The Mathematics and Statistics of Voting Power
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/√n, but under more general models, this variance can have the form cn^(−α) or √a−b log n. Voting power calculations undermore realistic models present research challenges in modeling and computation.
Additional Information© 2002 Institute of Mathematical Statistics. We thank Yuval Peres for alerting us to the Ising model on trees described in Section 5.1 and suggesting its application to votes. We also thank Peter Dodds, Amit Gandhi, Hal Stern, Jan Vecer, Tian Zheng and two referees for helpful discussions and comments. This work was supported in part by NSF Grants SES- 9987748 and SES-0084368.
Published - gelman+katz+tuerlinckx02.pdf