Extended Conversations in Sender-Receiver Games

Aumann and Hart (Econometrica, Nov. 2003) have shown that in games of onesided incomplete information, the set of equilibrium outcomes achievable can be expanded considerably if the players are allowed to communicate without exogenous time limits and completely characterise the equilibria from such communication. Their research provokes (at least) four questions. (i) Is it true that the set of equilibrium payoffs stabilises (i.e. remains unchanged) if there are sufficiently many rounds of communication? (ii) Is the set of equilibria from communication which is unbounded but finite with probability one is the same as equilibria from communication which is just unbounded? (iii) Are any of these sets of equilibria “simple” and if so, is there an algorithm to compute them? (iv) Does unbounded communication (of order type ω) exhaust all possibilities so that further communication is irrelevant? We show that in the context of finite Sender-Receiver games, the answer to all four is yes if the game satisfies a certain geometric condition. We then relate this condition to some geometric facts about the notion of bi-convexity and argue that if any of the questions has a negative answer then all three of the questions are likely to have a negative answer.