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Combinatorial Games with Restricted Options Under Normal and Misere Play

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Combinatorial Games with Restricted Options Under Normal and Misere Play Combinatorial Games with Restricted Options under Normal and Misere Play by Paul Ottaway Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Dalhousie University Halifax, Nova Scotia June 2009 © Copyright by Paul Ottaway, 2009 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de Tedition 395 Wellington Street 395, rue Wellington OttawaONK1A0N4 OttawaONK1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-56441-7 Our file Notre rSterence ISBN: 978-0-494-56441-7 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la lor canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. 1*1 Canada DALHOUSIE UNIVERSITY To comply with the Canadian Privacy Act the National Library of Canada has requested that the following pages be removed from this copy of the thesis: Preliminary Pages Examiners Signature Page (pii) Dalhousie Library Copyright Agreement (piii) Appendices Copyright Releases (if applicable) Table of Contents List of Tables vi List of Figures viii Abstract ix Acknowledgements x Chapter 1 Theory of Combinatorial Games 1 1.1 Overview 1 1.2 Properties 4 1.3 Game Representations 4 1.4 Outcome Classes 5 1.5 Structure of Games 8 1.6 NIM 15 1.7 Subtraction Games 19 Chapter 2 Option-Closed Games 22 2.1 Definitions and Examples 22 2.2 Reduced Canonical Form 23 2.3 ROLL THE LAWN 29 2.4 CRICKET PITCH 30 Chapter 3 Recent Results in Misere Play 35 3.1 Introduction 35 3.2 The Disjunctive Sum of Two Misere Games 36 3.3 Equivalence Classes of Misere Games 38 3.4 END-NIM 44 iv Chapter 4 Consecutive Move Ban 47 4.1 Motivation and Definitions 47 4.2 Normal Play 48 4.3 Misere Play 50 4.4 Two-handed Games 59 4.5 A Coin-Flipping Game 66 Chapter 5 The Short Disjunctive Sum 74 5.1 Definitions 74 5.2 Results 75 5.3 Regular Games 80 5.4 CLOBBER 87 5.5 MAZE and MAIZE 89 Chapter 6 Discussion 92 6.1 Option Closed Games 92 6.2 Misere games 92 6.3 Consecutive Move Ban 93 6.4 Short Disjunctive Sum 93 Appendix A Tables 95 Bibliography 100 v List of Tables Table 2.1 A short cricket pitch dictionary 31 Table 3.1 Comparison of normal and misere outcome classes 37 Table 4.1 Examples from each of the eight classes of one-handed games . 52 Table 4.2 Examples of all possible two-handed games 59 Table 4.3 The outcome class of a CMB game denoted by [a,b,c] 61 Table 4.4 The outcome of a CMB game denoted by [a, b, c, d, e] with (c — b) even and d = 0 63 Table 4.5 The outcome of a CMB game denoted by [a, 6, c, d, e] with (c — b) even and d = 1 63 Table 4.6 The outcome of a CMB game denoted by [a, 6, c, d, e] with (c — b) odd and d = 0 . 63 Table 4.7 The outcome of a CMB game denoted by [a, b, c, d, e] with (c — b) odd and d = 1 64 Table 4.8 The effect of a player's move on score and control values in a CMB game (part 1) 67 Table 4.9 The effect of a player's move on score and control values in a CMB game (part 2) 68 Table 4.10 The period lengths for CMB games with SL = SR = {a, 6, a + 6} 70 Table 4.11 The CMB class sequence for SL = SR = {2, 4, 7} 70 Table 4.12 The CMB class sequence for SL = SR = {3, 5, 8} 70 Table 4.13 The CMB class sequence for S^ = SR = {any subset of odd numbers} 71 Table 4.14 Periods and pre-periods for various subtraction sets 71 Table 4.15 The CMB class sequence for SL = {1, 2, 3}, SR = {2, 3, 4} . 72 Table 4.16 The CMB class sequence for SL = {1, 2, 4}, SR = {1, 3, 4} ... 73 Table 4.17 The CMB class sequence for SL = {1, 8}, SR = {3, 5} 73 Table 4.18 The CMB class sequence for SL = {1, 2, 5}, SR = {2, 3, 4} . 73 vi Table 5.1 The possible outcome classes of the short misere sum G + H . 77 Table 5.2 Games played with a short misere sum which are born by day 1 85 Table 5.3 A brief dictionary of misere clobber positions 88 Table A.l Examples of all possible outcomes of a misere sum (part 1) . 96 Table A.2 Examples of all possible outcomes of a misere sum (part 2) . 97 Table A.3 Examples of all possible outcomes of a short misere sum .... 98 Table A.4 A partial dictionary of Normal Play game values 99 vn List of Figures Figure 1.1 A typical game tree 5 Figure 1.2 Normal Play outcome analysis of a game tree 8 Figure 1.3 Misere Play outcome analysis of a game tree 9 Figure 1.4 The partial order of outcome classes 9 Figure 1.5 Normal Play value of a game tree 15 Figure 1.6 Example of the nim-sum operation 17 Figure 1.7 Finding a winning move in a game of NIM with four piles ... 18 Figure 1.8 The nim-sequence for S = {1, 3,4} 20 Figure 1.9 The sequence of values for SL = {1},SR = {2, 3} ....... 21 Figure 2.1 A game of ROLL THE LAWN 23 Figure 2.2 A game of CRICKET PITCH 23 Figure 3.1 Outcome analysis of NIM with one heap of size 2 35 Figure 4.1 Imposing a consecutive move ban 48 Figure 4.2 A consecutive move ban applied to a sum of components ... 50 Figure 5.1 A sample misere game with value 002 79 Figure 5.2 The games G and T(G) 81 Figure 5.3 Finding regular K where T(G) + T(H) = T(K) 83 Figure 5.4 The partial order of games played with a short misere sum born by day 1 86 Figure 5.5 A game of MAZE 89 Figure 5.6 Outcome analysis of misere MAZE 90 Figure 5.7 Values for MAZE played with a short misere sum 90 Figure 5.8 A game of MAIZE with T(G) = oo4 91 vm Abstract This thesis presents new results with respect to the analysis of combinatorial games played under both Normal and Misere Play ending conditions. We begin by demon­ strating how the concept of the reduced canonical form of a game can be applied to the analysis of option closed games such as CRICKET PITCH and ROLL THE LAWN. We continue with Misere Play games and show why their analysis is much more difficult than that of their Normal Play counterparts. In particular, we show that they do not form a group and admit no partial order which preserves outcome classes. In an effort to understand more regarding Misere Play games, we consider a specific restriction where a given player cannot play twice in the same component of a sum unless their opponent plays between those moves. This restricts the universe of different games to a finite number and we present the analysis of arbitrary sums under both Normal and Misere play rules. Finally, we consider the short disjunctive sum of games which is an alternate interpretation of disjunctive sum which is consistent with the traditional Normal Play interpretation but different when considering Misere Play games. Using this approach, we show that overriding moves are possible and develop a system for dealing with them in an arbitrary sum. We also show how the short disjunctive sum affects familiar games such as MAZE, MAIZE and CLOBBER when played using Misere Play rules. IX Acknowledgements I would like to thank Richard Nowakowski for his friendship, advice and support. I would also like to thank my family, friends and especially my wife, Heather, whose love has seen me through and made this possible. x Chapter 1 Theory of Combinatorial Games 1.1 Overview Combinatorial games have been formally studied for over a hundred years but have only recently become more popular since the publication of Winning Ways [4] and On Numbers and Games [8].
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