Stochastic Evolutionary Game Dynamics

Traditional game theory studies strategic interactions in which the agents make rational decisions. Evolutionary game theory differs in two key respects: the focus is on large populations of individuals who interact at random rather than on small numbers of players; and individuals are assumed to employ simple adaptive rules rather than to engage in perfectly rational behavior. In such a setting, an equilibrium is a rest point of the population-level dynamical process rather than a form of consistency between beliefs and strategies. This chapter shows how the theory of stochastic dynamical systems can be used to characterize the equilibria that are most likely to be selected when the evolutionary process is subject to small persistent perturbations. Such equilibria are said to be stochastically stable. The implications of stochastic stability are discussed in a variety of settings, including 2 × 2 games, bargaining games, public-goods games, potential games, and network games. Stochastic stability often selects equilibria that are familiar from traditional game theory: in 2 × 2 games one obtains the risk-dominant equilibrium, in bargaining games the Nash bargaining solution, and in potential games the potential-maximizing equilibrium. However, the justification for these solution concepts differs between the two approaches. In traditional game theory, equilibria are justified in terms of rationality, common knowledge of the game, and common knowledge of rationality. Evolutionary game theory dispenses with all three of these assumptions; nevertheless, some of the main solution concepts survive in a stochastic evolutionary setting.