Combinatorial games : tic-tac-toe theory

747 Pages·2008·3.036 MB·English

by  József Beck

Combinatorial Games
Traditionalgametheoryhasbeensuccessfulatdevelopingstrategyingamesofincomplete
information:whenoneplayerknowssomethingthattheotherdoesnot.Butithaslittleto
sayaboutgamesofcompleteinformation,forexampleTic-Tac-Toe,solitaire,andhex.This
isthesubjectofCombinatorialGameTheory.Mostboardgamesareachallengefor
mathematics:toanalyzeapositiononehastoexaminetheavailableoptions,andthenthe
furtheroptionsavailableafterselectinganyoption,andsoon.Thisleadstocombinatorial
chaos,wherebruteforcestudyisimpractical.
Inthiscomprehensivevolume,JózsefBeckshowsreadershowtoescapefromthe
combinatorialchaosviathefakeprobabilisticmethod,agame-theoreticadaptationofthe
probabilisticmethodincombinatorics.Usingthis,theauthorisabletodeterminetheexact
resultsaboutinfiniteclassesofmanygames,leadingtothediscoveryofsomestrikingnew
dualityprinciples.
József Beck isaProfessorintheMathematicsDepartmentofRutgersUniversity.He
hasreceivedtheFulkersonPrizeforresearchinDiscreteMathematicsandhaswritten
around100researchpublications.Heistheco-author,withW.L.Chen,ofthepioneering
monographIrregularitiesofDistribution.
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50 F.BorceuxHandbookofCategoricalAlgebraI
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52 F.BorceuxHandbookofCategoricalAlgebraIII
53 V.F.KolchinRandomGraphs
54 A.KatokandB.HasselblattIntroductiontotheModernTheoryofDynamicalSystems
55 V.N.SachkovCombinatorialMethodsinDiscreteMathematics
56 V.N.SachkovProbabilisticMethodsinDiscreteMathematics
57 P.M.CohnSkewFields
58 R.GardnerGeometricTomography
59 G.A.Baker,Jr.,andP.Graves-MorrisPadéApproximants,2ndedn
60 J.KrajicekBoundedArithmetic,PropositionalLogic,andComplexityTheory
61 H.GroemerGeometricApplicationsofFourierSeriesandSphericalHarmonics
62 H.O.FattoriniInfiniteDimensionalOptimizationandControlTheory
63 A.C.ThompsonMinkowskiGeometry
64 R.B.BapatandT.E.S.RaghavanNonnegativeMatriceswithApplications
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66 D.Cvetkovic,P.RowlinsonandS.SimicEigenspacesofGraphs
67 F.Bergeron,G.LabelleandP.LerouxCombinatorialSpeciesandTree-LikeStructures
68 R.GoodmanandN.WallachRepresentationsandInvariantsoftheClassicalGroups
69 T.Beth,D.Jungnickel,andH.LenzDesignTheory,I,2ndedn
70 A.PietschandJ.WenzelOrthonormalSystemsforBanachSpaceGeometry
71 G.E.Andrews,R.AskeyandR.RoySpecialFunctions
72 R.TicciatiQuantumFieldTheoryforMathematicians
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74 I.LasieckaandR.TriggianiControlTheoryforPartialDifferentialEquationsI
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87 B.MagurnAlgebraicIntroductiontoK-Theory
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100 E.OlivieriandM.EuláliaVaresLargeDeviationsandMetastability
102 L.W.Beinekeetal.(eds.)TopicsinAlgebraicGraphTheory
103 O.StaffansWell-PosedLinearSystems
105 M.LothaireAppliedCombinatoricsonWords
106 A.MarkoeAnalyticTomography
107 P.A.MartinMultipleScattering
108 R.A.BrualdiCombinatorialMatrixClasses
110 M.-J.LaiandL.L.SchumakerSplineFunctionsonTriangulations
111 R.T.CurtisSymmetricGenerationofGroups
112 H.Salzmann,T.Grundhöfer,H.HählandR.LöwenTheClassicalFields
113 S.PeszatandJ.ZabczykStochasticPartialDifferentialEquationswithLévyNoise
Combinatorial Games
Tic-Tac-Toe Theory
JÓZSEF BECK
Rutgers University
CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
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© Cambridge University Press 2008
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2008
A catalogue record for this publication is available from the British Library
ISBN 978-0-521-46100-9 hardback
Transferred to digital printing 2009
Cambridge University Press has no responsibility for the persistence or
accuracy of URLs for external or third-party Internet websites referred to in
this publication, and does not guarantee that any content on such websites is,
or will remain, accurate or appropriate. Information regarding prices, travel
timetables and other factual information given in this work are correct at
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the accuracy of such information thereafter.
Dedicated to
my mother who taught me how to play Nine Men’s Morris (“Mill”)

Contents
Preface page xi
A summary of the book in a nutshell 1
PART A WEAK WIN AND STRONG DRAW 15
Chapter I Win vs. Weak Win 17
1 Illustration: every finite point set in the plane is a
Weak Winner 19
2 Analyzing the proof of Theorem 1.1 32
3 Examples: Tic-Tac-Toe games 42
4 More examples: Tic-Tac-Toe like games 59
5 Games on hypergraphs, and the combinatorial chaos 72
ChapterII Themainresult:exactsolutionsforinfiniteclassesof
games 91
6 Ramsey Theory and Clique Games 92
7 Arithmetic progressions 106
8 Two-dimensional arithmetic progressions 118
9 Explaining the exact solutions: a Meta-Conjecture 131
10 Potentials and the Erdo˝s–Selfridge Theorem 146
11 Local vs. Global 163
12 Ramsey Theory and Hypercube Tic-Tac-Toe 172
PART B BASIC POTENTIAL TECHNIQUE – GAME-
THEORETIC FIRST AND SECOND MOMENTS 193
Chapter III Simple applications 195
13 Easy building via Theorem 1.2 196
14 Games beyond Ramsey Theory 204
15 A generalization of Kaplansky’s game 216
vii
viii Contents
Chapter IV Games and randomness 230
16 Discrepancy Games and the variance 231
17 Biased Discrepancy Games: when the extension from fair to
biased works! 245
18 A simple illustration of “randomness” (I) 260
19 A simple illustration of “randomness” (II) 270
20 Another illustration of “randomness” in games 286
PART C ADVANCED WEAK WIN – GAME-THEORETIC
HIGHER MOMENT 305
Chapter V Self-improving potentials 307
21 Motivating the probabilistic approach 308
22 Game-theoretic second moment: application to the
Picker–Chooser game 320
23 Weak Win in the Lattice Games 329
24 Game-theoretic higher moments 340
25 Exact solution of the Clique Game (I) 352
26 More applications 362
27 Who-scores-more games 372
Chapter VI What is the Biased Meta-Conjecture, and why is it
so difficult? 380
28 Discrepancy games (I) 381
29 Discrepancy games (II) 392
30 Biased Games (I): Biased Meta-Conjecture 400
31 Biased games (II): Sacrificing the probabilistic intuition to force
negativity 418
32 Biased games (III): Sporadic results 430
33 Biased games (IV): More sporadic results 439
PARTD ADVANCEDSTRONGDRAW–GAME-THEORETIC
INDEPENDENCE 459
Chapter VII BigGame–SmallGame Decomposition 461
34 The Hales–Jewett Conjecture 462
35 Reinforcing the Erdo˝s–Selfridge technique (I) 470
36 Reinforcing the Erdo˝s–Selfridge technique (II) 479
37 Almost Disjoint hypergraphs 485
38 Exact solution of the Clique Game (II) 492
Contents ix
Chapter VIII Advanced decomposition 504
39 Proof of the second Ugly Theorem 505
40 Breaking the “square-root barrier” (I) 525
41 Breaking the “square-root barrier” (II) 536
42 Van der Waerden Game and the RELARIN technique 545
Chapter IX Game-theoretic lattice-numbers 552
43 Winning planes: exact solution 553
44 Winning lattices: exact solution 575
45 I-Can-You-Can’t Games – Second Player’s Moral Victory 592
Chapter X Conclusion 610
46 More exact solutions and more partial results 611
47 Miscellany (I) 620
48 Miscellany (II) 634
49 Concluding remarks 644
Appendix A Ramsey Numbers 658
Appendix B Hales–Jewett Theorem: Shelah’s proof 669
Appendix C A formal treatment of Positional Games 677
Appendix D An informal introduction to game theory 705
Complete list of the Open Problems 716
What kinds of games? A dictionary 724
Dictionary of the phrases and concepts 727
References 730