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 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 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 Single Transferable Vote, Ranked Pairs, Schulze Method, 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.

Abstract. We consider the classical cake cutting problem where we wish to fairly divide a heterogeneous resource among interested agents. Work on this subject typically assumes that the cake is represented by an interval. We introduce a generalized setting where the cake is represented by an arbitrary undirected graph, which allows us to model the division of road networks. Unlike in the interval setting, common fairness criteria such as proportionality cannot always be satisfied in graphical cake cutting if each agent must receive a connected subgraph. We determine the optimal approximation of proportionality that can be obtained for any number of agents with additive valuations, and exhibit a tight guarantee for each graph in the case of two agents. We also study several variants and extensions, including when more than one connected piece per agent is allowed as well as when the item to be divided is undesirable.

Multiwinner voting captures a wide variety of settings, from parliamentary elections in democratic systems to product placement in online shopping platforms. There is a large body of work dealing with axiomatic characterizations, computational complexity, and algorithmic analysis of multiwinner voting rules. Although many challenges remain, significant progress has been made in showing existence of fair and representative outcomes as well as efficient algorithmic solutions for many commonly studied settings. However, much of this work focuses on single-shot elections, even though in numerous real-world settings elections are held periodically and repeatedly. Hence, it is imperative to extend the study of multiwinner voting to temporal settings. Recently, there have been several efforts to address this challenge. However, these works are difficult to compare, as they model multi-period voting in very different ways. We propose a unified framework for studying temporal fairness in this domain, drawing connections with various existing bodies of work, and consolidating them within a general framework. We also identify gaps in existing literature, outline multiple opportunities for future work, and put forward a vision for the future of multiwinner voting in temporal settings.

We consider binary group decision-making under a rich model of liquid democracy: agents submit ranked delegation options, where each option may be a function of multiple agents' votes; e.g., "I vote yes if a majority of my friends vote yes." Such ballots are unravelled into a profile of direct votes by selecting one entry from each ballot so as not to introduce cyclic dependencies. We study delegation via monotonic Boolean functions, and two unravelling procedures: MinSum, which minimises the sum of the ranks of the chosen entries, and its egalitarian counterpart, MinMax. We provide complete computational dichotomies: MinSum is hard to compute (and approximate) as soon as any non-trivial functions are permitted, and polynomial otherwise; for MinMax the easiness results extend to arbitrary-arity logical ORs and ANDs taken in isolation, but not beyond. For the classic model of delegating to individual agents, we give asymptotically near-tight algorithms for carrying out the two procedures and efficient algorithms for finding optimal unravellings with the highest vote count for a given alternative. These algorithms inspire novel tie-breaking rules for the setup of voting to change a status quo. We then introduce a new axiom, which can be viewed as a variant of the participation axiom, and use algorithmic techniques developed earlier in the paper to show that it is satisfied by MinSum and a lexicographic refinement of MinMax (but not MinMax itself).

We investigate a model of sequential decision-making where a single alternative is chosen at each round. We focus on two objectives—utilitarian welfare (UTIL) and egalitarian welfare (EGAL)—and consider the computational complexity of the associated maximization problems, as well as their compatibility with strategyproofness and proportionality. We observe that maximizing UTIL is easy, but the corresponding decision problem for EGAL is NP-complete even in restricted cases. We complement this hardness result for EGAL with parameterized complexity analysis and an approximation algorithm. Additionally, we show that, while a mechanism that outputs a UTIL outcome is strategyproof, all deterministic mechanisms for computing EGAL outcomes fail a very weak variant of strategyproofness, called non-obvious manipulability (NOM). However, we show that when agents have non-empty approval sets at each timestep, choosing an EGAL-maximizing outcome while breaking ties lexicographically satisfies NOM. Regarding proportionality, we prove that a proportional (PROP) outcome can be computed efficiently, but finding an outcome that maximizes UTIL while guaranteeing PROP is NP-hard. We also derive upper and lower bounds on the price of proportionality with respect to UTIL and EGAL.

We study two-stage committee elections where voters have dynamic preferences over candidates; at each stage, a committee is chosen under a given voting rule. We are interested in identifying a winning committee for the second stage that overlaps as much as possible with the first-stage committee. We show a full complexity dichotomy for the class of Thiele rules: this problem is tractable for Approval Voting (AV) and hard for all other Thiele rules (including, in particular, Proportional Approval Voting and the Chamberlin–Courant rule). We extend this dichotomy to the greedy variants of Thiele rules. We also explore this problem from a parameterized complexity perspective for several natural parameters. We complement the theory with experimental analysis: e.g., we investigate the average number of changes in the committee as a function of changes in voters’ preferences and the role of ties.

Selecting $k$ out of $m$ items based on the preferences of $n$ heterogeneous agents is a widely studied problem in algorithmic game theory. If agents have approval preferences over individual items and harmonic utility functions over bundles -- an agent receives $\sum_{j=1}^t\frac{1}{j}$ utility if $t$ of her approved items are selected -- then welfare optimisation is captured by a voting rule known as Proportional Approval Voting (PAV). PAV also satisfies demanding fairness axioms. However, finding a winning set of items under PAV is NP-hard. In search of a tractable method with strong fairness guarantees, a bounded local search version of PAV was proposed by Aziz et al. It proceeds by starting with an arbitrary size-$k$ set $W$ and, at each step, checking if there is a pair of candidates $a\in W$, $b\not\in W$ such that swapping $a$ and $b$ increases the total welfare by at least $\varepsilon$; if yes, it performs the swap. Aziz et al.~show that setting $\varepsilon=\frac{n}{k^2}$ ensures both the desired fairness guarantees and polynomial running time. However, they leave it open whether the algorithm converges in polynomial time if $\varepsilon$ is very small (in particular, if we do not stop until there are no welfare-improving swaps). We resolve this open question, by showing that if $\varepsilon$ can be arbitrarily small, the running time of this algorithm may be super-polynomial. Specifically, we prove a lower bound of~$Ω(k^{\log k})$ if improvements are chosen lexicographically. To complement our lower bound, we provide an empirical comparison of two variants of local search -- better-response and best-response -- on several real-life data sets and a variety of synthetic data sets. Our experiments indicate that, empirically, better response exhibits faster running time than best response.

AI agents are commonly trained with large datasets of demonstrations of human behavior. However, not all behaviors are equally safe or desirable. Desired characteristics for an AI agent can be expressed by assigning desirability scores, which we assume are not assigned to individual behaviors but to collective trajectories. For example, in a dataset of vehicle interactions, these scores might relate to the number of incidents that occurred. We first assess the effect of each individual agent's behavior on the collective desirability score, e.g., assessing how likely an agent is to cause incidents. This allows us to selectively imitate agents with a positive effect, e.g., only imitating agents that are unlikely to cause incidents. To enable this, we propose the concept of an agent's \textit{Exchange Value}, which quantifies an individual agent's contribution to the collective desirability score. The Exchange Value is the expected change in desirability score when substituting the agent for a randomly selected agent. We propose additional methods for estimating Exchange Values from real-world datasets, enabling us to learn desired imitation policies that outperform relevant baselines. The project website can be found at https://tinyurl.com/select-to-perfect.

We study a portioning setting in which a public resource such as time or money is to be divided among a given set of candidates, and each agent proposes a division of the resource. We consider two families of aggregation rules for this setting—those based on coordinate-wise aggregation and those that optimize some notion of welfare—as well as the recently proposed Independent Markets mechanism. We provide a detailed analysis of these rules from an axiomatic perspective, both for classic axioms, such as strategyproofness and Pareto optimality, and for novel axioms, which aim to capture proportionality in this setting. Our results indicate that a simple rule that computes the average of all proposals satisfies many of our axioms, including some that are violated by more sophisticated rules.

This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of proportionality is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the ordinal maximin share approximation, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions.

Justified representation (JR) is a standard notion of representation in multiwinner approval voting. Not only does a JR committee always exist, but previous work has also shown through experiments that the JR condition can typically be fulfilled by groups of fewer than k candidates, where k is the target size of the committee. In this paper, we study such groups—known as n/k-justifying groups—both theoretically and empirically. First, we show that under the impartial culture model, n/k-justifying groups of size less than k/2 are likely to exist, which implies that the number of JR committees is usually large. We then present efficient approximation algorithms that compute a small n/k-justifying group for any given instance, and a polynomial-time exact algorithm when the instance admits a tree representation. In addition, we demonstrate that small n/k-justifying groups can often be useful for obtaining a gender-balanced JR committee even though the problem is NP-hard.

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.

Elkind et al. (AAAI'21) introduced a model for deliberative coalition formation, where a community wishes to identify a strongly supported proposal from a space of alternatives, in order to change the status quo. In their model, agents and proposals are points in a metric space, agents' preferences are determined by distances, and agents deliberate by dynamically forming coalitions around proposals that they prefer over the status quo. The deliberation process operates via k-compromise transitions, where agents from k (current) coalitions come together to form a larger coalition in order to support a (perhaps new) proposal, possibly leaving behind some of the dissenting agents from their old coalitions. A deliberation succeeds if it terminates by identifying a proposal with the largest possible support. For deliberation in d dimensions, Elkind et al. consider two variants of their model: in the Euclidean model, proposals and agent locations are points in R^d and the distance is measured according to ||...||_2; and in the hypercube model, proposals and agent locations are vertices of the d-dimensional hypercube and the metric is the Hamming distance. They show that in the Euclidean model 2-compromises are guaranteed to succeed, but in the hypercube model for deliberation to succeed it may be necessary to use k-compromises with k >= d. We complement their analysis by (1) proving that in both models it is hard to find a proposal with a high degree of support, and even a 2-compromise transition may be hard to compute; (2) showing that a sequence of 2-compromise transitions may be exponentially long; (3) strengthening the lower bound on the size of the compromise for the d-hypercube model from d to 2^Ω(d).

We consider the setting where the members of a society (voters) have preferences over candidates, and the candidates can be ordered on an axis so that the voters’ preferences are single-peaked on this axis. We ask whether this axis can be identified by sampling the voters’ preferences. For several natural distributions, we obtain tight bounds on the number of samples required and show that, surprisingly, the bounds are independent of the number of candidates. We extend our results to the case where voters’ preferences are sampled from two different axes over the same candidate set (one of which may be known). We also consider two alternative models of learning: (1) sampling pairwise comparisons rather than entire votes, and (2) learning from equivalence queries.

We consider an agent community wishing to decide on several binary issues by means of issue-by-issue majority voting. For each issue and each agent, one of the two options is better than the other. However, some of the agents may be confused about some of the issues, in which case they may vote for the option that is objectively worse for them. A benevolent external party wants to help the agents to make better decisions, i.e., select the majority-preferred option for as many issues as possible. This party may have one of the following tools at its disposal: (1) educating some of the agents, so as to enable them to vote correctly on all issues, (2) appointing a subset of highly competent agents to make decisions on behalf of the entire group, or (3) guiding the agents on how to delegate their votes to other agents, in a way that is consistent with the agents' opinions. For each of these tools, we study the complexity of the decision problem faced by this external party, obtaining both NP-hardness results and fixed-parameter tractability results.

We study how to incentivize agents in a target subpopulation to produce a higher output by means of rank-order allocation contests, in the context of incomplete information. We describe a symmetric Bayes--Nash equilibrium for contests that have two types of rank-based prizes: (1) prizes that are accessible only to the agents in the target group; (2) prizes that are accessible to everyone. We also specialize this equilibrium characterization to two important sub-cases: (i) contests that do not discriminate while awarding the prizes, i.e., only have prizes that are accessible to everyone; (ii) contests that have prize quotas for the groups, and each group can compete only for prizes in their share. For these models, we also study the properties of the contest that maximizes the expected total output by the agents in the target group.

In some preference aggregation scenarios, voters' preferences are highly structured: e.g., the set of candidates may have one-dimensional structure (so that voters' preferences are single-peaked) or be described by a binary decision tree (so that voters' preferences are group-separable). However, sometimes a single axis or a decision tree is insufficient to capture the voters' preferences; rather, there is a small number K of axes or decision trees such that each vote in the profile is consistent with one of these axes (resp., trees). In this work, we study the complexity of deciding whether voters' preferences can be explained in this manner. For K=2, we use the technique developed by Yang [2020, https://doi.org/10.3233/FAIA200099] in the context of single-peaked preferences to obtain a polynomial-time algorithm for several domains: value-restricted preferences, group-separable preferences, and a natural subdomain of group-separable preferences, namely, caterpillar group-separable preferences. For K > 2, the problem is known to be hard for single-peaked preferences; we establish that it is also hard for value-restricted and group-separable preferences. Our positive results for K=2 make use of forbidden minor characterizations of the respective domains; in particular, we establish that the domain of caterpillar group-separable preferences admits a forbidden minor characterization.

We use the "map of elections" approach of Szufa et al. (AAMAS 2020) to analyze several well-known vote distributions. For each of them, we give an explicit formula or an efficient algorithm for computing its frequency matrix, which captures the probability that a given candidate appears in a given position in a sampled vote. We use these matrices to draw the "skeleton map" of distributions, evaluate its robustness, and analyze its properties. We further develop a general and unified framework for learning the distribution of real-world preferences using the frequency matrices of established vote distributions.

We study the problem of fairly allocating a divisible resource, also known as cake cutting, with an additional requirement that the shares that different agents receive should be sufficiently separated from one another. This captures, for example, constraints arising from social distancing guidelines. While it is sometimes impossible to allocate a proportional share to every agent under the separation requirement, we show that the well-known criterion of maximin share fairness can always be attained. We then provide algorithmic analysis of maximin share fairness in this setting—for instance, the maximin share of an agent cannot be computed exactly by any finite algorithm, but can be approximated with an arbitrarily small error. In addition, we consider the division of a pie (i.e., a circular cake) and show that an ordinal relaxation of maximin share fairness can be achieved. We also prove that an envy-free or equitable allocation that allocates the maximum amount of resource exists under separation.

Real-world security scenarios sometimes involve multiple defenders: security agencies of two or more countries might patrol the same border areas, and domestic security agencies might also operate in the same locations when their areas of jurisdiction overlap. Motivated by these scenarios and the observation that uncoordinated movements of the defenders may lead to an inefficient defense, we introduce a model of multi-defender security games and explore the possibility of improving efficiency by coordinating the defenders — specifically, by pooling the defenders' resources and allocating them jointly. The model generalizes the standard model of Stackelberg security games, where a defender (now a group of defenders) allocates security resources to protect a set of targets, and an attacker picks the best target to attack. In particular, we are interested in the situation with heterogeneous defenders, who may value the same target differently. Our task is twofold. First, we need to develop a good understanding of the uncoordinated situation, as the baseline to be improved. To this end we formulate a new equilibrium concept, and prove that an equilibrium under this concept always exists and can be computed efficiently. Second, to coordinate the heterogeneous defenders we take a mechanism design perspective and aim to find a mechanism to generate joint resource allocation strategies. We seek a mechanism that improves the defenders' utilities upon the uncoordinated baseline, achieves Pareto efficiency, and incentivizes the defenders to report their true incentives and execute the recommended strategies. Our analysis establishes several impossibility results, which indicate the intrinsic difficulties of defense coordination. Specifically, we show that even the basic properties listed above are in conflict with each other: no mechanism can simultaneously satisfy them all, or even some proper subsets of them. In terms of positive results, we present mechanisms that satisfy all combinations of the properties that are not ruled out by our impossibility results, thereby providing a comprehensive profile of the mechanism design problem with respect to the properties considered.

A preference profile is single-peaked on a tree if the candidate set can be equipped with a tree structure so that the preferences of each voter are decreasing from their top candidate along all paths in the tree. This notion was introduced by Demange (1982), and subsequently Trick (1989b) described an efficient algorithm for deciding if a given profile is single-peaked on a tree. We study the complexity of multiwinner elections under several variants of the Chamberlin–Courant rule for preferences single-peaked on trees. We show that in this setting the egalitarian version of this rule admits a polynomial-time winner determination algorithm. For the utilitarian version, we prove that winner determination remains NP-hard for the Borda scoring function; indeed, this hardness results extends to a large family of scoring functions. However, a winning committee can be found in polynomial time if either the number of leaves or the number of internal vertices of the underlying tree is bounded by a constant. To benefit from these positive results, we need a procedure that can determine whether a given profile is single-peaked on a tree that has additional desirable properties (such as, e.g., a small number of leaves). To address this challenge, we develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is single-peaked. We show how to use this representation to efficiently find the best tree for a given profile for use with our winner determination algorithms: Given a profile, we can efficiently find a tree with the minimum number of leaves, or a tree with the minimum number of internal vertices among trees on which the profile is single-peaked. We then explore the power and limitations of this framework: we develop polynomial-time algorithms to find trees with the smallest maximum degree, diameter, or pathwidth, but show that it is NP-hard to check whether a given profile is single-peaked on a tree that is isomorphic to a given tree, or on a regular tree.

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.

We study a setting in which a community wishes to identify a strongly supported proposal from a large space of alternatives, in order to change the status quo. We describe a deliberation process in which agents dynamically form coalitions around proposals that they prefer over the status quo. We formulate conditions on the space of proposals and on the ways in which coalitions are formed that guarantee deliberation to succeed, that is, to terminate by identifying a proposal with the largest possible support. Our results provide theoretical foundations for the analysis of deliberative processes in systems for democratic deliberation support, such as, e.g., LiquidFeedback or Polis.

We study the problem of fairly allocating a divisible resource, also known as cake cutting, with an additional requirement that the shares that different agents receive should be sufficiently separated from one another. This captures, for example, constraints arising from social distancing guidelines. While it is sometimes impossible to allocate a proportional share to every agent under the separation requirement, we show that the well-known criterion of maximin share fairness can always be attained. We then establish several computational properties of maximin share fairness---for instance, the maximin share of an agent cannot be computed exactly by any finite algorithm, but can be approximated with an arbitrarily small error. In addition, we consider the division of a pie (i.e., a circular cake) and show that an ordinal relaxation of maximin share fairness can be achieved.

We study the recently introduced cake-cutting setting in which the cake is represented by an undirected graph. This generalizes the canonical interval cake and allows for modeling the division of road networks. We show that when the graph is a forest, an allocation satisfying the well-known criterion of maximin share fairness always exists. Our result holds even when separation constraints are imposed; however, in the latter case no multiplicative approximation of proportionality can be guaranteed. Furthermore, while maximin share fairness is not always achievable for general graphs, we prove that ordinal relaxations can be attained.

This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of proportionality is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the ordinal maximin share approximation, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axes-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting.

Complexity of voting manipulation is a prominent topic in computational social choice. In this work, we consider a two-stage voting manipulation scenario. First, a malicious party (an attacker) attempts to manipulate the election outcome in favor of a preferred candidate by changing the vote counts in some of the voting districts. Afterwards, another party (a defender), which cares about the voters' wishes, demands a recount in a subset of the manipulated districts, restoring their vote counts to their original values. We investigate the resulting Stackelberg game for the case where votes are aggregated using two variants of the Plurality rule, and obtain an almost complete picture of the complexity landscape, both from the attacker's and from the defender's perspective.

We study a recently introduced class of strategic games that is motivated by and generalizes Schelling's well-known residential segregation model. These games are played on undirected graphs, with the set of agents partitioned into multiple types; each agent either occupies a node of the graph and never moves away or aims to maximize the fraction of her neighbors who are of her own type. We consider a variant of this model that we call swap Schelling games, where the number of agents is equal to the number of nodes of the graph, and agents may swap positions with other agents to increase their utility. We study the existence, computational complexity and quality of equilibrium assignments in these games, both from a social welfare perspective and from a diversity perspective.

In hedonic diversity games (HDGs), recently introduced by Bredereck, Elkind, and Igarashi (2019), each agent belongs to one of two classes (men and women, vegetarians and meat-eaters, junior and senior researchers), and agents' preferences over coalitions are determined by the fraction of agents from their class in each coalition. Bredereck et al. show that while an HDG may fail to have a Nash stable (NS) or a core stable (CS) outcome, every HDG in which all agents have single-peaked preferences admits an individually stable (IS) outcome, which can be computed in polynomial time. In this work, we extend and strengthen these results in several ways. First, we establish that the problem of deciding if an HDG has an NS outcome is NP-complete, but admits an XP algorithm with respect to the size of the smaller class. Second, we show that, in fact, all HDGs admit IS outcomes that can be computed in polynomial time; our algorithm for finding such outcomes is considerably simpler than that of Bredereck et al. We also consider two ways of generalizing the model of Bredereck et al. to k ≥ 2 classes. We complement our theoretical results by empirical analysis, comparing the IS outcomes found by our algorithm, the algorithm of Bredereck et al. and a natural better-response dynamics.

Obraztsova et al. (2013) have recently proposed an intriguing convexity axiom for voting rules. This axiom imposes conditions on the shape of the sets of elections with a given candidate as a winner. However, this new axiom is both too weak and too strong: it is too weak because it defines a set to be convex if for any two elements of the set some shortest path between them lies within the set, whereas the standard definition of convexity requires all shortest paths between two elements to lie within the set, and it is too strong because common voting rules do not satisfy this axiom. In this paper, we (1) propose several families of voting rules that are convex in the sense of Obraztsova et al.; (2) put forward a weaker notion of convexity that is satisfied by most common voting rules; (3) prove impossibility results for a variant of this definition that considers all, rather than some shortest paths.

In the multidimensional stable roommate problem, agents have to be allocated to rooms and have preferences over sets of potential roommates. We study the complexity of finding good allocations of agents to rooms under the assumption that agents have diversity preferences (Bredereck, Elkind, Igarashi, AAMAS'19): each agent belongs to one of the two types (e.g., juniors and seniors, artists and engineers), and agents' preferences over rooms depend solely on the fraction of agents of their own type among their potential roommates. We consider various solution concepts for this setting, such as core and exchange stability, Pareto optimality and envy-freeness. On the negative side, we prove that envy-free, core stable or (strongly) exchange stable outcomes may fail to exist and that the associated decision problems are NP-complete. On the positive side, we show that these problems are in FPT with respect to the room size, which is not the case for the general stable roommate problem.

We examine the problem of assigning plots of land to prospective buyers who prefer living next to their friends. In this setting, each agent's utility depends on the plot she receives and the identities of the agents who receive the adjacent plots. We are interested in mechanisms without money that guarantee truthful reporting of both land values and friendships, as well as Pareto optimality and computational efficiency. We explore several modifications of the Random Serial Dictatorship (RSD) mechanism, and identify one that performs well according to these criteria, We also study the expected social welfare of the assignments produced by our mechanisms when agents' values for the land plots are binary; it turns out that we can achieve good approximations to the optimal social welfare, but only if the agents value the friendships highly.

Integrity of elections is vital to democratic systems, but it is frequently threatened by malicious actors.The study of algorithmic complexity of the problem of manipulating election outcomes by changing its structural features is known as election control Rothe [2016].One means of election control that has been proposed, pertinent to the spatial voting model, is to select a subset of issues that determine voter preferences over candidates.We study a variation of this model in which voters have judgments about relative importance of issues, and a malicious actor can manipulate these judgments.We show that computing effective manipulations in this model is NP-hard even with two candidates or binary issues.However, we demonstrate that the problem becomes tractable with a constant number of voters or issues.Additionally, while it remains intractable when voters can vote stochastically, we exhibit an important special case in which stochastic voting behavior enables tractable manipulation.

In this paper, we study the problem of matching a set of items to a set of agents partitioned into types so as to balance fairness towards the types against overall utility/efficiency. We extend multiple desirable properties of indivisible goods allocation to our model and investigate the possibility and hardness of achieving combinations of these properties, e.g. we prove that maximizing utilitarian social welfare under constraints of typewise envy-freeness up to one item (TEF1) is computationally intractable. We also define a new concept of waste for this setting, show experimentally that augmenting an existing algorithm with a marginal utility maximization heuristic can produce a TEF1 solution with reduced waste, and also provide a polynomial-time algorithm for computing a non-wasteful TEF1 allocation for binary agent-item utilities.

We consider strategic games that are inspired by Schelling's model of residential segregation. In our model, the agents are partitioned into k types and need to select locations on an undirected graph. Agents can be either stubborn, in which case they will always choose their preferred location, or strategic, in which case they aim to maximize the fraction of agents of their own type in their neighborhood. We investigate the existence of equilibria in these games, study the complexity of finding an equilibrium outcome or an outcome with high social welfare, and also provide upper and lower bounds on the price of anarchy and stability. Some of our results extend to the setting where the preferences of the agents over their neighbors are defined by a social network rather than a partition into types.

We study the problem of computing committees that perform well according to several different criteria, which are expressed as committee scoring rules. We analyze the computational complexity of computing such committees and provide an experimental evaluation of the compromise levels that can be achieved between several well-known rules, including k-Borda, SNTV, Bloc, and the Chamberlin--Courant rule.

We use social choice theory to develop correlation coefficients between ranked preferences and an ordinal attribute such as educational attainment or income level. For example, such correlations could be used to formalise statements such as "voters' preferences over parties are better explained by age than by income level". In the literature, preferences that are perfectly explained by a single-dimensional agent attribute are commonly taken to be single-crossing preferences. Thus, to quantify how well an attribute explains preferences, we can order the voters by the value of the attribute and compute how far the resulting ordered profile is from being single-crossing, for various commonly studied distance measures (Kendall tau distance, voter/alternative deletion, etc.). The goal of this paper is to evaluate the computational feasibility of this approach. To this end, we investigate the complexity of computing these distances, obtaining an essentially complete picture for the distances we consider.

In voting theory, impossibility results and computational hardness results are often circumvented by recognising that voters' preferences are not arbitrary, but lie within a restricted domain. Uncovering the structure of the underlying domain often provides useful insights about the nature of the alternative space, and may be helpful in identifying a collective choice. Preferences single-peaked on a tree are an example of a relatively broad domain that nonetheless exhibits several desirable properties. We consider the setting where voters' preferences are independently sampled from rankings that are single-peaked on a given tree, and study the problem of reliably identifying the tree that generated the observed votes. We test our algorithm empirically; to this end, we develop an algorithm to uniformly sample preferences that are single-peaked on a given tree.

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.

If voters' preferences are one-dimensional, many hard problems in computational social choice become tractable. A preference profile can be classified as one-dimensional if it has the single-crossing property, which requires that the voters can be ordered from left to right so that their preferences are consistent with this order. In practice, preferences may exhibit some one-dimensional structure, despite not being single-crossing in the formal sense. Hence, we ask whether one can identify preference profiles that are close to being single-crossing. We consider three distance measures, which are based on partitioning voters or candidates or performing a small number of swaps in each vote. We prove that it can be efficiently decided if voters can be split into two single-crossing groups. Also, for every fixed k >= 1 we can decide in polynomial time if a profile can be made single-crossing by performing at most k candidate swaps per vote. In contrast, for each k >= 3 it is NP-complete to decide whether candidates can be partitioned into k sets so that the restriction of the input profile to each set is single-crossing.

A key issue in cooperative game theory is coalitional stability, usually captured by the notion of the core---the set of outcomes that are resistant to group deviations. However, some coalitional games have empty cores, and any outcome in such a game is unstable. We investigate the possibility of stabilizing a coalitional game by using subsidies. We consider scenarios where an external party that is interested in having the players work together offers a supplemental payment to the grand coalition, or, more generally, a particular coalition structure. This payment is conditional on players not deviating from this coalition structure, and may be divided among the players in any way they wish. We define the cost of stability as the minimum external payment that stabilizes the game. We provide tight bounds on the cost of stability, both for games where the coalitional values are nonnegative (profit-sharing games) and for games where the coalitional values are nonpositive (cost-sharing games), under natural assumptions on the characteristic function, such as superadditivity, anonymity, or both. We also investigate the relationship between the cost of stability and several variants of the least core. Finally, we study the computational complexity of problems related to the cost of stability, with a focus on weighted voting games.

We visualize aggregate outputs of popular multiwinner voting rules — SNTV, STV, Bloc, k-Borda, Monroe, Chamberlin–Courant, and PAV — for elections generated according to the two-dimensional Euclidean model. We consider three applications of multiwinner voting, namely, parliamentary elections, portfolio/movie selection, and shortlisting, and use our results to understand which of our rules seem to be best suited for each application. In particular, we show that STV (one of the few nontrivial rules used in real high-stake elections) exhibits excellent performance, whereas the Bloc rule (also often used in practice) performs poorly.

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.

We propose a new variant of the group activity selection problem (GASP), where the agents are placed on a social network and activities can only be assigned to connected subgroups. We show that if multiple groupscan simultaneously engage in the same activity, finding a stable outcome is easy as long as the networkis acyclic. In contrast, if each activity can be assigned to a single group only, finding stable outcomes becomes computationally intractable, even if the underlying network is very simple: the problem of determining whether a given instance of a GASP admits a Nash stable outcome turns out to be NP-hard when the social network is a path, a star, or if the size of each connected component is bounded by a constant.On the other hand, we obtain fixed-parameter tractability results for this problem with respectto the number of activities.

We consider voting under metric preferences: both voters and candidates are associated with points in a metric space, and each voter prefers candidates that are closer to her to ones that are further away. In this setting, it is often desirable to select a candidate that minimizes the sum of distances to the voters. However, common voting rules operate on voters' preference rankings and therefore may be unable to identify the best candidate. A relevant measure of the quality of a voting rule is then its distortion, defined as the worst-case ratio between the performance of a candidate selected by the rule and that of an optimal candidate. Anshelevich, Bhardwaj and Postl show that some popular rules such as Borda and Plurality do badly in this regard: their distortion scales linearly with the number of candidates. On the positive side, Anshelevich et al. identify a few voting rules whose distortion is bounded by a constant; however, these rules are rarely used in practice. In this paper, we analyze the distortion of two widely used (classes of) voting rules, namely, scoring rules and Single Transferable Vote (STV). We show that all scoring rules have super-constant distortion, answering a question that was left open by Anshelevich et al.; however, we identify a scoring rule whose distortion is asymptotically better than that of Plurality and Borda. For STV, we obtain an upper bound of O(log m), where m is the number of candidates, as well as a super-constant lower bound; thus, STV is a reasonable, though not a perfect rule from this perspective.

Suppose that a group of voters wants to select k 1 alternatives from a given set, and each voter indicates which of the alternatives are acceptable to her: the alternatives could be conference submissions, applicants for a scholarship or locations for a fast food chain. In this setting it is natural to require that the winning set represents the voters fairly, in the sense that large groups of voters with similar preferences have at least some of their approved alternatives in the winning set. We describe several ways to formalize this idea, and show how to use it to classify voting rules; surprisingly, two voting rules proposed in the XIXth century turn out to play an important role in our analysis.

We study two notions of stability in multiwinner elections that are based on the Condorcet criterion. The first notion was introduced by Gehrlein and is majoritarian in spirit. The second one, local stability, is introduced in this paper, and focuses on voter representation. The goal of this paper is to explore these two notions, their implications on restricted domains, and the computational complexity of rules that are consistent with them.

We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases wherethe underlying graph has simple structure, and/or the number of agents---or, less restrictively, the number of agent types---is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently.

We consider opinion diffusion in binary influence networks, where at each step one or more agents update their opinions so as to be in agreement with the majority of their neighbors. We consider several ways of manipulating the majority opinion in a stable outcome, such as bribing agents, adding/deleting links, and changing the order of updates, and investigate the computational complexity of the associated problems, identifying tractable and intractable cases.

We extend the principle of proportional representation to rankings: given approval preferences, we aim to generate aggregate rankings so that cohesive groups of voters are represented proportionally in each initial segment of the ranking. Such rankings are desirable in situations where initial segments of different lengths may be relevant, e.g., in recommender systems, for hiring decisions, or for the presentation of competing proposals on a liquid democracy platform. We define what it means for rankings to be proportional, provide bounds for well-known aggregation rules, and experimentally evaluate the performance of these rules.

Price of Anarchy measures the welfare loss caused by selfish behavior: it is defined as the ratio of the social welfare in a socially optimal outcome and in a worst Nash equilibrium. A similar measure can be derived for other classes of stable outcomes. In this paper, we argue that Pareto optimality can be seen as a notion of stability, and introduce the concept of Price of Pareto Optimality: this is an analogue of the Price of Anarchy, where the maximum is computed over the class of Pareto optimal outcomes, i.e., outcomes that do not permit a deviation by the grand coalition that makes all players weakly better off and some players strictly better off. As a case study, we focus on hedonic games, and provide lower and upper bounds of the Price of Pareto Optimality in three classes of hedonic games: additively separable hedonic games, fractional hedonic games, and modified fractional hedonic games; for fractional hedonic games on trees our bounds are tight.

Preference profiles that are single-peaked on trees enjoy desirable properties: they admit a Condorcet winner (Demange 1982), and there are hard voting problems that become tractable on this domain (Yu et al., 2013). Trick (1989) proposed a polynomial-time algorithm that finds some tree with respect to which a given preference profile is single-peaked. However, some voting problems are only known to be easy for profiles that are single-peaked on "nice" trees, and Trick's algorithm provides no guarantees on the properties of the tree that it outputs. To overcome this issue, we build on the work of Trick and Yu et al. to develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is single-peaked. We show how to use this representation to efficiently find the "best" tree for a given profile, according to a number of criteria; for other criteria, we obtain NP-hardness results. In particular, we show that it is NP-hard to decide whether an input profile is single-peaked with respect to a given tree. To demonstrate the applicability of our framework, we use it to identify a new class of profiles that admit an efficient algorithm for a popular variant of the Chamberlin-Courant rule.

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.

We study the complexity of deciding if a given profile of incomplete votes (i.e., a profile of partial orders over a given set of alternatives) can be extended to a single-crossing profile of complete votes (total orders). This problem models settings where we have partial knowledge regarding voters' preferences and we would like to understand whether the given preference profile may be single-crossing. We show that this problem admits a polynomial-time algorithm when the order of votes is fixed and the input profile consists of top orders, but becomes NP-complete if we are allowed to permute the votes and the input profile consists of weak orders or independent-pairs orders. Also, we identify a number of practical special cases of both problems that admit polynomial-time algorithms.

We investigate elections that are simultaneously single-peaked and single-crossing (SPSC). We show that the domain of 1-dimensional Euclidean elections (where voters and candidates are points on the real line, and each voter prefers the candidates that are close to her to the ones that are further away) is a proper subdomain of the SPSC domain, by constructing an election that is single-peaked and single-crossing, but not 1-Euclidean. We then establish a connection between narcissistic elections (where each candidate is ranked first by at least one voter), single-peaked elections and single-crossing elections, by showing that an election is SPSC if and only if it can be obtained from a narcissistic single-crossing election by deleting voters. We show two applications of our characterization.

Structured preference domains, such as, for example, the domains of single-peaked and single-crossing preferences, are known to admit efficient algorithms for many problems in computational social choice. Some of these algorithms extend to preferences that are close to having the respective structural property, i.e., can be made to enjoy this property by performing minor changes to voters' preferences, such as deleting a small number of voters or candidates. However, it has recently been shown that finding the optimal number of voters or candidates to delete in order to achieve the desired structural property is NP-hard for many such domains. In this paper, we show that these problems admit efficient approximation algorithms. Our results apply to all domains that can be characterized in terms of forbidden configurations; this includes, in particular, single-peaked and single-crossing elections. For a large range of scenarios, our approximation results are optimal under a plausible complexity-theoretic assumption. We also provide parameterized complexity results for this class of problems.

Overlapping Coalition Formation (OCF) games, introduced by Chalkiadakis, Elkind, Markakis, Polukarov and Jennings in 2010, are cooperative games where players can simultaneously participate in several coalitions. Capturing the notion of stability in OCF games is a difficult task:deviating players may continue to contribute resources to joint projects with non-deviators, and the crucial question is what payoffs the deviators expect to receive from such projects. Chalkiadakis et al. introduce three stability concepts for OCF games---the conservative core, the refined core, and the optimistic core---that are based on different answers to this question. In this paper, we propose a unified framework for the study of stability in the OCF setting, which encompasses the stability concepts considered by Chalkiadakis et al. as well as a wide variety of alternative stability concepts. Our approach is based on the notion of arbitration functions, which determine the payoff obtained by the deviators, given their deviation and the current allocation of resources. We provide a characterization of stable outcomes under arbitration. We then conduct an in-depth study of four types of arbitration functions, which correspond to four notions of the core; these include the three notions of the core considered by Chalkiadakis et al. Our results complement those of Chalkiadakis et al. and answer questions left open by their work. In particular, we show that OCF games with the conservative arbitration function are essentially equivalent to non-OCF games, by relating the conservative core of an OCF game to the core of a non-overlapping cooperative game, and use this result to obtain a strictly weaker sufficient condition for conservative core non-emptiness than the one given by Chalkiadakis et al.

We study the stability of cooperative games played over an interaction network, in a model that was introduced by Myerson ['77]. We show that the cost of stability of such games (i.e., the subsidy required to stabilize the game) can be bounded in terms of natural parameters of their underlying interaction networks. Specifically, we prove that if the treewidth of the interaction network H is k, then the relative cost of stability of any game played over H is at most k + 1, and if the pathwidth of H is k', then the relative cost of stability is at most k'. We show that these bounds are tight for all k≥ 2and all k' ≥ 1, respectively.

Complexity of voting manipulation is a prominent research topic in computational social choice. The voting manipulation literature usually assumes that the manipulator is only concerned with improving the outcome of the election from her perspective. However, in practice, the manipulator may also be reluctant to lie, i.e., she may have a preference for submitting a vote that does not deviate too much from her true ranking of the candidates. In this paper, we study the complexity of finding a manipulative vote that achieves the manipulator's goal yet is as close as possible to her true preference order. We analyze this problem for three natural notions of closeness, namely, swap distance, footrule distance, and maximum displacement distance, and a variety of voting rules, such as scoring rules, Bucklin, Copeland, and Maximin. For all three distances, we obtain polynomial-time algorithms for all scoring rules and Bucklin and hardness results for Copeland and Maximin.

The core is a central solution concept in cooperative game theory, and therefore it is important to know under what conditions the core of a game is guaranteed to be non-empty. Two notions that prove to be very useful in this context are Linear Programming (LP) duality and convexity. In this work, we apply these tools to identify games with overlapping coalitions (OCF games) that admit stable outcomes. We focus on three notions of the core defined in (Chalkiadakis et al. 2010) for such games, namely, the conservative core, the refined core and the optimistic core. First, we show that the conservative core of an OCF game is non-empty if and only if the core of a related classic coalitional game is non-empty. This enables us to improve the result of (Chalkiadakis et al. 2010) by giving a strictly weaker sufficient condition for the non-emptiness of the conservative core. We then use LP duality to characterize OCF games with non-empty refined core; as a corollary, we show that the refined core of a game is non-empty as long as the superadditive cover of its characteristic function is convex. Finally, we identify a large class of OCF games that can be shown to have a non-empty optimistic core using an LP based argument.

Algorithmic mechanism design is concerned with designing algorithms for settings where inputs are controlled by selfish agents, and the center needs to motivate the agents to report their true values. In this paper, we study scenarios where the center may be able to verify whether the agents report their preferences (types) truthfully. We first consider the standard model of mechanism design with partial verification, where the set of types that an agent can report is a function of his true type. We explore inherent limitations of this model; in particular, we show that the famous Gibbard--Satterthwaite impossibility result holds even if a manipulator can only lie by swapping two adjacent alternatives in his vote. Motivated by these negative results, we then introduce a richer model of verification, which we term mechanism design with probabilistic verification. In our model, an agent may report any type, but will be caught with some probability that may depend on his true type, the reported type, or both; if an agent is caught lying, he will not get his payment and may be fined. We characterize the class of social choice functions that can be truthfully implemented in this model. We then proceed to study the complexity of finding an optimal individually rational implementation, i.e., one that minimizes the center's expected payment while guaranteeing non-negative utility to the agent, both for truthful and for non-truthful implementation. Our hardness result for non-truthful implementation answers an open question recently posed by Auletta et al. [2011].

In elections, a set of candidates ranked consecutively (though possibly in different order) by all voters is called a clone set, and its members are called clones. A clone structure is the family of all clone sets of a given election. In this paper we study properties of clone structures. In particular, we give an axiomatic characterization of clone structures, show that they are organized hierarchically, and analyze clone structures in single-peaked and single-crossing elections. We describe a polynomial-time algorithm that finds a minimal collection of clones that need to be collapsed for an election to become single-peaked, and we show that this problem is NP-hard for single-crossing elections.

The conventional model of coalition formation considers every possible subset of agents as a potential coalition. However, in many real-world applications, there are inherent constraints on feasible coalitions: for instance, certain agents may be prohibited from being in the same coalition, or the coalition structure may be required to consist of coalitions of the same size. In this paper, we present the first systematic study of constrained coalition formation (CCF). We propose a general framework for this problem, and identify an important class of CCF settings, where the constraints specify which groups of agents should/should not work together. We describe a procedure that transforms such constraints into a structured input that allows coalition formation algorithms to identify, without any redundant computations, all the feasible coalitions. We then use this procedure to develop an algorithm for generating an optimal (welfare-maximizing) constrained coalition structure, and show that it outperforms existing state-of-the-art approaches by several orders of magnitude.

Approval-like voting rules, such as Sincere-Strategy Preference-Based Approval voting (SP-AV), the Bucklin rule (an adaptive variant of k-Approval voting), and the Fallback rule (an adaptive variant of SP-AV) have many desirable properties: for example, they are easy to understand and encourage the candidates to choose electoral platforms that have a broad appeal. In this paper, we investigate both classic and parameterized computational complexity of electoral campaign management under such rules. We focus on two methods that can be used to promote a given candidate: asking voters to move this candidate upwards in their preference order or asking them to change the number of candidates they approve of. We show that finding an optimal campaign management strategy of the first type is easy for both Bucklin and Fallback. In contrast, the second method is computationally hard even if the degree to which we need to affect the votes is small. Nevertheless, we identify a large class of scenarios that admit a fixed-parameter tractable algorithm.

An important task in the analysis of multiagent systems is to understand how groups of selfish players can form coalitions, i.e., work together in teams. In this paper, we study the dynamics of coalition formation under bounded rationality. We consider settings where each team's profit is given by a concave function, and propose three profit-sharing schemes, each of which is based on the concept of marginal utility. The agents are assumed to be myopic, i.e., they keep changing teams as long as they can increase their payoff by doing so. We study the properties (such as closeness to Nash equilibrium or total profit) of the states that result after a polynomial number of such moves, and prove bounds on the price of anarchy and the price of stability of the corresponding games.

Slinko and White, (2008) have recently introduced a new model of coalitional manipulation of voting rules under limited communication, which they call safe strategic voting. The computational aspects of this model were first studied by Hazon and Elkind, (2010), who provide polynomial-time algorithms for finding a safe strategic vote under k-approval and the Bucklin rule. In this paper, we answer an open question of Hazon and Elkind, (2010) by presenting a polynomial-time algorithm for finding a safe strategic vote under the Borda rule. Our results for Borda generalize to several interesting classes of scoring rules.

Computational complexity of voting manipulation is one of the most actively studied topics in the area of computational social choice, starting with the groundbreaking work of [Bartholdi et al., 1989]. Most of the existing work in this area, including that of [Bartholdi et al., 1989], implicitly assumes that whenever several candidates receive the top score with respect to the given voting rule, the resulting tie is broken according to a lexicographic ordering over the candidates. However, till recently, an equally appealing method of tiebreaking, namely, selecting the winner uniformly at random among all tied candidates, has not been considered in the computational social choice literature. The first paper to analyze the complexity of voting manipulation under randomized tiebreaking is [Obraztsova et al., 2011], where the authors provide polynomial-time algorithms for this problem under scoring rules and—under an additional assumption on the manipulator’s utilities— for Maximin. In this paper, we extend the results of [Obraztsova et al., 2011] by showing that finding an optimal vote under randomized tie-breaking is computationally hard for Copeland and Maximin (with general utilities), as well as for STV and Ranked Pairs, but easy for the Bucklin rule and Plurality with Runoff.

In their groundbreaking paper, Bartholdi, Tovey and Trick [1989] argued that many well-known voting rules, such as Plurality, Borda, Copeland and Maximin are easy to manipulate. An important assumption made in that paper is that the manipulator's goal is to ensure that his preferred candidate is among the candidates with the maximum score, or, equivalently, that ties are broken in favor of the manipulator's preferred candidate. In this paper, we examine the role of this assumption in the easiness results of [Bartholdi et al., 1989]. We observe that the algorithm presented in [Bartholdi et al., 1989] extends to all rules that break ties according to a fixed ordering over the candidates. We then show that all scoring rules are easy to manipulate if the winner is selected from all tied candidates uniformly at random. This result extends to Maximin under an additional assumption on the manipulator's utility function that is inspired by the original model of [Bartholdi et al., 1989]. In contrast, we show that manipulation becomes hard when arbitrary polynomial-time tie-breaking rules are allowed, both for the rules considered in [Bartholdi et al., 1989], and for a large class of scoring rules.

Weighted voting is a classic model of cooperation among agents in decision-making domains. In such games, each player has a weight, and a coalition of players wins the game if its total weight meets or exceeds a given quota. A player's power in such games is usually not directly proportional to his weight, and is measured by a power index, the most prominent among which are the Shapley-Shubik index and the Banzhaf index.In this paper, we investigate by how much a player can change his power, as measured by the Shapley-Shubik index or the Banzhaf index, by means of a false-name manipulation, i.e., splitting his weight among two or more identities. For both indices, we provide upper and lower bounds on the effect of weight-splitting. We then show that checking whether a beneficial split exists is NP-hard, and discuss efficient algorithms for restricted cases of this problem, as well as randomized algorithms for the general case. We also provide an experimental evaluation of these algorithms. Finally, we examine related forms of manipulative behavior, such as annexation, where a player subsumes other players, or merging, where several players unite into one. We characterize the computational complexity of such manipulations and provide limits on their effects. For the Banzhaf index, we describe a new paradox, which we term the Annexation Non-monotonicity Paradox.

We consider the problem of manipulating elections by cloning candidates. In our model, a manipulator can replace each candidate c by several clones, i.e., new candidates that are so similar to c that each voter simply replaces c in his vote with a block of these new candidates, ranked consecutively. The outcome of the resulting election may then depend onthenumberofclonesaswellasonhoweachvoterordersthecloneswithintheblock. We formalize what it means for a cloning manipulation to be successful (which turns out to be a surprisingly delicate issue), and, for a number of common voting rules, characterize the preference profiles for which a successful cloning manipulation exists. We also consider the model where there is a cost associated with producing each clone, and study the complexity of finding a minimum-cost cloning manipulation. Finally, we compare cloning with two related problems: the problem of control by adding candidates and the problem of possible (co)winners when new alternatives can join. 1.

We consider the problem of manipulating elections via cloning candidates. In our model, a manipulator can replace each candidate c by one or more clones, i.e., new candidates that are so similar to c that each voter simply replaces c in his vote with the block of c's clones. The outcome of the resulting election may then depend on how each voter orders the clones within the block. We formalize what it means for a cloning manipulation to be successful (which turns out to be a surprisingly delicate issue), and, for a number of prominent voting rules, characterize the preference profiles for which a successful cloning manipulation exists. We also consider the model where there is a cost associated with producing each clone, and study the complexity of finding a minimum-cost cloning manipulation. Finally, we compare cloning with the related problem of control via adding candidates.

We explore the relationship between two approaches to rationalizing voting rules: the maximum likelihood estimation (MLE) framework originally suggested by Condorcet and recently studied by Conitzer, Rognlie, and Xia, and the distance rationalizability (DR) framework of Elkind, Faliszewski, and Slinko. The former views voting as an attempt to reconstruct the correct ordering of the candidates given noisy estimates (i.e., votes), while the latter explains voting as search for the nearest consensus outcome. We provide conditions under which an MLE interpretation of a voting rule coincides with its DR interpretation, and classify a number of classic voting rules, such as Kemeny, Plurality, Borda and Single Transferable Vote (STV), according to how well they fit each of these frameworks. The classification we obtain is more precise than the ones that result from using MLE or DR alone: indeed, we show that the MLE approach can be used to guide our search for a more refined notion of distance rationalizability and vice versa.

We study the design of truthful mechanisms for set systems, i.e., scenarios where a customer needs to hire a team of agents to perform a complex task. In this setting, frugality [2] provides a measure to evaluate the "cost of truthfulness", that is, the overpayment of a truthful mechanism relative to the "fair" payment. We propose a uniform scheme for designing frugal truthful mechanisms for general set systems. Our scheme is based on scaling the agents' bids using the eigenvector of a matrix that encodes the interdependencies between the agents. We demonstrate that the r-out-of-k-system mechanism and the √-mechanism for buying a path in a graph [18] can be viewed as instantiations of our scheme. We then apply our scheme to two other classes of set systems, namely, vertex cover systems and k-path systems, in which a customer needs to purchase k edge-disjoint source-sink paths. For both settings, we bound the frugality of our mechanism in terms of the largest eigenvalue of the respective interdependency matrix. We show that our mechanism is optimal for a large subclass of vertex cover systems satisfying a simple local sparsity condition. For k-path systems, our mechanism is within a factor of k + 1 from optimal; moreover, we show that it is, in fact, optimal, when one uses a modified definition of frugality proposed in [10]. Our lower bound argument combines spectral techniques and Young's inequality, and is applicable to all set systems. As both r-out-of-k systems and single path systems can be viewed as special cases of k-path systems, our result improves the lower bounds of [18] and answers several open questions proposed in [18].

In the traditional voting manipulation literature, it is assumed that a group of manipulators jointly misrepresent their preferences to get a certain candidate elected, while the remaining voters are truthful. In this paper, we depart from this assumption, and consider the setting where all voters are strategic. In this case, the election can be viewed as a game, and the election outcomes correspond to Nash equilibria of this game. We use this framework to analyze two variants of Plurality voting, namely, simultaneous voting, where all voters submit their ballots at the same time, and sequential voting, where the voters express their preferences one by one. For simultaneous voting, we characterize the preference profiles that admit a pure Nash equilibrium, but show that it is computationally hard to check if a given profile fits our criterion. For sequential voting, we provide a complete analysis of the setting with two candidates, and show that for three or more candidates the equilibria of sequential voting may behave in a counterintuitive manner.

In the usual models of cooperative game theory, the outcome of a coalition formation process is either the grand coalition or a coalition structure that consists of disjoint coalitions. However, in many domains where coalitions are associated with tasks, an agent may be involved in executing more than one task, and thus may distribute his resources among several coalitions. To tackle such scenarios, we introduce a model for cooperative games with overlapping coalitions—or overlapping coalition formation (OCF) games. We then explore the issue of stability in this setting. In particular, we introduce a notion of the core, which generalizes the corresponding notion in the traditional (non-overlapping) scenario. Then, under some quite general conditions, we characterize the elements of the core, and show that any element of the core maximizes the social welfare. We also introduce a concept of balancedness for overlapping coalitional games, and use it to characterize coalition structures that can be extended to elements of the core. Finally, we generalize the notion of convexity to our setting, and show that under some natural assumptions convex games have a non-empty core. Moreover, we introduce two alternative notions of stability in OCF that allow a wider range of deviations, and explore the relationships among the corresponding definitions of the core, as well as the classic (non-overlapping) core and the Aubin core. We illustrate the general properties of the three cores, and also study them from a computational perspective, thus obtaining additional insights into their fundamental structure.

In voting theory, bribery is a form of manipulative behavior in which an external actor (the briber) offers to pay the voters to change their votes in order to get her preferred candidate elected. We investigate a model of bribery where the price of each vote depends on the amount of change that the voter is asked to implement. Specifically, in our model the briber can change a voter’s preference list by paying for a sequence of swaps of consecutive candidates. Each swap may have a different price; the price of a bribery is the sum of the prices of all swaps that it involves. We prove complexity results for this model, which we call swap bribery , for a broad class of voting rules, including variants of approval and k -approval, Borda, Copeland, and maximin. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Weighted voting games (WVG) are coalitional games in which an agent's contribution to a coalition is given by his weight, and a coalition wins if its total weight meets or exceeds a given quota. These games model decision-making in political bodies as well as collaboration and surplus division in multiagent domains. The computational complexity of various solution concepts for weighted voting games received a lot of attention in recent years. In particular, Elkind et al.(2007) studied the complexity of stability-related solution concepts in WVGs, namely, of the core, the least core, and the nucleolus. While they have completely characterized the algorithmic complexity of the core and the least core, for the nucleolus they have only provided an NP-hardness result. In this paper, we solve an open problem posed by Elkind et al. by showing that the nucleolus of WVGs, and, more generally, k-vector weighted voting games with fixed k, can be computed in pseudopolynomial time, i.e., there exists an algorithm that correctly computes the nucleolus and runs in time polynomial in the number of players n and the maximum weight W. In doing so, we propose a general framework for computing the nucleolus, which may be applicable to a wider of class of games.

The concept of distance rationalizability has several applications within social choice. In the context of voting, it allows one to define ("rationalize") voting rules via a consensus class (roughly, a set of elections in which it is obvious who should win) and a distance function: namely, a candidate is said to be an election winner if it is ranked first in one of the nearest (with respect to the given distance) consensus elections. It is known that many classic voting rules can be represented in this manner. In this paper, we provide new results on distance rationalizability of several well-known voting rules such as all scoring rules, Approval, Young's rule and Maximin. We also show that a previously published proof of distance rationalizability of Young's rule is incorrect: the consensus notion and the distance function used in that proof give rise to a voting rule that is similar to---but distinct from---the Young's rule. Finally, we demonstrate that some voting rules cannot be rationalized via certain notions of consensus. To the best of our knowledge, these are the first non-distance-rationalizability results for voting rules.

Weighted voting games are a popular model of collaboration in multiagent systems. In such games, each agent has a weight (intuitively corresponding to resources he can contribute), and a coalition of agents wins if its total weight meets or exceeds a given threshold. Even though coalitional stability in such games is important, existing research has nonetheless only considered the stability of the grand coalition. In this paper, we introduce a model for weighted voting games with coalition structures. This is a natural extension in the context of multiagent systems, as several groups of agents may be simultaneously at work, each serving a different task. We then proceed to study stability in this context. First, we define the CS-core, a notion of the core for such settings, discuss its non-emptiness, and relate it to the traditional notion of the core in weighted voting games. We then investigate its computational properties. We show that, in contrast with the traditional setting, it is computationally hard to decide whether a game has a non-empty CS-core, or whether a given outcome is in the CS-core. However, we then provide an efficient algorithm that verifies whether an outcome is in the CS-core if all weights are small (polynomially bounded). Finally, we also suggest heuristic algorithms for checking the non-emptiness of the CS-core.

This paper addresses the problem of fair equilibrium selection in graphical games. Our approach is based on the data structure called the best response policy, which was proposed by Kearns et al. [13] as a way to represent all Nash equilibria of a graphical game. In [9], it was shown that the best response policy has polynomial size as long as the underlying graph is a path. In this paper, we show that if the underlying graph is abounded-degree tree and the best response policy has polynomial size then there is an efficient algorithm which constructs a Nash equilibrium that guarantees certain payoffs to all participants. Another attractive solution concept is a Nash equilibrium that maximizes the social welfare. We show that, while exactly computing the latter is infeasible (we prove that solving this problem may involve algebraic numbers of an arbitrarily high degree), there exists an FPTAS for finding such an equilibrium as long as the best response policy has polynomial size. These two algorithms can be combined to produce Nash equilibria that satisfy various fairness criteria.

In set-system auctions, there are several overlapping teams of agents, and a task that can be completed by any of these teams. The auctioneer's goal is to hire a team and pay as little as possible. Examples of this setting include shortest-path auctions and vertex-cover auctions. Recently, Karlin, Kempe and Tamir introduced a new definition of frugality ratio for this problem. Informally, the "frugality ratio" is the ratio of the total payment of a mechanism to a desired payment bound. The ratio captures the extent to which the mechanism overpays, relative to perceived fair cost in a truthful auction. In this paper, we propose a new truthful polynomial-time auction for the vertex cover problem and bound its frugality ratio. We show that the solution quality is with a constant factor of optimal and the frugality ratio is within a constant factor of the best possible worst-case bound; this is the first auction for this problem to have these properties. Moreover, we show how to transform any truthful auction into a frugal one while preserving the approximation ratio. Also, we consider two natural modifications of the definition of Karlin et al., and we analyse the properties of the resulting payment bounds, such as monotonicity, computational hardness, and robustness with respect to the draw-resolution rule. We study the relationships between the different payment bounds, both for general set systems and for specific set-system auctions, such as path auctions and vertex-cover auctions. We use these new definitions in the proof of our main result for vertex-cover auctions via a bootstrapping technique, which may be of independent interest.

Graphical games have been proposed as a game-theoretic model of large-scale distributed networks of non-cooperative agents. When the number of players is large, and the underlying graph has low degree, they provide a concise way to represent the players' payoffs. It has recently been shown that the problem of finding Nash equilibria in a general degree-3 graphical game with two actions per player is complete for the complexity class PPAD, indicating that it is unlikely that there is any polynomial-time algorithm for this problem. In this paper, we study the complexity of graphical games with two actions per player on bounded-degree trees. This setting was first considered by Kearns, Littman and Singh, who proposed a dynamic programming-based algorithm that computes all Nash equilibria of such games. The running time of their algorithm is exponential, though approximate equilibria can be computed efficiently. Later, Littman, Kearns and Singh proposed a modification to this algorithm that can find a single Nash equilibrium in polynomial time. We show that this modified algorithm is incorrect-the output is not always a Nash equilibrium. We then propose a new algorithm that is based on the ideas of Kearns et al. and computes all Nash equilibria in quadratic time if the input graph is a path, and in polynomial time if it is an arbitrary graph of maximum degree 2. Moreover, our algorithm can be used to compute Nash equilibria of graphical games on arbitrary trees, but the running time can be exponential, even when the tree has bounded degree. We show that this is inevitable -- any algorithm of this type will take exponential time, even on bounded-degree trees with pathwidth 2. It is an open question whether our algorithm runs in polynomial time on graphs with pathwidth 1, but we show that finding a Nash equilibrium for a 2-action graphical game in which the underlying graph has maximum degree 3 and constant pathwidth is PPAD-complete (so is unlikely to be tractable).

We address the problem of lowering the buyer's expected payments in shortest path auctions, where the buyer's goal is to purchase a path in a graph in which edges are owned by selfish agents. We show that by deleting some of the edges of the graph, one can reduce the total payment of the VCG mechanism by a factor of θ(n). However, we prove that it is NP-hard to find the best subset of edges to delete, even if the edge costs are small integers, or the graph has very simple structure; in the former case, this problem is hard to approximate, too. On the positive side, we describe a pseudopolynomial time algorithm for series-parallel graphs and fixed edge costs. Also, we discuss the applicability of this algorithm for the case of general (probabilistic) costs and derive a general lower bound on the performance of algorithms that are based on expected edge costs.