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Standard Voting Power Indexes Don't Work: An Empirical Analysis Gelman, Andrew Katz, Jonathan Bafumi, Joseph Uploaded 11-02-2002 Keywords Banzhaf index decisive vote elections electoral college Shapley value voting power Abstract Voting power indexes such as that of Banzhaf (1965) are derived, explicitly or implicitly, from the assumption that all votes are equally likely (i.e., random voting). That assumption can be generalized to hold that the probability of a vote being decisive in a jurisdiction with $n$ voters is proportional to $1/sqrt{n}$. We test---and reject---this hypothesis empirically, using data from several different U.S. and European elections. We find that the probability of a decisive vote is approximately proportional to $1/n$. The random voting model (or its generalization, 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 our 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 $sqrt{n}$.