Replicator equation

In mathematics, the replicator equation is a type of dynamical system used in evolutionary game theory to model how the frequency of strategies in a population changes over time. It is a deterministic, monotone, non-linear, and non-innovative dynamic that captures the principle of natural selection in strategic interactions.^[1]

The replicator equation describes how strategies with higher-than-average fitness increase in frequency, while less successful strategies decline. Unlike other models of replication—such as the quasispecies model—the replicator equation allows the fitness of each type to depend dynamically on the distribution of population types, making the fitness function an endogenous component of the system. This allows it to model frequency-dependent selection, where the success of a strategy depends on its prevalence relative to others.

Another key difference from the quasispecies model is that the replicator equation does not include mechanisms for mutation or the introduction of new strategies, and is thus considered non-innovative. It assumes all strategies are present from the outset and models only the relative growth or decline of their proportions over time.

Replicator dynamics have been widely applied in fields such as biology (to study evolution and population dynamics), economics (to analyze bounded rationality and strategy evolution), and machine learning (particularly in multi-agent systems and reinforcement learning).