We conclude this book with a list of what we view as particularly important open problems and research directions. This is followed by a list of more specific or more technical open questions. These two lists are naturally far from being exhaustive; many more research directions remain to be explored.

7.1 Main Open Problems and Research Questions

  1. Q1

    Axiomatic characterisations: So far, only few axiomatic characterisations of ABC rules are known. Specifically, such characterisations are known only for ABC scoring rules and Thiele methods. Yet, axiomatic characterisations are essential if one wants to choose an ABC rule in a principled way. It is thus one of the major open problems to characterise other ABC rules, in particular, sequential Thiele methods, seq-Phragmén, the Method of Equal Shares, Monroe’s rule, Minimax Approval Voting, and Satisfaction Approval Voting. Further, almost no satisfiable proportionality-related axioms are known for the multi-attribute model (Sect. 4.7), let alone axiomatic characterisations.

  2. Q2

    Committee monotonicity and proportionality: The current state of research suggests that committee monotonic ABC rules are limited in how proportional they are, but there is no precise impossibility result known as of now. The main open question is whether there exist ABC rules that satisfy EJR and committee monotonicity. Only partial answers are known to this question. For example, it is known that such a rule can be defined for approval-based party-list elections (see the work of Brill et al. [5]; mentioned in Sect. 4.4), but there is no clear generalisation of this rule to the setting of ABC rules. In case such a rule does not exist, it might be easier to first show that committee monotonicity and the core property are incompatible.

  3. Q3

    The core property: Does there exist an ABC rule that satisfies the core property (Definition 4.10)? Equivalently, is the core always non-empty? In case the core can be empty, what is a sensible ABC rule that outputs a committee in the core whenever it exists? Can such a rule be computed in polynomial time?

  4. Q4

    Analysis beyond the worst-case: With a few notable exceptions, in Chaps. 3 and 4 we discussed axiomatic properties which are worst-case in spirit. A voting rule fails such an axiom even if there exist only few very unnatural election instances for which the property is not satisfied. An alternative approach would be to test if the properties hold for randomly generated instance of elections, or for elections from datasets containing real-life instance [14]. However, many common distributions of voters’ preferences are too simplistic and do not capture the complexity of the voters’ reasoning processes; the real election instances are rather scarce, and are collected in specific contexts, e.g., assuming that the voters’ know the election rule that will be used to select winners. It is an important task to develop intermediate approaches that allow for a more fine-grained analysis and allow to understand which of the rules exhibit most desired properties on election instances that are likely to occur in practice.

  5. Q5

    Relation between axiomatic properties and computability: It is still unclear which combinations of axiomatic properties of ABC rules can be achieved in polynomial time. It is known that some rules are \({{\textrm{NP}}}\)-hard to compute, but it is unclear which axiomatic properties of these rules cause computational hardness. For example, it is not known whether the axiom of FJR (see Definition 4.7) is satisfiable by a rule computable in polynomial time. Further, is there a polynomial-time computable ABC rule that is proportional (e.g., that satisfies PJR) and satisfies Pareto optimality? Or does there exist a polynomial-time rule that satisfies consistency and extends D’Hondt? (By Theorem 4.2, such a rule must violate either neutrality, anonymity, or continuity.)

  6. Q6

    Preference data from distribution: An important challenge is to prepare a representative database containing sample approval-based elections. Realistic probability distributions would allow for the automatic generation of synthetic (but meaningful) election instances, which are important for numerical simulations and performance tests of algorithms. In comparison to the ranking-based model, much fewer statistical models for generating approval-based elections are know. Further, it would be highly desirable to identify a set of distributions that are representative and that cover numerous potential types of voters and voting scenarios. A noteworthy attempt at creating such a representative collection of distributions has been made for the ranking-based model by Szufa et al. [19]. For ABC elections, this issue remains to be explored.

7.2 Further Open Problems

We continue with more specific or more technical open problems.

  1. Q7

    The key feature of Monroe’s rule is its underlying assumption that a committee member can represent only 1/k-th fraction of the voter population. Monroe’s rule could thus be generalised to many optimisation-based multi-winner rules by imposing the additional restriction that committee members can represent (i.e., derive score from) an \(\alpha \)-fraction of voters. This idea resembles the group activity selection problem, where a set of activities is chosen, each of which has a maximum number of participants, and agents are assigned to activities subject to their preferences; see the survey of Darmann and Lang [10]. More generally, adding this “Monroe-style” constraint can be seen as requiring a homogeneous representation load among chosen committee members. This is a sensible assumption whenever candidates can satisfy only a limited number of voters (e.g., if candidates represent consumable goods). This idea of committees with homogeneous representation loads is largely unexplored.

  2. Q8

    Most axiomatic notions for proportionality are only applicable to ABC rules that extend apportionment methods satisfying lower quota (see Fig. 4.1). This excludes, e.g., ABC rules that extend the Sainte-Laguë method. As the Sainte-Laguë method is in certain aspects superior to the D’Hondt method (Balinski and Young [2] discuss this in detail), it would be desirable to have notions of proportionality that are agnostic to the underlying apportionment method.

  3. Q9

    What is the proportionality degree of rev-seq-PAV?

  4. Q10

    Does there exist an ABC rule that satisfies priceability and Pareto efficiency?

  5. Q11

    What is the computational complexity of verifying whether a given committee belongs to the core? Is it possible to find a committee in the core in polynomial time (if it exists)? In case of computational hardness, can the methods presented in Chap. 5 be used to obtain algorithms that are fast in practice?Footnote 1

  6. Q12

    We have seen in Sect. 4.6 that proportionality and strategyproofness are typically incompatible. The corresponding impossibility result for arbitrary, i.e., irresolute, ABC rules [13] relies on Pareto efficiency. Since this is a property that many sensible ABC rules do not satisfy (see Sect. 3.2) it would be desirable to strengthen this result by relaxing this condition, e.g., by replacing Pareto efficiency with weak efficiency. Is this possible or are there ABC rules that are irresolute, strategyproof, proportional, but not Pareto efficient? Furthermore, both the result for irresolute [13] rules and for resolute rules [15, 16] rest on the assumption that the committee size k divides the number of voters. This assumption is unlikely to hold for large k and thus removing this assumption would be desirable.

  7. Q13

    A question related to monotonicity was asked by Sánchez-Fernández and Fisteus [17]: Is there an ABC rules that is proportional (even in a very weak sense, e.g., satisfying JR) and satisfies support monotonicity without additional voters (Definition 3.3)? As of now, AV and SAV are the only rules known to satisfy this property and both are not proportional.

  8. Q14

    Another question related to monotonicity concerns the Method of Equal Shares: while this method exhibits very strong proportionality guarantees (in particular EJR and priceability), it fails candidate monotonicity with additional voters (as discussed in Sect. 3.4). Is there an equally proportional ABC rule that also satisfies candidate monotonicity?

  9. Q15

    We mentioned in Sect. 3.1 that ABC rules that require tiebreaking do not satisfy neutrality (e.g., sequential and reverse sequential Thiele methods, Greedy Monroe, seq-Phragmén, and the Method of Equal Shares are not neutral). These rules can be made neutral with parallel universes tiebreaking: a committee is winning under the neutral variant if and only if it is winning for some tiebreaking order under the original rule. Parallel universes tiebreaking has been analysed for single-winner rules [4, 7, 11] but not for multi-winner rules. Such a modification will have an algorithmic impact (trying all permutations of candidates would require exponential time), but the exact computational complexity of these neutral rules is not settled. Further, under which conditions can these rules be computed in polynomial time?

  10. Q16

    In Sect. 5.1, we presented a coarse analysis of the computational complexity of ABC rules. This analysis could be refined by considering the Candidate Winner problem: given an election instance (A, k) and a candidate c, does there exist a winning committee W that contains c? This problem has recently be shown to be \(\Theta _2^p\)-complete for Monroe and CC by Sonar et al. [18]. A similar analysis for other computationally hard voting rules (such as PAV) is missing.

  11. Q17

    Sequential PAV approximates the optimal PAV-score by a factor of at least \(1-\frac{1}{e}\). What is the factor for Reverse Sequential PAV? Is it better? The same question can be asked for other Thiele methods.

  12. Q18

    Several approximation algorithms and heuristics have been proposed for PAV, including seq-PAV, rev-seq-PAV, the approximation algorithm based on dependent rounding ([6], discussed in Sect. 5.2.3), and a local-search algorithm used for finding EJR committees in polynomial time [1]. The difference between these algorithms has not been investigated from a practical point of view. The main question is which of these algorithms should be chosen to approximate PAV given a very large election?

  13. Q19

    Is it possible to compute Thiele methods and Monroe’s rule in polynomial time if the given preference profile belongs to the voter interval (VI) domain (see Sect. 5.2.4)?

  14. Q20

    The computation of some polynomial-time ABC rules can clearly be parallelised. For example, for AV each candidate can be processed independently of others. The framework of \({{\textrm{P}}}\)-completeness [12] can be used to determine which ABC rules are inherently sequential (by showing P-completeness) and which can be parallelised (by showing, e.g., \(\textrm{NL}\)-containment). Such work has been done for single-winner rules [3, 8, 9] but not for multi-winner rules.

  15. Q21

    In real-life elections, it is sometimes required that each voter can approve at most k candidates. It is interesting to see what are the consequences of such a requirement in terms of qualities of the committees produced by various rules. Sometimes, it is even possible to distribute up to k points to candidates, i.e., to approve candidates more than once. This is clearly beyond the ABC model, but some concepts and results may transfer to such voting systems.