Spatial Prisoner's Dilemma

On a 2D lattice, each cell plays the Prisoner's Dilemma with its neighbors and copies the strategy of the highest scorer. Spatial structure allows cooperators to form clusters, resisting invasion by defectors. A phase transition at b_c separates cooperation-dominated from defection-dominated regimes.

Payoff Matrix

CC→R=1, CD→S=0, DC→T=b, DD→P=0

Simulation

Generation0
Cooperators—
Defectors—
b_c (theory)≈ 1.8
Key result (Nowak & May 1992):
For 1 < b < 1.8: cooperators and defectors coexist — spatial clustering protects cooperators
For b > 1.8: defectors dominate
For b = 1: all cooperate

Fractal boundaries: The spatial patterns at coexistence have fractal cluster boundaries with Hausdorff dimension ≈ 1.76

Why cooperation persists: Cooperators earn more from C–C edges; spatial structure creates these edges preferentially.