We study the fair allocation of m indivisible items to n agents with additive utilities. In our setting, each indivisible item may be a good, yielding non-negative utility to some agents, or a chore, yielding negative utility to others. Whilst envy-free allocations may not exist in the indivisible-items setting, envy-freeness can be achieved if some amount of divisible good (i.e., money) is introduced. When each item's utility or disutility is bounded by one, we show that a subsidy of at most one dollar per agent suffices to guarantee the existence of an envy-free allocation, and that this bound is tight. Moreover, such an allocation can be computed in polynomial time. Since at least one agent need not receive any subsidy, our results imply that a total subsidy of at most n-1 dollars suffices to ensure envy-freeness.

One of the central questions in discrete fair division is whether fairness and efficiency can be achieved simultaneously. For indivisible goods, a canonical relaxation of envy-freeness is envy-freeness up to one good (EF1), while the standard efficiency benchmark is Pareto optimality (PO). In their seminal work, Caragiannis et al. showed that, for additive valuations, EF1 and PO are always compatible, and asked whether this compatibility extends to submodular valuations. This question has since become an important open problem in the study of fair division. In this paper, we settle the question in the negative. We construct an instance with two agents and eight goods, where both agents have submodular valuations, such that no EF1 allocation is even weakly Pareto optimal. Thus, the celebrated compatibility between EF1 and PO for additive valuations breaks down already for two agents under submodular valuations. We then map the boundary of this impossibility. On the negative side, we show that even for weighted matroid rank valuations, EF1 and fractional Pareto optimality (fPO) are incompatible. This rules out, in general, broad classes of weighted-welfare and Fisher-market-based approaches. On the positive side, we identify a common-envelope condition that restores compatibility. Under this condition, EF1+PO allocations exist for any number of agents. This yields new positive results showing that common-weight matroid-rank valuations always admit EF1+PO allocations. Finally, we quantify the efficiency loss that is unavoidable when insisting on EF1. Our submodular counterexample implies that there is a constant α<1 such that no EF1 allocation is α-. For the broader class of subadditive valuations, we prove a tight two-agent bound: for any ε>0, there exists an instance in which no EF1 allocation is (1/√(2)+ε)-PO.

We study the problem of assigning agents to the vertices of a graph such that no pair of neighbors can benefit from swapping assignments – a property we term neighborhood stability. We further assume that agents’ utilities are based solely on their preferences over the assignees of adjacent vertices and that those preferences are binary. Having shown that even this very restricted setting does not guarantee neighborhood stable assignments, we focus on special cases that provide such guarantees. We show that when the graph is a cycle or a path, a neighborhood stable assignment always exists for any preference profile. Furthermore, we give a general condition under which neighborhood stable assignments always exist. For each of these results, we give a polynomial-time algorithm to compute a neighborhood stable assignment.

We study fair division of indivisible mixed manna (items whose values may be positive, negative, or zero) among agents with additive valuations. In this context, we establish that fairness—in terms of a relaxation of envy-freeness—and Pareto efficiency can always be achieved together. Specifically, our fairness guarantees are in terms of envy-freeness up to k reallocations ( EFR-k ): An allocation 𝒜 of the indivisible items is said to be EFR-k if there exists a subset R of at most k items such that, for each agent i, we can reassign items from within R (in 𝒜 ) and obtain an allocation, 𝒜^i , which is envy-free for i. We establish that, when allocating mixed manna among n agents with additive valuations, an EFR-(n-1) and Pareto optimal ( PO ) allocation 𝒜 always exists. Further, the individual envy-free allocations 𝒜^i , induced by reassignments, are also PO . In addition, we prove that such fair and efficient allocations are efficiently computable when the number of agents, n, is fixed. We also obtain positive results focusing on EFR by itself (and without the PO desideratum). Specifically, we show that an EFR-(n-1) allocation of mixed manna can be computed in polynomial time. In addition, we prove that when all the items are goods, an EFR-⌊ n/2 ⌋ allocation exists and can be computed efficiently. Here, the (n-1) bound is tight for chores and ⌊ n/2 ⌋ is tight for goods. Our results advance the understanding of fair and efficient allocation of indivisible mixed manna and rely on a novel application of the Knaster-Kuratowski-Mazurkiewicz (KKM) Theorem in discrete fair division. We utilize weighted welfare maximization, with perturbed valuations, to achieve Pareto efficiency, and overall, our techniques are notably different from existing market-based approaches.

The existence of EFX allocations is a fundamental question in fair division. In this paper, we construct a three-agent, eight-good instance with monotone subadditive valuations such that no allocation satisfies $α$-EFX for any $α> \frac{1}{\sqrt[6]{2}} \approx 0.89$. We also provide a closely related three-agent, eight-good instance with submodular (in fact weighted coverage) valuations for which no EFX allocation exists. A key feature of our construction is its symmetry: the agents' valuations are identical up to a relabeling of the goods. Thus, EFX can fail even when agents differ only in how the goods are labeled. This symmetry makes the counterexamples compact and human-verifiable, yielding simple combinatorial obstructions to the existence of EFX.

We study the fundamental problem of fairly dividing indivisible items among agents with additive utilities. In our model, an item can be a good yielding non-negative utilities to some agents and simultaneously a chore yielding negative utilities to others. We take the best-of-both-worlds perspective and our goal is to construct a randomized allocation that is exactly fair ex ante while also being supported on ex post approximately fair allocations. The fairness notions examined in this paper are envy-freeness (EF) and its well-known relaxation envy-freeness up to one item (EF1). Our main result is that ex-ante EF and ex-post EF1 can be achieved simultaneously. To achieve this, we introduce a novel probabilistic Hall-type matrix decomposition that intricately correlates the fractional assignments of goods and chores. We resolve this decomposition problem by combining continuous minimax duality – via Sion's minimax theorem – with carefully designed biased flow networks.

We establish a compatibility between fairness and efficiency, captured via Nash Social Welfare (NSW), under the broad class of subadditive valuations. We prove that, for subadditive valuations, there always exists a partial allocation that is envy-free up to the removal of any good (EFx) and has NSW at least half of the optimal; here, optimality is considered across all allocations, fair or otherwise. We also prove, for subadditive valuations, the universal existence of complete allocations that are envy-free up to one good (EF1) and also achieve a factor 1/2 approximation to the optimal NSW. Our EF1 result resolves an open question posed by Garg, Husic, Li, Végh, and Vondrák (STOC 2023). In addition, we develop a polynomial-time algorithm which, given an arbitrary allocation A as input, returns an EF1 allocation with NSW at least 1/e^2/e≈1/2.08 times that of A. Therefore, our results imply that the EF1 criterion can be attained simultaneously with a constant-factor approximation to optimal NSW in polynomial time (with demand queries), for subadditive valuations. The previously best-known approximation factor for optimal NSW, under EF1 and among n agents, was O(n) – we improve this bound to O(1). It is known that EF1 and exact Pareto efficiency (PO) are incompatible with subadditive valuations. Complementary to this negative result, the current work shows that we regain compatibility by just considering a factor 1/2 approximation: EF1 can be achieved in conjunction with 1/2-PO under subadditive valuations. As such, our results serve as a general tool that can be used as a black box to convert any efficient outcome into a fair one, with only a marginal decrease in efficiency.

We study the strategyproof placement of k facilities on the real line for n agents who privately report their locations, without monetary transfers. For two facilities, the Proportional Mechanism of Lu, Sun, Wang, and Zhu (2010) is strategyproof in expectation and achieves a constant-factor approximation to the optimal social cost. Whether such a guarantee is possible for three facilities in the standard model, where each agent is served by her nearest open facility, has remained open. We resolve this question affirmatively by introducing the Random-Anchor Volume mechanism. The mechanism first opens a facility at the report of a uniformly random agent, called the anchor, and then jointly selects two additional reports, assigning each pair probability proportional to the product of the two consecutive gaps formed by the pair and the anchor. We prove that the mechanism is strategyproof in expectation and has expected social cost at most 8 OPT_3, where OPT_k denotes the minimum social cost achievable using at most k facilities. The mechanism naturally extends to every k≥ 2 by selecting k-1 additional reports with probability proportional to the product of the consecutive gaps among them and the anchor. Under truthful reporting, this generalization has expected social cost at most 4(k-1)OPT_k. Its incentive guarantee, however, has a sharp boundary: the mechanism is strategyproof in expectation for k∈{1,2,3}, but is manipulable for every k≥ 4.

We study a fair division problem in (multi)graphs where n agents (vertices) are pairwise connected by items (edges), and each agent is only interested in its incident items. We consider how to allocate items to incident agents in an envy-free manner, i.e., envy-free orientations, while minimizing the overall payment, i.e., subsidy. We first prove that computing an envy-free orientation with the minimum subsidy is NP-hard, even when the graph is simple and the agents have bi-valued additive valuations. We then bound the worst-case subsidy. We prove that for any multigraph (i.e., allowing parallel edges) and monotone valuations where the marginal value of each good is at most $1 for each agent, $1 each (a total subsidy of n-1, where n is the number of agents) is sufficient. This is one of the few cases where linear subsidy Θ(n) is known to be necessary and sufficient to guarantee envy-freeness when agents have monotone valuations. When the valuations are additive (while the graph may contain parallel edges) and when the graph is simple (while the valuations may be monotone), we improve the bound to n/2 and n-2, respectively. Moreover, these two bounds are tight.

In approval-based budget division, a budget needs to be distributed to some candidates based on the voters' approval ballots over these candidates. In the pursuit of a simple, consistent, and approximately fair rule for this setting, we introduce the maximum payment rule (MP). Under this rule, each voter controls a part of the budget and, in each step, the corresponding voters allocate their entire budget to the candidate approved by the largest number of voters with nonzero budget. We show that MP meets our criteria as it satisfies monotonicity and a demanding population consistency condition and gives a 2-approximation to a fairness notion called average fair share (AFS). Moreover, we generalize MP to the class of sequential payment rules and prove that it is the most desirable rule within this class: nearly all other sequential payment rules fail monotonicity while offering only small improvements in the approximation ratio to AFS.

We study the problem of fairly allocating indivisible items and a desirable heterogeneous divisible good (i.e., cake) to agents with additive utilities. In our paper, each indivisible item can be a good that yields non-negative utilities to some agents and a chore that yields negative utilities to the other agents. Given a fixed set of divisible and indivisible resources, we investigate almost envy-free allocations, captured by the natural fairness concept of envy-freeness for mixed resources (EFM). It requires that an agent $i$ does not envy another agent $j$ if agent $j$'s bundle contains any piece of cake yielding positive utility to agent $i$ (i.e., envy-freeness), and agent $i$ is envy-free up to one item (EF1) towards agent $j$ otherwise. We prove that with indivisible items and a cake, an EFM allocation always exists for any number of agents with additive utilities.

We explore solutions for fairly allocating indivisible items among agents assigned weights representing their entitlements. Our fairness goal is weighted-envy-freeness (WEF), where each agent deems their allocated portion relative to their entitlement at least as favorable as any others relative to their own. Often, achieving WEF necessitates monetary transfers, which can be modeled as third-party subsidies. The goal is to attain WEF with bounded subsidies. Previous work relied on characterizations of unweighted envy-freeness (EF), that fail in the weighted setting. This makes our new setting challenging. We present polynomial-time algorithms that compute WEF allocations with a guaranteed upper bound on total subsidy for monotone valuations and various subclasses thereof. We also present an efficient algorithm to compute a fair allocation of items and money, when the budget is not enough to make the allocation WEF. This algorithm is new even for the unweighted setting.

We propose a new model for aggregating preferences over a set of indivisible items based on a quantile value. In this model, each agent is endowed with a specific quantile, and the value of a given bundle is defined by the corresponding quantile of the individual values of the items within it. Our model captures the diverse ways in which agents may perceive a bundle, even when they agree on the values of individual items. It enables richer behavioral modeling that cannot be easily captured by additive valuation functions. We study the problem of maximizing utilitarian and egalitarian welfare within the quantile-based valuation setting. For each of the welfare functions, we analyze the complexity of the objectives. Interestingly, our results show that the complexity of both objectives varies significantly depending on whether the allocation is required to be balanced. We provide near-optimal approximation algorithms for utilitarian welfare, and for egalitarian welfare, we present exact algorithms whenever possible.

We consider the problem of fair allocation of indivisible items with subsidies when agents have weighted entitlements. Specifically, we extend the envy-freeability studied in the unweighted case to the weighted envy-freeability and delve deeper into its properties. We first highlight various important differences from the unweighted case, e.g., the sufficient conditions that lead to envy-freeability in the unweighted case do not lead to weighted envy-freeability in the weighted case. We then present various results concerningweighted envy-freeability including general characterizations, algorithms for achieving and testing weighted envy-freeability, lower and upper bounds of the amount of subsidies for weighted envy-freeable allocations. Additionally, we design algorithms that ensure weighted envy-freeability while incorporating other fairness properties, such as weighted envy-freeness up to one item transfer.

Approval voting is widely used for making multi-winner voting decisions. The canonical rule (also called Approval Voting) used in the setting aims to maximize social welfare by selecting candidates with the highest number of approvals. We revisit approval-based multi-winner voting in scenarios where the information regarding the voters' preferences is uncertain. We present several algorithmic results for problems related to social welfare maximization under uncertainty, including computing the social welfare probability distribution of a given outcome, computing the probability that a given outcome is social welfare maximizing, computing an outcome that is social welfare maximizing with the highest probability, and understanding how robust an outcome is with respect to social welfare maximization.

In pursuit of participatory budgeting (PB) outcomes with broader fairness guarantees, we initiate the study of lotteries over discrete PB outcomes. As the projects have heterogeneous costs, the amount spent may not be equal ex ante and ex post. To address this, we develop a technique to bound the amount by which the ex-post spend differs from the ex-ante spend -- the property is termed budget balanced up to one project (BB1). With respect to fairness, we take a best-of-both-worlds perspective, seeking outcomes that are both ex-ante and ex-post fair. Towards this goal, we initiate a study of ex-ante fairness properties in PB, including Individual Fair Share (IFS), Unanimous Fair Share (UFS) and their stronger variants, as well as Group Fair Share (GFS). We show several incompatibility results between these ex-ante fairness notions and existing ex-post concepts based on justified representation. One of our main contributions is a randomized algorithm which simultaneously satisfies ex-ante Strong UFS, ex-post full justified representation (FJR) and ex-post BB1 for PB with binary utilities.

We study the envy-free house allocation problem when agents have uncertain preferences over items and consider several well-studied preference uncertainty models. The central problem that we focus on is computing an allocation that has the highest probability of being envy-free. We show that each model leads to a distinct set of algorithmic and complexity results, including detailed results on (in-)approximability. En route, we consider two related problems of checking whether there exists an allocation that is possibly or necessarily envy-free. We give a complete picture of the computational complexity of these two problems for all the uncertainty models we consider.

Fair division considers the allocation of scarce resources among agents in such a way that every agent gets a fair share. It is a fundamental problem in society and has received significant attention and rapid developments from the game theory and artificial intelligence communities in recent years. The majority of the fair division literature can be divided along at least two orthogonal directions: goods versus chores, and divisible versus indivisible resources. In this survey, besides describing the state of the art, we outline a number of interesting open questions and future directions in three mixed fair division settings: (i) indivisible goods and chores, (ii) divisible and indivisible goods (mixed goods), and (iii) indivisible goods with subsidy which can be viewed like a divisible good.

In the committee voting setting, a subset of k alternatives is selected based on the preferences of voters. In this paper, our goal is to efficiently compute ex-ante fair probability distributions (or lotteries) over committees. Since it is not known whether a lottery satisfying the desirable fairness property of fractional core is polynomial-time computable, we introduce a new axiom called group resource proportionality (GRP), which strengthens other fairness notions in the literature. We characterize our fairness axiom by a correspondence with max flows on a network formulation of committee voting. Using the connection to flow networks revealed by this characterization, we then introduce voting rules which achieve fairness in conjunction with other desirable properties. The redistributive utilitarian rule satisfies ex-ante efficiency in addition to our fairness axiom. We also give a voting rule which maximizes social welfare subject to fairness by reducing to a minimum-cost maximum-flow problem. Lastly, we show our fairness property can be obtained in tandem with strong ex-post fairness properties – an approach known as best-of-both-worlds fairness. We strengthen existing best-or-both-worlds fairness results in committee voting and resolve an open question posed by Aziz et al. (2023). These findings follow from an auxiliary result which may prove useful in obtaining best-of-both-worlds type results in future research on committee voting.

We study approval-based committee voting in which a target number of candidates are selected based on voters' approval preferences over candidates. In contrast to most of the work, we consider the setting where voters express uncertain approval preferences and explore four different types of uncertain approval preference models. For each model, we study the problems such as computing a committee with the highest probability of satisfying axioms such as justified representation.