Values of Non-Atomic Games 9781400867080

Table of contents :
Cover
Contents
Preface
Introduction
I. The Axiomatic Approach
II. The Random Order Approach
III. The Asymptotic Approach
IV. Values and Derivatives
V. The Value and the Core
VI. An Application to Economic Equilibrium
VII. The Diagonal Property
VIII. Removal of the Standardness Assumption
Appendix A. Finite Games and Their Values
Appendix B. є-Monotonicity
Appendix C. The Mixing Value of Absolutely Continuous Set Functions
References
Index of Special Spaces and Sets
Index

Citation preview

VALUES OF NON-ATOMIC GAMES

A R a n d Corporation Research Study A list of other R a n d publications appears a t the back of this book.

Values of Non-Atomic Games BY R. J. A U M A N N A N D L. S. S H A P L E Y

Princeton University Press, Princeton, New Jersey

Copyright © 1974 by The Rand Corporation All rights reserved. No p a r t of this book may be reproduced in any form or by any electronic or mechanical means including information storage and retrieval systems without permission in writing from the publisher. L.C. Card No.: 72-4038 I S B N : 0-691-08103-4

This book is composed in Monotype Bodoni

Printed in the United States of America

To Esiher and

Marian

Contents

Preface

ix

Introduction

3

Chapter I. The Axiomatic Approach

11

§1. Preliminaries 11 §2. Definitions of Game and Value 13 §3. Statement of Chief Results 18 §4. Basic Properties of the Variation Norm and the Space BV 26 §5. The Space AC 35 §6. The Value of Scalar Measure Games 38 §7. The Value on p N A Ul §8. The Value on b v ' N A 56 §9. Functions of a Signed Measure 73 §10. A Variation on a Theme of Theorem B 82

Chapter II. The Random Order Approach

93

§11. Introduction 93 §12. The Random Order Impossibility Principle 9U §13. The Impossibility Principle: An Alternative Formulation 107 §14. The Mixing Value: Statement of Results 113 §15. T h e Mixing Value: Proof of Theorem E 116 §16. An Alternative Proof for Example 9.4 123

Chapter III. The Asymptotic Approach

126

§17. Introduction and Statement of Results 126 §18. Proof of Theorem F 128 §19. More on Example 9.4 and Related Set Functions 135

Chapter IV. Values and Derivatives

141

§20. Introduction IUl §21. Statement of Results IUU §22. Extensions: The Axiomatic Approach 1U5 §23. Proof of Theorem II 153 §24. Frechet Differentials 156 §25. Extensions: T h e Mixing and Asymptotic Approaches 162

Chapter V. The Value and the Core §26. Introduction and Statement of Results 167 I 170

167 §27. Proof of Theorem

Chapter VI. An Application to Economic Equilibrium

175

§28. Introduction 175 §29. Conceptual Preliminaries 176 §30. Description of the Model and Economic Interpretation 178 §31. Statement of Main Results 182 §32. The Competitive Equilibrium 18U §33. Examples 192 §34. Discussion of the Literature 208 §35. The Space 1U, 210 §36. Further Preparations 213 §37. Basic Properties of 8-Approximations 215 §38. The Derivatives of the Functions u s 22U §39. The Finite Type Case 232 §40. Proof of Theorem J 2Ul §41. Some Open Problems 250

vii

CONTENTS

Chapter VII. The Diagonal Property

252

§42. Introduction and Statement of Results 252 §43. The Diagonal Property for bv'NA, M I X , and A S Y M P 254 §44. Analogues of Theorems H and I 258 §45. The Asymptotic Value of a Transferable Utility Economy 269 §46. The Mixing Value of a Transferable Utility Economy 27U

Chapter YIII. Removal of the Standardness Assumption §47. Prelude 281

281

§48. Fugue 283

Appendix A. Finite Games and Their Values

295

Appendix B. «-Monotonicity

301

Appendix C. The Mixing Value of Absolutely Continuous Set Functions

304

References

315

Index of Special Spaces and Sets

321

Index

323

viii

Preface

This work had its inception in the summer of 1963. By that time, multiperson game theory had outgrown its early preoccupation with small numbers of players and with purely abstract models. The basic connection between the "core" concept and Edgeworthian economics had been made, and the study of the limiting behavior of the core, in economic games with many players, was well under way, as well as the study of related economic models with a continuum of economic agents. In a parallel development, the "value" concept had been applied to several types of games with many players—mainly voting games— including some special infinite-person situations. Thus, the time was ripe for a systematic attack on the "value" problem for economic and other games with a continuum of players. But many difficulties presented themselves. A rough working note, saved from that summer, is revealing: it is headed Eight ways to define values of general oceanic games. In fact, this note was a testament of frustration, as each "way" was beset by thorny counter-examples or crippling impossibility theorems or conceptual holes or tough, unsolved mathematical problems. It took a decade to resolve these difficulties and build a coherent mathematical theory, which we present here; except for Sections 17 and 18, none of the material has been published before. In this endeavor, we are indebted to the assistance of many people, whose specific contributions can only rarely be pin-pointed in the final text. From the outset our approach has been influenced by close personal contacts with those people involved with us in the preceding theoretical developments, as sketched above; we mention especially Gerard Debreu, John Milnor, Herbert Scarf, Norman Shapiro, and Martin Shubik. As the work progressed, many others became interested and "chipped in" with helpful ideas, references, corrections and provocative questions; these include S. Agmon, S. Amitsur, Y. Kannai, Y. Lindenstrauss, B. Peleg, B. Shitovitz, R. Weber, B. Weiss, and, surely, many more. Special thanks are due to Harry Furstenberg, who resolved a major ix

PREFACE

difficulty (thus, Lemma 8.6 and its proof are essentially his); to Uriel Rothblum, who went carefully over much of the text and suggested several important improvements (for example, the present definition of "mixing value" appeared first in his master's thesis); to Louis Billera, who conducted a graduate seminar at Cornell based on the manuscript, which generated a number of corrections and useful comments; and, finally, to Zvi Artstein, whose painstaking reading of the final text has done a great deal to improve the quality and consistency of the exposition. We would like to thank all the institutions and agencies that sup­ ported and funded this research. First and foremost, of course, we have been supported by our home institutions, The Hebrew University of Jerusalem and The Rand Corporation. Much of Aumann's research at the Hebrew University was funded by the Logistics and Mathematical Statistics Branch of the (United States) Office of Naval Research, and much of Shapley's at The Rand Corporation was funded by the United States Air Force Project RAND. Portions of Aumann's research were done also at The Rand Corporation, as consultant to the Mathematics Department; at Yale University, with the Department of Statistics and The Cowles Foundation for Research in Economics; at the Uni­ versity of California at Berkeley, as Ford Visiting Research Professor of Economics; at Stanford University, with the Institute for Mathe­ matical Studies in the Social Sciences; and at the Catholic University of Louvain, with the Center for Operations Research and Econometrics. The National Science Foundation funded a considerable part of the research, at many of the institutions listed. We are most grateful for all this support. In addition, our collaborative effort has benefitted appreciably from participation in several "workshops," namely: the First International Workshop in Game Theory, held in Jerusalem in 1965 and sponsored by the Israel Academy of Sciences and Humanities and the Hebrew University; the Workshop in Mathematical Economics and Game Theory, held in Santa Monica in 1969 and sponsored by the Mathe­ matical Social Sciences Roard and The Rand Corporation; the Second International Workshop in Game Theory, held in Berkeley in 1970 and sponsored by the National Science Foundation and the University of χ

PREFACE California; and the Workshop in Mathematical Economics, held in Luminy, France, in 1972 and sponsored by the University of Marseilles. Such gatherings of specialists and advanced students perform an important function in the growth and maturation of "young" fields of inquiry. Finally, we would be remiss if we did not salute Oskar Morgenstern, Albert W. Tucker, and John D. Williams for their indispensable service to the whole area of mathematical research exemplified by this book. They had steady confidence in the ultimate value of the "pure" mathematical approach to game theory, and this confidence took tangible form on many occasions during the early years of theoretical development, both through the institutional support that they repeatedly mobilized on behalf of mathematicians in the field and through their personal advice and encouragement. Jerusalem and Santa Monica December 1973

xi

VALUES OF NON-ATOMIC GAMES

Introduction

Interaction between people—as in economic or political activity— usually involves a subtle mixture of competition and cooperation. Thus bargaining for a purchase is cooperative, in that both sides want to consummate the transaction, but also competitive, in that each side wants terms that are more favorable to itself, and so less favorable to the other side. People cooperate to organize corporations, then compete with other corporations for business and with each other for positions of power within the corporation. Political parties compete for the voter's favor, but cooperate in forming ruling coalitions and in "log­ rolling." Often it is impossible to draw a clear border between "co­ operation" and "competition." When analyzed from a rational (as distinguished from psychological or sociological) viewpoint, such interactive situations have come to be known as games. The participants are called players, and the "winnings" of the players are called payoffs. The mathematical theory of games, founded by John von Neumann (1928), gained impetus with the publi­ cation of the classic book of von Neumann and Morgenstern (1944) and was subsequently developed by the contributions of many others. In its early years the theory was concerned largely with rc-player games for small values of η (e.g. η = 2, 3, or 4). Since about 1960, attention has focussed more and more on games with large masses of players, in which no individual player can affect the overall outcome. Such games arise naturally in the social sciences, as models for situations in which there are large numbers of very "small" individuals, like consumers in an economy or voters in an election. Mathematically, it is often con­ venient to represent these games with the aid of a "continuum" of players—like the continuum of points on a line or the continuum of drops in a liquid. Represented thus, such games are called non-atomic. One of the objects of the theory is to provide an a priori evaluation of games, i.e., to define an operator that assigns to each player of a game a number that purports to represent what he would be willing to pay in

3

INTRODUCTION order to participate. Such an operator is called a value. Value theory for finite games—i.e., η-player games with η finite—was first studied by Shapley (1953a), and is by now a well established branch of game theory. I t is the purpose of this book to develop a corresponding theory for non-atomic games. A game can be mathematically represented in many forms, cor­ responding to various different aspects of the game that one might wish to study. In this book we are interested in the processes of coalitionforming and payoff distribution. For this aspect the most appropriate is the "coalitional" form. When there are finitely many players, this consists of a function 1 ν that associates a real number ν (S) with each set S of players; intuitively, v(S) expresses the potential worth of S, should its members decide to act in concert. The set function υ is in general not additive; an example is the three-person "majority game," in which v(S) = 1 if S has at least two members, and 0 otherwise. Payoff distributions are represented by vectors with one component for each player, representing the amount he obtains under the particular distribution in question. A value for η-player games can then be defined as a mapping from set functions (games) to η-vectors (payoff distribu­ tions) that satisfies certain plausible or desirable conditions, like sym­ metry and joint optimality. For example, to the three-player majority game described above, the value assigns the payoff vector (^, | , ^ ) , as might have been expected. To generalize this coalitional form so that it will apply to non-atomic (and other infinite) games, we start with a measurable space (I, G), where I is interpreted as the space of players, and the σ-field 6 as the family of possible coalitions. A game is then a function from Q to the reals. As in the finite case, these set functions are typically non-additive; thus though a measure on (I, e) is an example of a game, a more interesting example is the square of a measure or the product of distinct measures. Since individual "winnings" are normally infinitesimal in a non-atomic situation, we represent payoff distributions by measures 1

Usually called "characteristic function" in the literature; see Note 1 to Appen­ dix A. A more general coalitional form in which v(S) is not a real number but a set of payoff distributions (intuitively, those "achievable" by S) has also been widely studied; we shall not be concerned with this in the present volume.

4

INTRODUCTION

on I, or, more precisely, by (finitely) additive functions on 6. A value is then a mapping, φ, from set functions to additive set functions; our "plausible conditions" are that φ be linear, monotonic (or continuous in an appropriate topology), symmetric in a natural sense, and, finally, "efficient" in the sense that always

and equality holds for all ν in Q if and only if Q is internal. The norm Il Il Q will be called the internal norm on Q or the Q-norm. Unless other­ wise specified, topological terms (such as "closed") will continue to refer to the variation norm. PBOPOSITION 4.9. Let Q be a closed subspace of BV, and let B = Q+ — Q+. Then B is reproducing, and is complete in the B-norm. Proof. We have Q + C β Π BV+ = δ + = (Q+ - Q+)+ C Q+,

(4.10) +

+

+

+

whence Q = B and B = Q -Q

30

+

+

= B -B ;

hence β is reproducing·

4. V A R I A T I O N N O R M A N D T H E S P A C E

BV

To prove the completeness, recall2 that a normed linear space is complete if and only if the absolute convergence of a series implies its convergence

implies that the partial sums of

verge in norm). Let »< that

B be such that

; we must show

converges in the B-norm. Let m and w, in

that

conbe such

v and

(4.11) Then

and

Ui and w„

i converge; hence by the monotonicity of the and

converge for all S: denote the sums by

u(S) and w(S), respectively. Then

since S«,(7) converges; since Q. Similarly w

u

that they are in

i.e.

and Q is closed, it follows that

Q. Since u and w are clearly monotonic, it follows

. Setting v = u — w, we deduce that v

B, and that

converges to v. This completes the proof of Proposition 4.9.

PROPOSITION 4 . 1 2 .

The closure of an internal space is internal.

Proof. Let A be internal, Q =

; by Proposition 4.9,

B is reproducing and is complete in the B-norm, and A

B. Now let

A * be the closure of A, as a subspace of B, in the B-norm; we have

Then we claim that (4.13)

Indeed, when v 2

A then (4.13) follows from

See, for example, Hoyden (1968), p. 116, Proposition 4.

31

THE AXIOMATIC APPROACH and the case υ £ A *\A follows from the case ν £ A by a simple limiting argument. Now A* is a closed subspace of B in the B-norm, and, since B is complete in this norm, it follows that A* also is. But then from (4.13) it follows that A* is complete in the variation norm as well, and hence it is a closed subspace of B V. Thus A*

= A* D A = Q D B D A*; +

hence equality holds throughout, and in particular Q = B = Q -

+

Q .

This proves that Q is reproducing. Hence the Q-norm is defined on Q, and from (4.13) and A* = B = Q we obtain

(4.14)

IHI = |M|g

for all ν Q Q. This completes the proof of Proposition 4.12. The remainder of this section will not be heavily used in the sequel; but it is of some interest in its own right. PROPOSITION 4.15. Let Q be a closed reproducing subspace of BV; then the variation norm and the internal norm on Q are equivalent, i.e. for some y, NIe ^ TIMI for all ν ξζ Q. Furthermore, if φ is a positive linear operator from Q into BV, then φ is continuous. Remark. The first sentence of this proposition is a kind of converse to the remark that every internal space is reproducing. Proof. Since Q is closed, it is complete in the variation norm. Since it is reproducing, Q = Q+-Q+;

hence by Proposition 4.9, Q is complete

in the Q-norm. Then since ||»|| g IMU, it follows3 from the interior map­ ping principle that the variation norm and the Q-norm are equivalent. To prove the second sentence of the proposition, we first show that (4.16)

(φν) (I)/v(I)

is bounded for ν G Q+-

If not, then for each n, we can find Vn in Q+ such that ((PVn)(I)/Vn(I) 3

> n.

See, for example, Royden (1968), p. 195, Proposition I I or, slightly less ex­ plicitly, Dunford and Schwartz (1958), Section II.2, p. 55 ff.

32

§4. VARIATION NORM AND THE SPACE BV Furthermore, we may choose Vn so that Vn(I) = 1/n2. Then ν = 2 " = 1 Vn G Q+; but from the positivity of φ it then follows that (φν)(Ι)

Z φ (Σ η = 1

Vn)(I)

> % 1/n η=1

for all k. Hence (φν)(Ι) is greater than any number—a contradiction. This proves (4.16); denote the bound by K. To complete the proof of the proposition, we proceed as in Proposition 4.7. Specifically, for ν £ Q, let u and w in Q+ be such that ν = u — w and 2||o|| g ^ u(I) + w(I). Then

HHI = Ik" - *>HI ^ UHI + IkHI = (φα)(Ι)

+ (vw)(I)

5Ξ K(U(I) + «(/))

^ 2/φ1| 0 ^ 2ZiTlIHI, by (4.16) and the first sentence of the proposition. This completes the proof of Proposition 4.15. In connection with values, the chief consequence of Proposition 4.15 is that, on reproducing spaces, every value is continuous. This is in­ teresting because the positivity condition in the definition of value has content only to the extent that Q is reproducing. Thus the positivity condition has no direct bearing on set functions υ in Q that are not the difference of monotonic functions in Q; and if Q has no non-trivial reproducing subspaces—i.e. Q+ = {0}—then the positivity condition is vacuous. Thus Proposition 4.15 can be interpreted to mean that whenever the positivity condition is fully effective, then it implies continuity. Continuity of the value is the chief tool in proving a number of uniqueness theorems, for example the uniqueness of the value on pNA (or bv'NA); the importance of knowing that a space is reproducing is therefore evident. But the condition is an elusive one because often we can establish it only via Proposition 4.12—i.e. by first proving internality; and, once we know the latter, we no longer need the reproducingness and can proceed directly (cf. the end of Section 7). Under certain conditions one can show directly (i.e. without considering in­ ternal norms) that a space is in a sense "almost reproducing"; this will be discussed in Appendix B. Unfortunately, we have not succeeded in

33

THE A X I O M A T I C APPROACH

deducing from this notion of "almost reproducing" that the value must be continuous. A good deal of this section has been devoted to finding conditions under which the value is continuous—in the variation norm. In Section 3, we saw that on the spaces that are of particular interest in this paper, such as bv'NA, the value is indeed continuous—again in the variation norm. It is therefore of some interest to note that the variation norm is crucial here; that, for example, the value is not continuous in the supremum norm:

Indeed, let

of polynomials with as n

be a sequence

I for all n, and such that

, uniformly for

0

1. For example, take

Suppose that I — [ — 1, 1], that

and that

Then

; but from formula (3.1)

applied to,

we deduce

, and hence 1

for all n, so

0.

It might be thought that the above phenomenon is a consequence of our indirectly building the variation into our value notion, via the positivity condition. But this is not the case. Indeed, on the space of all linear combinations of powers of measures in NA+—of

which the

above vn are members (see Lemma 7.2)—the value is uniquely determined even without the positivity condition.4 Thus, at least for this rather restricted space, considerations of positivity and the variation norm arise naturally out of the other conditions for the value. NOTES 1. Together, Propositions 4.7 and 4.15 demonstrate a close relationship between continuity and positivity. Such relationships have been known for 4

The value on this space is determined by Proposition 6.1, whose proof makes

no use of the positivity condition.

34

§5. T H E SPACE AC some time; an example is the following theorem of Bakhtin, Krasnoselskii, and Stetzenko (1962) : 5 Let Ei and E-; be two Banach spaces, and let and E2 respectively (i.e. if x, y i f , and numbers, then . Assume

and be cones in Ei are non-negative real

(i) Ki and K2 are closed; (ii) E1 =Kx - Ku (iii) x, y K'i implies Let be an operator from Ei to E% such that continuous.

Ki.

Then

is

Note that Proposition 4.15 is a consequence of this theorem. Indeed, for the first sentence of the proposition, we may take E\ to be Q with the variation norm, Et to be Q with the internal norm, to be the identity, and Kx = For the second sentence, take Ei = Q, E2 = FA (both with the variation norm), , and

5. The Space AC If v and w are set functions, then v is said to be absolutely continuous with respect to w (written

if for every

that for every chain 0 and every subchain

0 there is a

0 such

of

(5.1) Note that the relation

is transitive and that, if v and w are measures,

it coincides with the usual notion of absolute continuity. A set function v is said to be absolutely continuous if there is a measure such that

The set of all absolutely continuous set

functions in BV is denoted AC. PROPOSITION 5 . 2 .

AC is a closed subspace of BV.

Proof. AC is easily seen to be a linear space. To prove AC NA+, (5.1).

and let 8 correspond to

Given a chain fi, we claim that

BV, let

1 in accordance with 1.

Indeed,

because of the non-atomicity of /j, we may assume w.l.o.g. that

for all i. We may then partition 0 into subchains A, for example by taking sets of consecutive links, such that for all but the last subchain, 6

See also Krasnoselskii (1964), the footnote on p. 64.

35

T H E A X I O M A T I C APPROACH

we have

and for the last subchain we have

Since the variation of » over each such subchain is at most 1, and the number of such subchains is at most one more than is proved. Hence v

our claim

BV.

It remains to show that AC is closed (i.e. as a subspace of BV). Let where

and

. W.l.o.g. assume

= 1,

all i, and set

Then

for all i, and hence

Vi be such that

Then

for all i. Now for given e, let

and let S be such that for any subchain A,

implies

Thus v

and so v

AC. This completes the proof of the proposition.

COROLLARY 5 . 3 . pNA

AC.

Proof. Clearly AC contains all powers of non-atomic measures; the corollary then follows from Proposition 5.2. The next lemma is a simple consequence of Lyapunov's theorem (Proposition 1.3). LEMMA 5 . 4 .

let

S1

Let

be a vector of measures in NA, and

and S° in Q be such that

sets

S1

S°. Then we may construct a family of

, in such a way that

and that

implies

Proof. Apply Lyapunov's theorem to the measure space consisting of and its measurable subsets to obtain a set 36

such that

§5. THE SPACE AC

S* D S0 and

MS») = MS1) + MS"). Then apply the theorem again to S\Sl/i and S'A\S°. Continuing in this way, we may define S" for each dyadic rational a, in a way that satisfies the conditions of the lemma. Now if β is an arbitrary member of [0, 1], define S? = ^J S", the union being extended over all a that are dyadic rationale. This completes the proof of the lemma. We are now ready for the Proof of Theorem C (see Section 3). First suppose / is absolutely continuous. Then f(x) = JQ #(0 dt for all χ in the range R of μ, where g G L1 = L1^(R). I t is well known that g can be approximated in L 1 by polynomials; this follows from the fact that it can be approximated by step functions, which can be approximated by continuous functions, which can be approximated by polynomials. If q is a polynomial with JR |?(0 — g(t)\ dt < e, and if p(x) = JJ q(t) dt, then the variation of ρ — / is < e. Hence fix) can be approximated in variation by poly­ nomials in x, and so from the non-negativity of μ it follows that ν can be approximated in variation by the corresponding polynomials in μ. Hence ν G pNA. To complete the proof, let ν = f ° μ G pNA. By Corollary 5.3, / ο μ G AC, so there is a ν G ΛΓ^4+ such t h a t / » ^ « » . Without loss of generality, we may assume that μ(Ι) = v(I) = 1. Applying Lemma 5.4 to the vector measure (μ, ν), we may assign to each a G [0, 1] a set Sa in such a way that S0 = 0, S1 = I, a > β implies S" D &>, and 0 a /U(S ) = v(S ) = a. If we now apply the definition of/ ° μ