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Proportional Shapley levels values


  • Besner, Manfred


The proportional Shapley value (Besner 2016; Béal et al. 2017; Gangolly 1981) is an proportional counterpart to the Shapley value (Shapley 1953b) in cooperative games. As shown in Besner (2017a), the proportional Shapley value is a convincing non-linear alternative, especially in cost allocation, if the stand alone worths of the players are plausible weights. To enable similar properties for cooperative games with a level structure, we generalize this value. Therefore we adapt the proceeding applied to the weighted Shapley values in Besner (2017b). We present, analogous to the four classes of weighted Shapley levels values in Besner (2017b), four different values, the proportional Shapley hierarchy levels value, the proportional Shapley support levels value, the proportional Shapley alliance levels value and the proportional Shapley collaboration levels value, respectively.

Suggested Citation

  • Besner, Manfred, 2018. "Proportional Shapley levels values," MPRA Paper 87120, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:87120

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    References listed on IDEAS

    1. Gómez-Rúa, María & Vidal-Puga, Juan, 2010. "The axiomatic approach to three values in games with coalition structure," European Journal of Operational Research, Elsevier, vol. 207(2), pages 795-806, December.
    2. Frank Huettner, 2015. "A proportional value for cooperative games with a coalition structure," Theory and Decision, Springer, vol. 78(2), pages 273-287, February.
    3. Winter, Eyal, 1989. "A Value for Cooperative Games with Levels Structure of Cooperation," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(2), pages 227-240.
    4. Béal, Sylvain & Ferrières, Sylvain & Rémila, Eric & Solal, Philippe, 2018. "The proportional Shapley value and applications," Games and Economic Behavior, Elsevier, vol. 108(C), pages 93-112.
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    6. Tijs, Stef H. & Driessen, Theo S. H., 1986. "Extensions of solution concepts by means of multiplicative [var epsilon]-tax games," Mathematical Social Sciences, Elsevier, vol. 12(1), pages 9-20, August.
    7. Besner, Manfred, 2017. "Axiomatizations of the proportional Shapley value," MPRA Paper 82990, University Library of Munich, Germany.
    8. Calvo, Emilio & Javier Lasaga, J. & Winter, Eyal, 1996. "The principle of balanced contributions and hierarchies of cooperation," Mathematical Social Sciences, Elsevier, vol. 31(3), pages 171-182, June.
    9. María Gómez-Rúa & Juan Vidal-Puga, 2011. "Balanced per capita contributions and level structure of cooperation," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(1), pages 167-176, July.
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    11. Tijs, S.H. & Driessen, T.S.H., 1986. "Extensions of solution concepts by means of multiplicative å-games," Other publications TiSEM cfc61277-a471-446d-b8f0-c, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Besner, Manfred, 2018. "Player splitting, players merging, the Shapley set value and the Harsanyi set value," MPRA Paper 87125, University Library of Munich, Germany.

    More about this item


    Cooperative game · Level structure · (Proportional) Shapley (levels) value · Proportionality · Component substitution · Dividends;

    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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