Hausken, K. (2007) The impact of the future in games with multiple equilibria. Economics Letters , 96(2), pp. 183-188
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The impact of the future in games with multiple equilibria
Kjell Hausken
Abstract
The article shows that in a game with multiple equilibria, where one player estimates that there is at least a minuscule probability that the other player acquiesces, then conflict is inevitable if both players value the future sufficiently highly.
Keywords: conflict; repeatedgame; discounting
It is commonly believed that in repeated games where the threat of punishment is credible and immediate and future cooperation is collectively desirable, the more the players value their future interactions(thegreateristhediscountedbenefitoffuturecooperation)themorelikelycooperationwill betheequilibrium.Inthesegamescooperationisimmediateande.g.atriggerstrategycanbedesignedto continuethatcooperationintothefuture.Thisworksnicelywhenthereisanagreeduponoutcomethatis bestforeveryone(e.g.prisoner'sdilemma),wherecooperationbackedupbycredibleequilibriumthreats elicitcooperation.1Thethreatofconflictthusenforcescooperativerelations.Theapproachinthisarticle isdifferent. “Cooperation” isnotimmediate.Oneplayeristryingtogethispreferredequilibriumatthe
1 TheFolkTheorem(FudenbergandMaskin,1986:533)statesthat “anyindividuallyrationalpayoffvectorofaone-shotgame ofcompleteinformationcanariseinaperfectequilibriumoftheinfinitelyrepeatedgameifplayersaresufficientlypatient.”
Observealsothetit-for-tatstrategyanalyzedbyAxelrod(1984).
expense of the other. There is fundamental disagreement as to what is the best equilibrium. In addition, a player, by challenging the other player, is trying to induce him to give in to choosing the other equilibrium.
ConsiderthegameinTable1wherea1≥b1≥ t1,b2≥a2≥ t2,a1≥d1,b2N d2ora1N d1,b2≥d2.2The two purestrategy equilibria are(a1,a2) and(b1,b2). Rowplayer1 prefers(a1,a2), andcolumnplayer2 prefers(b1,b2).3
Considera “triggerstrategy” (OsborneandRubinstein,1994:143–154)forplayer2wheretheplayers movesimultaneouslyineachperiodandachallengelastsonlytwoperiods.Inperiod0player1getsto knowthatheischallenged,andinperiod1player2getstoknow whetherplayer1resistsoracquiesces:
Assumeplayof(I,I)inTable1inperiod−1 giving(a1,a2)andthatplayer1also choosesI inperiod0.
Challengeplayer1inperiods0and1ifplayer1choosesstrategyIinperiods −1and0.Thisgives(t1,t2) i n period0.Ifplayer1resists inperiod1giving(t1,t2),thenreverttostrategy Igiving(a1,a2).Ifplayer1 acquiescesinperiod1,thencontinuewithstrategyIIgiving(b1,b2)untilplayer1revertsbacktostrategyI.4
Table 1
Two-person two strategy game with two equilibria
I II
I a1,a2 t1,t2
II d1,d2 b1,b2
2 Table1encompassesgames64–69inRapoportandGuyer's(1966:213)ordinaltaxonomy.Themostwell-knownofthese are the Battleofthe Sexes (game68), Chicken(game 66),and game 69 withseveral namessuchas “LetGeorge doit”,
“Apology”, “Hero”, “Sacrificedleader”.Thegames64,65,67arehybridasymmetricgames.
3 InthefinitelyandinfinitelyrepeatedversionsofthegameinTable1thetwoNashequilibriaaresubgameperfect.Inthe infinitelyrepeated gamethe followingtwo strategiesconstitute asubgame perfectequilibrium withpayoff(a1,a2) in each period:Player1:ChoosestrategyIwhenchallenged,unlessstrategy2waschoseninthepast,thenalwayschoosestrategy II. Player 2: Choose strategy I unless player 1 failed to choose strategy I in the past, then always choose strategy II. The justificationforthesubgameperfectequilibriumwithpayoff(b1,b2)ineachperiodisanalogous.Forthesetwosubgameperfect equilibriaoneplayeracquiresareputationforrecalcitrance,theotherforacquiescence.Oneproblemwiththesetwoequilibriais thatthereputationisnevertested.Table1isequivalenttotheprobablymostwell-knownexampleofentrydeterrence,viz.the chainstoregameonnormalform,whena1 =d1 ≥b1 ≥ t1,b2 ≥a2 =d2 ≥ t2,whereplayer1istheincumbent(fight=strategyI, acquiesce=strategyII)andplayer2theentrant(stayout=strategyI,enter=strategyII).BothgameshavethesametwoNash equilibria,butthechainstoregameinitsfinitelyrepeatedversionhasonlyoneuniquesubgameperfectequilibrium(provedby backwardinduction);theentrantentersandtheincumbentdoesnotfight.KrepsandWilson(1982)andMilgromandRoberts (1982)werefirsttoformalizereputationeffects,whereasmallamountofincompleteinformationcanbesufficienttoovercome Selten's(1978)chainstoreparadox.AsKrepsandWilson(1982:255)pointout,thesecondequilibrium(theentrantstaysoutand theincumbentchoosesthestrategy “fightifentry”) i s“imperfect” and “notsoplausibleasthefirst.Itdependsonanexpectation bytheentrantofthebincumbent'sN behaviorthat,facedwiththefaitaccompliofentry,wouldbeirrationalbehaviorforthe bincumbentN.” Fortextbooktreatmentsseee.g.FudenbergandTirole(1991:369–374),OsborneandRubinstein(1994:105– 106,239–243),Rasmusen(1989:85–118,2001:110,129),Wilson(1985:31–33).
4A generalization is the following“trigger strategy”for player 2 where a challenge may lastf+ 1 periods,f≥1: Challenge player 1 in periods 0,1,2,…,f if player 1 chooses strategy I in periods−1,0,1,…,f−1. If player 1 resists in period f, then revert to strategy I giving (a1,a2). If player 1 acquiesces, continue with strategy II giving (b1,b2) until player 1 reverts back to strategy I. If this is before and including period f, continue to choose strategy II. If this is in periodf+ 1 or thereafter, then choose strategy I.
In an infinitely repeated game with discount factorsδ1andδ2for players 1 and 2, respectively, where player 1 resists, players 1 and 2 get
tiþd1t1þd21a1þd31a1þ: : :þdl1 a1¼t1þ ða1−t1Þd21
1−d1 ð1Þ
t2þd2t2þd22a2þd32a2þ: : :þdl2 a2¼t2þ ða2−t2Þd22
1−d2
ð2Þ respectively. If player 1 acquiesces, they get
t1þd1b1þd21b1þd31b1þ: : :þdl1 b1 ¼t1þ ðb1−t1Þd21
1−d1 ð3Þ
t2þd2b2þd22b2þd32b2þ: : :þdl2 b2 ¼t2þ ðb2−t2Þd22
1−d2 : ð4Þ
Player 1 resists in period 1 when his payoff in (1) is larger than that of (3), i.e.
t1þ ða1−t1Þd21
1−d1 Nt1þ ðb1−t1Þd1
1−d1 Zd1Nb1−t1
a1−t1¼d⁎1: ð5Þ
According to (5) player 1 is more likely to resist the more important the future is. In order for player 2's challenge in period 0 to be part of a subgame perfect equilibrium, her expected payoff from challenging must be larger than the payoffa2/(1−δ2) of not challenging. For this calculation player 2 needs to make a conjecture of the probability q1 that player 1 resists the challenge. Applying (2) and (4) player 2 challenges in period 0 when
q1
t2þ ða2−t2Þd22
1−d2 þ ð1−q1Þt2þ ðb2−t2Þd2
1−d2 N a2
1−d2
Zd2N −ð1−q1Þðb2−t2Þ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−q1Þ2ðb2−t2Þ2þ4q1ða2−t2Þ2 q
2q1ða2−t2Þ ¼d⁎2:
ð6Þ
Eqs. (5) and (6) imply that if sufficiently low weight is placed on the future (δiis small,i= 1,2), player 2 does not challenge, and neither does player 1 resist if there were a challenge, implying“peace”at (a1,a2).
Satisfaction of (5) but not (6) implies that player 2 does not challenge, implying (a1,a2). Conversely, satisfaction of (6) but not (5) implies that player 2 challenges and player 1 acquiesces, implying“peace”at (b1,b2). Finally, if sufficiently high weight is placed on the future (δiis large), player 2 challenges and player 1 resists, implying conflict at (t1,t2).
Proposition 1. 1. Ifδ1bδ1⁎andδ2bδ⁎2,then player 2 does not challenge,and player 1 does not resist, implying (a1,a2).2. Ifδ1Nδ1⁎andδ2bδ⁎2,then player 2 does not challenge,implying (a1,a2).3. Ifδ1bδ1⁎ andδ2Nδ⁎2,then player 2 challenges,and player 1 acquiesces,implying (b1,b2). 4. Ifδ1Nδ1⁎andδ2Nδ⁎2, then player 2 challenges,and player 1 resists,implying (t1,t2).
Proof. Follows from (5) and (6). □ Proposition 2. δ⁎2=(a2−t2)/(b2−t2) when q1=0,andδ⁎2=1 when q1=1.
Proof. Follows from (6) applying L'Hopital's rule for the first equality. □ Proposition 3. If q1b1 and a2bb2 and δi is sufficiently large, i=1,2, the threat point (t1,t2) is guaranteed.
Proof. Follows from Propositions 1 and 2. □
Proposition 3 states that if the future is sufficiently important, and player 2 estimates that player 1 is not 100% guaranteed to resist, then conflict is guaranteed. In other words, given that player 2 estimates at least a minuscule probability that player 1 acquiesces, sufficiently large emphasis on the future by both players makes conflict inevitable.
Fig.1illustratesthefourareasinProposition1assuming(a1,a2)=(4,3),(b1,b2)=(3,4),(t1,t2)=(0,2).
The horizontal axisis theprobability q1 estimated by player2 that player1 resists thechallenge. The verticalaxisisthediscountfactor δiwhichmaybedifferentforplayers1and2.Thismeansthatonefor eachvalueofq1canchooseonevalue δ1alongtheverticalaxisforplayer1'sdiscountfactor,andanother value δ2alongtheverticalaxisforplayer2'sdiscountfactor,andreadtheoptimalstrategyandpayofffor eachplayeroutofthediagram. Forexpositionalconvenience wefocuson onevalue alongthevertical axis,whichissufficientsinceallthefourareasarepresent.
Whenq1= 1, which means that player 2 estimates that player 1 is guaranteed to resist, then player 2 is best off not challenging, regardless of her discount factor, as specified in Proposition 2. Conversely, when q1= 0, which means that player 2 estimates that player 1 is guaranteed not to resist, then player 2 may
Fig. 1. The four areas in Proposition 1 dependent on the probabilityq1estimated by player 2 that player 1 resists the challenge, and the discount factorδi,i= 1,2 for both players. The payoffs within each area are shown in brackets.
challenge if the discount factor is large compared with the threat payoff t2 that has to be endured in period 0 because of the challenge, see (4). Hence when q1= 0 and a2bb2, δ2⁎b1 in accordance with Proposition 2.Fig. 1shows howδ⁎2 increases inq1, whileδ1⁎is constant in accordance with (5).
Below the curveδ⁎2 inFig. 1, the behavior of player 1 is irrelevant since player 2 does not challenge, which always causes (a1,a2). However, above the curve δ2⁎, the threat point (t1,t2) follows if player 1 resists, while (b1,b2) follows if player 1 acquiesces. When the parameters in (5) are adjusted so thatδ1⁎ decreases,Fig. 1reduces to three areas making (b1,b2) impossible. For example, increasingt1fromt1= 0 tot1= 2 causesδ1⁎= 1/2 and three areas.
The development above assumes that player 2 challenges the equilibrium (a1,a2). The development is analogous when player 1 challenges the equilibrium (b1,b2). In that case player 1 estimates a probability q2that player 2 resists the challenge.
Conflictatthethreatpoint(t1,t2)raisesthequestionofwhethertheoff-the-equilibrium-pathconjectures theplayersmakeofq2 andq1,andofeachothers'triggerstrategies,whenlocatedin(I,II)(whichisnotan equilibrium)inperiod1receiving(t1,t2)areincompatible.Eq.(5)assumesthatplayer1knowsplayer2's triggerstrategyofacquiescingwhenplayer1resists.Knowingthisconjecture,andassumingthat(5)is satisfied,player2mayrationallyconjecturethatplayer1iscertaintoresist,thatisestimateq1=1,which when insertedinto (6)gives δ2N 1.Henceplayer2 doesnot challengeifplayer1 isdeemedcertain to resist.Thisagainensures “peace” at(a1,a2).Butdeemingplayer1certaintoresistisproblematic.Itmeans thatplayer1commitsinadvancetothesequenceofactionshewilltake.Ifsufficientlychallengeditisnot rationalforplayer1tokeepsuchacommitment.NeitheraretheremediessuggestedbySchelling(1960) for ensuring that one's commitment is trustworthy present in this complete information game,5 and neitherarethereanysalientfocalpoints.Theanalysissuggestsononehandthatsufficientlyhighweight onthefutureincreasesthelikelihoodofconflict.Butontheotherhand,thispresupposesthatincompatible andsufficiently low off-the-equilibrium-pathconjecturesareadmissible.Theplayersareforcedtoplay thegameandcannotbargainthemselvesoutofthegame.Althoughthegamehascompleteinformation, theplayersmaywellchoosetoresolvetheirsituationbyresortingtoincompatibleconjectures.
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