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Almost Envy-Freeness with General Valuations

Authors

Benjamin Plaut, Tim Roughgarde
Type: Book-Chapter
Publication Date: 2018-01-01
Citations: 53
DOI: https://doi.org/10.1137/1.9781611975031.165

Abstract

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 contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular fair division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the fair division of indivisible goods with different classes of player valuations.

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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 … 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 contexts, sometimes in conjunction with Pareto optimality. For two players with valuations obeying a mild assumption, one of these results provides stronger guarantees than the currently deployed algorithm on Spliddit, a popular fair division website. Unfortunately, finding the leximin solution can require exponential time. We show that this is necessary by proving an exponential lower bound on the number of value queries needed to identify an EFX allocation, even for two players with identical valuations. We consider both additive and more general valuations, and our work suggests that there is a rich landscape of problems to explore in the fair division of indivisible goods with different classes of player valuations.
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Fair division of indivisible goods is a central challenge in artificial intelligence. For many prominent fairness criteria including envy-freeness (EF) or proportionality (PROP), no allocations satisfying these criteria might exist. … Fair division of indivisible goods is a central challenge in artificial intelligence. For many prominent fairness criteria including envy-freeness (EF) or proportionality (PROP), no allocations satisfying these criteria might exist. Two popular remedies to this problem are randomization or relaxation of fairness concepts. A timely research direction is to combine the advantages of both, commonly referred to as Best of Both Worlds (BoBW). We consider fair division with entitlements, which allows to adjust notions of fairness to heterogeneous priorities among agents. This is an important generalization to standard fair division models and is not well-understood in terms of BoBW results. Our main result is a lottery for additive valuations and different entitlements that is ex-ante weighted envy-free (WEF), as well as ex-post weighted proportional up to one good (WPROP1) and weighted transfer envy-free up to one good (WEF(1,1)). It can be computed in strongly polynomial time. We show that this result is tight - ex-ante WEF is incompatible with any stronger ex-post WEF relaxation. In addition, we extend BoBW results on group fairness to entitlements and explore generalizations of our results to instances with more expressive valuation functions.
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Cited by (35)

A Little Charity Guarantees Almost Envy-Freeness

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Bhaskar Ray Chaudhury , Telikepalli Kavitha , Kurt Mehlhorn +1 more
Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)A Little Charity Guarantees Almost Envy-FreenessBhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini SgouritsaBhaskar … Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)A Little Charity Guarantees Almost Envy-FreenessBhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini SgouritsaBhaskar Ray Chaudhury, Telikepalli Kavitha, Kurt Mehlhorn, and Alkmini Sgouritsapp.2658 - 2672Chapter DOI:https://doi.org/10.1137/1.9781611975994.162PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract 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 allocation always exists even when n = 3. We show there is always a partition of the set of goods into n + 1 subsets (X1, …, Xn, P) where for i ϵ [n], Xi is the bundle allocated to agent i and the set P is unallocated (or donated to charity) such that we have: (1)envy-freeness up to any good,(2)no agent values P higher than her own bundle, and(3)fewer than n goods go to charity, i.e., |P| < n (typically m ≫ n). Our proof is constructive. When agents have additive valuations and |P| is large (i.e., when |P| is close to n), our allocation also has a good maximin share (MMS) guarantee. Moreover, a minor variant of our algorithm also shows the existence of an allocation which is 4/7 groupwise maximin share (GMMS): this is a notion of fairness stronger than MMS. This improves upon the current best bound of 1/2 known for an approximate GMMS allocation. Previous chapter Next chapter RelatedDetails Published:2020eISBN:978-1-61197-599-4 https://doi.org/10.1137/1.9781611975994Book Series Name:ProceedingsBook Code:PRDA20Book Pages:xxii + 3011
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2020-04-03
Georgios Amanatidis , Evangelos Markakis , Apostolos Ntokos
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 … 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 (EFX) and a 2/φ+2 -approximation of groupwise maximin share fairness (GMMS), where φ is the golden ratio. The best known approximation factor, in polynomial time, for either one of these fairness notions prior to this work was 1/2. Moreover, the returned allocation achieves envy-freeness up to one good (EF1) and a 2/3-approximation of pairwise maximin share fairness (PMMS). While EFX is our primary focus, we also exhibit how to fine-tune our algorithm and improve further the guarantees for GMMS or PMMS.Finally, we show that GMMS—and thus PMMS and EFX—allocations always exist when the number of goods does not exceed the number of agents by more than two.
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Fair Division of Mixed Divisible and Indivisible Goods

2020-04-03
Xiaohui Bei , Zihao Li , Jinyan Liu +2 more
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) … 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 linear valuations over the divisible goods. Finally, we relax the envy-free requirement, instead asking for ϵ-envy-freeness for mixed goods (ϵ-EFM), and present an algorithm that finds an ϵ-EFM allocation in time polynomial in the number of agents, the number of indivisible goods, and 1/ϵ.
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Finding Fair and Efficient Allocations When Valuations Don’t Add Up

2020-01-01
Nawal Benabbou , Mithun Chakraborty , Ayumi Igarashi +1 more
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On the number of almost envy-free allocations

2020-04-07
Warut Suksompong
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Fair and Efficient Allocations under Subadditive Valuations

2021-05-18
Bhaskar Ray Chaudhury , Jugal Garg , Ruta Mehta
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 … 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 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. In case of additive valuations, an allocation that maximizes Nash welfare also satisfies fairness properties like Envy-Free up to one good (EF1). Although there is substantial work on approximating Nash welfare when agents have additive valuations, very little is known when agents have subadditive valuations. In this paper, we design a polynomial-time algorithm that outputs an allocation that satisfies either of the two approximations of EFX as well as achieves an O(n) approximation to the Nash welfare. Our result also improves the current best-known approximation of O(n log n) and O(m) to Nash welfare when agents have submodular and subadditive valuations, respectively. Furthermore, our technique also gives an O(n) approximation to a family of welfare measures, p-mean of valuations for p in (-\infty, 1], thereby also matching asymptotically the current best approximation ratio for special cases like p = -\infty while also retaining the remarkable fairness properties.
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On Fair and Efficient Allocations of Indivisible Goods

2021-05-18
Aniket Murhekar , Jugal Garg
We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one … We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation, and a non-constructive proof of existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO). We present a pseudo-polynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive, and that it can be computed in pseudo-polynomial time. We also consider the class of k-ary instances where k is a constant, i.e., each agent has at most k different values for the goods. We show that for such instances an EF1+fPO allocation can be computed in polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in polynomial time. Next, we consider instances where the number of agents is constant, and show that an EF1+PO (also EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations. Further, we show that the problem of computing an EF1+PO allocation polynomial-time reduces to a problem in the complexity class PLS. We also design a polynomial-time algorithm that computes Nash welfare maximizing allocations when there are constantly many agents with constant many different values for the goods.
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Achieving Proportionality up to the Maximin Item with Indivisible Goods

2021-05-18
Artem Baklanov , Pranav Garimidi , Vasilis Gkatzelis +1 more
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 … 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 up to the maximin item (PROPm) and show how to reach an allocation satisfying this notion for any instance involving up to five agents with additive valuations. PROPm provides a well-motivated middle-ground between PROP1 and PROPx, while also capturing some elements of the well-studied maximin share (MMS) benchmark: another relaxation of proportionality that has attracted a lot of attention.
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Protecting the Protected Group: Circumventing Harmful Fairness

2021-05-18
Omer Ben-Porat , Fedor Sandomirskiy , Moshe Tennenholtz
The recent literature on fair Machine Learning manifests that the choice of fairness constraints must be driven by the utilities of the population. However, virtually all previous work makes the … The recent literature on fair Machine Learning manifests that the choice of fairness constraints must be driven by the utilities of the population. However, virtually all previous work makes the unrealistic assumption that the exact underlying utilities of the population (representing private tastes of individuals) are known to the regulator that imposes the fairness constraint. In this paper we initiate the discussion of the \emph{mismatch}, the unavoidable difference between the underlying utilities of the population and the utilities assumed by the regulator. We demonstrate that the mismatch can make the disadvantaged protected group worse off after imposing the fairness constraint and provide tools to design fairness constraints that help the disadvantaged group despite the mismatch.
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Developments in Multi-Agent Fair Allocation

2020-04-03
Haris Aziz
Fairness is becoming an increasingly important concern when designing markets, allocation procedures, and computer systems. I survey some recent developments in the field of multi-agent fair allocation. Fairness is becoming an increasingly important concern when designing markets, allocation procedures, and computer systems. I survey some recent developments in the field of multi-agent fair allocation.
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Approximately EFX Allocations for Indivisible Chores

2022-07-01
Shengwei Zhou , Xiaowei Wu
In this paper we study how to fairly allocate a set of m indivisible chores to a group of n agents, each of which has a general additive cost function … In this paper we study how to fairly allocate a set of m indivisible chores to a group of n agents, each of which has a general additive cost function on the items. Since envy-free (EF) allocation is not guaranteed to exist, we consider the notion of envy-freeness up to any item (EFX). In contrast to the fruitful results regarding the (approximation of) EFX allocations for goods, very little is known for the allocation of chores. Prior to our work, for the allocation of chores, it is known that EFX allocations always exist for two agents, or general number of agents with identical ordering cost functions. For general instances, no non-trivial approximation result regarding EFX allocation is known. In this paper we make some progress in this direction by showing that for three agents we can always compute a 5-approximation of EFX allocation in polynomial time. For n&gt;=4 agents, our algorithm always computes an allocation that achieves an approximation ratio of 3n^2 regarding EFX. We also study the bi-valued instances, in which agents have at most two cost values on the chores, and provide polynomial time algorithms for the computation of EFX allocation when n=3, and (n-1)-approximation of EFX allocation when n&gt;=4.
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On Optimal Tradeoffs between EFX and Nash Welfare

2024-03-24
Michal Feldman , Simon Mauras , Tomasz Ponitka
A major problem in fair division is how to allocate a set of indivisible resources among agents fairly and efficiently. The goal of this work is to characterize the tradeoffs … A major problem in fair division is how to allocate a set of indivisible resources among agents fairly and efficiently. The goal of this work is to characterize the tradeoffs between two well-studied measures of fairness and efficiency --- envy freeness up to any item (EFX) for fairness, and Nash welfare for efficiency --- by saying, for given constants α and β, whether there exists an α-EFX allocation that guarantees a β-fraction of the maximum Nash welfare (β-MNW). For additive valuations, we show that for any α ∈ [0,1], there exists a partial allocation that is α-EFX and 1/(α+1)-MNW. This tradeoff turns out to be tight (for every α) as demonstrated by an impossibility result that we give. We also show that for α ∈ [0, φ-1 ≃ 0.618] these partial allocations can be turned into complete allocations where all items are assigned. Furthermore, for any α ∈ [0, 1/2], we show that the tight tradeoff of α-EFX and 1/(α+1)-MNW with complete allocations holds for the more general setting of subadditive valuations. Our results improve upon the current state of the art, for both additive and subadditive valuations, and match the best-known approximations of EFX under complete allocations, regardless of Nash welfare guarantees. Notably, our constructions for additive valuations also provide EF1 and constant approximations for maximin share guarantees.
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Competitive Equilibrium with Indivisible Goods and Generic Budgets

2021-01-27
Moshe Babaioff , Noam Nisan , Inbal Talgam-Cohen
Competitive equilibrium from equal incomes (CEEI) is a classic solution to the problem of fair and efficient allocation of goods (Foley 1967, Varian 1974). Every agent receives an equal budget … Competitive equilibrium from equal incomes (CEEI) is a classic solution to the problem of fair and efficient allocation of goods (Foley 1967, Varian 1974). Every agent receives an equal budget of artificial currency with which to purchase goods, and prices match demand and supply. However, a CEEI is not guaranteed to exist when the goods are indivisible even in the simple two-agent, single-item market. Yet it is easy to see that, once the two budgets are slightly perturbed (made generic), a competitive equilibrium does exist. In this paper, we aim to extend this approach beyond the single-item case and study the existence of equilibria in markets with two agents and additive preferences over multiple items. We show that, for agents with equal budgets, making the budgets generic—by adding vanishingly small random perturbations—ensures the existence of equilibrium. We further consider agents with arbitrary nonequal budgets, representing nonequal entitlements for goods. We show that competitive equilibrium guarantees a new notion of fairness among nonequal agents and that it exists in cases of interest (such as when the agents have identical preferences) if budgets are perturbed. Our results open opportunities for future research on generic equilibrium existence and fair treatment of nonequals.
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Maximum Nash Welfare and Other Stories About EFX

2020-07-01
Georgios Amanatidis , Georgios Birmpas , Aris Filos-Ratsikas +2 more
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 … 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 allocation for two possible values remains an open problem, we present a novel algorithm for directly constructing EFX allocations in this setting. Finally, we study the question of whether an MNW allocation implies any EFX guarantee for general additive valuation functions under a natural new interpretation of approximate EFX allocations.
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Almost Group Envy-free Allocation of Indivisible Goods and Chores

2020-07-01
Haris Aziz , Simon Rey
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 … 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 classes of additive utilities, we design polynomial-time algorithms to compute a GEF1 allocation. We also prove that checking whether a given allocation satisfies GEF1 is coNP-complete when there are either only goods, only chores or both.
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Almost Envy-Free Repeated Matching in Two-Sided Markets

2020-01-01
Sreenivas Gollapudi , Kostas Kollias , Benjamin Plaut
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Fair Division with Binary Valuations: One Rule to Rule Them All

2020-01-01
Daniel Halpern , Ariel D. Procaccia , Alexandros Psomas +1 more
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Fair Division of Indivisible Goods: A Survey

2022-07-01
Georgios Amanatidis , Georgios Birmpas , Aris Filos-Ratsikas +1 more
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 … 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 attention to the progress made in the last 10 years.
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High-Multiplicity Fair Allocation Using Parametric Integer Linear Programming

2023-09-28
Robert Bredereck , Andrzej Kaczmarczyk , Dušan Knop +1 more
Using insights from parametric integer linear programming, we improve the work of Bredereck et al. [Proc. ACM EC 2019] on high-multiplicity fair allocation. Answering an open question from their work, … Using insights from parametric integer linear programming, we improve the work of Bredereck et al. [Proc. ACM EC 2019] on high-multiplicity fair allocation. Answering an open question from their work, we proved that the problem of finding envy-free Pareto-efficient allocations of indivisible items is fixed-parameter tractable with respect to the combined parameter “number of agents” plus “number of item types.” Our central improvement, compared to their result, is to break the condition that the corresponding utility and multiplicity values have to be encoded in unary, which is required there. Concretely, we show that, while preserving fixed-parameter tractability, these values can be encoded in binary. Thus, we substantially expand the range of feasible utility and multiplicity values.
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When Do Envy-Free Allocations Exist?

2020-01-01
Pasin Manurangsi , Warut Suksompong
We consider a fair division setting in which $m$ indivisible items are to be allocated among $n$ agents, where the agents have additive utilities and the agents' utilities for individual … We consider a fair division setting in which $m$ indivisible items are to be allocated among $n$ agents, where the agents have additive utilities and the agents' utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when $m=\Omega(n\log n)$ but not when $m=n+o(n)$, and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition---instead, the transition is governed by the divisibility relation between $m$ and $n$. On the one hand, if $m$ is divisible by $n$, an envy-free allocation exists with high probability as long as $m\geq 2n$. On the other hand, if $m$ is not “almost” divisible by $n$, an envy-free allocation is unlikely to exist even when $m=\Theta(n\log n/\log\log n)$.
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Comparing Approximate Relaxations of Envy-Freeness

2018-07-01
Georgios Amanatidis , Georgios Birmpas , Vangelis Markakis
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 … 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 also been considered. In this work, we investigate further the connections between the four notions mentioned above and their approximate versions. We establish several tight or almost tight results concerning the approximation quality that any of these notions guarantees for the others, providing an almost complete picture of this landscape. Some of our findings reveal interesting and surprising consequences regarding the power of these notions, e.g., PMMS and EFX provide the same worst-case guarantee for MMS, despite PMMS being a strictly stronger notion than EFX. We believe such implications provide further insight on the quality of approximately fair solutions.
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Maximin-Aware Allocations of Indivisible Goods

2019-07-28
Hau Chan , Jing Chen , Bo Li +1 more
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 … 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 approximately satisfies MMA or its relaxations. Interestingly, the returned allocation is also 1/2-approximate EFX when all agents have sub- additive valuations, which improves the algorithm in [Plaut and Roughgarden, 2018].
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Almost Envy-Freeness in Group Resource Allocation

2019-07-28
Maria Kyropoulou , Warut Suksompong , Alexandros A. Voudouris
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 … 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). Moreover, we introduce a new model where the agents are not partitioned into groups in advance, but instead the partition can be chosen in conjunction with the allocation of the goods. In this model, we show that for agents with arbitrary monotonic valuations, there is always a partition of the agents into two groups of any given sizes along with an EF1 allocation of the goods. We also provide an extension of this result to any number of groups.
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Dynamic Fair Division Problem with General Valuations

2018-07-01
Bo Li , Wenyang Li , Yingkai Li
In this paper, we focus on how to dynamically allocate a divisible resource fairly among n players who arrive and depart over time. The players may have general heterogeneous valuations … In this paper, we focus on how to dynamically allocate a divisible resource fairly among n players who arrive and depart over time. The players may have general heterogeneous valuations over the resource. It is known that the exact envy-free and proportional allocations may not exist in the dynamic setting [Walsh, 2011]. Thus, we will study to what extent we can guarantee the fairness in the dynamic setting. We first design two algorithms which are O(log n)-proportional and O(n)-envy-free for the setting with general valuations, and by constructing the adversary instances such that all dynamic algorithms must be at least Omega(1)-proportional and Omega(n/log n)-envy-free, we show that the bounds are tight up to a logarithmic factor. Moreover, we introduce the setting where the players' valuations are uniform on the resource but with different demands, which generalize the setting of [Friedman et al., 2015]. We prove an O(log n) upper bound and a tight lower bound for this case.
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The Price of Fairness for Indivisible Goods

2019-07-28
Xiaohui Bei , Xinhang Lu , Pasin Manurangsi +1 more
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 … 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 mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions.
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Equitable Allocations of Indivisible Goods

2019-07-28
Rupert Freeman , Sujoy Sikdar , Rohit Vaish +1 more
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 … 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 good in pseudopolynomial time. Our experiments on real-world preference data reveal that approximate envy-freeness, approximate equitability, and Pareto optimality can often be achieved simultaneously.
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PROPm Allocations of Indivisible Goods to Multiple Agents

2021-08-01
Artem Baklanov , Pranav Garimidi , Vasilis Gkatzelis +1 more
We study the classic problem of fairly allocating a set of indivisible goods among a group of agents, and focus on the notion of approximate proportionality known as PROPm. Prior … We study the classic problem of fairly allocating a set of indivisible goods among a group of agents, and focus on the notion of approximate proportionality known as PROPm. Prior work showed that there exists an allocation that satisfies this notion of fairness for instances involving up to five agents, but fell short of proving that this is true in general. We extend this result to show that a PROPm allocation is guaranteed to exist for all instances, independent of the number of agents or goods. Our proof is constructive, providing an algorithm that computes such an allocation and, unlike prior work, the running time of this algorithm is polynomial in both the number of agents and the number of goods.
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An EF2X Allocation Protocol for Restricted Additive Valuations

2022-07-01
Hannaneh Akrami , Rojin Rezvan , Masoud Seddighin
We study the problem of fairly allocating a set of indivisible goods to a set of n agents. Envy-freeness up to any good (EFX) criterion (which requires that no agent … We study the problem of fairly allocating a set of indivisible goods to a set of n agents. Envy-freeness up to any good (EFX) criterion (which requires that no agent prefers the bundle of another agent after the removal of any single good) is known to be a remarkable analogue of envy-freeness when the resource is a set of indivisible goods. In this paper, we investigate EFX for restricted additive valuations, that is, every good has a non-negative value, and every agent is interested in only some of the goods. We introduce a natural relaxation of EFX called EFkX which requires that no agent envies another agent after the removal of any k goods. Our main contribution is an algorithm that finds a complete (i.e., no good is discarded) EF2X allocation for restricted additive valuations. In our algorithm we devise new concepts, namely configuration and envy-elimination that might be of independent interest. We also use our new tools to find an EFX allocation for restricted additive valuations that discards at most n/2 -1 goods.
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Almost Envy-Free Allocations of Indivisible Goods or Chores with Entitlements

2024-03-24
Max Springer , MohammadTaghi Hajiaghayi , Hadi Yami
We here address the problem of fairly allocating indivisible goods or chores to n agents with weights that define their entitlement to the set of indivisible resources. Stemming from well-studied … We here address the problem of fairly allocating indivisible goods or chores to n agents with weights that define their entitlement to the set of indivisible resources. Stemming from well-studied fairness concepts such as envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) for agents with equal entitlements, we present, in this study, the first set of impossibility results alongside algorithmic guarantees for fairness among agents with unequal entitlements. Within this paper, we expand the concept of envy-freeness up to any good or chore to the weighted context (WEFX and XWEF respectively), demonstrating that these allocations are not guaranteed to exist for two or three agents. Despite these negative results, we develop a WEFX procedure for two agents with integer weights, and furthermore, we devise an approximate WEFX procedure for two agents with normalized weights. We further present a polynomial-time algorithm that guarantees a weighted envy-free allocation up to one chore (1WEF) for any number of agents with additive cost functions. Our work underscores the heightened complexity of the weighted fair division problem when compared to its unweighted counterpart.
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Monotone and Online Fair Division

2019-01-01
Martin Aleksandrov , Toby Walsh
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Almost envy-free allocations of unweighted items

2022-03-10
智杰 张 , 东昊 应 , 家琳 张 +1 more
公平分配研究如何把m个不可分的物品公平地分配给n个玩家。每个玩家关于物品有一个可加的估值函数。物品是无权重的，如果他们的取值范围为{1, 0,−1}。非负估值的物品称为奖品，非正估值的物品称为苦差。本文考虑寻找无权重物品的分配，并满足“相差任意物品下是无忌妒的”（EFX0）。EFX0 是本领域内最受关注的公平性度量。一般可加估值函数下的EFX0 分配的存在性仍然是开放的。本文提出寻找无权重物品的EFX0 分配的多项式时间算法。为了达到这个目的，本文分别提出了寻找无权重奖品和无权重苦差的EFX0 分配的算法。接着，通过将二者小心地结合起来，本文得到了最终的算法。本文的结果完整刻画了无权重情况下寻找EFX0 分配的解决方案。 公平分配研究如何把m个不可分的物品公平地分配给n个玩家。每个玩家关于物品有一个可加的估值函数。物品是无权重的，如果他们的取值范围为{1, 0,−1}。非负估值的物品称为奖品，非正估值的物品称为苦差。本文考虑寻找无权重物品的分配，并满足“相差任意物品下是无忌妒的”（EFX0）。EFX0 是本领域内最受关注的公平性度量。一般可加估值函数下的EFX0 分配的存在性仍然是开放的。本文提出寻找无权重物品的EFX0 分配的多项式时间算法。为了达到这个目的，本文分别提出了寻找无权重奖品和无权重苦差的EFX0 分配的算法。接着，通过将二者小心地结合起来，本文得到了最终的算法。本文的结果完整刻画了无权重情况下寻找EFX0 分配的解决方案。

Well-behaved online load balancing against strategic jobs

2022-11-15
Bo Li , Minming Li , Xiaowei Wu
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Closing Gaps in Asymptotic Fair Division

2021-01-01
Pasin Manurangsi , Warut Suksompong
We study a resource allocation setting where $m$ discrete items are to be divided among $n$ agents with additive utilities, and the agents' utilities for individual items are drawn at … We study a resource allocation setting where $m$ discrete items are to be divided among $n$ agents with additive utilities, and the agents' utilities for individual items are drawn at random from a probability distribution. Since common fairness notions like envy-freeness and proportionality cannot always be satisfied in this setting, an important question is when allocations satisfying these notions exist. In this paper, we close several gaps in the line of work on asymptotic fair division. First, we prove that the classical round-robin algorithm is likely to produce an envy-free allocation provided that $m=\Omega(n\log n/\log\log n)$, matching the lower bound from prior work. We then show that a proportional allocation exists with high probability as long as $m\geq n$, while an allocation satisfying envy-freeness up to any item (EFX) is likely to be present for any relation between $m$ and $n$. Finally, we consider a related setting where each agent is assigned exactly one item and the remaining items are left unassigned, and show that the transition from non-existence to existence with respect to envy-free assignments occurs at $m=en$.
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Fairly Allocating Many Goods with Few Queries

2019-07-17
Hoon Oh , Ariel D. Procaccia , Warut Suksompong
We investigate the query complexity of the fair allocation of indivisible goods. For two agents with arbitrary monotonic valuations, we design an algorithm that computes an allocation satisfying envy-freeness up … We investigate the query complexity of the fair allocation of indivisible goods. For two agents with arbitrary monotonic valuations, we design an algorithm that computes an allocation satisfying envy-freeness up to one good (EF1), a relaxation of envy-freeness, using a logarithmic number of queries. We show that the logarithmic query complexity bound also holds for three agents with additive valuations. These results suggest that it is possible to fairly allocate goods in practice even when the number of goods is extremely large. By contrast, we prove that computing an allocation satisfying envyfreeness and another of its relaxations, envy-freeness up to any good (EFX), requires a linear number of queries even when there are only two agents with identical additive valuations.
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Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

2020-09-22
Paul W. Goldberg , Alexandros Hollender , Warut Suksompong
We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the … We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results extending and strengthening those from prior work. Finally, we investigate connections between approximate and exact envy-freeness, as well as between continuous and discrete cake cutting.
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