Fitness-based models and pairwise comparison models of evolutionary games are typically different—even in unstructured populations

Bin Wu,Tobias Galla,Arne Traulsen,Benedikt Bauer

doi:10.1088/1367-2630/17/2/023043

Abstract

The modeling of evolutionary game dynamics in finite populations requires microscopic processes that determine how strategies spread. The exact details of these processes are often chosen without much further consideration. Different types of microscopic models, including in particular fitness-based selection rules and pairwise comparison dynamics, are often used as if they were interchangeable. We challenge this view and investigate how robust these choices on the micro-level really are. We focus on a key macroscopic quantity, the probability for a single mutant to take over a population of wild-type individuals. We show that even in unstructured populations there is only one pair of a fitness-based process and a pairwise comparison process leading to identical outcomes for arbitrary games and for all intensities of selection. This strong restriction is not relaxed even when the class of pairwise comparison processes is broadened. This highlights the perils of making arbitrary choices at the micro-level without regard of the consequences at the macro-level.

Highlights

Evolutionary game theory is a powerful framework to model biological and social evolution when the success of an individual depends on the presence or absence of other strategies [1,2,3,4]
In summary we have challenged some of the key assumptions frequently made in modeling evolutionary dynamics
Fitness-based and pairwise comparison processes are often used as if these approaches were entirely exchangeable. This is appropriate—to a certain extent—when fitness is a positive constant as it is the case in many models of classical population genetics

Summary

Introduction

Evolutionary game theory is a powerful framework to model biological and social evolution when the success of an individual depends on the presence or absence of other strategies [1,2,3,4]. We show the choice of a fitness-based versus a pairwise comparison process is restricted to a unique pair if we require that for an arbitrary game, the two processes lead to identical fixation probabilities for all intensities of selection β This indicates that the choice of the microscopic process can make a difference even in unstructured populations. I.e. for payoff functions πAi and πBi, a fitness-based process assumes that at each time step an individual is selected for reproduction with a probability proportional to its fitness. This individual produces one identical offspring which replaces a randomly chosen individual in the population. 1, whereas the fixation probability for the pairwise comparison process reaches only discrete values, the behavior of the two processes has to be qualitatively different in terms of the fixation probability

Equivalence between two evolutionary process classes in fixation probability

Summary