The path integral formula for the stochastic evolutionary game dynamics

¹ School of Ecology and Environment, Northwestern Polytechnical University - Xian 710072, China
² School of Mathematics and Statistics, Northwestern Polytechnical University - Xian 710072, China
³ South China Institute of Environment Science, Ministry of Ecology and Environment of China Guangzhou 510530, China
⁴ Kansai Photon Science Institute, National Institutes for Quantum Science and Technology - Kyoto, 619-0215, Japan
⁵ Key Laboratory of Animal Ecology and Conservation Biology, Institute of Zoology, Chinese Academy of Sciences Beijing 100101, China

^(a) E-mail: chaowang@nwpu.edu.cn (corresponding author)
^(b) E-mail: wangrw@nwpu.edu.cn (corresponding author)

Received: 31 March 2023
Accepted: 30 May 2023

Abstract

Although the long-term behavior of stochastic evolutionary game dynamics in finite populations has been fully investigated, its evolutionary characteristics in a limited period of time is still unclear. In order to answer this question, we introduce the formulation of the path integral approach for evolutionary game theory. In this framework, the transition probability is the sum of all the evolutionary paths. The path integral formula of the transition probability is expected to be a new mathematical tool to explore the stochastic game evolutionary dynamics. As an example, we derive the transition probability for stochastic evolutionary game dynamics by the path integral in a limited period of time with the updating rule of the Wright-Fisher process.