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 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 simple, well-behaved, and approximately fair rules for this setting, we introduce the class of sequential payment rules, where each voter controls a part of the budget and repeatedly spends his share on his approved candidates to determine the final distribution. We show that all sequential payment rules satisfy a demanding population consistency notion and we identify two particularly appealing rules within this class called the maximum payment rule (MP) and the 1/3-multiplicative sequential payment rule (1/3-MSP). More specifically, we prove that (i) MP is, apart from one other rule, the only monotonic sequential payment rule and gives a 2-approximation to a fairness notion called average fair share, and (ii) 1/3-MSP gives a 3/2-approximation to average fair share, which is optimal among sequential payment rules.

Abstract We formalize a framework for coordinating funding and selecting projects, the costs of which are shared among agents with quasi-linear utility functions and individual budgets. Our model contains the discrete participatory budgeting model as a special case, while capturing other useful scenarios. We propose several important axioms and objectives and study how well they can be simultaneously satisfied. We show that whereas welfare maximization admits an FPTAS, welfare maximization subject to a natural and very weak participation requirement leads to a strong inapproximability. This result is bypassed if we consider some natural restricted valuations, namely laminar single-minded valuations and symmetric valuations. Our analysis for the former restriction leads to the discovery of a new class of tractable instances for the Set Union Knapsack problem, a classical problem in combinatorial optimization.

Envy-freeness is one of the most important fairness concerns when allocating items. We study envy-free house allocation 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.

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.

The fair allocation of resources to agents 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 in three mixed fair division settings: (i) indivisible goods and chores, (ii) divisible and indivisible goods (i.e., mixed goods), and (iii) fair division of indivisible goods with subsidy.

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. [2023a]. 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.

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.

Proportionality is an attractive fairness concept that has been applied to a range of problems including the facility location problem, a classic problem in social choice. In our work, we propose a concept called Strong Proportionality, which ensures that when there are two groups of agents at different locations, both groups incur the same total cost. We show that although Strong Proportionality is a well-motivated and basic axiom, there is no deterministic strategyproof mechanism satisfying the property. We then identify a randomized mechanism called Random Rank (which uniformly selects a number $k$ between $1$ to $n$ and locates the facility at the $k$'th highest agent location) which satisfies Strong Proportionality in expectation. Our main theorem characterizes Random Rank as the unique mechanism that achieves universal truthfulness, universal anonymity, and Strong Proportionality in expectation among all randomized mechanisms. Finally, we show via the AverageOrRandomRank mechanism that even stronger ex-post fairness guarantees can be achieved by weakening universal truthfulness to strategyproofness in expectation.

We study the fair division of a collection of mindivisible goods amongst a set of nagents. Whilst envy-free allocations typically do not exist in the indivisible-goods setting, envy-freeness can be achieved if some amount of a divisible good (money) is introduced. Specifically, Halpern and Shah (SAGT 2019, pp.374-389) showed that, given additive valuation functions where the marginal value of each good is at most one dollar for each agent, there always exists an envy-free allocation requiring a subsidy of at most (n-1)·m dollars. The authors also conjectured that a subsidy of $n-1$ dollars is sufficient for additive valuations. We prove this conjecture. In fact, a subsidy of at most one dollar per agent is sufficient to guarantee the existence of an envy-free allocation. Further, we prove that for general monotonic valuation functions an envy-free allocation always exists with a subsidy of at most 2(n-1) dollars per agent. In particular, the total subsidy required for monotonic valuations is independent of the number of goods.