Strategic Information Exchange

We study a class of two-player repeated games with incomplete information and informational externalities. In these games, two states are chosen at the outset, and players get private information on the pair, before engaging in repeated play. The payoff of each player only depends on his ‘own’ state and on his own action. We study to what extent, and how, information can be exchanged in equilibrium. We prove that provided the private information of each player is valuable for the other player, the set of sequential equilibrium payoffs converges to the set of feasible and individually rational payoffs as players become patient. Whether and how to acquire information is a question faced by most decision makers. In statistical decision problems, the decision maker tries to learn the value of an unknown parameter, and he can sample from an exogenous population at a fixed cost per draw. In other contexts, information is held by strategic agents. In signalling games for instance, a player holding payoff-relevant private information tries to influence the action choice of an uninformed party. There, the rationale for disclosing/hiding information is that the uninformed party’s action affects the payoff of the informed player. We here study a class of repeated games with private ∗Economics and Decision Sciences Department, HEC Paris, and GREG-HEC; . e-mail: [email protected] †The School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. e-mail: [email protected] ‡Economics and Decision Sciences Department, HEC Paris, and GREG-HEC. e-mail: [email protected]. 1 ar X iv :1 00 7. 44 27 v1 [ m at h. PR ] 2 6 Ju l 2 01 0 information, in which there is no such direct strategic interaction: as in setups of social learning, or of strategic experimentation, payoffs do not depend upon other players’ actions. However, players hold private information that has value to other players. Our goal is to understand to what extent information can be exchanged at equilibrium along the play, assuming communication is costly. This assumption of pure informational externalities plays a dual role. Obviously, it simplifies the analysis of the model and allows to study the exchange of information in isolation from other strategic considerations. But it leads to a game in which we might least expect exchange of information, and any positive result in this setup might potentially open the way for the analysis of other setups. As an illustration, consider the following game. There are two biased coins, C1 and C2 (say, with parameter 2 3 ), which are tossed independently once at the outset of the game. Each of two players, i = 1, 2, has to repeatedly guess the outcome of coin Ci. A correct guess yields a payoff of one, while an incorrect one yields zero, and successive payoffs are discounted. If past payoffs are not observed, there is no role for direct inference, and it is natural to expect that player i will repeatedly ‘guess’ the most likely outcome, for an expected payoff of 2 3 . Assume that, once coins are drawn, each player i gets to observe the outcome of coin Cj, where j 6= i, but that cheap talk is excluded. This private information has no ‘direct’ value, but it might have a strategic value, because it is valuable to the other player. In this game, is there an equilibrium payoff that improves upon ( 3 , 2 3 ) ? While stylized, this game is similar to the situation faced by executives of two different firms, who hold private information on their own firm. Trading the stock of one’s own firm is illegal, at times where private information is most valuable, and the executives may be tempted to implement some implicit collusive scheme of information exchange through time. We start with few simple observations. Since cheap talk is assumed away, exchange of information is to take place through actions, by ‘encoding’ privately held information into actions, and by conditioning the action choice of player i on the outcome of coin Cj. Plainly, there is no equilibrium in which each player i fully ‘discloses’ the outcome of Cj at stage 1. Indeed, once informed of the outcome of C2, player 2 can make the correct guess in all subsequent periods, and has no incentive whatsoever to incur the cost of disclosing information to player 1, including in stage 1. But then, player 1 would not be willing to play the myopically suboptimal action in stage 1. More generally, private information is here an asset to be exchanged for