Robust Virtual Implementation in Multi-Stage Mechanisms

Dynamic strategic distinguishability is a weak requirement in a simple auction environment with up to three agents and two types per agent, in which preferences are quasilinear, and an agent's valuation for the object equals the sum of everybody's type where the opponents' types are multiplied with a parameter measuring preference interdependence. I show that all types are always dynamically strategically distinguishable, except for the case in which agents put exactly the same weight on every agent's type (including their own). Thus, ex-post incentive compatibility is necessary and sufficient for robust virtual implementation except for the case in which the preference interdependence parameter equals one. Importantly, if there are three agents, a weak single-crossing condition holds and there is a sufficient degree of preference interdependence, types are not statically strategically distinguishable and the efficient allocation is not statically robustly virtually implementable (Bergemann and Morris, 2009). The efficient allocation rule is, however, ex-post incentive compatible (Dasgupta and Maskin, 2000) and robustly virtually implementable by a dynamic mechanism.