Although approximate notions of envy-freeness—such as envy-freeness up to one good (EF1)—have been extensively studied for indivisible goods, the seemingly simpler fairness concept of proportionality up to one good (PROP1) has received far less attention. For additive valuations, every EF1 allocation is PROP1, and well-known algorithms such as round-robin and envy-cycle elimination compute such allocations in polynomial time. PROP1 is also compatible with Pareto efficiency, as maximum Nash welfare allocations are EF1 and hence PROP1. We ask whether these favorable properties extend to non-additive valuations. We study a broad class of allocation instances with satiating goods, where agents have non-negative valuation functions that need not be monotone, allowing for negative marginal values. We present the following results: --EF1 implies PROP1 for submodular valuations over satiating goods, ensuring existence and efficient computation via envy-cycle elimination for monotone submodular valuations; --Round-robin computes a partial PROP1 allocation after the second-to-last round for satiating submodular goods and a complete PROP1 for submodular monotone valuations; --PROP1 allocations for satiating subadditive goods can be computed in polynomial-time; --Maximum Nash welfare allocations are PROP1 for monotone submodular goods, revealing yet another facet of their ``unreasonable fairness.''

Serial dictatorship is a simple mechanism for coordinating agents in solving combinatorial optimization problems according to their preferences. The most representative such problem is one-sided matching, in which a set of n agents have values for a set of n items, and the objective is to compute a matching of the agents to the items of maximum total value (a.k.a., social welfare). Following the recent framework of Caragiannis and Rathi (2023), we consider a model in which the agent-item values are not available upfront but become known by querying agent sequences. In particular, when the agents are asked to act in a sequence, they respond by picking their favorite item that has not been picked by agents who acted before and reveal their value for it. Can we compute an agent sequence that induces a social welfare-optimal matching? We answer this question affirmatively and present an algorithm that uses polynomial number (specifically, O(n^5) of queries). This solves the main open problem stated by Caragiannis and Rathi (2023). Our analysis uses a potential function argument that measures progress towards learning the underlying edge-weight information. Furthermore, the algorithm has a truthful implementation by adapting the paradigm of VCG payments.

The problem of identifying the satisfiability threshold of random 3-SAT formulas has received a lot of attention during the last decades and has inspired the study of other threshold phenomena in random combinatorial structures. The classical assumption in this line of research is that, for a given set of n Boolean variables, each clause is drawn uniformly at random among all sets of three literals from these variables, independently from other clauses. Here, we keep the uniform distribution of each clause, but deviate significantly from the independence assumption and consider richer families of probability distributions. For integer parameters n, m, and k, we denote by ℱ_k(n,m) the family of probability distributions that produce formulas with m clauses, each selected uniformly at random from all sets of three literals from the n variables, so that the clauses are k-wise independent. Our aim is to make general statements about the satisfiability or unsatisfiability of formulas produced by distributions in ℱ_k(n,m) for different values of the parameters n, m, and k. Our technical results are as follows: First, all probability distributions in ℱ₂(n,m) with m ∈ Ω(n³) return unsatisfiable formulas with high probability. This result is tight. We show that there exists a probability distribution 𝒟 ∈ ℱ₃(n,m) with m ∈ O(n³) so that a random formula drawn from 𝒟 is almost always satisfiable. In contrast, for m ∈ Ω(n²), any probability distribution 𝒟 ∈ ℱ₄(n,m) returns an unsatisfiable formula with high probability. This is our most surprising and technically involved result. Finally, for any integer k ≥ 2, any probability distribution 𝒟 ∈ ℱ_k(n,m) with m ∈ O(n^{1-1/k}) returns a satisfiable formula with high probability.

The design of energy markets is a subject of ongoing debate, particularly concerning the choice between the widely adopted Pay-as-Clear (PC) pricing mechanism and the alternative Pay-as-Bid (PB). These mechanisms determine how energy producers are compensated: under PC, all selected producers are paid the market-clearing price (i.e., the highest accepted bid), while under PB, each selected producer is paid their own submitted bid. The overarching objective is to meet the total demand for energy at minimal cost in the presence of strategic behavior. We present two key theoretical results. First, no mechanism can uniformly dominate PC or PB. This means that for any mechanism $$\mathcal {M}$$ , there exists a market configuration and a mixed-strategy Nash equilibrium of PC (respectively for PB) that yields strictly lower total energy costs than under $$\mathcal {M}$$ . Second, in terms of worst-case equilibrium outcomes, PB consistently outperforms PC: across all market instances, the highest possible equilibrium price under PB is strictly lower than that under PC. This suggests a structural robustness of PB to strategic manipulation. These theoretical insights are further supported by extensive simulations based on no-regret learning dynamics, which consistently yield lower average market prices in several energy market settings.

Motivated by recent work in computational social choice, we extend the metric distortion framework to clustering problems. Given a set of n agents located in an underlying metric space, our goal is to partition them into k clusters, optimizing some social cost objective. The metric space is defined by a distance function d between the agent locations. Information about d is available only implicitly via n rankings, through which each agent ranks all other agents in terms of their distance from her. Still, even though no cardinal information (i.e., the exact distance values) is available, we would like to evaluate clustering algorithms in terms of social cost objectives that are defined using d. This is done using the notion of distortion, which measures how far from optimality a clustering can be, taking into account all underlying metrics that are consistent with the ordinal information available. Unfortunately, the most important clustering objectives (e.g., those used in the well-known k-median and k-center problems) do not admit algorithms with finite distortion. To sidestep this disappointing fact, we follow two alternative approaches: We first explore whether resource augmentation can be beneficial. We consider algorithms that use more than k clusters but compare their social cost to that of the optimal k-clusterings. We show that using exponentially (in terms of k) many clusters, we can get low (constant or logarithmic) distortion for the k-center and k-median objectives. Interestingly, such an exponential blowup is shown to be necessary. More importantly, we explore whether limited cardinal information can be used to obtain better results. Somewhat surprisingly, for k-median and k-center, we show that a number of queries that is polynomial in k and only logarithmic in n (i.e., only sublinear in the number of agents for the most relevant scenarios in practice) is enough to get constant distortion.

Recent works have studied the design of algorithms for selecting representative sortition panels. However, the most central question remains unaddressed: Do these panels reflect the entire population's opinion? We present a positive answer by adopting the concept of metric distortion from computational social choice, which aims to quantify how much a panel's decision aligns with the ideal decision of the population when preferences and agents lie on a metric space. We show that uniform selection needs only logarithmically many agents in terms of the number of alternatives to achieve almost optimal distortion. We also show that Fair Greedy Capture, a selection algorithm introduced recently by Ebadian and Micha (2024), matches uniform selection's guarantees of almost optimal distortion and also achieves constant ex-post distortion, ensuring a ``best of both worlds'' performance.

We revisit the recently developed framework of proportionally fair clustering, where the goal is to provide group fairness guarantees that become stronger for groups of data points that are large and cohesive. Prior work applies this framework to centroid-based clustering, where points are partitioned into clusters, and the cost to each data point is measured by its distance to a centroid assigned to its cluster. However, real-life applications often do not require such centroids. We extend the theory of proportionally fair clustering to non-centroid clustering by considering a variety of cost functions, both metric and non-metric, for a data point to be placed in a cluster with other data points. Our results indicate that Greedy Capture, a clustering algorithm developed for centroid clustering, continues to provide strong proportional fairness guarantees for non-centroid clustering, although the guarantees are significantly different and establishing them requires novel proof ideas. We also design algorithms for auditing proportional fairness of a given clustering solution. We conduct experiments on real data which suggest that traditional clustering algorithms are highly unfair, while our algorithms achieve strong fairness guarantees with a moderate loss in common clustering objectives.

We study a participatory budgeting problem, where a set of strategic agents wish to split a divisible budget among different projects by aggregating their proposals on a single division. Unfortunately, the straightforward rule that divides the budget proportionally is susceptible to manipulation. Recently, a class of truthful mechanisms has been proposed, namely the moving phantom mechanisms. One such mechanism satisfies the proportionality property, in the sense that in the extreme case where all agents prefer a single project to receive the whole amount, the budget is assigned proportionally. While proportionality is a naturally desired property, it is defined over a limited type of preference profiles. To address this, we expand the notion of proportionality, by proposing a quantitative framework that evaluates a budget aggregation mechanism according to its worst-case distance from the proportional allocation. Crucially, this is defined for every preference profile. We study this measure on the class of moving phantom mechanisms, and we provide approximation guarantees. For two projects, we show that the Uniform Phantom mechanism is optimal among all truthful mechanisms. For three projects, we propose a new, proportional mechanism that is optimal among all moving phantom mechanisms. Finally, we provide impossibility results regarding the approximability of moving phantom mechanisms.

Abstract Motivated by the success of the serial dictatorship mechanism in social choice settings, we explore its usefulness in tackling various combinatorial optimization problems. We do so by considering an abstract model, in which a set of agents are asked to act in a particular ordering, called the action sequence. Each agent acts in a way that gives her the maximum possible value, given the actions of the agents who preceded her in the action sequence. Our goal is to compute action sequences that yield approximately optimal total value to the agents (a.k.a., social welfare). We assume query access to the value $$v_i(S)$$ v i ( S ) that the agent i gets when she acts after the agents in the ordered set S. We establish tight bounds on the social welfare that can be achieved using polynomially many queries. Even though these bounds show a marginally sublinear approximation of optimal social welfare in general, excellent approximations can be obtained when the valuations stem from an underlying combinatorial domain. Indicatively, when the valuations are defined using bipartite matchings, arborescences in directed graphs, and satisfiability of Boolean expressions, simple query-efficient algorithms yield 2-approximations. We discuss issues related to truthfulness and show how some of our algorithms can be implemented truthfully using VCG-like payments. Finally, we introduce and study the price of serial dictatorship, a notion that provides an optimistic measure of the quality of combinatorial optimization solutions generated by action sequences.

For the fundamental problem of fairly dividing a set of indivisible items among agents, envy-freeness up to any item (EFX) and maximin fairness (MMS) are arguably the most compelling fairness concepts proposed till now. Unfortunately, despite significant efforts over the past few years, whether EFX allocations always exist is still an enigmatic open problem, let alone their efficient computation. Furthermore, today we know that MMS allocations are not always guaranteed to exist. These facts weaken the usefulness of both EFX and MMS, albeit their appealing conceptual characteristics. We propose two alternative fairness concepts—called epistemic EFX (EEFX) and minimum EFX value fairness (MXS)---inspired by EFX and MMS. For both, we explore their relationships to well-studied fairness notions and, more importantly, prove that EEFX and MXS allocations always exist and can be computed efficiently for additive valuations. Our results justify that the new fairness concepts are excellent alternatives to EFX and MMS.

We study a scenario where an adjudication task (e.g., the resolution of a binary dispute) is outsourced to a set of agents who are appointed as jurors. This scenario is particularly relevant in a Web3 environment, where no verification of the adjudication outcome is possible, and the appointed agents are, in principle, indifferent to the final verdict. We consider simple adjudication mechanisms that use (1) majority voting to decide the final verdict and (2) a payment function to reward the agents with the majority vote and possibly punish the ones in the minority. Agents interact with such a mechanism strategically: they exert some effort to understand how to properly judge the dispute and cast a yes/no vote that depends on this understanding and on information they have about the rest of the votes. Eventually, they vote so that their utility (i.e., their payment from the mechanism minus the cost due to their effort) is maximized. Under reasonable assumptions about how an agent's effort is related to her understanding of the dispute, we show that appropriate payment functions can be used to recover the correct adjudication outcome with high probability. Our findings follow from a detailed analysis of the induced strategic game and make use of both theoretical arguments and simulation experiments.

Cut games are among the most fundamental strategic games in algorithmic game theory. It is well-known that computing an exact pure Nash equilibrium in these games is PLS-hard, so research has focused on computing approximate equilibria. We present a polynomial-time algorithm that computes 2.7371-approximate pure Nash equilibria in cut games. This is the first improvement to the previously best-known bound of 3, due to the work of Bhalgat, Chakraborty, and Khanna from EC 2010. Our algorithm is based on a general recipe proposed by Caragiannis, Fanelli, Gravin, and Skopalik from FOCS 2011 and applied on several potential games since then. The first novelty of our work is the introduction of a phase that can identify subsets of players who can simultaneously improve their utilities considerably. This is done via semidefinite programming and randomized rounding. In particular, a negative objective value to the semidefinite program guarantees that no such considerable improvement is possible for a given set of players. Otherwise, randomized rounding of the SDP solution is used to identify a set of players who can simultaneously improve their strategies considerably and allows the algorithm to make progress. The way rounding is performed is another important novelty of our work. Here, we exploit an idea that dates back to a paper by Feige and Goemans from 1995, but we take it to an extreme that has not been analyzed before.

We study the problem of impartial selection, a topic that lies at the intersection of computational social choice and mechanism design. The goal is to select the most popular individual among a set of community members. The input can be modeled as a directed graph, where each node represents an individual, and a directed edge indicates nomination or approval of a community member to another. An impartial mechanism is robust to potential selfish behavior of the individuals and provides appropriate incentives to voters to report their true preferences by ensuring that the chance of a node to become a winner does not depend on its outgoing edges. The goal is to design impartial mechanisms that select a node with an in-degree that is as close as possible to the highest in-degree. We measure the efficiency of such a mechanism by the difference of these in-degrees, known as its additive approximation.

A divisible public resource is to be divided among projects. We study rules that decide on a distribution of the budget when voters have ordinal preference rankings over projects. Examples of such portioning problems are participatory budgeting, time shares, and parliament elections. We introduce a family of rules for portioning, inspired by positional scoring rules. Rules in this family are given by a scoring vector (such as plurality or Borda) associating a positive value with each rank in a vote, and an aggregation function such as leximin or the Nash product. Our family contains well-studied rules, but most are new. We discuss computational and normative properties of our rules. We focus on fairness, and introduce the SD-core, a group fairness notion. Our Nash rules are in the SD-core, and the leximin rules satisfy individual fairness properties. Both are Pareto-efficient.

We study a participatory budgeting problem, where a set of strategic agents wish to split a divisible budget among different projects by aggregating their proposals on a single division. Unfortunately, the straightforward rule that divides the budget proportionally is susceptible to manipulation. Recently, a class of truthful mechanisms has been proposed, namely the moving phantom mechanisms. One such mechanism satisfies the proportionality property, in the sense that in the extreme case where all agents prefer a single project to receive the whole amount, the budget is assigned proportionally. While proportionality is a naturally desired property, it is defined over a limited type of preference profiles. To address this, we expand the notion of proportionality, by proposing a quantitative framework that evaluates a budget aggregation mechanism according to its worst-case distance from the proportional allocation. Crucially, this is defined for every preference profile. We study this measure on the class of moving phantom mechanisms, and we provide approximation guarantees. For two projects, we show that the Uniform Phantom mechanism is optimal among all truthful mechanisms. For three projects, we propose a new, proportional mechanism that is optimal among all moving phantom mechanisms. Finally, we provide impossibility results regarding the approximability of moving phantom mechanisms.

We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, n agents and m alternatives are located in an underlying metric space. The exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a committee of k alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a committee is her distance from the q-th closest alternative in the committee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of k and q: The distortion is unbounded when q

We study the PAC learnability of multiwinner voting, focusing on the class of approval-based committee scoring (ABCS) rules. These are voting rules applied on profiles with approval ballots, where each voter approves some of the candidates. According to ABCS rules, each committee of k candidates collects from each voter a score, that depends on the size of the voter's ballot and on the size of its intersection with the committee. Then, committees of maximum score are the winning ones. Our goal is to learn a target rule (i.e., to learn the corresponding scoring function) using information about the winning committees of a small number of sampled profiles. Despite the existence of exponentially many outcomes compared to single-winner elections, we show that the sample complexity is still low: a polynomial number of samples carries enough information for learning the target rule with high confidence and accuracy. Unfortunately, even simple tasks that need to be solved for learning from these samples are intractable. We prove that deciding whether there exists some ABCS rule that makes a given committee winning in a given profile is a computationally hard problem. Our results extend to the class of sequential Thiele rules, which have received attention due to their simplicity.

Motivated by fair division applications, we study a fair connected graph partitioning problem, in which an undirected graph with m nodes must be divided between n agents such that each agent receives a connected subgraph and the partition is fair. We study approximate versions of two fairness criteria: \alpha-proportionality requires that each agent receive a subgraph with at least (1/\alpha)*m/n nodes, and \alpha-balancedness requires that the ratio between the sizes of the largest and smallest subgraphs be at most \alpha. Unfortunately, there exist simple examples in which no partition is reasonably proportional or balanced. To circumvent this, we introduce the idea of charity. We show that by "donating" just n-1 nodes, we can guarantee the existence of 2-proportional and almost 2-balanced partitions (and find them in polynomial time), and that this result is almost tight. More generally, we chart the tradeoff between the size of charity and the approximation of proportionality or balancedness we can guarantee.

We consider the problem of fairly dividing a set of indivisible items. Much of the fair division literature assumes that the items are “goods” that yield positive utility for the agents. There is also some work in which the items are “chores” that yield negative utility for the agents. In this paper, we consider a more general scenario in which an agent may have positive or negative utility for each item. This framework captures, e.g., fair task assignment, where agents can experience both positive and negative utility for each task. We demonstrate that whereas some of the positive axiomatic and computational results extend to this more general setting, others do not. We present several new and efficient algorithms for finding fair allocations in this general setting. We also point out several gaps in the literature regarding the existence of allocations that satisfy certain fairness and efficiency properties and examine the complexity of computing such allocations.

Approval-based multiwinner voting rules have recently received much attention in the Computational Social Choice literature. Such rules aggregate approval ballots and determine a winning committee of alternatives. To assess effectiveness, we propose to employ new noise models that are specifically tailored for approval votes and committees. These models take as input a ground truth committee and return random approval votes to be thought of as noisy estimates of the ground truth. A minimum robustness requirement for an approval-based multiwinner voting rule is to return the ground truth when applied to profiles with sufficiently many noisy votes. Our results indicate that approval-based multiwinner voting can indeed be robust to reasonable noise. We further refine this finding by presenting a hierarchy of rules in terms of how robust to noise they are.

We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, n agents and m alternatives are located in an underlying metric space. The exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a committee of k alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a committee is her distance from the q-th closest alternative in the committee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of k and q: The distortion is unbounded when q⩽k/3, asymptotically linear in the number of agents when k/3 k/2.

Social networks on the Internet have seen an enormous growth recently and play a crucial role in different aspects of today's life. They have facilitated information dissemination in ways that have been beneficial for their users but they are often used strategically in order to spread information that only serves the objectives of particular users. These properties have inspired a revision of classical opinion formation models from sociology using game-theoretic notions and tools. We follow the same modeling approach, focusing on scenarios where the opinion expressed by each user is a compromise between her internal belief and the opinions of a small number of neighbors among her social acquaintances. We formulate simple games that capture this behavior and quantify the inefficiency of equilibria using the well-known notion of the price of anarchy. Our results indicate that compromise comes at a cost that strongly depends on the neighborhood size.

Impartial selection has recently received much attention within the multi-agent systems community. The task is, given a directed graph representing nominations to the members of a community by other members, to select a member with the highest number of nominations. This seemingly trivial goal becomes challenging when there is an additional impartiality constraint, requiring that no single member can influence her chance of being selected. Recent progress has identified impartial selection rules with optimal approximation ratios. Moreover, it was noted that worst-case instances are graphs with few vertices. Motivated by this fact, we propose the study of additive approximation, the difference between the highest number of nominations and the number of nominations of the selected member, as an alternative measure of the quality of impartial selection. Our positive results include two randomized impartial selection mechanisms which have additive approximation guarantees of ${\varTheta }(\sqrt {n})$ and ${\varTheta }(n^{2/3}\ln ^{1/3}n)$ for the two most studied models in the literature, where n denotes the community size. We complement our positive results by providing negative results for various cases. First, we provide a characterization for the interesting class of strong sample mechanisms, which allows us to obtain lower bounds of n − 2, and of ${\varOmega }(\sqrt {n})$ for their deterministic and randomized variants respectively. Finally, we present a general lower bound of 3 for all deterministic impartial mechanisms.

With very few exceptions, recent research in fair division has mostly focused on deterministic allocations. Deviating from this trend, we study the fairness notion of interim envy-freeness (iEF) for lotteries over allocations, which serves as a sweet spot between the too stringent notion of ex-post envy-freeness and the very weak notion of ex-ante envy-freeness. iEF is a natural generalization of envy-freeness to random allocations in the sense that a deterministic envy-free allocation is iEF (when viewed as a degenerate lottery). It is also certainly meaningful as it allows for a richer solution space, which includes solutions that are provably better than envy-freeness according to several criteria. Our analysis relates iEF to other fairness notions as well, and reveals tradeoffs between iEF and efficiency. Even though several of our results apply to general fair division problems, we are particularly interested in instances with equal numbers of agents and items where allocations are perfect matchings of the items to the agents. Envy-freeness can be trivially decided and (when it can be achieved, it) implies full efficiency in this setting. Although computing iEF allocations in matching allocation instances is considerably more challenging, we show how to compute them in polynomial time, while also maximizing several efficiency objectives. Our algorithms use the ellipsoid method for linear programming and efficient solutions to a novel variant of the bipartite matching problem as a separation oracle. We also study the extension of interim envy-freeness notion when payments to or from the agents are allowed. We present a series of results on two optimization problems, including a generalization of the classical rent division problem to random allocations using interim envy-freeness as the solution concept.

Fair division has emerged as a very hot topic in EconCS research, and envy-freeness is among the most compelling fairness concepts. An allocation of indivisible items to agents is envy-free if no agent prefers the bundle of any other agent to his own in terms of value. As envy-freeness is rarely a feasible goal, there is a recent focus on relaxations of its definition. An approach in this direction is to complement allocations with payments (or subsidies) to the agents. A feasible goal then is to achieve envy-freeness in terms of the total value an agent gets from the allocation and the subsidies. We consider the natural optimization problem of computing allocations that are envy-freeable using the minimum amount of subsidies. As the problem is NP-hard, we focus on the design of approximation algorithms. On the positive side, we present an algorithm which, for a constant number of agents, approximates the minimum amount of subsidies within any required accuracy, at the expense of a graceful increase in the running time. On the negative side, we show that, for a superconstant number of agents, the problem of minimizing subsidies for envy-freeness is not only hard to compute exactly (as a folklore argument shows) but also, more importantly, hard to approximate.

We study a generalization of the classical stable matching problem that allows for cardinal preferences (as opposed to ordinal) and fractional matchings (as opposed to integral). In this cardinal setting, stable fractional matchings can have much larger social welfare than stable integral ones. Our goal is to understand the computational complexity of finding an optimal (i.e., welfare-maximizing) stable fractional matching. We consider both exact and approximate stability notions, and provide simple approximation algorithms with weak welfare guarantees. Our main result is that, somewhat surprisingly, achieving better approximations is computationally hard. To the best of our knowledge, these are the first computational complexity results for stable fractional matchings in the cardinal model. En route to these results, we provide a number of structural observations that could be of independent interest.

We consider weighted congestion games with polynomial latency functions of maximum degree d ≥ 1 . For these games, we investigate the existence and efficiency of approximate pure Nash equilibria which are obtained through sequences of unilateral improvement moves by the players. By exploiting a simple technique, we firstly show that these games admit an infinite set of d -approximate potential functions. This implies that there always exists a d -approximate pure Nash equilibrium which can be reached through any sequence of d -approximate improvement moves by the players. As a corollary, we also obtain that, under mild assumptions on the structure of the players' strategies, these games also admit a constant approximate potential function. Secondly, using a simple potential function argument, we are able to show that a ( d + δ ) -approximate pure Nash equilibrium of cost at most ( d + 1 ) / ( d + δ ) times the cost of an optimal state always exists, for every δ ∈ [ 0 , 1 ] .

We consider the problem of fairly dividing a set of items. Much of the fair division literature assumes that the items are ``goods'' i.e., they yield positive utility for the agents. There is also some work where the items are ``chores'' that yield negative utility for the agents. In this paper, we consider a more general scenario where an agent may have negative or positive utility for each item. This framework captures, e.g., fair task assignment, where agents can have both positive and negative utilities for each task. We show that whereas some of the positive axiomatic and computational results extend to this more general setting, others do not. We present several new and efficient algorithms for finding fair allocations in this general setting. We also point out several gaps in the literature regarding the existence of allocations satisfying certain fairness and efficiency properties and further study the complexity of computing such allocations.

A public divisible resource is to be divided among projects. We study rules that decide on a distribution of the budget when voters have ordinal preference rankings over projects. Examples of such portioning problems are participatory budgeting, time shares, and parliament elections. We introduce a family of rules for portioning, inspired by positional scoring rules. Rules in this family are given by a scoring vector (such as plurality or Borda) associating a positive value with each rank in a vote, and an aggregation function such as leximin or the Nash product. Our family contains well-studied rules, but most are new. We discuss computational and normative properties of our rules. We focus on fairness, and introduce the SD-core, a group fairness notion. Our Nash rules are in the SD-core, and the leximin rules satisfy individual fairness properties. Both are Pareto-efficient.

Liquid democracy, which combines features of direct and representative democracy has been proposed as a modern practice for collective decision making. Its advocates support that by allowing voters to delegate their vote to more informed voters can result in better decisions. In an attempt to evaluate the validity of such claims, we study liquid democracy as a means to discover an underlying ground truth. We revisit a recent model by Kahng et al. [2018] and conclude with three negative results, criticizing an important assumption of their modeling, as well as liquid democracy more generally. In particular, we first identify cases where natural local mechanisms are much worse than either direct voting or the other extreme of full delegation to a common dictator. We then show that delegating to less informed voters may considerably increase the chance of discovering the ground truth. Finally, we show that deciding delegations that maximize the probability to find the ground truth is a computationally hard problem.

We study the following fundamental graph problem that models the important task of deanonymizing social networks. We are given a graph representing an eponymous social network and another graph, representing an anonymous social network, which has been produced by the original one after removing some of its nodes and adding some noise on the links. Our objective is to correctly associate as many nodes of the anonymous network as possible to their corresponding node in the eponymous network. We present two algorithms that attack the problem by exploiting only the structure of the two graphs. The first one exploits bipartite matching computations and is relatively fast. The second one is a local search heuristic which can use the outcome of our first algorithm as an initial solution and further improve it. We have applied our algorithms on inputs that have been produced by well-known random models for the generation of social networks as well as on inputs that use real social networks. Our algorithms can tolerate noise at the level of up to 10%. Interestingly, our results provide further evidence to which graph generation models are most suitable for modeling social networks and distinguish them from unrealistic ones.

We study the existence of allocations of indivisible goods that are envy-free up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph G. When the items are arranged in a path, we show that EF1 allocations are guaranteed to exist for arbitrary monotonic utility functions over bundles, provided that either there are at most four agents, or there are any number of agents but they all have identical utility functions. Our existence proofs are based on classical arguments from the divisible cake-cutting setting, and involve discrete analogues of cut-and-choose, of Stromquist's moving-knife protocol, and of the Su-Simmons argument based on Sperner's lemma. Sperner's lemma can also be used to show that on a path, an EF2 allocation exists for any number of agents. Except for the results using Sperner's lemma, all of our procedures can be implemented by efficient algorithms. Our positive results for paths imply the existence of connected EF1 or EF2 allocations whenever G is traceable, i.e., contains a Hamiltonian path. For the case of two agents, we completely characterize the class of graphs G that guarantee the existence of EF1 allocations as the class of graphs whose biconnected components are arranged in a path. This class is strictly larger than the class of traceable graphs; one can check in linear time whether a graph belongs to this class, and if so return an EF1 allocation.

Nowadays, several crowdsourcing projects exploit social choice methods for computing an aggregate ranking of alternatives given individual rankings provided by workers. Motivated by such systems, we consider a setting where each worker is asked to rank a fixed (small) number of alternatives and, then, a positional scoring rule is used to compute the aggregate ranking. Among the apparently infinite such rules, what is the best one to use? To answer this question, we assume that we have partial access to an underlying true ranking. Then, the important optimization problem to be solved is to compute the positional scoring rule whose outcome, when applied to the profile of individual rankings, is as close as possible to the part of the underlying true ranking we know. We study this fundamental problem from a theoretical point of view and present positive and negative complexity results. Furthermore, we complement our theoretical findings with experiments on real-world and synthetic data.

In the context of fair allocation of indivisible items, fairness concepts often compare the satisfaction of an agent to the satisfaction she would have from items that are not allocated to her: in particular, envy-freeness requires that no agent prefers the share of someone else to her own share. We argue that these notions could also be defined relative to the knowledge that an agent has on how the items that she does not receive are distributed among other agents. We define a family of epistemic notions of envy-freeness, parameterized by a social graph, where an agent observes the share of her neighbours but not of her non-neighbours. We also define an intermediate notion between envy-freeness and proportionality, also parameterized by a social graph. These weaker notions of envy-freeness are useful when seeking a fair allocation, since envy-freeness is often too strong. We position these notions with respect to known ones, thus revealing new rich hierarchies of fairness concepts. Finally, we present a very general framework that covers all the existing and many new fairness concepts.

We study the efficiency of mechanisms for allocating a divisible resource. Given scalar signals submitted by all users, such a mechanism decides the fraction of the resource that each user will receive and a payment that will be collected from her. Users are self-interested and aim to maximize their utility (defined as their value for the resource fraction they receive minus their payment). Starting with the seminal work of Johari and Tsitsiklis, a long list of papers studied the price of anarchy (in terms of the social welfare—the total users’ value) of resource allocation mechanisms for a variety of allocation and payment rules. Here, we further assume that each user has a budget constraint that invalidates strategies that yield a payment that is higher than the user’s budget. This subtle assumption, which is arguably more realistic, constitutes the traditional price of anarchy analysis meaningless as the set of equilibria may change drastically and their social welfare can be arbitrarily far from optimal. Instead, we study the price of anarchy using the liquid welfare benchmark that measures efficiency taking budget constraints into account. We show a tight bound of 2 on the liquid price of anarchy of the well-known Kelly mechanism and prove that this result is essentially best possible among all multiuser resource allocation mechanisms. This comes in sharp contrast to the no-budget setting where there are mechanisms that considerably outperform Kelly in terms of social welfare and even achieve full efficiency. In our proofs, we exploit the particular structure of worst-case games and equilibria, which also allows us to design (nearly) optimal two-player mechanisms by solving simple differential equations.

Nowadays, several crowdsourcing projects exploit social choice methods for computing an aggregate ranking of alternatives given individual rankings provided by workers. Motivated by such systems, we consider a setting where each worker is asked to rank a fixed (small) number of alternatives and, then, a positional scoring rule is used to compute the aggregate ranking. Among the apparently infinite such rules, what is the best one to use? To answer this question, we assume that we have partial access to an underlying true ranking. Then, the important optimization problem to be solved is to compute the positional scoring rule whose outcome, when applied to the profile of individual rankings, is as close as possible to the part of the underlying true ranking we know. We study this fundamental problem from a theoretical point of view and present positive and negative complexity results. Furthermore, we complement our theoretical findings with experiments on real-world and synthetic data.

We revisit fundamental problems in undirected and directed graphs, such as the problems of computing spanning trees, shortest paths, steiner trees, and spanning arborescences of minimum cost. We assume that there are d different cost functions associated with the edges of the input graph and seek for solutions to the resulting multidimensional graph problems so that the p-norm of the different costs of the solution is minimized. We present combinatorial algorithms that achieve very good approximations for this objective. The main advantage of our algorithms is their simplicity: they are as simple as classical combinatorial graph algorithms of Dijkstra and Kruskal, or the greedy algorithm for matroids.

Social networks on the Internet have seen an enormous growth recently and play a crucial role in different aspects of today's life. They have facilitated information dissemination in ways that have been beneficial for their users but it is also a common belief that they are often used strategically in order to spread information that only serves the objectives of particular users. These properties have inspired a revision of classical opinion formation models from sociology using game-theoretic notions and tools. We follow the same modeling approach, focusing on scenarios where the opinion expressed by each user is a compromise between her internal belief and the opinions of a small number of neighbors among her social acquaintances. We formulate simple games that capture this behavior and quantify the inefficiency of equilibria using the well-known notion of the price of anarchy. Our results indicate that compromise comes at a cost that strongly depends on the neighborhood size.

We consider a voting scenario where agents have opinions that are estimates of an underlying common ground truth ranking of the available alternatives, and each agent is asked to approve a set with her most preferred alternatives. We assume that estimates are implicitly formed using the well-known Mallows model for generating random rankings. We show that k-approval voting --- where all agents are asked to approve the same number k of alternatives and the outcome is obtained by sorting the alternatives in terms of their number of approvals --- has exponential sample complexity for all values of k. This negative result suggests that an exponential (in terms of the number of alternatives m) number of agents is always necessary in order to recover the ground truth ranking with high probability. In contrast, by just asking each agent to approve a random number of alternatives, the sample complexity improves dramatically: it now depends only polynomially on m. Our results may have implications on the effectiveness of crowdsourcing applications that ask workers to provide their input by approving sets of available alternatives.

We revisit the problem of designing optimal, individually rational matching mechanisms (in a general sense, allowing for cycles in directed graphs), where each player—who is associated with a subset of vertices—matches as many of his own vertices when he opts into the matching mechanism as when he opts out. We offer a new perspective on this problem by considering an arbitrary graph, but assuming that vertices are associated with players at random. Our main result asserts that, under certain conditions, any fixed optimal matching is likely to be individually rational up to lower-order terms. We also show that a simple and practical mechanism is (fully) individually rational, and likely to be optimal up to lower-order terms. We discuss the implications of our results for market design in general, and kidney exchange in particular.

How should one aggregate ordinal preferences expressed by voters into a measurably superior social choice? A well-established approach -- which we refer to as implicit utilitarian voting -- assumes that voters have latent utility functions that induce the reported rankings, and seeks voting rules that approximately maximize utilitarian social welfare. We extend this approach to the design of rules that select a subset of alternatives. We derive analytical bounds on the performance of optimal (deterministic as well as randomized) rules in terms of two measures, distortion and regret. Empirical results show that regret-based rules are more compelling than distortion-based rules, leading us to focus on developing a scalable implementation for the optimal (deterministic) regret-based rule. Our methods underlie the design and implementation of RoboVote.org, a not-for-profit website that helps users make group decisions via AI-driven voting methods.

We study the paradigmatic fair division problem of fairly allocating a divisible good among agents with heterogeneous preferences, commonly known as cake cutting. Classic cake cutting protocols are susceptible to manipulation. Do their strategic outcomes still guarantee fairness? To address this question we adopt a novel algorithmic approach, proposing a concrete computational model and reasoning about the game-theoretic properties of algorithms that operate in this model. Specifically, we show that each protocol in the class of generalized cut and choose (GCC) protocols --- which includes the most important discrete cake cutting protocols --- is guaranteed to have approximate subgame perfect Nash equilibria, or even exact equilibria if the protocol's tie-breaking rule is flexible. We further observe that the (approximate) equilibria of proportional protocols --- which guarantee each of the n agents a 1/n-fraction of the cake --- must be (approximately) proportional, thereby answering the above question in the positive (at least for one common notion of fairness).

Our aim with co-rank is to facilitate the grading of exams or assignments in massive open online courses (MOOCs).

We revisit the classic problem of estimating the population mean of an unknown single-dimensional distribution from samples, taking a game-theoretic viewpoint. In our setting, samples are supplied by strategic agents, who wish to pull the estimate as close as possible to their own value. In this setting, the sample mean gives rise to manipulation opportunities, whereas the sample median does not. Our key question is whether the sample median is the best (in terms of mean squared error) truthful estimator of the population mean. We show that when the underlying distribution is symmetric, there are truthful estimators that dominate the median. Our main result is a characterization of worst-case optimal truthful estimators, which provably outperform the median, for possibly asymmetric distributions with bounded support.

The maximum Nash welfare (MNW) solution --- which selects an allocation that maximizes the product of utilities --- is known to provide outstanding fairness guarantees when allocating divisible goods. And while it seems to lose its luster when applied to indivisible goods, we show that, in fact, the MNW solution is unexpectedly, strikingly fair even in that setting. In particular, we prove that it selects allocations that are envy free up to one good --- a compelling notion that is quite elusive when coupled with economic efficiency. We also establish that the MNW solution provides a good approximation to another popular (yet possibly infeasible) fairness property, the maximin share guarantee, in theory and --- even more so --- in practice. While finding the MNW solution is computationally hard, we develop a nontrivial implementation, and demonstrate that it scales well on real data. These results lead us to believe that MNW is the ultimate solution for allocating indivisible goods, and underlie its deployment on a popular fair division website.

Ordinal peer grading has been proposed as a simple and scalable solution for computing reliable information about student performance in massive open online courses. The idea is to outsource the grading task to the students themselves as follows. After the end of an exam, each student is asked to rank --- in terms of quality --- a bundle of exam papers by fellow students. An aggregation rule will then combine the individual rankings into a global one that contains all students. We define a broad class of simple aggregation rules and present a theoretical framework for assessing their effectiveness. When statistical information about the grading behaviour of students is available, the framework can be used to compute the optimal rule from this class with respect to a series of performance objectives. For example, a natural rule known as Borda is proved to be optimal when students grade correctly. In addition, we present extensive simulations and a field experiment that validate our theory and prove it to be extremely accurate in predicting the performance of aggregation rules even when only rough information about grading behaviour is available.

We study the truthful facility assignment problem, where a set of agents with private most-preferred points on a metric space are assigned to facilities that lie on the metric space, under capacity constraints on the facilities. The goal is to produce such an assignment that minimizes the social cost, i.e., the total distance between the most-preferred points of the agents and their corresponding facilities in the assignment, under the constraint of truthfulness, which ensures that agents do not misreport their most-preferred points. We propose a resource augmentation framework, where a truthful mechanism is evaluated by its worst-case performance on an instance with enhanced facility capacities against the optimal mechanism on the same instance with the original capacities. We study a well-known mechanism, Serial Dictatorship, and provide an exact analysis of its performance. Among other results, we prove that Serial Dictatorship has approximation ratio $$g/(g-2)$$ when the capacities are multiplied by any integer $$g \ge 3$$ . Our results suggest that even a limited augmentation of the resources can have wondrous effects on the performance of the mechanism and in particular, the approximation ratio goes to 1 as the augmentation factor becomes large. We complement our results with bounds on the approximation ratio of Random Serial Dictatorship, the randomized version of Serial Dictatorship, when there is no resource augmentation.

It is often observed that agents tend to imitate the behavior of their neighbors in a social network. This imitating behavior might lead to the strategic decision of adopting a public behavior that differs from what the agent believes is the right one and this can subvert the behavior of the population as a whole. In this paper, we consider the case in which agents express preferences over two alternatives and model social pressure with the majority dynamics: at each step an agent is selected and its preference is replaced by the majority of the preferences of her neighbors. In case of a tie, the agent does not change her current preference. A profile of the agents’ preferences is stable if the each agent’s preference coincides with the preference of at least half of the neighbors (thus, the system is in equilibrium). We ask whether there are network topologies that are robust to social pressure. That is, we ask whether there are graphs in which the majority of preferences in an initial profile $${\mathbf {s}}$$ always coincides with the majority of the preference in all stable profiles reachable from $${\mathbf {s}}$$ . We completely characterize the graphs with this robustness property by showing that this is possible only if the graph has no edge or is a clique or very close to a clique. In other words, except for this handful of graphs, every graph admits at least one initial profile of preferences in which the majority dynamics can subvert the initial majority. We also show that deciding whether a graph admits a minority that becomes majority is NP-hard when the minority size is at most 1 / 4-th of the social network size.

We present a novel extension of normal form games that we call biased games. In these games, a player's utility is influenced by the distance between his mixed strategy and a given base strategy. We argue that biased games capture important aspects of the interaction between software agents. Our main result is that biased games satisfying certain mild conditions always admit an equilibrium. We also tackle the computation of equilibria in biased games.

Motivated by applications to crowdsourcing, we study voting rules that output a correct ranking of alternatives by quality from a large collection of noisy input rankings. We seek voting rules that are supremely robust to noise, in the sense of being correct in the face of any "reasonable" type of noise. We show that there is such a voting rule, which we call the modal ranking rule. Moreover, we establish that the modal ranking rule is the unique rule with the preceding robustness property within a large family of voting rules, which includes a slew of well-studied rules.

In 1876, Charles Lutwidge Dodgson suggested the intriguing voting rule that today bears his name. Although Dodgson’s rule is one of the most well-studied voting rules, it suffers from serious deficiencies, both from the computational point of view—it is NP-hard even to approximate the Dodgson score within sublogarithmic factors—and from the social choice point of view—it fails basic social choice desiderata such as monotonicity and homogeneity. However, this does not preclude the existence of approximation algorithms for Dodgson that are monotonic or homogeneous, and indeed it is natural to ask whether such algorithms exist. In this article, we give definitive answers to these questions. We design a monotonic exponential-time algorithm that yields a 2-approximation to the Dodgson score, while matching this result with a tight lower bound. We also present a monotonic polynomial-time O(log m )-approximation algorithm (where m is the number of alternatives); this result is tight as well due to a complexity-theoretic lower bound. Furthermore, we show that a slight variation on a known voting rule yields a monotonic, homogeneous, polynomial-time O( m log m )-approximation algorithm and establish that it is impossible to achieve a better approximation ratio even if one just asks for homogeneity. We complete the picture by studying several additional social choice properties; for these properties, we prove that algorithms with an approximation ratio that depends only on m do not exist.

Sponsored search auctions are the main source of revenue for search engines. In such an auction, a set of utility maximizing advertisers competes for a set of ad slots. The assignment of advertisers to slots depends on the bids they submit; these bids may be different than the true valuations of the advertisers for the slots. Variants of the celebrated VCG auction mechanism guarantee that advertisers act truthfully and, under some assumptions, lead to revenue or social welfare maximization. Still, the sponsored search industry mostly uses generalized second price (GSP) auctions; these auctions are known to be nontruthful and suboptimal in terms of social welfare and revenue. In an attempt to explain this tradition, we study a Bayesian setting wherein the valuations of advertisers are drawn independently from a common regular probability distribution. In this setting, it is well known from the work of Myerson [1981] that the optimal revenue is obtained by the VCG mechanism with a particular reserve price that depends on the probability distribution. We show that, by appropriately setting the reserve price, the revenue over any Bayes-Nash equilibrium of the game induced by the GSP auction is at most a small constant factor away from the optimal revenue, improving previous results of Lucier et al. [2012]. Our analysis is based on the Bayes-Nash equilibrium conditions and the improved results are obtained by bounding the utility of each player at equilibrium using infinitely many deviating bids and also by developing novel prophet-like inequalities.

It is well known that strategic behavior in elections is essentially unavoidable; we therefore ask: how bad can the rational outcome be? We answer this question via the notion of the price of anarchy, using the scores of alternatives as a proxy for their quality and bounding the ratio between the score of the optimal alternative and the score of the winning alternative in Nash equilibrium. Specifically, we are interested in Nash equilibria that are obtained via sequences of rational strategic moves. Focusing on three common voting rules — plurality, veto, and Borda — we provide very positive results for plurality and very negative results for Borda, and place veto in the middle of this spectrum.

A well-studied approach to the design of voting rules views them as maximum likelihood estimators; given votes that are seen as noisy estimates of a true ranking of the alternatives, the rule must reconstruct the most likely true ranking. We argue that this is too stringent a requirement, and instead ask: How many votes does a voting rule need to reconstruct the true ranking? We define the family of pairwise-majority consistent rules, and show that for all rules in this family the number of samples required from the Mallows noise model is logarithmic in the number of alternatives, and that no rule can do asymptotically better (while some rules like plurality do much worse). Taking a more normative point of view, we consider voting rules that surely return the true ranking as the number of samples tends to infinity (we call this property accuracy in the limit); this allows us to move to a higher level of abstraction. We study families of noise models that are parametrized by distance functions, and find voting rules that are accurate in the limit for all noise models in such general families. We characterize the distance functions that induce noise models for which pairwise-majority consistent rules are accurate in the limit, and provide a similar result for another novel family of position-dominance consistent rules. These characterizations capture three well-known distance functions.

We present a new approximation algorithm for the Minimum Energy Broadcast Routing (MEBR) problem in ad hoc wireless networks that achieves an exponentially better approximation factor compared to the well-known Minimum Spanning Tree (MST) heuristic. Namely, for any instance where a minimum spanning tree of the set of stations is guaranteed to cost at most ρ ≥ 2 times the cost of an optimal solution for MEBR, we prove that our algorithm achieves an approximation ratio bounded by 2lnρ-2 ln 2 + 2. This result is particularly relevant for its consequences on Euclidean instances where we significantly improve previous results. In this respect, our experimental analysis confirms the better performance of the algorithm also in practice.

We adopt a utilitarian perspective on social choice, assuming that agents have (possibly latent) utility functions over some space of alternatives. For many reasons one might consider mechanisms, or social choice functions, that only have access to the ordinal rankings of alternatives by the individual agents rather than their utility functions. In this context, one possible objective for a social choice function is the maximization of (expected) social welfare relative to the information contained in these rankings. We study such optimal social choice functions under three different models, and underscore the important role played by scoring functions. In our worst-case model, no assumptions are made about the underlying distribution and we analyze the worst-case distortion---or degree to which the selected alternative does not maximize social welfare---of optimal social choice functions. In our average-case model, we derive optimal functions under neutral (or impartial culture) distributional models. Finally, a very general learning-theoretic model allows for the computation of optimal social choice functions (i.e., that maximize expected social welfare) under arbitrary, sampleable distributions. In the latter case, we provide both algorithms and sample complexity results for the class of scoring functions, and further validate the approach empirically.

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].

We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by potential function arguments. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. Do such equilibria exist? And if so, can we compute them efficiently?

The efficient design of networks has been an important engineering task that involves challenging combinatorial optimization problems. Typically, a network designer has to select among several alternatives which links to establish so that the resulting network satisfies a given set of connectivity requirements and the cost of establishing the network links is as low as possible. The Minimum Spanning Tree problem, which is well-understood, is a nice example. In this paper, we consider the natural scenario in which the connectivity requirements are posed by selfish users who have agreed to share the cost of the network to be established according to a well-defined rule. The design proposed by the network designer should now be consistent not only with the connectivity requirements but also with the selfishness of the users. Essentially, the users are players in a so-called network design game and the network designer has to propose a design that is an equilibrium for this game. As it is usually the case when selfishness comes into play, such equilibria may be suboptimal. In this paper, we consider the following question: can the network designer enforce particular designs as equilibria or guarantee that efficient designs are consistent with users' selfishness by appropriately subsidizing some of the network links? In an attempt to understand this question, we formulate corresponding optimization problems and present positive and negative results.

Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general PLS-complete. We present a surprisingly simple polynomial-time algorithm that computes O(1)-approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes (2 +ε)-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and 1/ε. It also applies to games with polynomial latency functions with constant maximum degree d: there, the approximation guarantee is do(d). The algorithm essentially identifies a polynomially long sequence of best-response moves that lead to an approximate equilibrium; the existence of such short sequences is interesting in itself. These are the first positive algorithmic results for approximate equilibria in non-symmetric congestion games. We strengthen them further by proving that, for congestion games that deviate from our mild assumptions, computing ρ-approximate equilibria is PLS-complete for any polynomial-time computable ρ.

In sponsored search auctions, advertisers compete for a number of available advertisement slots of different quality. The auctioneer decides the allocation of advertisers to slots using bids provided by them. Since the advertisers may act strategically and submit their bids in order to maximize their individual objectives, such an auction naturally defines a strategic game among the advertisers. In order to quantify the efficiency of outcomes in generalized second price auctions, we study the corresponding games and present new bounds on their price of anarchy, improving the recent results of Paes Leme and Tardos [16] and Lucier and Paes Leme [13]. For the full information setting, we prove a surprisingly low upper bound of 1.282 on the price of anarchy over pure Nash equilibria. Given the existing lower bounds, this bound denotes that the number of advertisers has almost no impact on the price of anarchy. The proof exploits the equilibrium conditions developed in [16] and follows by a detailed reasoning about the structure of equilibria and a novel relation of the price of anarchy to the objective value of a compact mathematical program. For more general equilibrium classes (i.e., mixed Nash, correlated, and coarse correlated equilibria), we present an upper bound of 2.310 on the price of anarchy. We also consider the setting where advertisers have incomplete information about their competitors and prove a price of anarchy upper bound of 3.037 over Bayes-Nash equilibria. In order to obtain the last two bounds, we adapt techniques of Lucier and Paes Leme [13] and significantly extend them with new arguments.

We consider approval voting elections in which each voter votes for a (possibly empty) set of candidates and the outcome consists of a set of k candidates for some parameter k, e.g., committee elections. We are interested in the minimax approval voting rule in which the outcome represents a compromise among the voters, in the sense that the maximum distance between the preference of any voter and the outcome is as small as possible. This voting rule has two main drawbacks. First, computing an outcome that minimizes the maximum distance is computationally hard. Furthermore, any algorithm that always returns such an outcome provides incentives to voters to misreport their true preferences. In order to circumvent these drawbacks, we consider approximation algorithms, i.e., algorithms that produce an outcome that approximates the minimax distance for any given instance. Such algorithms can be considered as alternative voting rules. We present a polynomial-time 2-approximation algorithm that uses a natural linear programming relaxation for the underlying optimization problem and deterministically rounds the fractional solution in order to compute the outcome; this result improves upon the previously best known algorithm that has an approximation ratio of 3. We are furthermore interested in approximation algorithms that are resistant to manipulation by (coalitions of) voters, i.e., algorithms that do not motivate voters to misreport their true preferences in order to improve their distance from the outcome. We complement previous results in the literature with new upper and lower bounds on strategyproof and group-strategyproof algorithms.

In cooperative multiagent systems an alternative that maximizes the social welfare — the sum of utilities — can only be selected if each agent reports its full utility function. This may be infeasible in environments where communication is restricted. Employing a voting rule to choose an alternative greatly reduces the communication burden, but leads to a possible gap between the social welfare of the optimal alternative and the social welfare of the one that is ultimately elected. Procaccia and Rosenschein have introduced the concept of distortion to quantify this gap. In this paper, we present the notion of embeddings into voting rules: functions that receive an agent's utility function and return the agent's vote. We establish that very low distortion can be obtained using randomized embeddings, especially when the number of agents is large compared to the number of alternatives. We investigate our ideas in the context of three prominent voting rules with low communication costs: Plurality, Approval, and Veto. Our results arguably provide a compelling reason for employing voting in cooperative multiagent systems.

In 1876 Charles Lutwidge Dodgson suggested the intriguing voting rule that today bears his name. Although Dodgson's rule is one of the most well-studied voting rules, it suffers from serious deficiencies, both from the computational point of view - it is NP-hard even to approximate the Dodgson score within sublogarithmic factors - and from the social choice point of view - it fails basic social choice desiderata such as monotonicity and homogeneity.

We study congestion games where players aim to access a set of resources. Each player has a set of possible strategies and each resource has a function associating the latency it incurs to the players using it. Players are non--cooperative and each wishes to follow a strategy that minimizes her own latency with no regard to the global optimum. Previous work has studied the impact of this selfish behavior on system performance. In this article, we study the question of how much the performance can be improved if players are forced to pay taxes for using resources. Our objective is to extend the original game so that selfish behavior does not deteriorate performance. We consider atomic congestion games with linear latency functions and present both negative and positive results. Our negative results show that optimal system performance cannot be achieved even in very simple games. On the positive side, we show that there are ways to assign taxes that can improve the performance of linear congestion games by forcing players to follow strategies where the total latency suffered is within a factor of 2 of the minimum possible; this result is shown to be tight. Furthermore, even in cases where in the absence of taxes the system behavior may be very poor, we show that the total disutility of players (latency plus taxes) is not much larger than the optimal total latency. Besides existential results, we show how to compute taxes in time polynomial in the size of the game by solving convex quadratic programs. Similar questions have been extensively studied in the model of non-atomic congestion games. To the best of our knowledge, this is the first study of the efficiency of taxes in atomic congestion games.

We present three new coordination mechanisms for scheduling n selfish jobs on m unrelated machines. A coordination mechanism aims to mitigate the impact of selfishness of jobs on the efficiency of schedules by defining a local scheduling policy on each machine. The scheduling policies induce a game among the jobs and each job prefers to be scheduled on a machine so that its completion time is minimum given the assignments of the other jobs. We consider the maximum completion time among all jobs as the measure of the efficiency of schedules. The approximation ratio of a coordination mechanism quantifies the efficiency of pure Nash equilibria (price of anarchy) of the induced game. Our mechanisms are deterministic, local, and preemptive in the sense that the scheduling policy does not necessarily process the jobs in an uninterrupted way and may introduce some idle time. Our first coordination mechanism has approximation ratio O(log m) and always guarantees that the induced game has pure Nash equilibria to which the system converges in at most n rounds. This result improves a recent bound of O(log2 m) due to Azar, Jain, and Mirrokni and, similarly to their mechanism, our mechanism uses a global ordering of the jobs according to their distinct IDs. Next we study the intriguing scenario where jobs are anonymous, i.e., they have no IDs. In this case, coordination mechanisms can only distinguish between jobs that have different load characteristics. Our second mechanism handles anonymous jobs and has approximation ratio although the game induced is not a potential gameand, hence, the existence of pure Nash equilibria is not guaranteed by potential function arguments. However, it provides evidence that the known lower bounds for non-preemptive coordination mechanisms could be beaten using preemptive scheduling policies. Our third coordination mechanism also handles anonymous jobs and has a nice “cost-revealing” potential function. Besides in proving the existence of equilibria, we use this potential function in order to upper-bound the price of stability of the induced game by O(log m), the price of anarchy by O(log2 m), and the convergence time to O(log2 m)-approximate assignments by a polynomial number of best-response moves. Our third coordination mechanism is the first that handles anonymous jobs and simultaneously guarantees that the induced game is a potential game and has bounded price of anarchy.

The voting rules proposed by Dodgson and Young are both designed to find the alternative closest to being a Condorcet winner, according to two different notions of proximity; the score of a given alternative is known to be hard to compute under either rule. In this paper, we put forward two algorithms for approximating the Dodgson score: an LP-based randomized rounding algorithm and a deterministic greedy algorithm, both of which yield an approximation ratio, where m is the number of alternatives; we observe that this result is asymptotically optimal, and further prove that our greedy algorithm is optimal up to a factor of 2, unless problems in have quasi-polynomial time algorithms. Although the greedy algorithm is computationally superior, we argue that the randomized rounding algorithm has an advantage from a social choice point of view. Further, we demonstrate that computing any reasonable approximation of the ranking produced by Dodgson's rule is -hard. This result provides a complexity-theoretic explanation of sharp discrepancies that have been observed in the Social Choice Theory literature when comparing Dodgson elections with simpler voting rules. Finally, we show that the problem of calculating the Young score is -hard to approximate by any factor. This leads to an inapproximability result for the Young ranking.

We study computationally hard combinatorial problems arising from the important engineering question of how to maximize the number of connections that can be simultaneously served in a wavelength division ultiplexing (WDM) optical network. In such networks, WDM technology can satisfy a set of connections by computing a route and assigning a wavelength to each connection so that no two connections routed through the same fiber are assigned the same wavelength. Each fiber supports a limited number of w wavelengths and in order to fully exploit the parallelism provided by the technology, one should select a set of connections of maximum cardinality which can be satisfied using the available wavelengths. This is known as the maximum routing and path coloring (maxRPC) problem. Our main contribution is a general analysis method for a class of iterative algorithms for a more general coloring problem. A lower bound on the benefit of the solution computed by such an algorithm in terms of the optimal benefit and the number of available wavelengths is given by a benefit-revealing linear program. We apply this method to maxRPC in both undirected and bidirected rings to obtain upper bounds on the approximation ratio of several algorithms. The best bounds obtained are $4/3$ and $103/73\approx1.41096$ for undirected and bidirected rings; these results improve known bounds of $3/2$ and $11/7\approx1.57143$, respectively. Our results extend to multifiber ring networks and also apply to the maximum path coloring (maxPC) problem, where paths instead of connection requests are given as part of the input. We also study the profit version of maxPC in rings where each path has a profit and the objective is to satisfy a set of paths of maximum total profit. We present an algorithm based on linear programming and randomized rounding with approximation ratio 1.49015, improving on the $\frac{e}{e-1}\approx1.58198$ bound obtained by a simple iterative algorithm.

We study the problem of maintaining connectivity in a wireless network where the network nodes are equipped with directional antennas. Nodes correspond to points on the plane and each uses a directional antenna modeled by a sector with a given angle and radius. The connectivity problem is to decide whether or not it is possible to orient the antennas so that the directed graph induced by the node transmissions is strongly connected. We present algorithms for simple polynomial-time-solvable cases of the problem, show that the problem is NP-complete in the $2$-dimensional case when the sector angle is small, and present algorithms that approximate the minimum radius to achieve connectivity for sectors with a given angle. We also discuss several extensions to related problems. To the best of our knowledge, the problem has not been studied before in the literature.

Abstract We address an important communication issue arising in wireless cellular networks that utilize frequency division multiplexing (FDM) technology. In such networks, many users within the same geographical region (cell) can communicate simultaneously with other users of the network using distinct frequencies. The spectrum of the available frequencies is limited; thus, efficient solutions to the call control problem are essential. The objective of the call control problem is, given a spectrum of available frequencies and users that wish to communicate, to maximize the benefit, i.e., the number of users that communicate without signal interference. We consider cellular networks of reuse distance k ≥ 2 and we study the online version of the problem using competitive analysis. In cellular networks of reuse distance 2, the previously best known algorithm that beats the lower bound of 3 on the competitiveness of deterministic algorithms, works on networks with one frequency, achieves a competitive ratio against oblivious adversaries, which is between 2.469 and 2.651, and uses a number of random bits at least proportional to the size of the network. We significantly improve this result by presenting a series of simple randomized algorithms that have competitive ratios significantly smaller than 3, work on networks with arbitrarily many frequencies, and use only a constant number of random bits or a comparable weak random source. The best competitiveness upper bound we obtain is 16/7 using only four random bits. In cellular networks of reuse distance k> 2, we present simple randomized online call control algorithms with competitive ratios, which significantly beat the lower bounds on the competitiveness of deterministic ones and use only O(log k) random bits. Also, we show new lower bounds on the competitiveness of online call control algorithms in cellular networks of any reuse distance. In particular, we show that no online algorithm can achieve competitive ratio better than 2, 25/12, and 2.5, in cellular networks with reuse distance k ε {2, 3, 4}, k = 5, and k ≥ 6, respectively. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008

In this paper we address an important communication issue arising in cellular (mobile) networks that utilize Frequency Division Multiplexing (FDM) technology. In such networks, many users within the same geographical region can communicate simultaneously with other users of the network using distinct frequencies. The spectrum of the available frequencies is limited; thus, efficient solutions to the call control problem are essential. The objective of the call control problem is, given a spectrum of available frequencies and users that wish to communicate, to maximize the number of users that communicate without signal interference. Using competitive analysis, we study the performance of algorithm p-RANDOM; an intuitive on-line randomized call control algorithm proposed previously for cellular networks. We give upper and lower bounds of its competitive ratio against oblivious adversaries as a function of the parameter p. Optimizing the upper bound function, we prove that there exists a 2.651-competitive randomized call control algorithm. In this way, we significantly improve the best known upper bound on the competitiveness of on-line randomized call control which was 2.934.

In this paper we consider communication issues arising in mobile networks that utilize Frequency Division Multiplexing (FDM) technology. In such networks, many users within the same geographical region can communicate simultaneously with other users of the network using distinct frequencies. The spectrum of available frequencies is limited; thus, efficient solutions to the frequency allocation and the call control problem are essential. In the frequency allocation problem, given users that wish to communicate, the objective is to minimize the required spectrum of frequencies so that communication can be established without signal interference. The objective of the call control problem is, given a spectrum of available frequencies and users that wish to communicate, to maximize the number of users served. We consider cellular, planar, and arbitrary network topologies.