Appendix G: Evolutionary Stable Strategies and Population Games
Definition G.1 Given the symmetric game (A, AT), pure strategy i∗ is evolutionary stable (ESS) if there is a 0 < p∗ < 1 so that
Definition G.2 A strategy X∗ is an ESS against (deviant strategy) strategies X1,…, Xs if either of (1) or (2) hold:
(1) u(x∗, x∗) > u(xk, x∗), for each k = 1, 2,…, s,
(2) for any xk such that u(x∗, x∗) = u(xk, x∗),we must have u(x∗, xj) > u(xk, xj), for all j = 1, 2,…, s.
Definition G.3 A strategy X∗ = (x∗, 1 − x∗) is an evolutionary stable strategy if for every strategy X = (x, 1 − x), with x ≠ x∗, there is some px (0, 1), which depends on the particular choice x, such that
G.1 Population Games
G.1.1 THE REPLICATOR EQUATIONS
or, equivalently,
Theorem G.4 Suppose that you have a system of differential equations
Assume that f : → and ∂f/∂pi are continuous. Then for any initial condition ...
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