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Optimizing the Efficiency of Weighted Voting Games



Having a group of voters endowed with weights, the simple weighted voting game (or system) represents a system of approving propositions in which the approved is only a proposition that is accepted by voters weighted to a number that is at least equal to a prescribed number called a quota. We call the system simple if there is only one set of weights and one quota, as opposed to the multi-rule systems that have more weights assigned to each voter and with more quotas. This paper presents an analysis of the efficiency of simple weighted voting systems. It assumes the Impartial Anonymous Culture (the probability of a single voter voting for a proposition is 1=2 and voters act independently). This culture is used for the general evaluationof voting systems, when no specific information about propositions and voters’ preferences are known, or when the voters’ preferences and proposition characteristics are not willing to be reflected in the voting system itself, keeping in mind its non-pragmatics, fairn ess and generality. The efficiency of a simple weighted voting system is defined as the probability of a proposition being approved. This paper focuses on efficiency maximization and minimization with respect to weights. We prove a theorem which enables the computing of the efficiency maximum and efficiency minimum with respect to weights, given the number of voters and quota in linear time.

Suggested Citation

  • Pavel Doležel, 2011. "Optimizing the Efficiency of Weighted Voting Games," Czech Economic Review, Charles University Prague, Faculty of Social Sciences, Institute of Economic Studies, vol. 5(3), pages 306-323, November.
  • Handle: RePEc:fau:aucocz:au2011_306

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    References listed on IDEAS

    1. Leech, Dennis & Aziz, Haris, 2007. "The Double Majority Voting Rule of the EU Reform Treaty as a Democratic Ideal for an Enlarging Union : an Appraisal Using Voting Power Analysis," The Warwick Economics Research Paper Series (TWERPS) 824, University of Warwick, Department of Economics.
    2. Algaba, E. & Bilbao, J. M. & Fernandez Garcia, J. R. & Lopez, J. J., 2003. "Computing power indices in weighted multiple majority games," Mathematical Social Sciences, Elsevier, vol. 46(1), pages 63-80, August.
    3. Prasad, K & Kelly, J S, 1990. "NP-Completeness of Some Problems Concerning Voting Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(1), pages 1-9.
    4. Leech, D., 2000. "Members' Voting Power in the Governance of the International Monetary Fund," The Warwick Economics Research Paper Series (TWERPS) 583, University of Warwick, Department of Economics.
    5. McIntyre, Elizabeth, 1954. "Weighted Voting in International Organizations," International Organization, Cambridge University Press, vol. 8(04), pages 484-497, November.
    6. Ines Lindner, 2008. "The power of a collectivity to act in weighted voting games with many small voters," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 30(4), pages 581-601, May.
    7. Dan S. Felsenthal & Moshé Machover, 2004. "A Priori Voting Power: What Is It All About?," Political Studies Review, Political Studies Association, vol. 2(1), pages 1-23.
    8. Aziz, Haris & Paterson, Mike & Leech, Dennis, 2007. "Combinatorial and computational aspects of multiple weighted voting games," The Warwick Economics Research Paper Series (TWERPS) 823, University of Warwick, Department of Economics.
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    More about this item


    Weighted voting game; integer programming; efficiency of voting; IAC models;

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior


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