Bargaining Solutions: Nash, Egalitarian and Kalai-Smorodinsky

A leading researcher in bargaining theory, Professor William Thomson, recently edited the notable collection of papers in axiomatic bargaining:

Bargaining and the Theory of Cooperative Games: John Nash and Beyond


The following is quoted from the Introduction written by the editor. The parts I refer below focus on three representative solutions of the bargaining problem(*) in the literature, while his Introduction covers much more materials. I strongly recommend those who are interested in bargaining theory to read this insightful survey article.

Almost sixty years ago, Nash (1950) published his seminal paper on what is now known as the "axiomatic theory of bargaining"./// He formulated a list of properties, or "axioms", that he thought a solution should satisfy, and he established the existence and the uniqueness of a solution satisfying all of the axioms; this solution is now called the "Nash solution".

Nash's model has been one of the most successful paradigms of game theory. His paper is the founding stone of a literature that now comprises several hundred theoretical papers. The Nash solution is presented in all game theory textbooks./// Together with the Shapley value (Shapley, 1953) and the core (Gillies, 1959), it constitutes the obligatory background on cooperative games in most economics graduate programs.

In spite of the large number of reasonable solutions that one can easily define, only three solutions and variants have consistently emerged: in addition to the Nash solution, they are egalitarian(**) and Kalai-Smorodinsky solutions(***)./// The dominance of these three solutions and these variants is a central conclusion to be drawn from the literature.

The Nash solution has come out somewhat more often than the other two, and the claim can perhaps be made that it is indeed special./// But the argument is a little dangerous. Earlier, we talked about the theorist's need to simplify and summarize in order to analyze, and in axiomatic analysis simplification often takes the form of independence and invariance axioms. The Nash solution satisfies many invariance conditions, thus it is not much of a surprise that it should have dome out often. On the other hand, the monotonicity axioms that have generally led  to the Kalai-Smorodinsky and egalitarian solutions are readily understood and endorsed by the man on the street.

It is mainly on the basis of monotonicity properties that the Kalai-Smorodinsky solution should e seen as an important challenger to the Nash solution, the egalitarian solution presenting another appealing choice. This latter solution enjoys even stronger monotonicity requirements and like the Nash solution, it satisfies strong independence conditions. Unlike both the Nash and Kalai-Smorodinsky solutions, it requires interpersonal comparisons of utility however, which, depending upon the context, may be seen as a desirable feature or a limitation.

(*) A bargaining problem consists of a pair (S, d) where S, the feasible set, is the subset of alternatives, and d, the disagreement point, is a point of S./// A bargaining solution defined on a class of problems is a function that associates with each problem (S, d) in the class a unique point of S, the solution outcome of (S, d).

(**) The egalitarian solution (Kalai, 1977a) selects the maximal point of S at which utility gains from d are equal. More generally, by making utility gains proportional to a fixed vector of weights, we obtain the weighted egalitarian solution relative to these weights (Kalai, 1977b)

(***) The Kalai-Smorodinsky solution (Kalai and Smorodinsky, 1975) selects the maximal point of S that is proportional to the profile of maximal payoffs that agents can separately reach among the points of S that dominate d.

References
Gillies (1959) "Solutions to general non-zero-sum games" in Contributions to the Theory of Games IV, Princeton University Press, 47-85.
Kalai (1977a) "Nonsymmetric Nash solutions and replications of 2-person bargaining", International Journal of Game Theory, Volume 6, Number 3, 129-133.
Kalai (1977b) "Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparisons", Econometrica, Vol. 45, No. 7 (Oct., 1977), 1623-1630.
Kalai and Smorodinsky (1975) "Other Solutions to Nash's Bargaining Problem", Econometrica, Vol. 43, No. 3 (May, 1975), 513-518.
Nash (1950) "The Bargaining Problem", Econometrica, Vol. 18, No. 2 (Apr., 1950), 155-162.
Shapley (1953) "A Value for n-Person Games" in Contributions to the Theory of Games II, Princeton University Press, 307-317.