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

Paul F Slade

doi:10.3390/math7060491

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.

Highlights

Selection between game strategies, on the order of magnitude as the reciprocal of population size, accords with partial differential equations, or diffusion theory, of population genetics [1,2,3,4]
Research on population genetics that mathematically reinvigorates evolutionary game theory intrigues the mainstream of theoretical biology [10,11,12]
Theorem 1 sharpens the calibration of the selection intensity obtained in preceding work [6], which concludes that the general validity of the rule required extremely weak selection, where p = 1

Summary

Introduction

On the order of magnitude as the reciprocal of population size, accords with partial differential equations, or diffusion theory, of population genetics [1,2,3,4]. Note that ρC (0) = ρD (0) = N1 , the singleton-type fixation probability with selective neutrality in a population of size N In this evolutionary rule, the population relative frequency of co-operation equilibrates to less than one-third. The calculations culminate in proof of Theorem 1 that describes the cubic limiting dominant term of the inequality from which the ‘1/3-rule’ derives This requires the Maclaurin series of ρC ($) up to third order. Inspection of Equation (7) yields the corresponding dominant coefficients in the Maclaurin series of singleton fixation probability at finite population size. These dominant cubic coefficients can be separated into four subgroups. N − 21 i2 − N − 12 i3 + i4 ∼ 3(N − 1)−3 N−1 i=1 N i − N i + N 4 − N 2 + N 2 − 8 h i

Dominant Triple Cubic Coefficients

Simplification of Third-Order Maclaurin Series

Non-Harmful Contests

Harmful Contests

Discussion