On the Effects of Priors in Weighted Voting Games Joel Oren, Yuval Filmus, Yair Zick and Yoram Bachrach 1

Motivation Weighted voting systems are a common decision making method. Parliaments The EU Council of Members The US Electoral College Each member (a state, a political party etc.) has a weight; in order to pass a bill, the total number of votes for the bill must exceed a given threshold. 2 Image: Wikipedia

Motivation A Central Question: how influential is a member? Voting power is not equal to voting weight. Example: there are three parties; need 60 votes to pass a bill. Party A – 58 votes Party B – 31 votes Party C – 31 votes Although party A is much bigger than the other two, it has the same decision making ability as the other two (any party needs the help of at least one other party in order to pass a bill). 3

The Shapley Value and WVGs q = 50 4

The Shapley Value and WVGs 5

Our Contribution 6

Balls and Bins Distributions

Balls and Bins: Uniform 8

Huge disparity at some thresholds Near Equality at others… Changing the threshold from 500 to 550 results in a huge shift in voting power 9

Balls and Bins: Uniform 10

Balls and Bins: Exponential 11

Balls and Bins: Exponential 12

13 Image: Wikipedia

Super-Increasing Weights 14

i.i.d distributed weights 15

Conclusions and Future Work We study how the Shapley value changes as a function of the threshold, under different weight distributions. Our results answer several open questions from Zick et al. (2011), Zuckerman et al. (2012) and Zick (2013). Future work: Proportional representation Probabilistic analysis of other cooperative game models? Have you seen this function? 16 Thank you! Questions?