Table Of ContentGame Theory
A Classical Introduction,
Mathematical Games and the Tournament
Andrew McEachern
Queen’sUniversity
SYNTHESISLECTURESONGAMESAND
COMPUTATIONAL INTELLIGENCE#1
M
&C Morgan&cLaypool publishers
Copyright©2017byMorgan&Claypool
GameTheory:AClassicalIntroduction,MathematicalGames,andtheTournament
AndrewMcEachern
www.morganclaypool.com
ISBN:9781681731582 paperback
ISBN:9781681731599 ebook
DOI10.2200/S00777ED1V01Y201705GCI001
APublicationintheMorgan&ClaypoolPublishersseries
SYNTHESISLECTURESONGAMESANDCOMPUTATIONALINTELLIGENCE
Lecture#1
SeriesEditor:DanielAshlock,UniversityofGuelph
SeriesISSN
ISSNpending.
ABSTRACT
Thisbookisaformalizationofcollectednotesfromanintroductorygametheorycoursetaught
at Queen’s University. The course introduced traditional game theory and its formal analysis,
butalsomovedtomoremodernapproachestogametheory,providingabroadintroductionto
the current state of the discipline. Classical games, like the Prisoner’s Dilemma and the Lady
andtheTiger,arejoinedbyaprocedurefortransformingmathematicalgamesintocardgames.
Includedisanintroductionandbriefinvestigationintomathematicalgames,includingcombi-
natorialgamessuchasNim.Thetextexaminestechniquesforcreatingtournaments,ofthesort
used in sports, and demonstrates how to obtain tournaments that are as fair as possible with
regardstoplayingoncourts.Thetournamentsaretestedasin-classlearningevents,providinga
novelcurriculumitem.Exampletournamentsareprovidedattheendofthebookforinstructors
interestedinrunningatournamentintheirownclassroom.Thebookisappropriateasatextor
companiontextforaone-semestercourseintroducingthetheoryofgamesorforstudentswho
wishtogetasenseofthescopeandtechniquesofthefield.
KEYWORDS
classicalgametheory,combinatorialgames,tournaments,deck-basedgames,tour-
namentdesign,graphgames
Contents
Preface ........................................................... xi
1 Introduction:ThePrisoner’sDilemmaandFiniteStateAutomata ...........1
1.1 Introduction ..................................................... 1
1.1.1 HelpfulSourcesofInformation ................................ 2
1.2 ThePrisoner’sDilemma............................................ 2
1.3 FiniteStateAutomata ............................................. 5
1.4 Exercises ........................................................ 8
2 GamesinExtensiveFormwithCompleteInformationandBackward
Induction ......................................................... 9
2.1 Introduction ..................................................... 9
2.2 TheLadyandtheTigerGame ..................................... 10
2.3 GamesinExtensiveFormwithCompleteInformation .................. 13
2.4 BackwardInduction .............................................. 14
2.5 TheUltimatumGame ............................................ 15
2.5.1 TheEnhancedUltimatumGame .............................. 16
2.5.2 WhatBackwardInductionSays ............................... 16
2.5.3 BackwardInductionisWrongAboutThisOne .................. 17
2.6 TheBoatCrashGame ............................................ 20
2.7 ContinuousGames............................................... 23
2.7.1 TwoModelsoftheStackelbergDuopology...................... 23
2.8 TheFailingsofBackwardInduction ................................. 27
2.9 Exercises ....................................................... 27
3 GamesinNormalFormandtheNashEquilibrium ......................31
3.1 IntroductionandDefinitions ....................................... 31
3.2 TheStagHunt .................................................. 31
3.3 DominatedStrategies............................................. 33
3.3.1 IteratedEliminationofDominatedStrategies.................... 33
3.4 TheNachoGame ................................................ 33
3.4.1 TheNachoGamewithKPlayers .............................. 34
3.5 NashEquilibria ................................................. 35
3.5.1 FindingtheNEbyIEDS .................................... 36
3.5.2 IEDSprocess.............................................. 36
3.6 TheVaccinationGame............................................ 37
3.6.1 TheN-playerVaccinationGame .............................. 39
3.7 Exercises ....................................................... 39
4 MixedStrategyNashEquilibriaandTwo-PlayerZero-SumGames.........43
4.0.1 TheFundamentalTheoremofNashEquilibria ................... 44
4.1 AnExamplewitha3-by-3PayoffMatrix............................. 45
4.2 Two-PlayerZeroSumGames ...................................... 47
4.2.1 TheGameofOddsandEvens ................................ 47
4.3 DominationofTwo-PlayerZeroSumGames ......................... 48
4.3.1 SaddlePoints.............................................. 48
4.3.2 SolvingTwo-by-TwoGames ................................. 49
4.4 Goofspiel ...................................................... 50
4.5 Exercises ....................................................... 50
5 MathematicalGames ...............................................53
5.1 Introduction .................................................... 53
5.1.1 TheSubtractionGame ...................................... 53
5.2 Nim........................................................... 55
5.2.1 Moore’sNim .............................................. 56
5.3 Sprouts ........................................................ 57
5.4 TheGraphDominationGame ..................................... 61
5.5 Deck-basedGames .............................................. 63
5.5.1 Deck-basedPrisoner’sDilemma............................... 64
5.5.2 Deck-basedRock-Paper-Scissors(-Lizard-Spock) ................ 66
5.5.3 Deck-basedDividetheDollar ................................ 68
5.5.4 FLUXX-likeGameMechanics ............................... 69
5.5.5 ANoteonAddingNewMechanicstoMathematicalGames ....... 72
5.6 Exercises ....................................................... 72
6 TournamentsandTheirDesign.......................................75
6.1 Introduction .................................................... 75
6.1.1 SomeTypesofTournaments ................................. 75
6.2 RoundRobinScheduling.......................................... 77
6.3 RoundRobinSchedulingwithCourts ............................... 78
6.3.1 BalancedTournamentDesigns................................ 78
6.3.2 Court-balancedTournamentDesigns .......................... 80
6.4 CyclicPermutationFractalTournamentDesign........................ 82
6.4.1 Case1:n 2k,k Z ..................................... 83
C
D 2
6.4.2 ARecursiveGenerationoftheMinimumNumberofRounds....... 88
6.5 Exercises ....................................................... 88
7 Afterword ........................................................91
7.1 ConclusionandFutureDirections................................... 91
A ExampleTournaments ..............................................93
A.1 ExampleTournaments ............................................ 93
A.1.1 TheEnhancedUltimatumGameTournament ................... 93
A.1.2 TheVaccinationGameTournament............................ 95
A.1.3 ADifferentKindofIPDTournament.......................... 97
A.2 SomeThingstoConsiderBeforeRunningaTournamentinaClassroom ... 98
Bibliography .....................................................101
Preface
Thisbookisnothinglikeatypicalintroductorygametheorybook.Whileitofferssomeofthe
same material you would find there, it takes a very different approach in an attempt at being
accessible to students and readers from a variety of backgrounds. It is intended for students
in the arts or sciences who are interested in a mathematics course that exposes them to the
interesting parts of game theory without being strongly focused on rigor. This book does not
require a strong mathematics background to follow, but it is meant to be a way to learn the
conceptsinadditiontoattendingclass.Duetothebriefnatureofthisbook,therewillbemany
parts that seem incomplete, or rather, like they could be expanded on a great deal. Attending
class is the best way to fill those gaps. This book also discusses many topics that have never
beenconsideredbygametheoryresearchers.Theremaybesomepeoplewhodisagreewithmy
inclusionofthesetopics,buttothemIsaythatthisbookisabouttakingastepinanewdirection
tobringsomelifeandinnovationtothegametheoryfield.Thechaptersonmathematicalgames
havebeen included because they encouragemathematical thinking while being accessible,and
becauseplayingactualgamesisfun.I’vealwaysthoughtclassicalgametheoryhassufferedfrom
not including the playing of games, but rather treating games as mathematical objects to be
analyzed, categorized, and then placed on the shelf. Finally, the mathematics of tournament
designisasubjectthathasbeenburieddeepinthecombinatoricsliterature.Itistimetobring
ittothelight,applyingitininterestingwaystocreatetournaments,andthenconductingthose
tournaments in all of their optimized glory. Competition is such a driving force in so many
aspectsofourlives,includingplayingsimplegames,thatitseemswrongsomehowthatwedonot
applyourmathematicaltechniquestoimprovingit.Asweapplythesemathematicaltechniques
to sharpening competition, competition sharpens our mathematical thinking, creating a cycle
thatcouldbringbenefitsofwhichwearenotyetaware.Thisbookisthefirststepinthedirection
ofusinggamestobroadenourintellectanddriveustohigherperformance.Muchofthisbook
featurestopicsfromactiveresearch,whichisanotherreasonwhyitdivergesratherstronglyfrom
atypicalgametheorytext.
AndrewMcEachern
May2017
C H A P T E R 1
Introduction: The Prisoner’s
Dilemma and Finite State
Automata
1.1 INTRODUCTION
Game theory is a multidisciplinary field that has been pursued independently in economics,
mathematics,biology,psychology,anddiversedisciplineswithincomputerscience(artificialin-
telligence,theoryofcomputerscience,videogamedevelopment,andhumancomputerinterface,
tonamejustafew).BeginningwiththeworkofJohnVonNeumannandOskarMorgenstern,
TheoryofGamesandEconomicBehavior [8],gametheoryhasfounditswayintoseveralsubjects.
Whyshouldthisbethecase,ifit’sinitialfocuswasonproblemsineconomics?
It may help if we define what we mean when we say game. After scouring both texts
and the Internet, I found many similar definitions of game theory, but none which captured
theessenceofthegame partofgametheory.Ioffermyowndefinition;agameisanysituation
involvingmorethanoneindividual,eachofwhichcanmakemorethanoneaction,suchthatthe
outcometoeachindividual,calledthepayoff,isinfluencedbytheirownaction,andthechoiceof
actionofatleastoneotherindividual.Onewaytothinkaboutgametheoryasafieldisthatitis
acollectionoftheoriesandtechniquesthathelpusthinkabouthowtoanalyzeagame.Taking
amomenttoreflectonthisdefinition,itshouldbeclearwhygametheoryhasseenapplication
insomanydiversefields.
Thepurposeoftreatingsituationslikegamesalsochangesfromresearchareatoresearch
area.Ineconomics,biology,andpsychology,thepurposeoftreatingsituationsinvolvingagents
withpossiblydifferinggoalsistopredicthumanoranimalbehavior.Bysolvingthegame,and
whatthatmeanswillbedefinedlaterinthischapter,aresearchermaybelievetheyhaveagood
idea of how an organism (animals, people, or companies) will behave when confronted with a
situationsimilartotheirgame.Foramathematician,thesolvingofagameusuallyreferstoan
impartialcombinatorialgame,asyouwillseeinChapter5,orprovingatheoremaboutaclass
of games that gives an overarching idea about what properties they may have. For a computer
scientist interested in artificial intelligence, games are an excellent way to study evolution of
competitiveorcooperativeAIunderveryspecificconditions.
2 1. INTRODUCTION:THEPRISONER’SDILEMMAANDFINITESTATEAUTOMATA
1.1.1 HELPFULSOURCESOFINFORMATION
Depending on the prerequisites of this course, it is possible that you have not been exposed to
muchofthebackgroundmaterialnecessarytosucceedinfullyunderstandingthemainideasin
thiscourse.Herearealistofwebsitesthateithercontainthesupportingmaterial,orwillatleast
pointyouintherightdirection.
• If you are not familiar with basic probability, or would like to brush up,
go to https://www.khanacademy.org/math/probability/probability-geometry
/probability-basics/v/basic-probabilityisanexcellentwebsite.
• Following on basic probability, if you are totally unfamiliar with decision theory, go to
http://people.kth.se/~soh/decisiontheory.pdftogetabasicintroduction.
• Togeta handle onthe Gaussian, or Normal, distribution,goto http://www.statisti
cshowto.com/probability-and-statistics/normal-distributions
• If it has been a while since you’ve seen calculus, mostly with regards to derivatives and
partialderivatives,http://tutorial.math.lamar.edu/maybethebestfreecollection
ofcalculusnotesIhaveseen.
• Ifyouareunfamiliarwiththeconceptofnoisewithregardstosignals, https://en.wik
ipedia.org/wiki/Noise_(signal_processing)isagoodplacetostart.
• For help with basic combinatorics and introductory graph theory concepts, you should
take a look at Chapters 1 and 4 of https://www.whitman.edu/mathematics/cgt_on
line/cgt.pdf.
1.2 THEPRISONER’SDILEMMA
WebeginourstudyofclassicalgametheorywithadiscussionofthePrisoner’sDilemma,since
itispossiblythemoststudiedgameacrossseveralareasinthefieldofgametheory.Thisislikely
due to the fact that it is fairly simple, and it can model a broad range of situations. To list all
ofthesituationsthePrisoner’sDilemmahasbeenusedtomodelisnotfeasible,asearchinthe
field in which you are interested can easily give you an idea of how often this game has been
used.Thisbringsupaninterestingpointtoconsider,whichweshouldkeepinmindfortherest
ofthetimewearestudyinggametheory:GeorgeBox,astatistician,wrotethat“essentially,all
modelsarewrong,butsomeareuseful.”Anytimewearemodelingasituationwithagame,we
havetomakesomesimplifyingassumptions.Whenwedothis,weloseinformationaboutthat
situation.Therearetimeswhengameshavebeenusedtomodelasituationinvolvingpeople,and
those games have entirely failed to predict how humans will behave. This is not to say that we
shouldneverusegamesasmodels,fortheyareuseful,butwemustbeawareoftheirlimitations.
The original Prisoner’s Dilemma revolves around the following story. Two criminals ac-
cused of burglary are being held by the authorities. The authorities have enough evidence to
1.2. THEPRISONER’SDILEMMA 3
convict either criminal of criminal trespass (a minor crime), but do not have enough evidence
toconvictthemofburglary,whichcarriesamuchheavierpenalty.Theauthoritiesseparatethe
accomplicesindifferentroomsandmakethefollowingoffertobothcriminals.Theauthorities
willdropthecriminaltrespasschargeandgiveimmunityfromanyself-incriminatingevidence
given,aslongasthesuspectgivesuphisorherpartner.Fourpossibleoutcomesresult.
1. Both suspects keep quiet, serve a short sentence for criminal trespassing, and divide the
money.
2. One suspect testifies against the other, but his or her partner says nothing. The testifier
goesfreeandkeepsallofthe money,whiletheonewhokeptquiet servesamuchlonger
sentence.
3. Theviceversaofpossibility2.
4. Both suspects give the other one up in an attempt to save themselves. They both receive
moderatesentencesand,assumingtheydon’tkilleachotherinprison,whentheygetout
theystillhaveachancetosplitthemoney.
It is convenient to quantify the outcomes as payoffs to either player. Since there are two
choicesforeachcriminal,wecallthefirstchoiceofnotspeakingwiththeauthoritiescooperate,
or C. There is a mismatch of terminology here, since we would think cooperate would involve
cooperating with the authorities, but instead think of it as cooperating with the other person
involved in the game, namely, your partner in crime. The other choice, which is testify against
yourpartner,iscalleddefect,orD.Wecancreatematrixtorepresentthesepayoffswithrespect
tothechoicesofbothcriminals.
Criminal 2
C D
Criminal 1 C (3,3) (0,5)
D (5,0) (1,1)
Figure1.1: ThepayoffmatrixforthePrisoner’sDilemma.
FromFigure1.1,wehavequantifiedthesituationasfollows:Ifbothcriminalskeepquiet,
or cooperate, they both receive a payoff of 3. If Criminal 1 defects and Criminal 2 cooperates,
then Criminal 1 receives a payoff of 5 and Criminal 2 receives a payoff of 0. If the choices are
reversed,soarethepayoffs.Ifbothattempttostabeachotherintheback,defect,thepayoffto
eachis1.FromnowonwewillcallthedecisionmakersinthePrisoner’sDilemmaplayers.Any
gamewithtwoplayersthathasthefollowingpayoffmatrixasdisplayedinFigure1.2iscalleda
Prisoner’sDilemma.C isthepayoffwhenbothplayerscooperate,S iscalledthesuckerpayoff,
T iscalledthetemptationpayoff,andDisthemutualdefectpayoff.
