More can be better: An analysis of single-mutant fixation probability functions under $2\times2$ games

Abstract

The self-organization of collective behaviour is a topic of interest in numerous research fields, and in this context, evolutionary game theory has proved to be a powerful probing tool. Even though initial evolutionary models with frequency-dependent fitness assumed infinite populations, it has been shown that the stability of a strategy may depend not only on the game's payoff matrix but on the size of a finite population. To perform a systematic analysis of 2x2 games in well-mixed finite populations, we start by proving that 9 of the 24 possible payoff orderings always lead to single mutant fixation probability functions decreasing monotonically with population size as they trivially do under fixed fitness scenarios. However, we observe a diversity of fixation functions with increasing regions under 12 other orderings, which included anti-coordination games (e.g. Hawk-Dove/Snowdrift game), the fixation of dominating strategies (e.g. defectors in the Prisoner's Dilemma), and the fixation of stag hunters under that game (the only exception in coordination games). Fixation functions that increase from a global minimum to a finite asymptotic value are pervasive. These may have been easily concealed by the weak selection limit. We prove under which payoff matrices it is possible to have fixation increasing for the smallest populations and find three different ways this can happen. Finally, we describe two distinct fixation functions having two local extremes and associate them with transitions from ones with one global extreme.