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The Asymmetric Leximin Solution

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  • Driesen, Bram W.

Abstract

In this article we define and characterize a class of asymmetric leximin solutions, that contains both the symmetric leximin solution of Imai[5] and the two-person asymmetric Kalai-Smorodinsky solution of Dubra [3] as special cases. Solutions in this class combine three attractive features: they are defined on the entire domain of convex n-person bargaining problems, they generally yield Pareto efficient solution outcomes, and asymmetries among bargainers are captured by a single parameter vector. The characterization is based on a strengthening of Dubra’s [3] property Restricted Independence of Irrelevant Alternatives (RIIA). RIIA imposes Nash’s IIA (Nash [9]), under the added condition that the contraction of the feasible set preserves the mutual proportions of players’ utopia values. Our axiom, entitled RIIA for Independent Players (RIP), says RIIA holds for a group of players, given that the contraction of the feasible set does not affect players outside that group.

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  • Driesen, Bram W., 2012. "The Asymmetric Leximin Solution," Working Papers 0523, University of Heidelberg, Department of Economics.
  • Handle: RePEc:awi:wpaper:0523
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    References listed on IDEAS

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    1. Thomson,William & Lensberg,Terje, 1989. "Axiomatic Theory of Bargaining with a Variable Number of Agents," Cambridge Books, Cambridge University Press, number 9780521343831, December.
    2. Driesen, Bram, 2012. "Proportional concessions and the leximin solution," Economics Letters, Elsevier, vol. 114(3), pages 288-291.
    3. Thomson, William, 1994. "Cooperative models of bargaining," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 35, pages 1237-1284 Elsevier.
    4. Britz, Volker & Herings, P. Jean-Jacques & Predtetchinski, Arkadi, 2010. "Non-cooperative support for the asymmetric Nash bargaining solution," Journal of Economic Theory, Elsevier, vol. 145(5), pages 1951-1967, September.
    5. Chih Chang & Yan-An Hwang, 1999. "A characterization of the leximin solution of the bargaining problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(3), pages 395-400, July.
    6. Thomson,William & Lensberg,Terje, 2006. "Axiomatic Theory of Bargaining with a Variable Number of Agents," Cambridge Books, Cambridge University Press, number 9780521027038, December.
    7. Bossert, Walter, 1993. "An alternative solution to bargaining problems with claims," Mathematical Social Sciences, Elsevier, vol. 25(3), pages 205-220, May.
    8. Kalai, Ehud & Smorodinsky, Meir, 1975. "Other Solutions to Nash's Bargaining Problem," Econometrica, Econometric Society, vol. 43(3), pages 513-518, May.
    9. Chang, Chih & Liang, Meng-Yu, 1998. "A characterization of the lexicographic Kalai-Smorodinsky solution for n=3," Mathematical Social Sciences, Elsevier, vol. 35(3), pages 307-319, May.
    10. Kalai, Ehud, 1977. "Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparisons," Econometrica, Econometric Society, vol. 45(7), pages 1623-1630, October.
    11. Anbarci, Nejat & Sun, Ching-jen, 2013. "Asymmetric Nash bargaining solutions: A simple Nash program," Economics Letters, Elsevier, vol. 120(2), pages 211-214.
    12. Chun, Youngsub & Thomson, William, 1992. "Bargaining problems with claims," Mathematical Social Sciences, Elsevier, vol. 24(1), pages 19-33, August.
    13. Laruelle, Annick & Valenciano, Federico, 2008. "Noncooperative foundations of bargaining power in committees and the Shapley-Shubik index," Games and Economic Behavior, Elsevier, vol. 63(1), pages 341-353, May.
    14. Mark A. Chen, 2000. "Individual monotonicity and the leximin solution," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 15(2), pages 353-365.
    15. Imai, Haruo, 1983. "Individual Monotonicity and Lexicographic Maxmin Solution," Econometrica, Econometric Society, vol. 51(2), pages 389-401, March.
    16. repec:spr:compst:v:49:y:1999:i:3:p:395-400 is not listed on IDEAS
    17. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
    18. Peters, Hans & Tijs, Stef & Zarzuelo, Jose, 1994. "A reduced game property for the Kalai-Smorodinsky and egalitarian bargaining solutions," Mathematical Social Sciences, Elsevier, vol. 27(1), pages 11-18, February.
    19. Dubra, Juan, 2001. "An asymmetric Kalai-Smorodinsky solution," Economics Letters, Elsevier, vol. 73(2), pages 131-136, November.
    20. Lensberg, Terje, 1988. "Stability and the Nash solution," Journal of Economic Theory, Elsevier, vol. 45(2), pages 330-341, August.
    21. Roth, Alvin E., 1977. "Independence of irrelevant alternatives, and solutions to Nash's bargaining problem," Journal of Economic Theory, Elsevier, vol. 16(2), pages 247-251, December.
    22. Chun, Youngsub & Peters, Hans, 1991. "The lexicographic equal-loss solution," Mathematical Social Sciences, Elsevier, vol. 22(2), pages 151-161, October.
    23. John C. Harsanyi & Reinhard Selten, 1972. "A Generalized Nash Solution for Two-Person Bargaining Games with Incomplete Information," Management Science, INFORMS, vol. 18(5-Part-2), pages 80-106, January.
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    Cited by:

    1. Jaume García-Segarra & Miguel Ginés-Vilar, 2013. "Stagnation proofness and individually monotonic bargaining solutions," Working Papers 2013/04, Economics Department, Universitat Jaume I, Castellón (Spain).

    More about this item

    Keywords

    Bargaining; asymmetric bargaining solution; leximin solution;

    JEL classification:

    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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