Maximin Share Guarantees via Limited Cost-Sensitive Sharing
Charles University（查尔斯大学）；Rutgers University（罗格斯大学）
Hana Salavcova, Martin Černý, Arpita Biswas
备注：
In Proc. of the 25th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2026), Paphos, Cyprus, May 25 - 29, 2026, IFAAMAS, 11 pages
We study the problem of fairly allocating indivisible goods when limited sharing is allowed, that is, each good may be allocated to up to $k$ agents, while incurring a cost for sharing. While classic maximin share (MMS) allocations may not exist in many instances, we demonstrate that allowing controlled sharing can restore fairness guarantees that are otherwise unattainable in certain scenarios. (1) Our first contribution shows that exact maximin share (MMS) allocations are guaranteed to exist whenever goods are allowed to be cost-sensitively shared among at least half of the agents and the number of agents is even; for odd numbers of agents, we obtain a slightly weaker MMS guarantee. (2) We further design a Shared Bag-Filling Algorithm that guarantees a $(1 - C)(k - 1)$-approximate MMS allocation, where $C$ is the maximum cost of sharing a good. Notably, when $(1 - C)(k - 1) \geq 1$, our algorithm recovers an exact MMS allocation. (3) We additionally introduce the Sharing Maximin Share (SMMS) fairness notion, a natural extension of MMS to the $k$-sharing setting. (4) We show that SMMS allocations always exist under identical utilities and for instances with two agents. (5) We construct a counterexample to show the impossibility of the universal existence of an SMMS allocation. (6) Finally, we establish a connection between SMMS and constrained MMS (CMMS), yielding approximation guarantees for SMMS via existing CMMS results. These contributions provide deep theoretical insights for the problem of fair resource allocation when a limited sharing of resources are allowed in multi-agent environments.
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.
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…
We study the recently introduced cake-cutting setting in which the cake is represented by an undirected graph. This generalizes the canonical interval cake and allows for modeling the division of road networks. We show that when the graph is a forest, an allocation satisfying the well-known criterion of maximin share fairness always exists. Our result holds even when separation constraints are imposed, in which case no multiplicative approximation of proportionality can be guaranteed. Furthermore, while maximin share fairness is not always achievable for general graphs, we prove that ordinal r…
Pennsylvania State University（宾夕法尼亚州立大学）；Rochester Institute of Technology（罗切斯特理工学院）
The maximin share (MMS) guarantee is a desirable fairness notion for allocating indivisible goods. While MMS allocations do not always exist, several approximation techniques have been developed to ensure that all agents receive a fraction of their maximin share. We focus on an alternative approximation notion, based on the population of agents, that seeks to guarantee MMS for a fraction of agents. We show that no optimal approximation algorithm can satisfy more than a constant number of agents, and discuss the existence and computation of MMS for all but one agent and its relation to approxim…
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）
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 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…
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…
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 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…
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 initiate the work on maximin share (MMS) fair allocation of m indivisible chores to n agents using only their ordinal preferences, from both algorithmic and mechanism design perspectives. The previous best-known approximation is 2-1/n by Aziz et al. [IJCAI 2017]. We improve this result by giving a simple deterministic 5/3-approximation algorithm that determines an allocation sequence of agents, according to which items are allocated one by one. By a tighter analysis, we show that for n=2,3, our algorithm achieves better approximation ratios, and is actually optimal. We also consider the set…
We study the problem of fairly allocating a divisible resource, also known as cake cutting, with an additional requirement that the shares that different agents receive should be sufficiently separated from one another. This captures, for example, constraints arising from social distancing guidelines. While it is sometimes impossible to allocate a proportional share to every agent under the separation requirement, we show that the well-known criterion of maximin share fairness can always be attained. We then provide algorithmic analysis of maximin share fairness in this setting -- for instance…
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…
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…
We study the problem of fair allocation of a set of indivisible items among agents with additive valuations, under cardinality constraints. In this setting, the items are partitioned into categories, each with its own limit on the number of items it may contribute to any bundle. We consider the fairness measure known as the maximin share (MMS) guarantee, and propose a novel polynomial-time algorithm for finding $1/2$-approximate MMS allocations for goods -- an improvement from the previously best available guarantee of $11/30$. For single-category instances, we show that a modified variant of …
It is often beneficial for agents to pool their resources in order to better accommodate fluctuations in individual demand. Many multi-round resource allocation mechanisms operate in an online manner: in each round, the agents specify their demands for that round, and the mechanism determines a corresponding allocation. In this paper, we focus instead on the offline setting in which the agents specify their demand for each round at the outset. We formulate a specific resource allocation problem in this setting, and design and analyze an associated mechanism based on the solution concept of lex…
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 …
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 consider the problem of allocating indivisible goods to agents with additive valuation functions. Kurokawa, Procaccia and Wang {[JACM, 2018]} present instances for which every allocation gives some agent less than her maximin share. We present such examples with larger gaps. For three agents and nine items, we design an instance in which at least one agent does not get more than a $\frac{39}{40}$ fraction of her maximin share. {Moreover, we show that there is no negative example in which the difference between the number of items and the number of agents is smaller than six, and that the ga…
