# Bargaining, conditional consistency, and weighted lexicographic Kalai-Smorodinsky Solutions

Original Paper

First Online:

Received:

Accepted:

- 143 Downloads

## Abstract

We reconsider the class of weighted Kalai-Smorodinsky solutions of Dubra (Econ Lett 73:131–136, 2001), and using methods of Imai (Econometrica 51:389–401, 1983), extend their characterization to the domain of multilateral bargaining problems. Aside from standard axioms in the literature, this result involves a new property that weakens the axiom Bilateral Consistency (Lensberg, J Econ Theory 45:330–341, 1988), by making the notion of consistency dependent on how ideal values in a reduced problem change relative to the original problem.

## Notes

### Acknowledgments

I thank Hans Peters, Michele Lombardi and Naoki Yoshihara for useful comments and suggestions. The usual caveat applies.

## References

- Anbarci N, Sun C (2013) Asymmetric Nash bargaining solutions: a simple Nash program. Econ. Lett. 120:211–214CrossRefGoogle Scholar
- Bossert W (1993) An alternative solution to bargaining problems with claims. Math. Soc. Sci. 25:205–220CrossRefGoogle Scholar
- Britz V, Herings JJ, Predtetchinski A (2010) Non-cooperative support for the asymmetric Nash bargaining solution. J. Econ. Theory 145:1951–1967CrossRefGoogle Scholar
- Chang C, Hwang Y (1999) A characterization of the leximin solution of the bargaining problem. Math. Methods Oper. Res. 49:395–400CrossRefGoogle Scholar
- Chang C, Liang M (1998) A characterization of the lexicographic Kalai-Smorodinsky solution for \(n=3\). Math. Soc. Sci. 35:307–319CrossRefGoogle Scholar
- Chen M (2000) Individual monotonicity and the leximin solution. Econ. Theory 15:353–365Google Scholar
- Chun Y, Peters H (1989a) The lexicographic egalitarian solution. Cahiers du C.E.R.O 30:149–156Google Scholar
- Chun Y, Peters H (1989b) Lexicographic monotone path solutions. OR Spektrum 11:43–47Google Scholar
- Chun Y, Peters H (1991) The lexicographic equal-loss solution. Math. Soc. Sci. 22:151–161CrossRefGoogle Scholar
- Chun Y, Thomson W (1992) Bargaining problems with claims. Math. Soc. Sci. 24:19–33CrossRefGoogle Scholar
- del Carmen MGM (1995) Efficient solutions for bargaining problems with claims. Math. Soc. Sci. 30:57–69CrossRefGoogle Scholar
- Driesen B (2012) Proportional concessions and the leximin solution. Econ. Lett. 114:288–291CrossRefGoogle Scholar
- Dubra J (2001) An asymmetric Kalai-Smorodinsky solution. Econ. Lett. 73:131–136CrossRefGoogle Scholar
- Harsanyi J (1959) A bargaining model for the cooperative \(n\)-person game. In: Tucker AW, Luce RD (eds) Contributions to the theory of games IV. Princeton University Press, PrincetonGoogle Scholar
- Imai H (1983) Individual monotonicity and lexicographic maxmin solution. Econometrica 51:389–401CrossRefGoogle Scholar
- Kalai E (1977a) Nonsymmetric Nash solutions and replications of 2-person bargaining. Int. J. Game Theory 6:129–133CrossRefGoogle Scholar
- Kalai E (1977b) Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45:1623–1630CrossRefGoogle Scholar
- Kalai E, Smorodinsky M (1975) Other solutions to Nash’s bargaining problem. Econometrica 43:513–518CrossRefGoogle Scholar
- Laruelle A, Valenciano F (2008) Non-cooperative foundations of bargaining power in committees and the Shapley–Shubik index. Games Econ. Behav. 63:341–353CrossRefGoogle Scholar
- Lensberg T (1988) Stability and the Nash solution. J. Econ. Theory 45:330–341CrossRefGoogle Scholar
- Luce D, Raiffa H (1957) Games and decisions: introduction and critical survey. Wiley, New YorkGoogle Scholar
- Nash J (1950) The bargaining problem. Econometrica 18:155–162CrossRefGoogle Scholar
- Nieto J (1992) The lexicographic egalitarian solution on economic environments. Soc. Choice Welf. 9:203–212CrossRefGoogle Scholar
- Peters H, Tijs S, Zarzuelo J (1994) A reduced-game property for the Kalai-Smorodinsky and egalitarian bargaining solutions. Math. Soc. Sci. 27:11–18CrossRefGoogle Scholar
- Raiffa H (1953) Arbitration schemes for generalized two-person games. In: Kuhn H, Tucker A (eds) Contributions to the Theory of Games II, vol 28., Annals of Mathematics StudiesPrinceton University Press, Princeton, pp 361–387Google Scholar
- Roth A (1977) Independence of irrelevant alternatives, and solutions to Nash’s bargaining problem. J. Econ. Theory 16:247–251CrossRefGoogle Scholar
- Roth A (1979) An impossibility result concerning \(n\)-person bargaining games. Int. J. Game Theory 8:129–132CrossRefGoogle Scholar
- Thomson W, Lensberg T (1989) Axiomatic theory of bargaining with a variable number of agents. Cambridge University Press, CambridgeCrossRefGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2015