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.

Negotiation set in two-person bargaining ( $B = {u \in F : u \geq d, u is Pareto optimal}$ )

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For a compact convex set of feasible payoff vectors

F

and a specified disagreement point

d

, the negotiation set is its individually rational Pareto frontier. It is the set of efficient agreements each player prefers at least weakly to disagreement. In a maximin bargaining solution,

d

is the vector of the two security level payoffs. This usage is distinct from objection-and-counterobjection bargaining sets in coalitional-game theory.

Joint dominance of payoff vectors ( $u^{'} \geq u, u^{'} \neq = u$ )

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A feasible payoff vector

u^{'}

jointly dominates

u

if every coordinate is at least as large and at least one is strictly larger. A Pareto optimal vector has no feasible joint dominator. Strict increase of every coordinate is not required. This compares each player's own utilities, not the numerical size of utilities across different players.

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

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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 ( $u_{i} \mapsto a_{i} u_{i} + b_{i}, a_{i} > 0$ )

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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.

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

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For a finite two-person game, let

d_{i}

be player

i

's security level payoff, obtained by maximizing its minimum expected payoff against the other player's actions. Apply the Nash bargaining solution to the convexified feasible payoff region with this disagreement point. This is the maximin bargaining solution. “Maximin” specifies the disagreement guarantees; the arbitration objective function remains the Nash product, not the minimum of two numerically incomparable utilities.

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 ( $(F, d)$ )

Negotiation set in two-person bargaining ( $B = {u \in F : u \geq d, u is Pareto optimal}$ )

Joint dominance of payoff vectors ( $u^{'} \geq u, u^{'} \neq = u$ )

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

Maximin 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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