Order on types based on monotone comparative statics

Abstract

This paper introduces a novel concept of orders on types by which the so-called monotone comparative statics is valid in all supermodular games with incomplete information. We fully characterize this order in terms of what we call common optimism, providing a sense in which our order has a sharp epistemic interpretation. We say that type ti′ is higher than type ti in the order of the common optimism if ti′ is more optimistic about state than ti; ti′ is more optimistic that all players are more optimistic about state than ti; and so on, ad infinitum. First, we show that whenever the common optimism holds, monotone comparative statics hold in all supermodular games. Second, we show the converse. We construct an “optimism-elicitation game” as a single supermodular game with the property that whenever the common optimism fails, monotone comparative statics fails as well.