Positional Games 3034808240, 978-3-0348-0824-8, 978-3-0348-0825-5, 3034808259

Table of contents :
Front Matter....Pages i-x
Introduction....Pages 1-12
Maker-Breaker Games....Pages 13-25
Biased Games....Pages 27-42
Avoider-Enforcer Games....Pages 43-60
The Connectivity Game....Pages 61-74
The Hamiltonicity Game....Pages 75-83
Fast and Strong....Pages 85-96
Random Boards....Pages 97-112
The Neighborhood Conjecture....Pages 113-139
Back Matter....Pages 141-146

Citation preview

Oberwolfach Seminars 44

Positional Games Dan Hefetz Michael Krivelevich Miloš Stojakovic´ Tibor Szabó

Oberwolfach Seminars Volume 44

Dan Hefetz Michael Krivelevich Miloš Stojakoviü Tibor Szabó

Positional Games

Dan Hefetz School of Mathematics University of Birmingham Birmingham, UK

Michael Krivelevich School of Mathematical Sciences Tel Aviv University Tel Aviv, Israel

Miloš Stojaković Department of Mathematics and Informatics University of Novi Sad Novi Sad, Serbia

Tibor Szabó Institut für Mathematik Freie Universität Berlin Berlin, Germany

ISSN 1661-237X ISSN 2296-5041 (electronic) ISBN 978-3-0348-0824-8 ISBN 978-3-0348-0825-5 (eBook) DOI 10.1007/978-3-0348-0825-5 Springer Basel Heidelberg New York Dordrecht London Library of Congress Control Number: 2014941607 Mathematics Subject Classification (2010): 05C57, 91A43, 91A46 © Springer Basel 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Author portraits on the back cover: Photo collection of the Mathematisches Forschungsinstitut Oberwolfach, Germany.

Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com)

Contents Preface 1

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2 Maker-Breaker Games 2.1 Maker-Breaker positional games . . . . . . . . 2.2 Coloring hypergraphs . . . . . . . . . . . . . 2.3 The Erd˝ os-Selfridge Criterion . . . . . . . . . 2.4 Applications of the Erd˝os-Selfridge Criterion 2.4.1 Clique Game . . . . . . . . . . . . . . 2.4.2 n-in-a-row . . . . . . . . . . . . . . . . 2.5 Exercises . . . . . . . . . . . . . . . . . . . .

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3 Biased Games 3.1 Background and motivation . . . . . . . . . 3.2 General criteria for biased games . . . . . . 3.3 The threshold bias of the connectivity game 3.4 Isolating a vertex and box games . . . . . . 3.4.1 Box games . . . . . . . . . . . . . . 3.5 Probabilistic intuition . . . . . . . . . . . . 3.6 Exercises . . . . . . . . . . . . . . . . . . .

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Avoider-Enforcer Games 4.1 Mis`ere is everywhere. . . . . . . . . . . . . . . . . . 4.2 Bias (non-)monotonicity and two sets of rules . . . 4.3 A couple of general criteria . . . . . . . . . . . . . 4.4 Some games whose losing sets are spanning graphs 4.5 Another encounter of non-planarity . . . . . . . . .

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Introduction 1.1 Examples of positional 1.2 General framework . . 1.3 Strong games . . . . . 1.4 Exercises . . . . . . .

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Games with losing sets of constant size . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The 5.1 5.2 5.3 5.4

Connectivity Game Probabilistic intuition . . . . The Connectivity Game . . . The Minimum Degree Game . Exercises . . . . . . . . . . .

The 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Hamiltonicity Game Problem statement, history . . . . . . . . The result . . . . . . . . . . . . . . . . . . Expanders, rotations and boosters . . . . Analysis of the minimum degree game and The proof . . . . . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . .

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7 Fast and Strong 7.1 Introduction . . . . . . . . . . . . . . . . . . 7.2 Winning weak games quickly . . . . . . . . 7.2.1 The weak perfect matching game . . 7.2.2 The weak Hamilton cycle game . . . 7.2.3 The weak k-connectivity game . . . 7.3 Explicit winning strategies in strong games 7.3.1 The strong perfect matching game . 7.3.2 The strong Hamilton cycle game . . 7.3.3 The strong k-connectivity game . . . 7.4 Exercises . . . . . . . . . . . . . . . . . . .

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8 Random Boards 8.1 Preliminaries . . . . . . . . . . . . . . . . . 8.2 Randomness in positional games . . . . . . 8.3 Threshold biases and threshold probabilities 8.4 Probabilistic intuition revisited . . . . . . . 8.5 Hitting time of Maker’s win . . . . . . . . . 8.6 Exercises . . . . . . . . . . . . . . . . . . .

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Neighborhood Conjecture Prologue . . . . . . . . . . . . . . The Local Lemma . . . . . . . . The Neighborhood Conjecture . . Game hypergraphs from trees . . How far can (k, d)-trees take us?

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Application to satisfiability . . . . . . . . . . . Towards improved (k, d)-trees . . . . . . . . . . 9.7.1 Leaf-vectors and constructibility . . . . 9.7.2 Not all parents are the same: operations Further Applications . . . . . . . . . . . . . . . 9.8.1 European Tenure Game . . . . . . . . . 9.8.2 Searching with lies . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . .

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Preface This monograph is an outcome of the Oberwolfach Seminar on Positional Games, given by the authors at the Mathematisches Forschungsinstitut Oberwolfach (MFO) in Oberwolfach, Germany, in May 2013. We are grateful to the MFO administration and scientific leadership for accepting our proposal to organize this seminar, and for actively encouraging us to develop our lecture notes, covering the material taught at the seminar, into the present text. Positional Games Theory is a branch of Combinatorics, the goal of which is to provide a solid mathematical footing for a variety of two-player games of perfect information, usually played on discrete objects, ranging from such popular recreational games as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. The field has experienced tremendous growth in recent years. The aim of this text is two-fold: to serve as a leisurely introduction to this fascinating subject and to treat recent exciting developments in the field. We strove to make our presentation relatively accessible, even for a reader with only basic prior knowledge of combinatorics. Each chapter is accompanied by a fair number of exercises, varying in difficulty. As the theory of Positional Games is obviously too extensive and diverse to be comprehensively covered in such a relatively short text, we settled on an admittedly subjective but hopefully representative selection of topics and results. We trust that this monograph will be suitable to provide the basis for a graduate or advanced undergraduate course in Positional Games. Many great mathematicians have contributed substantially to the development of the field over the years. However, if we were to name one person, it would certainly be J´ozsef Beck, the Grandmaster of Positional Games. It was him who infected us all with the (seemingly incurable) interest in the subject; he introduced us to its basics and advances through his courses, lectures and books. For all this, we are very grateful to him. We would like to thank Dennis Clemens who did a really wonderful job editing this book and putting various pieces, styles, and files together. We are indebted to our friends and colleagues for their sound advice and helpful comments. Those who have contributed (explicitly or implicitly) to this monograph

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Preface

include: Dennis Clemens, Asaf Ferber, Heidi Gebauer, Penny Haxell, P´eter Lengyel, Mirjana Mikalaˇcki, Marko Savi´c, and G´ abor Tardos. We wish to express our gratitude to the staff of the MFO for providing us with ideal conditions for running the seminar smoothly and for having – as usual – a very enjoyable experience staying at Oberwolfach. (Of course, the daily afternoon cake deserves to be mentioned and praised separately.) Last, but not least, we wish to thank the seminar participants, whose keen interest in the subject and active participation served as a wonderful catalyst to our writing and lecturing efforts. Dan Hefetz Michael Krivelevich Miloˇs Stojakovi´c Tibor Szab´o

Chapter 1

Introduction Winning isn’t everything; it’s the only thing. Henry “Red” Sanders

1.1 Examples of positional games Combinatorics is a very concrete science. Therefore, in many of its branches, one can allow oneself the pleasure of discussing concrete, down-to-earth examples first, before introducing an abstract framework and general tools. Positional Games is no exception in this sense, moreover the field has been – and is still – pretty much driven by practical motivation, and by our attempts (sometimes quite successful, sometime rather futile) to analyze formally concrete games, of which several are of substantial recreational value. Here too we start with several concrete examples and their analysis, before proposing a general framework. Example. Tic-Tac-Toe and its 2-dimensional generalizations. The classical TicTac-Toe certainly does not need a formal introduction – everyone played it in childhood. The game of Tic-Tac-Toe (or Crosses & Noughts) is played by two players, alternately claiming one unoccupied cell each from a 3-by-3 board; the player completing a winning line first wins, where winning lines are the three horizontal lines, three vertical lines, and two diagonals; if none of the lines is claimed in its entirety by either of the players by the end of the game, the game is declared a draw. The phrase “every child knows it is a draw” is right to the point here: this is indeed a draw (meaning that each player can play to prevent his opponent from winning), and the only known way to prove it is by case analysis (this could well be your first case analysis proof. . . ). Let us play with the setting a bit by looking at its immediate 2-dimensional generalizations. They are parameterized by n, the side of the square board of the

D. Hefetz et al, Positional Games, Oberwolfach Seminars 44, DOI 10.1007/978-3-0348-0825-5_1, © Springer Basel 2014

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Chapter 1. Introduction

game; we denote (with foresight) the game of Crosses and Naughts played on the n × n board by n2 . In this game, the winning lines are the n horizonal lines, n vertical lines, and two square diagonals, altogether 2n + 2 lines, all of them of size exactly n; we will also use the term “winning sets” instead of “winning lines”, having in mind a general setting of positional games, to be described later. The game is played by two players, alternately claiming one unoccupied cell each from the board; the player completing a winning line first wins, if none of the lines is entirely claimed by either of the players by the end of the game, the game is a draw. We assume that the first player to move marks his squares with Crosses, while the second one uses Naughts; for this reason we will also call the players (for now) Crosses and Naughts, respectively. The 22 game is easily seen to be Crosses’ win: his arbitrary first move creates a double trap (two lines occupied almost entirely by Crosses), and then in his second move he completes a winning line. The 32 game is the classical Tic-Tac-Toe, which is a draw; case analysis is the tool here. The 42 game is a draw. Let us show first that the first player can put his mark in every winning set. We do so by applying a variant of a pairing strategy: Crosses’ first move X1 is in the middle, thus catching (or blocking) three winning lines out of 10. For the remaining seven lines, it is possible to assign a pair of elements of the board to every line, with the assigned pairs being disjoint for distinct lines; then whenever the second player claims an element of the board, the first player answers by claiming its sibling from its pair. Such a pairing, together with the first move of Crosses, is depicted below: ⎤ ⎡ ∗ 1 2 1 ⎢ 3 X 1 4 5⎥ ⎥ ⎢ ⎣ 3 4 6 6⎦ 7 7 2 5 The first player does not even need the upper left corner, marked by asterisk. (If the second player claims the asterisk, the first player makes an arbitrary move.) Before proceeding further, let us say some more words about pairing strategy and pairing draw. Suppose that the n2 game (this discussion is in fact far more general and applies to a very large variety of games) is such that one can find a collection of pairwise disjoint pairs of elements of the board such that each winning line contains fully one of the pairs. Note that in principle one pair can serve more than one line. Then the game is a draw, or what we can call a pairing draw: each of the players can guarantee that he will not lose by making sure that by the end of the game he will possess at least one element from each pair. He always claims a sibling (an element from the same pair) of the last move of his opponent, and moves arbitrarily otherwise. The simplest way to show the existence of such a pairing would be to use each pair for one winning line only; moreover, in games where winning lines share at most one element – which is the case for n2 – this

1.1. Examples of positional games

3

is the only possible way. The obvious necessary condition for implementing this approach is to have at least twice as many elements of the board as the number of winning lines; then one can hope to apply some Hall-type arguments to show the existence of a desired pair assignment. Let us now look at 42 from the point of view of the second player. Since the total number of lines is 10, while the total number of squares is 42 = 16, which is less than twice the number of lines, he cannot hope to have one pairing strategy to bring him a draw immediately. Instead, he can come up with three pairing strategies, each for a different first move of Crosses (due to obvious symmetry, there are exactly three types of the opening move). These three pairing strategies are given below, with X1 marking the first move of Crosses, O1 the first move of Naughts. ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 1 X1 2 2 1 2 3 3 X1 1 2 1 ⎢ 3 O1 4 5⎥ ⎢1 O1 3 4⎥ ⎢4 X1 O1 5⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎣3 4 6 6⎦ ⎣5 3 6 5⎦ ⎣6 6 1 5⎦ 7 7 2 5 7 7 6 4 4 2 7 7 Starting from n = 5, the number of cells on the board, n2 , is at least twice the number of winning lines, which allows us to hope to get a pairing draw. Indeed, the next two cases, 52 and 62 , are pairing draws, as illustrated by the following tables: ⎡ ⎤ ⎡ ⎤ 13 1 9 10 1 14 11 1 8 1 12 ⎢ ∗ 12⎥ ⎥ ⎢ 6 2 2 9 10⎥ ⎢ 7 ∗ 2 2 ⎢ ⎥ ⎢ 3 8 ∗ ∗ 11 3 ⎥ ⎥ ⎢3 7 ∗ 9 3⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎣ 6 7 4 4 10⎦ ⎢ 4 8 ∗ ∗ 11 4 ⎥ ⎣ 7 ∗ 5 5 ∗ 12⎦ 12 5 8 5 11 14 6 9 10 6 13 The asterisk marked cells (52 − 2 · 12 = 1 in the first case and 62 − 2 · 14 = 8 in the second case) are not needed for these pairing strategies. Of course, the above tables supply a drawing strategy for either of the players. The following claim completes the analysis for all n ≥ 2. Proposition 1.1.1. Assume that n2 has a pairing strategy draw. Then (n + 2)2 also has a pairing strategy draw. Proof. Exercise.



Example. nd . This is a far reaching and extremely interesting (and complicated) generalization of Tic-Tac-Toe. Here the board is the d-dimensional cube X = [n]d (where [n] denotes as usually the set {1, . . . , n}), and the winning sets are the so-called geometric lines in X. A geometric line l is a family of n distinct points  (1) (2) (i) (i) a , a , . . . , a(n) of X, where a(i) = (a1 , . . . , ad ), such that for each coordi(1) (2) (n) nate 1 ≤ j ≤ d the sequence of corresponding coordinates (aj , aj , . . . , aj ) is

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Chapter 1. Introduction

either (1, 2, . . . , n) (increasing) or (n, n − 1, . . . , 1) (decreasing) or a constant (and of course, at least one of the coordinates should be non-constant). The winner is the player who occupies a whole geometric line first, otherwise the game ends in a draw. The familiar Tic-Tac-Toe is 32 in this notation. Our understanding of this family of games is rather limited. One thing we do know (and will explain it later) is that for fixed n, the first player is the one who wins for large enough d. On the other hand, for fixed d and n large enough, the game becomes a draw, even a pairing draw. Theorem 1.1.2. If n ≥ 3d − 1, then the nd game is a draw. Proof. Exercise.



Another peculiarity of sorts, showing the complexity of the game: the game 43 (marketed by Parker Brothers as Qubic) is known to be the first player’s win. However, as Oren Patashnik, one of the solvers, put it, the winning strategy of the first player, when written down fully, has the size of a phone book. . . Example. Hex. This game was apparently invented by the Danish scientist Piet Hein in 1942 and was played and researched by none other than John Nash in his student years. The game is played on a rhombus of hexagons of size n × n (in commercial/recreational versions n is usually 11), where two players, say, Blue and Red, take the two opposite sides of the board each, and then alternately mark unoccupied hexagons of the board with their own color. Whoever connects his own opposite sides of the board first, wins the game. Apparently thinking of Hex, Nash came up with the idea of Strategy Stealing (see Section 1.3) and proved that the first player is the winner. One can argue combinatorially that there is always a winner; proving that in every final position there is at most one player who has his two sides connected by a path of his hexagons is intuitively as clear as the Jordan Curve Theorem – and should, and can, be proven formally, see [39]. Example. Connectivity game. The board of the connectivity game is the edge set of a multigraph G. The players, called Connector and Disconnector, take turns in claiming one unoccupied edge of G each, till all edges of G have been claimed. Connector wins the game if in the end the set of his edges contains a spanning tree of G, Disconnector wins otherwise, i.e., if he manages to leave Connector with a non-connected graph. We assume that Disconnector starts the game, and Connector moves second. Observe the highly non-symmetric goals of the players here. This game was treated by Lehman, who proved: Theorem 1.1.3 ([64]). If a multigraph G has two edge disjoint spanning trees, then Connector wins the connectivity game played on G. Proof. The proof is by induction on |V (G)|. The base case |V (G)| = 2 is obvious. For the induction step, assume that T1 and T2 are edge disjoint spanning trees of G. If Disconnector claims an edge e belonging to one of the two trees of G, say, to T1 , this move cuts T1 into two connected parts. Connector then claims an

1.1. Examples of positional games

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edge f of T2 connecting these two parts, and then contracts f by identifying its two endpoints u and v. (This way he may introduce multiple edges, this is the reason the theorem is stated for multigraphs rather than for simple graphs.) Then Connector updates T1 and T2 accordingly (e is deleted, u and v are identified, and the remaining edges are updated accordingly), arriving at a smaller multigraph G with two edge disjoint spanning trees T1 and T2 . Applying induction to G gives a Connector’s spanning tree T  of G ; reversing then the contraction of f and adding f to T  produces a spanning tree of G by Connector. If Disconnector claims an edge outside of T1 ∪ T2 , then Connector can essentially ignore his opponent’s move by picking instead an arbitrary edge e of T1 , choosing f from T2 as before, deleting e, contracting f , etc.  We challenge the reader to give a thought to the version of the Connectivity game, where Connector is the first to move. Intuitively, this should make his life easier. We will return to this issue later. Example. Sim. Sim is also a well known recreational game, whose mathematical description is as follows. The game is played on the edge set of the complete graph K6 on six vertices. The two players change turns claiming one unoccupied edge each; the player who completes a triangle of his edges first, actually loses. (This is a reverse game, the player who completes a winning set first is the one to lose.) Due to the standard fact R(3, 3) = 6 from Ramsey theory, there is no drawing final position for the game, and hence for every game course one of the players wins. Sim too was solved using a computer, it turned out to be a second player’s win, with a fairly complicated winning strategy. Example. Hamiltonicity game. This, somewhat less popular commercially (why, you may wonder. . . ) game is set up as follows. The game is played by two players, who take turns claiming unoccupied edges of the complete graph Kn on n vertices, one edge each turn. The first player wins if by the end of the game he manages to make a Hamilton cycle (a cycle of length n) from his edges, the second player wins otherwise, i.e., if in the end he manages to put his edge, or to break, into every Hamilton cycle on n vertices. This game was introduced and analyzed by Chv´atal and Erd˝ os in their seminal paper [21]; it turned out to be a pretty easy win for the cycle maker for all large enough n. Example. Row-column game. Now the board of the game is the n × n square, and two players alternately claim its elements. The goal of the first player is to achieve a substantial advantage in some line, where a line means a row or a column, so altogether there are 2n lines; the question is how large an advantage the first player can achieve playing against a perfect opponent. If only rows (or only columns) are taken into account, then a simple pairing strategy shows that the first player can achieve nothing for even n, or 1 for odd n. However, when both rows and columns are important, first player can reach something of substance: √ Beck proved [8] that the first player has a strategy to end up with at least n/2 + 32 n elements in some line. The upper bound is due to Sz´ekely, who showed [86] that the second player

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Chapter 1. Introduction

√ has a strategy to restrict his opponent √ to at most n/2 + O( n log n) elements in each line; the multiplicative gap of c log n in the error term of the lower and the upper bounds still stands.

1.2

General framework

Having seen the above examples, we are now in position to define a general setting, providing a unifying framework for many games and game types. Positional games involve two players alternately occupying the elements of a given set X, the board of the game. We will assume X to be finite, unless it is obvious otherwise. The focus of players’ attention is a given family F = {A1 , . . . , Ak } ⊆ 2X of finite subsets of X, usually called the winning sets; the family F sometimes is called the hypergraph of the game. The players take turns, occupying previously unoccupied elements (vertices) of X. In the most general version, there are two additional parameters – positive integers p and q: the first player in his turn takes p unoccupied vertices, the second player responds by taking q unoccupied vertices (in the most basic version p = q = 1 – the so-called unbiased game). The game is completely specified by setting its outcome (first player’s win/second player’s loss, second player’s win/first player’s loss, or a draw) for every final position, or more generally for every possible game scenario (a sequence of alternating legal moves of the players). Observe that each game scenario has exactly one outcome, and we thus have a partition of game scenarios into three disjoint sets, corresponding to three possible game results. There is certain degree of vagueness in the way we described the outcome of the game above, but it allows us to accommodate a large variety of games under the same roof. Later we will be more specific when defining concrete game types. At this stage we can say formally, borrowing (real) Game Theory terminology, that positional games are two-player perfect-information zero sum games with no random moves. Let us briefly go over previously presented examples to see whether they fit into the above described framework. For nd , the board X is X = [n]d , and the family F of winning lines consists of all geometric lines in X; the player completing a winning line first wins, otherwise the game ends in a draw. (Notice the use of this general clause in our setting here – the result is not determined by the final position only, and we need to know the course of the game in order to establish it.) For the connectivity game, the board X is the edge set of a multigraph G = (V, E), and the winning sets are the subsets E0 ⊆ E containing spanning trees of G; Connector wins if in the end, and not necessarily first, he occupies completely one of the winning sets, and Disconnector wins otherwise. For Sim, the board is E(K6 ), the winning sets are triangles in K6 , and the player completing a winning set first now loses; if this has not happened by the end of the game, it is declared a draw. For the Hamiltonicity game, the board is E(Kn ), and the winning sets are graphs on n vertices, containing an n-cycle; the first player wins if he claims an

1.2. General framework

7

entire winning set by the end of the game, and the second player wins otherwise. Hex is a problem (of sorts)! Formally it does not fit our definition, as the winning sets of the players are different – they connect two different pairs of opposite sides of the board. However, the situation can be saved by proving the intuitively obvious, yet non-trivial, fact that a player wins in Hex if and only if he prevents his opponent from winning. This allows us to cast Hex in our framework too, by defining the winning sets to be the connecting sets of hexagons for the first player and making him the winner if he occupies one of them fully in the end of the game; the second player’s goal is redefined by assigning him instead the task of preventing the first player from occupying an entire winning set. Finally, the row-column game is essentially of our type too. For a given parameter k, if one needs to decide whether the first player can reach at least k elements in one of the 2n lines of the game, the way to proceed would be to define a game hypergraph whose board (set of vertices) consists of the n2 cells of the game, and whose winning sets are all k-subsets of the rows and columns. There is a very important distinction between casual games, where a more experienced/clever player has better chances to succeed, and the formal games we consider here. We assume that both players are all-powerful computationally. If so, each positional game is determined and has exactly one of the following three possible outcomes: 1. first player has a winning strategy; 2. second player has a winning strategy; 3. both players have drawing strategies. The proof is a straightforward formal exercise in logic (remember that the board X is assumed to be finite); all what it takes essentially is De Morgan’s laws. Here is the formal argument. The logical statement saying that the first player has a winning strategy can be written formally as: ∃x1 ∀y1 ∃x2 . . . such that the first player wins. Similarly, the statement about the second player having a winning strategy translates to: ∀x1 ∃y1 ∀x2 ∃y2 . . . such that the second player wins. The negation of the above two statements is  ¬ (∃x1 ∀y1 ∃x2 . . . first player wins) ∨ (∀x1 ∃y1 ∀x2 ∃y2 . . . second player wins) , which by De Morgan’s law is equivalent to: (∀x1 ∃y1 . . . first player loses or draw) ∧ (∃x1 ∀y1 . . . second player loses or draw), which exactly says that both players have strategies guaranteeing each at least a draw. When both players pursue their drawing strategies, the game ends in a draw. Clearly, the three options above are mutually exclusive. Of course, there is a crucial difference between knowing that a game is determined, and finding its actual outcome. In principle, every game can be described by a

8

Chapter 1. Introduction

tree of all possible plays, called the game tree. There is a vertex for every sequence of legal moves (a1 , b1 , a2 , b2 , . . .) of both players, including the empty sequence, corresponding to the root of the game tree; each sequence of moves is connected by an (incoming) edge to a sequence one move shorter, thus the leaves are exactly the final positions of the games. In order to solve the game, one backtracks one’s way from the leaves up the root in a pretty obvious way. The leaves are marked by the corresponding outcome of the game; if a non-leaf vertex corresponds to the position where it is the turn of player P to move, then P obviously chooses the best available alternative – if there is an outneighbor marked by P ’s win, then P chooses the corresponding move, and the node is marked by P ; if none of the outneighbors of the current node is marked by P , but there is a node marked as draw, P chooses the corresponding move, and the current node is marked by draw as well; finally (and regrettably for P ), if all of the outneighbors of the current node are marked by the other player’s name, then there is no way for P to escape, and the current node gets the mark of his opponent as well. In the end, the mark assigned to the root is the outcome of the game. Simple, isn’t it? The reality is of course much much more complicated; even for small and innocent looking games, their game trees are really huge, making the whole backtracking approach highly impractical and irrelevant. (Recall the example of 43 , or Qubic, and the phone book-sized winning strategy of the first player.) Thus, there is still room, and in fact plenty of it, for developing general theory and tools for positional games. Let us now say a couple of words on where it all belongs mathematically. The term “positional games” can be somewhat misleading. Classical Game Theory, initiated by John von Neumann, is largely based on the notions of uncertainty and lack of perfect information, giving rise to probabilistic arguments and the key concept of a mixed strategy. Positional games in contrast are perfect information games and as such can in principle be solved completely by an all-powerful computer and hence are categorized as trivial in classical Game Theory. In reality, this is not the case, due to the prohibitive complexity of the exhaustive search approach; this only stresses the importance of accessible mathematical criteria for analyzing such games. A probably closer relative is what is sometimes called “Combinatorial Game Theory”, popularized by John Conway and others, which includes such games as Nim; they are frequently heavily based on algebraic arguments and various notions of decomposition. Positional games are usually quite different and call for combinatorial arguments of various sorts. The fundamental monograph [10] of J´ ozsef Beck, the main proponent and contributor of the field for several decades, can serve as a thorough introduction to the subject, covering many of its facets, and posing many interesting problems (beware though that some of them have been solved since the first print!). Beck’s recent text [11] has a lot of stuff about games too.

1.3. Strong games

9

1.3 Strong games Strong games are probably the most natural type of games – these are the games played for fun by normal human beings. A strong game is played on a hypergraph (X, F) by two players, called First Player, or FP, and Second Player, or SP, or alternatively Red and Blue, who take turns in occupying previously unclaimed elements of the board, one element each time; Player 1 starts. The winner is the first player who completes a winning set A ∈ F; if this has not happened for the duration of the game, the game is declared a draw. Both Tic-Tac-Toe and nd are strong games. In principle, there can be three possible outcomes of a strong game: Red’s win, Blue’s win, or a draw. In reality, there are only two possible outcomes (again, assuming perfect play of both players). The most basic fact about strong games is the so-called strategy stealing principle, asserting formally the advantageous position of the first player; this is the reason for having only two possible outcomes. Theorem 1.3.1. In a strong game played on (X, F), First Player can guarantee at least a draw. Proof. Assume to the contrary that Second Player has a winning strategy S. The strategy is a complete recipe, prescribing SP how to respond to each move of his opponent, and to reach a win eventually. Now, First Player steals (or rather borrows for the duration of the game, in politically correct terms) this strategy S and adopts it as follows. He starts with an arbitrary move and then pretends to be Second Player (by ignoring his first move). After each move of SP, FP consults the strategy book S and responds accordingly. If he is told to claim an element of X which is still available, then he does so; if this element has been taken by him as his previously ignored arbitrary move, then he takes another arbitrary element instead. The important point to observe is that an extra move can only benefit First Player. Since S is a winning strategy, at some point in the game FP claims fully a winning set, even ignoring his extra move, before SP was able to do so. It thus follows that First Player has a winning strategy, excluding the possibility that Second Player has a winning strategy and thus providing the desired contradiction.  This is a very powerful result, due to its amazing applicability – it is valid for each and every strong game! Yet, it is a pretty useless statement at the same time – it is absolutely inexplicit and provides no clue for First Player on how to play for (at least) a draw. In order to show a perhaps less standard example of applying the strategy stealing argument, let us go back to the connectivity game and take another look at Theorem 1.1.3 of Lehman. We can now complement it by proving: Theorem 1.3.2. If Connector wins the connectivity game played on a multigraph G, then G has two edge disjoint spanning trees.

10

Chapter 1. Introduction

Proof. Assume that Connector (as a second player) has a strategy to build a spanning tree playing against any opponent on the edges of G. This strategy can easily be adapted to succeed in the situation when Connector starts the game; we leave this as an exercise to the reader. (Intuitively, in games of this sort it should be beneficial for either player to move first; in Chapter 2 we will justify this thesis formally for a large and well-defined class of games.) Now, sit two players at the graph G and make them occupy free edges of G alternatingly, so that each of them follows the winning strategy of Connector in the connectivity game (the first player actually follows the adapted winning strategy). By the time all edges of G have been claimed by the players, each player must own a spanning tree: indeed, both of their strategies succeed to build a spanning tree against any opponent. These two spanning trees of G are obviously edge disjoint, which proves the theorem.  Another general component of the theory of strong games is Ramsey-type results. This is usually based on the following immediate corollary. Corollary 1.3.3. If in a strong game played on (X, F) there is no final drawing position, then First Player has a winning strategy. Proof. The game is at least a draw for First Player, by Theorem 1.3.1; it cannot end in a draw by the corollary’s assumption. The only remaining possibility is for FP to win.  For example, we derive immediately that the clique game (KN , Kn ) (a strong game played on the edges of the complete graph KN on N vertices; the player completing a copy of Kn first wins) is Red’s win for N ≥ R(n, n), where R(n, n) is the usual Ramsey number defined as the minimal integer M such that every Red-Blue coloring of the edges of the complete graph KM on M vertices has a monochromatic copy of Kn ; we will also denote it by R(n) sometimes. This is simply because every Red-Blue coloring of KN for such N has a monochromatic Kn by the definition of the Ramsey number, implying that draw is not an option. The most striking example of the application of this tandem (Strategy Stealing + Ramsey) is probably for the nd game. Going back to the definition of nd , observe that as d grows while n is fixed, there are more and more winning lines, all of them of the same size n. We can thus expect that in such a case the game becomes more thrilling, and the chances for drawing outcome diminish. And indeed, Hales and Jewett, in one of the cornerstone papers of modern Ramsey theory [46] (notice its title – the paper was about positional games!), proved that for a given n and k, and a large enough d ≥ d0 (k, n), every k-coloring of [n]d contains a monochromatic geometric line. Thus, the strong game played on such a board cannot end in a draw (k = 2, and the two colors are marks of the two players), and we conclude that First Player wins! Hales-Jewett’s theorem has become so well known partly due to the fact that it generalizes an even more famous theorem, that of van der Waerden. (Van der

1.3. Strong games

11

extra set

Figure 1.1: Extra set paradox. Waerden’s theorem states that for all positive integers k and n, there exists an integer N such that any k-coloring of the first N positive integers contains a monochromatic n-term arithmetic progression.) In order to obtain van der Waerden from Hales-Jewett, let us look at the following natural bijection φ from [n]d to the set {0, . . . , nd − 1}: if a ¯ = (a1 , . . . , ad ) ∈ [n]d , then φ(¯ a) = (a1 − 1) + (a2 − 1)n + (a3 − 1)n2 + · · · + (ad − 1)nd−1 . One can easily check that the image of a geometric line in [n]d is an arithmetic progression of length n. Thus a monochromatic geometric line in a k-coloring of [n]d translates into a monochromatic arithmetic progression. Quite disappointingly, the above two statements constitute the whole contents of our bag of general tools available for tackling strong games. Strong games are notoriously hard to analyze, and rather few results are available at present. Trouble is everywhere here – Strategy Stealing is basically as implicit as a positive mathematical statement can be; Ramsey numbers and Ramsey-type bounds are frequently exceedingly large to provide tight upper bounds. (Indeed the gap in our knowledge, for a given n, between guaranteed Red’s win and a draw as a function of the dimension d in the nd game is embarrassingly huge.) Another problem in analyzing strong games is the fact that they are not hypergraph monotone. If First Player has a winning strategy in some game F, then adding another winning set A to F should seemingly only benefit him – there is one more potential winning set to occupy. In certain scenarios, however, this can in fact hurt him and change the game’s outcome to a draw. Beck calls this the Extra Set Paradox in [10]. A simple example is depicted in Figure 1.1: the winning sets, before the addition of an extra set, are all 8 full branches of the binary tree with four leaves, which is easily seen to be First Player’s win. However, adding one extra set of size 3 (depicted) turns the game into a draw, as one can verify through a straightforward case analysis. Problems abound here, let us mention one concrete example. Open Problem 1.3.4. Show that for every positive integer n there exist t and N0 such that for every N ≥ N0 , First Player can win in at most t moves the strong

12

Chapter 1. Introduction

game played on the edge set of the complete graph KN , where the goal is to create a copy of the complete graph Kn . In other words, we are basically asking for an explicit winning strategy for First Player in the (KN , Kn )-game. Of course, by Strategy Stealing+Ramsey, for N ≥ R(n, n) the game cannot end in a draw and is thus First Player’s win, but this is non-explicit. This question appears to be open even for the case n = 5.

1.4

Exercises

1. Compute the number of winning sets in nd . 2. Let F be a k-uniform hypergraph of maximum degree Δ. Prove that is Δ ≤ k/2, then the game F is a pairing draw. Hint: Use Hall’s theorem. 3. Prove that if n ≥ 3d − 1, then the nd game is a draw. 4. Assume that nd is a pairing draw. Prove that (n + 2)d is a pairing draw. 5. Prove that in the row-column game played on the d × n board, First Player can reach advantage d, if n is large enough compared to d. 6. Prove that the strong connectivity game, played on the edge set of the complete graph Kn (the player completing a spanning tree first wins), is First Player’s win. Remark: There are at least three different proofs of this statement: a direct one, one using Lehman’s theorem, and yet another one using strategy stealing. 7. By using the strategy stealing argument, show that First Player has a winning strategy for Hex.

Chapter 2

Maker-Breaker Games When you’re not concerned with succeeding, you can work with complete freedom. Larry David

2.1 Maker-Breaker positional games At any point during a strong positional game both players are required to do two things simultaneously: try to occupy a complete winning set for themselves, and prevent their opponent from occupying one for themselves. Apparently this dual job is a feature that we are not able to explicitly handle in any way other than by the brute force of a more or less complete case analysis of the game tree. Due to the exponential size of such game trees, this approach becomes impractical very quickly, and one must look for alternatives to say anything meaningful. Definition 2.1.1 below is one such attempt. It is motivated by the most important general result on strong games, formulated in Theorem 1.3.1 of Chapter 1. Applying the trick of Strategy Stealing, we proved there that SP (Second Player) cannot possibly have a winning strategy in a strong game, his best shot being to achieve a draw. Once SP comes to terms with this harsh reality and gives up his hopes for winning, he can concentrate fully on what he has at least a theoretical chance to achieve: prevent FP (First Player) from winning. It is then natural for SP to consider introducing a small change in his attitude towards the game and voluntarily resign his own power to threaten FP with occupying a winning set for himself. Say, SP could announce at the beginning of the game that he does not consider himself a winner in case he occupies a set for himself, but only if he is able to prevent FP from doing so. This might seem an unnecessary and silly move, for why should SP not threaten if he can? Well, in most games he is not, in any case, able to analyze this extra power of his, while resigning it will enable SP to

D. Hefetz et al, Positional Games, Oberwolfach Seminars 44, DOI 10.1007/978-3-0348-0825-5_2, © Springer Basel 2014

13

14

Chapter 2. Maker-Breaker Games

define a related, but simpler game type, the setup of which will make it possible to develop a mathematical theory. This theory will enable him to obtain winning strategies in many games of this new type, each leading to drawing strategies in the corresponding strong games. An added bonus of the mathematical theory includes numerous motivations and applications, both direct and indirect, discovered during the last half a century in combinatorics and theoretical computer science. Of course, once SP does not care about occupying a winning set for himself, FP can immediately forget about playing defense and concentrate on offense exclusively. This simplification of FP’s task is also captured in the definition of Maker-Breakertype positional games below. Definition 2.1.1. Let X be a finite set and F ⊆ 2X a family of subsets. In a Maker-Breaker game over the hypergraph (X, F) • the set X is called the board; the elements of F ⊆ 2X are the winning sets; • the players are called Maker and Breaker; • during a particular play, the players alternately occupy elements of X; as a default, we set Maker to start (unless stated otherwise); • the winner is – Maker if he occupies a winning set completely by the end of the game, – Breaker if he occupies an element in every winning set. In these lecture notes we deal almost exclusively with Maker-Breaker games for finite hypergraphs, hence it is convenient to require finiteness in the definition. Nevertheless, it is not much more complicated, and in fact quite intuitive, to define Maker-Breaker games over hypergraphs with finite winning sets but infinite board. Some examples will be discussed at the end of this section and in the exercises. Remark 2.1.2. The game is of perfect information with no chance moves, played on a finite hypergraph with players having complementary goals. Hence draw is impossible and, as we have seen in Chapter 1, exactly one of the players has a winning strategy. Furthermore, given the family F of winning sets, it is clear (at least to an all-powerful computer) which one of them does. As F fully determines the game, we often identify the two concepts and refer to a hypergraph as a Maker-Breaker game. Terminology: If Maker has a winning strategy in the game over the hypergraph F, then F is called a Maker’s win, otherwise (that is, when Breaker has a winning strategy) F is called a Breaker’s win. Remark 2.1.3. In the terminology of the monograph of Beck [10] Maker-Breaker games are also called weak games (as opposed to strong games). Maker’s win is also called Weak Win (since FP cannot necessarily convert the existence of Maker’s winning strategy into a winning strategy for himself in the corresponding strong game), while Breaker’s win is referred to as Strong Draw (since SP can always use

2.1. Maker-Breaker positional games

15

Breaker’s winning strategy, out of the box, for his purpose). An even stronger form of draw is the Pairing Draw, when a partition of (a subset of) the board into pairs is possible such that each winning set contains one of the pairs. Then a Breaker’s win/SP-draw is achieved easily by taking the other element of the pair in which Maker/FP has just occupied the first one. We have seen many examples of this pairing strategy in the first chapter, in the setup of the generalized Tic-Tac-Toe games. Remark 2.1.4. For a family F ⊆ 2X let F up := {F ⊆ X : ∃F  ∈ F, F  ⊆ F } be the up-set generated by F. It should be clear that the games over the families F and F up have the same winner. In particular, F and the family F min := {F ∈ F : F  ∈ F, F   F } of the inclusion minimal elements of F also have the same winner. Remark 2.1.5. In the definition we set, as a default, that Maker is the first player to move. Later we will sometimes need to consider the situation when Breaker moves first. Then the following simple proposition often comes in handy. Proposition 2.1.6. Let (X, F) be a hypergraph. (i) If Maker has a winning strategy in the game over F as the second player, then he also has a winning strategy if he starts the game. (ii) Similarly, if Breaker has a winning in the game over F as the second player, then he also has a winning strategy if he starts the game. Proof. We prove only (i), because (ii) is then a direct consequence. Maker starts the game with an arbitrary move v1 ∈ X. Then he ignores this move while continuing to play according to his winning strategy S over F as the second player. If at some point the strategy S calls for him to take v1 , he takes another arbitrary free element v2 ∈ X and now ignores v2 while continuing to play according to S. Formally, one defines the strategy of Maker inductively, such that at any point after Maker’s first move, the board of the game contains a particular play sequence Maker played against his opponent according to his winning strategy S as the second player, plus exactly one extra element occupied by Maker. Let us say that when Maker comes to make his next move, the play sequence on the board is b1 , m1 , b2 , m2 , . . . , mj−1 , bj , and the extra element occupied Maker is vi . Maker determines what move S calls for in the particular play sequence and he takes this move, unless it is the current extra element vi . In the latter case Maker designates mj = vi and takes another arbitrary free element vi+1 . This maintains the inductive statement. Now, we know that Maker would occupy a winning set fully in any particular play sequence he plays according to his winning strategy as the second player. So Breaker cannot possibly win, since by the inductive statement all his moves are part of the play sequence Maker is playing as the second player according to S and we know that Breaker cannot possibly put his mark in every winning set in that play sequence. Hence the described adaptation of S is a winning strategy for Maker.  Examples of Maker-Breaker games.

16

Chapter 2. Maker-Breaker Games

1. There are examples of games where FP (Maker) can actually benefit from the announcement of SP (Breaker) that SP does not care any more about occupying a winning set for himself. It is easy to check that the Maker-Breaker version of the usual Tic-Tac-Toe game on the 3-by-3 grid is a Maker’s win. That is, for SP it was crucial in the usual (strong) Tic-Tac-Toe that he, at some point in his drawing strategy, threatened FP with completely occupying a winning set himself. 2. The connectivity game of the first chapter is also a Maker-Breaker game. The members of the family C ⊆ 2E(G) of winning sets are the edge sets of the spanning trees of G. Connector (alias Maker) tries to occupy a spanning tree, while Disconnector (alias Breaker) tries to prevent Connector from doing so. A subtle difference compared to our definition was that in Chapter 1 we set Disconnector (Breaker) to start the game in order to be able to state the nice characterization of when Connector has a winning strategy (Theorems 1.1.3 and 1.3.2). The Hamiltonicity game and the row-column game of the first chapter can also be cast as Maker-Breaker games with the natural family of winning sets. The game Sim is not a Maker-Breaker game. 3. At first glance, the game of Hex seems to be a strong game: whichever player is the first to occupy his winning set is the winner. But of course there is this one caveat, that the families of winning sets are not the same for the two players. As discussed in Chapter 1, the situation can be remedied by the so-called Hex Theorem1 [39], which makes Hex into a Maker-Breaker game. The Hex Theorem asserts that in any red/blue-coloring of the hexagons of the Hex board, there is either a path of neighboring red hexagons connecting the two red sides of the board, or a path of neighboring blue hexagons connecting the two blue sides of the board. This makes Hex a Maker-Breaker game, where the winning sets are the paths between the two red sides of the board. Maker (Player RED) wins if he occupies one, Breaker (Player BLUE) wins if he prevents RED from doing so (and hence, by the Hex Theorem, builds his own blue path between the blue sides). 4. 5-in-a-row (Gomoku in Japanese, Am˝oba in Hungarian): The game of 5-in-a-row is a relative of the n2 Tic-Tac-Toe game (discussed in the first chapter), with the main difference being that the winning sets are of size 5 and not n. The players still put Noughts and Crosses alternately into the squares of a square grid and try to be the first to occupy a horizontal, vertical, or diagonal segment of 5 consecutive squares (i.e., a winning set). For the 15-by-15 board Victor Allis [2] gave a computer-aided proof that FP has a winning strategy. Competitively the game is mostly played on a 19-by-19 board (with some simple extra starting rule to even out the apparent advantage of FP). An interesting case is when the board is the infinite square grid.2 Experience shows 1 It

is interesting to note how the Brouwer fixed point theorem seems to have played a central role in the early work of John Nash. On the one hand, the Hex Theorem, which Nash must have been aware of, can be deduced from the fixed point theorem (and is in fact equivalent to it [39]). On the other hand, Nash used the fixed point theorem in his PhD thesis to show the existence of Nash equilibrium, his “other” invention during his graduate years. 2 To define the game on the infinite square grid, one must be a little careful. In the strong

2.2. Coloring hypergraphs

17

that actual games hardly ever last to the boundary of the chequered exercise book page they are customarily played on, not even close. Still for the “theory” of the game, this extension of the board size makes a huge difference: for one game we know the solution, for the other we do not. The 15-by-15 FP winning strategy of course implies that Maker wins in the Maker-Breaker version of the game. This in turn implies that Maker also has a winning strategy on the infinite board. Curiously, it is still not known whether FP can win the strong 5-in-a-row game on the infinite board. Why cannot FP just apply Allis’ 15-by-15 FP-strategy out of the box to occupy his winning set on the infinite board? Partly because of the dual task of simultaneous offense/defense FP has to perform in a strong game. Allis’ strategy will certainly create a 5-in-a-row eventually, while preventing SP from creating his own on the 15-by-15 board, but along the way it might also lead to a 4-in-a-row of SP at the boundary of the 15-by-15 board. This would not concern FP in the finite game, but in the infinite game he would be forced to play outside the 15-by-15 board and abandon his winning strategy. This is yet another manifestation of the extra set paradox of the first chapter, or rather of the difficulties it can cause. It is widely believed that the 5-in-row game on the infinite board is FP’s win, while 6-in-a-row is a draw. However, it is only known that 4-in-a-row is FP’s win (a trivial exercise) and that 8-in-a-row is a draw. In Section 2.4.2 we will see a proof that 40-in-a-row is a draw.

2.2

Coloring hypergraphs

 subsets of X. Let X k := {K ⊆ X : |K| = k} the set of all k-element  A hypergraph (X, F) is called k-uniform if F ⊆ X consists only of k-element k subsets. Sometimes we identify the hypergraph with its edge set F. A function f : X → {red, blue} is called a proper 2-coloring of the hypergraph (X, F) if every member of F has both a red and a blue colored vertex (that is, no edge is monochromatic). A hypergraph (X, F) is called 2-colorable if it has a proper 2-coloring. For a proper 2-coloring to exist we obviously need at least two vertices in each edge, so we assume k ≥ 2. For example a k-uniform hypergraph with exactly two distinct edges always has a proper 2-coloring: in each of the two edges take a vertex which is not part of the other edge and color it red, and color the rest of the vertices blue. It is a famous open problem to determine for each k the smallest number m(k) of edges in a non-2-colorable k-uniform hypergraph. The triangle graph shows that m(2) = 3. game the players’ goal is still to be the first to occupy a winning set fully, and a draw means that none of the players wins, hence the game goes on infinitely long. Strategy stealing still applies, so SP can only hope for a draw. In the Maker-Breaker game, Breaker wins if Maker does not occupy a winning set. This is not any more equivalent to Breaker putting his mark in every winning set: the point is that Breaker can force the game to last infinitely long without Maker winning.

18

Chapter 2. Maker-Breaker Games

The following result of Erd˝os [27], one of the first applications of the probabilistic method, provides an exponential lower bound. Claim 2.2.1. If |F| < 2k−1 , then F is 2-colorable. In particular, m(k) ≥ 2k−1 . Proof. Take a random 2-coloring f : V (F) → {red, blue}. That is: color all vertices x ∈ V (F) independently, uniformly at random such that Pr[f (x) = red] =

1 = Pr[f (x) = blue]. 2

For each A ∈ F, let YA be the characteristic random variable of the event that A is monochromatic. That is YA = 1 if A is monochromatic, otherwise YA = 0. Now,

  |F| YA = E [YA ] = k−1 < 1. E[#of monochromatic edges of F] = E 2 A∈F

A∈F

The random variable A∈F YA takes only non-negative integer values, so its average can only be strictly less than 1 if it also takes the value 0 at least once. Hence for sure, not just with some probability, but with 100% certainty, there exists a 2-coloring of F without monochromatic edges.  Remark 2.2.2. The classical term for a hypergraph F being 2-colorable is that F has property B. Remark 2.2.3. The best known lower  bound for m(k), due to Radhakrishnan and Srinivasan [76], is of the order 2k k/ log k. Their proof also starts with a random coloring of the vertices, but continues with a refined randomized recoloring procedure, which fixes the errors (the monochromatic sets). The best known upper bound, obtained by considering a random k-uniform hypergraph on roughly k 2 vertices, is of the order 2k k 2 . It was proved by Paul Erd˝ os [28] around the same time “Oh Pretty Woman” topped the US charts: not exactly yesterday. Improving these bounds remain intriguing open problems. The probabilistic method is a great tool to prove the existence of special objects, including, as the case may be, a coloring of the vertices of a hypergraph with red and blue so that in each hyperedge both red and blue occur. In our computeroriented world, however, a proof of the existence of something is of little value: We do not just want to know that this book exists and that some people actually own a copy, but we want to hold one in our hands. With such worldly possessions, this is usually easy to accomplish with enough money at hand. However, in the case of a proper coloring of a hypergraph, the existence of which is proved by the above claim, we need something else. When do we hold a 2-coloring of a hypergraph in our hand? We of course want that, given the hypergraph in some form (say, by its incidence matrix) we can construct a proper coloring, potentially with the help of a computer. So we need an algorithm for this task. This is easy, you might say, once we know the coloring

2.3. The Erd˝os-Selfridge Criterion

19

exists: the number of vertices is finite, the hypergraph is finite, so we could look at all possible 2-colorings of the vertex set and check for each whether it is proper. This solution, however, is not really satisfying, as the number of all colorings is 2|X| and no matter how super a computer you own now (or will own, ever) it will never even get close to finishing the check within the lifetime of our universe, on even relatively small problems where the vertex set X is, say, of size 100. In computer science one usually accepts algorithms as theoretically “fast” when their running time is polynomial in the size of the input. Is there a polynomial time algorithm which finds a proper 2-coloring of any given input hypergraph from Claim 2.2.1? Based on the proof of Claim 2.2.1, we can suggest a natural randomized algorithm: Color the vertices independently, uniformly at random and then check whether the obtained coloring is proper. If the answer is YES, then output this coloring and terminate, otherwise repeat the procedure with a new random coloring. Generating a random assignment involves |X| calls to a uniformly random source and the checking involves going through the colors of the k vertices of each of the |F| hyperedges. Altogether this represents |X| + k|F| = O(|X| · |F|) steps, which is just quadratic in the input data. Assuming |F| < 2k−2 , only a factor 2 stronger assumption than the one in Claim 2.2.1, we find that the probability of failure of the random coloring producing a proper coloring is at most 12 , by Markov’s Inequality. After just one hundred iterations, which shows up in the running time just as a multiplicative constant factor 100, the probability of failure to produce a proper coloring is at 1 , orders of magnitude smaller than the probability of hardware failure of most 2100 your supercomputer resulting in an incorrect outcome. Although, for all practical purposes, such an algorithm is satisfactory, still there is the dependence on a perfectly uniform random source and the loss of the constant factor 2 compared to the existence result of the claim. The question remains whether there is an efficient and possibly deterministic way to find a proper coloring whose existence is promised by Claim 2.2.1. The answer to this question is also YES and it was given in the context of positional games by Erd˝ os and Selfridge.

2.3

The Erd˝ os-Selfridge Criterion

The following proposition establishes the connection between 2-colorings and Maker-Breaker games. The argument is analogous to the one we saw in the proof of Theorem 1.3.2 of Chapter 1. Proposition 2.3.1. F is a Breaker’s win ⇒ F is 2-colorable.

20

Chapter 2. Maker-Breaker Games

Proof. Let us sit two players, FP and SP, to play on the board V (F) and give both of them the winning strategy S of Breaker for the game F (which exists by assumption). More precisely, give S to SP, as Breaker plays second in a MakerBreaker game, and give FP the winning strategy of Breaker as a first player, which exists by Proposition 2.1.6. Make both players play according to this strategy, such that FP colors his board elements with red and SP colors his with blue. Since FP plays according to a strategy which is a winning one for Breaker, he will put a red mark in every winning set by the end of the game. SP also plays according to Breaker’s winning strategy and hence by the end of the game a blue vertex as well will be in every winning set. Looking at the board at the end of the game, we see that every winning set contains both red and blue vertices: the coloring created by the two players during their play is proper.  By this proposition, the following fundamental result of Erd˝os and Selfridge is a strengthening of Claim 2.2.1. From its proof it will be obvious how to devise a very efficient deterministic algorithm which finds a proper 2-coloring of the underlying hypergraph. Theorem 2.3.2. Let F be a k-uniform hypergraph. Then |F| < 2k−1 ⇒ F is a Breaker’s win. This theorem is a corollary of the following more general one, dealing not only with uniform hypergraphs. Theorem 2.3.3 (Erd˝os-Selfridge Criterion, [31]). Let F be a hypergraph. Then  A∈F

2−|A|
2k−3 · Δ2 (F) · |X| ⇒ F is Maker’s win, where Δ2 (F) = max{deg(x, y) : x, y ∈ X, x = y} and deg(x, y) = |{A ∈ F : x, y ∈ A}|. 5. In the Maker-Breaker arithmetic progression game AP (n, k), the board is [n] and the winning sets are all arithmetic progressions of length k. Prove that there exist constants c1 , c2 > 0 such that if k < c1 log2 n, then the game is Maker’s win, and if k > c2 log2 n, then the game is Breaker’s win.

Chapter 3

Biased Games Life is never fair, and perhaps it is a good thing for most of us that it is not. Oscar Wilde

3.1 Background and motivation Before we discuss biased games, let us consider a few concrete examples of MakerBreaker games. 1. The triangle game FK3 (n). The board of this game is the edge set of Kn and the winning sets are all copies of K3 in Kn . We present an explicit very simple winning strategy for Maker, which works for every n ≥ 6 (a more careful analysis shows that Maker can win the game on K5 as well). In his first move, Maker claims an arbitrary edge uv. Let xy denote the edge Breaker claims in his first move. In his second move, Maker claims vw for some vertex w ∈ / {u, v, x, y}. We can assume that Breaker responds by claiming uw, as otherwise Maker will claim it in his next move and thus win. In his third move Maker claims vz for some vertex z ∈ / {u, v, x, y, w}. This creates the double threat uz and wz. Since Breaker cannot claim both edges in one move, Maker will claim one of them in his fourth move and thus win. 2. The connectivity game C(n). The board of this game is the edge set of Kn and the winning sets are all spanning connected subgraphs of Kn . We present an explicit very simple winning strategy for Maker, which works for every n. For every 1 ≤ i ≤ n − 1, in his ith move, Maker claims an arbitrary free edge which does not close a cycle in his graph. In every subsequent move, he plays arbitrarily. It is evident that, if Maker can play his first n−1 moves according to this strategy, then he wins the game. It thus remains to show that he can

D. Hefetz et al, Positional Games, Oberwolfach Seminars 44, DOI 10.1007/978-3-0348-0825-5_3, © Springer Basel 2014

27

28

Chapter 3. Biased Games indeed do so. Suppose, to the contrary, that there exists some 1 ≤ i ≤ n−1 for which Maker cannot play his ith move according to the proposed strategy. It follows that immediately after his (i − 1)th move, Breaker has already claimed all edges of some cut of Kn . However, this is clearly impossible, as every cut contains at least n − 1 > i − 1 edges.

Since the triangle game and the connectivity game (as well as many other MakerBreaker games) are an easy win for Maker, one would like to give Breaker more power so that he has a better chance of winning. One way to do so, first suggested by Chv´ atal and Erd˝ os [21], is by allowing Breaker to claim several board elements per move rather than just one. This leads to the following definition. Definition 3.1.1. Let p and q be positive integers, let X be a finite set, and let F ⊆ 2X be a family of subsets of X. The biased (p : q) Maker-Breaker game (X, F) is the same as the Maker-Breaker game (X, F), except that Maker claims p free board elements per move and Breaker claims q free board elements per move. The integers p and q are referred to as the bias of Maker and Breaker, respectively. In the last move of the game, if there are less free board elements than his bias, a player claims every free board element. Remark 3.1.2. The Maker-Breaker games we have seen before form the special case p = q = 1 of Definition 3.1.1. We refer to such games as fair or unbiased. Example. The triangle game FK3 (n). As shown above, this game is an easy win for Maker. It is therefore sensible to consider a biased version of this game. The following result is a slightly weakened version of a theorem of Chv´ atal and Erd˝os [21]. It determines up to a constant factor the amount of extra power Breaker needs in order to win this game. Theorem 3.1.3. √ Maker has a winning strategy in the (1 : b) triangle game FK3√(n) for every b ≤ n/2. On the other hand, Breaker has a winning strategy if b ≥ 2 n. √ Proof. Assume first that b ≤ n/2. We present a strategy for Maker and then prove that, by following this strategy, he wins the game. Let u ∈ V (Kn ) be an arbitrary vertex. For every positive integer i, Maker plays his ith move as follows. If there exists a free edge which closes a triangle in his graph, then Maker claims one such edge and thus wins. If no such edge exists, then he claims a free edge uvi , where vi ∈ V (Kn ) is an arbitrary vertex. If this is not possible either, then he forfeits the game. We claim that, by following this strategy, Maker wins the game. Suppose, to the  contrary, that he does not. It follows that there exists an integer d = d(n) ≥ n−1 b+1 such that for every 1 ≤ i ≤ d, Maker has claimed uvi in his ith move and Breaker has claimed every edge of {vi vj : 1 ≤ i < j ≤ d} during those d moves. Clearly,  this is not possible since d2 > bd holds (with room to spare) by the assumed upper bound on b.

3.1. Background and motivation

29

√ Next, assume that b ≥ 2 n. We present a strategy for Breaker and then prove that, by following this strategy, he wins the game. For every positive integer i, let ui vi denote the edge claimed by Maker in his ith move. In his ith move, Breaker claims b/2 free edges which are incident to ui and b/2 free edges which are incident to vi (if only t < b/2 such edges remain, then he claims all of these and additional b/2 − t arbitrary free edges). When claiming edges at ui (respectively at vi ), he starts by claiming free edges which pose an immediate threat, that is, edges ui z (respectively vi z) such that vi z (respectively ui z) was previously claimed by Maker. We claim that, by following this strategy, Breaker wins the game. Note first that, at any point during the game, the maximum degree in Maker’s graph is at most b/2 + 1. Indeed, by the proposed strategy (with the possible exception of the last time), whenever Maker claims an edge which is incident to a vertex x, Breaker claims at least b/2 edges which are incident to x. It follows that Maker’s degree n−1 ≤ b/2 + 1. The same argument applies to any vertex at x is at most 1 + b/2+1 of Kn and thus to the maximum degree in Maker’s graph. Suppose, to the contrary, that Maker wins the game. Let u, v, w be vertices of Kn such that uv, vw and uw were all claimed by Maker; assume without loss of generality they were claimed in this order. As soon as Maker has claimed vw, Breaker has responded according to his strategy by claiming, in particular, at least b/2 edges at w. Since he did not claim uw, it follows by his strategy that there were vertices z1 , z2 , . . . , zb/2 ∈ V (Kn ) \ {u, w} such that vzi was previously claimed by Maker for every 1 ≤ i ≤ b/2. The degree of v in Maker’s graph is thus at least b/2 + 2, contrary to the aforementioned upper bound on the maximum degree in Maker’s graph.  Remark 3.1.4. A more careful analysis of Maker’s strategy described in√the proof of Theorem 3.1.3 shows that it is in fact a winning strategy for every b < 2n + 2− 5/2; we leave this as an exercise to the reader. Moreover, an improved strategy for Breaker, due to Balogh and Samotij √ [4], shows that the (1 : b) triangle game is Breaker’s win for every b ≥ (2 − 1/24) n. As noted (and proved) above, Maker wins the (1 : 1) triangle game. On the other  hand, it is obvious that Breaker wins the biased (1 : n2 ) version of this game. Since increasing Breaker’s bias favors him whereas decreasing it favors Maker, it follows that there exists a unique integer 1 < b∗ ≤ n2 such that Breaker wins the (1 : b) triangle game if and only √ if b ≥ b∗ . We did not determine the precise value ∗ ∗ of b , but proved that b = Θ( n). The existence of such an integer b∗ is not unique to the triangle game. In fact, it holds for (essentially) every Maker-Breaker game. This leads to the following important definition. Definition 3.1.5. Let X be a finite set and let F ⊆ 2X be a family of subsets of X such that F = ∅ and min{|A| : A ∈ F} ≥ 2. The unique positive integer bF such that Breaker wins the (1 : b) game (X, F) if and only if b ≥ bF is called the threshold bias of (X, F).

30

Chapter 3. Biased Games

Proving that the threshold bias is well defined for every Maker-Breaker game (X, F) (provided that F =  ∅ and min{|A| : A ∈ F} ≥ 2) is left as an exercise to the reader. A central goal of the theory of biased Maker-Breaker games is to determine (or at least approximate) the threshold bias of natural games such as the connectivity game, the Hamiltonicity game, etc. We will address this issue in the current chapter, as well as in some of the subsequent ones. The currently best known upper and lower bounds on the threshold bias of the triangle game were stated in Remark 3.1.4. Improving these is an interesting open problem. Open Problem 3.1.6. Find a real number c such that Maker has a winning strategy √ in the triangle game FK3 (n) for every b√≤ (c − o(1)) n, whereas Breaker has a winning strategy for every b ≥ (c + o(1)) n.

3.2

General criteria for biased games

We proved Theorem 3.1.3 by providing both players with ad hoc strategies. It would of course be better to have some general strategy which is applicable to a wide variety of games. As seen in Chapter 2, such general strategies do exist. Breaker’s strategy is described in the proof of the Erd˝os-Selfridge Theorem and Maker’s strategy is described in the proof of Beck’s weak win criterion (see Exercise 4 in Chapter 2). The main result of this section is a biased version of the Erd˝ osSelfridge Theorem due to Beck [5] (see also [10] and [85]). Theorem 3.2.1. Let X be a finite set, let F be a family of subsets of X, and let p and q be positive integers. If  A∈F

(1 + q)−|A|/p
· Δ2 (F) · |X|, p (p + q)3

A∈F

then Maker (as the first player) has a winning strategy for the (p : q) game (X, F).

3.3. The threshold bias of the connectivity game

3.3

33

The threshold bias of the connectivity game

A fundamental task in the theory of biased games is that of determining the threshold bias of natural games. We illustrate this here with the important example of the connectivity game. For a positive integer n, the board of the connectivity game is E(Kn ) and the family of winning sets C = C(n) consists of all connected spanning subgraphs of Kn . The threshold bias of this game was first studied by Chv´atal and Erd˝os [21], who determined its order of magnitude. In this section we will prove (a slightly stronger version of) their result. For every n ≥ 4, it readily follows from Lehman’s Theorem [64] (see also Chapter 1) that Maker has a winning strategy for the connectivity game (E(Kn ), C) simply because Kn admits two edge disjoint spanning trees. In fact, for every n, Kn admits n/2 pairwise edge disjoint spanning trees. It is thus tempting to try and prove a biased version of Lehman’s Theorem, namely, to prove that for every positive integer b, Maker has a winning strategy for the (1 : b) connectivity game, played on the edge set of a graph G, provided that G admits b + 1 pairwise edge disjoint spanning trees. The following claim shows that this is false even if one replaces b + 1 trees by f (b) trees for an arbitrary function f . Proposition 3.3.1. For every positive integer k there exists a graph Gk which admits k pairwise edge disjoint spanning trees and yet Breaker has a winning strategy for the (1 : 2) connectivity game, played on the edge set of Gk . Our proof of Proposition 3.3.1 will make use of the following lemma, whose proof is left as an exercise to the reader (it will also be discussed in greater detail in Section 3.4.1, where we will consider the so-called Box Games). Lemma 3.3.2. Let A1 , . . . , An be pairwise disjoint sets of size m each and let X =  n i=1 Ai . If m ≤ (p − 1) ln n, then Maker, as first or second player, has a winning strategy for the (p : 1) game (X, {A1 , . . . , An }). Proof of Proposition 3.3.1. Fix a positive integer k and let n = ek . Let Gk be the graph consisting of n+1 pairwise vertex disjoint cliques C0 , . . . , Cn of order 2k each and a matching Mi = {ei1 , . . . , eik } between Ci and Ci+1 for every 0 ≤ i < n. It is well known and easy to see that K2k admits k pairwise edge disjoint spanning trees and thus so does Gk (for every 1 ≤ j ≤ k and every 0 ≤ i < n, the jth tree will contain eij ). In order to prove that Breaker has a winning strategy for the (1 : 2) connectivity game, played on the edge set of Gk , it is sufficient to show that he can claim all edges of Mi for some 0 ≤ i < n. This is equivalent to proving that  Maker, as the second player, has a winning strategy for the (2 : 1) game n−1 i=0 Mi , {M1 , . . . , Mn−1 } . Since Mi ∩ Mj = ∅ holds for every 0 ≤ i = j < n, |Mi | = k for every 0 ≤ i < n, n = ek , and k ≤ lnek , this follows from Lemma 3.3.2. 

34

Chapter 3. Biased Games

Despite the lack of a biased version of Lehman’s Theorem, we are able to determine the correct order of magnitude of the threshold bias of the connectivity game. Theorem 3.3.3. For every ε > 0 there exists an integer n0 such that (ln 2 − ε)

n n ≤ bC ≤ (1 + ε) ln n ln n

holds for every n ≥ n0 . atal and Erd˝os [21], will be The upper bound on bC in Theorem 3.3.3, due to Chv´ proved in the next section. The remainder of this section is devoted to the proof of the lower bound, due to Beck [5]. It is based on Theorem 3.2.1 and on Beck’s building via blocking technique. A slightly weaker lower bound was previously proved by Chv´atal and Erd˝os [21]. Proof of the lower bound in Theorem 3.3.3. Fix some integer b ≤ (ln 2 − ε)n/ ln n; we will prove that Maker has a winning strategy for the (1 : b) game (E(Kn ), C). Consider the following auxiliary game, which we refer to as the Cut game. The board is again E(Kn ) and the family of winning sets Fn consists of all cuts of Kn , that is, Fn = {EKn (S, V (Kn ) \ S) : S ⊆ V (Kn ), S = ∅, S = V (Kn )}. To avoid confusion, we refer to the two players of the Cut game as CutMaker and CutBreaker. It is evident that Maker wins the (1 : b) game (E(Kn ), C) if and only if he claims at least one edge in every cut of Kn , that is, if and only if CutBreaker wins the (b : 1) game (E(Kn ), Fn ). This is an example of Beck’s building via blocking technique – instead of trying to build (make) a spanning tree, Maker tries to block (break) every cut. Equivalently, instead of trying to win as Maker in the (1 : b) connectivity game, he tries to win as CutBreaker in the (b : 1) Cut game. In order to prove the lower bound it thus suffices to prove that CutBreaker has a winning strategy for the (b : 1) game (E(Kn ), Fn ). We will do so using Theorem 3.2.1. Indeed, we have 

−|A|/b

2

=

A∈Fn

≤

n/2    n

k

k=1 √  n 





k=1

+

(n−k) ln n

n · 2− (ln 2−ε)n

k=1 √  n 

≤

2−k(n−k)/b k +

n/2  √ k= n+1

 en k

(n−k) ln n

· 2− (ln 2−ε)n

k

 k √ (1 + ε)(n − n) ln n exp ln n − n

n/2  √ k= n+1



 k en (1 + ε)(n − n/2) ln n √ · exp − n n

3.4. Isolating a vertex and box games

35

√  n

≤



[exp {ln n − (1 + ε/2) ln n}]

k

k=1 n/2 

+

√

[exp {1 + ln n/2 − (1/2 + ε/2) ln n}]

k

k= n+1

≤

n/2 

n−εk/3

k=1

= o(1) . It thus follows by Theorem 3.2.1 that CutBreaker has a winning strategy for the (b : 1) game (E(Kn ), Fn ) and therefore Maker has a winning strategy for the  (1 : b) game (E(Kn ), C), as claimed.

3.4

Isolating a vertex and box games

Our first task in this section is to complete the proof of Theorem 3.3.3. Proof of the upper bound in Theorem 3.3.3. Fix some integer b ≥ (1 + ε)n/ ln n. We will describe an explicit strategy for Breaker and prove that, by following this strategy, he wins the connectivity game. In fact, he will isolate a vertex in Maker’s graph. Breaker’s strategy is divided into the following two stages. Stage I. Breaker builds a clique C of order b/2 such that all of its vertices are isolated in Maker’s graph. Stage II. Breaker isolates one of the vertices of C in Maker’s graph. It is evident that, if Breaker can follow the proposed strategy, then he wins the game. It thus remains to prove that he can indeed do so. We consider each stage separately. Stage I. For every positive integer i, if Breaker plays his ith move in Stage I, then immediately before this move let Ci denote Breaker’s clique; in particular, C1 = ∅. In his ith move in Stage I Breaker plays as follows. Let Si ⊆ V (Kn ) \ Ci be a set of maximum size such that every v ∈ Si is isolated in Maker’s graph and |{uv : u ∈ Si , v ∈ Si ∪ Ci and uv is free}| ≤ b. Breaker claims all edges of {uv : u ∈ Si , v ∈ Si ∪ Ci and uv is free}. If needed, he claims additional arbitrary edges. Let xi yi be the edge claimed by Maker in his subsequent move. We set Ci+1 = (Ci ∪ Si ) \ {xi , yi }. Since Ci is a clique all of whose vertices are isolated in Maker’s graph, it follows by Breaker’s strategy that so is Ci+1 . Moreover, it follows that |(Ci ∪ Si ) ∩ {xi , yi }| ≤ 1 and thus |Ci+1 | ≥ |Ci | + |Si | − 1. In particular, |Ci+1 | > |Ci | holds whenever |Si | ≥ 2. Note that, if |Ci | ≤ b/2, then |Si | ≥ 2 and thus Breaker can indeed enlarge his clique until it reaches the desired order. This

36

Chapter 3. Biased Games

stage clearly lasts at most b/2 < n/ ln n moves and thus the required sets Si exist for each of Breaker’s moves. Stage II. Let C denote Breaker’s clique at the end of Stage I, let t = |C|, and let u1 , . . . , ut denote its vertices. Breaker restricts his attention to the family of sets FC = {Ai : 1 ≤ i ≤ t}, where Ai = {ui v : v ∈ V (Kn ) \ C}. If he can claim all elements of some Ai ∈ FC , then he isolates the corresponding vertex claim some Ai ∈ FC , Breaker can ui and thus wins the game. In order to fully  t assume the role of Maker in the (b : 1) game i=1 Ai , FC . Since Ai ∩ Aj = ∅ for every 1 ≤ i = j ≤ t, inorder to prove  that Maker indeed has a winning t strategy for the (b : 1) game i=1 Ai , FC , one would like to use Lemma 3.3.2. However, the Ai ’s might have different sizes. Let m = max{|A1 |, . . . , |At |} and let m each. It is evidentthat if B1 , . . . , Bt be arbitrary pairwise disjoint sets of size  t Maker has a winning strategy for the (b : 1) game i=1 Bi , {B1 , . . . , Bt } , then   t he also has one for the (b : 1) game i=1 Ai , {A1 , . . . , At } . Since m ≤ n ≤ (1 + 2ε/3)n/ ln n · (1 − ε/3) ln n ≤ (b − 1) ln t, we can now apply Lemma 3.3.2 to complete the proof. 

3.4.1 Box games In the proofs of Proposition 3.3.1 and of the upper bound in Theorem 3.3.3 we used Lemma 3.3.2. This lemma states a sufficient condition for Maker’s win in a game (X, F), where F consists of pairwise disjoint sets of equal size. One can of course apply Theorems 3.2.1 and 3.2.2 to study such games, but the results obtained will not be very strong. It seems that the condition that all winning sets are pairwise disjoint should make the analysis of these games much easier. Indeed, such games were studied by Chv´ atal and Erd˝os [21] several years before Theorems 3.2.1 and 3.2.2 were proved. Before discussing their results, we introduce their appealing terminology. n Let A1 , . . . , An be finite pairwise disjoint sets and let X = i=1 Ai . One can view the (p : q) Maker-Breaker game (X, {A1 , . . . , An }) in the following way. There are n boxes, where for every 1 ≤ i ≤ n, the ith box contains |Ai | balls. In each of his moves Maker removes balls from these boxes, p balls in total. In each of his moves Breaker destroys q boxes. Maker wins this game if and only if he manages to empty one of the boxes before it is destroyed by Breaker. Such games were thus coined box games by Chv´ atal and Erd˝ os [21], who denoted the (p : q) game (X, {A1 , . . . , An }) by Box(p, q; |A1 |, . . . , |An |) and referred to the players as BoxMaker and BoxBreaker, respectively. Studying box games, one quickly realizes what the optimal strategies of both players should look like. BoxBreaker should always destroy the smallest boxes, whereas BoxMaker should try to balance the sizes of all boxes, thus voiding the strength of BoxBreaker’s strategy (the latter is not completely accurate, if certain

3.4. Isolating a vertex and box games

37

boxes are too large compared to others, then BoxMaker should in fact ignore these boxes and balance the rest). If this is indeed true, then we immediately obtain an algorithm to determine the winner of Box(p, q; a1 , . . . , an ) for all positive integers p, q, a1 , . . . , an . All we have to do is simulate the game where both players use their proposed strategies and see who wins. Moreover, a careful analysis of these strategies gives rise to a sufficient and necessary condition for a player’s win in Box(p, q; a1 , . . . , an ) for all positive integers p, q, a1 , . . . , an . Unfortunately, the resulting condition is quite technical and so we will not present it here. The interested reader can find all the details in [47]. The special case in which q = 1 atal and Erd˝ os [21]. A proof of and a1 ≤ . . . ≤ an ≤ a1 + 1 was solved by Chv´ their sufficient condition for Maker’s win in this case is left as an exercise. Their proof of the matching sufficient condition for Breaker’s win contained a technical error which was subsequently fixed by Hamidoune and Las Vergnas [47]. Below, we state and prove two simple and easy to use sufficient conditions for Breaker’s win in Box(p, 1; a1 , . . . , an ) which are not far from being best possible.

n Theorem 3.4.1. If ai = m, for every 1 ≤ i ≤ n, and m > p · i=1 1/i, then BoxBreaker has a winning strategy for Box(p, 1; a1 , . . . , an ). Remark 3.4.2. The sufficient condition for BoxBreaker’s win given in Theorem 3.4.1 is almost the exact complement of the sufficient condition for BoxMaker’s win given in Lemma 3.3.2 (at least for large n and p). This shows that, for the special case of boxes of equal size, Theorem 3.4.1 is essentially best possible. Proof of 3.4.1. At any point during the game a box is called surviving if it was not previously destroyed by BoxBreaker; we denote the set of surviving boxes by S. At any point during the game the size of a box is the number of balls it still contains. BoxBreaker employs the obvious strategy (as indicated above): in every move he destroys a box i ∈ S whose size is minimal (breaking ties arbitrarily). We will prove that this is a winning strategy for BoxBreaker. Suppose, to the contrary, that BoxMaker wins the game; assume further that he wins it in his kth move, for some 1 ≤ k ≤ n. Assume without loss of generality (by relabeling the boxes) that, for every 1 ≤ i ≤ k − 1, BoxBreaker destroys box i in his ith move, and that in his kth move BoxMaker fully claims box k. For every i ∈ S ∩ {1, . . . , k} and at any point during the game, let ci denote the remaining size of box i. For every 1 ≤ j ≤ k, let  1 ci k − j + 1 i=j k

Φ(j) :=

denote the potential of the game just before BoxMaker’s jth move. Note that Φ(k) = ck ≤ p by our assumption that BoxMaker wins the game in his kth move, and that Φ(1) = m. For every 1 ≤ j ≤ k − 1, in his jth move BoxMaker decreases

38

Chapter 3. Biased Games

Φ(j) by at most p/(k − j + 1). In his (j + 1)st move, BoxBreaker destroys the smallest surviving box and thus Φ(j + 1) ≥ Φ(j) − p/(k − j + 1). It follows that !   k  p p p + + ... + 1/i − 1 Φ(k) ≥ m − =m−p k k−1 2 i=1 ! n  ≥m−p 1/i − 1 > p , i=1

where the last inequality follows by the assumed lower bound on m. This contradicts our assumption that BoxMaker wins and concludes the proof of the theorem.  The following criterion for BoxBreaker’s win in the Box Game is more general than the one given in Theorem 3.4.1, but is slightly weaker in the uniform case (that is, when all boxes have the same size). Theorem 3.4.3. If

n 

e−ai /p