Nash Implementation and Uncertain Renegotiation

This paper studies Nash implementation when the outcomes of the mechanism can be renegotiated among the agents but the planner does not know the renegotiation function that they will use. We characterize the social objectives that can be implemented in Nash equilibrium when the same mechanism must work for every admissible renegotiation function. The constrained Walrasian correspondence, the core correspondence, and the Pareto-efficient and envy-free correspondence satisfy the necessary and sufficient conditions for this form of implementation if and only if freedisposal of the commodities is allowed. The uniform rule, on the other hand, is not Nash implementable for some admissible renegotiations functions.