How small are small mutation rates?

Area of Science:

• Evolutionary Game Theory
• Population Genetics
• Mathematical Biology

Background:

• Evolutionary game dynamics in finite populations are often approximated using Markov chains when mutation rates are low.
• The accuracy of this approximation depends critically on the mutation rate relative to population size and game structure.

Purpose of the Study:

• To determine the precise mutation rate thresholds required for accurate Markov chain approximations of evolutionary game dynamics.
• To quantify the difference between true stationary distributions and approximated ones under varying mutation rates.

Main Methods:

• Analysis of evolutionary game dynamics in a finite population of size N.
• Calculation of the difference between real and approximated stationary distributions.
• Embedded Markov chain analysis on pure states under small mutation rates.

Main Results:

• For coexistence games, mutation rate (μ) must be < N^(-1/2)exp[-N] for accurate Markov chain approximation.
• For other game types, mutation rate < (N ln N)^(-1) is sufficient for good approximation.
• Results are applicable to imitation processes with arbitrary selection intensity.

Conclusions:

• The study provides critical mutation rate bounds for the validity of Markov chain approximations in evolutionary game theory.
• These findings are essential for accurately modeling evolutionary dynamics in finite populations, especially under rare mutation scenarios.