A Simple Necessary Condition for Partially Honest Nash Implementation

This paper belongs to the recent literature which explores the consequence(s) of allowing some player(s) to be minimally honest in the standard Nash implementation. In this literature, Dutta and Sen(2012) give sufficient conditions for the implementation of social choice correspondences in Nash equilibria provided at least one individual is partially honest, that is Maskin Monotonicity is no longer required. They did not present any necessary condition. Thus, in this paper, we seek to fill the gap by deriving a simple necessary condition called, Partial-Equivalency. This condition allows us to lift the silence on the implementability of several social choice correspondences where Dutta and Sen's theorem (2012) does not give any answer. We apply our result to domains of private good economies with single-peaked preferences and we give examples of solutions of the problem of fair division that are not partially honest Nash implementable. We introduce the mild property of citizen sovereignty and we prove that Partial-Equivalency is not only necessary, but becomes also sufficient. Also, we extend our result to partially honest environment with incomplete information.