Bilateral Bayesian Truth Serum : The n x m Signals Case

We consider the following ’truth serum’ mechanism problem: Two players observe signals drawn from a common prior on n x m. They are risk neutral, care only about mechanism payoffs, and will not provide information without incentives. A counterexample shows that one cannot construct a robust (i.e., detail-free) mechanism by eliciting signals and first order beliefs; players must provide something more. In the ’basic mechanism,’ the further input is the prior probability of their signal. In the ’self-prediction mechanism,’ players estimate each other’s first-order beliefs. In the ’information gain mechanism,’ players predict each other’s signal twice, once before and once after they have reported their signal. Correct reporting of own signals and beliefs are a strict Bayesian Nash equilibrium in each case, with equilibrium scores aligned with Shannon information