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Betting on Others' Bets: Unions of Surplus Extraction Mechanisms

We construct the generalized Cremer-McLean mechanism where agent i's participation fee depends not only on the valuations reported by -i agents at the second stage, but also on the choice of the participation fee by -i at the first stage. Such construction allows to exploit the convex hull property of beliefs whenever it appears in beliefs about beliefs rather than in beliefs about preferences. As such betting retrieves agents' entire hierarchies of beliefs, it reveals what is common knowledge among them. Hence for any given finite or countable collection of type spaces where each type spaces verifies the convex hull property within itself, the designer can propose a union of GCM mechanisms and extract the surplus across type spaces, i.e., regardless of absence of knowledge by the designer which type space agents share (and without relying on the shoot-the-liar mechanism). We discuss when the technique of using a union of individual mechanisms is extendable to more general cases.