Introduction to Evolutionary Game Theory

This chapter begins with some basic terminology, introducing elementary game theoretic notions such as payoff, strategy, best reply, Nash equilibrium pairs etc. Players who use strategies which are in Nash equilibrium have no incentive to deviate unilaterally. Next, a population viewpoint is introduced. Players meet randomly, interact according to their strategies, and obtain a payoff. This payoff determines how the frequencies in the strategies will evolve. Successful strategies spread, either (in the biological context) through inheritance or (in the cultural context) through social learning. The simplest description of such an evolution is based on the replicator equation. The basic properties of replicator dynamics are analyzed, and some low-dimensional examples such as the Rock-Scissors-Paper game are discussed. The relation between Nash equilibria and rest points of the replicator equation are investigated, which leads to a short proof of the existence of Nash equilibria. We then study mixed strategies and evolutionarily stable strategies. This introductory chapter continues with a brief discussion of other game dynamics, such as the best reply dynamics, and ends with the simplest extension of replicator dynamics to asymmetric games. 1. Predictions and Decisions Predictions can be difficult to make, especially, as Niels Bohr quipped, if they concern the future. Reliable forecasts about the weather or about some social development may seem to offer comparable challenges, at first sight. But there is a fundamental difference: a weather forecast does not influence the weather, whereas a forecast on economy can influence the economic outcome. Humans will react if they learn about the predictions, and they can anticipate that others will react, too. When the economist Oskar Morgenstern, in the early ’thirties, became aware of the problem, he felt that he had uncovered an ’impossibility theorem’ of a similarly fundamental nature as the incompleteness theorem of his friend, the logician Kurt Gödel. Morgenstern was all the more concerned about it as he was director of the Vienna-based Institut für Konjunkturforschung, the Institute for Business Cycles Research, whose main task was actually to deliver predictions on the Austrian economy. Oscar Morgenstern expained his predicament in many lectures and publications, using as his favorite example the pursuit of Sherlock Holmes by the 2000 Mathematics Subject Classification. Primary 91A22. The author wants to thank Ross Cressman for his helpful comments.