simultaneously

Prisoner'sdilemma

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1988novel,ePrisoner's

Dilemma(novel).FortheDoctorWhoaudiobook,eThePrisoner'sDilemma.

Forthe2001play,eThePrisoner'sDilemma(play).

erencesud

maybemadeclearerwithadifferentorconsistentstyleof

citation,footnoting,orexternallinking.(October2012)

Theprisoners'dilemmaisacanonicalexampleofagameanalyzedingame

theorythatshowswhytwopurely"rational"individualsmightnot

cooperate,evenifitappearsthatitisintheirbestinterests[citationneeded]

riginallyframedbyMerrillFloodandMelvinDresher

formalizedthegamewithprison

ntencerewardsandgaveitthename"prisoner'sdilemma"(Poundstone,

1992),prentingitasfollows:

prisonerisinsolitaryconfinementwithnomeansofspeakingto

iceadmittheydon't

haveenoughevidencetoconvictthepairontheprincipalcharge.

Theyplantontencebothtoayearinprisononalesrcharge.

Simultaneously,thepoliceoffereachprisoneraFaustianbargain.

Eachprisonerisgiventheopportunityeithertobetraytheother,

bytestifyingthattheothercommittedthecrime,ortocooperate

'showitgoes:

IfAandBbothbetraytheother,eachofthemrves2years

inprison

IfAbetraysBbutBremainssilent,Awillbetfreeand

Bwillrve3yearsinprison(andviceversa)

IfAandBbothremainsilent,bothofthemwillonlyrve

1yearinprison(onthelesrcharge)

Itisimpliedthattheprisonerswillhavenoopportunitytorewardor

punishtheirpartnerotherthantheprisonntencestheyget,andthat

e

betrayingapartneroffersagreaterrewardthancooperatingwiththem,

allpurelyrationallf-interestedprisonerswouldbetraytheother,and

sotheonlypossibleoutcomefortwopurelyrationalprisonersisforthem

tobetrayeachother.[1]Theinterestingpartofthisresultisthat

pursuingindividualrewardlogicallyleadsbothoftheprisonerstobetray,

ity,

humansdisplayasystematicbiastowardscooperativebehaviorinthisand

similargames,muchmoresothanpredictedbysimplemodelsof"rational"

lf-interestedaction.[2][3][4][5]Amodelbadonadifferentkindof

rationality,wherepeopleforecasthowthegamewouldbeplayedifthey

formedcoalitionsandthentheymaximizetheirforecasts,hasbeenshown

tomakebetterpredictionsoftherateofcooperationinthisandsimilar

gamesgivenonlythepayoffsofthegame.[6]

Thereisalsoanextended"iterated"versionofthegame,wheretheclassic

gameisplayedoverandoverbetweenthesameprisoners,andconquently,

bothprisonerscontinuouslyhaveanopportunitytopenalizetheotherfor

umberoftimesthegamewillbeplayedisknown

totheplayers,then(bybackwardinduction)twoclassicallyrational

playerswillbetrayeachotherrepeatedly,forthesamereasonsasthe

finiteorunknownlengthgamethereisno

fixedoptimumstrategy,andPrisoner'sDilemmatournamentshavebeenheld

tocompeteandtestalgorithms.

Theprisoner'sdilemmagamecanbeudasamodelformanyrealworld

alusage,thelabel

"prisoner'sdilemma"maybeappliedtosituationsnotstrictlymatching

theformalcriteriaoftheclassicoriterativegames:forinstance,tho

inwhichtwoentitiescouldgainimportantbenefitsfromcooperatingor

sufferfromthefailuretodoso,butfinditmerelydifficultorexpensive,

notnecessarilyimpossible,tocoordinatetheiractivitiestoachieve

cooperation.

Contents

[hide]

1Strategyfortheclassicprisoners'dilemma

2Generalizedform

o2.1Specialca:Donationgame

3Theiteratedprisoners'dilemma

o3.1Strategyfortheiteratedprisoners'dilemma

o3.2Stochasticiteratedprisoner'sdilemma

3.2.1Zero-determinantstrategies

o3.3Continuousiteratedprisoners'dilemma

o3.4EmergenceofStableStrategies

4Real-lifeexamples

o4.1Inenvironmentalstudies

o4.2Inanimals

o4.3Inpsychology

o4.4Ineconomics

o4.5Insport

o4.6Multiplayerdilemmas

o4.7Armsraces

5Relatedgames

o5.1Clod-bagexchange

o5.2FriendorFoe?

o5.3Iteratedsnowdrift

6Seealso

7References

8Furtherreading

9Externallinks

Strategyfortheclassicprisoners'

dilemma[edit]

Thenormalgameisshownbelow:

PrisonerBstayssilent

(cooperates)

PrisonerBbetrays

(defects)

PrisonerAstayssilent

(cooperates)

Eachrves1year

PrisonerA:3years

PrisonerB:goes

free

PrisonerAbetrays

(defects)

PrisonerA:goesfree

PrisonerB:3years

Eachrves2years

Here,regardlessofwhattheotherdecides,eachprisonergetsahigher

pay-offbybetrayingtheother("defecting").Thereasoninginvolvesan

argumentbydilemma:perates,

Ashoulddefect,

defects,Ashouldalsodefect,sincerving2yearsisbetterthanrving

erway,elreasoningwillshowthatB

shoulddefect.

Intraditionalgametheory,someveryrestrictiveassumptionsonprisoner

sumedthatbothunderstandthenatureofthe

game,andthatdespitebeingmembersofthesamegang,theyhavenoloyalty

toeachotherandwillhavenoopportunityforretributionorreward

portantly,averynarrowinterpretationof

"rationality"isappliedindefiningthedecision-makingstrategiesof

heconditionsandthepayoffsabove,prisonerA

eissymmetric,soPrisonerBshouldact

oth"rationally"decidetodefect,eachreceivesa

ionalgametheory

resultsinbothplayersbeingworoffthanifeachchotolesnthe

ntenceofhisaccompliceatthecostofspendingmoretimeinjail

himlf.

Generalizedform[edit]

ThestructureofthetraditionalPrisoners’Dilemmacanbegeneralized

ethatthetwoplayersare

reprentedbythecolors,redandblue,andthateachplayerchoosto

either"Cooperate"or"Defect".

Ifbothplayerscooperate,theybothreceivethereward,R,for

defectswhileRedcooperates,thenBluereceivesthe

temptation,TpayoffwhileRedreceivesthe"sucker's",S,payoff.

Similarly,ifBluecooperateswhileReddefects,thenBluereceivesthe

sucker'

playersdefect,theybothreceivethepunishmentpayoffP.

Thiscanbeexpresdinnormalform:

CanonicalPDpayoffmatrix

CooperateDefect

CooperateR,RS,T

DefectT,SP,P

andtobeaprisoner'sdilemmagameinthestrongn,thefollowing

conditionmustholdforthepayoffs:

T>R>P>S

ThepayoffrelationshipR>Pimpliesthatmutualcooperationissuperior

tomutualdefection,whilethepayoffrelationshipsT>RandP>Simply

,mutual

defectionistheonlystrongNashequilibriuminthegame(i.e.,theonly

outcomefromwhicheachplayercouldonlydoworbyunilaterally

changingstrategy).Thedilemmathenisthatmutualcooperationyields

abetteroutcomethanmutualdefectionbutitisnottherationaloutcome

becauthechoicetocooperate,attheindividuallevel,isnotrational

fromalf-interestedpointofview.

Specialca:Donationgame[edit]

The"donationgame"[7]isaformofprisoner'sdilemmainwhichcooperation

correspondstoofferingtheotherplayerabenefitbatapersonalcost

cwithb>offmatrixisthus

CooperateDefect

Cooperateb-c,b-c-c,b

Defectb,-c0,0

Notethat2R>T+S(i.e.2(b-c)>b-c)whichqualifiesthedonationgameto

beaniteratedgame(enextction).

eXgrowsoranges,Y

ginalutilityofanappletotheorange-growerXis

b,whichishigherthanthemarginalutility(c)ofanorange,sinceX

rly,forapple-growerY,

themarginalutilityofanorangeisbwhilethemarginalutilityofan

Ycontracttoexchangeanappleandanorange,and

eachfulfillstheirendofthedeal,theneachreceiveapayoffofb-c.

Ifone"defects"anddoesnotdeliveraspromid,thedefectorwill

receiveapayoffofb,defect,

thenneitheronegainsorlosanything.

Theiteratedprisoners'dilemma[edit]

helpimprovethisarticlebyaddingcitationstoreliable

cedmaterialmaybechallengedandremoved.

(November2012)

Iftwoplayersplayprisoners'dilemmamorethanonceinsuccessionand

theyrememberpreviousactionsoftheiropponentandchangetheirstrategy

accordingly,thegameiscallediteratedprisoners'dilemma.

Inadditiontothegeneralformabove,theiterativeversionalsorequires

that2R>T+S,topreventalternatingcooperationanddefectiongiving

agreaterrewardthanmutualcooperation.

Theiteratedprisoners'dilemmagameisfundamentaltocertaintheories

ssumptionthatthegamecanmodel

transactionsbetweentwopeoplerequiringtrust,cooperativebehaviour

inpopulationsmaybemodeledbyamulti-player,iterated,versionofthe

,conquently,

1975,GrofmanandPoolestimatedthecountofscholarlyarticlesdevoted

toitatover2,ratedprisoners'dilemmahasalsobeenreferred

toasthe"Peace-Wargame".[8]

IfthegameisplayedexactlyNtimesandbothplayersknowthis,then

y

ofisinductive:

onemightaswelldefectonthelastturn,sincetheopponentwillnot

ore,bothwilldefectonthe

,theplayermightaswelldefectonthecond-to-last

turn,sincetheopponentwilldefectonthelastnomatterwhatisdone,

eappliesifthegamelengthisunknownbuthasaknown

upperlimit.

Unlikethestandardprisoners'dilemma,intheiteratedprisoners'

dilemmathedefectionstrategyiscounter-intuitiveandfailsbadlyto

standardeconomictheory,

though,errationalstrategyin

theiteratedprisoners'dilemmawithfixedNistocooperateagainsta

superrationalopponent,andinthelimitoflargeN,experimentalresults

onstrategiesagreewiththesuperrationalversion,notthe

game-theoreticrationalone.

Forcooperationtoemergebetweengametheoreticrationalplayers,the

totalnumberofroundsNmustberandom,oratleastunknowntotheplayers.

Inthisca'alwaysdefect'maynolongerbeastrictlydominantstrategy,

tresultsshownbyRobertAumannina1959

paper,rationalplayersrepeatedlyinteractingforindefinitelylong

gamescansustainthecooperativeoutcome.

Strategyfortheiteratedprisoners'dilemma[edit]

Interestintheiteratedprisoners'dilemma(IPD)waskindledbyRobert

AxelrodinhisbookTheEvolutionofCooperation(1984).Inithereports

onatournamentheorganizedoftheNstepprisoners'dilemma(withNfixed)

inwhichparticipantshavetochootheirmutualstrategyagainandagain,

dinvitedacademic

colleaguesallovertheworldtodevicomputerstrategiestocompete

gramsthatwereenteredvariedwidelyin

algorithmiccomplexity,initialhostility,capacityforforgiveness,and

soforth.

Axelroddiscoveredthatwhentheencounterswererepeatedoveralong

periodoftimewithmanyplayers,eachwithdifferentstrategies,greedy

strategiestendedtodoverypoorlyinthelongrunwhilemorealtruistic

strategiesdidbetter,this

toshowapossiblemechanismfortheevolutionofaltruisticbehaviour

frommechanismsthatareinitiallypurelylfish,bynaturallection.

Thewinningdeterministicstrategywastitfortat,whichAnatolRapoport

hesimplestofany

programentered,containingonlyfourlinesofBASIC,andwonthecontest.

Thestrategyissimplytocooperateonthefirstiterationofthegame;

afterthat,theplayerdoeswhathisorheropponentdidontheprevious

ingonthesituation,aslightlybetterstrategycanbe"tit

fortatwithforgiveness."Whentheopponentdefects,onthenextmove,

theplayersometimescooperatesanyway,withasmallprobability(around

1–5%).Thisallowsforoccasionalrecoveryfromgettingtrappedina

ctprobabilitydependsontheline-upof

opponents.

Byanalysingthetop-scoringstrategies,Axelrodstatedveral

conditionsnecessaryforastrategytobesuccessful.

Nice

Themostimportantconditionisthatthestrategymustbe"nice",

thatis,itwillnotdefectbeforeitsopponentdoes(thisis

sometimesreferredtoasan"optimistic"algorithm).Almostallof

thetop-scoringstrategieswerenice;therefore,apurelylfish

strategywillnot"cheat"onitsopponent,forpurely

lf-interestedreasonsfirst.

Retaliating

However,Axelrodcontended,thesuccessfulstrategymustnotbea

pleofa

averybad

choice,as"nasty"strategieswillruthlesslyexploitsuchplayers.

Forgiving

playerswill

retaliate,theywillonceagainfallbacktocooperatingifthe

opslongrunsof

revengeandcounter-revenge,maximizingpoints.

Non-envious

Thelastqualityisbeingnon-envious,thatisnotstrivingtoscore

morethantheopponent.

Theoptimal(points-maximizing)strategyfortheone-timePDgameis

simplydefection;axplainedabove,thisistruewhateverthe

r,intheiterated-PDgamethe

optimalstrategydependsuponthestrategiesoflikelyopponents,andhow

mple,considera

populationwhereeveryonedefectverytime,exceptforasingle

dividualisata

a

population,theoptimalstrategyforthatindividualistodefectevery

ulationwithacertainpercentageofalways-defectorsand

therestbeingtitfortatplayers,theoptimalstrategyforanindividual

dependsonthepercentage,andonthelengthofthegame.

InthestrategycalledPavlov,win-stay,lo-switch,Ifthelastround

outcomewasP,P,aPavlovplayerswitchesstrategythenextturn,which

meansP,Pwouldbeconsideredasafailuretocooperate.[citationneeded]Fora

certainrangeofparameters[specify],Pavlovbeatsallotherstrategiesby

givingpreferentialtreatmenttoco-playerswhichremblePavlov.

Derivingtheoptimalstrategyisgenerallydoneintwoways:

anNashEquilibrium:Ifthestatisticaldistributionof

opposingstrategiescanbedetermined(e.g.50%titfortat,50%

alwayscooperate)anoptimalcounter-strategycanbederived

analytically.[9]

arlosimulationsofpopulationshavebeenmade,where

individualswithlowscoresdieoff,andthowithhighscores

reproduce(ageneticalgorithmforfindinganoptimalstrategy).

Themixofalgorithmsinthefinalpopulationgenerallydependson

roductionofmutation

(randomvariationduringreproduction)lesnsthedependencyon

theinitialpopulation;empiricalexperimentswithsuchsystems

tendtoproducetitfortatplayers(eforinstanceChess1988),

butthereisnoanalyticproofthatthiswillalwaysoccur.

Althoughtitfortatisconsideredtobethemostrobustbasicstrategy,

ateamfromSouthamptonUniversityinEngland(ledbyProfessorNicholas

JenningsandconsistingofRajdeepDash,SarvapaliRamchurn,AlexRogers,

PerukrishnenVytelingum)introducedanewstrategyatthe

20th-anniversaryiteratedprisoners'dilemmacompetition,whichproved

rategyreliedoncooperation

betweenprogramstoachievethehighestnumberofpointsforasingle

versitysubmitted60programstothecompetition,which

weredesignedtorecognizeeachotherthroughariesoffivetotenmoves

atthestart.[10]Oncethisrecognitionwasmade,oneprogramwouldalways

cooperateandtheotherwouldalwaysdefect,assuringthemaximumnumber

rogramrealizedthatitwasplaying

anon-Southamptonplayer,itwouldcontinuouslydefectinanattemptto

ult,[11]thisstrategy

endeduptakingthetopthreepositionsinthecompetition,aswellas

anumberofpositionstowardsthebottom.

Thisstrategytakesadvantageofthefactthatmultipleentrieswere

allowedinthisparticularcompetitionandthattheperformanceofateam

wasmeasuredbythatofthehighest-scoringplayer(meaningthattheu

oflf-sacrificingplayerswasaformofminmaxing).Inacompetition

whereonehascontrolofonlyasingleplayer,titfortatiscertainly

eofthisnewrule,thiscompetitionalsohas

littletheoreticalsignificancewhenanalysingsingleagentstrategies

ascomparedtoAxelrod'r,itprovidedthe

frameworkforanalysinghowtoachievecooperativestrategiesin

multi-agentframeworks,,long

beforethisnew-rulestournamentwasplayed,RichardDawkinsinhisbook

TheSelfishGenepointedoutthepossibilityofsuchstrategieswinning

ifmultipleentrieswereallowed,butheremarkedthatmostprobably

reliesoncircumventingrulesabouttheprisoners'dilemmainthatthere

isnocommunicationallowedbetweenthetwoplayers,whichtheSouthampton

programsarguablydidwiththeiropening"tenmovedance"torecognize

oneanother;thisonlyreinforcesjusthowvaluablecommunicationcanbe

inshiftingthebalanceofthegame.

Stochasticiteratedprisoner'sdilemma[edit]

Inastochasticiteratedprisoner'sdilemmagame,strategiesare

specifiedbyintermsof"cooperationprobabilities".[12]Inanencounter

betweenplayerXandplayerY,X'sstrategyisspecifiedbyatof

nctionoftheoutcomes

function

ofonlytheirmostrecentnencounters,itiscalleda"memory-n"strategy.

Amemory-1strategyisthenspecifiedbyfourcooperationprobabilities:

,whereistheprobabilitythatXwill

cooperateintheprentencountergiventhatthepreviouncounterwas

characterizedby(ab).Forexample,ifthepreviouncounterwasonein

whichXcooperatedandYdefected,thenistheprobabilitythatXwill

oftheprobabilitiesare

either1or0,pleofa

deterministicstrategyisthe"titfortat"strategywrittenas

P={1,0,1,0},inwhichXrespondsasYdidinthepreviouncounter.

Anotheristhewin–stay,lo–switchstrategywrittenasP={1,0,0,1},

inwhichXrespondsasinthepreviouncounter,ifitwasa"win"(i.e.

ccordc)butchangesstrategyifitwasaloss().Ithas

beenshownthatforanymemory-nstrategythereisacorresponding

memory-1strategywhichgivesthesamestatisticalresults,sothatonly

memory-1strategiesneedbeconsidered.[12]

IfwedefinePastheabove4-elementstrategyvectorofXand

asthe4-elementstrategyvectorofY,a

transitionmatrixMmaybedefinedforXwhoijthentryisthe

probabilitythattheoutcomeofaparticularencounterbetweenXandY

willbejgiventhatthepreviouncounterwasi,whereiandjareone

ofthefouroutcomeindices:cc,cd,dc,mple,fromX's

pointofview,theprobabilitythattheoutcomeoftheprentencounter

isccgiventhatthepreviouncounterwascdiqualto

.(NotethattheindicesforQarefromY'spoint

ofview:acdoutcomeforXisadcoutcomeforY.)Underthedefinitions,

theiteratedprisoner'sdilemmaqualifiesasastochasticprocessandM

isastochasticmatrix,allowingallofthetheoryofstochasticprocess

tobeapplied.[12]

Oneresultofstochastictheoryisthatthereexistsastationaryvector

tlossofgenerality,it

maybespecifiedthatvisnormalizedsothatthesumofitsfourcomponents

hentryinwillgivetheprobabilitythattheoutcome

ofanencounterbetweenXandYwillbejgiventhattheencounternsteps

imitasnapproachesinfinity,Mwillconvergeto

amatrixwithfixedvalues,givingthelong-termprobabilitiesofan

rwordsthe

rowsofwillbeidentical,givingthelong-termequilibriumresult

probabilitiesoftheiteratedprisonersdilemmawithouttheneedto

eenthat

visastationaryvectorforandparticularly,sothateachrow

estationaryvectorspecifiesthe

ngand

astheshort-termpayoffvectorsforthe{cc,cd,dc,dd}

outcomes(FromX'spointofview),theequilibriumpayoffsforXandY

cannowbespecifiedasand,allowingthetwo

strategiesPandQtobecomparedfortheirlongtermpayoffs.

Zero-determinantstrategies[edit]

Therelationshipbetweenzero-determinant(ZD),cooperatingand

defectingstrategiesintheIteratedPrisoner’sDilemma(IPD).

Cooperatingstrategiesalwayscooperatewithothercooperating

strategies,anddefectingstrategiesalwaysdefectagainstother

ntainsubtsofstrategiesthatarerobust

understronglection,meaningnoothermemory-1strategyislected

cooperatingstrategiescontainasubtthatarealwaysrobust,meaning

thatnoothermemory-1strategyislectedtoinvadeandreplacesuch

strategies,erction

betweenZDandgoodcooperatingstrategiesisthetofgenerousZD

ionstrategiesaretheinterctionbetweenZDand

-for-tatliesattheinterctionof

cooperating,defectingandZDstrategies.

In2012,ndFreemanDysonpublishedanewclassof

strategiesforthestochasticiteratedprisoner'sdilemmacalled

"zero-determinant"(ZD)strategies.[12]Thelongtermpayoffsfor

encountersbetweenXandYcanbeexpresdasthedeterminantofamatrix

whichisafunctionofthetwostrategiesandtheshorttermpayoffvectors:

and,whichdonotinvolvethe

hedeterminantfunctionis

linearinf,itfollowsthat

(whereU={1,1,1,1}).Any

strategiesforwhichisbydefinitiona

ZDstrategy,andthelongtermpayoffsobeytherelation

.

Tit-for-tatisaZDstrategywhichis"fair"inthenofnotgaining

r,theZDspacealsocontains

strategiesthat,inthecaoftwoplayers,canallowoneplayerto

unilaterallyttheotherplayer'sscoreoralternatively,forcean

evolutionaryplayertoachieveapayoffsomepercentagelowerthanhis

ortedplayercoulddefectbutwouldtherebyhurthimlfby

,extortionsolutionsturntheiterated

prisoner'ically,Xisable

tochooastrategyforwhich,unilaterally

ttingtoaspecificvaluewithinaparticularrangeofvalues,

independentofY'sstrategy,offeringanopportunityforXto"extort"

playerY(andviceversa).(ItturnsoutthatifXtriestottoa

particularvalue,therangeofpossibilitiesismuchsmaller,only

consistingofcompletecooperationorcompletedefection.[12])

AnextensionoftheIPDisanevolutionarystochasticIPD,inwhichthe

relativeabundanceofparticularstrategiesisallowedtochange,with

ocessmaybe

accomplishedbyhavinglesssuccessfulplayersimitatethemore

successfulstrategies,orbyeliminatinglesssuccessfulplayersfromthe

game,eenshownthat

intuitionis

thatanevolutionarilystablestrategymustnotonlybeabletoinvade

anotherpopulation(whichextortionaryZDstrategiescando)butmustalso

performwellagainstotherplayersofthesametype(whichextortionary

ZDplayersdopoorly,becautheyreduceeachother'ssurplus).[13]

Theoryandsimulationsconfirmthatbeyondacriticalpopulationsize,

ZDextortionlosoutinevolutionarycompetitionagainstmore

cooperativestrategies,andasaresult,theaveragepayoffinthe

tion,there

aresomecasinwhichextortionersmayevencatalyzecooperationby

helpingtobreakoutofaface-offbetweenuniformdefectorsandwin–stay,

lo–switchagents.[14]

WhileextortionaryZDstrategiesarenotstableinlargepopulations,

anotherZDclasscalled"generous"strategiesisbothstableandrobust.

Infact,whenthepopulationisnottoosmall,thestrategiescan

supplantanyotherZDstrategyandevenperformwellagainstabroadarray

ofgenericstrategiesforiteratedprisoner'sdilemma,including

win–stay,lo–sprovenspecificallyforthedonation

gamebyAlexanderStewartandJoshuaPlotkinin2013.[15]Generous

strategieswillcooperatewithothercooperativeplayers,andintheface

ofdefection,thegenerousplayerlosmoreutilitythanitsrival.

GenerousstrategiesaretheinterctionofZDstrategiesandso-called

"good"strategies,whichweredefinedbyAkin(2013)[16]tobethofor

whichtheplayerrespondstopastmutualcooperationwithfuture

cooperationandsplitxpectedpayoffquallyifshereceivesatleast

oodstrategies,thegenerous(ZD)

populationisverysmall,defectionstrategiestendtodominate.[15]

Continuousiteratedprisoners'dilemma[edit]

Mostworkontheiteratedprisoners'dilemmahasfocudonthediscrete

ca,inwhichplayerithercooperateordefect,becauthismodelis

r,somerearchershavelookedat

modelsofthecontinuousiteratedprisoners'dilemma,inwhichplayers

oyd[17]

foundthatinsuchsituations,cooperationismuchhardertoevolvethan

inthediscreteiteratedprisoners'icintuitionforthis

resultisstraightforward:inacontinuousprisoners'dilemma,ifa

populationstartsoffinanon-cooperativeequilibrium,playerswhoare

onlymarginallymorecooperativethannon-cooperatorsgetlittlebenefit

rast,inadiscreteprisoners'

dilemma,titfortatcooperatorsgetabigpayoffboostfromassorting

withoneanotherinanon-cooperativeequilibrium,relativeto

aturearguablyoffersmoreopportunitiesfor

variablecooperationratherthanastrictdichotomyofcooperationor

defection,thecontinuousprisoners'dilemmamayhelpexplainwhy

real-lifeexamplesoftitfortat-likecooperationareextremelyrarein

nature(stein[18])eventhoughtitfortatemsrobustin

theoreticalmodels.

EmergenceofStableStrategies[edit]

Playerscannotemtocoordinatemutualcooperation,thusoftenget

way,

iteratedroundsfacilitatetheevolutionofstablestrategies.[19]Iterated

roundsoftenproducenovelstrategies,whichhaveimplicationstocomplex

strategyoutperformsasimpleTit-For-Tatstrategy-thatis,ifyoucan

getawaywithcheating,repeatthatbehavior,howeverifyougetcaught,

switch.[20]

Real-lifeexamples[edit]

Theprisonerttingmayemcontrived,butthereareinfactmany

examplesinhumaninteractionaswellasinteractionsinnaturethathave

soner'sdilemmaisthereforeofinterest

tothesocialsciencessuchaconomics,politics,andsociology,aswell

astothebiologicalsciencessuchathologyandevolutionarybiology.

Manynaturalprocesshavebeenabstractedintomodelsinwhichliving

beingsareengagedinendlessgamesofprisoner'de

applicabilityofthePDgivesthegameitssubstantialimportance.

Inenvironmentalstudies[edit]

Inenvironmentalstudies,thePDividentincrissuchasglobal

guedallcountrieswillbenefitfromastable

climate,butanysinglecountryisoftenhesitanttocurbCO

ediatebenefittoanindividualcountrytomaintain

currentbehaviorisperceivedtobegreaterthanthepurportedeventual

benefittoallcountriesifbehaviorwaschanged,thereforeexplaining

thecurrentimpasconcerningclimatechange.[21]

Animportantdifferencebetweenclimatechangepoliticsandthe

prisoner'sdilemmaisuncertainty;theextentandpaceatwhichpollution

emmafacedbygovernmentis

thereforedifferentfromtheprisoner'sdilemmainthatthepayoffsof

fferencesuggestsstateswillcooperate

muchlessthaninarealiteratedprisoner'sdilemma,sothatthe

probabilityofavoidingapossibleclimatecatastropheismuchsmaller

thanthatsuggestedbyagame-theoreticalanalysisofthesituationusing

arealiteratedprisoner'sdilemma.[22]

OsangandNandyprovideatheoreticalexplanationwithproofsfora

regulation-drivenwin-winsituationalongthelinesofMichaelPorter's

hypothesis,inwhichgovernmentregulationofcompetingfirmsis

substantial.[23]

Inanimals[edit]

Cooperativebehaviorofmanyanimalscanbeunderstoodasanexampleof

theprisoner'nimalngageinlongtermpartnerships,

whichcanbemorespecificallymodeledasiteratedprisoner'sdilemma.

Forexample,guppiesinspectpredatorscooperativelyingroups,andthey

arethoughttopunishnon-cooperativeinspectorsbytitfortat

strategy.[citationneeded]

Vampirebatsaresocialanimalsthatengageinreciprocalfoodexchange.

Applyingthepayoffsfromtheprisoner'sdilemmacanhelpexplainthis

behavior:[24]

C/C:"Reward:Igetbloodonmyunluckynights,whichsavesmefrom

ogivebloodonmyluckynights,whichdoesn't

costmetoomuch."

D/C:"Temptation:nIget

theaddedbenefitofnothavingtopaytheslightcostoffeeding

youonmygoodnight."

C/D:"Sucker'sPayoff:Ipaythecostofsavingyourlifeonmygood

ybadnightyoudon'tfeedmeandIrunarealrisk

ofstarvingtodeath."

D/D:"Punishment:Idon'thavetopaytheslightcostsoffeeding

narealriskofstarvingonmypoor

nights."

Inpsychology[edit]

Inaddictionrearch/behavioraleconomics,GeorgeAinsliepointsout[25]

thataddictioncanbecastasanintertemporalPDproblembetweenthe

ca,defectingmeans

relapsing,anditiasytoethatnotdefectingbothtodayandinthe

futureisbyfarthebestoutcome,andthatdefectingbothtodayandin

ewhereoneabstainstodaybut

relapsinthefutureisclearlyabadoutcome—insomenthe

disciplineandlf-sacrificeinvolvedinabstainingtodayhavebeen

"wasted"becauthefuturerelapmeansthattheaddictisrightback

wherehestartedandwillhavetostartover(whichisquitedemoralizing,

andmakesstartingovermoredifficult).Thefinalca,whereoneengages

intheaddictivebehaviortodaywhileabstaining"tomorrow"willbe

blemhere

isthat(asinotherPDs)thereisanobviousbenefittodefecting"today",

buttomorrowonewillfacethesamePD,andthesameobviousbenefitwill

beprentthen,ultimatelyleadingtoanendlessstringofdefections.

JohnGottmaninhisrearchdescribedin"thescienceoftrust"defines

goodrelationshipsasthowherepartnersknownottoenterthe(D,D)

celloratleastnottogetdynamicallystuckthereinaloop.

Ineconomics[edit]

Advertisingissometimescitedasareallifeexampleoftheprisoner’s

garetteadvertisingwaslegalintheUnitedStates,

competingcigarettemanufacturershadtodecidehowmuchmoneytospend

ectivenessofFirmA’sadvertisingwaspartially

,theprofit

derivedfromadvertisingforFirmBisaffectedbytheadvertising

FirmAandFirmBchotoadvertiduring

agivenperiodtheadvertisingcancelsout,receiptsremainconstant,and

rmswouldbenefit

r,shouldFirmBchoonotto

adverti,heless,the

optimalamountofadvertisingbyonefirmdependsonhowmuchadvertising

eststrategyisdependentonwhattheother

firmchoosthereisnodominantstrategy,whichmakesitslightly

differentfromaprisoner'comeissimilar,though,in

thatbothfirmswouldbebetteroffweretheytoadvertilessthanin

mescooperativebehaviorsdoemergeinbusiness

tance,cigarettemanufacturerndordthecreation

oflawsbanningcigaretteadvertising,understandingthatthiswould

reducecostsandincreaprofitsacrosstheindustry.[citationneeded][26]This

analysisislikelytobepertinentinmanyotherbusinesssituations

involvingadvertising.[citationneeded]

Withoutenforceableagreements,membersofacartelarealsoinvolvedin

a(multi-player)prisoners'dilemma.[27]'Cooperating'typicallymeans

keepingpricesatapre-agreedminimumlevel.'Defecting'meanslling

underthisminimumlevel,instantlytakingbusiness(andprofits)from

-trustauthoritieswantpotentialcartel

memberstomutuallydefect,ensuringthelowestpossiblepricesfor

consumers.

Insport[edit]

Dopinginsporthasbeencitedasanexampleofaprisoner'sdilemma.[28]

Iftwocompetingathleteshavetheoptiontouanillegalanddangerous

drugtoboosttheirperformance,thentheymustalsoconsiderthelikely

herathletetakesthedrug,then

onedoes,thenthatathletegainsa

significantadvantageovertheircompetitor(reducedonlybythelegal

ormedicaldangersofhavingtakenthedrug).Ifbothathletestakethe

drug,however,thebenefitscanceloutandonlythedrawbacksremain,

puttingthembothinaworpositionthanifneitherhaduddoping.[28]

Multiplayerdilemmas[edit]

Manyreal-lifedilemmasinvolvemultipleplayers.[29]Although

metaphorical,Hardin'stragedyofthecommonsmaybeviewedasanexample

ofamulti-playergeneralizationofthePD:Eachvillagermakesachoice

lectiverewardforunanimous(or

evenfrequent)defectionisverylowpayoffs(reprentingthe

destructionofthe"commons").Acommonsdilemmamostpeoplecanrelate

ashingdishesan

individualcangainbysavinghistime,butifthatbehaviorisadopted

byeveryresidentthecollectivecostisnocleanplatesforanyone.

Thecommonsarenotalwayxploited:WilliamPoundstone,inabookabout

theprisoner'sdilemma(eReferencesbelow),describesasituationin

ssiblefor

peopletotakeapaperwithoutpaying(defecting)butveryfewdo,feeling

thatiftheydonotpaythenneitherwillothers,destroyingthesystem.

SubquentrearchbyElinorOstrom,winnerofthe2009SverigesRiksbank

PrizeinEconomicSciencesinMemoryofAlfredNobel,hypothesizedthat

thetragedyofthecommonsisoversimplified,withthenegativeoutcome

tcomplicatingpressures,groups

communicateandmanagethecommonsamongthemlvesfortheirmutual

benefit,enforcingsocialnormstoprervetheresourceandachievethe

maximumgoodforthegroup,anexampleofeffectingthebestcaoutcome

forPD.[30]

Armsraces[edit]

TheColdWarandsimilararmsracescanbemodeledasaPrisoner'sDilemma

situation.[31]DuringtheColdWartheopposingalliancesofNATOandthe

chside'spoint

ofview,disarmingwhilsttheiropponentcontinuedtoarmwouldhaveled

ly,arming

sideschotoarm,neithercouldaffordtoattacktheother,butatthe

sides

chotodisarm,warwouldbeavoidedandtherewouldbenocosts.

Althoughthe'best'overalloutcomeisforbothsidestodisarm,the

rationalcourforbothsidesistoarm,andthisisindeedwhathappened.

Bothsidespouredenormousresourcesintomilitaryrearchandarmament

forthenextthirtyyearsuntilSovietPresidentMikhailGorbachevand

USPresidentRonaldReagannegotiatedarmsreductionsandreforminthe

SovietUnioncaudideologicaldifferencestoabate.

Relatedgames[edit]

Clod-bagexchange[edit]

Hofstadter[32]oncesuggestedthatpeopleoftenfindproblemssuchasthe

PDproblemeasiertounderstandwhenitisillustratedintheformofa

simplegame,everalexamplesheudwas"clod

bagexchange":

Twopeoplemeetandexchangeclodbags,withtheunderstanding

thatoneofthemcontainsmoney,andtheothercontainsapurcha.

Eitherplayercanchootohonorthedealbyputtingintohisor

herbagwhatheorsheagreed,orheorshecandefectbyhanding

overanemptybag.

Inthisgame,defectionisalwaysthebestcour,implyingthatrational

r,inthiscabothplayerscooperating

andbothplayersdefectingactuallygivethesameresult,assumingthere

arenogainsfromtrade,sochancesofmutualcooperation,eveninrepeated

games,arefew.

FriendorFoe?[edit]

FriendorFoe?isagameshowthatairedfrom2002to2005ontheGame

exampleoftheprisoner'sdilemmagame

testedonrealpeople,ameshow,

airiliminated,theyplayagame

similartotheprisoner'sdilemmatodeterminehowthewinningsaresplit.

Iftheybothcooperate(Friend),theysharethewinnings50–

cooperatesandtheotherdefects(Foe),thedefectorgetsallthewinnings

defect,bothleavewithnothing.

Noticethatthepayoffmatrixisslightlydifferentfromthestandardone

givenabove,asthepayoutsforthe"bothdefect"andthe"cooperatewhile

theopponentdefects"kesthe"bothdefect"

caaweakequilibrium,comparedwithbeingastrictequilibriuminthe

standardprisoner'testantknowthattheiropponentis

goingtovote"Foe",thentheirownchoicedoesnotaffecttheirown

tainn,FriendorFoehasapayoffmodelbetween

prisoner'sdilemmaandthegameofChicken.

Thepayoffmatrixis

CooperateDefect

Cooperate1,10,2

Defect2,00,0

ThispayoffmatrixhasalsobeenudontheBritishtelevisionprogrammes

TrustMe,Shafted,TheBankJobandGoldenBalls,andontheAmericanshows

tafromtheGoldenBallsrieshas

beenanalyzedbyateamofeconomists,whofoundthatcooperationwas

"surprisinglyhigh"foramountsofmoneythatwouldemconquential

intherealworld,butwerecomparativelylowinthecontextofthegame.[33]

Iteratedsnowdrift[edit]

RearchersfromtheUniversityofLausanneandtheUniversityof

Edinburghhavesuggestedthatthe"IteratedSnowdriftGame"maymore

ghthismodelis

actuallyachickengame,model,the

riskofbeingexploitedthroughdefectionislower,andindividualsalways

wdriftgameimaginestwo

driverswhoarestuckonoppositesidesofasnowdrift,eachofwhomis

giventheoptionofshovelingsnowtoclearapath,orremainingintheir

r'shighestpayoffcomesfromleavingtheopponenttoclear

allthesnowbythemlves,buttheopponentisstillnominallyrewarded

fortheirwork.

Thismaybetterreflectrealworldscenarios,therearchersgivingthe

exampleoftwoscientistscollaboratingonareport,bothofwhomwould

benefitiftheotherworkedharder."Butwhenyourcollaboratordoesn’t

doanywork,it’sprobablybetterforyoutodoalltheworkyourlf.

You’llstillendupwithacompletedproject."[34]

ExampleSnowdriftPayouts(A,B)

BcooperatesBdefects

Acooperates200,200100,300

Adefects300,1000,0

ExamplePDPayouts(A,B)

BcooperatesBdefects

Acooperates200,200-100,300

Adefects300,-1000,0

Seealso[edit]

Centipedegame

Christmastruce

Evolutionarilystablestrategy

Folktheorem(gametheory)

Innocentprisoner'sdilemma

Nashequilibrium

Prisoner'sdilemmaandcooperationanexperimentalstudy

Publicgoodsgame

Reciprocalaltruism

Swifttrusttheory

Warofattrition(game)

Hobbesiantrap

References[edit]

^Milovsky,Nicholas."TheBasicsofGameTheoryandAssociatedGames".

Retrieved11February2014.

^Fehr,Ernst;Fischbacher,Urs(Oct23,2003)."TheNatureofhuman

altruism".Nature(NaturePublishingGroup)425(6960):785–791.

Bibcode:2003Natur.425..:10.1038/14574401.

RetrievedFebruary27,2013.

^Tversky,Amos;Shafir,Eldar(2004).Preference,belief,andsimilarity:

lectedwritings..MassachuttesInstituteofTechnologyPress.

vedFebruary27,2013.

^Toh-Kyeong,Ahn;Ostrom,Elinor;Walker,James(Sep5,2002).

"IncorporatingMotivationalHeterogeneityintoGame-TheoreticModelsof

CollectiveAction".PublicChoice117(3–4).RetrievedFebruary27,2013.

^Oosterbeek,Hesl;Sloof,Randolph;VandeKuilen,Gus(Dec3,2003).

"CulturalDifferencesinUltimatumGameExperiments:Evidencefroma

Meta-Analysis".ExperimentalEconomics(SpringerScienceandBusinessMediaB.V)

7(2):171–:10.1023/B:vedFebruary27,

2013.

^Capraro,V(2013)."AModelofHumanCooperationinSocialDilemmas".

PLoSONE8(8)::10.1371/.0072427.

^Hilbe,Christian;ndKarlSigmund(April2013).

"EvolutionofextortioninIteratedPrisoner’sDilemmagames".PNAS110(17):

:10.1073/ved25November2013.

^Shy,Oz(1995).IndustrialOrganization:TheoryandApplications.

ved

February27,2013.

^Forexampleethe2003study"BayesianNashequilibrium;astatistical

testofthehypothesis"fordiscussionoftheconceptandwhetheritcanapply

inrealeconomicorstrategicsituations(fromTelAvivUniversity).

^::UniversityofSouthampton

^The2004Prisoners'DilemmaTournamentResultsshowUniversityof

Southampton'sstrategiesinthefirstthreeplaces,despitehavingfewerwins

andmanymorelossthantheGRIMstrategy.(NotethatinaPDtournament,the

aimofthegameisnotto"win"matches–thatcaneasilybeachievedbyfrequent

defection).Itshouldalsobepointedoutthatevenwithoutimplicitcollusion

betweensoftwarestrategies(exploitedbytheSouthamptonteam)titfortatis

notalwaystheabsolutewinnerofanygiventournament;itwouldbemorepreci

tosaythatitslongrunresultsoverariesoftournamentsoutperformits

rivals.(Inanyoneeventagivenstrategycanbeslightlybetteradjustedto

thecompetitionthantitfortat,buttitfortatismorerobust).Thesameapplies

forthetitfortatwithforgivenessvariant,andotheroptimalstrategies:on

anygivendaytheymightnot'win'againstaspecificmixofcounter-strategies.

asimulation,titfortatwillalmostalwayscometodominate,thoughnasty

strategieswilldriftinandoutofthepopulationbecauatitfortat

populationispenetrablebynon-retaliatingnicestrategies,whichinturnare

dDawkinsshowedthathere,nostatic

mixofstrategiesformastableequilibriumandthesystemwillalwaysoscillate

betweenbounds.

12.^Jumpupto:abcdePress,WilliamH.;(2012)."Iterated

Prisoner’sDilemmacontainsstrategiesthatdominateanyevolutionary

opponent".ved26November2013.

^Adami,Christoph;ArendHintze(2013).Evolutionaryinstabilityof

ZeroDeterminantstrategiesdemonstratesthatwinningisn'teverything.p.3.

arXiv:1208.2666.

^Hilbe,Christian;,andKarlSigmund(April2013).

"EvolutionofextortioninIteratedPrisoner’sDilemmagames".PNAS110(17):

6915–ved25November2013.

15.^Jumpupto:abStewart,AlexanderJ.;n(2013)."Fromextortion

togenerosity,evolutionintheIteratedPrisoner’sDilemma".PNASEarly

ved25November2013.

^Akin,Ethan(2013).StableCooperativeSolutionsfortheIterated

Prisoner':1211.0969.

^Le,S.,Boyd,R.(2007)."EvolutionaryDynamicsoftheContinuous

IteratedPrisoner'sDilemma".JournalofTheoreticalBiology245(2):258–267.

doi:10.1016/17125798.

^Hammerstein,P.(2003).Whyisreciprocitysorareinsocialanimals?

:stein,Editor,GeneticandCulturalEvolution

ofCooperation,.83–94.

^Spaniel,William(2011).GameTheory101:TheCompleteTextbook.

^Nowak,Martin;KarlSigmund(1993)."Astrategyofwin-stay,lo-shift

thatoutperformstit-for-tatinthePrisoner'sDilemmagame".Nature364.

^"Markets&Data".TheEconomist.2007-09-27.

^Rehmeyer,Julie(2012-10-29)."Gametheorysuggestscurrentclimate

negotiationswon'tavertcatastrophe".yforScience&the

Public.

^OsangandNandy2003

^Dawkins,Richard(1976).UniversityPress.

^GeorgeAinslie(2001).0-521-59694-7.

^Thisargumentforthedevelopmentofcooperationthroughtrustisgiven

inTheWisdomofCrowds,whereitisarguedthatlong-distancecapitalismwas

abletoformaroundanucleusofQuakers,whoalwaysdealthonourablywiththeir

businesspartners.(Ratherthandefectingandrenegingonpromis–a

phenomenonthathaddiscouragedearlierlong-termunenforceableoveras

contracts).Itisarguedthatdealingswithreliablemerchantsallowedthememe

forcooperationtospreadtoothertraders,whospreaditfurtheruntilahigh

degreeofcooperationbecameaprofitablestrategyingeneralcommerce

^Nicholson,Walter(2000).IntermediateMicroeconomics(8thed.).

Harcourt.

28.^Jumpupto:abSchneier,Bruce(2012-10-26)."LanceArmstrongandthePrisoners'

DilemmaofDopinginProfessionalSports|WiredOpinion".ved

2012-10-29.

^GokhaleCS,ionarygamesinthemultiver.

ProceedingsoftheNationalAcademyofSciences.2010Mar23;107(12):5500–4.

^"TheVolokhConspiracy"ElinorOstromandtheTragedyoftheCommons".

ved2011-12-17.

^i(1984)."Armsracesasiteratedprisoner'sdilemma

games".MathematicalandSocialSciences7(3):253–266.

doi:10.1016/0165-4896(84)90022-2.

^Hofstadter,DouglasR.(1985).MetamagicalThemas:questingforthe

0-465-04566-9.–e

Ch.29ThePrisoner'sDilemmaComputerTournamentsandtheEvolutionof

Cooperation.

^VandenAsm,MartijnJ.(January2012)."SplitorSteal?Cooperative

BehaviorWhentheStakesAreLarge".ManagementScience58(1):2–20.

doi:10.1287/mnsc.1110.1413.

^Kümmerli,Rolf."'Snowdrift'gametops'Prisoner'sDilemma'in

explainingcooperation".Retrieved11April2012.

Furtherreading[edit]

RobertAumann,"Acceptablepointsingeneralcooperativen-persongames",

(eds.),ContributionstotheTheory23

ofGamesIV,AnnalsofMathematicsStudy40,287–324,Princeton

UniversityPress,PrincetonNJ.

Axelrod,R.(1984).0-465-02121-2

Bicchieri,Cristina(1993).dge

UniversityPress.

(1988).Simulatingtheevolutionofbehavior:theiterated

prisoners'xSystems,2:663–670.

Dresher,M.(1961).TheMathematicsofGamesofStrategy:Theoryand

ApplicationsPrentice-Hall,EnglewoodCliffs,NJ.

Greif,A.(2006).InstitutionsandthePathtotheModernEconomy:Lessons

dgeUniversityPress,Cambridge,UK.

Rapoport,h(1965).Prisoner'sDilemma.

UniversityofMichiganPress.

Externallinks[edit]

Listentothisarticle(info/dl)

Sorry,yourbrowreitherhasJavaScriptdisabledordoesnothaveanysupportedplayer.

Youcandownloadtheclipordownloadaplayertoplaytheclipinyourbrowr.

Thisaudiofilewascreatedfromarevisionofthe"Prisoner'sdilemma"articledated2007-06-25,anddoesnotreflect

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Morespokenarticles

Prisoner'sDilemma(StanfordEncyclopediaofPhilosophy)

TheBowerbird'sDilemmaThePrisoner'sDilemmainornithology–

mathematicalcartoonbyLarryGonick.

Dixit,Avinash;Nalebuff,Barry(2008)."Prisoner'sDilemma".InDavid

son(ed.).ConciEncyclopediaofEconomics(2nded.).

Indianapolis:978-.

OCLC237794267.

GameTheory101:Prisoner'sDilemma

Dawkins:NiceGuysFinishFirst

[hide]

v

t

e

Topicsingametheory

Definitions

Normal-formgame

Extensive-formgame

Graphicalgame

Cooperativegame

Succinctgame

Informationt

Hierarchyofbeliefs

Preference

Equilibriumconcepts

Nashequilibrium

Subgameperfection

Mertens-stableequilibrium

Bayesian-Nash

PerfectBayesian

Tremblinghand

Properequilibrium

Epsilon-equilibrium

Correlatedequilibrium

Sequentialequilibrium

Quasi-perfectequilibrium

Evolutionarilystablestrategy

Riskdominance

Core

Shapleyvalue

Paretoefficiency

Quantalresponequilibrium

Self-confirmingequilibrium

StrongNashequilibrium

Markovperfectequilibrium

StrategiesDominantstrategies

Purestrategy

Mixedstrategy

Titfortat

Grimtrigger

Collusion

Backwardinduction

Forwardinduction

Markovstrategy

Classofgames

Symmetricgame

Perfectinformation

Simultaneousgame

Sequentialgame

Repeatedgame

Signalinggame

Cheaptalk

Zero-sumgame

Mechanismdesign

Bargainingproblem

Stochasticgame

LargePoissongame

Nontransitivegame

Globalgames

Games

Prisoner'sdilemma

Traveler'sdilemma

Coordinationgame

Chicken

Centipedegame

Volunteer'sdilemma

Dollarauction

Battleofthexes

Staghunt

Matchingpennies

Ultimatumgame

Rock-paper-scissors

Pirategame

Dictatorgame

Publicgoodsgame

Blottogames

Warofattrition

ElFarolBarproblem

Cakecutting

Cournotgame

Deadlock

Diner'sdilemma

Guess2/3oftheaverage

Kuhnpoker

Nashbargaininggame

Screeninggame

Prisonersandhatspuzzle

Trustgame

Princessandmonstergame

MontyHallproblem

Theorems

Minimaxtheorem

Nash'stheorem

Purificationtheorem

Folktheorem

Revelationprinciple

Arrow'simpossibilitytheorem

Keyfigures

KennethArrow

RobertAumann

KennethBinmore

SamuelBowles

MelvinDresher



DrewFudenberg

s

JohnHarsanyi

LeonidHurwicz



DanielKahneman



EricMaskin

Jean-FrançoisMertens

PaulMilgrom

OskarMorgenstern

RogerMyerson

JohnNash

JohnvonNeumann

ArielRubinstein

ThomasSchelling

ReinhardSelten

HerbertSimon

LloydShapley

JohnMaynardSmith

JeanTirole



WilliamVickrey



PeytonYoung

Seealso

Tragedyofthecommons

Tyrannyofsmalldecisions

All-payauction

Listofgamesingametheory

Confrontationanalysis

Listofgametheorists

Combinatorialgametheory

Retrievedfrom

"/w/?title=Prisoner%27s_dilemma&oldid

=615571415"

Categories:

Gametheory

Thoughtexperiments

Dilemmas

Environmentalstudies