In multiwinner elections with many candidates, as in participatory budgeting or large-scale recommendation, voters cannot plausibly evaluate every candidate, yet standard proportional-fairness guarantees such as EJR+ are stated for fully specified approval ballots. We ask whether strong proportional representation can still be guaranteed while eliciting only a little from each voter. We study this in a spatial model, the Axis-aligned Random Rectangle Voter (ARRV) model, in which candidates occupy a d-dimensional issue space and each voter approves an axis-aligned hyper-rectangle: a tolerance interval on every issue. Preferences are revealed only through Planar queries, each comparing a voter's tolerance to a candidate on a single issue. We give an algorithm returning an EJR+ committee for any distribution over rectangular preferences, using only 𝒪(dlog dk) Planar queries per voter in expectation given a sufficiently large electorate, independent of the number of candidates m, where d is the number of issues and k the committee size. The algorithm rests on a dimension-agnostic verify-or-fallback framework whose query cost is governed by two properties supplied by interchangeable modules. We describe such modules, yielding end-to-end guarantees for known, unknown, and smooth distributions.

Popular centroid-based clustering methods are typically optimized for global objectives and may fail to adequately represent large groups of datapoints. To address this concern, recent work puts forward clustering analogs of social choice proportionality concepts, such as Proportionally Representative Fairness (also known as mPJR). For proportionality guarantees to be useful in practice, they must be (a) achievable and (b) efficiently auditable, so that one can check whether standard approaches, such as k-means, which are not guaranteed to provide proportional representation in general, nevertheless output proportional solutions on specific inputs. In this work, we study the computational complexity of verifying proportional representation in clustering. We first show that verifying mPJR is coNP-hard. Inspired by PJR+ – a strengthening of PJR that is polynomial-time verifiable in the committee voting setting – we introduce mPJR+ as its metric analog. However, verifying mPJR+ relies on repeated submodular minimization, rendering it impractical at scale. Hence, we introduce Default Coalitions mPJR+ (DC-mPJR+): a new proportionality concept that offers representation guarantees to a restricted set of coalitions around unselected centers, and as a result, admits an O(mn log n + mnk) verification algorithm. DC-mPJR+ is satisfied by SEAR and remains a meaningful proxy for global fairness: any solution satisfying γ-DC-mPJR+ also satisfies (γ+ 2)-mPJR+. Together, our results identify a practical and theoretically grounded path for auditing proportional representation in clustering.

We study a stylized model of infrastructure investment in public transportation. In our model, each agent travels between a pair of terminals in a network captured by a weighted graph, where edge weights represent distances. The central planner can improve the travel time along a fixed number of edges, with the goal of maximizing the utilitarian or egalitarian welfare. When there is only one agent, we provide a polynomial-time algorithm that combines Dijkstra's algorithm with a dynamic program. We then demonstrate how to use this algorithm as a subroutine to solve the problem for two agents. Generalizing this idea, we present an XP algorithm parameterized by the number of agents. However, our problem turns out to be W[1]-hard with respect to the number of agents. Nevertheless, we obtain a fixed-parameter tractability result for the special case where all agents travel to a common hub. If the number of agents is variable, we obtain NP-completeness and inapproximability results. We discuss implications of our results for a related model of railway network design.

The goal of multi-winner elections is to choose a fixed-size committee based on voters’ preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. Recently, Aziz et al. proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). In this paper, we extend the work of Aziz et al. in several directions. First, we answer an open question of Aziz et al., by showing that Reweighted Approval Voting satisfies JR for k = 3; 4; 5, but fails it for k >= 6. Second, we observe that EJR is incompatible with the Perfect Representation criterion, which is important for many applications of multi-winner voting, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with perfect representation, and a committee that provides PJR can be computed in polynomial time if the committee size divides the number of voters. Moreover, just like EJR, PJR can be used to characterize the classic PAV rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.

The study of proportionality in multiwinner voting with approval ballots has received much attention in recent years [23]. Typically, proportionality is captured by variants of the Justified Representation axiom [1], which say that cohesive groups of at least ℓ·n/k voters (where n is the total number of voters and k is the desired number of winners) deserve ℓ representatives. The quantity n/k is known as the Hare quota in the social choice literature. Another—more demanding—choice of quota is the Droop quota, defined as ⌊n/k+1⌋ +1 . This quota is often used in multiwinner voting with ranked ballots: in algorithms such as Single Transferable Voting, and in proportionality axioms, such as Droop’s Proportionality Criterion. A few authors have considered it in the context of approval ballots [5, 10, 17, 30, 31], but the existing analysis is far from comprehensive. The contribution of our work is a systematic study of JR-style axioms (and voting rules that satisfy them) defined using the Droop quota instead of the Hare quota. For each of the standard JR axioms (namely, JR, PJR, EJR, FPJR, FJR, PJR+ and EJR+), we identify a voting rule that satisfies the Droop version of this axiom. In some cases, it suffices to consider known rules (modifying the corresponding Hare proof, sometimes quite substantially), and in other cases it is necessary to modify the rules from prior work. Each axiom is more difficult to satisfy when defined using the Droop quota, so our results expand the frontier of satisfiable proportionality axioms. We complement our theoretical results with an experimental study, showing that for many probabilistic models of voter approvals, Droop JR/EJR+ are considerably more demanding than standard (Hare) JR/EJR+.

We study the complexity of determining a winning committee under the Chamberlin-Courant voting rule when voters' preferences are single-crossing on a line, or, more generally, on a tree. For the line, Skowron et al. (2015) describe an O(n^2mk) algorithm (where n, m, k are the number of voters, the number of candidates and the committee size, respectively); we show that a simple tweak improves the time complexity to O(nmk). We then improve this bound for k=Ω(log n) by reducing our problem to the k-link path problem for DAGs with concave Monge weights, obtaining a nm2^O(√(log k log log n)) algorithm for the general case and a nearly linear algorithm for the Borda misrepresentation function. For trees, we point out an issue with the algorithm proposed by Clearwater, Puppe and Slinko (2015), and develop a O(nmk) algorithm for this case as well.

Recent breakthroughs in generative artificial intelligence (AI) and large language models (LLMs) unravel new capabilities for AI personal assistants to overcome cognitive bandwidth limitations of humans, providing decision support or even direct representation of abstained human voters at large scale. However, the quality of this representation and what underlying biases manifest when delegating collective decision making to LLMs is an alarming and timely challenge to tackle. By rigorously emulating more than >50K LLM voting personas in 363 real-world voting elections, we disentangle how AI-generated choices differ from human choices and how this affects collective decision outcomes. Complex preferential ballot formats show significant inconsistencies compared to simpler majoritarian elections, which demonstrate higher consistency. Strikingly, proportional ballot aggregation methods such as equal shares prove to be a win-win: fairer voting outcomes for humans and fairer AI representation, especially for voters likely to abstain. This novel underlying relationship proves paramount for building democratic resilience in scenarios of low voters turnout by voter fatigue: abstained voters are mitigated via AI representatives that recover representative and fair voting outcomes. These interdisciplinary insights provide decision support to policymakers and citizens for developing safeguards and policies for risks of using AI in democratic innovations.

Participatory Budgeting (PB) is commonly studied from an axiomatic perspective, where the aim is to design procedurally fair and economically efficient rules for voters with full information regarding their preferences. In contrast, we take an epistemic perspective and consider a framework where PB projects have different levels of underlying quality, indicating how well the project will take effect, which cannot be directly observed before implementation. Agents with noisy information cast votes to aggregate their information, and aim to elect a high-quality set of projects. We evaluate the performance of common PB rules by measuring the expected utility of their outcomes, compared to the optimal set of projects. We find that the quality of approximation improves as the range of project costs shrinks. When projects have unit cost, these common rules can identify the ``best'' set with probability converging to 1. We also study whether strategic agents have incentives to honestly convey their information in the vote. We find that it happens only under very restrictive conditions. We also run numerical experiments to examine the performance of different rules empirically and support our theoretical findings.

In strategic candidacy games, both voters and candidates have preferences regarding the election results, and candidates strategically decide whether to participate or abstain, in order to sway the outcome in their preferred direction. Often, it is assumed that candidates are keen to be elected or even just show up for visibility purposes, unless their presence is harmful. However, there may be candidates who find it costly to run an electoral campaign in the first place and therefore prefer to withdraw if their participation has no effect on the election outcome (a.k.a. lazy candidates), or those who would rather not win the election due to perceived interpersonal or reputational risks, unless they see no better alternative (a.k.a. reluctant leaders). In this work, we open the study of reluctance in strategic candidacy, and extend the standard candidacy game model to incorporate the concepts of reluctant participation (lazy candidacy) and reluctant leadership. We examine the pure-strategy Nash equilibria and the outcomes of natural iterative dynamic processes in such games, both from a normative and from a computational perspective, and compare them with their counterparts in the conventional model.

Shortlisting is the process of selecting a subset of alternatives from a larger pool for further consideration or final decision-making. It is widely applied in social choice and multi-agent system scenarios. The growing demand for participatory decision-making and the continuously expanding space of candidates create an urgent need for efficient and fair shortlisting procedures. However, little principled study has been done on this problem. This blue-sky paper aims to highlight the overlooked significance of shortlisting, distinguish it from related problems, provide initial thoughts, and, more importantly, serve as a call to arms. We envision that principled shortlisting can reduce cognitive burden, enable fair collective decisions, encourage broader participation, and ultimately build trust in democratic systems.

We study a temporal voting model where voters have dynamic preferences over a set of public chores-projects that benefit society, but impose individual costs on those affected by their implementation. We investigate the computational complexity of optimizing utilitarian and egalitarian welfare. Our results show that while optimizing the former is computationally straightforward, minimizing the latter is computationally intractable, even in very restricted cases. Nevertheless, we identify several settings where this problem can be solved efficiently, either exactly or by an approximation algorithm. We also examine the effects of enforcing temporal fairness and its impact on social welfare, and analyze the competitive ratio of online algorithms. We then explore the strategic behavior of agents, providing insights into potential malfeasance in such decision-making environments. Finally, we discuss a range of fairness measures and their suitability for our setting.

We consider the complexity of maximizing egalitarian welfare in Friends and Enemies Games – a subclass of hedonic games in which every agent partitions other agents into friends and enemies. We investigate two classic scenarios proposed in the literature, namely, Friends Appreciation (𝖥𝖠) and Enemies Aversion (𝖤𝖠): in the former, each agent primarily cares about the number of friends in her coalition, breaking ties based on the number of enemies, while in the latter, the opposite is true. For 𝖤𝖠, we show that our objective is hard to approximate within O(n^1-ε), for any fixed ε>0, and provide a polynomial-time (n-1)-approximation. For 𝖥𝖠, we obtain an NP-hardness result and a polynomial-time approximation algorithm. Our algorithm achieves a ratio of 2-Θ(1/n) when every agent has at least two friends; however, if some agent has at most one friend, its approximation ratio deteriorates to n/2. We recover the 2-Θ(1/n) approximation ratio for two important variants: when randomization is allowed and when the friendship relationship is symmetric. Additionally, for both 𝖤𝖠 and 𝖥𝖠 we identify special cases where the optimal egalitarian partition can be computed in polynomial time.

We study contests where the designer's objective is an extension of the widely studied objective of maximizing the total output: The designer gets zero marginal utility from a player's output if the output of the player is very low or very high. We consider two variants of this setting, which correspond to two objective functions: binary threshold, where the designer's utility is a non-decreasing function of the number of players with output above a certain threshold; and linear threshold, where a player's contribution to the designer's utility is linear in her output if the output is between a lower and an upper threshold, and becomes constant below the lower and above the upper threshold. For both of these objectives, we study rank-order allocation contests and general contests. We characterize the contests that maximize the designer's objective and indicate techniques to efficiently compute them.

We study a model of temporal voting where there is a fixed time horizon, and at each round the voters report their preferences over the available candidates and a single candidate is selected. Prior work has adapted popular notions of justified representation as well as voting rules that provide strong representation guarantees from the multiwinner election setting to this model. In our work, we focus on the complexity of verifying whether a given outcome offers proportional representation. We show that in the temporal setting verification is strictly harder than in multiwinner voting, but identify natural special cases that enable efficient algorithms.

We study stability in additively separable hedonic games when coalition sizes have to respect fixed size bounds. We consider four classic notions of stability based on single-agent deviations, namely, Nash stability, individual stability, contractual Nash stability, and contractual individual stability. For each stability notion, we consider two variants: in one, the coalition left behind by a deviator must still be of a valid size, and in the other there is no such constraint. We provide a full picture of the existence of stable outcomes with respect to given size parameters. Additionally, when there are only upper bounds, we fully characterize the computational complexity of the associated existence problem. In particular, we obtain polynomial-time algorithms for contractual individual stability and contractual Nash stability, where the latter requires an upper bound of 2. We obtain further results for Nash stability and contractual individual stability, when the lower bound is at least 2.

Polarization is a major concern for a well-functioning society. Often, mass polarization of a society is driven by polarizing political representation, even when the latter is easily preventable. The existing computational social choice methods for the task of committee selection are not designed to address this issue. We enrich the standard approach to committee selection by defining two quantitative measures that evaluate how well a given committee interconnects the voters. Maximizing these measures aims at avoiding polarizing committees. While the corresponding maximization problems are NP-complete in general, we obtain efficient algorithms for profiles in the voter-candidate interval domain. Moreover, we analyze the compatibility of our goals with other representation objectives, such as excellence, diversity, and proportionality. We identify trade-offs between approximation guarantees, and describe algorithms that achieve simultaneous constant-factor approximations.

We study two axioms for social choice functions that capture the impact of similar candidates: independence of clones (IoC) and composition consistency (CC). We clarify the relationship between these axioms by observing that CC is strictly more demanding than IoC, and investigate whether common voting rules that are known to be independent of clones (such as STV, Ranked Pairs, Schulze, and Split Cycle) are composition-consistent. While for most of these rules the answer is negative, we identify a variant of Ranked Pairs that satisfies CC. Further, we show how to efficiently modify any (neutral) social choice function so that it satisfies CC, while maintaining its other desirable properties. Our transformation relies on the hierarchical representation of clone structures via PQ-trees. We extend our analysis to social preference functions. Finally, we interpret IoC and CC as measures of robustness against strategic manipulation by candidates, with IoC corresponding to strategy-proofness and CC corresponding to obvious strategy-proofness.

Participatory budgeting (PB) is a form of citizen participation that allows citizens to decide how public funds are spent. Through an election, citizens express their preferences on various projects (spending proposals). A voting mechanism then determines which projects will be approved. The Method of Equal Shares (MES) is the state of the art algorithm for a proportional, voting based approach to participatory budgeting and has been implemented in cities across Poland and Switzerland. A significant drawback of MES is that it is not exhaustive meaning that it often leaves a portion of the budget unspent that could be used to fund additional projects. To address this, in practice the algorithm is combined with a completion heuristic - most often the “add-one" heuristic which artificially increases the budget until a heuristically chosen threshold. This heuristic is computationally inefficient and will become computationally impractical if PB is employed on a larger scale. We propose the more efficient add-opt heuristic for Exact Equal Shares (EES), a variation of MES that is known to retain many of its desirable properties. We solve the problem of identifying the next budget for which the outcome for EES changes in O(mn) time for cardinal utilities and O(m^2n) time for uniform utilities, where m is the number of projects and n is the number of voters. Our solution to this problem inspires the efficient add-opt heuristic which bypasses the need to search through each intermediary budget. We perform comprehensive experiments on real-word PB instances from Pabulib and show that completed EES outcomes usually match the proportion of budget spent by completed MES outcomes. Furthermore, the add-opt heuristic matches the proportion of budget spend by add-one for EES.

In multiwinner approval elections with many candidates, voters may struggle to determine their preferences over the entire slate of candidates. It is therefore of interest to explore which (if any) fairness guarantees can be provided under reduced communication. In this paper, we consider voters with one-dimensional preferences: voters and candidates are associated with points in $\mathbb R$, and each voter's approval set forms an interval of $\mathbb R$. We put forward a probabilistic preference model, where the voter set consists of $σ$ different groups; each group is associated with a distribution over an interval of $\mathbb R$, so that each voter draws the endpoints of her approval interval from the distribution associated with her group. We present an algorithm for computing committees that provide Proportional Justified Representation + (PJR+), which proceeds by querying voters' preferences, and show that, in expectation, it makes $\mathcal{O}(\log( σ\cdot k))$ queries per voter, where $k$ is the desired committee size.

We study proportional representation in the framework of temporal voting with approval ballots. Prior work adapted basic proportional representation concepts – justified representation (JR), proportional JR (PJR), and extended JR (EJR) – from the multiwinner setting to the temporal setting. Our work introduces and examines ways of going beyond EJR. Specifically, we consider stronger variants of JR, PJR, and EJR, and introduce temporal adaptations of more demanding multiwinner axioms, such as EJR+, full JR (FJR), full proportional JR (FPJR), and the Core. For each of these concepts, we investigate its existence and study its relationship to existing notions, thereby establishing a rich hierarchy of proportionality concepts. Notably, we show that two of our proposed axioms – EJR+ and FJR – strengthen EJR while remaining satisfiable in every temporal election.