Evolutionary Game Theory

Evolutionary game theory applies the notion of equilibrium to populations without assuming rationality. Replicator dynamics explains how strategies become fixed through selection - a process that optimizes without planning.

• **Biology:** ritualised fights, 1:1 sex ratios, altruism in bee colonies - all predicted by ESS analysis without assuming rational agents
• **TCP/IP:** the TCP congestion control algorithm implements replicator dynamics for network throughput allocation
• **Behavioral economics:** imitation dynamics models of bounded rationality converge to evolutionary equilibrium rather than Nash
• **Language evolution:** Nowak-Komarova models show grammar stabilizes as an ESS in a population of speakers

Предварительные знания

• Nash equilibrium in mixed strategies
• Repeated games
• Basic probability theory

• Repeated games and the folk theorem

Replicator equations

John Maynard Smith introduced the evolutionarily stable strategy (ESS) in 1973 to explain ritualised combat in animals: why do deer use antlers in fights but rarely kill each other? The answer is not compassion but evolutionary equilibrium. In a population of 10,000 ostriches, Hawks and Doves maintain precisely the proportions predicted by the ESS model. No planner decides this - selection pressure does.

Evolutionarily stable strategies (ESS)

The Hawk-Dove game models resource competition. Hawk always fights; Dove retreats against Hawk, shares with Dove. When V > C, Hawks dominate (fighting always pays); when V < C, a mixed ESS at p* = V/C arises. The payoff matrix: (H,H) = ((V-C)/2, (V-C)/2), (H,D) = (V, 0), (D,H) = (0, V), (D,D) = (V/2, V/2).

Applications: hawk-dove, cooperation, ML

ESS applies beyond biology: TCP congestion control implements replicator dynamics for bandwidth allocation, behavioral economics models bounded rationality as an approximation to ESS dynamics, and language evolution models (Nowak, Komarova 2001) show grammar stabilizes as an ESS in a population of speakers.

Relations between ESS and classical game theory concepts: ESS implies NE (the first ESS condition is Nash's). Strict NE implies ESS (strictness guarantees the second condition). Mixed NE may or may not be ESS depending on whether the second condition holds.

Connections to other fields

Evolutionary game theory bridges mathematical biology, economics, and learning theory.

• Repeated Games and the Folk Theorem — Evolutionary dynamics select among the equilibria the folk theorem allows
• Evolutionary Game Theory — Introductory course on evolutionarily stable strategies and replicator dynamics
• Game Theory in ML — Multi-agent learning and GAN training are modelled by replicator-style dynamics

Итоги

• ESS: sigma* is evolutionarily stable if no mutant can invade; implies NE but not conversely
• Replicator dynamics: dx_i/dt = x_i*(f_i - f_bar); strategies with above-average fitness grow in frequency
• Hawk-Dove: ESS Hawk fraction = V/C when V < C; perturbations return the system to p*
• Fundamental theorem: mean fitness f_bar increases monotonically along replicator trajectories (Taylor-Jonker)
• Applications: biology (fights, sex ratios), networks (TCP), behavioral economics, language evolution

Вопросы для размышления

• Why does every ESS imply a Nash equilibrium but not every Nash equilibrium imply ESS?
• How is replicator dynamics related to reinforcement learning algorithms in multi-agent settings?
• Why is the 1:1 sex ratio an ESS in most populations, even though it might seem 'wasteful' to have so many males?

Связанные уроки

• la-25-quadratic-forms