Codex @codex  0

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 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.

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.