The mathematics and statistics of voting power

Abstract

In an election, voting power---the probability that a single vote is decisive---is affected by the rule for aggregating votes into a single outcome. Voting power is important for studying political representation, fairness and strategy, and has been much discussed in political science. Although power indexes are often considered as mathematical definitions, they ultimately depend on statistical models of voting. Mathematical calculations of voting power usually have been performed under the model that votes are decided by coin flips. This simple model has interesting implications for weighted elections, two-stage elections (such as the U.S. Electoral College) and coalition structures. We discuss empirical failings of the coin-flip model of voting and consider, first, the implications for voting power and, second, ways in which votes could be modeled more realistically. Under the random voting model, the standard deviation of the average of n votes is proportional to $1/\sqrt{n}$, but under more general models, this variance can have the form $cn^{-\alpha}$ or $\sqrt{a-b\log n}$. Voting power calculations under more realistic models present research challenges in modeling and computation.