EXPECTED WORTH FOR 2 × 2 MATRIX GAMES WITH VARIABLE GRID SIZES

By

Michael R. Powers, Martin Shubik, and Wen Wang

May 2016

COWLES FOUNDATION DISCUSSION PAPER NO. 2039

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY

Box 208281 New Haven, Connecticut 06520-8281

http://cowles.yale.edu/

1

ExpectedWorthfor2×2MatrixGameswithVariableGridSizes

MichaelR.Powers,1MartinShubik,2andWenWang3

May17,2016

Abstract

Weoffer adetailed examinationof a broad class of 2× 2matrix games as a first step towardconsideringmeasuresofresourcedistributionandefficiencyofoutcomes.Inthepresentessay,onlynoncooperativeequilibriaandentropicoutcomesareconsidered,andacrudemeasureofefficiencyemployed.Othersolutionconceptsandtheformalconstructionofanefficiencyindexwillbeaddressedinacompanionpaper.

JELClassifications:C63,C72,D61Keywords:2×2matrixgames,efficiency,coordination,worthofcoordination.

12×2MatrixGameswithCardinalPayoffs

Inthefolkloreandelementarypedagogyofgametheory,the2×2matrixgameplaysaspecialrole.

Severalofthesegamesbearwell‐knownnames,suchasthePrisoner’sDilemma,StagHunt,andBattle

of theSexes.Although thereareonly144strategicallydifferent2×2gameswithstrictlyordinal

preferences,oneoftenisinterestedinconsideringrelatedgameswithcardinalpreferences,whose

number is unbounded. The present paper is devoted to addressing applications in which it is

desirabletoexaminealargebutfinitesetof2×2gameswithcardinalpreferences.

1.1 OutcomeSets

Ageneric2×2gameisdescribedbythematrixshowninTable1.Here,therowplayerhasthetwo

strategies,“Up”and“Down”(correspondingtorowsi=1,2,respectively),whereasthecolumnplayer

1 Zurich Group Professor of Risk Mathematics, School of Business and Management, and Professor, Schwarzman

ScholarsProgram,TsinghuaUniversity.2SeymourKnoxProfessorofMathematicalInstitutionalEconomics(Emeritus),YaleUniversity,andExternalFaculty,

SantaFeInstitute.3InternationalBusinessSchool,NankaiUniversity.

2

has “Left” and “Right” (corresponding to columns j= 1,2, respectively). This yields four possible

payoffpairs(outcomes),(ai,j,bi,j),fori=1,2andj=1,2.

Table1:AGeneric2×2MatrixGame

  Left(j=1) Right(j=2)

  Up(i=1) a1,1,b1,1 a1,2,b1,2

  Down(i=2) a2,1,b2,1 a2,2,b2,2

Theuniverseofallstrictlyordinalgamesiseasilydenumeratedbynotingthateachofthepayoff

vectors [a11,a12,a21,a22] and [b11,b12,b21,b22] must be permutations of the ordinal vector [1,2,3,4],

yielding a total of 4! × 4! = 576different outcomes.Thisnumber canbedividedby 2 to remove

duplicationsarisingfrominterchangingrows,andbyanother2toaccountforinterchangingcolumns,

leaving the canonical 144 strategically distinct games shown in Appendix 1. Topologically, the

outcomesetsofthesegamesmaybecharacterizedbyasmallersetof24distinctshapes,22ofwhich

aretwo‐dimensional(i.e.,gamesofopposition)andtheremaining2one‐dimensional(i.e.,gamesof

coordination).TheseshapesareshownindetailinAppendix1,wheretheyareassociatedwiththe

144games.

Inconsideringcardinalgames,weassumethatthepayoffpairs,(ai,j,bi,j),maybeexpressedinwell‐

definedunitsofmoneyorgold,withafixedminimalleveloffinenessthatcanbeperceivedand/or

traded.4Isthereanupperboundonhowlargeanindividualpayoffcanbe?Philosophically,onecould

argueineitherdirection;butforallpracticalpurposes,onecanimposealargeenoughupperbound

thatencompassesallpossibleobservationsforagivensociety.Wethereforeinvestigate

aclosedsetof2×2matrixgameswithpayoffsgivenbyelementsintheset ,

for k ∈ {1,2,...}, with a grid size of ∆ = 1/2k−1. Equivalently, one might choose the payoff set

,withagridsizeof∆=1.Althoughthelatterapproachoffersthesimplicityofan

easilycomprehensiblefixedgridsize,theformerprovidesbothaboundedmaximumpayoffsizeand

anintuitivelystraightforwardlimitingprocesstoassesstheimpactofgridsizeonplayerbehavior.

4Wedonotconcernourselveswithindividualpreferencesdirectly,butallowforthepossibilitythateachamountof

money/goldismappedontoindividualpreferencesinsomerisk‐aversemanner.

3

Intheremainderofthepaper,westudyvariouspropertiesofcardinalgamesarrangedinto144

categories associatedwith their corresponding strictlyordinal games.Our investigation relieson

both analytical and simulationmethods. In the latter case, we employ a computer program that

carriesoutthefollowingstepsforeachofgamesG=1,2,...,100,000,foragivenvaluek∈{1,2,...}:

1. Foreachofthefourcells,(i,j)=(1,1),(1,2),(2,1),(2,2),generatetwoindependentrandom

  variables,ai,j∼Uniform andbi,j∼Uniform ,

wherethefourpairs(ai,j,bi,j)aremutuallyindependent.

2. Ifeitherai,j=ai0,j0orbi,j=bi0,j0forany(i,j)=(i0,j0),thenrejectthegameandreturntostep(1).

3. Definethecardinal2×2gameGbythefourpayoffpairs(ai,j,bi,j).

4. Separatelyorderthefourai,jandfourbi,jfromlowesttohighest,andlet̃ ai,j=rank(ai,j)∈{1,2,3,4}

and˜bi,j=rank(bi,j)∈{1,2,3,4}forall(i,j).

5. Definetheordinal2×2gameGbythefourpayoffpairs ,andmatchthisgameto

oneofthe144canonicalstrictlyordinalgames.

Bysymmetry,weknowthatthenumberofordinalgamesgeneratedforeachofthe144canonical

formswillbeapproximatelythesame.

1.2 MassProperties

Thegenerationofa largenumberofdistinctcardinalgames,eachassociatedwithoneof the144

canonicalordinalgames,providesthemeanstoconsiderthemasspropertiesofseveralapproaches

togameplay.Asolutionistheoutcome(orsetofoutcomes)derivedbytheselectionofastrategyby

eachofthegame’splayers,andmaybebaseduponawidearrayofindividualplayercharacteristics.

Forthepresent,however,wewilllimitconsiderationto(1)noncooperative,individuallyoptimizing

players,and(2)entropyplayersselectingeachroworcolumnrandomly,withprobability1/2.

4

2 JointMaximumPayoffs

Givenasetofcardinalgamesgeneratedrandomly,asabove,itisnaturaltoconsiderthedistribution

ofthejointmaximumpayoff, ,foragivenk∈{1,2,...},andeasytoseethatthesamplespaceofJMkisgivenbythesetofvalues .

2.1 Distributionfork=1

For the caseofk=1,5this sample space is simply the setof integers {5,6,7,8}, and it isuseful to

associatethesejointmaximawitheachofthreegamecategories:(1)gamesofcoordination,forwhich

JM1=8;(2)mixed‐motivegames,forwhichJM1∈{7,6};and(3)gamesofopposition,forwhichJM1=

5.Forthisbaselinecase,onecanworkoutthedistributionofthejointmaximumasinTable2,from

which it is clear that JM1 is negatively skewed,withmean,median, andmode of 6.875, 7, and 7,

respectively.

Table2:DistributionofJM1

  Value #ofGames  8 36  7 60  6 42  5 6  Total 144

2.2 Distributionfork>1

Fork>1,thedistributionofJMkismorecomplex,butmuchcanbelearnedsimplybyconsideringthe

limitingcaseask→∞.Lettingai,jandbi,jbeindependentandidenticallydistributed(contin‐

uous)Uniform(0,4]randomvariablesforall(i,j),onecandefine ,andobservethatJM∞=max{X1,X2,X3,X4},whereX`∼i.i.d.Triangular[0,8];i.e.,

5Wenotethatfork=1,thesamplespaceoftherandomcardinalgamesisidenticaltothesetof144canonicalordinal

games.

5

x2/32

FX(x)=

−x2/32+x/2−1

forx∈[0,4].

forx∈(4,8]

Itthenfollowsthat

fory∈[0,4]

fory∈(4,8]

and

  4) fory∈[0,4].

2) fory∈(4,8]

Thisprobabilitydensityfunction,plottedinFigure1,showsthatthedistributionofJM∞isnegatively

skewed,withmean,median,andmodegivenapproximatelyby5.6968,5.7436,and

5.8619,respectively.3

Figure1:ProbabilityDensityFunctionofJM∞

7436; and Mode = argmax{fJM∞(y)}=y∈[0,8]

.The above analysis reveals that the shape of the distribution of the jointmaximum remains

negativelyskewedforlargevaluesofk,justasitisfork=1.Onenoteworthydifference,however,is

6

thatgamesofcoordinationandgamesofoppositionbecomelessandlessprobable,approachingsets

ofmeasurezeroask→∞.

3 NoncooperativeEquilibrium

Theconceptofnoncooperativeequilibriumhasexistedintheeconomicliteraturesincetheworkof

AugustinCournot (1836),butwasmathematically fully formalizedandgeneralizedby JohnNash

(1952).Wedefinetheoutcomeofa2×2matrixgameasanorderedpairofstrategies,(sR,sC),inwhich

the first element denotes the rowplayer’smethodof selecting a row (i∈ {1,2}), and the second

elementdenotesthecolumnplayer’smethodofselectingacolumn(j∈{1,2}).6Wefurtherdefinea

noncooperativeequilibriumasanoutcomeinwhicheachplayerhasnomotivationtochangehisor

herstrategy,giventheindicatedstrategyoftheotherplayer.Restrictingattentiontopurestrategies,

inwhicheachplayer’sdecisionconsistsofafixed(asopposedtorandom)choiceofroworcolumn,

onecanseethat( )constitutesapure‐strategynoncooperativeequilibrium(PSNE)if

andonlyif

i∗=argmax{ai,j∗}i∈{1,2}

and

j∗=argmax{bi∗,j}.j∈{1,2}

A simple illustration of PSNE is given by the ordinal Prisoner’s Dilemma of Table 3. If both

prisonersremainsilent,theneachwillbegivenonlyaminorpenalty(of3);however,ifoneconfesses

andtheotherdoesnot,thentheformerreceivesaverylightpenalty(of4),whereasthelatterreceives

amoreseverepenalty(of1)thanthepenaltyifbothhadconfessed(of2).Forthisgame,itiseasyto

confirmthatthestrategypair(Down,Right)formsaPSNEwithpayoffs(2,2).

Table3:Prisoner’sDilemma

  Left(“RemainSilent”) Right(“Confess”)  Up(“RemainSilent”) 3,3 1,4

6Inamatrixornormal‐formgame,anymoveisequivalenttoastrategybecausetheplayersfacenocontingencies.

7

  Down(“Confess”) 4,1 2,2

Oneof themost importantcontributionsofNash(1952)wastheextensionofnoncooperativeequilibrium from games with only pure strategies to games allowing each player to select aprobabilitydistributionoverhisorherpossiblechoices.Thus,insteadofjustthepure‐strategypairs,(sR,sC)=(i,j),wecanconsiderrandom‐strategypairs,(sR,sC)=(x,y),inwhichx∈(0,1)denotestherowplayer’s(non‐trivial)probabilityofselectingUp(i=1),andy∈(0,1)denotesthecolumnplayer’s(likewisenon‐trivial)probabilityofselectingLeft(j=1).Anoncooperativeequilibriumthatinvolvesrandomstrategiesisreferredtoasamixed‐strategynoncooperativeequilibrium(MSNE).Onecansolveforagame’sMSNEsfromthetwoconditions:

x∗=argmax{a1,1xy∗+a1,2x(1−y∗)+a2,1(1−x)y∗+a2,2(1−x)(1−y∗)}x∈(0,1)

(4.1)

y∗=argmax{b1,1x∗y+b1,2x∗(1−y)+b2,1(1−x∗)y+b2,2(1−x∗)(1−y)}. (4.2)y∈(0,1)

ThesimplegameofMatchingPennies,showninTable4,providesanintuitivelyreasonableuse

ofmixedstrategies.Forthemixed‐strategypair(x,y)=(1/2,1/2),eachplayerreceivesanexpected

payoffof0.However,ifeitherplayerselectedapurestrategy,thentheotherplayer’sbestresponse

wouldcausethefirstplayertolose1unitwithcertainty.7Althoughcointossesarecommonlyused

by individuals in certain decision‐making settings, the use ofmore complicatedmixed strategies

appearstodependheavilyonplayersophisticationandproblemcontext.

Table4:MatchingPennies

  Heads Tails  Heads 1,‐1 ‐1,1  Tails ‐1,1 1,‐1

3.1 PopulationofPSNEs

Inthestudyofmatrixgames,itisusefultoseparatePSNEsandMSNEsbecausetheformerdepend

onlyonpayoffordinalities,whereasthelatteraresensitivetocardinaldifferences.Forthesetof144

strictlyordinal2×2games,thedistributionofthenumberofPSNEsisgiveninTable5.

7Alternatively,onecouldconsideranordinalversionofthisgameinwhichthepayoffs−1and1arereplacedbythe

payoffranks1and2,respectively.Intheordinalgame,usingthemixedstrategy(x,y)=(1/2,1/2)giveseachplayerthesameopportunitytoreceive1or2.

8

Table5:DistributionofPSNEs

  Value #ofGames0 181 1082 18

  Total 144

3.2 PopulationofMSNEs

FromthelistinAppendix2,onecanseethatthesetof144canonicalgamescontains36gameswith

exactlyoneMSNE,andnogameswithmorethanoneMSNE.These36gamesmatchexactlywiththe

setofgameshavingeitherzeroortwoPSNEs,fromwhichitfollowsthatall144ordinalgamescontain

atleastonenoncooperativeequilibrium.

3.3 SomeEasyCalculations

Mostfeaturesandsolutionsof2×2matrixgames–whetherstrictlyordinalorcardinal–areeasyto

calculate. Indevelopingprocedures forsuchcalculations,dominantrowsand/orcolumnsplayan

importantrole.

Arow[column]inamatrixgameissaidtodominate(strictly)anotherrow[column]ifandonly

ifeachpayoffinthefirstrow[column]isgreaterthanthecorrespondingpayoffinthesecondrow

[column].Ina2×2game,itiswellknownthat:

• If a gamepossessesexactlyonedominant rowor column, then itmustpossessexactlyone

PSNE.

• Ifagamepossesseszerodominantrowsorcolumns, then itmustpossesszeroPSNEs(and

thereforeexactlyoneMSNE).

• Ifagamepossessesonedominantrowandonedominatingcolumn,thenitmaypossesseither

(a)exactlyonePSNEor(b)twoPSNEsandexactlyoneMSNE.

TheabovefactsenableustoconstructthefollowingalgorithmforcomputingallPSNEsandMSNEs:

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1. Checkeachrowandcolumnforthedominanceproperty,andletd∈{0,1,2}denotethetotal

numberofdominantrows/columns.

2. Ifd=1,thensolvefortheuniquePSNEbyidentifyingthecellinthedominantrow[column]

thathasthegreaterpayoffforthecolumn[row]player.

3. Ifd=0,thensolvefortheMSNEexplicitlyas

.

4. If d = 2, then solve for each of the two PSNEs using the method described in step (1)

immediatelyabove,andsolvefortheMSNEasinstep(2)immediatelyabove.(Thetwo

PSNEsalsocanbefoundascornersolutionsofthesystemofequationsin(4.1)and(4.2).)

4 PopulationsofThreeSpecialGames

InAppendix3,weprovidetheC++programusedtogeneratecardinal(andstrictlyordinal)gamesas

describedinsteps(1)through(5)ofSubsection1.1.Appendix4containsvarioussamplemeansand

variancesassociatedwith100,000cardinalgamesgeneratedwithagridsizeof∆=1/256(i.e.,fork

=9).

Inthepresentsection,weconsiderthepopulationsofthreespecialgamesdiscussedwidelyinthe

behavioralscienceliterature:

• Prisoner’sDilemma(game1ofAppendix2);

• StagHunt(game41ofAppendix2);and• BattleoftheSexes(game10ofAppendix2).

The commonnames of these games attach considerable context to the abstract payoff structure,

whichmayormaynotbejustified.Inparticular,thereislittleindicationthatanyonewhodoesnot

know these common nameswould associate themwith the specific games involved (see, e.g., I.

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PowersandShubik,1991).Nevertheless,thestructuralfeaturesofthesethreesettingsallowusto

illustrateseveralimportantaspectsof2×2games.

4.1 Prisoner’sDilemma

PossiblythemoststudiedofallgamesisthePrisoner’sDilemma,whosepopularityarisesatleastin

partbecauseitsordinalformistheonlygamewithinthecanonical144forwhich:(a)thereisaunique

PSNE, (Down,Right), that is strictlydominatedbyanother feasibleoutcome, (Up,Left); and (b)all

otheroutcomesareParetooptimal.Figure22ofAppendix1presentsthepayoffset forthiswell‐

knowngame.

InexploringthepopulationofcardinalPrisoner’sDilemmagamesforagivenchoiceofgridsize,

∆=1/2k−1, it ishelpful to thinkof theentiredomainofpossiblepayoffs, from thegamewith the

smallestpayoff values, inTable6, to thatwith the largestpayoff values, inTable7.Naturally, an

ordinaltreatmentofpreferencesrecognizesnodifferencebetweenthesetwogames.

Table6:Prisoner’sDilemma,SmallestPayoffs

  Left RightUpDown

Table7:Prisoner’sDilemma,LargestPayoffs

  Left Right

  Up 4

  Down 4

Askincreasesandthegridbecomesfiner,thepopulationofcardinalPrisoner’sDilemmagames

coversmoreandmoreoftheentiresquareinterval(0,4]2.Inallcases,thegamehasonlythesingle

PSNE(Down,Right),whosepayoffpairdependsontheparticularvaluesofai,jandbi,jgeneratedin

step(1)ofSubsection1.1.

Asabaseline,wenotethatfork=1,thePSNEpayoffpairisalways(2,2),whereasentropyplayers

do better on average, obtaining an expected payoff of (2.5,2.5). Figures 2 and 3 provide scatter

11

diagramsofpayoffpairsfrom100,000gamesfork=9and∆=1/256.Figure2showsthePSNEs,and

Figure3showsthecorrespondingentropicoutcomes.Ineachcase,thesample‐meanpayoffpairis

indicatedbyareddiamondnearthecenterofthepointcluster.

Figure2:Prisoner’sDilemma,NoncooperativeEquilibrium

Figure3:Prisoner’sDilemma,EntropyPlayers

4.2 StagHunt

TheStagHunt(Table8)possessestwoPSNEs,oneofwhich,(Up,Left),strictlydominatestheother,

(Down,Right).However, dependingonunderlyingassumptions, it canbe argued that the smaller

payoffpairsometimeswillbechosenbyrationalplayers.Inparticular,iftherow[column]player

believesthatthecolumn[row]playerwillchooseRight[Down]withprobabilitygreaterthan1/2,

thenheorshewillbemotivatedtochooseDown[Right].

Table8:StagHunt

  Left Right

12

  Up 4,4 1,3  Down 3,1 2,2

Fork=1,thetwoPSNEshavepayoffpairs(4,4)and(2,2),respectively,andtheMSNE,(x∗,y∗)=

(1/2,1/2),yieldsexpectedpayoffsof(2.5,2.5).Giventhevalueof(x∗,y∗),onecanseethattheMSNE

yieldsidenticalstrategiesandpayoffsasthegamewithentropyplayers.

Figures4and5providescatterdiagramsofpayoffpairsfrom100,000gamesfork=9and∆=

1/256.Figure4includesthePSNEsandMSNE,eachwithequalfrequency,andFigure5showsthe

corresponding entropic outcomes. As before, the sample‐mean payoff pairs are indicated by red

diamondsnearthecentersofthepointclusters.

Figure4:StagHunt,NoncooperativeEquilibrium

Figure5:StagHunt,EntropyPlayers

13

4.3 BattleoftheSexes

UnlikethePrisoner’sDilemmaandStagHunt,theBattleoftheSexes(Table9)isnotasymmetric

game(i.e.,ai,j6=bj,iforsome(i,j)).LiketheStagHunt,however,itpossessestwoPSNEs,(Up,Left)and

(Down,Right),and1MSNE,(x∗,y∗)=(1/2,1/2).

Table9:BattleoftheSexes

  Left Right  Up 4,3 2,2  Down 1,1 3,4

Fork=1, the twoPSNEshavepayoff pairs (4,3) and (3,4), respectively, and theMSNEgives

expectedpayoffsof(2.5,2.5).Thus,asinthecaseoftheStagHunt,thegamewithentropyplayers

yieldsidenticalstrategiesandpayoffsasnoncooperativeplayersusingtheMSNE.

Figures5and6providescatterdiagramsofpayoffpairsfrom100,000gamesfork=9and∆=

1/256.Figure5includesthePSNEsandMSNE,eachwithequalfrequency,andFigure6showsthe

correspondingentropicoutcomes.Onceagain, thesample‐meanpayoffpairsare indicatedbyred

diamonds.

Figure6:BattleoftheSexes,NoncooperativeEquilibrium

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Figure7:BattleoftheSexes,EntropyPlayers

4.4 CorrelatedStrategiesandEfficiency

InexaminingtheresultsoftheStagHuntandBattleoftheSexes,weobservethattheexpectedpayoffs

oftheMSNEarestrictlylowerthanatleastonepairofPSNEpayoffsinbothgames.Thissuggestsa

potentialproblemofineffectivecoordination.Inotherwords,theplayersmayarriveata

noncooperativeequilibriumforwhichtheindividualand/orjointpayoffs(i.e.,ai∗,j∗,bi∗,j∗,and/or

ai∗,j∗+bi∗,j∗)arelessthanoptimal.Therefore,thevalueofbeingabletocoordinatestrategiesmaybe

substantial.

Table10offerssomequantificationofthedeficienciesattributabletoineffectivecoordinationfor

thethreegamesdiscussedabove.Foreachgame,thesecondcolumnpresentstheaverageofthejoint

payoffsoverallnoncooperativeequilibria,givingequalweighttoeachPSNEandMSNE.Forthe

Prisoner’sDilemma,thisis2+2=4;fortheStagHunt,itis[(4+4)+(2+2)+(2.5+2.5)]/3≈5.6667;andfortheBattleoftheSexes,itis[(4+3)+(3+4)+(2.5+2.5)]/3≈6.3333.Inthethird

column, we construct a simple efficiency measure by dividing the average joint payoff by the

maximumpossiblejointpayoffthatcanbeachievedbyeitherapure‐ormixed‐strategypairimposed

byanexogenousagency(custom,law,privateintermediation,etc.).Sincetheagentisexogenous,it

canexpandthedomainofmixedstrategiestocorrelatedstrategies,forwhichtheplayers’random

selectionsofUpandDownarestatisticallydependent.ForthePrisoner’sDilemma,thisyields3+3=

6;fortheStagHunt,ityields4+4=8;andfortheBattleoftheSexes,ityieldseither4+3=3+4=7

or[(4+3)p+(3+4)(1−p)]=7,wherethelattervaluecomesfromanycorrelatedmixedstrategy

15

thatchooses(Up,Left)and(Down,Right)withprobabilitiespand1−p,respectively.8Thefourthand

fifthcolumnspresentcorrespondingcalculationsforgameswithentropyplayers.

Table10:JointPayoffsfork=1

Game Avg.ofNEPayoffs NEEfficiency EntropyPayoffs EntropyEfficiency

Prisoner’sDilemma 4.0000 0.6667 5.0000 0.8333StagHunt 5.6667 0.7083 5.0000 0.6250

BattleoftheSexes 6.3333 0.9048 5.0000 0.7143

5 Discussion

5.1 EfficiencyAnalysisofAll2×2Games

Table 11 provides efficiency measures – as defined in the previous subsection – for the entire

populationofcardinalgamesgeneratedinsteps(1)through(5)ofSubsection1.1fork=1andk=9.

Thistableaddressesthenatureoftheoptimalityofindividualbehaviorwithinallpossible

2×2‐gamestructures,subdividedbythevaluesofJM1intheassociatedcanonicalordinalgame.

Thus,forclarity,wewouldnotethat:(a)Prisoner’sDilemmagamesareincludedinthecategoryof

JM1=6;(b)StagHuntgamesareincludedinJM1=7;and(c)BattleoftheSexesgamesareincluded

inJM1=8.

Table11:EfficienciesofAllGames

k=1 k=1 k=9 k=9

NE Entropy NE Entropy

JM1=8 0.9410 0.6250 0.7532 0.5000JM1=7 0.9127 0.7143 0.7305 0.5716JM1=6 0.9352 0.8333 0.7491 0.6669JM1=5 NA NA NA NAAverage 0.9300 0.7386 0.7445 0.5910

As is often true in game theory, behavioral paradoxes abound. We purposely indicate that

efficiencymeasuresare“notapplicable”forthecaseofJM1=5,becausethesegamesofoppositionare

qualitativelydifferentfromallothers.Formally,theefficiencymeasurescouldbedefinedas1.0,but

8WithintheconventionalBattleoftheSexesstoryline,themanandwomanwhoaretryingtodecidewhichmovietosee

couldchoosebetween“his”movieand“her”moviebytossingacoin(forwhichp=1/2).

16

suchcalculationswouldbemisleadingbecausethesegamesdonotreflect thecharacteristicsofa

society.Specifically, thereisnoroomforcooperation,coordination,oranyformofdiscourse,and

whateverone individualgains theother loses.Asnotedbefore,suchstructuraldystopiasbecome

increasinglyrareinthe“Flatland”(seeAbbott,1952[1884])ofmatrixgamesthatarisesfor large

valuesofk.

By definition, the category of JM1 = 8 comprises games of coordination. These games always

includeanoutcomewithpayoffpair(4,4),andsuchoutcomesmustbePSNEs.Inthiscase,thereason

whyefficiencyisnotexactly1.0isthatsomegameshavetwoPSNEsand1MSNE,andtheselowerthe

average.

Somewhatsurprisingly,thefall‐offintheefficiencyofnoncooperativeequilibriaask increases

from1to9isratherlargeforallgamecategories.Fork=1,theefficiencylossingamesofcoordination

(JM1=8)andmixed‐motivegames(JM1=7,6)isbetween6and9percent.Fork=9,thisgrowsto

between25and27percent,andresultsfork=10indicatethatk=9isveryclosetothelimit,with

differencesinefficiencyoflessthan0.01percent.(SeeTable12.)Partofthedecreaseinefficiencyis

attributabletothefactthatE[JMk]→E[JM∞]≈5.6968forlargek

,whichissubstantiallylessthanE[JM1]=6.875.Table12:Efficienciesfork=9andk=10

k=9 k=9 k=10 k=10

NE Entropy NE Entropy

JM1=8 0.7532 0.5000 0.7528 0.5004JM1=7 0.7305 0.5716 0.7310 0.5717JM1=6 0.7491 0.6669 0.7489 0.6672JM1=5 NA NA NA NAAverage 0.7445 0.5910 0.7447 0.5914

Wealsoconsiderthecompletelydifferentbehaviorofentropyplayers,whomaybeviewedas

“know‐nothing”or“zero‐intelligence”decisionmakers.Althoughitiseasytoconstructgames(e.g.,

thePrisoner’sDilemma)inwhichtheentropyplayers’expectedpayoffsof(2.5,2.5)aregreaterthan

thoseofnoncooperativeplayers,Table11showsthatforbothk=1andk=9,entropyplayersperform

worseonaveragethannoncooperativeplayers.Thisisbecausetheyareunabletotakeadvantageof

certaingamestructures,suchasrowand/orcolumndominance,thatassistcoordination.Eveninthis

17

highly constrained environment, differences in the cardinalmeasuresof payoffs yield far greater

variabilityandinequalitywhenkislargethanwhenk=1.Themeaningofthischangeisthatasthe

varietyofoutcomesgrows,theworthofcoordinationorcollaborationgrowsaswell.

5.2 Whythe2×2CaseIsSoImportant

Therearemanyreasonswhy2×2gamesarecrucialbothtothestudyofgametheoryspecifically,and

behavioralsciencemoregenerally.Theseinclude:

1. They offer a highly useful starting point for illustrating and contrasting many problems

andparadoxesinstrategicanalysis.

2. Theyarewidelyusedbyintroductoryinstructorsofgametheoryinthebehavioralsciences.(Is

thispedagogicalusejustified?Wewouldarguethatitis,withappropriatequalifications.)

3. They greatly facilitate analogy generation and storytelling in connecting specific real‐

worldproblemstoabstractmodels.Hence,theyprovidevaluableexercisesintyingthephysical

worldtomathematics.

4. They offer minimal repeated‐game models for the dynamics of learning, signaling, and

othercomplexhumanbehaviors.

5. Throughahandfulofspecialcases(e.g.,thePrisoner’sDilemma,StagHunt,andBattleofthe

Sexes),theysuccessfullyillustratefundamentalproblemsinstrategyandsociety.

Items(1),(2),and(4)arehighlyrelatedforbothteachingandresearch,especiallyifwebelievethat

dyadicrelationsareofconsiderableimportanceindescribinghumanandotheranimalbehavior.Itis

thusforgoodreasonthatelementarytextbooksinthebehavioralsciencesaboundwith2×2‐game

examples.

6 Forn>3,aBasicChangeintheParadigm

Onthewhole,wewouldarguethatthestudyof2×2gamesisimportantbecausesomuchofhuman

activityiswellmodeledbyinteractionsbetweentwoindividualsorbetweenoneindividualandan

18

institution,withrelativelyfewchoicesforeachpartyintheshortterm.Thisperspective,however,

canblindusagainsttheenormouscomplexitiesthatarisewhenincreasingmatrixsize.Forexample,

inthecaseofa3×3matrix,thenumberofdistinctstrategiccasesrisesto(9!)2/(3!)2≈

3.6578×109;andinthecaseof3playerswith2strategieseach,wehavea2×2×2matrixwith(8!)3

/(2!)3≈8.1935×1012strategicallydifferentpossibilities.

Inthecaseof3×3games,thesimpleexampleofTable13issufficienttodestroythehopesof

those interested in developing plausible dynamic strategic solutions. One natural candidate for

simpledynamicsisoptimalresponse;thatis,playersconsiderwheretheyhavebeeninaprevious

playofthegame,andusethatasthebasisfortheircurrentoptimization.However,abriefglanceat

Table13 shows that if the rowand columnplayersbeginwith strategies (sC,sR) = (1,1), then the

column player will move to sC = 3, and a 4‐cycle will emerge that never converges to the joint

maximumPSNEat(sC,sR)=(2,2).(Notethatifthepayoffat(sC,sR)=(2,2)were(1,1)insteadof(9,9),

thatparticular outcomewould still be aPSNE.)Quint, Shubik, andYan (1995)demonstrated the

extensive potential for cycling in a large class of n × n games involving both sequential and

simultaneousmoves.

Table13:ASimple3×3MatrixGame

  sC=1 sC=2 sC=3 sR=1 4,1 0,0 1,4

sR=2 0,0 9,9 0,0

sR=3 1,4 0,0 4,1

6.1 SmoothandRoughGames

Inany2×2game,itispossibletocompute4firstdifferencesbetweencellpayoffs.Inthe3×3case,

onecancompute12firstdifferencesand4seconddifferences,andthepayoffsetsstillhaveonlyfew

hillsandvalleys.However, formatricesof4×4andabove, thepotential roughnessof thepayoff

structureincreasesrapidly.Althoughthemathematicalstructureisclearlydefined,theanalysisof

applied decision problems becomes extremely difficult unless some appropriate smoothing

mechanism is imposed on the payoff surfaces. In corporate, military, and political planning, the

adjustment,evaluation,refinement,anddiscardingofmanyfeaturesofastrategicprocesstendto

19

narrowthefinalchoicestoasetthatincludescertainlylessthanten,andoftennotmorethantwoor

three,alternatives.

Thesesomewhatterseremarkswillbeenlargedinfurtherwork.

6.2 Coordination,Opposition,andNoncooperativeEquilibria

As we increase the number of strategies, or players, or both, the relative numbers of games of

coordinationandoppositionbecomevanishinglysmall.

Ifweconsiderm×nmatrixgameswithm,n≥3,therelativenumberofPSNEsdrops,andMSNEs

proliferate.Thereisaliteratureillustratingthis,whichincludeslimitingformulasfortheprobability

ofencounteringaPSNEinalarge,randommatrixgame.(SeeGoldberg,Goldman,andNewman,1968,

Dresher,1970,andI.Powers,1990.)

6.3 LifeIsaSetofMeasureZero

Aguidingprincipleforexploringtheenormousuniverseofmatrixgamesistoselectappropriate

limitingprocessestoobtainrobustsetsofgamesaddressingimportantquestionsofinterest.Wedo

thathereintakingthelimitofallcardinal2×2gameswithaminimalgridsize.

Ingeneral,manyproblemsofinterestrequiresuchspecificationthattheycanberegardedassets

ofmeasurezerowithrespecttofarlargerabstractcategoriestowhichtheybelong.Animportant

example(towhichwewillreturninasubsequentpaper)isthepresenceofties.Inmanycontextsof

humanactivity,tiesinperceivedvaluationarepresent.Furthermore,whenoneindividualhasfiner

perceptionsthananother,theincreasedprecisionalmostalwaysworkstohisorheradvantage.In

thepresentessay,wehaveruledouttiesforsimplicityandmanageability.Ifwehadpermittedthem,

thenumberofcanonicalordinalgameswouldhaveincreasedfrom144to726(NEEDREFERENCE

HERE).

20

7 ConcludingRemarks

Theprincipalpurposeofthispaperwastofindareasonablynaturalwaytoconsiderallcardinal2×2

gameswithinagivenfinitegrid.Theessentiallycombinatoricaspectsoftheinvestigationcalledfor

simulationtoevaluatestructuralaspectsthataredifficulttoseeusingonlyanalyticmethods.Our

study of this structure provided a sufficiently rich background for examining in detail both

noncooperative equilibria and entropy‐player solutions. In a subsequent paper,wewill consider

othersolutionconceptsthatenableonetoinvestigatetheinfluenceofstructureonbehaviorwith

variousintents.

Theamountof“fat left inthesystem”dependsonthesolutionused. Inmanycases,areferee,

government,orotheroutsideagencycouldbeusedtoguidethesystemtoasuperioroutcome,while

consuming fewer resources than it adds. The gap between the current solution and the joint

maximumisthemaximumworthofthecoordinator.Giventhebasicanalysisofthepresentpaper,we

nowareinapositiontoconsiderdevelopingimprovedmeasuresofefficiencyandsymmetryforany

outcomeinamatrixgame.Weplantodiscussthistopicaswellinafuturepaper.

Inshort,thisfirstessaywasaimedatprovidingasimpleideaoftheworthofgovernment,witha

quickandcrudeestimate.Asecondessaywillbedevotedto themanyproblemsofstructureand

behaviorarisingfromnoncooperativeequilibriain2×2games.Finally,athirdessaywilladdressthe

developmentofmoresophisticatedefficiencymeasuresbasedupontheanalysesofthepriorwork.

21

References

[1] Abbott,E.A.,1952[1884],Flatland:ARomanceofManyDimensions(6thed.),NewYork:Dover.

[2] Dresher, M., 1970, “Probability of a Pure Equilibrium Point in n‐Person Games”, Journal of

CombinatorialTheory,8,134‐145.

[3] Goldberg,K.,Goldman,A. J.,andNewman,M.,1968,“TheProbabilityofanEquilibriumPoint”,

JournalofResearchofNationalBureauofStandardsU.S.A.,72,B,93‐101.

[4] Kilgour,D.M,andN.M.Frazer.1988.ATaxonomyofAllOrdinal2x2GamesTheoryanddecision

24:99‐117

[5] Powers,I.Y.,1990,“LimitingDistributionsoftheNumberofPureStrategyNashEquilibriainn‐

PersonGames”,InternationalJournalofGameTheory,19,3,277‐286.

[6] Powers,I.Y.andShubik,M.,1991,“TheNamesoftheGames”,InternationalSecurityandArms

ControlDiscussionPaper,No.7,YaleUniversity.

[7] Quint,T.,Shubik,M.,andYan,D.,1996,“DumbBugsandBrightNoncooperativePlayers:

Games,Context,andBehavior”,inUnderstandingStrategicInteraction:EssaysinHonorof

ReinhardSelten,(W.Albers,etal.,eds.),Berlin:Springer‐Verlag,pp.185‐197.

22

Appendix1

  FIGURE 1 FIGURE 2

3, 4 2 ,34,2 , 132 , 43 3, 4 1,,4 43,24, 2

,44,31,13,21,11,1 3 2,412,3 ,2,12ºªºª ººªª ºª»« »« »« »« »« ¼¼ ¼¬ ¼ ¼

,4 2,34 4,4 3,,4 4, 4 2,323,2 4,1 1,1 ,3 1,12,3 3,221,1 3,2 1

ºªª ººªºª »« »« »« »«¼ ¼¬ ¼ ¼2P

3

42P

4

3

1P1

22

1 1P2 3 41 1 2 3 4

19,3,93,122 33,80,105,76,125

23

  FIGURE 3 FIGURE 4

20

1 1  2 3 4 3 4

24

1 2 3 4 1 2 3 4

101 35,53,21,103

  FIGURE 5 FIGURE 6

1 1  2 3 4 3 4

25

  FIGURE 7 FIGURE 8

21

ª4,4 1,2ºª4,4 3,1ºª4,4 2,3º ª3,4 4,1ºª3,4 1,2ºª3,4 4,1ºª3,4 2,3ºª2,3 4,1º«¬3,1 2,3»«¼¬1,22,3»«¼¬3,11,2¼» «¬2,31,2»«¼¬2,34,1»«¼¬1,22,3»«¼¬1,2

4,1»«¼¬1,2 3,4»¼

1 1  2 3 4 3 4

26

3

2P4 4

3

2P

22

1P12 3 4

1P11 1 2 3 4

140,29,106 30,128,16,71,110

1 1  2 3 4 3 4

27

  FIGURE 9 FIGURE 10

  FIGURE 11 FIGURE 12

22

ª4,4 3,2ºª4,4 3,2ºª4,4 2,1ºª4,4 2,1º ª3,4 4,2ºª3,4 2,1ºª3,4 4,2ºª3,4 2,1º«¬1,3 2,1»«¼¬2,11,3»«¼¬3,21,3¼¬»«1,3 3,2»¼ ¬«1,3 2,1»«¼¬1,34,2»«¼¬2,11,3»«¼¬4,2

1,3»¼

1 1  2 3 4 3 4

28

  FIGURE 13 FIGURE 14

3

2P

4 4

3

2P

22

1P12 3 4

1P1

1 1 2 3 4

28,85,113,132 86,63,96,44

1 1  2 3 4 3 4

29

  FIGURE 15 FIGURE 16

23

  1 1  2 3 4 3 4

30

  M M

  FIGURE 17 FIGURE 18

  M M  ª3,2 2,1º ª3,3 4,2ºª3,3 4,2ºª3,3 2,1º  «¬4,31,4»¼ «¬1,42,1»«¼¬2,11,4»«¼¬4,21,4»¼

22 13,1, ,43 24 ,4, 4,2413 ,2 ,2 1,3 1,3 1,34,

ºª ººªª »« »« »«134,2,4 , ,4, 14412 2,, 32 2,4

2, 131 4,3, 2 ,3 4,1 3,2 1,3 1,3ºª ºª ºªºª »« »«« »»«

3

¬ ¼¼¼ ¼ ¼¬ ¼ ¼2P

4 4

3

2P

22

1P12 3 4

1P1

1 1 2 3 4

43,64,112 114,45,6,88

  1 1 2

  143 89,24,142

  FIGURE 19 FIGURE 20

24

  M M

  ª2,2 3,1º ª3,3 4,1ºª3,3 4,1ºª2,2 4,1ºª2,2 4,1º  «¬4,3 1,4»¼ «¬1,42,2»«¼¬2,21,4»«¼¬3,31,4»«¼¬1,4 3,3»¼

  1 2 3 4 1 2 3 4

  57 12,99,138,1

  FIGURE 21 FIGURE 22

2P 2P

3

4 4

3

1P 1P

22

3

2P

4 4

3

2P

22

1P1 1P1

32

  M M  ª2,3 4,2ºª2,3 3,1º ª2,3 4,1ºª3,2 4,1ºª2,3 4,1º  «¬1,43,1»«¼¬4,21,4»¼ «¬1,43,2»«¼¬2,31,4¼¬»«3,21,4»¼

  FIGURE 23 FIGURE 24

25

Appendix2WhatisaSolutiontoaMatrixGame?  Appendix2  MartinShubik

Game# Payoff 

Matrix

Shape

Joint

Max PSNEs Symmetric

NashPayoff

Dom.

Pareto 

Optima Transpose Row Col.

1 (1,4) (3,3)

22 6 1 Sym 2 2 2 3 NA (2,2) (4,1)

2 (1,2) (3,1)

11 7 1

2 4 2 2 115 (2,4) (4,3)

3 (1,1) (3,2)

3 8 1 Sym 4 4 2 1 NA (2,3) (4,4)

4 (1,4) (3,3)

20 6 1

4 2 1 3 108 (4,2) (2,1)

2P 2P

3

4 4

3

1P1 1P

22

12 3 41 1 2 3 4

118,144 120,100,136

5 (1,3) (3,4)

16 7 1

3 4 1 2 117 (4,1) (2,2)

6 (1,3) (3,2)

18 6 0 2.5 2.5 0 3 134 (4,1) (2,4)

7 (1,2) (3,3)

7 8 2 Sym

4 4

0 1 NA (4,4) (2,1) 3 3

8 (1,2) (3,1)

9 8 1

4 4 1 1 113 (4,4) (2,3)

9 (1,1) (3,2)

5 7 1

3 2 1 2 105 (4,3) (2,4)

10 (1,1) (3,4)

2 7 2 Sym

3 4

0 2 NA (4,3) (2,2) 4 3

11 (1,4) (2,3)

24 5 1

3 2 2 4 120 (3,2) (4,1)

12 (1,4) (2,2)

22 6 1 Sym 3 3 2 3 NA (3,3) (4,1)

13 (1,4) (2,1)

19 7 1

4 3 1 2 128 (3,2) (4,3)

14 (1,3) (2,4)

17 6 1

4 2 2 2 112 (3,1) (4,2)

15 (1,3) (2,2)

15 8 1

4 4 1 1 131 (3,1) (4,4)

16 (1,2) (2,3)

10 7 1

3 4 1 2 137 (3,4) (4,1)

34

17 (1,2) (2,4)

11 7 1

4 3 2 2 86 (3,1) (4,3)

18 (1,2) (2,1)

8 7 1

3 4 2 2 109 (3,4) (4,3)

19 (1,1) (2,3)

3 8 1 Sym 4 4 2 1 NA (3,2) (4,4)

20 (1,1) (2,2)

1 8 1

4 4 2 1 102 (3,3) (4,4)

21 (1,1) (2,4)

6 6 1

29

3 3 1 3 123 (3,3) (4,2)

22 (1,4) (2,3)

23 6 1

4 2 2 3 114 (4,2) (3,1)

Game#

Payoff 

Matrix

Shape

Joint

Max PSNEs Symmetric

NashPayoff

Dom.

Pareto 

Optima Transpose Row Col.

23 (1,4) (2,2)

21 7 1

4 3 2 2 87 (4,3) (3,1)

24 (1,4) (2,1)

20 6 1

3 3 1 3 127 (4,2) (3,3)

25 (1,3) (2,4)

18 6 1

3 2 2 3 118 (4,1) (3,2)

26 (1,3) (2,2)

15 8 1 Sym 4 4 2 1 NA (4,4) (3,1)

27 (1,3) (2,2)

16 7 1

3 4 1 2 135 (4,1) (3,4)

28 (1,3) (2,1)

13 8 1

4 4 2 1 83 (4,4) (3,2)

29 (1,2) (2,3)

9 8 1

4 4 1 1 132 (4,4) (3,1)

30 (1,2) (2,3)

10 7 1

3 4 2 2 98 (4,1) (3,4)

31 (1,2) (2,4)

12 6 1

3 3 2 3 89 (4,1) (3,3)

32 (1,2) (2,1)

7 8 1

4 4 2 1 107 (4,4) (3,3)

33 (1,1) (2,3)

3 7 1

3 4 2 2 81 (4,2) (3,4)

34 (1,1) (2,2)

2 7 1 3 4 2 2 104 (4,3) (3,4)

35 (1,1) (2,4)

6 6 1 Sym 3 3 2 3 NA (4,2) (3,3)

36 (1,1) (2,4)

5 7 1

4 3 1 2 125 (4,3) (3,2)

37 (1,4) (4,3)

21 7 1

2 2 1 2 116 (2,2) (3,1)

38 (1,4) (4,2)

23 6 1

2 3 1 3 88 (2,3) (3,1)

39 (1,4) (4,1)

22 6 0

2.5 2.5 0 3 138 (2,2) (3,3)

40 (1,4) (4,1)

24 5 1 2 3 1 4 100 (2,3) (3,2)

41 (1,3) (4,4)

15 8 2 Sym

4 4

0 1 NA (2,2) (3,1) 2 2

42 (1,3) (4,4)

13 8 1

4 4 1 1 106 (2,1) (3,2)

43 (1,3) (4,2)

17 6 1

2 4 1 2 97 (2,4) (3,1)

44 (1,3) (4,2)

14 7 0

2.5 2.5 0 2 126 (2,1) (3,4)

30

Game#

Payoff 

Matrix

Shape

Joint

Max PSNEs Symmetric

NashPayoff

Dom.

Pareto 

Optima Transpose Row Col.

45 (1,3) (4,1)

18 6 1

2 4 1 3 91 (2,4) (3,2)

46 (1,2) (4,3)

11 7 2 4 3

0 2 133 (2,4) (3,1) 2 4

47 (1,2) (4,3)

8 7 1

4 3 1 2 94 (2,1) (3,4)

48 (1,2) (4,4)

7 8 1

4 4 1 1 82 (2,1) (3,3)

49 (1,2) (4,1)

12 6 1

2 4 1 3 121 (2,4) (3,3)

50 (1,2) (4,1)

10 7 0

2.5 2.5 0 2 143 (2,3) (3,4)

51 (1,1) (4,3)

2 7 1

4 3 1 2 79 (2,2) (3,4)

52 (1,1) (4,2)

4 7 1

4 2 1 2 101 (2,3) (3,4)

53 (1,1) (4,2)

6 6 2 Sym

4 2

0 3 NA (2,4) (3,3) 2 4

54 (1,1) (4,4)

1 8 1

4 4 1 1 92 (2,2) (3,3)

55 (1,1) (4,4)

3 8 2 4 4

0 1 122 (2,3) (3,2) 2 3

56 (1,4) (4,3)

19 7 1

3 2 1 2 110 (3,2) (2,1)

57 (1,4) (4,3)

21 7 0

2.5 2.5 0 2 141 (3,1) (2,2)

58 (1,4) (4,2)

20 6 1

3 3 1 3 84 (3,3) (2,1)

59 (1,4) (4,1)

24 5 0

2.5 2.5 0 4 136 (3,2) (2,3)

60 (1,4) (4,1)

22 6 1

3 3 1 3 99 (3,3) (2,2)

61 (1,3) (4,4)

13 8 2 4 4

0 1 140 (3,2) (2,1) 3 2

62 (1,3) (4,4)

15 8 1

4 4 1 1 111 (3,1) (2,2)

63 (1,3) (4,2)

14 7 1

3 4 1 2 95 (3,4) (2,1)

64 (1,3) (4,2)

17 6 0

2.5 2.5 0 2 130 (3,1) (2,4)

65 (1,3) (4,1)

18 6 0

2.5 2.5 0 3 144 (3,2) (2,4)

66 (1,3) (4,1)

16 7 1

3 4 1 2 90 (3,4) (2,2)

31

Game#

Payoff 

Matrix

Shape

Joint

Max PSNEs Symmetric

NashPayoff

Dom.

Pareto 

Optima Transpose Row Col.

67 (1,2) (4,3)

8 7 2 4 3

0 2 129 (3,4) (2,1) 3 4

68 (1,2) (4,3)

11 7 1

4 3 1 2 96 (3,1) (2,4)

69 (1,2) (4,4)

7 8 2 Sym

4 4

0 1 NA (3,3) (2,1) 3 3

70 (1,2) (4,4)

9 8 1

4 4 1 1 85 (3,1) (2,3)

71 (1,2) (4,1)

10 7 1

3 4 1 2 119 (3,4) (2,3)

72 (1,2) (4,1)

12 6 0

2.5 2.5 0 3 142 (3,3) (2,4)

73 (1,1) (4,3)

5 7 1

4 3 1 2 80 (3,2) (2,4)

74 (1,1) (4,3)

2 7 2 Sym

4 3

0 2 NA (3,4) (2,2) 3 4

75 (1,1) (4,2)

6 6 1

4 2 1 3 103 (3,3) (2,4)

76 (1,1) (4,2)

4 7 2 3 4

0 2 139 (3,4) (2,3) 4 2

77 (1,1) (4,4)

3 8 1

4 4 1 1 93 (3,2) (2,3)

78 (1,1) (4,4)

1 8 2 4 4

0 1 124 (3,3) (2,2) 3 3

79 (1,1) (2,2)

2 7 1

3 4 1 2 51 (3,4) (4,3)

80 (1,1) (2,3)

4 7 1

3 4 1 2 73 (3,4) (4,2)

81 (1,1) (2,4)

5 7 1

4 3 2 2 33 (3,2) (4,3)

82 (1,2) (2,1)

7 8 1

4 4 1 1 48 (3,3) (4,4)

83 (1,2) (2,3)

9 8 1 4 4 2 1 28 (3,1) (4,4)

84 (1,2) (2,4)

12 6 1

3 3 1 3 58 (3,3) (4,1)

85 (1,3) (2,1)

13 8 1

4 4 1 1 70 (3,2) (4,4)

86 (1,3) (2,1)

14 7 1

3 4 2 2 17 (3,4) (4,2)

87 (1,3) (2,2)

16 7 1

3 4 2 2 23 (3,4) (4,1)

88 (1,3) (2,4)

18 6 1

3 2 1 3 38 (3,2) (4,1)

32

Game#

Payoff 

Matrix

Shape

Joint

Max PSNEs Symmetric

NashPayoff

Dom.

Pareto 

Optima Transpose Row Col.

89 (1,4) (2,1)

20 6 1

3 3 2 3 31 (3,3) (4,2)

90 (1,4) (2,2)

21 7 1

4 3 1 2 66 (3,1) (4,3)

91 (1,4) (2,3)

23 6 1

4 2 1 3 45 (3,1) (4,2)

92 (1,1) (2,2)

1 8 1

4 4 1 1 54 (4,4) (3,3)

93 (1,1) (2,3)

3 8 1

4 4 1 1 77 (4,4) (3,2)

94 (1,2) (2,1)

6 7 1

3 4 1 2 47 (4,3) (3,4)

95 (1,2) (2,4)

11 7 1

4 3 1 2 63 (4,3) (3,1)

96 (1,3) (2,1)

14 7 1

3 4 1 2 68 (4,2) (3,4)

97 (1,3) (2,4)

17 6 1 4 2 1 2 43 (4,2) (3,1)

98 (1,4) (2,1)

19 7 1

4 3 2 2 30 (4,3) (3,2)

99 (1,4) (2,2)

22 6 1

3 3 1 3 60 (4,1) (3,3)

100 (1,4) (2,3)

24 5 1

3 2 1 4 40 (4,1) (3,2)

101 (1,1) (3,2)

5 7 1

2 4 1 2 52 (2,4) (4,3)

102 (1,1) (3,3)

1 8 1

4 4 2 1 20 (2,2) (4,4)

103 (1,1) (3,3)

6 6 1

2 4 1 3 75 (2,4) (4,2)

104 (1,1) (3,4)

2 7 1

4 3 2 2 34 (2,2) (4,3)

105 (1,1) (3,4)

4 7 1

2 3 1 2 9 (2,3) (4,2)

106 (1,2) (3,1)

9 8 1

4 4 1 1 42 (2,3) (4,4)

107 (1,2) (3,3)

7 8 1

4 4 2 1 32 (2,1) (4,4)

108 (1,2) (3,3)

12 6 1

2 4 1 3 4 (2,4) (4,1)

109 (1,2) (3,4)

8 7 1

4 3 2 2 18 (2,1) (4,3)

110 (1,2) (3,4)

10 7 1

2 3 1 2 56 (2,3) (4,1)

33

Game#

Payoff 

Matrix

Shape

Joint

Max PSNEs Symmetric

NashPayoff

Dom.

Pareto 

Optima Transpose Row Col.

111 (1,3) (3,1)

15 8 1

4 4 1 1 62 (2,2) (4,4)

112 (1,3) (3,1)

17 6 1

2 4 2 2 14 (2,4) (4,2)

113 (1,3) (3,2)

13 8 1

4 4 1 1 8 (2,1) (4,4)

114 (1,3) (3,2)

18 6 1

2 4 2 3 22 (2,4) (4,1)

115 (1,3) (3,4)

14 7 1

4 2 2 2 2 (2,1) (4,2)

116 (1,3) (3,4)

16 7 1

2 2 1 2 37 (2,2) (4,1)

117 (1,4) (3,1)

21 7 1

4 3 1 2 5 (2,2) (4,3)

118 (1,4) (3,1)

23 6 1

2 3 2 3 25 (2,3) (4,2)

119 (1,4) (3,2)

19 7 1

4 3 1 2 71 (2,1) (4,3)

120 (1,4) (3,2)

24 5 1

2 3 2 4 11 (2,3) (4,1)

121 (1,4) (3,3)

20 6 1

4 2 1 3 49 (2,1) (4,2)

122 (1,1) (3,2)

3 8 2 4 4

0 1 55 (4,4) (2,3) 3 2

123 (1,1) (3,3)

6 6 1

3 3 1 3 21 (4,2) (2,4)

124 (1,1) (3,3)

1 8 2 4 4

0 1 78 (4,4) (2,2) 3 3

125 (1,1) (3,4)

4 7 1

3 4 1 2 36 (4,2) (2,3)

126 (1,2) (3,1)

11 7 0

2.5 2.5 0 2 44 (4,3) (2,4)

127 (1,2) (3,3)

12 6 1 3 3 1 3 24 (4,1) (2,4)

128 (1,2) (3,4)

10 7 1

3 4 1 2 13 (4,1) (2,3)

129 (1,2) (3,4)

8 7 2 4 3

0 2 67 (4,3) (2,1) 3 4

130 (1,3) (3,1)

17 6 0

2.5 2.5 0 2 64 (4,2) (2,4)

131 (1,3) (3,1)

15 8 1

4 4 1 1 15 (4,4) (2,2)

132 (1,3) (3,2)

13 8 1

4 4 1 1 29 (4,4) (2,1)

34

45

Game#

Payoff 

Matrix

Shape

Joint

Max PSNEs Symmetric

NashPayoff

Dom.

Pareto 

Optima Transpose Row Col.

133 (1,3) (3,4)

14 7 2 3 4

0 2 46 (4,2) (2,1) 4 2

134 (1,4) (3,1)

23 6 0

2.5 2.5 0 3 6 (4,2) (2,3)

135 (1,4) (3,1)

21 7 1

4 3 1 2 27 (4,3) (2,2)

136 (1,4) (3,2)

24 5 0

2.5 2.5 0 4 59 (4,1) (2,3)

137 (1,4) (3,2)

19 7 1

4 3 1 2 16 (4,3) (2,1)

138 (1,4) (3,3)

22 6 0

2.5 2.5 0 3 39 (4,1) (2,2)

139 (1,1) (4,3)

5 7 2 4 3

0 2 76 (2,4) (3,2) 2 4

140 (1,2) (4,4)

9 8 2 4 4

0 1 61 (2,3) (3,1) 2 3

141 (1,3) (4,1)

16 7 0

2.5 2.5 0 2 57 (2,2) (3,4)

142 (1,4) (4,2)

20 6 0

2.5 2.5 0 3 72 (2,1) (3,3)

143 (1,4) (4,3)

19 7 0

2.5 2.5 0 2 50 (2,1) (3,2)

144 (1,4) (4,2)

23 6 0

2.5 2.5 0 3 65 (3,1) (2,3)

Game#correspondstothenumberingsystemestablishedinthecompanionpaper

PayoffMatrixgivesthenormalformofeachgamewithpayoffslistedas(rowpayoff,columnpayoff)

Shapecorrespondstotheshapeofthepayoffset'sconvexhullasshowninAppendix1

JointMaxgivesthehighestpossiblecombinedpayoffforthetwoplayers

Symmetricismarked"Sym"ifthegameissymmetric,otherwiseitisleftblank

NashPayoffliststhepayoffsofthenoncooperativeequilibrium

iftherearetwoequilibriawithdifferentpayoffsums,theonewiththehighestsumislistedfirst

Dom.specifiesthenumberofrowandcolumnstrategiesthatarestrictlydominated

ParetoOptimagivesthenumberofpayoffpairsthatareParetooptimal

Transposeliststhegamenumbercorrespondingtothetransposeofthegameshown

Appendix3

#include <iostream>#include <fstream>#include <stdio.h>#include <time.h>#include  <stdlib.h> 

#include  <iomanip> 

using  namespace 

std;

const int maxnum=1<<11; 

const double di=1<<9; 

const int loop=300000; 

class cmatr

47

{ public: int aul,bul,aur,bur,adl,bdl,adr,bdr; public: int 

d1,d2,d3,d4,d5,d6,d7,d8; public: int nash; public: double 

na1,na2,na3,na4; public: void setnum(int a1,int a2,int a3,int a4,int 

a5,int a6,int a7,int a8){

aul=a1;bul=a2;aur=a3;bur=a4; 

adl=a5;bdl=a6;adr=a7;bdr=a8; 

d1=a1;d2=a2;d3=a3;d4=a4; 

d5=a5;d6=a6;d7=a7;d8=a8;}public: void print(){

cout<<"cm"<<endl<<"aul "<<aul<<" bul "<<bul<<" aur "<<aur<<" bur 

"<<bur<<endl; cout<<"adl "<<adl<<" bdl "<<bdl<<" adr "<<adr<<" bdr 

"<<bdr<<endl; cout<<d1<<' '<<d2<<' '<<d3<<' '<<d4<<endl<<d5<<' '<<d6<<' 

'<<d7<<' '<<d8<<endl;}public: bool eql(cmatr a){

for (int i=1;i<=4;i++){

if ((aul==a.aul)&&(bul==a.bul)&&(aur==a.aur)&&(bur==a.bur)&&(adl==a.adl)&&(bdl==a.bdl)&&(adr==a.adr)&&(bdr==a.bdr)) return true;

a.setnum(a.adr,a.bdr,a.aul,a.bul,a.aur,a.bur,a.adl,a.bdl)

; }return false;

}public: void rep(){

aul=1;aur=1;bul=1;bur=1;adl=1;adr=1;bdl=1;bdr=1;36

if (d1>d3) aul++; if (d1>d5) aul++; if (d1>d7) aul++;if (d3>d1) aur++; if (d3>d5) aur++; if (d3>d7) aur++; 

if (d5>d1) adl++; if (d5>d3) adl++; if (d5>d7) adl++; 

if (d7>d1) adr++; if (d7>d3) adr++; if (d7>d5) adr++; 

if (d2>d4) bul++; if (d2>d6) bul++; if (d2>d8) bul++; 

if (d4>d2) bur++; if (d4>d6) bur++; if (d4>d8) 

bur++; if (d6>d2) bdl++; if (d6>d4) bdl++; if (d6>d8) 

bdl++; if (d8>d2) bdr++; if (d8>d4) bdr++; if (d8>d6) 

bdr++;}

};class matr{ public: int ul,ur,dl,dr; 

public: int 

d1,d2,d3,d4; public: 

void randnum(){

ul=rand()%maxnum+1;d1=ul; 

ur=rand()%maxnum+1;d2=ur; 

dl=rand()%maxnum+1;d3=dl; 

dr=rand()%maxnum+1;d4=dr;}public: bool eql(){

if ((ul==ur)||(ul==dl)||(ul==dr)) return 

true; if ((ur==dl)||(ur==dr)) return true; if 

(dl==dr) return true; return false;}public: void rep(){

int a1=1,a2=1,a3=1,a4=1; if (ul>ur) a1++; if 

(ul>dl) a1++; if (ul>dr) a1++; if (ur>ul) a2++; if 

(ur>dl) a2++; if (ur>dr) a2++; if (dl>ul) a3++; 

if (dl>ur) a3++; if (dl>dr) a3++; if (dr>ul) 

a4++; if (dr>ur) a4++; if (dr>dl) a4++; 

ul=a1;ur=a2;dl=a3;dr=a4;}public: cmatr combine(matr b){

cmatr c;c.setnum(ul,b.ul,ur,b.ur,dl,b.dl,dr,b.dr); 

return c;}public: void print(){

49

cout<<ul<<' '<<ur<<' '<<dl<<' '<<dr<<endl;}

};double  a[500000],b[500000],a1[500000],b1[500000];  double 

oa[150][4000],ob[150][4000],oa1[150][4000],ob1[150][4000];  int 

soa[150],sob[150],soa1[150],sob1[150];  int  ty[500000];  char 

nu[4]=""; string s="out"; string s2="solution"; string s1; int m=0;int main(){ srand((int)time(0)); int aul,bul,aur,bur,adl,bdl,adr,bdr; int i,j,k,t,tp; int res[150]; double 

mr[150],mc[150],vr[150],vc[150],sr[150],sc[150],ms[150],vs[150],ss[150]; double 

mr1[150],mc1[150],vr1[150],vc1[150],sr1[150],sc1[150],ms1[150],vs1[150],ss1[150]; double 

mmr,mmc,vvr,vvc,ssr,ssc,mms,vvs,sss; double 

mmr1,mmc1,vvr1,vvc1,ssr1,ssc1,mms1,vvs1,sss1; int rc[150],rc1[150]; cmatr samp[150]; 

cmatr c,d; cmatr *p; matr pa,pb;

FILE * fp=NULL; 

fp=fopen("www.txt","r"); 

for (i=1;i<=144;i++){

fscanf(fp,"%d%d%d%d%d%d%d%d",&aul,&bul,&aur,&bur,&adl,&bdl,&adr,&bdr

); samp[i].setnum(aul,bul,aur,bur,adl,bdl,adr,bdr); res[i]=0;}for (i=1;i<=144;i++){

fscanf(fp,"%d",&samp[i].nash); if 

((samp[i].nash==1)||(samp[i].nash==0)) 

fscanf(fp,"%lf%lf",&samp[i].na1,&samp[i].na2);if (samp[i].nash==2) 

fscanf(fp,"%lf%lf%lf%lf",&samp[i].na1,&samp[i].na2,&samp[i].na3,&samp[i].na4);} fclose(fp); 

fp=NULL; for 

(i=1;i<=loop;i++){

pa.randnum(); while (pa.eql()) 

pa.randnum(); pb.randnum(); while 

(pb.eql()) pb.randnum(); 

c=pa.combine(pb); c.rep(); 

a1[i]=(double) 

(c.d1+c.d3+c.d5+c.d7)/4; 

b1[i]=(double) 

(c.d2+c.d4+c.d6+c.d8)/4; for 

(k=1;k<=144;k++){

if (samp[k].eql(c))

50

{ res[k]++; ty[i]=k; a[i]=0; 

b[i]=0; if 

(samp[k].nash==0){

a[i]=c.d1+c.d3+c.d5+c.d7; 

a[i]=a[i]/4; 

b[i]=c.d2+c.d4+c.d6+c.d8; 

b[i]=b[i]/4;}else if (samp[k].nash==1){

if ((c.aul==samp[k].na1)&&(c.bul==samp[k].na2)){ a[i]=c.d1;b[i]=c.d2;};if ((c.aur==samp[k].na1)&&(c.bur==samp[k].na2)){ a[i]=c.d3;b[i]=c.d4;};if ((c.adl==samp[k].na1)&&(c.bdl==samp[k].na2)){ a[i]=c.d5;b[i]=c.d6;};if ((c.adr==samp[k].na1)&&(c.bdr==samp[k].na2)){ a[i]=c.d7;b[i]=c.d8;};

}else if (samp[k].nash==2){

if ((c.aul==samp[k].na1)&&(c.bul==samp[k].na2)){ a[i]=c.d1;b[i]=c.d2;};if ((c.aur==samp[k].na1)&&(c.bur==samp[k].na2)){ a[i]=c.d3;b[i]=c.d4;};if ((c.adl==samp[k].na1)&&(c.bdl==samp[k].na2)){ a[i]=c.d5;b[i]=c.d6;};if ((c.adr==samp[k].na1)&&(c.bdr==samp[k].na2)){ a[i]=c.d7;b[i]=c.d8;};if ((c.aul==samp[k].na3)&&(c.bul==samp[k].na4)){ a[i]+=c.d1;b[i]+=c.d2;};if ((c.aur==samp[k].na3)&&(c.bur==samp[k].na4)){ a[i]+=c.d3;b[i]+=c.d4;};if ((c.adl==samp[k].na3)&&(c.bdl==samp[k].na4)){ a[i]+=c.d5;b[i]+=c.d6;

51

};if ((c.adr==samp[k].na3)&&(c.bdr==samp[k].na4)){ a[i]+=c.d7;b[i]+=c.d8;};a[i]+=(double) (c.d1+c.d3+c.d5+c.d7)/4; 

b[i]+=(double) 

(c.d2+c.d4+c.d6+c.d8)/4; 

a[i]=(double)a[i]/3; b[i]=(double)b[i]/3;};break;

}}

}for (i=1;i<=144;i++){ mr[i]=0;mc[i]=0;vr[i]=0;vc[i]=0;sr[i]=0;sc[i]=0; rc[i]=0; 

mr1[i]=0;mc1[i]=0;vr1[i]=0;vc1[i]=0;sr1[i]=0;sc1[i]=0

; rc1[i]=0; soa[i]=0;sob[i]=0; soa1[i]=0;sob1[i]=0;}mmr=0;mmc=0;vvr=0;vvc=0;ssr=0;ssc=0;mms=0;vvs=0;sss=0; 

mmr1=0;mmc1=0;vvr1=0;vvc1=0;ssr1=0;ssc1=0;mms1=0;vvs1=0;sss1=0;

for (i=1;i<=loop;i++){ a[i]=a[i]/di; 

b[i]=b[i]/di; 

a1[i]=a1[i]/di; 

b1[i]=b1[i]/di

;}for (i=1;i<=loop;i++){ tp=ty[i]; rc[tp]++; rc1[tp]++; 

sr[tp]+=a[i];sr1[tp]+=a1[i]; 

sc[tp]+=b[i];sc1[tp]+=b1[i]; 

ss[tp]+=a[i]+b[i];ss1[tp]+=a1[i]+b1[i]; 

ssr+=a[i];ssc+=b[i];sss+=a[i]+b[i]; 

ssr1+=a1[i];ssc1+=b1[i];sss1+=a1[i]+b1[i]

; soa[tp]++; oa[tp][soa[tp]]=a[i]; 

sob[tp]++; ob[tp][sob[tp]]=b[i]; 

soa1[tp]++; oa1[tp][soa1[tp]]=a1[i]; 

sob1[tp]++; ob1[tp][sob1[tp]]=b1[i];}for (i=1;i<=144;i++){ mr[i]=sr[i]/rc[i];mr1[i]=sr1[i]/rc1[i]; 

mc[i]=sc[i]/rc[i];mc1[i]=sc1[i]/rc1[i]; 

ms[i]=ss[i]/rc[i];ms1[i]=ss1[i]/rc1[i]; 

52

sr[i]=0;sc[i]=0;ss[i]=0;sr1[i]=0;sc1[i]=0;ss1[i]=0

;}mmr=ssr/loop;mmc=ssc/loop;mms=sss/loop; 

mmr1=ssr1/loop;mmc1=ssc1/loop;mms1=sss1/loop

; ssr=0;ssc=0;sss=0; ssr1=0;ssc1=0;sss1=0; for 

(i=1;i<=loop;i++){  tp=ty[i];  sr[tp]+=(a[i]‐mr[tp])*(a[i]‐mr[tp]);sr1[tp]+=(a1[i]‐mr1[tp])*(a1[i]‐

mr1[tp]);  sc[tp]+=(b[i]‐mc[tp])*(b[i]‐mc[tp]);sc1[tp]+=(b1[i]‐

mc1[tp])*(b1[i]‐mc1[tp]);

ss[tp]+=(a[i]+b[i]‐ms[tp])*(a[i]+b[i]‐ms[tp]);ss1[tp]+=(a1[i]+b1[i]‐ms1[tp])*(a1[i]+b1[i]‐ms1[tp]); 

ssr+=(a[i]‐mmr)*(a[i]‐mmr);ssr1+=(a1[i]‐mmr1)*(a1[i]‐mmr1); ssc+=(b[i]‐mmc)*(b[i]‐

mmc);ssc1+=(b1[i]‐mmc1)*(b1[i]‐mmc1); sss+=(a[i]+b[i]‐mms)*(a[i]+b[i]‐

mms);sss1+=(a1[i]+b1[i]‐mms1)*(a1[i]+b1[i]‐mms1);}vvr=ssr/loop;vvc=ssc/loop;vvs=sss/loop; 

vvr1=ssr1/loop;vvc1=ssc1/loop;vvs1=sss1/loop; 

for (i=1;i<=144;i++){ vr[i]=sr[i]/rc[i];vr1[i]=sr1[i]/rc1[i]; 

vc[i]=sc[i]/rc[i];vc1[i]=sc1[i]/rc1[i]

; 

vs[i]=ss[i]/rc[i];vs1[i]=ss1[i]/rc1[i];} cout.setf(ios::fixed); 

fp=fopen("out.txt","w")

;fprintf(fp,"r mean %.3lf c mean %.3lf sum mean %.3lf r var %.3lf c var %.3lf sum var 

%.3lf\n",mmr,mmc,mms,vvr,vvc,vvs);fprintf(fp,"r  mean  %.3lf  c  mean  %.3lf  sum  mean  %.3lf  r  var  %.3lf  c  var  %.3lf  sum  var 

%.3lf\n",mmr1,mmc1,mms1,vvr1,vvc1,vvs1); for (i=1;i<=144;i++){

//cout<<setprecision(3)<<"BOX  "<<i<<":"<<"r  mean:"<<mr[i]<<"  c  mean"<<mc[i]<<" 

sum mean:"<<ms[i]<<"  r  var"<<vr[i]<<"  c  var"<<vc[i]<<"  sum var:"<<vs[i]<<endl;  fprintf(fp,"BOX 

%d:r mean %.3lf c mean %.3lf sum mean %.3lf r var %.3lf c var %.3lfsum var %.3lf\n",i,mr[i],mc[i],ms[i],vr[i],vc[i],vs[i]);

fprintf(fp,"BOX %d:r mean %.3lf c mean %.3lf sum mean %.3lf r var %.3lf c var %.3lfsum var %.3lf\n",i,mr1[i],mc1[i],ms1[i],vr1[i],vc1[i],vs1[i]);

} fclose(fp); for 

(i=1;i<=144;i++){

m++;itoa(m,nu,10); s1=s+nu+".txt"; 

fp=fopen(s1.c_str(),"w"); fprintf(fp,"BOX 

53

%d\n",i); for (k=1;k<=soa[i];k++) 

fprintf(fp,"%.3lf  %.3lf\n",oa[i][k],ob[i][k]);fclose(fp);

}m=0;for (i=1;i<=144;i++){

m++;itoa(m,nu,10); s1=s2+nu+".txt"; 

fp=fopen(s1.c_str(),"w"); fprintf(fp,"BOX %d\n",i); 

for (k=1;k<=soa1[i];k++) fprintf(fp,"%.3lf

  %.3lf\n",oa1[i][k],ob1[i][k]);fclose(fp);

}return 0;

}

Game# Type Rowmean Columnmean Summean Rowvar Columnvar Sumvar

1NE

Entropy1.5791.989

1.5771.995

3.1563.984

0.6580.350

0.6180.315

1.2970.684

2NE

Entropy1.6091.996

3.2021.994

4.8113.990

0.6230.324

0.4280.324

1.0620.662

3NE

Entropy3.1971.994

3.2282.011

6.4254.005

0.4290.330

0.4030.325

0.8200.667

4NE

Entropy3.1981.992

1.5851.996

4.7843.987

0.4150.319

0.6290.329

1.0440.647

5NE

Entropy2.3771.986

3.1842.002

5.5613.987

0.6390.321

0.4370.341

1.0390.654

6NE

Entropy1.9951.994

2.0011.999

3.9963.993

0.3360.336

0.3470.347

0.6690.669

7NE

Entropy2.5181.983

2.5361.993

5.0533.976

0.3700.327

0.3730.318

0.7410.645

8NE

Entropy3.2172.023

3.2242.011

6.4414.034

0.4220.333

0.4050.332

0.8480.678

9NE

Entropy2.3911.998

1.6032.011

3.9944.009

0.6580.347

0.6130.314

1.2440.641

10NE

Entropy2.5431.999

2.5382.001

5.0814.001

0.3770.327

0.3660.320

0.7320.630

11NE

Entropy2.3982.004

1.6052.000

4.0034.004

0.6310.335

0.6440.328

1.2600.666

12NE

Entropy2.4092.004

2.3991.994

4.8083.999

0.6470.340

0.6390.338

1.2930.673

13NE

Entropy3.1922.007

2.4102.012

5.6014.019

0.4150.324

0.6340.336

1.0510.669

14NE

Entropy3.2001.998

1.6061.995

4.8073.993

0.4260.329

0.6480.328

1.0580.668

15NE

Entropy3.1922.010

3.2011.986

6.3933.996

0.4300.342

0.4200.319

0.8660.666

16NE

Entropy2.3901.989

3.2342.014

5.6244.003

0.6550.335

0.4140.331

1.0340.664

17NE

Entropy3.2062.014

2.4202.015

5.6254.029

0.4350.341

0.6070.320

1.0480.660

18NE

Entropy2.3781.986

3.2172.013

5.5953.998

0.6410.341

0.4240.334

1.0360.663

Appendix4

Table14:Theoutputfork=9forthe144games

19NE

Entropy3.1701.970

3.1991.999

6.3693.969

0.4400.325

0.4320.327

0.8680.643

20NE

Entropy3.2092.005

3.1962.008

6.4054.013

0.4350.332

0.4160.312

0.8410.638

21NE

Entropy2.4042.004

2.392  2.004 44

4.7964.008

0.6200.337

0.6410.341

1.3030.695

22NE

Entropy3.1991.990

1.5991.995

4.7983.985

0.4190.326

0.6450.342

1.0500.652

23NE

Entropy3.2362.025

2.3891.985

5.6254.010

0.4340.329

0.6490.328

1.0900.653

Game# Type Rowmean Columnmean Summean Rowvar Columnvar Sumvar

26NE

Entropy3.1921.995

3.1841.989

6.3753.984

0.4430.335

0.4300.340

0.8480.675

27NE

Entropy2.3951.988

3.2042.000

5.5993.988

0.6480.329

0.4430.356

1.1010.673

28NE

Entropy3.2081.999

3.2232.017

6.4314.016

0.4250.324

0.4240.343

0.8400.692

29NE

Entropy3.2152.010

3.2032.002

6.4174.012

0.4180.331

0.4200.328

0.8360.634

30NE

Entropy2.3791.987

3.2072.006

5.5863.993

0.6440.337

0.4070.333

1.0550.666

31NE

Entropy2.4122.010

2.3761.988

4.7883.997

0.6430.331

0.6300.326

1.3020.674

32NE

Entropy3.1921.984

3.1981.991

6.3903.975

0.4200.319

0.4170.332

0.8220.637

33NE

Entropy2.3861.991

3.2061.995

5.5923.987

0.6290.326

0.4200.303

1.1120.636

34NE

Entropy2.4132.015

3.2092.010

5.6214.024

0.6280.333

0.4050.321

1.0420.630

35NE

Entropy2.4021.989

2.4302.011

4.8324.000

0.6180.320

0.6480.324

1.2830.642

36NE

Entropy3.1931.993

2.4182.003

5.6113.996

0.4390.351

0.6440.331

1.0850.701

37NE

Entropy1.5931.997

1.6162.017

3.2094.015

0.6200.326

0.6460.323

1.2620.661

38NE

Entropy1.5801.988

2.4062.004

3.9863.992

0.6230.318

0.6390.325

1.2180.619

39NE

Entropy1.9991.997

2.0202.019

4.0194.016

0.3260.326

0.3340.334

0.6630.663

40NE

Entropy1.6042.000

2.4122.006

4.0164.005

0.6630.347

0.6470.336

1.3090.672

41NE

Entropy2.2782.008

2.2681.998

4.5464.005

0.3630.346

0.3320.315

0.6950.655

42NE

Entropy3.2152.011

3.2131.988

6.4284.000

0.4130.326

0.4140.319

0.8530.668

43NE

Entropy1.6202.005

3.2041.994

4.8243.999

0.6120.334

0.4290.333

1.0630.670

44NE

Entropy2.0292.028

2.0172.016

4.0464.044

0.3270.327

0.3310.331

0.6450.645

45NE

Entropy1.5871.995

3.2182.013

4.8054.008

0.6340.328

0.4060.314

1.0450.645

46NE

Entropy2.2692.003

2.5562.023

4.8254.025

0.3490.331

0.3720.339

0.7150.669

47NE

Entropy3.2021.998

2.4161.991

5.6183.989

0.4450.339

0.6050.313

1.0400.624

48NE

Entropy3.1961.983

3.193  2.007 45

6.3893.989

0.4170.327

0.4180.325

0.8090.618

49NE

Entropy1.5921.992

3.1931.979

4.7853.971

0.6380.338

0.4400.331

1.0700.663

50NE

Entropy1.9891.987

2.0042.002

3.9923.990

0.3320.332

0.3350.335

0.6720.672

Game# Type Rowmean Columnmean Summean Rowvar Columnvar Sumvar

51NE

Entropy3.2072.000

2.3731.991

5.5793.991

0.4060.317

0.6260.328

1.0820.657

52NE

Entropy3.2012.011

1.5801.995

4.7814.005

0.4190.333

0.6380.332

1.0900.674

53NE

Entropy2.2782.009

2.2501.982

4.5283.991

0.3400.325

0.3440.323

0.6630.635

54NE

Entropy3.2022.007

3.1772.006

6.3804.013

0.4370.331

0.4390.339

0.8890.664

55NE

Entropy2.2531.985

2.5211.994

4.7743.980

0.3650.345

0.3970.347

0.7390.670

56NE

Entropy2.4332.024

1.6002.002

4.0344.026

0.6230.342

0.6330.325

1.3080.696

57NE

Entropy1.9891.988

2.0092.008

3.9993.996

0.3630.363

0.3300.330

0.6810.681

58NE

Entropy2.4112.002

2.4312.024

4.8414.026

0.6200.310

0.6550.346

1.3150.668

59NE

Entropy2.0082.006

2.0152.014

4.0234.020

0.3130.313

0.3290.329

0.6560.656

60NE

Entropy2.3781.992

2.4101.995

4.7883.988

0.6390.328

0.6610.344

1.2770.677

61NE

Entropy2.5271.991

2.2792.010

4.8064.001

0.3690.330

0.3630.341

0.7240.659

62NE

Entropy3.1871.981

3.2101.992

6.3973.973

0.4320.324

0.4140.323

0.8320.646

63NE

Entropy2.3591.972

3.2102.006

5.5693.978

0.6330.328

0.4050.330

1.0280.660

64NE

Entropy2.0222.020

2.0072.005

4.0284.025

0.3400.340

0.3440.344

0.6690.669

65NE

Entropy1.9941.992

1.9861.985

3.9803.977

0.3150.315

0.3540.354

0.6430.643

66NE

Entropy2.4152.016

3.1781.982

5.5943.998

0.6440.340

0.4350.338

1.0730.692

67NE

Entropy2.5572.015

2.5291.999

5.0864.014

0.3710.331

0.3790.340

0.7630.685

68NE

Entropy3.1931.994

2.4202.008

5.6144.001

0.4340.335

0.6040.316

1.0570.655

69NE

Entropy2.5221.984

2.5622.021

5.0844.005

0.3790.329

0.3560.317

0.7340.657

70NE

Entropy3.1951.989

3.2142.000

6.4093.989

0.4350.331

0.4250.324

0.8850.658

71NE

Entropy2.4272.018

3.2042.000

5.6314.018

0.6330.336

0.4000.313

1.0790.656

72NE

Entropy2.0072.006

1.9971.996

4.0044.002

0.3400.340

0.3230.323

0.6380.639

73NE

Entropy3.1961.996

2.382  1.989 46

5.5783.984

0.4240.330

0.6590.342

1.0570.670

74NE

Entropy2.5362.003

2.5502.015

5.0864.017

0.3730.329

0.3610.328

0.7310.666

75NE

Entropy3.1891.983

1.6182.007

4.8073.990

0.4390.337

0.6250.316

1.0360.640

Game# Type Rowmean Columnmean Summean Rowvar Columnvar Sumvar

76NE

Entropy2.5322.001

2.2772.005

4.8094.006

0.3800.340

0.3400.321

0.7230.658

77NE

Entropy3.2122.002

3.1971.995

6.4093.996

0.4290.332

0.4260.332

0.8690.657

78NE

Entropy2.5041.968

2.5372.001

5.0413.969

0.3790.334

0.3780.332

0.7560.669

79NE

Entropy2.4272.011

3.1982.003

5.6254.014

0.6370.325

0.4130.319

1.0540.649

80NE

Entropy2.4122.025

3.2011.999

5.6134.024

0.6330.328

0.4220.333

1.0500.651

81NE

Entropy3.1961.986

2.4082.006

5.6053.992

0.4330.332

0.6370.337

1.0740.676

82NE

Entropy3.1902.009

3.1781.990

6.3683.999

0.4330.345

0.4540.344

0.8770.684

83NE

Entropy3.2132.021

3.2051.992

6.4194.013

0.4490.349

0.4260.326

0.8670.678

84NE

Entropy2.3982.001

2.3831.988

4.7803.988

0.6330.329

0.6420.339

1.3070.686

85NE

Entropy3.1881.994

3.1982.000

6.3863.994

0.4300.326

0.4330.327

0.8850.653

86NE

Entropy2.4372.022

3.1801.974

5.6173.997

0.6490.336

0.4500.337

1.0570.664

87NE

Entropy2.3992.003

3.2022.002

5.6024.005

0.6100.327

0.4240.333

1.0150.646

88NE

Entropy2.3871.994

1.6212.012

4.0084.006

0.6320.324

0.6360.333

1.2570.661

89NE

Entropy2.4172.011

2.4202.007

4.8374.019

0.6300.325

0.6490.340

1.3350.688

90NE

Entropy3.1941.988

2.3882.000

5.5823.988

0.4430.335

0.6330.338

1.1200.696

91NE

Entropy3.2112.003

1.6242.020

4.8354.023

0.4240.321

0.6440.333

1.0920.666

92NE

Entropy3.1781.979

3.1941.990

6.3723.969

0.4600.339

0.4190.322

0.9050.664

93NE

Entropy3.2112.000

3.2172.028

6.4284.028

0.4000.318

0.4270.335

0.8320.660

94NE

Entropy2.4001.993

3.1982.004

5.5983.997

0.6470.335

0.4250.341

1.0770.683

95NE

Entropy3.1861.980

2.4172.010

5.6023.991

0.4220.329

0.6510.342

1.0580.654

96NE

Entropy2.3931.985

3.1681.970

5.5613.956

0.6240.330

0.4430.341

1.0670.646

97NE

Entropy3.2092.007

1.5811.980

4.7903.988

0.4160.328

0.6220.322

1.0520.656

98NE

Entropy3.1932.002

2.404  2.007 47

5.5974.009

0.4220.336

0.6120.324

1.0240.643

99NE

Entropy2.3911.994

2.4232.018

4.8144.012

0.6510.337

0.6230.328

1.2600.660

100NE

Entropy2.3721.984

1.6021.993

3.9743.977

0.6630.344

0.6510.337

1.3080.677

Game# Type Rowmean Columnmean Summean Rowvar Columnvar Sumvar

101NE

Entropy1.5941.997

3.2061.995

4.8003.992

0.6330.331

0.4220.330

1.0380.676

102NE

Entropy3.2092.025

3.2012.011

6.4114.036

0.4510.350

0.4370.344

0.8960.685

103NE

Entropy1.6032.002

3.1761.998

4.7794.000

0.6580.346

0.4190.323

1.0840.661

104NE

Entropy3.2062.010

2.3721.989

5.5783.999

0.4250.327

0.6600.345

1.1160.677

105NE

Entropy1.5681.974

2.4012.022

3.9693.997

0.6170.320

0.6280.329

1.2850.660

106NE

Entropy3.2142.015

3.2152.009

6.4304.024

0.4220.338

0.4310.337

0.8530.681

107NE

Entropy3.1861.990

3.2282.019

6.4144.009

0.4310.329

0.4260.335

0.8600.696

108NE

Entropy1.6182.013

3.2061.993

4.8244.006

0.6360.338

0.4110.316

1.0310.665

109NE

Entropy3.2052.003

2.4052.012

5.6104.016

0.4270.332

0.6510.339

1.0530.649

110NE

Entropy1.5711.989

2.4031.991

3.9743.980

0.6710.351

0.6450.327

1.2680.652

111NE

Entropy3.2262.019

3.2051.987

6.4314.006

0.4100.325

0.4300.333

0.8340.656

112NE

Entropy1.6122.005

3.2152.021

4.8274.026

0.6330.333

0.4200.333

0.9950.637

113NE

Entropy3.1921.990

3.2061.994

6.3983.984

0.4360.335

0.4230.335

0.8270.664

114NE

Entropy1.6111.996

3.2092.006

4.8204.002

0.6410.349

0.4110.334

1.0400.677

115NE

Entropy3.2092.001

1.6302.027

4.8394.028

0.4200.325

0.6230.332

1.0410.654

116NE

Entropy1.6001.998

1.5971.991

3.1973.989

0.6510.336

0.6370.340

1.2520.664

117NE

Entropy3.2121.999

2.3882.000

5.6003.998

0.4000.316

0.6520.342

1.0890.676

118NE

Entropy1.6282.017

2.4092.002

4.0374.018

0.6390.344

0.6400.338

1.2780.700

119NE

Entropy3.1811.975

2.3891.996

5.5703.971

0.4520.344

0.6490.345

1.1150.688

120NE

Entropy1.6172.004

2.4052.007

4.0224.012

0.6730.341

0.6220.325

1.3010.664

121NE

Entropy3.1951.993

1.5531.968

4.7483.961

0.4340.323

0.5990.327

1.0780.662

122NE

Entropy2.5522.014

2.2541.990

4.8064.004

0.3820.344

0.3500.331

0.7190.658

123NE

Entropy2.3591.969

2.395  1.996 48

4.7543.965

0.6310.325

0.6490.336

1.2910.645

124NE

Entropy2.5322.002

2.5392.009

5.0714.011

0.3690.332

0.3780.335

0.7540.680

125NE

Entropy2.3841.987

3.1952.007

5.5793.994

0.6240.320

0.4220.335

1.0320.634

Game# Type Rowmean Columnmean Summean Rowvar Columnvar Sumvar

126NE

Entropy2.0032.002

2.0001.999

4.0034.000

0.3390.339

0.3350.335

0.6880.688

127NE

Entropy2.4082.011

2.4262.016

4.8344.027

0.6540.342

0.6410.332

1.2840.666

128NE

Entropy2.3831.970

3.2022.008

5.5853.978

0.6720.342

0.4100.330

1.0320.656

129NE

Entropy2.5372.001

2.5512.022

5.0884.023

0.3820.345

0.3750.335

0.7740.695

130NE

Entropy2.0022.000

2.0202.019

4.0224.019

0.3330.333

0.3370.337

0.6720.672

131NE

Entropy3.2182.017

3.2051.991

6.4234.008

0.4140.335

0.4260.325

0.8170.671

132NE

Entropy3.1891.999

3.2001.997

6.3893.996

0.4350.338

0.4280.332

0.8700.679

133NE

Entropy2.5452.007

2.2872.020

4.8324.027

0.3540.322

0.3480.327

0.7130.658

134NE

Entropy2.0212.020

1.9961.995

4.0184.015

0.3330.333

0.3280.328

0.6480.649

135NE

Entropy3.2252.014

2.3961.997

5.6214.011

0.4190.327

0.6420.337

1.0340.634

136NE

Entropy1.9991.998

1.9811.979

3.9803.977

0.3460.346

0.3210.321

0.6790.679

137NE

Entropy3.2122.002

2.3841.994

5.5963.995

0.4160.320

0.6450.343

1.0700.657

138NE

Entropy2.0042.002

1.9991.997

4.0023.999

0.3400.340

0.3340.334

0.7030.703

139NE

Entropy2.2712.003

2.5431.998

4.8144.001

0.3500.327

0.3760.333

0.7370.667

140NE

Entropy2.2852.017

2.5211.998

4.8054.015

0.3450.331

0.3720.330

0.7100.651

141NE

Entropy1.9771.975

1.9911.990

3.9683.965

0.3200.320

0.3440.344

0.6740.674

142NE

Entropy2.0072.005

2.0022.001

4.0094.006

0.3480.348

0.3230.323

0.6790.679

143NE

Entropy2.0112.009

2.0092.008

4.0204.017

0.3280.328

0.3270.327

0.6450.645

144NE

Entropy2.0142.013

2.0172.016

4.0314.028

0.3190.319

0.3290.329

0.6410.642

49