Replicator Dynamics

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

Consider a population of

n

types. Let

A

be the

n	imes n

Payoff Matrix defining the payoffs in the game. Let

x

be a vector of size

n

such that

x_i

denotes the frequency of type

i

in the population. Because of the assumption of infinitely large populations, all possible population states can be mapped to a population vector

x

, and vice versa.

Since individuals meet randomly (complete mixing assumption), an individual's fitness, or expected payoff can be written as

\left(Ax
ight)_i

. The mean fitness of the population as a whole can be written as

x^TAx

.

The replicator equation can now be written as

\dot{x_i}=x_i\left(\left(Ax
ight)_i-x^TAx
ight)

, defining the per capita rate of growth for type

i

.

REFERENCES

Taylor, P. D. (1979). ''Evolutionarily Stable Strategies with Two Types of Players'' J. Appl. Prob. 16, 76-83.

Taylor, P. D., and Jonker, L. B. (1978). ''Evolutionarily Stable Strategies and Game Dynamics'' Math. Biosci. 40, 145-156.

	CATEGORIES ABOUT REPLICATOR EQUATION

	game theory

	differential equations

	evolutionary game theory

	evolutionary dynamics

Information About