Mean-field interactions in evolutionary spatial games

Physical Review Research. 2021. Vol. 3. Article L032072.

Antonov D., Burovskiy E., Shchur L.

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.

Keywords: evolutionary games, prisoner's dilemma, дилемма заключенного, mean-field approach, пространственные эволюционные игры, приближение среднего поля