Econometrica Supplementary Material

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1 Econometrica Sulementary Material SUPPLEMENT TO WEAKLY BELIEF-FREE EQUILIBRIA IN REPEATED GAMES WITH PRIVATE MONITORING (Econometrica, Vol. 79, No. 3, May 2011, ) BY KANDORI,MICHIHIRO IN THIS SUPPLEMENT, I formally show that the weakly belief-free equilibria identified in Section 4 lie above the Pareto frontier of the belief-free equilibrium ayoff set. The roof is based on the characterization theorem by Ely, Horner, and Olszewski (2005) (referred to as EHO hereafter). To exlain their characterization of the belief-free equilibrium ayoffs, I first introduce the notion of regime A and an associated value. Using these concets, I then find an uer bound for the belief-free equilibrium ayoffs. AregimeA = A 1 A 2 is a roduct of nonemty subsets of the stage game action sets A i A i, A i, i = 1 2. In each eriod of a belief-free equilibrium, layers tyically have multile best-rely actions and they are layed with ositive robabilities. A regime corresonds to the set of such actions. For each regime A,defineanumber = su v i such that for some mixed action α i whose suort is A i and x i : A i Ω i R +, v i g(a i α i ) x i ( ) i (ω i a i a i )α i (a i ) for all a i with equality if a i A i,where i (ω i a i a i ) is the marginal distribution of ω i given action rofile (a i a i ). Intuitively, the ositive number x i reresents the reduction in layer i s future ayoffs. Note that a belief-free equilibrium has the roerty that layer i s ayoff is solely determined by the oonent s strategy. This is why the reduction in i s future ayoffs, x i,deends on the oonent s action and signal ( ). Note also that the oonent s action a i is restricted to the comonent A i of the current regime A = A i A i. The above set of inequalities ensures that layer i s best rely actions in the current eriod corresond to set A i, a comonent of the regime A = A i A i. Hence, the value is closely related to the best belief-free ayoff when the current regime is A (a more recise exlanation will be given below). Now let V be the limit set of belief-free equilibrium ayoffs when δ 1. EHO rovided an exlicit formula to comute V. For our urose here, I only sketch the relevant art of their characterization to obtain a bound for V.In Section 4.1, EHO artitioned all games into three classes: the ositive, the negative, and the abnormal cases (for our urose here, we do not need to know 2011 The Econometric Society DOI: /ECTA8480

2 2 KANDORI, MICHIHIRO their definitions). Their Proosition 6 shows that the abnormal case obtains only if one of the layers has a dominant action in the stage game that yields the same ayoff against all actions of the other layer. Clearly, this is not the case in our examle with the risoner s dilemma stage game, so our examle is in either the ositive or the negative case. 1 If it is in the negative case, EHO s Proosition 5 shows that the only belief-free equilibrium is the reetition of the stage game Nash equilibrium, yielding (0 0) in our examle. If our examle is in the ositive case, Proosition 5 in EHO imlies that the limit set of belief-free equilibrium ayoffs can be calculated as V = [ (A)m A i ] (S1) (A) A A i=12 where m A i is some number (for our urose here, we do not need to know its definition) and is a robability distribution over regimes A. Theunion is taken with resect to all robability distributions such that the intervals in formula (S1) are well defined (i.e., A (A)mA i (A)M A A i, i = 1 2). The oint to note is that V is a union of roduct sets (rectangles), and the efficient oint (uer-right corner) of each rectangle is a convex combination of (M A 1 MA). 2 The characterization (S1) ofv imlies, in the ositive case, the belief-free equilibrium ayoffs satisfy the bound (S2) (v 1 v 2 ) V v 1 + v 2 max M A 1 A + M A 2 where maximum is taken over all ossible regimes (i.e., for all A = A 1 A 2 such that A i A i, A i, i = 1 2). In what follows, I estimate M A + M A 1 2 for each regime A. In our examle, A i ={CD}, so that A i ={C} {D}, or{cd}. Before examining each regime, I first derive some general results. Consider a regime A where C A i. In this case, the incentive constraint in the definition of reduces to (S3) (S4) v i = g(cα i ) x i ( ) i (ω i Ca i )α i (a i ) g(dα i ) x i ( ) i (ω i D a i )α i (a i ) 1 With some calculation, we can determine which case alies to our examle, but this is not necessary to derive our uer bound ayoff.

3 EQUILIBRIA IN REPEATED GAMES 3 This inequality (S4) can be rearranged as ( ) i (ω i D a i ) (S5) x i ( ) i (ω i Ca i ) (ω i Ca i ) 1 α i (a i ) i Now let g(dα i ) g(cα i ) L = max ω i a i i (ω i D a i ) i (ω i Ca i ) be the maximum likelihood ratio to detect layer i s deviation from C to D. The inequality (S5)andL 1 > 0imly 2 x i ( ) i (ω i Ca i )α i (a i ) g(dα i) g(cα i) L 1 Plugging this into the definition (S3)ofv i,weobtain v i g(cα i ) g(dα i) g(cα i ) L 1 This is essentially the formula identified by Abreu, Milgrom, and Pearce (1991). The reason for welfare loss (the second term on the right hand side) is that layers are sometimes unished simultaneously in belief-free equilibria. Recall that is obtained as the suremum of v i with resect to x i and α i whose suort is A i. (Note that the right hand side of the above inequality, in contrast, does not deend on x i.) Hence, we have su g(cα i ) g(dα i) g(cα i ) (S6) L 1 where the suremum is taken over all α i whose suort is A i. Now we calculate the maximum likelihood ratio L and determine the right hand side of the inequality (S6). In our examle, when a i = C, max i (ω i Da i ) ω i i (ω i Ca i is equal to (as our examle is symmetric, consider i = 2 ) without loss of generality) 2 (ω 2 = B D C) 2 (ω 2 = B CC) = /3 = Note that as long as layer i s action affects the distribution of the oonent s signal (which is certainly the case in our examle), there must be some ω i which becomes more likely when layer i deviates from C to D.Hence,wehaveL > 1.

4 4 KANDORI, MICHIHIRO When a i = D, max i (ω i Da i ) ω i i (ω i Ca i is equal to ) 2 (ω 2 = B D D) 2/5 + 1/5 = 2 (ω 2 = B CD) 1/4 + 1/8 = 8 5 As the former is larger, we conclude L = 15. Plugging this into (S6), we obtain 8 the following uer bounds of. (a) When C A i and A i ={C}, g(cc) = 1 1/2 = (b) When C A i and A i ={D}, g(cd) g(dc) g(cc) g(d D) g(cd) = 1 6 1/6 = (c) When C A i and A i ={CD}, the larger uer bound in the above two cases alies, so that we have 3 7 Given those bounds, we are ready to estimate M A 1 + M A 2 for each regime A. Case (i), where C A i for i = 1 2: The above analysis (cases (a) and (c)) shows M A 1 + M A Case (ii), where C A i and A i ={D}: Our case (b) shows 5.In 14 contrast, s simly achieved by x i i 0(asD is the dominant strategy in the stage game) so that M A = su i g(dα i ) = g(dc) = 3 α i 2

5 EQUILIBRIA IN REPEATED GAMES 5 Hence, we have M A 1 + M A = 8 7 Case (iii), where A ={D} {D}: SinceD is the dominant action in the stage game, is achieved by x i 0. Moreover, the oonent s action is restricted to A i ={D}, so that we have = g(d D) = 0. Hence, M A 1 + M A 2 = 0 Let me summarize our discussion above. If our examle is in the negative case as defined by EHO, the only belief-free equilibrium ayoff is (0 0). Otherwise, our examle is in the ositive case, where the sum of belief-free equilibrium ayoffs v 1 + v 2 (in the limit as δ 1) is bounded above by the maximum of the uer bounds found in Cases (i) (iii), which is equal to 8.Altogether, those results show that any limit belief-free equilibrium ayoff rofile 7 (as δ 1) (v 1 v 2 ) V satisfies v 1 + v To comlete our argument, I now examine the belief-free equilibrium ayoffs fora fixed discount factor δ<1. Let V(δ)be the set of belief-free equilibrium ayoff rofiles for discount factor δ<1. The standard argument 3 shows that this is monotone increasing in δ (i.e., V(δ) V(δ ) if δ<δ ). Hence, we have V(δ) V, so that for any discount factor δ, all belief-free equilibrium ayoffs (v 1 v 2 ) V(δ)satisfy v 1 + v 2 8. Now recall that in our examle, our 7 one-eriod memory transition rule is an equilibrium if δ , with reduced game given by (S7) C D C x x α β D β α yy Numerical comutation shows x y α β > 06 forδ Hence, the total ayoff in any entry in our reduced game ayoff table (S7) exceeds 12, which 3 The roof is as follows. Suose we terminate the reeated game under δ >δrandomly in each eriod with robability 1 δ and start a new game (and reeat this rocedure). In this way, δ we can decomose the reeated game under δ into a series of randomly terminated reeated games, each of which has effective discount factor equal to δ δ = δ. Hence,anyequilibrium δ (average) ayoff under δ canalsobeachievedunderδ >δ. This argument resuoses that ublic randomization is available (to terminate the game). Even without ublic randomization, however, our conclusion V(δ) V also holds, because (i) the set of belief-free ayoff rofiles V(δ)is smaller without ublic randomization and (ii) the same limit ayoff set V obtains with or without ublic randomization (see Ely, Horner, and Olszewski (2004), theonlineaendixto EHO).

6 6 KANDORI, MICHIHIRO is larger than the uer bound for the total ayoffs associated with the belieffree equilibria, 8. This imlies that all of our equilibria lie above the Pareto 7 frontier of the belief-free equilibrium ayoff set. REFERENCES ABREU, D., P. MILGROM, AND D. PEARCE (1991): Information and Timing in Reeated Partnershis, Econometrica, 59, [3] ELY, J. C., J. HORNER, AND W. OLSZEWSKI (2004): Disensing With Public Randomization in Belief-Free Equilibria, Mimeo, available at htts://sites.google.com/site/jo4horner/home/ ublications.[5] (2005): Belief-Free Equilibria in Reeated Games, Econometrica, 73, [1] Faculty of Economics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo , Jaan; kandori@e.u-tokyo.ac.j. Manuscrit received March, 2009; final revision received October, 2010.