arXiv subjects

Edith Elkind

Publications and source records attributed to Edith Elkind.

At least 73 records · Page 4Linked to original sources

arXiv2019

Single-crossing Implementation

An election over a finite set of candidates is called single-crossing if, as we sweep through the list of voters from left to right, the relative order of every pair of candidates changes at most once. Such elections have many attractive properties: e.g., their majority relation is transitive and they admit efficient algorithms for problems that are NP-hard in general. If a given election is not single-crossing, it is important to understand what are the obstacles that prevent it from having this property. In this paper, we propose a mapping between elections and graphs that provides us with a convenient encoding of such obstacles. This mapping enables us to use the toolbox of graph theory in order to analyze the complexity of detecting nearly single-crossing elections, i.e., elections that can be made single-crossing by a small number of modifications.

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arXiv2019

Protecting Elections by Recounting Ballots

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.

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arXiv2019

Hedonic Diversity Games

We consider a coalition formation setting where each agent belongs to one of the two types, and agents' preferences over coalitions are determined by the fraction of the agents of their own type in each coalition. This setting differs from the well-studied Schelling's model in that some agents may prefer homogeneous coalitions, while others may prefer to be members of a diverse group, or a group that mostly consists of agents of the other type. We model this setting as a hedonic game and investigate the existence of stable outcomes using hedonic games solution concepts. We show that a core stable outcome may fail to exist and checking the existence of core stable outcomes is computationally hard. On the other hand, we propose an efficient algorithm to find an individually stable outcome under the natural assumption that agents' preferences over fractions of the agents of their own type are single-peaked.

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arXiv2019

Schelling Games on Graphs

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.

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arXiv2019

What Do Multiwinner Voting Rules Do? An Experiment Over the Two-Dimensional Euclidean Domain

We visualize aggregate outputs of popular multiwinner voting rules--SNTV, STV, Bloc, k-Borda, Monroe, Chamberlin--Courant, and HarmonicBorda--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.

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arXiv2017

Group Activity Selection on Social Networks

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 (gGASP). We show that if multiple groups can simultaneously engage in the same activity, finding a stable outcome is easy as long as the network is 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 gGASP admits a Nash stable outcome turns out to be NP-hard when the social network is a path or a star, or if the size of each connected component is bounded by a constant. We then study the parameterized complexity of finding outcomes of gGASP that are Nash stable, individually stable or core stable. For the parameter `number of activities', we propose an FPT algorithm for Nash stability for the case where the social network is acyclic and obtain a W[1]-hardness result for cliques (i.e., for standard GASP); similar results hold for individual stability. In contrast, finding a core stable outcome is hard even if the number of activities is bounded by a small constant, both for standard GASP and when the social network is a star. For the parameter `number of players', all problems we consider are in XP for arbitrary social networks; on the other hand, we prove W[1]-hardness results with respect to the parameter `number of players' for the case where the social network is a clique.

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arXiv2017

Cooperative Games with Bounded Dependency Degree

Cooperative games provide a framework to study cooperation among self-interested agents. They offer a number of solution concepts describing how the outcome of the cooperation should be shared among the players. Unfortunately, computational problems associated with many of these solution concepts tend to be intractable---NP-hard or worse. In this paper, we incorporate complexity measures recently proposed by Feige and Izsak (2013), called dependency degree and supermodular degree, into the complexity analysis of cooperative games. We show that many computational problems for cooperative games become tractable for games whose dependency degree or supermodular degree are bounded. In particular, we prove that simple games admit efficient algorithms for various solution concepts when the supermodular degree is small; further, we show that computing the Shapley value is always in FPT with respect to the dependency degree. Finally, we note that, while determining the dependency among players is computationally hard, there are efficient algorithms for special classes of games.

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arXiv2017

Cognitive Hierarchy and Voting Manipulation

By the Gibbard--Satterthwaite theorem, every reasonable voting rule for three or more alternatives is susceptible to manipulation: there exist elections where one or more voters can change the election outcome in their favour by unilaterally modifying their vote. When a given election admits several such voters, strategic voting becomes a game among potential manipulators: a manipulative vote that leads to a better outcome when other voters are truthful may lead to disastrous results when other voters choose to manipulate as well. We consider this situation from the perspective of a boundedly rational voter, and use the cognitive hierarchy framework to identify good strategies. We then investigate the associated algorithmic questions under the k-approval voting rule. We obtain positive algorithmic results for k=1 and 2, and NP- and coNP-hardness results for k>3.

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arXiv2017

Fair Division of a Graph

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 where the 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.

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arXiv2017

Committees providing EJR can be computed efficiently

We identify a whole family of approval-based multi-winner voting rules that satisfy PJR. Moreover, we identify a subfamily of voting rules within this family that satisfy EJR. All these voting rules can be computed in polynomial time as long as the subalgorithms that characterize each rule within the family are polynomial. One of the voting rules that satisfy EJR can be computed in $O(n m k)$.

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arXiv2017

Optimal Average Satisfaction and Extended Justified Representation in Polynomial Time

In this short note, we describe an approval-based committee selection rule that admits a polynomial-time algorithm and satisfies the Extended Justified Representation (EJR) axiom. This rule is based on approximately maximizing the PAV score, by means of local search. Our proof strategy is to show that this rule provides almost optimal average satisfaction to all cohesive groups of voters, and that high average satisfaction for cohesive groups implies extended justified representation.

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arXiv2017

On Parameterized Complexity of Group Activity Selection Problems on Social Networks

In Group Activity Selection Problem (GASP), players form coalitions to participate in activities and have preferences over pairs of the form (activity, group size). Recently, Igarashi et al. have initiated the study of group activity selection problems on social networks (gGASP): a group of players can engage in the same activity if the members of the group form a connected subset of the underlying communication structure. Igarashi et al. have primarily focused on Nash stable outcomes, and showed that many associated algorithmic questions are computationally hard even for very simple networks. In this paper we study the parameterized complexity of gGASP with respect to the number of activities as well as with respect to the number of players, for several solution concepts such as Nash stability, individual stability and core stability. The first parameter we consider in the number of activities. For this parameter, we propose an FPT algorithm for Nash stability for the case where the social network is acyclic and obtain a W[1]-hardness result for cliques (i.e., for classic GASP); similar results hold for individual stability. In contrast, finding a core stable outcome is hard even if the number of activities is bounded by a small constant, both for classic GASP and when the social network is a star. Another parameter we study is the number of players. While all solution concepts we consider become polynomial-time computable when this parameter is bounded by a constant, we prove W[1]-hardness results for cliques (i.e., for classic GASP).

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arXiv2017

The Condorcet Principle for Multiwinner Elections: From Shortlisting to Proportionality

We study two notions of stability in multiwinner elections that are based on the Condorcet criterion. The first notion was introduced by Gehrlein: A committee is stable if each committee member is preferred to each non-member by a (possibly weak) majority of voters. The second notion is called local stability (introduced in this paper): A size-$k$ committee is locally stable in an election with $n$ voters if there is no candidate $c$ and no group of more than $\frac{n}{k+1}$ voters such that each voter in this group prefers $c$ to each committee member. We argue that Gehrlein-stable committees are appropriate for shortlisting tasks, and that locally stable committees are better suited for applications that require proportional representation. The goal of this paper is to analyze these notions in detail, explore their compatibility with notions of proportionality, and investigate the computational complexity of related algorithmic tasks.

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arXiv2016

Proportional Rankings

In this paper we extend the principle of proportional representation to rankings. We consider the setting where alternatives need to be ranked based on approval preferences. In this setting, proportional representation requires that cohesive groups of voters are represented proportionally in each initial segment of the ranking. Proportional rankings are desirable in situations where initial segments of different lengths may be relevant, e.g., hiring decisions (if it is unclear how many positions are to be filled), the presentation of competing proposals on a liquid democracy platform (if it is unclear how many proposals participants are taking into consideration), or recommender systems (if a ranking has to accommodate different user types). We study the proportional representation provided by several ranking methods and prove theoretical guarantees. Furthermore, we experimentally evaluate these methods and present preliminary evidence as to which methods are most suitable for producing proportional rankings.

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arXiv2016

Group Activity Selection on Social Networks

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arXiv2016

Justified Representation in Approval-Based Committee Voting

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 agreement 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. However, it turns out that several prominent approval-based voting rules may fail to output such a committee. In particular, while Proportional Approval Voting (PAV) always outputs a committee that provides JR, Reweighted Approval Voting (RAV), a tractable approximation to PAV, does not have this property. 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 we consider do not; indeed, EJR can be used to characterize PAV within the class of weighted PAV rules. We also consider several other questions related to JR and EJR, including the relationship between JR/EJR and core stability, and the complexity of the associated algorithmic problems.

cs.MA↗

arXiv2016

Hedonic Games with Graph-restricted Communication

We study hedonic coalition formation games in which cooperation among the players is restricted by a graph structure: a subset of players can form a coalition if and only if they are connected in the given graph. We investigate the complexity of finding stable outcomes in such games, for several notions of stability. In particular, we provide an efficient algorithm that finds an individually stable partition for an arbitrary hedonic game on an acyclic graph. We also introduce a new stability concept -in-neighbor stability- which is tailored for our setting. We show that the problem of finding an in-neighbor stable outcome admits a polynomial-time algorithm if the underlying graph is a path, but is NP-hard for arbitrary trees even for additively separable hedonic games; for symmetric additively separable games we obtain a PLS-hardness result.

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arXiv2016

Structure in Dichotomous Preferences

Many hard computational social choice problems are known to become tractable when voters' preferences belong to a restricted domain, such as those of single-peaked or single-crossing preferences. However, to date, all algorithmic results of this type have been obtained for the setting where each voter's preference list is a total order of candidates. The goal of this paper is to extend this line of research to the setting where voters' preferences are dichotomous, i.e., each voter approves a subset of candidates and disapproves the remaining candidates. We propose several analogues of the notions of single-peaked and single-crossing preferences for dichotomous profiles and investigate the relationships among them. We then demonstrate that for some of these notions the respective restricted domains admit efficient algorithms for computationally hard approval-based multi-winner rules.

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