Multiwinner voting rules can be used to select a fixed-size committee from a larger set of candidates. We consider approval-based committee rules, which allow voters to approve or disapprove candidates. In this setting, several voting rules such as Proportional Approval Voting (PAV) and Phragm\'en's rules have been shown to produce committees that are proportional, in the sense that they proportionally represent voters' preferences; all of these rules are strategically manipulable by voters. On the other hand, a generalisation of Approval Voting gives a non-proportional but strategyproof voting rule. We show that there is a fundamental tradeoff between these two properties: we prove that no multiwinner voting rule can simultaneously satisfy a weak form of proportionality (a weakening of justified representation) and a weak form of strategyproofness. Our impossibility is obtained using a formulation of the problem in propositional logic and applying SAT solvers; a human-readable version of the computer-generated proof is obtained by extracting a minimal unsatisfiable set (MUS). We also discuss several related axiomatic questions in the domain of committee elections.

In participatory budgeting (PB), voters decide through voting which subset of projects to fund within a given budget. Proportionality in the context of PB is crucial to ensure equal treatment of all groups of voters. However, pure proportional rules can sometimes lead to suboptimal outcomes. We introduce the Method of Equal Shares with Bounded Overspending (BOS Equal Shares), a robust variant of Equal Shares that balances proportionality and efficiency. BOS Equal Shares addresses inefficiencies inherent in strict proportionality guarantees yet still provides good proportionality similar to the original Method of Equal Shares. In the course of the analysis, we also discuss a fractional variant of the method which allows for partial funding of projects.

In this survey, we review the literature investigating participatory budgeting as a social choice problem. Participatory Budgeting (PB) is a democratic process in which citizens are asked to vote on how to allocate a given amount of public money to a set of projects. From a social choice perspective, it corresponds then to the problem of aggregating opinions about which projects should be funded, into a budget allocation satisfying a budget constraint. This problem has received substantial attention in recent years and the literature is growing at a fast pace. In this survey, we present the most important research directions from the literature, each time presenting a large set of representative results. We only focus on the indivisible case, that is, PB problems in which projects can either be fully funded or not at all. The aim of the survey is to present a comprehensive overview of the state of the research on PB. We aim at providing both a general overview of the main research questions that are being investigated, and formal and unified definitions of the most important technical concepts from the literature.

Citizen-focused democratic processes where participants deliberate on alternatives and then vote to make the final decision are increasingly popular today. While the computational social choice literature has extensively investigated voting rules, there is limited work that explicitly looks at the interplay of the deliberative process and voting. In this paper, we build a deliberation model using established models from the opinion-dynamics literature and study the effect of different deliberation mechanisms on voting outcomes achieved when using well-studied voting rules. Our results show that deliberation generally improves welfare and representation guarantees, but the results are sensitive to how the deliberation process is organized. We also show, experimentally, that simple voting rules, such as approval voting, perform as well as more sophisticated rules such as proportional approval voting or method of equal shares if deliberation is properly supported. This has ramifications on the practical use of such voting rules in citizen-focused democratic processes.

Epistemic voting interprets votes as noisy signals about a ground truth. We consider contexts where the truth consists of a set of objective winners, knowing a lower and upper bound on its cardinality. A prototypical problem for this setting is the aggregation of multi-label annotations with prior knowledge on the size of the ground truth. We posit noise models, for which we define rules that output an optimal set of winners. We report on experiments on multi-label annotations (which we collected).

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.

Participatory Budgeting (PB) is a process in which voters decide how to allocate a common budget; most commonly it is done by ordinary people -- in particular, residents of some municipality -- to decide on a fraction of the municipal budget. From a social choice perspective, existing research on PB focuses almost exclusively on designing computationally-efficient aggregation methods that satisfy certain axiomatic properties deemed "desirable" by the research community. Our work complements this line of research through a user study (N = 215) involving several experiments aimed at identifying what potential voters (i.e., non-experts) deem fair or desirable in simple PB settings. Our results show that some modern PB aggregation techniques greatly differ from users' expectations, while other, more standard approaches, provide more aligned results. We also identify a few possible discrepancies between what non-experts consider \say{desirable} and how they perceive the notion of "fairness" in the PB context. Taken jointly, our results can be used to help the research community identify appropriate PB aggregation methods to use in practice.

The executive branch, or government, is typically not elected directly by the people, but rather formed by another elected body or person such as the parliament or the president. As a result, its members are not directly accountable to the people, individually or as a group. We consider a scenario in which the members of the government are elected directly by the people, and wish to achieve proportionality while doing so. We propose a formal model consisting of $k$ offices, each with its own disjoint set of candidates, and a set of voters who provide approval ballots for all offices. We wish to identify good aggregation rules that assign one candidate to each office. As using a simple majority vote for each office independently might result in disregarding minority preferences altogether, here we consider an adaptation of the greedy variant of Proportional Approval Voting (GreedyPAV) to our setting, and demonstrate -- through computer-based simulations -- how voting for all offices together using this rule overcomes this weakness. We note that the approach is applicable also to a party that employs direct democracy, where party members elect the party's representatives in a coalition government.

Approval ballot based committee formation is concerned with aggregating individual approvals of voters. Voters submit their approvals of candidates and these approvals are aggregated to arrive at the optimal committee of specified size. There are several aggregation techniques proposed in the literature and these techniques differ among themselves on the criterion function they optimize. Voters preferences for a candidate is based on his/her opinion on candidate suitability. We note that candidates have attributes that make him/her suitable or otherwise. Hence, it is relevant to approve attributes and select candidates who have the approved attributes. This paper addresses the committee selection problem when voters submit their approvals on attributes. Though attribute based preference is addressed in several contexts, committee selection problem with attribute approval has not been attempted earlier. We note that extending the theory of candidate approval to attribute approval in committee selection problem is not trivial. In this paper, we study different aspects of this problem and show that none of the existing aggregation rules satisfies Unanimity and Justified Representation when attribute based approvals are considered. We propose a new aggregation rule that satisfies both the above properties. We also present other analysis of committee selection problem with attribute approval.

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 (JR). This axiom requires that if a large enough group of voters exhibits agree- ment 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 JR. We then check if this axiom is fulfilled by well-known approval-based voting rules. We show that the answer is negative for most of the rules we consider, with notable exceptions of PAV (Proportional Approval Voting), an extreme version of RAV (Reweighted Approval Voting), and, for a restricted preference domain, MAV (Minimax Approval Voting). We then introduce a stronger version of the JR axiom, which we call extended justified representation (EJR), and show that PAV satisfies EJR, while other rules do not. We also consider several other questions related to JR and EJR, including the relationship between JR/EJR and unanimity, and the complexity of the associated algorithmic problems.