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The Banzhaf Value for Cooperative and Simple Multichoice Games

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  • Josep Freixas

    (Universitat Politècnica de Catalunya (Campus Manresa))

Abstract

This article proposes a value which can be considered an extension of the Banzhaf value for cooperative games. The proposed value is defined on the class of j-cooperative games, i.e., games in which players choose among a finite set of ordered actions and the result depends only on these elections. If the output is binary, only two options are available, then j-cooperative games become j-simple games. The restriction of the value to j-simple games leads to a power index that can be considered an extension of the Banzhaf power index for simple games. The paper provides an axiomatic characterization for the value and the index which is closely related to the first axiomatization of the Banzhaf value and Banzhaf power index in the respective contexts of cooperative and simple games.

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  • Josep Freixas, 2020. "The Banzhaf Value for Cooperative and Simple Multichoice Games," Group Decision and Negotiation, Springer, vol. 29(1), pages 61-74, February.
    • Handle: RePEc:spr:grdene:v:29:y:2020:i:1:d:10.1007_s10726-019-09651-4
    DOI: 10.1007/s10726-019-09651-4
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    References listed on IDEAS

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    1. Josep Freixas, 2005. "Banzhaf Measures for Games with Several Levels of Approval in the Input and Output," Annals of Operations Research, Springer, vol. 137(1), pages 45-66, July.
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    3. Matthew Gould & Matthew D. Rablen, 2017. "Reform of the United Nations Security Council: equity and efficiency," Public Choice, Springer, vol. 173(1), pages 145-168, October.
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    8. Freixas, Josep, 2012. "Probabilistic power indices for voting rules with abstention," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 89-99.
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    Cited by:

    1. Josep Freixas & Montserrat Pons, 2022. "A critical analysis on the notion of power," Annals of Operations Research, Springer, vol. 318(2), pages 911-933, November.
    2. Denis Bouyssou & Thierry Marchant & Marc Pirlot, 2021. "The size of the maximum antichains in products of linear orders," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(3), pages 648-659, October.
    3. Izabella Stach, 2022. "Reformulation of Public Help Index θ Using Null Player Free Winning Coalitions," Group Decision and Negotiation, Springer, vol. 31(2), pages 317-334, April.
    4. Josep Freixas & Montserrat Pons, 2021. "An Appropriate Way to Extend the Banzhaf Index for Multiple Levels of Approval," Group Decision and Negotiation, Springer, vol. 30(2), pages 447-462, April.
    5. Denis Bouyssou & Thierry Marchant & Marc Pirlot, 2021. "The size of the maximum antichains in products of linear orders," Post-Print hal-03047087, HAL.
    6. Boyang Dai & Xiangfeng Yang & Xiaoyue Liu, 2022. "Shapley Value of Uncertain Coalitional Game based on Hurwicz Criterion with Application to Water Resource Allocation," Group Decision and Negotiation, Springer, vol. 31(1), pages 241-260, February.
    7. Bertrand Mbama Engoulou & Pierre Wambo & Lawrence Diffo Lambo, 2023. "Banzhaf–Coleman–Dubey–Shapley sensitivity index for simple multichoice voting games," Annals of Operations Research, Springer, vol. 328(2), pages 1349-1364, September.
    8. Bertrand Mbama Engoulou & Pierre Wambo & Lawrence Diffo Lambo, 2023. "A Characterization of the Totally Critical Raw Banzhaf Power Index on Dichotomous Voting Games with Several Levels of Approval in Input," Group Decision and Negotiation, Springer, vol. 32(4), pages 871-888, August.
    9. 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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