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Minimal winning coalitions in weighted-majority voting games

Author

Listed:
  • Steven J. Brams

    (Department of Politics, New York University, New York, NY 10003, USA)

  • Peter C. Fishburn

    (AT&T Bell Laboratories, Murray Hill, NJ 07974, USA)

Abstract

Riker's size principle for n-person zero-sum games predicts that winning coalitions that form will be minimal in that any player's defection will negate the coalition's winning status. Brams and Fishburn (1995) applied Riker's principle to weighted-majority voting games in which players have voting weights w1\geqw2\geq\dots\geqwn, and a coalition is winning if its members' weights sum to more than half the total weight. We showed that players' bargaining power tends to decrease as their weights decrease when minimal winning coalitions obtain, but that the opposite trend occurs when the minimal winning coalitions that form are "weight-minimal", referred to as least winning coalitions. In such coalitions, large size may be more harmful than helpful. The present paper extends and refines our earlier analysis by providing mathematical foundations for minimal and least winning coalitions, developing new data to examine relationships between voting weight and voting power, and applying more sophisticated measures of power to these data. We identify all sets of minimal and least winning coalitions that arise from different voting weights for n\leq6 and characterize all coalitions that are minimal winning and least winning for every n. While our new analysis supports our earlier findings, it also indicates there to be less negative correlation between voting weight and voting power when least winning coalitions form. In this context, players' powers are fairly insensitive to their voting weights, so being large or small is not particularly important for inclusion in a least winning coalition.

Suggested Citation

  • Steven J. Brams & Peter C. Fishburn, 1996. "Minimal winning coalitions in weighted-majority voting games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 13(4), pages 397-417.
  • Handle: RePEc:spr:sochwe:v:13:y:1996:i:4:p:397-417
    Note: Received: 14 October 1994 / Accepted : 22 August 1995
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    Cited by:

    1. Frits Hof & Walter Kern & Sascha Kurz & Kanstantsin Pashkovich & Daniël Paulusma, 2020. "Simple games versus weighted voting games: bounding the critical threshold value," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 54(4), pages 609-621, April.
    2. Michela Chessa, 2014. "A generating functions approach for computing the Public Good index efficiently," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 658-673, July.
    3. Campbell, Donald E. & Graver, Jack & Kelly, Jerry S., 2012. "There are more strategy-proof procedures than you think," Mathematical Social Sciences, Elsevier, vol. 64(3), pages 263-265.
    4. Geir Asheim & Carl Claussen & Tore Nilssen, 2006. "Majority voting leads to unanimity," International Journal of Game Theory, Springer;Game Theory Society, vol. 35(1), pages 91-110, December.
    5. Mika WidgrÚn, 2002. "On the Probablistic Relationship between the Public Good Index and the Normalized Bannzhaf Index," Homo Oeconomicus, Institute of SocioEconomics, vol. 19, pages 373-386.
    6. Xavier Molinero & Maria Serna & Marc Taberner-Ortiz, 2021. "On Weights and Quotas for Weighted Majority Voting Games," Games, MDPI, vol. 12(4), pages 1-25, December.
    7. Sascha Kurz & Nikolas Tautenhahn, 2013. "On Dedekind’s problem for complete simple games," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(2), pages 411-437, May.
    8. Manfred J. Holler, 1998. "Two Stories, One Power Index," Journal of Theoretical Politics, , vol. 10(2), pages 179-190, April.
    9. Thomas Bräuninger & Thomas König, 2000. "Making Rules for Governing Global Commons," Journal of Conflict Resolution, Peace Science Society (International), vol. 44(5), pages 604-629, October.
    10. Manfred Holler & Rie Ono & Frank Steffen, 2001. "Constrained Monotonicity and the Measurement of Power," Theory and Decision, Springer, vol. 50(4), pages 383-395, June.
    11. Kurz, Sascha & Mayer, Alexander & Napel, Stefan, 2020. "Weighted committee games," European Journal of Operational Research, Elsevier, vol. 282(3), pages 972-979.
    12. José María Alonso‐Meijide & Manfred J. Holler, 2009. "Freedom Of Choice And Weighted Monotonicity Of Power," Metroeconomica, Wiley Blackwell, vol. 60(4), pages 571-583, November.

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