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A Stability Index for Local Effectivity Functions

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  • Joseph M. Abdou

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris sciences et lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

Abstract

We study the structure of unstable local effectivity functions defined for n players and p alternatives. A stability index based on the notion of cycle is introduced. In the particular case of simple games, the stability index is closely related to the Nakamura Number. In general it may be any integer between 2 and p. We prove that the stability index for maximal effectivity functions and for maximal local effectivity functions is either 2 or 3.

Suggested Citation

  • Joseph M. Abdou, 2008. "A Stability Index for Local Effectivity Functions," Post-Print halshs-00331223, HAL.
    • Handle: RePEc:hal:journl:halshs-00331223
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00331223
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    References listed on IDEAS

    as
    1. Moulin, H. & Peleg, B., 1982. "Cores of effectivity functions and implementation theory," Journal of Mathematical Economics, Elsevier, vol. 10(1), pages 115-145, June.
    2. Rosenthal, Robert W., 1972. "Cooperative games in effectiveness form," Journal of Economic Theory, Elsevier, vol. 5(1), pages 88-101, August.
    3. Abdou, J, 1995. "Nash and Strongly Consistent Two-Player Game Forms," International Journal of Game Theory, Springer;Game Theory Society, vol. 24(4), pages 345-356.
    4. Eyal Winter & Bezalel Peleg, 2002. "original papers : Constitutional implementation," Review of Economic Design, Springer;Society for Economic Design, vol. 7(2), pages 187-204.
    5. Abdou, Joseph & Keiding, Hans, 2003. "On necessary and sufficient conditions for solvability of game forms," Mathematical Social Sciences, Elsevier, vol. 46(3), pages 243-260, December.
    6. repec:dau:papers:123456789/13220 is not listed on IDEAS
    7. J. Abdou, 1998. "Rectangularity and Tightness: A Normal Form Characterization of Perfect Information Extensive Game Forms," Mathematics of Operations Research, INFORMS, vol. 23(3), pages 553-567, August.
    8. Peleg, Bezalel, 2004. "Representation of effectivity functions by acceptable game forms: a complete characterization," Mathematical Social Sciences, Elsevier, vol. 47(3), pages 275-287, May.
    9. Abdou, J., 2000. "Exact stability and its applications to strong solvability," Mathematical Social Sciences, Elsevier, vol. 39(3), pages 263-275, May.
    10. Hervé Moulin, 1981. "The Proportional Veto Principle," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 48(3), pages 407-416.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Joseph Abdou, 2012. "The structure of unstable power mechanisms," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(2), pages 389-415, June.
    2. Joseph Abdou, 2012. "Stability and index of the meet game on a lattice," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(4), pages 775-789, November.
    3. Joseph M. Abdou, 2009. "The Structure of Unstable Power Systems," Post-Print halshs-00392515, HAL.
    4. repec:hal:wpaper:halshs-00633589 is not listed on IDEAS

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    More about this item

    Keywords

    Effectivity function; local effectivity function; acyclicity; stability index; Nakamura Number; acyclicity.; Fonction d'effectivité; acyclicité; fonction d'effectivité locale; indice de stabilité; nombre de Nakamura.;
    All these keywords.

    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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