Abstract
In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters cast approval ballots over parties, such that each voter can support multiple parties. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates. Using techniques from both apportionment and multiwinner elections, we are able to provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Second, we demonstrate that extended justified representation is compatible with committee monotonicity.
Highlights
The fundamental fairness principle of proportional representation is relevant in a variety of applications ranging from recommender systems to digital democracy (Skowron et al 2017)
Proportional representation prescribes that the number of representatives championing a particular opinion in a legislature be proportional to the number of voters who favor that opinion
We show that it becomes tractable to check whether a committee provides extended justified representation or the weaker axiom of proportional justified representation
Summary
The fundamental fairness principle of proportional representation is relevant in a variety of applications ranging from recommender systems to digital democracy (Skowron et al 2017) It features most explicitly in the context of political elections, which is the language we adopt for this paper. Most studies of approval-based multiwinner elections assume that voters directly express their preference over individual candidates; we refer to this setting as candidate-approval elections This assumption runs counter to widespread democratic practice, in which candidates belong to political parties and voters indicate preferences over these parties (which induce implicit preferences over candidates). Using approval ballots could reveal that the blocks jointly approve a party of more general appeal; allocating more seats to this party leads to mutual gain This cooperation is necessary for small minority opinions that are not centrally coordinated. In contrast to approval voting over individual candidates, party-approval voting does not require a break with the current role of political parties—it can be combined with both “open list” and “closed list” approaches to filling the seats allocated to a party
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