We study the problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations. Envy-freeness up to any good (EFX) is arguably the most compelling fairness notion in this context. However, the existence of EFX allocations has not been settled and is one of the most important problems in fair division. Towards resolving this problem, many impressive results show the existence of its relaxations, e.g., the existence of $0.618$-EFX allocations, and the existence of EFX at most $n-1$ unallocated goods. The latter result was recently improved for three agents, in which the two unallocated goods are allocated through an involved procedure. Reducing the number of unallocated goods for arbitrary number of agents is a systematic way to settle the big question. In this paper, we develop a new approach, and show that for every $\varepsilon \in (0,1/2]$, there always exists a $(1-\varepsilon)$-EFX allocation with sublinear number of unallocated goods and high Nash welfare.
For this, we reduce the EFX problem to a novel problem in extremal graph theory. We introduce the notion of rainbow cycle number $R(\cdot)$. For all $d \in \mathbb{N}$, $R(d)$ is the largest $k$ such that there exists a $k$-partite digraph $G =(\cup_{i \in [k]} V_i, E)$, in which
1) each part has at most $d$ vertices, i.e., $\lvert V_i \rvert \leq d$ for all $i \in [k]$,
2) for any two parts $V_i$ and $V_j$, each vertex in $V_i$ has an incoming edge from some vertex in $V_j$ and vice-versa, and
3) there exists no cycle in $G$ that contains at most one vertex from each part.
We show that any upper bound on $R(d)$ directly translates to a sublinear bound on the number of unallocated goods. We establish a polynomial upper bound on $R(d)$, yielding our main result. Furthermore, our approach is constructive, which also gives a polynomial-time algorithm for finding such an allocation.
A Counterexample to EFX $n \ge 3$ Agents, $m \ge n + 5$ Items, Submodular Valuations via SAT-Solving
Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Saarland Informatics Campus（萨尔兰信息学校园）；University of Bonn（波恩大学）；Hertz Chair for Algorithms and Optimization（赫茨算法与优化讲席）；Saarbrücken Graduate School for Computer Science（萨尔布吕肯计算机科学研究生院）；CISPA Helmholtz Center for Information Security（CISPA赫尔霍兹信息安全中心）
We consider fair allocation of indivisible items in a model with goods, chores, and copies, as a unified framework for studying: (1)~the existence of EFX and other solution concepts for goods with copies; (2)~the existence of EFX and other solution concepts for chores. We establish a tight relation between these issues via two conceptual contributions: First, a refinement of envy-based fairness notions that we term envy \emph{without commons} (denoted $\efxwc$ when applied to EFX). Second, a formal \emph{duality theorem} relating the existence of a host of (refined) fair allocation concepts fo…
We consider the classic problem of fairly allocating indivisible goods among agents with additive valuation functions and explore the connection between two prominent fairness notions: maximum Nash welfare (MNW) and envy-freeness up to any good (EFX). We establish that an MNW allocation is always EFX as long as there are at most two possible values for the goods, whereas this implication is no longer true for three or more distinct values. As a notable consequence, this proves the existence of EFX allocations for these restricted valuation functions. While the efficient computation of an MNW a…
Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a $(ϕ-1)$-approximation of envy-freeness up to any good…
Simultaneous Ordinal Maximin Share and Envy-Based Guarantees
Max Planck Institute for Informatics（马克斯·普朗克 informatics 研究所）；Universität des Saarlandes（萨尔兰州立大学）；Hertz Chair for Algorithms and Optimization, University of Bonn（波恩大学赫兹算法与优化讲席）；University of Bonn（波恩大学）
Envy-free up to one good (EF1) and envy-free up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations. In sharp contrast, it is unknown whether or not an EFX allocation always exists, even for four agents and additive valuations. In addition, the best approximation guarantee for EFX is $(ϕ-1) \simeq 0.61$ by Amanitidis et al.. In order to find a middle ground to bridge this gap, in this paper we suggest another fairness criterion, namely envy-freeness up to a rando…
Envy-freeness up to any good (EFX) provides a strong and intuitive guarantee of fairness in the allocation of indivisible goods. But whether such allocations always exist or whether they can be efficiently computed remains an important open question. We study the existence and computation of EFX in conjunction with various other economic properties under lexicographic preferences--a well-studied preference model in artificial intelligence and economics. In sharp contrast to the known results for additive valuations, we not only prove the existence of EFX and Pareto optimal allocations, but in …
When allocating a set of indivisible items among agents, the ideal condition of envy-freeness cannot always be achieved. Envy-freeness up to any good (EFX), and envy-freeness with $k$ hidden items (HEF-$k$) are two very compelling relaxations of envy-freeness, which remain elusive in many settings. We study a natural relaxation of these two fairness constraints, where we place the agents on the vertices of an undirected graph, and only require that our allocations satisfy the EFX (resp. HEF) constraint on the edges of the graph. We refer to these allocations as graph-EFX (resp. graph-HEF) or s…
In fair division problems with indivisible goods it is well known that one cannot have any guarantees for the classic fairness notions of envy-freeness and proportionality. As a result, several relaxations have been introduced, most of which in quite recent works. We focus on four such notions, namely envy-freeness up to one good (EF1), envy-freeness up to any good (EFX), maximin share fairness (MMS), and pairwise maximin share fairness (PMMS). Since obtaining these relaxations also turns out to be problematic in several scenarios, approximate versions of them have been considered. In this wor…
Pushing the Frontier on Approximate EFX Allocations
Department of Informatics, Athens University of Economics and Business（阿提卡经济与商业大学信息学院）；Archimedes, Athena Research Center（阿提卡研究中心）；School of Informatics, University of Edinburgh（爱丁堡大学信息学院）
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…
The Power of Share-Based Notions in Proving Envy-Based Fairness Guarantees
Max Planck Institute Informatics（马克斯·普朗克 informatics 研究所）；Universität des Saarlandes（萨尔兰大学）；Hertz Chair for Algorithms and Optimization, University of Bonn（波恩大学赫兹算法与优化讲席）；Weizmann Institute（魏茨曼科学研究所）；The University of Tokyo（东京大学）；University of Warsaw（华沙大学）
Several fairness concepts have been proposed recently in attempts to approximate envy-freeness in settings with indivisible goods. Among them, the concept of envy-freeness up to any item (EFX) is arguably the closest to envy-freeness. Unfortunately, EFX allocations are not known to exist except in a few special cases. We make significant progress in this direction. We show that for every instance with additive valuations, there is an EFX allocation of a subset of items with a Nash welfare that is at least half of the maximum possible Nash welfare for the original set of items. That is, after d…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We study the problem of finding fair and efficient allocations of a set of indivisible items to a set of agents, where each item may be a good (positively valued) for some agents and a bad (negatively valued) for others, i.e., a mixed manna. As fairness notions, we consider arguably the strongest possible relaxations of envy-freeness and proportionality, namely envy-free up to any item (EFX and EFX$_0$), and proportional up to the maximin good or any bad (PropMX and PropMX$_0$). Our efficiency notion is Pareto-optimality (PO). We study two types of instances: (i) Separable, where the item set …
The goal of fair division is to distribute resources among competing players in a "fair" way. Envy-freeness is the most extensively studied fairness notion in fair division. Envy-free allocations do not always exist with indivisible goods, motivating the study of relaxed versions of envy-freeness. We study the envy-freeness up to any good (EFX) property, which states that no player prefers the bundle of another player following the removal of any single good, and prove the first general results about this property. We use the leximin solution to show existence of EFX allocations in several con…
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…
Allocating Indivisible Goods to Strategic Agents: Pure Nash Equilibria and Fairness
School of Mathematics, Statistics and Actuarial Science, University of Essex（埃塞克斯大学数学、统计与精算科学学院）；Archimedes/Athena RC（阿基米德/雅典娜研究中心）；Department of Computer Science, University of Liverpool（利物浦大学计算机科学系）；Department of Computer, Control, and Management Engineering, Sapienza University of Rome（罗马大学计算机、控制与管理工程系）；Input Output Global (IOG)（输入输出全球公司）；Institute for Logic, Language and Computation, University of Amsterdam（阿姆斯特丹大学逻辑、语言与计算研究所）
Fair division of indivisible goods is a very well-studied problem. The goal of this problem is to distribute $m$ goods to $n$ agents in a "fair" manner, where every agent has a valuation for each subset of goods. We assume general valuations. Envy-freeness is the most extensively studied notion of fairness. However, envy-free allocations do not always exist when goods are indivisible. The notion of fairness we consider here is "envy-freeness up to any good" (EFX) where no agent envies another agent after the removal of any single good from the other agent's bundle. It is not known if such an a…
We study the problem of fairly allocating indivisible goods between groups of agents using the recently introduced relaxations of envy-freeness. We consider the existence of fair allocations under different assumptions on the valuations of the agents. In particular, our results cover cases of arbitrary monotonic, responsive, and additive valuations, while for the case of binary valuations we fully characterize the cardinalities of two groups of agents for which a fair allocation can be guaranteed with respect to both envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX). Mo…
