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. 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).
Equation The most general continuous form of the replicator equation is given by the differential equation:where x i {\displaystyle x_{i}} is the proportion of type i {\displaystyle i} in the population, x = ( x 1 , … , x n ) {\displaystyle x=(x_{1},\ldots ,x_{n})} is the vector of the distribution of types in the population, f i ( x ) {\displaystyle f_{i}(x)} is the fitness of type i {\displaystyle i} (which is dependent on the population), and ϕ ( x ) {\displaystyle \phi (x)} is the average population fitness (given by the weighted average of the fitness of the n {\displaystyle n} types in the population). Since the elements of the population vector x {\displaystyle x} sum to unity by definition, the equation is defined on the n-dimensional simplex. The replicator equation assumes a uniform population distribution; that is, it does not incorporate population structure into the fitness. The fitness landscape does incorporate the population distribution of types, in contrast to other similar equations, such as the quasispecies equation. In application, populations are generally finite, making the discrete version more realistic. The analysis is more difficult and computationally intensive in the discrete formulation, so the continuous form is often used, although there are significant properties that are lost due to this smoothing. Note that the continuous form can be obtained from the discrete form by a limiting process. To simplify analysis, fitness is often assumed to depend linearly upon the population distribution, which allows the replicator equation to be written in the form:
x i ˙ = x i ( ( A x ) i − x T A x ) {\displaystyle {\dot {x_{i}}}=x_{i}\left(\left(Ax\right)_{i}-x^{T}Ax\right)}
… excerpt ends here. Continue reading the full article.
