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Register a new userEvolutionary and asymptotic stability in symmetric multi-player games
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We provide a classification of symmetric three-player games with two strategies and investigate evolutionary and asymptotic stability (in the replicator dynamics) of their Nash equilibria. We discuss similarities and differences between two-player and multi-player games. In particular, we construct examples which exhibit a novel behavior not found in two-player games.
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Animal behavior and evolution can often be described by game-theoretic models. Although in many situations, the number of players is very large, their strategic interactions are usually decomposed into a sum of two-player games. Only recently evolutionarily stable strategies were defined for multi-player games and their properties analyzed (Broom et al., 1997). Here we study the long-run behavior of stochastic dynamics of populations of randomly matched individuals playing symmetric three-player games. We analyze stochastic stability of equilibria in games with multiple evolutionarily stable strategies. We also show that in some games, a population may not evolve in the long run to an evolutionarily stable equilibrium.
Interactions among individuals in natural populations often occur in a dynamically changing environment. Understanding the role of environmental variation in population dynamics has long been a central topic in theoretical ecology and population biology. However, the key question of how individuals, in the middle of challenging social dilemmas (e.g., the tragedy of the commons), modulate their behaviors to adapt to the fluctuation of the environment has not yet been addressed satisfactorily. Utilizing evolutionary game theory and stochastic games, we develop a game-theoretical framework that incorporates the adaptive mechanism of reinforcement learning to investigate whether cooperative behaviors can evolve in the ever-changing group interaction environment. When the action choices of players are just slightly influenced by past reinforcements, we construct an analytical condition to determine whether cooperation can be favored over defection. Intuitively, this condition reveals why and how the environment can mediate cooperative dilemmas. Under our model architecture, we also compare this learning mechanism with two non-learning decision rules, and we find that learning significantly improves the propensity for cooperation in weak social dilemmas, and, in sharp contrast, hinders cooperation in strong social dilemmas. Our results suggest that in complex social-ecological dilemmas, learning enables the adaptation of individuals to varying environments.
How cooperation can evolve between players is an unsolved problem of biology. Here we use Hamiltonian dynamics of models of the Ising type to describe populations of cooperating and defecting players to show that the equilibrium fraction of cooperators is given by the expectation value of a thermal observable akin to a magnetization. We apply the formalism to the Public Goods game with three players, and show that a phase transition between cooperation and defection occurs that is equivalent to a transition in one-dimensional Ising crystals with long-range interactions. We then investigate the effect of punishment on cooperation and find that punishment plays the role of a magnetic field that leads to an alignment between players, thus encouraging cooperation. We suggest that a thermal Hamiltonian picture of the evolution of cooperation can generate other insights about the dynamics of evolving groups by mining the rich literature of critical dynamics in low-dimensional spin systems.
The Letter presents a novel way to connect random walks, stochastic differential equations, and evolutionary game theory. We introduce a new concept of potential function for discrete-space stochastic systems. It is based on a correspondence between one-dimensional stochastic differential equations and random walks, which may be exact not only in the continuous limit but also in finite-state spaces. Our method is useful for computation of fixation probabilities in discrete stochastic dynamical systems with two absorbing states. We apply it to evolutionary games, formulating two simple and intuitive criteria for evolutionary stability of pure Nash equilibria in finite populations. In particular, we show that the $1/3$ law of evolutionary games, introduced by Nowak et al [Nature, 2004], follows from a more general mean-potential law.
Population structure induced by both spatial embedding and more general networks of interaction, such as model social networks, have been shown to have a fundamental effect on the dynamics and outcome of evolutionary games. These effects have, however, proved to be sensitive to the details of the underlying topology and dynamics. Here we introduce a minimal population structure that is described by two distinct hierarchical levels of interaction. We believe this model is able to identify effects of spatial structure that do not depend on the details of the topology. We derive the dynamics governing the evolution of a system starting from fundamental individual level stochastic processes through two successive meanfield approximations. In our model of population structure the topology of interactions is described by only two parameters: the effective population size at the local scale and the relative strength of local dynamics to global mixing. We demonstrate, for example, the existence of a continuous transition leading to the dominance of cooperation in populations with hierarchical levels of unstructured mixing as the benefit to cost ratio becomes smaller then the local population size. Applying our model of spatial structure to the repeated prisoners dilemma we uncover a novel and counterintuitive mechanism by which the constant influx of defectors sustains cooperation. Further exploring the phase space of the repeated prisoners dilemma and also of the rock-paper-scissor game we find indications of rich structure and are able to reproduce several effects observed in other models with explicit spatial embedding, such as the maintenance of biodiversity and the emergence of global oscillations.
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