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2014 | OriginalPaper | Chapter

Voting Power Techniques: What Do They Measure?

Author : Sreejith Das

Published in: Voting Power and Procedures

Publisher: Springer International Publishing

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Abstract

Voting power science is a field of co-operative game theory concerned with calculating the influence a voter can exert on the outcome of a voting game. The techniques used to calculate voting power have names like the Shapley-Shubik index, and the Banzhaf measure. They are invaluable when used to design democratically fair voting games.

In this paper we examine these different techniques, with the specific aim of trying to understand what they are measuring. Many commentators have argued that the techniques are similar, albeit with different probability models. But by focusing upon the less well know differences that exist in the underlying measures themselves, it soon becomes apparent that the dissimilarities between the techniques extend far beyond their methods of counting voting coalitions.

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For the purposes of this example, we shall ignore how you can come to know \(\frac{1} {\vert \Omega \vert }\) before all the blocks have been counted!

It is customary to write ∫ _x y(x) dx instead of ∫ _x y(x) x.

Shapley and Shubik (1954) understood it was possible to be Decreasingly Critical, but they did not appreciate that this was materially different to being Increasingly Critical (examine their comments regarding their proposed “blocking index”).

The actual fraction that is added is inversely proportional to the number of voters that express full support in ω. Hence, the probability model of the modified index implies that coalitions with more voters expressing full support are less likely to occur.

The actual fraction that is added in the Deegan-Packel index is inversely proportional to the number of voters that express support in ω. Hence, like the modified Johnston index, the probability model of the Deegan-Packel index implies that coalitions with more voters expressing support are less likely to occur.

The author would like to point out that Holler doesn’t advocate this as a realistic assumption, but acknowledges its usefulness in voting power calculations.

The new “weighing” machines have been simplified for the purposes of this example, they are actually sigma finite marginal measures, and are more correctly given by \(\mu _{\omega ^{N\setminus \{i\}}}(dx_{i})\;\lambda (d\omega ^{N\setminus \{i\}}).\) (See Appendix 3.)

It should be noted that the vast majority of real life voting games are structured to ensure that this is the case. For example, games where the votes are cast simultaneously, or games where they are cast anonymously. The key requirement is that the other voters cannot observe the actual event of voter i voting, and then react. We do not preclude scenarios in which voter i tells everyone how it intends to vote, providing the others do not actually see the vote taking place.

Abstention, is not the same as “maybe”, see Das (2008) for details.

This could only be avoided with the use of a biased probability distribution which imposed a disproportionately high likelihood of the voter voting “no”.

The development of more realistic probability models will no doubt become a huge challenge for the future.

Banzhaf, J. (1965). Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review, 19, 317–342.

Coleman, J. S. (1971). Control of collectives and the power of a collectivity to act. In B. Lieberman (Ed.), Social choice (pp. 192–225). New York: Gordon and Breach.

Das, S. (2008). Class conditional voting probabilities. Ph.D. thesis, Birkbeck College, University of London, London.

Das, S. (2011). Criticality in games with multiple levels of approval. Social Choice and Welfare, 37, 373–395. doi:10.1007/s00355-010-0488-zCrossRef

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Felsenthal, D., & Machover, M. (1998). The measurement of voting power. Cheltenham: Edward Elgar.CrossRef

Holler, M. (1982). Forming coalitions and measuring voting power. Political Studies, 30, 262–271.CrossRef

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Pollard, D. (2003). A user’s guide to measure theoretic probability. Cambridge: Cambridge University Press.

Shapley, L., & Shubik, M. (1954). A method for evaluating the distribution of power in a committee system. American Political Science Review, 48, 787–792.CrossRef

Straffin, P. (1977). Homogeneity, independence and power indices. Public Choice, 30, 107–118.CrossRef

Straffin, P. (1978). Probability models for power indices. In P. C. Ordeshook (Ed.), Game theory and political science (pp. 477–510). New York: New York University Press.

Title: Voting Power Techniques: What Do They Measure?
Author: Sreejith Das
Publisher: Springer International Publishing
Book: Voting Power and Procedures
Print ISBN: 978-3-319-05157-4

Electronic ISBN: 978-3-319-05158-1

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-319-05158-1_5