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Multiwinner Voting with Fairness Constraints L Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-18) Multiwinner Voting with Fairness Constraints L. Elisa Celis, Lingxiao Huang, Nisheeth K. Vishnoi EPFL, Switzerland first.last@epfl.ch Abstract committee of size k that maximizes the score. Different views on the desired properties of the selection process have led Multiwinner voting rules are used to select a s- to a number of different scoring rules and, consequently, to mall representative subset of candidates or item- a variety of different algorithmic problems. Prevalent ex- s from a larger set given the preferences of voter- amples include committee scoring rules [Aziz et al., 2017b; s. However, if candidates have sensitive attributes Elkind et al., 2014], approval-based rules [Aziz et al., 2017a], such as gender or ethnicity (when selecting a com- OWA-based rules [Skowron et al., 2015b], variants of the mittee), or specified types such as political leaning Monroe rule [Betzler et al., 2013; Monroe, 1995; Skowron (when selecting a subset of news items), an algo- et al., 2015a] and the goalbase rules [Uckelman et al., 2009]. rithm that chooses a subset by optimizing a mul- tiwinner voting rule may be unbalanced in its se- However, it has been shown that voting rules, in the most lection – it may under or over represent a particu- general sense, can create or propagate biases; e.g., negatively lar gender or political orientation in the examples affecting the fraction of women in the US legislature [Rep- above. We introduce an algorithmic framework for resentation2020, 2017], and resulting in an electorate that multiwinner voting problems when there is an ad- under-represents minorities [Faliszewski et al., 2016]. Fur- ditional requirement that the selected subset should thermore, such algorithmic biases have been shown to influ- be “fair” with respect to a given set of attributes. ence and reinforce human stereotypes [Kay et al., 2015]. An Our framework provides the flexibility to (1) speci- increasing awareness of such problems has led governments fy fairness with respect to multiple, non-disjoint at- to generic [Bundy, 2017] and specific [Zealand, 1986] rec- tributes (e.g., ethnicity and gender) and (2) spec- ommendations that aim to ensure sufficient representation of ify a score function. We study the computational minority populations. complexity of this constrained multiwinner voting In response, “proportional representation” rules [Mon- problem for monotone and submodular score func- roe, 1995], that represent the electorate proportionately in tions and present several approximation algorithms the elected body, have been developed. Formally, let and matching hardness of approximation results for P1;:::;Pp ⊆ [m] be p disjoint groups of candidates where various attribute group structure and types of score i 2 f1; : : : ; pg represents a given group. For any i 2 [p], let functions. We also present simulations that suggest fi be the fraction of voters who belong to (or, more generally, that adding fairness constraints may not affect the prefer) group i. Then, a voting rule achieving full propor- scores significantly when compared to the uncon- tionality would ensure that the selected committee S satisfies strained case. k·fi ≤ jS\Pij ≤ k·fi ; [Brill et al., , Definition 5]. Pro- portional representation schemes include the proportional ap- proval voting rule [Brill et al., ] and the Chamberlain Courant 1 Introduction rule [1983]. Other approaches consider different notions of The problem of selecting a committee from a set of candi- “fairness” in representation; e.g., in degressive proportionali- dates given the preferences of voters is called multiwinner ty it is argued that minorities should have disproportionately voting and arises in various social, political, and e-commerce many representatives. settings; from electing a parliament, to choosing a committee, We present the first algorithmic framework for multiwin- to selecting products to display. Formally, there is a set C of ner voting problems that can (1) incorporate general notions m “candidates” that can be selected (i.e., people, products, or of fairness with respect to arbitrary group structures (e.g., sat- other items) and a set A of n “voters” who have (possibly in- isfying fairness across multiple attributes simultaneously) and complete) preference list over the m candidates; we assume (2) outputs a subset that maximizes the given score function these lists are given. The goal is to select a subset of C of size subject to these constraints. This requires the development k based on these preferences. of new approximate algorithms (Section 6) and hardness of It remains to specify how the selection will be made. approximation results (Section 5) which depend on both the One common approach is to define a “total” score function score function and the structure of the fairness constraints. C scoreA : 2 ! R≥0 which gives a score that depends on the Empirically, we show that existing multiwinner voting rules voters’ preferences to each potential committee. If the voter may introduce bias and that our approach not only ensures set A is clear from the context, we use score for short. This fairness, but does so with a score that is close to the (uncon- reduces the selection to an optimization problem: choose a strained) optimal (see Section 7). 144 Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-18) 2 Our Contributions committee of size k by “dependent rounding”. In the case Model. In a multiwinner voting setting, we are given m of ∆ ≥ 2 (Theorem 11), we use a swap randomized round- [ ] candidates, n voters that each has a (potentially incomplete ing procedure introduced by Chekuri et al., 2010 . In the and/or non-strict) preference list over the m candidates, a case of ∆ = 1 (Theorem 14) we design a two-layered de- pendent rounding procedure that runs in linear time. Some score function score : 2[m] ! R defined by these list- ≥0 of the algorithmic results are achieved by reduction to well- s, and a desired number k 2 [m] of winners. In addition, studied problems, like the monotone submodular maximiza- to consider fair solutions, we are given arbitrary (potential- tion problem with ∆-extendible system (Theorem 12) and ly non-disjoint) groups of candidates P ;:::;P ⊆ C, and 1 p constrained set multi-cover (Theorem 13). The hardness re- fairness constraints on the selected winner set S of the for- sults follow from reductions from well-known NP-hard prob- m: ` ≤ jS \ P j ≤ u ; 8i 2 [p] for given numbers i i i lems, including ∆-hypergraph matching, 3-regular vertex ` ; : : : ; ` ; u ; : : : ; u 2 Z . 1 p 1 p ≥0 cover, constrained set multi-cover and independent set, and The goal of the constrained multiwinner voting problem is borrow techniques from a recent work on fairness for ranking to select a committee of size k that maximizes score(S) and problems [Celis et al., 2018b]. satisfies all fairness constraints. If the score function is mono- tone and submodular we call the problem the constrained MS We also study special cases; setting in which the fairness multiwinner voting problem. This includes the well-studied constraints involve only lower bounds (Theorem 13) or only Chamberlin-Courant (CC) rule, the Monroe rule, the OWA- upper bounds (Theorem 12). We further consider a certain [ ] based rules and the goalbase rules. specific MS score functions such as SNTV Cox, 1994 , or α-CC and β-CC [Chamberlin and Courant, 1983] where the Results. The algorithmic problems that arise largely remain unconstrained problem has recently received considerable at- NP-hard, hence we focus on developing approximation algo- tention. See Table 2 for a summary of these results, and nat- rithms. An important practical parameter, that also plays a ural corollaries which are described in the full version of the role in the complexity of the constrained multiwinner voting paper [Celis et al., 2017b]. problem, is the maximum number of groups to which a candi- date can belong; we denote it by ∆: In real-world situations, Generality. This approach is general in that 1) it can handle we expect ∆ to be a small constant, i.e., each candidate on- arbitrary MS score functions, 2) multiple sensitive attributes ly belongs to a few groups. Our main results (classified by which can take on arbitrary group structures, 3) interval con- the kind of fairness parameters) are summarized in Tables 1 straints that need not specify exact probabilities of represen- and 2. tations for each group, and in doing so 4) can satisfy many When ∆ = 1, e.g., when the groups partition the candi- different existing notions of fairness. dates, we present a (1 − 1=e)-approximation algorithm (The- Note that fairness can be simultaneously ensured across orem 14) which is optimal given the (1−1=e−")-hardness of multiple sensitive attributes (e.g., ethnicity and political par- approximation result by [Nemhauser et al., 1978]. However, ty) – the number of attributes a single candidate can have is when ∆ ≥ 3; unlike the unconstrained case, even checking captured by ∆. For example, in the New Zealand parliamen- whether there is a feasible solution becomes NP-hard (Theo- tary election [Zealand, 1986] the parliament is required to rem 6). The problem of finding a solution that violates cardi- include sufficient representation across 3 types of attributes: nality or fairness constraints up to any multiplicative constant political parties, special interest groups, and Maori represen- factor remains NP-hard (Theorem 7 and 8). Moreover, even tation. Each candidate has an identity under each attribute, if the feasibility is guaranteed, the problem remains hard to i.e., ∆ = 3. approximate within a factor of Ω(log ∆=∆) (Theorem 9). This type of constraints generalize notions of proportion- To bypass this issue, we assume that the problem instance ality that have arisen in the voting literature such as fully always has a feasible solution, and suggest natural sufficient proportional representation [Monroe, 1995], by letting `i = conditions that guarantee feasibility.
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