In this chapter, we discuss connections of approval-based committee voting with a number of other applications and formalisms.

6.1 Ranking-Based Multi-Winner Elections

Besides ABC voting, the other classic multi-winner election model is when voters provide a ranking of candidates from the most to the least preferred one. That is, in the ranking-based model a voter’s preference is expressed as a linear order of all candidates instead of a subset of candidates, as it is the case in the ABC model. As it is the case with approval-based multi-winner elections, also the ranking-based model has attracted much attention in recent years. Alas, at the point of writing this book, there does not exist a comprehensive overview of this field of research. However, a very helpful introduction to multi-winner voting in general (with a focus on the ranking-based model) can found in a book chapter by Faliszewski et al. [32].

When comparing approval-based and ranking-based multi-winner rules, it is worth mentioning that the class of ABC scoring rules (Definition 3.5) has a very close analogue in the ranking-based model, namely the class of committee scoring rules [28]. Indeed, committee scoring rules admit a very similar axiomatic characterisation to the one given in Theorem 3.2 for ABC scoring rules [64]. The class of committee scoring rules has been explored in depth by Faliszewski et al. [33]. In particular, the subclass of OWA-based committee scoring rules corresponds to the class of Thiele methods in the approval-based model. Other subclasses of committee scoring rules can be analogously defined for approval ballots, but to the best of our knowledge they have not been considered in the context of approval-based elections.

The approval-based and ranking-based model can be generalised to the model where voters provide weak orders over candidates, i.e., ranking with ties. In this model, approval ballots correspond to a ranking with two levels (approved and disapproved candidates). This variant has been considered, e.g., by Aziz and Lee [4], but generally attracted much less attention so far. This is due to the fact that the concepts discussed in this book (e.g., notions of proportionality) do not easily generalise to this more expressive setting and require substantial conceptual developments. Further work is required to consolidate the literature from the approval-based and ranking-based model in a systematic and notationally concise form.

6.2 Trichotomous Preferences and Incomplete Information

In this book we consider the variant of the multi-winner election model where agents vote by specifying sets of approved candidates. Several recent (mostly algorithmic) works study an extended variant of this model, where the ballots are trichotomous, i.e., where each voter can approve, disapprove or remain neutral with regard to a candidate. This model is discussed in detail by Brams and Fishburn [17] and Lines [56]. Baumeister and Dennisen [10] and Baumeister et al. [11] generalise AV and MAV to trichotomous votes and explore related algorithmic questions. This line of work has been continued by Liu and Guo [58]. Further, Baumeister et al. [12] extend MAV to the case where each voter assigns each candidate to one of \(\ell \) predefined buckets, where \(\ell \) is a parameter. Zhou et al. [68] introduce variants of CC, PAV, and SAV for trichotomous ballots and study questions regarding parameterized complexity. Finally, Talmon and Page [66] define and study notions of proportionality in the trichotomous setting. In general, many questions regarding the trichotomous model remain unanswered. In particular, an axiomatic analysis is mostly missing (with work of Alcantud and Laruelle [1] and Gonzalez et al. [40] as notable exceptions).

A model closely related to trichotomous preferences arises if approval ballots are incomplete due to missing information. In this model, the middle, “neutral” option corresponds to “unknown”. In practice, voting rules often have to be computed given incomplete information (such as missing ballots or incomplete ballots; see the handbook chapter of Boutilier and Rosenschein [16] for a broader discussion). For ABC rules, a first analysis with focus on AV is due to Barrot et al. [9]. A more comprehensive treatment by Imber et al. [45] considers the class of Thiele methods and focuses on computational problems related to incomplete information. Apart from the three-valued model of incomplete information, as discussed here, they also propose models where “unknown” candidates are ordered by preference but it is unclear where to separate them in approved and disapproved candidates. Finally, Terzopoulou et al. [67] study structured preference domains (cf. Sect. 5.2.4) in connection with incomplete information.

6.3 A Variable Number of Winners

Throughout this paper, we assume that the committee size is fixed. In the literature on multi-winner voting with a variable number of winners [48, 49] (also known as social dichotomy functions [26]), this assumption is dropped and a voting rule can return an arbitrary number of candidates—depending on the given election instance. An example for such a rule, based on approval ballots, is the mean rule, which returns all candidates with an above-average number of approvals (introduced by Duddy et al. [27], further analysed by Brandl and Peters [19]). Another example is Minimax Approval Voting (MAV), as discussed in Sect. 2.7. In this setting, MAV returns all candidate subsets that minimise the largest Hamming distance among all voters. Other ABC rules do not easily translate to this setting. For example, Thiele methods always achieve a maximum score for the complete set of all alternatives. Consequently, the formulation of such voting rules often contains a penalty mechanism for larger sets.

More details, in particular a computational view point and an experimental evaluation, can be found in the work of Faliszewski et al. [34]. Further, the special case of shortlisting rules has been analysed by Lackner and Maly [54]; this work includes recommendations which voting rules are particularly suitable for shortlisting scenarios. Shortlisting in a proportional fashion was studied by Freeman et al. [36]; their focus lies on proportionality guarantees (related to the ones introduced in Sect. 4.2) for variable-sized sets of candidates. Finally, Allouche et al. [2] consider an epistemic scenario where a “correct” selection of candidates has to be identified; approval ballots are viewed as noisy estimates of a ground truth.

6.4 Participatory Budgeting

In participatory budgeting (PB), we assume that candidates come with different costs, and that the sum of the costs of the selected candidates cannot exceed a given budget. Thus, multi-winner elections can be viewed as a special case of PB, where the costs of the candidates are all equal. Typically, candidates correspond to projects in this setting, each of which has an associated cost to be implemented. For an overview of different models and approaches to PB, we refer the reader to a recent survey by Aziz and Shah [5].

Participatory budgeting based on approval ballots is one of the standard models and is often used in real-world PB referenda. Knapsack voting suggested by Goel et al. [39] closely resembles AV. Peters et al. [60] showed that the Method of Equal Shares preserves its proportionality properties in the setting of PB—it satisfies an adapted version of EJR, and a logarithmic approximation of the core. Aziz et al. [6] provide a taxonomy of axioms aimed at formalising proportionality in PB; those axioms are adaptations of JR and PJR (see Sect. 4.2). Talmon and Faliszewski [65] study other axioms, mostly pertaining to different forms of monotonicity (see Sects. 3.3 and 3.4) and through experiments provide visualisations of the kind of committees returned by different participatory budgeting rules. Baumeister et al. [13] consider the computational complexity of strategic voting. Generally, the assumption is that projects are independent of each other; Jain et al. [47] study participatory budgeting without this assumption. Finally, Rey et al. [61] connect participatory budgeting based on approval ballots with judgement aggregation (see Sect. 6.7), which offers another possibility to include constraints.

6.5 Budget Division and Probabilistic Social Choice

The goal of a probabilistic social choice function is to divide a single unit of a global resource between the candidates. Thus, multi-winner elections can be viewed as instances of probabilistic social choice with the additional requirement that each candidate gets either 1/k-th fraction of the global resource, or nothing. For an overview of results on probabilistic social choice functions, we refer to a book chapter by Brandt [21].

Several works [7, 15, 20, 25, 31, 59] study probabilistic social choice functions for approval votes. The particular focus of some of these works is put on formalising the concepts of fairness and proportionality. Some of these concepts closely resemble the ones that we discussed in the context of approval-based multi-winner elections (Sect. 4). For example, Aziz et al. [7] and Fain et al. [31] study the concept of the core (Sect. 4.4), Aziz et al. [7] additionally consider the axioms of average fair share, group fair share, and individual fair share—the properties that closely resemble—respectively—proportionality degree, PJR, and JR (Sect. 4.2), Michorzewski et al. [59] show the relation between these fairness properties and the utilitarian welfare of outcomes (cf. Sect. 4.5.2). Bogomolnaia et al. [15] focuses on mechanisms which are strategyproof, and Duddy [25] proves that strategyproofness is incompatible with certain forms of proportionality—an impossibility result similar to the ones that we discuss in Sect. 4.6.

6.6 Voting in Combinatorial Domains

Multi-winner rules output fixed-size subsets of available candidates. An alternative way of thinking of such rules is that (1) for each candidate c they make a decision whether c should be selected to the winning committee or not, and (2) there is a constraint which specifies that exactly k decisions must be positive. Thus, with m candidates there are m dependent binary decisions (each decision is of the form “include a candidate in the winning committee or not”) that are made by a multi-winner rule. These decisions are dependent (related) because of the constraint on the number of positive decisions.

The literature on voting in combinatorial domains studies a more general setup, where a number of decisions (not necessarily binary) need to be made, and where there exist (possibly complex) relations between the decisions. Similarly, the preferences of the voters might have complex forms. For example, consider two issues—\(I_1\) with two possible decisions \(Y_1\) and \(N_1\), and \(I_2\) with two possible decisions \(Y_2\) and \(N_2\). A voter might prefer decision \(Y_2\) only if the decision with respect to issue \(I_1\) is \(Y_1\); otherwise this voter might prefer \(N_2\) over \(Y_2\) (see the work of Brams et al. [18] for a detailed discussion of this example). Various languages have been studied that allow voters to express such complex combinatorial preferences. For example, in the context of approval-based multi-winner elections, some of these languages would allow voters to express the view that a certain group of candidates works particularly well together, so they should either be all selected as members of the winning committee or none of them should be chosen, or the view that some candidates should never be chosen together. In the literature on multi-winner elections, on the other hand, it is assumed that the preferences of the voters are separable, thus the voters can only make statements about their levels of appreciation for different candidates. An interesting middle ground between very general forms of combinatorial preferences and simple (i.e., separable) preferences was proposed by Barrot and Lang [8]: conditional approval ballots allow voters to specify their approval ballots conditional on whether certain candidates are to be included in the committee.

A comprehensive overview of the literature on voting in combinatorial domains can be found in a book chapter by Lang and Xia [55]. We highlight three works from this literature that deal with models particularly related to the model of approval-based multi-winner elections. In public decision making, as studied by Conitzer et al. [23], the decisions are not related, the preferences of the voters with respect to decisions on various issues are separable, thus the model closely resembles the one studied in this book. The main difference is that in the model for public decisions there is no constraint specifying the number of decisions that can be positive. There, the authors focus on designing fair (i.e., proportional) rules. The model of sub-committee elections, due to Aziz and Lee [3], generalises the ones of multi-winner elections and public decisions. There, it is assumed that the set of candidates is partitioned and for each group of candidates there is a threshold bounding the number of candidates selected from this group.

Another formalism closely related to ABC voting is perpetual voting, introduced by Lackner [53]. Here, instead of a committee we have time steps and in each step one candidate is selected. Hence, after k rounds k candidates are picked, which can be viewed as a committee. The main difference is that the set of available candidates and voters’ preferences can change each round. The goal is to provide proportionality over time, which requires that the decision in round k is made under consideration of the voters’ satisfaction in previous rounds. This formalism can be viewed as a special case of voting in combinatorial domains (with a very specific sequentiality constraint). Further, due to the sequential structure imposed by time, perpetual voting rules have close connections with committee monotonic ABC rules (such as seq-Phragmén and seq-PAV). Similar questions in a utility-based model have been studied by Freeman et al. [35]. A voting rule related to the setting of perpetual voting is due to Gottlob FregeFootnote 1 [37, 38]. The main difference is that the set of candidates remains the same in each round and the goal is to achieve a proportionally fair outcome for candidates (instead of voters). An analysis of this voting system is due to Harrenstein et al. [44].

6.7 Judgment Aggregation and Propositional Belief Merging

In judgment aggregation, we are given a set of logical propositions and a set of voters providing true/false valuations for these propositions; the goal is to find a collective, aggregated valuation. Sometimes it is also required that the collective valuation must be consistent with exogenous logical constraints. Multi-winner elections can be represented as instances of judgment aggregation, where for each candidate we have a single Boolean variable representing whether the candidate is elected or not; the exogenous constraints can be used to enforce that exactly k from these variables are set true. A chapter by Endriss [29] in the Handbook of Computational Social Choice discusses this framework in detail and reviews judgment aggregation rules; see also the survey by List and Puppe [57].

Propositional belief merging [50,51,52] is a very general framework, which allows agents to aggregate their individual positions (beliefs, preferences, judgements, goals) on a set of issues. Also here this combined, collective outcome has to satisfy given exogenous logical constraints. Approval-based committee voting can be seen as a special case of propositional belief merging, although the focus of these two directions of research has little overlap: belief merging operators are analysed with respect to a set of postulates that are only partially relevant in a voting context. A few works have made an explicit effort to connect voting and belief merging. A particular focus in this regard has been the study of belief merging and strategyproofness [22, 30, 41]. Further, Haret et al. [42] consider classic axioms from social choice theory in the context of belief merging. Finally, Haret et al. [43] introduce and analysed proportional belief merging operators.

6.8 Proportional Rankings

The theory of multi-winner elections can be applied in a seemingly unrelated setting, where the goal is to find a ranking of candidates based on voters’ preferences. One can observe that every committee monotonic (Definition 3.2), resolute ABC rule \(\mathcal {R}\) can be used to obtain a ranking of candidates: we put in the first position in the ranking the candidate that \(\mathcal {R}\) returns for \(k=1\); call this candidate c. Committee monotonicity guarantees that the set of two candidates returned by \(\mathcal {R}\) for \(k=2\) contains c; the other candidate is put in the second position in the ranking, etc.

In particular, if we use a proportional committee-monotonic rule (for example, seq-Phragmén or seq-PAV) then the obtained ranking will proportionally reflect the views of the voters in the sense that each prefix of such a ranking, viewed as a committee, will be proportional; this idea has been studied in detail by Skowron et al. [63]. Proportional rankings are desirable, e.g., when one wants to provide a list of recommendations or search results that accommodate different types of users (cf. diversifying search results [24, 62]), or in the context of liquid democracy [14], where an ordered list of proposals is presented to voters for their consideration.

Proportional rankings in a dynamic setting, where the rankings also take previously selected (and now unavailable) alternatives into account, have been studied by Israel and Brill [46]. This setting arises, e.g., in dynamic Q&A platforms, where questions are proposed and upvoted. The authors argue that questions that already have been asked should be taken into account when choosing the next question(s).