Replicator Dynamics
Let $x_i(t)$ be the population share playing strategy $i$, with $x \in \Delta^{n-1}$ (the simplex). Fitness of strategy $i$ against the population is $f_i(x) = (Ax)_i$ for payoff matrix $A$. Mean fitness is $\bar{f}(x) = x^\top A x$. The replicator equation is:
\[\dot{x}_i = x_i\bigl[f_i(x) - \bar{f}(x)\bigr]\]Strategies with above-average fitness grow; below-average fitness shrinks. The simplex $\Delta^{n-1}$ is forward-invariant and all interior equilibria are Nash equilibria of the underlying game.
Evolutionarily Stable Strategy
A strategy $\sigma^$ is an evolutionarily stable strategy (ESS) if it can resist invasion by any rare mutant $\sigma \neq \sigma^$:
\[u(\sigma^*, \sigma^*) > u(\sigma, \sigma^*) \quad \text{or} \quad \bigl[u(\sigma^*, \sigma^*) = u(\sigma, \sigma^*) \text{ and } u(\sigma^*, \sigma) > u(\sigma, \sigma)\bigr]\]Every ESS corresponds to a Nash equilibrium, but not every Nash equilibrium is an ESS. ESS are asymptotically stable fixed points of the replicator dynamics under mild conditions.
Hawk-Dove Example
With resource value $V$ and injury cost $C > V$, the payoff matrix is:
| Hawk | Dove | |
|---|---|---|
| Hawk | $(V-C)/2$ | $V$ |
| Dove | $0$ | $V/2$ |
The unique interior Nash equilibrium is the mixed strategy $p^* = V/C$ (probability of Hawk). This mixed equilibrium is the unique ESS, and the replicator dynamics converge to it from any interior initial condition.