Reputation, incomplete information, and differences in patience in repeated games with multiple equilibria

Reputation, incomplete information, and differences in patience in repeated games with multiple equilibria

Kjell Hausken

Abstract

A game with multiple equilibria and incomplete information, which allows for reputation building, is repeated infinitelymanytimes.Increasingdifferencesinpatiencecontributetoagreaterlikelihoodofcooperation.Asone playerbecomessufficientlymorepatientthantheotherplayer,bothplayersbenefit,andbothplayers'risklimits, andtheconflictbetween theplayers,decrease.

Keywords:Conflict; Risk limits; Repeated game; Discounting; Reputation; Incomplete information

Theliteraturehasproducedmixedresultsregardingwhetherplayersshouldchooseconflicttodaytoreap benefitstomorrow.First,theFolktheorem(FudenbergandMaskin,1986)isoftentakentoimplycooperationin long-termrelationships,whichiscorrectwhentheprisoner'sdilemmaisplayedrepeatedly.Second,Hausken (2005)hasforthebattleofthesexeswhereplayer1valuesthefutureandplayer2ismyopic,shownthatplayer 1prefersconflictinthepresentwhenthefutureisimportant.Similarly,SkaperdasandSyropoulos(1996)equip each agentwitha resourcewhichcan beallocated into productionversus arms.Theyshow thatincreased importanceofthefuturemayharmcooperation.Thisarticleconsidersabroadclassofgameswithmultiple equilibria,andintroducesincompleteinformationwhichallowstheplayerstobuildreputations.Theobjective ofthearticleistounderstandhowtheplayers'payoffs,risklimits,andtheconflictbetweenthemareinfluenced by different emphases on the future. The players are concerned about how the conflict between them evolves. The notion of risk limits was essential in Zeuthen's (1930) work. He originated the principle of risk dominance as a dominance relation based on comparing the various players' risk limits.¹

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ConsiderthegameinTable1wherea₁≥ b₁≥ t₁,b₂≥ a₂≥ t₂,a₁≥ d₁,b₂N d₂ora₁N d₁,b₂≥ d₂.²The twopure strategyequilibriaare(a₁,a₂) and (b₁,b₂).Rowplayer1prefers(a₁,a₂), and col umn player 2 prefers(b₁,b₂).³

Inthe finitely andinfinitely repeatedversions of the gamein Table1the twoNash equilibria are subgameperfect.⁴ In the infinitelyrepeatedgame the followingtwo strategies constitutea subgame

Table 1

Two-person two-strategy game with two equilibria

I II

I a₁,a₂ t₁,t₂

II d1,d2 b1,b2

1 SubsequentlyEllsberg(1961)discussedtheprinciplerelatedtohisparadoxes.ThereafterHarsanyi(1977)analyzedZeuthen's principle,andHarsanyiandSelten(1988:90)appliedthenotionofriskdominanceasacriterionforequilibriumselection.

2 Table1encompassesgames64–69inRapoportandGuyer's(1966:213)ordinaltaxonomy.Themostwell-knownofthesearethe BattleoftheSexes(game68),Chicken(game66),andgame69withseveralnamessuchas “LetGeorgedoit”, “Apology”, “Hero”,

“Sacrificedleader”.Thegames64,65,67arehybridasymmetricgames.

3 InthefinitelyandinfinitelyrepeatedversionsofthegameinTable1thetwoNashequilibriaaresubgame perfect.Inthe infinitelyrepeatedgame the followingtwostrategies constitutea subgame perfectequilibriumwith payoff(a1,a2) in each period:Player1:ChoosestrategyIwhenchallenged,unlessstrategy2waschoseninthepast,thenalwayschoosestrategyII.

Player 2: Choose strategy I unless player 1failed 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(theentrantstaysout and the incumbent chooses the strategy “fight if entry”) i s “imperfect” and “not soplausible as the first. It depends on anexpectation by the entrant of the bincumbent'sN behavior that, faced with the fait accompli of entry, would be irrationalbehavior for the bincumbentN.” For text book treatments see e.g. Fudenberg and Tirole (1991:369–374), Osborne andRubinstein(1994:105–106,239–243),Rasmusen(1989:85–118,2001:110,129),Wilson(1985:31–33).

4 Table1isequivalenttotheprobablymostwell-knownexampleofentrydeterrence,viz.thechainstoregameonnormalform,when a1=d1≥b1≥t1,b2≥a2=d2≥t2,whereplayer1istheincumbent(fight=strategyI,acquiesce=strategyII)andplayer2theentrant(stay out=strategyI,enter=strategyII).BothgameshavethesametwoNashequilibria,butthechainstoregameinitsfinitelyrepeated versionhasonlyoneuniquesubgameperfectequilibrium(provedbybackwardinduction);theentrantentersandtheincumbentdoes notfight.KrepsandWilson(1982)andMilgromandRoberts(1982)werefirsttoformalizereputationeffects,whereasmallamountof incompleteinformationcanbesufficienttoovercomeSelten's(1978)chainstoreparadox.AsKrepsandWilson(1982:255)pointout, thesecondequilibrium(theentrantstaysoutandtheincumbentchoosesthestrategy“fightifentry”) i s “imperfect” and“not so plausibleasthefirst.Itdependsonanexpectationbytheentrantofthe bincumbent'sN behaviorthat,facedwiththefaitaccompliof entry,wouldbeirrationalbehaviorforthebincumbentN.” Fortextbooktreatmentsseee.g.FudenbergandTirole(1991:369–374), OsborneandRubinstein(1994:105–106,239–243),Rasmusen(1989:85–118,2001:110,129),Wilson(1985:31–33).

perfect equilibrium with payoff (a₁,a₂) in each period: Player 1: Choose strategy I when challenged, unless strategy 2 was chosen in the past, then always choose strategy II. Player 2: Choose strategy I unless player 1 failed to choose strategy I in the past, then always choose strategy II. The justification for the subgame perfect equilibrium with payoff (b1,b2) in each period is analogous.

For these two subgame perfect equilibria one player acquires a reputation for recalcitrance, the other for acquiescence. One problem with these two equilibria is that the reputation is never tested.

Onewayaroundthisproblemistointroduceincompleteinformationsothatreputationscanbebuilt.⁵ Aliteratureonreputationboundshasemerged,expressingtheaveragediscountedpayoffstheplayerscan guaranteetothemselves. Thefirstsystematic treatmentwas presentedby FudenbergandLevine(1989, 1992).⁶Player1preferstheequilibrium(a1,a2),andgetst1inthethreatpoint,sowedefinea1∞ asplayer 1'slowerbound.Analogously,player2preferstheequilibrium(b1,b2),andgetst2inthethreatpoint,so we define b_2∞ as player2's lower bound. For players involved inreputation building Schmidt (1993) determinesforinfinitelyrepeatedgameswithconflictinginterestsandsimultaneousmovesineachperiod thetwolowerbounds

a1l¼ 1−l⁰₂d^k₁^{1ðl⁎1;d2Þ}

t1þl⁰₂d^k₁^{1ðl⁎1;d2Þ}a1; b2l¼ 1−l⁰₁d^k₂^{2ðl⁎2;d1Þ}

t2þl⁰₁d^k₂^{2ðl⁎2;d1Þ}b2; ð1Þ expressed as average discounted payoffs, where⁷

k1ðl^⁎1;d2Þ ¼ð½N1 þ1Þlnl^⁎₁

lnð1−e1Þ ; k2ðl^⁎2;d1Þ ¼ð½N2 þ1Þlnl^⁎₂ lnð1−e2Þ ;

N1¼lnð1−d2Þ þlnða2−t2Þ−lnðb2−t2Þ lnd2

; N2¼lnð1−d1Þ þlnðb1−t1Þ−lnða1−t1Þ lnd1

;

e1¼ð1−d2Þ²ða2−t2Þ

ðb2−t2Þ þd^ð½2^N1þ1Þð1−d2Þ; e2¼ð1−d1Þ²ðb1−t1Þ

ða1−t1Þ þd^ð½1^N2þ1Þð1−d1Þ;

ð2Þ

where [Ni] is the integer value ofNi, andμi0N0,μ⁎i=1−μi0N0 are the probabilities of the“normal”and

“committed” (always challenges) types, respectively, of player i, i= 1,2. Eq. (1) states that if the probability μ20

of the normal type of player 2 is close to one, and if player 1 is very patient, then the lower bound0_1∞for player 1 is close to his commitment payoffa₁. The bounda_1∞for player 1 (w.l.o.g.) is valid and reputation building has impact only when player 1 is sufficiently more patient than player 2.

5 Toallowaroleforreputationatleastoneplayermusthaveprivateinformationthatpersistsovertime,thisplayermustbe likelytotakeseveralactionsinsequence,andtheplayermustbeunabletocommitinadvancetothesequenceofactionsshe willtake(Wilson,1985;KrepsandWilson,1982).

6 SeeCelentanietal.(1996),Crippsetal.(1996),Sorin(1999),andWatson(1996)forsubsequenttreatments.

7 FortheliteratureonreputationboundstreatedsystematicallyfirstbyFudenbergandLevine(1989,1992),seee.g.Celentani et al. (1996), Cripps et al. (1996), Sorin (1999), Watson (1996). See also Fudenberg and Levine (1989, 1992), Eq. (1) correspondsinSchmidt's(1993)articleto(30)inTheorem3,Eq.(2)correspondsto(22),(17),(18),andEq.(3)correspondsto (28).(Thereisaprintingerrorin(23)whichdoesnotfollowfrom(18)and(37)).

¯ ¯

Thatis(Schmidt,1993:337),forany δ2b 1, μ1⁎N 0, ε1N 0, ∃δ1(μ1⁎,δ2,ε1)b1 s . t . f o r a n yδ1≥δ1

(μ1⁎,δ2,ε1) t h e averagepayoffofthenormaltypeofplayer1isatleasta1∞, w h e r e⁸

Pd₁ðl⁎

1;d2;e1Þ ¼ a1−t1−e1

a1−t1

1

k1ðl⁎

1;d2Þ; _Pd₂ðl⁎

2;d1;e2Þ ¼ b2−t2−e2

b2−t2

1

k2ðl⁎

2;d1Þ; ð3Þ

⁎=μ⁎

illustrated in Fig. 1 when μ1 2= 0.05 and μ1⁎ = μ2⁎=0.3, assuming (a₁,a₂)=(4,3), (b₁,b₂)=(3,4), (t1,t2)=(2,2).⁹ Theareawherethe boundsdonot holdissubstantial.[N1]jumpsdiscretelyfrom0to 1 w h e n δ2= 0.33,from 1 to2when δ2= 0.5,from 2to3 when δ2= 0.59,from 3to 4when δ2= 0.65,etc.,whichexplainthediscretejumpsinFig.1.

When the bounda_1∞(b_2∞) is valid for player 1 (2), the associated payoff to player 2 (1) compatible with (1) is

a2l¼ 1−l⁰2d^k1^{1ðl⁎1;d2Þ}

t2þl⁰2d^k1^{1ðl⁎1;d2Þ}a2; b1l¼ 1−l⁰1d^k2^{2ðl⁎2;d1Þ}

t1þl⁰1d^k2^{2ðl⁎2;d1Þ}b1: ð4Þ

illustrated inFig. 2forδ2= 0.2. That is, whenμ1⁎=μ⁎2= 0.3 and δ2= 0.2, player 1's discount factor must be aboveδ_¹(0.3, 0.2,ε1) = 0.861519 for the reputation bounda1∞to be valid. The reputation bounda1∞

increases from a_1∞= 3.44 when δ1= 0.861519 to a_1∞= 3.90 when δ1= 1, which means that player 1 benefits from getting more patient. Player 2's associated payoff a2∞ is determined from Eq. (4), and increases froma_2∞= 2.72 toa_2∞= 2.95 in the same interval. In other words, as δ1becomes sufficiently

Fig. 1.δ

¯¹(μ⁎1,δ2,ε1) andδ

¯²(μ⁎2,δ1,ε2) forμ⁎1=μ⁎2= 0.05 andμ1⁎=μ⁎2= 0.3.

8 Eq.(1)correspondsinSchmidt's(1993)articleto(30)inTheorem3,Eq.(2)correspondsto(22),(17),(18),andEq.(3)

correspondsto(28).(Thereisaprintingerrorin(23)whichdoesnotfollowfrom(18)and(37)).

9 Schmidt (1993:334)defines “conflictinginterests” sothatthe commitmentstrategy ofone player holdstheother player downto hisminmaxpayoff.Thisissatisfiedwhend₂ N a₂ ord₁ N b₁.SeeCrippsetal.(1996)for ananalysisofgamesalso withoutconflictinginterests,whichgivesweakerbounds.IthankLarrySamuelsonforadiscussionaboutthispoint.

larger than δ2, player 1 can increase his lower bounda_1∞ toward his most preferred payoffa₁, which also increases player 2's payoff towarda_2∞. Let us formulate this as a property.

Property 1. As one player becomes sufficiently more patient than the other player, both players benefit.

Let us defineplayer 1's risk limit as r₁ = (a₁ − b₁) / (a₁ − t₁) in the static game in Table 1, which reachesits minimum at r₁= 0 when player1 isindifferent between the two equilibria, andreachesits maximum at r1 = 1 when player 1 is indifferent between the threat point and the non-preferred equilibrium.Intheinfinitelyrepeatedgameplayer1canguaranteea_1∞ ratherthanb₁tohimself.Wethus define player 1's risk limit in the infinitely repeated game by replacing player 1's least preferred equilibriumpayoff b₁ witha_1∞.Player 1strives towardthepayoff a₁,andasthedifference betweena₁ anda_1∞ decreases,player1'srisklimitdecreases.Weanalogouslydefineplayer2'srisklimitasr2= (b2− a₂) / (b₂ − t₂)inthestaticgame,replacinga₂ witha_2∞ intherepeatedgame.Hencewhenplayer1'slower bounda_1∞ isvalid,wedefinetherisklimitsandconflictmeasure

r11l¼a1−a1l

a1−t1

¼1−l⁰₂d^k₁^{1ðl⁎1;d2Þ}; r21l¼b2−a2l

b2−t2

¼1−ða2−t2Þl⁰2d^k₁^{1ðl⁎1;d2Þ} b2−t2

; cr1l¼r11lr21l; d1z_Pd₁ðl^⁎1;d2;e1Þ

ð5Þ

Definingconflictastheproductoftheplayers'risklimitshasnotbeendoneearlierintheliterature,but Axelrod(1970) hasdefinedconflictinastaticgamesuchasTable1as

c¼ða1−b1Þðb2−a2Þ

ða1−t1Þðb2−t2Þ ð6Þ

which happens to equal the product r₁r₂. In a two-dimensional utility diagram for the two players, Axelrod defines conflict as the relation of the small rectangle (a1−b1)(b2−a2) of conflictful behavior to

Fig. 2. The bounda_1∞and payoffa_2∞as functions ofδ1≥_δ1(0.3,0.2,ε1) = 0.861519.

thelargerectangle(a₁ − t₁)(b₂ − t₂) ofjointdemand.Axelrod(1970:57) referstothesmallrectangle as

“theproportionofthejointdemandareawhichisinfeasible,”¹⁰ andtothelargerectangleastheareaof jointdemandspannedoutbythethreatpoint(t1,t2)andtheoutmostpointdeterminedbythebestpayoff (a₁,b₂) eachplayercanpossibly obtainunder hismostfavorable circumstances.

When player 2's lower boundb_2∞is valid, we analogously define

r12l¼1−ðb1−t1Þl⁰₁d^k₂^{2ðl⁎2;d1Þ}

a1−t1 ; r22l¼b2−b2l

b2−t2 ¼1−l⁰1d^k2^{2ðl⁎2;d1Þ};

cr2l¼r12lr22l; d2z_Pd₂ðl^⁎₂;d1;e2Þ: ð7Þ Eq. (5) is illustrated inFig. 3forδ2= 0.2. As player 1's patience increases beyondδ1≥δ

¯¹(0.3,0.2,ε1), both risk limits and the conflict measure decrease. Increased difference between the players' emphasis on the future causes them to be more inclined to “cooperate” on the equilibrium preferred by the patient player. Conversely, as the players' emphasis on the future gets more similar, the conflict between them increases. Only the player who most successfully engages in costly reputation building in the present, which involves insisting on playing his preferred equilibrium to deter the other player from getting his preferred equilibrium, and which requires a high emphasis on the future, increases his chances to get his preferred equilibrium in the long run.

Thebenefitsof “playinghard” areprimarilyinthefutureandcanbegainedonlybychoosingconflict today.Withoutexplicitconflict inthepresentthathigherpayoffcannotbe obtained.Playerswhoplace greatervalueonthefuturearemorelikelytochooseconflictinordertoreapthosefuturebenefits.Note thesimilarity between this reasoning and that ofSkaperdas and Syropoulos (1996) andGarfinkel and Skaperdas (2000), where, despite a short-run incentive to settle a conflict, there can be long-term

10 It is the jointly infeasible expectation of an additional gain in case of a conflict.

Fig. 3. The risk limits and conflict measure whenδ1≥δ

¯¹(0.3,0.2,ε1) = 0.861519.

“compound rewards to cheating”, or “long-term compounding rewards to going to war.” If these compounded gains are large enough then conflictful and not“cooperative”behavior is the equilibrium.

Property 2. As oneplayer becomes sufficientlymore patient than theother player,both players' risk limits, and the conflict between the players, decrease.

The Properties 1 and 2 jointly mean that both players benefit from an increasing discrepancy in the players' discount factors, which causes lower risk limits and reduced conflict. That is, increasing differences in patience contribute to a greater likelihood of cooperation.

References

Axelrod,R.,1970. Conflict of Interest. MarkhamPublishers,Chicago.

Celentani,M.,Fudenberg, D.,Levine,D.K.,Pesendorfer,W.,1996.Maintainingareputationagainstalong-livedopponent.

Econometrica 64,691–704.

Cripps,M.W.,Schmidt,K.M.,Thomas,J.P.,1996.Reputationinperturbedrepeatedgames. Journal of Economic Theory 69,387– 410.

Ellsberg,D.,1961.Risk,ambiguityandthesavageaxioms. Quarterly Journal of Economics 75,643–649.

Fudenberg,D.M.,Levine,E.,1989.Reputationandequilibriumselectioningameswithapatientplayer.Econometrica57,759–778.

Fudenberg,D.M.,Levine,E.,1992.Maintainingareputationwhenstrategiesareimperfectlyobserved.Review of Economic Studies 59,561–579.

Fudenberg, D.M.,Maskin, E.,1986.Thefolktheoremin repeatedgameswithdiscounting orwithincompleteinformation.

Econometrica 54,533–554.

Fudenberg,D.M.,Tirole,J.,1991. Game Theory. MITPress,Cambridge.

Garfinkel, M.R., Skaperdas,S., 2000. Conflictwithout misperceptions or incompleteinformation: how the futurematters.

Journal of Conflict Resolution 44(6),793–807.

Harsanyi,J.C.,1977.Rational Behavior and Bargaining Equilibrium in Games and Social Situations.Cambridge.

Harsanyi,J.C.,Selten,R.,1988.A General Theory of Equilibrium Selection in Games. MITPress,Cambridge.

Hausken,K.,2005.Thebattleofthesexeswhenthefutureisimportant. Economics Letters 87(1),89–93.

Kreps,D.M.,Wilson,R.,1982.Reputationandimperfectinformation.JournalofEconomicTheory27,253–279.

Milgrom,P.,Roberts,J.,1982.Predation,reputationandentrydeterrence. Journal of Economic Theory 27,280–312.

Osborne,M.J.,Rubinstein,A.,1994.ACourseinGameTheory.MITPress,Cambridge.

Rapoport,A.,Guyer,M.,1966. A Taxonomy of 2 × 2 Games, GeneralSystems11,203–214.

Rasmusen,E.,1989.Games and Information.BasilBlackwell,Inc.,Cambridge.

Rasmusen,E.,2001. Games and Information. BasilBlackwell,Inc.,Cambridge.

Schmidt,K.M.,1993.Reputationandequilibriumcharacterizationinrepeatedgameswithconflictinginterests.Econometrica 61, 325–351.

Selten,R.,1978.Thechainstoreparadox. Theory and Decision 9,127–159.

Skaperdas, S.,Syropoulos, C.,1996. Canthe shadowofthe futureharm cooperation?Journal of Economic Behavior and Organization 29,355–372.

Sorin,S.,1999.Merging,reputation,andrepeatedgameswithincompleteinformation.GamesandEconomicBehavior29,274–

308.Watson,J.,1996.Reputationinrepeatedgameswithnodiscounting. Games and Economic Behavior 15,82–109.

Wilson,R.,1985.Reputationsingamesandmarkets.In:Roth,A.E.(Ed.),Game-theoretic Models of Bargaining.Cambridge UniversityPress,Cambridge,pp.27–62.

Zeuthen,F.,1930. Problems of Monopoly and Economics. Routledge,London.