Monotone equilibria in signalling games

We study the monotonicity of sender's equilibrium strategy with respect to her type in signalling games. We use counterexamples to show that when the sender's payoff is non-separable, the Spence-Mirrlees condition cannot rule out equilibria in which the sender uses non-monotone strategies. These equilibria can survive standard refinements as incentives are strict and the sender plays every action with positive probability. We provide sufficient conditions under which the sender's strategy is monotone in every Nash equilibrium. Our conditions require the sender's payoff to have strictly increasing differences between the state and the action profile and monotone with respect to each player's action. We also identify and fully characterize a novel property on the sender's payoff that we call increasing absolute differences over distributions, under which every pair of distributions over the receiver's actions can be ranked endogenously. Our sufficient conditions fit into a number of applications, including advertising, warranty provision, education and job assignment, etc.