Large language models (LLMs) are increasingly used for decision-making tasks where fairness is an essential desideratum. But what does fairness even mean to an LLM? To investigate this, we conduct a comprehensive evaluation of how LLMs perceive fairness in the context of resource allocation, using both synthetic and real-world data. We find that several state-of-the-art LLMs, when instructed to be fair, tend to prioritize improving collective welfare rather than distributing benefits equally. Their perception of fairness is somewhat sensitive to how user preferences are represented, but less so to the real-world context of the decision-making task. Finally, we show that the best strategy for aligning an LLM's perception of fairness to a specific criterion is to provide it as a mathematical objective, without referencing "fairness", as this prevents the LLM from mixing the criterion with its own prior notions of fairness. Our results provide practical insights for understanding and shaping how LLMs interpret fairness in resource allocation problems.

We introduce a model of fair division with market values, where indivisible goods must be partitioned among agents with (additive) subjective valuations, and each good additionally has a market value. The market valuation can be viewed as a separate additive valuation that holds identically across all the agents. We seek allocations that are simultaneously fair with respect to the subjective valuations and under the market valuation. We show that an allocation that satisfies stochastically-dominant envy-freeness up to one good (SD-EF1) with respect to both the subjective valuations and the market valuation does not always exist, but the weaker guarantee of EF1 with respect to the subjective valuations along with SD-EF1 with respect to the market valuation can be guaranteed. We also study a number of other guarantees such as Pareto optimality, EFX, and MMS. In addition, we explore non-additive valuations and extend our model to cake-cutting. Along the way, we identify several tantalizing open questions.

Participatory budgeting (PB) is an increasingly popular tool for democratically allocating limited budgets to public-good projects. In PB, constituents vote on their preferred projects via ballots, and then an aggregation rule selects a set of projects whose total cost fits within the budget. Recent work studies how to design PB ballots and aggregation rules that yield low-distortion outcomes (informally, outcomes with high social welfare). Existing distortion bounds, however, rely on strong assumptions that restrict voters' latent utilities. We prove that low distortion PB outcomes can be achieved by dropping these assumptions and instead leveraging the established idea that voters can be public-spirited: they may consider others' interests alongside their own when voting. Flanigan, Procaccia, and Wang (2023) prove that in public-spirited single-winner voting (the special case of PB where exactly one project can be funded) with ranking ballots, deterministic aggregation rules can achieve constant distortion. Our first contribution is to extend this analysis to PB; there, we prove that the best distortion permitted by deterministic rules with ranking ballots grows linearly in the number of projects. We find that this impossibility --- a problem in practice, where m is often large --- holds for other known ballots as well. Our second contribution is the design of a new PB ballot format that breaks this linear distortion barrier. This ballot asks voters to rank a predetermined set of entire feasible bundles of projects. We design multiple protocols for implementing these ballots, each striking a different trade-off between the number of bundles voters must rank and the distortion: with m bundles, we get sublinear distortion; with polynomial bundles, we get logarithmic distortion; and with pseudopolynomial bundles, we get constant distortion.

Fairness and efficiency have become the pillars of modern fair division research, but prior work on achieving both simultaneously is largely limited to the unconstrained setting. We study fair and efficient allocations of indivisible goods under additive valuations and various types of allocation feasibility constraints, and demonstrate the unreasonable effectiveness of the maximum Nash welfare (MNW) solution in this previously uncharted territory. Our main result is that MNW allocations are 1/2-envy-free up to one good (EF1) and Pareto optimal under the broad family of (arbitrary) matroid constraints. We extend these guarantees to complete MNW allocations for base-orderable matroid constraints, and to a family of non-matroid constraints (which includes balancedness). We establish tightness of our results by providing counterexamples for the satisfiability of certain stronger desiderata, but show an improved result for the special case of goods with copies (Gafni et al. 2023). Finally, we also establish novel best-of-both-worlds guarantees for goods with copies and balancedness.

We study temporal fair division, whereby a set of agents are allocated a (possibly different) set of goods on each day for a period of days. We study this setting, as well as a number of its special cases formed by the restrictions to two agents, same goods on each day, identical preferences, or combinations thereof, and chart out the landscape of achieving two types of fairness guarantees simultaneously: fairness on each day (per day) and fairness over time (up to each day, or the weaker version, overall). In the most general setting, we prove that there always exists an allocation that is stochastically-dominant envy-free up to one good (SD-EF1) per day and proportional up to one good (PROP1) overall, and when all the agents have identical preferences, we show that SD-EF1 per day and SD-EF1 overall can be guaranteed. For the case of two agents, we prove that SD-EF1 per day and EF1 up to each day can be guaranteed using an envy balancing technique. We provide counterexamples for other combinations that establish our results as among the best guarantees possible, but also leave open some tantalizing questions.

A fundamental task in multi-agent systems is to match n agents to n alternatives (e.g., resources or tasks). This is often done by eliciting agents' ordinal rankings over the alternatives rather than their exact numerical utilities. While this simplifies elicitation, the incomplete information leads to inefficiency, captured by a worst-case measure called distortion. Recent work shows that making just a few cardinal utility queries per agent can significantly improve the distortion, with Amanatidis et al. (2024) achieving O(√n) distortion with two queries per agent. We generalize their result by achieving O(n^(1/λ)) distortion with λ queries per agent, for any constant λ, which is optimal up to a constant factor given a previous lower bound by Amanatidis et al. (2022). We extend this finding to the general social choice problem of selecting one of m alternatives based on n agents' preferences, achieving O((min{n, m})^(1/λ)) distortion with λ queries per agent, for any constant λ, which is also optimal given prior results. Thus, our work settles open questions regarding the optimal distortion achievable with a fixed number of cardinal value queries in both settings.

Proportional fairness criteria inspired by democratic ideals of proportional representation have received growing attention in the clustering literature. Prior work has investigated them in two separate paradigms. Chen et al. [ICML 2019] study _centroid clustering_, in which each data point's loss is determined by its distance to a representative point (centroid) chosen in its cluster. Caragiannis et al. [NeurIPS 2024] study _non-centroid clustering_, in which each data point's loss is determined by its maximum distance to any other data point in its cluster. We generalize both paradigms to introduce _semi-centroid clustering_, in which each data point's loss is a combination of its centroid and non-centroid losses, and study two proportional fairness criteria---the core and, its relaxation, fully justified representation (FJR). Our main result is a novel algorithm which achieves a constant approximation to the core, in polynomial time, even when the distance metrics used for centroid and non-centroid loss measurements are different. We also derive improved results for more restricted loss functions and the weaker FJR criterion, and establish lower bounds in each case.

Stability has been a foundational concept in the two-sided matching literature. For fractional matchings under strict preferences, it is commonly formalized as ex ante stability. When agents on one side have weak preferences involving indifferences, the seminal work of Kesten and Ünver [Theoretical Economics, 2015] introduces strong ex ante stability, which combines ex ante stability with a fairness criterion that ensures no discrimination among (equally-preferred) agents on one side. They propose the Fractional Deferred Acceptance (FDA) algorithm, which is guaranteed to converge to a strongly ex ante stable (fractional) matching, but may require infinitely many iterations to do so. To address this, they propose another algorithm, which we call FDA-Cycle, that computes the matching to which FDA converges; they further claim that FDA-Cycle terminates in polynomial time.

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 initiate the study of fairness among classes of agents in online bipartite matching where there is a given set of offline vertices (aka agents) and another set of vertices (aka items) that arrive online and must be matched irrevocably upon arrival. In this setting, agents are partitioned into a set of classes and the matching is required to be fair with respect to the classes. We adopt popular fairness notions (e.g. envy-freeness, proportionality, and maximin share) and their relaxations to this setting and study deterministic and randomized algorithms for matching indivisible items (leading to integral matchings) and for matching divisible items (leading to fractional matchings). For matching indivisible items, we propose an adaptive-priority-based algorithm, MATCH-AND-SHIFT, prove that it achieves (1/2)-approximation of both class envy-freeness up to one item and class maximin share fairness, and show that each guarantee is tight. For matching divisible items, we design a water-filling-based algorithm, EQUAL-FILLING, that achieves (1-1/e)-approximation of class envy-freeness and class proportionality; we prove (1-1/e) to be tight for class proportionality and establish a 3/4 upper bound on class envy-freeness.

When an agent votes, she typically ranks the set of available alternatives. Occasionally, she may also wish to report the intensity of her preferences by indicating adjacent pairs of alternatives in her ranking between which her preference is acutely decisive; for instance, she may suggest that she likes alternative a more than b, but b much more than c. We design near-optimal voting rules which aggregate such preference rankings with intensities using the recently-popular distortion framework. We also show that traditional voting rules, which aggregate preference rankings while ignoring (or not eliciting) intensities, can incur significant welfare loss.

We consider the problem of partitioning n agents in an undirected social network into k almost equal in size (differing by at most one) groups, where the utility of an agent for a group is the number of her neighbors in the group. The core and envy-freeness are two compelling axiomatic fairness guarantees in such settings. The former demands that there be no coalition of agents such that each agent in the coalition has more utility for that coalition than for her own group, while the latter demands that no agent envy another agent for the group they are in. We provide (often tight) approximations to both fairness guarantees, and many of our positive results are obtained via efficient algorithms.

Designing provably fair decision-making algorithms is a task of growing interest and importance. In this article, I argue that preference-based notions of fairness proposed decades ago in the economics literature and subsequently explored in-depth within computer science (specifically, within the field of computational social choice) are aptly suited for a wide range of modern decision-making systems, from conference peer review to recommender systems to participatory budgeting.

We design online algorithms for fair allocation of public goods to a set of N agents over a sequence of T rounds and focus on improving their performance using predictions. In the basic model, a public good arrives in each round, and every agent reveals their value for it upon arrival. The algorithm must irrevocably decide the investment in this good without exceeding a total budget of B across all rounds. The algorithm can utilize (potentially noisy) predictions of each agent’s total value for all remaining goods. The algorithm's performance is measured using a proportional fairness objective, which informally demands that every group of agents be rewarded proportional to its size and the cohesiveness of its preferences. We show that no algorithm can achieve better than Θ(T/B) proportional fairness without predictions. With reasonably accurate predictions, the situation improves significantly, and Θ(log(T/B)) proportional fairness is achieved. We also extend our results to a general setting wherein a batch of L public goods arrive in each round and O(log(min(N,L)T/B)) proportional fairness is achieved. Our exact bounds are parameterized as a function of the prediction error, with performance degrading gracefully with increasing errors.

Large conferences such as NeurIPS and AAAI serve as crossroads of various AI fields, since they attract submissions from a vast number of communities. However, in some cases, this has resulted in a poor reviewing experience for some communities, whose submissions get assigned to less qualified reviewers outside of their communities. An often-advocated solution is to break up any such large conference into smaller conferences, but this can lead to isolation of communities and harm interdisciplinary research. We tackle this challenge by introducing a notion of group fairness, called the core, which requires that every possible community (subset of researchers) to be treated in a way that prevents them from unilaterally benefiting by withdrawing from a large conference. We study a simple peer review model, prove that it always admits a reviewing assignment in the core, and design an efficient algorithm to find one such assignment. We use real data from CVPR and ICLR conferences to compare our algorithm to existing reviewing assignment algorithms on a number of metrics.

With a rapid growth in the deployment of AI tools for making critical decisions (or aiding humans in doing so), there is a growing demand to be able to explain to the stakeholders how these tools arrive at a decision. Consequently, voting is frequently used to make such decisions due to its inherent explainability. Recent work suggests that using randomized (as opposed to deterministic) voting rules can lead to significant efficiency gains measured via the distortion framework. However, rules that use intricate randomization can often become too complex to explain to the stakeholders; losing explainability can eliminate the key advantage of voting over black-box AI tools, which may outweigh the efficiency gains. We study the efficiency gains which can be unlocked by using voting rules that add a simple randomization step to a deterministic rule, thereby retaining explainability. We focus on two such families of rules, randomized positional scoring rules and random committee member rules, and show, theoretically and empirically, that they indeed achieve explainability and efficiency simultaneously to some extent.

We study the problem of designing voting rules that take as input the ordinal preferences of n agents over a set of n alternatives and output a single alternative, aiming to optimize the overall happiness of the agents. The input to the voting rule is each agent's ranking of the alternatives from most to least preferred, yet the agents have more refined (cardinal) preferences that capture the intensity with which they prefer one alternative over another. To quantify the extent to which voting rules can optimize over the cardinal preferences given access only to the ordinal ones, prior work has used the distortion measure, i.e., the worst-case approximation ratio between a voting rule's performance and the best performance achievable given the cardinal preferences.

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

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.

A fundamental question in social choice and multi-agent systems is aggregating ordinal preferences expressed by agents into a measurably prudent collective choice. A promising line of recent work views ordinal preferences as a proxy for underlying cardinal preferences. It aims to optimize distortion, the worst-case approximation ratio of the (utilitarian) social welfare. When agents rank the set of alternatives, prior work identifies near-optimal voting rules for selecting one or more alternatives. However, ranking all the alternatives is prohibitive when there are many alternatives. In this work, we consider the setting where each agent ranks only her t favorite alternatives and identify almost tight bounds on the best possible distortion when selecting a single alternative or a committee of alternatives of a given size k. Our results also extend to approximating higher moments of social welfare. Along the way, we close a gap left open in prior work by identifying asymptotically tight distortion bounds for committee selection given full rankings.

We consider the allocation of homogeneous divisible goods to agents with linear additive valuations. Our focus is on the case where some agents are secretive and reveal no preference information, while the remaining agents reveal full preference information. We study distortion, which is the worst-case approximation ratio when maximizing social welfare given such partial information about agent preferences. As a function of the number of secretive agents k relative to the overall number of agents n, we identify the exact distortion for every p-mean welfare function, which includes the utilitarian welfare (p=1), the Nash welfare (p -> 0), and the egalitarian welfare (p -> -Inf).

Sortition is a form of democracy built on random selection of representatives. Two of the key arguments in favor of sortition are that it provides representation (a random panel reflects the composition of the population) and fairness (everyone has a chance to participate). Uniformly random selection is perfectly fair, but is it representative? Towards answering this question, we introduce the notion of a representation metric on the space of individuals, and assume that the cost of an individual for a panel is determined by the $q$-th closest representative; the representation of a (random) panel is measured by the ratio between the (expected) sum of costs of the optimal panel for the individuals and that of the given panel. For $k/2 < q \le k-\Omega(k)$, where $k$ is the panel size, we show that uniform random selection is indeed representative by establishing a constant lower bound on this ratio. By contrast, for $q \leq k/2$, no random selection algorithm that is almost fair can give such a guarantee. We therefore consider relaxed fairness guarantees and develop a new random selection algorithm that sheds light on the tradeoff between representation and fairness.

A voting rule decides on a probability distribution over a set of m alternatives, based on rankings of those alternatives provided by agents. We assume that agents have cardinal utility functions over the alternatives, but voting rules have access to only the rankings induced by these utilities. We evaluate how well voting rules do on measures of social welfare and of proportional fairness, computed based on the hidden utility functions.

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.

We study the classical problem of matching n agents to n objects, where the agents have ranked preferences over the objects. We focus on two popular desiderata from the matching literature: Pareto optimality and rank-maximality. Instead of asking the agents to report their complete preferences, our goal is to learn a desirable matching from partial preferences, specifically a matching that is necessarily Pareto optimal (NPO) or necessarily rank-maximal (NRM) under any completion of the partial preferences. We focus on the top-k model in which agents reveal a prefix of their preference rankings. We design efficient algorithms to check if a given matching is NPO or NRM, and to check whether such a matching exists given top-k partial preferences. We also study online algorithms for eliciting partial preferences adaptively, and prove bounds on their competitive ratio.

We consider approval-based committee elections, in which a size-k subset of available candidates must be selected given approval sets for each voter, indicating the candidates approved by the voter. A number of axioms capturing ideas of fairness and proportionality have been proposed for this framework. We argue that even the strongest of them, such as priceability and the core, only rule out certain undesirable committees, but fail to ensure that the selected committee is fair in all cases. We propose two new solution concepts, stable priceability and balanced stable priceability, and show that they select arguably fair committees. Our solution concepts come with a non-trivial-to-construct but easy-to-understand market-based explanation for why the chosen committee is fair. We show that stable priceability is closely related to the notion of Lindahl equilibrium from economics.

We study the fundamental problem of allocating indivisible goods to agents with additive preferences. We consider eliciting from each agent only a ranking of her k most preferred goods instead of her full cardinal valuations. We characterize the amount of preference information that must be elicited in order to satisfy envy-freeness up to one good and approximate maximin share guarantee, two widely studied fairness notions. We also analyze the multiplicative loss in social welfare incurred due to the lack of full information with and without fairness requirements.

The notion of distortion in social choice problems has been defined to measure the loss in efficiency---typically measured by the utilitarian social welfare, the sum of utilities of the participating agents---due to having access only to limited information about the preferences of the agents. We survey the most significant results of the literature on distortion from the past 15 years, and highlight important open problems and the most promising avenues of ongoing and future work.

We introduce a new model for two-sided matching which allows us to borrow popular fairness notions from the fair division literature such as envy-freeness up to one good and maximin share guarantee. In our model, each agent is matched to multiple agents on the other side over whom she has additive preferences. We demand fairness for each side separately, giving rise to notions such as double envy-freeness up to one match (DEF1) and double maximin share guarantee (DMMS). We show that (a slight strengthening of) DEF1 cannot always be achieved, but in the special case where both sides have identical preferences, the round-robin algorithm with a carefully designed agent ordering achieves it. In contrast, DMMS cannot be achieved even when both sides have identical preferences.

The wisdom of the crowd has long become the de facto approach for eliciting information from individuals or experts in order to predict the ground truth. However, classical democratic approaches for aggregating individual \emph{votes} only work when the opinion of the majority of the crowd is relatively accurate. A clever recent approach, \emph{surprisingly popular voting}, elicits additional information from the individuals, namely their \emph{prediction} of other individuals' votes, and provably recovers the ground truth even when experts are in minority. This approach works well when the goal is to pick the correct option from a small list, but when the goal is to recover a true ranking of the alternatives, a direct application of the approach requires eliciting too much information. We explore practical techniques for extending the surprisingly popular algorithm to ranked voting by partial votes and predictions and designing robust aggregation rules. We experimentally demonstrate that even a little prediction information helps surprisingly popular voting outperform classical approaches.

We propose a multi-agent variant of the classical multi-armed bandit problem, in which there are $N$ agents and $K$ arms, and pulling an arm generates a (possibly different) stochastic reward for each agent. Unlike the classical multi-armed bandit problem, the goal is not to learn the "best arm"; indeed, each agent may perceive a different arm to be the best for her personally. Instead, we seek to learn a fair distribution over the arms. Drawing on a long line of research in economics and computer science, we use the Nash social welfare as our notion of fairness. We design multi-agent variants of three classic multi-armed bandit algorithms and show that they achieve sublinear regret, which is now measured in terms of the lost Nash social welfare. We also extend a classical lower bound, establishing the optimality of one of our algorithms.

Facility location is the problem of locating a public facility based on the preferences of multiple agents. In the classic framework, where each agent holds a single location on a line and can misreport it, strategyproof mechanisms for choosing the location of the facility are well-understood.We revisit this problem in a more general framework. We assume that each agent may hold several locations on the line with different degrees of importance to the agent. We study mechanisms which elicit the locations of the agents and different levels of information about their importance. Further, in addition to the classic manipulation of misreporting locations, we introduce and study a new manipulation, whereby agents may hide some of their locations. We argue for its novelty in facility location and applicability in practice. Our results provide a complete picture of the power of strategyproof mechanisms eliciting different levels of information and with respect to each type of manipulation. Surprisingly, we show that in some cases hiding locations can be a strictly more powerful manipulation than misreporting locations.

In the standard model of voting, it is assumed that a voting rule observes the ranked preferences of each individual over a set of alternatives and makes a collective decision. In practice, however, not every individual votes. Is it possible to make a good collective decision for a group given the preferences of only a few of its members? We propose a framework in which we are given the ranked preferences of k out of n individuals sampled from a distribution, and the goal is to predict what a given voting rule would output if applied on the underlying preferences of all n individuals. We focus on the family of positional scoring rules, derive a strong negative result when the underlying preferences can be arbitrary, and discover interesting phenomena when they are generated from a known distribution.

We study the following metric distortion problem: there are two finite sets of points, V and C, that lie in the same metric space, and our goal is to choose a point in C whose total distance from the points in V is as small as possible. However, rather than having access to the underlying distance metric, we only know, for each point in V, a ranking of its distances to the points in C. We propose algorithms that choose a point in C using only these rankings as input and we provide bounds on their distortion (worst-case approximation ratio). A prominent motivation for this problem comes from voting theory, where V represents a set of voters, C represents a set of candidates, and the rankings correspond to ordinal preferences of the voters. A major conjecture in this framework is that the optimal deterministic algorithm has distortion 3. We resolve this conjecture by providing a polynomial-time algorithm that achieves distortion 3, matching a known lower bound. We do so by proving a novel lemma about matching voters to candidates, which we refer to as the ranking-matching lemma. This lemma induces a family of novel algorithms, which may be of independent interest, and we show that a special algorithm in this family achieves distortion 3. We also provide more refined, parameterized, bounds using the notion of decisiveness, which quantifies the extent to which a voter may prefer her top choice relative to all others. Finally, we introduce a new randomized algorithm with improved distortion compared to known results, and also provide improved lower bounds on the distortion of all deterministic and randomized algorithms.

In this work, we study fairness in centroid clustering. In this problem, k cluster centers must be placed given n points in a metric space, and the cost to each point is its distance to the nearest cluster center. Recent work of Chen et al. [Chen et al., 2019] introduces the notion of a proportionally fair clustering, in which no group of at least n/k points can find a new cluster center which provides lower cost to each member of the group. They propose a greedy capture algorithm which provides a 1+√2 approximation of proportional fairness for any metric space, and derive generalization bounds for learning proportionally fair clustering from samples in the case where a cluster center can only be placed at one of finitely many given locations in the metric space. We focus on the case where cluster centers can be placed anywhere in the (usually infinite) metric space. In case of the L² distance metric over ℝ^t, we show that the approximation ratio of greedy capture improves to 2. We also show that this is due to a special property of the L² distance; for the L¹ and L^∞ distances, the approximation ratio remains 1+√2. We provide universal lower bounds which apply to all algorithms. We also consider metric spaces defined on graphs. For trees, we show that an exact proportionally fair clustering always exists and provide an efficient algorithm to find one. The corresponding question for general graph remains an interesting open question. Finally, we show that for the L² distance, checking whether a proportionally fair clustering exists and implementing greedy capture over an infinite metric space are NP-hard problems, but (approximately) solvable in special cases. We also derive generalization bounds which show that an approximately proportionally fair clustering for a large number of points can be learned from a small number of samples. Our work advances the understanding of proportional fairness in clustering, and points out many avenues for future work.

We study the problem of allocating indivisible goods among agents. When randomization is allowed, it is possible to achieve compelling fairness guarantees such as envy-freeness (Foley, 1967), which states that no agent should (in expectation) prefer any other agent's allocation to her own. For instance, we can simply allocate each good, independently of the other goods, to an agent chosen uniformly at random. However, while this scheme is fair ex-ante, it may produce outcomes that are very unfair ex-post, such as by chance assigning all the goods to a single agent. On the other hand, in the absence of randomization, some amount of ex-post unfairness is unavoidable. Nevertheless, it is possible to guarantee relaxations of envy-freeness that bound the maximum level of envy. One popular relaxation is envy-freeness up to one good (Lipton et al., 2004; Budish, 2011), which requires that the envy of any agent toward another agent can be removed by the elimination of at most one good from the envied agent's bundle.

In recent work, Mandal et al. [2019] study a novel framework for the winner selection problem in voting, in which a voting rule is seen as a combination of an elicitation rule and an aggregation rule. The elicitation rule asks voters to respond to a query based on their preferences over a set of alternatives, and the aggregation rule aggregates voter responses to return a winning alternative. They study the tradeoff between the communication complexity of a voting rule, which measures the number of bits of information each voter must send in response to its query, and its distortion, which measures the quality of the winning alternative in terms of utilitarian social welfare. They prove upper and lower bounds on the communication complexity required to achieve a desired level of distortion, but their bounds are not tight. Importantly, they also leave open the question whether the best randomized rule can significantly outperform the best deterministic rule.

The past decade has witnessed a rapid growth of research on fairness in machine learning. In contrast, fairness has been formally studied for almost a century in microeconomics in the context of resource allocation, during which many general-purpose notions of fairness have been proposed. This paper explore the applicability of two such notions — envy-freeness and equitability — in machine learning. We propose novel relaxations of these fairness notions which apply to groups rather than individuals, and are compelling in a broad range of settings. Our approach provides a unifying framework by incorporating several recently proposed fairness definitions as special cases. We provide generalization bounds for our approach, and theoretically and experimentally evaluate the tradeoff between loss minimization and our fairness guarantees.

Bitcoin, a cryptocurrency built on the blockchain data structure, has generated significant academic and commercial interest. Contrary to prior expectations, recent research has shown that participants of the protocol (the so-called “miners”) are not always incentivized to follow the protocol. We study the game induced by one such attack – the pool block withholding attack – in which mining pools (groups of miners) attack other mining pools. We focus on the case of two pools attacking each other, with potentially other mining power in the system.We show that this game always admits a pure Nash equilibrium, and its pure price of anarchy, which intuitively measures how much computational power can be wasted due to attacks in an equilibrium, is at most 3. We conjecture, and prove in special cases, that it is in fact at most 2. Our simulations provide compelling evidence for this conjecture, and show that players can quickly converge to the equilibrium by following best response strategies.

Much of the social choice literature examines direct voting systems, in which voters submit their ranked preferences over candidates and a voting rule picks a winner. Real-world elections and decision-making processes are often more complex and involve multiple stages. For instance, one popular voting system filters candidates through primaries: first, voters affiliated with each political party vote over candidates of their own party and the voting rule picks a candidate from each party, which then compete in a general election.We present a model to analyze such multi-stage elections, and conduct the first quantitative comparison (to the best of our knowledge) of the direct and primary voting systems with two political parties in terms of the quality of the elected candidate. Our main result is that every voting rule is guaranteed to perform almost as well (i.e., within a constant factor) under the primary system as under the direct system. Surprisingly, the converse does not hold: we show settings in which there exist voting rules that perform significantly better under the primary system than under the direct system.

We consider the problem of fairly dividing a collection of indivisible goods among a set of players. Much of the existing literature on fair division focuses on notions of individual fairness. For instance, envy-freeness requires that no player prefer the set of goods allocated to another player to her own allocation. We observe that an algorithm satisfying such individual fairness notions can still treat groups of players unfairly, with one group desiring the goods allocated to another. Our main contribution is a notion of group fairness, which implies most existing notions of individual fairness. Group fairness (like individual fairness) cannot be satisfied exactly with indivisible goods. Thus, we introduce two “up to one good” style relaxations. We show that, somewhat surprisingly, certain local optima of the Nash welfare function satisfy both relaxations and can be computed in pseudo-polynomial time by local search. Our experiments reveal faster computation and stronger fairness guarantees in practice.

We take an unorthodox view of voting by expanding the design space to include both the elicitation rule, whereby voters map their (cardinal) preferences to votes, and the aggregation rule, which transforms the reported votes into collective decisions. Intuitively, there is a tradeoff between the communication requirements of the elicitation rule (i.e., the number of bits of information that voters need to provide about their preferences) and the efficiency of the outcome of the aggregation rule, which we measure through distortion (i.e., how well the utilitarian social welfare of the outcome approximates the maximum social welfare in the worst case). Our results chart the Pareto frontier of the communication-distortion tradeoff.

Gerrymandering is the process by which parties manipulate boundaries of electoral districts in order to maximize the number of districts they can win. Demographic trends show an increasingly strong correlation between residence and party affiliation; some party’s supporters congregate in cities, while others stay in more rural areas. We investigate both theoretically and empirically the effect of this trend on a party's ability to gerrymander in a two-party model ("urban party" and "rural party"). Along the way, we propose a definition of the gerrymandering power of a party, and an algorithmic approach for near-optimal gerrymandering in large instances. Our results suggest that beyond a fairly small concentration of urban party's voters, the gerrymandering power of a party depends almost entirely on the level of concentration, and not on the party's share of the population. As partisan separation grows, the gerrymandering power of both parties converge so that each party can gerrymander to get only slightly more than what its voting share warrants, bringing about, ultimately, a more representative outcome. Moreover, there seems to be an asymmetry between the gerrymandering power of the parties, with the rural party being more capable of gerrymandering.

This paper is part of an emerging line of work at the intersection of machine learning and mechanism design, which aims to avoid noise in training data by correctly aligning the incentives of data sources. Specifically, we focus on the ubiquitous problem of linear regression, where strategyproof mechanisms have previously been identified in two dimensions. In our setting, agents have single-peaked preferences and can manipulate only their response variables. Our main contribution is the discovery of a family of group strategyproof linear regression mechanisms in any number of dimensions, which we call generalized resistant hyperplane mechanisms. The game-theoretic properties of these mechanisms --- and, in fact, their very existence --- are established through a connection to a discrete version of the Ham Sandwich Theorem.

We consider the problem of fairly allocating indivisible public goods. We model the public goods as elements with feasibility constraints on what subsets of elements can be chosen, and assume that agents have additive utilities across elements. Our model generalizes existing frameworks such as fair public decision making and participatory budgeting. We study a groupwise fairness notion called the core, which generalizes well-studied notions of proportionality and Pareto efficiency, and requires that each subset of agents must receive an outcome that is fair relative to its size. In contrast to the case of divisible public goods (where fractional allocations are permitted), the core is not guaranteed to exist when allocating indivisible public goods. Our primary contributions are the notion of an additive approximation to the core (with a tiny multiplicative loss), and polynomial time algorithms that achieve a small additive approximation, where the additive factor is relative to the largest utility of an agent for an element. If the feasibility constraints define a matroid, we show an additive approximation of 2. A similar approach yields a constant additive bound when the feasibility constraints define a matching. For feasibility constraints defining an arbitrary packing polytope with mild restrictions, we show an additive guarantee that is logarithmic in the width of the polytope. Our algorithms are based on the convex program for maximizing the Nash social welfare, but differ significantly from previous work in how it is used. As far as we are aware, our work is the first to approximate the core in indivisible settings.

Participatory budgeting enables the allocation of public funds by collecting and aggregating individual preferences; it has already had a sizable real-world impact. But making the most of this new paradigm requires a rethinking of some of the basics of computational social choice, including the very way in which individuals express their preferences. We analytically compare four preference elicitation methods -- knapsack votes, rankings by value or value for money, and threshold approval votes -- through the lens of implicit utilitarian voting, and find that threshold approval votes are qualitatively superior. This conclusion is supported by experiments using data from real participatory budgeting elections.

Peer prediction mechanisms incentivize agents to truthfully report their signals, in the absence of a verification mechanism, by comparing their reports with those of their peers. Prior work in this area is essentially restricted to the case of homogeneous agents, whose signal distributions are identical. This is limiting in many domains, where we would expect agents to differ in taste, judgment and reliability. Although the Correlated Agreement (CA) mechanism [30] can be extended to handle heterogeneous agents, the new challenge is with the efficient estimation of agent signal types. We solve this problem by clustering agents based on their reporting behavior, proposing a mechanism that works with clusters of agents and designing algorithms that learn such a clustering. In this way, we also connect peer prediction with the Dawid and Skene [5] literature on latent types. We retain the robustness against coordinated misreports of the CA mechanism, achieving an approximate incentive guarantee of ε-informed truthfulness. We show on real data that this incentive approximation is reasonable in practice, and even with a small number of clusters.

We generalize the classic problem of fairly allocating indivisible goods to the problem of fair public decision making, in which a decision must be made on several social issues simultaneously, and, unlike the classic setting, a decision can provide positive utility to multiple players. We extend the popular fairness notion of proportionality (which is not guaranteeable) to our more general setting, and introduce three novel relaxations --- proportionality up to one issue, round robin share, and pessimistic proportional share --- that are also interesting in the classic goods allocation setting. We show that the Maximum Nash Welfare solution, which is known to satisfy appealing fairness properties in the classic setting, satisfies or approximates all three relaxations in our framework. We also provide polynomial time algorithms and hardness results for finding allocations satisfying these axioms, with or without insisting on Pareto optimality.

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.

A paradigmatic problem in social choice theory deals with the aggregation of subjective preferences of individuals --- represented as rankings of alternatives --- into a social ranking. We are interested in settings where individuals are uncertain about their own preferences, and represent their uncertainty as distributions over rankings. Under the classic objective of minimizing the (expected) sum of Kendall tau distances between the input rankings and the output ranking, we establish that preference elicitation is surprisingly straightforward and near-optimal solutions can be obtained in polynomial time. We show, both in theory and using real data, that ignoring uncertainty altogether can lead to suboptimal outcomes.

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.

We present the first model of optimal voting under adversarial noise. From this viewpoint, voting rules are seen as error-correcting codes: their goal is to correct errors in the input rankings and recover a ranking that is close to the ground truth. We derive worst-case bounds on the relation between the average accuracy of the input votes, and the accuracy of the output ranking. Empirical results from real data show that our approach produces significantly more accurate rankings than alternative approaches.

Some crowdsourcing platforms ask workers to express their opinions by approving a set of k good alternatives. It seems that the only reasonable way to aggregate these k-approval votes is the approval voting rule, which simply counts the number of times each alternative was approved. We challenge this assertion by proposing a probabilistic framework of noisy voting, and asking whether approval voting yields an alternative that is most likely to be the best alternative, given k-approval votes. While the answer is generally positive, our theoretical and empirical results call attention to situations where approval voting is suboptimal.

As part of a collaboration with a major California school district, we study the problem of fairly allocating unused classrooms in public schools to charter schools. Our approach revolves around the randomized leximin mechanism. We extend previous work to the classroom allocation setting, showing that the leximin mechanism is proportional, envy-free, efficient, and group strategyproof. We also prove that the leximin mechanism provides a (worst-case) 4-approximation to the maximum number of classrooms that can possibly be allocated. Our experiments, which are based on real data, show that a nontrivial implementation of the leximin mechanism scales gracefully in terms of running time (even though the problem is intractable in theory), and performs extremely well with respect to a number of efficiency objectives. We take great pains to establish the practicability of our approach, and discuss issues related to its deployment.

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.

We investigate the limiting behavior of trader wealth and prices in a simple prediction market with a finite set of participants having heterogeneous beliefs. Traders bet repeatedly on the outcome of a binary event with fixed Bernoulli success probability. A class of strategies, including (fractional) Kelly betting and constant relative risk aversion (CRRA) are considered. We show that when traders are willing to risk only a small fraction of their wealth in any period, belief heterogeneity can persist indefinitely; if bets are large in proportion to wealth then only the most accurate belief type survives. The market price is more accurate in the long run when traders with less accurate beliefs also survive. That is, the survival of traders with heterogeneous beliefs, some less accurate than others, allows the market price to better reflect the objective probability of the event in the long run.

We investigate synergy, or lack thereof, between agents in cooperative games, building on the popular notion of Shapley value. We think of a pair of agents as synergistic (resp., antagonistic) if the Shapley value of one agent when the other agent participates in a joint effort is higher (resp. lower) than when the other agent does not participate. Our main theoretical result is that any graph specifying synergistic and antagonistic pairs can arise even from a restricted class of cooperative games. We also study the computational complexity of determining whether a given pair of agents is synergistic. Finally, we use the concepts developed in the paper to uncover the structure of synergies in two real-world organizations, the European Union and the International Monetary Fund.

Mean proximity rules provide a simple geometric framework to achieve consensus among a collection of rankings (votes) over a set of alternatives. They embed all rankings into a Euclidean space, take the mean of the embeddings of the input votes, and return the ranking whose embedding is closest to the mean. Previous work on mean proximity rules has not integrated an important axiom---neutrality---into the framework. By drawing on ideas from the representation theory of finite groups, we show that integrating neutrality actually helps achieve a succinct representation for every mean proximity rule. Various connections are drawn between mean proximity rules and other prominent approaches to social choice.

Recently fair division theory has emerged as a promising approach for allocation of multiple computational resources among agents. While in reality agents are not all present in the system simultaneously, previous work has studied static settings where all relevant information is known upfront. Our goal is to better understand the dynamic setting. On the conceptual level, we develop a dynamic model of fair division, and propose desirable axiomatic properties for dynamic resource allocation mechanisms. On the technical level, we construct two novel mechanisms that provably satisfy some of these properties, and analyze their performance using real data. We believe that our work informs the design of superior multiagent systems, and at the same time expands the scope of fair division theory by initiating the study of dynamic and fair resource allocation mechanisms.

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 consider Markov decision processes (MDPs) with specifications given as Büchi (liveness) objectives. We consider the problem of computing the set of almost-sure winning vertices from where the objective can be ensured with probability 1. We study for the first time the average case complexity of the classical algorithm for computing the set of almost-sure winning vertices for MDPs with Buchi objectives. Our contributions are as follows: First, we show that for MDPs with constant out-degree the expected number of iterations is at most logarithmic and the average case running time is linear (as compared to the worst case linear number of iterations and quadratic time complexity). Second, for the average case analysis over all MDPs we show that the expected number of iterations is constant and the average case running time is linear (again as compared to the worst case linear number of iterations and quadratic time complexity). Finally we also show that given that all MDPs are equally likely, the probability that the classical algorithm requires more than constant number of iterations is exponentially small.

We study the problem of allocating multiple resources to agents with heterogeneous demands. Technological advances such as cloud computing and data centers provide a new impetus for investigating this problem under the assumption that agents demand the resources in fixed proportions, known in economics as Leontief preferences. In a recent paper, Ghodsi et al. [2011] introduced the dominant resource fairness (DRF) mechanism, which was shown to possess highly desirable theoretical properties under Leontief preferences. We extend their results in three directions. First, we show that DRF generalizes to more expressive settings, and leverage a new technical framework to formally extend its guarantees. Second, we study the relation between social welfare and properties such as truthfulness; DRF performs poorly in terms of social welfare, but we show that this is an unavoidable shortcoming that is shared by every mechanism that satisfies one of three basic properties. Third, and most importantly, we study a realistic setting that involves indivisibilities. We chart the boundaries of the possible in this setting, contributing a new relaxed notion of fairness and providing both possibility and impossibility results.