### Robust Implementation in Rationalizable Strategies in General Mechanisms

*Takashi Kunimoto*, Rene Saran*

Last modified: 2022-04-17

#### Abstract

A social choice function (SCF) is robustly implementable in rationalizable strategies (RoRat-implementable) if every rationalizable strategy profile on every type space results in outcomes consistent with it. First, we establish the equivalence between RoRat-implementation and “weak rationalizable implementation”, the latter being a ``type-free'' concept. Second, using the equivalence result, we identify weak robust monotonicity as a necessary and almost sufficient condition for RoRat-implementation. This exhibits a contrast with robust implementation in interim equilibria (RoEq-implementation), i.e., every equilibrium on every type space must achieve outcomes consistent with the SCF. Bergemann and Morris (2011) show that strict robust monotonicity is a necessary and almost sufficient condition for RoEq-implementation. We argue that strict robust monotonicity is strictly stronger than weak robust monotonicity, which further implies that, within general mechanisms, RoRat-implementation is more permissive than RoEq-implementation. The gap between RoRat-implementation and RoEq-implementation stems from the strictly stronger nonemptiness requirement inherent in the latter concept.