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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 $\DistFamily_k(n,m)$ 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 $\DistFamily_k(n,m)$ for different values of the parameters $n$, $m$, and $k$., Comment: 26 pages, 1 fugure

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 (agents) that are large and cohesive. Prior work applies this framework to centroid clustering, where the loss of an agent is its distance to the centroid assigned to its cluster. We expand the framework to non-centroid clustering, where the loss of an agent is a function of the other agents in its cluster, by adapting two proportional fairness criteria -- the core and its relaxation, fully justified representation (FJR) -- to this setting. We show that the core can be approximated only under structured loss functions, and even then, the best approximation we are able to establish, using an adaptation of the GreedyCapture algorithm developed for centroid clustering [Chen et al., 2019; Micha and Shah, 2020], is unappealing for a natural loss function. In contrast, we design a new (inefficient) algorithm, GreedyCohesiveClustering, which achieves the relaxation FJR exactly under arbitrary loss functions, and show that the efficient GreedyCapture algorithm achieves a constant approximation of FJR. We also design an efficient auditing algorithm, which estimates the FJR approximation of any given clustering solution up to a constant factor. Our experiments on real data suggest that traditional clustering algorithms are highly unfair, whereas GreedyCapture is considerably fairer and incurs only a modest loss in common clustering objectives., Comment: A preliminary version appeared at NeurIPS 2024

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 [10], 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 (n^5) of queries. This solves the main open problem stated by Caragiannis and Rathi [CR23]. 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.

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 & 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., Comment: Accepted at ICML'24

We consider the assignment problem, where $n$ agents have to be matched to $n$ items. Each agent has a preference order over the items. In the serial dictatorship (SD) mechanism the agents act in a particular order and pick their most preferred available item when it is their turn to act. Applying SD using a uniformly random permutation as agent ordering results in the well-known random serial dictatorship (RSD) mechanism. Accurate estimates of the (expected) efficiency of its outcome can be used to assess whether RSD is attractive compared to other mechanisms. In this paper, we explore whether such estimates are possible by sampling a (hopefully) small number of agent orderings and applying SD using them. We consider a value setting in which agents have values for the items as well as a metric cost setting where agents and items are assumed to be points in a metric space, and the cost of an agent for an item is equal to the distance of the corresponding points. We show that a (relatively) small number of samples is enough to approximate the expected social welfare of RSD in the value setting and its expected social cost in the metric cost setting despite the #P-hardness of the corresponding exact computation problems., Comment: 17 pages

Multi-winner approval-based voting has received considerable attention recently. A voting rule in this setting takes as input ballots in which each agent approves a subset of the available alternatives and outputs a committee of alternatives of given size $k$. We consider the scenario when a coalition of agents can act strategically and alter their ballots so that the new outcome is strictly better for a coalition member and at least as good for anyone else in the coalition. Voting rules that are robust against this strategic behaviour are called strongly group-strategyproof. We prove that, for $k\in \{1,2, ..., m-2\}$, strongly group-strategyproof multi-winner approval-based voting rules which furthermore satisfy the minimum efficiency requirement of unanimity do not exist, where $m$ is the number of available alternatives. Our proof builds a connection to single-winner voting with ranking-based ballots and exploits the infamous Gibbard-Satterthwaite theorem to reach the desired impossibility result. Our result has implications for paradigmatic problems from the area of approximate mechanism design without money and indicates that strongly group-strategyproof mechanisms for minimax approval voting, variants of facility location, and classification can only have an unbounded approximation ratio., Comment: 16 pages, 1 table

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, 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 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., Comment: to appear in AAAI 2024

We consider the fundamental problem of designing a truthful single-item auction with the challenging objective of extracting a large fraction of the highest agent valuation as revenue. Following a recent trend in algorithm design, we assume that the agent valuations belong to a known interval, and a (possibly erroneous) prediction for the highest valuation is available. Then, auction design aims for high consistency and robustness, meaning that, for appropriate pairs of values $\gamma$ and $\rho$, the extracted revenue should be at least a $\gamma$- or $\rho$-fraction of the highest valuation when the prediction is correct for the input instance or not. We characterize all pairs of parameters $\gamma$ and $\rho$ so that a randomized $\gamma$-consistent and $\rho$-robust auction exists. Furthermore, for the setting in which robustness can be a function of the prediction error, we give sufficient and necessary conditions for the existence of robust auctions and present randomized auctions that extract a revenue that is only a polylogarithmic (in terms of the prediction error) factor away from the highest agent valuation.

We study the classic problem of dividing a collection of indivisible resources in a fair and efficient manner among a set of agents having varied preferences. Pareto optimality is a standard notion of economic efficiency, which states that it should be impossible to find an allocation that improves some agent's utility without reducing any other's. On the other hand, a fundamental notion of fairness in resource allocation settings is that of envy-freeness, which renders an allocation to be fair if every agent (weakly) prefers her own bundle over that of any other agent's bundle. Unfortunately, an envy-free allocation may not exist if we wish to divide a collection of indivisible items. Introducing randomness is a typical way of circumventing the non-existence of solutions, and therefore, allocation lotteries, i.e., distributions over allocations have been explored while relaxing the notion of fairness to ex-ante envy freeness. We consider a general fair division setting with $n$ agents and a family of admissible $n$-partitions of an underlying set of items. Every agent is endowed with partition-based utilities, which specify her cardinal utility for each bundle of items in every admissible partition. In such fair division instances, Cole and Tao (2021) have proved that an ex-ante envy-free and Pareto-optimal allocation lottery is always guaranteed to exist. We strengthen their result while examining the computational complexity of the above total problem and establish its membership in the complexity class PPAD. Furthermore, for instances with a constant number of agents, we develop a polynomial-time algorithm to find an ex-ante envy-free and Pareto-optimal allocation lottery. On the negative side, we prove that maximizing social welfare over ex-ante envy-free and Pareto-optimal allocation lotteries is NP-hard., Comment: 22 pages

{\em Distortion} is a well-established notion for quantifying the loss of social welfare that may occur in voting. As voting rules take as input only ordinal information, they are essentially forced to neglect the exact values the agents have for the alternatives. Thus, in worst-case electorates, voting rules may return low social welfare alternatives and have high distortion. Accompanying voting rules with a small number of cardinal queries per agent may reduce distortion considerably. To explore distortion beyond worst-case conditions, we use a simple stochastic model according to which the values the agents have for the alternatives are drawn independently from a common probability distribution. This gives rise to so-called {\em impartial culture electorates}. We refine the definition of distortion so that it is suitable for this stochastic setting and show that, rather surprisingly, all voting rules have high distortion {\em on average}. On the positive side, for the fundamental case where the agents have random {\em binary} values for the alternatives, we present a mechanism that achieves approximately optimal average distortion by making a {\em single} cardinal query per agent. This enables us to obtain slightly suboptimal average distortion bounds for general distributions using a simple randomized mechanism that makes one query per agent. We complement these results by presenting new tradeoffs between the distortion and the number of queries per agent in the traditional worst-case setting., Comment: 34 pages, 2 figures, to appear in the proceedings of the 20th Conference on Web and Internet Economics (WINE-24)

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 consider a novel setting where a set of items are matched to the same set of agents repeatedly over multiple rounds. Each agent gets exactly one item per round, which brings interesting challenges to finding efficient and/or fair {\em repeated matchings}. A particular feature of our model is that the value of an agent for an item in some round depends on the number of rounds in which the item has been used by the agent in the past. We present a set of positive and negative results about the efficiency and fairness of repeated matchings. For example, when items are goods, a variation of the well-studied fairness notion of envy-freeness up to one good (EF1) can be satisfied under certain conditions. Furthermore, it is intractable to achieve fairness and (approximate) efficiency simultaneously, even though they are achievable separately. For mixed items, which can be goods for some agents and chores for others, we propose and study a new notion of fairness that we call {\em swap envy-freeness} (swapEF)., Comment: 20 pages

We consider the problem of allocating indivisible goods among $n$ agents in a fair manner. For this problem, one of the best notions of fairness is envy-freeness up to any good (EFX). However, it is not known if EFX allocations always exist. Hence, several relaxations of EFX allocations have been studied. We propose another relaxation of EFX, called epistemic EFX (EEFX). An allocation is EEFX iff for every agent $i$, it is possible to shuffle the goods of the other agents such that agent $i$ does not envy any other agent up to any good. We show that EEFX allocations always exist for additive valuations, and give a polynomial-time algorithm for computing them. We also show how EEFX is related to some previously-known notions of fairness., Comment: Edit: include additional authors and some minor changes

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 straight-forward rule that divides the budget proportionally is susceptible to manipulation. In a recent work, Freeman et al. [arXiv:1905.00457] proposed a class of truthful mechanisms, called moving phantom mechanisms. Among others, they propose a proportional mechanism, 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 which 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 the optimal among all truthful mechanisms. For three projects, we propose a new, proportional mechanism which is virtually optimal among all moving phantom mechanisms. Finally, we provide impossibility results regarding the approximability of moving phantom mechanisms., Comment: 30 pages, 12 figures, 2 tables. A preliminary version appeared in the 36th AAAI Conference in Artificial Intelligence

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 $v_i(S)$ 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.

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 $q \leq k/3$, asymptotically linear in the number of agents when $k/3 < q \leq k/2$, and constant when $q > k/2$., Comment: A preliminary version of this paper appears in AAAI 2022

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.

The problem of fair division known as "cake cutting" has been the focus of multiple papers spanning several decades. The most prominent problem in this line of work has been to bound the query complexity of computing an envy-free outcome in the Robertson-Webb query model. However, the root of this problem's complexity is somewhat artificial: the agents' values are assumed to be additive across different pieces of the "cake" but infinitely complicated within each piece. This is unrealistic in most of the motivating examples, where the cake represents a finite collection of homogeneous goods. We address this issue by introducing a fair division model that more accurately captures these applications: the value that an agent gains from a given good depends only on the amount of the good they receive, yet it can be an arbitrary function of this amount, allowing the agents to express preferences that go beyond standard cake cutting. In this model, we study the query complexity of computing allocations that are not just envy-free, but also approximately Pareto optimal among all envy-free allocations. Using a novel flow-based approach, we show that we can encode the ex-post feasibility of randomized allocations via a polynomial number of constraints, which reduces our problem to solving a linear program.

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, which 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 recently due to their simplicity., Comment: 28 pages, 1 figure

We study the problem of {\em 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 {\em 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 {\em additive} approximation. In particular, we study the extent to which prior information on voters' preferences could be useful in the design of efficient deterministic impartial selection mechanisms with good additive approximation guarantees. We consider three models of prior information, which we call the {\em opinion poll}, the {\em a prior popularity}, and the {\em uniform} model. We analyze the performance of a natural selection mechanism that we call {\em approval voting with default} (AVD) and show that it achieves a $O(\sqrt{n\ln{n}})$ additive guarantee for opinion poll and a $O(\ln^2n)$ for a priori popularity inputs, where $n$ is the number of individuals. We consider this polylogarithmic bound as our main technical contribution. We complement this last result by showing that our analysis is close to tight, showing an $\Omega(\ln{n})$ lower bound. This holds in the uniform model, which is the simplest among the three models.

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.

Selling a perfectly divisible item to potential buyers is a fundamental task with apparent applications to pricing communication bandwidth and cloud computing services. Surprisingly, despite the rich literature on single-item auctions, revenue maximization when selling a divisible item is a much less understood objective. We introduce a Bayesian setting, in which the potential buyers have concave valuation functions (defined for each possible item fraction) that are randomly chosen according to known probability distributions. Extending the sequential posted pricing paradigm, we focus on mechanisms that use linear pricing, charging a fixed price for the whole item and proportional prices for fractions of it. Our goal is to understand the power of such mechanisms by bounding the gap between the expected revenue that can be achieved by the best among these mechanisms and the maximum expected revenue that can be achieved by any mechanism assuming mild restrictions on the behavior of the buyers. Under regularity assumptions for the probability distributions, we show that this revenue gap depends only logarithmically on a natural parameter characterizing the valuation functions and the number of agents. Our results follow by bounding the objective value of a mathematical program that maximizes the ex-ante relaxation of optimal revenue under linear pricing revenue constraints., Comment: 23 pages, 1 figure

Fair division has emerged as a very hot topic in multiagent systems, 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 {\em 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 that, 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 super-constant 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.

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 is always robust to reasonable noise. We further refine this finding by presenting a hierarchy of rules in terms of how robust to noise they are., Comment: Preliminary version appeared in IJCAI 2020

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 the 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 $\Theta(\sqrt{n})$ and $\Theta(n^{2/3}\ln^{1/3}n)$ 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 $\Omega(\sqrt{n})$ for their deterministic and randomized variants respectively. Finally, we present a general lower bound of $2$ for all deterministic impartial mechanisms., Comment: 18 pages, 7 figures. A preliminary version appeared in SAGT '19. Corrected a minor bug in Theorem 3 and typos. Theory Comput Syst (2022)

We study natural improvement dynamics in weighted congestion games with polynomial latencies of maximum degree $d\geq 1$. We focus on two problems regarding the existence and efficiency of approximate pure Nash equilibria, with a reasonable small approximation factor, in these games. By exploiting a simple technique, we firstly show that such a game always admits a $d$-approximate potential function. This implies that every sequence of $d$-approximate improvement moves by the players leads to a $d$-approximate pure Nash equilibrium. As a corollary, we also obtain that, under mild assumptions on the structure of the players' strategies, the game always admits a constant approximate potential function. Secondly, using a simple potential function argument, we are able to show that a $(d+\delta)$-approximate pure Nash equilibrium of cost at most $(d+1)/(d+\delta)$ times the cost of an optimal state always exists, for $\delta\in [0,1]$.

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). After observing that, in this cardinal setting, stable fractional matchings can have much higher social welfare than stable integral ones, our goal is to understand the computational complexity of finding an optimal (i.e., welfare-maximizing) or nearly-optimal stable fractional matching. We present simple approximation algorithms for this problem with weak welfare guarantees and, rather unexpectedly, we furthermore show that achieving better approximations is hard. This computational hardness persists even for approximate stability. To the best of our knowledge, these are the first computational complexity results for stable fractional matchings. En route to these results, we provide a number of structural observations., Comment: Accepted to Artificial Intelligence (AIJ)

Several fairness concepts have been proposed recently in attempts to approximate envy-freeness in settings with indivisible goods. Among them, the concept of envy-freeness up to any item (EFX) is arguably the closest to envy-freeness. Unfortunately, EFX allocations are not known to exist except in a few special cases. We make significant progress in this direction. We show that for every instance with additive valuations, there is an EFX allocation of a subset of items with a Nash welfare that is at least half of the maximum possible Nash welfare for the original set of items. That is, after donating some items to a charity, one can distribute the remaining items in a fair way with high efficiency. This bound is proved to be best possible. Our proof is constructive and highlights the importance of maximum Nash welfare allocation. Starting with such an allocation, our algorithm decides which items to donate and redistributes the initial bundles to the agents, eventually obtaining an allocation with the claimed efficiency guarantee. The application of our algorithm to large markets, where the valuations of an agent for every item is relatively small, yields EFX with almost optimal Nash welfare. To the best of our knowledge, this is the first use of large market assumptions in the fair division literature. We also show that our algorithm can be modified to compute, in polynomial-time, EFX allocations that approximate optimal Nash welfare within a factor of at most $2\rho$, using a $\rho$-approximate allocation on input instead of the maximum Nash welfare one., Comment: 19 pages, 1 figure

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. If the graph is a path and the utility functions are monotonic over bundles, we show the existence of EF1 allocations for at most four agents, and the existence of EF2 allocations for any number of agents; our proofs involve discrete analogues of the Stromquist's moving-knife protocol and the Su--Simmons argument based on Sperner's lemma. For identical utilities, we provide a polynomial-time algorithm that computes an EF1 allocation for any number of agents. For the case of two agents, we characterize the class of graphs that guarantee the existence of EF1 allocations as those whose biconnected components are arranged in a path; this property can be checked in linear time., Comment: Accepted journal version

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., Comment: This article is a complete version of a conference paper, which appeared in IJCAI 2019. The conference version contains an error in the proof of Theorem 2 regarding EF1 existence for arbitrary utility functions

When a company undergoes a merger or transfers its ownership, the existing governing body has an opinion on which buyer should take over as the new owner. Similar situations occur while assigning the host of big sports tournaments, like the World Cup or the Olympics. In all these settings, the values of the external bidders are as important as the opinions of the internal experts. Motivated by such scenarios, we consider a social welfare maximizing approach to design and analyze truthful mechanisms in {\em hybrid social choice} settings, where payments can be imposed to the bidders, but not to the experts. Since this problem is a combination of mechanism design with and without monetary transfers, classical solutions like VCG cannot be applied, making this a novel mechanism design problem. We consider the simple but fundamental scenario with one expert and two bidders, and provide tight approximation guarantees of the optimal social welfare. We distinguish between mechanisms that use ordinal and cardinal information, as well as between mechanisms that base their decisions on one of the two sides (either the bidders or the expert) or both. Our analysis shows that the cardinal setting is quite rich and admits several non-trivial randomized truthful mechanisms, and also allows for closer-to-optimal welfare guarantees.

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 [Mathematics of Operations Research, 2004], 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 multi-user 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.

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., Comment: 36 pages, 10 figures

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 viewpoint and present positive and negative complexity results and, furthermore, complement our theoretical findings with experiments on real-world and synthetic data., Comment: Accepted to Artificial Intelligence Journal. A preliminary version appeared in Proceedings of the 31st AAAI Conference on Artificial Intelligence (AAAI), pages 430-436, 2017

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.