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

Abstract

Evolutionary game theory has proved to be a powerful tool to probe the self-organization of collective behaviour by considering frequency-dependent fitness in evolutionary processes. It has shown that the stability of a strategy depends not only on the payoffs received after each encounter but also on the population’s size. Here, we study 2 × 2 games in well-mixed finite populations by analyzing the fixation probabilities of single mutants as functions of population size. We proved that nine of the 24 possible games always lead to monotonically decreasing functions, similarly to fixed fitness scenarios. However, fixation functions showed increasing regions under 12 distinct anti-coordination, coordination and dominance games. Perhaps counter-intuitively, this establishes that single-mutant strategies often benefit from being in larger populations. Fixation functions that increase from a global minimum to a positive asymptotic value are pervasive but may have been easily concealed by the weak selection limit. We obtained sufficient conditions to observe fixation increasing for small populations and three distinct ways this can occur. Finally, we describe fixation functions with the increasing regions bounded by two extremes under intermediate population sizes. We associate their occurrence with transitions from having one global extreme to other shapes.

Network topology and movement cost, not updating mechanism, determine the evolution of cooperation in mobile structured populations

“…Initial models often assumed infinite populations [ 8 , 10 , 11 ] due to the simplicity that this assumption offers and their resulting mathematical tractability. However, this choice leaves out the impact that the finiteness of real populations has on emerging behaviour [ 12 , 13 ], and may make models inflexible to incorporating realistic aspects of populations [ 14 ]. Given this, a growing interest in finite population models emerged and established them as an essential part of the field [ 12 , 13 , 15 ].…”

“…However, this choice leaves out the impact that the finiteness of real populations has on emerging behaviour [ 12 , 13 ], and may make models inflexible to incorporating realistic aspects of populations [ 14 ]. Given this, a growing interest in finite population models emerged and established them as an essential part of the field [ 12 , 13 , 15 ].…”