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Characterization of the Shapley-Shubik Power Index Without the Efficiency Axiom

Author

Listed:
  • Ezra Einy

    (Department of Economics, Ben-Gurion University of the Negev, Israel)

  • Ori Haimanko

    (Department of Economics, Ben-Gurion University of the Negev, Israel)

Abstract

We show that the Shapley-Shubik power index on the domain of simple (voting) games can be uniquely characterized without the e¢ ciency axiom. In our axiomatization, the efficiency is replaced by the following weaker require- ment that we term the gain-loss axiom: any gain in power by a player implies a loss for someone else (the axiom does not specify the extent of the loss). The rest of our axioms are standard: transfer (which is the version of additivity adapted for simple games), symmetry or equal treatment, and dummy

Suggested Citation

  • Ezra Einy & Ori Haimanko, 2010. "Characterization of the Shapley-Shubik Power Index Without the Efficiency Axiom," Working Papers 1004, Ben-Gurion University of the Negev, Department of Economics.
    • Handle: RePEc:bgu:wpaper:1004
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    References listed on IDEAS

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    1. Dubey, Pradeep & Einy, Ezra & Haimanko, Ori, 2005. "Compound voting and the Banzhaf index," Games and Economic Behavior, Elsevier, vol. 51(1), pages 20-30, April.
    2. Roth, Alvin E., 1977. "Utility functions for simple games," Journal of Economic Theory, Elsevier, vol. 16(2), pages 481-489, December.
    3. Blair, Douglas H. & McLean, Richard P., 1990. "Subjective evaluations of n-person games," Journal of Economic Theory, Elsevier, vol. 50(2), pages 346-361, April.
    4. Ezra Einy, 1987. "Semivalues of Simple Games," Mathematics of Operations Research, INFORMS, vol. 12(2), pages 185-192, May.
    5. Pradeep Dubey & Abraham Neyman & Robert James Weber, 1981. "Value Theory Without Efficiency," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 122-128, February.
    6. Shapley, L. S. & Shubik, Martin, 1954. "A Method for Evaluating the Distribution of Power in a Committee System," American Political Science Review, Cambridge University Press, vol. 48(3), pages 787-792, September.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Sylvain Béal & Marc Deschamps & Mostapha Diss & Rodrigue Tido Takeng, 2024. "Cooperative games with diversity constraints," Working Papers hal-04447373, HAL.
    2. Stefano Benati & Giuseppe Vittucci Marzetti, 2013. "Probabilistic spatial power indexes," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 40(2), pages 391-410, February.
    3. Shan, Erfang & Cui, Zeguang & Lyu, Wenrong, 2023. "Gain–loss and new axiomatizations of the Shapley value," Economics Letters, Elsevier, vol. 228(C).
    4. Hans Peters & José M. Zarzuelo, 2017. "An axiomatic characterization of the Owen–Shapley spatial power index," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(2), pages 525-545, May.
    5. Qianqian Kong & Hans Peters, 2021. "An issue based power index," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(1), pages 23-38, March.
    6. Casajus, André, 2014. "The Shapley value without efficiency and additivity," Mathematical Social Sciences, Elsevier, vol. 68(C), pages 1-4.
    7. Albizuri, M.J. & Goikoetxea, A., 2022. "Probabilistic Owen-Shapley spatial power indices," Games and Economic Behavior, Elsevier, vol. 136(C), pages 524-541.
    8. André Casajus & Frank Huettner, 2019. "The Coleman–Shapley index: being decisive within the coalition of the interested," Public Choice, Springer, vol. 181(3), pages 275-289, December.
    9. Ritu Dutta & Souvik Roy & Surajit Borkotokey, 2023. "The Generalized Shapley Value of Cooperative Games as a Social Preference Function," Group Decision and Negotiation, Springer, vol. 32(2), pages 277-300, April.
    10. Karos, Dominik & Peters, Hans, 2015. "Indirect control and power in mutual control structures," Games and Economic Behavior, Elsevier, vol. 92(C), pages 150-165.
    11. Ritu Dutta & Rajnish Kumnar & Surajit Borkotokey, 2023. "How to choose a Compatible Committee?," Papers 2308.03507, arXiv.org.
    12. Gusev, Vasily V., 2020. "The vertex cover game: Application to transport networks," Omega, Elsevier, vol. 97(C).
    13. Vasily V. Gusev, 2021. "Set-weighted games and their application to the cover problem," HSE Working papers WP BRP 247/EC/2021, National Research University Higher School of Economics.
    14. Cesarino Bertini & Josep Freixas & Gianfranco Gambarelli & Izabella Stach, 2013. "Comparing Power Indices," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 15(02), pages 1-19.
    15. M. J. Albizuri & A. Goikoetxea, 2021. "The Owen–Shapley Spatial Power Index in Three-Dimensional Space," Group Decision and Negotiation, Springer, vol. 30(5), pages 1027-1055, October.

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    More about this item

    Keywords

    Simple Games; Shapley-Shubik Power Index; Effciency Axiom;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior

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