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In a two-person Nash bargaining problem, an agreement specifies a jointly feasible pair of payoffs $(u,v)$, measured in each player's own utility function. For a finite game, binding agreements can randomize jointly among action pairs, so the feasible payoff region $F$ is the convex hull of the pure-action payoff. This permits correlated joint actions; it is generally larger than the payoff region generated by independent mixed strategies. A disagreement point $d=(d_1,d_2)$ specifies what each player receives if negotiations fail. Assume a compact convex set of feasible agreements and at least one agreement strictly improving both disagreement payoffs.

A vector $(u^{\prime },v^{\prime })$ jointly dominates $(u,v)$ if $u^{\prime }\ge u$, $v^{\prime }\ge v$, and at least one inequality is strict. This is joint dominance of payoff vectors; strict improvement for both is unnecessary. A Pareto optimal agreement is one that has no feasible joint dominator. A jointly dominated agreement wastes a possible gain, but Pareto efficiency alone does not settle how gains should be divided. The negotiation set in two-person bargaining isIts individual-rationality constraints keep either player from accepting less than disagreement; its Pareto efficiency constraint rules out wasted opportunities. This negotiation set is distinct from objection-and-counterobjection bargaining sets for coalitional games.

The Nash arbitration procedure selects the Nash bargaining solution by maximizing gains relative to disagreement:The objective function is the Nash product. Essentiality makes its maximum positive; compactness gives attainment. On positive gains, its logarithm is $\log (u-d_1)+\log (v-d_2)$, whose Hessian is diagonal and negative definite. Thus strict concavity on the feasible convex set makes the maximizing payoff vector unique. Positive gain in either coordinate increases the product, so the solution is Pareto optimal.

The procedure respects feasibility, bargaining individual rationality, Pareto efficiency, bargaining symmetry, positive affine invariance in bargaining, and bargaining independence of irrelevant alternatives. In particular, replacing each player's utility by a separate positive affine rescaling multiplies the product by a positive constant; it does not compare one player's raw utility scale to the other's. Deleting other feasible agreements cannot change a unique maximizer that remains feasible. Symmetry under interchange of the players gives equal selected coordinates when the bargaining problem itself is symmetric.

These properties also explain why the product rule is distinguished. Normalize a positive-gain maximizer to $(1,1)$ and disagreement to $(0,0)$ by positive affine maps. The first-order inequality along a segment from $(1,1)$ to any feasible $(a,b)$ gives $a+b\le 2$. A sufficiently large symmetric triangle, bounded below in each coordinate and above by $a+b=2$, contains the normalized feasible region. Symmetry and Pareto efficiency select $(1,1)$ in that triangle. Independence of irrelevant alternatives then selects the same point in the original normalized region, and affine invariance transforms it back. This is the supporting triangle for Nash bargaining argument; the assumptions of convexity and a strict joint improvement are important. Without strict improvement, a zero product need not select a unique agreement.

For a finite game, a natural disagreement convention is each player's security level payoff:where $A,B$ are the two payoff matrices and $p,q$ are mixed strategies. Each player can independently guarantee at least its own security level. The maximin bargaining solution applies the Nash product rule using this $d$. In this convention, “maximin” determines the disagreement payoffs; it does not replace the arbitration objective function by $\min (u,v)$, which would depend improperly on interpersonal utility scales. The resulting agreement can strictly benefit both players relative to their guarantees, yet need not benefit both relative to a particular noncooperative Nash equilibrium.