Bernstein Polynomial Approximation of Fixation Probability in Finite Population Evolutionary Games

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 <= w <= 1, where w is the selection pressure parameter, and converges uniformly to the exact solution as d -> infinity. We compare it to a widely used non-uniform approximation formula in the weak-selection limit (w similar to 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 similar to 1) (requiring the use of fewer modes to obtain the same level of accuracy) than in the weak selection regime.