Multiwinner approval rules as apportionment methods
| DOI | 10.1177/0951629818775518 |
| Date | 01 July 2018 |
| Published date | 01 July 2018 |
| Author | Piotr Skowron,Markus Brill,Jean-François Laslier |
| Subject Matter | Articles |
Article
Multiwinner approval rules as
apportionment methods
Journal of Theoretical Politics
2018, Vol. 30(3) 358–382
©The Author(s) 2018
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DOI:10.1177/0951629818775518
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Markus Brill
Technische Universität Berlin, Berlin, Germany
Jean-François Laslier
Paris School of Economics, Paris, France
Piotr Skowron
University of Warsaw, Warszawa, Poland
Abstract
We establish a link between multiwinner elections and apportionment problems by showing
how approval-based multiwinner election rules can be interpreted as methods of apportion-
ment. We consider several multiwinner rules and observe that some, but not all, of them induce
apportionment methods that are well-established in the literature and in the actual practice
of representation, be it proportional or non-proportional. For instance, we show that propor-
tional approval voting induces the D’Hondt method and that Monroe’s rule induces the largest
remainder method. Our approach also yields apportionment methods implementing degressive
proportionality. Furthermore, we consider properties of apportionment methods and exhibit
multiwinner rules that induce apportionment methods satisfying these properties.
Keywords
Apportionment; multiwinner elections; panachage; personalized voting
1. Introduction
In a parliamentary election, the candidates are traditionally organized in political parties
and the election determines how many parliamentary seats each party is allocated. Under
so-called ‘closed party list’ systems of representation, a voter is allowed to give her vote
to one and only one party. In a sense, this forces the voter to approve all candidates from
Corresponding author:
Piotr Skowron, University of Warsaw, Warszawa, Poland.
Email: p.skowron@mimuw.edu.pl
Brill et al. 359
one party and no candidate from any other party. Counting the ballots under a closed-list
system amounts to deciding how many seats each party gets based on the ‘size’ of each
party in the electorate, and is therefore formally an apportionment problem, identical, for
instance, to the problem of allocating seats to the states of a union based on the population
figures.
Though widely used (Farrell, 2001), closed-list systems havea number of drawbacks.
For instance, it is a known feature of these systems that candidates tend to campaign
within their parties (for being placed on a good position on the party list), rather than
campaigning for the citizens’ votes. Closed-list systems thus favor party discipline, at
the potential expense of alienating the political elites from the citizens (e.g. see Ames,
1995; André et al., 2015; Chang, 2005; Colomer, 2011).
In an attempt to overcome these drawbacks, many countries use various forms of
‘open-list’ system, keeping the idea of party lists but giving some flexibility to the voters
by allowing each voter to vote for specific candidates inside a chosen party list. In some
(rare) cases, voters are given even more freedom. Under so-called ‘panachage’ systems,
sometimes used in Luxembourg and in France, voters can vote for candidates from dif-
ferent parties. Personalization is complete when a rule allows voters to vote directly for
individual candidates, and the outcome of the election does not depend on how candidates
are grouped into parties. Such is the case for some elections in Switzerland, were vari-
ants of multiwinner approval voting are used in several cantons (see Laslier and Van der
Straeten, 2016). In a recent book, Renwick and Pilet (2016) extensivelyexamine the trend
toward greater personalization, which they see as ‘one of the key shifts in contemporary
politics.’
This paper will take as its object such fully open voting systems. In the literature,
they are called ‘multiwinner’ rules because they elect a fixed number of candidates, for
instance a whole parliament. In a multiwinner election, we are given a set of voters who
entertain preferences over a set of alternatives. Based on these preferences, the goal is to
select a committee, i.e. a (fixed-size) subset of the alternatives. Preferences are usually
specified either as rankings, i.e. linear orders over the set of all alternatives (e.g. Brams
and Fishburn, 2002; Elkind et al., 2017), or as approval votes, i.e., yes/no assessments of
the alternatives (e.g. Kilgour and Marshall, 2012). We are particularly interested in the
latter variant, where each voter can be thought of as specifying a subset of alternatives
that she finds ‘acceptable’.
The decision scenario modeled by multiwinner elections—selecting a subset of
objects from a potentially much larger pool of available objects—is ubiquitous, and
includes picking players to form a sports team, selecting items to display in an online
shop, choosing the board of directors of a company, etc. (for a more detailed discussion
on the applications of multiwinner election rules, we refer the reader to the recent survey
by Faliszewski et al. 2017), but we are here chiefly interested in the political context. In
this context, even if the voting rule allows personal candidacyand votes, most candidates
do belong to political parties. It therefore makes sense to study, as a theoretical bench-
mark, the behavior of a multiwinner rule in the specific cases where voter preferences
perfectly reflect some underlying party structure.
To do so, we imagine that the ballots cast are such that, when a voter votes for a
candidate belonging to some party, she also votes for all the candidates of that party,
and for no other candidates. As we have already mentioned, counting such ballots, and
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