Evolutionary Game Theory

Evolutionary Game Theory applies game theory to model strategy evolution in biological and social populations, pioneered by John Maynard Smith and George R. Price. Unlike classical game theory, it assumes strategies are inherited, driven by natural selection rather than rational choice. This MathMultiverse guide explores Evolutionary Stable Strategies (ESS), replicator dynamics, and applications in ecology, economics, and social behavior, with interactive visualizations.

From animal conflicts to human cooperation, it explains stable behaviors in dynamic systems, using mathematical models like the Hawk-Dove game or Prisoner’s Dilemma.

Core Concepts

Evolutionary Stable Strategy (ESS)

An ESS resists invasion by rare mutant strategies:

\[ u(s^*, s^*) > u(s, s^*) \text{ or } u(s^*, s^*) = u(s, s^*), u(s^*, s) > u(s, s) \]

Payoff and Fitness

Fitness reflects payoff plus baseline:

\[ F_i(s_i, s_{-i}) = u_i(s_i, s_{-i}) + b \]

Strategy Frequency

Frequencies sum to 1:

\[ \sum_{i=1}^n x_i = 1, \ x_i \geq 0 \]

Replicator Dynamics

Frequency change:

\[ \dot{x_i} = x_i [u_i(x) - \bar{u}(x)] \]

Game Examples

Hawk-Dove Game

Payoff (V = 50, C = 100):

\[ \begin{array}{c|cc} & H & D \\ \hline H & (-25, -25) & (50, 0) \\ D & (0, 50) & (25, 25) \end{array} \] \[ x_H^* = \frac{V}{C} = 0.5 \]

Hawk Frequency Dynamics

Evolution of Hawk strategy frequency.

Prisoner’s Dilemma

Payoff:

\[ \begin{array}{c|cc} & C & D \\ \hline C & (3, 3) & (0, 5) \\ D & (5, 0) & (1, 1) \end{array} \]

Rock-Paper-Scissors

Payoff:

\[ \begin{array}{c|ccc} & R & P & S \\ \hline R & (0, 0) & (-1, 1) & (1, -1) \\ P & (1, -1) & (0, 0) & (-1, 1) \\ S & (-1, 1) & (1, -1) & (0, 0) \end{array} \]

Evolutionary Dynamics

Replicator Equation

For Hawk-Dove:

\[ \dot{x_H} = x_H (1 - x_H) \left[ \frac{50 - 100}{2} x_H + 50 (1 - x_H) - \frac{50}{2} (1 - x_H) \right] \]

Stability

Stable if eigenvalues of Jacobian \( J = \frac{\partial \dot{x_i}}{\partial x_j} \) are negative.

Moran Process

Fixation probability:

\[ \phi_i = \frac{1}{1 + \sum_{k=1}^{N-1} \prod_{j=1}^k \frac{F_j}{F_{j+1}}} \]