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The two-person bargaining problem studies how two agents share a surplus that they can jointly generate. It is in essence a payoff selection problem. In many cases, the surplus created by the two players can be shared in many ways, forcing the players to negotiate which division of payoffs to choose. There are two typical approaches to the bargaining problem. A normative approach that studies how the surplus should be shared. The normative approach formulates appealing axioms that the solution to a bargaining problem should satisfy. The positive approach answers the question how the surplus will be shared. Under the positive approach, the bargaining procedure is modeled in detail as a non-cooperative game.

The bargaining game

The Nash bargaining solution is the unique solution to a two-person bargaining problem that satisfies the axioms of scale invariance, symmetry, efficiency, and independence of irrelevant alternatives. According to Walker,[1] Nash's bargaining solution was shown by John Harsanyi to be the same as Zeuthen's solution[2] of the bargaining problem.

The Nash bargaining game is a simple two-player game used to model bargaining interactions. In the Nash bargaining game, two players demand a portion of some good (usually some amount of money). If the total amount requested by the players is less than that available, both players get their request. If their total request is greater than that available, neither player gets their request.

Nash (1953) presents a non-cooperative demand game with two players who are uncertain about which payoff pairs are feasible. In the limit as the uncertainty vanishes, equilibrium payoffs converge to those predicted by the Nash bargaining solution.

Rubinstein also modelled bargaining as a non-cooperative game in which two players negotiate on the division of a surplus known as the alternating offers bargaining game. The players take turns acting as the proposer. The division of the surplus in the unique subgame perfect equilibrium depends upon how strongly players prefer current over future payoffs. In the limit as players become perfectly patient, the equilibrium division converges to the Nash bargaining solution.

Formal description

A two-person bargain problem consists of:

• A feasibility set F {\displaystyle F} , a closed subset of R 2 that is often assumed to be convex {\displaystyle \mathbb {R} ^{2}} , the elements of which are interpreted as agreements. Set F {\displaystyle F} is convex because an agreement could take the form of a correlated combination of other agreements.
• A disagreement, or threat, point d = ( d 1 , d 2 ) {\displaystyle d=(d_{1},d_{2})} , where d 1 {\displaystyle d_{1}} and d 2 {\displaystyle d_{2}} are the respective payoffs to player 1 and player 2.

The problem is nontrivial if agreements in F {\displaystyle F} are better for both parties than the disagreement point. A solution to the bargaining problem selection an agreement ϕ {\displaystyle \phi } in F {\displaystyle F}.

Equilibrium analysis

Strategies are represented in the Nash demand game by a pair (x, y). x and y are selected from the interval [d, z], where d is the disagreement outcome and z is the total amount of good. If x + y is equal to or less than z, the first player receives x and the second y. Otherwise both get d; often d = 0 {\displaystyle d=0} .

There are many Nash equilibria in the Nash demand game. Any x and y such that x + y = z is a Nash equilibrium. If either player increases their demand, both players receive nothing. If either reduces their demand they will receive less than if they had demanded x or y. There is also a Nash equilibrium where both players demand the entire good. Here both players receive nothing, but neither player can increase their return by unilaterally changing their strategy.

In Rubinstein’s alternating offers bargaining game, players take turns acting as the proposer for splitting some surplus. The division of the surplus in the unique subgame perfect equilibrium depends upon how strongly players prefer current over future payoffs. In particular, let d be the discount factor, which refers to the rate at which players discount future earnings.  That is, after each step the surplus is worth d times what it was worth previously. Rubinstein showed that if the surplus is normalized to 1, the payoff for player 1 in equilibrium is 1/(1+d), while the payoff for player 2 is d/(1+d). In the limit as players become perfectly patient, the equilibrium division converges to the Nash bargaining solution.

Kalai–Smorodinsky bargaining solution

Main article: Kalai–Smorodinsky bargaining solution

Independence of Irrelevant Alternatives can be substituted with a Resource monotonicity axiom. This was demonstrated by Ehud Kalai and Meir Smorodinsky.[6] This leads to the so-called Kalai–Smorodinsky bargaining solution: it is the point which maintains the ratios of maximal gains. In other words, if we normalize the disagreement point to (0,0) and player 1 can receive a maximum of g 1 {\displaystyle g_{1}} with player 2’s help (and vice versa for g 2 {\displaystyle g_{2}} ), then the Kalai–Smorodinsky bargaining solution would yield the point ϕ {\displaystyle \phi } on the Pareto frontier such that ϕ 1 / ϕ 2 = g 1 / g 2 {\displaystyle \phi _{1}/\phi _{2}=g_{1}/g_{2}} .