An Evolutionary Interpretation of Mixed-strategy Equilibria*

One of the more convincing interpretations of mixed strategy equilibria describes a mixed equilibrium as a steady state in a large population in which all players use pure strategies but the population as a whole mimics a mixed strategy. To be complete, however, this interpretation requires a good story about how the population arrives at the appropriate distribution over pure strategies. In this paper I attempt to give an explanation based on an evolutionary, stochastic learning process. Convergence properties of these processes have been studied extensively but almost exclusively for the case of convergence to pure Nash equilibria. Here I study the conditions under which an evolutionary process converges to population mixed-strategy equilibria. I find that not all mixed equilibria can be justified as the result of the evolutionary learning process even if the equilibrium is unique. For symmetric 2x2 and 3x3 games I give necessary and sufficient conditions for convergence and for n×n games I give a sufficient condition. For cases in which the conditions are not satisfied counterexamples are given, in which the process enters a limit cycle. * I am indebted to Susan Elmes for numerous discussions and very helpful comments. ** Department of Economics, Columbia University, New York, NY 10027, USA. (212) 749-1578, email: [email protected]