Standard Voting Power Indexes Do Not Work: An Empirical Analysis
ANDREW GELMAN a1, JONATHAN N. KATZ a2andJOSEPH BAFUMI a1a a1 Department of Political Science, Columbia University (Gelman is also in the Department of Statistics) a2 Division of the Humanities and Social Sciences, California Institute of Technology
Voting power indexes such as that of Banzhaf are derived, explicitly or implicitly, from the assumption that all votes are equally likely (i.e., random voting). That assumption implies that the probability of a vote being decisive in a jurisdiction with n voters is proportional to 1/[surd radical]n. In this article the authors show how this hypothesis has been empirically tested and rejected using data from various US and European elections. They find that the probability of a decisive vote is approximately proportional to 1/n. The random voting model (and, more generally, the square-root rule) overestimates the probability of close elections in larger jurisdictions. As a result, classical voting power indexes make voters in large jurisdictions appear more powerful than they really are. The most important political implication of their result is that proportionally weighted voting systems (that is, each jurisdiction gets a number of votes proportional to n) are basically fair. This contradicts the claim in the voting power literature that weights should be approximately proportional to [surd radical]n.
a We thank David Park, the Editor of the Journal and several reviewers for helpful comments and the US National Science Foundation for grants SES-9987748, SES-0084368 and SES-0318115.