Cooperative bargaining is a process in which two people decide how to share a surplus that they can jointly generate.
It studies how the surplus should be shared, by formulating appealing axioms that the solution to a bargaining problem should satisfy.
Under the positive approach, the bargaining procedure is modeled as a non-cooperative game.
Often, the feasible set is restricted to include only payoffs that have a possibility of being better than the disagreement point for both agents.
This could be some focal equilibrium that both players could expect to play, or zero if no agreement is reached.
It is the unique solution to a two-person bargaining problem that satisfies the axioms of scale invariance, symmetry, efficiency, and independence of irrelevant alternatives.
In the Nash bargaining game, two players demand a portion of some good (usually some amount of money).
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.
x and y are selected from the interval [d, z], where d is the disagreement outcome and z is the total amount of good.
In Rubinstein's alternating offers bargaining game,[5] 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 the limit as players become perfectly patient, the equilibrium division converges to the Nash bargaining solution.
Various solutions have been proposed based on slightly different assumptions about what properties are desired for the final agreement point.
[7]: 15–16 Independence of irrelevant alternatives can be substituted with a resource monotonicity axiom, as suggested by Ehud Kalai and Meir Smorodinsky.
[8] This leads to the Kalai–Smorodinsky rule, which selects the point which maintains the ratio of maximal gains.
The egalitarian bargaining solution, introduced by Ehud Kalai, is a third solution which drops the condition of scale invariance while including both the axiom of independence of irrelevant alternatives, and the axiom of resource monotonicity.
Kalai notes that this solution is closely related to the egalitarian ideas of John Rawls.
[9] A series of experimental studies[10] found no consistent support for any of the bargaining models.
Although some participants reached results similar to those of the models, others did not, focusing instead on conceptually easy solutions beneficial to both parties.
[11] In real-world negotiations, participants often first search for a general bargaining formula, and then only work out the details of such an arrangement, thus precluding the disagreement point and instead moving the focal point to the worst possible agreement.
Kenneth Binmore has used the Nash bargaining game to explain the emergence of human attitudes toward distributive justice.
Herbert Gintis supports a similar theory, holding that humans have evolved to a predisposition for strong reciprocity but do not necessarily make decisions based on direct consideration of utility.
[14] Some economists have studied the effects of risk aversion on the bargaining solution.