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Using the Multilinear Extension to Study Some Probabilistic Power Indices

Author

Listed:
  • Josep Freixas

    (Universitat Politècnica de Catalunya (Barcelona–Tech))

  • Montserrat Pons

    (Universitat Politècnica de Catalunya (Barcelona–Tech))

Abstract

We consider binary voting systems modeled by a simple game, in which voters vote independently of each other, and the probability distribution over coalitions is known. The Owen’s multilinear extension of the simple game is used to improve the use and the computation of three indices defined in this model: the decisiveness index, which is an extension of the Banzhaf index, the success index, which is an extension of the Rae index, and the luckiness index. This approach leads us to prove new properties and inter-relations between these indices. In particular it is proved that the ordinal equivalence between success and decisiveness indices is achieved in any game if and only if the probability distribution is anonymous. In the anonymous case, the egalitarianism of the three indices is compared, and it is also proved that, for these distributions, decisiveness and success indices respect the strength of the seats, whereas luckiness reverses this order.

Suggested Citation

  • Josep Freixas & Montserrat Pons, 2017. "Using the Multilinear Extension to Study Some Probabilistic Power Indices," Group Decision and Negotiation, Springer, vol. 26(3), pages 437-452, May.
  • Handle: RePEc:spr:grdene:v:26:y:2017:i:3:d:10.1007_s10726-016-9514-6
    DOI: 10.1007/s10726-016-9514-6
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    References listed on IDEAS

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    1. Annick Laruelle & Federico Valenciano, 2005. "Assessing success and decisiveness in voting situations," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 24(1), pages 171-197, January.
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    4. Freixas, Josep & Pons, Montserrat, 2008. "Circumstantial power: Optimal persuadable voters," European Journal of Operational Research, Elsevier, vol. 186(3), pages 1114-1126, May.
    5. René Brink & Frank Steffen, 2014. "On the Measurement of Success and Satisfaction," Studies in Choice and Welfare, in: Rudolf Fara & Dennis Leech & Maurice Salles (ed.), Voting Power and Procedures, edition 127, pages 41-64, Springer.
    6. Federico Valenciano & Annick Laruelle & Ricardo Martínez, 2004. "Success Versus Decisiveness: Conceptual Discussion And Case Study," Working Papers. Serie AD 2004-30, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
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    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.

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    More about this item

    Keywords

    Game theory; Voting; Simple games; Power indices; Multilinear extension;
    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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