An analysis of standard evolutionary dynamics adapted to extensive form games. Evolutionary game theory attempts to predict individual behavior (whether of humans or other species) when interactions between individuals are modeled as a noncooperative game. Most dynamic analyses of evolutionary games are based on their normal forms, despite the fact that many interesting games are specified more naturally through their extensive forms. Because every extensive form game has a normal form representation, some theorists hold that the best way to analyze an extensive form game is simply to ignore the extensive form structure and study the game in its normal form representation. This book rejects that suggestion, arguing that a game's normal form representation often omits essential information from the perspective of dynamic evolutionary game theory. The book offers a synthesis of current knowledge about extensive form games from an evolutionary perspective, emphasizing connections between the extensive form representation and dynamic models that traditionally have been applied to biological and economic phenomena. It develops a general theory to analyze dynamically arbitrary extensive form games and applies this theory to a range of examples. It lays the foundation for the analysis of specific extensive form models of behavior and for the further theoretical study of extensive form evolutionary games.

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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In a pairwise interaction, an individual who uses costly punishment must pay a cost in order that the opponent incurs a cost. It has been argued that individuals will behave more cooperatively if they know that their opponent has the option of using costly punishment. We examined this hypothesis by conducting two repeated two-player Prisoner's Dilemma experiments, that differed in their payoffs associated to cooperation, with university students from Beijing as participants. In these experiments, the level of cooperation either stayed the same or actually decreased when compared with the control experiments in which costly punishment was not an option. We argue that this result is likely due to differences in cultural attitudes to cooperation and punishment based on similar experiments with university students from Boston that found cooperation did increase with costly punishment.antisocial punishment ͉ cultural effects ͉ experimental outcome ͉ Prisoner's Dilemma repeated game ͉ reputation

This article verifies that the ideal free distribution (IFD) is evolutionarily stable, provided the payoff in each patch decreases with an increasing number of individuals. General frequency-dependent models of migratory dynamics that differ in the degree of animal omniscience are then developed. These models do not exclude migration at the IFD where balanced dispersal emerges. It is shown that the population distribution converges to the IFD even when animals are nonideal (i.e., they do not know the quality of all patches). In particular, the IFD emerges when animals never migrate from patches with a higher payoff to patches with a lower payoff and when some animals always migrate to the best patch. It is shown that some random migration does not necessarily lead to undermatching, provided migration occurs at the IFD. The effect of population dynamics on the IFD (and vice versa) is analyzed. Without any migration, it is shown that population dynamics alone drive the population distribution to the IFD. If animal migration tends (for each fixed population size) to the IFD, then the combined migrationpopulation dynamics evolve to the population IFD independent of the two timescales (i.e., behavioral vs. population).

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Abstract. The measure dynamics approach to modelling single-species coevolution with a one-dimensional trait space is developed and compared to more traditional methods of adaptive dynamics and the Maximum Principle. It is assumed that individual fitness results from pairwise interactions together with a background fitness that depends only on total population size. When fitness functions are quadratic in the real variables parameterizing the one-dimensional traits of interacting individuals, the following results are derived. It is shown that among monomorphisms (i.e. measures supported on a single trait value), the CSS (Continuously Stable Strategy) characterize those that are Lyapunov stable and attract all initial measures supported in an interval containing this trait value. In the cases where adap- *

Two most influential models of evolutionary game theory are the Hawk-Dove and Prisoner's dilemma models. The Hawk-Dove model explains evolution of aggressiveness, predicting individuals should be aggressive when the cost of fighting is lower than its benefit. As the cost of aggressiveness increases and outweighs benefits, aggressiveness in the population should decrease. Similarly, the Prisoner's dilemma models evolution of cooperation. It predicts that individuals should never cooperate despite cooperation leading to a higher collective fitness than defection. The question is then what are the conditions under which cooperation evolves? These classic matrix games, which are based on pair-wise interactions between two opponents with player payoffs given in matrix form, do not consider the effect that conflict duration has on payoffs. However, interactions between different strategies often take different amounts of time. In this article, we develop a new approach to an old idea that opportunity costs lost while engaged in an interaction affect individual fitness. When applied to the Hawk-Dove and Prisoner's dilemma, our theory that incorporates general interaction times leads to qualitatively different predictions. In particular, not all individuals will behave as Hawks when fighting cost is lower than benefit, and cooperation will evolve in the Prisoner's dilemma.