Truncation selection and payoff distributions applied to the replicator equation

Abstract

The replicator equation has been frequently used in the theoretical literature to explain a diverse array of biological phenomena. However, it makes several simplifying assumptions, namely complete mixing, an infinite population, asexual reproduction, proportional selection, and mean payoffs. Here, we relax the conditions of mean payoffs and proportional selection by incorporating payoff distributions and truncation selection into extensions of the replicator equation and agent-based models. In truncation selection, replicators with fitnesses above a threshold survive. The reproduction rate is equal for all survivors and is sufficient to replace the replicators that did not survive. We distinguish between two types of truncation: independent and dependent with respect to the fitness threshold. If the payoff variances from all strategy pairing are the same, then we recover the replicator equation from the independent truncation equation. However, if all payoff variances are not equal, then any boundary fixed point can be made stable (or unstable) if only the fitness threshold is chosen appropriately. We observed transient and complex dynamics in our models, which are not observed in replicator equations incorporating the same games. We conclude that the assumptions of mean payoffs and proportional selection in the replicator equation significantly impact replicator dynamics.