Two-agent Nash implementation with partially-honest agents: Almost Full Characterizations

In a two-agent society with partially-honest agents, we extend Dutta and Sen (2009)'s results of Nash implementation to the domain of weak orders. We identify the class of Nash implementable social choice correspondences with a "gap" between necessary and sufficient conditions, both when exactly one agent is partially-honest and when both agents are partially-honest. We also show that, on the domain of linear orders, the "gap" between our conditions gets closed and they become equivalent to those devised by Dutta and Sen. New implementing mechanisms are devised. In line with earlier works, the classic condition of monotonicity is no longer required, whereas a weak version of the standard punishment condition is required even when both agents are known to be partially-honest. We derive simpler sufficient conditions that are satisfied in a wide range of applications.