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Bernstein Polynomial Approximation of Fixation Probability in Finite Population Evolutionary Games

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  • Jiyeon Park

    (University of Southern California)

  • Paul K. Newton

    (University of Southern California
    University of Southern California
    University of Southern California)

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.

Suggested Citation

  • Jiyeon Park & Paul K. Newton, 2024. "Bernstein Polynomial Approximation of Fixation Probability in Finite Population Evolutionary Games," Dynamic Games and Applications, Springer, vol. 14(3), pages 686-696, July.
    • Handle: RePEc:spr:dyngam:v:14:y:2024:i:3:d:10.1007_s13235-023-00509-8
    DOI: 10.1007/s13235-023-00509-8
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    References listed on IDEAS

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    1. Martin A. Nowak & Akira Sasaki & Christine Taylor & Drew Fudenberg, 2004. "Emergence of cooperation and evolutionary stability in finite populations," Nature, Nature, vol. 428(6983), pages 646-650, April.
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