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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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  • 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.
    • Handle: RePEc:spr:jogath:v:41:y:2012:i:4:p:851-884
    DOI: 10.1007/s00182-010-0246-6
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    Cited by:

    1. Cheng-Zhong Qin & Shuzhong Shi & Guofu Tan, 2015. "Nash bargaining for log-convex problems," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(3), pages 413-440, April.
    2. Yongsheng Xu & Naoki Yoshihara, 2020. "Nonconvex Bargaining Problems: Some Recent Developments," Homo Oeconomicus: Journal of Behavioral and Institutional Economics, Springer, vol. 37(1), pages 7-41, November.
    3. Youngsub Chun, 2020. "Some Impossibility Results on the Converse Consistency Principle in Bargaining," Homo Oeconomicus: Journal of Behavioral and Institutional Economics, Springer, vol. 37(1), pages 59-65, November.
    4. 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.
    5. Yongsheng Xu & Naoki Yoshihara, 2019. "An equitable Nash solution to nonconvex bargaining problems," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(3), pages 769-779, September.
    6. János Flesch & Dries Vermeulen & Anna Zseleva, 2019. "Catch games: the impact of modeling decisions," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(2), pages 513-541, June.
    7. William Thomson, 2022. "On the axiomatic theory of bargaining: a survey of recent results," Review of Economic Design, Springer;Society for Economic Design, vol. 26(4), pages 491-542, December.
    8. Zambrano, Eduardo, 2016. "‘Vintage’ Nash bargaining without convexity," Economics Letters, Elsevier, vol. 141(C), pages 32-34.
    9. Qin, Cheng-Zhong & Tan, Guofu & Wong, Adam Chi Leung, 2019. "Implementation of Nash bargaining solutions with non-convexity," Economics Letters, Elsevier, vol. 178(C), pages 46-49.
    10. Luís Carvalho, 2014. "A Constructive Proof of the Nash Bargaining Solution," Working Papers Series 2 14-01, ISCTE-IUL, Business Research Unit (BRU-IUL).

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    More about this item

    Keywords

    Nash bargaining solutions; Non-convex bargaining problems; C72; D44;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions

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