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Justified representation in approval-based committee voting

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Abstract

We consider approval-based committee voting, i.e. the setting where each voter approves a subset of candidates, and these votes are then used to select a fixed-size set of winners (committee). We propose a natural axiom for this setting, which we call justified representation (\(\mathrm {JR}\)). This axiom requires that if a large enough group of voters exhibits agreement by supporting the same candidate, then at least one voter in this group has an approved candidate in the winning committee. We show that for every list of ballots it is possible to select a committee that provides \(\mathrm {JR}\). However, it turns out that several prominent approval-based voting rules may fail to output such a committee. In particular, while Proportional Approval Voting (\(\mathrm {PAV}\)) always outputs a committee that provides \(\mathrm {JR}\), Sequential Proportional Approval Voting (\(\mathrm {SeqPAV}\)), which is a tractable approximation to \(\mathrm {PAV}\), does not have this property. We then introduce a stronger version of the \(\mathrm {JR}\) axiom, which we call extended justified representation (\(\mathrm {EJR}\)), and show that \(\mathrm {PAV}\) satisfies \(\mathrm {EJR}\), while other rules we consider do not; indeed, \(\mathrm {EJR}\) can be used to characterize \(\mathrm {PAV}\) within the class of weighted \(\mathrm {PAV}\) rules. We also consider several other questions related to \(\mathrm {JR}\) and \(\mathrm {EJR}\), including the relationship between \(\mathrm {JR}\)/\(\mathrm {EJR}\) and core stability, and the complexity of the associated computational problems.

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Notes

  1. \(\mathrm {SAV}\) is equivalent to equal and even cumulative voting (e.g., see Alcalde-Unzu and Vorsatz 2009), where for each voter, a score of 1 is divided evenly among all candidates in the voter’s approval set, and the k candidates with the highest total score are selected (see also Brams and Kilgour 2014, p. 328).

  2. We are grateful to Svante Janson and Xavier Mora for pointing this out to us.

  3. It is convenient to think of \({\mathbf {w}}\) as an infinite vector; note that for an election with m candidates only the first m entries of \({\mathbf {w}}\) matter. To analyze the complexity of \({\mathbf {w}}\)-\(\mathrm {PAV}\) rules, one would have to place additional requirements on \({\mathbf {w}}\); however, we do not consider computational properties of such rules in this paper.

  4. \({\mathbf {w}}\)-\(\mathrm {PAV}\) rules with \(w_1=0\) have been considered by Fishburn and Pekec (2004) and Skowron et al. (2016). We note that such rules do not satisfy justified representation (as defined in Sect. 3).

  5. For readability, we use the Hare quota \(\left\lceil \frac{n}{k}\right\rceil \); this choice of quota motivates our name for this rule. However, all our proofs go through if we use the Droop quota \(\left\lfloor \frac{n}{k+1}\right\rfloor +1\) instead. For a discussion of differences between these two quotas, see the article of Tideman (1995).

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Acknowledgements

The authors thank the anonymous reviewers of the Multidisciplinary Workshop on Advances in Preference Handling (MPREF 2014), and the Twenty-ninth AAAI Conference (AAAI 2015) for their helpful feedback on earlier versions of the paper. We further thank Svante Janson, Martin Lackner, Xavier Mora, and Piotr Skowron for valuable discussions. Brill, Conitzer, and Freeman were supported by NSF and ARO under Grants CCF-1101659, IIS-0953756, IIS-1527434, CCF-1337215, W911NF-12-1-0550, and W911NF-11-1-0332, by a Feodor Lynen research fellowship of the Alexander von Humboldt Foundation, and by COST Action IC1205 on Computational Social Choice. Aziz is supported by a Julius Career Award. Brill and Elkind were partially supported by ERC-StG 639945. Conitzer is supported by a Guggenheim Fellowship. Walsh also receives support from the Asian Office of Aerospace Research and Development (AOARD 124056) and the German Federal Ministry for Education and Research through the Alexander von Humboldt Foundation.

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Authors and Affiliations

  1. Data61, CSIRO and UNSW Australia, Sydney, 2052, Australia

    Haris Aziz & Toby Walsh

  2. University of Oxford, Oxford, OX1 3QD, UK

    Markus Brill & Edith Elkind

  3. Duke University, Durham, NC, 27708, USA

    Vincent Conitzer & Rupert Freeman

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  1. Haris Aziz
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  2. Markus Brill
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Correspondence to Haris Aziz.

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Aziz, H., Brill, M., Conitzer, V. et al. Justified representation in approval-based committee voting. Soc Choice Welf 48, 461–485 (2017). https://doi.org/10.1007/s00355-016-1019-3

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