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Extending the Nash solution to choice problems with reference points

Author

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  • Sudhölter, Peter

    () (Department of Business and Economics)

  • Zarzuelo, José M.

    () (Faculty of Economics and Business Administration)

Abstract

In 1985 Aumann axiomatized the Shapley NTU value by non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives. We show that, when replacing unanimity by "unanimity for the grand coalition" and translation covariance, these axioms characterize the Nash solution on the class of n-person choice problems with reference points. A classical bargaining problem consists of a convex feasible set that contains the disagreement point here called reference point. The feasible set of a choice problem does not necessarily contain the reference point and may not be convex. However, we assume that it satisfies some standard properties. Our result is robust so that the characterization is still valid for many subclasses of choice problems, among those is the class of classical bargaining problems. Moreover, we show that each of the employed axioms – including independence of irrelevant alternatives – may be logically independent of the remaining axioms.

Suggested Citation

  • Sudhölter, Peter & Zarzuelo, José M., 2012. "Extending the Nash solution to choice problems with reference points," Discussion Papers of Business and Economics 13/2012, University of Southern Denmark, Department of Business and Economics.
  • Handle: RePEc:hhs:sdueko:2012_013
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    References listed on IDEAS

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    1. Hans Peters & Dries Vermeulen, 2012. "WPO, COV and IIA bargaining solutions for non-convex bargaining problems," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(4), pages 851-884, November.
    2. Rubinstein, Ariel, 1999. "Experience from a Course in Game Theory: Pre- and Postclass Problem Sets as a Didactic Device," Games and Economic Behavior, Elsevier, pages 155-170.
    3. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
    4. Marco Mariotti & Antonio Villar, 2005. "The Nash rationing problem," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(3), pages 367-377, September.
    5. Rubinstein, Ariel & Zhou, Lin, 1999. "Choice problems with a 'reference' point," Mathematical Social Sciences, Elsevier, vol. 37(3), pages 205-209, May.
    6. Bezalel Peleg & Peter Sudhölter & José Zarzuelo, 2012. "On the impact of independence of irrelevant alternatives: the case of two-person NTU games," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 3(1), pages 143-156, March.
    7. Aumann, Robert J, 1985. "An Axiomatization of the Non-transferable Utility Value," Econometrica, Econometric Society, vol. 53(3), pages 599-612, May.
    8. Peters Hans & Vermeulen Dries, 2006. "WPO, COV and IIA bargaining solutions," Research Memorandum 021, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    9. Herrero, Maria Jose, 1989. "The nash program: Non-convex bargaining problems," Journal of Economic Theory, Elsevier, vol. 49(2), pages 266-277, December.
    10. Mariotti, Marco, 1998. "Extending Nash's Axioms to Nonconvex Problems," Games and Economic Behavior, Elsevier, vol. 22(2), pages 377-383, February.
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    12. Maschler, M & Owen, G, 1989. "The Consistent Shapley Value for Hyperplane Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(4), pages 389-407.
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    Cited by:

    1. Subiza Begoña & Peris Josep E., 2014. "A Solution for General Exchange Markets with Indivisible Goods when Indifferences are Allowed," Mathematical Economics Letters, De Gruyter, pages 1-5.
    2. José-Manuel Giménez-Gómez & António Osório & Josep E. Peris, 2015. "From Bargaining Solutions to Claims Rules: A Proportional Approach," Games, MDPI, Open Access Journal, vol. 6(1), pages 1-7, March.

    More about this item

    Keywords

    Bargaining problem; Nash set; Shapley NTU value;

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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