= Solution
Use the usual essential two-person <Nash bargaining problem>: $F\subset\mathbb R^2$ is a <compact convex set>, $d\in F$ is the <disagreement point>, and some $u\in F$ satisfies $u_i>d_i$ for both players. The <Nash bargaining solution> is
$$
\boxed{N(F,d)=\mathop{\operatorname{argmax}}_{u\in F,\ u\geq d}
(u_1-d_1)(u_2-d_2).}
$$
The positive maximum exists by compactness and essentiality. On positive gains, maximizing this <Nash product> is equivalent to maximizing $\log(u_1-d_1)+\log(u_2-d_2)$, a <strictly concave function>. Convexity then gives a unique maximizer. The essentiality and compactness hypotheses matter: for example, with $F=[0,1]\times\{0\}$ and $d=0$, the product is zero everywhere and its argmax alone is not a single-valued definition.
The rule satisfies all four axioms. \b[<Pareto efficiency>:] a feasible vector dominating the chosen vector with at least one strict improvement would increase its positive <Nash product>. \b[<Bargaining symmetry>:] if the problem is unchanged by swapping players, uniqueness makes the answer unchanged, so the two payoffs agree. \b[<Positive affine invariance in bargaining>:] for $v_i=a_i u_i+b_i$ with $a_i>0$, gains transform to $a_i(u_i-d_i)$ and the product is multiplied by the positive constant $a_1a_2$, preserving its maximizer. \b[<Bargaining independence of irrelevant alternatives>:] if $G\subseteq F$ is another admissible feasible set containing the chosen vector and the same disagreement point, that vector remains the unique product maximizer over $G$.
To prove characterization, let $f$ be any feasible single-valued rule satisfying these axioms, and let $u^*=N(F,d)$. Normalize payoffs by the positive affine transformation
$$
z_i=\frac{u_i-d_i}{u_i^*-d_i}.
$$
The transformed set $H$ has disagreement point zero and product maximizer $e=(1,1)$. For any $z\in H$, convexity puts $e+t(z-e)$ in $H$; for sufficiently small $t>0$ both gains remain positive. The one-sided <derivative> of the product at its maximum is therefore nonpositive:
$$
\left.\frac{d}{dt}\prod_{i=1}^2(1+t(z_i-1))\right|_{t=0}
=z_1+z_2-2\leq0.
$$
Thus $H\subseteq\{z:z_1+z_2\leq2\}$. By compactness choose $M\geq0$ so every coordinate of every $z\in H$ is at least $-M$. The <supporting triangle for Nash bargaining> is
$$
T_M=\{z:z_1\geq-M,\ z_2\geq-M,\ z_1+z_2\leq2\}.
$$
It contains $H$, is compact, convex, symmetric and essential, and contains disagreement zero. Symmetry forces $f(T_M,0)$ onto the diagonal; <Pareto efficiency> then forces it to be $e=(1,1)$. Since $e\in H\subseteq T_M$, <bargaining independence of irrelevant alternatives> gives $f(H,0)=e$. Undoing the normalization by <positive affine invariance in bargaining> gives $f(F,d)=u^*$. Hence the four axioms uniquely characterize the <Nash bargaining solution> on this domain.
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