Ex Post Implementation

We analyze the problem of fully implementing a social choice set in ex post equilibrium. Weidentify 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 monotonicityno veto condition that is sufficient. Ex post monotonicity is satisfied in all single crossing environments with strict ex post incentive constraints. In many economically significant environments, ex post implementation can be achieved in the direct mechanism. We show by means of two classic examples that ex post monotonicity does not imply nor is it implied by Maskin monotonicity (necessary and almost sufficient for complete information implementation). The single unit auction with interdependent valuations is shown to satisfy ex post monotonicity but not Maskin monotonicity. Ex post implementation in the direct mechanism is also possible in this case. We describe an example where the Pareto correspondence fails ex post monotonicity but Maskin monotonicity is satisfied.Ex post equilibrium, Implementation, Single crossing, Interdependent values

The limits of ex post implementation

The sensitivity of Bayesian implementation to agents' beliefs about others suggests the use of more robust notions of implementation such as ex-post implementation, which requires that each agent' s strategy be optimal for every possible realization of the types of other agents. We show that the only deterministic social choice functions that are ex-post implementable in generic mechanism design frameworks with multi-dimensional signals, interdependent valuations and transferable utilities, are constant functions. In other words, deterministic ex-post implementation requires that the same alternative must be chosen irrespective of agents' signals. The proof shows that ex-post implementability of a non-trivial deterministic social choice function implies that certain rates of information substitution coincide for all agents. This condition amounts to a system of differential equations that are not satis�ed by generic valuation functions

Ex-post Implementation with Interdependent Valuations

We consider a social choice setting with multidimensional signals and interdependent valuations. Such frameworks have been recently and increasingly used in order to study multi-object auctions. We obtain concise characterizations of ex-post implementable (not necessarily efficient) social choice functions in terms of affine functions that associate a weight to each agent and to each alternative. These characterizations can greatly reduce the complexity of the search for a constrained efficient (i.e., second best) mechanism in the generic cases where efficient outcomes cannot be implemented.

Robust Mechanism Design

The mechanism design literature assumes too much common knowledge of the environment among the players and planner. We relax this assumption by studying implementation on richer type spaces. We ask when ex post implementation is equivalent to interim (or Bayesian) implementation for all possible type spaces. The equivalence holds in the case of separable environments; examples of separable environments arise (1) when the planner is implementing a social choice function (not correspondence); and (2) in a quasilinear environment with no restrictions on transfers. The equivalence fails in general, including in some quasilinear environments with budget balance. In private value environments, ex post implementation is equivalent to dominant strategies implementation. The private value versions of our results offer new insights into the relation between dominant strategy implementation and Bayesian implementation.Mechanism design, Common knowledge, Universal type space, Interim equilibrium, Ex-post equilibrium, Dominant strategies

Robust Implementation in General Mechanisms

A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with it. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation. Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces.Mechanism design, Implementation, Robustness, Common knowledge, Interim equilibrium, Dominant strategies

Robust Implementation in General Mechanisms

A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with it. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation. Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces.Mechanism design, Implementation, Robustness, Common knowledge, Interim equilibrium, Dominant strategies

Ex post implementation in environments with private goods

We prove by construction that ex post incentive compatible mechanisms exist in a private goods setting with multi-dimensional signals and interdependent values. The mechanism shares features with the generalized Vickrey auction of one-dimensional signal models. The construction implies that for environments with private goods, informational externalities (i.e., interdependent values) are compatible with ex post equilibrium in the presence of multi-dimensional signals.Ex post incentive compatibility, multi-dimensional information, interdependent values