# Replicator equation

The replicator equation is a differential equation that defines the dynamics of evolutionary games. The standard form of the replicator equation assumes

• Infinite population size: Assuming an infinitely large population makes the equation easier to understand and analyze, as the dynamics of populations become deterministic when an infinite number of individuals is assumed. In the case of a finite population size, the dynamics of populations become stochastic
• Continuous time: The dynamics in a replicator equation are defined by the rates of birth and death of individuals, resulting in differential equations
• Complete mixing: It is assumed that all individuals in the population have an equal chance to meet each other in a game to determine their interdependent payoff or fitness
• Strategies breed true: Strategies are assumed to be inherited into the next population dependent on their expected payoff

Consider a population of [itex]n[itex] types. Let [itex]A[itex] be the [itex]n\times n[itex] payoff matrix defining the payoffs in the game. Let [itex]x[itex] be a vector of size [itex]n[itex] such that [itex]x_i[itex] denotes the frequency of type [itex]i[itex] in the population. Because of the assumption of infinitely large populations, all possible population states can be mapped to a population vector [itex]x[itex], and vice versa.

Since individuals meet randomly (complete mixing assumption), an individual's fitness, or expected payoff can be written as [itex]\left(Ax\right)_i[itex]. The mean fitness of the population as a whole can be written as [itex]x^TAx[itex].

The replicator equation can now be written as [itex]\dot{x_i}=x_i\left(\left(Ax\right)_i-x^TAx\right)[itex], defining the per capita rate of growth for type [itex]i[itex].

### References

• Hofbauer, J. and Sigmund, K. (1998) Evolutionary game dynamics

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