Hawk-Dove — Evolutionarily Stable Strategy
A population of animals contests resources. Hawks fight to win (high reward, but injury cost). Doves share or flee (modest reward, no injury). When two hawks meet, they both pay injury cost C; against a dove, hawk takes V; doves split V; two doves split with cost 0. The frequency of hawks evolves toward an ESS — a stable equilibrium John Maynard Smith identified in 1973.
Replicator dynamics — frequency space
why the ESS is V/C
Payoffs (V = resource value, C = injury cost from a fight):
• Hawk vs Hawk: fight → expected (V − C)/2 each
• Hawk vs Dove: hawk takes everything → V for hawk, 0 for dove
• Dove vs Hawk: dove flees → 0 for dove, V for hawk
• Dove vs Dove: split → V/2 each
Let p be the fraction of hawks. Expected payoff:
E_H = p · (V−C)/2 + (1−p) · V
E_D = p · 0 + (1−p) · V/2
Setting E_H = E_D and solving: p* = V/C (when V < C). This is the evolutionarily stable strategy (ESS). Below p*, hawks do better and grow; above p*, doves do better and grow. The system converges to V/C.
When V > C (the resource is worth the cost of fighting): hawks always win — the entire population becomes hawks. When V < C: the ESS is mixed, with hawks proportional to V/C.
Replicator dynamics: dp/dt = p · (1 − p) · (E_H − E_D) — strategies that out-perform the population mean grow proportionally to their advantage. This is the deterministic limit of a large random-matching population. The same equation governs population genetics (allele frequencies under selection) and language evolution (competing dialects).
Why this matters: ESS is the bridge between game theory and Darwin. Strategies don't need rational agents — only differential reproduction. Selection finds Nash equilibria automatically.