Stochastic dynamics of invasion and fixation

We study evolutionary game dynamics in finite populations. We analyze an evolutionary process, which we call pairwise comparison, for which we adopt the ubiquitous Fermi distribution function from statistical mechanics. The inverse temperature in this process controls the intensity of selection, leading to a unified framework for evolutionary dynamics at all intensities of selection, from random drift to imitation dynamics. We derive a simple closed formula that determines the feasibility of cooperation in finite populations, whenever cooperation is modeled in terms of any symmetric two-person game. In contrast with previous results, the present formula is valid at all intensities of selection and for any initial condition. We investigate the evolutionary dynamics of cooperators in finite populations, and study the interplay between intensity of selection and the remnants of interior fixed points in infinite populations, as a function of a given initial number of cooperators, showing how this interplay strongly affects the approach to fixation of a given trait in finite populations, leading to counterintuitive results at different intensities of selection.

Figure 1

(Color online) Fixation probabilities for the three qualitatively different dilemmas of cooperation describable by $2\times 2$ games. Simulation results (filled symbols) coincide perfectly with the theoretical results Eqs. (7, 8) (solid lines). For comparison, the theoretical result for the Moran process at maximum selection intensity ($w=1$, open circles) and the neutral selection result ($\beta =0$, diagonal line) are shown. (a) In the prisoner’s dilemma, defection is dominant. As a result, only for very weak selection $(\beta =0.01)$ do cooperators acquire significant chances of fixation. (b) The stag-hunt game has an unstable equilibrium at $k^{*}=(N+2)∕2$, and the transition across this point depends sensitively on the intensity of selection. (c) The snowdrift game exhibits an internal equilibrium at $k^{*}=(N-2)∕2$, which is an inflection point in the theoretical curves. A small population size of $N=20$ has been chosen on purpose to stress the quality of the theoretical result related to Eqs. (7, 8). Each simulation result corresponds to the fraction of fixation of cooperators in ${10}^4$ independent realizations.