Mean-Potential Law in Evolutionary Games

The Letter presents a novel way to connect random walks, stochastic differential equations, and evolutionary game theory. We introduce a new concept of a potential function for discrete-space stochastic systems. It is based on a correspondence between one-dimensional stochastic differential equations and random walks, which may be exact not only in the continuous limit but also in finite-state spaces. Our method is useful for computation of fixation probabilities in discrete stochastic dynamical systems with two absorbing states. We apply it to evolutionary games, formulating two simple and intuitive criteria for evolutionary stability of pure Nash equilibria in finite populations. In particular, we show that the $1/3$ law of evolutionary games, introduced by Nowak et al. [Nature, 2004], follows from a more general mean-potential law.

Figure 1

Potential function, guessed from Eq. (22) (approximate) and computed numerically (exact), for two models with linear advantage update: (top) a stag-hunt game with $x^{*}=0.25$, (bottom) a three-player game with the following payoffs: $A$ receives 4 when playing against $AA$ or $BB$ and 0 otherwise, $B$ gets 17 against $AB$ or $BA$, otherwise nothing. The number of players and selection pressure are $N=10$ and $w=0.1$. In the first model, both strategies are pure Nash equilibria. $A$ is evolutionarily stable in finite populations because at the point when all play $A$, the potential (i) has a local minimum and (ii) is below the mean potential. In the second model, both pure strategies are unstable—$A$ does not satisfy condition (ii), whereas $B$ fails to meet condition (i).