Bernstein Polynomial Approximation of Fixation Probability in Finite Population Evolutionary Games

Jiyeon Park () and Paul K. Newton ()
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Jiyeon Park: University of Southern California
Paul K. Newton: University of Southern California

Dynamic Games and Applications, 2024, vol. 14, issue 3, No 7, 686-696

Abstract: 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 $$0 \le w \le 1$$ 0 ≤ w ≤ 1 , where w is the selection pressure parameter, and converges uniformly to the exact solution as $$d \rightarrow \infty $$ d → ∞ . We compare it to a widely used non-uniform approximation formula in the weak-selection limit ( $$w \sim 0$$ w ∼ 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 ( $$w \sim 1$$ w ∼ 1 ) (requiring the use of fewer modes to obtain the same level of accuracy) than in the weak selection regime.

Keywords: Finite-population evolutionary games; Fixation probability; Boundary layers; Rate of evolution; Bernstein polynomials; Markov processes (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13235-023-00509-8

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