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Composition Independence In Compound Games: A Characterization Of The Banzhaf Power Index And The Banzhaf Value

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  • Ori Haimanko

    (BGU)

Abstract

We introduce the axiom of composition independence for power indices and value maps. In the context of compound (two-tier) voting, the axiom requires the power attributed to a voter to be independent of the second-tier voting games played in all constituencies other than that of the voter. We show that the Banzhaf power index is uniquely characterized by the combination of composition independence, four semivalue axioms (transfer, positivity, symmetry, and dummy), and a mild efficiency-related requirement. A similar characterization is obtained as a corollary for the Banzhaf value on the space of all finite games (with transfer replaced by additivity).
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Suggested Citation

  • Ori Haimanko, 2017. "Composition Independence In Compound Games: A Characterization Of The Banzhaf Power Index And The Banzhaf Value," Working Papers 1713, Ben-Gurion University of the Negev, Department of Economics.
    • Handle: RePEc:bgu:wpaper:1713
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    References listed on IDEAS

    as
    1. Haimanko, Ori, 2018. "The axiom of equivalence to individual power and the Banzhaf index," Games and Economic Behavior, Elsevier, vol. 108(C), pages 391-400.
    2. Ori Haimanko, 2019. "Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(3), pages 755-768, September.
    3. M. Josune Albizuri & Luis M. Ruiz, 2001. "A new axiomatization of the Banzhaf semivalue," Spanish Economic Review, Springer;Spanish Economic Association, vol. 3(2), pages 97-109.
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    5. Dan S. Felsenthal & Moshé Machover, 1998. "The Measurement of Voting Power," Books, Edward Elgar Publishing, number 1489.
    6. Marcin Malawski, 2002. "Equal treatment, symmetry and Banzhaf value axiomatizations," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 47-67.
    7. Annick Laruelle & Federico Valenciano, 2001. "Shapley-Shubik and Banzhaf Indices Revisited," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 89-104, February.
    8. 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.
    9. Ezra Einy, 1987. "Semivalues of Simple Games," Mathematics of Operations Research, INFORMS, vol. 12(2), pages 185-192, May.
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    11. André Casajus, 2012. "Amalgamating players, symmetry, and the Banzhaf value," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(3), pages 497-515, August.
    12. 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.
    13. 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.
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    1. Ori Haimanko, 2019. "Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(3), pages 755-768, September.
    2. Jilei Shi & Erfang Shan, 2021. "The Banzhaf value for generalized probabilistic communication situations," Annals of Operations Research, Springer, vol. 301(1), pages 225-244, June.

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    More about this item

    Keywords

    Simple Games; Compound Games; Banzhaf Power Index; Banzhaf value; Composition Property; Semivalues; Transfer; Symmetry; Positivity; Dummy;
    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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