1、Folk Theorem with Communication1Ichiro ObaraDepartment of EconomicsUniversity of Minnesota and University of California Los AngelesFirst Version August 5, 2005This version February 7, 20081I am grateful to the associate editor, an anonymous referee, Galit Ashkenazi-Golan,Michihiro Kandori, and Georg
2、e Mailath for their helpful comments. I also thank the seminarparticipants at 2005 SAET conference at Vigo and University of Southern California.AbstractThis paper proves a new folk theorem for repeated games with private monitoringand communication, extending the idea of delayed communication in Co
3、mpte 6 tothecasewhereprivatesignalsarecorrelated.The sucient condition for the folk theorem is generically satisfied with morethan two players, even when other well-known conditions are not. The folk theoremalso applies to some two-players repeated games.Keywords: Communication, Folk Theorem, Privat
4、e Monitoring, Repeated GamesJEL classification codes: C72, C73, D821IntroductionWe have observed a significant progress in repeated games with private monitoringinthelastfewyears. Itstartedwithaseriesofpaperswhichprovedafolktheo-rem with communication, such as Ben-Porath and Kahneman 4, Compte 6, an
5、dKandori and Matsushima 11. They are very important contributions to the theoryof long term relationships, especially because repeated games with private moni-toring are very dicult to analyze without communication. These folk theorems,however, do not cover all the interesting cases because of the s
6、pecific assumptionsthey require on private monitoring structure. This note proves a new folk theoremwith communication to expand the range of environments to which the folk theoremapplies.The main contribution of this note is to extend the idea of delayed communi-cation in Compte 6 to the case where
7、 private signals are correlated. Compte 6focuses on Tpublic equilibria in which players play the same action for T periodsand announce their accumulated private signals truthfully only every T periods.These private signals in each T-period block are used to “test” whether each playerhas deviated or
8、not within the same block. A player is punished at the end of theblock with a lower continuation payo when the private signals reported by theother players look “bad”. In conducting this statistical test, it is important that aplayer does not learn the likelihood of her punishment from her own priva
9、te signals.If she is confident that she will not be punished in the end of a T-period block, shemay start deviating toward the end of the block. Compte 6 avoids this problemby assuming conditional independence between players private signals. However,conditional independence is a nongeneric assumpti
10、on.1Furthermore, it is dicultto introduce even a slight correlation of private signals. This is because T must goto infinity to obtain the exact folk theorem, hence players may be able to obtaina large amount of information from their accumulated private signals even if eachsignal has a limited info
11、rmation.2This note proposes a new condition which serves the same purpose as conditionalindependence, thus making it possible to apply the idea of delayed communicationeven when private signals are correlated. The condition is generically satisfied formost of stage games when the number of players i
12、s more than two. When the numberof actions and signals is the same across players, this condition is generically satisfiedeven more easily than the sucient condition proposed by Kandori and Matsushima11.There are a few recent contributions proving folk theorems for repeated gameswith private monitor
13、ing and communication. They pay extra attention to the casewith two players: the case that was not extensively analyzed in the initial contribu-1Compte 6 does allow for some correlation of private signals o the equilibrium path.2As observed by Abreu, Milgrom and Pearce 1, this type of statistical te
14、st becomes moreeective as T becomes large. In fact, the cost of expected punishment converges to 0 as T goes toinfinity and players become infinitely patient.1tions.3Fudenberg and Levine 9 proves a Nash-threat folk theorem when playersprivate signals are highly correlated. Ashkenazi-Golan 2 assumes
15、that deviationsare perfectly observable by at least one player with positive probability and proves aNash-threat folk theorem. These results, as well as the result of this note, apply torepeated games with two or more players. Finally, McLean, Obara and Postlewaite15 proves a folk theorem when priva
16、te signals are correlated and can be treatedlike a public signal once aggregated. But this result requires at least three players.I also should mention that many folk theorem results without communicationhave been obtained recently. However, most of them assume almost perfect moni-toring (Bhaskar an
17、d Obara 5, Ely and Valimaki 7, Horner and Olszewski 10, andMailath and Morris 12).4One exception is Matsushima 14 that allows for noisyprivate monitoring. However he assumes a certain type of conditional independenceof private signals as in Compte 6. The result of this note may be useful to dealwith
18、 noisy correlated private signals even without communication, but that is leftfor future research.The next section presents the model briefly. Section 3 introduces the assumptionson monitoring structure. Section 4 presents the main result and Section 5 discussessome extension.2ModelStage gameG =(I,A
19、,g)isdefined as follows. The set of players isI = 1,2,.,n,n2. In each period, player i I chooses an action from a finite action set Ai(simul-taneously with the other players) and observes a private signal sifrom a finiteset Si.Bothaiand siare private information, observable only to player i. Letp(s|
20、a) be the probability of s S =Qni=1Sigiven a A =Qni=1Ai. It is as-sumed that p(s|a) has full support on S for every a A. The signal distributionon Si= j6=iSjfor player i given a A (and si Si) is denoted by pi(si|a)(pi(si|a,si). Player is (expected) stage game payo gi: A g().This stage game G is play
21、ed repeatedly over time. In the end of each period,players send messages m =(m1,.,mn) M =Qni=1Misimultaneously, which3However, Theorem 2 in Compte 6 applies to two-player games. Kandori and Matsushima 11spends one section to prove a folk theorm for the repeated prisoners dilemma game with conditiona
22、lindependence and private monitoring.4Also see Mailath and Morris 135For any X XsiSipisi|a0i,aixi(si)The next condition is the key for the folk theorem. Fix a A and si Siandconsider all the conditional beliefs on Siwhen player i observes a dierent signal or6I follow the convention and set hti= ht= i
23、n t =1.7It is possible to generate any public randomization device endogenously through (direct) com-munication by redefining is message space as Mi0,1 for each i I and using a jointly-controlledlottery (Aumann, Maschler and Stearns 3).3deviates from ai. Let Ri(a,si) be the convex-hull of all such c
24、onditional probabilityvectors, i.e. conpi(|a0i,ai,s0i) maxnp2y2|CC,y1,p2(y2|DC,y1),p2y2|DC,y1op2y2|CC,y1 maxnp2y2|CC,y1,p2y2|DC,y1,p2y2|DC,y1oNote that player 1 cannot be an informed player at (D,C) when she is informed at(C,C).8This condition is satisfied in the following example. Suppose that ther
25、e is a hid-den signal y y,y. Each players private signal si,i=1,2 is a noisy conditionallyindependent observation of y.Playeris private signal siis correct (yifor y andyifor y) with probability 1 0when ai= C and correct with probability 1 00when ai= D. Assume also that 00. Then this eqi(s)stillsatis
26、fies all the strictinequalities and lies in (0,1) if z is large enough.For this qi(s), (2) is satisfied becauseEqi(s)|a=XsiEqi(s)|a,siPr(si|a)=XsiPr(si|a)= .The above strict inequalities imply thatPsiqi(si),si)pi(si|(a0i,ai),si)isstrictly larger than when either a0i6= aior si6= (si). This implies th
27、at (1) holdswhen either a0i6= aior : Si Siis not an identity function, becauseEqi(si),si)|a0i,ai=XsiXsiqi(si),si)pisi|a0i,ai,siPr(si|a0i,ai) = Eqi(s)|a.Conversely, suppose that player i can be secretly screened at a A. Then, foreach si, (ai,si) is the unique solution ofmax(a0i,s0i)AiSiXsiqi(s)pisi|a
28、0i,ai,s0i.Thenpi(si|a,si)isanextremepointofconpi(|a0i,ai,s0i) 0. The incentive of playerj 6= i is provided through transfers xj(sj)(byLemma1)periodbyperiod.Notethat player js revelation constraints are satisfied by definition because xjdoes notdepend on sj.Toavoideciency losses, jixj(sj) is added to
29、 player is transfer.This is as if player is stage game payo is transformed into gi(a)Pj6=ijixj(sj).Note that this creates an incentive problem for player i because she may have in-centive to misrepresent her signal to control the transfers from j. This problem canbe taken care of by using qias expla
30、ined below. This is another reason (in additionto “no learning”) why it is useful to make qidependent on siin addition to si.Compte 6 avoids this problem by making xj(si,j) independent of si, but this doesnot allow the case with two players.Player is incentive is provided through punishments. This i
31、s where (1) is used.The probability of a bad signal, hence the probability of punishment, increases whenthe distribution of sisuggested by player is message does not match with thetrue distribution of si.9This punishment is costly because it occurs with positiveprobability (and i 0). To reduce the e
32、xpected cost of punishments, privatesignals are stored and revealed only in every T periods. The eciency of monitoringis improved by using such accumulated private signals. Indeed the expected cost ofpunishment goes to 0 as T (and 1). For this type of “review strategies”to work, player i must be kep
33、t from learning the likelihood of punishment withineach T period review phase. Condition (2) guarantees that such learning is nothappening.It is also possible to use player i with i0. Suppose that Assumption 1 is satisfied and D()is full dimensional. Then, for any smooth set W in the interior of D()
34、, thereexists (0,1) and an integer T such that W E(,T) for all (,1).Proof. See Appendix.An immediate corollary of this theorem is the following Nash-threat folk theorem.Corollary 1 Suppose that Assumption 1 is satisfied and VNEis full dimensional.Also suppose that every player is informed at every a
35、ction profile that achieves anextreme point of V .Then,foranyv intVNE, there exists (0,1) and an integerT such that v E(,T) for all (,1).Proof. First note that VNE lim0D(). For each v intVNE. PickasmoothsetW s.t. v W intVNE. Then pick small enough 0 such thatW intD() and apply the above theorem.The
36、proof of the theorem consists of two steps. The first step is to transforman infinitely repeated game into a T-period game with side transfers. Considerthe best stationary T-period equilibrium in the direction of for this T-periodgame. The equilibrium payo provides an upper bound of the best station
37、ary T-public equilibrium of the original infinitely repeated game. It is well known thatthis bound is indeed tight, i.e. the area surrounded by a collection of such boundsindexed by is approximately the set of stationary Tpublic equilibrium payosofthe original infinitely repeated game as 1. In the s
38、econd step, it is shown thatthis area contains D() for any 0asT .8There is nothing new in the first step. Here I brieflycopytheresultsfromCompte6. Consider the followingT-period game with side transfers: stage gameGis playedT times and players announce their private signals mi=(si,1,.,si,T),i=1,.,na
39、tthe end of the T periods, on which their side transfers xi(m),i I are based. Leta,Tibe player i0s T-period action strategy and m,Tibe player is report strategy,which are essentially a T-period truncated version of aiand mirespectively. LetTi=a,Ti,m,Tibe player is T-period strategy. Player is payo f
40、rom thisTperiod game given 0 (0,1) is defined bygT,0iT+Exi(m)|TwheregT,0iT=sup0,1(1)PTt=1t1Egi(at)|T1T(3)One can regard gT,0iTas player is average payo within the first T periodsof the original infinitely repeated game and xi(m) as the variation of continuationpayos.11A Tperiod strategy profile Tis
41、called stationary if it specifies the same(mixed) action profile every period and all private signals are announced truth-fully at the end of the T periods. Player is stationary strategy with i 4Aiis denoted Ti(i). Let T(a)beaprofile of stationary strategies. Note thatgT,0iT()= gi()forany. Isayv 0 a
42、nd 0, thereexists 0and T0such that k,0,T0k() for all and 0,1Proof. See the appendix.Here I just provide a rough sketch of this second step. Fix .Itiseasytoshow that D() D(,T)whenk()isachievedby NE.There exists a T-period equilibrium with zero transfer where is played in every period. So supposethat
43、k() is achieved by a non-Nash pure strategy profile a A(). Below Iconstruct transfers xi,i I, for which T(a)isastationaryTperiod equilibriumand the eciency loss is small (EhPjIjxj(m)|T(a)i).Let player i be an informed player at a with i . For j 6= i, since Assumption1issatisfied, there exists transf
44、er xjthat provides the incentive for player j to playaj(by Lemma 1). Define xj(m)=1TPTt=1xj(sj,t). These transfers take care ofthe incentive of player j 6= i.To keep “the budget balanced”, exi,j(m)=jixj(m) is added to player istransfer.13Then player i may have incentive to deviate or send a false me
45、ssage tomanipulate xj(m),j 6= i. To address this problem, I follow Compte 6 to use thefollowing scheme. Private signal profiles (s1,.,sT)reportedintheendoftheTperiods are translated into T binary signals c =(c1,.,cT) g,bT. The proba-bility of ct= b at period t is given by some function qi(st). This
46、is where a publicrandomization device is used. Then player i is punished if and only if ct= b fort =1,2,.,T. The sum of this punishment andPj6=iexi,j(m)isthetotalsidetransfer for player i.12Note that k(,0,T) is monotonically increasing in 0.13Theroleof 0istoobtainalowerboundfori, which provides an u
47、pper bound for thesetransfers.10Since player i can be secretly screened, this qican satisfy (1) and (2). The firstcondition (1) guarantees that any deviation by player i increases the probability ofb in that period.I like to make the expected probability of punishment as small as possible toavoid ec
48、iency losses. To meet this goal, first I single out the binding incentiveconstraint out of many incentive constraints. If player i can learn the likelihood ofher punishment from her private information, then the binding incentive constraintcan be with respect to a very complicated contingent deviation. The second con-dition (2) is useful here to exclude this possibility: such learning does not occur inequilibrium. Then it turns out that the bindi
