A FOLK THEOREM FOR COMPETING MECHANISMS

Abstract. We provide a partial characterization of the set of out-come functions that can be supported as perfect Bayesian equilib-rium in the recommendation game described in Yamashita (Econo-metrica 2010). We show that the set of outcome functions that can be supported is at least as large as the set supportable by a mech-anism designer in the sense of Myerson (Myerson 1979). We show how to support random and correlated outcomes as equilibrium outcomes in the recommendation game. Many outcome functions can typically be supported as equilibria in competing mechanism games. Some of these outcomes look quite ’collusive’. The reason for this is that competing mechanism games often provide players the opportunity to make what they do conditional on what other players do. This allows players to support collusive outcomes by writing contracts that commit them to react whenever an opponent deviates from a putative equilibrium outcome. A complete characterization of supportable outcomes in regular contracting games is provided in Peters (2010). He shows that an equilibrium outcome function is supportable as a perfect Bayesian equilibrium in a regular contracting game only if it is supportable in a particular reciprocal contracting game in which players contracts condition directly on other players ’ contracts. In most of the literature on common agency and competing auctions, contracts cannot condition directly on other contracts. It is natural to ask whether this feature could be used to limit the large set of sup-portable outcomes. Yamashita (2010) suggested a contracting game in which contracts condition on one another indirectly through commu-nication with agents. The logic of his game is straightforward. Each principal commits to a mechanism that simply asks agents what he should do. If the majority of the agents ’ recommendations agree, the principal commits himself to carry out the recommendation