The Thirty-Fourth AAAI Conference on Artificial Intelligence (AAAI-20) Approval-Based Apportionment Markus Brill,1 Paul Golz,¨ 2 Dominik Peters,2 Ulrike Schmidt-Kraepelin,1 Kai Wilker1 1Technische Universitat¨ Berlin, Chair of Efficient Algorithms 2Carnegie Mellon University, Computer Science Department {brill, u.schmidt-kraepelin}@tu-berlin.de, {pgoelz, dominikp}@cs.cmu.edu, [email protected] Abstract equally well represented by several political parties, there is no way to express this preference within the voting system. In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters In the context of single-winner elections, approval voting supporting each party. We study a generalization of this setting, has been put forward as a solution to this problem as it strikes in which voters cast approval ballots over parties, such that an attractive compromise between simplicity and expressiv- each voter can support multiple parties. This approval-based ity (Brams and Fishburn 2007; Laslier and Sanver 2010). apportionment setting generalizes traditional apportionment Under approval voting, each voter is asked to specify a set and is a natural restriction of approval-based multiwinner elec- of candidates she “approves of,” i.e., voters can arbitrarily tions, where approval ballots range over individual candidates. partition the set of candidates into approved candidates and Using techniques from both apportionment and multiwinner disapproved ones. Proponents of approval voting argue that elections, we are able to provide representation guarantees its introduction could increase voter turnout, “help elect the that are currently out of reach in the general setting of multi- winner elections: First, we show that core-stable committees strongest candidate,” and “add legitimacy to the outcome” of are guaranteed to exist and can be found in polynomial time. an election (Brams and Fishburn 2007, pp. 4–8). Second, we demonstrate that extended justified representation Due to the practical and theoretical appeal of approval vot- is compatible with committee monotonicity. ing in single-winner elections, a number of scholars have sug- gested to also use approval voting for multiwinner elections, in which a fixed number of candidates needs to be elected 1 Introduction (Kilgour and Marshall 2012). In contrast to the single-winner The fundamental fairness principle of proportional represen- setting, where the straightforward voting rule “choose the tation is relevant in a variety of applications ranging from candidate approved by the highest number of voters” enjoys recommender systems to digital democracy (Skowron et al. a strong axiomatic foundation (Fishburn 1978), several ways 2017). It features most explicitly in the context of political of aggregating approval ballots have been proposed in the elections, which is the language we adopt for this paper. In multiwinner setting (e.g., Aziz et al. 2017; Janson 2016). this context, proportional representation prescribes that the Most studies of approval-based multiwinner elections as- number of representatives championing a particular opinion sume that voters directly express their preference over individ- in a legislature be proportional to the number of voters who ual candidates; we refer to this setting as candidate-approval favor that opinion. elections. This assumption runs counter to widespread demo- In most democratic institutions, proportional represen- cratic practice, in which candidates belong to political parties tation is implemented via party-list elections: Candidates and voters indicate preferences over these parties (which in- are members of political parties and voters are asked to duce implicit preferences over candidates). In this paper, we indicate their favorite party; each party is then allocated therefore study party-approval elections, in which voters ex- a number of seats that is (approximately) proportional to press approval votes over parties and a given number of seats the number of votes it received. The problem of transform- must be distributed among the parties. We refer to the process ing a voting outcome into a distribution of seats is known of allocating these seats as approval-based apportionment. as apportionment. Analyzing the advantages and disadvan- We believe that party-approval elections are a promising tages of different apportionment methods has a long and framework for legislative elections in the real world. Allow- illustrious political history and has given rise to a deep ing voters to express approval votes over parties enables the and elegant mathematical theory (Balinski and Young 1982; aggregation mechanism to coordinate like-minded voters. For Pukelsheim 2014). example, two blocks of voters might currently vote for par- Unfortunately, forcing voters to choose a single party pre- ties that they mutually disapprove of. Using approval ballots vents them from communicating any preferences beyond could reveal that the blocks jointly approve a party of more their most preferred alternative. For example, if a voter feels general appeal; allocating more seats to this party leads to Copyright c 2020, Association for the Advancement of Artificial mutual gain. This cooperation is particularly necessary for Intelligence (www.aaai.org). All rights reserved. small minority opinions that are not centrally coordinated. In 1854 such cases, finding a commonly approved party can make the Candidate-approval elections difference between being represented or votes being wasted (Kilgour and Marshall 2012; Aziz et al. 2017) because the individual parties receive insufficient support. In contrast to approval voting over individual candidates, (ii) party-approval voting does not require a break with the cur- Party-approval elections rent role of political parties—it can be combined with both (iii) [approval-based apportionment] “open list” and “closed list” approaches to filling the seats allocated to a party. (i) 1.1 Related Work Party-list elections [apportionment] (Balinski and Young 1982; Pukelsheim 2014) To the best of our knowledge, this paper is the first to for- mally develop and systematically study approval-based ap- portionment. That is not to say that the idea of expressing and Figure 1: Relations between the different settings of multi- aggregating approval votes over parties has not been consid- winner elections. An arrow from X to Y signifies that X is a ered before. Indeed, several scholars have explored possible generalization of Y . The relationship corresponding to arrow generalizations of existing aggregation procedures. (iii) has been explored by Brill, Laslier, and Skowron (2018). For instance, Brams, Kilgour, and Potthoff (2019) study We establish and explore the relationship (i) in Section 3 and multiwinner approval rules that are inspired by classical ap- the relationship (ii) in Section 4. portionment methods. Besides the setting of candidate ap- proval, they explicitly consider the case where voters cast party-approval votes. They conclude that these rules could embedded in this setting by replacing each party by multi- “encourage coalitions across party or factional lines, thereby ple candidates belonging to this party, and by interpreting diminishing gridlock and promoting consensus.” a voter’s approval of a party as approval of all of its candi- Such desire for compromise is only one motivation for con- dates. This embedding establishes party-approval elections as sidering party-approval elections, as exemplified by recent a subdomain of candidate-approval elections (see arrow (ii) work by Speroni di Fenizio and Gewurz (2019). To allow for in Figure 1). In Section 4, we explore the axiomatic and more efficient governing, they aim to concentrate the power computational ramifications of this domain restriction. of a legislature in the hands of few big parties, while nonethe- less preserving the principle of proportional representation. 1.3 Contributions To this end, they let voters cast party-approval votes and trans- In this paper, we formally introduce the setting of approval- form these votes into a party-list election by assigning each based apportionment and explore different possibilities of voter to one of her approved parties. One method for doing constructing axiomatically desirable aggregation methods for this (referred to as majoritarian portioning later in this paper) this setting. Besides its conceptual appeal, this setting is also assigns voters to parties in such a way that the strongest party interesting from a technical perspective. has as many votes as possible. Exploiting the relations described in Section 1.2, we re- Several other papers consider extensions of approval-based solve problems that remain open in the more general setting voting rules to accommodate party-approval elections (Brams of approval-based multiwinner voting. First, we prove that and Kilgour 2014; Mora and Oliver 2015; Janson 2016; committee monotonicity is compatible with extended justi- Janson and Oberg¨ 2019). All of these papers have in common fied representation (a representation axiom proposed by Aziz that they study specific rules or classes of rules, rather than et al. 2017) by providing a rule that satisfies both properties. exploring the party-approval setting in its own right. Second, we show that the core of an approval-based appor- tionment problem is always nonempty and that core-stable 1.2 Relation to Other Settings committees can be found in polynomial time. Party-approval