The Robustness of Sequential Implementation under Information Perturbations

Bayesian monotonicity is a necessary condition for full implementation in Bayes-Nash equilibrium. Under the sequential equilibrium refinement, however, multi-stage mechanisms can expand the scope for implementation. We show that this additional implementation power is fragile. Whenever a multi-stage mechanism sequentially implements a social choice correspondence that is not Bayesian monotone, we construct arbitrarily small "information perturbations" and show that there are sequential equilibria under these perturbations that yield inadmissible decisions with probability bounded away from zero. Under these perturbations, players' types are not independent conditional on the state and some player is sometimes uncertain of their payoff type. We show by example that the general result fails if perturbations are required to satisfy conditional independence or "known payoff types."