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Dummy Players and the Quota in Weighted Voting Games: Some Further Results

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Evaluating Voting Systems with Probability Models

Part of the book series: Studies in Choice and Welfare ((WELFARE))

Abstract

This chapter is a companion paper of Barthélémy et al. (2019) which studies the role of the quota on the occurrence of “dummy” players in small weighted voting games (i.e., in voting games with 3, 4 or 5 players). We here extend the results obtained in this paper by considering voting games with a larger number of players (up to 15). It is shown that the probability of having a player without voting power is very sensitive to the choice of the quota and the quota values that minimize this probability are derived.

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Notes

  1. 1.

    In the particular case where \(q=q_{maj}\), not only player 2 but also player 3 cannot be a dummy; see Proposition 1 in Barthélémy et al. (2013). In this case, the maximum number of dummy players is \(n-3\).

  2. 2.

    For a clear and simple presentation, see Straffin (1994).

  3. 3.

    This is an arbitrary choice. Any number of players, reasonably large, can be studied.

  4. 4.

    See Straffin (1994) for a presentation of this power index.

  5. 5.

    Obviously, other values of w or Q lead to the same kind of curves.

  6. 6.

    Note that \(Q=0.5\) and \(Q=0.6\) correspond to the same quota \(q=6\).

  7. 7.

    Note however that for high values of Q, the convergence is not clear and more players are needed in order to recover Penrose’s law.

  8. 8.

    The results we have obtained for smaller values of w are available from the authors upon request.

References

  • Barthélémy F, Lepelley D, Martin M (2013) On the likelihood of dummy players in weighted majority games. Social Choice Welf 41(2):263–279

    Article  Google Scholar 

  • Barthélémy F, Lepelley D, Martin M (2019) Dummy players and the quota in weighted voting games. THEMA working paper

    Google Scholar 

  • Barthélémy F, Martin M, Tchantcho B (2011) Some conjectures on the two main power indices. THEMA working paper 2011–2014

    Google Scholar 

  • Chang P-L, Chua VCH, Machover M (2006) L.S. Penrose’s limit theorem\(:\) tests by simulation. Math Social Sci 51(1):90–106

    Google Scholar 

  • Courtin S, Mbih B, Moyouwou I (2014) Are Condorcet procedures so bad according to the reinforcement axiom? Social Choice Welf 42(4):927–940

    Article  Google Scholar 

  • Diss M, Louichi A, Merlin V, Smaoui H (2012) An example of probability computations under the IAC assumption: the stability of scoring rules. Math Social Sci 64(1):57–66

    Article  Google Scholar 

  • Felsenthal D, Machover M (1998) The measurement of voting power. Publishing, UK, Theory and Practice, Problems and Paradoxes

    Google Scholar 

  • Gerhlein WV, Lepelley D (2017) Voting rules and paradoxical outcomes. Studies in choice and welfare, Springer

    Google Scholar 

  • Penrose LS (1946) The elementary statistics of majority voting. J R Stat Soc 109:53–57

    Article  Google Scholar 

  • Lepelley D, Louichi A, Smaoui H (2008) On Ehrhart polynomials and probability calculations in voting theory. Social Choice Welf 30(3):363–383

    Article  Google Scholar 

  • Lindner I, Machover M (2003) L.S. Penrose’s limit theorem: proof of some special cases. Math Social Sci 47(1):37–49

    Google Scholar 

  • Nijenhuis A, Wilf H (1978) Combinatorial algorithms for computers and calculators, 2nd edn. Academic Press, New York

    Google Scholar 

  • Penrose LS (1952) On the objective study of crowd behavior. H.K. Lewis and Co, London

    Google Scholar 

  • Straffin PD (1994) Power and stability in politics. Chapter 32, Aumann RJ, Hart S (eds) Handbook of game theory, vol \(2\). North-Holland, Amsterdam, The Netherlands

    Google Scholar 

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Acknowledgements

Financial support from the ANR SOLITER is gratefully acknowledged. This research has been also developed within the center of excellence MME-DII (ANR-11-LBX-0023-01). Warm thanks are due to Dominique Lepelley for his valuable help.

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Authors and Affiliations

  1. Department of Social Sciences, Université de Versailles Saint-Quentin en Yvelines, CEMOTEV, Guyancourt Cedex, 78047, France

    Fabrice Barthélémy

  2. Department of Economics, CY Cergy Paris UniversitHEMA, Cergy-Pontoise Cedex, 95011, France

    Mathieu Martin

Authors
  1. Fabrice Barthélémy
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  2. Mathieu Martin
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Corresponding author

Correspondence to Fabrice Barthélémy .

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Editors and Affiliations

  1. Department of Economics, CRESE EA3190, Université Bourgogne Franche-Comté, Besançon, France

    Mostapha Diss

  2. Faculty of Economics, Management, and Geography, CNRS and Université de Caen Normandie, Caen, France

    Vincent Merlin

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Barthélémy, F., Martin, M. (2021). Dummy Players and the Quota in Weighted Voting Games: Some Further Results. In: Diss, M., Merlin, V. (eds) Evaluating Voting Systems with Probability Models. Studies in Choice and Welfare. Springer, Cham. https://doi.org/10.1007/978-3-030-48598-6_13

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