Francoise Forges
email: francoise.forges@gmail.com

Françoise Forges

Antoine Salomon
email: antoine.salomon@dauphine.fr

Gorkem Celik

Martin Cripps

Gaël Giraud

Johannes Hörner

Vincent Iehlé

Frédéric Koessler

Marcin Peski

Péter

Bayesian repeated games and reputation

Keywords: Bayesian game, incentive compatibility, individual rationality, in…nitely repeated game, private values, public good, reputation. JEL classi…cation: C73, C72, C71, D82, H41

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Bayesian repeated games and reputation

Introduction

We consider in…nitely repeated games in which the players privately know their own payo¤s but are uncertain about the payo¤s (i.e., the types) of the other players. We assume that the players start with given beliefs over each other's types and observe each other's decisions at every stage of the game. We are looking for a characterization of the perfect Bayesian equilibrium payo¤s of the game, when the players are very patient. In other words, our goal is to understand how the well-known folk theorem extends in repeated games with private values and full monitoring.

The folk theorem goes back to the 1970's and the extension that we seek is very natural. So one would expect it to be already part of the literature. However, as we shall show below, only partial answers are available. Before surveying them, we …rst brie ‡y describe our own contribution without reference to any earlier work.

We make the further assumption that, in the one-shot Bayesian game, uniform punishment strategies are available against every player. This property means that there is no need to know a player's type to punish him in the harshest possible way, i.e., at his ex post individually rational level. The assumption holds in a number of economic applications, including auctions and oligopoly.

Both properties, private values and uniform punishment strategies, are also satis…ed in a class of familiar public good games, in which every player is endowed with some private good. Every player can either enjoy his endowment or devote it to a public good, which is produced if and only if enough players contribute. Individuals di¤er in their relative preferences for the private and the public goods. Such games will be our reference model.

Hoping for some intuition on how the equilibrium payo¤s of the Bayesian discounted in…nitely repeated game look like when players become very patient, we …rst characterize the set N [ 1 ] of equilibrium payo¤s of the game tive compatible and individually rational payo¤s in the one-shot game.

The similarity with the complete information case stops essentially there. Applying the previous characterization to our reference public good games, we …nd that Bayesian undiscounted in…nitely repeated games may not have any equilibrium, i.e., it may happen that N [ 1 ] = ;. This result is stated as Proposition 2. By contrast, Nash's theorem guarantees existence of an equilibrium in every discounted in…nitely repeated game , for every …xed discount factor . But the problem is then to describe the solutions as players become very patient, namely as the discount factor goes to 1. The fact that the undiscounted game does not have any equilibrium makes useless the techniques that are often applied to check the non-emptiness of relevant limit sets like lim inf !1 N [ ].

Given this di¢culty, we pursue with the study of a particular model, namely, the two-person Bayesian discounted in…nitely repeated version of the public good games mentioned above. It turns out that, with discounting, Proposition 1 (all equilibria are payo¤-equivalent to completely revealing ones) and Proposition 2 (the slightest doubt on the players' types can lead to non existence of equilibrium) no longer hold, even when players become increasingly patient. More importantly, for every discount factor , we explicitly construct a family of perfect Bayesian equilibria in speci…c strategies, in which the players behave as in a war of attrition. We prove that the pay-o¤s associated with these war of attrition equilibria converge as the discount factor goes to 1. As a consequence, the (inferior) limit of the corresponding sets of perfect Bayesian equilibrium payo¤s, i.e., lim inf !1 N [ ], is nonempty (and even has a nonempty interior). These results are summarized in Proposition 7.

The war of attrition equilibria are …rst constructed in an auxiliary reputation game, in which normal players play a game with strictly competitive interests (SCI) and possibly face automata. In this auxiliary game, we allow the players to have di¤erent discount factors. Propositions 4, 5 and 6 propose three di¤erent sets of su¢cient conditions (on the parameters of the SCI game, the strategies of the automata and the discount factors) that guarantee the existence of perfect Bayesian equilibria in war of attrition strategies in the auxiliary game. We believe that this analysis, which is entirely performed in the discrete time, discounted reputation game, deserves interest for its own sake.

Time has come to describe how our results are connected to three popular themes in the repeated games literature: information transmission, cooperation and reputation.

Information transmission

Aumann and Maschler started to analyze two-person in…nitely repeated games with incomplete information in the 1960's (see [START_REF] Aumann | Repeated Games of Incomplete Information[END_REF]). They mostly studied the zero-sum case, in which the assumption of known-own payo¤s is obviously not meaningful. They provided a characterization of the value v 1 of the undiscounted zero-sum in…nitely repeated game and showed that v 1 might not exist when both players have private information. [START_REF] Mertens | The value of two-person zero-sum repeated games with lack of information on both sides[END_REF] proved that the values v of the discounted games always converge as goes to 1; they characterized lim !1 v and showed that it coincides with v 1 when the latter exists. Our main results (Propositions 1, 2 and 7) partially extend the previous ones to non-zero-sum games.

Together with Stearns, Aumann and Maschler also started to investigate two-person non-zero-sum undiscounted in…nitely repeated games in which only one of the players has private information (see [START_REF] Aumann | Repeated games with incomplete information: an approach to the nonzero sum case[END_REF] and [START_REF] Aumann | Repeated Games of Incomplete Information[END_REF]). In this framework, a full characterization of Nash equilibrium payo¤s was obtained by [START_REF] Hart | Nonzero-sum two-person repeated games with incomplete information[END_REF] and existence was established by [START_REF] Sorin | Some results on the existence of Nash equilibria for non-zero sum games with incomplete information[END_REF] and Simon et al. (1995). As the previous authors, [START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF] considers two-person games but, instead of assuming that only one player is privately informed, he assumes that every player knows his own payo¤. He shows that the Nash equilibrium of the undiscounted in…nitely repeated game 1 are payo¤-equivalent to completely revealing equilibria of the in…nitely repeated game 1 . 1In Proposition 1, we add an assumption to Koren (1992)'s ones, namely, that uniform punishment strategies are available. This enables us to formulate the characterization by simple inequalities, entirely in terms of the one-shot game. By contrast, Koren (1992)'s characterization makes use of [START_REF] Blackwell | An analog of the minimax theorem for vector payo¤s[END_REF]'s approachability, i.e., of punishments that are typical to the in…nitely repeated game. Our new assumption enables us to de…ne individual rationality in the in…nitely repeated game exactly as in the one-shot game, so that we can dispense with approachability. As a by-product, our re-sult immediately goes through in the case of n players. 2 Known-own payo¤s are crucial in [START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF] and in Proposition 1. Without this assumption, the Nash equilibria of 1 are no longer payo¤-equivalent to completely revealing equilibria, even if there are two players and only one of them has private information (see [START_REF] Hart | Nonzero-sum two-person repeated games with incomplete information[END_REF] and [START_REF] Aumann | Repeated Games of Incomplete Information[END_REF]). In the appendix, we illustrate the further role of uniform punishment strategies in Proposition 1. [START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF] already proposed an example of an undiscounted in…nitely repeated game 1 with known-own payo¤s that did not have any Nash equilibrium. We recall it as Example 1. Proposition 2, together with a complement in the appendix, further shows that the emptiness of N [ 1 ] is quite robust. The phenomenon arises in a large class of public good games (see, e.g., Fudenberg and Tirole (1991, example 6.1) and [START_REF] Palfrey | Repeated play, cooperation and coordination: an experimental study[END_REF]). It does not rely on the fact that some players' types have a dominant strategy in the repeated game (in other words, it does not rely on the fact that for some given types, the set of feasible and individually rational payo¤s has an empty interior).

The papers on non-zero-sum games surveyed up to now all deal with undiscounted in…nitely repeated games. A natural next step is to check whether [START_REF] Mertens | The value of two-person zero-sum repeated games with lack of information on both sides[END_REF]'s result for zero-sum games survives, namely, whether an appropriate limit of the sets of equilibrium payo¤s N [ ] as goes to 1 is not empty and can be characterized. 3 Restricting to public good games, Proposition 7 gives a positive answer to the …rst question. [START_REF] Peski | Repeated games with incomplete information on one side[END_REF] focuses on two-person discounted repeated games with lack of information on one side and known-own payo¤s, when the informed player has two types (namely, the discounted version of [START_REF] Shalev | Nonzero-sum two-person repeated games with incomplete information and known-own payo¤s[END_REF] in the case of two types). In this very particular model, he shows that lim [START_REF] Peski | Repeated games with incomplete information on one side[END_REF] characterizes the set lim !1 N [ ] and con…rms a phenomenon that was already suggested in [START_REF] Cripps | Some asymptotic results in discounted repeated games of one-sided incomplete information[END_REF], namely, that more equilibrium pay-o¤s can be achieved in the limit discounted case than in the undiscounted case, namely, that lim !1 N [ ] % N [ 1 ] (where N [ 1 ] 6 = ;, as shown by [START_REF] Shalev | Nonzero-sum two-person repeated games with incomplete information and known-own payo¤s[END_REF]).

!1 N [ ] exists (in the sense that lim inf !1 N [ ] = lim sup !1 N [ ]).
Peski (2013) extends the previous results to a larger class of n person discounted repeated games with known-own payo¤s, in which several players may possess private information. More precisely, [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF] makes an "open thread assumption" which, in the two-person case, amounts to the existence of a belief-free equilibrium in the sense of [START_REF] Hörner | Belief-free equilibria in games with incomplete information[END_REF].

As already in [START_REF] Peski | Repeated games with incomplete information on one side[END_REF], the non emptiness of limit sets of Bayesian perfect Nash equilibrium payo¤s as players become increasingly patient is not an issue in [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF], at least in the two player case, since belief free equilibria are assumed to exist at the outset. The main result in [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF] is the full characterization of lim !1 N [ ] as the set of payo¤s of special equilibria, the …nitely revealing ones, which have the property that the players update their beliefs …nitely many times.

As shown by [START_REF] Hörner | Belief-free equilibria in games with incomplete information[END_REF], two-person discounted repeated games with lack of information on one side always have a belief-free equilibrium when players are patient enough. 4 Hence Peski ( 2013)'s open thread assumption is always satis…ed in this particular case. However, the assumption easily fails in familiar applications like our public good games, in which both players have private information. Indeed, it appears that, as soon as the undiscounted in…nitely repeated good game has no equilibrium for some speci…cation of the players' beliefs, the discounted in…nitely repeated game cannot have any belief free equilibrium.

Our Proposition 7 shows that nonetheless, in every discounted repeated version of our public good game with a su¢ciently high discount factor, there exist perfect Bayesian equilibrium payo¤s which converge as the players become increasingly patient. We thus show in particular that Peski (2013)'s open thread assumption is not necessary for the convergence of equilibrium payo¤s, i.e., for lim inf !1 N [ ] to be nonempty. As already mentioned above, we establish Proposition 7 by explicitly constructing equilibria, in which the players's strategies follow a war of attrition. These equilibria di¤er from the ones that are used in Peski (2013)'s characterization, as they are not …nitely revealing. However, our construction can be adapted to show that, in the public good game, lim inf !1 N [ ] contains …nitely revealing equilibrium payo¤s.

Cooperation

The tractable characterization in Proposition 1 facilitates the comparison between the solutions of the undiscounted repeated game and the cooperative solutions of the initial Bayesian game. More precisely, Proposition 1 leads to a partial version of the folk theorem, which we state as Proposition 3: for any n person Bayesian game with known-own payo¤s and uniform punishments, the set N [ 1 ] of Nash equilibrium payo¤s of the undiscounted in…nitely repeated game 1 is contained in the set F [ ] of interim cooperative solutions of the Bayesian game, as de…ned in [START_REF] Myerson | Game theory: analysis of con ‡ict[END_REF] and [START_REF] Forges | A folk theorem for Bayesian games with commitment[END_REF]. By de…nition, the set F [ ] consists of the payo¤s that are feasible (i.e., achievable by means of a mechanism), incentive compatible and (interim) individually rational. The fact that N [ 1 ] F [ ] con…rms that the repetition of the game enables the players to cooperate, as in the folk theorem with complete information. However, it may happen that N

[ 1 ] $ F [ ],
even when N [ 1 ] is not empty. Hence, according to a common, strict interpretation, we could say that the folk theorem does not hold.

Our characterization shows that, under incomplete information, the cooperative solutions of the one-shot game (i.e., F [ ]) and the non-cooperative solutions of the undiscounted repeated game (i.e., N [ 1 ]) mostly di¤er in the individual rationality levels of the players. Under our assumptions of private values and uniform punishments, the ex post individual rationality level v i ( i ) of player i, namely the level at which the other players can punish him when they know his type i , is relevant in the in…nitely repeated game. Interim individually rational payo¤s in the sense of [START_REF] Myerson | Game theory: analysis of con ‡ict[END_REF] are always ex post individually rational. When there exist uniform punishment strategies, the reverse also holds: this is the key of Proposition 3. However, if the assumption of uniform punishments is relaxed, individual rationality in the in…nitely repeated game relies on Blackwell (1956)'s approachability strategies. As a consequence, Proposition 3 is no longer true, while Koren (1992)'s characterization still holds, at least in the two-person case. To sum up, the inclusion

N [ 1 ]
F [ ], which does not seem very surprising at …rst sight, crucially depends on the fact that uniform punishment strategies are available.

A recent paper of Wiseman (2012) o¤ers a di¤erent perspective on the previous topic. He establishes a partial folk theorem in discounted repeated games where the players have the same initial information and get private and public signals along the play. While his model captures in particular known own payo¤s (see his example 3), he makes an assumption that ensures "gradual public learning" which has no counterpart in in…nitely repeated games like the ones considered here. As a consequence, Wiseman (2012)'s folk theorem can be formulated in terms of feasible, ex post individually rational payo¤s, without any requirement of incentive compatibility. By contrast, incentive compatibility is crucial in this paper and in [START_REF] Peski | Repeated games with incomplete information on one side[END_REF][START_REF] Peski | Repeated games with incomplete information and discounting[END_REF].

Reputation

An extensive literature shows how a player can use the other players' uncertainty on his payo¤ to establish a reputation (see the references in [START_REF] Mailath | Repeated Games and Reputations[END_REF]). 5 The repeated games of this paper share a property with reputation models: players know their own payo¤s. However, our basic model is not necessarily generated by perturbing a game with complete information and may not involve any commitment type. Many other di¤erences could be mentioned (e.g., most reputation models deal with a single informed player 6 and imperfect monitoring) but the main one lies in the motivations. We are interested in the characterization and the existence of equilibrium payo¤s while a reputation e¤ect is identi…ed by a form of equilibrium payo¤ uniqueness. Loosely speaking, a reputation e¤ect obtains when all appropriately re…ned equilibria of the perturbed game guarantee a high payo¤ to the player who is possibly committed to some behavior (see, e.g., [START_REF] Schmidt | Reputation and equilibrium characterization in repeated games with con ‡icting interests[END_REF] for a more precise, yet informal, de…nition).

The previous comparison is relevant for characterization results like Proposition 1. More must be said for the results that are established for public good games. In these, there are two players, who can both be "normal" or "greedy" and have two actions ("contribute" or "do not contribute"). When both players are normal, they play a particular game with strictly competitive interests (SCI), a notion de…ned in [START_REF] Cripps | Reputation with equal discounting in repeated games with strictly con ‡icting interests[END_REF] and Atakan and Ekmekci (2013). The di¤erence with the latter papers is that our greedy players are not committed to a speci…c strategy. Proposition 2, which says that, for a substantial set of parameters, the undiscounted repeated game has no Nash equilibrium at all, implies that, without discounting, no twosided reputation e¤ect is achievable. While Cripps and Thomas (1995) and [START_REF] Israeli | Sowing doubt optimally in two-person repeated games[END_REF] acknowledge this phenomenon (as a direct consequence of Ko-ren (1992)'s example), they do not demonstrate its robustness, as we do here (especially in the three action version of the public good game given in the appendix).

We also construct sequences of equilibrium payo¤s in every discounted public good game which all converge as goes to 1 (Proposition 7). These equilibria are constructed in two steps. First, we assume that the greedy players are committed to some strategies, while the normal players have complete freedom to pick theirs. This de…nes an auxiliary reputation game. Then we show that it is a best reply for the greedy players to behave as automata. The auxiliary game is at the same time more restrictive and more general than the SCI model of [START_REF] Atakan | A two-sided reputation result with long-run players[END_REF]. We assume that the players have only two actions7 but, to account for the strategic possibilities of our greedy players, we are led to consider a richer class of automata.

We derive explicit conditions (on the parameters of the SCI game, the strategies of the automata and the normal players' discount factors 8 ) which guarantee that the auxiliary game has a perfect Bayesian equilibrium in war of attrition strategies. Atakan and Ekmekci (2013) make use of a war of attrition game in continuous time that was proposed in [START_REF] Abreu | Bargaining and reputation[END_REF] to study reputation in bargaining. We perform our analysis in the repeated (i.e., discrete time) game with a …xed discount factor and generate a war of attrition through appropriate players' strategies. By proceeding in this way, we can compute the exact expression of a family of equilibrium payo¤s (each equilibrium corresponding to a speci…cation of the automata) and show that they do converge as players become increasingly patient.

Atakan and Ekmekci (2013) are rather interested in a reputation result. Having identi…ed relevant reputation payo¤s for each normal player in the SCI game, they allow the normal players to face particular, elementary automata and they prove that, when the stage game is played arbitrarily frequently, the normal players get approximately their reputation payo¤s at any Bayesian perfect equilibrium. Our results are consistent with theirs, in the sense that, when we restrict to elementary automata, we recover their reputation payo¤s as limit payo¤s.

As suggested in the previous paragraphs, the paper develops as follows: Section 2 describes the one-shot Bayesian game and the reference public good games; Section 3 proposes a characterization of the set of equilibrium payo¤s in the Bayesian undiscounted repeated game and demonstrates its possible emptiness in public good games; Section 4 deduces a partial folk theorem; Section 5 goes on with the analysis of equilibrium payo¤s in Bayesian discounted repeated games; Section 5.1 is devoted to auxiliary reputation games, which are based on SCI games with two actions and rich automata; Section 5.2 concentrates on the public good games; Section 6 is an appendix wich contains supplementary material and proofs. Sections 5.1 and 6.4 (resp., 5.2 and 6.5) contain precise elements of comparison with Atakan and Ekmekci (2013) (resp., [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF]).

Basic Bayesian game 2.1 De…nition

Let us …x n players and, for every player i, i = 1; :::n, a …nite set of types i a probability distribution q i over i a …nite set of actions A i , with jA i j j i j a utility function u i : i A ! R, where A = Y

1 i n A i .
This de…nes a (one-shot) Bayesian game with independent, private values 9 , which we denote as (q), with q = (q i ) 1 i n . 10 Without loss of generality, we assume that q i ( i ) > 0 for every i 2 i . The interpretation is that types i , i = 1; :::; n, are …rst chosen in , independently of each other, according to q. At the interim stage, player i is only informed of his own type i . The players then choose simultaneously an action.

For any …nite set E, let us denote as (E) the set of probability distributions over E. A mixed strategy 11 of player i in (q) is a mapping from i 9 While private values (i.e., "known own payo¤s") are crucial for our results, the independence assumption can be relaxed in most of them. 10 We only recall the parameter q in the notation (q) for the Bayesian game, because it will often happen, e.g., in the examples, that the beliefs q vary while all other parameters are …xed. 11 More correctly, "behavior strategy".

to (A i ). Similarly, a correlated strategy for players j 6 = i is a mapping from

i = Q j6 =i j to (A i ), where A i = Q j6 =i A j .
We keep the notation u i for the (multi)linear extension of utility functions over mixed and/or correlated strategies. Hence we write, for every i = 1; :::; n, i 2 i , 2 (A),

u i ( i ; ) = X a (a)u i ( i ; a)
In particular, for every i = 1; ::

:; n, i 2 i , i 2 (A i ), i 2
(A i ), u i ( i ; i ; i ) = X a i ;a i i (a i ) i (a i )u i ( i ; a i ; a i )
We introduce a new property, which we call "uniform punishment strategies". To de…ne it, consider the (type-dependent) individually rational level v i ( i ) of player i, for every possible type i . We assume that the other players, j 6 = i, have a strategy pro…le (of course independent of player i's type) which enable them to keep player i's payo¤ below v i ( i ), for every possible type i . So, while player i can make use of his information i to guarantee himself the level v i ( i ), the other players can guarantee that his payo¤ does not exceed this level, even if they do not know his type i .

More precisely, for every player i, i = 1; :::; n, and i 2 i , the individually rational level v i ( i ) is

v i ( i ) = min i 2 (A i ) max i 2 (A i ) u i ( i ; i ; i ) = min i 2 (A i ) max a i 2A i u i ( i ; a i ; i ) (1)
In the previous expression, the probability distribution i achieving the "min" possibly depends on i , which is …xed in the underlying optimization problem. v i ( i ) can thus be interpreted as the ex post individual rationality level of player i, namely, the lowest amount at which players j 6 = i can hold player i when they know his type i and correlate their strategies.

Our assumption of "uniform punishment strategies" can be formalized as follows:

8i 9 i 2 Y j6 =i (A j ) s.t. 8 i 2 i 8a i 2 A i u i ( i ; a i ; i ) v i ( i ) (2)
When (2) holds, i de…nes independent 12 punishment strategies which enable players j 6 = i to punish player i uniformly, i.e., whatever his type i is, but even more, to keep player i's payo¤ below his ex post individual rationality level. 13 Assumption (2) is satis…ed in many applications, in particular in the public good games below. 14 We will clarify its role in the appendix.

A public good game

The example belongs to a class of public good games that was studied, e.g., by [START_REF] Palfrey | Repeated play, cooperation and coordination: an experimental study[END_REF]. There are two players. The private information of player i, i = 1; 2, is the value i that he attributes to his endowment of a single unit of the private good. The private endowment values i are chosen independently of each other, according to a probability distribution q i . Player i has two possible actions a i : "contribute" (c) and "do not contribute" (d). A public good is produced as soon as one of the players contributes. The value of the public good is normalized to 1 for both players.

The payo¤ matrix associated with the pair of types

( 1 ; 2 ) is thus c d c 1; 1 1; 1 + 2 d 1 + 1 ; 1 1 ; 2
where we always assume i 0 but can have i < 1 or i > 1. Fudenberg and Tirole (1991, example 6.1, p. 211) propose the following interpretation: player 1 and player 2 belong to a group (say, the members of some university department) and each of them can represent the group at a committee (say, the scienti…c board of the university). To attend the committee is time consuming and it is enough that one player attends the committee meeting to defend the interests of the group. The whole problem is to decide which one of the players will go to the meeting, given that the value of time for each player is private information.

In this public good game, by playing d, player i guarantees himself i while by playing c, he guarantees himself 1. Hence, by playing according to 13 As a slight weakening, v i ( i ) could just be de…ned as

min i2 Q j6 =i (Aj ) max i2 (Ai) u i ( i ; i ; i ):
his type, player i can guarantee himself max f i ; 1g. By not contributing, namely by playing a j = d, player j guarantees that player i's payo¤ does not exceed max f i ; 1g. Hence, v i ( i ) = max f i ; 1g and a j = d is a uniform punishment strategy of player j against player i.

We will speci…cally consider a Bayesian public good game in which each player has two possible types: i = f!; zg, i = 1; 2, where 0 < ! < 1 and z > 2: ! represents a "normal" type, who values the public good more than his initial endowment, while z represents a "greedy" type. The possible payo¤ matrices are thus

2 = ! 2 = z c d c d 1 = ! c 1; 1 1; 1 + ! 1; 1 1; 1 + z d 1 + !; 1 !; ! 1 + !; 1 !; z 1 = z c 1; 1 1; 1 + ! 1; 1 1; 1 + z d 1 + z; 1 z; ! 1 + z; 1 z; z (3)
We denote as p i , 0 < p i < 1, the probability that player i is normal (i = 1; 2) and we refer to this game as P G(p; !; z), with p = (p 1 ; p 2 ).

Bayesian undiscounted repeated game

Let us turn to in…nitely repeated versions of the Bayesian game (q), starting with the undiscounted one, which we denote as 1 (q). The players' types are intrinsic characteristics, which are …xed throughout the game. More precisely, 1 (q) is played as follows:

-at a virtual stage (stage 1): the types i , i = 1; :::; n, are chosen in = Y

1 i n i independently of each other, according to q. Player i is only informed of his own type i .

-at every stage t (t = 0; 1; :::): every player i chooses an action in A i . The choices are made simultaneously and revealed publicly right after stage t.

Payo¤s in 1 (q) are evaluated as limits of means, namely as (Banach) limits of arithmetic averages (see [START_REF] Hart | Nonzero-sum two-person repeated games with incomplete information[END_REF], [START_REF] Forges | Repeated games of incomplete information : nonzero-sum[END_REF]). In Section 5, we shall rather consider the discounted version of the in…nitely repeated game.

Characterization of Nash equilibrium payo¤s

Let us write q i ( i ) for Q j6 =i q j ( j ) and let us denote as N [ 1 (q)] the set of all Nash equilibrium interim expected payo¤s of 1 (q). The next Proposition provides a tractable characterization of N [ 1 (q)]. 15Proposition 1 Let (q) be a Bayesian game with independent private values in which uniform punishment strategies are available. Let x = (x i )

1 i n = ((x i ( i )) i 2 i ) 1 i n .
The payo¤ vector x is a Nash equilibrium interim expected payo¤ in 1 (q) if and only if there exist probability distributions ( ) 2 (A), 2 , such that for every i = 1; ::

:; n, i , 0 i 2 i x i ( i ) = X i 2 i q i ( i )u i ( i ; ( i ; i )) X i 2 i q i ( i ) max fu i ( i ; ( 0 i ; i )); v i ( i )g (4)
In the case of complete information, namely if the prior probability distribution q is degenerate, Proposition 1 reduces to the standard folk theorem: x = (x i ) 1 i n 2 R n is a Nash equilibrium payo¤ of the in…nitely repeated game if and only if x is feasible (i.e., achieved by means of a probability distribution 2 (A)) and individually rational (i.e., x i is larger than player i's minmax level).

The interpretation of Proposition 1, under incomplete information, is that all Nash equilibria of 1 (q) are payo¤ equivalent to completely revealing equilibria, in which -at stage 0, every player i truthfully reveals his type i to the other players -at every stage t 1, given the reported types 0 = ( 0 i ) 1 i n , every player i plays according to ( 0 ) 2 (A) provided that ( 0 ) has been followed at every previous stage 1; :::; t 1. Otherwise, if player i does not follow ( 0 ) at some stage t 1, players j 6 = i punish player i by using independent uniform punishment strategies i holding player i at v i ( i ) at every stage t + 1, t + 2, ... whatever his type i and action are.

According to the nondeviation condition (4), player i, of type i , takes for granted that players j 6 = i follow the equilibrium strategies. He can report a type 0 i , which is possibly di¤erent from i . At the end of stage 0, player i learns the true types i of the other players and can then either follow ( 0 i ; i ) or not. In the former case, he fully mimics the equilibrium strategy of type 0 i . In the latter case, the other players notice the deviation and player i is punished at the level v i ( i ).

Condition ( 4) is thus both an incentive compatibility condition and an individual rationality condition. Even under our strong assumptions, it is not possible to separate these two aspects of player i's nondeviating condition. Obviously, for 0 i = i , ( 4) is equivalent to For every i and

= ( i ; i ) 2 : u i ( i ; ( )) v i ( i ) (5)
which implies that For every i and i 2 i :

x i ( i ) v i ( i ) (6)
With some abuse of language, we will refer to the latter property as x is ex post individually rational and will denote as EXP IR i [ (q)] the set of all vector payo¤s which satisfy it for player i. Proposition 1 is established in the appendix. Intuitively, three properties play a key role to show that an arbitrary equilibrium is payo¤ equivalent to a completely revealing one. First, the posterior probability distributions of every player on the other players' types converge, so that the players' behavior on the equilibrium path becomes nonrevealing. Let us make things extreme and assume that at equilibrium, no information is revealed from some stage T on; then, from T on, the players' behavior on the equilibrium path is described by a probability distribution T over A (which depends on the history of moves up to stage T ). Second, once the players have settled on T , every player i may as well learn the types of the other players, since, thanks to private values, his payo¤ only depends on i and T . Finally, because there is no discounting, the information that is transmitted in T stages can as well be transmitted at once, at the …rst one. In the appendix, we also show the role of uniform punishment strategies.

Existence of Nash equilibrium

Thanks to Proposition 1, the set N [ 1 (q)] of Nash equilibrium payo¤s of the undiscounted repeated game has a tractable representation so that it is rela-tively easy to check whether it is empty or not. Let us denote as P G 1 (p; !; z) the in…nitely repeated version of the public good game P G(p; !; z) introduced in Section 2.2. We will show that P G 1 (p; !; z) has no Nash equilibrium as soon as z, p 1 and p 2 are su¢ciently large. [START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF] already illustrated the same phenomenon in the two-player game of Example 1 below. However, in the latter, as soon as one of the players is greedy, the set of feasible individually rational payo¤s has an empty interior. The same happens in P G 1 (p; !; z), when both players are greedy, but only in that case. As shown in the appendix, our example can be modi…ed slightly so as to completely get rid of this peculiarity.

Proposition 2 Consider the undiscounted in…nitely repeated public good game P G 1 (p; !; z). If the priors p 1 ; p 2 and the payo¤ z are su¢ciently large, P G 1 (p; !; z) has no Nash equilibrium, i.e., N [P G 1 (p; !; z)] = ;. The same holds in a variant of P G 1 (p; !; z) in which, for every pair of types, the interior of the set of feasible individually rational payo¤s is nonempty.

Proof: Let us consider an interim expected equilibrium payo¤ x of the game P G 1 (p; !; z). According to Proposition 1, x is described by probability distributions ( ) over A = fc; dg fc; dg for every pair of types 2 fn; gg fn; gg, which must satisfy the equilibrium conditions (4) in Proposition 1, in particular the ex post individual rationality conditions [START_REF] Blackwell | An analog of the minimax theorem for vector payo¤s[END_REF]. From a geometric representation of the feasible, individually rational payo¤s when one of the players is normal and the other one is greedy, it appears that if z is su¢ciently large, z z + 1. The ex post individual rationality conditions thus imply that ((c; d) j n; g) 1 ", ((d; c) j g; n) 1 "

where we write (a j ) for ( )(a), for every a 2 A. A formal proof of these inequalities is given in the appendix. Condition (4) for player 1 of type n implies that he should not bene…t from the following deviation: pretend to be of type g at stage 0; thereafter, mimic type g when the other player turns out to be normal and defend his individual rationality level (namely play c) when the other player turns out to be greedy:

p 2 u 1 (n; (n; n)) + (1 p 2 )u 1 (n; (n; g)) p 2 u 1 (n; (g; n)) + (1 p 2 ) 1 (8)
The latter condition amounts to ex post individual rationality when p 2 = 0.

Let us set 1 = ((c; d) j n; g) and 2 = ((d; c) j g; n). Feasibility implies that u 1 (n; (n; g))

1 1 + (1 1 )(1 + !) (9) u 1 (n; (g; n)) 2 (1 + !) (10)
Hence, [START_REF] Cripps | Reputation and commitment in twoperson repeated games without discounting[END_REF] implies that

p 2 u 1 (n; (n; n)) + (1 p 2 )(1 1 )! p 2 2 (1 + !) so that u 1 (n; (n; n)) 2 (1 + !) 1 p 2 p 2 (1 1 )! if p 2 > 0
and, if z is su¢ciently large, from [START_REF] Chan | On the non-existence of reputation e¤ects in twoperson in…nitely repeated games[END_REF],

u 1 (n; (n; n)) (1 + !) "(1 + ! p 2 ):
If in addition p 2 is su¢ciently large, say p 2 !, we must have

u 1 (n; (n; n)) (1 + !) 2
By proceeding similarly for player 2,

u 2 (n; (n; n)) (1 + !) 2
which contradicts feasibility for " su¢ciently small (since

u 1 (n; (n; n)) + u 2 (n; (n; n)) 2 + !).
The variant of P G 1 (p; !; z) is described in the appendix.

The previous …nding should be contrasted with what happens in one-sided reputation models. If only one player is uncertain of the type of the other, there exist equilibria accounting for a reputation e¤ect, even if the game is not discounted (see [START_REF] Shalev | Nonzero-sum two-person repeated games with incomplete information and known-own payo¤s[END_REF], [START_REF] Cripps | Reputation and commitment in twoperson repeated games without discounting[END_REF] and [START_REF] Israeli | Sowing doubt optimally in two-person repeated games[END_REF]). However, as soon as both players believe with arbitrarily small probability that the other player could be (very) greedy, the game has no equilibrium at all. Two features of the previous example are important: lack of information on both sides and no discounting. The analysis of the discounted game, when the players become in…nitely patient, leads to di¤erent conclusions, as we will see in Section 5.

Repetition and cooperation

Under complete information, the set of feasible and individually rational pay-o¤s of a one-shot game coincides not only with the set of Nash equilibrium payo¤s of the in…nitely repeated game but also with a well founded set of cooperative solutions of the one-shot game (see e.g. [START_REF] Kalai | A commitment folk theorem[END_REF]). The inequalities (4) in Proposition 1 can be interpreted as describing cooperative solutions for the one-shot Bayesian game (q), so that Proposition 1 establishes a relationship between "repetition" and "cooperation". However, under incomplete information, a more natural candidate for the set of cooperative solutions is Myerson (1991)'s set of feasible, incentive compatible and interim individually rational payo¤s in the (one-shot) Bayesian game (q) (see [START_REF] Forges | A folk theorem for Bayesian games with commitment[END_REF]). We denote this set as F [ (q)] and de…ne it precisely below. 16 Myerson (1991)'s de…nitions take a simpler form in our framework of independent private values. We then establish a partial analog of the folk theorem, namely that F [ (q)] contains N [ 1 (q)], the set of Nash equilibrium payo¤s of the in…nitely repeated game 1 (q).

A payo¤ x = (x i )

1 i n = ((x i ( i )) i 2 i ) 1 i n is feasible in (q) if there exists a correlated strategy ( ) 2 (A), 2 , achieving x, namely x i ( i ) = X i q i ( i )u i ( i ; ( i ; i )) i = 1; :::; n; i 2 i (11)
A feasible payo¤ x achieved through (as in [START_REF] Forges | Communication equilibria in repeated games with incomplete information[END_REF]) is incentive compatible

if x i ( i ) X i q i ( i )u i ( i ; ( 0 i ; i )) for every i; i ; 0 i 2 i (12)
A payo¤ x is interim individually rational if, for every player i, there exists a correlated strategy i 2 (A i ) of players j 6 = i such that 17

x i ( i ) max 16 Under an assumption that is similar to our uniform punishment strategies, Peters and Szentes (2012) also …nd that F [ (q)] corresponds to a set of reasonable cooperative solutions of (q), which are achieved through interim commitment. 17 Literally, Myerson (1991)'s interim individual rationality condition requires that there exists a type dependent correlated strategy of players

a i 2A i u i ( i ; a i ; i ) for every i 2 i ( 13
)
j 6 = i, i ( i ) 2 (A i ), i 2 i , such that x i ( i ) max ai2Ai P Let IN T IR i [ (q)
] be the set of all vector payo¤s satisfying the previous property for player i. Observe that the previous de…nition describes a set of vector payo¤s which cannot be reduced to a "corner set" (of the form x i ( i ) w i ( i ), i 2 i , for some well-de…ned individually rational level w i ( i )). By contrast, ex post individually rational payo¤s are described by a "corner set", since (v i ( i )) i 2 i is de…ned without ambiguity by [START_REF] Abreu | Bargaining and reputation[END_REF].

The set F [ (q)] is formally de…ned as the set of payo¤s satisfying ( 11), ( 12) and [START_REF] Forges | A folk theorem for Bayesian games with commitment[END_REF]. F [ (q)] contains the set of Nash equilibrium payo¤s of (q) and is thus not empty.

In the next two statements, we make use of uniform punishment strategies.

Lemma 0 Let (q) be a Bayesian game with independent private values and let x be a feasible payo¤ in (q). If x is interim individually rational (namely, (13)), x is ex post individually rational (namely, ( 6)): IN T IR i [ (q)] EXP IR i [ (q)] for every player i. If there exist uniform punishment strategies, namely (2), then the reverse also holds:

IN T IR i [ (q)] = EXP IR i [ (q)]
for every player i.

The proof of Lemma 0 is straightforward and therefore omitted. The intuition behind the …rst part is that players j 6 = i can impose a harder punishment to player i if they know player i's type i (i.e., ex post). For the second part, a uniform punishment strategy of players j 6 = i against player i provides an appropriate correlated strategy i in [START_REF] Forges | A folk theorem for Bayesian games with commitment[END_REF].

Proposition 3 Let (q) be a Bayesian game with independent private values in which uniform punishment strategies are available:

N [ 1 (q)] F [ (q)].
Proof: The proposition readily follows from the characterizations of N [ 1 (q)] (in Proposition 1) and F [ (q)] (( 11), ( 12) and ( 13) above): the equality in ( 4) is [START_REF] Forges | Communication equilibria in repeated games with incomplete information[END_REF], the inequality in (4) implies ( 12) and ( 6), which in turn implies (13) by Lemma 0.

As stated in Proposition 2, unlike F [ (q)], N [ 1 (q)] can be empty. The characterization in Proposition 1 indeed makes clear that repetition is just equivalent to a very demanding form of cooperation in the one-shot game, in which every player can decide to participate (and get his individually rational level if he does not) after having learnt the types of the other players. This participation constraint makes incentive compatibility harder in N [ 1 (q)] than in F [ (q)]. Example 1 below, taken from [START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF], further illustrates that N [ 1 (q)] can be strictly included in F [ (q)]. 18 Example 1: A game in which N [ 1 (q)] is not empty and strictly included in F [ (q)]

We will study a variant of the well-known battle of the sexes. Each player has two possible types: i = fn; gg, i = 1; 2, and two possible actions: A i = fc; dg, i = 1; 2. We denote as p i 2 [0; 1] the probability that player i's type is n (namely, q i = (p i ; 1 p i )). Payo¤s are described by the following matrices:

2 = n 2 = g c d c d 1 = n c 3; 1 0; 0 3; 1 0; 3 d 0; 0 1; 3 0; 1 1; 3 1 = g c 3; 1 3; 0 3; 1 3; 3 d 1; 0 1; 3 1; 1 1; 3
When 1 = n, player 1 prefers c to d, but also prefers to make the same choice as the other player. When 1 = g, player 1 just prefers c to d, independently of the choice of the other player. The preferences of player 2 are similar. In this game, v i (n) = 3 4 , v i (g) = 3, i = 1; 2. A uniform punishment strategy of player 1 (resp., 2) is to play c with probability 3 4 (resp., 1 4 ). Let us consider the (ex post e¢cient) correlated strategy ( ), 2 , de…ned by

2 = n 2 = b c d c d 1 = n c 1 2 0 0 0 d 0 1 2 0 1 1 = g c 1 0 0 1 d 0 0 0 0 (14)
It is easily checked that ( ) satis…es ( 12) and ( 13), namely, induces a payo¤ in F [ (q)], if and only if p i 1 2 , i = 1; 2. Similarly, in order to induce 18 For appropriate values of the prior p, it also happens in the public good game that N [P G 1 (p; !; z)] is not empty and is strictly included in F [P G(p; !; z)]. However, a full characterization of N [P G 1 (p; !; z)] seems much harder in this game than in Koren (1992)'s example. 20 a payo¤ in N [ 1 (q)], ( ) must satisfy (4); in particular, player 1 of type 1 = n cannot gain by pretending to be of type 0 1 = g, namely,

p 2 + 1 p 2 max 3; 3 4 + (1 p 2 ) max 0; 3 4 , p 2 1 5
The previous condition illustrates that, as expected, player 1 has more deviation possibilities at a (completely revealing) Nash equilibrium of 1 (q) than at an interim cooperative solution of (q). Imagine that player 1 is of type n but pretends to be of type g at the …rst stage of 1 (q). Then he learns player 2's type 2 and faces (g; 2 ). If 2 = n, player 1 gets the best payo¤ 3 by playing according to (g; n). However, if 2 = g, player 1 gets 0 by playing according to (g; g). In this case, he should not play according to (g; g) but rather play c with probability 3 4 at every stage in order to guarantee himself 3 4 . By checking the other equilibrium conditions in (4), we get that ( ) induces a payo¤ in N [ 1 (q)] if and only if p i 1 5 , i = 1; 2. On the other hand, as already pointed out in [START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF], the correlated strategy de…ned by

2 = n 2 = g c d c d 1 = n c 0 0 0 0 d 0 1 0 1 1 = g c 3 4 1 4 0 1 d 0 0 0 0 induces a payo¤ in N [ 1 (q)] if and only if p 2 1
6 . There are thus many probability distributions q 2 ( ) for which ( ) de…ned by ( 14) induces a payo¤ in F [ (q)], and at the same time, N [ 1 (q)] is not empty but does not contain the payo¤ de…ned by [START_REF] Fudenberg | Game theory[END_REF].

Bayesian discounted repeated game

In this section, we focus on two players and we allow every player to evaluate his payo¤ sequence with a discount factor 2 (0; 1). The discounted version (q) of the in…nitely repeated game is played as in Section 3, but a sequence of actions a = (a t ) t 0 2 A N leads now to the payo¤ 1)

U i ( i ; a) =(
1 X t=0 t u i ( i ; a t ) for every i, i
Let us denote as N [ (q)] the set of all (interim expected) Nash equilibrium payo¤s of (q). By the same arguments as under complete information, N [ (q)] is nonempty and compact, for every 2 [0; 1). An interesting question is whether lim !1 N [ (q)] is also nonempty, for some appropriate de…nition of the limit of a sequence of sets.

The equilibrium payo¤s of the undiscounted repeated game 1 (q) are natural candidates that could belong to lim !1 N [ (q)]. However, as stated in Proposition 2, the undiscounted repeated public good game P G 1 (p; !; z) may have no equilibrium. Under the assumption that there exist belief-free equilibria (in the sense of Hörner and Lovo (2009)), Peski (2013) characterizes the set lim !1 N [ (q)] as the set of …nitely revealing equilibrium payo¤s, in which beliefs are updated …nitely many times. This is again of little use in our public good game. Indeed, Hörner and Lovo (2009)'s characterization of belief-free equilibrium payo¤s and Proposition 1 imply that there cannot be any belief-free equilibrium when the undiscounted repeated game has no equilibrium. 19Let P G (p; !; z) be the discounted in…nitely repeated game associated with the public good game P G(p; !; z) introduced in Section 2.2. Even if there are no equilibria that would easily be shown to belong to lim !1 N [P G (p; !; z)] when p 1 ; p 2 and z are large, we establish below that (the interior of) this set is always nonempty. 20The proof of this result will be fully constructive: we will show that if both players are su¢ciently patient, P G (p; !; z) has a family of perfect Bayesian equilibria of the form of a war of attrition, in which the normal players do not contribute until one of them gives up and does contribute. Then, the one who has …rst given up goes on by contributing at most stages. The contributing player thus reveals that his type is normal, whereas the other player only contributes occasionally and keeps his type unknown. We will …rst de…ne precisely the war of attrition equilibria in auxiliary reputation games, in which the payo¤s of the normal players have a general structure and the greedy players reduce to automata. We consider a rather large class of automata to re ‡ect the strategic possibilities of the greedy players.

A family of reputation games

We introduce a family of reputation games G (p; i; ), where = ( 1 ; 2 ), p = (p 1 ; p 2 ), i = 1 or 2, = ( 1 ; 2 ), p k , k 2 [0; 1], k 2 [0; 1), k = 1; 2, and 1 + 2 < 1. There are two agents. At stage 1, according to independent moves of nature, agent k (k = 1; 2) is a player, with probability p k , or an automaton, with probability 1 p k . At every stage t = 0; 1; :::, every agent chooses c or d. The moves are made simultaneously and are observed after every stage. As usual, automata do not have payo¤s. The stage payo¤s of the players are described by

c d c m 1 ; m 2 0; g 2 d g 1 ; 0 l 1 ; l 2
where 0 m k < g k and 0 < l k , k = 1; 2. 21 The stage game between the players is a game with strictly competitive interests in the sense of [START_REF] Cripps | Reputation with equal discounting in repeated games with strictly con ‡icting interests[END_REF] and Atakan and Ekmekci (2013). Player k discounts streams of payo¤s using a personal discount factor k . 22 We also assume that at every stage t = 0; 1; :::, independently of past events, a random variable R t is drawn according to a uniform distribution over [0; 1] and is publicly revealed. 23Finally, the behavior of the automata depends on the parameters i and . Let j 6 = i, i.e., j = i + 1 mod 2. At every stage t = 0; 1; :::, automaton i plays d unless the next three conditions all hold 24agent j played c for the …rst time at an odd stage s < t at any stage u 2 fs + 1; : : : ; t 1g, agent j played d if and only if

R u < j , R t < j .
In other words, automaton i may consent to an agreement that gives the average payo¤ j g j to player j. If agent j did not initiate the agreement by playing c at an odd stage, or if the agreement was broken, automaton i always plays d.

The behavior of automaton j is similar, and is de…ned by inverting i and j and by replacing "odd" by "even" in the former description.

We go on by identifying speci…c strategies in G (p; i; ), in which the players behave as in a war of attrition. We …rst de…ne a war of attrition protocol, which depends on i and and also on a positive integer parameter T determining the horizon of the war.

War of attrition protocol: Before stage 0, every player chooses the …rst stage at which he will play c if the other agent did not play c yet. To this aim, player i (resp., player j 6 = i) chooses i 2 f0; 2; : : : ; 2T g (resp., j 2 f1; 3; : : : ; 2T + 1g) according to some probability distribution. If player i (resp., j) plays c for the …rst time at an even (resp., odd) stage s 2T + 1, the players initiate an agreement in which player i (resp., j) chooses c most of the time. More precisely, if player k (k = i; j) is the one who gives in, then, at every stage u s + 1, if R u k , player k plays c and the other player plays d and if R u < k , player k plays d and the other player plays c. 25The previous description tells what a player should do at every stage as long as he follows the protocol and faces either an automaton or a player who also follows the protocol. A complete description of the players' strategies, which take account of all histories that are not consistent with the protocol, will be given in the appendix. In particular, if a player adopts a war of attrition strategy, he follows the war of attrition protocol as long as the other agent's moves are consistent with it and behaves as his associated automaton otherwise.

The next three propositions give precise, easy to check, conditions that guarantee that G (p; i; ) has a perfect Bayesian equilibrium in war of attrition strategies. The proofs are given in the appendix, together with the explicit expression of the equilibrium payo¤s. The condition in the …rst proposition is particularly simple: it just says that the automata cooperate with an arbitrary small positive probability after a player has …rst cooperated. Proposition 4 Let p and be …xed, with 1 ; 2 > 0. There exists 2 (0; 1) such that, for any 1 > and any 2 > , G (p; i; ) has a perfect Bayesian equilibrium in war of attrition strategies for some i 2 f1; 2g.

As many other results of the reputation literature (see Mailath and Samuelson ( 2006)), the following proposition makes use of the fact that a player may be signi…cantly more patient than the other. Proposition 5 Fix p, and i. There exists 2 (0; 1) and c > 0 such that, for every 1 ; 2 2 (0; 1) satisfying i > and 1 j c( 1i ), G (p; i; ) has a perfect Bayesian equilibrium in war of attrition strategies.

The next proposition assumes that the players have the same discount factor and identi…es a su¢cient condition, condition (15) below, which is satis…ed as soon as -the players have the same payo¤s (g 1 = g 2 , l 1 = l 2 ) 26 , -automata always play d (i.e., = 0) and -p i > p j (which of course always holds, for i = either 1 or 2, unless p 1 = p 2 ).

Proposition 6 Consider the game with equal discount factors = 1 = 2 and assume that p, and i satisfy

log(1 p i ) l i + i g i g i < log(1 p j ) l j + j g j g j : (15)
Then there exists 2 (0; 1) such that, for any > , G (p; i; ) has a perfect Bayesian equilibrium in war of attrition strategies.

Proposition 6 is consistent with Atakan and Ekmekci (2013)'s theorem 2. They specially study a reputation game in which the players of the basic game, with strictly competitive interests, have an arbitrary number of actions (as opposed to two, c and d, here) but they focus on very simple automata, which would always play d in our framework. In other words, they set 1 = 2 = 0, while we allow for 1 ; 2 > 0. Assuming that both players have the same discount factor , they show that if condition [START_REF] Hart | Nonzero-sum two-person repeated games with incomplete information[END_REF] holds with 1 = 2 = 0, then, when is su¢ciently large, at any perfect Bayesian equilibrium of the reputation game, one of the players (player i) 27 gets approximately his minmax payo¤ 0 while the other player (player j) bene…ts from a reputation e¤ect, namely gets a substantial part of g j . As already suggested in the introduction, the main di¤erence between our results and those of Atakan and Ekmekci ( 2013) is that, being interested in a reputation result, they establish the asymptotic uniqueness of a very special kind of equilibria, while we are interested in characterizing a larger class of equilibria, which exist for every su¢ciently large discount factor. We will pursue the comparison in the appendix, when the expression of the equilibrium payo¤s is available.

War of attrition equilibria in the public good game

Let us go back to the public good game P G (p; !; z), in which there are two players and no automaton. When both players are normal, they play a game with strictly competitive interests in which (after subtracting 1 to all payo¤s) m k = 0, g k = !, l k = 1 !, k = 1; 2. Building on the previous section, we consider speci…c strategies for the players of P G (p; !; z), which we call war of attrition strategies, as in the previous reputation games. Let i = 1 or 2, j 6 = i and k 2 [0; 1 z ), k = 1; 2. In a war of attrition strategy pro…le parametrized by i and = ( 1 ; 2 ), player k behaves as automaton k in G (p; i; ) if his type is greedy and as player k in G (p; i; ) if his type is normal. 28The next proposition states that equilibria in war of attrition strategies yield explicit elements in lim !1 N [P G (p; !; z)].

Proposition 7 Consider the discounted in…nitely repeated public good game P G (p; !; z). For every 1 ; 2 2 (0; 1 z ), there exists 2 (0; 1) such that, for every > , P G (p; !; z) has a perfect Bayesian equilibrium in war of attrition strategies parametrized by i = 1 or 2 and = ( 1 ; 2 ). Furthermore, if the priors are such that p 1 6 = p 2 , the same holds for 1 = 2 = 0. The corresponding equilibrium payo¤s converge, as ! 1, to

x i (n; ) = 1 + i !; x j (n; ) = 1 + 0 (1 i )! + 0 j !; x i (g; ) = z + 1 j + j (1 z) a i ; x j (g; ) = z + (1 i + i (1 z)) ( 0 + (1 0 )a j ) ;
where i = 1 or 2, j 6 = i, 0 = 1

1 p i (1 p j ) 1 !+ j ! 1 !+ i ! , a i = 1 !+ i ! 1 !+(1 j )! 1 (1 p j ) 1 !+(1 j )! 1 !+ i ! , and a j = 1 !+ j ! 1 !+(1 i )! 1 (1 p j ) 1 !+(1 i )! 1 !+ i ! . In particular, lim inf !1 N [P G (p; !; z)]
has a nonempty interior.

It may be useful to compare the latter result with our …ndings in the undiscounted case, namely, Propositions 1 and 2. Note that in a war of attrition equilibrium, once one of the players has revealed that his type is normal, the players follow a cooperative agreement, as in the standard proof of the folk theorem with complete information. Under our assumption of private values, given such an agreement, the player who has not revealed his type would not mind revealing it (provided that the agreement is not modi…ed, of course).

However, the previous equilibria cannot be reduced to completely revealing ones (from the beginning of the game). Indeed, in the discounted game, the time before revelation is costly and matters at equilibrium. The fact that payo¤s are discounted is thus critical in war of attrition equilibria. In the undiscounted game, a normal player is always better o¤ waiting for the other player revealing his type, because waiting is free.

Note also that the previous equilibria are not …nitely revealing in the sense of [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF], namely, they involve a number of changes of the players' posteriors that increases with . However, we can re…ne our construction of equilibria by introducing, in the war of attrition, an arbitrary number of nonrevealing stages between each stage of information transmission. It is then possible to keep the number of information transmission stages bounded as ! 1 and to obtain similar limit payo¤s, so that lim inf !1 N [P G (p; !; z)] contains a set (with a non-empty interior) of …nitely revealing equilibrium payo¤s (see the remark after the proof of Proposition 7 in the Appendix for details).

6 Appendix: proofs and complements 6.1 Undiscounted game (Proposition 1)

Strategies and payo¤ functions

A strategy of player i in 1 (q) is a sequence of mappings i = ( t i ) t 0 ,

t i : i A t 1 ! (A i ).
The n tuple of prior probability distributions q = (q i ) 1 i n and an n tuple of strategies = ( i ) 1 i n induce a probability distribution over

A N , where A N is the set of all in…nite sequence of moves. We denote as E q; the corresponding expectation. Given a = (a t ) t 0 2 A N , let us de…ne

U T +1 i ( i ; a) = 1 T + 1 T X t=0 u i ( i ; a t )
for every i, i and T = 0; 1; ::: [START_REF] Hart | Nonzero-sum two-person repeated games with incomplete information[END_REF] (see also [START_REF] Forges | Repeated games of incomplete information : nonzero-sum[END_REF], [START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF], Shalev (1994)), we de…ne the interim payo¤s associated with an n tuple of strategies as

As in
U i ( i ; ) =L h E q; (U T i ( i ; e a) j i ) i
where L is a Banach limit and e a denotes the sequence of moves as a random variable.

Su¢cient conditions for an equilibrium

Let us assume that the conditions (4) hold. Then we can construct an n tuple of strategies = ( i ) 1 i n in 1 (q) which achieve the interim payo¤s x i ( i ) (namely, such that x i ( i ) = U i ( i ; ) for every i, i ) and which de…ne a Nash equilibrium of 1 (q). For every player i, i is described as follows:

at the …rst stage (t = 0): choose a i so as to reveal type i (which is possible since jA i j j i j)

at every stage t 1: given the n tuple of reported types 0 , play according to ( 0 ) if ( 0 ) was chosen at every previous stage; otherwise, play a punishment strategy in order to keep the …rst player j who did not follow ( 0 ) below his ex post individually rational level v j ( j ).

Necessary conditions for an equilibrium

Let us start with an arbitrary Nash equilibrium

= ( i ) 1 i n in 1 (q). Let i ( i ) be the associated strategy of player i of type i , namely, i ( i ) = ( t i ( i )) t 0 , with t i ( i ) : A t 1 ! (A i ). Let x i ( i ) = U i ( i ;
) be the associated interim equilibrium payo¤ of player i of type i . Let us show that the conditions (4) hold, namely, that the same payo¤s can be achieved by a completely revealing equilibrium.

In order to get some intuition, let us assume that, at equilibrium, there is a …nite, possibly very long, phase of information transmission (say, until stage t 0 ) and that afterwards (thus, at stages t 0 + 1, t 0 + 2,...), the players play independently of their types. Since is an equilibrium, player i of type i cannot bene…t from playing according to i ( 0 i ), with 0 i possibly di¤erent from i , until stage t 0 and then, from stage t 0 + 1 on, by either continuing to play i ( 0 i ) or just guaranteeing himself v i ( i ) (i.e., by playing optimally in "his true one-shot game", with payo¤s u i ( i ; ), at every stage t 0 + 1, t 0 + 2,...). 29More precisely, the equilibrium strategies i ( i ) generate probability distributions ( j 1 ; :::; n ) over the limit frequencies of moves, i.e., over (A) (see [START_REF] Hart | Nonzero-sum two-person repeated games with incomplete information[END_REF] or [START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF] for details). Together with the prior q, these probability distributions generate a probability distribution P q; over (A) such that

x i ( i ) = U i ( i ; ) = E q; (u i ( i ; e ) j i ) for every i; i (16)
where E q; is the expectation with respect to P q; and e stands for the frequency of move as a random variable. 30By considering the previous speci…c deviations of player i of type i (namely, mimic type 0 i and/or play optimally in the one-shot game), we obtain that

x i ( i ) E q; (max fu i ( i ; e ); v i ( i )g j 0 i ) for every i; i ; 0 i (17)
We can also rely on a variant of the revelation principle to see that ( 16) and ( 17) must be satis…ed as soon as is an equilibrium. Let us imagine that a fully reliable mediator asks the players to report their types and then given the n tuple of reported types 0 2 , chooses a frequency of moves 2 (A) according to ( j 0 ) and recommends to all players 31 . In other words, when the players report 0 = ( 0 i ) 1 i n , the mediator selects exactly as the players themselves do at the equilibrium . ( 16) says that by telling the truth and following the recommendation of the mediator, the players get the same interim payo¤ as by playing . [START_REF] Hörner | Belief-free equilibria in games with incomplete information: characterization and existence[END_REF] says that if players j 6 = i tell the truth to the mediator, follow the recommendation as long as every player follows and punish any deviator at his ex post minmax level, then player i of type i cannot bene…t from reporting type 0 i to the mediator and/or not following .

Conditions ( 16) and ( 17) di¤er from (4) in two respects. ( 16) and ( 17) involve (type dependent) probability distributions over (A), while (4) is formulated in terms of deterministic distributions ( ), 2 . Moreover, in [START_REF] Hörner | Belief-free equilibria in games with incomplete information: characterization and existence[END_REF], the probability distribution is not necessarily completely revealing 32 . By construction, and recalling that types are independent of each other, for any function f over (A), the probability P q; satis…es E q; (f (e ) j i ) = X i q i ( i )E (f (e ) j i ; i ) for every i, i

Hence, for every i, i , ( 16) can be rewritten as

x i ( i ) = X i q i ( i )E (u i ( i ; e ) j i ; i ) Recalling that u i ( i ;
) is linear, we get

x i ( i ) = X i q i ( i )u i i ; E (e j i ; i )
which is the …rst part of (4) if we set ( ) = E (e j ).

By proceeding similarly and using in addition that "max" is convex, for every i; i ; 0 i , ( 17) can be rewritten as

x i ( i ) X i q i ( i )E (max fu i ( i ; e ); v i ( i )g j 0 i ; i ) X i q i ( i ) max E (u i ( i ; e ) j 0 i ; i ) ; v i ( i ) X i q i ( i ) max u i i ; E (e j 0 i ; i ) ; v i ( i ) X i q i ( i ) max fu i ( i ; ( 0 i ; i )) ; v i ( i )g
The last expression is the inequality in (4).

Generalization of the public good game, complements on Proposition 2

The main goal of this section is to formally establish inequalities [START_REF] Chan | On the non-existence of reputation e¤ects in twoperson in…nitely repeated games[END_REF]. We actually consider a variant of the public good game P G(p; !; z) of Section 2.2, in which every player has an extra action e, with the following payo¤s:

2 = n 2 = g c d e c d e c (1; 1) (1; 1
+ !) (1; 0) c (1; 1) (1; 1 + z) (1; 0) 1 = n d (1 + !; 1) (!; !) (0; 0) d (1 + !; 1) (!; z) (0; 0) e (0; 1) (0; 0) (0; 0) e (0; 0) (0; z) (0; 1 + z) c d e c d e c
(1; 1)

(1; 1 + !) (0; 0) c (1; 1) (1; 1 + z) (0; 0) 1 = g d (1 + z; 1) (z; !) (z; 0) d (1 + z; 1) (z; z) (z; 0) e (0; 1) (0; 0) (1 + z; 0) e (0; 0) (0; z) (1 + z; 1 + z)
In the new game, for every pair of types ( 1 ; 2 ), the set of feasible, individually rational payo¤s of the public good game with complete information, in which 1 and 2 are common knowledge, has a nonempty interior, a property that is not satis…ed when 1 = 2 = g in the initial game.

In the new game, the ex post individual rationality levels are still

v i (n) = 1, v i (g) = z, i = 1; 2
uniform punishments still exist (play d) for every ( 1 ; 2 ) 6 = (g; g), the set of feasible, individually rational pay-o¤s is the same as in P G(p; !; z). In particular, there exist feasible, strictly individually rational payo¤s for every ( 1 ; 2 ) 6 = (g; g). But now, there also exist strictly individually rational payo¤s for ( 1 ; 2 ) = (g; g).

Let us show that, without discounting, if z, p 1 and p 2 are su¢ciently large, there are no equilibria. As in Section 3.2, let ( j ) be the equilibrium distributions over A = fc; d; eg fc; d; eg when is reported and consider the equilibrium conditions in Proposition 1 (namely, ( 12)). We …rst prove that:

Claim: if z is su¢ciently large, the feasibility and ex post individual rationality conditions imply that ((d; c) j g; n) 1 " and ((c; d) j n; g) 1 ".

Proof of the claim: we establish the second inequality, the …rst one can be established in the same way. The ex post individual rationality condition for player 1, of greedy type, implies that z + d (z 1) + z z 1

(write the ex post individual rationality condition for the payo¤s 1 and forget about some negative weights). Similarly, the ex post individual rationality condition for player 2, of normal type, implies that

! + d + (! + 1) ! (19)
(proceed as for player 1, but also replace by 1 all the other probabilities in ( j g; n)).

The feasibility constraints are 0, d 0, 0 and + d + 1. If the latter constraint is binding, namely if + d + = 1, then ! + d + (! + 1) !, so that, by [START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF], ! + d + (! + 1) = ! and = 1, which establishes the claim (( 18) is also satis…ed).

The constraint 0 cannot be binding as soon as z 1 + 1 ! (,

z z 1
! + 1). Indeed, if = 0, (18) and ( 19) imply then that 1 ! z z 1 (! + 1) 0 contradicting ! 1. Hence the relevant extreme points are determined by the following binding constraints:

-(18), (19), d = 0: = ! z , = 1 1+! z . -(18), (19), = 0: = 1 1 z(1 !)+! . -(18), d = = 0: = 1 1 z . -(19), d = = 0: = 1.
In all cases, the claim is satis…ed.

If all kl 's are set to 0, the proof of the claim establishes the inequalities [START_REF] Chan | On the non-existence of reputation e¤ects in twoperson in…nitely repeated games[END_REF] in the original public good game.

Furthermore, in the new game, the nondeviation condition (8) must be satis…ed and feasibility still implies ( 9) and [START_REF] Cripps | Reputation with equal discounting in repeated games with strictly con ‡icting interests[END_REF]. Hence, by proceeding exactly as in Section 3.2, one shows that the set of Nash equilibrium payo¤s is also empty in the new game.

Role of uniform punishments (Propositions 1 and 3)

In the case of two players, if values are private and independent in (q), Koren (1992) proves that the Nash equilibria of 1 (q) are payo¤ equivalent to completely revealing equilibria without assuming uniform punishments (i.e., (2)). However, in this more general case, the equilibrium conditions can take a more complex form than (4). Examples 2 and 3 below illustrate how the absence of uniform punishments modi…es the results. In example 2, the conditions (4) of Proposition 1 are no longer su¢cient for an equilibrium. Proposition 3 does not hold either: we construct an equilibrium payo¤ in 1 (q) which does not belong to F [ (q)], i.e., cannot be achieved through commitment in (q).

In example 3, an assumption weaker than uniform punishments holds, which guarantees that the Nash equilibrium payo¤s of 1 (q) can be characterized exactly as in Proposition 1, by (4). However, Proposition 3 still fails.

In both examples 2 and 3, there are two players and only player 1 has private information (j 2 j = 1, A = A 1 A 2 ), so that the conditions in Proposition 1 reduce to: there exists ( 1 ) 2 (A), 1 2 1 , such that, for player 1,

x 1 ( 1 ) = u 1 ( 1 ; ( 1 )) u 1 ( 1 ; ( 0 1 )) 8 1 ; 0 1 2 1 i.e.
, incentive compatibility (20) v 1 ( 1 ) 8 1 2 1 i.e., ex post individual rationality [START_REF] Kalai | A commitment folk theorem[END_REF] and, for player 2,

x 2 = u 2 ( ( 1 )) v 2 8 1 2 1 i.e.
, ex post individual rationality [START_REF] Mailath | Repeated Games and Reputations[END_REF] As shown by [START_REF] Hart | Nonzero-sum two-person repeated games with incomplete information[END_REF], in order to characterize the equilibrium payo¤s of 1 (q), ex post individual rationality (namely, ( 6) or (21) above) is not su¢cient. A stronger condition, which makes full use of the fact that 1 (q) is an in…nitely repeated game, is needed. This condition is formally stated below, in the current framework of lack of information on one side. 33 Let val 1 [u] denote the value to player 1 of the one-shot game with payo¤ function u.

De…nition A vector payo¤ x 1 = (x 1 ( 1 )) 1 2 1 is individually rational for player 1 in the in…nitely repeated game 1 (q) if and only if

8p 1 2 ( 1 ), X 1 p 1 ( 1 )x 1 ( 1 ) val 1 " X 1 p 1 ( 1 )u 1 ( 1 ; ) # (23)
Let IN T IR 1 [ 1 (q)] be the set of vector payo¤ that are individually rational for player 1 in the in…nitely repeated game 1 (q). The previous de-…nition is justi…ed by Blackwell (1956)'s approachability theorem: condition ( 23) is necessary and su¢cient for player 2 to have a strategy in the in…nitely repeated game 1 (q) such that player 1's payo¤ cannot exceed x 1 ( 1 ) when he is of type 1 .

Let us compare IN T IR 1 [ 1 (q)] with the two sets of individually rational payo¤s introduced for the one-shot game (q), namely, EXP IR 1 [ (q)] and IN T IR 1 [ (q)]. First of all, player 2 can use a punishment strategy of the one-shot game at every stage of the in…nitely repeated game: as a consequence of Blackwell (1956)'s characterization, [START_REF] Forges | A folk theorem for Bayesian games with commitment[END_REF] implies [START_REF] Mertens | The value of two-person zero-sum repeated games with lack of information on both sides[END_REF]. Furthermore, [START_REF] Mertens | The value of two-person zero-sum repeated games with lack of information on both sides[END_REF] holds in particular at the extreme points of ( 1 ), so that it implies ex post individual rationality (i.e., [START_REF] Kalai | A commitment folk theorem[END_REF]). To sum up,

IN T IR 1 [ (q)] IN T IR 1 [ 1 (q)] EXP IR 1 [ (q)]
These inclusions hold in two-person games with independent private values, even if player 2 also has private information (see [START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF]). From Lemma 0, under the assumption of uniform punishments, the three sets coincide. In examples 2 and 3 below, this assumption does not hold. In example 2, the two inclusions are strict. In example 3, the …rst inclusion is strict but

IN T IR 1 [ 1 (q)] = EXP IR 1 [ (q)].

Example 2

Let n = 2, 1 = fh; lg, j 2 j = 1: only player 1 has private information. Here, the prior probability distribution is fully described by the probability that player 1's type is h, which we still denote as q 2 [0; 1]. Let jA 1 j = jA 2 j = 2 and the utility functions be described by

u 1 (h; ) = 1 0 0 0 u 1 (l; ) = 0 0 0 1 u 2 ( ) = 0 2 0 0
The assumption of uniform punishments is clearly not satis…ed: player 2 must play right in order to hold player 1 of type h at his value level v 1 (h) = 0 and must play left to hold him at v 1 (l) = 0. Consider the probability distribution

(h) = (l) = = 1 4 1 2 0 1 4 2 (A 1 A 2 )
Let us check that it de…nes an equilibrium of 1 (q), for every p 2 (0; 1), namely that the associated payo¤s, x 1 (h) = x 1 (l) = 1 4 , x 2 = 1, verify the above conditions (including [START_REF] Mertens | The value of two-person zero-sum repeated games with lack of information on both sides[END_REF]). Player 2's payo¤ x 2 = 1 is individually rational since the value of player 2's game is v 2 = 0.

is clearly incentive compatible since it is nonrevealing. According to [START_REF] Mertens | The value of two-person zero-sum repeated games with lack of information on both sides[END_REF], a vector payo¤ (x 1 (h); x 1 (l)) is individually rational for player 1 in 1 (q) if and only if

8p 2 [0; 1] , px 1 (h) + (1 p)x 1 (l) val 1 p 0 0 1 p = p(1 p)
This function is convex so that a vector payo¤ (x 1 (h); x 1 (l)) is individually rational for player 1 in the sense of [START_REF] Mertens | The value of two-person zero-sum repeated games with lack of information on both sides[END_REF] if and only if it is ex post individually rational (namely, (21): x 1 (h) 1 and x 1 (l) 1):

IN T IR 1 [ 1 (q)] = EXP IR 1 [ (q)].
In particular, in this example, the equilibrium conditions in 1 (q) are correctly described in Proposition 1, namely by ( 20), ( 21) and [START_REF] Mailath | Repeated Games and Reputations[END_REF]. 34In spite of the previous property, Proposition 3 fails. The probability distributions

(h) = 1 0 0 0 (l) = 0 0 0 1
lead to an equilibrium in 1 (q), with payo¤ ((1; 2), but (1; 1) is not interim individually rational for player 1 in the sense of ( 13): let = ( ; 1 ); max a 1 u 1 (h; a 1 ; ) = 2 1 is incompatible with max a 1 u 1 (l; a 1 ; ) = + 1 1.

In both examples 2 and 3, interim individual rationality takes a di¤erent form in the one-shot game and in the in…nitely repeated game. In example 2, in order to defend himself, player 1 must play in a non-revealing way in the repeated game. In example 3, player 1 bene…ts from revealing his information to player 2.

The phenomena described in the previous examples were …rst identi…ed in the study of zero-sum in…nitely repeated games with incomplete information (see [START_REF] Aumann | Repeated Games of Incomplete Information[END_REF]).

Reputation games (Propositions 4, 5 and 6)

Propositions 4, 5 and 6 state that, under appropriate assumptions, the reputation game G (p; i; ) has a perfect Bayesian equilibrium in war of attrition strategies, which are based on a war of attrition protocol. We start by giving a full description of war of attrition strategies. Recall that, in a war of attrition protocol, player i (resp., player j 6 = i) chooses the …rst stage i 2 f0; 2; : : : ; 2T g (resp., j 2 f1; 3; : : : ; 2T + 1g) at which he will play c if the other agent did not play c yet. If player i (resp., j) plays c for the …rst time at an even (resp., odd) stage s 2T + 1, the players initiate an agreement in which player i (resp., j) chooses c with frequency 1 i (resp., 1 j ). A player can observe three kinds of histories that are inconsistent with the protocol I c has been played for the …rst time at a "wrong" stage, namely, I' by agent i at an odd stage t or by agent j at an even stage t (t < 2T + 2 or t 2T + 2), I" by player i only at an even stage t or by player j only at an odd stage t, with t 2T + 2;

II c is has been played for the …rst time at a "right" stage but the agreement described by i or j is not followed by some player at some stage;

III the stage is 2T + 2 and c has never been played.

War of attrition strategies: A war of attrition strategy for player k (k = 1; 2) in G (p; i; ) is de…ned from a war of attrition protocol (which depends on the parameters of the game, T and appropriate probability distributions) as follows: at every stage t = 0; 1; :::, if the history up to stage t 1 is consistent with the protocol, play according to it.

Otherwise, on type I' histories, if c has been played for the …rst time by player i (resp., j) at an odd (resp., even) stage, player i (resp., j) chooses c and player j (resp., i) chooses d.

on type I" histories, players initiate an agreement, exactly as in the protocol before 2T + 1.

on type II histories, if player i (resp., only player j) did not follow the agreement, player i (resp., j) chooses c and player j (resp., i) chooses d from then on.

on type III histories, player i (resp., j) plays a best response to automaton j (resp., i) (and the history becomes of type I after one or two stages).

Propositions 4, 5 and 6 will result from Lemma 8 below. To prepare for it, let us set " 1 and v 2 such that (" 1 ; v 2 ) is the mean stage payo¤ in the sequence of actions that begins when player 1 concedes …rst, i.e. " 1 = 1 1 g 1 and v 2 = ( 12 )g 2 + 2 (1 1 )g 2 . We de…ne " 1 and v 2 symmetrically. Note that

v 1 " 1 g 1 = 1 1 ( 1 + 2 ) and v 2 " 2 g 2 = 1 2 ( 1 + 2 )
, so that the fact that 1 + 2 1 implies that " 1 < v 1 and " 2 < v 2 . We then set

1 = (1 2 1 )(l 1 + " 1 ) 1 (l 1 + v 1 ) 2 1 (l 1 + " 1 ) ; and T 1 = log(1 p 2 ) log( 1 1 )
; with the convention that T 1 = 0 if 1 1.

We also de…ne 2 and T 2 , by inverting the roles of player 1 and player 2.

Lemma 8 Assume that the following conditions hold:

T i T j ; (24)
(1 i )( i i m i l i ) " i ; (25) (1 j )( j j m j l j ) " j ; (26) 1 p i (1 j ) T i v j (1 j )m j v j + l j j (l j + " j ) (1 j )m j : (27)
Then the reputation game G (p; i; ) has a perfect Bayesian equilibrium in war of attrition strategies with payo¤s

x i = " i ; x j = 1 1 p i (1 j ) T i v j + 1 p i (1 j ) T i ( l j + j (l j + " j )) :
Before establishing the lemma, it may be useful to understand how it applies in a particular case where the automata always play d ( = 0), while the players have the same discount factor k = and the same payo¤s g k = g, l k = l, m k = 0, k = 1; 2. In this case, we can set = k , k = 1; 2;

1 as soon as l l+g . Condition [START_REF] Myerson | Game theory: analysis of con ‡ict[END_REF], which …xes the index i, is satis…ed if agent i is more likely to be a player than agent j (namely, i is such that p i p j ).

In particular, G i (0) = " i . So, the law of j has to be such that " i = G i (2) = : : : = G i (2T ). From equality G i (2) = " i , we get P( j = 1) = i . Note that if i > p j (which is equivalent to T i = 0), the latter equality can not hold. In this case, T = T i = 0 and P( j = 1) = p j . Now, let us show by induction that: 8t 2 f0; 1; : : : ; T 1g; P(

j = 2t + 1) = i (1 i ) t :
The property is obvious for t = 0. Then, if the property holds for each s 2 f0; 1; : : : ; t 1g (t T 1), let us show that it also holds for s = t. From the de…nition of G i , one has:

G i (2t + 2) G i (2t) = P( j = 2t + 1) l i + 2t+1 i (l i + v i ) +P( j > 2t + 1) l i + 2t+2 i (l i + " i ) P( j > 2t 1) l i + 2t i (l i + " i ) = P( j = 2t + 1) l i + 2t+1 i (l i + v i ) + (P( j > 2t 1) P( j = 2t + 1)) l i + 2t+2 i (l i + " i ) P( j > 2t 1) l i + 2t i (l i + " i ) = P( j = 2t + 1) (l i + v i i (l i + " i )) 2t+1 i P( j > 2t 1)(l i + " i )(1 2 i ) 2t i : As P( j > 2t 1) = 1 t 1 X s=0 P ( j = 2s + 1) = 1 t 1 X s=0 (1 i ) s P( j = 1) = 1 i 1 (1 i ) t 1 (1 i ) = (1 i ) t ; (28)
one …nally has

G i (2t + 2) G i (2t) = (l i + v i i (l i + " i )) 2t+1 i P( j = 2t + 1) i (1 i ) t : (29)
The fact that G i (2t + 2) = G i (2t) then implies that

P( j = 2t + 1) = i (1 i ) t ;
which ends the induction.

Our construction must ensure that player i does not prefer conceding at stage 2T + 2, i.e. G i (2T + 2) G i (2T ). In fact, we construct an equilibrium such that this preference is strict, i.e. G i (2T +2) < G i (2T ). Because equality (29) still holds for t = T this is equivalent to

P( j = 2T + 1) < i (1 i ) T : (30)
Moreover P( j 2T + 1) = p j , and we have

T 1 X t=0 i (1 i ) t = 1 (1 i ) T p j < T X t=0 i (1 i ) t = 1 (1 i ) T +1 :
This implies that T = T i .

Probability distribution of i

Here, the point is that player j has to be indi¤erent between conceding at stages t = 1; 3; : : : ; 2T + 1. Let us denote G j (t) the overall payo¤ of player j when he is willing to concede at stage t, the law of i being …xed. Again, "2t + 1" refers to a regular war of attrition strategy, whereas "2t" refers to an arbitrary strategy that dictates to contribute at stage 2t if i did not contribute before. We have, for any t 0:

G j (2t+1) = t X s=0 P( i = 2s)( l j + 2s j (l j +v j ))+P( i > 2t)( l j + 2t+1 j (l j +" j )):
The law of i has to be such that G j (1) = : : : = G j (2T i + 1). By the same means as in the former case, the fact that G j (1) = G j (3) implies that

P( i = 2) = j (1 P( i = 0)):
Again, this equality can not hold if j (1 P( i = 0)) > p i , but this can not happen. Indeed, if j > p i then T j = 0, and T i = 0 because of [START_REF] Myerson | Game theory: analysis of con ‡ict[END_REF].

As mentioned before, we then have T = 0. Thus P( i = 0) = p i (which is consistent with equality (31) below) and P( i = 2t) = 0 for any t 1.

One can prove by induction that: 8t 2 f1; : : : ; T g;

P( i = 2t) = (1 j ) t 1 P( i = 2)
Then, the fact that P( i = +1) = 1 p i is equivalent to:

p i = T i X t=0 P( i = 2t);
and to

p i = P( i = 0) + T i X t=1 P( i = 2t) = P( i = 0) + P( i = 2) T i X t=1 (1 j ) t 1 = P( i = 0) + j (1 P( i = 0)) T i 1 X t=0 (1 j ) t = P( i = 0) + j (1 P( i = 0)) 1 (1 j ) T i 1 (1 j ) = P( i = 0) + (1 P( i = 0)) 1 (1 j ) T i so that P( i = 0) = 1 1 p i (1 j ) T i : (31)
Thus, we de…ne a proper distribution of i , because then P( i = 0) < 1 and

P( i = 0) 1 1 p i (1 j ) T j 1 1 p i (1 j ) log(1 p i ) log(1 j ) = 0:
Type dependent probabilities de…ning the equilibrium strategies

The values of P( j = 2t + 1) and of P( i = 2t) enable us to specify player i and j's strategies: player i chooses stage 0 with probability

i 0 = P( i = 0) p i = 1 p i 1 1 p i (1 j ) T i ! and stage 2t (1 t T i ) with probability i 2t = P( i = 2t) p i = 1 p i (1 j ) t 1 P( i = 2) = 1 p i (1 j ) t 1 j (1 P( i = 0)) = 1 p i p i (1 j ) t 1 T i j ;
Player j chooses stages 2t + 1 (0 t T i 1) with probability

j 2t+1 = P( j = 2t + 1) p j = 1 p j (1 i ) t i ;
and stage 2T i + 1 with probability

j 2T i +1 = P( j = 2T i + 1) p j = 1 p j p j T i 1 X t=0 P( j = 2t + 1) ! = 1 p j p j 1 + (1 i ) T i :

Equilibrium conditions

Let us check that this strategic pro…le is indeed an equilibrium. First, any player k 2 f1; 2g is right to follow an agreement that was previously initiated by himself or by the other player. Indeed, if he plays c instead of d, he will get m k instead of g k and if he plays d instead of c he will get l k instead of 0. For the rest of the stages, he will be punished and win at most 0, which is less than what the agreement would have given him. Note that this covers the case of I" histories. Now, let us show that player i's strategy is a best response to j's one. We have G i (2T + 2) G i (2T ) by construction, meaning that player i has no interest in starting contributing at stage 2T +2. He obviously does not prefer to start contributing at stage t, t > 2T + 2, because if j did not contribute before stage 2T + 1, j is necessarily an automaton and then there is no more reason for i to wait for starting contributing.

Let us now show that he had not better start contributing at odd stages 2t + 1, t 2 f0; 1; : : : ; T g. We compare the payo¤ of an arbitrary strategy that starts contributing 2t+1 to the regular pure strategy that starts contributing at stage 2t. Note that starting contributing at stage 2t + 1 is punished by agent j. We have:

G i (2t + 1) G i (2t) P( j 2t) 0 +P( j 2t + 1) l i + 2t+1 i l i l i + 2t i (l i + " i ) +P( j = 2t + 1)(1 i ) 2t+1 i m i P( j 2t + 1) 2t i [ i l i (l i + " i )] +P( j = 2t + 1)(1 i ) 2t+1 i m i
For any t 2 f0; : : : ; T 1g, P( j = 2t + 1) = i (1 i ) t , and

P( j = 2T + 1) i (1 i ) T (cf. ( 30
)), so that:

G i (2t + 1) G i (2t) (1 i ) t 2t i [ i l i (l i + " i )] + i (1 i ) t (1 i ) 2t+1 i m i (1 i ) t 2t i [ i l i (l i + " i ) + i i (1 i )m i ];
for any t 2 f0; 1; : : : ; T g. By means of inequality (25), we then have G i (2t + 1) G i (2t).

Finally, player i is indi¤erent between 0; 2; : : : ; 2T by construction, so his strategy is a best response.

Similar arguments show that player j's strategy is a best response to i's one. In particular, one can show that G i (2t+2) G i (2t+1) (1 P( i = 0))(1 j ) t 2t+1 j [ j l j (l j +" j )+ j j (1 j )m j ]; for any t 2 f0; 1; : : : ; T g. So G i (2t + 2) G i (2t + 1) by inequality [START_REF] Peski | Repeated games with incomplete information on one side[END_REF]. Yet, it remains to show that player j should not concede at stage 0, i.e.: G j (1) P( i = 0)(1 j )m j + (1 P( i = 0)) 0:

One can check that this is equivalent to inequality [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF].

2T if c has not been played before, and we impose the preference to be strict, as in the former proof. This is equivalent to inequality i p 2T +1 j > 1.

if c has never been played before stage 2T , player i plays c with probability 1 at stage 2T . Player j must then be indi¤erent between c to d at stage 2T if c has not been played before. This is equivalent to inequality j

p 2T i = 1.
These two constraints enable us to determine T and P( i = 0).

We will now prove Propositions 4, 5 and 6. They all consists in giving simple su¢cient condition for inequalities [START_REF] Myerson | Game theory: analysis of con ‡ict[END_REF], ( 25), ( 26), [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF], so that Lemma 8 holds.

Proof of Proposition 4

As k > 0, one has k (l k + " k ) > l k as long as k is close enough to 1. Thus we can choose close enough to 1 so that, for any 1 ; 2 > , one has 1 (l 1 + " 1 ) > l 1 and 2 (l 2 + " 2 ) > l 2 , and so that Inequalities ( 25) and ( 26) hold. Then, for any of these 1 ; 2 , one can choose i 2 f1; 2g such that T i T j . The left hand side of inequality ( 27) is lower than 1:

1 p i (1 j ) T i 1 p i (1 j ) T j 1;
whereas the inequality j (l j + " j ) > l j ensures that the right hand side is greater than 1. Therefore, inequalities (24), ( 25), ( 26), [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF] of Lemma 8 all hold.

Proof of Proposition 5 One can check that k k !1

(1 k ) 2(l k + k g k ) g k (1 1 2 )
for any k 2 f1; 2g, so that

T k = k 1 k + k (1 k ) 1 k ; ( 32
)
where k = log(1 p l ) g k (1 1 2 ) 2(l k + k g k ) (with l 2 f1; 2g, l 6 = k) and k is a function such that k (x) !

T i = i 1 i + i (1 i ) 1 i c i 1 j + c i (1 i ) 1 j = 1 2 j 1 j + c i (1 i ) 1 j T j 1 2 j 1 j j (1 j ) 1 j + c i (1 i ) 1 j = T j 1 1 j j 2 + j (1 j ) c i (1 i ) :
Moreover, if 1 j c(1 i ), j goes to 1 when i goes to 1, so that j (1 j ) and i (1 i ) are arbitrary small if i is close enough to 1. This ensures that T i < T j if i is close enough to 1. Then we can …nd a simple upper bound for the right hand side of inequality [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF]:

1 p i (1 j ) T i 1 p i (1 j ) T j 1 = (1 j ) 1 p i (1 j ) T j 1 j :
If j > 0, the left hand side of inequality ( 27) tends to 1 i 1 i j > 1 as j goes to 1, so that (27) holds as long as j is close enough to 1. If j = 0, one can check that the left hand side of inequality [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF] equals 1 l j g j (1 i ) (1 j )+o(1 j ), whereas 1 j = 1 2 l j g j (1 i ) (1 j )+o(1 j ). Thus, [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF] is always valid as long as j is close enough to 1. Finally, inequalities [START_REF] Palfrey | Repeated play, cooperation and coordination: an experimental study[END_REF] and [START_REF] Peski | Repeated games with incomplete information on one side[END_REF] are always valid when i and j are close enough to 1. This means that if 1 j c(1 i ) and if i is close enough to 1 (i.e. i for some 2 f0; 1g), then j is also close to 1, and the four equations of Lemma 8 all hold.

Proof of Proposition 6 This proof is similar to the former proof of Proposition 5. Equation (32) still holds, and the inequality log(1 p i ) l i + i g i g i < log(1 p j ) l j + j g j g j is equivalent to i < j . We have

T i = T j 1 1 j i + j (1 ) i (1 )
; so that T i < T j if is close enough to 1. The rest follows as in the proof of Proposition 5.

Further elements of comparison between Proposition 6 and Atakan and Ekmekci (2013)'s theorem 2

We can now make more precise the comparison that we started at the end of Section 5.1. In their theorem 2, Atakan and Ekmekci (2013) assume without loss of generality that the basic payo¤s satisfy g 1 = g 2 and focus on the case = 0. Under the latter assumption, both automata always play d, but the war of attrition strategies (described at the beginning of Section 6.4) still identify one of the players, player i, as the one who can start to cooperate at even stages (in particular at stage 0), while the other (player j 6 = i) can start to cooperate at odd stages. This construction has no counterpart in Atakan and Ekmekci (2013) who rely on an auxiliary war of attrition in continuous time. When g 1 = g 2 = g and = 0, condition (15) in Proposition 6, which identi…es player i, reduces to l i log(1 p i ) < l j log(1 p j ). Furthermore, in this case, by proceeding as in the proofs of Propositions 5 and 6, one can show that, when ! 1, the players' equilibrium payo¤s, given in Lemma 8, converge to These payo¤s are exactly the same as the ones that are implicitly obtained in Atakan and Ekmekci (2013)'s theorem 2, as the limit payo¤s of any perfect Bayesian equilibrium of the reputation game with = 0. Proposition 6 is not only consistent with Atakan and Ekmekci (2013)'s result but also provides an explicit construction of a larger family of equilibria (namely, with > 0), for any su¢ciently large discount factor, in the original discrete time framework. 35

x i = 0 x j = lim

Public good game (Proposition 7)

Proposition 7 results from the following lemma.

Breaking an agreement initiated by the other player by playing c instead of d: he will get 1 and then will be punished and win at most z for all the remaining stages, which is always less than he would have got by always playing d.

Breaking an agreement initiated by the other player by playing d instead of c: he will win z and then be punished and win at most z for all remaining stages; by following the agreement, he …rst get 1 and then get averagely l + (1 l )(z + 1), where l 2 f1; 2g, l 6 = k. This means that k is right to follow the agreement if:

z (1 )1 + ( l :1 + (1 l )(z + 1));
and this is equivalent to the second inequality in [START_REF] Sorin | Merging, reputation, and repeated games with incomplete information[END_REF].

The expression of limit payo¤s is a straightforward application of the formulas that can be found in the proof of Lemma 8. For a …xed , payo¤s are given by: x i (n; ) = 1 + i !; x j (n; ) = 1 + P( i = 0)(1 i )! + P( i = 0) j !;

x i (g; ) = z + T X t=0 P( j = 2t + 1)v i 2t+1 ;

x j (g; ) = z + T X t=0 P( j = 2t)v j 2t :

and the result comes from letting go to 1.

Elements of comparison with Peski (2013)

Following Peski (2013)'s ideas, we could also exhibit limit payo¤s of …nitely revealing equilibria. To this aim, one needs to re…ne the construction of the war of attrition equilibria by adding an arbitrary number of non-revealing stages between the information transmission stages. This means, e.g., that player i (resp., j) possibly concedes at stages 2tk (resp., (2t + 1)k), for some k 1, instead of stages 2t (resp., 2t + 1), the additional stages being devoted to playing (d; d) with probability 1 if no one conceded before. With an appropriate choice of k as an increasing function of , one can construct

Let 1 A

 1 and set = 23 + 33 .

without discounting, in which the payo¤s are evaluated by the limit of means criterion. The characterization (stated as Proposition 1) turns out to be amazingly tractable: a vector payo¤ is an equilibrium of the Bayesian undiscounted in…nitely repeated game 1 if and only if it is a particular, completely revealing equilibrium payo¤, which can be entirely described in terms of the one-shot game. A partial analog to the standard folk theorem (stated as Proposition 3) follows: the equilibrium payo¤s of the Bayesian undiscounted in…nitely repeated game can be interpreted as feasible, incen-

[START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF]'s result generalizes Shalev (1994)'s characterization of Nash equilibrium payo¤s in undiscounted repeated games with lack of information on one side and knownown payo¤s (see[START_REF] Forges | Repeated games of incomplete information : nonzero-sum[END_REF] for a survey of results on non-zero sum in…nitely repeated games with incomplete information).

See Hörner et al. (2011) for a generalization of[START_REF] Blackwell | An analog of the minimax theorem for vector payo¤s[END_REF] to an arbitrary number of players.

[START_REF] Bergin | A characterization of sequential equilibrium strategies in in…nitely repeated incomplete information games[END_REF] proposes a characterization of equilibrium strategies in discounted in…nitely repeated games but does not consider lim !1 N [ ].

This result can be deduced from Shalev (1994)'s characterization of the set of Nash equilibrium payo¤s N [ 1 ] of any two-person undiscounted in…nitely repeated game 1 with lack of information on one side and known-own payo¤s, which implies that N [ 1 ] is always nonempty.

See also[START_REF] Sorin | Merging, reputation, and repeated games with incomplete information[END_REF] for a synthetic presentation of various related models, including in…nitely repeated games with known own payo¤s.

There are exceptions, though, e.g.,[START_REF] Kreps | Reputation and imperfect information[END_REF],[START_REF] Chan | On the non-existence of reputation e¤ects in twoperson in…nitely repeated games[END_REF] and[START_REF] Atakan | A two-sided reputation result with long-run players[END_REF].

We allow for arbitrary payo¤s as long as the SCI property is satis…ed.

The normal players may have di¤erent discount factors in this part of the paper.

Independent punishment strategies are important for proposition 1.

As a recent reference, Peters and Szentes (2012)'s assumption 1 (p. 397) takes exactly the form of (2) if values are private and independent and mixed strategies are allowed.

[START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF] already established a version of this result in the case of only two players and without assuming uniform punishments. The latter assumption greatly facilitates the formulation of the equilibrium conditions and the extension to n players.

i q i ( i )u i (t i ; a i ; i ( i )) for every i 2 i . But, with independent private values, (13) is an equivalent formulation, since u i ( i ; ) is linear.

[START_REF] Hörner | Belief-free equilibria in games with incomplete information[END_REF] show that all belief-free equilibrium payo¤s must belong to the set V of ex post incentive compatible and ex post individually rational payo¤s. By Proposition 1, V \ q N [ 1 (q)].

More precisely, lim inf !1 N [P G (p; !; z)] 6 = ;, with the de…nition x 2 lim inf !1 N [ (q)] , 8 n ! 1 9x n 2 N n (q) such that x n ! x.

The reputation game G (p; i; ) is described by the following parameters: k , p k , m k , g k , l k , k , k = 1; 2 and i. The notation just keeps track of the main ones.There is a slight abuse of notation in using a single matrix: the payo¤s of a player are the same, whether he faces another player or an automaton.

In this section, we follow the tradition of the reputation literature and allow the players to use di¤erent discount factors.

As usual (see[START_REF] Peski | Repeated games with incomplete information and discounting[END_REF] for a recent reference), the public random device is introduced to simplify the exposition but is not necessary.

[START_REF] Myerson | Game theory: analysis of con ‡ict[END_REF] If one of the conditions does not hold, the automaton plays c.

This latter part is similar to the standard construction to prove the folk theorem under complete information.

As observed by[START_REF] Atakan | A two-sided reputation result with long-run players[END_REF], g 1 = g 2 can be achieved through a normalization and is thus without loss of generality.

In the particular case g 1 = g 2 , l 1 = l 2 , agent i is the one who is the more likely to be a player rather than an automaton, namely, p i > p j .

Observe that given the description of automata, the behavior of greedy players is de…ned on every possible history. The behavior of normal players is fully described in the appendix.

Note that player i may reveal further information on his type by playing so as to guarantee himself v i ( i ). This typically happens out of equilibrium.

If information transmission ends up after …nitely many stages t 0 , e can be interpreted as the frequency of moves from stage t 0 + 1 on.

As in the standard proof of the folk theorem under complete information, we interpret a distribution of moves as a deterministic sequence of moves (in A) which achieves the frequency of . This interpretation is straightforward if the components of are rational (in Q).

The above reliable mediator selects as a random function of the players' reported types but does not reveal these reported types.

The same condition holds as well in two-person games with independent private values (see[START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF]).

The simpli…cation of the equilibrium conditions in the case of convex value functions (which give rise to a linear concavi…cation) is acknowledged in[START_REF] Koren | Two-person repeated games where players know their own payo¤s[END_REF], remark 4. A similar condition is considered in[START_REF] Forges | Communication equilibria in repeated games with incomplete information[END_REF].

There is an additional subtle point. As observed above, inequality[START_REF] Hart | Nonzero-sum two-person repeated games with incomplete information[END_REF] in Proposition 6 must be strict, which excludes for instance the case p 1 = p 2 and l 1 = l 2 . No such restriction is needed in Atakan and Ekmekci (2013)'s continuous time framework. We can recover their result in the border case by deriving the equilibrium payo¤s for k > 0 and letting k ! 0, k = 1; 2.

these payo¤s can be chosen arbitrarily close to the payo¤s of Proposition 7.

so that ( 1 4 ; 1 4 ) is indeed individually rational for player 1 in 1 (p), for every p 2 (0; 1). Hence (( 14 ; 1 4 ); 1) 2 N [ 1 (q)] for every q 2 (0; 1). However, (( 14 ; 1 4 ); 1) = 2 F [ (q)] because ( 1 4 ; 1 4 ) is not interim individually rational in the sense of ( 13): let = ( ; 1 ); max a 1 u 1 (h; a 1 ; ) = 1 4 is incompatible with max a 1 u 1 (l; a 1 ; ) = 1 satis…es the equilibrium conditions of Proposition 1 (namely [START_REF] Kreps | Reputation and imperfect information[END_REF], ( 21) and ( 22) above) but the vector payo¤ of player 1 is (0; 0) and is not individually rational for player 1 in 1 (q), namely does not satisfy [START_REF] Mertens | The value of two-person zero-sum repeated games with lack of information on both sides[END_REF]. Hence does not de…ne an equilibrium of 1 (q).

Example 2 illustrates that player 1 can bene…t from not revealing his information to player 2, if player 2 intends to punish him. Of course, when uniform punishments are available, the revelation of information does not matter.

Example 3

The framework is the same as in example 2 but the utility functions are described by

. As in the previous example, the assumption of uniform punishments is not satis…ed. Let p 2 [0; 1].

Conditions ( 25) and ( 26) are satis…ed, for every , since m k = 0, k = 1; 2.

The whole point is to guarantee [START_REF] Peski | Repeated games with incomplete information and discounting[END_REF], which reduces to

) T i g g + ( 1)l

The right hand side is < 1 for every < 1 and can be made arbitrarily close to 1 by choosing su¢ciently close to 1. The left hand side is

1 p j by de…nition of T i . Hence, if p i > p j (i.e., if p 1 6 = p 2 and index i is chosen to satisfy (24)), ( 27) is satis…ed for su¢ciently large.

Proof:

In order to show that the strategies described above de…ne an equilibrium, we …rst construct the unique distributions for i and j that make player i (resp. j) indi¤erent between playing c for the …rst time at stage 2t (resp. 2t + 1) for any t 2 f0; 1; : : : ; T g. Then we check that the derived strategic pro…le is indeed an equilibrium. We also show that T necessarily equals T i . We adopt the convention that i = +1 if agent i is an automaton. Let us compute the distribution P that an outside observer will assign to i and j , prior to stage 0. Note that player i (resp. j) is an outside observer of j (resp. i ). Note also that P( j = t) = 0 if t is an even number or if t > 2T + 1, and P( j = +1) = 1 p j . Similarly, P( i = t) = 0 if t is an odd number or if t > 2T , and P( i = +1) = 1 p i .

Probability distribution of j

The fact that player i has to be indi¤erent between starting contributing at any stage t = 0; 2; : : : ; 2T will enable us to …nd the entire distribution of j . We denote by G i (t) the overall payo¤ of player i when he starts to contribute at stage t, the law of j being …xed with the former restrictions. Note that G i (2t) refers to the expected payo¤ of the strategy that dictates to start contributing at stage 2t if player j did not contribute before and that initiate an agreement. G i (2t + 1) is the expected payo¤ of an arbitrary strategy which is only required to start contributing at stage 2t + 1 if agent j did not contribute before. We have, for any t 0:

Equilibrium payo¤s One has x i (n; ) = G i (0) and x j (n; ) = G j (1), the expressions in the lemma follow easily.

Bayesian perfectness

We now check that the players' strategies are best responses to each other, for appropriate beliefs, on histories that are not consistent with the war of attrition protocol. The case of I" histories has already been considered above.

On I' and II histories, the player, say k, who is supposed to play c has revealed earlier that he is not an automaton, which determines the belief of the other player, `6 = k. Given this belief and the fact that player k plays c, it is a best reply for player `to play d. For player k, whatever his beliefs on agent `(which may be a player or an automaton), playing c is a best response to d.

Finally, if a normal player gets to see a III history, then the other agent is an automaton with probability 1. It is then optimal for this normal player to react accordingly.

Remark:

The previous proof is written in terms of the players ex ante, overall payo¤s. Starting with [START_REF] Kreps | Reputation and imperfect information[END_REF], the reputation literature rather considers conditional expected payo¤s at every stage. We can as well construct our war of attrition equilibria inductively. By doing so, we …nd that player i is indi¤erent between playing c …rst at a stage 2t and waiting two more stages (hoping that agent j will concede at stage 2t + 1) if and only if the probability that agent j plays c at stage 2t + 1 given past history is i . That determines strategy of player j at stages 2t + 1, t 2 f0; : : : ; T 1g: play c with probability i p 2t+1 j and d otherwise, where p 2t+1 j is the probability that player i assigns to j being a player (as opposed to an automaton) given that c has not been played before stage 2t + 1. The same holds for player i's strategy at stages 2t, t 2 f1; : : : ; T g: play c with probability j p 2t i and d otherwise, where p 2t i is the probability that player j assigns to i being a player given that c has not been played before stage 2t. For the construction to be complete, it remains to see that: if c has never been played before stage 2T + 1, player j plays c with probability 1 at stage 2T + 1. Player i must then prefer c to d at stage Lemma 9 Let us assume that the discounted in…nitely repeated public good game P G (p; !; z) and = ( 1 ; 2 ) satisfy the following conditions:

and inequalities [START_REF] Myerson | Game theory: analysis of con ‡ict[END_REF] and ( 27)

, for i = 1 or 2 and j 6 = i. Then P G (p; !; z) has a perfect Bayesian equilibrium in war of attrition strategies parametrized by i and . The corresponding equilibrium payo¤s converge, as ! 1, to

where 0 = 1

, and

Proof:

The situation of normal players i and j in P G (p; !; z) is the same as in the reputation game G (p; i; ) with m k = 0, g k = !, l k = 1 ! because a greedy player behaves as an automaton. Thanks to Lemma 8, we mostly need to check that playing as an automaton is indeed a best response for a greedy player. A greedy player k has basically four ways to deviate from the strategy of an automaton.

Playing c …rst, when not supposed to (namely, at an odd (resp., even) stage if k = i (resp., k = j)) : the player will gain 1 and will then be punished and gain at most z for the remaining stages, whereas he could have won at least z all along by simply always playing d.

Playing c …rst, when supposed to: this will be interpreted as the initiation of an agreement by the other player. Depending on whether and when he will break this agreement, the player will …rst gain 1 and then his average gain will be between (1 k )1 + k (z + 1) and z. As (1 k )1 + k (z + 1) 2, this is again beaten by always playing d.

a class of equilibria such that the number of information transmission stages is bounded as goes to 1, and such that equilibrium payo¤s still converge 36 . To be more precise, such a construction would basically change into k in the inequalities that drive the existence of war of attrition equilibria (Lemma 8). To ensure this existence, it is then su¢cient to choose k such that k tends to a limit that is arbitrarily close to 1. This can be achieved by choosing values of k of order c log , with c > 0 and small enough. With such a choice, one can check that the corresponding value of T i is bounded, and so is the number of information transmission stages. Note that, in this construction, even if the number of information transmission stages is bounded, the horizon of the war of attrition (i.e., the last stage where some information may be revealed) still goes to in…nity as ! 1.