Evolutionarily Stable Strategies and Replicator Dynamics

Evolution Instead of Rationality

Classical game theory assumes fully rational players who know the equilibrium. Evolutionary game theory (EGT) offers an alternative foundation: instead of rationality—selection. Individuals with higher fitness (payoff) reproduce faster. Equilibrium emerges not from reasoning, but from evolutionary dynamics.

Maynard Smith and Price (1973) developed EGT to explain animal behavior. Later, the concepts were transferred to economics, sociology, computer science. The main question: which strategies “survive” in the population?

Evolutionarily Stable Strategies (ESS)

Definition: A strategy σ* is an ESS if, for a sufficiently small invasion of mutants with strategy σ ≠ σ*, the “resident” population σ* “displaces” the mutants:

u(σ*, εσ + (1−ε)σ*) > u(σ, εσ + (1−ε)σ*) for all σ ≠ σ* and small ε > 0

ESS Conditions (sufficient):

1. u(σ*, σ*) > u(σ, σ*) for all σ ≠ σ* — strict Nash equilibrium, OR
2. u(σ*, σ*) = u(σ, σ*) AND u(σ*, σ) > u(σ, σ) — in case of a tie against residents, σ* is better against mutants

Connection with Nash Equilibrium: ESS → Nash equilibrium (in mixed strategies). The reverse is not true: not every Nash equilibrium is an ESS.

Replicator Dynamics

In continuous time, let xᵢ be the share of the population with strategy i. Replicator dynamics:

ẋᵢ = xᵢ · [uᵢ(x) − ū(x)]

Where uᵢ(x) = Σⱼ xⱼ aᵢⱼ — fitness of strategy i, ū(x) = Σᵢ xᵢ uᵢ — average fitness.

Meaning: strategies whose fitness is above average grow; below average—decline. The vector (xᵢ) remains on the simplex Σxᵢ = 1.

Connection with ESS: ESS is an asymptotically stable equilibrium of replicator dynamics (stable to small perturbations). Nash equilibria that are not ESS may be neutrally stable or unstable.

Numerical Example: The “Hawk–Dove” Game

A population with strategies: Hawk (H) — aggressive, Dove (D) — peaceful. Payoff matrix for V=4, C=6:

If two hawks meet: both fight, expected payoff (4−6)/2 = −1 each. Hawk–Dove: hawk takes V=4, dove retreats (0). Two doves: divide V/2=2 each.

Let x be the share of hawks, (1−x) — doves.

u(H, x) = −1·x + 4·(1−x) = 4 − 5x. u(D, x) = 0·x + 2·(1−x) = 2 − 2x. ū = x·u(H) + (1−x)·u(D).

Equilibrium ẋ = 0: u(H, x) = u(D, x) → 4 − 5x = 2 − 2x → x* = 2/3. Share of hawks at equilibrium: 2/3 ≈ 67%.

Checking ESS at x* = 2/3: this is an internal mixed strategy equilibrium—and it is an ESS when V < C. When V > C: “Hawk” is the pure ESS (aggression is optimal).

Biological meaning: with high conflict cost C > V, evolution maintains a mixed population. Territorial disputes among birds, competition for mates—this is exactly such a mixed strategy.

Multipopulation Games and Applications

In asymmetric games (buyers–sellers, males–females): replicator dynamics for two populations: ẋᵢ = xᵢ(uᵢ(x,y) − ū_x), ẏⱼ = yⱼ(vⱼ(x,y) − v̄_y). Stability—through the Jacobian at the equilibrium point.

Economic Applications:

Evolution of standards (QWERTY vs Dvorak): Two keyboard standards. Coordination game: if everyone uses QWERTY, it’s best to use QWERTY too. QWERTY is an ESS (even if Dvorak is “objectively better”). This is “path dependence”—dependence on the initial path.

Price wars in an industry: Hawk = price cutting, Dove = maintaining high prices. If C > V → mixed equilibrium: some firms aggressively cut prices, some do not.

Biological Applications:

Sex ratio: Fisher proved that 1:1 is an ESS. If males < females → a male has higher reproductive success → selection in favor of male birth → returns to 1:1.

Altruism: Hamilton’s rule: altruistic behavior is ESS-stable if r · Δbenefit > Δcost (r is the relatedness coefficient). The “altruism gene” spreads through relatives.

Evolutionary Game Theory in Biology and Economics

Evolutionary game theory explains how strategies spread in populations without rational choice. In biology, replicator dynamics model frequency-dependent natural selection: strategies with higher fitness grow faster. Multilevel selection—the dilemma between within-group selection (driving towards selfishness) and between-group selection (driving towards cooperation)—is described by generalized replicators with hierarchical populations. In behavioral economics, Simon’s “bounded rationality” concept converges with evolutionary equilibrium: people use rules of thumb (heuristics) that are stable against “invasion” of non-standard strategies in the ESS sense. Signaling games in evolutionary biology—self-advertisement: peacock tail, bird song—are stable signals of gene quality; ESS predicts the level of signaling cost. Diffusion of innovations in social networks is described by replicator dynamics on graphs: a new technology spreads if it is an ESS in local interactions. In finance, technical traders using heuristics survive only if their strategies are close to ESS in an environment with other players—this is exactly what explains the “technical magic” of the market.

Assignment: (a) For V=2, C=6: find the ESS, equilibrium share of hawks x*. (b) Construct the replicator dynamics equation ẋ = x(1−x)(u(H, x)−u(D, x)) and show that x* is stable. (c) For V=6, C=4: what is the ESS? How does the biological interpretation change?