Replicator Dynamics

ODEs describing how strategy frequencies evolve under selection proportional to fitness.

Ask the Game theory assistant1 min read · Updated September 9, 2026

Definition

$\dot x_i = x_i\bigl(f_i(x)-\bar f(x)\bigr)$ , where $f_i(x)=(Ax)_i$ is the fitness of pure strategy $i$ and $\bar f(x)=x^\top A x$ is the average.

The simplex $\Delta(S)$ and its faces are invariant under the flow.

Intuition

Strategies doing better than average grow; those doing worse decline.

No rationality is assumed — the dynamics are a pure selection model.

Worked example

Rock-Paper-Scissors under replicator dynamics exhibits closed orbits around the center $(1/3,1/3,1/3)$ .

In the Hawk-Dove game trajectories converge to the mixed ESS $x_H=v/c$ .

The math

For symmetric games, every ESS is asymptotically stable; strict Nash equilibria are asymptotically stable; interior Nash equilibria are Lyapunov-stable. Relative entropy to a Nash equilibrium is a Lyapunov function for doubly symmetric games.

Related to Lotka-Volterra equations via a nonlinear change of variables.

Where it is used

Used in population genetics, cultural learning, congestion adaptation, and multi-agent RL convergence analysis.

Models of social learning via imitation yield the same ODE.

More in Game theory

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