LectureIII:NormalFormGames,RationalityandIteratedDeletionofDominatedStrategies
MarkusM.M¨obiusFebruary19,2004
Readings:
?Gibbons,sections1.1.Aand1.1.B?Osborne,sections2.1-2.5andsection2.9
1De?nitionofNormalFormGame
Gametheorycanberegardedasamulti-agentdecisionproblem.It’suseful
tode?ne?rstexactlywhatwemeanbyagame.
Everynormalform(strategicform)gamehasthefollowingingredients.1.ThereisalistofplayersD={1,2,..,I}.Wemostlyconsidergameswithjusttwoplayers.AsanexampleconsidertwopeoplewhowanttomeetinNewYork.2.EachplayericanchooseactionsfromastrategysetSi.Tocontinueourexample,eachoftheplayershastheoptiontogotheEmpireStatebuildingormeetattheoldoaktreeinCentralPark(whereeverthatis...).SothestrategysetsofbothplayersareS1=S2={E,C}.3.Theoutcomeofthegameisde?nedbythe’strategypro?le’whichconsistsofallstrategieschosenbytheindividualplayers.Forexample,inourgametherearefourpossibleoutcomes-bothplayersmeetattheEmpirestatebuilding(E,E),theymiscoordinate,(E,C)and(C,E),or
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theymeetinCentralPark(C,C).Mathematically,thesetofstrategypro?les(oroutcomesofthegame)isde?nedas
S=S1×S2
Inourcase,Shasorder4.Ifplayer1cantake5possibleactions,andplayer2cantake10possibleactions,thesetofpro?leshasorder50.4.Playershavepreferencesovertheoutcomesoftheplay.Youshouldrealizethatplayerscannothavepreferencesovertheactions.Inagamemypayo?dependsonyouraction.InourNewYorkgameplayersjustwanttobeabletomeetatthesamespot.Theydon’tcareiftheymeetattheEmpireStatebuildingoratCentralPark.IftheychooseEandtheotherplayerdoesso,too,?ne!IftheychooseEbuttheotherplayerchoosesC,thentheyareunhappy.Sowhatmatterstoplayersareoutcomes,notactions(ofcoursetheiractionsin?uencetheoutcome-butforeachactiontheremightbemanypossibleoutcomes-inourexampletherearetwopossibleoutcomesperaction).Recall,thatwecanrepresentpreferencesoveroutcomesthroughautilityfunction.Mathematically,preferencesoveroutcomesarede?nedas:
ui:S→R
Inourexample,ui=1ifbothagentschoosethesameaction,and0otherwise.
Allthisinformationcanbeconvenientlyexpressedinagamematrixasshownin?gure1:
Amoreformalde?nitionofagameisgivenbelow:De?nition1Anormal(strategic)formgameGconsistsof?A?nitesetofagentsD={1,2,..,I}.?StrategysetsS1,S2,..,SI
?Payo?functionsui:S1×S2×..SI→R(i=1,2,..,n)
We’llwriteS=S1×S2×..×SIandwecalls∈Sastrategypro?le(s=(s1,s2,..,sI)).Wedenotethestrategychoicesofallplayersexceptplayeriwiths?ifor(s1,s2,..,si?1,si+1,..sI).
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Figure1:General2by2gameECE1,10,0C0,01,12SomeImportantGames
Wealreadydiscussedcoordinationgames.Theseareinterestinggames,be-causeplayershaveanincentivetoworktogetherratherthanagainsteachother.The?rstgamesanalyzedbygametheoristswerejusttheopposite-zerosumgames,wherethesumofagents’utilitiesineachoutcomesumsuptozero(oraconstant).
2.1Zero-SumGames
Zero-sumgamesaretruegamesofcon?ict.Anygainonmysidecomesattheexpenseofmyopponents.Thinkofdividingupapie.Thesizeofthepiedoesn’tchange-it’sallaboutredistributionofthepiecesbetweentheplayers(taxpolicyisagoodexample).
Thesimplestzerosumgameismatchingpennies.Thisisatwoplayergamewhereplayer1getaDollarfromplayer2ifbothchoosethesameaction,andotherwiselosesaDollar:
HTH1,?1?1,1T?1,11,?13
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