Coordination failure in repeated games with private monitoring

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Journal of Economic Theory 148 (2013) 1891–1928 www.elsevier.com/locate/jet

Coordination failure in repeated games with private monitoring Takuo Sugaya a,∗ , Satoru Takahashi b a Stanford Graduate School of Business, Knight Management Center, 655 Knight Way, Stanford, CA 94305, USA b Department of Economics, National University of Singapore, AS2 Level 6, 1 Arts Link, Singapore 117570,

Republic of Singapore Received 11 October 2011; final version received 7 February 2013; accepted 1 April 2013 Available online 26 July 2013

Abstract Players coordinate continuation play in repeated games with public monitoring. We investigate the robustness of such equilibrium behavior with respect to ex-ante small private-monitoring perturbations. We show that with full support of public signals, no perfect public equilibrium is robust if it induces a “regular” 2 × 2 coordination game in the continuation play. This regularity condition is violated in all belief-free equilibria. Indeed, with an individual full rank condition, every interior belief-free equilibrium is robust. We also analyze block belief-free equilibria and point out that the notion of robustness is sensitive to whether we allow for uninterpretable signals. © 2013 Elsevier Inc. All rights reserved. JEL classification: C72; C73; D82; D83 Keywords: Repeated game; Almost-public monitoring; Private monitoring; Belief-free equilibrium; Global game; Robustness

1. Introduction In repeated games with public monitoring, the players have common knowledge about past signals and coordinate continuation strategies contingent on the public histories of past signals. In many cases, however, players observe noisy signals that are slightly different for different players. The purpose of this paper is to investigate whether and how equilibrium construction * Corresponding author. Fax: +1 650 723 4315.

E-mail addresses: [email protected] (T. Sugaya), [email protected] (S. Takahashi). 0022-0531/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jet.2013.07.017

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in public-monitoring repeated games depends on the common knowledge assumption about past signals. More specifically, we investigate if players can coordinate approximately in the same way if the monitoring structure is perturbed so that there is no longer common knowledge about past signals. We formalize the question about coordination as follows. Fix a repeated game with public monitoring. Perturb the game so that players observe noisy private signals. Each player has an “interpretation” of signals, which is a function that maps her private signals to public signals. A private-monitoring structure is close to the public-monitoring structure under a profile of interpretations if, conditional on each action profile, the probability that all players observe private signals interpreted as a common public signal y under private monitoring is close to the probability that they observe y under public monitoring. Note that we measure proximity between the two monitoring structures only in the ex-ante sense, and we do not impose any restriction on a player’s interim beliefs about the opponents’ signals conditional on her own signals. For example, Hörner and Olszewski [16] analyze repeated games with almost-perfect monitoring, which is close to perfect monitoring in the ex-ante sense. For another example, consider the Cournot oligopoly game where firms produce almost homogeneous outputs, and “market price” θ is drawn from a continuous distribution. Each firm i does not observe θ directly, but observes a firm-specific price: ω i = θ + εi . This monitoring structure is close to the public monitoring where firms observe binary public ¯ or {θ < θ¯ }, as long as noise terms εi ’s concentrate signals about high or low prices, {θ  θ} around 0.1 Given the above class of perturbations, we say that an equilibrium of a public-monitoring repeated game is robust to private-monitoring perturbations if, for every private-monitoring perturbation close to the original public-monitoring structure under a profile of interpretations, there exists an equilibrium of the perturbed game close to the translation of the original equilibrium into private strategies via the same or similar profile of interpretations. In two-player games with full support of public signals, we show that no perfect public equilibrium (henceforth PPE) is robust to private-monitoring perturbations under a certain regularity condition that excludes belief-free equilibria and in particular, repetitions of static Nash equilibria (Theorem 1). The analysis of PPE in games with public monitoring has flourished since Abreu, Pearce, and Stacchetti [1] and Fudenberg, Levine, and Maskin [13]. Theorem 1 reveals that such analysis depends critically on the common knowledge assumption regarding past histories. Except for PPEs that violate the regularity condition, no PPE can be translated into equilibria in all perturbed games; either (i) a perturbed game has no equilibrium that is close to PPEs or (ii) the way to translate a PPE depends on fine details of perturbations. We prove Theorem 1 as follows. Suppose, for simplicity, that there are two public signals y¯ and y. Take any PPE, and imagine we are at the beginning of period 2. By the definition of PPE, it is optimal for each player to follow the continuation strategy after observing y¯ (resp. y) if the opponent also takes the continuation strategy after observing y¯ (resp. y). Now perturb the monitoring structure. At the beginning of period 2, each player chooses an interpretation of 1 Mailath and Morris [20,21] analyze a closely related notion of almost-public monitoring. They, however, exclude almost-perfect monitoring by assuming full support of the original monitoring structure. They also exclude the Cournot example because their notion of closeness measures distances between two monitoring structures from the interim perspective. See Section 1.1.

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her private signal in period 1, and follows the continuation strategy that the player would take in public monitoring after observing the public signal corresponding to the interpretation. This induces a 2 × 2 coordination game, where the players simultaneously choose their interpretations with payoffs given by corresponding continuation payoffs in the repeated game. Then, under a regularity condition, either (y, ¯ y) ¯ or (y, y) is a strictly risk-dominant profile in the induced 2 × 2 game, and the profile that is risk-dominant prevails for all private signals in period 1 when players are uncertain about each other’s choice of interpretations, as in the contagion argument given by Rubinstein [29] and Carlsson and van Damme [5]. Thus the basic part of the proof is simple. At the technical level, however, we have two more issues. First, the PPE may prescribe players to play mixed actions in period 1. A mixed strategy induces a rather complicated prior over the product set of actions and private signals. Here we use the full-support assumption and show that the contagion argument applies to the signal space independently of the realization of the action profile. Second, since signals do not carry payoffrelevant information in the repeated game, a dominance region, where a particular interpretation of signals is a dominant choice in the 2 × 2 game, does not exist in our setting. For this reason, we show the existence of a region with non-trivial size where the players follow the risk-dominant interpretation with high probability; the existence follows from the requirement of the closeness of the strategy. These two issues are addressed in Lemma 3. Note that Theorem 1 depends on the regularity condition: The regularity condition identifies which interpretation, y¯ or y, is strictly risk-dominant in the 2 × 2 game. The regularity condition is satisfied in many PPEs, but violated in all belief-free equilibria, where each player is indifferent between the two interpretations independently of the opponent’s interpretation. Indeed, we show that, in two-player games with an individual rank condition on public signals, every interior belief-free equilibrium is robust to private-monitoring perturbations (Theorem 2). Belief-free equilibria are arguably one of the most important findings in the literature on private-monitoring repeated games, and have been extensively investigated in the last decade.2 Given the widespread use of belief-free equilibria, the result in Theorem 2 is probably as expected and hardly surprising. For example, Theorem 2 can be seen as a generalization of Ely, Hörner, and Olszewski [8]. Nevertheless, it should be noted that Theorem 2 assures that the notion of robustness, as defined in this paper, is not absurdly restrictive. Combining Theorems 1 and 2, our exercise can be seen as a first step toward characterizing belief-free equilibria in terms of their robustness properties. That is, we show that our notion of robustness is “almost” necessary and “almost” sufficient for an equilibrium to be belief-free (as long as we regard both (i) non-interior belief-free equilibria and (ii) non-regular non-belief-free equilibria as negligible). We should, however, point out a subtlety in the definition of robustness. That is, in Theorem 1, we implicitly require that every private signal must be interpreted as a public signal with probability 1. Instead suppose that, after observing private signals that occur with a small probability, players can play continuation strategies that do not resemble original continuation strategies contingent on any public signal. Then the proof of Theorem 1 fails and the contagion of risk-dominant interpretation stops near such “uninterpretable” signals after which the 2 Belief-free equilibria are first constructed by Piccione [28] in the repeated prisoners’ dilemma, and simplified by Ely and Välimäki [9]. Kandori and Obara [19] consider belief-free equilibria in private strategies in the prisoners’ dilemma with public-monitoring. Ely, Hörner, and Olszewski [8] give a general definition of belief-free equilibrium and characterize the equilibrium payoff set in two-player games. Yamamoto [30] extends the characterization to general n-player games.

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players can play strategies completely different from the original ones. Indeed, we show that any block belief-free equilibrium, which may satisfy the regularity condition, can be approximated by equilibria in perturbed games if we allow for uninterpretable signals with small probability (Theorem 3). This suggests that our non-robustness result is not as straightforward an application of the contagion argument as it might appear to be at the first glance. 1.1. Literature review Most closely related to our paper is research by Mailath and Morris [20,21].3 In those papers, Mailath and Morris require that private-monitoring perturbations be small not only in the ex-ante sense but also in the interim sense. That is, conditional that a player observes a private signal interpreted as a public signal, the probability is close to 1 that the other players observe private signals interpreted as the same public signal. Observing that the interim closeness renders almost-common knowledge about relevant histories of finite length, Mailath and Morris [20,21] extend Monderer and Samet [25] to dynamic environments and show that a strict PPE is robust to almost-public-monitoring perturbations if and “almost” only if the PPE has bounded recall. In contrast, we require that private-monitoring perturbations be small only in the ex-ante sense: Conditional that a player observes a private signal interpreted as a public signal, the probability that the others observe private signals interpreted as the same public signal may be bounded away from 1. Using the larger class of perturbations, we can show that strict PPEs with bounded recall may not be robust in our sense. P˛eski [27] shows that a repeated game with private monitoring has no equilibrium other than repetitions of static equilibria if (i) the private-monitoring structure is sufficiently “connected” in the sense that updates of a player’s belief about the opponents’ signals from her signal changes are continuous (with respect to her signal), (ii) the continuation strategies depend only on finite partitions of the past histories, and (iii) there are smooth i.i.d. payoff shocks. There are several differences from our non-robustness result, but probably the most important is the existence of payoff shocks, which P˛eski uses to show that players’ equilibrium strategies depend smoothly on past histories. In contrast, our non-robustness result does not rely on such payoff perturbations, and hence does not eliminate belief-free PPEs. Also, our contagion argument is fundamentally different from his smoothness argument.4 Fudenberg and Olszewski [14] analyze repeated games with a long-run player against a sequence of short-run players under asynchronous monitoring, where each player observes underlying public signals of actions at random and privately known times. They show that the best “cutoff” equilibrium payoff can be strictly lower under asynchronous monitoring than under synchronous monitoring. To see this result, suppose that the long-run player observes a signal near but above the cutoff. Since short-run players who observe signals at different times than the long-run player observes signals with a different noise, the long-run player believes with probability close to 1/2 that the next short-run player has observed a signal below the cutoff and has switched to the punishment phase. This possibility of the short-run player’s switch diminishes the long-run player’s incentive to maintain the cooperation phase. This intuition of coordination 3 See also Hörner and Olszewski [17] and Mailath and Olszewski [22]. 4 In his companion paper, P˛eski [26] shows that in two-player repeated games with alternating moves and “rich” private

monitoring but without payoff shocks, any equilibrium with a finite past satisfies a version of the belief-free property. In contrast, we analyze repeated games with simultaneous moves.

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failure between the long-run and short-run players is similar to the contagion argument we use in our non-robustness result. In the class of repeated games with perfect monitoring, Ely [7] introduces the notion of weak robustness, and shows in his Theorem 1 that no strict equilibrium is weakly robust, except for repetitions of static equilibria.5 This non-robustness result relies on conditionally independent monitoring and complements our non-robustness result, which assumes full-support public monitoring, and can apply to some of the mixed-strategy (hence non-strict) equilibria. The rest of the paper is organized as follows. Section 2 defines repeated games with public monitoring and their private-monitoring perturbations. Section 3 gives two examples that lead us to formalize the notion of robustness to private-monitoring perturbations in Section 4. Sections 5 and 6 show Theorems 1 (non-robustness) and 2 (robustness), respectively. Section 7 introduces a weaker notion of robustness and shows another robustness result (Theorem 3). Section 8 concludes the paper. 2. Framework 2.1. Repeated games with public monitoring We begin our analysis by describing a repeated game with public monitoring. Let I = {1, . . . , n} be the set of players with n  2. For each i ∈ I , Ai is a finite set of actions available to player i in each period. Given each player i’s action ai ∈ Ai , the action profile a = (a1 , . . . , an ) ∈ A := A1 × · · · × An is determined. Given this action profile, a public signal y is drawn from a finite set Y with probability ρ(y | a), and  each player i receives ex-post payoff u∗i (ai , y), which induces ex-ante expected payoff ui (a) := y∈Y ρ(y | a)u∗i (ai , y). A repeated game with public monitoring is an infinite repetition of the stage game defined above. A history of player i at the beginning of period t is given by h˜ i,t = (ai,1 , y1 , . . . , ai,t−1 , yt−1 ) ∈ H˜ i,t := (Ai × Y )t−1 . A public component is given by h˜ t = (y1 , . . . , yt−1 ) ∈ H˜ t := Y t−1 , which is commonly known at the beginning of period t . A strategy of player i is a function s˜i  from H˜ i := t H˜ i,t to Ai . The continuation strategy of s˜i after h˜ i ∈ H˜ i is denoted by s˜i | h˜ i . Let Ui (˜s) denote the discounted average expected payoff of player i with discount factor δ (normalized by 1 − δ) when players obey a profile s˜ = (˜s1 , . . . , s˜n ). A strategy profile s˜ is a sequential equilibrium if, for each i and h˜ i ∈ H˜ i , 1. there exists a belief over the other players’ histories h˜ −i (conditional on h˜ i ) that is consistent with s˜ ; 2. given this belief, the continuation strategy s˜i | h˜ i is a best response to the distribution of s˜−i | h˜ −i := (˜s1 | h˜ 1 , . . . , s˜i−1 | h˜ i−1 , s˜i+1 | h˜ i+1 , . . . , s˜n | h˜ n ). ˜  A strategy is public if it depends only on the public component of the history: s˜i : H := ˜ ˜ → A . A public-strategy profile s is a perfect public equilibrium (PPE) if, for each H i t t ˜h ∈ H˜ , the induced continuation strategy profile s˜ | h˜ = (˜s1 | h, ˜ . . . , s˜n | h) ˜ is a Nash equilibrium ˜  Ui (˜s  , s˜−i | h) ˜ for all i ∈ I , h˜ ∈ H˜ , and s˜  . of the repeated game, i.e., Ui (˜s | h) i i The public-monitoring structure (Y, ρ) has full support if ρ(y | a) > 0 for all a ∈ A and y ∈ Y . 5 See also Matsushima [23].

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2.2. Private-monitoring perturbations The objective of this paper is to check the robustness of an equilibrium in the repeated game with public monitoring to private-monitoring perturbations. For this purpose, we consider private-monitoring perturbations as follows. A perturbed game is a repeated game with private monitoring. In each period, if the players play an action profile a ∈ A, a profile ω = (ω1 , . . . , ωn ) of private signals is drawn from a product Ω := Ω1 × · · · × Ωn of measurable spaces according to joint probability π(· | a). Each player i observes ωi ∈ Ωi , interprets it as fi (ωi ) ∈ Y , and receives ex-post payoff u∗i (ai , fi (ωi )), where fi is a measurable function from Ωi to Y .6 We call fi player i’s signal interpretation. Let f = (f1 , . . . , fn ).7 A private history of player i at the beginning of period t is given by hi,t = (ait−1 , ωit−1 ) ∈ Hi,t := (Ai × Ωi )t−1 , which consists of player i’s past actions ait−1 = (ai,1 , . . . , ai,t−1 ) and private signals ωit−1 = (ωi,1 , . . . , ωi,t−1 ). A strategy si of player i is a measurable function from  t Hi,t to Ai . With slight abuse of notations, we extend the domain of fi to all private histories in the perturbed game. That is, for each hi,t = (ait−1 , ωit−1 ) ∈ Hi,t in the perturbed game, we denote by   fi (hi,t ) = ai,1 , fi (ωi,1 ), . . . , ai,t−1 , fi (ωi,t−1 ) ∈ H˜ i,t the corresponding (private) history in the original game. We define the closeness of private monitoring to public monitoring in the following sense. Definition 1. For ε  0, a private-monitoring structure (Ω, π, f) is ex-ante ε-close to the publicmonitoring structure (Y, ρ) if      π ω ∈ Ω: fi (ωi ) = y for all i ∈ I  a − ρ(y | a)  ε for all a ∈ A and y ∈ Y . This definition says that the probability that every player observes a signal interpreted as y in the private-monitoring structure is close to the probability that every player observes y in the public-monitoring structure. Note that the definition of ex-ante closeness does not impose any restriction on a player’s interim beliefs about the opponents’ signals, conditional on her own signals. The possibility of non-closeness of interim beliefs is in contrast with the Mailath and Morris [20,21] approach, which requires both ex-ante closeness (Definition 1) and interim closeness, that is,    π ω−i ∈ Ω−i : fj (ωj ) = y for all j = i  a, ωi  1 − ε for all i ∈ I , a ∈ A, and ωi ∈ fi−1 (y) whenever the conditional probability is well defined.8 ,9 6 Mailath and Morris [20,21] allow for small payoff differences from u∗ (a , f (ω )) in the perturbed game. If we allow i i i i for such payoff perturbations, then Proposition 1 does not hold anymore, but Theorems 2 and 3 are intact. 7 We focus on stationary perturbations for notational simplicity, but all the results in this paper extend to non-stationary perturbations, where the monitoring structure (Ωt , πt , ft ) is indexed by t . 8 Mailath and Morris [21] extend the definition of interpretations so that f is a measurable function from Ω to Y ∪ {∅}. i i Here, fi (ωi ) = ∅ means that ωi is an “uninterpretable” signal with no corresponding public signal. We will discuss the issue of uninterpretable signals in Section 7. 9 Mailath and Morris [20] assume that Ω = Y and f is the identity function for every i ∈ I . In this case, ex-ante i i closeness implies interim closeness if (Y, ρ) has full support.

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3. Examples In this section, we give two examples to illustrate how private-monitoring perturbations prevent players from coordinating their future play. These examples also lead us to the formal definition of robustness in Section 4. 3.1. One-period punishment The first example is the prisoners’ dilemma game with two signals, taken from Mailath and Morris [20, Section 3.1]. Let I = {1, 2}, A1 = A2 = {C, D}, and Y = {y, y}. Signals occur with probabilities ⎧ ⎪ ⎨ p if a = CC, ρ(y | a) = q if a = CD or DC, ⎪ ⎩ r if a = DD, ρ(y | a) = 1 − ρ(y | a). We assume that 0 < p < 1 so that both signals occur with positive probabilities when the players play CC. We also assume p = q = r, and let the ex-post utility function u∗i (ai , y) be represented by the following matrix:

C D

y¯

y

3−p−2q p−q 3(1−r) q−r

− p+2q p−q 3r − q−r

Then, ex-ante stage-game payoffs are given by C

C 2, 2

D 3, −1

D −1, 3 0, 0

We consider the following public strategy:

C if t = 1 or yt−1 = y, s˜i (ht ) = D if t  2 and yt−1 = y. Under this strategy, players punish each other for one period immediately after y is observed (tit-for-tat). We assume r > q and δ  (3p − 2q − r)−1 > 0 (and hence p > q) so that the profile s˜ = (˜s1 , s˜2 ) is a PPE. Given the repeated prisoners’ dilemma and the one-period-punishment strategy s˜ , let V (yi , yj ) := Ui (˜si | yi , s˜j | yj ) denote the continuation payoff that player i receives from period 2 onward in the repeated prisoners’ dilemma if she follows the continuation strategy s˜i | yi , and the opponent j = i follows the continuation strategy s˜j | yj .10 (Note that the values with yi = yj are the continuation payoffs in the repeated prisoners’ dilemma after corresponding public histories, whereas the values with yi = yj are the “counterfactual” continuation payoffs as if the players had observed different signals in period 1.) We have 10 We omit the player index in V because of symmetry.

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  V (y, y) = 2(1 − δ) + δ pV (y, y) + (1 − p)V (y, y) ,   V (y, y) = δ rV (y, y) + (1 − r)V (y, y) , thus V (y, y) = 2 −

2δ(1 − p) , 1 − δ(p − r)

V (y, y) =

2δr . 1 − δ(p − r)

Also we have   V (y, y) = −(1 − δ) + δ qV (y, y) + (1 − q)V (y, y) ,   V (y, y) = 3(1 − δ) + δ qV (y, y) + (1 − q)V (y, y) . With these values, we define the following symmetric 2 × 2 game with action set {y, y} for each player: y

y

G(˜s) = y V (y, y), V (y, y)

V (y, y), V (y, y)

y V (y, y), V (y, y)

V (y, y), V (y, y)

Since s˜ is a symmetric PPE, G(˜s) is a symmetric coordination game with two pure-strategy equilibria (y, y) and (y, y). We assume that one of these equilibria is strictly risk-dominant: V (y, y) + V (y, y) = V (y, y) + V (y, y), i.e., δ = (2p − 2r)−1 . We perturb the monitoring structure as follows. Each action profile a ∈ A generates a real number θ , which is uniformly distributed on [p − 1, p] if a = CC, on [q − 1, q] if a = CD or DC, and on [r − 1, r] if a = DD. As in Carlsson and van Damme [5], each player i observes a noisy signal ωi about θ : ωi = θ + ξi , where ξ1 and ξ2 are uniformly distributed on [−ε/2, ε/2] independently of θ . Player i interprets the signal ωi as y if ωi  0, and as y if ωi < 0.11 Note that this perturbation is ex-ante ε-close to the original public-monitoring structure. Call this perturbation the global-game perturbation (with ε). We will show that the perturbed game has no equilibrium that approximates the one-periodpunishment strategy. For the sake of concreteness, we focus on the following threshold strategy: ¯ In period 1, play C; in period t  2, play C if and only if ωi,t−1  ω. Claim 1. Suppose that 0 < p < 1, r > q, δ  (3p − 2q − r)−1 > 0, and δ = (2p − 2r)−1 . Then there exist η > 0 and ε¯ > 0 such that, for any ε ∈ (0, ε¯ ] and ω¯ ∈ [−η, η], the threshold strategy with threshold ω¯ is not an equilibrium of the perturbed game. Proof. Pick sufficiently small η, ε¯ > 0. Suppose that (y, y) is strictly risk-dominant in G(˜s). (A similar argument holds if (y, y) is strictly risk-dominant in G(˜s).) Consider player i’s private 11 Here, players receive continuous signals. One can instead use discrete signals drawn from finite or countably infinite signal spaces, similar to those in Rubinstein’s [29] e-mail game.

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history at the end of period 1 such that she observes ωi,1 below, but sufficiently close to, ω. ¯ She believes that the opponent j observes ωj,1 above ω¯ with probability close to 1/2, and hence expects that the opponent follows s˜j | y and s˜j | y with almost equal probabilities in the future. Since (y, y) risk-dominates (y, y) in G(˜s), player i strictly prefers s˜i | y to s˜i | y, which implies that the threshold strategy is not an equilibrium. 2 Claim 1 assumes that either (y, y) or (y, y) is strictly risk-dominant in G(˜s). To see that this assumption is indispensable, note that if δ = (2p − 2r)−1 , then the threshold strategy with ω¯ = 0 is an equilibrium of the perturbed game.12 Other than this regularity condition, however, the proof of Claim 1 does not depend on particular choices of payoffs or strategies. Indeed, in Section 5, we extend Claim 1 to more general games and equilibrium strategies, and we even allow for non-threshold strategies in perturbed games. Note that if we require private-monitoring perturbations to be both ex-ante and interim close to public-monitoring in the sense of Mailath and Morris [20,21], then the one-period-punishment strategy is robust to such perturbations since it has bounded recall. Thus, it is sensitive to the class of perturbations we consider whether or not an equilibrium is robust. 3.2. The Ely–Välimäki equilibrium Consider the repeated prisoners’ dilemma with the same ex-ante payoff matrix as before, but assume perfect monitoring this time. The following strategy is adapted from Ely and Välimäki [9]:  αC + (1 − α)D if t = 1 or aj,t−1 = C, ˜ s˜i (ht ) = βC + (1 − β)D if t  2 and aj,t−1 = D. In this strategy, if a player plays C, it costs her 1 in the current period, but induces the opponent to play C in the next period with additional probability α − β. Therefore, if α − β = 1/(3δ), then she is indifferent between C and D. Hence this strategy becomes a PPE. This equilibrium is belief-free: For each i and t , whether player j will take αC + (1 − α)D or βC + (1 − β)D in period t , as long as the ex-ante probability is 1 that player j will play αC + (1 − α)D in period t + 1 after observing ai,t = C and the ex-ante probability is 1 that player j will play βC + (1 − β)D in period t + 1 after observing ai,t = D, player i is indifferent between C and D in period t . That is, player i has (weak) incentives to follow the equilibrium without coordinating her play with player j . We have the following. Claim 2. Suppose that δ > 1/3. Then the Ely–Välimäki equilibrium s˜ = (˜s1 , s˜2 ) with α, β ∈ (0, 1) and α − β = 1/(3δ) is robust to private-monitoring perturbations in the following sense: For every T < ∞, there exists ε > 0 such that, for every private-monitoring perturbation (Ω, π, f) that is ex-ante ε-close to (Y, ρ), the perturbed game admits an equilibrium s = (s1 , s2 ) such that si (hi,t ) = s˜i (fi (hi,t )) for all i ∈ I , t  T , and hi,t ∈ Hi,t . 12 The threshold is exactly equal to 0 because in our example, u (C, a ) − u (D, a ) is independent of a , and the i j i j j global-game perturbation does not alter marginal distributions of interpreted signals. Without these simplifying assumptions, the threshold might be history-dependent and slightly different from 0.

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Sketch of proof. As explained above, in the Ely–Välimäki equilibrium, each player faces the trade-off between payoffs in the current and the following periods. This trade-off is independent of whether the opponent is expected to play C with probability α or β (or an in-between probability in expectation) in the current period, as long as the ex-ante probability of taking αC + (1 − α)D is 1 after observing ai,t = C and the ex-ante probability of taking βC + (1 − β)D is 1 after observing ai,t = D. Therefore, even in the perturbed game, the player remains approximately indifferent between C and D regardless of her beliefs about the opponent’s continuation strategies. We need a slightly more careful argument, however. In the perturbed game, the distribution of fj (ωj ) can slightly differ from the distribution of y. In that case, ex-ante payoffs also slightly differ from those in the original game. Also, with probabilities slightly different from those under perfect monitoring, a player’s action leads to rewards or punishments in the next period. Therefore, to keep the opponent exactly indifferent between C and D, her continuation payoffs must be adjusted appropriately. We make this adjustment by modifying continuation strategies after period T . This modification is possible as long as ε is small, since assuming α, β ∈ (0, 1) (and hence δ > 1/3) guarantees that we have enough freedom to change the continuation payoffs compared to small ε. 2 For a complete proof, see Theorem 2, which generalizes Claim 2. The intuition behind the robustness is straightforward: As we have seen in Claim 1, it is difficult for the players to coordinate their future play based on past signals in the perturbed game. The Ely–Välimäki equilibrium, on the other hand, does not require any coordination based on past signals. Only the ex-ante probabilities of signals for each action profile are important. This is why the Ely–Välimäki equilibrium is robust to private-monitoring perturbations that are ex-ante close to the public monitoring structure. However, as the second paragraph of the sketch indicates, we need to allow for small departures from the original equilibrium to adjust continuation payoffs. This leads us to introduce (η, T )-closeness in the next section, which requires that the equilibrium be close until period T but after period T , the continuation play can be completely different from the original strategy. In addition, Theorem 2 considers the interior belief-free equilibrium: The continuation payoff is in the interior of the equilibrium payoff set in the original public monitoring game. This interiority gives us enough freedom of the continuation payoffs for the small adjustment of the equilibrium payoff. 4. Definition of robustness In this section, we define the robustness of PPEs to private-monitoring perturbations. Using this definition, we will examine PPEs in Sections 5 and 6. We also discuss other possible definitions. To motivate our definition of robustness, let us begin with the definition by Mailath and Morris [20,21]. Given a strict PPE s˜ = (˜s1 , . . . , s˜n ) of the original game, they focus on the strategy profile s = (s1 , . . . , sn ) of the perturbed game translated via f = (f1 , . . . , fn ), i.e., si (hi,t ) = s˜i (fi (hi,t )) for all i ∈ I , t  1, and hi,t ∈ Hi,t . Then they show that the translated strategy profile s is a (sequential) equilibrium of all perturbed games that are close to the original game in both ex-ante and interim senses if and “almost” only if s˜ has bounded recall. This definition of robustness, however, is too restrictive in our setup. For example, suppose that (Y, ρ) has full support, and that s˜ is a strict PPE that is history-dependent, i.e., there exist

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i ∈ I , t  1, and h˜ t , h˜ t ∈ H˜ t such that s˜i (h˜ t ) = s˜i (h˜ t ). Then consider a perturbation (Ω, π) such that Ωi = Y × Y , Ωj = Y for j = i, and ⎧ ⎨ ρ(y | a) − ε π(ω | a) = ε/(|Y | − 1) ⎩ 0

if ωi = (y, y), ωj = y for all j = i, if ωi = (y, y  ), y  = y, ωj = y for all j = i, otherwise.

Then, (Ω, π, f) is ex-ante ε-close to (Y, ρ) with the interpretation f such that player i interprets ωi = (y, y  ) as y  , and the other players interpret their signals literally. Now if player i observes a private history whose first components indicate h˜ t while the second components indicate h˜ t , then the strategy si translated via fi prescribes her to play s˜i (h˜ t ), but she knows that the other players follow s˜−i | h˜ t and strictly prefers s˜i (h˜ t ). Thus the translated strategy profile s is not an equilibrium in the perturbed game.13 In general, each player wants to predict how other players interpret the signals in order to coordinate future play, but his own interpretation may not be a good predictor of other players’ interpretations. In the above example, however, this problem is purely nominal. As long as player i interprets ωi = (y, y  ) as y, and plays the perturbed game according to the first components of her private histories, the translated strategy profile becomes an equilibrium of the perturbed game for small ε. On the other hand, the example in Section 3.1 cannot be easily fixed. To make our non-robustness result less trivial, and in particular, to distinguish substantial issues from nominal ones, we introduce the notion of endogenous interpretations {gt }, which allows for more flexible interpretations and makes the notion of robustness more permissive.14 Definition 2. Fix a private-monitoring perturbation (Ω, π, f), η  0, and T  ∞. 1. For each t  T − 1, a profile gt = (g1,t , . . . , gn,t ) of measurable functions from Hi,t+1 to Y is η-close to f if       π ω ∈ Ω: gi,t ωi,t  ait−1 , ai,t , ωit−1 = fi (ωi,t )  ai,t , a−i,t  1 − η for all i ∈ I , (ai,t , a−i,t ) ∈ A, and (ait−1 , ωit−1 ) ∈ Hi,t . −1 2. A profile s = (s1 , . . . , sn ) of strategies in the perturbed game is (η, T )-close to s˜ via {gt }Tt=1 if    si (hi,t ) − s˜i gi (hi,t )   η for all i ∈ I , t  T , and hi,t = (ait−1 , ωit−1 ) ∈ Hi,t , where     gi (hi,t ) = ai,1 , gi,1 (ωi,1 | ai,1 ), . . . , ai,t−1 , gi,t−1 ωi,t−1  ait−1 , ωit−2 . Definition 3. A sequential equilibrium s˜ of a public-monitoring repeated game is robust to private-monitoring perturbations if, for every η > 0 and T < ∞, there exists ε > 0 such that, for every private-monitoring perturbation (Ω, π, f) that is ex-ante ε-close to (Y, ρ), there exist a 13 Moreover, the Ely–Välimäki equilibrium considered in Section 3.2 is not robust in this restrictive sense. 14 Note that endogenous interpretations do not affect the physical structure of the perturbed game. For example, stagegame payoffs u∗i (ai , fi (ωi )) are still determined by exogenous interpretations fi (ωi ).

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sequence {gt } of interpretations and a Nash equilibrium s of the perturbed game such that each gt is η-close to f and s is (η, T )-close to s˜ via {gt }. In the above definition of robust equilibria, we can choose {gt } to construct an equilibrium in the perturbed game that approximates the original equilibrium. This is why {gt } is called “endogenous” interpretations. We require that those endogenous interpretations be equal to exogenous interpretations f with large probability for the first T − 1 periods. Endogenous interpretations generalize threshold strategies in Section 3.1, where the endogenous threshold ω¯ can be different from, but must be close to, the exogenous threshold 0. We also allow for gi,t to depend on her entire private history (ωit−1 , ait−1 , ai,t ). Note that the equilibrium s of the perturbed game can be slightly different from the PPE s˜ of the original game translated via {gt } for the first T − 1 periods, and completely different after period T . We need such flexibility to make small adjustments and offset small perturbations in continuation payoffs, as we discussed in Section 3.2. Such flexibility makes our notion of robustness more permissive, our non-robustness result (Theorem 1) stronger, but our robustness results (Proposition 1 and Theorem 2) weaker. As we will see in the proofs, however, Proposition 1 holds even without any flexibility. Also, Theorem 2 holds with η = 0, i.e., without flexibility for the first T − 1 periods. Note also that for the first T periods, each player must interpret all of her private histories as public histories and follow the original equilibrium strategy according to the interpretations. If we relaxed this requirement to “most of her private histories”, our proof of Theorem 1 would fail, and more equilibrium strategies would become robust. We will discuss this issue in Section 7. An important implication of our definition of robustness is the following. Suppose that the perturbed game has an equilibrium that approximates the original equilibrium. Then player i’s equilibrium payoff in the perturbed game is close to her equilibrium payoff in the original game. Moreover, conditional that each player j takes aj and interprets her signal as yj , player i’s continuation payoff in the perturbed game is close to Ui (˜s1 | a1 , y1 , . . . , s˜n | an , yn ). It is obvious from the definition, but worth stating explicitly. Lemma 1. Fix a strategy profile s˜ in the public-monitoring repeated game. For every γ > 0, there exist η > 0, T < ∞, and ε > 0 such that, if a private-monitoring perturbation (Ω, π, f) is ex-ante ε-close to (Y, ρ), a profile gt of endogenous interpretations is η-close to f for every t  T − 1, and a profile s of strategies in the perturbed game is (η, T )-close to s˜ via {gt }, then player i’s continuation payoff in the perturbed game conditional on a1 and ω1 is within distance γ of Ui (˜s1 | a1,1 , g1 (ω1,1 | a1,1 ), . . . , s˜n | an,1 , gn (ωn,1 | an,1 )). 5. A non-robustness result In this section, we extend Claim 1 and provide a sufficient condition for non-robustness. For simplicity, we assume that the public-monitoring repeated game has two players I = {1, 2} and two public signals Y = {y, y}, and the public-monitoring structure has full support. We concentrate on PPEs. (We will discuss extensions to general stage games and private strategies at the end of this section.) Fix a PPE s˜ . Recall that Ui (˜si | yi , s˜j | yj ) denotes player i’s continuation payoff in the repeated game from period 2 onward if she plays s˜i | yi and the opponent plays s˜j | yj . As in Section 3.1, s˜ induces the following 2 × 2 game with action set {y, y} for each player:

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y

G(˜s) = y U1 (˜s1 | y, s˜2 | y), U2 (˜s1 | y, s˜2 | y)

U1 (˜s1 | y, s˜2 | y), U2 (˜s1 | y, s˜2 | y)

y U1 (˜s1 | y, s˜2 | y), U2 (˜s1 | y, s˜2 | y)

U1 (˜s1 | y, s˜2 | y), U2 (˜s1 | y, s˜2 | y)

Since s˜ is a PPE, G(˜s) has two pure-strategy equilibria (y, y) and (y, y). We impose a regularity condition on the 2 × 2 coordination game G(˜s) in the following sense. Definition 4. A 2 × 2 coordination game α2 u1 , u2

β2 v1 , v2

β1 w1 , w2

x1 , x2

α1

with two pure-strategy equilibria (α1 , α2 ) and (β1 , β2 ) is regular if (α1 , α2 ) and (β1 , β2 ) have different Nash products, i.e., (u1 − w1 )(u2 − v2 ) = (x1 − v1 )(x2 − w2 ). The regularity condition is satisfied in many cases. For example, in Section 3.1, the 2 × 2 game induced by the one-period-punishment strategy is regular if and only if δ = (2p − 2r)−1 . More generally, we can show that the regularity condition is generic in the following sense: Fix stage-game payoffs u∗i (ai , y) and signal probabilities ρ(y | a), and take a public strategy profile s˜ that is a strict PPE for δ = δ0 ∈ (0, 1) and that induces different continuation plays after different public signals in period 1. Since continuation payoffs are analytic functions of δ, the set of δ such that the induced 2 × 2 game G(˜s) is not regular does not have an accumulation point in (0, 1). Hence, there is an open neighborhood of δ0 in which s˜ is still an equilibrium and G(˜s) satisfies the regularity condition, possibly except for δ = δ0 . Similar genericity results hold for other parameters. Note, however, that the regularity condition excludes all belief-free equilibria and in particular, repetitions of static Nash equilibria. (We will discuss more about belief-free equilibria in Section 6.) The next theorem generalizes Claim 1, stating that no PPE is robust to private-monitoring perturbations if it induces a regular coordination game. Note that this is consistent with Claim 2 since we exclude the Ely–Välimäki equilibrium by the regularity condition. Theorem 1. Consider a public-monitoring repeated game with two players, two public signals, and full support. If s˜ is a PPE such that G(˜s) is regular, then s˜ is not robust to private-monitoring perturbations. Proof. Suppose that s˜ is a PPE such that G(˜s) is regular. For each ε > 0, we will construct a private-monitoring perturbation (Ω, π, f) that is ex-ante ε-close to (Y, ρ) such that the perturbed game has no equilibrium close to s˜ , contradictory to our assumption. By the regularity of G(˜s), after changing the labels of y and y if necessary, there exist p1 , p2 > 0 with p1 + p2 < 1 such that (y, y) is a strict (p1 , p2 )-dominant equilibrium in G(˜s), i.e., pi Ui (˜si | y, s˜j | y) + (1 − pi )Ui (˜si | y, s˜j | y) > pi Ui (˜si | y, s˜j | y) + (1 − pi )Ui (˜si | y, s˜j | y)

(1)

for all i ∈ I . The following lemma shows that a similar risk-dominance property holds for the coordination game where we replace public signals with the endogenous interpretations.

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Lemma 2. There exist η > 0, T < ∞, and ε > 0 such that the following is true. Suppose that (Ω, π, f) is ex-ante ε-close to (Y, ρ), gt is η-close to f for every t  T − 1, and s is an equilibrium of the perturbed game that is (η, T )-close to s˜ via {gt }. Then, for every i ∈ I and ai,1 ∈ supp si (∅), if ωi,1 satisfies π({ωj,1 ∈ Ωj : gj,1 (ωj,1 | aj,1 ) = y} | ai,1 , aj,1 , ωi,1 ) > pi for all aj,1 ∈ supp sj (∅), then gi,1 (ωi,1 | ai,1 ) = y. Proof. Pick any i ∈ I and ai,1 ∈ supp si (∅). If player i believes that, regardless of aj,1 ∈ supp sj (∅), player j interprets gj,1 (ωj,1 | aj,1 ) = y with a probability no less than pi , then, by Lemma 1 and (1), player i receives a higher payoff if she interprets ωi,1 as y and follows s˜i | y than s˜i | y. 2 For the rest of the proof, we take the global-game perturbation as discussed in Section 3.1. More precisely, let Ω1 = Ω2 = R and π = π ε , where, conditional on each a ∈ A, π ε (· | a) denotes the joint distribution of ω = (ω1 , ω2 ) induced by the uniform distribution of (θ, ξ1 , ξ2 ) on [−ρ(y | a1 ), ρ(y, | a1 )] × [−ε/2, ε/2]2 and ωi = θ + ξi for each i ∈ I . Let fi (ωi ) = y if ωi  0 and fi (ωi ) = y if ωi < 0. Since (Y, ρ) has full support, let B := mina,y ρ(y | a) > 0. The following lemma follows from the characteristics of the perturbation in period 1. (The proof is in Appendix A.1.) Lemma 3. There exist η, ε, d, C1 , C2 , D > 0 such that the following are true. (i) For every η < η and ε < ε, if g1,1 satisfies    π ε ω1 ∈ R2 : g1,1 (ω1,1 | a1,1 ) = f1 (ω1,1 )  a1,1 , a2,1  1 − η

(2)

for all (a1,1 , a2,1 ) ∈ A, then there exists x0 ∈ [Dε, B − Dε] such that    π ε ω1,1 ∈ R: g1,1 (ω1,1 | a1,1 ) = y  a1,1 , a2,1 , ω2,1 > p2 for all (a1,1 , a2,1 ) ∈ A, and ω2,1 ∈ [x0 − C2 ε, x0 + C2 ε]. (ii) For every ε > 0 and x ∈ [−B + Dε, B − Dε], we have    π ε ωj,1 ∈ [x − Cj ε, x + Cj ε]  a1 , ωi,1 > pi for all i ∈ I , a1 ∈ A, and ωi,1 ∈ [x − (Ci + d)ε, x + Ci ε]. Fix any η < min{B, η, η} and T > T . By the robustness of s˜ , there exists εη,T > 0 such that, for every ε < min{εη,T , ε, ε}, there exist a sequence {gt } of endogenous interpretations and an equilibrium s of the perturbed game such that each gt is η-close to f, and s is (η, T )-close to s˜ via {gt }. This implies (2) is satisfied. By Lemma 3 (i), there exists x0 ∈ [Dε, B − Dε] such that player 2 with private signal ω2,1 ∈ [x0 − C2 ε, x0 + C2 ε] believes that player 1 interprets ω1,1 as y with probability more than p2 . By Lemma 2, player 2 interprets ω2,1 as y on the equilibrium path. By Lemma 3 (ii), player 1 with private signal ω1,1 ∈ [x0 − (C1 + d)ε, x0 + C1 ε] believes that player 2 interprets ω2,1 as y with probability more than p1 , and, by Lemma 2, player 1 interprets ω1,1 as y on the equilibrium path. Once again, by Lemma 3 (ii), player 2 with private signal ω2,1 ∈ [x0 − (C2 + 2d)ε, x0 + C2 ε] believes that player 1 interprets ω1,1 as y with probability more than p2 , and, by Lemma 2, player 2 interprets ω2,1 as y on the equilibrium path. By induction, each player i interprets ωi,1 ∈ [−B + (D + d)ε, x0 + Ci ε] as y on the equilibrium path. Since this is true for all ε < min{εη,T , ε, ε}, this is a contradiction. 2

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In the rest of this section, we investigate whether and how we can relax the assumptions in Theorem 1. First, Theorem 1 extends easily to games with more than two public signals as follows. Assume that there are two public signals, say y and y, such that the regularity condition is satisfied in the corresponding 2 × 2 coordination game G(˜s; y, y), where each player follows continuation strategy either s˜i | y or s˜i | y.15 Then consider private-monitoring perturbations such that, if a public signal y = y, ¯ y occurs, both players observe y without noise, and if y¯ or y occurs, then the players observe noisy private signals as in the global-game perturbation. We impose an extra condition on endogenous interpretations such that the players interpret these private signals as either y or y.16 Corollary 1. Consider a public-monitoring repeated game with two players and full support. If s˜ is a PPE such that G(˜s; y, y) is regular for some pair of signals y and y, then s˜ is not robust to private-monitoring perturbations under the extra condition (described above) imposed on endogenous interpretations. Second, Theorem 1 extends to games with more than two players under an additional condition. (Here we assume two public signals or use the same trick as above.) For a PPE s˜ , let G(˜s) be the n-player game, where each player i’s action set is {y, y}, and her payoff is Ui (˜s1 | y1 , . . . , s˜n | yn ) when players play (y1 , . . . , yn ) ∈ {y, y}n . The induced game G(˜s) is supermodular if Ui (˜si | y, (˜sj | yj )j =i )−Ui (˜si | y, (˜sj | yj )j =i ) is weakly increasing as each player j = i switches from yj = y to y. A supermodular game G(˜s) is regular if either (y, . . . , y) or (y, . . . , y) is a contagious equilibrium in an open neighborhood of G(˜s).17 Under these supermodularity and regularity conditions, Theorem 1 is generalized to games with more than two players.18 Third, Theorem 1 holds for equilibria in private (non-public) strategies if the players’ action profile in period 1, s˜(∅) = (˜s1 (∅), s˜2 (∅)), is pure and the 2 × 2 coordination game given by the continuation payoffs after the equilibrium action profile, G(˜s1 | s˜1 (∅), s˜2 | s˜2 (∅)), is regular. Corollary 2. Consider a public-monitoring repeated game with two players, two public signals, and full support. If s˜ is a sequential equilibrium such that s˜(∅) is pure and G(˜s1 | s˜1 (∅), s˜2 | s˜2 (∅)) is regular, then s˜ is not robust to private-monitoring perturbations. Fourth, Theorem 1 assumes the regularity condition for the continuation payoffs starting in ˜ for period 2. This assumption can be relaxed to any later period, i.e., the regularity of G(˜s | h) ˜ ˜ some h ∈ H if we allow for non-stationary perturbations {(Ωt , πt , ft )} indexed by t and do not ˜ (The original condition corresponds to h˜ = ∅.) perturb monitoring structures until h. 15 Note that we require the regularity condition for some pair of public signals, which is “easier” to be satisfied if the

game has more public signals. 16 Otherwise, if we had a region of player i’s private signals which are interpreted as y  = y, y, it might stop the contagion argument. Restricting the interpretation to y and y is a strong condition for games with more than two signals, but we do not know much about contagion in games with more than two actions. 17 An equilibrium of a static game G is contagious if it is selected in a global game with some noise distribution. Frankel, Morris, and Pauzner [11] show that every supermodular game has at least one (and only one, generically) contagious equilibrium. 18 Supermodularity is obviously a strong condition for games with more than two players, but we do not know much about contagion in non-supermodular global games.

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Fifth, we assume that the public signals have full support to derive B > 0. Intuitively, the full-support assumption plays a role similar to the connectedness assumption in P˛eski [27]: both full supportness and connectedness smoothly relate players’ incentives after y with those after y. In some cases, even if this assumption is violated, one may be able to use different perturbations, such as conditionally independent perturbations used by Matsushima [23] and Ely [7], to show different non-robustness results. Exploring such possibilities is left for future research. 6. A robustness result In this section, we show that belief-free equilibria are robust to private-monitoring perturbations. Here we focus on two-player games. Definition 5. (See Ely, Hörner, and Olszewski [8].) A pair of strategies s˜ = (˜s1 , s˜2 ) in the repeated game with public monitoring is a belief-free equilibrium if, for every i ∈ I , h˜ i,t ∈ H˜ i,t , and h˜ j,t ∈ H˜ j,t , Ui (˜si | h˜ i,t , s˜j | h˜ j,t )  Ui (˜si , s˜j | h˜ j,t ) for all s˜i . The Ely–Välimäki equilibrium we discussed in Section 3.2 is a belief-free equilibrium. Note that no belief-free public equilibrium induces a regular 2 × 2 game, since both s˜i | y¯ and s˜i | y are best responses to both s˜j | y¯ and s˜j | y. We also define belief-free equilibria for the perturbed game. Namely, a pair s = (s1 , s2 ) of strategies in the perturbed game is a belief-free equilibrium if, for every i ∈ I , hi,t ∈ Hi,t , and hj,t ∈ Hj,t , si | hi,t is a best response for player i against sj | hj,t in the perturbed game starting in period t. 6.1. Unperturbed marginal distributions Let us begin our analysis with the simple case where private-monitoring perturbations do not alter marginal distributions of interpreted signals. For example, Y = Ω1 = Ω2 = {y, y} with identity interpretations f, and π(· | a) is given by y

y

y ρ(y | a) − ε y

ε

ε ρ(y | a) − ε

In this case, every belief-free public equilibrium remains a belief-free equilibrium in the perturbed game under the translation of the strategy via exogenous interpretations for the entire horizon of the repeated game. This follows immediately from the definition of belief-free equilibria. Proposition 1. Let s˜ be a belief-free equilibrium in (Y, ρ). If π({ω ∈ Ω: fi (ωi ) = y} | a) = ρ(y | a) for all i ∈ I , a ∈ A, and y ∈ Y , then the pair of private strategies, s = (s1 , s2 ), defined by si (hi,t ) = s˜i (fi (hi,t )) for all i ∈ I , t  1, and hi,t = (ait−1 , ωit−1 ) ∈ Hi,t , is a belief-free equilibrium of the perturbed game. 6.2. General perturbations In general, private-monitoring perturbations may alter marginal distributions of interpreted signals. In this case, a belief-free equilibrium may not be an equilibrium in the perturbed game via

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exogenous interpretations for all periods. We will show, however, that under mild assumptions on the original repeated game and a given belief-free equilibrium, we can restore players’ incentives by modifying future strategies after period T . A pure action ai ∈ Ai of player i has individual full rank in (Y, ρ) if the collection (ρ(· | ai , aj ))aj ∈Aj of |Y |-dimensional vectors is linearly independent. If some action of player i has individual full rank, then |Y |  |Aj |. We assume the following. Assumption 1. Every pure action has individual full rank. One implication of Assumption 1 is that since player j knows her own action aj and can statistically identify player i’s action ai though signal y, player j can use the realization of (aj , y) to control player i’s expected payoff in any way. A similar property also holds even if the monitoring structure is slightly perturbed. Lemma 4. Suppose that (Y, ρ) satisfies Assumption 1. Then for any i ∈ I and any function V˜i : A → R, there exists w˜ i : Aj × Y → R such that    Eρ w˜ i (aj , y)  ai , aj = V˜i (ai , aj ) for every (ai , aj ) ∈ A, where the expectation Eρ [· | ai , aj ] is taken under ρ(· | ai , aj ). Moreover, given any such w˜ i , as we perturb (Y, ρ) and V˜i so that (Ω, π, f) is ex-ante ε-close to (Y, ρ) and Vi (a) → V˜i (a) as ε → 0 for every a ∈ A, there exists wi : Aj × Y → R such that     Eπ wi aj , fj (ωj )  ai , aj = Vi (ai , aj ) for every (ai , aj ) ∈ A, where the expectation Eπ [· | ai , aj ] is taken under π(· | ai , aj ), and that wi (aj , y) → w˜ i (aj , y) as ε → 0 for any aj ∈ Aj and y ∈ Y . See Appendix A.2 for the proof. For a strategy s˜j , let Vi (˜sj ) := maxs˜i Ui (˜si , s˜j ). A regime is a product of nonempty subsets of A1 and A2 , A = A1 × A2 . For a nonempty subset Ai of Ai , we say that player i’s payoff vi is Ai -enforced by αj ∈ (Aj ) and wi : Aj × Y → R if we have    vi  (1 − δ)ui (ai , αj ) + δEρ wi (aj , y)  ai , αj for all ai ∈ Ai with equality if ai ∈ Ai . Definition 6. A belief-free equilibrium s˜ is compatible with a sequence {At } of regimes if the following conditions are satisfied for all i ∈ I , t  1, h˜ i,t ∈ H˜ i,t , and h˜ j,t ∈ H˜ j,t : 1. supp s˜i (h˜ i,t ) ⊆ Ai,t . 2. Vi (˜sj | h˜ j,t ) is Ai,t -enforced by s˜j (h˜ j,t ) and {Vi (˜sj | (hj,t , aj,t , yt ))}aj,t ∈Aj , yt ∈Y . Every belief-free equilibrium is compatible with some sequence of regimes.19 For simple notation, we focus on a fixed (stationary) regime: At = A for all t  1.20 For a given regime A, the set of all belief-free equilibria that are compatible with A satisfies the following properties. 19 According to our definition, a belief-free equilibrium may be compatible with multiple regime sequences, one of which is constructed by Ely, Hörner, and Olszewski [8] and another by Yamamoto [30]. 20 Extensions to non-stationary regimes are straightforward.

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Exchangeability If (˜s1 , s˜2 ) and (˜s1 , s˜2 ) are belief-free equilibria compatible with A, then so are (˜s1 , s˜2 ) and (˜s1 , s˜2 ) (Ely, Hörner, and Olszewski [8, Proposition 1]). Recursion If (˜s1 , s˜2 ) is a belief-free equilibrium compatible with A, then so is (˜s1 | h˜ 1,t , s˜2 | h˜ 2,t ) for every h˜ 1,t ∈ H˜ 1,t and h˜ 2,t ∈ H˜ 2,t . Let W (A) be the set of payoff profiles sustained by belief-free equilibria compatible with A. By the exchangeability property, W (A) is a product of two compact intervals, W (A) = [w 1 (A), w1 (A)] × [w2 (A), w2 (A)] (Ely, Hörner, and Olszewski [8, Corollary 1]). A product set W = W1 × W2 is self-A-generating if, for every i ∈ I and vi ∈ Wi , vi is Ai -enforced by some αj ∈ (Aj ) and wi : Aj × Y → Wi . By the recursion property, W (A) is self-A-generating. Also, every bounded self-A-generating product set is a subset of W (A). We say that W (A) is properly self-A-generating if, for every i ∈ I , wi (A) is Ai -enforced by some α j ∈ (Aj ) and wi : Aj × Y → [w i (A), wi (A)) (excluding the right endpoint), and wi (A) is Ai -enforced by some α j ∈ (Aj ) and wi : Aj × Y → (w i (A), wi (A)] (excluding the left endpoint). We say that a belief-free equilibrium s˜ = (˜s1 , s˜2 ) compatible with A is interior if w i (A) < Vi (˜sj | h˜ j,t ) < wi (A) for all i ∈ I and h˜ j,t ∈ H˜ j,t . For example, consider the repeated prisoners’ dilemma we analyzed in Section 3.2, and take the regime A = {C, D} × {C, D}. Then for any δ  1/3, we have W (A) = [0, 2] × [0, 2], which is properly self-A-generating if and only if δ > 1/3. Furthermore, the Ely–Välimäki equilibrium is interior if and only if α, β ∈ (0, 1). The next theorem gives a set of sufficient conditions for a belief-free equilibrium to be robust to private-monitoring perturbations. The condition of proper self-generation is weak; for a sufficiently large discount factor, either W (A) is properly self-A-generating or there is no interior point in W (A). In addition, in terms of equilibrium payoffs, the interiority condition eliminates only boundary points of W (A). Indeed, Ely, Hörner, and Olszewski [8, Proposition 5 and Lemma 3] show that almost all belief-free equilibrium payoffs are robust to private-monitoring perturbations in the positive case with patient players.21 Nevertheless, we impose these conditions to discuss the robustness of equilibrium strategies (not payoffs) under fixed δ. Theorem 2. Consider a two-player public-monitoring repeated game that satisfies Assumption 1. If W (A) is properly self-A-generating, then any interior A-compatible belief-free equilibrium is robust to private-monitoring perturbations. Proof. Let s˜ be an interior A-compatible belief-free equilibrium. Fix T < ∞. We will show that there exists ε > 0 such that, if a private-monitoring perturbation (Ω, π, f) is ex-ante ε-close to (Y, ρ), then the perturbed game admits a belief-free equilibrium that is (0, T )-close to s˜ via f. Since s˜ is an interior belief-free equilibrium compatible with A, for every t  1, i ∈ I , h˜ i,t ∈ H˜ i,t , and h˜ j,t ∈ H˜ j,t , we have        Vi (˜sj | h˜ j,t )  (1 − δ)ui ai,t , s˜j (h˜ j,t ) + δEρ Vi s˜j  (h˜ j,t , aj,t , yt )  ai,t , s˜j (h˜ j,t )      = Eρ (1 − δ)u∗i (ai,t , yt ) + δVi s˜j  (h˜ j,t , aj,t , yt )  ai,t , s˜j (h˜ j,t ) 21 Ely, Hörner, and Olszewski [8, Lemma 3 (iii) and (iv)] show the continuity result only for the case with almost-perfect monitoring, but it can be easily extended to the case under Assumption 1.

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for all ai,t ∈ Ai with equality if ai,t ∈ Ai , and w i (A) < Vi (˜sj | h˜ j,t ) < wi (A). (Note that both aj,t and yt are random variables inside Eρ .) Fix i ∈ I . We define wi : H˜ j,t → (w i (A), wi (A)) for each t  T + 1 recursively as follows. Let wi (∅) = Vi (˜sj ) ∈ (w i (A), wi (A)). For each t = 1, . . . , T and h˜ j,t ∈ H˜ j,t , given that wi (h˜ j,t ) ∈ (w i (A), wi (A)), it follows from Lemma 4 that for sufficiently small ε > 0, there exists {wi (h˜ j,t , aj , y)}aj ∈Aj , y∈Y such that       wi (h˜ j,t )  Eπ (1 − δ)u∗i ai,t , fj (ωj,t ) + δwi h˜ j,t , aj,t , fj (ωj,t )  ai,t , s˜j (h˜ j,t ) for all ai,t ∈ Ai with equality if ai,t ∈ Ai and wi (h˜ j,t , aj,t , yt ) ∈ (w i (A), wi (A)) for all aj,t ∈ Aj and yt ∈ Y . Further, we have wi (h˜ j,t , aj,t , yt ) → Vi (h˜ j,t , aj,t , yt ) as ε → 0. We can proceed until period T + 1. We are left to show the continuation payoff wi (h˜ j,T +1 ) can be implemented. The following lemma shows that the set of belief-free equilibrium payoffs compatible with A is lower hemicontinuous with respect to private-monitoring perturbations. (The proof is in Appendix A.3.) Lemma 5. If Assumption 1 is satisfied and W (A) is properly self-A-generating, then there exists γ¯ > 0 such that, if 0 < γ < γ¯ , then there exists ε > 0 such that [w 1 (A) + γ , w1 (A) − γ ] × [w 1 (A) + γ , w1 (A) − γ ] is self-A-generating in any perturbed game that is ex-ante ε-close to (Y, ρ). For each h˜ 1,T +1 ∈ H˜ 1,T +1 and h˜ 2,T +1 ∈ H˜ 2,T +1 , since w1 (h˜ 2,T +1 ) ∈ (w 1 (A), w1 (A)) and w2 (h˜ 1,T +1 ) ∈ (w 2 (A), w2 (A)), by Lemma 5, for sufficiently small ε > 0, there exists a beliefh˜

h˜

free equilibrium (σ1 1,T +1 , σ2 2,T +1 ) of the perturbed game starting in period T + 1 that sustains (w1 (h˜ 2,T +1 ), w2 (h˜ 1,T +1 )). Note that, by the exchangeability of belief-free equilibria, we can assume without loss of generality that the choice of player 1’s strategy is independent of h˜ 2,T +1 , and that the choice of player 2’s strategy is independent of h˜ 1,T +1 . For each t  1, i ∈ I , and hi,t = (ait−1 , ωit−1 ) ∈ Hi,t , let

s˜i (fi (hi,t )) if t  T , si (hi,t ) = f (h ) σi i i,T +1 (T hi,t ) if t  T + 1, where T hi,t is the truncation of hi,t by removing the first T -period history. Then s = (s1 , s2 ) is a belief-free equilibrium of the perturbed game that is (0, T )-close to s˜ via f. 2 We can strengthen this robustness result by allowing for a larger class of perturbations. For example, the same set of conditions is sufficient to show the robustness of the belief-free equilibrium to private-monitoring perturbations where monitoring and payoff structures are nonstationary and slightly different from those in the original game. Conversely, we can ask what happens if we use more restrictive notions of proximity between the original equilibrium and private-monitoring strategies. For example, one can require that an equilibrium in the private-monitoring repeated game be (η, T )-close to the original equilibrium via exogenous interpretations with η = 0 or with T = ∞. Here, η = 0 means that each player’s action is exactly the same as her action in the original equilibrium, and T = ∞ means that the two strategy profiles prescribe similar actions uniformly over all histories. So far, we prove the robustness under η = 0 and T = ∞ in Proposition 1, and the current proof of Theorem 2 uses η = 0. We conjecture that Theorem 2 would hold for any η > 0 with T = ∞ under additional regularity conditions.

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7. Weak robustness In Sections 5 and 6, we showed that an equilibrium in public monitoring is robust “almost” if and only if the equilibrium is belief-free. A key in the proof of the non-robustness result (Theorem 1) is the contagion argument, which implicitly relies on the requirement in Definition 2 that in the perturbed game, each player always interprets her private signal as y or y and follows a continuation strategy similar to that in the original game via the interpretation. Then a natural question arises whether there is a non-belief-free equilibrium that is robust in a weaker sense, which allows the players to play completely different strategies from the original equilibrium strategies after some rare signal observations. The set of such signal observations works as a “buffer area” between the region where the signals are interpreted as y¯ and the region where the signals are interpreted as y, which impedes the contagion argument in the proof of Theorem 1. In this section, we introduce the notion of weak robustness, and show that block belief-free equilibria, a strictly more general class of equilibria than belief-free equilibria, are weakly robust to private-monitoring perturbations. As in Section 6, we focus on two-player games. We also maintain Assumption 1 so that by Lemma 4, player j can control player i’s continuation payoffs. 7.1. Definition of weak robustness To understand the role of “buffer areas” more specifically, consider a public-monitoring structure with two public signals Y = {y, ¯ y} and its global-game perturbation with ε as in Sections 3.1 and 5. Let Ω1ε = Ω2ε = R \ (−ε/2, ε/2), and call (−ε/2, ε/2) the buffer area. Then for any ωi ∈ Ωiε , player i is certain that if ωj ∈ Ωjε (not in the buffer area), then fj (ωj ) = fi (ωi ). The following lemma generalizes this observation to all perturbations. Lemma 6. If (Ω, π, f) is ex-ante ε-close to (Y, ρ), then there exist Ω1ε ⊆ Ω1 and Ω2ε ⊆ Ω2 such that for any i ∈ I and a ∈ A, √ 1. π(Ωiε | a)  1 − 4|A| ε; √ 2. π({ωj ∈ Ωj : fj (ωj ) = fi (ωi )} | a, ωi , Ωjε )  1 − 3|A| ε for all ωi ∈ Ωiε . The first property means that player i’s signal belongs to Ωiε with a high probability; the second means that, conditional that player i observes ωi ∈ Ωiε and player j ’s signal belongs to Ωjε , player j ’s interpretation is the same as player i’s interpretation with high probability. See Appendix A.4 for the proof. ε := (A × Ω ε )t−1 be the set of player i’s histories where player i has always observed Let Hi,t i i signals in Ωiε . We define a weak version of closeness between two strategy profiles as follows. Definition 7. A strategy profile s in the perturbed game is weakly (η, T )-close to s˜ if    si (hi,t ) − s˜i fi (hi,t )   η ε . for all i ∈ I , t  T , and hi,t ∈ Hi,t

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Unlike Definition 2, Definition 7 requires that each player interprets her private histories only with high probability.22 Signals outside Ωiε are similar to “uninterpretable” signals in Mailath and Morris [21], but we choose those signals endogenously by Lemma 6. Definition 8. A sequential equilibrium s˜ of a public-monitoring repeated game is weakly robust to private-monitoring perturbations if, for every η > 0 and T < ∞, there exists ε > 0 such that, for every private-monitoring perturbation (Ω, π, f ) that is ex-ante ε-close to (Y, ρ), there exists a Nash equilibrium s of the perturbed game that is weakly (η, T )-close to s˜ . 7.2. Block belief-free equilibria Given the definition of weak robustness, we want to find a class of equilibria that are weakly robust to private-monitoring perturbations. To get a better sense of this exercise, we pick an arbitrary equilibrium in public monitoring and try to construct an equilibrium in private monitoring that is weakly close to the original equilibrium. Note that after observing ωi ∈ Ωiε , player i is required to play a continuation strategy similar to the original one. So we want to make sure that player i’s incentive after ωi ∈ Ωiε is close to her incentive in public monitoring. A difficulty is that after observing ωi ∈ Ωiε , player i may believe with a non-negligible probability that ωj ∈ / Ωjε , and player j ’s continuation strategies after such ωj can be very different from the original ones. For example, in the global-game perturbation with Ω1ε = Ω2ε = R \ (−ε/2, ε/2), the probability that ωj ∈ (−ε/2, ε/2) conditional on ωi = ±ε/2 is as high as 1/2. One way to control player i’s incentive after such histories is to make player i indifferent among all of her actions if player j observes ωj ∈ / Ωjε . This indifference implies that player i ε should condition that ωj ∈ Ωj when she calculates the best response. We can do so by choosing player j ’s continuation strategies after ωj ∈ / Ωjε so that the differences in player i’s continuation payoffs offset the differences in player i’s stage-game payoffs. Recall belief-free equilibria in Section 6. Suppose that player j has two continuation strategies, one “generous” and the other “harsh”,23 and that both are optimal regardless of player i’s private history (the belief-free property). Then, without coordinating with player i, player j can control player i’s continuation payoffs by switching between these two strategies. Actually, we do not need this belief-free property every period; it is enough to have it sufficiently often. Below, we introduce “block belief-free equilibria”, which satisfy the belief-free property every L periods and some additional conditions.24 ,25 To define a block belief-free equilibrium, we regard a repeated game as an infinite repetition of L-period blocks. In each block, each player i takes either a generous strategy s˜iG or a harsh  G ˜ B ˜ ˜ ˜ ˜ strategy s˜iB such that s˜iG , s˜iB : L l=1 Hl → (Ai ) and s˜i (hl ) = s˜i (hl ) for all l  2 and hl ∈ Hl . 22 One could further weaken Definition 7 by allowing for endogenous interpretations as in Definition 2. We do not do so since we prove only a robustness result (Theorem 3). 23 Player j ’s strategy is generous if it gives a high payoff to player i following the equilibrium strategy. On the other hand, player j ’s strategy is harsh if it gives a low payoff to player i regardless of player i’s strategy. 24 It is straightforward to generalize our proof of robustness to the case where the distance between two belief-free periods is different for different blocks but bounded from above. 25 One of the referees suggested us to show the weak robustness of other equilibria such as strict equilibria with bounded recall. At this moment, we cannot prove or disprove this conjecture because without belief-free periods, player j may or may not be willing to control player i’s continuation payoffs after observing ωj ∈ / Ωjε .

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With abuse of notation, given a block, we use period l for the lth period in the block and h˜ l ∈ H˜ l (h˜ i,l ∈ H˜ i,l , respectively) for a public (private, respectively) history from the beginning of the block to the lth period. We say that player i is in state xi ∈ {G, B} in a block if she plays s˜ixi . The initial state is arbitrarily determined. Given the current state xi and history h˜ i,L+1 at the end of the current block, transition probabilities of states are given by p˜ iG : H˜ L+1 → (0, 1) and p˜ iB : H˜ i,L+1 → (0, 1). Here, p˜ iG (h˜ L+1 ) (p˜ iB (h˜ i,L+1 ), respectively) is the probability that the state in the next block is G given that player i is in state G (B, respectively) and observes public history h˜ L+1 (private history h˜ i,L+1 , respectively) in the current block. We impose the following optimality conditions: For each i ∈ I , 1. conditional on xj = G, x (a) in period l  2, for any public history h˜ l , s˜i i (h˜ l ) is a unique (hence pure) best response for player i to player j ’s continuation strategy (that is, taking s˜jG | h˜ l in the current block x

and transiting s˜jG or s˜jB according to transition probabilities p˜ j j in the future blocks), given that player i plays optimally from the next period on26 ; (b) in the initial period, both s˜iG (∅) and s˜iB (∅) are best responses for player i to player j ’s continuation strategy, given that player i plays optimally from the next period on; 2. conditional on xj = B, for any private history h˜ j,l of player j , player i is indifferent among all of her actions, given player j ’s continuation strategy. By Conditions 11-(a), 1-(b), and 2, player i’s value at the beginning of a block is determined x solely by player j ’s state xj ∈ {G, B} in the current block. Let vi j be player i’s value when player j ’s state is xj ∈ {G, B}. Definition 9. A block belief-free equilibrium with L < ∞ and  >0 specifies initial states ˜ xi ∈ {G, B}, L-period finitely-repeated game strategies s˜iG , s˜iB : L l=1 Hl → (Ai ) with G G B ˜ ˜ ˜ ˜ s˜i (hl ) = s˜i (hl ) for all l  2 and hl ∈ Hl , and transition probabilities p˜ i : H˜ L+1 → (0, 1) and p˜ iB : H˜ i,L+1 → (0, 1) that satisfy Conditions 1-(a), 1-(b), and 2 as well as viG − viB   for i ∈ I. In the above definition, the following technical conditions should be highlighted. (Details are discussed after Theorem 3 and in Appendix A.5.) First, we assume p˜ jG (h˜ L+1 ), p˜ jB (h˜ j,L+1 ) ∈ (0, 1), for the same reason as in Theorem 2. That is, the interiority of p˜ jG and p˜ jB allows player j to slightly adjust player i’s continuation payoffs in private monitoring. Second, s˜iG and s˜iB depend only on public histories and are the same except in the initial period, and p˜ iG depends only on public histories.27 We will explain in (iii) after Theorem 3 26 Since player i is indifferent between s˜ G and s˜ B at the beginning of each block, the condition “that player i plays i i x optimally from the next period on” is equivalent to the condition “that player i takes s˜i i | h˜ l+1 in the current block and G either s˜i or s˜iB in the future blocks”. 27 Note that s˜ G (∅) = s˜ B (∅) in the initial period, and hence the distributions of public histories in the current block i i can be different between the two states. This difference in the distributions can make s˜iG generous and s˜iB harsh, i.e., vjG > vjB . Note also that p˜ iB may depend on private histories. Thus, even though s˜iG and s˜iB depend only on public histories, if player i experiences xi = B in past blocks, then her strategy in the current block may depend on her private actions in these blocks through state transitions.

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why we want both s˜iG and p˜ iG to depend only on public histories. Given this, it follows from Condition 1-(a) that s˜iB must be the same as s˜iG after period 2, and hence depend only on public histories. Last, Condition 1-(a) requires that player i has strict incentives from period 2 to period L if player j is in state G, whereas Condition 2 requires that player i be indifferent among all of her actions if player j is in state B. See (i) after Theorem 3 for why we want indifference in state xj = B, and (iv) for why we want strict incentives from period 2 to period L in state xj = G. We also define the following property of block belief-free equilibria. Definition 10. A block belief-free equilibrium has one-period memory if, for each i ∈ I and x l  2, both s˜i i (h˜ l−1 , yl−1 ) and p˜ iG (h˜ l−1 , yl−1 , yl , . . . , yL ) − p˜ iG (h˜ l−1 , yl−1 , yl , . . . , yL ) are independent of h˜ l−1 for all yl−1 , yl , . . . , yL , and yl , . . . , yL . One-period memory means that player i’s action and the marginal effect of future histories on state transitions p˜ iG depend only on calendar time l (in each block) and the signal in the previous period. The latter part is equivalent to having an additively separable representation p˜ iG (h˜ L+1 ) = L G  l=2 p˜ i,l (yl−1 , yl ), i.e., zero cross derivatives with respect to yl and yl  with |l − l |  2. Note that any block belief-free equilibrium with L  2 has one-period memory. We show that any block belief-free equilibrium with one-period memory is weakly robust for a sufficiently large discount factor. Theorem 3. Fix a two-player stage game and a public-monitoring structure (Y, ρ) that satisfies Assumption 1. For any L < ∞ and  > 0, there exists δL, < 1 such that if δ > δL, , then any block belief-free equilibrium with L, , and one-period memory is weakly robust to privatemonitoring perturbations. Note that a block belief-free equilibrium with L = 1 corresponds to a belief-free equilibrium (with two states), whose robustness follows from Theorem 2. On the other hand, a block belieffree equilibrium with L  2 may not be belief-free, since it allows the players to coordinate their strategies within each block. Indeed, in Appendix A.7, we offer an example of a block belieffree equilibrium s˜ with L = 2 (and so with one-period memory) such that the players play pure actions in the initial period and that the induced coordination game G(˜s) satisfies the regularity condition. By Corollary 2 and Theorem 3, such s˜ highlights the importance of buffer areas, as it is not robust, but weakly robust. Let us comment on the relationship between our notion of block belief-free equilibria and “block constructions” in the existing literature on private-monitoring repeated games. Hörner and Olszewski [16] divide the repeated game into infinitely many L-period blocks, where the belief-free property holds at the beginning of each block. They introduce blocks to expand the set of equilibrium payoffs: Without the block construction, the set of belief-free equilibrium payoffs may be strictly smaller than the set of feasible and individually rational payoffs. Their equilibrium does not guarantee strict incentives against s˜jG after “erroneous histories”, and hence may not be block belief-free in our sense.28 Matsushima [24] and Fong, Gossner, Hörner, and Sannikov [10] use blocks to construct “review phases”, where players aggregate information over L periods and improve monitoring accuracy. Matsushima’s equilibrium is block belief-free in our 28 An exception is Hörner and Olszewski’s [16] equilibrium with L = 2, where our Theorem 3 applies.

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sense if we specify strategies in state B appropriately, but it has multi-period memory because of the review process. Nevertheless, we can apply Corollary 3 below, which does not require one-period memory. On the other hand, Fong–Gossner–Hörner–Sannikov’s equilibrium may not be block belief-free for a reason similar to Hörner–Olszewski’s equilibrium. We relegate the proof of Theorem 3 to Appendix A.5, and give only a sketch here. Sketch of proof. In each perturbed game, we want to construct an equilibrium that is weakly close to s˜ . We also see the private-monitoring repeated game as a repetition of L-period blocks, x and each player i has two possible states xi ∈ {G, B}. In each block, given xi , player i takes si i . Given the initial state being the same as in the public monitoring, it suffices to show the existence of an equilibrium where, in each block, given xi , player i takes sixi (hi,l ) = s˜ixi (fi (hi,l )) as long ε (that is, ω  ∈ Ω ε for all l   l − 1) and transits to the next state with a probability as hi,l ∈ Hi,l i,l i xi ε close to p˜ i (fi (hi,L+1 )) as long as hi,L+1 ∈ Hi,L+1 .29 ε so that By Lemma 4, we choose player j ’s transition probabilities after observing hj,l ∈ / Hj,l player i becomes indifferent among all action profiles given such hj,l . The condition δ > δL, ensures that switching between sjG and sjB at the end of a block offers incentives large enough to offset any difference in stage-game payoffs within the block. Since player i is indifferent among all of her actions given xj = B in public monitoring, by Lemma 4 and the interiority of pjB , we adjust player j ’s transition probabilities slightly to keep player i indifferent in private monitoring. Thus, together with the previous paragraph, we can ε without loss of generality when we discuss player i’s incenassume xj = G and hj,l ∈ Hj,l tives. Now consider player i’s incentives at hi,l ∈ Hi,l . By the one-period memory property, player i only needs to infer fj (ωj,l−1 ) in order to calculate her best responses. Lemma 6 guarantees that, ε (and so ω ε ε if hi,l ∈ Hi,l i,l−1 ∈ Ωi,l−1 ), then regardless of aj,l−1 , conditional on hj,l ∈ Hj,l (and ε so ωj,l−1 ∈ Ωj,l−1 ), player i believes that fj (ωj,l−1 ) = fi (ωi,l−1 ) with high probability. Since player i has strict incentives in public monitoring in state xj = G, it remains optimal to play sixi (hi,l ) = s˜ixi (fi (hi,l )) in private monitoring. (But in period 1 with xj = G, we adjust player j ’s continuation strategies to make both s˜iG (∅) and s˜iB (∅) optimal.) ε , we simply let player i take a best response to player / Hi,l In the remaining case where hi,l ∈ ε (and choose transition probabilities to make j ’s strategy conditional on xj = G and hj,l ∈ Hj,l player j indifferent among all action profiles, as in the second paragraph). 2 In the above sketch, various assumptions are used rather implicitly. Below, we discuss in details how we use these assumptions in our proof. (i) We assume that player i is indifferent among all of her actions given xj = B. To see the importance of this assumption, recall that we want the belief-free property in both public and private monitoring, i.e., we want player i to be indifferent between the generous and harsh strategies independently of her belief about xj ∈ {G, B}. In particular, we want player i’s equilibrium action ε in private monitoring to be optimal conditional on each x ∈ {G, B}. Player i’s / Hi,l after hi,l ∈ j ε in state x = G, however, depends on fine details of perturbations, and / Hi,l best response at hi,l ∈ j is “unlikely” to be a best response in state xj = B. (Note that Lemma 6 does not help us specify ε .) To make player i’s best response / Hi,l player i’s belief about player j ’s histories after hi,l ∈ 29 We write s˜ xi (f (h )) and p˜ G (f (h i i,l i i,L+1 )) with abuse of notations: They depend only on the public components of i i private histories fi (hi,l ) and fi (hi,L+1 ).

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against xj = G also optimal against xj = B, we specify player j ’s strategy in state xj = B so that player i is indifferent among all of her strategies in state xj = B.30 ,31 (ii) We assume one-period memory, and hence in each period l, player i only needs to make inferences about fj (ωj,l−1 ) given xj = G, and Lemma 6 guarantees that regardless of their action profile in period l − 1, player i believes that fj (ωj,l−1 ) = fi (ωi,l−1 ) with high probability conditional on ωj,l−1 ∈ Ωjε . In contrast, in the case of multi-period memory, all fj (ωj,1 ), . . . , fj (ωj,l−1 ) become relevant for player i’s incentives in period l. Then, by Lemma 6, when observing ωi,l  ∈ Ωiε in period l  , player i originally believes that fj (ωj,l  ) = fi (ωi,l  ) with high probability conditional on ωj,l  ∈ Ωjε . However, the conditional probability that fj (ωj,l  ) = fi (ωi,l  ) can be positive √ (Lemma 6 only guarantees that the probability is no more than 3|A| ε ), and player i’s signal observations ωi,l  +1 , . . . , ωi,l−1 depend on player j ’s actions in period l  + 1, . . . , l − 1, which in turn depend on fj (ωj,l  ). Hence, player i updates the belief about fj (ωj,l  ) from ωi,l  +1 , . . . , ωi,l−1 . If player i observes signals that strongly suggest that player j ’s actions in period l  + 1, . . . , l − 1 contradict to fj (ωj,l  ) = fi (ωi,l  ), then player i starts to believe that fj (ωj,l  ) = fi (ωi,l  ) with non-negligible probability. There are at least two ways to prevent such strong updating. One is to require that player i originally believe with certainty that fj (ωj,l  ) = fi (ωi,l  ) conditional on ωj,l  ∈ Ωjε . The globalgame perturbation we used in Sections 3.1 and 5 is one of such examples. Another is to require that likelihood ratios among player j ’s actions given player i’s signal be uniformly bounded. Putting them together, we can show the following. Corollary 3. Fix a two-player stage game and a public-monitoring structure (Y, ρ) that satisfies Assumption 1. For any L < ∞ and  > 0, there exists δL, < 1 such that if δ > δL, , then any block belief-free equilibrium with L and  is weakly robust to private-monitoring perturbations if we restrict the class of perturbations (Ω, π, f ) such that either (a) Condition 2 of Lemma 6 is strengthened to π({ωj ∈ Ωj : fj (ωj ) = fi (ωi )} | a, ωi , Ωjε ) = 1 for all a ∈ A and ωi ∈ Ωiε , or (b) likelihood ratios are uniformly bounded, i.e., there exists M < ∞ such that π(ωi | ai , aj )/π(ωi | ai , aj ) < M for all ωi ∈ Ωi , ai ∈ Ai , and aj , aj ∈ Aj .32 The proof of Corollary 3 is given in Appendix A.6. (iii) We assume that both s˜jG and p˜ jG depend only on public histories. Also, these properties are approximately maintained for (sjG , pjG ) in the perturbed game. If instead sjG or pjG depended heavily on player j ’s actions, player i would need to make inferences about these actions. Suppose that player j plays some mixed action αj ex-ante in the first period. Then, after taking ai and observing ωi , player i calculates the belief about player j ’s action by π(ωi | ai , aj )αj (aj ) Pr(aj | ai , αj , ωi ) =    a  ∈Aj π(ωi | ai , aj )αj (aj ) j

in private monitoring, which can be significantly different from the belief 30 This is a well-known problem in private monitoring when there arise “erroneous histories” such as in Hörner and

Olszewski [16] and in Fong, Gossner, Hörner and Sannikov [10], both of which adapt the same treatment as ours. 31 Symmetrically, we could assume that player i has strict incentives in state x = B, and are indifferent among all j strategies in state xj = G. 32 A public-monitoring structure (Y, ρ) admits private-monitoring perturbations with uniformly bounded likelihood ratios only if (Y, ρ) has full support.

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ρ(y | ai , aj )αj (aj )   a  ∈Aj ρ(y | ai , aj )αj (aj )

Pr(aj | ai , αj , y) = 

j

calculated in public monitoring even if fi (ωi ) = y and ωi ∈ Ωiε .33 (iv) We assume that player i has strict incentives to play equilibrium actions from period 2 ε , player i at h ∈ H ε to period L when xj = G. By Lemma 6, conditional on xj = G and Hj,l i,l i,l ε differently from may place a small probability on the event that player j interprets hj,l ∈ Hj,l fi (hi,l ), but with strict incentives, such a small probability does not affect the optimality of player i’s strategy. Without strict incentives, fine details of beliefs (and so player i’s histories) would affect the set of player i’s optimal actions, and player j could not control player i’s incentives well.34 8. Conclusion We analyzed the robustness of equilibrium strategies with respect to private-monitoring perturbations. We showed that an equilibrium is robust “almost” if and only if it is a belief-free equilibrium. We also discussed a weakened notion of robustness and its implications. To conclude the paper, we want to emphasize that different classes of perturbations give rise to different notions of robustness and different classes of robust equilibria. For example, boundedrecall equilibria are robust in the sense of Mailath and Morris [20,21], but not necessarily robust in our sense. In turn, belief-free equilibria are robust in our sense, but not necessarily robust to other types of perturbations. Indeed, applying the purification exercise à la Harsanyi [15] to the repeated prisoners’ dilemma, Bhaskar, Mailath, and Morris [3] argue that the belief-free equilibrium constructed by Ely and Välimäki [9] is not robust to smooth i.i.d. payoff shocks.35 We need further research to investigate the relationship between various classes of perturbations and the corresponding classes of robust equilibria. Appendix A A.1. Proof of Lemma 3 √ Let ζ := (1 − p1 − p2 )/4√> 0. Let C1 := 2 − 2(p1 + 2ζ ) ∈ [1, 2) and C2 := 1 if p1  p2 , and C1 := 1 and C2 := 2 − 2(p2 + 2ζ ) ∈ [1, 2) if p1 > p2 . Let D := max{C1 , C2 } + 1. Let λ denote the Lebesgue measure on R. The following lemmas are useful. Lemma 7. There exists d > 0 such that the following is true. For every ε > 0 and x ∈ [−B + Dε, B − Dε], we have    π ε ωj,1 ∈ [x − Cj ε, x + Cj ε]  a1 , ωi,1  pi + ζ for all i ∈ I , a1 ∈ A and ωi,1 ∈ [x − (Ci + d)ε, x + Ci ε]. 33 In contrast, s˜ B does not need to be a public strategy. Given x = B, we assume that player i is indifferent among j j

all of her actions given any private history of player j in public monitoring, and we maintain this indifference in private monitoring. Therefore, player i does not need to make inferences about player j ’s private histories in state xj = B. 34 In contrast, we do not need to require strict incentives in period 1 or when x = B. In both cases, given player j ’s j state, player i’s best responses are independent of player j ’s private history in public monitoring. Thus, as we did in Theorem 2 for belief-free equilibria, we can easily maintain player i’s incentives in private monitoring. 35 See also Bhaskar [2], Bhaskar, Mailath, and Morris [4], and P˛eski [26,27]. Chassang and Takahashi [6] consider payoff shocks that are correlated among players but independent across periods.

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Proof. Take d > 0 sufficiently small. Then, noting that, for ωi,1 ∈ [−B + ε/2, B − ε/2], the probability density of ωj,1 with respect to the Lebesgue measure conditional on a1 and ωi,1 is given by  dπ ε (ωj,1 | a1 , ωi,1 ) 1/ε − |ωj,1 − ωi,1 |/ε 2 if ωj,1 ∈ [ωi,1 − ε, ωi,1 + ε], = dλ 0 otherwise, a simple computation yields the result.

2

Lemma 8. There exist ε, K > 0 such that, for every η > 0 and ε < ε, if g1,1 satisfies    π ε ω1 ∈ R2 : g1,1 (ω1,1 | a1,1 ) = f1 (ω1,1 )  a1,1 , a2,1  1 − η

(3)

for all (a1,1 , a2,1 ) ∈ A, then there exists x0 ∈ [Dε, B − Dε] such that   λ ω1,1 ∈ [x0 − C1 ε, x0 + C1 ε]: g1,1 (ω1,1 | a1,1 ) = y < Kηε

(4)

for all a1,1 ∈ A1 . Proof. Fix any K > 2C1 |A1 |/B. Suppose that there exist η > 0, sufficiently small ε > 0, and g1,1 such that (3) holds, but (4) does not. By (3), we have   λ ω1,1 ∈ [Dε, B − Dε]: g1,1 (ω1,1 | a1,1 ) = y for some a1,1 ∈ A1     λ ω1,1 ∈ [Dε, B − Dε]: g1,1 (ω1,1 | a1,1 ) = y a1,1 ∈A1

=



   π ε ω1,1 ∈ [Dε, B − Dε]: g1,1 (ω1,1 | a1,1 ) = y  a1,1 , a2,1

a1,1 ∈A1

 |A1 |η

(5)

for any a2,1 ∈ A2 . Let Mε be the largest integer smaller than (B − 2Dε)/(2C1 ε). Then there ε exists {xm }M m=1 such that [x1 − C1 ε, x1 + C1 ε], . . . , [xMε − C1 ε, xMε + C1 ε] are disjoint and included in [Dε, B − Dε]. Since no xm satisfies (4), we have   λ ω1,1 ∈ [Dε, B − Dε]: g1,1 (ω1,1 | a1,1 ) = y for some a1,1 ∈ A1 

Mε    λ ω1,1 ∈ [xm − C1 ε, xm + C1 ε]: g1,1 (ω1,1 | a1,1 ) = y for some a1,1 ∈ A1 m=1

 Mε Kηε.

(6)

Since Mε ε converges to B/(2C1 ) as ε goes to 0, (5) and (6) contradict to each other for sufficiently small ε. 2 Proof of Lemma 3. We will show that Lemma 3 holds for d > 0 in Lemma 7, ε > 0 in Lemma 8, and some η > 0. Since Lemma 3 (ii) is a corollary of Lemma 7, what remains to show is Lemma 3 (i). Pick K > 0 as in Lemma 8. Suppose that η > 0, ε < ε, and g1,1 satisfies    π ε ω1 ∈ R2 : g1,1 (ω1,1 | a1,1 ) = f1 (ω1,1 )  a1,1 , a2,1  1 − η

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for all (a1,1 , a2,1 ) ∈ A. Pick x0 ∈ [Dε, B − Dε] as in Lemma 8. For every ω2,1 ∈ [x0 − C2 ε, x0 + C2 ε], note that the probability density of ω1,1 with respect to the Lebesgue measure conditional on a1 , {ω1,1 : ω1,1 ∈ [x0 − C1 ε, x0 + C1 ε]}, and ω2,1 is bounded from above by   2 1 E max 1, =: . 2 ε ε (1 + C1 − C2 ) Thus, by Lemma 8, we have      π ε ω1,1 : g1,1 (ω1,1 | a1,1 ) = y  a1,1 , a2,1 , ω1,1 : ω1,1 ∈ [x0 − C1 ε, x0 + C1 ε] , ω2,1  1 − EKη for all (a1,1 , a2,1 ) ∈ A. By Lemma 7, we have    π ε ω1,1 : g1,1 (ω1,1 | a1,1 ) = y  a1,1 , a2,1 , ω2,1     π ε ω1,1 ∈ [x0 − C1 ε, x0 + C1 ε]  a1,1 , a2,1 , ω2,1   × π ε ω1,1 : g1,1 (ω1,1 | a1,1 ) = y  a1,1 , a2,1 ,    ω1,1 : ω1,1 ∈ [x0 − C1 ε, x0 + C1 ε] , ω2,1  (p1 + ζ )(1 − EKη) > p1 for all (a1,1 , a2,1 ) ∈ A and sufficiently small η.

2

A.2. Proof of Lemma 4 ˜ i (aj ) denote the |Ai | × |Y | matrix that stacks row vectors Fix aj ∈ Aj arbitrarily. Let Q (ρ(· | ai , aj ))ai ∈Ai . We see w˜ i (aj , ·) as a |Y |-dimensional column vector and V˜i (·, aj ) as an |Ai |-dimensional column vector. Define Qi (aj ), wi (aj , ·) and Vi (·, aj ) similarly, but with ρ(y | ai , aj ) replaced with π({ωj ∈ Ωj : fj (ωj ) = y} | ai , aj ). ˜ i (aj )−1 such that Q˜ i (aj )Q˜ i (aj )−1 By Assumption 1, there exists a |Y | × |Ai | matrix Q is the identity matrix of size |Ai |. Also, for sufficiently small ε, there exists a |Y | × |Ai | matrix Qi (aj )−1 such that Qi (aj )Qi (aj )−1 is the identity matrix of size |Ai | and ˜ i (aj )−1 for any aj ∈ Aj . Then w˜ i (aj , ·) = Q ˜ i (aj )−1 V˜i (·, aj ) and limε→0 Qi (aj )−1 = Q wi (aj , ·) = Qi (aj )−1 Vi (·, aj ) satisfy the desired properties. A.3. Proof of Lemma 5 Since W (A) is properly self-A-generating, take γ¯ > 0 such that, for every i ∈ I , w i (A) is Ai -enforced by some α j ∈ (Aj ) and wi : Aj × Y → [w i (A), wi (A) − (1 + 1/δ)γ¯ ], and wi (A) is Ai -enforced by some α j ∈ (Aj ) and wi : Aj × Y → [w i (A) + (1 + 1/δ)γ¯ , wi (A)]. For any γ ∈ (0, γ¯ ], by translating continuation payoffs by ±γ /δ, we have that every vi ∈ [w i (A) + γ , wi (A) − γ ] is Ai -enforced by some αj ∈ (Aj ) and wi : Aj × Y → [w i (A) + γ /δ, wi (A) − γ /δ]. By Lemma 4, we can modify wi into wi such that vi is Ai -enforced by αj and wi in the perturbed game and wi (aj , y) → wi (aj , y) as ε → 0. Thus, for sufficiently small ε, [w 1 (A) + γ , w1 (A) − γ ] × [w2 (A) + γ , w2 (A) − γ ] is self-A-generating in the perturbed game.

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A.4. Proof of Lemma 6 Let α ∗ be the uniform distribution over A, which takes each pure action profile with probability 1/|A|. For each player i ∈ I , let  √     Ei = ωi ∈ Ωi : π ωj ∈ Ωj : fj (ωj ) = fi (ωi )  α ∗ , ωi  ε . √ Since (Ω, π, f) is ex-ante ε-close to (Y, ρ), we have π(Ei | α ∗ )  1 − ε for i ∈ I . (Other∗ wise, the ex-ante probability √ of fi (ωi ) = fj (ωj ) with respect to π(· | α ) exceeds ε.) Hence ∗ π(E1 × E2 | α )  1 − 2 ε. Let Ω1ε × Ω2ε ⊆ E1 × E2 be the event that the players have common 1/3-belief about E1 × E2 with respect to π(· | α ∗ ). By Kajii and Morris [18, Proposition 4.2], we have           √ π Ωiε  α ∗  π Ω1ε × Ω2ε  α ∗  1 − 2 1 − π E1 × E2  α ∗  1 − 4 ε, √ which implies π(Ωiε | a)  1 − 4|A| ε for any a ∈ A.36 Also, since π(Ωjε | α ∗ , ωi )  1/3 for ε any ωi ∈ Ωi , we have    π ωj ∈ Ωj : fj (ωj ) = fi (ωi )  α ∗ , ωi , Ωjε 

√ π({ωj ∈ Ωj : fj (ωj ) = fi (ωi )} | α ∗ , ωi ) 3 ε ε ∗ π(Ωj | α , ωi )

√ for any ωi ∈ Ωiε , which implies π({ωj ∈ Ωj : fj (ωj ) = fi (ωi )} | a, ωi , Ωjε )  3|A| ε for any a ∈ A and ωi ∈ Ωiε . A.5. Proof of Theorem 3 Fix (Y, ρ), L, and . For each i, by Lemma 4 and subtraction or addition of large constants, we construct ϕ˜i− : Aj × Y → (−∞, 0) and ϕ˜i+ : Aj × Y → (0, ∞) such that Eρ [u∗i (ai , y) + ϕ˜i− (aj , y) | ai , aj ] and Eρ [u∗i (ai , y) + ϕ˜i+ (aj , y) | ai , aj ] are both independent of (ai , aj ) ∈ A. Also, let ϕi− , ϕi+ : Aj × Ωj → R be such that Eπ [u∗i (ai , fi (ωi )) + ϕi− (aj , ωj ) | ai , aj ] and Eπ [u∗i (ai , fi (ωi )) + ϕi+ (aj , ωj ) | ai , aj ] are both independent of (ai , aj ) ∈ A, and limε→0 (ϕi− (aj , ωj ) − ϕ˜ i− (aj , fj (ωj ))) = limε→0 (ϕi+ (aj , ωj ) − ϕ˜ i+ (aj , fj (ωj ))) = 0 for any aj ∈ Aj and ωj ∈ Ωj .37 Let |ϕ| ˜ := maxi,aj ,y max(−ϕ˜i− (aj , y), ϕ˜i+ (aj , y)). We pick δL, < 1 such that δ > δL,

⇒

˜ 1 − δ L−1 |ϕ| < 1. L−1  δ

(7)

Take any δ > δL, and any s˜ that is a block belief-free equilibrium with L, , and one-period memory. We will construct an equilibrium in private monitoring that is weakly (η, T )-close to s˜ for any small η and large T . 36 Although Kajii and Morris [18] assume countable state spaces, their proof easily extends to uncountable ones. 37 In fact, we can assume without loss of generality that ϕ − (a , ω ) and ϕ + (a , ω ) depend on ω only through j j j j j i i fj (ωj ).

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A.5.1. Reward functions in public monitoring Following Fudenberg and Levine [12] and Hörner and Olszewski [16], we simplify our exposition by rewriting player i’s payoffs in terms of the movement of her continuation payoffs x x θ˜i j (h˜ j,L+1 ) rather than player j ’s transition probabilities p˜ j j (h˜ j,L+1 ):   δL  1 − p˜ jG (h˜ L+1 ) viG − viB , θ˜iG (h˜ L+1 ) ≡ − 1−δ   δL B ˜ p˜ j (hj,L+1 ) viG − viB . θ˜iB (h˜ j,L+1 ) ≡ 1−δ To see the equivalence between the two notations, recall the definition of a block belief-free equilibrium: For all i ∈ I , Conditions 1-(a), 1-(b), and 2 of Section 7.2 are satisfied; player i’s x value conditional on xj is equal to vi j ; p˜ jG and p˜ jB take values in (0, 1). We can rewrite Conditions 1-(a) and 1-(b) as follows: 1-(a) for each period l  2, for any history h˜ l , s˜ixi (h˜ l ) is a unique maximizer for    v˜ G (h˜ l ) ≡ max Eρ (1 − δ)u∗ (ai,l , yl ) + δ v˜ G (h˜ l+1 )  ai,l , s˜ G (h˜ l ), h˜ l i

i

ai,l

i

j



L    1−δ   = l−1 max Eρ δ l −1 u∗i (ai,l  , yl  ) + θ˜iG (h˜ L+1 )  ai,l , s˜ixi , s˜jG , h˜ l ai,l δ 



l =l

+ δ L−l+1 viG ,

(8)

where v˜iG (h˜ l ) is player i’s normalized value in period l given xj = G, v˜iG (h˜ L+1 ) ≡ δ −L (1 − δ)θ˜iG (h˜ L+1 ) + viG , and (ai,l , s˜ixi ) is the strategy that plays ai,l in period l and follows s˜ixi elsewhere38 ; 1-(b) any action in supp s˜iG (∅) ∪ supp s˜iB (∅) is a maximizer for (8) with l = 1. Similarly, we can rewrite Condition 2 as follows: For any private history h˜ j,l of player j , player i’s value given h˜ j,l ,    v˜iB (h˜ j,l ) ≡ Eρ (1 − δ)u∗i (ai,l , yl ) + δ v˜iB (h˜ j,l+1 )  ai,l , s˜jB (h˜ j,l ), h˜ j,l  L     1−δ  l −1 ∗ B B δ ui (ai,l  , yl  ) + θ˜i (h˜ j,L+1 )  s˜i , s˜j , h˜ j,l = l−1 Eρ δ  l =l

+ δ L−l+1 viB ,

(9)

is independent of ai,l and s˜i , where v˜iB (h˜ j,L+1 ) ≡ δ −L (1 − δ)θ˜iB (h˜ j,L+1 ) + viB . 38 The second and third lines of (8) follow from using the first line of (8) recursively and the definition of θ˜ G : i

   Eρ (1 − δ)u∗i (ai,l , yl ) + δ v˜iG (h˜ l+1 )  ai,l , s˜jG (h˜ l ), h˜ l    x  = Eρ (1 − δ)u∗i (ai,l , yl ) + (1 − δ)δu∗i (ai,l+1 , yl+1 ) + δ 2 v˜iG (h˜ l+2 )  ai,l , s˜i i , s˜jG , h˜ l   L    −l ∗ xi  G ˜  l L−l+1 G ˜ δ u (ai,l  , yl  ) + δ v˜ (hL+1 )  ai,l , s˜ , s˜ , hl = · · · = Eρ (1 − δ) 1−δ = l−1 Eρ δ

 L  l  =l

l  =l

i

i

i

j

   x   δ l −1 u∗i (ai,l  , yl  ) + θ˜iG (h˜ L+1 )  ai,l , s˜i i , s˜jG , h˜ l + δ L−l+1 viG .

Hereafter, we will sometimes use objective functions of the form “

L

l  =l δ

l  −1 (payoff in period l  ) + reward”.

T. Sugaya, S. Takahashi / Journal of Economic Theory 148 (2013) 1891–1928 x

1921

x

Also, using vi j = v˜i j (∅), we have  L    1−δ xj xj  xi xj l−1 ∗ ˜ ˜ vi = Eρ δ ui (ai,l , yl ) + θi (hj,L+1 )  s˜i , s˜j 1 − δL

(10)

l=1

for xi , xj ∈ {G, B}.39 Thus a block belief-free equilibrium is defined as follows: For all i, 1. conditional on xj = G, (a) in period l  2, for any h˜ l , s˜ixi (h˜ l ) is a unique maximizer for (8); (b) in period l = 1, any action in supp s˜iG (∅) ∪ supp s˜iB (∅) is a maximizer for (8); 2. conditional on xj = B, for any h˜ j,l , player i’s value (9) is independent of ai,l and s˜i ; 3. (10) is satisfied for xi , xj ∈ {G, B}; 4. −θ˜iG (h˜ L+1 ), θ˜iB (h˜ j,L+1 ) ∈ (0, δ L (viG − viB )/(1 − δ)) for any h˜ L+1 ∈ H˜ L+1 and h˜ j,L+1 ∈ H˜ j,L+1 . A.5.2. Reward functions in private monitoring Given a block belief-free equilibrium with (˜siG , s˜iB , p˜ iG , p˜ iB ) and values viG and viB , let θ˜iG and θ˜iB be the reward functions defined in Appendix A.5.1. For any ε  > 0, there exists ε > 0 such  G B that we can construct siG , siB : L l=1 Hi,l → (Ai ), θi , θi : Hj,L+1 → R in private monitoring such that x

1. conditional on xj = G, for any hi,l , si i (hi,l ) is a maximizer for      viG (hi,l ) ≡ max Eπ (1 − δ)u∗i ai,l , fi (ωi,l ) + δviG (hi,l+1 )  ai,l , sjG , hi,l ai,l  L  l  −1 u∗ (a  , f (ω  ))   1−δ i i,l  xi  G l  =l δ i i,l = l−1 max Eπ  ai,l , si , sj , hi,l ai,l δ + θiG (hj,L+1 ) + δ L−l+1 viG ,

(11)

where viG (hi,L+1 ) ≡ Eπ [δ −L (1 − δ)θiG (hj,L+1 ) + viG | sjG , hi,L+1 ]40 ; 2. conditional on xj = B, for any hj,l , player i’s value,      viB (hj,l ) ≡ Eπ (1 − δ)u∗i ai,l , fi (ωi,l ) + δviB (hj,l+1 )  ai,l , sjB (hj,l ), hj,l  L       1−δ  l −1 ∗ B B δ ui ai,l  , fi (ωi,l  ) + θi (hj,L+1 )  si , sj , hj,l = l−1 Eπ δ  l =l L−l+1 B vi , +δ

(12)

is independent of ai,l and si , where viB (hj,L+1 ) ≡ δ −L (1 − δ)θiB (hj,L+1 ) + viB ; 39 For x = G, θ˜ G depends only on the public component of h˜ j j,L+1 . i 40 Note that player i calculates the expectation in (11) based on her belief about player j ’s action s G (h ), which, in j,l j turn, is based on her belief about hj,l conditional that player j follows sjG and player i observes hi,l up to period l − 1. Similarly, the calculation of viG (hi,L+1 ) involves player i’s belief about hj,L+1 conditional that player j follows sjG and player i observes hi,L+1 in the current block.

1922

3. 4. 5. 6.

T. Sugaya, S. Takahashi / Journal of Economic Theory 148 (2013) 1891–1928 x

x

vi j (∅) = vi j for any xj ∈ {G, B}; −θiG (hj,L+1 ), θiB (hj,L+1 ) ∈ [0, δ L (viG − viB )/(1 − δ)] for any hj,L+1 ∈ Hj,L+1 ; x x ε ; si i (hi,l ) = s˜i i (fi (hi,l )) for any xi ∈ {G, B} and hi,l ∈ Hi,l xj xj ε . |θi (hj,L+1 ) − θ˜i (fj (hj,L+1 ))| < ε  for any xj ∈ {G, B} and hj,L+1 ∈ Hj,L+1

To see that Conditions 1–6 are sufficient for the weak robustness of the block belief-free equilibrium s˜ , consider the strategy profile s in the perturbed game such that in each block, given state xi , player i plays sixi and transits to the next state according pjG (hj,L+1 ) ≡ 1 + pjB (hj,L+1 ) ≡

1 − δ θiG (hj,L+1 ) , δ L viG − viB

1 − δ θiB (hj,L+1 ) . δ L viG − viB x

By Condition 4, s is well defined. By Conditions 1–3 and the definition of (pjG , pjB ), vi j is indeed player i’s value in the infinite-horizon game given that player j is currently in state xj , and conditional on each xj , both siG and siB are best responses for player i to player j ’s continuation strategy. Thus s is an equilibrium of the perturbed game. Further, Conditions 5, 6, and the definition of (pjG , pjB ) imply that, for any η > 0 and T < ∞, there exists ε > 0 such that s is weakly (η, T )-close to s˜ . Therefore, we are left to construct (siG , siB , θiG , θiB ) satisfying Conditions 1–6. A.5.3. Construction of (siG , siB , tθiG , θiB ) and verification ε , but We define player i’s strategies by sixi (hi,l ) = s˜ixi (fi (hi,l )) for xi ∈ {G, B} and hi,l ∈ Hi,l xi ε / Hi,l for a while. postpone the construction of si (hi,l ) for hi,l ∈ For each hj,L , let lj∗ be the first period l  L − 1 where player j observes a signal outside Ωjε . If there is no such period, then let lj∗ = L. (Note that lj∗ is independent of ωj,L .) We define reward functions by L     θiG (hj,L+1 ) = 1{lj∗ =L} θ˜iG fj (hj,L+1 ) + 1{lj∗ L−1} δ l −1 ϕi− (aj,l  , ωj,l  ) l  =2

+ ΦiG (aj,1 , ωj,1 ), L     θiB (hj,L+1 ) = 1{lj∗ =L} θ˜iB fj (hj,L+1 ) + 1{lj∗ L−1} δ l −1 ϕi+ (aj,l  , ωj,l  ) l  =2

l∗

+

j 



δ l −1 ΦiB [hj,l  ](aj,l  , ωj,l  ),

l  =1

where ΦiG and ΦiB are small adjustments that will be determined later. (The indicator 1{P } for statement P is equal to 1 if P is true and 0 otherwise.) Plugging these reward functions to (11) and (12), player i’s incentives are given as follows: 1. Conditional on xj = G,

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(a) for any hi,l with l  2, player i chooses ai,l to maximize a weighted average of ⎤ ⎡ L l  −1 ∗ ui (ai,l  , fi (ωi,l  )) l  =l δ  ⎥ ⎢  x  + 1{lj∗ =L} θ˜iG (fj (hj,L+1 )) Eπ ⎢  ai,l , si i , sjG , hi,l , lj∗  l ⎥ ⎦ ⎣ L  −1 − l ∗   + 1{lj L−1} l  =2 δ ϕi (aj,l , ωj,l ) (13) (note that

ΦiG (aj,1 , ωj,1 )

is sunk) and

 L

l  −1 (u∗ (a  , f (ω  )) i i,l l  =l δ i i,l − + ϕi (aj,l  , ωj,l  ))

Eπ

   xi  G  ai,l , si , sj , hi,l , hj,l

(14)

for various hj,l with lj∗  l − 1 (note that ΦiG (aj,1 , ωj,1 ) and ϕi− (aj,l  , ωj,l  ) for l   l − 1 are sunk); (b) in the initial period, player i chooses ai,1 to maximize ⎡ L ⎤ l  −1 u∗ (a  , f (ω  )) + 1 ∗ ˜G i i,l {lj =L} θi (fj (hj,L+1 )) l  =1 δ i i,l  ⎢ ⎥ L  x  l  −1 ϕ − (a  , ω  ) Eπ ⎢  ai,1 , si i , sjG ⎥ j,l j,l ⎣ + 1{lj∗ L−1} l  =2 δ ⎦. i + ΦiG (aj,1 , ωj,1 ) (15) 2. Conditional on xj = B, for any hj,l , player i’s value, which is equal to ⎡ L l  −1 ∗ ⎤ ui (ai,l  , fi (ωi,l  )) + 1{lj∗ =L} θ˜iB (fj (hj,L+1 )) l  =l δ  ⎢ ⎥ L  +  ⎢ ⎥ Eπ ⎢ + 1{lj∗ L−1} l  =2 δ l −1 ϕi (aj,l  , ωj,l  )  si , sjB , hj,l ⎥ ⎣ ⎦ l ∗  + lj =l δ l −1 ΦiB [hj,l  ](aj,l  , ωj,l  )

(16)

if lj∗  l and  Eπ

L  l  =l

δ

l  −1

 ∗    ui ai,l  , fi (ωi,l  ) + ϕi+ (aj,l  , ωj,l  )  si , sjB , hj,l

 (17)

if lj∗  l − 1 (with l  2), is independent of si . Now we verify player i’s incentives. ε with l  2 Note first that, by the definition of ϕ − , (14) is Conditional on xj = G and hi,l ∈ Hi,l i independent of ai,l . Hence we can assume without loss of generality that player i maximizes (13). Now compare (13) in private monitoring and (8) in public monitoring with h˜ l being the public component of fi (hi,l ). Then we can find the following differences in player i’s incentives.

1. Given ωi,l−1 ∈ Ωiε , player i believes with small probability that player j observed fj (ωj,l−1 ) = fi (ωi,l−1 ). 2. With small probability, player i will newly observe ωi,l  ∈ / Ωiε in future l   l. (In this case, her continuation strategies are yet to be determined.)

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3. With small probability, player j will newly observe ωj,l  ∈ / Ωjε in future l   l.

4. Even the same action profile a may induce distributions π({ωi ∈ fi−1 (y)} | a) and π({ωj ∈ fj−1 (y)} | a) that are slightly different from the original distribution ρ(y | a). By ex-ante closeness between (Y, ρ) and (Ω, π, f) and Lemma 6, none of these differences is significant. Recall that s˜jG and p˜ jG (and hence θ˜iG ) have one-period memory. Thus, without these differences, even if player i believes fj (hj,l−1 ) = fi (hi,l−1 ) with significant probability, player i’s payoff differences between two actions in (13) are similar to those in (8). Since player i has strict incentives to play s˜ixi (h˜ l ) in the original game, sixi (hi,l ) = s˜ixi (fi (hi,l )) is optimal in the perturbed game. x

ε with l  2 In this case, we choose s i (h ) as an action / Hi,l Conditional on xj = G and hi,l ∈ i,l i that maximizes (11) (or (13), equivalently). Such a maximizer is well defined although we have ε because, by the definition of ϕ − , player / Hj,l not yet specified player j ’s strategies sjG after hj,l ∈ i ε . / Hj,l i’s value is independent of how player j plays after hj,l ∈

Conditional on xj = G and period 1 Compare (15) in private monitoring and (8) in public monitoring with l = 1. By Lemma 6, we have  L  l  −1 u∗ (a  , f (ω  )) + 1 ∗ ˜G i i,l {lj =L} θi (fj (hj,L+1 ))   l  =1 δ i i,l xi  G Eπ  ai,1 , si , sj  l  −1 ϕ − (a  , ω  ) + 1{lj∗ L−1} L j,l j,l l  =2 δ i  L      G  x l −1 ∗ G i → Eρ δ ui (ai,l  , yl  ) + θ˜ (h˜ L+1 )  ai,1 , s˜ , s˜ i

l  =1

i

j

as ε → 0. Thus, by Lemma 4, there exists ΦiG : Aj × Ωj → R such that (15) is exactly equal to the maximand in the second line of (8) for any ai,1 ∈ Ai , and limε→0 ΦiG (aj , ωj ) = 0 for any aj ∈ Aj and ωj ∈ Ωj . This implies that any action in supp s˜iG (∅) ∪ supp s˜iB (∅) is a maximizer for (15), and that viG (∅) = viG . Conditional on xj = B and hj,l ∈ Hj,l Similarly to the case of xj = G, by the definition of ϕi+ , ε (i.e., l ∗  l). (17) is independent of si . Hence, we only analyze hj,l ∈ Hj,l j B Suppose that we have constructed Φi [hj,l  ] for any l   l + 1 and hj,l  ∈ Hj,l  such that player i’s value in period l + 1 is independent of si , and we construct ΦiB [hj,l ] with hj,l ∈ Hj,l as follows. By Lemma 6 and the induction hypothesis, we have ⎡ L l  −1 ∗ ⎤ ui (ai,l  , fi (ωi,l  )) + 1{lj∗ =L} θ˜iB (fj (hj,L+1 )) l  =l δ ⎢ ⎥  L ⎢ ⎥  l  −1 ϕ + (a  , ω  ) Eπ ⎢ + 1{lj∗ L−1} l  =2 δ j,l j,l  si , sjB , hj,l ⎥ i ⎣ ⎦ lj∗  −1 B l + l  =l+1 δ Φi [hj,l  ](aj,l  , ωj,l  )   L     δ l −1 u∗i (ai,l  , yl  ) + θ˜iB (h˜ j,L+1 )  s˜i , s˜jB , fj (hj,l ) → Eρ l  =l

independently of si and s˜i . Thus, by Lemma 4, there exists ΦiB [hj,l ] : Aj × Ωj → R such that (16) is exactly equal to what is in the bracket in the second line of (9) independently of si

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and s˜i , and that limε→0 ΦiB [hj,l ](aj , ωj ) = 0 for any aj ∈ Aj and ωj ∈ Ωj .41 This implies that viB (∅) = viB . Thus, we have constructed (siG , siB , θiG , θiB ). It is easy to verify Conditions 1–6 in Appendix A.5.2. In particular, to verify Condition 4, we consider the following two cases. In the case of x x lj∗ = L, since θi j (hj,L+1 ) → θ˜i j (fj (hj,L+1 )) as ε → 0 and −θ˜iG (fj (hL+1 )), θ˜iB (fj (hj,L+1 )) ∈ (0, δ L (viG − viB )/(1 − δ)), we have −θiG (hj,L+1 ), θiB (hj,L+1 ) ∈ (0, δ L (viG − viB )/(1 − δ)) for sufficiently small ε. In the case of lj∗  L − 1, by (7), we have −θiG (hj,L+1 ), θiB (hj,L+1 ) ∈ (0, δ L (viG − viB )/(1 − δ)) for sufficiently small ε. A.6. Proof of Corollary 3 The proof of Corollary 3 differs from that of Theorem 3 only in the case of xj = G and ε with l  2 in Appendix A.5.3. Note that s˜ G and θ˜ G depend only on public histories, hi,l ∈ Hi,l j i and as explained in Section 7.2, if either (a) or (b) in Corollary 3 is satisfied, then player i in period l believes that fj (ωj,l  ) = fi (ωi,l  ) for all l   l − 1 with high probability conditional on lj∗  l. Again, since player i has strict incentives to play s˜ixi (h˜ l ) in the original game, sixi (hi,l ) = s˜ixi (fi (hi,l )) is optimal in the perturbed game. A.7. An example of block belief-free equilibria Here, we offer an example of a block belief-free equilibrium in the two-player prisoners’ dilemma with two public signals, as in Section 3.1. We assume p 1+r  , (18) p−q q −r which are satisfied in a nonempty open set of parameter values, say, in a neighborhood of (p, q, r) = (.9, .3, 0). Fix η > 0 sufficiently small and  = (1 − r)/[2(q − r)] − η > 0. For any sufficiently large δ, we will construct a class of block belief-free equilibria with L = 2 and . Thus Theorem 3 applies to these equilibria and guarantees weak robustness. On the other hand, as we will see, these equilibria prescribe pure actions in the first period, and generically satisfy the regularity condition. Hence, by Corollary 1, these equilibria are not robust without buffer areas. We will construct (˜siG , s˜iB , θ˜iG , θ˜iB ) that satisfies Conditions 1–4 in Appendix A.5.1. Let s˜iG and s˜iB be 1 > p > q > r,

3p > 2q + 2,

2+

s˜iG (∅) = C,

s˜iB (∅) = D,

C if y1 = y, ¯ G B s˜i (y1 ) = s˜i (y1 ) = D if y1 = y,

and θ˜iG and θ˜iB be G G (y1 ) + δ θ˜i,2 (y1 , y2 ) − η, θ˜iG (y1 , y2 ) = θ˜i,1 B B θ˜iB (aj,1 , y1 , aj,2 , y2 ) = θ˜i,1 (y1 ) + δ θ˜i,2 (aj,2 , y2 ) + η

41 Although it is not essential for our proof, we can assume without loss of generality that Φ B [h ] depends on h j,l j,l i only through fj (hj,l ).

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with

G (y1 ) = θ˜i,1

1 p−q

− δ(2 −

1−p p−q ) + κ

if y1 = y,

0

G (y1 , y2 ) = θ˜i,2

B (y1 ) = θ˜i,1

if y1 = y, ¯

if y1 = y2 = y¯ or y1 = y,

0 1 − − p−q

if y1 = y, ¯ y2 = y,

κ δ(1−p)

if y1 = y, ¯

0 p 1+δr p−q ) − q−r

δ(2 + ⎧ 1 ⎪ ⎪ ⎨ p−q B 1 θ˜i,2 (aj,2 , y2 ) = q−r ⎪ ⎪ ⎩ 0

if y1 = y,

if aj,2 = C and y2 = y, ¯ if aj,2 = D and y2 = y, ¯

(19)

if y2 = y.

Pick κ ∈ (0, (3p − 2q − 2)/(p − q)] arbitrarily. ((18) guarantees the existence of such κ.) Then we can verify Conditions 1–4 in Appendix A.5.1 as follows. 1. Conditional on xj = G, G (y ) are sunk, (8) is equivalent to (a) in period 2, since η and θ˜i,1 1    G max Eρ u∗i (ai,2 , y2 ) + θ˜i,2 (y1 , y2 )  ai,2 , s˜jG (y1 ), y1 . ai,2

G (y, i. If y1 = y, ¯ then player i strictly prefers C to D since 2 + (1 − p)θ˜i,2 ¯ y) > 3 + G ¯ y) by (18), (19), and κ > 0. (1 − q)θ˜i,2 (y, ii. If y1 = y, then player i strictly prefers D to C by (19); (b) in period 1, player i is indifferent between C and D since, except for constant η,

   G G 2 + p θ˜i,1 (y) ¯ + δ p 2 + (1 − p)θ˜i,2 (y, ¯ y) + (1 − p)   continuation value after y1 =y¯



(0)  continuation value after y1 =y

   G G (y) ¯ + δ q 2 + (1 − p)θ˜i,2 (y, ¯ y) + (1 − q) = 3 + q θ˜i,1   continuation value after y1 =y¯

(0) 



continuation value after y1 =y

by (19). 2. Conditional on xj = B, B (y ) are sunk, (9) is equivalent to (a) in period 2, since η and θ˜i,1 1   B Eρ u∗i (ai,2 , y2 ) + θ˜i,2 (aj,2 , y2 ) | ai,2 , s˜jB (y1 ), (aj,1 , y1 ) being independent of ai,2 . Conditional on (aj,1 , y1 ), player i is indifferent between C and D by (19);

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1927

(b) in period 1, player i is indifferent between C and D since, except for constant η,      B B B −1 + (1 − q)θ˜i,1 (y) + δ q 2 + p θ˜i,2 (C, y) ¯ + (1 − q) 0 + r θ˜i,2 (D, y) ¯     continuation value after y1 =y¯

continuation value after y1 =y

     B B B (y) + δ r 2 + p θ˜i,2 (C, y) ¯ + (1 − r) 0 + r θ˜i,2 (D, y) ¯ = 0 + (1 − r)θ˜i,1     continuation value after y1 =y¯

continuation value after y1 =y

by (19). 3. By (10) and (19), we have      1−δ  G G 2 + p θ˜i,1 (y) ¯ + δ p 2 + (1 − p)θ˜i,2 (y, ¯ y) + (1 − p)0 − η 2 1−δ ! " p η 1 2+ − , = 1+δ p−q 1+δ    1−δ  B B −1 + (1 − q)θ˜i,1 (y) + δ q 2 + p θ˜i,2 (C, y) ¯ viB = 1 − δ2    B + (1 − q) 0 + r θ˜i,2 (D, y) ¯ +η ! " δ p 1 1−r η = 2+ − + . 1+δ p−q 1+δ q −r 1+δ

viG =

(Here, since both ai,1 = C and D are optimal, we use ai,1 = C for explicit calculation.) By (18), we have ! " ! " 1−δ p 1 1−r 1−r viG − viB = 2+ + − 2η > − η = . 1+δ p−q 1+δ q −r 2(q − r) 4. For sufficiently large δ, we have −θ˜iG (y1 , y2 ), θ˜iB (aj,1 , y1 , aj,2 , y2 ) ∈ (0, δ 2 (viG − viB )/ (1 − δ)) since η > 0 and ! " 1 1−p G ˜ lim θi,1 (y) ¯ = − 2− + κ  0, δ→1 p−q p−q p 1+r B (y) = 2 + lim θ˜i,1 − 0 δ→1 p−q q −r by (18), (19), and κ  (3p − 2q − 2)/(p − q). ¯ (˜s G , s˜ B , θ˜ G , θ˜ B ) constitutes a block Therefore, there exists δ¯ < 1 such that for any δ > δ, i i i i belief-free equilibrium. By choosing x1 = x2 = G as initial states, the players play pure actions C in period 1, and the regularity condition is satisfied if and only if δ = κ(p − q)/(1 − p). References [1] D. Abreu, D. Pearce, E. Stacchetti, Toward a theory of discounted repeated games with imperfect monitoring, Econometrica 58 (1990) 1041–1063. [2] V. Bhaskar, Informational constraints and the overlapping generations model: folk and anti-folk theorems, Rev. Econ. Stud. 65 (1998) 135–149. [3] V. Bhaskar, G.J. Mailath, S. Morris, Purification in the infinitely-repeated prisoners’ dilemma, Rev. Econ. Dynam. 11 (2008) 515–528.

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[4] V. Bhaskar, G.J. Mailath, S. Morris, A foundation for Markov equilibria with finite social memory, Rev. Econ. Stud. 80 (3) (2013) 925–948. [5] H. Carlsson, E. van Damme, Global games and equilibrium selection, Econometrica 61 (1993) 989–1018. [6] S. Chassang, S. Takahashi, Robustness to incomplete information in repeated games, Theor. Econ. 6 (2011) 49–93. [7] J.C. Ely, Correlated equilibrium and trigger strategies with private monitoring, Working paper, 2002. [8] J.C. Ely, J. Hörner, W. Olszewski, Belief-free equilibria in repeated games, Econometrica 73 (2005) 377–415. [9] J.C. Ely, J. Välimäki, A robust folk theorem for the prisoner’s dilemma, J. Econ. Theory 102 (2002) 84–105. [10] K. Fong, O. Gossner, J. Hörner, Y. Sannikov, Efficiency in the repeated prisoner’s dilemma with imperfect private monitoring, Working paper, 2010. [11] D. Frankel, S. Morris, A. Pauzner, Equilibrium selection in global games with strategic complementarities, J. Econ. Theory 108 (2003) 1–44. [12] D. Fudenberg, D.K. Levine, Efficiency and observability with long-run and short-run players, J. Econ. Theory 62 (1994) 103–135. [13] D. Fudenberg, D.K. Levine, E. Maskin, The folk theorem with imperfect public information, Econometrica 62 (1994) 997–1039. [14] D. Fudenberg, W. Olszewski, Repeated games with asynchronous monitoring of an imperfect signal, Games Econ. Behav. 72 (2011) 86–99. [15] J.C. Harsanyi, Games with randomly disturbed payoffs: a new rationale for mixed-strategy equilibrium points, Int. J. Game Theory 2 (1973) 1–23. [16] J. Hörner, W. Olszewski, The folk theorem for games with private almost-perfect monitoring, Econometrica 74 (2006) 1499–1544. [17] J. Hörner, W. Olszewski, How robust is the folk theorem?, Quart. J. Econ. 124 (2009) 1773–1814. [18] A. Kajii, S. Morris, The robustness of equilibria to incomplete information, Econometrica 65 (1997) 1283–1309. [19] M. Kandori, I. Obara, Efficiency in repeated games revisited: the role of private strategies, Econometrica 74 (2006) 499–519. [20] G.J. Mailath, S. Morris, Repeated games with almost-public monitoring, J. Econ. Theory 102 (2002) 189–228. [21] G.J. Mailath, S. Morris, Coordination failure in repeated games with almost-public monitoring, Theor. Econ. 1 (2006) 311–340. [22] G.J. Mailath, W. Olszewski, Folk theorems with bounded recall under (almost) perfect monitoring, Games Econ. Behav. 71 (2011) 174–192. [23] H. Matsushima, On the theory of repeated games with private information: part I: anti-folk theorem without communication, Econ. Letters 35 (1992) 253–256. [24] H. Matsushima, Repeated games with private monitoring: two players, Econometrica 72 (2004) 823–852. [25] D. Monderer, D. Samet, Approximating common knowledge with common beliefs, Games Econ. Behav. 1 (1989) 170–190. [26] M. P˛eski, Asynchronous repeated games with rich private monitoring and finite past, Working paper, 2009. [27] M. P˛eski, An anti-folk theorem for finite past equilibria in repeated games with private monitoring, Theor. Econ. 7 (2012) 25–55. [28] M. Piccione, The repeated prisoner’s dilemma with imperfect private monitoring, J. Econ. Theory 102 (2002) 70–83. [29] A. Rubinstein, The electronic mail game: a game with almost common knowledge, Amer. Econ. Rev. 79 (1989) 389–391. [30] Y. Yamamoto, A limit characterization of belief-free equilibrium payoffs in repeated games, J. Econ. Theory 144 (2009) 802–824.