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 exist problem instances to which adding some divisible resources would strictly decrease the MMS approximation ratio of the instance. On the algorithmic front, we propose a constructive algorithm that will always produce an $α$-MMS allocation for any number of agents, where $α$ takes values between $1/2$ and $1$ and is a monotone increasing function determined by how agents value the divisible goods relative to their MMS values.
We consider the problem of approximate maximin share (MMS) allocation of indivisible items among three agents with additive valuation functions. For goods, we show that an $\frac{11}{12}$ - MMS allocation always exists, improving over the previously known bound of $\frac{8}{9}$ . Moreover, in our allocation, we can prespecify an agent that is to receive her full proportional share (PS); we also present examples showing that for such allocations the ratio of $\frac{11}{12}$ is best possible. For chores, we show that a $\frac{19}{18}$-MMS allocation always exists. Also in this case, we can presp…
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…
We consider the problem of fair allocation of indivisible goods to agents with submodular valuation functions, where agents may have either equal entitlements or arbitrary (possibly unequal) entitlements. We focus on share-based fairness notions, specifically, the maximin share (MMS) for equal entitlements and the anyprice share (APS) for arbitrary entitlements, and design allocation algorithms that give each agent a bundle of value at least some constant fraction of her share value. For the equal entitlement case (and submodular valuations), Ghodsi, Hajiaghayi, Seddighin, Seddighin, and Yami …
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…
One of the important yet insufficiently studied subjects in fair allocation is the externality effect among agents. For a resource allocation problem, externalities imply that a bundle allocated to an agent may affect the utilities of other agents. In this paper, we conduct a study of fair allocation of indivisible goods when the externalities are not negligible. We present a simple and natural model, namely \emph{network externalities}, to capture the externalities. To evaluate fairness in the network externalities model, we generalize the idea behind the notion of maximin-share ($\MMS$) to a…
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…
The problem of fair division of indivisible goods has been receiving much attention recently. The prominent metric of envy-freeness can always be satisfied in the divisible goods setting (see for example \cite{BT95}), but often cannot be satisfied in the indivisible goods setting. This has led to many relaxations thereof being introduced. We study the existence of {\em maximin share (MMS)} allocations, which is one such relaxation. Previous work has shown that MMS allocations are guaranteed to exist for all instances with $n$ players and $m$ goods if $m \leq n+4$. We extend this guarantee to t…
In this paper, we consider the problem of how to fairly dividing $m$ indivisible chores among $n$ agents. The fairness measure we considered here is the maximin share. The previous best known result is that there always exists a $\frac{4}{3}$ approximation maximin share allocation. With a novel algorithm, we can always find a $\frac{11}{9}$ approximation maximin share allocation for any instances. We also discuss how to improve the efficiency of the algorithm and its connection to the job scheduling problem.
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 fair and economically efficient allocation of indivisible goods among agents whose valuations are rank functions of matroids. Such valuations constitute a well-studied class of submodular functions (i.e., they exhibit a diminishing returns property) and model preferences in several resource-allocation settings. We prove that, for matroid-rank valuations, a social welfare-maximizing allocation that gives each agent her maximin share always exists. Furthermore, such an allocation can be computed in polynomial time. We establish similar existential and algorithmic results for the pairwis…
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…
We investigate the problem of fairly allocating indivisible goods among interested agents using the concept of maximin share. Procaccia and Wang showed that while an allocation that gives every agent at least her maximin share does not necessarily exist, one that gives every agent at least $2/3$ of her share always does. In this paper, we consider the more general setting where we allocate the goods to groups of agents. The agents in each group share the same set of goods even though they may have conflicting preferences. For two groups, we characterize the cardinality of the groups for which …
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 …
We study a fair division problem with indivisible items, namely the computation of maximin share allocations. Given a set of $n$ players, the maximin share of a single player is the best she can guarantee to herself, if she would partition the items in any way she prefers, into $n$ bundles, and then receive her least desirable bundle. The objective then is to find an allocation, so that each player is guaranteed her maximin share. Previous works have studied this problem mostly algorithmically, providing constant factor approximation algorithms. In this work we embark on a mechanism design app…
We study individual rational, Pareto optimal, and incentive compatible mechanisms for auctions with heterogeneous items and budget limits. For multi-dimensional valuations we show that there can be no deterministic mechanism with these properties for divisible items. We use this to show that there can also be no randomized mechanism that achieves this for either divisible or indivisible items. For single-dimensional valuations we show that there can be no deterministic mechanism with these properties for indivisible items, but that there is a randomized mechanism that achieves this for either …
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 study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study the envy-free division of chores, and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete, even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up…
We consider the problem of allocating indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, that is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focused on approximation results that aim to find allocations in which the value of the bundle…
In fair division of indivisible goods, $\ell$-out-of-$d$ maximin share (MMS) is the value that an agent can guarantee by partitioning the goods into $d$ bundles and choosing the $\ell$ least preferred bundles. Most existing works aim to guarantee to all agents a constant fraction of their 1-out-of-$n$ MMS. But this guarantee is sensitive to small perturbation in agents' cardinal valuations. We consider a more robust approximation notion, which depends only on the agents' \emph{ordinal} rankings of bundles. We prove the existence of $\ell$-out-of-$\lfloor(\ell+\frac{1}{2})n\rfloor$ MMS allocati…
The study of matching theory has gained importance recently with applications in Kidney Exchange, House Allocation, School Choice etc. The general theme of these problems is to allocate goods in a fair manner amongst participating agents. The agents generally have a unit supply/demand of a good that they want to exchange with other agents. On the other hand, Bochet et al. study a more general version of the problem where they allow for agents to have arbitrary number of divisible goods to be rationed to other agents in the network. In this current work, our main focus is on non-bipartite netwo…
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）
We study the problem of fairly allocating a set of indivisible items to a set of agents with additive valuations. Recently, Feige et al. (WINE'21) proved that a maximin share (MMS) allocation exists for all instances with $n$ agents and no more than $n + 5$ items. Moreover, they proved that an MMS allocation is not guaranteed to exist for instances with $3$ agents and at least $9$ items, or $n \ge 4$ agents and at least $3n + 3$ items. In this work, we shrink the gap between these upper and lower bounds for guaranteed existence of MMS allocations. We prove that for any integer $c > 0$, there e…
We study the problem of allocating indivisible items to agents with additive valuations, under the additional constraint that bundles must be connected in an underlying item graph. Previous work has considered the existence and complexity of fair allocations. We study the problem of finding an allocation that is Pareto-optimal. While it is easy to find an efficient allocation when the underlying graph is a path or a star, the problem is NP-hard for many other graph topologies, even for trees of bounded pathwidth or of maximum degree 3. We show that on a path, there are instances where no Paret…
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…
We study the problem of fairly allocating a set of m indivisible chores (items with non-positive value) to n agents. We consider the desirable fairness notion of 1-out-of-d maximin share (MMS) -- the minimum value that an agent can guarantee by partitioning items into d bundles and receiving the least valued bundle -- and focus on ordinal approximation of MMS that aims at finding the largest d <= n for which 1-out-of-d MMS allocation exists. Our main contribution is a polynomial-time algorithm for 1-out-of-floor(2n/3) MMS allocation, and a proof of existence of 1-out-of-floor(3n/4) MMS allocat…
Fair division is a fundamental problem in various multi-agent settings, where the goal is to divide a set of resources among agents in a fair manner. We study the case where m indivisible items need to be divided among n agents with additive valuations using the popular fairness notion of maximin share (MMS). An MMS allocation provides each agent a bundle worth at least her maximin share. While it is known that such an allocation need not exist, a series of work provided approximation algorithms for a 2/3-MMS allocation in which each agent receives a bundle worth at least 2/3 times her maximin…
We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of $n$ agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into $n$ bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, so we resort to approximation algorithms. Our …
Many allocation problems in multiagent systems rely on agents specifying cardinal preferences. However, allocation mechanisms can be sensitive to small perturbations in cardinal preferences, thus causing agents who make ``small" or ``innocuous" mistakes while reporting their preferences to experience a large change in their utility for the final outcome. To address this, we introduce a notion of algorithmic stability and study it in the context of fair and efficient allocations of indivisible goods among two agents. We show that it is impossible to achieve exact stability along with even a wea…
