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Public Choice

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27-07-2019

Extensions of the Simpson voting rule to the committee selection setting

Authors: Daniela Bubboloni, Mostapha Diss, Michele Gori

Published in: Public Choice | Issue 1-2/2020

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Abstract

Committee selection rules are procedures selecting sets of candidates (committees) of a given size on the basis of the preferences of the voters. Two natural extensions of the well-known single-winner Simpson voting rule to the multiwinner setting have been identified in the literature. We propose an in-depth analysis of those committee selection rules, assessing and comparing them with respect to several desirable properties, among which are unanimity, fixed majority, non-imposition, stability, local stability, Condorcet consistency, some kinds of monotonicity, resolvability and consensus committee. We also investigate the probability that the two methods are resolute and suffer the reversal bias, the Condorcet loser paradox and the leaving member paradox. We compare the results obtained with the ones related to further well-known committee selection rules. The probability assumption on which our results are based is the widely used Impartial Anonymous Culture.

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The Simpson voting rule also is known as the Simpson–Kramer, the Condorcet and the Minimax voting rule (see Kramer 1977; Simpson 1969).

Obviously, the concept is a generalization to the committee selection setting of the well-known concept of weak Condorcet winner.

In Coelho (2004) $\mathfrak {M}$ is denoted by SEO and it is defined as

$$\begin{aligned} \mathfrak {M}(C,V,p,k)=\underset{W\in 2_{k}^{C}}{\mathrm {arg\,min}}\,\max _{y\not \in W,x\in W}c_{p}(y,x). \end{aligned}$$

Those properties largely are studied in the literature in the setting of voting rules. See, for instance, Bubboloni and Gori (2016), Diss and Gehrlein (2012), Diss and Tlidi (2018), Duggan and Schwartz (2000), Fishburn and Gehrlein (1976), Gehrlein and Lepelley (2010), Jeong and Ju (2017) and Saari and Barney (2003). In the committee selection setting, resoluteness and immunity to the reversal bias are studied in Bubboloni and Gori (2019) for a fixed number of voters and alternatives; the Condorcet loser paradox is introduced here for the first time.

See, for instance, Diss and Doghmi (2016), Kamwa and Merlin (2015) and Staring (1986).

Note that if k were allowed to be 1, then every csr would suffer the leaving member paradox.

Note that when, $m\le 2$ or $n=1$, the analysis of the considered properties turns out to be straightforward.

See Diss and Doghmi (2016) and Elkind et al. (2017) for the definitions of the csrs.

For a detailed description of algorithms computing Ehrhart polynomials, we recommend the report by Verdoolaege et al. (2005).

Possible for all of the scoring csrs we are considering here but, unfortunately, that is not true when we consider $\mathfrak {S}$ and $\mathfrak {M}$. For those csrs exact probabilities can be obtained only for $n\in \{2,3,4,5\}$.

The MATLAB code of our simulations is available upon request.

Once again the technique cannot be used for $\mathfrak {S}$ and $\mathfrak {M}$ for which obtaining the probabilities in the limiting case is possible only with computer simulations. We consider for them a number of voters $n=100{,}000$.

This is another assumption widely used for analyzing the probability of electoral events. Under this assumption, the preference relation of each voter is drawn uniformly at random from the set of all possible linear orders.

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Title: Extensions of the Simpson voting rule to the committee selection setting
Authors: Daniela Bubboloni
Mostapha Diss
Michele Gori
Publication date: 27-07-2019
Publisher: Springer US
Published in: Public Choice / Issue 1-2/2020
Print ISSN: 0048-5829
Electronic ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-019-00692-6

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