Bounded depths of rationality and implementation with complete information

We move towards a more robust theory of individual behavior in mechanism design by allowing hierarchies of players with varying but bounded “depths of rationality”. Specifically, we study (full) implementation with complete information when players are at least rational and at most k-rational, where k≥2. We show that the set of implementable social choice functions (SCFs) is independent of the upper bound k. In “independent domain of preferences” (IDP) environments, we obtain a revelation principle: An SCF is implementable if and only if it is implementable by its associated direct mechanism. Two conditions characterize such SCFs: Strategy-proofness and “strong non-bossiness”. For non-IDP environments, we provide a characterization and simple necessary and sufficient conditions. We also analyze the case when rationality is at least mutual knowledge. There is a significant gap in implementation when rationality is mutual knowledge and when it is not. However, it does not make a significant difference whether rationality is mutual knowledge or common knowledge.