In the allocation of resources to a set of agents, how do fairness guarantees impact the social welfare? A quantitative measure of this impact is the price of fairness, which measures the worst-case loss of social welfare due to fairness constraints. While initially studied for divisible goods, recent work on the price of fairness also studies the setting of indivisible goods.
In this paper, we resolve the price of two well-studied fairness notions for the allocation of indivisible goods: envy-freeness up to one good (EF1), and approximate maximin share (MMS). For both EF1 and 1/2-MMS guarantees, we show, via different techniques, that the price of fairness is $O(\sqrt{n})$, where $n$ is the number of agents. From previous work, it follows that our bounds are tight. Our bounds are obtained via efficient algorithms. For 1/2-MMS, our bound holds for additive valuations, whereas for EF1, our bound holds for the more general class of subadditive valuations. This resolves an open problem posed by Bei et al. (2019).
We study a fair allocation problem of indivisible items under additive externalities in which each agent also receives values from items that are assigned to other agents. We propose several new fairness concepts. We extend the well-studied envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) to this setting, and we propose a new fairness concept called general fair share (GFS). We undertake a detailed study and present algorithms for finding fair allocations.
The leximin solution -- which selects an allocation that maximizes the minimum utility, then the second minimum utility, and so forth -- is known to provide EFX (envy-free up to any good) fairness guarantee in some contexts when allocating indivisible goods. However, it remains unknown how fair the leximin solution is when used to allocate indivisible chores. In this paper, we demonstrate that the leximin solution can be modified to also provide compelling fairness guarantees for the allocation of indivisible chores. First, we generalize the definition of the leximin solution. Then, we show th…
We study fair resource allocation when the resources contain a mixture of divisible and indivisible goods, focusing on the well-studied fairness notion of maximin share fairness (MMS). With only indivisible goods, a full MMS allocation may not exist, but a constant multiplicative approximate allocation always does. We analyze how the MMS approximation guarantee would be affected when the resources to be allocated also contain divisible goods. In particular, we show that the worst-case MMS approximation guarantee with mixed goods is no worse than that with only indivisible goods. However, there…
We investigate the efficiency of fair allocations of indivisible goods using the well-studied price of fairness concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envy-freeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We also introduce the concept of strong price of fairness, wh…
We study several fairness notions in allocating indivisible chores (i.e., items with non-positive values) to agents who have additive and submodular cost functions. The fairness criteria we are concern with are envy-free up to any item (EFX), envy-free up to one item (EF1), maximin share (MMS), and pairwise maximin share (PMMS), which are proposed as relaxations of envy-freeness in the setting of additive cost functions. For allocations under each fairness criterion, we establish their approximation guarantee for other fairness criteria. Under the additive setting, our results show strong conn…
We initiate the study of indivisible chore allocation for agents with asymmetric shares. The fairness concept we focus on is the weighted natural generalization of maxmin share: WMMS fairness and OWMMS fairness. We first highlight the fact that commonly used algorithms that work well for the allocation of goods to asymmetric agents, and even for chores to symmetric agents do not provide good approximations for allocation of chores to asymmetric agents under WMMS. As a consequence, we present a novel polynomial-time constant-approximation algorithm, via linear program, for OWMMS. For two specia…
Fair allocation of indivisible goods is a well-explored problem. Traditionally, research focused on individual fairness - are individual agents satisfied with their allotted share? - and group fairness - are groups of agents treated fairly? In this paper, we explore the coexistence of individual envy-freeness (i-EF) and its group counterpart, group weighted envy-freeness (g-WEF), in the allocation of indivisible goods. We propose several polynomial-time algorithms that provably achieve i-EF and g-WEF simultaneously in various degrees of approximation under three different conditions on the age…
The theory of algorithmic fair allocation is within the center of multi-agent systems and economics in the last decade due to its industrial and social importance. At a high level, the problem is to assign a set of items that are either goods or chores to a set of agents so that every agent is happy with what she obtains. Particularly, in this survey, we focus on indivisible items, for which absolute fairness such as envy-freeness and proportionality cannot be guaranteed. One main theme in the recent research agenda is about designing algorithms that approximately achieve the fairness criteria…
We study how to fairly allocate a set of indivisible chores to a group of agents, where each agent $i$ has a non-negative weight $w_i$ that represents its obligation for undertaking the chores. We consider the fairness notion of weighted envy-freeness up to one item (WEF1) and propose an efficient picking sequence algorithm for computing WEF1 allocations. Our analysis is based on a natural and powerful continuous interpretation for the picking sequence algorithms in the weighted setting, which might be of independent interest. Using this interpretation, we establish the necessary and sufficien…
Allocating resources to individuals in a fair manner has been a topic of interest since the ancient times, with most of the early rigorous mathematical work on the problem focusing on infinitely divisible resources. Recently, there has been a surge of papers studying computational questions regarding various different notions of fairness for the indivisible case, like maximin share fairness (MMS) and envy-freeness up to any good (EFX). We survey the most important results in the discrete fair division literature, focusing on the case of additive valuation functions and paying particular attent…
In AAMAS 2014, Bouveret and Lemaitre (2014) presented a hierarchy of fairness concepts for allocation of indivisible objects. Among them CEEI (Competitive Equilibrium with Equal Incomes) was the strongest. In this note, we settle the complexity of computing a discrete CEEI assignment by showing it is strongly NP-hard. We then highlight a fairness notion (CEEI-FRAC) that is even stronger than CEEI for discrete assignments, is always Pareto optimal, and can be verified in polynomial time. We also show that computing a CEEI-FRAC discrete assignment is strongly NP-hard in general but polynomial-ti…
In this paper, we study the classic problem of fairly allocating indivisible items with the extra feature that the items lie on a line. Our goal is to find a fair allocation that is contiguous, meaning that the bundle of each agent forms a contiguous block on the line. While allocations satisfying the classical fairness notions of proportionality, envy-freeness, and equitability are not guaranteed to exist even without the contiguity requirement, we show the existence of contiguous allocations satisfying approximate versions of these notions that do not degrade as the number of agents or items…
In this paper, we study how to fairly allocate m indivisible chores to n (asymmetric) agents. We consider (weighted) proportionality up to any item (PROPX) and show that a (weighted) PROPX allocation always exists and can be computed efficiently. For chores, we argue that PROPX might be a more reliable relaxation for proportionality by the facts that any PROPX allocation ensures 2-approximation of maximin share (MMS) fairness [Budish, 2011] for symmetric agents and of anyprice share (APS) fairness [Babaioff et al, 2021] for asymmetric agents. APS allocations for chores have not been studied be…
NICTA（澳大利亚国家信息与通信技术研究院）；University of New South Wales（新南威尔士大学）
We consider the discrete assignment problem in which agents express ordinal preferences over objects and these objects are allocated to the agents in a fair manner. We use the stochastic dominance relation between fractional or randomized allocations to systematically define varying notions of proportionality and envy-freeness for discrete assignments. The computational complexity of checking whether a fair assignment exists is studied for these fairness notions. We also characterize the conditions under which a fair assignment is guaranteed to exist. For a number of fairness concepts, polynom…
Higher School of Economics（高等经济大学）；Conestoga High School（康内斯托加高中）；Drexel University（德雷塞尔大学）
We study the problem of fairly allocating indivisible goods and focus on the classic fairness notion of proportionality. The indivisibility of the goods is long known to pose highly non-trivial obstacles to achieving fairness, and a very vibrant line of research has aimed to circumvent them using appropriate notions of approximate fairness. Recent work has established that even approximate versions of proportionality (PROPx) may be impossible to achieve even for small instances, while the best known achievable approximations (PROP1) are much weaker. We introduce the notion of proportionality u…
In the allocation of indivisible goods, the maximum Nash welfare (MNW) rule, which chooses an allocation maximizing the product of the agents' utilities, has received substantial attention for its fairness. We characterize MNW as the only additive welfarist rule that satisfies envy-freeness up to one good. Our characterization holds even in the simplest setting of two agents.
We study the fair division of items to agents supposing that agents can form groups. We thus give natural generalizations of popular concepts such as envy-freeness and Pareto efficiency to groups of fixed sizes. Group envy-freeness requires that no group envies another group. Group Pareto efficiency requires that no group can be made better off without another group be made worse off. We study these new group properties from an axiomatic viewpoint. We thus propose new fairness taxonomies that generalize existing taxonomies. We further study near versions of these group properties as allocation…
Competitive equilibrium with equal incomes (CEEI) is a well known fair allocation mechanism; however, for indivisible resources a CEEI may not exist. It was shown in [Budish '11] that in the case of indivisible resources there is always an allocation, called A-CEEI, that is approximately fair, approximately truthful, and approximately efficient, for some favorable approximation parameters. This approximation is used in practice to assign students to classes. In this paper we show that finding the A-CEEI allocation guaranteed to exist by Budish's theorem is PPAD-complete. We further show that f…
We study the problem of fairly allocating indivisible goods to groups of agents. Agents in the same group share the same set of goods even though they may have different preferences. Previous work has focused on unanimous fairness, in which all agents in each group must agree that their group's share is fair. Under this strict requirement, fair allocations exist only for small groups. We introduce the concept of democratic fairness, which aims to satisfy a certain fraction of the agents in each group. This concept is better suited to large groups such as cities or countries. We present protoco…
We study the problem of dynamically allocating $T$ indivisible items to $n$ agents with the restriction that the allocation is fair all the time. Due to the negative results to achieve fairness when allocations are irrevocable, we allow adjustments to make fairness attainable with the objective to minimize the number of adjustments. For restricted additive or general identical valuations, we show that envy-freeness up to one item (EF1) can be achieved with no adjustments. For additive valuations, we give an EF1 algorithm that requires $O(mT)$ adjustments, improving the previous result of $O(nm…
We consider a multi-agent resource allocation setting in which an agent's utility may decrease or increase when an item is allocated. We take the group envy-freeness concept that is well-established in the literature and present stronger and relaxed versions that are especially suitable for the allocation of indivisible items. Of particular interest is a concept called group envy-freeness up to one item (GEF1). We then present a clear taxonomy of the fairness concepts. We study which fairness concepts guarantee the existence of a fair allocation under which preference domain. For two natural c…
We study fair division of indivisible goods in a single-parameter environment. In particular, we develop truthful social welfare maximizing mechanisms for fairly allocating indivisible goods. Our fairness guarantees are in terms of solution concepts which are tailored to address allocation of indivisible goods and, hence, provide an appropriate framework for fair division of goods. This work specifically considers fairness in terms of envy freeness up to one good (EF1), maximin share guarantee (MMS), and Nash social welfare (NSW). Our first result shows that (in a single-parameter environment)…
We study the problem of fair division when the resources contain both divisible and indivisible goods. Classic fairness notions such as envy-freeness (EF) and envy-freeness up to one good (EF1) cannot be directly applied to the mixed goods setting. In this work, we propose a new fairness notion envy-freeness for mixed goods (EFM), which is a direct generalization of both EF and EF1 to the mixed goods setting. We prove that an EFM allocation always exists for any number of agents. We also propose efficient algorithms to compute an EFM allocation for two agents and for $n$ agents with piecewise …
In fair division applications, agents may have unequal entitlements reflecting their different contributions. Moreover, the contributions of agents may depend on the allocation itself. Previous fairness notions designed for agents with equal or pre-determined entitlement fail to characterize fairness in these collaborative allocation scenarios. We propose a novel fairness notion of average envy-freeness (AEF), where the envy of agents is defined on the average value of items in the bundles. Average envy-freeness provides a reasonable comparison between agents based on the items they receive an…
Indian Institute of Science（印度科学理工学院）；Chennai Mathematical Institute（金奈数学研究所）
We study the problem of allocating indivisible goods among n agents in a fair manner. For this problem, maximin share (MMS) is a well-studied solution concept which provides a fairness threshold. Specifically, maximin share is defined as the minimum utility that an agent can guarantee for herself when asked to partition the set of goods into n bundles such that the remaining (n-1) agents pick their bundles adversarially. An allocation is deemed to be fair if every agent gets a bundle whose valuation is at least her maximin share. Even though maximin shares provide a natural benchmark for fairn…
We study fair allocation of indivisible chores (i.e., items with non-positive value) among agents with additive valuations. An allocation is deemed fair if it is (approximately) equitable, which means that the disutilities of the agents are (approximately) equal. Our main theoretical contribution is to show that there always exists an allocation that is simultaneously equitable up to one chore (EQ1) and Pareto optimal (PO), and to provide a pseudopolynomial-time algorithm for computing such an allocation. In addition, we observe that the Leximin solution---which is known to satisfy a strong fo…
We consider Max-min Share (MmS) allocations of items both in the case where items are goods (positive utility) and when they are chores (negative utility). We show that fair allocations of goods and chores have some fundamental connections but differences as well. We prove that like in the case for goods, an MmS allocation does not need to exist for chores and computing an MmS allocation - if it exists - is strongly NP-hard. In view of these non-existence and complexity results, we present a polynomial-time 2-approximation algorithm for MmS fairness for chores. We then introduce a new fairness…
In fair division, equitability dictates that each participant receives the same level of utility. In this work, we study equitable allocations of indivisible goods among agents with additive valuations. While prior work has studied (approximate) equitability in isolation, we consider equitability in conjunction with other well-studied notions of fairness and economic efficiency. We show that the Leximin algorithm produces an allocation that satisfies equitability up to any good and Pareto optimality. We also give a novel algorithm that guarantees Pareto optimality and equitability up to one go…
In this work, we revisit the problem of fairly allocating a number of indivisible items that are located on a line to multiple agents. A feasible allocation requires that the allocated items to each agent are connected on the line. The items can be goods on which agents have non-negative utilities, or chores on which the utilities are non-positive. Our objective is to understand the extent to which welfare is inevitably sacrificed by enforcing the allocations to be fair, i.e., price of fairness (PoF). We study both egalitarian and utilitarian welfare. Previous works by Suksompong [Discret. App…
We study envy-free allocations of indivisible goods to agents in settings where each agent is unaware of the goods allocated to other agents. In particular, we propose the maximin aware (MMA) fairness measure, which guarantees that every agent, given the bundle allocated to her, is aware that she does not envy at least one other agent, even if she does not know how the other goods are distributed among other agents. We also introduce two of its relaxations and discuss their egalitarian guarantee and existence. Finally, we present a polynomial-time algorithm, which computes an allocation that a…
In fair division of indivisible goods, using sequences of sincere choices (or picking sequences) is a natural way to allocate the objects. The idea is the following: at each stage, a designated agent picks one object among those that remain. This paper, restricted to the case where the agents have numerical additive preferences over objects, revisits to some extent the seminal paper by Brams and King [9] which was specific to ordinal and linear order preferences over items. We point out similarities and differences with this latter context. In particular, we show that any Pareto-optimal alloca…
We study fair allocation of indivisible public goods subject to cardinality (budget) constraints. In this model, we have n agents and m available public goods, and we want to select $k \leq m$ goods in a fair and efficient manner. We first establish fundamental connections between the models of private goods, public goods, and public decision making by presenting polynomial-time reductions for the popular solution concepts of maximum Nash welfare (MNW) and leximin. These mechanisms are known to provide remarkable fairness and efficiency guarantees in private goods and public decision making se…
