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A payoff is individually rational for a Nash bargaining problem when neither player receives less than its disagreement point payoff. Essentiality permits strict improvement for both. Maximizing a positive Nash product therefore selects a strictly individually rational payoff.

The disagreement point specifies each player's payoff if bargaining fails. It supplies the baseline from which the Nash product measures gains. In a game-based example it may be set to the vector of security level payoffs; this choice must be recomputed when the underlying payoff matrices change.

For a finite two-person game, let $d_i$ be player $i$'s security level payoff, obtained by maximizing its minimum expected payoff against the other player's actions. Apply the Nash bargaining solution to the convexified feasible payoff region with this disagreement point. This is the maximin bargaining solution. “Maximin” specifies the disagreement guarantees; the arbitration objective function remains the Nash product, not the minimum of two numerically incomparable utilities.

For an essential Nash bargaining problem, maximizing the Nash product gives a unique vector: on positive gains its logarithm is a strictly concave function. The rule satisfies the four bargaining axioms, and the supporting triangle for Nash bargaining proves their converse characterization. The nonessential case requires additional conventions, since a zero product may have many maximizers.

The Pareto frontier consists of feasible payoff vectors that no other feasible vector weakly improves in every coordinate and strictly improves in at least one. A Nash bargaining solution with positive gains lies on this frontier, because increasing either gain without decreasing the other raises its Nash product.

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 $(u^{\prime },v^{\prime })$ jointly dominates $(u,v)$ if $u^{\prime }\ge u$, $v^{\prime }\ge v$, 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 isIts 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: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 $a+b\le 2$. 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: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.

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.

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 isThe 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. Pareto efficiency: a feasible vector dominating the chosen vector with at least one strict improvement would increase its positive Nash product. Bargaining symmetry: if the problem is unchanged by swapping players, uniqueness makes the answer unchanged, so the two payoffs agree. 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 $a_1{a}_2$, preserving its maximizer. 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 $u^{\ast }=N(F,d)$. Normalize payoffs by the positive affine transformationThe 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:Thus $H\subseteq \{z:z_1+z_2\le 2\}$. By compactness choose $M\ge 0$ so every coordinate of every $z\in H$ is at least $-M$. The supporting triangle for Nash bargaining isIt 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 $f(F,d)=u^{\ast }$. Hence the four axioms uniquely characterize the Nash bargaining solution on this domain.

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.

Transforming each utility by an independent positive affine map must transform the chosen payoff by the same map. The Nash product changes only by a positive factor, so the Nash bargaining solution has this invariance.

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 $z_1+z_2\le 2$. 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.