• QUADRATIC COEFFICIENTS OF THE ‘1/3-RULE’ WITH FINITE POPULATION SIZE CONVERGENCE

PAUL F. SLADE*

Abstract


Frequency-dependent selection between two non-mutating strategies in a Moran model of random genetic drift yields a well-known evolutionary rule. When the fixation probability of one strategy exceeds the selectively neutral value, being the reciprocal of population size, its relative frequency in the population equilibrates to less than ⅓. Maclaurin series of the singleton type fixation probability function calculated at second order enables the convergent domain of the payoff matrix to be obtained exactly. Results include identification of the dominant payoff matrix entries at second order with respect to potential influence on the stochastic evolution of the game. The extent of violation of this evolutionary rule can be shown to depend on the values in the payoff matrix and selection intensity.  Second order coefficients obtained by direct calculus and precise algebra yield functions of population size and payoff matrix entries.  The calculations herein clarify the resultant sensitivity to selection intensity when compared to previous work. Finite population size convergence quantifies the applicability of the asymptotic inequalitiy from which the rule derives.


Keywords


Co-operation; Fixation probability; Frequency-dependent selection; Maclaurin series; Moran model.

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