Uniqueness in Two-type Signalling Games: Finite Response Sets vs. Continuum Response Sets

The paper considers signalling games with two types of informed player. When the informed player's payoff function satisfies certain sorting and type monotonicity conditions, and when the uninformed player's response is a smooth strictly monotone mapping from posterior beliefs, the results of Cho and Sobel (1990) establish existence of a unique refined equilibrium outcome which is separating. However, when the response set is finite, refined pooling equilibria generally exist. We consider the finite-response case, and our results are as follows: For every prior probability assessment over the two types there is a finite number of responses ¯n(?0), where ?0 is the prior weight on the "good" type, such that for every n = ¯n(?0) no pooling can be part of a refined equilibrium. I.e., if Nature chooses ?0 from some continuous probability distribution on the unit interval, then the probability that refined pooling equilibria exist goes to zero when n increases withut bound. In the limit the measure is zero as shown by Cho and Sobel.