Bernstein Polynomial Approximation of Fixation Probability in Finite Population Evolutionary Games

Abstract

We use the Bernstein polynomials of degree d as the basis for constructing a uniform approximation to the rate of evolution (related to the fixation probability) of a species in a two-component finite-population, well-mixed, frequency-dependent evolutionary game setting. The approximation is valid over the full range $0 \le w \le 1$, where w is the selection pressure parameter, and converges uniformly to the exact solution as $d \rightarrow \infty$. We compare it to a widely used non-uniform approximation formula in the weak-selection limit ($w \sim 0$) as well as numerically computed values of the exact solution. Because of a boundary layer that occurs in the weak-selection limit, the Bernstein polynomial method is more efficient at approximating the rate of evolution in the strong selection region ($w \sim 1$) (requiring the use of fewer modes to obtain the same level of accuracy) than in the weak selection regime.