= Solution
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> $(u',v')$ jointly dominates $(u,v)$ if $u'\ge u$, $v'\ge v$, 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> is
$$
B_d=\{(u,v)\in F:u\ge d_1,\ v\ge d_2,\ (u,v)\text{ is Pareto optimal}\}.
$$
Its 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:
$$
\boxed{(u_*,v_*)=\operatorname*{argmax}_{(u,v)\in F,\ u\ge d_1,\ v\ge d_2}
(u-d_1)(v-d_2).}
$$
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 $a+b\le2$. 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>:
$$
d_1=\max_p\min_q p^TAq,\qquad
d_2=\max_q\min_p p^TBq,
$$
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>.
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