Non-Gaussian fluctuations arising from finite populations: Exact results for the evolutionary Moran process

The appropriate description of fluctuations within the framework of evolutionary game theory is a fundamental unsolved problem in the case of finite populations. The Moran process recently introduced into this context in Nowak et al., [Nature (London) 428, 646 (2004)] defines a promising standard model of evolutionary game theory in finite populations for which analytical results are accessible. In this paper, we derive the stationary distribution of the Moran process population dynamics for arbitrary $2\times 2$ games for the finite-size case. We show that a nonvanishing background fitness can be transformed to the vanishing case by rescaling the payoff matrix. In contrast to the common approach to mimic finite-size fluctuations by Gaussian distributed noise, the finite-size fluctuations can deviate significantly from a Gaussian distribution.

Figure 1

Stationary probability distribution for different evolutionary dynamics depending on the distance to the maximum $(N=100)$. For comparison, also the slow decay for neutral evolution is shown. The decay of the distribution can be fitted by a stretched exponential $\exp (-b{x}^{\gamma })$ with $\gamma =2.06$ (anticoordination game), $\gamma =0.87$ (constant fitness), and $\gamma =0.63$ (Prisoner’s Dilemma). The inset shows the same data where both axes are logarithmized, thus stretched exponentials appear as straight lines. The decay deviates significantly from a Gaussian distribution for constant fitness and Prisoner’s Dilemma, corresponding to a random motion in an anharmonic potential.

Figure 2

Scaling of the variance, normalized by $N$, of the finite-size fluctuations for anticoordination game (slope $-1∕2$), constant fitness (slope -1), and Prisoner’s dilemma (slope $-3∕2$). For neutral evolution (not shown) the variance increases faster than $N$.