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How to share when context matters : The Mobius value as a generalized solution for cooperative games

Author

Listed:
  • Jacques-François Thisse

    (CERAS - Centre d'enseignement et de recherche en analyse socio-économique - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique, CORE - Center of Operation Research and Econometrics [Louvain] - UCL - Université Catholique de Louvain = Catholic University of Louvain)

  • Antoine Billot

    (CORE - Center of Operation Research and Econometrics [Louvain] - UCL - Université Catholique de Louvain = Catholic University of Louvain, UP2 - Université Panthéon-Assas)

Abstract

All quasivalues rest on a set of three basic axioms (efficiency, null player, and additivity), which are augmented with positivity for random order values, and with positivity and partnership for weighted values. We introduce the concept of Möbius value associated with a sharing system and show that this value is characterized by the above three axioms. We then establish that (i) a Möbius value is a random order value if and only if the sharing system is stochastically rationalizable and (ii) a Möbius value is a weighted value if and only if the sharing system satisfies the Luce choice axiom.

Suggested Citation

  • Jacques-François Thisse & Antoine Billot, 2005. "How to share when context matters : The Mobius value as a generalized solution for cooperative games," Post-Print halshs-00754051, HAL.
    • Handle: RePEc:hal:journl:halshs-00754051
    DOI: 10.1016/j.jmateco.2004.12.008
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    References listed on IDEAS

    as
    1. Itzhak Gilboa & Dov Monderer, 1989. "A Game-Theoretic Approach to the Binary Stochastic Choice Problem," Discussion Papers 854, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. repec:cor:louvrp:-1434 is not listed on IDEAS
    3. Michel Grabisch & Jean-Luc Marichal & Marc Roubens, 2000. "Equivalent Representations of Set Functions," Mathematics of Operations Research, INFORMS, vol. 25(2), pages 157-178, May.
    4. Ehud Kalai & Dov Samet, 1983. "On Weighted Shapley Values," Discussion Papers 602, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    5. Chun, Youngsub, 1991. "On the Symmetric and Weighted Shapley Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(2), pages 183-190.
    6. Chateauneuf, Alain & Jaffray, Jean-Yves, 1989. "Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 263-283, June.
    7. Marc Roubens & Michel Grabisch, 1999. "An axiomatic approach to the concept of interaction among players in cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(4), pages 547-565.
    8. Nowak Andrzej S. & Radzik Tadeusz, 1994. "The Shapley Value for n-Person Games in Generalized Characteristic Function Form," Games and Economic Behavior, Elsevier, vol. 6(1), pages 150-161, January.
    9. Fabien Lange & Michel Grabisch, 2006. "Interaction transform for bi-set functions over a finite set," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00186891, HAL.
    10. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series dp421, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    11. Monderer, Dov & Samet, Dov, 2002. "Variations on the shapley value," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 54, pages 2055-2076, Elsevier.
    12. BILLOT, Antoine & THISSE, Jacques-François, 1999. "A discrete choice model when context matters," LIDAM Reprints CORE 1434, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Citations

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    Cited by:

    1. Rene (J.R.) van den Brink & Rene Levinsky & Miroslav Zeleny, 2018. "The Shapley Value, Proper Shapley Value, and Sharing Rules for Cooperative Ventures," Tinbergen Institute Discussion Papers 18-089/II, Tinbergen Institute.
    2. Demuynck, Thomas & Rock, Bram De & Ginsburgh, Victor, 2016. "The transfer paradox in welfare space," Journal of Mathematical Economics, Elsevier, vol. 62(C), pages 1-4.
    3. Fabien Lange & László Kóczy, 2013. "Power indices expressed in terms of minimal winning coalitions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(2), pages 281-292, July.
    4. Manfred Besner, 2020. "Parallel axiomatizations of weighted and multiweighted Shapley values, random order values, and the Harsanyi set," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 55(1), pages 193-212, June.
    5. Manfred Besner, 2020. "Value dividends, the Harsanyi set and extensions, and the proportional Harsanyi solution," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(3), pages 851-873, September.
    6. Besner, Manfred, 2019. "Value dividends, the Harsanyi set and extensions, and the proportional Harsanyi payoff," MPRA Paper 92247, University Library of Munich, Germany.
    7. Pierre Dehez, 2017. "On Harsanyi Dividends and Asymmetric Values," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-36, September.
    8. Besner, Manfred, 2021. "Disjointly productive players and the Shapley value," MPRA Paper 108241, University Library of Munich, Germany.
    9. Besner, Manfred, 2021. "Disjointly and jointly productive players and the Shapley value," MPRA Paper 108511, University Library of Munich, Germany.
    10. René Brink & René Levínský & Miroslav Zelený, 2015. "On proper Shapley values for monotone TU-games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(2), pages 449-471, May.

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    More about this item

    Keywords

    Shapley value; Quasivalue; Möbius inverse;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D46 - Microeconomics - - Market Structure, Pricing, and Design - - - Value Theory
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement

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