The Nash Equilibria of Long Cheap Talk Games
cheap talk games
In the associated one-shot communication game the expert learns his type and sends a message to the decision maker, who then chooses an action. Such games are sometimes called persuasion or disclosure games (see, e.g., Milgrom, 1981; Milgrom and Roberts, 1986;Seidmann and Winter, 1997). To the best of our knowledge, this literature has always focused on one-shot information revelation with very specific assumptions on players’ preferences,like single-peakedness, strict concavity and monotonicity. Our first result (Theorem 1) is a full characterization of Nash equilibrium payoffs of one-shot communication games with certifiable information. Roughly, equilibrium payoff vectors are obtained by convexifying the graph of an extended set of equilibrium payoffs of the basic game without communication (the silent game), by keeping the payoff of the informed player constant and individually rational.Several geometric illustrations involving full, partial and/or no information revelation are provided.In a multistage communication game, the talking phase has an arbitrary large number of periods. In each communication period both players simultaneously send a message that depends on the history of play up to that period. The informed player’s message may alsodepend on his private information. As in Hart (1985) and Aumann and Hart (2003), our equilibrium characterization makes use of the mathematical concepts of diconvexification and dimartingale. In Theorem 2 we show that the set of equilibrium payoffs of any multi-stage communication game can be characterized in terms of starting points of dimartingalesconverging to the graph of an extended set of equilibrium payoffs of the silent game, andstaying in an adapted set of individually rational payoffs during the whole process.
In the associated one-shot communication game the expert learns his type and sends a message to the decision maker, who then chooses an action. Such games are sometimes called persuasion or disclosure games (see, e.g., Milgrom, 1981; Milgrom and Roberts, 1986;Seidmann and Winter, 1997). To the best of our knowledge, this literature has always focused on one-shot information revelation with very specific assumptions on players’ preferences,like single-peakedness, strict concavity and monotonicity. Our first result (Theorem 1) is a full characterization of Nash equilibrium payoffs of one-shot communication games with certifiable information. Roughly, equilibrium payoff vectors are obtained by convexifying the graph of an extended set of equilibrium payoffs of the basic game without communication (the silent game), by keeping the payoff of the informed player constant and individually rational.Several geometric illustrations involving full, partial and/or no information revelation are provided.In a multistage communication game, the talking phase has an arbitrary large number of periods. In each communication period both players simultaneously send a message that depends on the history of play up to that period. The informed player’s message may alsodepend on his private information. As in Hart (1985) and Aumann and Hart (2003), our equilibrium characterization makes use of the mathematical concepts of diconvexification and dimartingale. In Theorem 2 we show that the set of equilibrium payoffs of any multi-stage communication game can be characterized in terms of starting points of dimartingalesconverging to the graph of an extended set of equilibrium payoffs of the silent game, andstaying in an adapted set of individually rational payoffs during the whole process.
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