Robust Implementation: The Role of Large Type Spaces∗

We analyze the problem of fully implementing a social choice function when the planner does not know the agents’ beliefs about other agents’ types. We identify an ex post monotonicity condition that is necessary and in economic environments sufficient for full implementation in ex post equilibrium; we also identify an ex post monotonicity no veto condition that is sufficient. These results are the ex post equilibrium analogues of Jackson’s (1991) results about Bayesian implementation. We show by example that ex post monotonicity implies neither Maskin monotonicity (necessary and almost sufficient for complete information implementation) nor for some type spaces interim monotonicity (i.e., the Bayesian monotonicity condition that is necessary and almost sufficient for Bayesian implementation). We identify a robust monotonicity condition that is equivalent to interim monotonicity on all type spaces; robust monotonicity implies both Maskin monotonicity and ex post monotonicity. These results follow the implementation literature in focussing on pure strategy equilibria and allowing a finite mechanism to be chosen after the finite type space is chosen. We say that there is uniform implementation if there exists a finite mechanism that fully implements a social choice function for every finite type space that could be constructed for a fixed set of payoff types (under this concept there can be no gap between pure and mixed strategy implementation). We show that uniform implementation is equivalent to an ex post version of dominance solvability and show by example that uniform implementation may be impossible even though implementation is possible on every type space.