22.3. , 23.

CHAPTER VI FORMULATION OF THE GENERAL THEORY, ZERO-SUM n-PERSON GAMES 26. The Characteristic Function 26.1. Motivation aDd Definition

26.1.1. We now tUrn to the zero-sum n-person game for general n. The experience gained in Chapter V concerning the case n = 3 suggests that.

the possibilities of coalitions betwecn players will playa decisive role in the general theory which we are developing. It is therefore important to evolve a mathematical tool which expresses these "possibilities" in a quantitative way. Since we have an exact concept of "value" (of a play) for the zero-sum two-person game, we can also attribute a "value" to any given group of players, provided that it is opposed by the coalition of all the other players . We shall give these rather heuristic indications an exact meaning in what follows. The important thing is, at any rate, that we shall thus reach a mathematical concept on which one can try to basc a general theory-and that the attempt will, in fine, prove successful. Let us now state the exact mathematical definitions which carry out this program. 25.1.2. Suppose then that we have a game r of n players who, for the sake of brevity, will be denoted by I, 2, . . . ,n. It is convenient to introduce the set I = (1, 2, . . . , n) of all these players. Without yet making any predictions or assumptions about the course a play of this game is likely to take, we observe this: if we group the players into two parties, and treat each party as an absolute coalition- Le. if we assume full cooperation within each party-then a zero-sum two-person game results.' Precisely: Let S be any given subset of I, -8 its complement in I. We consider the zero-sum two-person game which results when all players k belonging to 8 cooperate with each other on the one hand, and all players k belonging to -8 cooperate with each other on the other hand. Viewed in this manner r falls under the theory of the zero-sum twoperson game of Chapter III. Each play of this game has a well defined value (we mean the v' defined in 17.8.1.). Let us denote by v(S) the value of a play for the coalition of all k belonging to S (wh ich, in our prescnt interpretation, is one of the players). I This ill exactly what we did in the CIlI!e n _ 3 in 23.1 .1. already alluded to at the beginning of 24.1. 238

The general poB8ibility

Willi

THE CHARACTERISTIC FUNCTION

239

Mathematical expressions for v(S) obtain as follows:' 26.1.3. Assume the zero-sum n-person game r in the normalized form of 11.2.3. There each player k = 1, 2, . . . ,n chooses flo variable Tk (each one uninformed about the n - 1 other choices) and gets the amount Of course (the game is zero-sum):

.-. •

~ X!(TI, . . . , T.. )

== o.

The domains of the variables are: k = 1,2, . . . ,n.

T k = l , " ' , {Jk

Now in the two-person game which arises between an absolute coalition of all players k belonging to S (player 1') and that one of all players k belonging to -8 (player 2'), we have the following situation: The composite player l' has the aggregate of variables Tk where k runs over all elements of S. It is necessary to treat this aggregate as one variable and we shall therefore dcsignate it by one symbol T' . The composite player 2' has the aggregate of variables Tk where k runs over all elemente of -So This aggregate too is one variable, which we designate by the symbol T-S • The player I' gets the amount (25:2)

L

:K(T S, T- S) =

L

3Ck(Tl, . . . ,T~) = -

3C.(TI,·· · , T..)j2

tin -8

kill8

the player 2' gets the negative of this amount. ~

A mixed strategy of the player l ' is a vector ~

of which we denote by

trl. Thus the t of

t .•

~

~

S6a,1

the components

S,. are characterized by

0, ~

A mixed strategy of the player 2' is a vector ~

of which we denote by '1.-"

of

Thus the

'I

'I

of S,_,,4 the components

of S,-. are characterized by

}; ,.-' ~ 1.

.-. I

This is a repetition of the construction of 23.2., whicb applied only to the special

cue n - 3.

T-"

1 The T ~ , of the first expression form together the aggregate of the TI, . . . ,T ~ of the two other exprC38ions; ~o T S, T- S deteTmine tho!lC Tit • • • , To. The equality of the two l8Jlt exprC38ions is, of course, only a restatement of the zero-sum property. I fj!l is the number of possihle aggregates .. ", i.e. the product of all~} where k runa over all elements of S . • ~-If is the number of possible aggregates T- If , i.e. the product of all ~. where k ruOB over all elementa of -So

240

GENERAL THEORY: ZERO-SUM n-PERSONS ~

~

The bilinear form K{ ~, 7)) of (17:2) in ]7.4. 1. is therefore ~~

K { ~, 7)) =

~ X(T 3 , T- .'IH •• 7). -",

..... -.

and finally yeS)

=

Max .... Min.... K ( I

~

-.-. ~,

7)) = Mi n.... l\'fax .... K ( •

I

-.~ ,

7)).

26.11. Discussion of th e Concept

26.2.1. The above function v(S) is defined for all subsets S of 1 and has real numbers as values. Thus it is, in the sense of 13.1.3., a numerical set junction. We call it the characteristic junction oj the game r. As we have repeatedly indicated, we expect to ba.'le the entire theory of the zero-sum n-person game on t his function . It is well to visuali ze what this claim involves. We propose to determine evt:rythi ng that can be said about coalit ions between players, compensations between partners in every coalition, mergers or fights between coalitions, etc., in terms of the characteristic function yeS) alone. Prima jacie, this program may seem unreasonable, particularly in view of these two facts : (a) An altogether fictitious two-person game, which is related to the real n-person game only by a theoretical construction, was used to define v(S). Thus v(S) is based on a hypothetical situation, and not strictly on the n-person game itself. (b) v(S) describes what a given coali tion of players (specifically, t he set S) t)an obtain from their opponents (the set -S)- but it fails to describe bow the proceeds of the enterprise are to be d ivided among the partners k belonging to S. This div ision, the " imputation ," is indeed directly determined by the indivirlual functions Xk{TI,' " T k belonging to S, while v(S) depends on mu ch less. I ndeed, v(8) is determined by their partial sum dC(rll', 1'- .'1) alone, and even by less than that since it is R ),

~

~

the sa.ddle value oi the biline!i.r form K( ~, 7) based on jC{T'1, 1'- .'1) (d. the formulae of 25.1.3.). 26.2.2. I n spite of t hese considerations we expect to find that the characteristic funct.ion yeS) determines everythi ng, including the" imputation" (d. (b) above). The analysis of t he zero-sum three-person game in C ha.pter V indicates thllt the di rect distribution (i.e., " imputation ") by means of the Jek(TI, . . . , Tn) is necessarily offset by some systel"!l of "compensations" which the players must make to each other before coalitions can be formed . The" compensations" should be determined essentially by the possibilities which exist for each partner in the coalition S (Le. for each k belonging to S), to forsake it and to joi n some other coalition 7'. (One m:l.y have to consider also t he influence of possible simultaneous and cOl1certed desertions by sets of several partners in S etc.) I.e. the "imputation" of yeS) to t he players k belonging to S should be determined

THE CHARACTERIST IC FUNCTION

24 1

by the other v(T)I-ar.d not by the JCl(Th . . . ,Tn). We have demonstrated this fo r the zero-sum three-person game in Chapter V. One of the main objectives of the theory we are tryi ng to build up is to establish t.he same t hing for the general n-person game. 26.3. Fundamental Properties

26.3.1. Before we undertake to elucidate the importance of the characteristic function v(S} for the general theory of games, we shall investigate t his function as a mathematical entity in itself. We know t hat it is a numerical set function, defined for all subsets 8 of I = (1, 2, . . . , n) and we now propose to determinc its essential properties. It will turn out that they a re the following: (25 'h) (25,3,b) (25,3,,)

vee)

~

0,

v( -8) = -v(S), y(S u T) ;; v(S) + veT) , if

SnT

~

e.

We prove first that the characteristic set function v(8} of every game fulfills (25:3:a}-(25:3:c). 26.3.2. The simplest proof is a conceptual one, which can be carried out with practically no mathematical formulae. However, since we gave exact mathematical expressions for v(8) in 25.1.3. , one might desire a strictly mathematical, formalistic proof- in terms of the operations Max and Min and the appropriate vectorial variables. We emphasize therefore that ou!' conceptual pronf is strictly equivalent to the desired formalistic, mathematical one, and t hat the translation ca!l be carri~d out without any real difficulty. But since the conceptual proof makes the essential ideas clearer, and in a briefer and simpler way, while the formalistic proof would involve a certain amount of cumbersome notations, we prefer to give the forme r. The reader who is interested may find it a good exercise to construct the forma listic proof by translating our conceptual one. 26.3.3. Proof of (25:3:a):2 The coalition e has no members, so it always gets the amount zero, therefore vee) = o. Proof of (25:3:b): v(S) and v( -8) originate from the same (fictitious) zero-sum two-person ga.me,- the one played by the coalition 8 against I All thil! ia very much in the scnse of the remarks in 4.3.3. on the role of "virtu,,"l" existence. I Observe that we are treating even the empty set 113 a oosJition. The reader should tbink this over carefully. In spite of its strange appearance, the step is harmle--...s-and quite in the spirit of general set theory. Indeed, it would be teebnically quite a nuisance to exclude the empty set from consideration. Of course this empty coalition has no moves, no variables, no influenc'l, no gains, and no losses. But thia is immaterial. The complementary set of the set of all players /, will also be treated as a po8llible coalition. This too is the convenient procedure from the set.-theoretical point of view. To a lesser extent this coalition alllO may appear to be strange, aince it bas no opponents. Although it hM an abundance of members--and hence cf moves and variables-it will (in a zero~um game) equally have nothing to influence, and no gains or l038es. But thiJI too is immaterial.

e

e,

242

GENERAL THEORY: ZERO-SUM n-PERSONS

the coalition -So The value of & play of this game for it.layer may playa lone hand, since there is no advaTltage in any coalition. Indeed, every player can get. the amount zero for himsell irrespective of what the others are doing. And in no coalition can all its members together get more than zero. Thus the value of a play of this game is zero for every player, in an absolutely unequivocal way. If a general characteristic function v(S) is in strategic equivaience with such a Y(S)- i.e. if its reduced form is yeS) =: O-then we have the same conditions, only shifted by for the player k. A play of a game r with this characteristic function yeS) bas unequivocally the value for the player k: he can get this amount even alone, irrespective of what the others are doing. No coalition could do better in toto. We call a game r, the characteristic function v(S) of which has such a reduced form yeS) == 0, ine88ential. l 27.3.2. Second case: "I > O. By a change in unitt we could make "I = 1. ~ This obviously affects none of the strategically significant aspects of t.he game, and it is occasionally quite convenient to do. At this moment, however, we do not propose to do this. In the present case, at any rate, the players will have good reasons to want to form coalitions. Any player who is left to himself loses the amount "I (Le. he gets -"I, cf. (27:5·) or (27 :7"'», while any n - 1 players who

a;

a:

, Tha.t this coincides With the meaning given to the word inessential in 23.1.3. (in the case of a zero-sum three-person game) will be seen at the end of 27.4.1. • Since payments are made, we mean the monetary unit. I n a wider sense it might be the unit of utility. Cf. 2.1.1. I This would not have been possible in the first case, where "I - O.

~peeial

250

GENERAL THEORY, ZERO-SUM n-PERSONS

cooperate win together the amount 'Y (i.e. their coalition gets 'Y,

cr.

(27:5··)

or (27:7··»,1

Hence an appropriate strategy of coalitions is now of great importance. We call & game r eS8ential when its characteristic function v(S) has a reduced form v (S) not == 0. 1 2T.4.. Variou. Criteria.

Non-additive Utilitiel

27 .•. 1. Given a characteristic function v(S), we wish to have an explicit 'Y of its reduced form ,, (8). (ef. above.) Now -y is the joint value of the v«k», Le. of the + aZ, and this

expression for the

is by (27:4)

~

v«k»

L:" v«J1),'

Hence

i-I

- ~ 2:"

v((j)).

i - I

Consequently we have: (27:B) The game r is inessential if and only if (Le.

'Y =

0) ,

and it is essential if and only if

"

~ v((i))

vetil). So

Proof: If a

S #-

277

L

L

As S is fiat, this means

Q;

>

But S must

v(S).

iinS

ai;::::! v(S), which is a contradiction.

lin S

S is certainly necessary if - S is flat and S #-

e.

Explanation: A coalition must be considered if it (is not empty and) opposes one of the kind described in (31 :F). ~

Proof: The preliminary conditions are fulfilled for all imputations Ad (30:4:a): S #was postulated.

e

Ad (30:4:b): Always ai

~

v«i», so

•

L a; =

. '" ,

• not In S

L

0, the left-hand side is equal to -

-

ai.

v((i».

Q.

Since

Since -S is flat, the

iin S

right-hand side is equal to v(-S), i.e. (use (25:3:b) in 25.3.1.) to -v(S). So Q ;;:;:; -v(S), ~ aj ~ v(S), i.e. S is effective.

L.

lin S

iinS

From (31 :F), (31:G) we obtain by specialization : (31 :H)

A p-element set is certainly necessary if p certainly unnecessary if p = 0, I, n.

=

n - I, and

Explanation: A coalition must be considered if it has only one opponent.. A coalition need not be considered, if it is empty or consists of one player only (!), or if it has no opponents. Proof: p = n - 1: -8 has only one element, hence it is flat by (31 :0) above. The assertion now follows from (3 1:G). p = 0, 1: Immediate by (31 :0) and (31 :F). p = n: In this case necessarily S = I = ( 1, . . . , n) rendering the main condition unfulfillable. Indeed, that now requires Qj > f3. for all i = 1, . . . ,n, hence

L" ai > k" f3i.

~~

But as

a, (1

are imputations, both

sides vanish,-and this is a contradiction. Thus those p for which the necessity of 8 is in doubt, are restricted to p ;r! 0, 1, n - 1, n, i.e. to the interval (3LB)

2~p~n-2.

This interval plays a role only when n ;:;:; 4. The situation discussed is similar to that at the end of 27.5.2. and in 27.5.3. , and the case n = 3 appears once more as one of particular simplicity. 31.11:. The System of All Imputation..

One-element Solution.

31.2.1. We now discuss the structure of the system of all imputations. (31:1)

For an inessential game there exists precisely one imputation :

278 (3 1:9)

GENERAL THEORY : ZERO-SUM n-PERSONS

-

- - 1-,,' .','"-.} , .......

=

OJ

i

v«i»

=

1, . . . , n.

For an essential game there exist infinitely many imputations -an (n - l )-dimcIUlional continuum~but (3 1 :9) is not one of

them. Proof: Consider an imputation

and put

v«i» +

(3. =

i = I , . . . ,n.

fo.

fj

Then the characteristic conditions (30 :1), (30 :2) of 30.1.1. become (3 1 :10)

i ". 1, . . . ,

for

to ?; 0

•

~ '; - -

(3 1:11)

._1

n.

•

~ v((i)). i_ I

•

~

If r is inessential , then (27:B) in 27.4.1. gives (31:10), (3 1:11) amount to imputation.

=

EI

... =

t ..

=

If r is essential, the (27 :B) in 27.4.1. gives -

v((.)) _ 0; so

0, i.e. (31:9) is the unique

r" v«i»

> 0, so (3 1:10),

- -

(31 :11 ) possess infinitely many solutions, which form an (n - 1)-dimensional cont inuum;1 80 the same is t rue for the imputations fJ. But t he a of (3 1 :9) is not one of them, because ~I = . . . - ~~ ,.,. 0 now violate (3 1 :11 ).

An immediate consequence: (3 1:J)

A solution V is never empty.

- e --

e is not a solution.

Proof: I.e. the empty set

imputation {j ,- there exists at least one by (3 1 :1).

e

-

Indeed: Consider any {j

is not in e and no

a in has a f-o {j. So violates (30 :5:b) in 30.1.1. 2 31.2.2. We have pointed out before that the simultaneous validity of

(3 1 :12)

is by no means impossible. J (3 1 :K)

However :

--

Never a

I'-


Q. for

aU i

~

io, i.e. for all i in the Ret S = - (io). ~

This set has n - 1 elements and it fulfills the main condition (for

--

~

hence (31 :H) gives {J

~

~

a.

(J,

~

a),

--

Not a'" {J : Assume that a ... {J. ThenasetSfulfillingthemaincondition must exist, which is not excluded by (31 :H). So S must have ~ 2 elements. So an i ~ i. in S must exist. The former implies {Ji > ai ~

~

'Hencea".(J. I For i _ i i , we have actually I

L"

.

;-1

PI -

L"

ai

(J •• _

v«i,».

be

al ~

u.. for one value of i (i -

v{(i)). i l ) and

GE~ERAL

280

THEORY: ZERO-SUM n-PERSOXS ~

(by the construction of (3); the latter implies (l"i > {3i (owing to the main condilion)-and this is a contradiction. 31.2.3. We can draw the conclusions in which we were primarily interested :

-

.

--

An imputation (1", for which never a' only if the game is inessentiaL I

f-

a, exists if and

Proof: Sufficiency: If the game is inessential, then it possesses by (3 1 :1) ~

precisely one imputation 0:, and this has the desired property by (3l:K). ~

~

~

Necessity: If the game is essential, and a is an impu tation, then a' = {J ~

of (3LL) gives (1"'

~

=

~

{3 f- (1".

A game which possesses a one-element solution' is necessarily inessential. ~

Proof: Denole the one-clement solution in question by V = «(1"). This V must satisfy (30:5:b) in 30.1.1. This means under our present circum~

~

stances: Every {3 other than

(I"

~

is dominated by (1".

I.e.:

implies ~

Now if the game is essential, then (31 :L) provides a {J which violates this condition. An inessential game possesses precisely one solution V.

(3LO)

~

This

~

is the one-element set V = (0:) with the

(I"

of (31 :1). ~

Proof: By (31 :1) there exists precisely one imputation, the (I" of (31 :1). A solution V cannot be empty by (31 :.1 ); hence the only possibility is ~

V

= (

cr).

~

Now V = ( 0: ) is indeed a solution, i.e. it fulfills (30:5:a), (30:5:b) ~

in 30.1.1.: the former by (31 :l{), the latter because a is the only imputation by (3 U ). We can now answer completely the first question of 30.4.1.: A game possesses a one-element solution (cf. footnote 2 above) if and only if it is inessential; and then it possesses no other solutions. Proof: This is just a combinat ion of the results of (31 :N) and (31 :0). Cf. the considerations of (30:A:a ) in 30.2.2., and particularly footnote 2 on p. 265. We do not exclude the possibility that this game may possess other solutionS!l8 well, which mayor may not be one-element sets. Actually this never happens (unde r our present hypothesis), as the combination of the rC1lu!t of (31:N) with that of (31 :O)--or the result of (31 :P)- shows. But the present consideration is independent of all this. I

t

FIRST CONSEQUENCES

281

31.S. The Isomorphism Which Corresponds to Strategic Equivalence

31.3.1. Consider two games rand r ' with the characteristic functions v(S) and v'(S) which arc strat.egically equivalent in the sense of 27.1. We propose to prove t hat. they arc really equivalent from the point of view of the concepts defined in 30.1.1. This will be done by establishing an isomorphic correspondence between the entities which form the substratum of the definitions of 30.1.1., i.c. the imputations. That. is, we wish to establish a one-to-one correspondence, between the imputations of rand those of r', which is isomorphic with respect to those concepts, Le. which carries effective sets, domination, and solutions for r into those for r ', The considerations are merely a n exact elaboration of the heuristic indications of 27.1.1., hence the reader may find them unnecessary. However, they give quite an instructive instance of an " isomorphism proof," and , besides, our previous remarks on the relationship of verbal and of exact proofs may be applied again. 31.3.2. Let the strategic equivalence be given by aY, . . . ,a~ in the sense ~

of (27: 1), (27:2) in 27.1.1.

Consider all imputations a = 1011, ... ,

~

of r and all imputations a' = one correspondence

la~,··· ,a~lofr '.

a~1

We look for a one-to-

with the specified properties. What (3 1 :15) ought to be is easily guessed from the motivation at the beginning of 27.1.1. We described there the passage from rI,o r ' by adding to the game a fixed payment of a~ to the player k. Applying this principle to the imputations means (3 1:16) a~ = Uk + aZ for k = I , . . . ,n.' Accordingly we define the correspondence (31 :15) by the equations (3 1 :16). 31.3.3. We now verify the asserted properties of (3 1 :15), (31 :16). The imputations of r are mapped on the imputations of r': This means by (30: 1), (30:2) in 30.1.1., that (3 1:17) a, ~ v«i» for i = 1, . . . . 11.

2:" a, =

0,

i_ I

go over into (3 Ll7')

a;

~

v'«i»

i

fo,

=

1, . . .

,n,

(3L I8') ~

-- -- + --

'If wt! introduce the (fixed) vector

written vecw riall y a' of the imputations.

a

a".

a

0 -

t a~J

a:1

then (31: 16) may be

--

I.e. it is a translation (by a ) in the veCWT space

282

GENERAL THEORY, ZERO-SUM n-PERSONS

This is so for (3 1 :17), (3 1:17*) because v'«i» = v«i»

+ 0:: (by (27:2) in

"

27.1.1.), and for (3 1:18), (3 1:18·) because ~ a? = 0 (by (27:1) id .), i-I

Effectivity for r goes over into effectivity for r': This means by (30 :3) in 30.1.1., that ~ .,,, v(S) i in S

goes over into

~ .: " V'(S). iin 8

This becomes evident by comparison of (3 1 :16) with (27 :2). Domination for r goes over into domination for r': This means the same thing for (30:4 :&)-(30:4 :c) in 30.1.1. (30 :4 :a) is trivial; (30:4:b) is effectivity, which we settled: (30:4:c) asserts that a; > (3; goes over into a; > fJ: which is obvious. The solutions of r arc mapped on the solutions of r': This means the same for (30 :5:&), (30:5:b) (or (30:5:c» in 30.1.1. These conditions involve only domination, which we settled. We resta.te these results:

If two zero-sum games rand r ' are strategically equivalent, then there exists an isomorphism between their imputationsi.e. a one-to-one mapping of those of r on those of r ' which lea.ves the concepts defined in 30.1.1. invariant. 32. Determination of an Solutions of the Essential Zero-sum Three-person Game 811.1. Fonnulation of the MathematiCal Problem.

The Graphical Method

32.1.1. We now turn to the second problem formulated in 30.4.1.: The determination of all solutions for the essential zero-sum three-person games. We know that we may consider this game in the reduced form and that we can choose 'Y = I.' The characteristic function in this case is completely determined as we have discussed before;1

(3H)

v(S) ~

0 - I ~

1

when S has

1~ elements.

An imputation is a vector

I Cf. the diBeu!lfJion at the beginning of 29.1., or the references there given: thee.nd of 27.1. and the aecond remark in 27.3. I Cf. the diseu88ion at the beginning of 29.1., or the second case of 27.5.

ALL SOLUTIONS OF THE THREE-PERSON GAME

283

whose three components must fulfill (30: 1), (30:2) in 30.1.1.- These con· ditions now become (considering (30:1» al~

a,~

-I, a1

- 1, = O.

+ a2 + a,

a,~

- I,

We know, from (31 :1) in 31.2.1., that these ai, a2, a, form only a twodimensional continuum -i.e. that they should be representable in the plane. Indeed, (32 :3) makes a very simple plane representation possible .

...' , •

,

0

"0° \

\;

., > 0 _ _

.

.:~;-1", , .•..•'1

~,"'"

/

' :.,

. ,-

_ _ ., > 0 G, _

-I as far as possible.

0 - 1

0 1

2., (36'1)

v(S)

~

1 0

(and 4 belongs to S) elements 2 (and 4 does not belong to S) 3 4

2

when S has -2xl

(ef. Figure 63.)

(Observe that this gives (35: 1) in 35.1.1. for Xl = 1 a.nd (35:6) in 35.2.1. for = - 1.) We assume that Xl > -I but not by too much,-just how much excess is to be permitted will emerge Ia.ter. Let us first consider this situation heuristically. Since Xl is supposed to be not very far from -1, the discussion of 35.2. may still give some guidance. A coalition of two players from among Xl

I

This will be done by one of us in subaequent mathematical publica.tiona.

ZERO-RUr-"l FOUR-PERSON GAMES

306

I ,2,3 may even now be the most. important. strategic a im , but. it is no longer the only nne : the formula (35:7) of 35.2.1. is not true, but instead (36,2)

v(S u 1')

>

v(S)

+ v(T)

;f

Sn7'=8

whon '1' = (4) .1 Indeed, it is easily verified from (36: 1) that this excess is alw3.,Y5 2 2(1 + Xl)' For x . = -1 this vanishes, but we have Xl slightly > - 1, so the exprcsEion i~ slightly > O. Observe that for t he preceding coalition of t,tlO play -1, it will be only slightly < 4. Thus the first coalition (between two players, other than player 4), is of a much stronger texture than any other (where player 4 enters into the picture),- but the latter cannot be disregarded nevertheless. Since the fi rst coalition is the stronger one, it may be suspected that it will form first and that once it is formed it will act as olle pJnyer in its dealings with the two others. Theil some kind of a. three-rerson game may be expected to take place for the final crystallization. 36.1.3. Taking, e.g. (1,2) for this "fir!;t" coalition, the surmised threeperson game is bet.ween t he players (1,2), 3,4. 1 I n this game the a, b, c of Z3. 1. are a = v«3,4» = ZXI, b = v«1,2,4» = I , c = v((l,2,3» = 1.$ Hence, if we may apply t he results obtained there (all of t.his is extremely heuristic!) t.he player (l,Z) gets t he amount. a = -a +Zb

if successfu l (in joining the lafit. coalition), and -a player 3 gets the amount f1 if defeat.ed .

=

a - b+c

Z

=

The player 4 gets the amount

Xl

= -2xI

+c =

if defeat.ed .

.

If successful, a nd -b

'Y =

1 -

a+b-c Z =

XI

Xl,

T he =

- I

. If suecefiS-

ful, and -c = -1 if defeated. Since " fi rst" coalitions (l,3), (2,3) ma.y form, just as well as (1,Z) , there are the same heuristic reasons as in the fi rst discussion of t he threeperson game (in 21.-22.) to expect t.hat t.he partners of these coalitions \\.iI\ split even. Thus, when such a coalition is successful (cf. above), its members may be expected to get

.!....~

XI

each, and when it is defeated

each . 36.1.4. Sum ming up: If these surmises prove correct., the sit.uat.ion is ns follows: XI

e

, Unless S or - T, in which caae there is always" in (36:2) . I.e. in the prCli' cnt situation S must have onc or 11'0'0 elements. I Dy footnote I above, S has one or two elements and it does not contain 4. I I.e., now S, T are two one-element sets, not containing player 4 . • One might eay that 0,2) is a juridical person, while 3,4 are, in our picture, natural persons. I In all the formulae which follow, remembe r that X, is nea r to -i,-i.e. presumably negative; hcnee -x, ie a gain, lind x, is I!. loss.

DISCUSSION OF THE MAIN D IAGONALS

307

If the "first" coalition is (1,2), and if it is successful in finding an ally, and if the player who joins it in the final coalition is player 3, t.hen the 1 - XI 1 - XL players 1,2,3,4 get the amounts ~-2~ ' ~-2- ' XL, -I respectively. If the

player who joins the final coal ition is player 4, then these amounts are replaced by

I -

Xl

I -

Xl

~2~ ' ~-2~ '

- I,

Xl.

If the "first" coalition (1,2) is not

successful, Le. if the players 3,4 combine again:;t it, then the players get the amounts -Xl, -XL, Xl, XI respectively . If the "first" coalition is (1,3) or (2,3), then the corresponding permutation of the players 1,2,3 must be applied to the above. 36.2. Tbe P art Adjacent to tbe Comer V J II.:

Exact Discussion

36.2.1. I t is now necessary to submit all this to an exact check. heuristic suggestion manifestly corresponds to the following surmise: Let V be the set of these imputations:

-

I - XL 1 - XL ( - 2- ' ~-2- '

II,

=

{-XL, -XI, XI,

I I

- 1

I - XI 1 - XI I 2- ' - 2- ' - ,XI (--

0"

a

XL,

The

and the imputations which originate from these by permuting the players, (Le. the components) 1,2,3.

xd

(Cf. footnote 5, p. 306.) Wc expect that this V is a solution in the rigorous sense of 30.1. if XI is near to -\ and we must determine whether this is so, and precisely in what inten:al of the XL . This determination, if carried out, yields the following result: The set V of (36:3) is a solution if and only if -1 ~

This then is the answer to the = -I, i.e. t he corner V Ill) a correct result. L 36.2.2. The proof of (36:A) significant technical difficulty. Xl

XI;;;;; -

!.

question, how far (from t he starting point the above heuristic con'Sideration guides to can be carried out rigorously without any It consists of a rather mechanical disposal

I We wish to emphnsize that (36:A) does not assert that V is (in the specified range of z,) the only 8OIution of the game in question. However, attempts with numerous similarly built sets failed to disclOlie further solutions for x, ~ - ! (Le. in the range of (36:A)). Fer z, slightly> - i (i.e. slightly outside the range of (36:A)), where the V of (36 :A) is no longer a solution, the same is true for the solution which replaces it. er. (36:8) in 36.3.1. We do not question, of course, that olher solutions of the "discriminatory" type, R$ repeatedly discussed before, always exist. But they are fundamentally different from the finite solutions V which are now under consideration. These nre the arguments which seem to justify oLlr view that some qualitative change in the nature of the solutions occurs at

z,

~

-

~ (011

the diagonall-cente r- 1T/lI).

308

ZERO-SUM FOUR-PERSON GAMES

of a series of special cases, and docs not contribute 1tnything to the clarification of a ny question of principle,' The reader may t herefore omit reading it if he feels so disposed, without losing the connection with the main course of the exposition. He should remember only the statement of the results in (36,A). Nevertheless we give the proof in full for the following reason : The set V of (36:3) was found by heuristic considerations, i.e. without using the exact theory of 30.1. at all. The rigorous proof to be given is based on 30.1. alone, and thereby brings us back to t he only ultimately satisfactory standpoint, that of the exact theory. The heuristic considerations were only a device to guess the solution, for want of any better technique; and it is a fortunate feature of the exact theory that its solutions can occasionally be guessed in this way. But such a guess must afterwards be justified by the exact method, or rather that method must be used to determine in what domain (of the parameters involved) the guess was admissible. We give the exact proof in order to enable the reader to contrast and to compare explicitly these two procedu res,-the heuristic and the rigorous. 36.2.3. The proof is as follows: If x, = -1, then we are in the corner VIII, and the V of (36:3) coincides with the set which we introduced heuristically (as a solu tion) in 35.2.3., and which can easily be justified rigorously (cf. also footnote 2 on p. 299). Therefore we disregard this case now, and assume that (36,4)

We must first establish which sets S ~ 1 = (1,2,3,4) are certainly necessary or certainly unnecessary in the sense of 31. 1.2.-since we sre carrying out a proof which is precisely of t he type considered t here. The following observations a re immediate: (36,5)

(36,6)

By virtue of (3 1:H) in 31.1.5., three-elemcnt sets S are certainly necessary, two-clement sets are dubious, and all other sets are certainly unn ecc5Sary.~ Whenever a two-element set turns out to be certainly necessary, we may disregard all those three-element sets of which it is a subset, owing to (31:C) in 31.1.3.

Consequently we shall now examine the two-clement sets.

This of course

~

must be done for all the a in the set V of (36:3). I The reader may .:lontrast this proof with some given in connection with the theory of t he zero·8um tWO-pe/"801l game, c.g. the combination of 16.4. with 17.6. Such a proof i9 more transparent, it usually COVCf8 more ground, and gives some qualitative elucidation of t he subject and its relation to other parts of mathematics. Illsomillater parts of this theory such proofs have been found, e.g. iu 46. But much of it is slill in the primitive and technically unsatiafactory state of which the considerations which follow are typical. 'Tbia iedue to n _ 4.

DISCUSSION OF THE MAIN DIAGONALS

309

Consider first those two-element sets S which occur in conjunction ~

a;

with a',l As = - 1 we may exclude by (3 1:A) in 31.1.3. t he possibility that S contains 4. S = (1,2) would be effective if a; + a; ;:;; v«I,2», i.e. 1 - X I ;:;; - 2x\, XI ;;;;: -1 which is not t he case by (36 :4). S = (1,3) 1 + Xl is effective if + a~ v« 1,3», i.e. 2- ;:;; -2XI , Xl ;:;; - t. Thus

a;

;:; :

the condition XI;:;;

!

-

which we assume to be satisfied makes its fi rst appearance.

S = (2,3) we

~

do not need , since 1 and 2 play the same role in a' (d. footnote 1 above) . ~

We now pass to a:

As a~' = - 1 we now exclude theS which contains 3

/1.

~

~

(eL above). S = (1,2) is disposed of as before, since a I and a" agree in these com ponents. S = ( 1,4) would be effective if a~' + a~' ~ v« 1,4)), i.c. 1

~

XI ;;;;; 2XI, XI

~ ! which, by (36:7), is not the case. S

=

(2,4) is

discarded in the same way. ~

Finally we take 0: "'. S = (1,2) is effective: o:~" + o:~" = v« 1,2» i.e. -2XI = -2xl. S = (1,3) need not be oonsidered for the following ~

reason: We are already considering S = (1,2) for 0: ''', if we interchange 2 and 3 (d. footnote I above) t his goes over into (1,3), with the components ~

-Xl, -Xl. Our original S = (1,3) for a'" with the components -XI, XI is thus rendered unnecessary by (3 LB) in 31.1.3., as -Xl $;;: X I owing to (36:7). S = (2,3) is discarded in the same way. S = (1,4) would be effective if o:~" + o:~" ;;;: v«(l,4» i.e. 0 ~ 2X I, XI $;;: 0, which, by (36:7), is not the case. S = (2,4) is discarded in the same way. S = (3,4) is effective: a;" + a~" = v«3,4», i.e. 2xI = 2xl . Summing up : Among the two-element sets S the three given below are certainly necessary, and all others are certai nly unnecessary: ~

~

(1,3) for a', (1,2) and (3,4) for a"'. Concerning t hree-element sets S: By (31 :A) in 31.1.3. we may exclude ~

~

those containing 4 for a' and 3 for a". ~

for

0:'

Consequently only (1,2,3) is left

~

and (1,2,4) for

0: ".

Of these the former is excluded by (36:6), ~

as it contains the set (1,3) of (36:8) .

For a'" every three-element set

, Here, and in the entire discussion which follow~, we shall make use of the freedom to apply permutations of 1,2,3 &II stated in (36:3), in order to abbreviate the argumentation. Hence the reader must afterwards apply these permutations of 1,2,3 to our re8ults.

ZERO-SUM FOUR-PERSON GAMES

310

contains the set (1,2) or the set (3,4) of (36:8); hence we may exclude it by (36,6), Summing up: Among the three-element sets S, the one given below is certainiy necessary, and all others are certainly unnecessary:! ~

(1,2,4) for a

/I. ~

36.2.4. We now verify (30:5:a) in 30.1.l., i.e. that no a of Y dominates ~

any fj of V. ~

~

~

a = a': By (36:8), (36:9) we must use S = (1,3), ~

~

Can a

with this S any 1,2,3 permutation of a I or a" or a "'1

< Xl

the existence of a component

dominate

I

~

This requires first ~

(this is the 3 component of a') among ~

the 1,2,3 components of the imputation in question.

~

Thus a' and a'"

~

are excluded.! Even in a" the 1,2 components are excluded (eL footnote 2) but the 3 component will do. But now another one of the 1,2,3 components of this imputation

-;11 must be < 1 ~

Xl

(this is the 1 component of -; '),

-;11 are both =

and this is not the case; the 1,2 components of ~

~

1

~

Xl.

~

a = a": By (36:8), (36:9) we must useS = (1,2,4). ~

~

Can a" dominate

~

with this S any 1,2,3 permutation of a 'or a or a '''? This requires first that the 4 component of the imputation in question be < Xl (this is the

-

4 component of

a ").

Thus

a

/I

and

a

/II

/I

are excluded .

require further that two of its 1,2,3 components be


o-i.e. the half Ccnter-I of the diagonal- as well. It turns out that on this half, qualitative changes occur of the same sort as in the fi rst half covered by (36:A) and (36:B). Actually three such intervals exist, namely: (36 ,C) (36 ,D ) (36,E)

o~

XI

i ;:g;

XI ;;;;

0 t his means only that - z ~ - 1, i.e. we must have (3S,2)

O i - i.e. when (38':2) is replaced by (38 ,3)

i -I. 38.1.3. Now we shall go one step furthe r a nd drop thc requi rement of reduction, i.e. (38:5). So we demand of v(S) only (38:4) , restricting ils values for two-element sels S. We restate t he final form of ou r question: (38,A)

Consider all zero-sum four·pcrso n games where v(S)

(38,6)

~

0

for all two·element sets S.

.F or whi ch among these is the set V of (37:2) in 37.3.2. a solut ion ? It will be not.ed t hat since we have dropr:e:i t he requirements of normal· ization and reduction of v(S) all connections with t he geometrical represcn· tation in Q are severed. A special manipu lation will be necessary t.herefore at. the end, in order to put t.he results which we shall obtai n back into the framework of Q. 38.2. Eu or = - I, according to whether z is < or = Ut. Now U" U~, U3, u~ arc the four numbers of (38:14), the smallest of which is v. By (38:15) z ~ v, i.e. always -z + Uk - 1 ~ -·1 , and = occurs only for the greatest possible value of z, z = v, - c.ud then only for those k for which Uk attains its minimum, V. , If x" X I , x, diffcr from 0 uy < 1', thcn cach of the four numhers II" III, II" II, llf (38 : 14 ) 1 + ,'~ _ land> 1 - ,', - 1; hCII(;c on a relativc size they vRry by < ! : I .. I. So wc !Ire still in Z. In othcr words: Z contains II cuhc with thc same ccntcr as Q, but with ,'~ of Q's (linear) size. Actually Z is somewhat bigger than this, its.volume is about, rlo~ of the volume of Q. .. On that diagonlll x, .. x, .. ;r " so the II" Ill, u~, u. are : (tb rce times) 1 - x, and I + 3z , . So for x , ~ 0, v _ I - x" w - 1 + 3L" hence (38 :16) ueromes x, < i. And for x , ~ fl, v _ I + 3l." IV - 1 - x " h(>nc~ (38: 16) becomes> - -?,. &I (he inl("TSCction is thi~: o ~ x, < : (this is Ilreeiscly C) o ~ x, > - ,', (Bis O~L ,> -!).

is


0, i.e.

i_I

everye > e*, is the excess of a fully detached imputation- henec a fortiori of a detached one; and e* is, of course, the excess of a detachcd imputation ~

" '.

Thus all parts of (45:B:a), (45:B:b) hold for (45:6). 46.2.4. The fu lly detached and the detached extended imputations are also closely connected with the concept of domination. The properties involved are given in (45:C) and (45:0) below. They form a peculiar anti· thesis to each other. This is remarkable, since our two concepts are strongly analogous to each other-indeed, the second one arises from the first one by the inclusion of its limiting cases. 'This continuity a rgument is valid because the _

~ign

is included in (45:4 ).

372

COMPOS ITIO N AND DECOMPOS ITION OF GAMES ~

(45:C)

A fully detached extended imputation a dominates no other ~

extended imputation f3. ~

Proof:]f a

~

f-o

~

fj, then a must possess a non-empty effective set. ~

(45,D)

An extended imputation a is detached if nnd only if it is ~

dominated by no other extended imputation {3. ~

Proof: Sufficiency of being detached: Let a = lab' ~

.

J

« .. I

be detached .

~

Assume a contrario {J ~ a , with the effective set S. Then S is not. empty : ai < Pi for i in S. So ~ a; < ~ (ji & v(S) contradicting (45:4). 'in S

i in S ~

Necessity of being detached: Assume that a = la[, .. , a .. I is not detached. Let S be a (necessarily non-empty) set for which (45:4) fails, i.c. ~ CI, < v(S), Then for a sufficiently small 6 > 0, even itn S

~ (0'

+ ')

~ v(S).

i in S ~

Put f3 =

l,s[, ... ,

.B~I ~

and S is effective for

(j :

= {OJ

r

+ 6,'

"

a"

+ 61. ~

(j,

:i v (S) .

Th us

then always

ai


0, Irlr > O.

Proof: The statements concerning Ir l l , which is = n1' by (45:2) of 45. 1., coincide with the defi nitions of inessentiality and essentiality of 27.3. , as reasserted in 42.5.1. The statements concerning Ir l1 follow from these concerning Ir l l , by means of the inequalities of (45:F), which we can use here. 46.3.2. The quantitative relationship of Ir l! and Irl 1 is characterIZed as follows: Always (45,F)

STRUCTURE OF THE EXTENDED THEORY

373

Proof; As we know, Ir l. and Ir l. are invariant under strategic equivalence, hence we may assume ihe game r to be zero-sum, and even reduced in the sense of 27.1.4. We can now use the notations and relat.ions of 27.2. Since WI. "" ny. we want. to prove that. _ n_

(45,7)

n _ 1 "1

~

C!

Ir lI s-

n(n - 2)

2

-yo ~

Proof of the fi rst inequality of (45:7) : Let a = jal, ... , « .. I be detached . Then (45:4) gives for the (n - l)-element set S = I - (k) ,

•

L

L.

a~ =

(:Ii -

Qi

~ v(S) = 'Y, Le.

'In 8

(45,S)

Summing (45:8) over k i.e. (n - 1)

•

L

(:Ii

~

I, .

n, gives n

•

L

(:Ii-

•

L a. ~ n ~ 1 'Y.

~ ny,

•

L ai.

Now v(I) = 0, so e =

i- I

Thus e ;;:; ~I

n-

'Y

for all detached imput.ations ; hence

Proof of the second inequality of (45 :7): P ut a Oll "" n - -2- 2 1', and -a 00 = loro, ...

,

-

a:~1

Irl t !i:i n~I ,. = ja", . . .

This a Ot is detached, i.e. it fulfills (45 :4) for all S s;; I . the number of elements of S. Now we have : p "'" 0 : S ... e, (45:4) is trivial. p "" I : S = (i), (45 :4) becomes aGO iii: v«i»,

I

aool.

Indeed : Let p be

.I.e. -n -2- 2 'Y Ii: -,. -'Y w h'IC h l' S 0 b' VIOU8. P ;;: 2 : (45:4) becomes paM i?:, v(S), but by (27 :7) in 27.2.

v(8) " (n - ph, so it sufficp.s to prove pa GO

~

n- 2

(n - ph i.e. P - 2- Y Ir lt is "too great" in that sense. This view will be corroborated in a much more precise sense in 46.8. , They are"', - t fo r n - I ; I, 0 for n "" 2.

KOle also the paradoxical values ... and

-11 1

•n _ 1

n - 2

< - ,-

means 2 < (n - l )(n - 2) which is clearly the case for all n

• For" _ 4 our inequality is! Cl;Sential game with

11'1. -

~

11'1,

11'1, and

;i

Ir lo :i Ir h.

also one with

~

4.

Ali mentioned above, we know an

11'1. - i

11'1..

STRUCTURE OF TH E EXTEN DED TH EORY

'!i.'.

375

Detached Imputation. and Varioul Solutions.

The Theorem CODDectine E(e, ), F (e. )

45.4.1. (44:E:c) in the definition of a. solution in 44.7.3. and our result (45: 0 ) in 45.2.4. give immediately: (45,0)

A solut ion V for E(~.) [F(e . )) must. contain every detached extended imputation of E (te) (J"(eo)].

The importance of t his result is due to its role in the followi ng consideration. After what was said at the beginning of 44 .7.2. about the roles of E (eo) and F(eo), the importance of establishing t he complete intcNcia.tiooship between t hese two cases will be obvious. I. e. we must determine the connection between the solutions for E(eo) and P(e.}. Now the whole difference between E(e,) and P(to} and thei r solutions is not easy to appraise in an in t.ui t.ive way. It is difficult to see a priori why t here should be any difference at all : In the first case t he " gift.," made to the players from t he outside, has the prescribed value e., in t he second case it. has the prescribed maximum value eo. It is difficult to see how t he "outside source," which is willing to contribut e up to eo can ever be allowed to contribute less t han eo in a "stable" standard of behavior (Le. solu t.ion). Howcvcr, our past cxperience will caution us against rash conclusions in this respect. Thus we saw in 33. 1. and 38.3. t hat already thrce and four·person games possess solutions in which an isolated and defeated player is not "exploited" up to the limit of the physical possibilities- and the present case bears some analogy to t hat. 46.4.2. (45:G) permits us to make a more specific statement: ~

A detached extended imputation

(I

belongs by (4 5:G) to every solution for ~

F(eo), if it belongs to F(eo). On the other hand, a clearly cannot belong to any solution for E(co) if it does not belong to E(eo). We now defin e: (45,10)

~

D* {e o) is the set of all detac hed extended imputations a in F{eo), but not in E (eo).

So we see : Any solu t ion of P(eo) contains all elements of D· (eo); a ny solut ion of E (to) contains no elemcnt. of D·(e.). Consequently F(e. ) and E(eo) havc certainly no solution in common if D*(eo) is not empty. ~

Now the dctached a of D*(eo) a re characterized by having an excess e ;:;; eo, but not e = eo-i.e. by

(45,") From th is we conclude: (45 ,H)

e

< eo.

D·(eo) is empty if and only if

e. :1! 1rlt·

376

COMPOS ITION AND DECOMPOS ITION OF GAMES

Proof : Owing to (45:B) and to (45:1 1) above, the non-empt iness of D +{ea) is equivalent to the existence of an e with Ir l, ~ e < co- i.e. to eo > Irlt. Hcncc the emptiness of D·(eo} amounts to eo & Ir l,. Thus the solutions for F(eo) and for E(e~) are Sllre to differ, when eo> Irk This is further evidence that eo is "too large" fo r nor mal beha vior when it is

> Ir l,· 45.4.3. Now we can prove that the difference indicated above is the only one between the solutions for E{eo) and for P(eo). 1\'l ore precisely: (4.oJ)

T he relationship

V+=!: W

(45' 12)

=

V

U

D +(co)

establishes a one-to-one relationship between all solutions V for E(co) and all solutions W for F(co). T his will be demonstrated in the next section. 46.6. Proof of t he Theorem

45.5.1. We begi n by proving some auxiliary lemmas. T he first one cor.sists of a perfectly obv ious observation, but of wide applicability: ~

Let the two extended imputations "Y = I"YI' . . . , "Y .. I and

(45,J) ~

.5 =

/.5 . ., "

.5~1

( 15013)

bear the relationship

for all then for every

--- --- ---"Y implies 0:,

!-

(X

--

i= l ,···,'lj 0:

!-

.5.

The meaning of t his result is, of course, that (-45;13) expresses some ~

~

kind of inferiority of .5 to "Y - in spite of the intransitivity of domination. T his inferiori ty is, however, not as complete as one might expect. Thus

---

one cannot mn.ke the plausible inference of"Y

!-

-..

--

{3 from .5 !- {3, because the

~

~

effectiv ity of a set S for {j may not imply the same for "y . (T he reader should recall the basic definit ions of 30.1.1.) It should also be observed, that (4S:J) emerges only because we have extended the concept of imputations. For our older definitior,s (d. 42.4. 1.) we would have had

L." 1'. =

"

~

{ji;

hence "Y; ;:;; O. fo r alt i ~

=

1, ..

,n

~

nece.~ i tatcs

"Y. = {j , for alt i = 1, ., n, i.e. "Y = {j. 45.5.2. Now four lemmas leading directly to the desired proof of (-45:1).

(-45:K)

If

- 0:

!-

-

---P with a detached and in F(eo) and (3 in E(eQ), then ~

there exists an

0: J!-

~

~

fj with a' detached and in E(eo).

STRUCTURE OF THE EXTENDED THEORY

377

Proof: Let S be the set of (30:4:a)-(30:4:c) in 30.1.1. for the domination ~

a

~

f-o

S = I would imply

(J.

OIi

> Pi for all

"

~ .; - v(l) ~

>

"

~

a'

= fa~,

...

,a~1

f

~

=

" v(1) :i eo = ~ ~; - viI),

1, . . . ,

fl.,

not in S.

Define

with a~

choosing

v(l).

L." a. -

contradicting the above. So S ¢ 1. Choose, therefore, an io ~

-

~~;

But as a is in F(eo) and fJ in E(eo), so

i = 1, . . . • n so

0 so that

=

a:. = ClIi

L" a~ -

ai. +~,

i # io,

for

Thus all a~ !1:;:

v(l) = eo·

;-1

is detached and it is clearly in E(ea). ........

................

for all i in S, so our ex

f-o

fJ implies a'

.

~

a;j

henCe

'

Again, as a~ = "' for i ¢ io, hence ........ f-o

fJ.

Every solution W for F(ea) has the form (45:12) of (45:1) for a unique V s; E(eo),l Pruu!: Obviously the V in question- if it exists at all-is the intersection W n E(to), SO it is unique. In order that (45:12) should hold for

v=

WnE(eo),

we need only that the remainder of W be equal to D*(eo), Le.

Let us therefore prove (45:14). Every element of D*{eo) is detached and in F(eo)-so it is in W by (45:0). Again, it is not in E(eo), so it is in W - E(eo). Thus (45,15)

If also (45,16)

then (45:15), (45:16) together give (45:14), that (45:16) is not true.

8.8

desired.

Assume therefore,

~

lal, ... , a.1 in W - E(eo) and not " Then a is in F(eo), but not in E(eo), so L a. - vel) < eo. As

Accordingly, consider an

a

=

~

in D*(eo). I

We do not yet assert that tbia V is a solution for H(,.l-that will come in (45:M).

COMPOSITION AND DECOM POS ITION OF GAMES

378

-

o is not in D'* (en), thia excludes its being de tached .

-

non-e mpty set S with

L

" , 0~1

0; =

Oi

=

Oi

,

(:Ii

f

>0

v(S).

ii .. S

Now form 0' = lo~,

choosing




~,

Oi

(:I;

-

ii:; ai, this implies -f3 f-o -a by (45 :J ).

a belong to (the solution) W. He nce (:I' for all i in S, a nd (:I: :;;: v(S). So

L

;,n S

--> (:I '

f-o

-

-->

(:I.

-

But as both a " a belong to (the solution) "Ii I this is a contra-

diction. The V of (4 5 :L) is a solution for E (eo).

Proof: V S; E (eo ) is clear, and V fulfill s (44:E:a) of 44.7.3. along with W (which is a solution for F(eo»), since V s;; W. So we need only verify (44 :E:b) of 44. 7.3.

- - -- -- - - -- -Considcr a

~

in E (eo), but not in

W, hence there exists a n If t his

an

(:I

a in

V.

W with

Then (J is also in F(e.) but not in

a f-o ~

(W is a solution for F(eD)!).

belongs to E (eD), then it belongs to W n E (eD)

(:I

in E (e.) with

If

0

0

f.o

=

V I i.c. we have

f3.

docs not belong to E(eo}, then it belongs to W - E (eo) - D· (eo},

and so it is de tached.

Thus a

f.o

f3,

(:I

detached and in F(eo ).

H ence there

exists by (45 :K) a n -(:I' f-o -~, -a' detached and in E (eo). By (45 :G) t his --> (:I' belongs t o W, (E (eo) so F(eo ), W is a solution for F(eo )!); hence it belongs to W n E (eo) = V. So we have an (:I' in E(e.) wiLh a' T hus (44:E:b) of 44 .7.3. holds a t a ny rate. ( .t5 : ~ )

f-o (J.1

If V i ~ a solution fol' Eh), t hcn the W of (45 :12 ) in (45 :1) is a solution for F(eo) .

---

Proof: W soF(eo) is clea r, so we mllst prove (44 :E :a), (44:E :b) of 44 .7.3. Ad (44 :E :a): Assume

(:I

f-o

exclude t hat f3 be detached.

~

for two a,

~

inW.

a f-o ~ a nd (45:D)

So f3 is not in D· (eo), hence it is in

W - D· C,.) -

v.

STRUCTURE OF THE EXTENDED THEORY --+

--+

Hence a

--+

fl excludes that

f-
O.

For 0 < E < ft, (J et) is not detached, so there exists a. non-empty S s;; I with ~ PiCE) < v(S). Hence there exists by continuity a non-empty i in S

S s;; I even with

-

L.

- -

I3;(Et) ~ veS).

Besides, always Pi(E,)

And (J eft) belongs to W.

{:J (ft) f-< {J

- --

Summing up: In every case there exists an

-. a,

> Pi,

hence

'in S

-(t

f-o

{J

in W.

(This

(l

was

fj (fl), a, {J (ft) above, respectively.) So (44:E:b) is fulfilled. We can now give the promised proof: Pro%f (45:1): Immediate, by combining (45:L), (45:M ), (45 :N). 4li.6. Summary and ConclusioDs

4.5.6.1.

OUT

(45,0) (45,0,,)

main results, obtained so far, can be summarized as follows:

If

eo
,

(45,0,b)

then E(eo), F(eo) are not empty, both have the same solutions, which arc all not empty. If (45:0:c) then E(eo), F(eo) are not empty. they have no solution in common, all their solutions are not empty.

Proof: Immediate by combining (45:A) , (45: 1) and (45:H). This result makes the critical character of the points eo = -Irb. Ir !, quite clear and it further strengthens the views expressed at the end of 45.1. and following (45: H) in 45.4.2. concerning these points: That it is here where eo becomes" too small" or "too large" in the sense of 44.6.1. 45.6.2. We are also able now to prove some relations which will be useful later (in 46.5.).

eo

Let W be a non-em pty solution for F(eo), i.e. assume that -I r l l . Then

~

-

Max. w e(o:) = eo 0,"

381

DETERMINATION OF ALL SOLUTIONS

Also ~

Max-.

amW

~

w e( a) m

e( a) - Min- . ..

= Max (0, eo -

Jrlt)·'

Proof: (45: P:c) follows from (4 5:P:a), (45:P:b) since

eo - Min (eo,

Ir lt)

Max (eo - eo, eo -

=

Ir lf )

Max (0, eo -

=

Irl,).

We now prove (45:P:a) , (45: P:b). Write W = V u D*(eo), V a solution for E(eo) , following (45:1).

eo

~

As

~

-l r lL, so Vis not empty (hy (45:A) or (45:0».

As weknow e(a) =eo

~

throughout V and e( a) < eo t hroughout D* (eo}. Now for eo ~ Ir lt , D* (eo) is em pty (by (45:H», so ~

~

Max- . e( a ) = Max-. e( a) = Co, "mW "mY

(45"9)

~

Min-.

.. 'nW

~

eta) = Min-.

.. mY

eta) = eo.

> Ir l" D* (eo) is not empty (again by (45:H» , it is the set of all detached a with e( Cl) < eo. Hence by (45:B:b) in 45.2.3. these e( a) have And for eo

~

a minimum ,

~

Ir lf .

So

~

we have in

this case:

~

w e( a) .. m

Max-.

~

= Max-. V e( a) = eo, .. m

~

Min- .

" mW

~

eta) = Min.......

" m

D·(~ o)

eta) =

Ir lt .

(·15 : 19), (45:19 *) together give our (.15:1':0.), and ("5:20), (45:20*) give

together our (45:P:b). 46. Determination of All Solutions in a Decomposable Game l6.1. Elementary Properties of Decompositions

46.1.1. Let us now return to the decc:nposition of a game l'. Let r be decomposable for J, K ( = I - J) with 6., H as its J-, stituents. ~

Given any extended imputation a ~

J-, K-constituf'nts excesses

~

(3, l'

«(:J.

=

a.

=

lal, . .. , a~1 for

for i in J, 1'.

=

0;

K~co n­

I , we form its

for i in K) , and their

'Our assertion includes the claim that t hese Max..... and Min_ exist. "inW "inW t Verbally: The maximum exceS:! in the solution W is the maximum excess allowed in F (eo): ~ •• The minimum excess in the solution W is again ~o, unless t~ > Ir b, in which case it is only Ir l,. I.e. the min imum is as ncarly e.!I.II po~ib!e, considering that it must never exceed Ir lt. The "width" of the interval of excesses in W is the excess of~. over Irl., if any.

382

COM POS ITION AND DECOMPOSITION OF GAMES ~

~"

Excess of a in I: e = e(a)

=

~ ai - V( I ),

i. L ~

(4601)

r

~

Excess of (3 in J:! =/(a)

=

U; -

v(J),

iinJ ~

Excessof

'Y

~

inK: g = g( 0:) =

L

a; - V(K).L

iin K

Since v(J)

(46,2)

+ v(K)

~

v(l)

(by (42:6:b) in 42.3.2., or equally by (41:6) in 41.3.2. with S = J, T = K), therefore '~f+g

(46,3)

We have

(46,A) ('6,A,,) ('6,A,b)

Ir l, Ir l,

~ ~

1"1, + IHI., 1" 1, + IHI>.

r is inessential if and only if .0., H are both inessential.

(46,A,,)

Proof: Ad (46:A:a): Apply the definition (45:2) in 45.1. to f, A, H in turn .

Ir l,

(4604)

~ v(I) -

L v( (,) , L v((i)), L v((i)).

; in I

(46,5)

1"1.

~ v(J) -

(46,6)

IHI.

~ v(K) -

tin J

i in K

Comparing (46:4) wit.h the sum of (46:5) and (46:6) gives (46:A:a), owi ng to

---

(46'2) .

Ad (46 :A :b): Let a, {J, detached (in I) if

L

'Y

be as above (before (46: 1».

Q, '" v(R)

Then ....... a is

R r;. I.

for all

lin R

Recalling (4C:6) in 41.3.2. we may write for th is

L

(46,7)

,inS ~

Q,

+ L

Q, '"

v(S)

+ v(T)

for all

S

~J,

T.s;;K.

iinT

~

Again {J, 'Yare detached (in J, K) if

L L

(46,8)

Q, '"

v(S)

for all

S!;.J,

Q, '"

v(T)

for all

T~K.

;in S

(46,9)

;;n T

, Up to this point it was not necessary to givc explicit ex pression to the dependence ~

~

or a's I'xceS'! e upon a .

We do this now for e as wen as for j, IJ.

383

DETERMINATION OF ALL SOLUTIONS

Now (46:7) is equivalent. to (46:8), (46:9). Indeed: (46:7) obt.ains by adding to (46 :8) and for S = (46 :8) and (46:9); and (46 :7) specializes for T = to (46:9).

e

e

~

~~

Thus a is detached, if and only if its (J-, K-) constituents fJ, l' are both detached. As their excesses e and f, g are correlated by (46:3), this gives for t heir minima (cf. (45:B:b»

Ir l.

~

lal. + IHI.,

i.e. our formula (46:A:b). Ad (46:A :c); Immediate by combining (46:A:a) or (46:A:b) with (45 :E) as applied to r, fl. , H. The quantities Ir ll , Irl, are both quantitative measures of the essentiality of the game r, in the sense of 45.3.1. Our above result states that both are additive for the composition of games. 46.1.2. Another lemma which will be useful in our further discussions: ~

~

If a f-< 11 (for r), then t he set S of 30.1.1. for t his domination can be chosen with S ~ J or S ~ K without any loss of generality.! ~

~

Proof: Consider the set S of 30.1.1. for the domination a f-< {J. If accidentally S.s;; J or S J;; K , then there is nothing to prove, so we may asl:!ume that neither S.s;; J nor S.s;; K. Consequently S = SI UTI, where SIS; J, TI.s;; K, and neither SI nor TI is empty. We have (1., > (J, for all i in 8, i.e. for all i in 81, as well as for all i in T I • Finally

L ',,, v(S). i in S

The left hand side is clearly equal to side is equal to V(SI)

L

a.:

+ L

U;,

while the right hand

+ v(TI) by (4 1:6) in 41.3.2. Thus ~ a, + L Cl, ~ v(Sd + v(TI)'

;inS.

iinT.

hence at least one of

L ',,, V(S,), ;in 5,

L ',,, v(T,) 'in T,

must be true. ~

~

Thus of the three conditions of domination in 30.1.1. (for a r< (J) (30:4:a), (30:4 :c) holds for both of 8 1, Tl and (30:4 :b) for at least one of them. Hence, we may replace our original S by either 81 (; J) or TI(s; K ). This completes the proof. I I .e. this extra restriction on S does not (in thia casel) modify the concept of domination.

384

COMPOSI TION AKD DECOM POS ITI ON OF GAM ES

46.2. Decom position and Its R elati on t o th e Solutions : First Re5ult. Coucu ninl f(h)

46.2.1. We now di rect our course towards t he main objective of this part of t he theory: The determ ination of all solutions U, of t he decomposable game r . This will be achieved in 46.6. , concluding a chain of seven lemmas. We begin with some purely descri ptive observations. Consider a solution U, fo r F(e.) of r . If U, is empty, t here is nothing more to say. Let us assume, t herefore, t hat. U, is not empty-owing to (45:A) (or equally to (45:0» th is is equ ivalent to

-\r h ""

co;;:;

- IHiJ.

-I~b

Using t he notntions of (46: 1) in 46. 11. we form:

-

Max-. U I ( a) = ~, .. In

I

Min-< . u / (-;;)=!' .. In

I

-

Max-. U g( a) = .. In

I

-

-It,

1 T hat all these quantities call be forme d, i.e. that t he maxima and the minima in question exist and are lI.!l8umed, call be IlScertained by a simple continuity consideration.

-

Indeed f( G ) -

-

L

L

G, - v(J) and g( O.

Now the definition (46:23) becomes

and so the full condit.ion (46:16)1 becomes

1.0.12+ IHI2 + w

=

eo,

I.e.

eo

(46,33)

=

Ir lt

+ w.

(46:31) and (46:33) give also (46,34)

46.5.3. Summing up: (46 ,H )

The condit.ions (4() :IG) , (4(i : 17 ) of (46:G) amount to this: One of the two fo\lowin~ cases must hold: Case (a): (I) - Ir ll ~ ec ~ Ir IJ , toget.her wit.h

- I.o.h ~ ~ ~ 1.0.11, - IHI, " " " IHI.,

(2) 1(3)

and (4) Case (b) , (1) toget.her with (2) (3) a nd (4)

~+It=eo.

eo > Irl

l ,

~ >

161" " > IHI" "-

Ir l,

~ ~

- 161,

~

" - IH(,.' - Irh and ~

Proof: Case (a): We knew all along, that. eo ~ ~ -Iah, -I H ll. The ot.her condit.ions coincide with (46:28), (46:25), (46:27) which contain the complet.e description of this case. Case (b): These conditions coincide with (46:34), (46:29), (46 :30) , (46 :33) which contain the complete description of the case (after elimination of w which subsumes (46:31) under (1H3». ~ ~

Ct footnote 2 on p. 391. • T he reader will note tbat wbile (1)-(3) for (a) and for (b) show a strong analogy, the final condition (4) is entirely different for (a ) Bnd for (b). NevertheleSll, all this wal obtained by the rigorous discuuion of one consistent t heory t More will be uid about thie later. I

DETERMINATION OF ALL SOLUTIONS

393

'6.6. The Complete ReBult iD E(t . )

46.6. (46:G) and (46:H) characterize the solutions of r for F(eo) in a complete and explicit. way. It is now apparent, too, that the cases (a), (b) of (46:H) coincide with (45:0:b), (45 :0:c) in 45.6. L: Indeed (a), (b) of (46: H) are distinguished by their conditions (I), and these are precisely (4 5,0,b), (45,04 We now combine the results of (46:0), (46: H) with those of (45: 1), (45:0). This will give us a comprehensive picture of the situation , utilizio'g all OUf information.

(46J) (46,h)

(46,U)

If (I)

then the empty set is the only solution of r , for E(eo) as well as for F(co). If (1) then r has the same solutions 0, for E(eo) and for F (eo). These which obtain in the following manner : Choose any two ip, (f so that

0, are precisely those sets, (2)

and

(3)

-1"1, " • " 1"1" - IHI, " f " IHI..

(4) Choose any two solutions VIt W ~ of 6, H for E{ip), E«(f). Then Of is t.he composition of VI and W K in the sense of 44.7.4.

If

(46,,,,)

(1)

then r does not have the same solutions 0, for E(eo) and U, for F(eo). These A, and U, are prccisely thosc sets which obtain in the following manner : Form the two numbers ip, (f with (2)

(3) which are defined by (4) " - Ir l. ~

-

• - 1"1, ~ f - IHI,· Choose any two solutions VI. WK of 6 , H for E(ip), E(y,). Then Of is the sum of the following sets: The composition of ~

~

V I and of the set of all detached f1' (in K) with e( f1 ') = IHI2; the ~

composition of the set of all detached a

~

I

(in J) with e( a ') = 16 12

394

COMPOSITION AND DECOMPOSITION OF GAMES ~

and of

WK; the composition of the set of all detached a' (in J) ~

with e( a ')

~

=

Ir ll , respectively- are the" too small" or "too large" values of eo in the sense of 44.6.1.; i.e., that case (46:I:b), - Irb ~ eo ~ Ir l., is in some way the normal zone. Now our picture shows that when the excess eo of r lies in the normal zone, then the corresponding excesses i1J, ~ of ~, H lie also within their respective normal zones. 2 In other words : The normal behavior (position of the excess in (46:1 :b)) is hereditary from rto~, H. Second: In the case (46:1 :b)- the normal zone-~, ~ are not completely determined by eo, as we repeatedly saw before. In case (46:I:c), on the other hand, they are. This is pictured by the fact that the former domain is the rectangle (b) in the ~,f-plane, while the latter domain is only a line (c). It is worth noting, however, that at the two ends of the case (46:1:b)for eo = - Irl lt Ir!:r- the interval available for ~, ~ is constricted to 9. point. ~ Thus the transition from the variable ~, ~ of (46:l:b) to the fixed ones of (46:1 :c) is continuous. Third: Our first remark stated that normal behavior (Le., that the position of the excess corresponds to (46:I:b)) is hereditary from r to~, H. It is remarkable that, in general, no such heredity holds for the vanishing of the excess, i.e. that eo = 0 4 does not in general imply II = 0, ~ = O. It is precisely the vanishing of the excess which specializes our present theory (of 44.7.) to the older form (of 42.4.1. which, as we know, is equivalent to the original one of 30.1.1.). We will examine the variability of ~, ~ when eo = 0 more closely in the last (sixth) remark. Before we do that, however, we give our attention to the connection between our prcsent theory and the older form. , We leave to the reader the simple verification that the geometrical arrangements of Fig. 69. express, indeed, the condition of (46:1:b), (46:1:c). f I.e.- Ir \. ;:l to ~ Ir\. implies -I&\' :s ;p ;S 1&11, - IH!. ;S " ;S IHI., d. (46 :1:b) . • This is one ca.se of degeneration, alluded to It. t.he end of 46.7 . • Of coul1le, t . - Oiies in the normal ca.se (46:I:b): - Irl! ;S 0 ;S 1rl,.

DETERMINATION OF ALL SOLUTIONS

397

Fourth : It is now evident that the present, wider form of the theory must of necessity receive consideration, even if ou r primary interest is with the original form a lone. Indeed: in order to find the solutions of a decomposable game r in the original sense (for eo - 0), we need solutions for the constituent games to., H in the wider, new sense (for i/J, J- which may not be zero) . This gives the remarks of 44.6.2. a more precise meaning: It is now specifically apparent how the passage from the old theory to the new one becomes necessary when the game (6 or H) is looked upon as non-isolated. The exact formulation of this idea will come in 46.10. 46.8.2. Fifth: We can now justify the final statementz about (44:D) in 44.3.2. and (44:0:a), (44;0:h) in 44.3.3. (46: I :h) shows th8.t (44:D) is true in thc case (44: I:b), if we relinquish the old theory for the new one; (44:l:c) shows that (41:0) i8 not true in the ea.se (44:l:c) even at. that price. Thus the desire to secure t.he validity of t.he plausible scheme of (44:0) motivatcs the passagc to the new theory as well as the restriction to case (44:I:b)- the normal case. If we insist interpreting (44:0), (44:0:a), (44:0:b) by the old theory, t.hen (-14:0), (44:0:a) fail,L while t.he conditional statement of (44:0:b) remains t.ruc. 2 46.8.3. Sixth: We saw thnt e. = 0 does not. in general imply i/J = 0, f = O. What does this " in general " mean? '1>, Vt arc subject. to the conditions (2)-(4) of (46:I:b). As eo = 0, so (4) means that Vt = -i/J and permits us to express the remaining (2), (3) in terms of i/J alone. They becomc t·his:

Now apply (45:E) to 6 , H. Then we see: If 6, H a re both essential, then the lower limits of (46:35) are < 0 and the upper limits are> 0, so i/J can really be ¢ O. If either 6 or H is inessential, then (46:35) implies i/J = 0 and hence Vt = O. We state this explicitly: (46J)

eo = 0 implies 'I> = 0, Vt = 0, Le. (44;0) of 44.3.2. holds even in the sense of old theory if and only if either 6 or H is inessential. 46.9. Dummiel

46.9.1. We can now dispose of the narrower type of decomposition, described in footn ote 1 on p. 3'ID-the addition of "dummies" to the game. Consider the game 6 of the players I', . . . ,k'" "Inflate" it by adding to it a series of "dummies" K; i.e. compose 6 with an inessential game H of the players I", . . . ,IN. Then the composite game is r. L tU we may have e. - 0, ~ ~ 0, I ~ O. Then the decomposability requirement (44:8:11.) of 44.3. 1. is violateU, IllJ stated in 44.3.3. I Repreaenting the special ease e. - 0, 0; - 0, ~ - O. • It is now convenient to reintroduce tho: notations of 41 .3. 1. for the players.

398

COMPOSITION AND DECOMPOSITION OF GAMES

We will use the old theory for all these games. By (3 1:1) in 31.2.1. there exists precisely one imputation for the inessential game H- say ~

1'~ =

hh ... ,'Yr"I·' By (3 1:0) or (31:P) in 31.2.3. H possesses a ~

unique solution: The one element set (1' ~). Now by (46:J) and (46 :1 :b) the general solution of r obtains by com· posing the general solution of 6 with the general solution of H-and the latter one is unique! In other words: "Inflate" every imputation /31 = 1/3" •... ,/3k'\ of J (i.e. 6) to an ~

~

~

imputation a I of I (i.e. r) by composing it with

l' ~.

i.e. by adding to it the

~

components 1'h ... , 'Yr . : a 1 = 1,6" •...• fJk'. 'Yf", ... ,1'f"l. Then this process of " inflation "-i.e. of composition-produces the general80lu· tion of r from the general solution of 6. This result can be summed up by saying that the " inflation" of a game by the addition of "dummies" does not affect its solution essentially- it is only necessary to add to every imputation components representing the "dummies," and the values of these components are the plausible ones: What each "dummy" would obtain in the inessential game H, which describes their relationship to each other. 4.6.9.2. We conclude by adding that (46:J) states that if and only if the composition is not of the special type discussed above, the old theory ceases to have the simple properties of the new one, and its hereditary character fails, Il8 indicated in the thi rd remark of 46.8.l. "8.10. blibedd.iDc of • Game

4.6.10.1. In the fourth remark of 46.8.1. we reaffirmed the indications of 44.6.2., according to which the passage from the old theory to the new one becomes necessary when the game is looked upon as non·isolated. We will now give this idea its final and exact expression. I t is more convenient this time to denote the game under consideration by 6 and the set of its players by J. It ought to be understood that this 6 is perfectly general-no decomposability of .1 is assumed. We begin by introducing the concepts which are needed to treat a given game 6 as a non·isolated occurrence: This amounts to imbedding it without modifying it, into a wider setup, which it is convenient to describe as another game r . We define accordingly: 6 is imbedded into r , or r is an imbeddi11fl of 6, if r is the composition or .1 with another game H.I In other words, 6 is imbedded in all those games of which it is a constituent.' Recall the notations of 44.2. The game H and the set of itll playef"ll K are perfectly arbitrary, except that K and J must be diejunct. I Since a conatituent of a constituent ia itself a constituent (recall the appropriate definitions, in particular (43: D) in 43.3.1.), an imbedding of an imbedding ie again an imbedding. In other words: Imbedding is a transitive relation. Thie relieves ue from eonsidering any indirect relationships baaed upon it. I

I

DETERMINATION OF ALL SOLUTIONS

399

4.6.10.2. Let us now investigate the solutions of .:l viewing .6 as a. nonisolated occurrence. In the light of the above, this amounts to enumerating all solutions of all imbeddings r of 6 , and interpreting them, as far as a is concerned. The last operation must be the taking of the J -constituent in the sense of 44 .7.4. We know from the fifth remark in 46.8.2. that this is only feasible, if we consider no solutions from outside the normal zone (b). One might hesitate whether the solu tions of r should be taken in the sense of the old or the new theory. The former may appear to be morc justified on the standpoint of 44 .6.2.: The outside influences upon the game having been accounted for by the passage from .6. to H, there is no longer any excuse for going outside t he old theory.' It happens, however, that we need not settle this point at all, because the result for 6 will be the same, irrespectively of which theory we use for r . But if we use the new theory for r , we must restrict ourselves to the CIiSe (46: I :b), as discussed above. Thus t he question presents itself ultimately in this form : Consider all imbeddings r of .1., and all solutions of these r : (a) in the sense 0; the old theory, i.e. for E(O), (b) in t he sense of the new theory in the normal zone, Le. for any E(eo) of (46:I :b). Which are the J·constitucnts of the solu tions? 46.10.3. The answer is very simple : The J·constituents (of the r solutions) referred to in (46:K) are precisely the following set.s: All solutions for A in the normal zone, Le. for any E«(1I) of (46 :1 :b). This is true for both (a) a nd (b) 01 (46,K). Proof: eo ... 0 belongs to case (46 :l:b) (cr. footnote 4 on p. 396), hence (a) is narrower t han (b) . Thcrefore, we need only show that all the sets obtained from (b) are among the oues described above, and that a ll these sets can even be obtained with the help of (a). The first assertion is only that of the hereditary character of the normal zone (b). The second assertion foll ows from (46:I:b), if we can do this: Given a (11 with - lAb :Ii (11 ~ lAh, find a game H and y, with - IHh :;; ~ S; IHlt, such that ifJ + J. = 0 and that H possesses solutions fo r E (J.) . Now such an H exists, and it can even be chosen as a three--person game. Indeed: Let H be the essential t hree--person game with general..,. > O. Then by (45'2) ;n 45.1. IHb ~ 3> and by (45,9) ;n 45.3.3. IHI. ~ tlHI, ~ h. We have required y, = -(11 and what we know now amounts to

'.Besides, the tran8itil'ity poin led out in footnote 3 on p. 398, show. tha t any furthe r imbedding of r ean be regarded directly all an imbedding of 6..

400

COMPOSITION AND DECOMPOSITION OF GAMES

This can clearly be met by choosing",( s ufficiently great. Then we also need a solution of H for E(y,). The existence of such a solution (for - 3")' ;::i! Y, ;;;;: h) will be shown in 47. 4.6.10.4. To this result two more remarks should be added: First: If we wanted to handle the process of imbedding in such a manner that the old theory remains hereditary, we would have to see to this: The composition of r from .6. and H has to be such that eo = 0 implies i/J = 0 (and hence y, = 0). By (46:T) this means that either .6. or H are inessential. The latter means (cf. eod), that only "dummies" are added to A. Summing up:

The old theory remains hereditary if and only if either the original game II is inessential, or the imbedding is restricted to the addition of "dummies" to 11. Second: It was suggested already in 44.6.2. to treat the outside source, which creates the excesses and paves the way for the transition of our old theory to the new one, as another player. Our above result (44:L) justifies a slightly modified view: The outside source of 44.6.2. is the game H which is added to .6.--or rather the set K of its players. Now we have seen that the game H must be essential, in order to achieve the desired result. Furthermore we know that an essential game must have n ~ 3 participants, and the proof of (44:L) showed that a suitable H with n = 3 participants does indeed exist. So we see: The outside source of 44.6.2. can be regarded as a group of new players- but not as one player. Indeed, the minimum effective number of members of this group is 3. 4S.10.6. The foregoing considerations have justified our passage from the old theory to the new one (within the normal zone (b» and clarified the nature of this transition. We see now that the "common sense" surmise of 44.3. fails to hold in the old theory, but that it is true in precisely that new domain to whiCh we changed. This rounds out the theory in a satisfactory manner. The leading principle of the discussions of 44.4.3.-46.10.4. was this: The game under consideration was originally viewed as an isolated occurrence, but then removed from this isolation and imbedded, without modification, in all possible ways into a greater game. This order of ideas is not alien to the natural sciences, particularly to mechanics. The first standpoint corresponds to the analysis of t he so-called closed systems, the second to their imbcdding, without interaction, into all possible greater closed systems.

DETERMINATION OF ALL SOLUTIONS

401

The methodical importance of this procedure has been variously empha3ized in the modern li terature on theoretical physics, particularly in the a.nalysis of t he structure of Quantum Mechanics. It is remarkable that it. could be made use of so essentiaJly in our present investigation . • 6.11. Signitleaace of the Normal ZODe

46.11.1. The result (46:I :b) defines for every solution of the composite game r in the normal zone-i.e. a fortiQri for every solution in t he sense of the old theory- numbers i/J,~. This and the immediate properties of i/J, ~ in connection with the solut ion, appear to be so fundamental, as to deserve a fuller non-mathematical exposition. We are considering two games.6., H played by two disjunct sets of players J and K. The rules of these games stipulate absolutely no physical connection between them. We view them nevertheless as one game r - but this game, of course, is composite, with the two isolated constituents A, H. Let us now find all solutions of the entire arrangement, i.e. of t he composite game r. Since it is not desired to consider anything outside of r , we adhere to the original theory of 30.1.1. and 42.4.1.' Then we have shown t hat any such solution UI determines a number ~ 2 with the following prop~

erty: For every impu tation (I of UI the players of l:J. (i .e. in J) obtain together the amount ~, and the players of H (Le. in K ) obtain together t he amount --~. Thus t he principle of organization embodied in U, must ~ti pulate (among other things) that the players of H transfer under all conditions the amount ~ to the players of 1::... The remainder of the characterization of Ur-ie. of the principle of organization or standard of behavior embodied in it- is this: First: The players of 6, in their relationship with each other, must be regulated by a standard of behavior which is stable, provided that t he transfer of ~ from the other group is placed beyond dispute. ! Second: The players of H, in t hcir relationshi p with each other, must be regulated by a standard of behav ior which is stable, prov ided that the tran sfer of '" to the other group is placed beyond di sputc. ~ Third: The octroyed transfer ~ must lie between the limits (46:35) of 46.8.3. (46,35)

I::l~d

&

~ & I I ~iJ

46,11,2. The meaning of these rules is clearly that a ny solution, i.e. any stable social order of r is based upon payment of a definite tribute by one of the two groups to the other. The amount of t his tribute is an integral part of the solution. The possible amounts, i.e. those which can I.e. t. _ O. • Since ¥> + y, - t. _ 0, we do not introduce y, - -¥> . • I.e. that the J-constituent VJ of Ut is a solution of d for E(¥». 'I .e. that the K-constituent W. of UJ is a solution of H for E(-~). I

402

COMPOS ITIO N AND DECOMPOSITION OF GAMES

occur in solutions, are strictly determined by (46:35) above. This condition shows in particular: First: The t ribute zero, i.e. the absence of a tribute is always among the possibilities. Second: The tribute zero is the only possible one if and on ly if one of the two games 6, H is inessential (cf. the sixth remark in 46.8.3.). Third: In all ot.her cases both positive and negative tributes are possible - i.e. both the players of 6. and the players of H may be the tribute paying group. The limits of (46:35) are set h.v both games 6, H , i.e., by the objective physical possibilities of both groups.' These limits express that each group has a minimum below which no form of social organization can depress it: -16h, -IHI,; and, each group has a maximum, above which it cannot raise itself under any form of social organization: 1611, IHII . Thus, for a particular physical background, i.e. a game, say 6, the two numbers 161" 1611can be interpreted this way: - 1611is the worst that will be endured under any conditions and 1.6.1 1 is the maximum claim which may find outside acceptance under any conditions. I The results (45:E) and (45:F) of 45.3.1.-2. now acquire a new significance: According to these the two numbers can only vanish together (when 6. is inessential) and their ratio always lies between definite limits. '6.U. First Occurrence of the Phenomenon of Truafer: n _ 6

46.12. We have seen repeatedly (thus in (46:J) in 46.8.3. and in the second and third remarks in 46.11.2.) that the characteristic new element of the theory of a composite game r manifests itself only when both constituents 6, H are essential. This is the occurrence of eo = 0, but

i.e. a non-zero tribute in t.he sense of 46.11. Now we know that in order to be essential a game must have ~ 3 players. If this is to be true for both 6, H, then the composite game r must have ~ 6 players. Six players are actually enough as the following consideration shows: Let A, H both be the essential three-person games with 'Y = 1. Then 161. ~ IHI, ~ 3, 161. ~ IHI. ~ l. (ef. in46.10.3.). Hence for both ifJ and 1- = -ifJ lie between -3 and i. This implies, as will be shown in 47., the existence ofsolutions VI, W K of 6, H for E(ifJ)' E(y,). Their com-

-I"." I,

I But where the actual amount ill lies between those limits, is not determined by those objective data, but by the solution, i.e. the fltsndard of behavior which happens to be generally accepted. I It must be recalled that all this takes the value of the coalition of all playerB of a., v(J), as zero; i.e., we are discussing the 10000B which are purely attributable to lack of co-operation among the group, and unfavorable general social organizations-and gains, which are purely attributable to lack of co-ope:ration in outside groupa and favorable general social organizationa.

THREE-PERSON GAME

403

position UI is then a. solution of the composite game r with the given VJ· Since VJ was only restricted by - I ;2 VJ ;2 I, we can choose it non-zero. Thus we have demonstrated: n = 6 is the smallest number of players for which t he characteristic new element of our theory of composite games (the possibility of eo = 0 with VJ = -It r! 0, cf. above) can be observed in a suitable game.

We have repeatedly expressed the belief that an increase in the number of players need not only cause a more involved operation of the concepts which occurred for smaller numbers, but that it also may originate qualitatively new phenomena. Specifically such occurrences were observed as the number of players successively increased to 2,3,4. It is, therefore, of interest that the same happens now as the number of players reaches six.' 47. The Essential Three-person Game in the New Theory 47.1. Need for This DiscussioD

47.1. It remains for us to discuss the solutions of the essential threeperson game, according to the new theory. This is necessary, since we have already made use of the existence of these solutions in 46.10. and 46.12., but the discussion possesses also an interest of its own. In view of the interpretation which we were induced to put on these solutions in 46.12. and also of their central role in the theory of decomposition,2 it seems desirable to acquire a detailed knowledge of their structure. Furthermore, a familiarity with these details will lead to other interprctations of some significance. (Cf. 47.8. and 47.9.) Finally, we shall find that the principles used in determining the solutions in question are of wider applicability. (Cf. 60.3.2., GO.3.3.) 47.2. Prepa.ratory Considerations

47.2.1. We consider the essential three-person game, to be denoted by r , in the normalization 'Y = I. Thus Irh = 3, Irl, = i (cf. 46.12.). We wish to determine the solutions of this r for E(eo). ~ In the applications, referred to above we needed only t he normal zone -3 & eo & .;. but we prefer to discuss now all eo. This discussion will be carried out with the graphical method, which we used in treating the old theory in 32. We will, therefore, follow the scheme of 32. in several respects. I For 801M other qualitatively new phenomena which emerge only when there are six players, cf. 53.2. I This is the only problem of absolutely general charaeter, of whieh we have a complete solution at present! ~ We are writing r, to although the applieations employed the notations ~, "and H, il-( - -,,). or eourse, the present I' has nothing to do with the de fJ" (l'j > flit i.e. that

+

.+

(l'

and a l

>

r:r, a; > (Ji,

.- 1

01' '"

-

2,_ ,

-

3

By (47:3 *) the first condition may be writu-n

We restate this: Domination

means that either a 1 > {11, (47,5)

0'

011

0'

0:

2

>

{p

and

a' ~

> (31,

a' > fJ'

and

0/

2

~

>

01 '

>

and

011

6:

a2

{P,

(31

47.3. Tbe Sil: Casu of th e D h;(:ussioD..

_(I _2;_)' -(l-~-).

_(1_2;)

Cases (1) - (111)

4.1.3.1. After these prepa:'ations we can proceed to discuss the solutions V of r for E(eo), for all values of eo. It will be found convenient to distinguish six cases. Of these Case (1)

corresponds to (45:0:a), Cases (U)-(IV) and one point of (V) to (45:0:b) (the normal zone), and Cases (V) and (VI) (without that point) to (45:0:c) (all in 45.6.1.).

47.3.2. ('ase (I):

CG


~ elements. 4

The simple game which obtains in t his way,5 will be called t he direct maiority

game. I Compari80n of (49;1) and (49:.1) ehowe that theSor Wr - Wj!are eimult.aneously certainly ne"2 19 - 2- (0 odd!), we may also say; S must have

n +l • - 2- elemenUl. t Precilely ; The class of strategically equivalent ones (of

11

participants).

432

SI MPLE GAMES

T he smallest n for which this can be done,' is 3. We know t hat there exists only one essent.ial three-persoll game, and t hat. for th is TV it consists precisely of the 2 and 3-element sets- i.e. of the sets with> t elements.

So we see: (W,A)

T he (unique) essential three-person game is simple; it is t he direct majority game of three participants.

For the subsequent n which are eligible, n = 5, 7, , the di rect majority game is merely one possibility among many. 60.1.2. The direct. majority game is only available, when n is odd, and yet simple games exist for even n as well- indeed our prototype of simple games (cf. 48. 1. 2., 48.1.3.) had n = 4. However, t he concept of majority is easily extended to cover the case of even n as well. To this end we introduce weigh led majorities in the following manner: Let each one of the players 1, " , n he given a numerical weight, say WI, • , w .. respectively. Defi ne IV as t he set of all those 8 which contai n a majority of totul weight. This means:

(SO,I)

L w, • in S

>. L •

w,'

i _I

or equivalently,

LW,> LW"

(50,2)

'inS

'in-S

We must again t.ake care to exclude ties. However, owing to t he greater generality of our present setup, it is better to proceed immediately to a complete discussion of (49:W*). 60.1.3. Let us see, therefore, what restriction (-19:W*) imposes upon the WI, " , w~. Ad (49 :W *:a): Since we can express that S belongs to IV by (50:2), so -8 belongs to TV when

L

(SO ,3)

w,
U.s unless Wi - 0 for all i not in S.

MAJORITY GAMES AND THE MAIN SOLUTION

435

of W. In the terminology introduced in 49.6.3.: 0., is required to be constant for the minimal elements of lV-i.e. the elements of Woo,

We define accordingly: The weights

WI, • • . , W R

are homogeneous, if the as of

(50:6) have a common value, to be denoted by a J for all S of

W". Whenever (50:E) is valid we shall indicate this by writing (WI, . . • , W..JII instead of [WI, ... ,w..J. Clearly a > O. A common positive factor affects none of the significant properties of WI, . • • , w .. , therefore we can use t his in the case of homogeneity for a final normalization: Making a = 1. We conclude by observing that the games mentioned at the end of SO.l.3. are homogeneous and normalized by a = 1. These are the direct majority games of an odd number of participants [I, . . . , II, . and the corner I of Q [1,1,1,2j- which can accordingly be written {I, . . . , Ij~ and 11 ,1,1,2]A. Indeed, the reader will verify with ease that a" = I for all S of W'" in both instances. 6O.S. A More Direct Use of the Concept of Imputstion ill Fonnillg Solutions

60.3.1. The homogeneous case introduced above is closely connected with the ordinary economic concept of imputation. We propose to show this now. More precisely: We defined in 30.1.1. a general concept of imputations and based on it a concept of solutions. In forming these we were led by the same principles of judgment which are used in economics, and therefore some relationship with the ordinary economic concept of imputation must be expected. However, our considerations took us rather far from that concept. This applies especially to the constructions which were necessary when we found that sets of imputations- i.e. solutions-and not single imputations must be the subject of our theory. It will now appear that for certain simple games the connection with the ordinary economic concept of imputation can be establishcd somewhat more directly. One might say that for the special games in question the connection between this primitive concept and our solutions can be directly established. Actually it will provide a simple method to find a particular solution for each one of those games. 60.3.2. The two concepts of solution, i.e. the two procedures, support each other quite effectively. The ordinary economic concept provides a useful surmise as to the form of a certain solution. And then our mathematical theory may be used to determine the solutions in question and to make the requirements of the ordinary approach complete. (eL 50.4. on the one hand, and SO.5. et sequ. on the other.) These considerations also serve another end: They bring out the limitations of the ordinary approach with great clarity. The ordinary approach

436

SIMPLE GAMES

functi ons in t his form only for the simple games, and even t here not always and not entirely unaided by our mathematical theory. Besides , it does not disclose all t he solutions for the games to which it appiie!'!. (Further remarks on this subj ect occur throughout the discussion , and particularly in 50.8.2.) In this connection we emphasize again that any game is a model of a possible social or economic organization a nd any soiution is Il. possible stable standard of behavior in it. And the games and solutions not covered by t he method referred to- i.e. by the unimproved economic concept of imputation- will prove to be quite vital ones for social or economic theory. It will be seen that the simple games which can be treated by this special method are closely connected with t he homogeneous weighted majority games of which they are a generalization. 60.'. Discussion of This Direct Approaeh

60.4.1. Consider a simple game r which we assume in the reduced form with "Y ""' I, but which we do not yet restrict any further. Let us t ry to discuss it in the sense of the ordinary econom ic ideas without making use of our systematic theory. Clearly, in this game the sole aim of playcrs is to form a winning coalition, and once a minimal coalition of this kind is formed , there is no motive for its participants to admit additional membcrs. Consequently one can assume that the minimal winning coalitions-thc S of W"'- are the structures that will form . It is therefore plausiblc to assume that a player's fate presents only two significant alternatives: He either succeeds in joining one of the desirablc coali tions or he docs not. In the lattcr case he is defeated, hcnce he obtains the amount - I. In the former case he is successfu l and according to ordinary ideas onc ought to ascribe to this success a value. This value may ,'ary n·om one player to a nother; for player i we denote it by - I + Xi so that Xi is the margin between defeat and success for playcr i. 1 60.4.2. Lct us now formulate the requirements which must be imposed on these XI , • • . , x" in t he course of a conventional econom ic discussion. First: By the very meaning of the Xi necessarily Xi ~

o.

Second: If it happens that no minimal winni ng coalition con tai ns a certain pta)'er i, then there exists for him no alternative to the value - I , and so we need not define any Xi for him .! I We lUIsu me here that the re is only one way of winning, i.e. that the margin l:, is the sa me whiche ver (minimal winning) coaliti on the player succeeds in joining. This is plausible since there is only one kind of succe!S!l ill a simple game : the complete oneevery coalition heing either fully defO

(combine (50:D) (50:E) of 50.2.)

i. I

Wi ~

O.

Actually, there is more than sim ilarity. Thus, if a system of Wi, fulfilling (50: 18), (50:19) is given, a system Xi fulfilling (50: 17), (50:7) obtains as follows: The quantity b of (50:18) is positive. L Multiplication of all Wi by a common positive factor leaves everytHng unaffected, and by choosing this factor as nib we can replace b in (50: 18) by n. Now we can simply put Xi 5i Wi and (50: 18), (50:19) become (50:17), (50:7). If conversely a system of Xi fulfilling (50: 17), (50:7) is given, there is an extra difficulty. We may put Wi"'" Xi. 1 Then (50:7) becomes (50: 19) and (50: 17) yields (50:18) with b

= n,

i.e. a

= 2n

question arises whether the last requirement a

-

"

~

Wi.

But now the

> 0 is fulfilled- i.e. whether

Summing up: Every homogeneous, weighted majority game possesses a main simple solution. Conversely, if a (simple) game possesses a main simple solution, homogeneous weights for the game can be derived from it if and only if (50:20) is fulfilled. 50.8.2. This conncction between homogeneous weights and main simple solutions is significant. But it must be stressed that a homogeneous, weighted majority game will in general have other solutions besides the I Otherwise all i occurring in the S of W" would have Wi _ 0 by (50:18) and (50:1 9). T hen (50:6) of 50.2.1. and (50:19) gives a. :a: 0 for the S of W .., hence a ~ 0, which is not the case. I Thc i which bclong to no minimal winniug set cause II slight disturbance, sillce thcy have no Xi (cr. the sccond remark in 50.4.), while we require their Wi. H owever, the contingency is unimportant (cr. loc. cit.) and we can put these w. - 0 as is easily concluded from the references of footnote 2 011 p. 436.

ENUMERATION OF ALL SI MPLE GAMES

445

main simple one. l And a game wit.h a main simple solution may not fulfill (50:20), i.e. t.here need not be < in

"




Beyond all this, finally , we must not. lose sight of the main limit.ation of t hese considerations: Whether we take the concept of "ordinary" imputation in its narrower fo rm of 50.B. J. (i.e. U = W ... ) or in its widcr original form of 50.6., 50.7. 1. (i.e. U ~ W"' , d. (50: 1) in 50.6.2.), it is certainly restricted to simple games. That. it is necessary to go beyond these, and beyond the special solutions described here, and t.hat t.his forces us to fall back completely on the systematical theory of 30.1. 1., was pointed out at the end of 50.3. 61. Methods for the Enumeration of All Simple Games 61.1 . Preliminary Remarks

61.1.1. Beginning with 50.1 .1. we int.roduced specific simple games which permitted characterization by numerical criteria instead of the original set theoretical ones (ef. the beginning of 50.2.1.). We saw, however, that these numerical procedures could be carried out in several ways a nd that there was no cert.ai nty that all simple games could be accounted for with their help. It is therefore desirable to devise combinatorial (set theoretical) methods that produce systematic enumeration of all simple games. This is, indeed , indispensable in order to gain an insight into the possibilities of simple games and particularly to see how far t he above mentioned numerical procedures carry us. It will appear that the decisive examples of the non-obvious possibilities obtain only for relatively high numbers of players,3 so that a mere verbal analysis ctlnnot be very effective. 61.1.2. We pointed out. at the end of 49.6.3. t hat the enumeration of all simple games is equivalent to the enumeration of their sets 11', i.e. of all sets 1V which fu lfill (49: W·) in 49.6.2. We also noted there that it may be advantageous to replace the use of IV (all winning coalitions), by W" (all minimal winning coalitions). Either procedure provides an enumeration of all simple games. The use of IV is preferable from the conceptual standpoint since IV has the I The main simple solution or the essen tial three-pc["!;on game ([1,1, 11 ., d. the cnd of .50.2.) is thc original solution or 29.1.2., i.e. (32:ll) of 32.2.3. We know from 32.2.3. and 33.1. that other solutions exist. The main simple solution of eorller I of Q ([1, 1,1 ,2)., d. the end of .50.2.) i~ the original solution of 3.5.1.3. We will discuss this game, togdhcr with the mo re ~ellcral one [I, ... , I, n - 21. (1\ participants) in .5.5. and obtain nil solutions. All these references makc it clell r that the solutions other thlln the main simpl!' one are quite signili,·ant, d. 33.1 and 54.1. I _ occu rs for the fi["!;t time for 1\ _ 6, d. the fourth renlark of 53. 2.4 . > O(" ; llt~ for the first time for II _ 6 or 7, d. the sixth ff'mark of ;)3.2.6. Doth these l\ xam]tl,~s !lrf' Iluite illtcrc~tillg in Ill('ir OWII rigl'\. ' n - 6, 7 d . .53.2.

446

SIMPLE GAMES

simpler definition and W" was introduced indirectly with the help of W. For a practical enumeration of all simple games-which is our present aimthe use of W'" is preferable since W" is a smaller set than JV I and therefore morc readily described . We will give both procedures successively. It will appear t hat these discussions provide a natural application of the concepts of satisfactoriness and saturat ion introduced in 30.3. 61.2. The Saturation Method: Enumeration by

MeiDl

of 1V

61.2.1. The sets TV are characterized by (49:W·) in 49.6.2. Le. by the conditions (49 :W· :a)-{49: W· :c) which constitute (49: W .). Let us for a moment disregard (49:W*:c), and consider (49:W*:a), (49:W*:b). These two conditions imply that no two elements of TV can be disjunct. 2 In other words: Denote the negation of disjunctness-i.e. of S n T ;:o!' 9-hy Sffi,T. Then (49:W·:a), (49:W·:b,) imply Gh-satisfactoriness. 3 A more exhaustive statement along these lines is this:

(49: W·:a), (49: W· :b) are equivalent to

.

Indeed:

14 = 2n.

i_ I

As t he games discussed in the second, third and fifth remarks, this one too conesponds to an organizational principle that deserves closer study. In this game the sets of IV"', i.e. the decisive winning coal itions are always minorities (three-element sets) . Nevert.heless, no player has any advantage over any other: Figure 89 a nd its discussion s how that any cyclic permutat ion of the piayers I, , 7- i. e. any rotation of the circle of Figure 89leaves the structure of t he game unaffected . Any player can be carried in t his manner into any other player's place. \ Thus the structure of the game

, •

,

• Figure 89.

.~----~~,~~--~~,

Figure 90.

is determined not. by t he individual propert.ies of the playerst- all a re, as wc saw, in exact.ly the same position- hut. by the relation among the players. It is, indeed , the understanding rcached among 3 players who arc corrclated by (S .) I which decides about victory or defeat. 64, Determination of All Solutions in Suitable Games 64..1.

RUilODi

to Consider O ther Sol utions than the MaiD SolutioD in Simple Games

64.1.1. Our discussion of simple games thus far placed moat emphasis upon the spccial kind of solutions discussed in 5O.S.l.-5O.7.2. and particularly on t he main simple solut.ion of 50.8.1. On the basis of what we have leamed in t he previous sections- es pecially from the examples of 53.2.-this approach docs not aplJear to do justice to all aspects of our problem. I T he game i~ nevertheless IIOt fair in the s~nse of 28.2. 1., since e.g. the two thrcce\ementseUi ( 1,2,4) and ( 1,3,4) net dilfcrently: The former belongs to W, the latter to L. (So in the reduced form of the l1;an"l, with 'Y - I , th e v(S) of the former is 4, and that of the latter is -3.} 1 Wh ich the ru les of the game might gIVe them. ' There eXist in this game no significant relatiolls between any two p layers: It is possible to carry any two given players into any two given ones by 11. ~ui table permutation (of all players I, ' ' , 7) which It:a\'cs the game invariant.

ALL SOLUTIONS IN SUITABLE GAMES

47 1

To begin with, we have seen that we cannot expect all simple games to have f a, Ii in V, hence (30::;:a) is automaticaily i3atisficd. Therefore we need only investigate (3D:5:b) of 30.1. 1. I.e. we must, seCHre this property: ~

If

{3~ ~

W, then a

~

f-o

~

{J for som e a with an = w.

More explicitly: We must uclerminc what limitations (55:5) Imposes upon w. The (3R '#- w of (55:5) cnn be clas.'·;ificd:

> w,

(55,6)

(j,.

(55,;)

fJ.


O.

I.e. (n - l )(n - 2 - w) - I

> 0, n - 2 - w > -n -'- , , and so

w
{n

I, n -

>

l)a., and hence (55:N) gives

-

(n -

2J~

489

I)a • • a.

i - I

-

,

< --,

n- '

as desired.

An a in (t has p fi xed components: the a;(= a ; = a . ), i in S.; and n - p variable ones: the ai, i in ( I, . n) - S . . T hese are subject to the conditions fo r i in (1, . . . ,n) - S . ,

ai ~ - I

and

•

L ai ""

0, i.e.

;_1

ai = -pa • . lin (L .

. .. )-S.

T he lower limits in (55:2 1) add up to less t han t he sum prescri bed in n - p n (55:22), i.e.- (n - p) < -pa.. Indeed, t his means a. < - - = - - l.

P

And by (55:!\') p guarantees a.

I
o.

for i in S*, foriin( I ,·

-

, I l - I , n) -8 *.

-

, Comp:uison of this definition with (55:0') shows that th is rz ' is an rz that it belongs to V -if Ilnd only if the condition of (55 :U) is fulfilled. ~

Since a ' belongs to a this is in agrecment with the resul t of (55 :U) .

i,

i in S-i.e.

THE SIMPLE GAME [I, . . . , 1, n ~

This 80

belongs to a.

I)

I)~

We have

>

1' .. =

n - l ) - S. , 1);>1';=a, or (j" ~

~

~

~

1) in V' with Now by (55:Z)

--

A !oTtiori 1)

~

1) ..

----

a .. = {J .. , and for i

hence

I)

~

Since a, (J belong to V', this excludes

or

491

is clearly an imputation, and as i in S. gives 1);=1';=a;={J, =q;= a.,

I)

(1,

an

21~

>

I)"

1) 1)ft

~

>

or 0

~

{j.

Hence there exists

I).

and

I) ..

- -> 1' ..

r

a

~

1'i

>

1),

= aft = fJ ...

13· As a, fJ,

Consequently

from V'.

I)

~

in

1)

I),

1)i

for an i in (1, .

>

I);

,n- l )-S•. ~

>

1'i

=

or

ai

Thus

(Ji.

1)

~

~

a

=

for

all belong to V', t his is a contradiction.

< 0 is impossible, so

; -1

~ y, ~ O. ;-1

Now 1', ;;i

ai, 1'.

;;i {j; and

L." a,

L."

fJi

=

O.

Hence (55:25) yields ~

all these ;;i relations, i.e. 1'; =

a,

= {J,.

This proves a =

~

{J

as desi red.

~

The values of the a .. fo r all a in V' make up precisely the interval -l;;ia,,;;iw·. ~

Proof: For an a

(55 :E').

In V', a" ;;: -\ is e\'ident, and aft :;i w· follows from Hence we need only exclude the existence of a YI in

-1 ;;i YI ;;i w+, ~

such that a .. # YL for all a in V'. ~

~

There exist certainly elements a of V' with aft to V' by (55:0'), and = w· ~ Y,. Form

a:

Min_

",in V ' with a. ;;:; y,

~

YL: Indeed a· belongs

a .. = Y1, I

, In this ease it is not nece!lSllry to form the exact minimum, but the procedure whieh achieves this is somewhat longer than the olle used below. That this minimum can be fo rmed, i.e. that it exillts and ill assumed, can be aseertained in the same way as in footnote 1 on p. 384. Cf. in particular ( *) loe. cit. What is stated there for V is equally true fo r the analogous set V' in a and for the intersection of V' with the closed set of ~

the a with a. ?; y,. Beeause of this need for closure we must use the condition a. s: y, and not a. > y, although we are really aiming at the latter. But the two will be seen to be equivalent in the ease unde r consideration. (Cf. (55:26) be1ow.)

SI MPLE GAMES

492

and choose a n a + in V ' \\·it.h at

~

YI for which t.his minimum is ass umed:

By (55: I1 ') this a + is unique. and si nce necessari ly at ~ YI, so Y2

~ Yl,

It follows from the definition of Y2 that ~

for no a in V ',

(55 :27)

Now put Yl

"'ith (3.

,

a:t-

=

Yt -

=

o;!" -

t

,

+Ii -;----::p

~,

> 0 and form the imputation

t

= Yt -

E

= YI> {3, == at =

= a. for i in S . , (Ji ~

for i in (J, ..

-

,n -

I) - S. .

Clearly {J belongs to

~

~

a and 13" = YI excludes fJ from V',

-

0';

Hence there exists a "\' in V ' with

'Y 'l- {3.

By (55:Z) this means 1'. > fl. a nd ;i > tJ, for a n i in ( 1, . ,n- l) -S • . ~O\\' 'Y. > {in = Yl nCl'c;;sitatcs by (55:27) 'Y. ~ Yt . 1'. = Yt would

- -

imply i' = a above i ill ( I, )'h

>

+

(by (55:1l') , cr. above). H('ll cc 'Y; = a,+ < (3; for t he , Il - I) - S .. , and not 'Yi > (3. as required. He nce

y,.

-

Thus

'YH

> Yt =

a! and )"

> (3, > at for t he above j in ( I,

--

,11- 1)-S • .

So Y f- a +, and as y , a + b a~(Yt), Yl > Yt, and a,(Yl) > a, (Y2) for a suitable i in (I, ',11. - I) - S., contradicting (55:l{':b). -a

in (a), -{J in (b): I.e. -+ a = -a (y), -{J = -a

~

i

(i in S.), and so

~

a (yj f-o a '. Now (55: 1) is exclude1, since a~(y) = y :;; w· ~ W = a~, and (55:2) is excluded, since ai(Y) = aj = C!-i = a.. So \\'e have a contrad iction.

- -

-

in (b), {J in (a): I.e. --. a = a

a

~

i

(i in S.),

-(J =

a (y), and so

~

a i ~ a (y). Now aj = a,{y) = l'J, = a. and for j r! i, Il, a; = -1 ~ a.(y), i.e. a) ~ a;(y) for all j = I, ' " 11 - 1. T his excludes both (55: 1), (55:2), and gives a contradiction. --

a,

Now

a~

---+ ill (IJ): I.e. ---. a = a'. {J = uk (i, k in S.), and so = a~ = w, thus contradicting (55:B)

{J

-;

a i~

-

a*.

~

Ad (30 :5:b): Assume that (3 is undominated by the clements of V. ~

We \\"ish to prove that this implies t hat (3 belongs to V- which establisll('s (30,5 ,b) . Assume fi rst that 13n ~ w. If (3, < q; = a, for all i = I,' " 11 - I,

-

-

then a (-1) ~ {J, contradicting our assumption. Hence {Ji ~ q; for some i = 1, . . . ,'7J - 1. f\01l' the argument used in the proof of (55:H) ~

~

shows that necess[trily i in S . and (3 = a .,} this casco

'w

obt~jn8 frvl!'l (b), ':!' h

(55 ;1J;c).

~

i.

Therefore (3 hclongs to V

. , ,!n_: from (a) with y ." -I, Ilnd then "" S. from

2J~

THE S I MPLE GAl\'IE (1, . . . , 1, n -

Assume next that ~

clearly Ct'

{jn


ome y > (J ,.) will not affcct this relation {J, < O',(y).1 For this new y we have Y

> {J",

> {Ji, and therefore

~

~

(y) f-< f3 contradicting our assumption. Thus all possibilities are accounted for. O!;(y}

0'

66.11. Reformulation of the Complete Rellult

6fi.11.1. These three cases (I), (II'), (IIU)- into which we subdivided our problem- have been completely settled by (55:0), (55:W), (55:1\-1') rcspectively. Let us now see to what extent these three classes of solutions are related to ea~h other. Among the undetermined parameters occurring in (55:L')- i.e. in (,55:M'), describing case (Il U)- is the set S •. According to (55:L':a) this is any set!;;; (1, . . . ,n ~~ I) with the exception of ertec\ to hrin~ us illto rlo"er contact with qlw"tions of til!' famili!!!" economic t.\·pe. In the di>'CUS5ions which follow, the reader Kill soon ohsen'e' a chaage in the trend of the illust rutivc cxamples and of t he interpretations: wc shall he,~ i n to tlNl t with rl'..lCstio!ls of bilateral monopoly, oli/,!"opot.v, rnn1"kci~ , cl " . 66.1 .2. Complete abandonment of t he 1.("·O-~UIll rcstrirtion fo r our !;amC5 means, :1:; \\":.1>; pointed oul in -12.1., that the fl\nrtions J(\ ( ~" , ~ ~) \\"hich ('haf(lct('l"ized it in :hc sense of 11.2.3. al"c now entirely unrestril'ted. I.e., tll'lt the requirement

.-, "

,~ J(' ~(r" ~

.

. . r.) =: 0

1 ~! 5:,,,,,1iderably more special than the jl:eDerDl ('.omoosit;ons deJ\lt wit.h loc. cit. The speci fic rel>ults used could 'Th'! np"I'!l~il\" of rl'!lTr'ctinll: 11 WIIS dra! II-PCl'liOIl game 1' , it s zero:;um extension r , and the new treat me n t of imputations as int roduced

in 50.S. Certainly all solutions of r in general camlOt be used to define a satisfactory concept of solutions for r. This was established by the consideration of a special case- Leo by casuistic procedure~in 56.5.-5G.6. Let us now approach this problem systematically; i.e. a pply to the game the formal dcfinition of a solution as given in 30.1.1. and try to determine in full generality which of its features a re unsatisfactory and require modification. I n doing this we shall lise the concept of imputation (of r) in the new arrangement (56:7) in 5G.8.1. T he important point about this arrangement is that it s tresses ab illitio the primary importance of t he real players in r,~ i.e. di rects our attention more to I' than to r. This does not impair, of course, the fa ct that we apply the forma l theory of 30.1.1. to the zero-sum n + I-person game r, and not to the general n-person game r (which would not be possible). The cor.cepts of 30.1. 1. are all based on that of domination. We t herefore begin b y expressing the meani ng of domination as defincd loco cit. for imputations (of r) with the new arrangement of (56 :7) in 56.8.1. Consider two imputations

a

~

11 a l,

-

, a. ll,

P

liP"

~

, P.II·

- ,.. -

Domina.tion

P

a

-

means that there cxist s a non-empty se t S

~

( 1,

II, 'II

+ 1)

whic h is

effective for a , i.e.

L

(56 'li)

iill S

such t hat (56018)

O:i

> Pi

a,

"

,'(5) ,

for all i in S.

We wish to ex press this in terms of the 0:., /3, with i = 1, . . . , I t is I herefore necesSary to distinguis h betwcen l\ro possibili t ies : 66.10.2. Firs t: S docs not contain n

s ~ ( 1,

..

, 1/) ,

+ 1.

T hen

S not empty.

11

alone.

EXTE:\SION OF THE THEORY

521

The conditions (56:17), (56 :18) above need not be reformulated si nc~ they invol ve only the a" (3, with i = 1,' " n. Besides S s;. (IJ . . . , n) in the v(S) of (56:17). Second: S does contain n + 1. Put T = S - (n + I). Then (56,20)

T may be empty.

Ts; ( I , " , n ) ,

The conditions (56: 17), (56: 18) above must be reformulated sinee they involve a~+I, (3,,+1. It is natural to form -8 in ( I , .. ,n, II + I ), i.e. as ( 1, .. . , n, n + 1) - 8j and - T in ( I , , II); i.e. as (1, , n) - T. These two sets arc clearly equal, but it is nevertheless ulleful to have symbols for both. We denote the first by 1.8 and the second by -1' . .,+1

Si nce

L

aj

= 0, so

i_ I

~

a,.

L

= -

ilnS

O'i

iin lo S

L

= -

0';,

iin - T

v(S) ~ -v(1.S) ~ -v(- T ).

Hence (56:17) becomes

L

(56,21)

.;;;,v(-7').

;in -T

T his involves only the 0', with i = 1, ' " n. Besides - 1' S; ( I, in the v(-1') of (56:21). Nex t (56: 18) becomes

,n)

for all i in 1',

(56,22) and This last inequality means that (56,23)

(56:22), (56:23) also involve only the Summing up:

(56 ,D) (56,D,")

O'i,

(3,. with i = 1, ..

, n.

--

E-o (3 means that therc exists either an S with (56: 19) and (56:17), (56:18);

0'

0'

(56,D,b)

a T with (56,20) and (50,21), (56 ,22), (56,23).

Note that these criteria involve only sets 8, T , -1'.& ( I , ' " n) and the (3,. with i = I, . . . ,n. I.e. they refer only to the original game r and to the real players I, . . . • n.

OIi,

522

GENERAL NON-ZERO-SLJM GA NIt;S

66.10.3 . The criterion (56:D) of domulatioll was Obtltillell LJy a IIlt:fal application of t he original defi nition of 30.1. I., the application llell'),!; JII(lde di rectly to r and then t ranslated in terms of r . This rigorous operalion hav ing been carried out, let us now exami ne the res ult from the point oi view of interpretation ; i.c. let us scc whether the conditions of (56 :0) produce a reasonable definitio n of domInation fo r tile pn;1;eUL cw;c. Accordi ng to (56: D) domination holds in two Ca"..:.') (5 6:D:II ) aud (56,D,b). (56: D :a) is merely a restatement of the original definition of JO.l.1. I It expresses that there exist s a grou p of (real) players (the set:S of (56:IH» ,

each of whom prefers his individual situation in a to that in fJ (this is (56: 18», and who know that they are able as a group, i.e. u.s un alliunce, to e nfo rce this preference of theirs (this is (56:17)). (56:D:b) on the other hand is, wnen viewed in terms of r ar.d of the real pl ayers alone, something entirely new. It reqllil'cs agai n that there exist a group of (real) players (the set T of (5G:20» eacn of wnom prclcrs ~

his own individual situation in a to t ilat in (3 (this is ( bG:~2 ». '!'he ability of this g roup to enforce the preference in queStIOn (i.e. (56:ri») is not required. I nstead we hu\·e the condition t11at tile real jJlayers left ou t of this group must not be able to block the preierred imputation in question, that is insofar as it affects them (thi.':> is ( 51):21 ).~ Finally there is the peculiar condition that the tOtality of aU (real) players- i.e. society as a \\·hole- must be worse off under the (preferred) ~

~

regime a than under the (rejected) regime {J (this is (56:23»). 66.10.4. T his strange alternative (56: D :b) was, of course, obtained by t reating the ficti tious player n 1 as a reai entity. If we refrain from

+

'Applied, howeve r, to the gene ral game I', for whieh that theo ry was Ilotintcndt!d! • The (real) pl9.yers left out, i.e. those of -T, eould bloek the prderred imputation ~

I», hener'

~

i = I, .

{J,=a.+ -

" = a .... 1 -

+

t

=

V«II

, n;

+ 1» .

I It may seem peculiar tha.t it took U~ so Ion!!; to rea.ch this lIi mple princillle,- in fllct we need the further considcrations of 56. 11.2. b("fo re we acce pt it finally . il owever, the act of taking Ol'er the definition of 30. 1.1. without allY Iliternllli"e~, in spite of the eltremely wide generalization whic h j~ now performed, requires mOl!t careful attention. The detailed inductive approach give n in these psragraph~ lICemed to bc best suitt.' 0, i.e. essentiality, is among the possibilities, as asserted. In the case of essentiali ty we may further normalize ")' = 1, thereby completely determining (60:2). Thus there exists only one type of essential general two-person games. Note that while Irl l = 2"), may t hus be > 0, there is always (for n = 2) Ir lt = O. It suffices to prove this for t he reduced form, i.e. for (60:2). Indeed: Recalling the definitions of 45.2.1. and 45.2.3. we see that ~

+

a = ! 10'" a~11 is detached when ai, at ~ -")',0'1 at 5;:; 0, and that the minimum of t he corresponding e = 0'1 + at is 0. 1 Hence \rj2 = 0 as desireJ. , It is assumed e.g. for 0:, - 0:, - O.

550

GENERAL NON-ZERO-SUM GAMES

Summing up: :1o'or n = 2 a. zero.sum game must be inessential, a general game nood not be. Accordingly the former must have Irl l = 0; the latter may have Irl l > 0 too. But both have always Irl, = O. We leave it to the reader to interpret this result in the light of previous discussions, and particularly of 45.3.4. 60.2.2. The solutions for a general game r with n = 2 are easily determined. By the valid part of (31:H) in 31.1.5. (cf. the pertinent observations in 59.3.2.) all sets S!; 1 with 0, 1 or n elements are certainly unnecessa.rybut since n = 2, these exhaust all subsets. Hence we may determine the solutions of r as if domination never held. Consequently a solution is simply defined by the property that no imputation can be outside of it. I.e. there exists precisely one solution: the set of all imputations. ~

The general imputation is given in this case as a = ({ai, a111. subject to the conditions (57:15), (57:16) in 57.5.1., which now become:

(60,3) (60'4)

" " v«I», " " v«2)), " + " ~ v«I,2)) ~ v(/).

We restate the result:

r possesses precisely one solution, the set of all imputations. These are the ~

a = {{al,all}

with the ai, a, of (60:3), (60:4). ~

Note that (60:3), (60:4) determine a unique pair al, al (i.e. a) if and only if

(60,5)

v«l))

+ v«2))

~

v«I,2)).

By the criteria of 27.4. this expresses precisely the inessentiality of r. This result is, as it should be, in harmony with (31 :P) in 31.2.3. (cf. 59.3.2.). Otherwise

v«l))

+ v«2))
0 and Irl l > 0 (cf. 45.3.3.). So we see: For n = 3 a zero-sum game as well as a general game may he essential, and both Irl l > 0 and Irl. > 0 are possibilitie!!.

SOLUTIONS FOR n

~

3

551

The case whcre r is inessential is taken care of by (31 :0) or (3 1 :P) in 3 1. 2.3. (d. 59.3.2.) . We assume therefore that r is essential. Use the reduced form of r in the normali1lation l' = 1. Then we can describe its characteristic function ;; (S) wit.h the help of (59 :1 6) and (59: 17) in 59.2.3. as fo llows:

0(8) - (

-!

when S has

and (60,S)

0 ((2,3)) - a"

v « l ,3»

= a~,

;;«1,2»

=

a3

whenS has 2 elements.

And it is verified immediately that a ;;(S) of (60:7), (60:8) fu lfill s the conditions (57:2:a), (57:2:c) of 57 .2. 1. , i.e. thfl.t it is the characteristic funct ion of a suitable r (cL 57.3...1 .), if and only if

Note t hat t his r can be chosen zero-sum , I.e. that (25:3:b) of 25.3 .1. holds if and only if

In other words : The domain (60:~) represents all general games, while its upper boundary point (60:10) reoresents the (unique) zero-s um game of our case. 60.3.2. Let us now determine the solutions of this (essential) general t hree-perRon game.

-

The geneml imputation is given :n this C[l se as a = {{aI, a2, a,1 j, su bject 1,0 t he cond itions (S"/:15), (57:16) in ,,)7.1U. , which now become : O'I ~

- I, al

cr~ ;;;;

rI~~

- 1,

+ £1', + a, =

- I,

Q.

These conditions a re precisely tltose of :l2.1.1 . for OIl, OI~, 03 (d. (32:2) , (32:3) there), Le. those used in the theory of the essential r.cro-sum thrf~eperson game.

T hey agree also, apart from the fact.or 1 +~, wit.h the

conditions of 47.2.2. for al, ",1 , 0 3 (cL (.J7 :2*), (-!7 :3*) t here) , i.e. with those used in the theory of the essentialr.cro-Sllm t hree-octf;on ga.me with excess. Consequently we CRn lise the graphical representll.tion d~sc r ih...u in 32. 1. 2. , in particular in Figure 52. We obtain t he domain of the a as the fundamental triangle in 32. 1.2. in Figure 53. It is also similar to t ha.t in 47.2.2. in Figu re 70. We express the relationship of domination in t his graphical repres(!ntation.

Concern ing the set S of 30.1 . 1. for a domination a

f-


{J;.

-ak.

- ... ~

a

means that

>

a,

> PI > t33 > (33

and and and

o:~

;;;,

- a ~;

0:2

~ -'Il2;

0"1

S;;;

-al.!

The circumstances described in (60: 13) can now be added to the pictu re of the fund amental t riangle. The similarity is now more with 47. than with 32. The operation corresponds to the t ransition from Figure 70 to Figures 71 ,72, or to Figures 84,85, or to Figures 87, 88. Indeed, the difference as against Figures 71 , 84, 87 (which all describe t he same operation, in t he successi\'C Cases (IV) , (V), (VI») is only t his: The six lines

(60014)

which form the configuration there, are now replaced by the six lines

(60,15) respectively. Hence the second triangle (formed by the three last lines) which appears in the fundamental triangle (formed by t he three first lines) need not be placed symmet l'ically with respect to the latter, as it is in t he three figures mentioned. I i, j, k a permutation of 1,2,3. 1

This is quite similar to (47:5) in 47.2.3., except that we have there 1

of all three a" at, a •.

There

• IS

also the change of scale by the factor

1

2" -""3 in place

+• 3" referred to

a fter (60:11 ), (60: 12). The relation to (32 :4) in 32.1.3. is the same as for (47:5) in 47.2.3., cf. footnote 2 on p. 406.

SOLUTI ONS FOR n ;:;; 3

553

60.3.3. It. is convenient to distinguish two cas€S, according to whether the

(60,16) sides of t he t hree last lines of (60:15) (where the three domination relations of (60: 13) are valid) intersect in a common area, or not. Owing to (60:12) the forme r means that (50",,")

while the latter means that

(60"7,b) We call these cases (a) and (b), respectively. Case (a): We have the conditions of F igures 71, 72, except that the inner triangle need not be placed symmetrically wit h respect to the fundamental triangle, as it is there. If this is borne in mind, then the discussion of Case (I V), as given in 47.4.-47.5. can be repeated literally. The solutions are therefore, with the same qualificatio n, t hose depicted in Figures 82, 83. We note that if an ai = 1, then the corresponding sides of t he inner and the fundamental triangle coi ncide (ef. (60: 15», and t he corresponding curve disappears. I Case (b): We have essentially the conditions of Figures 84, 85---of which those of Figures 87, 88 are but a variant-with t he same proviso for asymmetry as in Case (a) above. We redraw the arrangement of Figure 84, t he funda mental t riangle being marked by I and the inner t riangle by \: F igure 92. The a r rangement has several variant.s, because the inner triangle can stick out from the fu ndamental triangle in various ways.2 Figures 92-95 depict t hese variants. J ,4 If these circumstances are borne in mind, then t he discussion of Case (V) as given in 47.6. can be repeated literally.6 The solutions are therefore, 'Thus ill the ~ero-sum ease, where a, - a. - al - I, none of these curves oceurin acco rd with the result of 32. • By (60:9) ·-2 :iii aj ~ I. This means, as thE; reader may eal!ily verify for himself, that each sioJe of the "inner" triangle must PI\.Slj between the corresponding side of the fundamentallriangle and its opposite vertex. Our Figurcs 92-95 exhaust all possibilities within th is restriction. 1 The only ones which can oceur in a zero-sum gamt', i.e. ror a, _ al _ a. _ 1, are those which can be symmctric: Figures 92, 95. Of these, Figure 92 eorresponds to Figure 84, and Figure 95 corresponds to Figure 87. • The Figures 92- 95 differ from each other by the successive disappearance of the areal! @, @, to,

v.,)

(tI ,'-;+ l - U,_i) ~ (to - t)(U._ I,+1 - tI._I,)

fort < 10.

ence now (61 :14), (61 :15) imply (61 :16), (61 :17).

Consequcntly (61 :14),

562

GENERAL NON-ZERO-SUM GAMES

(61 :15) too are necessary and sufficient. Combining (6 1 :14), (61 :15) with part of (6 1 :18). (6 1 :19) we may a lso write: Each one of

(6 L21 )

l

U._I. -

U'_I, _I, VI, -

VC , _I

is greater than each onc of U._I , +! -

U._I"

V',+I -

vl ..

1

According to the usual ideas, the maximizing t = to IS the number of units actually transferred. We have shown that it is characterized by (6 1 :21), and the reader will verify that (61 :21) is precisely Bohm-Rawcrk's definition of the "marginal pairs."~ So we see: (61 'A)

The size of the transaction, i.e. the number to of units trans-

ferred, is determined in accord with Bohm-Bawerk's criterion of the" marginal pairs." To this extent we may say that the ordinary common-sense result has been reproduced by our theory. It may be noted, to conclude, that the case when this game is inessential has a simple meaning. Inessentiality means here v((1,2))

~

v((I))

+ v((2)),

i.e. by (61:9) equality in (61: 11 ). Considering (6 1:9), (61:10) this means that the maximum in the latte r is assumed at t = 0, i.e. that to = O. So we see: (6LB)

Our game is inessential if and only if no transfers take place in it-i.e. when to = 0. 3 61.6. The Pri(:e.

Dis(:u5Sion

61.6.1. Let us now pass to the determination of the price in this set·up. In order to provide an interpretation in this respect we must consider more closely the (unique) solution of our game, as provided by the considerations of 60.2.2. Mathematically the present set-up is no more general than the earlier one analyzed in 61.2.-61.4.: both represent essential general two-person games, and we know that there exists only one such game. Nevertheless, that set-.up was only a special case of our present one: Corresponding to 8 = 1. This difference will be felt as we now pass to the interpretation. I C v, > Wi -

VI

WI

> > v. - V,_I, > ... > w. - W,_I.

In the immediate application only (63 :16) witl be required.

T his is it:

v+w>z+ u.%

(63019)

Proof: Owing to (63:0), (63:7) and (63:13) , (63 :14), the assertion (H:l :19) can be written as follows: l\·rIlX,_O. I.....• (U._I

+ v,) + Max._o. 1. .. .. • (u._. + w.) > Max, .• _ 0. I ... . .•

(U. _I_.

1+ . :;; .

+ v, + W.) + U•.

Consider the I, r for which the onaximum Oil the right-hand side is assumed. Since we have (63:B :b), i.e. < in the second incqualitif:s of (63 :8), (63:9), we l:un conclude from the argumentation of &3.1.2. that these i, T arc ~ O. We denote t hem by 10, roo Hence our nSOiertion is this Max,_o.

1. . . . • •

(U. _I

+ v,) + l\ lax._o. I.....•

I. e., we claim: There exist two t, u .~,

T

(u.~.

+ w.)

> u.-s._,.

+ VI , + w'. + U,.

with

+ v, + u._. + w. > u'~'._ '. + v,, + w" + u•.

Now t.his is actu ally t·he caOiC for t = 10, may then be written as

T

=

TO.

The above inequality

It should be eonceptuai!y ciear that this folluws frv m our assumption of decreasing utiliticl;. Formally it obtains from (63:16) in this way : (63:20) states t hat I But not ncce:s.'Sa.ry; the absence of this pro!,erty would only complicate the discussion somewhat. • In 62.1.2. this was trivia.lly true. Indeed using (63:13), (63:14) we obtain in that ,~,

and this gIves (63: 19) immediately.

ECONOMIC I NTERPRETATION FOR n

L'.

(63,21)

(U.-T. __+ I - u._,.--i)

.-1

>

=

'.

~ (U._i+1 -

._1

3: GENERAL

577

U. __ ).

(63:16) implies U.' -

whenever

8'

ti"_1

>

U." -

!t,"_1

< s", hence in part.icular

and from this (63:21) follows, 63.S. Prelimiuary Discussion

63.3, We now apply 60.3.1., 60.3.2. t.o t.he present. set.up. This will prove t.o be quite similar to the application carried out in 62.3. for the set.up of 62.1.2. The exposition which follows will therefore be more concise, and is best read parallel with the corresponding parts of 62.3. As to the comparison of the mathematical result with that of the ordi· nary, common sense approach, the remarks of 62.2. apply again. We indir,ated already there what complications the present setup produces. We shall consider the situation only briefly, although it is a rather important one. The general viewpoints Wflre sufficiently illustrated by our earlier, simpler examples, and the specific, detailed interpretative analysis of this setup--and other, even more gell,eral ones-will be taken up suo jure in a subsequent publicat.ion. 63.4. The Solution.

63.4.1. The imputations in the present setup are the

with (63,22) (63,23)

al ;;;. ti, 0:1

a~ ;;;.

+ o:~ +

0, a ,

0:3 ;;;.

0,

= t.

It is again necessary to introduce the reduced form. transformation

This amounts to a

(63,24)

as described in 62.3. The processes discussed there determine the a~, so that (63:24) now becomes (63:25)

,

01 1

=

al -

+

z 2u - 3- '

a~ = a t _ z ~

a~,

ag,

ti,

The correspond ing changes on v(S) are again given by (59: 1) in 59.2.1.; they carry (63:4), (63:13), (63:14) into

GENERAL NON-ZERO-SUM GAMES

578

v'«I» Y'«1,2)) =

~

~

v'«2»

~

v'«3))

3u - ;z - u,

, - u - - 3- '

v'«1,3»

3w-2z-u 3 v'«2,3)) ~ _ 2(, ~ u),

v'«I,2,3» = O. Thus

'Y =

z; u,

and we again refrain from passing to the normalization

'Y = I.

Hence we must again insert a proportionality factor when applying 60.3.1., 60.3.2. as described in 62.3. This proportionality factor is now

,-u

-3-' Comparison with (60:8) in 60.3.1. shows that now 2 (0: - u)

3

«\ = -

'

a1 =

3w-2z-u 3

a,

=

3v-2z-u 3

The six lines of (60: 15) in 60.3.2. which describe the triangles from which we derived our solutions, become now :

,-u

- - 3, OJ

=

a~ =

'

, - u

- - 3-

'

3w-2z-u

2(0: - u) 3 '

"

,

CI', =

3v-2z-u 3

63.4.2. Applying the criterium of 60.3.3., we find that aj

+ a1 + a, = v + w

- 2z

~

O.

Hence we have again (60: 17 :b) loc. cit.-i.e. the Case (b) id" and it remains to be decided which one of its four subcases, represented by Figures 92-95, is present. Following t he same procedure of graphical representation as in 62.3., we obtain Figure 101. The qualitative features of this figure foHow from the following considerations:

The second a;-line goes through the intersect.ion of the first Indeed;

a;- and a;-lines.

2(z - u)

3

(63,C,b) (63,C,,)

z - u

- -3-

z - u

- -3-

~

O.

The second a~- [a;-] li ne is to the right [left] of the first one. Indeed: It has a greater a;- [aH value, since

ECONOMIC INTERPRETATION FOR n

(63,C,d)

-

3w-2z-u 3

+ -z-u 3-

=

-

3v-2z-u 3

+ z-u - 3-

= z - v

The first

and

a~-line

% -

W

~

3, GENERAL 579

> 0, > O.

lies below the intersection of the second

a~

a~-lines.

Indeed: z-u 3w-2z-u 3v-2z-u - - 3- 3 3

=

z+u - v - w

< 0,

by (63,19) in 63.2.2.

Comparison of this figure with Figures 92-95 shows that it is again a (rotated and) degenerate form of Figure 94 (cf. footnote 1 on p. 568), although less degenerate than the corresponding Figure 96 in 62.3.: The

®

I

.

;--'~~

Figure 102.

Figure 101. V , Th • • ". _ and 1M .u,,~

J

Figure 103.

area 0 is again degenerated to a point (the upper vertex of the fundamental triangle .6), but the areas CD, 0 , @, 0 arc still undegenerated (the four areas into which the fundamental triangle is divided in our figure). This disposition of the five areas of Figure 94 is shown in Figure 102. The general solution now obtains as stated at the end of 60.3.3. , by fitting the picture of Figure 86 into the situation described by Figure 102. Figure 103 shows the result (cf. footnote I on p. 569).

580

~ON-ZERO-SU~'I

GE:"JEHAL

GA:\'IES

Summing up :

Assuming (63:B :a), (63:B :b) and (63:16) , the general solut ion V is given by Figure 103. Comparison of t his figure with t hose of 62.3.---4. shows that Figure 103 is a form intermediate between those of F igures 98"":100, a nd those figures are in turn degenerate forms of Figure 103. 63.6. Algebraic Form 01 tb e Result

63.6. The result expressed by Figure 103 can be stated algebraically , in the same way as was dOlLe for F igure 98 in 62.5. 1. In Figure 103 the soiution V consists of the area E and the curve '"'-' . The first part of V is characterized by

3w-2z-u 3 E;

0;

~

Z-ll

-

- 3-

'

31J-2z-u ..... 3 E,

'>

0I 3 =

Owing to (63:25) in 63...1.1., t his means that Z -

10 ~

at ;;;

0,

Now (63:23) in 03.4.1. gives and so the exact range of

£t' ]

II

is

+ 10 -

Z ~ OIL ~

z.

(Recall that II + w - z > u by (63: 19) in 03.2.2.) We state all t hese conditions together, the result being somewhat more ccmplicatcd than its analogue (02 :1 8) in 62.5.1. !t is t his: (63:30)

+W

( II a,

-

z~

al

+ a1 + al =

o~

~ z,

0:3 ~ Z -

v,

z.

The ranges 1Il t he first line of (63 :30) are the precise oncl' for a" at, a,. The second part of V (the curve) can be discussed li terally as in 62.5. 1.: al varies from its minimum in (63:30) above (v W - z) to its absolute minimum (u), and a2 , a3 are monotonic decreasing functions of a,. So wc have:

+

(63:3 1) u

~

a,

~

v'+ w - z,

a2, a3 arc monotonic decreasing functions

of

al.,,1

Thus the general solu tion V is thc sum of the two sets given by (63:30) and (63:3 1). It will be notcd that the role of thc functio ns in (63:3 1) is the same as that discusscd at. the cnd of 02.5. 1. 'They must, of course, fulfill (03:22), (G3:23) in G3.4.l. , As Figure 103 shows, the lowest point in thc arc" Iii coincides with tht! highel:lt point on lhe curve. I.e. tile point a, - V + W - lof (03 :30) and of (63:3 1) is the same. Hence we could exclude a, - 11 + W - l from tither one (bt:t not from both!) of (63:30), (63:3 1).

ECONOl\HC I NTERPRETATION FOR n

=

3: Of-NERAL 581

Summing up: Assuming (63:B:a), (63:B:b) and (63: 16), the general solution V is given by (63:30), (63:31). 63.6. Diseussion

63.6.1. Let us now perform the equivalent of 62.6. and apply the ordinary common-sense analysis to the market of one seller and two buyers and 8 indivisible units of a particular good, in order to compare its result with the mathematical olle stated in (63 :E). Actually the interpretation which ought to be carried out now must combine the ideas of 61.5 .2.-61.6.3. with those of 62.6.: the former apply because we havc divisibility into 8 units; the latter because the market is one of three persons. As indicated in 63.3., we do not propose to go fully into all details on this occasion. The t\~o parts (63:30), (63:31), of which our present solution consists are closely similar to the two parts (62: 18), (62: 19) (or (62:20), (62: 19), or (62:21), (62:23» obtained in 62.5. (Cf. also (63:E) in 63.5. with (62:C) in 62.5.2.) It appears most reasonable, therefore, to interpret them in the same way as we did in the corresponding situation in 62.6.2.: (63 :30) describes the situation whcre the two buyers compete for the 8 units in the seller's possession, while (63:31) describes t he situation where they have formed a coalition and face the seller united. The reader wi!1 have no difficulty in amplifyin~ the details, in parallel to 62.6.2. These being accepted, there is nothing new to be said about (63:3 1), the situation in which the buyers have combined and do not compete. (63 :30) howevcr, wh ich describes their competition, still deserves some attention. Let us consider the imputations belonging to (63:30), and let us formulate their contents in terms of the ordinary concept of prices. This is t he equivalent of what we did at the corresponding point in 61.6.1., 61.6.2. We introduce again the t, T for which the maximum in v« 1,2,3)

=

Max, .• _o.!. ....• (U.-- (31 °

when when

I = 1,

m=1.

The remarkable thing is that both (64:12) and (64: 13) are transitive ~

~

relations, while domination a ~ fJ is not. There is, of course, no con· tradiction in this,~(64:12) or (64: 13) is merely a necessary condition for ~

~

a: ~ fJ.

But it is nevertheless the first time that the domination concept in an actual game is so closely linked to a transitive relation. This connection seems to be a quite essential feature of the monopolistic (or monopsonistic) situations.l It will playa role of some importance in 65.9.1. I.e. S must contain both sellers and buyers. The verbal interpretation of (64:12), (64:13) is simple and plausible: No effective domination b~ possible without the monopolist (o r monopsonist). L

1

CHAPTER XII EXTENSIONS OF THE CONCEPTS OF DOMINATION AND SOLUTION 66. The Extension.

Special Cases

611.1. Formulation of the Problem

66.1.1. Our mathematical considerations of the n-person game beginning with the definitions of 30.1.1. made use of the concepts of imputation, domination and solu tion, which were then unambiguously established . Nevertheless in the subsequent development of the theory there occurred repeatedly instances where these concepts underwent variations. These instances were of three kinds: First: It happened in t he course of our mathematical deductions, based strictly on the original definitions, that concepts rose to importance which were obviously analogous to the original ODes (of imputation, domination, sobtion) but not exacUy identical with them. In this case it was convenient to designate them by t hose names, necessarily remembering the differences. Examples of this are to be found in the investigation of t he essential t hree-person game with excess in 47.3.-47.7. where the discussion of the fundamental t riangle is reduced to t hat of one of the various smaller t riangles in it. Another example is offered by the investigation of a special simple n-person game in 55.2.-55.11. , where t he discussion of the original domain is reduced to t hat one of V ' in a (cf. the analysis of 55.8.2., 55.8.3.). Second: In the course of our considerations on decomposabi!ity in Chapter IX, we explicitly re-defincd (generalized) the concepts of imputation, domination and solution in 44.4.2..44.7.4 . This corresponded to an extension of t he t heory from zero-sum to constant-sum games. Throughout what followed we emphasized t hat we were investigating a new theory, analogous to, but not identical with, the original one of 30.1.1. Actually these two types of variations of our concepts are not fundamentally different: T he second type can be subsumed under the fi rst one. Indeed, the new theory was introduced in order to handle t he problem of decomposition of the original one more effectively . This motive was stressed throughout the heuristic considerations which led to this generalization. In the analysis of imbedding in 46.10., pa rticularly in (46:K) a nd (46:L) there, we established rigorously t hat t he new t heory can be subordinated to the original one precisely in this sense. Third: The concepts of imputation, domination Rnd solut ions were again re-defined (generalized) in Chapter XI, specifically in 56.8., 56.11., 56.12. S87

588

EXTE:,SIQ:\,S OF TI-IE CO:':CF.PTS

This corrc:;pomled to the final cxtCll1sion of the theory to general games. We again emphasized that from there on we were ill\"Cf;tigating a new theory analogous to, but not identical with, the prcccdiug onc~. This generalization was, however, fundamentally different from the two preceding ones: It represcntcri a rcat conceptual widening of the theory and not a mere technical con ven icncc. 65.1.2. Throughout t,he changes referred to above it was in evidence that while the concepts of imputation, domination and solution varied (particularly regarding extcnsiOl:')' some connection among them remained invariant. In order to acquire a general im,ight into thcse changes-and other analogous ones which may follow- it is ne('essar.y to find a precise formulation of this invariant connection. When this is done we can permit complete generality in all respects and reformulate the theory on that basis. By recalling the insta.nccs enumerated in 65.1.1., it will appear that this invariant connection is the process by which the concept of a solution is derived from those of imputation and domination. This is the condition (30:5:c) {or the equ ivalcnt ones (30:5:a) and (30 :5:b) in 30.1. 1. Hence we reach perfect generality if we release the notions of imputation and domination from all restrictions, but define the solutions in the way indicated. In accordance with this program we proceed as follo\\'s: Instead of imputations we consider thf' {'I{'menb; of a n arbitrar.\' but fixed domain (set) D. Instead of domination we consider an arhitrary bl:t fixcd relation S between the elements x, y of D.! Now a solution (in D for S) is 11 set V s; D which fulfills thc following condition:

The clements of V arc precisely those elcments. y of D for which xSy holds for no clement x of V. ~ 66.2. General Remarks

66.2. These definitions provide the basis for a morc gencral theory in the sense indicated. It should be noted that ou r present concept of solution bears the same relation to that one of saturation analyzed in 30.3. and particularly in 30.3.5., as the original concept of 30.1.1. In particular our ((;5: 1) should be compared with the fourth example in 30.3.3., our present. S corresponding to the negation of the Sxo, contmdicting

(65,B,.). I Note that the word partial is used in the neutral sense, i.e., a complete ordering is a special case of the partial ones. since (65:A :11.) implies (65:B:II.) . • Note thllt this is close to a plausible type of partially ordered utilitics in the sense of the last remark of 3.7.2. Each imagined event may be affeeted with two numerical characteristics, both of which must be inercased in orJ{'r to produce a cl{'ar and reproducible preference. 'Some other important relations, nota.t all in the nature of all ordering, also possess this prop{'rty: E.g. equality, Z - y.

THE EXTENSION.

SPECIAL CASES

59 1

Thh, result suggests considering the total aggregate of all cond itions They arc implied by (65:B:a), (65:A:b), Le. by partial ordering, and represent, as will appear, a further weakening of this property. We define accordingly: ( AI) , (At). ( A 3), .. ' .

(65:D:c)

A relation S is acyclic if it fulfill s all conditions (AI), (At ), ( A ,), . . .

The feader will understand why we call t his acyc\icity: If any {A",} should fail , there would be a chain of relations which is a cycle, since its last element, x .. , coincides with its firs t one, :to. We have already remarked t hat acyclicity is implied by partial ordering (this is, of cou rse, the content of (65 :D :b)), and hence afortiQri by complete ordering. It remains to show that it is actually a broader concept t han pa rtial ordering, i.e., that a relation can be acyclieal without being an ordering (partial or complete). T hese are examples of the latter phenomenon : Let D be t he set of all positive integers, and xSy the relation of immediate succession , i.e., x = y + 1. Or, let D be the set of all real numbers, and xSy the rela tion of being greater t han, but not by too much- say by no more than I- i.e., the relation y + I ~ x > y . We conclude this section by observing that our examples oi complete and of partial orderings and of acyclical relations could easily be multiplied . Space forbids us LO go into this here, but it may be suggested to t he reader 8.:i a useful exercise. The references to the literatu re in footnote I on puge 62 and footnote 5 on page 589 can also be consulted to advantage. 66.4.. The Sotulions: }o"or a Symmetri(: Relalion.

For a Complete Ordering

66.4.1. Let us now d iscuss t he schemes of specialization rClorred to at the end of 65.2. First: S is sym met ri c in the sense of 30.3.2. In this case it is expedient to go back to the connection with saturation, pointed out at the beginning of 65.2. Owing to t he symmetry of S it will provide all information about solutIOns which we desire. Second: S is a complete ordering. In this case we define as usual: x is a maximum oi D if no y with ySx exists. It is sometimes convenient to indicate the connection with a complete ordering by calling it an absolute maximum of D. (Cf. t his with the corresponding place in the next remark.) Clearly D has either no maximum or preciseiy one. I Now we have:

V is a solution if a nd only ii it is a one-element set, consisting of the maximum of D. IProof: If :l:, 11 nre both maxima oi D, then ySx and xSy being excluded, (65:A :a) neceasitatcs x - y.

592

EXTE~S I Ol\"S

OF T H E

CO~CE PTS

Proof: Necessity: Let V be a solut.ion. Since D is not e mpty, V is not. empty either. Consider a y in V. If xSy, then x canllot be in V, hence a u in V with u$;x exists. The transitivity gives uSy which is impossible, since u, y ure hoth ill V. So n o x (in D!) exists wit.h xSy, I and y must be a maximum of D. So D has a maximum whic h must be unique (cf. above). Hence V is a one-clement set, consisting of it. Sufficiency: Let Io be the maximum of D, V = (xo). Given a y (of D1), the validity of xSy for no x of V amounts simply to the negation of xoSy. Since ySxo is excluded, t his negation is equivalent to y = Xo. So these y fon,) the set V. Hence V is a solution. 66.4.2. Thus there exists no solution V if D has no maxi mum, whiie 0. solution exist.s and is unique if D has a maximum . If D is finite, then the latter is certainly the case. T his is intuitively q uite plausible and also easy to prove. For the sake of completeness a.nd also to make the parallelism with the corrcspDlH.l ing parts of the next. remark more cvident , wc nc,·crthelcss gil'c the proof in fll!!: (G5,F)

If D is finite, thcn it. has a m:lximum.

Proof: A ~,s\lme the opposite, i.e., that D has no maximum. C hoose allY XL III D, t hcn an X2 with X:STL, thcn an T3 with XaSX1 etc., etc. By (G5:A :b) x ...~x" for III > 11, hcnce by (G5:A :a) x .. ~ x~. I. e., the XL, XI, X3, are all distinct from each other, and so D is infinite. T hese results show that bot.h thc existencc and uniqueness of V parallel those of the maximum of D. 66.6. The Solutions ; For a Partial Ord ering

66.5.1. Third: S is a pa!"tial ordering. In this casc we t.ake over literaUy t.he definition of a maximum of D from t.he prcccding remark. It is sometimes I'om·cnient t.o indicate t.he connection II·ith a partia l ordering by calling it (L rdlliil'C maximum of D . (C"f. this wit.h the correspo:lding placc in t.he preceding remark. T his contrast. is quite useful, footnole 2 below noiwithsl anding.) /) may ba'·c no maXimll!TI , it may ha\·e one and it may haye sev('ral.~ TilliS rcbti\·c m:lxima arc not ncc('ss:ll"ilr uniquc, while tile absolute ones arc. 1 LA similar ~itllatioll wa~ :1.lrendr Llis"usscd in 'i .G.2. I The arj.\u!!L en t of footnote I on I' . ii91 fails, sinc(' it d epend~ 011 (65: .01. :a) whieh is now weakened 10 (I;;.:ll:a). E.g., Ink!. for I) the unit ~qLLan' in t h(' plan(' and Ilcfin{' in it ll. pnrtial ordering by citlo('r ene of thc two pro('css.... s in tIl!" two first examptf's at the .... nd of f"i."I.3.2. Theil the ma .~ima of [) form it~ cnlire UppCT cd!!;", or the upper and the ri~ht eil!!;es togr.lher. r('sp .... ctivcly . • Th(' reader is warnNI ag:linsl mixin)::; lip OLLr notion of a rf'lati,·c maximum wilh lh:1.1 onf' whi,·h ocears in Ihr: thco ry 01 f:Ln("liuns: Th('Tf' a lo,·al nl:lx;mnm is fr" (I'IPI111} called a rel:l.live one. Sinf'e tIl!' qU:lnlili,.s inl"ol,·c(\ IIo"rp :Lrf' lluln(·rirnl, hl'!!"1' .·omplrtely ordered, this has nothing to do with ollr pre$Cnt consid{'tntions.

THE EXTENSION.

SPECIAL CASES

593

The question of existence also plays a different role for relative maxima than for absolute ones. It will appear that the decisive property now is this: If Y in D is not a maximum, then a maximum x with x20y

exists. For absolute maxima- Le ., if S is a. complete ordering- (65:G) expresses precisely the existence of one, I For relative maxima this need not be the case, i.c. for a partial ordering the mere existence of some (relative) maxima

need not imply (65:0).

Examples of this are easy to give, but we will not

pursue this matter further. Suffice it to say, that (65:0) will prove to be the proper extension of the existence of an absolute maximum (eL the preceding remark) to t he case of relative maxima (eL below). Now we have: V is a solution if and only if (65:G) is fulfilled (by D and 51) ann V is t he set of all (relat ive) maxima.

Proof: Necessity: Let V be a solution . If y is not in V, then an x in V with xSy exists, hence y is not a maximum. So all maxima belong to V. If y is in V, then the a rgument given in the proof of (65:E) in the preeed· ing remark can be repeated literally, showing that y is.!l. maximum. So V is precisely the set of all maxima. If y is not a maximum, i.e., not in V, then an x in V, i.e., a maximum, with xSy exists, so (65:G) is fulfilled (by D and 5). Sufficiency: Assume that (65:G) is fulfilled, and let V be the set of all maxIma. For x, y in V, xSy is impossible, si nce y is a maximum. If y is not in V, i.e., not a maximum, then by (65:G) an x which is a maximum, i.e., in V, with xSy exists. So V is a solution by (65: 1). The reader should verify how this result (65:H) specializes to (65:E) of t he preceding remark when the ordering is complete. Our result (65:H) shows that there exists no solution V if D and oS do not fulfill the condition (65:G), while a solution exists and is unique if this condition is fulfilled. 66.6.2. If D is finite then the latter is certainly the case. We give the proof in full:

(65 J)

If D is finite, then it fulfills the condition (65:G).

Proof: Assume the opposite, i.e., that D does not fulfill (65 :G). Call a y exceptional, if it is not a maximum and xSy holds for no maximum x. The failure of (65:G) means that exceptional yexist. Consider an exceptiona l y. Since it is not a maximum, an x with xSy exists. Since y is exceptional , this x is not a maximum. If a maximum u Proof: Since D is not empty (65:0) implies the existence of a maximum. Convers n, hence by (65:B:a) z .. r! x n _ I. e., XI, Xt, X3, ". . . are all distinct. from each other and so D is infin ite. (CC. the l!lSt part. of this a rgument with t he proof of (65:F) in the preceding remark . Observe, that we CQuid replace its (65:A:a) by the weaker (65,B,.).)

These resu lts show t hat the existence of a solution now does not correspond to thc existence of It maximum, but to the cond ition (65:0) . This is quite remarkable considering the concluding part of the preceding remark in 6,s.4.2. It corroborates our earlier observation that in t he present case of partial ordering (65:0) is t he proper substitute for t he existence of a maximu m. The uniqueness of the solution is even more remarkable. Ir: the light of the last part of our preceding remark, it wou ld have seemed natural for th is uniqueness to be connected wi th that one of the maximum. But we see now that the solution is unique, whi le the (relative) maximum need not be, as was already mentioned.' 66.6. Acyclicity and Strict Acyclicity

66.6.1. Fourth: S is acyclic. We know that this case comprises the two preceding ones, i.e., that it is more general than both. I n t hose two cases we determined the necessary and sufficient conditions for the existence of a solution and we also found t hat when they are sat.isfied t he solution is unique. (Cf. (65:E) and (65 :H).) Furthermore, it was seen that when D is finit.e these cond itions are certainly satisfied. (Cf. (65:F) and (6501).) If! the acyclic case we will fin d condit-ions which are similar to these in many ways and in some respects we will gain deeper insights than before. It will be necessary, however, to vary our standpoint somewhat in the course of our discussion a nd our resu lts will be subject to certain limitations. The case of a finite D will again be settled in an exhaustive and satisfactory way. It is again convenient to introduce the concept of maxima,' and not only for D itself but a lso for its subsets. So we define: x is a maximum of E(s; D) if x belongs to E and if no y in E with ySx exists. We denote the set of all maxima of E by E"'(.s. E) . I (6S:H ) shows that the 90lution V is not connected with any plIrticular (non-unique) maximum, but with the (unique) sct of all maxima . • Sin(:e we used the qualification ,. ab~oll\tc" in the second, and" relative" in the third remArk, we should now employ another still weake r one. I t seems unnecessary, howev~r, to bring in such a terminological innova.tion at this occasion.

THE EXTENSION. ~

SPECIAL CASES

595

Our discussions will show that it is of decisive importance whether D a.nd possess this property: E

~

e (for E ~ D)

implies E'"

~

e.

I.e.: Every non-empty subset of D possesses maxima. I Prima facie (65:K) does not appear to be related in any way to acyciicity, but there exists actually a very close connection. Before we attack our proper objective, the role of solutions in the present case, we investiga.te this connection. 66.6.2. For this purpose we drop all restrictions concern ing D and~, even that of acyclicity. It is convenient to introduce a property which is a variation of the (A ... ) of (65:0) in 65.3.3., and which will turn out intrinsically connected with them: (A.,) Never XI~XO, X,sXl, x~x!, . . . , where xo, Xl, Xt, . • • belong to D .2 We define, for reasons which shall appear soon: A relation ~ is strictly acyclic if it fulfills the condition (A .. ). We now clarify the relationship of strict acyclicity- i.e. of (A ..)-hoth to (65:K) and to acyclicity, by proving the five lemmas which follow. The essential results are (65:0) and (65:P); (65:L)-(65:N) are preparatory for (65 ,0). Strict acyclicity implies acyclicity. Proof: Assume that S is not acyclic. Then there exist Xo, XI, . . . ,X... _ I and x .. = Xo in D, such that xlSxo, XtSxl, •• ,X,,~X"'_I. Now extend this sequence X~, Xl, . • . ,X",_I to an infinite one Xo, Xl, Xt, • • • by putting Xo

X .. _I

Then clearly X1SXo, X2~Xl'

= x'"

=

Xt..

=

=

=

X~"'_I

=

Xt .. _l

X,SX2, . • •

etc., etc., and so strict acyc1icity fails.

Acyclicity without strict acyclicity implies this: There exists a sequence Xo, Xl, X2, X" • . • , in D with this property: I Even if ~ is a complete ordering, this property (65:K) is of great importance in set theory. T hose readers who are familia r with that theory will observe, that (65:K) is precisely the funciamenlal concept of well order-i7lg. (In this case 5 must be interpreted aa "before" inSlead of "greater. ") For litcrature cf. A. Framktl, loc. cit. p. 1951'1', and 299ff, and i". J[a~dorff, loco cit. p. 55ff, both in footnote 1 on p. 61 ; also E. Zerme/o loco cit. in footnote 2 on p. 269. It is remarkable that the same property plays a role in connection with our concept 01 solution for arbitrary relations. The major part of the considerations which make up the remainder of this chapter dcals with this property and its consequcnces. ActulLlly this subject and its ramifications appear to deserve considerable further study from the mathematical point of vicw. 1 The sequence :to, 2'" 2'1, •• should be infinite in the sense that thc indices must go on ad infinitum, but the:t; themselves need not all be different from each other . • Cf. this with footnote 2 above, and the laat part of this lemma.

EXTEl\'310KS OF THE COXCE PTS

596

For x,,s.r 9' P = q + 1 is sufficient a nd 11 > q is necessary. \ (n:) implies that the In, Xl, :1"2, arc pairwhie difTNcni from each othe r and t herefore D must be infinite in this casco P roof: Since S is not strictly acyclic, there exist Xo, X" X2, in D, such that x1SZu, X2SX1, X3SX2, . Hence p = q + 1 is sufficient for xpSXq. Now assume that xpSx q . We wish to prove the necessity of p > q. Assume the opposite: p ;;;: q. Now Xp+ 1SXp, Xp + 2SXNI, . . . , X qSX q _ L,2 X~SX 9 and these rela tions contradict (A "') with m = q - p + I: It suffices to replace its xo, Xl, , X", _ l and x", = Xo by our Xp , Xp+ l , " , Xv and Xpo This conflicts with the acyclicity of s. Thus all parts of (B! ) are established. Now the consequences of (B:) : If the xo, XL , X 2, • • • were not r-airwise distinct, t hen xI' = XQ would occu r for some p > q. By (n! ) xQ+lSx q , hence Xq+iSXp; by (n:) this implies q 1 > p, i. e. q ~ p , but q < p . So the xo, XL, X2, . • . are pairwise distinct and thcrefore D must be infinite.

+

(65 :N)

Non-acyei icity implies this: For some m(= 1, 2,'" ) we have: There exist xo, Xl, . • ,X .. _ L and x .. = Xo III D with this property : For x"sx q , p = q 1 is necessl1ry a nd sufficicnt ..

(8:)

+

P roof: Since S is not acyclic, there cxist Xo, XL,' , Z", _ I and z .. = Xo in D such that XLSX~, xtSx L, ' " X"SX ... _L' C hoose such a system with its m( = 1, 2, . . . ) as small as possible. Clearly p = q + 1 is su fficien t for X,.SX q. We wish to prove that it is necessa.'Y too. Assumc therefore x,.Sx. but p ;.e q I. Now a cyclical rearrangement of the Io, XL, . . • , X ..._L, X ... = Xo does not affect their properties a nd we can apply this so as to ma ke XI' the last element- i.c., to carry pinto m. I.e., t he re is no loss of generali ty in assuming p = m. Now p ~ q 1, i.e., q ~ m - I. We can also assume that q ~ Tn, s ince q = Tn could be replaced by q = O. So q ~ m - 2. After these preparations we can replace Xo, XL, . . . , X .. _I, Z .. = Xo by Xo, XI, • • . ,X q , x .. = Io ~ without affecting thcir propcrties. This replaces m by q I , which is < m, and t his contradicts the assumcd minimum property of 1n. Thus all parts of (n! ) a rc established. 65.6.3. Summ ing up:

+

+

+

(65,0) (65 ,0 ,,)

Acyclicity is equivalent to the negation of all (Bt), (B: ),

I In connection with this result. cf. alse 65.8.3 . • These are precisely q - 11 relations, hence they do not a ppear if p - q . • Observe that the characte rization or the Interrelatedness or the %0, Zl, complete III (8:'), bu t not in (8:). This will be of im portnnce below. ' I.e., omit %1+'" " Z.. _ I·

Z"

•

is

THE EXTENSION.

SPECIAL CASES

597

Strict acyciicity is equivalent to the negaLioll of all (B;), and of (lJ:). Strict acyclicity implies acyclicity for all D but it is equivalent to it for the finite D. (lJ;),'

(65,0,,)

Proof: Ad (65 :O:a): The condition is necessary since (B!) contradicts (A ..), hence acyclicity. The condition is sufficient by (65:N).

Ad (65:0:b): The condition is necessary since non-aeycl icity contradicts strict acyc\icity by (65:L), and (B:) contradicts (A .. ), hence strict E.cyclity. The condition is sufficient since the negation of strict acyclicity permits the application of (65:M) in case of acyclicity, and the application of (65:0:a) above in case of non-acyclicity. Ad (65:0:c): The forward implication was stated in (65:L) . If D is finite the reverse implication- and hence the equivalcnce-results from the last remark in (65:M). Finally we establish the connection with (65 :K): (65,1')

(65:K) is equivalent to strict acyclicity.

Proof: Necessity: Assume that S is not strictly acyclic. Choose :to, Xl, X t , . in D with x1Sxo, X,SXI, X , SX2, . . ' . Then E = (xo, Xl, Xt, . . . ) is.s;;: D and ¢ e , and it possesses clearly no maxima. So (65:K) fails . Sufficiency: Assume that (65:1 j, then x is in Bi !; Ai c: Ai. Y is a maximum in Ai, hence xSy is impossible. Tbus we have a eontradiction in any event.

600

EX T ENSIONS OF T H E CONCEPTS

If y is not in V Q, then xSy for some x in Y o: y is in - VOl hence in some C,. Hence xSy for an x in B i , and this x is in consequ.ence in V &0-

This completes the proof. Combining (65:V) and (65:W) we call state: There exists one and on ly one solution (in D for S) , the Yo of (65:2) above. 66.8. Uniqueness of the SOl:!tiODS, Aeyclicity aI!d Strict Atyclicity

65.8.1. Let us reconsider the last three remarks, still retaining for a moment the assumption of finiteness, in order to avoid further complications. It is conspicuolls that they all yielded the same result, although under varying assumptions. In each case we proved the existence of a unique solution, but the hypothesis was first complete ordering, then partial ordering, and finally (ordinary or strict) acyclicity~i.e., it was weakened at every step. This being so, it is na'ural to ask whether we have reached with the last r>:lmark the limit of this weakening~or whether acyclicity cou ld be replaced by even less \\'ithout impairing the existence of a unique solution. It must be admitted, that th is line of investigaUon takes us away from the theory of games. Indeed, ill that theory the existence of solutions was of primary importa.nce, but we have learned that there could be no question of uniqu·elless. Nevertheless, since we now have some results on existence with uniqueness, we will continue to study this CMe. We will see later, that it has even indirectly a certain bearing on the theory of games. (Cf. 67.) I n the sense o... tlined we should ask therefore this: Which properties of the relation S are necessary and sufficient in order that there ex ist a unique solution? 1t. is easy to sec, however, that this question is not likely to have a simple and sati&factory answer. I ndeed, the solution (in D for S) discloses only little about the structure of D (together with S). The acyclicnl case is less suited to judge this, since it is somewhat complicated, b ut the cases of complete or partial ordering make the point quite clear. There the solution is only related to the rr:axima of D and it docs not express at all what the properties of the other elements of Dare. It is not difficult to el iminate this objection. Consider a set E s;. D instead of D. The relation S in D is also :l relation in E and if it was a complete ordering or a partial ordering or (ordinarily or strictly) acyclic in D, then it. will be the same in E.l Hence our result (65:X) implies that in every E s;. D there exists a unique solution (for S). Now these solutions, when formed for all E £ D, tell much more about the structure of D. It is best to restrict ourselves agll.in to the cases of (complete or partial) ordering. Clearly the knowledge of the maxima of E for all sets E!; D gives a very detailed information about the structure of D (together with $). , I.e., at least the sarue ~it can happen that a partial ordering in D is complete in or that an acycli Wo, i.e. by (3:C:a) (1 - a )tto + avo> woo Hence a = ),ao belongs to II . However a < ao, although we should have a i;;; au. Thus we obtain a contrad iction in each case. T here fore the original assumption is impossible , and the desired property is cstablished . A.2.2. It is worth while to stop for a moment at this point. (A:B) and (A :C) have effected a one-to-one mapping of the uti lity interval Uo < w < Vo (uo, Vo fixed with Uo < Vo, otherw ise arbitrary !) on the numerical interval o < a < I. This is clearly the first step towards establ ishi ng a numerical representation of utilities. Howeve r, the result is still significantly incomplete in several respects. These seem to be the major limitations: First: T he numerical represent.ation was obtained for a uti lity interval Uo < W < Vo only, not for all util ities w simultaneously. Nor is it. clear, how the mappings which go with d ifferent pairs Uo, Vo fit together. Second: The numerical represe ntation of (A:B), (A:C) has not yet been correlated with ou r requirements (3: I :a), (3:1 :b). Now (3 :I :a) is clearly J T his is intuitively fairly plausible. I t is, furthermore, a perfectly rigorous inference. Indeed. it coincides with one of the eIassical theorems effecting the introductio n of irrational numbers, the theorem concerning the Dedekind cut. Details can be foun d in texts on real function theory or on the foundations of analysis. Cf. e.g. C. Caralhtodory loc. cit. footnote I on p. 343. Cf. there p. 11, Axiom VII. Our class I should be substitu ted for the set Ia 1 mentioned there. The set 1A! mentioned there then contains our class II .

620

THE AXIOMAT I C TREATMEKT OF UT I LITY

sa tisfied: It is just a nother wily of ['xpressi ng the monotony t hat is secured by (A:B). Howev('f t.he nlidit y of (3:1 :0) remains to be established . We will fulfill all these requi reme nts jointly, The procedure will primarily foll ow a course suggested by t he firs t re m!lrk, but in the process the requirements of the second remark a nd the appropriate uniqueness results will also be established. We begin by proving a group of lemmata which is more in the spiri t of t he second re mark and of the uniqueness inquiry; however it is basic in orde r to make progress towards the objectives of the firs t remark too.

Let Vo, Va be as above: Un, Vo fixed, Uo < Va. For all w in t he interval Uo < W < Va define t he numerical function f(w) = i ......(w) as follows: (i) f(u .) ~ O. (ii ) !(vo) = I. (iii ) f ew) for w ~ Uo, Vo, i.e . for Uo < w < Vo, is t.he number a in o < a < 1 which corresponds to 10 in the sense of (A :B ), (A :C). The mapping w - f (w)

has the following properties: (i') It is mon otone. (i i' ) For 0 < (3 < I and IV ~ ~)u.

f((l -

(iii') For 0

~

< {3 < I and w

f (( 1 -

(A :F )

+ Ow)

~),.

110

+ ~w)

~

~f(w) .

~ Vo

1-

~

+ Bf (w).

A mapping of all w with Ito ~ w ~ Vo on any set of nu mbers, which possesses t he properties (i), (ii) and either (ii') or (iii'), is identical with the mapping of (.0\ :0 )

Proo!: (A :D) is a definition; \\·e must proye (A:E) and (A: F ). Ad (A :E ) : Ad (i' ) : F or 1I~ < w < ~'o t he mapping is monotone by (A :B). All 10 of t,his interval are mapped on numbers> 0, < I, i.e. on numbers> than the map of Ito and < than t he map of Va. HencE' we have monotony t hroughout "110 :;; w & Va. Ad (ii'): For w = Vo : The statement isf« 1 - {i)lto + (Jvo) = {i, and t hi s coincides with t he definition in (A :B) (with (J in place of a ). For w ~ Vo, i.e. !to < w < Vo: Pu t f (w) = a, i.e . by (A:B) tv = (I -

Then by (3 :C:b) (\\"i th l'o,

a)1/0

+ avo.

!lo, {J, a in place of 11, tI, a, {J, and using (3:C:a» = ( I - (3)110 (i« 1 - a)1Io a vo) = ( I - {Ja)1/o (3avo. He nce by (A :B) f « 1 - muo (iw) = (ia = f3f(w), as desired. Ad (iii'): For w = 110: The statement is !«I - (i)!!o (iua) = I - {J, and t his coincides with the definition in (A:B) (with 1 - {J in place of a and

(l - {i )uo

+ (3w

lIsing (3:C:a.».

+ +

+

+

+

DERIVATION FROM THE AXIOMS

621

F or W?'! '!to, Le. lto < w < Vo: Pu t f (w) = a , i.e. by (A:H)

w

= (I -

a) /t o

+ a vo.

The n by (3: (::1» (wit h Uo, va, fJ, 1 - a in place of u, v, a, fJ, a nd using (3 :C:a»

( 1 - fJ)vo

+ fJw

= (I -

fJ)vo

+ fJ«

+ avo)

1 - a )uo

= fJ (l -

he nce by (A :B)

+

[ « I - ~),. ~w) ~ I - ~( I - a ) as desired. Ad (A.:F): Conside r a ma pping

w

(A:I)

+ (1 -

- ~ + pa ~

~

a )uo fJ ( 1 - a » vo,

I .-

P + P[(w),

- h ew )

wit h (i) , (ii) and eithe r (ii') or (iii').

The mapping

w~[(w)

(A,2)

is a one·to-one mapping of lto ~ w ~ Vo on 0 ;;;;; a ;;;;; I , hence it can be iU\'e rted: (A :3) a _ "'(a ). N ow combine (A:!) wit h (A:3), i.e. wit h the inverse of (A:2) : (A A)

a

~ [,(f ( a )) ~

. (a ).

Since both (A: I) and (A:2) fulfill (i), (ii ), we obtain fo r (A:4) (A,S)

.(0)

~

.(1 )

0,

~

I.

If (A:I ) fulfill s (ii') or (iii') , the n, as (A:2) fulfill s both (ii') and (iii'), we obta in for (A:4 ) (A ,6) . (Pa) ~ P. (a), 0'

.0 - P + pal

(A ,7)

~

I -

P + p. (a).

Now putt ing a = I in (A:6) and using (A :5) gives . (P)

( A ,8)

~

p,

and putting a = 0 in (A :7) and using (A :5) gives hold in ( I - 6)u + 6u S ( I - 'Y )u "YU . This is a contradiction. Therefore both cl asses r and I I must be e mpty. Consequently never u S (1 - 'Y )u ')'u, i.e. always (I - 'Y )u -yu = u, as desired.

+

+

+

Always h« 1 - ,)u (0

+ ,,)

~

+ ,h(,)

( I - ,)h(u)

v it obtains from the former by putting v, u, 1 - 'Y in place of u, v, "y. For 'U = v it follows from

(A,T) . We can now prove the existence and uniqueness theorem in t he desired form , i.e. corresponding to (3: 1 ::1) and (3 :1 :b). At this point we a lso drop the assumed fixed choice of u·, v·, which was introduced before (A:M).

There exists a mapping w _ v(w)

of all w on a set of numbers possessing thf' following properties: (i ) Monotony. (ii ) For 0 < 'Y < 1 and any 11 . v v((i - 'Y)

1t

+ 'Y v)

'Y )v(u)

= (I -

+ 'Yv(v).

For any two mappings yew) and v'ew ) pOi'.'SCssing t he propties (i), (ii), we have v'ew) = wov(w )

with two suitable but fixed

+ w"

Wo,

w, and

Wu

> O.

Proof: Let u·, v· be two different utili ties,' It · S v*. If u · > v· , then inte rchange u · and v· . Thus at a ny rate

Use t.hese u·, v· for t.he construct.ion of hew), i.e. for (A :L)-(A:U). prove: Ad (A :V); Thf> mapping W - "+ hew) fulfills (i) by (A :R), (iii), and (ii ) by (A :U) . Ad (A:W). Consider yew) first. . By (i) v(u·)


0,

+

WI ,

a:~ > 0,

a.

This is

the desired result. A.3. Concluding Remarks A.3.L (A:V ) and (A: W) are clearly the existe nce and uniqlLeness theorems called for in 3.5.1. Conseque ntly the assertions of 3.5.- 3.6. are established in t heir e ntirety. At t his point the reader is advised to reread the analysis of the concept of utility and of its numerical inte rpretation, as gi\'en in 3.3. and 3.8. There are two points, both of which have been considered or at least referred to loc. cit., but. which seem worth reemphasizing now. A.3.2 . The first one deals with the relationship betwccn our procedure a nd the concept of complementarity. Simply additive formulae , lik(' (3:1 :b) , would seem to indicate that we are assuming absence of any form of comple me ntarity bet.ween the things the utilities of which we are combining. i t is important to realize, that \\'c are doing this solely in a situation where there can indeed be no complementarity. As pointed out in the first part of 3.3.2., our U, II are the utilities not of definite- and possibly coexiste nt- goods or services, but of imagined events. The u, v of (3:1 :b) in particular refer to alternatively conceived events tI, v, of wh ich only onc can and \\.i\1 become real. I.e. (3:1 :b) deals with either ha vi ng tI (with t.he probability a:) or v (with the remain ing probability I - a:)- bllt sincc the two are in no case conceived as taking place together, they can ne\'cr complement each other in the ordinary sense. It should be noted that the theory or games does offer an adequate way of dealing with complementarity . when this concept is legitimately applicable: In calcu lating the value \,(8) of a coalition S (in an n-person game ), as described in 25., all possible forms of complementarity between goods or between service:;;, which may int.en·ene , must be taken into account. Furthermore, t he formu la (25:3:c) e xpresses that the coal ition 8 u T may be worth more than the su m of the \'alues of its two constituent coalitions 8 T , and hence it expresses the possible complementarity between the services

CO:,\CLUDINCl RE:\IARKS

629

of the members of the coalition S and those of the memhers of the coalition 7'. (Cf. a.lso 27.4.3. ) A. 3,3. The second remark ticals with the question, whethf'r our approach forces olle to value a loss ('xactly as much as a (monetarily ) equal gain, whether it permits to attarh a utility or a dislltilit." to gamhling (eve n when the expectation values baJa.nce ), etc. We ha\'c already touched upon these questions in the last part of 3.7. 1. (d. also the footnotes 2 and 3 eod.), However, some add it ional a.nd mon' sjX'f'ific remarks may be useful. Consider the following exam pIt' : Daniel Bem oulli proposed (cf. foot note 2 on p. 28), that the utility of a monetary gain dx should Ilot only be proportional to the gain d.r, hut a lso (a.ssu ming t,he gain to he infinitesimalthat is, asymptotically for \'c ry small gains d.l") inversely proportional to the amount x of the owne r's total posses!o f -dX

x = In...2· x x~ T he excess utility of gaining the (fi nite ) amount" over losi ng the same

T he excess utlltty of

amount

.IS In ,. +-, .f

o\'('r oll"nlllg

" )' - In -x- = In ( I - -. X - If

x-

then

This is

z,

< 0, i.e. of equal

gttins and losses the latter nre more strongly fclt than the former. A 50 );,-50 % gamble \\'ith ('qUill risks, is definite ly Jisauvuntageous. :\"e\'e rt heless l3el"lloll lli 's ut ility sat isfies ou r axioms and obeys ollr results: Ho\\'eyer, the utility of possessing;r units of money is proportional to In x, and not to x!U Thus a suitable detinition of utilit y (\\"hid) in such (l situation is essentially uniquely determined by ou r axioms ) elimi nates in this case t he specific utility or disutility of gambling, which prima facie appeared to exist. We have s tresse d 13ernoulli's utilit.y, 1I0t because we t hink that it is particularly significant, or mu ch nearc r to reality than many other morc or less similar constructions. The purpose was solely to demonstrate , that the lise of numerical utilities does not nccessarily ill\'ol"c assuming that 50 %-50 % gambles with equal monetary risks must be treated as indifferent, and the like .' It const itutes a much deeper prohlem to formu late a system, in which gambling has unde r all conditions a definitc utility or disutility , where nurnericalutilities fu lfilling the calculus of mathematical expectations cannot be defined by ally process, direct or indirect, In such a system some of our , The 50 %- 50 % gamble discussed above involved equal risks in te rms of z, but not in terms of In %, • T hat the utility of % units of money may be measurable, but not proportior.al to x, was pointed out in footnote 3 on p. 18 . • As stated in remark (I) in 3.7.3" we are disregarding transfers of utilities between several persons. The stricter standpoint used elsewhere in this book, as outlined in 2.1. 1. , specifically, the fr ee transferabilit y of utilities between persons, docs force one to 8..Ssume proportionality between utility and monetary mellsures. However, this is not relevant at the present stage of the discussion.

630

THE AXIOMATIC TREATMENT OF UTILITY

axioms must be necessarily invalid . It is difficult to foresee at t his time, which axiom or group of axioms is most likely to undergo such a modification . A.3.4. There are nevertheless some observations which suggest them· selves in this respect. First : The axiom (3:A )-or, more specifically, (3:A:a)- expresses the completeness of the ordering of all utilities, i.e. the completeness of the individual's system of preferences. It is very dubious, whether the ideali!.:ation of reality which treats th is postulate as a valid one, is appropriate or even convenient. I.e. one might want to allow for two utilities u, v the relationship of incomparability, denoted by u II v, which means that neither U = IJ nor u > v nor u < II. It should be noted that the current method of indifference curves does not properly correspond to this po~sibility. Indeed, in that case the conj unction of "neither u > II nor u < v," correspondi ng to the disjunction of "either u = v or u II v," and to be denoted. by u v means t hat u has a greater ordinate than v as well as a greater abscissa than v.) Second: In the group (3:B) the axioms (3:B:a) and (3:B:b) expre3S a property of monotony which it would be hard to abandon. The axioms (3:B:c) and (3:B:d), on the other hand express what is known in geometrical axiomatics as the Archimedean property: No matter how much the utility I' exceeds (or is exceeded by) t he utility u , and no matter how little the utility w exceeds (or is exceeded by) the utility u, if II is admixed to u wit.h a sufficiently small numerical probability, the difference that this admixture makes from u will be less than the difference of w from u. It is probably desirable to require this property under all conditions, since its abandonment would be tantamount to introducing infinite utility differences. L 1 For a statement of the Archimedean property in an axiomatization of geometry, where it originated , cf. e.g. D. Hilb"t, loco cit. footnote I on page 74 . Cf. there ~iom V.1. The Archimedean property has since heen widely used in aziomatizations of number systems and of algebras. There is a slight difference between our treatment of the Archimedean property and its treatment in most of the literat ure we are referring to. We are making free use of the concept of the real number, while this is usually avoided in the literature in question. Therefore the conventional approach is to "majorise" the" larger " quantity by successive

CONCLUDING REMARKS

631

rn this connection it is also worth while to make the following observa. tion: Let any completely ordered system of utilities 'U be given, whi ch does not allow the combination of events with probabilities, and where the utilities are not numerically interpreted. (E.g. a system based on the familiar ordering by indifference curves. Completeness of this ordering obtains, as indicated in the first remark above, by extendi ng the concept of equality- i.e. by treating the concept u "'" v, that we introduced there, as equality. In this case u "'" v means, of cou rse, that u and v lie on the same indifference curve.) Now introduce events affected with probabilities. This means that one introduces combinations of, say, n (= 1,2, .. ) events with respective probabilities

alJ

. , a" (a!, ..

,a.. ~ 0,

L"

a, = I).

i _I

This requires the introd uction of the corresponding (symbolic) utility combinations a,u, + . . . + a "u" (Ul , .. . , u .. in '\I). It is possible to order these atU, +. . + a ..u" (any n = I, 2, . and any ai, . . ,a" and Ul, , u .. subject to the above conditions) completely, and without making them numerical- if the ordering is allowed to be non-Archimedean. Indeed, comparing, say, atU t + .. + a"U.. and P,V, +. . + P.. v... WE' ma:y assume that n = m and that the tt" . . . , u" and the Vh • . • , v.. coincide (write a,u, + .. + a ..u .. + OV, + ... + Ov .. and Ou! + .. . + Ou" + fJ,V, + ... + fJ ..v." for a,Ut + ... + a"u .. and fJ\V, + ... + {J",v ... , and then replace n + m; Uh , U .. , v" . . . , v... ; a" , a", 0, 0; 0, " , 0, {j" . . ,{j,. by n; u" . . . , u .. ; a,,' , a,,; {jh . . . , (j,,). Then we compare alU\ + a"U" and {J\UI P"u ... Next make, by an appropriate permutation of I, . . . , n, u, > > u... After t hese preparations define (I,U, + ... + a ..U .. > p,u, + + p"u" as meaning that for the smallest i(= I, " , n) for which (Ii ~ Pi, say i = io, there is ai, > {ji,. It is clear that these utilities are non-nu merical. Their non-Archimedean character becomes clear if one visualizes that here an !l.rbitrary small excess probability a" - Pi, affecting Ui, will outweigh any potential opposite excess probabilities {j; - ai of t he remaining Ui, i = ill + I, ' " n , i.e. of utilities < u', ' (Th is then excludes the application of criteria like that one in footnote 1 on page \8.) Obviously, they violate our axioms (3,B,c) and (3,B,d ). Such a non-Archimedean ordering is clearly in confl.ict with our normal ideas concerning the nature of utility and of preference. If, on the other

+ ...

+

+

additions of the" smaller " one (cL e.g. Hilbert's procedure loc. cit .), while we "minorise" the "smaller" entity (the utility discrepancy betwei!n 10 and u in ou r case) by a suitable small numerical multiple (the ",-fold in our caee) of t he "larger" entity (the utility discrepancy between ~ a nd u in our case). This difference in treatment is purely technica l and does not affect the conceptual situation. The reader will also note that we are talking of entiti~ like "the excess of ~ over u" or the "excess of u over v" or (to combine the two former) the "discrepancy of u and II" (u, v, being utilities) merely to facilitate the verbal dillCu88ion-they are not part of our rigorous, axiomatic system.

li3:2

T ll f. AXIOMATIC THEATl\'I ENT OF UTILITY

hand, one desires to define utilities (and their ordering) for the probabilityincluding system, satisfying our axioms (3:A)-(3:C)- and hence possessi ng the Archimcdeull property- then the utilities would have to be numerica l, si nce our deduction of A.2. applics. Third: It secms probable, t.hat the really critical group of axioms is (3:C)- or, mOl"e specifically, the axiom (3:C:b ). This axiom expresses the combination rule for mu ltiple chance alternatives, and it is plausible, that a specific utility or disutility of gambling can only exist if thIS simple com bination rule is abandoned . 30me change of the system (3 :A)-(3:C), at. allY rate involving the abandonment or at least a radiceJ.! mod ificat ion of (3 :C:b), may perhaps lead to a mathematically complete and satisfactory calcu lus of utilities, which allows for the possibi lity of a specific utility or disutility of gambling. It is hoped that a way will be found to achieve this, but the mathematical difficulties seem to be considerable. Of cou rse, this makes the fulfillment of the hope of a successful approach by purely verbal means appear even more remote, It will be clear from the above remarks, that the current method of using indifference cu rves offers no heJp in the attempt to 0', ercome these difficu lties. It merely broadens the concept of equality lc.f. the first remark above), but it gives no useful indications- and a/ortiori no specific instructions-as to how one should treat situations that in \'ol"e probahil ities, which are inevitably a!>Socia.tcd with expected utilities.

Afterword AR I EL R U B I NS TE I N

During the past ten years Princeton University Press has done a remarkable job of republishing, in a beaUliful and eye-catching fo rmal, many of the sem inal works from the early d ays of game theory al Princeto n. This new priming of Theory of Call11'S and Economic Behavior, ma rking the book's sixtieth annive rsary, continues the celebration of ga me theo ry. Since the original publication of the book, gam e theory has moved from the fringe of econom ics into its mainstream. The distin ction between economic theorist a nd game th eorist has virtually disappeared . The 1994 Nobel Prize awarded to J o hn Nash , J ohn Harsa n)'i, and Re inhard Sehe n \..- tAl. Th e new bounds become

"

vi

= mill

max

Keg, 1))

max

K(g, 11).

"

(

and

tAl

= min

"

(

It is easily shown that VI ;;;2 vi ;;;2 v2 ;;;2 t.lb that is, that each player is at least as well off as before the p ro babilities were introduced. Moreover it can be shown that v;=t.I~=V

and h en ce that the game is determined. The p roof of the latter result depe nds on the fact that the numbers x,.1 == !T ,~( TI' T2)~., are components of a vector X which d e pe nds on ~ and that th e tips of the vectors X for all possible fs constitute a co nvex se t of points. Next conside r an n-playe r game in which the players divide into twO hostile gmups called Sand -So This can be interpreted as a 2-player game between the players Sand -s. Ifpmbabiliues are employed in the manner described above, lhen S wi ll receive

vi = tAl=

V(S) =

v

and -Swill rece ive v(-5)

~

-v( 5).

If I is the set of all playe rs, then v(J) = 0, that is, the game is ze ro-sum . Finally v(S+ T)

~

v(S)

+

veT)

;f 5 and T ace mUlually exclu,;ve gmup'. Thal ;s, lhe playccs of 5 + T can obtain at least as much by cooperating as they can by splitling up into two groups Sand T. The function v(S) sa tisfying the above relations

COPELAND

644

is ca lled a c haraClcrislic functi o n. Corresponding to any funClion satisfying these relations the re e xists a game having this v(S) as its characteris-tic function. The constru ct.ion of suc h a game involves partitions of I in to subsets ca lled rings and so lo sets. If the equali ty v(S + T ) = v(S) + v{T) always holds, that is, if v(S) is additive, then the coalitions will be ineffective and the game will be dete rmi ned. This is U1C case for /I = 2. Moreover two charJ.Cleristic fu nctions (whe ther addiuvc or not) which d iffe r by an additive function wi ll produce the same strategies of coalitio ns. If v(S) is not additive, it can be modified by a suitable additive fUllClion and a suitable scale faeLOr so lhat v(S) = - I for all l-elemem sets. Thus for 11 = 3, v{ S) is gh'e n by th e following table

o v(S)

- I

+1

o

Jor the

O-elelll('//t s('1 (- / or tll(' (olIIplemflll oj l) I-element sels 2-elemelll sets (colII/liemellts oj I-element sets) 3-ele/llflll sel ( I).

For /I ~ 4, v{ S) is no lo nger d e termin ed and the numbe r of p ossibilities becomes almost bewilde rin g. T he reade r will begin to realize that there is never a dull mome nt with these games. We h ave seen that fo r each o f the cases n = 1,2, 3, 4 a new silUation appears. For n = 5 no n ew phenom e non has as yet been discovc red but for /I ~ 6 we firs t meet the possibility of a game whic h splits in to two o r more games which are in some respects quite distinct but whic h n e,'c rtheless exert potent in n uences on o n e another. This ph e n omenon has th e cou ntc r part of nations whose economies are distinct yet interdependent. II remain s to consi de r what coal ition s can be expected to form in a g ive n game and how the slakes wi ll be di" ided in th e presence o f suc h coa li tions. A divisio n of stakes is ca lled an im/mla lioll and is represen ted by a vector 0. with components 0 ]. Cl2, ' . . , a n where Cl k is the amount the kth player receives. One cou ld imagine lhal if a group of novices we re playin g o ne of these games a cenai n c haos wou ld result. Coalitions would be made and broke n as eac h player soug ht to improve his own status. Finally as the p layers became more acquainted with the game certa in imputati o ns would co me to be trusted because of the stability of the corresponding coalitions and because of the profi tableness to a n effective group of playe rs. T here would thus e m e rge a sel V oj {rusted illl/mia/iolls. Th e,"c would of course be players who we re dissatisfi ed with any g i\'en trusted imputation but they would not be stron g e nough to force a change unless they cou ld bribe some o f the favored p layers to desert their coalitio ns. No r would su ch bribe ry be effective sin ce th e potential recipient of the bribe would realize that the c haos produced by his d ese rtion would eventually leave him in a less favorabl e

645

REVIEW S

posi tion. Thus F correspo nds to a group behavior pattern. It is an institution or a morality arising from enl ightened self interest. But how can Vbe descri bed mathematically? We begin with a definition. We Sfl)' Ihal an illl/mialion. 0'. dOlllinales an impulalioll f3 if there is an ef fecliue {,,17"01lp ofpla),ers ea(.1i ofwhirh is beller o1fw/derO'. Ihan llllder{3. nIP group is effeelive proT!ided it a m guara Tllee for ils members Ihe stakes prescribed &)1 0: against an)' opposition from withoul Ihe 617"011/). A se l V of impUlations is call ed a solmion provided ever}' imputat ion oUlSide of Vis dominated by some impm ation of Vand no imputation in V is dominated by any o th e r imputation in V Thus V is II maxilllal set of '/HI/tuail)' lWliominaiel/ imputaliolls. Unfortuna tely d ominan ce d oes not produce even a partial o rdering of the se t of a ll imputa tions. It is not a transitive relation. T h is ma kes the d iscove ry of solu tions a d iffic ult task. \Ve shall however o u tline a method of fin d ing solu tions for the case 11 = 3. Wh e n "11 = 3 we ha\'e

that is, U1 C game is zero-sum. T h us the tip o f 0'. lies in a plane wh ich passes throug h the origi n and is equally inclin ed to the coordinate axes. This plane is divid ed into six congruent sectors by th c traces o f the coordinate planes. Next O'. k 5;;: - I (lor k = I, 2, 3) since each player can obtain at least - I without the ben efit of any coali tion (see th e above table). These inequali ties require the tip of 0'. to li e within an equilate ral triangle whose cenler is at the common in tersection of the traces of the coordinate planes and whose sides are parallel to these traces. An impuwtion 0: dominates those impulatio ns which are re presented by points imerior to three parallelograms each of which has two sides in common with the above equilatcml triangle and on e "enex at the tip of 0:. On the basis of these geome LJical considerations it is easy to find solutions V We first look for a Vwhose imputations do not all lie on a line O'. k = a constant (that is, a line parallel to a trace). Th ere is o nly o ne such solution , nam ely, V, ( 1/ 2, 1/ 2, 0) ,

(1 / 2,0, 1/ 2) ,

(0, 1/ 2, 1/ 2),

We next look fo r a Vwhose imputa.tions do lie on a lin e, say, co rresponding so lutions are

0'. 3

= c. The

where a and ca re required to satisfy certain inequalities. Thus v,. co ntains a continuum of solutions corresponding to values of th e parameter a. This ex hauslS the possible solutions. The first solution V seems quite

646

H U RWICZ

reasonable whereas Vr seems unnaLUrai and difficult to imc rprct btU let liS return to this question latc r.

Let us conside r the following nOIHcro-sum

2~pla}'e r

game. Each player

(I or 2) c hooses either the number I or the number 2. If both p layers

c hoose I, the n each receives the stake 1/ 2. O therwise each recei\'cs - I. If we reduce th is game to a zcro-sum 3-playcr game by the introduction of a fiCli liolis player 3, then the charact.cristic fun ction becomes lhe one given in the above table. Now if we lake the first solution V. we discover th at the fiClilious player may pia)' an aclive part in the f'Ormation of coal itions. H ence if we wish to retain the 2-player ch aracter of the game, we must c hoose the solution v,. and it is reasonable to assign to c the value - I. The authors appl)' this theo ry of games LO the a nal),sis of a ma rket consisti ng of one bu)'er and one seller and also of a marke t consisting of two buyers and o ll e selle r. T he book leaves m uch to be done but th is fact an I)' e n hances its illleresl. It should be product ive of many extensions along the li nes o f economic interpretauon as well as of mathematical researc h . In fact the authors suggest a number of d irections in wh ich research might profi tably be pursued.

The A .,neri.can Economic Review (Decem bel' 1945 ) I L EON ID H UR\V1CZ Volume 35, No.5. pp. 909-25

H ad it merely called LO o ur attention the existence and exact nature of certain fundamenta l gaps in econom ic theory, the TileD,)' of Cames and ECOIlOmic BehaVIor by \'on Neumann and !\'lorgenstern would have bee n a boo k of outstanding importance. But it does marc than that. It is Cowles Commission I'apers. New Series, No. 13A. TIlt: illlthor. on leal'e from Iowa State College where he is associate professor, is now o n il Guggenheim MenlO ri al Fcllow A2 , a nd A.~, a nd those ope n to duo polist B by B l, B." a nd B3 . The pro fit mad e by A, to be d enoted by a, obviously is d e te rmi ned by the c hoices of strategy made by the two duo polists. This d e pende nce wi ll be in dicated by subscrip ts allach ed to a, with the first subsCli pt refe lTing to A's strategy a nd th e second subscript to that of B; thus, e.g., (l13 is the pro fi t wh ich will be made by A if he c hooses strdteb'Y Aj while B c hooses the strdtegy B~, Simila rly, bl~ wo uld de note the profi ts by B u nder the same ci rc umsta nces. The possible o utcomes of the ~d u opolis ti c com petition" may be re p resented in the fo llowi ng two tables: Table l a shows Ihe pro h ls A will ma ke depe nd ing o n h is own a nd B's c ho icc of stratcgies. T he first row cor respo nds to th e c ho ice of AI' e tc.; colu m ns correspond to B's strategies. Ta bl e I b gives a n alogous information rega rding B's p ro fi ts.

6A side-issue of consider.lble interest discussed ill the "1"111'0')· ofGal/l($ is that of measurabi lity of the utilit y function . T he authors need llIeasl!l1l.bility in ordo.:r to be able to SCI up mbles of the type 10 be presented later in the case where milil}' r,Hher than profit is being maximized. The proof of measurability is not given: howe,·er. an article gh·ing the p roof is p romised for thc neoiislic competition. 696--9Y, 704. 705-6 MOllOI)()l isls.474 Monopoly, 13, 474, 543, 584, 586. 602.

603, 662. Sn (I/So Duopoly Monopsony. 58'1. 586. 602. 603, 662 Monotone u'Hlsformalions. 23 Monlmon, Pierre-Remond dc. x Morgenstern. 0 .. 176. 178: ~Dcmand Theory RCCollsidcrcd.~ 724: dismissal (luring Nazi regime. 7 15; influences on. 7 1~-13: M"I/",,,,nlio.l Theal)' oj t;xpolUlillg (!lid Collfmaillg l:Jollflmif'S. i 19: 011 Nash. xiii; Pndir/Ilbilil)' ojS/(}("k MfII'k,/ P,ices. 724: at PrillCetol,. x. xii. 715: ~Profes:>or Hicks o n Valuc and c..pital: 718-19: reputation of. ti68. 686: "Voll komm cllc Vor"us:>icht und Wirtschaflli ches Gldchgewicht.·· 713- 14; von Neum;ulIl's collaboration with. vii. 71 2. 7101-25: Wir/srlmjtsplOgnose. 712. i 13. Sec also Throl}' of GrImes 01111 ffOllomic &II(IIIior Mor~. M .. 95 Morse. Marston. 717 "·i orse. Philip M .. 709 "·ioti\"alion.43 Moves: chance. :)0. 69. 75. 80. 83.112. 118,122,124,126,183.185.190,517. 604.64 1; dummy, 127: first kind. 50: in a !rime, '19. 55, ,,8. 59. 72,98,109. III: impossiblc. 72: personal. 50. 55. 70.75.112.122.126.183.185.190.

223,508,510: removing of. 183: second kind. 50 Nash.John. xii. xiii. 633 111111 Disullil)' (ZipO. 638 Negation. 66 Ncgotiatiolls. 263. 534. 541 Ncum3.nn,j. '"On. I. 154. 186: computer designed hr. 724: and Damlig. xi; illuess/ de;tlh of, 719, 72'1; Morge nstern's (01lahOl:0rm;tllOnc, 396, 399. 401. '117 Numerical uli!ities, 17 IT.. 157,605.606. 640; axiomatic treatment of, 2'1. 26 fT.. 617fT.. 670. 717-18; and preferences. consiste nt / o rdered. 697: substi tutability, 60" Numeric;tl wcight. 432 Nnliollnl Unil)"

OfTcnsivc Slmtegics. 164 , 205 Oligo l}Olr. l. 13,47.504.6'17.662,690, 696 OppenJwimer.j. Robert. ix 0pl}Osition of interest, I J, 220, 484 OptimalilY, pcnnanent, 162.205. Sunlso Strategy Optimum. 38 Optimum behavior. 34 Order ofsociet}', 41. 43. Su alsoOrganization: Standards of beha,ior Ordering. 37, 38: complete. 19,26,28. 589,591,593.595.600.617.670: parti23. 540 Sociolo!,'}·. mathem,uics in. 637-40. 668 Solitaire, 86, 698 Solo SCts. 244. 530. 531. 644 Solutions. 102. 350. 367. 368. 478. 527. 588: for;)n acyclic relation. 597; ;)reas (two-dimellsiollal parts) in. 4 18; as)'rnmetric. 315. 362; for a complete ordering. 591; composition of, 361; concept of, 36. 661. 671: co ncept of. e:>:teTlsion of. 587; conce pt of im putation in formillg, 435, 660-61. 684-85; cul,'es (oned ime nsional parL~) in . 417; decompos."I ble. 362: decomposition. 36 1: definition of. 39. 264. 660. 670; definition of. new, 526: discrimin;) tol)', 301. 307, 318, 320. 329. 442, 51 1. 512, 662; essential zero-sum three-person game. 282; existence of. 42. 66Q.....61; families of. 329. 603; finite, 307. 500; finite sets of imputatio ns. 328: for r in £("1)),393 fT.: for r in fl",,). 384 fT.; ge lleral games with n ;i 3. orall. 548; generalthree·pe r:son g'dllle. 551; illdecomposable, 362: inobjectil'e (discriminator)'). 290; main simple. 444, 464.467,469: nlultiplicityof. 266. 288. 661: natura l, 465; non-discriminator),. 290,475,51 1; objective (nond iscriminatory), 290; one-element. 277. 280: for a panial ordering. 592; set of ;)11. 44: of simple brames. 430. 672; as

737

smmlards of behavior. 661. 685: superIltlmerary. 288: symmetric, 315: for a symmetric reiatiO Il . 591; unique. 594, 60(}, 60 1. 603: unsymmetrical centra!. 3 19; usc of term. 635 Sorokin. Pitirim. 638 Sound ness. 265 Space: Euclidean. 21, 128, 129: half, 137; linear. 157: ,K1illlensioll;.llim·ar. 128; positive "cctor, 254 Special form ofdel)Cnde nce. 56 S])C;ser. A., 256 Splitting the personali ty. ::'3 St. Petersb urg Problem. 28. 83. 718 Stability. 36, 26 I. 263. 266. 365. 639; iu ner. 42. 43. 265 Srackelberg, H. "011, 657n.17. 663 Stakes, dil'ision of. SN Imputations Stalin.Joseph. 697. 705 Standards ofbch;rvior. 31,40, 41. 265, 266. 271.289,361.365.401,418.472.478. 50 I. 512. 513; discriminato ry. 290; llIultiplicity (of stable, o r accepted). 42. 44. 417; non·discriminatOlY. ~'90; solutions as. 661. 685; stability/ instabilit), of. 639 Statics. 44. 45,1 '17.189.290 Statistics. 10. 12, 14. 144 Steinhaus. H .. vii Stochastic g'dmes. xii Stone,J, R.N .. "iii StOne. Paper. Scissors, III, 143. 144. 164. 185 Stop rule, 59. 60 SIIt:r5et, 61 Symmetry, 104, 109, 16.'"., 166, 190,224 , 2"'•. 256.258.267.3 ]5. 44fi fT., .'iOO, 591. Su also Groups

Tarski, A., 62 Tautology, 8, 40 Temperatu re. 17,21 Theory, extended, structure of, 368 Theory, ne\\'. 526. 528, Su flim New Theory ThNH)' ()fGames ami Ec()II()mi( Btlwvior (I'on Neumann and ~Iorge nstern ): collaboration on, I'ii, 712. 714-25: difficulty/ readability of, 668. 6H2. 685. 686; elephant diagram in. 720: gener.llity of approach in . 662. 687; humorous account of. 678-79; influence of, 633. 646-47, 669.674-75.693,723-25: pulr lication of, 633. 686, 722-23; reception of. lii-ix. xiii, 661, 669. 686, 690-9 1, 693. 723: research foll owing publication of. ix-xi: scope of, 647, 687, 723: title, choice o f, 722: tra nslations of, 724. Sn also Game theory

Theory of Logica] Re pulsion. 679 Thellnooynamics. 23, 675 Thermometry. 22 Thompson, Gera]d L., i l9 Tic-we-toe, 6i6 Tics, 125.3 15 Time-series analys is. 724 T intner, G., 28 Tor.o logy. 154.384 Total I'alm:, 251 Trade unions. 705 Transfer. 30. 364, 365, 401. 402 l hnsfembility of utility, 8. 608 lhnsfinite induction, 269 Transformation. 22. 23 Transiti"i ty, 38. 39, 51. 589. 590. 672 Tnlllsl>orlatio n Problem . xi Trees. 66, 67 Tri bute, 30. 402 Tucke r; A. W" xi, xiii Tug-of-war, 100 Tu key. J ohn W .. 706-8 Tyeho de Brahe, 4 Umpires. 69, 72. 84 Uncertaint y. 35 Unde rstandings. 223. 224, 237 Utilities: corn p" .... bility of, 29; complete ordcling of. 19.26.29, 604,6 17 tT.; differences of. 18, 63 [: dOlllain of, 23, 607: non-numerical, 16, 606,607; nonadditive. 250; numerical (suN um e lical uli[ities); partially ordered. ]9.590; system of, 26: rransferabili ty, 8, 604, 606, 608, 629: variable. 560 Utilitr.8, [5,23,33,47.83, 156,556, 563, 565. 569. 572. 573,583, 585,608: axiomalie Iremme nt, 26 fr.. 6 17 tT.; decreasing, 56 1. 576; discrcte, 613: cXI>t:c ted , 30: gc nclcrson, 220 IT., 260 fT.• 64S--46. 673. 681: thrcc-I>Cl"Son , solu ti on ofessentia!. 282 fT.: tWOperson, xi. 48. 85 IT., 116, 169 If .• 176. 646,672-73,681. 684: usc oflcrm. 635 Zero-sum restrictio n, 84, 504 Zipe George K.. 6:18 "Zur Thcorie de l' Gescllschaflssp;cle~ (Neumann). 687

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CREDITS

Review, by Herben A. Simon. Reprinted from n,e A mericolljournaloJSociolog)' (May 1945) 50 (6): 558-60.

Review, by Arthur H. Copeland. Reprinted by permission of the America n Mathematical Society from Bulletin of Ill" American Molilemalirfll Sociel)' (July 1945) 51 (7): 498-504.

Review (The Theo!"y of Economic Behavior), by Leon id Hur. ... icz. Re printed by permission of the A.Jllcrican Economic Association fro m 'I'll" Ameri((w Ecol/omic Ueview (Decembe r 1945) 35 (5): 909-2~). Review, byT. Barna. Repri nted from /'lollomiw (l\olay 1946) n.s. 13 (50): 136-38. Review, by \-Valler A. Roscn blith. Reprinted with permission from PSJChomerrika (March 1951) 16 ( I ): 141-46.

Heads I Win, and Tails, You Lose, by Pa ul Samuelson. Repri nted from Book H'I?ek by permission of Pa ul Samuelson. Big D, by Palll Crume. Repri nted wit h permission from Dallas Morning NeloS (December 5, 1957). Mathematics of Games a nd Economics, by E. Rowland. Repri nted with pcrmission from Na/llre (Februal,), 16, 1946) 157: 172-73. Copyright © 1946 Macmillan Publishe rs Ltd. TheOl,), of Games, hy Claude Chc\'allcy. Re printed from Fino (March 1945). Mathcmatical Theory of Po ker Is Applicd to I~us in css Problcms, by '..viII Lissne r. Reprinted with pe rm ission from Th e Nelv York Til!l(,.~ (March 10, 1946). Copyright © 1946 The New York Ti mes Co. A Thcory of Strategy. by J ohn McDonald. Rcpri ntcd with permission from For/ll/l f. ( 1949): 100-110. Copyright © 1949 Timc Inc. The Collaboration between Oskar Morgcnstcm and J o hn \'on Neumann on the Theory of Games. by Oskar Morgcnstem. Reprinted by pcmlission of Am edcan Economic Association fromjollrrla/ o/Economic Litera/ure (September 1976) 14 (3) , 80£>-- 16.