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1、RepeatedGames(II)13.1InfinitelyRepeatedPrisoners’Dilemma3.1.1InfinitelyRepeatedPrisoners’DilemmaunderALLCandALLD2CD35ToshowthatstrategypairisNOTNEQ,C30onlyneedtoidentifyonestrategythateither01Dplayerwouldstrictlyprefer.51Ex:(ALLC,ALLC)∞3tRememberthatU(ALLC,ALLC)=∑δ(3)=it=01−δ
2、Is(ALLC,ALLC)NEQ?NO,becausethereisincentivetodefect.If1isplayingALLC,then2wouldprefertoplayALLDbecauseshewouldreceivepayoffof5eachperiod1:CCCCCC…2:DDDDDD…∞t53∴(ALLC,ALLC)isU(ALLD
3、ALLC)=∑δ(5)=>2notNEQt=01−δ1−δ3CD35ToshowthatstrategypairisNOTNEQ,C30onlyneedtoidentifyonestrategy
4、thateither01Dplayerwouldstrictlyprefer.51Note:Whenidentifyingastrategythateitherplayerwouldprefer,donotchangebothplayers’strategiesatthesametime.Ex:Don’ttry(ALLD,ALLD)todeterminewhether(ALLC,ALLC)isNEQ.4CD35ToshowthatstrategypairISNEQ,mustC30showthatthereisNOotherstrategythat
5、01Deitherplayerwouldstrictlyprefer.51Ex:(ALLD,ALLD)∞t1RememberthatUi(ALLD
6、ALLD)=∑δ(1)=t=01−δIs(ALLD,ALLD)NEQ?Considerplayer2.Isthereanydeviationon2’spartthatwouldeverbetempting,giventhat1playsALLD?5CD35ToshowthatstrategypairISNEQ,mustC30showthatthereisNOotherstrategythat01Dei
7、therplayerwouldstrictlyprefer.51Ex:(ALLD,ALLD)Whatif2playsALLC?1:DDDDDD…2:CCCCCC…U(ALLC
8、ALLD)0(0)2(0)3(0)02=+δ+δ+δ+=1>0foranyvalueofδ1-δ∴2willneverwanttochangefromALLDtoALLC6CD35ToshowthatstrategypairISNEQ,mustC30showthatthereisNOotherstrategythat01Deitherplayerwouldstrictly
9、prefer.51Ex:(ALLD,ALLD)Whatif2onlycooperatesonceinsomeperiodL?012…LL+1L+2L+3…1:DDD…DDDD…2:DDD…CDDD…2’sPayoff:11δ1δ20δL1δL+11δL+21δL+3…Notethat2’spayoffisthesameas(ALLD,ALLD)exceptinL.Since1δL>0δLalways,2willneverwanttocooperateonce.7CD35ToshowthatstrategypairISNEQ,mustC30show
10、thatthereisNOotherstrategythat01Deitherplayerwouldstrictlyprefer.51Ex:(ALLD,ALLD)•ALLCandcooperatingonceinperiodLcanbethoughtofasrepresentinganinfinitenumberofstrategiesthatresultanywherefrom1to∞defectionsagainstALLD•Iftheplayer1isplayingALLD,thenplayer2willneverhaveanincenti
11、vetocooperateinanyperiod.∴(ALLD,ALLD)isalwaysNEQ83.1.2InfinitelyRepe
