In approval-based budget division, the task is to allocate a divisible resource to the candidates based on the voters' approval preferences over the candidates. For this setting, Brandl et al. (2021) have shown that no distribution rule can be strategyproof, efficient, and fair at the same time. In this paper, we aim to circumvent this impossibility theorem by focusing on approximate strategyproofness. To this end, we analyze the incentive ratio of distribution rules, which quantifies the maximum multiplicative utility gain of a voter by manipulating. While it turns out that several classical rules have a large incentive ratio, we prove that the Nash product rule (NASH) has an incentive ratio of 2, thereby demonstrating that we can bypass the impossibility of Brandl et al. by relaxing strategyproofness. Moreover, we show that an incentive ratio of 2 is optimal within three natural classes of rules and that the positive result for the Nash product rule even holds when voters may report arbitrary concave utility functions. Finally, we complement our results with an experimental analysis.

In rank aggregation, the goal is to combine multiple input rankings into a single output ranking. In this paper, we analyze rank aggregation methods, so-called social welfare functions (SWFs), with respect to strategyproofness, which requires that no agent can misreport his ranking to obtain an output ranking that is closer to his true ranking in terms of the Kemeny distance. As our main result, we show that no anonymous SWF satisfies unanimity and strategyproofness when there are at least four alternatives. This result is proven by SAT solving, a computer-aided theorem proving technique, and verified by Isabelle, a highly trustworthy interactive proof assistant. Further, we prove by hand that strategyproofness is incompatible with majority consistency, a variant of Condorcet-consistency for SWFs. Lastly, we show that all SWFs in two natural classes have a large incentive ratio and are thus highly manipulable.

Social decision schemes (SDSs) map the voters’ preferences over multiple alternatives to a probability distribution over these alternatives. In a seminal result, Gibbard (Econometrica 45(3):665–681, 1977) has characterized the set of SDSs that are strategyproof with respect to all utility functions and his result implies that all such SDSs are either unfair to the voters or alternatives, or they require a significant amount of randomization. To circumvent this negative result, we propose the notion of U-strategyproofness which postulates that only voters with a utility function in a predefined set U cannot manipulate. We then analyze the tradeoff between U-strategyproofness and various decisiveness notions that restrict the amount of randomization of SDSs. In particular, we show that if the utility functions in the set U value the best alternative much more than other alternatives, there are U-strategyproof SDSs that choose an alternative with probability 1 whenever all but k voters rank it first. On the negative side, we demonstrate that U-strategyproofness is incompatible with Condorcet-consistency if the set U satisfies minimal symmetry conditions. Finally, we show that no ex post efficient and U-strategyproof SDS can be significantly more decisive than the uniform random dictatorship if the voters are close to indifferent between their two favorite alternatives.

A central problem in multiagent systems is the fair assignment of objects to agents. In this paper, we initiate the analysis of classic majoritarian social choice functions in assignment. Exploiting the special structure of the assignment domain, we show a number of surprising results with no counterparts in general social choice. In particular, we establish a near one-to-one correspondence between preference profiles and majority graphs. This correspondence implies that key properties of assignments – such as Pareto-optimality, least unpopularity, and mixed popularity – can be determined solely by the associated majority graph. We further show that all Pareto-optimal assignments are semi-popular and belong to the top cycle. Elements of the top cycle can thus easily be found via serial dictatorships. Our main result is a complete characterization of the top cycle, which implies the top cycle can only consist of one, two, all but two, all but one, or all assignments. By contrast, we find that the uncovered set contains only very few assignments.

In rank aggregation, the task is to aggregate multiple weighted input rankings into a single output ranking. While numerous methods, so-called social welfare functions (SWFs), have been suggested for this problem, all of the classical SWFs tend to be majoritarian and are thus not acceptable when a proportional ranking is required. Motivated by this observation, we will design SWFs that guarantee that every input ranking is proportionally represented by the output ranking. Specifically, our central fairness condition requires that the number of pairwise comparisons between candidates on which an input ranking and the output ranking agree is proportional to the weight of the input ranking. As our main contribution, we present a simple SWF called the Proportional Sequential Borda rule, which satisfies this condition. Moreover, we introduce two variants of this rule: the Ranked Method of Equal Shares, which has a more utilitarian flavor while still satisfying our fairness condition, and the Flow-adjusting Borda rule, which satisfies an even stronger fairness condition. Many of our axioms and techniques are inspired by results on approval-based committee voting and participatory budgeting, where the concept of proportional representation has been studied in depth.

In approval-based budget division, a budget needs to be distributed to some candidates based on the voters' approval ballots over these candidates. In the pursuit of a simple, consistent, and approximately fair rule for this setting, we introduce the maximum payment rule (MP). Under this rule, each voter controls a part of the budget and, in each step, the corresponding voters allocate their entire budget to the candidate approved by the largest number of voters with nonzero budget. We show that MP meets our criteria as it satisfies monotonicity and a demanding population consistency condition and gives a 2-approximation to a fairness notion called average fair share (AFS). Moreover, we generalize MP to the class of sequential payment rules and prove that it is the most desirable rule within this class: nearly all other sequential payment rules fail monotonicity while offering only small improvements in the approximation ratio to AFS.

The metric distortion of a randomized social choice function (RSCF) quantifies its worst-case approximation ratio to the optimal social cost when the voters' costs for alternatives are given by distances in a metric space. This notion has recently attracted significant attention as numerous RSCFs that aim to minimize the metric distortion have been suggested. Since such tailored voting rules have, however, little normative appeal other than their low metric distortion, we will study the metric distortion of well-established RSCFs. Specifically, we first show that C1 maximal lottery rules, a well-known class of RSCFs, have a metric distortion of 4, which is optimal within the class of majoritarian RSCFs. Secondly, we conduct extensive computer experiments on the metric distortion of RSCFs to obtain insights into their average-case performance. These computer experiments are based on a new linear program for computing the metric distortion of a lottery and reveal that the average-case metric distortion of some classical RSCFs is often only slightly worse than that of RSCFs tailored to minimize the metric distortion. Finally, we also analytically study the expected metric distortion of RSCFs for the impartial culture distribution. Specifically, we show that, under this distribution, every reasonable RSCF has an expected metric distortion close to 2 when the number of voters is large.

An important - but very demanding - property in collective decision-making is strategyproofness, which requires that voters cannot benefit from submitting insincere preferences. Gibbard (1977) has shown that only rather unattractive rules are strategyproof, even when allowing for randomization. However, Gibbard's theorem is based on a rather strong interpretation of strategyproofness, which deems a manipulation successful if it increases the voter's expected utility for at least one utility function consistent with his ordinal preferences. In this paper, we study weak strategyproofness, which deems a manipulation successful if it increases the voter's expected utility for all utility functions consistent with his ordinal preferences. We show how to systematically design attractive, weakly strategyproof social decision schemes (SDSs) and explore their limitations for both strict and weak preferences. In particular, for strict preferences, we show that there are weakly strategyproof SDSs that are either ex post efficient or Condorcet-consistent, while neither even-chance SDSs nor pairwise SDSs satisfy both properties and weak strategyproofness at the same time. By contrast, for the case of weak preferences, we discuss two sweeping impossibility results that preclude the existence of appealing weakly strategyproof SDSs.

In this paper, we study voting rules on the interval domain, where the alternatives are arranged according to an externally given strict total order and voters report intervals of this order to indicate the alternatives they support. For this setting, we introduce and characterize the class of position-threshold rules, which compute a collective position of the voters with respect to every alternative and choose the left-most alternative whose collective position exceeds its threshold value. Our characterization of these rules mainly relies on reinforcement, a well-known population consistency condition, and robustness, a new axiom that restricts how the outcome is allowed to change when a voter removes the left-most or right-most alternative from his interval. Moreover, we characterize a generalization of the median rule to the interval domain, which selects the median of the endpoints of the voters' intervals.

We introduce and analyze distance preservation games (DPGs). In DPGs, agents express ideal distances to other agents and need to choose locations in the unit interval while preserving their ideal distances as closely as possible. We analyze the existence and computation of location profiles that are jump stable (i.e., no agent can benefit by moving to another location) or welfare optimal for DPGs, respectively. Specifically, we prove that there are DPGs without jump stable location profiles and identify important cases where such outcomes always exist and can be computed efficiently. Similarly, we show that finding welfare optimal location profiles is NP-complete and present approximation algorithms for finding solutions with social welfare close to optimal. Finally, we prove that DPGs have a price of anarchy of at most 2.

We study committee voting rules under ranked preferences, which map the voters' preference relations to a subset of the alternatives of predefined size. In this setting, the compatibility between proportional representation and committee monotonicity is a fundamental open problem that has been mentioned in several works. We address this research question by designing a new committee voting rule called the Solid Coalition Refinement (SCR) rule that simultaneously satisfies committee monotonicity and Dummett's PSC as well as one of its variants called inclusion PSC. This is the first rule known to satisfy both of these properties. Moreover, we show that this is effectively the best that we can hope for as other fairness notions adapted from approval voting are incompatible with committee monotonicity. Finally, we prove that, for truncated preferences, the SCR rule still satisfies PSC and a property called independence of losing voter blocs, thereby refuting a conjecture of Graham-Squire et al. (2024).

In approval-based committee (ABC) elections, the goal is to select a fixed-size subset of the candidates, a so-called committee, based on the voters' approval ballots over the candidates. One of the most popular classes of ABC voting rules are ABC scoring rules, for which voters give points to each committee and the committees with maximal total points are chosen. While the set of ABC scoring rules has recently been characterized in a model where the output is a ranking of all committees, no full characterization of these rules exists in the standard model where a set of winning committees is returned. We address this issue by characterizing two important subclasses of ABC scoring rules in the standard ABC election model, thereby both extending the result for ABC ranking rules to the standard setting and refining it to subclasses. In more detail, by relying on a consistency axiom for variable electorates, we characterize (i) the prominent class of Thiele rules and (ii) a new class of ABC voting rules called ballot size weighted approval voting. Based on these theorems, we also infer characterizations of three well-known ABC voting rules, namely multi-winner approval voting, proportional approval voting, and satisfaction approval voting.

Social decision schemes (SDSs) map the preferences of a group of voters over some set of m alternatives to a probability distribution over the alternatives. A seminal characterization of strategyproof SDSs by Gibbard implies that there are no strategyproof Condorcet extensions and that only random dictatorships satisfy ex post efficiency and strategyproofness. The latter is known as the random dictatorship theorem. We relax Condorcet-consistency and ex post efficiency by introducing a lower bound on the probability of Condorcet winners and an upper bound on the probability of Pareto-dominated alternatives, respectively. We then show that the SDS that assigns probabilities proportional to Copeland scores is the only anonymous, neutral, and strategyproof SDS that can guarantee the Condorcet winner a probability of at least 2/m. Moreover, no strategyproof SDS can exceed this bound, even when dropping anonymity and neutrality. Secondly, we prove a continuous strengthening of Gibbard's random dictatorship theorem: the less probability we put on Pareto-dominated alternatives, the closer to a random dictatorship is the resulting SDS. Finally, we show that the only anonymous, neutral, and strategyproof SDSs that maximize the probability of Condorcet winners while minimizing the probability of Pareto-dominated alternatives are mixtures of the uniform random dictatorship and the randomized Copeland rule.

This paper studies social preference functions (SPFs), which map the voters' ordinal preferences over a set of alternatives to a non-empty set of strict rankings over the alternatives. Maybe the most prominent SPFs are positional scoring rules and Kemeny's rule. While these two types of rules behave intuitively quite differently, they are axiomatically surprisingly similar and we thus provide a joint axiomatic characterization of these SPFs. To this end, we introduce the class of bivariate scoring rules which generalize Kemeny's rule by weighting comparisons between alternatives depending on their positions in the voters' preference relations. In particular, this class contains both Kemeny's rule and all positional scoring rules. As our main result, we then characterize the set of bivariate scoring rules as the only SPFs that satisfy—aside from some standard axioms—mild consistency conditions for variable electorates and variable agendas. As corollaries of this result, we also derive variants of the well-known characterizations of positional scoring rules by Smith (1973) and of Kemeny's rule by Young (1988). Thus, our result unifies the independent streams of research on positional scoring rules and Kemeny's rule by giving a joint axiomatic basis of these SPFs.

For the problem of aggregating several rankings into one ranking, Kemeny (1959) proposed two methods: the median rule which selects the ranking with the smallest total swap distance to the input rankings, and the mean rule which minimizes the squared swap distances to the input rankings. The median rule has been extensively studied since and is now known simply as Kemeny's rule. It exhibits majoritarian properties, so for example if more than half of the input rankings are the same, then the output of the rule is the same ranking. We observe that this behavior is undesirable in many rank aggregation settings. For example, when we rank objects by different criteria (quality, price, etc.) and want to aggregate them with specified weights for the criteria, then a criterion with weight 51% should have 51% influence on the output instead of 100%. We show that the Squared Kemeny rule (i.e., the mean rule) behaves this way, by establishing a bound on the distance of the output ranking to any input rankings, as a function of their weights. Furthermore, we give an axiomatic characterization of the Squared Kemeny rule, which mirrors the existing characterization of the Kemeny rule but replaces the majoritarian Condorcet axiom by a proportionality axiom. Finally, we discuss the computation of the rule and show its behavior in a simulation study.

In approval-based committee (ABC) voting, the goal is to choose a subset of predefined size of the candidates based on the voters' approval preferences over the candidates. While this problem has attracted significant attention in recent years, the incentives for voters to participate in an election for a given ABC voting rule have been neglected so far. This paper is thus the first to explicitly study this property, typically called participation, for ABC voting rules. In particular, we show that all ABC scoring rules even satisfy group participation, whereas most sequential rules severely fail participation. We furthermore explore several escape routes to the impossibility for sequential ABC voting rules: we prove for many sequential rules that (i) they satisfy participation on laminar profiles, (ii) voters who approve none of the elected candidates cannot benefit by abstaining, and (iii) it is NP-hard for a voter to decide whether she benefits from abstaining.

Gibbard and Satterthwaite have shown that the only single-valued social choice functions (SCFs) that satisfy non-imposition (i.e., the function's range coincides with its codomain) and strategyproofness (i.e., voters are never better off by misrepresenting their preferences) are dictatorships. In this paper, we consider set-valued social choice correspondences (SCCs) that are strategyproof according to Fishburn's preference extension and, in particular, the top cycle, an attractive SCC that returns the maximal elements of the transitive closure of the weak majority relation. Our main theorem implies that, under mild conditions, the top cycle is the only non-imposing strategyproof SCC whose outcome only depends on the quantified pairwise comparisons between alternatives. This result effectively turns the Gibbard-Satterthwaite impossibility into a complete characterization of the top cycle by moving from SCFs to SCCs. It is obtained as a corollary of a more general characterization of strategyproof SCCs.

Approval-based committee (ABC) voting rules elect a fixed size subset of the candidates, a so-called committee, based on the voters' approval ballots over the candidates. While these rules have recently attracted significant attention, axiomatic characterizations are largely missing so far. We address this problem by characterizing ABC voting rules within the broad and intuitive class of sequential valuation rules. These rules compute the winning committees by sequentially adding candidates that increase the score of the chosen committee the most. In more detail, we first characterize almost the full class of sequential valuation rules based on mild standard conditions and a new axiom called consistent committee monotonicity. This axiom postulates that the winning committees of size k can be derived from those of size k-1 by only adding candidates and that these new candidates are chosen consistently. By requiring additional conditions, we derive from this result also a characterization of the prominent class of sequential Thiele rules. Finally, we refine our results to characterize three well-known ABC voting rules, namely sequential approval voting, sequential proportional approval voting, and sequential Chamberlin-Courant approval voting.

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.

One of the central economic paradigms in multi-agent systems is that agents should not be better off by acting dishonestly. In the context of collective decision-making, this axiom is known as strategyproofness and turns out to be rather prohibitive, even when allowing for randomization. In particular, Gibbard's random dictatorship theorem shows that only rather unattractive social decision schemes (SDSs) satisfy strategyproofness on the full domain of preferences. In this paper, we obtain more positive results by investigating strategyproof SDSs on the Condorcet domain, which consists of all preference profiles that admit a Condorcet winner. In more detail, we show that, if the number of voters $n$ is odd, every strategyproof and non-imposing SDS on the Condorcet domain can be represented as a mixture of dictatorial SDSs and the Condorcet rule (which chooses the Condorcet winner with probability $1$). Moreover, we prove that the Condorcet domain is a maximal connected domain that allows for attractive strategyproof SDSs if $n$ is odd as only random dictatorships are strategyproof and non-imposing on any sufficiently connected superset of it. We also derive analogous results for even $n$ by slightly extending the Condorcet domain. Finally, we also characterize the set of group-strategyproof and non-imposing SDSs on the Condorcet domain and its supersets. These characterizations strengthen Gibbard's random dictatorship theorem and establish that the Condorcet domain is essentially a maximal domain that allows for attractive strategyproof SDSs.