On Random Quotas and Proportional Representation in Weighted Voting Games / 432
Weighted voting games (WVGs) model decision making bodies such as parliaments and councils. In such settings, it is often important to provide a measure of the influence a player has on the vote. Two highly popular such measures are the Shapley-Shubik power index, and the Banzhaf power index. Given a power measure, proportional representation is the property of having players' voting power proportional to the number of parliament seats they receive. Approximate proportional representation (w.r.t. the Banzhaf power index) can be ensured by changing the number of parliament seats each party receives; this is known as Penrose's square root method. However, a discrepancy between player weights and parliament seats is often undesirable or unfeasible; a simpler way of achieving approximate proportional representation is by changing the quota, i.e. the number of votes required in order to pass a bill. It is known that a player's Shapley-Shubik power index is proportional to his weight when one chooses a quota at random; that is, when taking a random quota, proportional representation holds in expectation. In our work, we show that not only does proportional representation hold in expectation, it also holds for many quotas. We do so by providing bounds on the variance of the Shapley value when the quota is chosen at random, assuming certain weight distributions. We further explore the case where weights are sampled from i.i.d. binomial distributions; for this case, we show good bounds on an important parameter governing the behavior of the variance, as well as substantiating our claims with empirical analysis.