"Evolutionary dynamics and efficient equilibria in games"

Robert Molzon
University of Kentucky

Monday, November 14, 2011
4:00 pm
845 POT

Abstract:

Classical Game Theory as developed by von Neumann and Morgenstern relies in a fundamental way on the assumption of rational behavior by players or agents. In 1973 the biologist John Maynard Smith developed an approach to the evolution of strategic behavior in animals that helped explain certain traits that could not be based on an assumption of rational behavior. It is now generally recognized that homo sapiens (in spite of the name) are no great shakes when it comes to rational behavior either. This enlightenment prompted the application of evolutionary game theory to the study of social behavior of humans in fields such as economics, linguistics, and political science.

Evolutionary game theory studies the dynamics and equilibria of strategies of agents who interact in a manner that generally involves a stochastic component. Because the resulting stochastic dynamical systems are often complex to the point of intractability, it is often assumed that population size large enough so that the random factors somehow cancel out through a "Law of Large Numbers". I study this assumption in the context of a well known model for learning in games. The model consists of a large number of players who are pairwise matched to play a two strategy game that has two Nash equilibria. Players update their strategy by imitation. I show that the limiting distribution of strategies as population size becomes arbitrarily large is quite diferent than it would be for an infinite population. The primary mathematical tools needed to obtain the resuls are discrete probability, stochastic processes, and estimates for combinatorial expressions.