Rationalizable incentives: Interim rationalizable implementation of correspondences

When the normative goals for a set of agents can be summarized in a set-valued rule and agents take actions that are rationalizable, a new theory of incentives emerges in which standard Bayesian incentive compatibility (BIC) is relaxed significantly. The paper studies the interim rationalizable implementation of social choice sets with a Cartesian product structure, a leading example thereof being ex-post efficiency. Setwise incentive compatibility (setwise IC), much weaker than BIC, is shown to be necessary for implementation. Setwise IC enforces incentives flexibly within the entire correspondence, instead of the pointwise enforcement entailed by BIC. Sufficient conditions, while based on the existence of SCFs in the correspondence that make truthful revelation a dominant strategy, are shown to be permissive to allow the implementation of ex-post efficiency in many settings where equilibrium implementation fails (e.g., bilateral trading, multidimensional signals). Furthermore, this success comes at little cost: all our mechanisms are well behaved, in the sense that best responses always exist.