= Solution
A <Nash bargaining problem> consists of a <compact convex set> $S$ of feasible two-player <payoff> <vectors> and a <disagreement point> $d\in S$. If bargaining fails, the players receive $d$; agreement selects a feasible <vector>. The <convex set> allows jointly agreed <correlated payoff lotteries> over outcomes. Assume essentiality: some feasible <vector> strictly improves both coordinates of $d$. The admissible answers satisfy <bargaining individual rationality>, $u_i\geq d_i$.
The four characterizing axioms are <Pareto efficiency> (no feasible <vector> weakly improves both chosen <payoffs> and strictly improves one), <bargaining symmetry> (interchanging identical player roles leaves the answer unchanged), <positive affine invariance in bargaining> (independently replacing utilities by $a_i u_i+b_i$, $a_i>0$, transforms the answer by the same maps), and <bargaining independence of irrelevant alternatives> (restricting to an admissible smaller feasible set that still contains the chosen <vector> leaves the choice unchanged). The last axiom keeps the <disagreement point> fixed.
The arbitration procedure maximizes the <Nash product> over individually rational <vectors>:
$$
f(S,d)=\operatorname*{argmax}_{u\in S,\ u\geq d}(u_1-d_1)(u_2-d_2).
$$
A maximizer exists by <compactness> and has positive gains by essentiality. It is unique, since the <logarithm> of the product is a <strictly concave function> of the two gains on their positive domain. The product is strictly increasing in either positive gain, giving <Pareto efficiency>. A positive affine utility change multiplies the product by $a_1a_2$, giving <positive affine invariance in bargaining>. Uniqueness gives <bargaining symmetry> and <bargaining independence of irrelevant alternatives>.
For completeness, these axioms also force the procedure. Normalize the product-maximizing <vector> to $(1,1)$ and $d$ to zero using <positive affine invariance in bargaining>. For any normalized feasible <vector> $z$, the <directional derivative> of the product along the feasible segment from $(1,1)$ to $z$ is $z_1+z_2-2$. The initial part of that segment has positive gains, so optimality implies $z_1+z_2\leq2$. Choose $M$ sufficiently large that the transformed feasible set is contained in the symmetric triangle
$$
T_M=\{z:z_1,z_2\geq-M,\ z_1+z_2\leq2\}.
$$
On this triangle, <bargaining symmetry> requires equal coordinates and <Pareto efficiency> then forces $(1,1)$. By <bargaining independence of irrelevant alternatives>, restricting from $T_M$ to the original normalized set still selects $(1,1)$. Undoing the normalization proves the <Nash bargaining solution> characterization. This is the <supporting triangle for Nash bargaining> argument.
For the numerical game, the maximin <disagreement point> is the <vector> of the players' separate <security level payoffs>. If player I uses its first action with <probability> $p$, its worst expected <payoff> is
$$
\min\{4-2p,\ 2+6p\}.
$$
The two lines meet at $p=1/4$; before that point the increasing second line is smaller and afterwards the decreasing first line is smaller. Hence $d_1=7/2$. If player II uses its first action with <probability> $q$, its worst <payoff> is
$$
\min\{2+2q,\ 3+2q\}=2+2q,
$$
maximized at $q=1$, giving $d_2=4$. Thus \b[$d=(7/2,4)$]. These are separate worst-case guarantees; they need not equal the actual <payoffs> when both security strategies are used together.
The cooperative feasible set is the <convex hull> of the four outcome <vectors>. The outcomes $(2,4)$ and $(2,3)$ are dominated by $C=(4,5)$. Its <Pareto frontier> is the segment joining $C$ to $B=(8,2)$, with equation $y=8-3x/4$. Any maximizer of the positive <Nash product> is on this frontier. <Bargaining individual rationality> restricts its relevant portion to $4\leq x\leq16/3$. There we maximize
$$
g(x)=(x-7/2)(4-3x/4),\qquad g'(x)=\frac{53}{8}-\frac32x,\qquad g''(x)=-\frac32.
$$
The stationary point lies strictly inside that interval and is the unique maximum. Therefore \b[the <Nash bargaining solution> is]
$$
\boxed{(x,y)=\left(\frac{53}{12},\frac{75}{16}\right).}
$$
The gains are $11/12$ and $11/16$, with <Nash product> $121/192$. The arbitration can implement the <vector> by the agreed <correlated payoff lottery> assigning <probability> $5/48$ to outcome $B$ and $43/48$ to outcome $C$. Indeed $4+4(5/48)=53/12$ and $5-3(5/48)=75/16$. This is a <correlated payoff lottery>; independent <mixed strategies> are not required to implement the cooperative agreement.
Back to article page
