Nash bargaining problem ( $(F, d)$ )

The essential two-person bargaining domain consists of a compact convex set of feasible utility vectors

F

, a disagreement point

d \in F

, and at least one feasible vector strictly exceeding

d

coordinatewise. The Nash bargaining solution selects a jointly feasible improvement using a rule characterized by Pareto efficiency, bargaining symmetry, positive affine invariance in bargaining and bargaining independence of irrelevant alternatives.

Table of contents

Bargaining individual rationality ( $u \geq d$ )

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A payoff is individually rational for a Nash bargaining problem when neither player receives less than its disagreement point payoff. Essentiality permits strict improvement for both. Maximizing a positive Nash product therefore selects a strictly individually rational payoff.

Bargaining independence of irrelevant alternatives

 0  0

If an admissible smaller feasible set with the same disagreement point contains the original chosen vector, deleting the other outcomes must leave the choice unchanged. The Nash bargaining solution satisfies this axiom because its unique product maximizer remains feasible.

Positive affine invariance in bargaining ( $u_{i} \mapsto a_{i} u_{i} + b_{i}, a_{i} > 0$ )

 0  0

Transforming each utility by an independent positive affine map must transform the chosen payoff by the same map. The Nash product changes only by a positive factor, so the Nash bargaining solution has this invariance.

Bargaining symmetry

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When the feasible payoff set and disagreement point are invariant under interchange of players, the solution must give both players the same payoff.

Nash bargaining solution ( $argmax_{u \in F, u \geq d} (u_{1} - d_{1}) (u_{2} - d_{2})$ )

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For an essential Nash bargaining problem, maximizing the Nash product gives a unique vector: on positive gains its logarithm is a strictly concave function. The rule satisfies the four bargaining axioms, and the supporting triangle for Nash bargaining proves their converse characterization. The nonessential case requires additional conventions, since a zero product may have many maximizers.

Supporting triangle for Nash bargaining ( $z_{1}, z_{2} \geq - M, z_{1} + z_{2} \leq 2$ )

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Normalize a positive Nash product maximizer and the disagreement point to

(1, 1)

and zero. The first-order product inequality along every feasible segment places the transformed set in

z_{1} + z_{2} \leq 2

. Compactness allows a symmetric containing triangle with finite lower coordinate bounds. Bargaining symmetry and Pareto efficiency choose

(1, 1)

in that triangle; bargaining independence of irrelevant alternatives transfers this choice to the original normalized set. Positive affine invariance in bargaining then proves the full characterization.

Nash product ( $(u_{1} - d_{1}) (u_{2} - d_{2})$ )

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The product of the two players' gains over the disagreement point is maximized over the individually rational feasible set in the Nash bargaining solution. Positive affine changes of utility multiply this product by a positive constant and leave the selected payoff correspondence unchanged after transforming coordinates.

Disagreement point ( $d$ )

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The disagreement point specifies each player's payoff if bargaining fails. It supplies the baseline from which the Nash product measures gains. In a game-based example it may be set to the vector of security level payoffs; this choice must be recomputed when the underlying payoff matrices change.

Nash bargaining problem ( $(F, d)$ )

Bargaining individual rationality ( $u \geq d$ )

Bargaining independence of irrelevant alternatives

Positive affine invariance in bargaining ( $u_{i} \mapsto a_{i} u_{i} + b_{i}, a_{i} > 0$ )

Bargaining symmetry

Nash bargaining solution ( $argmax_{u \in F, u \geq d} (u_{1} - d_{1}) (u_{2} - d_{2})$ )

Supporting triangle for Nash bargaining ( $z_{1}, z_{2} \geq - M, z_{1} + z_{2} \leq 2$ )

Nash product ( $(u_{1} - d_{1}) (u_{2} - d_{2})$ )

Disagreement point ( $d$ )

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