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:
Payoff and Fitness
Fitness reflects payoff plus baseline:
Strategy Frequency
Frequencies sum to 1:
Replicator Dynamics
Frequency change:
Game Examples
Hawk-Dove Game
Payoff (V = 50, C = 100):
Hawk Frequency Dynamics
Evolution of Hawk strategy frequency.
Prisoner’s Dilemma
Payoff:
Rock-Paper-Scissors
Payoff:
Evolutionary Dynamics
Replicator Equation
For Hawk-Dove:
Stability
Stable if eigenvalues of Jacobian \( J = \frac{\partial \dot{x_i}}{\partial x_j} \) are negative.
Moran Process
Fixation probability: