Abstract
We introduce a mean-field term to an evolutionary spatial game model. Namely, we consider the game of Nowak and May, based on the Prisoner's dilemma, and augment the game rules by a self-consistent mean-field term. This way, an agent operates based on local information from its neighbors and nonlocal information via the mean-field coupling. We simulate the model and construct the steady-state phase diagram, which shows significant new features due to the mean-field term: while for the game of Nowak and May, steady states are characterized by a constant mean density of cooperators, the mean-field game contains steady states with a continuous dependence of the density on the payoff parameter. Moreover, the mean-field term changes the nature of transitions from discontinuous jumps in the steady-state density to jumps in the first derivative. The main effects are observed for stationary steady states, which are parametrically close to chaotic states: the mean-field coupling drives such stationary states into spatial chaos. Our approach can be readily generalized to a broad class of spatial evolutionary games with deterministic and stochastic decision rules.
Highlights
The mean-field approximation, initially devised in 1907 in the context ofmagnetic properties of materials [1], is since playing a truly central role in a variety of branches of physics, ranging from superconductivity [2] to ferromagnetism of disordered metals [3]; from ultracold quantum gases [4] to spin glasses [5], to name just a few
We simulate the model and construct the steady-state phase diagram, which shows significant new features due to the mean-field term: while for the game of Nowak and May, steady states are characterized by a constant mean density of cooperators, the mean-field game contains steady states with a continuous dependence of the density on the payoff parameter
The main effects are observed for stationary steady states, which are parametrically close to chaotic states: the mean-field coupling drives such stationary states into spatial chaos
Summary
The mean-field approximation, initially devised in 1907 in the context of (ferro)magnetic properties of materials [1], is since playing a truly central role in a variety of branches of physics, ranging from superconductivity [2] to ferromagnetism of disordered metals [3]; from ultracold quantum gases [4] to spin glasses [5], to name just a few. We demonstrate how mean-field ideas can be applied to a system that does not have an explicit representation in the language of statistical mechanics Instead, it is a deterministic dynamical system with a steady-state, i.e., a long-lived state with a well-defined “thermodynamic limit” of the infinite system size. One limitation of the replicator equation approach is that it does not allow for structured populations, where the interaction range is restricted to a local neighborhood Such populations are known to exhibit long-time behavior beyond standard replicator dynamics [19]. We construct an evolutionary game model which features both kinds of effects: purely local interactions with nearest neighbors, and a self-consistent, mean-field-type coupling to the order parameter.
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