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A note on the ordinal equivalence of power indices in games with coalition structure

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  • Sébastien Courtin
  • Bertrand Tchantcho

Abstract

The desirability relation was introduced by Isbell (A class of simple games 25: 423–439, 1958 ) to qualitatively compare the a priori influence of voters in a simple game. In this paper, we extend this desirability relation to simple games with coalition structure. In these games, players organize themselves into a priori disjoint coalitions. It appears that the desirability relation defined in this paper is a complete preorder in the class of swap-robust games. We also compare our desirability relation with the preorders induced by the generalizations to games with coalition structure of the Shapley–Shubik and Banzhaf–Coleman power indices (Owen, Lecture notes in economics and mathematical systems: Essays in honor of Oskar Morgenstern. In: Hernn R, Moschlin O (eds.) Values of games with a priori unions, 1977 ; Owen, Power, voting and voting power. In: Holler MJ (ed.) Modification of the Banzhaf–Coleman index for games with a priori unions, 1981 ). It happens that in general they are different even if one considers the subclass of weighed voting games. However, if structural coalitions have equal size then both Owen–Banzhaf and the desirability preordering coincide. Copyright Springer Science+Business Media New York 2015

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  • Sébastien Courtin & Bertrand Tchantcho, 2015. "A note on the ordinal equivalence of power indices in games with coalition structure," Theory and Decision, Springer, vol. 78(4), pages 617-628, April.
    • Handle: RePEc:kap:theord:v:78:y:2015:i:4:p:617-628
    DOI: 10.1007/s11238-014-9445-0
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    References listed on IDEAS

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    1. Roland Pongou & Bertrand Tchantcho & Lawrence Diffo Lambo, 2011. "Political influence in multi-choice institutions: cyclicity, anonymity, and transitivity," Theory and Decision, Springer, vol. 70(2), pages 157-178, February.
    2. Tchantcho, Bertrand & Lambo, Lawrence Diffo & Pongou, Roland & Engoulou, Bertrand Mbama, 2008. "Voters' power in voting games with abstention: Influence relation and ordinal equivalence of power theories," Games and Economic Behavior, Elsevier, vol. 64(1), pages 335-350, September.
    3. Hamiache, Gerard, 1999. "A new axiomatization of the Owen value for games with coalition structures," Mathematical Social Sciences, Elsevier, vol. 37(3), pages 281-305, May.
    4. Parker, Cameron, 2012. "The influence relation for ternary voting games," Games and Economic Behavior, Elsevier, vol. 75(2), pages 867-881.
    5. Laruelle,Annick & Valenciano,Federico, 2011. "Voting and Collective Decision-Making," Cambridge Books, Cambridge University Press, number 9780521182638.
    6. Dominique Lepelley & N. Andjiga & F. Chantreuil, 2003. "La mesure du pouvoir de vote," Post-Print halshs-00069255, HAL.
    7. Lawrence Diffo Lambo & Joël Moulen, 2002. "Ordinal equivalence of power notions in voting games," Theory and Decision, Springer, vol. 53(4), pages 313-325, December.
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    Cited by:

    1. Sylvain Béal & Eric Rémila & Philippe Solal, 2019. "Coalitional desirability and the equal division value," Theory and Decision, Springer, vol. 86(1), pages 95-106, February.
    2. Joseph Armel Momo Kenfack & Bertrand Tchantcho & Bill Proces Tsague, 2019. "On the ordinal equivalence of the Jonhston, Banzhaf and Shapley–Shubik power indices for voting games with abstention," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(2), pages 647-671, June.
    3. Courtin, Sébastien & Nganmeni, Zéphirin & Tchantcho, Bertrand, 2017. "Dichotomous multi-type games with a coalition structure," Mathematical Social Sciences, Elsevier, vol. 86(C), pages 9-17.
    4. Sébastien Courtin & Zéphirin Nganmeni & Bertrand Tchantcho, 2017. "Dichotomous multi-type games with a coalition structure," Post-Print halshs-01545772, HAL.

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    More about this item

    Keywords

    Voting games; Coalition structure; Power indices; Desirability relation; C71; D72;
    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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