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Group Decision and Negotiation

Erschienen in:

01.09.2015

Positional Voting Systems Generated by Cumulative Standings Functions

verfasst von: Bonifacio Llamazares, Teresa Peña

Erschienen in: Group Decision and Negotiation | Ausgabe 5/2015

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Abstract

Positional voting systems are a class of voting systems where voters rank order the candidates from best to worst and a set of winners is selected using the positions of the candidates in the voters’ preference orders. Although scoring rules are the best known positional voting systems, this class includes other voting systems proposed in the literature as, for example, the Majoritarian Compromise or the \(q\)-Approval Fallback Bargaining. In this paper we show that some of these positional voting systems can be integrated in a model based on cumulative standings functions. The proposed model allows us to establish a general framework for the analysis of these voting systems, to extend to them some results in the literature for the particular case of the scoring rules, and also facilitates the study of the social choice properties considered in the paper: monotonicity, Pareto-optimality, immunity to the absolute winner paradox, Condorcet consistency, immunity to the absolute loser paradox and immunity to the Condorcet loser paradox.

Vorheriger Artikel Lattice Order Group Decision Making with Interval Probability Based on Prospect Theory

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As far as we know this term first appears in Gärdenfors (1973).

It is worth noting that there exist other PVSs based on mathematical programming techniques that have an important drawback from the point of view of social choice theory: The relative order between two candidates may be altered when the number of first, second, ..., \(m\)th ranks obtained by other candidates changes, although there are not any variations in the number of first, second, ..., \(m\)th ranks obtained by both candidates (see Llamazares and Peña 2009). The PVSs with this drawback are not considered in this paper.

Although the cumulative standings of each candidate depend on the profile \(\varvec{p}\), in order to avoid cumbersome notation we shall omit \(\varvec{p}\) in the notation of these values when there is no possible confusion. When it will be necessary, we will use the notation \(V_{k}^{i}\) for the profile \(\varvec{p}\), \(V_{k}^{\prime i}\) for the profile \(\varvec{p}'\) and so on.

Nevertheless, for convenience we continue to use the notation \(V_{m}^{i}\) instead of \(n\) when appropriate.

Taking into account that the score of each candidate only depends on the number of first, second, ..., \(m\)th ranks obtained by him/her, the use of CSFs excludes the PVSs that have the drawback mentioned in the introduction.

It is easy to check that the functions considered in this section are monotonic and, therefore, are CSFs.

Notice that, although the majoritarian Compromise historically precedes \(q\)-approval fallback bargaining, we first show this PVS for the sake of simplicity of the CSF.

The best known special cases of \(k\)-approval voting are plurality (\(k=1\)) and antiplurality (\(k=m-1\)).

This corollary extends to all PVSs generated by CSFs the result obtained for the scoring rules by Saari (1992, 1994) (see also Merlin et al. 2000).

It is worth noting that the same set of winning candidates can be also obtained through a model proposed by Llamazares and Peña (2013).

Excepting the scoring rules because the results relative to these voting systems are already known (see Llamazares and Peña 2014).

It is worth noting that Condorcet-consistency is a controversial property: Although some authors such as Felsenthal (2012) hold that a Condorcet winner, if one exists, ought always to be elected, others, such as Fishburn (1974), question the importance of this property.

In this case the cumulative standing function will be denoted by \(F_n\) instead of \(F_q\).

Notice that, when \(m=3\), the third column of Table 2 is \(V_{2}^{i}\) and not \(V_{m-1}^{i}\).

It is worth noting that, when \(m=3\), the third column of Table 3 is \(V_{2}^{i}\) and not \(V_{m-1}^{i}\).

It is worth noting that Brams and Kilgour (2001) and Sertel and Yilmaz (1999) use examples with four and five candidates, respectively, to show that their methods are not Condorcet consistent. Our examples show this result independently of the number of the candidates. In this respect, for \(m\ge 3\), the no-Condorcet consistency of the Condorcet’s practical method and the Majoritarian Compromise has been noted by Merlin et al. (2006) using the geometry of voting.

According to Fishburn (1974), the forward cyclic list of orders generated by \(A_{1}\,A_{2}\cdots A_{m}\) is the \(m\)-tuple of orders \((A_{1}\,A_{2}\cdots A_{m-1}\,A_{m},\; A_{2}\,A_{3}\cdots A_{m}\,A_{1},\; A_{3}\,A_{4}\cdots A_{1}\,A_{2},\; \dots ,\; A_{m}\,A_{1}\cdots A_{m-2}\,A_{m-1})\).

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Titel: Positional Voting Systems Generated by Cumulative Standings Functions
verfasst von: Bonifacio Llamazares
Teresa Peña
Publikationsdatum: 01.09.2015
Verlag: Springer Netherlands
Erschienen in: Group Decision and Negotiation / Ausgabe 5/2015
Print ISSN: 0926-2644
Elektronische ISSN: 1572-9907
DOI: https://doi.org/10.1007/s10726-014-9412-8

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