Abstract:
Evolutionary game theory is a powerful framework for studying evolution in populations of interacting individuals. A common assumption in evolutionary game theory is that interactions are symmetric, which means that the players are distinguished by only their strategies. In nature, however, the microscopic interactions between players are nearly always asymmetric due to environmental effects, differing baseline characteristics, and other possible sources of heterogeneity. To model these phenomena, we introduce into evolutionary game theory two broad classes of asymmetric interactions: ecological and genotypic. Ecological asymmetry results from variation in the environments of the players, while genotypic asymmetry is a consequence of the players having differing baseline genotypes. We develop a theory of these forms of asymmetry for games in structured populations and use the classical social dilemmas, the Prisoner’s Dilemma and the Snowdrift Game, for illustrations. Interestingly, asymmetric games reveal essential differences between models of genetic evolution based on reproduction and models of cultural evolution based on imitation that are not apparent in symmetric games.

Abstract:
Cooperative behavior among unrelated individuals in human and animal societies represents a most intriguing puzzle to scientists in various disciplines. Here we present a simple yet effective mechanism promoting cooperation under full anonymity by allowing for voluntary participation in public goods games. This natural extension leads to rock--scissors--paper type cyclic dominance of the three strategies cooperate, defect and loner i.e. those unwilling to participate in the public enterprise. In spatial settings with players arranged on a regular lattice this results in interesting dynamical properties and intriguing spatio-temporal patterns. In particular, variations of the value of the public good leads to transitions between one-, two- and three-strategy states which are either in the class of directed percolation or show interesting analogies to Ising-type models. Although volunteering is incapable of stabilizing cooperation, it efficiently prevents successful spreading of selfish behavior and enables cooperators to persist at substantial levels.

Abstract:
Competition among cooperators, defectors, and loners is studied in an evolutionary prisoner's dilemma game with optional participation. Loners are risk averse i.e. unwilling to participate and rather rely on small but fixed earnings. This results in a rock-scissors-paper type cyclic dominance of the three strategies. The players are located either on square lattices or random regular graphs with the same connectivity. Occasionally, every player reassesses its strategy by sampling the payoffs in its neighborhood. The loner strategy efficiently prevents successful spreading of selfish, defective behavior and avoids deadlocks in states of mutual defection. On square lattices, Monte Carlo simulations reveal self-organizing patterns driven by the cyclic dominance, whereas on random regular graphs different types of oscillatory behavior are observed: the temptation to defect determines whether damped, periodic or increasing oscillations occur. These results are compared to predictions by pair approximation. Although pair approximation is incapable of distinguishing the two scenarios because of the equal connectivity, the average frequencies as well as the oscillations on random regular graphs are well reproduced.

Abstract:
The recent discovery of zero-determinant strategies for the repeated Prisoner's Dilemma sparked a surge of interest in the surprising fact that a player can exert control over iterated interactions regardless of the opponent's response. These remarkable strategies, however, are known to exist only in games in which players choose between two alternative actions such as "cooperate" and "defect." Here we introduce a broader class of $\textit{autocratic strategies}$ by extending zero-determinant strategies to iterated games with more general action spaces. We use the continuous Donation Game as an example, which represents an instance of the Prisoner's Dilemma that intuitively extends to a continuous range of cooperation levels. Surprisingly, despite the fact that the opponent has infinitely many donation levels from which to choose, a player can devise an autocratic strategy to enforce a linear relationship between his or her payoff and that of the opponent even when restricting his or her actions to merely two discrete levels of cooperation throughout the course of the interaction. In particular, a player can use such a strategy to extort an unfair share of the payoffs from the opponent. Therefore, although the action space for the continuous Donation Game dwarfs that of the classical Prisoner's Dilemma, players can still devise relatively simple autocratic and, in particular, extortionate strategies.

Abstract:
In evolutionary game theory, an important measure of a mutant trait (strategy) is its ability to invade and take over an otherwise-monomorphic population. Typically, one quantifies the success of a mutant strategy via the probability that a randomly occurring mutant will fixate in the population. However, in a structured population, this fixation probability may depend on where the mutant arises. Moreover, the fixation probability is just one quantity by which one can measure the success of a mutant; fixation time, for instance, is another. We define a notion of homogeneity for evolutionary games that captures what it means for two single-mutant states, i.e. two configurations of a single mutant in an otherwise-monomorphic population, to be "evolutionarily equivalent" in the sense that all measures of evolutionary success are the same for both configurations. Using asymmetric games, we argue that the term "homogeneous" should apply to the evolutionary process as a whole rather than to just the population structure. For evolutionary matrix games in graph-structured populations, we give precise conditions under which the resulting process is homogeneous. Finally, we show that asymmetric matrix games can be reduced to symmetric games if the population structure possesses a sufficient degree of symmetry.

Abstract:
Evolutionary graph theory is a well established framework for modelling the evolution of social behaviours in structured populations. An emerging consensus in this field is that graphs that exhibit heterogeneity in the number of connections between individuals are more conducive to the spread of cooperative behaviours. In this article we show that such a conclusion largely depends on the individual-level interactions that take place. In particular, averaging payoffs garnered through game interactions rather than accumulating the payoffs can altogether remove the cooperative advantage of heterogeneous graphs while such a difference does not affect the outcome on homogeneous structures. In addition, the rate at which game interactions occur can alter the evolutionary outcome. Less interactions allow heterogeneous graphs to support more cooperation than homogeneous graphs, while higher rates of interactions make homogeneous and heterogeneous graphs virtually indistinguishable in their ability to support cooperation. Most importantly, we show that common measures of evolutionary advantage used in homogeneous populations, such as a comparison of the fixation probability of a rare mutant to that of the resident type, are no longer valid in heterogeneous populations. Heterogeneity causes a bias in where mutations occur in the population which affects the mutant's fixation probability. We derive the appropriate measures for heterogeneous populations that account for this bias.

Abstract:
Coevolving and competing species or game-theoretic strategies exhibit rich and complex dynamics for which a general theoretical framework based on finite populations is still lacking. Recently, an explicit mean-field description in the form of a Fokker-Planck equation was derived for frequency-dependent selection with two strategies in finite populations based on microscopic processes [A.Traulsen, J.C. Claussen, and C.Hauert, Phys. Rev. Lett. 95, 238701 (2005)]. Here we generalize this approach in a twofold way: First, we extend the framework to an arbitrary number of strategies and second, we allow for mutations in the evolutionary process. The deterministic limit of infinite population size of the frequency dependent Moran process yields the adjusted replicator-mutator equation, which describes the combined effect of selection and mutation. For finite populations, we provide an extension taking random drift into account. In the limit of neutral selection, i.e. whenever the process is determined by random drift and mutations, the stationary strategy distribution is derived. This distribution forms the background for the coevolutionary process. In particular, a critical mutation rate $u_c$ is obtained separating two scenarios: above $u_c$ the population predominantly consists of a mixture of strategies whereas below $u_c$ the population tends to be in homogenous states. For one of the fundamental problems in evolutionary biology, the evolution of cooperation under Darwinian selection, we demonstrate that the analytical framework provides excellent approximations to individual based simulations even for rather small population sizes. This approach complements simulation results and provides a deeper, systematic understanding of coevolutionary dynamics.

Abstract:
Traditionally, frequency dependent evolutionary dynamics is described by deterministic replicator dynamics assuming implicitly infinite population sizes. Only recently have stochastic processes been introduced to study evolutionary dynamics in finite populations. However, the relationship between deterministic and stochastic approaches remained unclear. Here we solve this problem by explicitly considering large populations. In particular, we identify different microscopic stochastic processes that lead to the standard or the adjusted replicator dynamics. Moreover, differences on the individual level can lead to qualitatively different dynamics in asymmetric conflicts and, depending on the population size, can even invert the direction of the evolutionary process.

Abstract:
Frequency dependent selection and demographic fluctuations play important roles in evolutionary and ecological processes. Under frequency dependent selection, the average fitness of the population may increase or decrease based on interactions between individuals within the population. This should be reflected in fluctuations of the population size even in constant environments. Here, we propose a stochastic model, which naturally combines these two evolutionary ingredients by assuming frequency dependent competition between different types in an individual-based model. In contrast to previous game theoretic models, the carrying capacity of the population and thus the population size is determined by pairwise competition of individuals mediated by evolutionary games and demographic stochasticity. In the limit of infinite population size, the averaged stochastic dynamics is captured by the deterministic competitive Lotka-Volterra equations. In small populations, demographic stochasticity may instead lead to the extinction of the entire population. As the population size is driven by the fitness in evolutionary games, a population of cooperators is less prone to go extinct than a population of defectors, whereas in the usual systems of fixed size, the population would thrive regardless of its average payoff.