Game theory

Berge Equilibrium

An equilibrium where each player maximizes the payoff of all others, assuming they do not deviate from their Berge strategies.

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

Definition

A strategy profile s∗s^* is a Berge equilibrium if for each player ii, ui(si∗,s−i∗)≥ui(si,s−i∗)u_i(s_i^*,s_{-i}^*)\ge u_i(s_i,s_{-i}^*) for all si∈Sis_i\in S_i (Nash condition), and additionally ui(si∗,s−i∗)≥ui(si∗,s−i)u_i(s_i^*,s_{-i}^*)\ge u_i(s_i^*,s_{-i}) for all s−i∈S−is_{-i}\in S_{-i} (Berge condition).

The second condition says player ii's payoff is maximized at s∗s^* given ii plays their part — the opponent's choices do not harm ii at the equilibrium.

Intuition

In a Berge equilibrium, each player takes the action that is best for the others, creating a kind of altruistic balance.

It contrasts with Nash (self-interested) and provides a model of "I'll do what's good for you, trusting you'll do what's good for me."

Worked example

In a two-player coordination game with Pareto-ranked equilibria, the payoff-dominant equilibrium is often also a Berge equilibrium.

Team sports: each player positions themselves to benefit teammates, relying on teammates to do the same.

The math

Every Nash equilibrium that is also Pareto efficient and satisfies ui(si∗,s−i∗)=max⁡s−iui(si∗,s−i)u_i(s_i^*,s_{-i}^*)=\max_{s_{-i}}u_i(s_i^*,s_{-i}) for all ii is a Berge equilibrium.

Berge (1957) introduced the concept; Colman (2006) revived it as a model of team reasoning.

Where it is used

Team decision-making, cooperative behavior in social dilemmas, and evolutionary models of altruism.

Alternative to Nash in settings with group-oriented reasoning.

More in Game theory

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