Mean-Potential Law in Evolutionary Games.

Area of Science:

• Mathematical Biology
• Evolutionary Game Theory
• Stochastic Processes

Background:

• Connecting random walks, stochastic differential equations, and evolutionary game theory is complex.
• Discrete-state stochastic systems lack general potential functions for analysis.
• Calculating fixation probabilities in evolutionary dynamics is computationally challenging.

Purpose of the Study:

• To introduce a novel potential function for discrete-space stochastic systems.
• To establish a framework connecting stochastic differential equations and random walks in finite spaces.
• To develop new criteria for evolutionary stability in finite populations.

Main Methods:

• Developing a potential function for discrete-state stochastic systems.
• Establishing an exact correspondence between one-dimensional stochastic differential equations and random walks.
• Applying the potential function to compute fixation probabilities in systems with two absorbing states.

Main Results:

• A new potential function for discrete stochastic systems is introduced.
• The method provides an exact correspondence between stochastic differential equations and random walks in finite spaces.
• New criteria for evolutionary stability of pure Nash equilibria in finite populations are formulated, including a generalized mean-potential law.

Conclusions:

• The novel potential function offers a unified approach to random walks, stochastic differential equations, and evolutionary game theory.
• The method simplifies the computation of fixation probabilities in discrete stochastic dynamical systems.
• The findings provide a more general theoretical basis for established evolutionary game theory laws, such as the 1/3 law.