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Dummy Players and the Quota in Weighted Voting Games

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Abstract

In a weighted voting game, each voter has a weight and a proposal is accepted if the sum of the weights of the voters in favor of that proposal is at least as large as a certain quota. It is well-known that, in this kind of voting process, it can occur that the vote of a player has no effect on the outcome of the game; such a player is called a “dummy” player. This paper studies the role of the quota on the occurrence of dummy players in weighted voting games. Assuming that every admissible weighted voting game is equally likely to occur, we compute the probability of having a player without voting power as a function of the quota for three, four and five players. It turns out that this probability is very sensitive to the choice of the quota and can be very high. The quota values that minimize (or maximize) the likelihood of dummy players are derived (Some technical details are voluntarily omitted in this version of our study. These details can be found in the online appendix associated with this paper at https://bit.ly/2MVVuBW).

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Notes

  1. The reader is referred to Taylor and Zwicker (1999) for a general presentation of voting games.

  2. It does not mean however that we discard the possibility that the game coincides with the unanimity game: a game with \(q < w\) can be isomorphic to the unanimity game.

  3. 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 \(n-3\).

  4. For a general background on Ehrhart theory and on the general problem of counting integer points in polytopes, see for example Beck and Robins (2007).

  5. For a rigorous description of this algorithm and for implementation details, see Verdoolaege et al. (2004). Note also that these results could have been obtained by using the last version of the software [Normaliz] (2018), based on an algorithm different from Barvinok’s one.

  6. A quasi polynomial is a polynomial the coefficients of which are rational periodic numbers. Periodic numbers are usually made explicit by a list of rational numbers enclosed in square brackets. For example, \(U(x)=[1/2,3/4,1]_x\) is a periodic number with period equal to 3 and with \(U(x)=1/2\) if \(x=0\) mod 3, \(U(x)=3/4\) if \(x=1\) mod 3 and \(U(x)=1\) if \(x=2\) mod 3.

  7. When Qw is not an integer, we have computed the probability with q equal to the smallest integer higher than Qw.

  8. We are very grateful to Sascha Kurz for having provided us with the exhaustive lists of minimum integer representations of weighted voting games up to seven players. ‘Minimum integer’ representation means that the weights are integers and every other integer representation is at least as large in each component. Note also that for \(n \le 7\), each weighted voting game admits one and only one minimum integer representation (see for example Freixas and Molinero 2009).

  9. Notice that, as \(q \ge q_{maj}\), we can omit the inequalities associated with \(T \in \mathcal L\) when \(N\backslash T\) is winning. This happens for the game [2; 1, 1, 1, 0].

  10. The particular shape of the curve for the three-player case is strongly impacted by the fact that the probability of having a dummy player is, in this case, equal to zero when \( Q=1/2\).

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Acknowledgements

Financial support from the ANR SOLITER is gratefully acknowledged. Warm thanks are due to Sascha Kurz for his valuable help.

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Authors and Affiliations

  1. CEMOTEV, Université de Saint Quentin, Economics, Guyancourt Cedex, France

    Fabrice Barthelemy

  2. CEMOI, Université de La Réunion, 97715, Saint-Denis Cedex 9, France

    Dominique Lepelley & Hatem Smaoui

  3. THEMA, Université de Cergy-Pontoise, 33 boulevard du Port, 95011, Cergy-Pontoise Cedex, France

    Mathieu Martin

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  1. Fabrice Barthelemy
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  2. Dominique Lepelley
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  4. Hatem Smaoui
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Correspondence to Dominique Lepelley.

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Barthelemy, F., Lepelley, D., Martin, M. et al. Dummy Players and the Quota in Weighted Voting Games. Group Decis Negot 30, 43–61 (2021). https://doi.org/10.1007/s10726-020-09705-y

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