Evolutionarily Stable Sets

A set of strategies such that every strategy in the set resists invasion by any outside mutant.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

A set $A\subseteq \Delta(S)$ is evolutionarily stable if (i) for every $x,y\in A$ , $u(x,x)=u(y,x)=u(y,y)$ (equal payoffs within the set), and (ii) for every $x\in A$ , $y\notin A$ , there exists $\bar\varepsilon>0$ with $u(x,\,\varepsilon y+(1-\varepsilon)x)>u(y,\,\varepsilon y+(1-\varepsilon)x)$ for all $\varepsilon\in(0,\bar\varepsilon)$ .

Thomas (1985) introduced ESSets to handle games where a single ESS does not exist but a set is collectively stable.

Intuition

When no single strategy is unbeatable, a pool of behavior may collectively resist invasion.

Think of a set of social norms or behavioral conventions that mutually reinforce each other.

Worked example

In Rock-Paper-Scissors, the entire interior of the simplex is an evolutionarily stable set under certain payoff variants.

In the War of Attrition, any symmetric distribution over waiting times can form part of an ESSet.

The math

An ESSet is a closed singleton if and only if it contains an ESS.

Neutrally stable strategies (NSS) form a weaker version: $u(x,y)\ge u(y,y)$ for all $y$ near $x$ .

Where it is used

Understanding polymorphism in biology, cultural norms, and convention formation.

Models situations where multiple behaviors co-exist stably.

Go deeper

More in Game theory

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