Nash Bargaining Solution

Maximize the product of gains over disagreement to split a surplus.

The idea

The Nash bargaining solution predicts the terms on which two players settle a negotiation from which both can gain. Model the negotiation by a feasible set $S$ of utility pairs $(u_1, u_2)$ the players could agree on, together with a disagreement point $d = (d_1, d_2)$ giving what each receives if the talks collapse. No player signs an agreement paying less than their disagreement payoff, so we restrict to pairs with $u_i \ge d_i$.

Nash derived the answer from axioms rather than from a model of the haggling. He required four properties of the chosen pair: it is Pareto efficient; identical players receive identical payoffs (symmetry); rescaling one player's utility, whose units are arbitrary, does not move the choice (scale invariance); and deleting unchosen alternatives from $S$ does not change it (independence of irrelevant alternatives).

Exactly one rule satisfies all four: select the feasible pair maximizing the Nash product

$N(u_1, u_2) = (u_1 - d_1)(u_2 - d_2),$

the product of the two players' gains over disagreement. The product form is what delivers scale invariance: rescaling one player's utility multiplies every candidate's product by the same positive constant, which cannot change which candidate is largest. Geometrically, the level curves of the Nash product are hyperbolas bending around $d$, and the solution is the point where the highest level curve that still reaches $S$ touches the frontier.

Ways to work on it

• Walkthrough. Split a bargaining surplus by maximizing the Nash product.
• Practice. Find the Nash split on a linear frontier with a disagreement point.
• Hardest. Maximize the Nash product on a curved frontier.

Not sure where to start? Take the ten-question placement test.