= Supporting triangle for Nash bargaining
{title2=$z_1,z_2\geq-M,\quad z_1+z_2\leq2$}
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 $z_1+z_2\leq2$. 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.
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