Fair division of indivisible items is a well-studied topic in Economics and Computer Science. The objective is to allocate items to agents in a fair manner, where each agent has a valuation for each subset of items. Envy-freeness is one of the most widely studied notions of fairness. Since complete envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling one is envy-freeness up to any item (EFX), where no agent envies another agent after the removal of any single item from the other agent's bundle. However, despite significant efforts by many researchers for several years, it is known that a complete EFX allocation always exists only in limited cases. In this paper, we show that a complete EFX allocation always exists when each agent is of one of two given types, where agents of the same type have identical additive valuations. This is the first such existence result for non-identical valuations when there are any number of agents and items and no limit on the number of distinct values an agent can have for individual items. We give a constructive proof, in which we iteratively obtain a Pareto dominating (partial) EFX allocation from an existing partial EFX allocation.
The existence of EFX allocations is a major open problem in fair division, even for additive valuations. The current state of the art is that no setting where EFX allocations are impossible is known, and EFX is known to exist for ($i$) agents with identical valuations, ($ii$) 2 agents, ($iii$) 3 agents with additive valuations, ($iv$) agents with one of two additive valuations and ($v$) agents with two-valued instances. It is also known that EFX exists if one can leave $n-1$ items unallocated, where $n$ is the number of agents. We develop new techniques that allow us to push the boundaries of …
Envy-freeness is one of the most widely studied notions in fair division. Since envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling concept is envy-freeness up to any item (EFX). We study the existence of EFX allocations for general valuations. The existence of EFX allocations is a major open problem. For general valuations, it is known that an EFX allocation always exists (i) when $n=2$ or (ii) when all agents have identical valuations, where $n$ is the number of agents. it is also known that …
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…
The existence of EFX allocations is a fundamental open problem in discrete fair division. Given a set of agents and indivisible goods, the goal is to determine the existence of an allocation where no agent envies another following the removal of any single good from the other agent's bundle. Since the general problem has been illusive, progress is made on two fronts: $(i)$ proving existence when the number of agents is small, $(ii)$ proving existence of relaxations of EFX. In this paper, we improve results on both fronts (and simplify in one of the cases). We prove the existence of EFX allocat…
We study the problem of allocating a set of indivisible goods among agents with subadditive valuations in a fair and efficient manner. Envy-Freeness up to any good (EFX) is the most compelling notion of fairness in the context of indivisible goods. Although the existence of EFX is not known beyond the simple case of two agents with subadditive valuations, some good approximations of EFX are known to exist, namely $\tfrac{1}{2}$-EFX allocation and EFX allocations with bounded charity. Nash welfare (the geometric mean of agents' valuations) is one of the most commonly used measures of efficiency…
Czech Technical University in Prague（布拉格捷克理工大学）；University of West Attica（西阿提卡大学）；Athens University of Economics and Business（雅典经济与商业大学）；Archimedes/Athena RC（阿基米德/雅典研究中心）
We study the problem of fairly allocating a set of $m$ indivisible goods to a set of $n$ agents. Envy-freeness up to any good (EFX) criteria -- which requires that no agent prefers the bundle of another agent after removal of any single good -- is known to be a remarkable analogous of envy-freeness when the resource is a set of indivisible goods. In this paper, we investigate EFX notion for the restricted additive valuations, that is, every good has some non-negative value, and every agent is interested in only some of the goods. We introduce a natural relaxation of EFX called EFkX which requi…
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）
A Simple Polynomial-Time EFX Repair for Cancelable Valuations
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（赫兹算法与优化讲席）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
EFX Allocations Exist on Triangle-Free Multi-Graphs
Department of Computer Engineering, Sharif University of Technology（德黑兰理工大学计算机工程系）；Max Planck Institute for Informatics, Saarland Informatics Campus (SIC)（萨尔兰信息学园区马克斯·普朗克计算机科学研究所）
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 consider a fair division model in which agents have positive, zero and negative utilities for items. For this model, we analyse one existing fairness property - EFX - and three new and related properties - EFX$_0$, EFX$^3$ and EF1$^3$ - in combination with Pareto-optimality. With general utilities, we give a modified version of an existing algorithm for computing an EF1$^3$ allocation. With $-α/0/α$ utilities, this algorithm returns an EFX$^3$ and PO allocation. With absolute identical utilities, we give a new algorithm for an EFX and PO allocation. With $-α/0/β$ utilities, this algorithm a…
We study the problem of distributing a set of indivisible items among agents with additive valuations in a $\mathit{fair}$ manner. The fairness notion under consideration is Envy-freeness up to any item (EFX). Despite significant efforts by many researchers for several years, the existence of EFX allocations has not been settled beyond the simple case of two agents. In this paper, we show constructively that an EFX allocation always exists for three agents. Furthermore, we falsify the conjecture by Caragiannis et al. by showing an instance with three agents for which there is a partial EFX all…
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）
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…
On the existence of EFX allocations in multigraphs
Athens University of Economics and Business（雅典经济与商业大学）；Archimedes/Athena RC（阿基米德/雅典娜研究中心）；National and Kapodistrian University of Athens（雅典国立与卡波迪斯特里亚大学）；National Technical University of Athens（雅典国立技术大学）
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…
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）
