Interim Rationalizable Implementation of Functions

This paper investigates rationalizable implementation of social choice functions (SCFs) in incomplete information environments. We identify weak interim rationalizable monotonicity (weak IRM) as a novel condition and show it to be a necessary and almost sufficient condition for rationalizable implementation. We show by means of robust examples that interim rationalizable monotonicity (IRM), found in the literature, is strictly stronger than weak IRM and that IRM is not necessary for rationalizable implementation, as had been previously claimed. These examples also demonstrate that Bayesian monotonicity, the key condition for full Bayesian implementation, is not necessary for rationalizable implementation. That is, rationalizable implementation can be more permissive than Bayesian implementation. We revisit well-studied classes of economic environments and show that the SCFs considered there are interim rationalizable implementable. A comprehensive discussion of related issues, including well-behaved mechanisms, mechanisms satisfying the best response property, double implementation, and responsive SCFs is also provided. Funding: This work was supported by Ministry of Education, Singapore [Grant MOE Academic Research Fund Tier 2/MOE-T2EP402A20-0].