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, with intermediate types choosing higher actions than higher and lower types. We provide an algorithm to systematically construct an equilibrium and thus establish equilibrium existence for this general class of preferences with an arbitrary discrete-type distribution. Our analysis sheds light on the connection between discrete-type and continuous-type models and clarifies robust predictions of signaling under double-crossing preferences.