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WPO, COV and IIA bargaining solutions for non-convex bargaining problems

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  • Hans Peters

    ()

  • Dries Vermeulen

    ()

Abstract

We characterize all n-person multi-valued bargaining solutions, defined on the domain of all finite bargaining problems, and satisfying Weak Pareto Optimality (WPO), Covariance (COV), and Independence of Irrelevant Alternatives (IIA). We show that these solutions are obtained by iteratively maximizing nonsymmetric Nash products and determining the final set of points by so-called LDR decompositions. If, next, we assume the (set-theoretic) Axiom of Determinacy, then this class coincides with the class of iterated Nash bargaining solutions; but if we assume the Axiom of Choice then we are able to construct an additional large set of discontinuous and even nonmeasurable solutions. We show however that none of these nonmeasurable solutions can be defined in terms of set theoretic formulae. We next show that a number of existing results in the literature as well as some new results are implied by our approach. These include a characterization of all WPO, COV and IIA solutions—including single-valued ones—on the domain of all compact bargaining problems, and an extension of a theorem of Birkhoff characterizing translation invariant and homogeneous orderings. Copyright The Author(s) 2012

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File URL: http://hdl.handle.net/10.1007/s00182-010-0246-6
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Bibliographic Info

Article provided by Springer in its journal International Journal of Game Theory.

Volume (Year): 41 (2012)
Issue (Month): 4 (November)
Pages: 851-884

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Handle: RePEc:spr:jogath:v:41:y:2012:i:4:p:851-884

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Related research

Keywords: Nash bargaining solutions; Non-convex bargaining problems; C72; D44;

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References

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  1. Vincenzo Denicolò & Marco Mariotti, 2000. "Nash Bargaining Theory, Nonconvex Problems and Social Welfare Orderings," Theory and Decision, Springer, vol. 48(4), pages 351-358, June.
  2. 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.
  3. Conley, John P. & Wilkie, Simon, 1996. "An Extension of the Nash Bargaining Solution to Nonconvex Problems," Games and Economic Behavior, Elsevier, vol. 13(1), pages 26-38, March.
  4. Zame, William R., 2007. "Can intergenerational equity be operationalized?," Theoretical Economics, Econometric Society, vol. 2(2), June.
  5. Kaneko, Mamoru & Nakamura, Kenjiro, 1979. "The Nash Social Welfare Function," Econometrica, Econometric Society, vol. 47(2), pages 423-35, March.
  6. Mariotti, Marco, 1998. "Extending Nash's Axioms to Nonconvex Problems," Games and Economic Behavior, Elsevier, vol. 22(2), pages 377-383, February.
  7. Chris Starmer, 2000. "Developments in Non-expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk," Journal of Economic Literature, American Economic Association, vol. 38(2), pages 332-382, June.
  8. Naumova, Natalia & Yanovskaya, Elena, 2001. "Nash social welfare orderings," Mathematical Social Sciences, Elsevier, vol. 42(3), pages 203-231, November.
  9. Machina, Mark J, 1982. ""Expected Utility" Analysis without the Independence Axiom," Econometrica, Econometric Society, vol. 50(2), pages 277-323, March.
  10. Herrero, Maria Jose, 1989. "The nash program: Non-convex bargaining problems," Journal of Economic Theory, Elsevier, vol. 49(2), pages 266-277, December.
  11. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
  12. Marco Mariotti, 1998. "Nash bargaining theory when the number of alternatives can be finite," Social Choice and Welfare, Springer, vol. 15(3), pages 413-421.
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Cited by:
  1. Luís Carvalho, 2014. "A Constructive Proof of the Nash Bargaining Solution," Working Papers Series 2 14-01, ISCTE-IUL, Business Research Unit (BRU-IUL).
  2. Sudhölter, Peter & Zarzuelo, José M., 2013. "Extending the Nash solution to choice problems with reference points," Games and Economic Behavior, Elsevier, vol. 80(C), pages 219-228.

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