Evolutionarily Stable Strategy

A strategy that, if adopted by the population, cannot be invaded by any rare mutant.

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

Definition

$x^*$ is an ESS if for every $y\ne x^*$ there exists $\bar\varepsilon>0$ such that $u(x^*,\,\varepsilon y+(1-\varepsilon)x^*)>u(y,\,\varepsilon y+(1-\varepsilon)x^*)$ for all $\varepsilon\in(0,\bar\varepsilon)$ .

Maynard Smith-Price equivalent: $u(x^*,x^*)\ge u(y,x^*)$ for all $y$ , and if $u(x^*,x^*)=u(y,x^*)$ then $u(x^*,y)>u(y,y)$ .

Intuition

Once nearly everyone plays $x^*$ , no small group of mutants can earn strictly more and thus they cannot spread.

ESS is a local stability refinement of symmetric Nash.

Worked example

Hawk-Dove: the mixed ESS is $x_H^*=v/c$ when $c>v$ .

In Rock-Paper-Scissors, the unique symmetric Nash $(1/3,1/3,1/3)$ is not an ESS — populations cycle under replicator dynamics.

The math

Every ESS is a Nash equilibrium; not every Nash equilibrium is an ESS. An ESS is asymptotically stable under replicator dynamics (Taylor-Jonker).

For matrix games, ESS $\Leftrightarrow$ $x^*$ is a local strict maximum of $u(x,x)$ on an invariant face.

Where it is used

Explains animal contests, plant seed strategies, honest signaling, and robust equilibrium selection.

Widely used to motivate equilibrium refinements in economics.

More in Game theory

Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.