Dynamic Stability of Nash Equilibria in Tester Games

Abstract

Tester games are population games in which the agents assist a social planner. The strategies available to the agents are indexed by the individuals affected by the planner's decision, and the associated rewards are ordered oppositely to the evaluations by these individuals of the decision. When the decision made by the planner depends on the choices made by the agents in a suitable way, Nash equilibria of the tester game correspond to maxmin solutions of the planner's decision problem. Dynamical systems designed to find Nash equilibria in population games can therefore be used as numerical algorithms to solve certain classes of maxmin problems. For this approach to be successful, Nash equilibria should be dynamically stable. We prove stability of Nash equilibria of tester games under generically fulfilled conditions for the exponential multiplicative weights algorithm (MWA) when the step size is sufficiently small. We also give bounds for admissible step sizes. The exponential MWA is a discrete-time version of the standard replicator equation of evolutionary game theory.

Keywords: Maxmin optimization, population games, collective decision, Nash equilibrium, local stability

JEL Classification: C61, C73, D70, D81, G11