The Kalai–Smorodinsky (KS) bargaining solution is a solution to the Bargaining problem. It was suggested by Ehud Kalai and Meir Smorodinsky, as an alternative to Nash's bargaining solution suggested 25 years earlier. The main difference between the two solutions is that the Nash solution satisfies independence of irrelevant alternatives, while the KS solution instead satisfies resource monotonicity. The solution builds on earlier work by Howard Raiffa on discrete bargaining procedures.
Setting A two-person bargain problem consists of a pair ( F , d ) {\displaystyle (F,d)} :
A feasible agreements set F {\displaystyle F} . This is a closed convex subset of R 2 {\displaystyle \mathbb {R} ^{2}} . Each element of F {\displaystyle F} represents a possible agreement between the players. The coordinates of an agreement are the utilities of the players if this agreement is implemented. The assumption that F {\displaystyle F} is convex makes sense, for example, when it is possible to combine agreements by randomization. A disagreement 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 when the bargaining terminates without an agreement. It is assumed that the problem is nontrivial, i.e., the agreements in F {\displaystyle F} are better for both parties than the disagreement. A bargaining solution is a function f {\displaystyle f} that takes a bargaining problem ( F , d ) {\displaystyle (F,d)} and returns a point in its feasible agreements set, f ( F , d ) ∈ F {\displaystyle f(F,d)\in F} .
Requirements from bargaining solutions The Nash and KS solutions both agree on the following three requirements: Pareto optimality is a necessary condition. For every bargaining problem, the returned agreement f ( F , d ) {\displaystyle f(F,d)} must be Pareto-efficient. Symmetry is also necessary. The names of the players should not matter: if player 1 and player 2 switch their utilities, then the agreement should be switched accordingly. Invariant to positive affine transformations also seems like a necessary condition: if the utility function of one or more players is transformed by a linear function, then the agreement should also be transformed by the same linear function. This makes sense if we assume that the utility functions are only representations of a preference relation, and do not have a real numeric meaning. In addition to these requirements, Nash requires Independence of irrelevant alternatives (IIA). This means that, if the set of possible agreements grows (more agreements become possible), but the bargaining solution picks an agreement that was contained in the smaller set, then this agreement must be the same as the agreement reached when only the smaller set was available, since the new agreements are irrelevant. For example, suppose that in Sunday we can agree on option A or option B, and we pick option A. Then, in Monday we can agree on option A or B or C, but we do not pick option C. Then, Nash says that we must pick option A. The new option C is irrelevant since we do not select it anyway. Kalai and Smorodinsky differ from Nash on this issue. They claim that the entire set of alternatives must affect the agreement reached. In the above example, suppose the preference relation of player 2 is: C>>B>A (C is much better than B, which is somewhat better than A) while the preference relation of 1 is reversed: A>>B>>C. The fact that option C becomes available allows player 2 to say: "if I give up my best option - C, I have a right to demand that at least my second-best option will be chosen". Therefore, KS remove the IIA requirement. Instead, they add a monotonicity requirement. This requirement says that, for each player, if the utility attainable by this player for each utility of the other player is weakly larger, then the utility this player gets in the selected agreement should also be weakly larger. In other words, a player with better options should get a weakly-better agreement. The formal definition of monotonicity is based on the following definitions.
B e s t i ( F ) {\displaystyle Best_{i}(F)} - the best value that player i can expect to get in a feasible agreement.
… excerpt ends here. Continue reading the full article.
