Abstract
We construct a new measure of voting power that yields reasonable measurements even if the individual votes are not cast independently. Our measure hinges on probabilities of counterfactuals, such as the probability that the outcome of a collective decision would have been yes, had a voter voted yes rather than no as she did in the real world. The probabilities of such counterfactuals are calculated on the basis of causal information, following the approach by Balke and Pearl. Opinion leaders whose votes have causal influence on other voters’ votes can have significantly more voting power under our measure. But the new measure of voting power is also sensitive to the voting rule. We show that our measure can be regarded as an average treatment effect, we provide examples in which it yields intuitively plausible results and we prove that it reduces to Banzhaf voting power in the limiting case of independent and equiprobable votes.
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Balke, A., & Pearl, J. (1994). Probabilistic evaluation of counterfactual queries. Proceedings of the Twelfth National Conference on Artificial Intelligence, AAAII-94, 220–237.
Beisbart, C. (2010). Groups can make a difference. Voting power measures extended. Theory and Decision, 69, 469–488; a former version is available in the Discussion Paper Series of the CPNSS at the LSE. http://www.lse.ac.uk/collections/CPNSS/CPNSS-DPS/LSEPhilosopyPapersList.htm.
Beisbart C., Bovens L. (2008) A power measure analysis of Amendment 36 in Colorado. Public Choice 124: 231–246
Brams S. J., Davis M. D. (1974) The 3/2’s rule in presidential campaigning. American Political Science Review 68: 113–134
Dretske F. (1988) Explaining behaviour. MIT Press, Cambridge, MA
Dubey P., Shapley L. S. (1979) Mathematical properties of the Banzhaf power index. Mathematics of Operations Research 4: 99–131
Felsenthal D. S., Machover M. (1998) The measurement of voting power: Theory and practice, problems and paradoxes. Edward Elgar, Cheltenham
Gelman A., Katz J. N., Bafumi J. (2004) Standard voting power indexes don’t work: An empirical analysis. British Journal of Political Science 34: 657–674
Gelman A., Katz J. N., Tuerlinckx F. (2002) The mathematics and statistics of voting power. Statistical Science 17: 420–435
Kaniovski S., Leech D. (2009) A behavioural power index. Public Choice, 141: 17–29
King G., Keohane R. O., Verba S. (1994) Designing social inquiry. Scientific inference in qualitative research. Princeton University Press, Princeton, NJ
Laruelle A., Valenciano F. (2005) Assessing success and decisiveness in voting situations. Social Choice and Welfare 24: 171–197
Lewis, D. (1979). Counterfactual dependence and time’s arrow. Noũs; also in Lewis, D. (1986). Philosophical papers (Vol. II, pp. 32–52). New York: Oxford University Press.
Machover, M. (2007). Discussion topic: Voting power when voters’ independence is not assumed. Mimeo. http://eprints.lse.ac.uk/2966/.
Morgan S. L., Winship C. (1999) Counterfactuals and causal inference. Cambridge University Press, Cambridge
Morriss P. (1987). Power. A philosophical analysis (2nd ed., 2002). Manchester: Manchester University Press.
Perl J. (2000) Causality. Models, reasoning, and inference. Cambridge University Press, New York
Spirtes P., Glymour C., Scheines R. (2000) Causation, prediction, and search, (2nd ed.). MIT Press, Boston, MA
Winship C., Morgan S. L. (2007) The estimation of causal effects from observed data. Annual Review Sociology 25: 659–707
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Bovens, L., Beisbart, C. Measuring voting power for dependent voters through causal models. Synthese 179 (Suppl 1), 35–56 (2011). https://doi.org/10.1007/s11229-010-9854-8
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DOI: https://doi.org/10.1007/s11229-010-9854-8