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).

在加权投票游戏中，如果支持该提议的选民的加权总和至少等于某个配额，则每个投票者都有权重并接受提议。众所周知，在这种投票过程中，可能会发生玩家的投票对游戏结果没有影响的现象。这样的玩家称为“虚拟”玩家。本文研究了配额对加权投票游戏中虚拟玩家的出现的作用。假设每个可接受的加权投票游戏均等可能发生，我们根据三，四和五名玩家的配额来计算拥有无投票权的玩家的概率。事实证明，此概率对配额的选择非常敏感，并且可能很高。