Multi-strategy evolutionary games: A Markov chain approach

Interacting strategies in evolutionary games is studied analytically in a well-mixed population using a Markov chain method. By establishing a correspondence between an evolutionary game and Markov chain dynamics, we show that results obtained from the fundamental matrix method in Markov chain dynamics are equivalent to corresponding ones in the evolutionary game. In the conventional fundamental matrix method, quantities like fixation probability and fixation time are calculable. Using a theorem in the fundamental matrix method, conditional fixation time in the absorbing Markov chain is calculable. Also, in the ergodic Markov chain, the stationary probability distribution that describes the Markov chain's stationary state is calculable analytically. Finally, the Rock, scissor, paper evolutionary game are evaluated as an example, and the results of the analytical method and simulations are compared. Using this analytical method saves time and computational facility compared to prevalent simulation methods.

Automated summary

This study analytically examines the interacting strategies in evolutionary games using a Markov chain method in a well-mixed population. By establishing a correspondence between an evolutionary game and Markov chain dynamics, the study demonstrates the equivalence of results obtained from the fundamental matrix method in Markov chain dynamics to those in the evolutionary game. The study shows that the fundamental matrix method allows for the calculation of quantities such as fixation probability and fixation time. Furthermore, the method enables the calculation of conditional fixation time in the absorbing Markov chain and the analytical calculation of the stationary probability distribution in the ergodic Markov chain. The study evaluates the Rock, scissor, paper evolutionary game as an example and compares the results of the analytical method with simulations, showing that the analytical method saves time and computational resources compared to prevalent simulation methods.