The limits of weak selection and large population size in evolutionary game theory

TL;DR

This paper investigates how the order of applying weak selection and large population limits affects mathematical results in evolutionary game theory, revealing differences in fixation probabilities but not in strategy success conditions.

Contribution

It provides formal definitions of the Nw and wN limits and analyzes their impact on fixation probabilities in the Moran process.

Findings

Asymptotic fixation probabilities differ between Nw and wN limits.

Conditions for strategy success are unaffected by the order of limits.

Mathematical expressions for fixation probabilities are derived for both limits.

Abstract

Evolutionary game theory is a mathematical approach to studying how social behaviors evolve. In many recent works, evolutionary competition between strategies is modeled as a stochastic process in a finite population. In this context, two limits are both mathematically convenient and biologically relevant: weak selection and large population size. These limits can be combined in different ways, leading to potentially different results. We consider two orderings: the wN limit, in which weak selection is applied before the large population limit, and the Nw limit, in which the order is reversed. Formal mathematical definitions of the Nw and wN limits are provided. Applying these definitions to the Moran process of evolutionary game theory, we obtain asymptotic expressions for fixation probability and conditions for success in these limits. We find that the asymptotic expressions for fixation probability, and the conditions for a strategy to be favored over a neutral mutation, are different in the Nw and wN limits. However, the ordering of limits does not affect the conditions for one strategy to be favored over another.