Continuous Implementation with Simple Mechanisms

We revisit the idea that robustness concerns can provide a foundation for simple mechanisms. Formally, we consider the robust implementation notion of continuous implementation (Oury and Tercieux, 2012), where continuity is with respect to players' hierarchy of beliefs. We define simplicity as (the truthful equilibrium of) a direct revelation mechanism. We show that these two desiderata are at odds. For continuity with respect to a finer topology on the hierarchy of beliefs (which is a weaker requirement), the uniform-weak topology, we show that there is a revelation principle. Conversely, for continuity with respect to the coarser product topology (a stronger requirement), we show that no such revelation principle holds. At a technical level, our characterization of continuous implementation in the product topology uses a novel contagion argument which does not require the existence of “crazy types” as is commonly assumed, and may be of independent interest