Abstract
We introduce a method of measuring voting power in simple voting games with correlated votes using the Bahadur parameterisation. With a method for measuring voting power with correlated votes, we can address a question of practical importance. Given that most of the applied power analysis is carried out with either the Penrose–Banzhaf or the Shapley–Shubik measures of power, what happens when you use these two measures in games with correlated votes? Simulations of all possible voting games with up to six players show that both measures tend to overestimate power when the votes are positively correlated. Yet, in most voting scenarios, the Shapley–Shubik index is closer to the probability of criticality than the Penrose–Banzhaf measure. This also holds for the power distribution in the EU Council of Ministers. Based on these simulations, we conclude that, while the Penrose–Banzhaf measure may be ideal for designing constitutional assemblies, the Shapley–Shubik index is better suited for the analysis of power distributions beyond the constitutional stage.
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Notes
The number of distinct SVGs equals the number of positively monotonic Boolean functions—the Dedekind number, less one to exclude the empty set being a winning coalition. The Dedekind numbers form a rapidly-growing sequence of integers, with only the first nine terms computed to date.
George and Bowman (1995) provide a more compact representation of the probabilities for exchangeable binary random variables. However, the George and Bowman parametrisation is unsuitable for our simulations because it is not formulated in terms of marginals and correlation coefficients. Kaniovski (2008) formulates a quadratic optimization problem for finding a distribution with given marginals and correlations. For a recent survey of algorithms for generating correlated binary random variables, see Preisser and Qaqish (2012).
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We thank the editor and two anonymous reviewers for their constructive comments, which helped us to improve the manuscript.
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Kaniovski, S., Das, S. Measuring voting power in games with correlated votes using Bahadur’s parametrisation. Soc Choice Welf 44, 349–367 (2015). https://doi.org/10.1007/s00355-014-0831-x
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DOI: https://doi.org/10.1007/s00355-014-0831-x