On The Non-Robustness Of Nash Implementation

I consider the implementation problem under complete information and employ Nash equilibrium as a solution concept. A socially desirable rule is given as a correspondence from the set of states to the set of outcomes. This social choice rule is said to be implementable in Nash equilibrium if there exists a mechanism such that all (pure strategy)Nash equilibrium outcomes are socially desirable. Maskin (1999) clarified the conditions on the social choice rules under which he was able to construct a general Nash implementing mechanism when there are at least three players. This paper identifies the minimal amount of incomplete information which prevents the Maskinian mechanism from being robustly implemented in Nash equilibrium. More precisely, with mild constrains on the social choice rules, one can construct a canonical perturbation of the complete information structure under which a sequence of Bayesian Nash equilibria of the Maskinian mechanism supports a non-Nash equilibrium outcome in the limit of the space of the original complete information structure. Therefore, there is a minimal sense in which the Maskinian mechanism is not robust to incomplete information. This non-robustness result is also extended to the environment with two players.