The replicator equation and other game dynamics

Abstract

The replicator equation is the first and most important game dynamics studied in connection with evolutionary game theory. It was originally developed for symmetric games with finitely many strategies. Properties of these dynamics are briefly summarized for this case, including the convergence to and stability of the Nash equilibria and evolutionarily stable strategies. The theory is then extended to other game dynamics for symmetric games (e.g., the best response dynamics and adaptive dynamics) and illustrated by examples taken from the literature. It is also extended to multiplayer, population, and asymmetric games.Nash equilibrium | evolutionarily stable strategy (ESS) | dynamic stability G ame dynamics model how individuals or populations change their strategy over time based on payoff comparisons. This contrasts with classical noncooperative game theory that analyzes how rational players will behave through static solution concepts such as the Nash equilibrium (NE) (i.e., a strategy choice for each player whereby no individual has a unilateral incentive to change his or her behavior). In general, game dynamics assume that strategies with higher payoff do better. As we will see, the limiting behavior of these dynamics (i.e., the evolutionary outcome) is often a NE with additional stability properties.The most important game dynamics is the replicator equation, defined for a single species by Taylor and Jonker (1) and named by Schuster and Sigmund (2). The replicator equation is the first game dynamics studied in connection with evolutionary game theory, a theory that was developed by Maynard Smith and Price (3) (see also ref. 4) from the biological perspective in order to predict the evolutionary outcome of population behavior without a detailed analysis of such biological factors as genetic or population size effects. With payoff translated as fitness (i.e., reproductive success), the frequency of a strategy in a large, well-mixed single species changes under the (continuous-time) replicator equation at a per capita rate equal to the difference between its expected payoff and the average payoff of the population (Eq. 1). If each strategy payoff is constant (in particular, independent of strategy frequency), the ultimate outcome of evolution is that everyone plays the strategy with highest payoff, a result that is true for all game dynamics and not only the replicator equation. In biological terms, we have Darwin's survival of the fittest through natural selection.Of more interest is what happens when individual payoff depends on the actions of others (i.e., when there is an actual game involved). An early success of evolutionary game theory is then the result that an evolutionarily stable strategy (ESS) is dynamically stable for the single species replicator equation described above (1, 5). The ESS, an intuitive concept of uninvadability originally generalizing the "unbeatable" sex ratios analyzed by Hamilton (6), can be defined solely in terms of payoff comparisons (4, 7). Strategic (i.e., game-theoretic) ...

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