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The proportional Shapley value and applications

Author

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  • Philippe Solal

    (GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne - ENS de Lyon - École normale supérieure de Lyon - UL2 - Université Lumière - Lyon 2 - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique)

  • Sylvain Béal

    (CRESE - Centre de REcherches sur les Stratégies Economiques (UR 3190) - UFC - Université de Franche-Comté - UBFC - Université Bourgogne Franche-Comté [COMUE])

  • Sylvain Ferrières

    (CRESE - Centre de REcherches sur les Stratégies Economiques (UR 3190) - UFC - Université de Franche-Comté - UBFC - Université Bourgogne Franche-Comté [COMUE])

  • Éric Rémila

    (GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne - ENS de Lyon - École normale supérieure de Lyon - UL2 - Université Lumière - Lyon 2 - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique)

Abstract

We study a non linear weighted Shapley value (Shapley, 1953b) for cooperative games with transferable utility, in which the weights are endogenously given by the players' stand-alone worths. We call it the proportional Shapley value since it distributes the Harsanyi dividend (Harsanyi, 1959) of all coalitions in proportion to the stand-alone worths of its members. We show that this value recommends appealing payoff distributions in several applications among which a land production economy introduced in Shapley and Shubik (1967). Although the proportional Shapley value does not satisfy the classical axioms of linearity and consistency (Hart and Mas-Colell, 1989), the main results provide comparable axiomatic characterizations of our value and the Shapley value by means of weak versions of these two axioms. These characterizations rely on another result, which states that there exists a unique extension of a value defined on games that are additive except, possibly, for the grand coalition to the set of all games in the much larger class we consider. Moreover, our value inherits several well-known properties of the weighted Shapley values.
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Suggested Citation

  • Philippe Solal & Sylvain Béal & Sylvain Ferrières & Éric Rémila, 2017. "The proportional Shapley value and applications," Post-Print halshs-01644830, HAL.
    • Handle: RePEc:hal:journl:halshs-01644830
    DOI: 10.1016/j.geb.2017.08.010
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    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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