Dominant Cubic Coefficients of the ‘1/3-Rule’ Reduce Contest Domains

Paul F. Slade
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Paul F. Slade: Computational Biology and Bioinformatics Unit, Research School of Biology, Australian National University, Acton 2601, Australia

Mathematics, 2019, vol. 7, issue 6, 1-11

Abstract: Antagonistic exploitation in competition with a cooperative strategy defines a social dilemma, whereby eventually overall fitness of the population decreases. Frequency-dependent selection between two non-mutating strategies in a Moran model of random genetic drift yields an evolutionary rule of biological game theory. When a singleton fixation probability of co-operation exceeds the selectively neutral value being the reciprocal of population size, its relative frequency in the population equilibrates to less than 1/3. Maclaurin series of a singleton type fixation probability function calculated at third order enables the convergent domain of the payoff matrix to be identified. Asymptotically dominant third order coefficients of payoff matrix entries were derived. Quantitative analysis illustrates non-negligibility of the quadratic and cubic coefficients in Maclaurin series with selection being inversely proportional to population size. Novel corollaries identify the domain of payoff matrix entries that determines polarity of second order terms, with either non-harmful or harmful contests. Violation of this evolutionary rule observed with non-harmful contests depends on the normalized payoff matrix entries and selection differential. Significant violations of the evolutionary rule were not observed with harmful contests.

Keywords: contested payoff; co-operation; fixation probability; frequency-dependent selection; Moran model (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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