Mechanism design with level-k types: Theory and an application to bilateral trade

We develop necessary and sufficient conditions for level-k implementation that apply in independent private value environments. These conditions establish a set of level-k incentive constraints that are analogous to Bayesian incentive constraints. We show that in two special environments, the level-k incentive constraints collapse down to Bayesian incentive constraints. We then show, via a bilateral trade application, that this is not a general implication. Bilateral trade is ex post efficient under level-k implementation while it is not Bayesian implementable. We also address a robustness question concerning the common prior assumption embedded in level-k implementation by developing the concept of ex post level-k implementation. We develop necessary and sufficient conditions for ex post level-k implementation and show the relationship between ex post level-k and ex post implementation is analogous to the relationship between level-k and Bayesian implementation.