Nash Bargaining Solution

The unique allocation satisfying Pareto-efficiency, symmetry, scale invariance, and IIA.

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

Definition

Given feasible set $F\subseteq\mathbb{R}^2$ and disagreement point $d$ , $\phi(F,d)=\arg\max_{x\in F,\,x\ge d}(x_1-d_1)(x_2-d_2).$

Generalizes to $n$ players via the weighted Nash product $\prod_i(x_i-d_i)^{\alpha_i}$ .

Intuition

Maximize the product of gains from agreement relative to the outside option.

It is the unique "fair" solution under Nash's four axioms.

Worked example

Splitting $1 with $d=(0,0)$ : the Nash solution is $(0.5,0.5)$ ; if $d=(0.3,0)$ , it becomes $(0.65,0.35)$ .

Wage bargaining with threat points gives each side a share of the surplus proportional to bargaining strength.

The math

Nash (1950) axiomatization: Pareto-efficiency + symmetry + invariance to affine utility transformations + IIA uniquely identifies the product-maximizer.

Rubinstein (1982) provided a non-cooperative microfoundation via alternating-offers bargaining.

Where it is used

Labor-management negotiations, trade agreements, and divorce settlements.

Workhorse in matching/search models of the labor market.

More in Game theory

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