Solutions Manual to Accompany Game Theory: An Introduction, 2nd Edition

Appendix G: Evolutionary Stable Strategies and Population Games

Definition G.1   Given the symmetric game (A, A^T), 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,…, X_s if either of (1) or (2) hold:

(1) u(x^∗, x^∗) > u(x_k, x^∗), for each k = 1, 2,…, s,

(2) for any x_k such that u(x^∗, x^∗) = u(x_k, x^∗),we must have u(x^∗, x_j) > u(x_k, x_j), 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 p_x (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/∂p_i are continuous. Then for any initial condition ...