Correlated equilibria and communication equilibria in all-pay auctions

We study cheap-talk pre-play communication in static all-pay auctions. For the complete information case of two bidders, all correlated equilibria are payoff equivalent to the Nash equilibrium if there is no reserve price, or if it is commonly known that one bidder has a strictly higher value. Similarly, for the independent private values case of two bidders with no reserve price, all communication equilibria are payoff equivalent to the Bayesian Nash equilibrium. Hence, in such environments the Nash equilibrium and the Bayesian Nash equilibrium predictions are robust to pre-play communication between the bidders. On the other hand, if there are three or more symmetric bidders, or two symmetric bidders and a positive reserve price, then with complete information there may exist correlated equilibria such that the bidders’ payoffs are higher than in any Nash equilibrium, and with independent private values there may exist communication equilibria such that the bidders’ payoffs are higher than in any Bayesian Nash equilibrium. In these cases, pre-play cheap talk may affect the outcomes of the game, since the bidders have an incentive to coordinate on such equilibria.