Implementation with Uncertain Evidence

We study a full implementation problem with a state unknown to the designer but known to agents, where agents have uncertain evidence privately drawn from state-dependent distributions. Stochastic evidence enables perfect deceptions,'' where agents' reports can mimic the evidence distribution of a false state, making differentiation impossible for any mechanism. This yields our main result: a necessary and sufficient condition, No Perfect Deceptions (NPD), for implementation in (mixed-strategy) Bayesian Nash equilibria. The solution requires novel techniques like belief elicitation via competing scoring rules, and an endogenous test allocation’’ using the evidence structure. For informationally small agents (McLean and Postlewaite (2002)), a generalized condition (GNPD) is sufficient. Our mechanisms work for two or more agents, avoid integer/modulo games, and use limited liability transfers that vanish in equilibrium.