Approval Ballots Research Articles

The Price of Justified Representation

In multiwinner approval voting, the goal is to select a k -member committee based on voters’ approval ballots. A well-studied concept of proportionality in this context is the justified representation (JR) axiom, which demands that no large cohesive group of voters remains unrepresented. However, the JR axiom may conflict with other desiderata, such as social welfare (the number of approvals obtained by committee members) or coverage (the number of voters who approve at least one committee member). In this article, we investigate the price of imposing the JR axiom (as well as the more demanding extended justified representation (EJR) axiom) on social welfare and coverage. Our approach is twofold: we derive worst-case bounds on the loss of welfare/coverage caused by imposing JR, and study the computational complexity of finding committees with high welfare that provide JR (obtaining hardness results, approximation algorithms, and an exact algorithm for one-dimensional preferences).

Theory of Ranked-choice and Approval Ballots for Delegate-allocation Tasks and Analysis of the Ranked-choice Delegate Allocation System Introduced in the 2020 US Democratic Primary Presidential Primary

Theory of Ranked-choice and Approval Ballots for Delegate-allocation Tasks and Analysis of the Ranked-choice Delegate Allocation System Introduced in the 2020 US Democratic Primary Presidential Primary

Strategyproofness and Proportionality in Party-Approval Multiwinner Elections

In party-approval multiwinner elections the goal is to allocate the seats of a fixed-size committee to parties based on the approval ballots of the voters over the parties. In particular, each voter can approve multiple parties and each party can be assigned multiple seats. Two central requirements in this setting are proportional representation and strategyproofness. Intuitively, proportional representation requires that every sufficiently large group of voters with similar preferences is represented in the committee. Strategyproofness demands that no voter can benefit by misreporting her true preferences. We show that these two axioms are incompatible for anonymous party-approval multiwinner voting rules, thus proving a far-reaching impossibility theorem. The proof of this result is obtained by formulating the problem in propositional logic and then letting a SAT solver show that the formula is unsatisfiable. Additionally, we demonstrate how to circumvent this impossibility by considering a weakening of strategyproofness which requires that only voters who do not approve any elected party cannot manipulate. While most common voting rules fail even this weak notion of strategyproofness, we characterize Chamberlin-Courant approval voting within the class of Thiele rules based on this strategyproofness notion.

Phragmén’s voting methods and justified representation

In the late 19th century, Swedish mathematician Edvard Phragmén proposed a load-balancing approach for selecting committees based on approval ballots. We consider three committee voting rules resulting from this approach: two optimization variants—one minimizing the maximum load and one minimizing the variance of loads—and a sequential variant. We study Phragmén ’s methods from an axiomatic point of view, focusing on properties capturing proportional representation. We show that the sequential variant satisfies proportional justified representation, which is a rare property for committee monotonic methods. Moreover, we show that the optimization variants satisfy perfect representation. We also analyze the computational complexity of Phragmén ’s methods and provide mixed-integer programming based algorithms for computing them.

Approval-based apportionment

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 can support multiple parties by casting approval ballots. 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 instead of parties. Using techniques from both apportionment and multiwinner elections, we identify rules that generalize the D’Hondt apportionment method and that satisfy strong axioms which are generalizations of properties commonly studied in the apportionment literature. In fact, the rules we discuss 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 (also known as house monotonicity).

The Price of Justified Representation

In multiwinner approval voting, the goal is to select k-member committees based on voters' approval ballots. A well-studied concept of proportionality in this context is the justified representation (JR) axiom, which demands that no large cohesive group of voters remains unrepresented. However, the JR axiom may conflict with other desiderata, such as coverage (maximizing the number of voters who approve at least one committee member) or social welfare (maximizing the number of approvals obtained by committee members). In this work, we investigate the impact of imposing the JR axiom (as well as the more demanding EJR axiom) on social welfare and coverage. Our approach is threefold: we derive worst-case bounds on the loss of welfare/coverage that is caused by imposing JR, study the computational complexity of finding 'good' committees that provide JR (obtaining a hardness result, an approximation algorithm, and an exact algorithm for one-dimensional preferences), and examine this setting empirically on several synthetic datasets.

The Complexity of Learning Approval-Based Multiwinner Voting Rules

We study the PAC learnability of multiwinner voting, focusing on the class of approval-based committee scoring (ABCS) rules. These are voting rules applied on profiles with approval ballots, where each voter approves some of the candidates. According to ABCS rules, each committee of k candidates collects from each voter a score, that depends on the size of the voter's ballot and on the size of its intersection with the committee. Then, committees of maximum score are the winning ones. Our goal is to learn a target rule (i.e., to learn the corresponding scoring function) using information about the winning committees of a small number of sampled profiles. Despite the existence of exponentially many outcomes compared to single-winner elections, we show that the sample complexity is still low: a polynomial number of samples carries enough information for learning the target rule with high confidence and accuracy. Unfortunately, even simple tasks that need to be solved for learning from these samples are intractable. We prove that deciding whether there exists some ABCS rule that makes a given committee winning in a given profile is a computationally hard problem. Our results extend to the class of sequential Thiele rules, which have received attention due to their simplicity.

Truth-Tracking via Approval Voting: Size Matters

Epistemic social choice aims at unveiling a hidden ground truth given votes, which are interpreted as noisy signals about it. We consider here a simple setting where votes consist of approval ballots: each voter approves a set of alternatives which they believe can possibly be the ground truth. Based on the intuitive idea that more reliable votes contain fewer alternatives, we define several noise models that are approval voting variants of the Mallows model. The likelihood-maximizing alternative is then characterized as the winner of a weighted approval rule, where the weight of a ballot decreases with its cardinality. We have conducted an experiment on three image annotation datasets; they conclude that rules based on our noise model outperform standard approval voting; the best performance is obtained by a variant of the Condorcet noise model.

Evaluating approval-based multiwinner voting in terms of robustness to noise

Approval-based multiwinner voting rules have recently received much attention in the Computational Social Choice literature. Such rules aggregate approval ballots and determine a winning committee of alternatives. To assess effectiveness, we propose to employ new noise models that are specifically tailored for approval votes and committees. These models take as input a ground truth committee and return random approval votes to be thought of as noisy estimates of the ground truth. A minimum robustness requirement for an approval-based multiwinner voting rule is to return the ground truth when applied to profiles with sufficiently many noisy votes. Our results indicate that approval-based multiwinner voting can indeed be robust to reasonable noise. We further refine this finding by presenting a hierarchy of rules in terms of how robust to noise they are.

Consistent approval-based multi-winner rules

This paper is an axiomatic study of consistent approval-based committee (ABC) rules. These are multi-winner voting rules that select a committee, i.e., a fixed-size group of candidates, based on approval ballots. We introduce the class of ABC scoring rules and provide an axiomatic characterization of this class based on the consistency axiom. Building upon this result, we axiomatically characterize three important consistent multi-winner rules: Proportional Approval Voting, Multi-Winner Approval Voting and the Approval Chamberlin–Courant rule. Our results demonstrate the variety of ABC scoring rules and illustrate three different, orthogonal principles that multi-winner voting rules may represent: proportionality, diversity, and individual excellence.

Approval-Based Apportionment

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.

On the Complexity of Extended and Proportional Justified Representation

We consider the problem of selecting a fixed-size committee based on approval ballots. It is desirable to have a committee in which all voters are fairly represented. Aziz et al. (2015a; 2017) proposed an axiom called extended justified representation (EJR), which aims to capture this intuition; subsequently, Sanchez-Fernandez et al. (2017) proposed a weaker variant of this axiom called proportional justified representation (PJR). It was shown that it is coNP-complete to check whether a given committee provides EJR, and it was conjectured that it is hard to find a committee that provides EJR. In contrast, there are polynomial-time computable voting rules that output committees providing PJR, but the complexity of checking whether a given committee provides PJR was an open problem. In this paper, we answer open questions from prior work by showing that EJR and PJR have the same worst-case complexity: we provide two polynomial-time algorithms that output committees providing EJR, yet we show that it is coNP-complete to decide whether a given committee provides PJR. We complement the latter result by fixed-parameter tractability results.

Chamberlin--Courant Rule with Approval Ballots: Approximating the MaxCover Problem with Bounded Frequencies in FPT Time

We consider the problem of winner determination under Chamberlin--Courant's multiwinner voting rule with approval utilities. This problem is equivalent to the well-known NP-complete MaxCover problem and, so, the best polynomial-time approximation algorithm for it has approximation ratio 1 - 1/e. We show exponential-time/FPT approximation algorithms that, on one hand, achieve arbitrarily good approximation ratios and, on the other hand, have running times much better than known exact algorithms. We focus on the cases where the voters have to approve of at most/at least a given number of candidates.

A Representative Committee by Approval Balloting

A new voting rule for electing committees is described. Specifically, we use approval balloting and propose a new voting procedure that guarantees that if there is a committee that represents (with a given proportion of representatives) all of the existing voters, then the selected committee has to represent all of voters in at least the same proportion. This property is a way of selecting a committee that represents completely all of voters when such a committee exists. The usual voting rules in this context do not satisfy this condition.

Phragmén’s Voting Methods and Justified Representation

In the late 19th century, Lars Edvard Phragmén proposed a load-balancing approach for selecting committees based on approval ballots. We consider three committee voting rules resulting from this approach: two optimization variants one minimizing the maximal load and one minimizing the variance of loads —and a sequential variant. We study Phragmén's methods from an axiomatic point of view, focussing on justified representation and related properties that have recently been introduced by Aziz et al. (2015a) and Sánchez-Fernández et al. (2017). We show that the sequential variant satisfies proportional justified representation, making it the first known polynomial-time computable method with this property. Moreover, we show that the optimization variants satisfy perfect representation. We also analyze the com- putational complexity of Phragmén's methods and provide mixed-integer programming based algorithms for computing them.

Strategic voting in multi-winner elections with approval balloting: a theory for large electorates

We propose a theory of strategic voting in multi-winner elections with approval balloting. With a tiny probability that any vote might be misrecorded, best responses involve voting by pairwise comparisons. Two candidates play a critical role: the weakest expected winner and the strongest expected loser. Expected winners are approved if and only if they are preferred to the strongest expected loser and expected losers are approved if and only if they are preferred to the weakest expected winner. At equilibrium, if any, a candidate is elected if and only if he is approved by at least half of the voters. With single-peaked preferences, an equilibrium always exists, in which the first candidates according to the majority tournament relation are elected. The theory is applied to individual data from the 2011 Regional Government election in Zurich.

Approval elections with a variable number of winners

Multi-winner elections, for example, the election of members to a committee, are now quite common, and include the interesting subclass of elections with a variable number of winners, or VNW elections. In VNW elections, voters determine how many winners there are, as well as which candidates win. Common VNW elections include elections to bestow honorary status, such as enshrinement in a hall of fame, and elections to determine a shortlist of, say, job candidates for interviews. Such elections are VNW elections whenever the number of winners is not specified in advance. Multi-winner elections are often conducted by adapting standard procedures for single-winner elections. Approval balloting is particularly appropriate for multi-winner elections, as every ballot, and the set of winners, are subsets of the set of all candidates. Many methods of counting approval ballots appropriate to VNW elections are reviewed and illustrated, and their properties—desirable and undesirable—discussed.

Fully Proportional Representation with Approval Ballots: Approximating the MaxCover Problem with Bounded Frequencies in FPT Time

We consider the problem of winner determination under Chamberlin--Courant's multiwinner voting rule with approval utilities. This problem is equivalent to the well-known NP-complete MaxCover problem (i.e., a version of the SetCover problem where we aim to cover as many elements as possible) and, so, the best polynomial-time approximation algorithm for it has approximation ratio 1 - 1/e. We show exponential-time/FPT approximation algorithms that, on one hand, achieve arbitrarily good approximation ratios and, on the other hand, have running times much better than known exact algorithms. We focus on the cases where the voters have to approve of at most/at least a given number of candidates.

Collective approval

We consider the problem of aggregating individual approval ballots into one collective approval ballot. An approval ballot is simply a subset of a given set of alternatives. An individual may approve of as many alternatives as he or she wishes. Each approval is counted as a vote. We show that if an aggregation rule is neutral, consistent and discerning, then an alternative is collectively approved of if it receives a number of votes greater than the mean number of votes received by the alternatives and is not approved of if it receives a number of votes less than the mean.

Pareto efficiency in multiple referendum

We consider situations of multiple referendum: finitely many yes-or-no issues have to be socially assessed from a set of approval ballots, where voters approve as many issues as they want. Each approval ballot is extended to a complete preorder over the set of outcomes by means of a preference extension. We characterize, under a mild richness condition, the largest domain of top-consistent and separable preference extensions for which issue-wise majority voting is Pareto efficient, i.e., always yields out a Pareto-optimal outcome. Top-consistency means that voters’ ballots are their unique most preferred outcome. It appears that the size of this domain becomes negligible relative to the size of the full domain as the number of issues increases.