Evolutionary Dynamics of Signals with Type-specific Payoff Increments

This paper studies the replicator dynamics and the best-response dynamics of a signaling game with type-specific preferences over signals—a generalization of the beer-quiche game (Cho and Kreps in Q J Econ 102(2):179–221, 1987). When the prior probability of the high type is below the critical value at which player 2 is indifferent between accepting and not accepting, there is a unique, partially revealing equilibrium with partial pooling in the signal that the high type prefers. Under the replicator dynamics, this equilibrium is (Lyapunov) stable but not asymptotically stable. It is surrounded by periodic orbits each of which attracting a three-dimensional stable manifold from the interior of the state space. When the prior probability of the high type is above the critical value (the case usually considered), there are two equilibrium outcomes, each with pooling in one of the two signals. Under the replicator dynamics, the equilibrium outcome with pooling in the signal that the high type prefers is stable but not asymptotically stable. The equilibrium outcome with pooling in the signal that the low type prefers is unstable. Still, both components have basins of attraction with nonempty interior. The proofs use center manifold theory and chain recurrence. Throughout, results of the dynamic analysis are compared to equilibrium selection based on the intuitive criterion and index theory.