Dummy Players and the Quota in Weighted Voting Games
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Dummy Players and the Quota in Weighted Voting Games Fabrice Barthelemy1 · Dominique Lepelley2 · Mathieu Martin3 · Hatem Smaoui2 Accepted: 2 September 2020 © Springer Nature B.V. 2020
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). Keywords Cooperative game theory · Weighted voting games · Dummy player · Probability of voting paradoxes JEL Classification C7 · D7
* Dominique Lepelley dominique.lepelley@univ‑reunion.fr Fabrice Barthelemy [email protected] Mathieu Martin mathieu.martin@u‑cergy.fr Hatem Smaoui hatem.smaoui@univ‑reunion.fr 1
CEMOTEV, Université de Saint Quentin, Economics, Guyancourt Cedex, France
2
CEMOI, Université de La Réunion, 97715 Saint‑Denis Cedex 9, France
3
THEMA, Université de Cergy-Pontoise, 33 boulevard du Port, 95011 Cergy‑Pontoise Cedex, France
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1 Introduction In cooperative game theory, the power of a player in a voting game is defined as the probability to be decisive in the collective choice process. In a weighted voting game, if each player is given a weight that is both strictly positive and strictly lower than the quota (defined as the total weight needed to form a winning coalition), it could be expected that the voting power of every player is different from 0, that is that there is no dummy player. However, several (real) examples are given in the literature, showing that dummy players do exist. One of the most famous occurrences of a dummy player is offered by Luxembourg in the Council of Ministers of the EU between 1958 and 1973. Luxembourg held one vote, whereas the quota for a proposition to be approved was 12 out of 17. Since other member states held an even number of votes (4 for Germany, France and Italy, 2 for Belgium and The Netherlands), Luxembourg formally was never able to make any difference in the voting process and was a dummy. Such situations are obviously extremely undesirable but we should not worry about them if it could be shown that their occurrence is rare. Unfortunately, it is demonstrated in Bart
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