Nash bargaining solution (${\mathrm{argmax}}_{u\in F,u\ge d}(u_1-d_1)(u_2-d_2)$)

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.

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 $z_1+z_2\le 2$. 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.

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.