Signaling under double-crossing preferences: The case of discrete types

The class of double-crossing preferences, where signaling is cheaper for higher types than for lower types at low signaling levels and the opposite is true at high signaling levels, underlines the phenomenon of countersignaling. We show that under the D1 refinement, the equilibrium signaling action must be quasi-concave in type and generally exhibits pooling, with intermediate types choosing higher actions than higher and lower types. We provide an algorithm to systematically construct an equilibrium and use this algorithm to establish its existence for this general class of preferences with an arbitrary discrete-type distribution.