Lessons in play : an introduction to combinatorial game theory

346 Pages·2019·7.688 MB·English

by  Richard J. Nowakowski, Michael H. Albert, David Wolfe

Lessons in Play
Second Edition

Lessons in Play
An Introduction to
Combinatorial Game Theory
Second Edition
Michael H. Albert
Richard J. Nowakowski
David Wolfe
CRCPress
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Library of Congress Cataloging-in-PublicationData
Names: Albert,MichaelH.,author. |Nowakowski,RichardJ.,author. |Wolfe,David,
1964- author.
Title: Lessonsinplay: anintroductiontocombinatorialgametheory/MichaelAlbert,
RichardNowakowski,DavidWolfe.
Description: Secondedition. |BocaRaton,Florida: CRCPress,[2019]|
Includes bibliographicalreferencesandindex.
Identifiers: LCCN2018060986| ISBN9781482243031 (hardback: alk. paper)|
ISBN9781482243048 (ebook: alk. paper)
Subjects: LCSH:Gametheory--Textbooks. |Combinatorialanalysis--Textbooks.
Classification: LCCQA269.A4252019|DDC519.3--dc23
LCrecordavailableathttps://lccn.loc.gov/2018060986
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Contents
Preface xi
0 CombinatorialGames 1
0.1 BasicTerminology 3
Problems 7
1 BasicTechniques 11
1.1 Greedy 11
1.2 Symmetry 14
1.3 Parity 16
1.4 GiveThemEnoughRope! 16
1.5 StrategyStealing 17
1.6 ChangetheGame! 20
1.7 CaseStudy:LongChainsinDots&Boxes 21
Problems 29
2 OutcomeClasses 35
2.1 OutcomeFunctions 36 ∗
2.2 GamePositionsandOptions 37
2.3 ImpartialGames:MindingYourPsandNs 42
2.4 CaseStudy:RolltheLawn 45 ∗
2.5 CaseStudy:Timber 47 ∗
2.6 CaseStudy:PartizanEndnim 51
Problems 54
3 MotivationalInterlude:SumsofGames 59
3.1 Sums 59
3.2 Comparisons 61
3.3 EqualityandIdentity 66
3.4 CaseStudy:DomineeringRectangles 68
Problems 71
vii
viii Contents
4 TheAlgebraofGames 75
4.1 TheFundamentalDefinitions 75
4.2 GamesFormaGroupwithaPartialOrder 84
4.3 CanonicalForm 89
4.4 CaseStudy:CricketPitch 96 ∗
4.5 Incentives 99
Problems 100
5 ValuesofGames 105
5.1 Numbers 105
5.2 CaseStudy:Shove 114 ∗
5.3 Stops 116
5.4 AFewAll-Smalls: Up,Down,andStars 118
5.5 Switches 124
5.6 CaseStudy:Elephants&Rhinos 125 ∗
5.7 TinyandMiny 127
5.8 CaseStudy:TopplingDominoes 128
5.9 ProofsofEquivalenceofGamesandNumbers 131
Problems 133
6 Structure 139
6.1 GamesBornbyDay2 139
6.2 ExtremalGamesBornbyDayn 141
6.3 MoreAboutNumbers 146
6.4 TheDistributiveLatticeofGamesBornbyDayn 148
6.5 GroupStructure 150
Problems 151
7 ImpartialGames 155
7.1 AStar-StuddedGame 156
7.2 TheAnalysisofNim 158
7.3 AddingStars 160
7.4 AMoreSuccinctNotation 161
7.5 Taking-and-BreakingGames 163
7.6 SubtractionGames 165
7.7 KeypadGames 175 ∗
Problems 178
8 HotGames 181
8.1 ComparingGamesandNumbers 182
8.2 CopingwithConfusion 185
8.3 CoolingThingsDown 188
Contents ix
8.4 StrategiesforPlayingHotGames 195
8.5 NortonProducts 197
8.6 DomineeringSnakes 200
Problems 202
9 All-SmallGames 207
9.1 CastofCharacters 208
9.2 Motivation:TheScaleofUps 215
9.3 EquivalenceUnder 216
(cid:6)
9.4 AtomicWeight 219
9.5 All-SmallShove 223
9.6 MoreTopplingDominoes 224
9.7 Clobber 225
Problems 229
10TrimmingGameTrees 233
10.1Introduction 233 ∗
10.2ReducedCanonicalForm 234 ∗
10.3Hereditary-TransitiveGames 237 ∗
10.4Maze 238
10.5OrdinalSum 239 ∗
10.6Stirling-Shave 242 ∗
10.7EvenMoreTopplingDominoes 246 ∗
Problems 248
ω FurtherDirections 251
ω.1 TransfiniteGames 251
ω.2 AlgorithmsandComplexity 252
ω.3 LoopyGames 253
ω.4 Kos:RepeatedLocalPositions 254
ω.5 Top-DownThermography 254
ω.6 EnrichedEnvironments 255
ω.7 Idempotents 255
ω.8 Mise`rePlay 255
ω.9 ScoringGames 256 ∗
A Top-DownInduction 259
A.1 Top-DownInduction 259
A.2 Examples 260
A.3 WhyIsTop-DownInductionBetter? 263
A.4 StrengtheningtheInductionHypothesis 265
A.5 InductiveReasoning 267
Problems 268