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For a compact convex set of feasible payoff vectors $F$ and a specified disagreement point $d$, the negotiation set is its individually rational Pareto frontier. It is the set of efficient agreements each player prefers at least weakly to disagreement. In a maximin bargaining solution, $d$ is the vector of the two security level payoffs. This usage is distinct from objection-and-counterobjection bargaining sets in coalitional-game theory.

Use the two-by-two payoff entries visible in the original PDF. If $q$ is the probability that II uses its first action, I's expected payoffs from its two pure strategies are $4-q$ and $2q$. Their difference $4-3q$ is strictly positive for every $q\in [0,1]$. Thus I's first action strict dominance its second, and every Nash equilibrium has I use the first action with probability one. Against that action, II receives 8 from its first action and 4 from its second, so its first action is uniquely optimal. ThereforeThis excludes nontrivial mixed equilibria as well as the other pure pairs.

For the maximin bargaining solution, first compute the two security level payoffs. If I uses its first action with probability $p$, its payoffs against II's two pure strategies are $2+p$ and $4p$. Both increase with $p$, so its best worst-case payoff is attained at $p=1$ and is $d_1=3$. If II uses its first action with probability $q$, its payoffs against I's two actions are $4+4q$ and $6-6q$. The lower of these is maximized at their intersection,These are separate security calculations, not the expected payoffs from a pair of security strategies.

Cooperation permits jointly chosen lotteries, giving the convex hull of the four payoff. The vector $(3,8)$ dominates $(2,0)$ and $(0,6)$, so the Pareto frontier is the segment from $(3,8)$ to $(4,4)$, with equation $v=20-4u$. Individual rationality relative to $d=(3,24/5)$ restricts this toOn this negotiation set, the Nash product isIts unique maximizer is interior, givingOne implementing binding agreement always uses I's first action and uses II's first action with probability $3/5$, its second with probability $2/5$. This agreement raises both payoffs above their security levels, but II receives $32/5<8$, less than at the noncooperative equilibrium. Thus player II prefers the noncooperative game under the specified maximin bargaining rule. The cooperative feasible set contains the equilibrium payoff; it is the particular arbitration rule, not an inability to cooperate at that payoff, that produces II's lower allocation.

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, $u_i\ge d_i$.

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: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 $a_1{a}_2$, 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 $z_1+z_2\le 2$. Choose $M$ sufficiently large that the transformed feasible set is contained in the symmetric triangleOn 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 isThe 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 ismaximized at $q=1$, giving $d_2=4$. Thus $d=(7/2,4)$. 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 $4\le x\le 16/3$. There we maximizeThe stationary point lies strictly inside that interval and is the unique maximum. Therefore the Nash bargaining solution isThe 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.

A security level payoff maximizes what a player guarantees against the opponent. Let $p$ be the row player's probability of the first action. Its guaranteed payoff isThe decreasing and increasing terms cross at $p=1/5$, attaining $d_1=12/5$; moving either way lowers the smaller term. For the column player, any mixture has zero payoff against the first row, while its second-row payoff is nonnegative. Therefore $d_2=0$ andThe feasible set is the convex hull of the four joint pure-action payoff vectors. Its upper Pareto frontier connects $(4,0)$ to $(3,2)$ to $(2,3)$. Write the row payoff as $u$ and column payoff as $v$. On the first Pareto frontier segment, $v=8-2u$ for $3\le u\le 4$. The Nash product isa concave quadratic with derivative $64/5-4u$, maximized at $u=16/5$, $v=8/5$, giving product $32/25$. On the portion of the other Pareto frontier segment satisfying bargaining individual rationality, $12/5\le u\le 3$ and $v=5-u$. Its product derivative $37/5-2u$ is positive throughout, so its largest product is $6/5$ at $(3,2)$, smaller than $32/25$. All dominated points can be discarded by Pareto efficiency. ConsequentlyThis payoff is implemented by a correlated payoff lottery choosing the payoff $(4,0)$ with probability $1/5$ and $(3,2)$ with probability $4/5$. The convex hull permits such lotteries over joint outcomes; it is not restricted to independent mixed strategies.

Transposing the second player's matrix leaves the row player's security level payoff at $12/5$. The column player's second action now yields payoffs $2$ and $3$, while its first yields zero for either row. Its security level payoff is therefore $2$, guaranteed by the second action; against the first row no mixture can guarantee more. Thus $d=(12/5,2)$.

The relevant upper Pareto frontier joins the payoff vectors $(4,2)$ and $(2,3)$, so $v=4-u/2$. On the segment satisfying bargaining individual rationality $12/5\le u\le 4$, the Nash product becomesIts derivative is $16/5-u$ and its second derivative is $-1$. HenceThe implementing correlated payoff lottery chooses $(4,2)$ with probability $3/5$ and $(2,3)$ with probability $2/5$. Both players' gains are strictly positive: $4/5$ and $2/5$. The new disagreement point must be recomputed after transposition; reusing the previous column security payoff would solve a different Nash bargaining problem.