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Incentive Compatibility in Signaling Games with a Continuum of Types

George J. Mailath

Abstract

This paper provides two results that are useful in proving the existence of and characterizing separating equilibria in signaling games. These are games in which a privately informed agent takes an action which is observed by uniformed agents before they in turn choose actions. Depending on the equilibrium, the privately informed agent's action could reveal the private information, partially or completely. If the private information is completely revealed, the equilibrium is called separating. A key element in the analysis of separating equilibria is the examination of the implied incentive compatibility constraints (namely, that no type should have an incentive to take an action other than the prescribed equilibrium action). It is shown that these constraints together with one of two conditions imply differentiability of strategies when the set of possible types is an interval. The first condition, implied by sequentiality in many games, is an initial value condition on the informed agent's strategy. The second condition is a monotonicity condition which is similar to the familiar single crossing condition. The monotonicity condition turns out to be necessary and sufficient for there to be a strictly monotonic strategy satisfying the incentive compatibility constraints. As a direct consequence of these two results, the analysis of Milgrom and Roberts (1982) is considerably strengthened.