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Axiomatisation of the Shapley value and power index for bi-cooperative games

Author

Listed:
  • Christophe Labreuche

    (Laboratoire Albert Fert (ex-UMPhy Unité mixte de physique CNRS/Thales) - THALES [France] - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique)

  • Michel Grabisch

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

Bi-cooperative games have been introduced by Bilbao as a generalization of classical cooperative games, where each player can participate positively to the game (defender), negatively (defeater), or do not participate (abstentionist). In a voting situation (simple games), they coincide with ternary voting game of Felsenthal and Mochover, where each voter can vote in favor, against or abstain. In this paper, we propose a definition of value or solution concept for bi-cooperative games, close to the Shapley value, and we give an interpretation of this value in the framework of (ternary) simple games, in the spirit of Shapley-Shubik, using the notion of swing. Lastly, we compare our definition with the one of Felsenthal and Machover, based on the notion of ternary roll-call, and the Shapley value of multi-choice games proposed by Hsiao and Ragahavan.

Suggested Citation

  • Christophe Labreuche & Michel Grabisch, 2006. "Axiomatisation of the Shapley value and power index for bi-cooperative games," Post-Print halshs-00113340, HAL.
    • Handle: RePEc:hal:journl:halshs-00113340
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00113340v1
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    References listed on IDEAS

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    1. Felsenthal, Dan S & Machover, Moshe, 1996. "Alternative Forms of the Shapley Value and the Shapley-Shubik Index," Public Choice, Springer, vol. 87(3-4), pages 315-318, June.
    2. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
    3. repec:cup:apsrev:v:48:y:1954:i:03:p:787-792_00 is not listed on IDEAS
    4. Mustapha Ridaoui & Michel Grabisch & Christophe Labreuche, 2019. "Interaction indices for multichoice games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02353519, HAL.
    5. Hsiao Chih-Ru & Raghavan T. E. S., 1993. "Shapley Value for Multichoice Cooperative Games, I," Games and Economic Behavior, Elsevier, vol. 5(2), pages 240-256, April.
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    2. Christophe Labreuche & Michel Grabisch, 2008. "A value for bi-cooperative games," Post-Print halshs-00308738, HAL.
    3. Fabien Lange & Michel Grabisch, 2011. "New axiomatizations of the Shapley interaction index for bi-capacities," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00625355, HAL.

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    JEL classification:

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

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