(*) Contraction independence is often called the axiom of independence of irrelevant alternatives (IIA), which is defined as follows:
Contraction independence If, keeping the disagreement point constant, the feasible set contracts but the alternative chosen as solution outcome remains feasible, then it should remain the solution outcome.
On contraction independence
Contraction independence has been the object of the sharpest criticisms. Nash himself expressed some misgivings about it and Luce and Raiffa's (1957) objections are well known. In evaluating a bargaining situation, it is unavoidable and probably desirable that it be simplified and summarized, that it be reduced to its essential features. The issue is how much and what information should be discarded in the process, and one can make a convincing case that contraction independence ignores too much. Indeed, it covers contractions that affect the shape of the feasible in ways that seem relevant, for instance, the elimination of the only alternatives at which a particular agent's payoff is higher than at the initial solution outcome and the other agent's payoff lower than at the initial solution outcome; contraction independence prevents solutions from responding to such eliminations.
The main counterargument to this criticism was made by Nash himself: in practice, information is usually lacking about which alternatives are truly available, and a compromise under evaluation only competes against others that are not too different from itself. Modeling this lack or information explicitly is what an investigator should perhaps do, but there are also advantages to keeping the model simple; contraction independence is a formal way to express the idea.
Whether a solution is implementable depends on the type of game forms that are used, and on the behavioral assumptions made about how agents confronted with such games behave. For implementation by normal form games and when agents calculate best responses taking as given the choices made by the other agents, a critical property for what is then called Nash-implementability is Maskin-monotonicity (actually an invariance property with respect to enlargements of the lower contour set at the chosen alternative). Most solutions are not Maskin-monotonic and therefore not Nash-implementable by normal form games. However, an implementation of the Kalai-Smorodinsky solution by a sequential game is possible (Moulin, 1984). Later contributions delivered the Nash solution (Howard, 1992), the egalitarian solution (Bossert and Tan, 1995)./// For subgame perfect implementation, a general theorem is offered by Miyagawa (2002). It covers all solutions obtained, after normalizing problems so that the ideal point has equal coordinates, by maximizing a monotone and quasi-concave function of the agents' payoffs. Implementation is by means of a stage game and the equilibrium notion is subgame perfection.
Bossert and Tan (1995) "An arbitration game and the egalitarian bargaining solution", Social Choice and Welfare, Volume 12, Number 1, 29-41.
Luce and Raiffa (1957), Games and Decisions: Introduction and Critical Survey, Wiley.
Miyagawa (2002) "Subgame-perfect implementation of bargaining solutions", Games and Economic Behavior, Volume 41, Issue 2, 292-308.
Moulin (1984) "Implementing the Kalai-Smorodinsky bargaining solution", Journal of Economic Theory, Volume 33, Issue 1, 32-45.