Confession and Pardon in Repeated Games with Private Monitoring and Communication

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Confession and Pardon in Repeated Games with Private Monitoring and Communication
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  Confession and Pardon in Repeated Gameswith Private Monitoring and Communication Galit Ashkenazi-GolanJuly 26, 2004 Abstract We investigate multi-player discounted repeated games with pri-vate monitoring and communication. After each period, every playerobserves a random private signal whose distribution depends on thecommon action taken. We do not assume that all signal-profiles arealways observed with a positive probability. However, we do assumethat deviating from certain actions might reduce the contents of in-formation received by the deviator. Under this assumption we obtain,via sequential equilibria, a folk theorem with Nash threats. In equilib-rium players are provided with incentives to report a deviation whenthey detect one. Moreover, in equilibrium, the deviating player hasan incentive to confess his deviation. This is done by making a pun-ishment that follows a confession lighter than a punishment that doesnot follow a confession. Thus, a confession induces a pardon.For getting other results, the proof method is combined with theresults of Compte(1998) and Kandori and Matsushima(1998) who as-sumed that there are at least three players and full support of the sig-nal profiles (i.e., that every signal profile has a positive probability).The combined method provides a larger set of sequential equilibriumoutcomes than each method separately.  1 Introduction The literature on discounted repeated games can be divided into two branches:games with perfect monitoring and games with imperfect monitoring. In themodel of perfect monitoring each player observes after each period the ac-tions played by all the others. Aumann and Shapley (1994) and Rubinstein(1994) characterize the equilibrium payoffs in such games with no discount.They state that every feasible and individually rational payoff vector is anequilibrium payoff. This result is known as Folk Theorem. Fudenberg andMaskin (1986, 1991) analyzed discounted games and obtained a Folk theoremfor perfect equilibrium.Games with imperfect monitoring are further divided into games withpublic monitoring and games with private monitoring. Games with publicmonitoring are games where the players observe after each period a commonlyknown random signal − a public signal. Players are assumed to have aperfect recall and they are allowed to condition their actions on previousdata, including their own payoffs and signals. A player’s strategy specifies1  how she should choose an action at any time and after every eventuality. Apublic strategy is such that actions are conditioned only on the history of public signals and not on the player’s own previous actions.In a perfect public equilibrium players are restricted to employ only publicstrategies. The set of perfect public equilibria payoffs of those games hasbeen thoroughly investigated (Green and Porter(1984), Abreu et al (1990),Fudenberg et al(1994)). When the strategies of the players are not restrictedto public strategies, and the players are allowed to condition their choiceson the private histories of their own actions, the set of equilibria payoffs canbe strictly larger than with public strategies. This phenomenon might occurwhen private histories can serve as a correlation means between the players’actions (see for instance, Mailath et al(2002)).Games with imperfect private monitoring are games where each playerobserves a random private signal after each period. Such games present newdifficulties. One of the prominent ones is to precisely characterize the abilityto correlate between players who try to punish a deviator by using theirprivate signals.One way to overcome these difficulties is to allow the players to publiclycommunicate, as done by Matsushima (1990). After every stage the players2  are allowed to convey public messages that may depend on the history of theirprivate signals. Ben-Porath and Kahneman (1996) proved a Folk theorem fora model where each player can be perfectly observed by at least two others.In this model the players use a communication channel to report a deviationwhen they detect one. Compte (1998) and Kandori and Matsushima (1998)obtained Folk theorems for games with three players or more, when theplayers are allowed to communicate. The latter two papers assume a fullsupport on the set of signals. That is, each signal profile is observed with apositive probability after every history of actions. In addition, they assumedthat any deviation induced a distribution over signal-profiles that allowedthe non-deviating players to statistically detect the deviation and, moreover,the identity of the deviator.In this paper we investigate multi-player repeated games with imperfectprivate monitoring where the players are allowed to communicate. The mainresults of the paper are based on an assumption that refers only to payoffs,called extreme payoffs, which are extreme points of the set of all possiblepayoffs (the feasible set). We assume that every deviation from a commonaction, whose payoff is extreme, is detectable in one of two ways. The firstway is to directly ”observe” it: the deviation induces a positive probability3  for at least one profile of private signals (of the conforming players) whichis assigned a zero probability under the distribution corresponding to theagreed upon joint action.The second way to detect a deviation is the indirect one: the deviationmay cause a loss of information that the deviator would otherwise receive.In equilibrium paths that we later construct, the players are supposed topublicly report their private signal after each period. Any report of a sig-nal profile that is not consistent with the equilibrium path, signifies that adeviation had occurred to the players .A signal of a deviator is called sufficiently informative if it has two prop-erties. First, it lets her know that her opponents’ signals are consistent withthe equilibrium path; and second, it allows her to complete these signals witha signal of her own, so that all together they are consistent with the equi-librium path. A sufficiently informative signal enables a player to get awaywith a deviation. We assume that, having deviated from an action profilewhose payoff is extreme, a deviator observes a sufficiently informative signalwith a probability strictly less than 1.A model in which every profitable deviation is detected with a positiveprobability, is a version of the model of standard-trivial observation, intro-4
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