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Social Choice and Welfare

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26-11-2018 | Original Paper

The theoretical Shapley–Shubik probability of an election inversion in a toy symmetric version of the US presidential electoral system

Authors: Olivier de Mouzon, Thibault Laurent, Michel Le Breton, Dominique Lepelley

Published in: Social Choice and Welfare | Issue 2-3/2020

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Abstract

In this article, we evaluate asymptotically the probability \(\phi \left( n\right) \) of an election inversion in a toy symmetric version of the US presidential electoral system. The novelty of this paper, in contrast to all the existing theoretical literature, is to assume that votes are drawn from an IAC (Impartial Anonymous Culture)/Shapley–Shubik probability model. Through the use of numerical methods, it is conjectured, that \(\sqrt{n}\)\( \phi \left( n\right) \) converges to 0.1309 when n (the size of the electorate in one district) tends to infinity. It is also demonstrated that \( \phi \left( n\right) =o\left( \sqrt{\frac{ln(n)^{3}}{n}}\right) \) and \(\phi \left( n\right) =\Omega \left( \frac{1}{\sqrt{n}}\right) \).

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Available only for authorised users

See Nurmi (1999).

Miller (2012b) contains an insightful presentation of the Electoral College that he qualifies as “a terrific boon for political science (and public choice) research (and teaching)”.

If there are m feasible orderings, this means that each marginal distribution is (1 / m, 1 / m, ..., 1 / m).

This prior is a special case of a Dirichlet distribution.

See Straffin (1988) for a very nice united presentation of IC and IAC.

In the two-alternative case, the IC and IAC model have been extensively used to study the fairness of voting bodies from the point of view of the power defined as the expected propensity for a voter to be influent/pivotal in the decision making process. The computation of the Banzhaf and Shapley–Shubik power indices is one of the main area of applied research.

Saying that the beta parameter of the distribution tends to \(+\infty \) amounts to say that the probability that the election is tied in any given district tends to 1, which is what postulates the IC model.

While this paper is about election inversions, the same apparatus could be used to estimate the votes/seats relationship (see e.g. Tufte 1973) and the likelihood of any event like ‘The left won \(x\%\) of the popular vote and \(y\%\) of the seats/electoral votes’ where both x and y are numbers in \( \left[ 0,1\right] \). If you divide the unit square \(\left[ 0,1\right] ^{2}\) into the four squares \(\left[ 0,\frac{1}{2}\right] \times \left[ 0,\frac{1}{2 }\right] ,\left[ 0,\frac{1}{2}\right] \times \left[ \frac{1}{2},1\right] , \left[ \frac{1}{2},1\right] \times \left[ 0,\frac{1}{2}\right] \) and \(\left[ \frac{1}{2},1\right] \times \left[ \frac{1}{2},1\right] \), the probability of election inversions is the probability of the union of the two anti-diagonal squares. Obviously, knowing the totality of the joint distribution on \(\left[ 0,1\right] ^{2}\) has some intrinsic value.

Owen computes the Shapley–Shubik and the Banzhaf powers of the US citizens as a function of the state where they vote. He shows, surprisingly, that the relative powers of the citizen of any state (the denominator is the power of a citizen from the District of Columbia) are about the same for the two models.

In Sect. 3, we sketch the general principles of our numerical analysis. The reader is refered to the online appendix for the full details. In this, we discuss the asymptotic behavior of estimates resulting from approximations together with two bounds.

Most authors in the theoretical vein have considered toy symmetric versions of the Electoral College. Recent developments aiming to an evaluation of the probability of an election inversion in the general case are Kikuchi (2017) and Kaniovski and Zaigraev (2017).

An immediate corollary of these results is that the probability of an election inversion tends to 0 when the number of voters per district tends to \(\infty \). Kikuchi (2018) proves that this convergence to 0 holds true for any number of districts and any profile of populations and electoral votes across districts.

So malapportionment is excluded from the scope of our analysis.

A more pedestrian and illustrative exposition of these probability models emphasizing their differences in terms of correlation between votes is presented in the following subsection.

In particular, \(\Gamma \left( n\right) =(n-1)!\) if n is a positive integer.

In our paper, we do not need to define the majority mechanism when m is an even integer.

This illustration remains valid for any number of districts and any number of voters per district.

Other ways to generate correlation among Bernoulli variables exist. For instance, we could use a beta distribution instead of this truncated uniform distribution or the Gaussian distribution as in Le Breton et al. (2016).

Strictly speaking the density is not defined for \(\delta =0\). IC is the limit when \(\delta \rightarrow 0\).

In the conclusion, we report computations of the probability of election inversions obtained from simulations for that one-parameter model.

To compute \(\phi \), we need therefore to know the values of \(\ln \Gamma \) for the integers ranging from 1 to \(3n+2\). For large values of n, this is done by the use of Stirling’s formula.

Proofs are provided in appendix A.

The number of computations for \(\phi _{maj}\) and \(\phi _{min}\) is of order n (in contrast to \(n^{3}\) for \(\phi \)).

If we pile up the r levels, we obtain indeed, a geometric pattern close to a tetrahedron. It is represented in Fig. 2. When we compute \(\phi \) for the following value of n, we add to the tetrahedron a ground level with a number of new entries to compute equal to \(\frac{n+1}{2}\times (\frac{n+1}{2}-1)/2\). Note that when n increases, the values \(B^{k,l,r}\) will decrease and converge to 0. For instance, on the top \(B^{\frac{n+1}{2},\frac{n+1}{2},\frac{n-3}{2}}=O(n^{-2})\). The values on low levels converge even more rapidly towards 0. For instance, for the ground level (\(r=0\)), \(B^{\frac{n+1}{2},\frac{n+1}{2},0}\) behaves as \(n^{-1.5}\left( \frac{16}{27}\right) ^{n} \).

The approximation error resulting from \({\tilde{\phi }}_{min}\) rather than \( \phi _{min}\) is evaluated and analysed in the online appendix.

We refer to the online appendix for the details and in particular for an evaluation of the ultimate number of computations needed to compute \(\tilde{ \phi }(n)\) and of the approximation error.

We refer to the online appendix for an evaluation of the ultimate number of computations needed to compute \({\tilde{\phi }}(n)\) and of the approximation error.

Omitted proofs appear in appendix B.

This restriction on the set of values of n which are considered makes the writing easier (which would otherwise call for the introduction of the integer parts).

Precisely, \(n=2\times 10^{6}+1\) and 10\(^{6}\) simulations.

This exercise is theoretical and is by no means intended to suggest that this model fits the data. It simply points out the impact of the specific value of the correlation coefficient on the probability of an election inversion.

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Title: The theoretical Shapley–Shubik probability of an election inversion in a toy symmetric version of the US presidential electoral system
Authors: Olivier de Mouzon
Thibault Laurent
Michel Le Breton
Dominique Lepelley
Publication date: 26-11-2018
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2-3/2020
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-018-1162-0

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