Designing and analyzing voting rules for Participatory Budgeting (PB) elections is an active research area in computational social choice. Many PB voting rules aim to optimize a specific objective. For instance, the ubiquitous Greedy rule attempts to maximize utilitarian welfare, while the Method of Equal Shares (MES) aims to achieve proportional representation. However, it is often desirable to achieve good outcomes on multiple objectives rather than a close-to-perfect outcome for one. Inspired by mixed-member systems for parliamentary elections, we introduce mixed voting rules for PB. These are composed of a sequence of two or more rules that can each spend some fraction of the overall budget in order to add projects to the set selected by earlier rules. We develop a theoretical framework for formulating and analyzing mixed PB voting rules, and explore how existing rules can be adapted to this framework. We particularly focus on MES and its potential to address imbalances in representation created by earlier rules. We propose different ways to adjust MES voter budgets based on how satisfied voters are with previously chosen projects, and examine how well the resulting rules approximate well-known proportionality axioms such as EJR+. In particular, we show that one of these methods improves upon a natural proportionality baseline. We also extend our main positive result to general additive satisfaction functions. We complement our theoretical results with an extensive empirical analysis of real-world PB elections. Our experiments show that mixed rules can achieve favorable trade-offs between utilitarian welfare and proportionality. We identify several refinements that further improve their performance, and apply our framework to PB rules beyond Greedy and MES.

We introduce River, a novel Condorcet-consistent voting method that is based on pairwise majority margins and can be seen as a simplified variation of Tideman's Ranked Pairs method. River is simple to explain, simple to compute even "by hand," and gives rise to an easy-to-interpret certificate in the form of a directed tree. Like Ranked Pairs and Schulze's Beat Path method, River is a refinement of the Split Cycle method and shares with those many desirable properties, including independence of clones. Unlike the other three methods, River satisfies a strong form of resistance to agenda-manipulation that is known as independence of Pareto-dominated alternatives.

A common criticism of liquid democracy within the relevant academic literature is that delegation cycles can occur, seemingly resulting in unused voting power. Yet, practitioners argue that delegation cycles are not only unproblematic but are even formed intentionally by participants. To bring theory closer to reality, we introduce a model that captures this strategic behavior under uncertainty. We study the existence, structure and quality of Nash equilibria, revealing that delegation cycles naturally emerge. To complement these findings, we perform computational experiments using best-response dynamics.

In recent years, research in Participatory Budgeting (PB) has put a greater emphasis on rules satisfying notions of fairness and proportionality, with the Method of Equal Shares (MES) being a prominent example. However, proportionality can come at a cost to the total utilitarian welfare. Our work formalizes this relationship, by deriving minimum utilitarian welfare guarantees for MES for a subclass of satisfaction functions called DNS functions, which includes two of the most popular ways of measuring a voter's utility in the PB setting: considering (1) the total cost of approved projects or (2) the total number of those projects. Our results are parameterized in terms of minimum and maximum project costs, which allows us to improve on the mostly negative results found in prior studies, and reduce to the existing multiwinner guarantee when project costs are equal. We show that our guarantees are asymptotically tight for rules satisfying Extended Justified Representation up to one project, showing that no proportional rule can achieve a better utilitarian guarantee than MES.

Proportional representation plays a crucial role in electoral systems. In ordinal elections, where voters rank candidates based on their preferences, the Single Transferable Vote (STV) is the most widely used proportional voting method. STV is considered proportional because it satisfies an axiom requiring that large enough "solid coalitions" of voters are adequately represented. Using real-world data from local Scottish elections, we observe that solid coalitions of the required size rarely occur in practice. This observation challenges the importance of proportionality axioms and raises the question of how the proportionality of voting methods can be assessed beyond their axiomatic performance. We address these concerns by developing quantitative measures of proportionality. We apply these measures to evaluate the proportionality of voting rules on real-world election data. Besides STV, we consider SNTV, the Expanding Approvals Rule, and Sequential Ranked-Choice Voting. We also study the effects of ballot truncation by artificially completing truncated ballots and comparing the proportionality of outcomes under complete and truncated ballots.

Proportional representation plays a crucial role in electoral systems. In ordinal elections, where voters rank candidates based on their preferences, the Single Transferable Vote (STV) is the most widely used proportional voting method. STV is considered proportional because it satisfies an axiom requiring that large enough solid coalitions of voters are adequately represented. Using real-world data from local Scottish elections, we observe that solid coalitions of the required size rarely occur in practice. This observation challenges the importance of proportionality axioms and raises the question of how the proportionality of voting methods can be assessed beyond their axiomatic performance. We address these concerns by developing quantitative measures of proportionality. We apply these measures to evaluate the proportionality of voting rules on real-world election data. Besides STV, we consider SNTV, the Expanding Approvals Rule, and Sequential Ranked-Choice Voting. We also study the effects of ballot truncation by artificially completing truncated ballots and comparing the proportionality of outcomes under complete and truncated ballots.

We introduce River, a novel Condorcet-consistent voting method that is based on pairwise majority margins and can be seen as a simplified variation of Tideman's Ranked Pairs method. River is simple to explain, simple to compute even 'by hand', and gives rise to an easy-to-interpret certificate in the form of a directed tree. Like Ranked Pairs and Schulze's Beat Path method, River is a refinement of the Split Cycle method and shares with those many desirable properties, including independence of clones. Unlike the other three methods, River satisfies a strong form of resistance to agenda-manipulation that is known as independence of Pareto-dominated alternatives.

When selecting committees based on preferences of voters, a variety of different criteria can be considered. Two natural objectives are maximizing the utilitarian welfare (the sum of voters' utilities) and coverage (the number of represented voters) of the selected committee. Previous work has studied the impact on utilitarian welfare and coverage when requiring the committee to satisfy minimal requirements such as justified representation or weak proportionality. In this paper, we consider the impact of imposing much more demanding proportionality axioms. We identify a class of voting rules that achieve strong guarantees on utilitarian welfare and coverage when combined with appropriate completions. This class is defined via a weakening of priceability and contains prominent rules such as the Method of Equal Shares. We show that committees selected by these rules (i) can be completed to achieve optimal coverage and (ii) can be completed to achieve an asymptotically optimal approximation to the utilitarian welfare if they additionally satisfy EJR+. Answering an open question of Elkind et al. (2022), we use the Greedy Justified Candidate Rule to obtain the best possible utilitarian guarantee subject to proportionality. We also consider completion methods suggested in the participatory budgeting literature and other objectives besides welfare and coverage.

We provide the first large-scale data collection of real-world approval-based committee elections. These elections have been conducted on the Polkadot blockchain as part of their Nominated Proof-of-Stake mechanism and contain around one thousand candidates and tens of thousands of (weighted) voters each. We conduct an in-depth study of application-relevant questions, including a quantitative and qualitative analysis of the outcomes returned by different voting rules. Besides considering proportionality measures that are standard in the multiwinner voting literature, we pay particular attention to less-studied measures of overrepresentation, as these are closely related to the security of the Polkadot network. We also analyze how different design decisions such as the committee size affect the examined measures.

When selecting multiple candidates based on approval preferences of agents, the proportional representation of agents' opinions is an important and well-studied desideratum. Existing criteria for evaluating the representativeness of outcomes focus on groups of agents and demand that sufficiently large and cohesive groups are "represented" in the sense that candidates approved by some group members are selected. Crucially, these criteria say nothing about the representation of individual agents, even if these agents are members of groups that deserve representation. In this paper, we formalize the concept of individual representation (IR) and explore to which extent, and under which circumstances, it can be achieved. We show that checking whether an IR outcome exists is computationally intractable, and we verify that all common approval-based voting rules may fail to provide IR even in cases where this is possible. We then focus on domain restrictions and establish an interesting contrast between "voter interval" and "candidate interval" preferences. This contrast can also be observed in our experimental results, where we analyze the attainability of IR for realistic preference profiles.

As the world's democratic institutions are challenged by dissatisfied citizens, political scientists and also computer scientists have proposed and analyzed various (innovative) methods to select representative bodies, a crucial task in every democracy. However, a unified framework to analyze and compare different selection mechanisms is missing, resulting in very few comparative works. To address this gap, we advocate employing concepts and tools from computational social choice in order to devise a model in which different selection mechanisms can be formalized. Such a model would allow for desirable representation axioms to be conceptualized and evaluated. We make the first step in this direction by proposing a unifying mathematical formulation of different selection mechanisms as well as various social-choice-inspired axioms such as proportionality and monotonicity.

We consider spatial voting where candidates are located in the Euclidean d-dimensional space, and each voter ranks candidates based on their distance from the voter's ideal point. We explore the case where information about the location of voters' ideal points is incomplete: for each dimension, we are given an interval of possible values. We study the computational complexity of finding the possible and necessary winners for positional scoring rules. Our results show that we retain tractable cases of the classic model where voters have partial-order preferences. Moreover, we show that there are positional scoring rules under which the possible-winner problem is intractable for partial orders, but tractable in the one-dimensional spatial setting. We also consider approval voting in this setting. We show that for up to two dimensions, the necessary-winner problem is tractable, while the possible-winner problem is hard for any number of dimensions.

This white paper outlines a long-term scientific vision for the development of digital-democracy technology. We contend that if digital democracy is to meet the ambition of enabling a participatory renewal in our societies, then a comprehensive multi-methods research effort is required that could, over the years, support its development in a democratically principled, empirically and computationally informed way. The paper is co-authored by an international and interdisciplinary team of researchers and arose from the Lorentz Center Workshop on “Algorithmic Technology for Democracy” (Leiden, October 2022).

When selecting a subset of candidates (a so-called committee) based on the preferences of voters, proportional representation is often a major desideratum. When going beyond simplistic models such as party-list or district-based elections, it is surprisingly challenging to capture proportionality formally. As a consequence, the literature has produced numerous competing criteria of when a selected committee qualifies as proportional. Two of the most prominent notions are Dummett's proportionality for solid coalitions (PSC) and Aziz et al.'s extended justified representation (EJR). Both guarantee proportional representation to groups of voters who have very similar preferences; such groups are referred to as solid coalitions by Dummett and as cohesive groups by Aziz et al. However, these notions lose their bite when groups are only almost solid or almost cohesive. In this paper, we propose proportionality axioms that are more robust: they guarantee representation also to groups that do not qualify as solid or cohesive. Further, our novel axioms can be easily verified: Given a committee, we can check in polynomial time whether it satisfies the axiom or not. This is in contrast to many established notions like EJR, for which the corresponding verification problem is known to be intractable. In the setting with approval preferences, we propose a robust and verifiable variant of EJR and a simply greedy procedure to compute committees satisfying it. In the setting with ranked preferences, we propose a robust variant PSC, which can be efficiently verified even for general weak preferences. In the special case of strict preferences, our notion is the first known satisfiable proportionality axiom that is violated by the Single Transferable Vote (STV). We also discuss implications of our results for participatory budgeting, querying procedures, and to the notion of proportionality degree.

Given a set of agents with approval preferences over each other, we study the task of finding k matchings fairly representing everyone’s preferences. To formalize fairness, we apply the concept of proportional representation as studied in approval-based multiwinner elections. To this end, we model the problem as a multiwinner election where the set of candidates consists of matchings of the agents, and agents’ preferences over each other are lifted to preferences over matchings. Due to the exponential number of candidates in such elections, standard algorithms for classical sequential voting rules (such as those proposed by Thiele and Phragmén) are rendered inefficient. We show that the computational tractability of these rules can be regained by exploiting the structure of the approval preferences. Moreover, we establish algorithmic results and axiomatic guarantees that go beyond those obtainable in the classical approval-based multiwinner setting: Assuming that approvals are symmetric, we show that Proportional Approval Voting (PAV), a well-established but computationally intractable voting rule, becomes polynomial-time computable, and that its sequential variant, which does not provide any proportionality guarantees in general, fulfills a rather strong guarantee known as extended justified representation. Some of our algorithmic results extend to other types of compactly representable elections with an exponential candidate space.

Selecting a committee that meets diversity and proportionality criteria is a challenging endeavor that has been studied extensively in recent years. This task becomes even more challenging when some of the selected candidates decline the invitation to join the committee. Since the unavailability of one candidate may impact the rest of the selection, inviting all candidates at the same time may lead to a suboptimal committee. Instead, invitations should be sequential and conditional on which candidates invited so far accepted the invitation: the solution to the committee selection problem is a query policy. If invitation queries are binding, they should be safe: one should not query a candidate without being sure that whatever the set of available candidates possible at that stage, her inclusion will not jeopardize committee optimality. Assuming approval-based inputs, we characterize the set of rules for which a safe query exists at every stage. In order to parallelize the invitation process, we investigate the computation of safe parallel queries, and show that it is often hard. We also study the existence of safe parallel queries with respect to proportionality axioms such as extended justified representation.

We consider spatial voting where candidates are located in the Euclidean $d$-dimensional space and each voter ranks candidates based on their distance from the voter's ideal point. We explore the case where information about the location of voters' ideal points is incomplete: for each dimension, we are given an interval of possible values. We study the computational complexity of computing the possible and necessary winners for positional scoring rules. Our results show that we retain tractable cases of the classic model where voters have partial-order preferences. Moreover, we show that there are positional scoring rules under which the possible-winner problem is intractable for partial orders, but tractable in the one-dimension spatial setting (while intractable in higher fixed number of dimensions).

We consider a multi-issue election setting over a set of possibly interdependent issues with the goal of achieving proportional representation of the views of the electorate. To this end, we employ a proportionality criterion suggested recently in the literature, that guarantees fair representation for all groups of voters of sufficient size. For this criterion, there exist rules that perform well in the case where all the issues have a binary domain and are independent of each other. In particular, this has been shown for Proportional Approval Voting (PAV) and for the Method of Equal Shares (MES). In this paper, we go two steps further: we generalize these guarantees for issues with a non-binary domain, and, most importantly, we consider extensions to elections with dependencies among issues, where we identify restrictions that lead to analogous results. To achieve this, we define appropriate generalizations of PAV and MES to handle conditional ballots. In addition to proportionality considerations, we also examine the computational properties of the conditional version of MES. Our findings indicate that the conditional case poses additional challenges and differs significantly from the unconditional one, both in terms of proportionality guarantees and computational complexity.

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.

The ability to measure the satisfaction of (groups of) voters is a crucial prerequisite for formulating proportionality axioms in approval-based participatory budgeting elections. Two common -- but very different -- ways to measure the satisfaction of a voter consider (i) the number of approved projects and (ii) the total cost of approved projects, respectively. In general, it is difficult to decide which measure of satisfaction best reflects the voters' true utilities. In this paper, we study proportionality axioms with respect to large classes of approval-based satisfaction functions. We establish logical implications among our axioms and related notions from the literature, and we ask whether outcomes can be achieved that are proportional with respect to more than one satisfaction function. We show that this is impossible for the two commonly used satisfaction functions when considering proportionality notions based on extended justified representation, but achievable for a notion based on proportional justified representation. For the latter result, we introduce a strengthening of priceability and show that it is satisfied by several polynomial-time computable rules, including the Method of Equal Shares and Phragmén's sequential rule.