Robustness Foundations for B-IC

Standard Mechanism Design has generated many powerful predictions. These predictions however often rely on strong knowledge assumption imposed about agents beliefs on the designer and are not robust to changes in those beliefs. At the other extreme, the belief-free approach has maximally relaxed these strong knowledge assumptions by instead imposing no restrictions. Belief Restrictions (Ollár and Penta, 2017) provide a unified framework for robust mechanism design that accommodates varying degrees of robustness with respect to agents’ beliefs, encompassing both the belief-free and Bayesian settings of the classical literature as special cases. This paper studies general problems of partial implementation under a natural equilibrium concept tailored to environments with belief restrictions. This concept generalizes both ex-post and Bayes-Nash equilibria – each emerging as a special case when the belief restrictions coincide with the belief-free or standard Bayesian settings, respectively – and imposes intermediate constraints in all other cases. This yields a corresponding notion of incentive compatibility, which we refer to as BR-IC (for Belief Restrictions). Our main results identify the environments where implementation with respect to this concept is equivalent to Bayes-Nash implementation in all type spaces that are consistent with the Belief Restrictions. In this sense, the results provide a foundation to the adoption of BR-IC as a short-cut to study robust Bayesian implementation, thereby generalizing the results that Bergemann and Morris (2005, Ecta) obtained for ex-post equilibrium, from belief-free to general belief-restrictions .