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An Improved Approximation Algorithm for Maximin Shares

Authors

Jugal Garg, Setareh Taki
Type: Article
Publication Date: 2020-07-09
Citations: 73
DOI: https://doi.org/10.1145/3391403.3399526

Abstract

We study the problem of fair allocation of m indivisible items among n agents with additive valuations using the popular notion of maximin share (MMS) as our measure of fairness. 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 [5, 7], a series of remarkable work [1-3, 6, 7] provided 2/3 approximation algorithms in which each agent receives a bundle worth at least 2/3 times her maximin share. More recently, [4] showed the existence of 3/4 MMS allocations and a PTAS to find a 3/4 - ε MMS allocation. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain.

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An Improved Approximation Algorithm for Maximin Shares

2019-01-01
Jugal Garg , Setareh Taki
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 … 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 share. More recently, Ghodsi et al. [EC'2018] showed the existence of a 3/4-MMS allocation and a PTAS to find a (3/4-\epsilon)-MMS allocation for an \epsilon > 0. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain. In this paper, we develop a new approach that gives a simple algorithm for showing the existence of a 3/4-MMS allocation. Furthermore, our approach is powerful enough to be easily extended in two directions: First, we get a strongly polynomial-time algorithm to find a 3/4-MMS allocation, where we do not need to approximate the MMS values at all. Second, we show that there always exists a (3/4 + 1/(12n))-MMS allocation, improving the best previous factor. This improves the approximation guarantee, most notably for small n. We note that 3/4 was the best factor known for n> 4.

Breaking the 3/4 Barrier for Approximate Maximin Share

2024-01-01
Hannaneh Akrami , Jugal Garg
We study the fundamental problem of fairly allocating a set of indivisible goods among n agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is … We study the fundamental problem of fairly allocating a set of indivisible goods among n agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her MMS value. An allocation is called MMS if all agents receive at least their MMS value. However, since MMS allocations need not exist when n > 2, a series of works showed the existence of approximate MMS allocations with the current best factor of . The recent work [3] showed the limitations of existing approaches and proved that they cannot improve this factor to 3/4 + Ω(1). In this paper, we bypass these barriers to show the existence of ()-MMS allocations by developing new reduction rules and analysis techniques.
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Breaking the $3/4$ Barrier for Approximate Maximin Share

2023-01-01
Hannaneh Akrami , Jugal Garg
We study the fundamental problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is … We study the fundamental problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her MMS value. An allocation is called MMS if all agents receive at least their MMS value. Since MMS allocations need not exist when $n>2$, a series of works showed the existence of approximate MMS allocations with the current best factor of $\frac34 + O(\frac{1}{n})$. However, a simple example in [DFL82, BEF21, AGST23] showed the limitations of existing approaches and proved that they cannot improve this factor to $3/4 + \Omega(1)$. In this paper, we bypass these barriers to show the existence of $(\frac{3}{4} + \frac{3}{3836})$-MMS allocations by developing new reduction rules and analysis techniques.

Approximation Algorithms for Computing Maximin Share Allocations

2017-10-31
Georgios Amanatidis , Evangelos Markakis , Afshin Nikzad +1 more
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 … 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 main result is a $2/3$-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang, which also produces a $2/3$-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of their algorithm. Furthermore, motivated by the apparent difficulty, both theoretically and experimentally, in finding lower bounds on the existence of approximate solutions, we undertake a probabilistic analysis. We prove that in randomly generated instances, with high probability there exists a maximin share allocation. This can be seen as a justification of the experimental evidence reported in relevant works. Finally, we provide further positive results for two special cases that arise from previous works. The first one is the intriguing case of $3$ agents, for which it is already known that exact maximin share allocations do not always exist (contrary to the case of $2$ agents). We provide a $7/8$-approximation algorithm, improving the previously known result of $3/4$. The second case is when all item values belong to $\{0, 1, 2\}$, extending the $\{0, 1\}$ setting studied in Bouveret and Lema\^itre. We obtain an exact algorithm for any number of agents in this case.
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Improving Approximation Guarantees for Maximin Share

2023-01-01
Hannaneh Akrami , Jugal Garg , Setareh Taki
We consider fair division of a set of indivisible goods among $n$ agents with additive valuations using the fairness notion of maximin share (MMS). MMS is the most popular share-based … We consider fair division of a set of indivisible goods among $n$ agents with additive valuations using the fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her ($1$-out-of-$n$) MMS value. An allocation is called MMS if all agents receive their MMS values. However, since MMS allocations do not always exist, the focus shifted to investigating its ordinal and multiplicative approximations. In the ordinal approximation, the goal is to show the existence of $1$-out-of-$d$ MMS allocations (for the smallest possible $d>n$). A series of works led to the state-of-the-art factor of $d=\lfloor3n/2\rfloor$ [Hosseini et al.'21]. We show that $1$-out-of-$4\lceil n/3\rceil$ MMS allocations always exist, thereby improving the state-of-the-art of ordinal approximation. In the multiplicative approximation, the goal is to show the existence of $\alpha$-MMS allocations (for the largest possible $\alpha < 1$), which guarantees each agent at least $\alpha$ times her MMS value. We introduce a general framework of "approximate MMS with agent priority ranking". An allocation is said to be $T$-MMS, for a non-increasing sequence $T = (\tau_1, \ldots, \tau_n)$ of numbers, if the agent at rank $i$ in the order gets a bundle of value at least $\tau_i$ times her MMS value. This framework captures both ordinal approximation and multiplicative approximation as special cases. We show the existence of $T$-MMS allocations where $\tau_i \ge \max(\frac{3}{4} + \frac{1}{12n}, \frac{2n}{2n+i-1})$ for all $i$. Furthermore, we can get allocations that are $(\frac{3}{4} + \frac{1}{12n})$-MMS ex-post and $(0.8253 + \frac{1}{36n})$-MMS ex-ante. We also prove that our algorithm does not give better than $(0.8631 + \frac{1}{2n})$-MMS ex-ante.
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Approximation Algorithms for Maximin Fair Division

2017-06-20
Siddharth Barman , Sanath Kumar Krishna Murthy
We consider the problem of dividing indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin … We consider the problem of dividing 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 focussed on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee.
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On Lower Bounds for Maximin Share Guarantees

2023-01-01
Halvard Hummel
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 … 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 exists a number of agents $n_c$ such that an MMS allocation exists for any instance with $n \ge n_c$ agents and at most $n + c$ items, where $n_c \le \lfloor 0.6597^c \cdot c!\rfloor$ for allocation of goods and $n_c \le \lfloor 0.7838^c \cdot c!\rfloor$ for chores. Furthermore, we show that for $n \neq 3$ agents, all instances with $n + 6$ goods have an MMS allocation.
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Simplification and Improvement of MMS Approximation

2023-08-01
Hannaneh Akrami , Jugal Garg , Eklavya Sharma +1 more
We consider the problem of fairly allocating a set of indivisible goods among n agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations … We consider the problem of fairly allocating a set of indivisible goods among n agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The Garg-Taki algorithm gives the current best approximation factor of (3/4 + 1/12n). Most of these results are based on complicated analyses, especially those providing better than 2/3 factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of (3/4 + min(1/36, 3/(16n-4))). For small n, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques.
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Guaranteeing Maximin Shares: Some Agents Left Behind

2021-05-19
Hadi Hosseini , Andrew Searns
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 … 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 approximate MMS guarantees. We then prove the existence of allocations that guarantee MMS for $\frac{2}{3}$ of agents, and devise a polynomial time algorithm that achieves this bound for up to nine agents. A key implication of our result is the existence of allocations that guarantee $\text{MMS}^{\lceil{3n/2}\rceil}$, i.e., the value that agents receive by partitioning the goods into $\lceil{\frac{3}{2}n}\rceil$ bundles, improving the best known guarantee of $\text{MMS}^{2n-2}$. Finally, we provide empirical experiments using synthetic data.

Guaranteeing Maximin Shares: Some Agents Left Behind

2021-01-01
Seyed Hossein Hosseini , Andrew Searns
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 … 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 approximate MMS guarantees. We then prove the existence of allocations that guarantee MMS for $\frac{2}{3}$ of agents, and devise a polynomial time algorithm that achieves this bound for up to nine agents. A key implication of our result is the existence of allocations that guarantee $\text{MMS}^{\lceil{3n/2}\rceil}$, i.e., the value that agents receive by partitioning the goods into $\lceil{\frac{3}{2}n}\rceil$ bundles, improving the best known guarantee of $\text{MMS}^{2n-2}$. Finally, we provide empirical experiments using synthetic data.
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Guaranteeing Maximin Shares: Some Agents Left Behind

2021-08-01
Hadi Hosseini , Andrew Searns
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 … 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 approximate MMS guarantees. We then prove the existence of allocations that guarantee MMS for 2/3 of agents, and devise a polynomial time algorithm that achieves this bound for up to nine agents. A key implication of our result is the existence of allocations that guarantee the value that an agent receives by partitioning the goods into 3n/2 bundles, improving the best known guarantee when goods are partitioned into 2n-2 bundles. Finally, we provide empirical experiments using synthetic data.
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Simplification and Improvement of MMS Approximation

2023-01-01
Hannaneh Akrami , Jugal Garg , Eklavya Sharma +1 more
We consider the problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations … We consider the problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The Garg-Taki algorithm gives the current best approximation factor of $(\frac{3}{4} + \frac{1}{12n})$. Most of these results are based on complicated analyses, especially those providing better than $2/3$ factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of $(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4}))$. For small $n$, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques.

Approximation Algorithms for Maximin Fair Division

2020-02-29
Siddharth Barman , Sanath Kumar Krishnamurthy
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 … 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 , which 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 allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations, a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations . Specifically, we show that when the valuations of the agents are nonnegative , monotone , and submodular, then a 0.21-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the article is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions .
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An improved approximation algorithm for maximin shares

2021-06-08
Jugal Garg , Setareh Taki
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Groupwise Maximin Fair Allocation of Indivisible Goods

2018-04-25
Siddharth Barman , Arpita Biswas , Sanath Kumar Krishnamurthy +1 more
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 … 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 fairness, it has its own drawbacks and, in particular, it is not sufficient to rule out unsatisfactory allocations. Motivated by these considerations, in this work we define a stronger notion of fairness, called groupwise maximin share guarantee (GMMS). In GMMS, we require that the maximin share guarantee is achieved not just with respect to the grand bundle, but also among all the subgroups of agents. Hence, this solution concept strengthens MMS and provides an ex-post fairness guarantee. We show that in specific settings, GMMS allocations always exist. We also establish the existence of approximate GMMS allocations under additive valuations, and develop a polynomial-time algorithm to find such allocations. Moreover, we establish a scale of fairness wherein we show that GMMS implies approximate envy freeness. Finally, we empirically demonstrate the existence of GMMS allocations in a large set of randomly generated instances. For the same set of instances, we additionally show that our algorithm achieves an approximation factor better than the established, worst-case bound.
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Groupwise Maximin Fair Allocation of Indivisible Goods

2017-01-01
Siddharth Barman , Arpita Biswas , Sanath Kumar Krishnamurthy +1 more
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 … 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 fairness, it has its own drawbacks and, in particular, it is not sufficient to rule out unsatisfactory allocations. Motivated by these considerations, in this work we define a stronger notion of fairness, called groupwise maximin share guarantee (GMMS). In GMMS, we require that the maximin share guarantee is achieved not just with respect to the grand bundle, but also among all the subgroups of agents. Hence, this solution concept strengthens MMS and provides an ex-post fairness guarantee. We show that in specific settings, GMMS allocations always exist. We also establish the existence of approximate GMMS allocations under additive valuations, and develop a polynomial-time algorithm to find such allocations. Moreover, we establish a scale of fairness wherein we show that GMMS implies approximate envy freeness. Finally, we empirically demonstrate the existence of GMMS allocations in a large set of randomly generated instances. For the same set of instances, we additionally show that our algorithm achieves an approximation factor better than the established, worst-case bound.
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Improved maximin fair allocation of indivisible items to three agents

2022-01-01
Uriel Feige , Alexey Norkin
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 … 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 prespecify an agent that is to receive no more than her PS, and we present examples showing that for such allocations the ratio of $\frac{19}{18}$ is best possible.
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On Lower Bounds for Maximin Share Guarantees

2023-08-01
Halvard Hummel
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 … 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 ≥ 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 &gt; 0, there exists a number of agents n_c such that an MMS allocation exists for any instance with n ≥ n_c agents and at most n + c items, where n_c ≤ ⌊0.6597^c · c!⌋ for allocation of goods and n_c ≤ ⌊0.7838^c · c!⌋ for chores. Furthermore, we show that for n ≠ 3 agents, all instances with n + 6 goods have an MMS allocation.
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Approximation Algorithms for Maximin Fair Division

2017-01-01
Siddharth Barman , Sanath Kumar Krishnamurthy
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 … 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 allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 0.21-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the paper is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions.
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Maximin Share Guarantees for Few Agents with Subadditive Valuations

2025-02-07
George Christodoulou , Vasilis Christoforidis , Symeon Mastrakoulis +1 more
We study the problem of fairly allocating a set of indivisible items among a set of agents. We consider the notion of (approximate) maximin share (MMS) and we provide an … We study the problem of fairly allocating a set of indivisible items among a set of agents. We consider the notion of (approximate) maximin share (MMS) and we provide an improved lower bound of $1/2$ (which is tight) for the case of subadditive valuations when the number of agents is at most four. We also provide a tight lower bound for the case of multiple agents, when they are equipped with one of two possible types of valuations. Moreover, we propose a new model that extends previously studied models in the area of fair division, which will hopefully give rise to further research. We demonstrate the usefulness of this model by employing it as a technical tool to derive our main result, and we provide a thorough analysis for this model for the case of three agents. Finally, we provide an improved impossibility result for the case of three submodular agents.
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Maximum Nash welfare and other stories about EFX

2021-02-09
Georgios Amanatidis , Georgios Birmpas , Aris Filos-Ratsikas +2 more
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On Best-of-Both-Worlds Fair-Share Allocations

2022-01-01
Moshe Babaioff , Tomer Ezra , Uriel Feige
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Approximate and strategyproof maximin share allocation of chores with ordinal preferences

2022-07-05
Haris Aziz , Bo Li , Xiaowei Wu
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Existence and Computation of Maximin Fair Allocations Under Matroid-Rank Valuations

2020-01-01
Siddharth Barman , Paritosh Verma
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 … 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 pairwise maximin share guarantee as well. To complement these results, we show that if the agents have binary XOS valuations or weighted-rank valuations, then maximin fair allocations are not guaranteed to exist. Both of these valuation classes are immediate generalizations of matroid-rank functions.

A Little Charity Guarantees Almost Envy-Freeness

2019-12-23
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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Ordinal Maximin Share Approximation for Goods

2022-05-26
Hadi Hosseini , Andrew Searns , Erel Segal-Halevi
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 … 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 allocations of goods for any integer $\ell\geq 1$, and present a polynomial-time algorithm that finds a $1$-out-of-$\lceil\frac{3n}{2}\rceil$ MMS allocation when $\ell = 1$. We further develop an algorithm that provides a weaker ordinal approximation to MMS for any $\ell > 1$.
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Fair division of mixed divisible and indivisible goods

2021-01-05
Xiaohui Bei , Zihao Li , Jinyan Liu +2 more
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Multiple Birds with One Stone: Beating 1/2 for EFX and GMMS via Envy Cycle Elimination

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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Allocating Indivisible Goods to Strategic Agents: Pure Nash Equilibria and Fairness

2022-01-01
Georgios Amanatidis , Georgios Birmpas , Federico Fusco +3 more
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Fair-Share Allocations for Agents with Arbitrary Entitlements

2023-10-26
Moshe Babaioff , Tomer Ezra , Uriel Feige
We consider the problem of fair allocation of indivisible goods to n agents with no transfers. When agents have equal entitlements, the well-established notion of the maximin share (MMS) serves … We consider the problem of fair allocation of indivisible goods to n agents with no transfers. When agents have equal entitlements, the well-established notion of the maximin share (MMS) serves as an attractive fairness criterion for which, to qualify as fair, an allocation needs to give every agent at least a substantial fraction of the agent’s MMS. In this paper, we consider the case of arbitrary (unequal) entitlements. We explain shortcomings in previous attempts that extend the MMS to unequal entitlements. Our conceptual contribution is the introduction of a new notion of a share, the AnyPrice share (APS), that is appropriate for settings with arbitrary entitlements. Even for the equal entitlements case, this notion is new and satisfies [Formula: see text], for which the inequality is sometimes strict. We present two equivalent definitions for the APS (one as a minimization problem, the other as a maximization problem) and provide comparisons between the APS and previous notions of fairness. Our main result concerns additive valuations and arbitrary entitlements, for which we provide a polynomial-time algorithm that gives every agent at least a [Formula: see text] - fraction of the agent’s APS. This algorithm can also be viewed as providing strategies in a certain natural bidding game, and these strategies secure each agent at least a [Formula: see text] - fraction of the agent’s APS. Funding: T. Ezra’s research is partially supported by the European Research Council Advanced [Grant 788893] AMDROMA “Algorithmic and Mechanism Design Research in Online Markets” and MIUR PRIN project ALGADIMAR “Algorithms, Games, and Digital Markets.” U. Feige’s research is supported in part by the Israel Science Foundation [Grant 1122/22].
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Approximating Nash Social Welfare Under Binary XOS and Binary Subadditive Valuations

2022-01-01
Siddharth Barman , Paritosh Verma
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Maximin Shares Under Cardinality Constraints

2022-01-01
Halvard Hummel , Magnus Lie Hetland
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Unified Fair Allocation of Goods and Chores via Copies

2023-09-22
Yotam Gafni , Xin Huang , Ron Lavi +1 more
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 … 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 without commons (denoted EFX WC when applied to EFX). Second, a formal duality theorem relating the existence of a host of (refined) fair allocation concepts for copies to their existence for chores. We demonstrate the usefulness of our duality result by using it to characterize the existence of EFX for chores through the dual environment, as well as to prove EFX existence in the special case of leveled preferences over the chores. We further study the hierarchy among envy-freeness notions without commons and their α-MMS guarantees, showing, for example, that any EFX WC allocation guarantees at least \(\frac{4}{11}\) -MMS for goods with copies.
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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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Fairness Criteria for Allocating Indivisible Chores: Connections and Efficiencies

2021-01-01
Ankang Sun , Bo Chen , Xuan Vinh Doan
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 … 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 connections between these fairness criteria and, at the same time, reveal intrinsic differences between goods allocation and chores allocation. However, such strong relationships cannot be inherited by the submodular setting, under which PMMS and MMS are no longer relaxations of envy-freeness and, even worse, few non-trivial guarantees exist. We also investigate efficiency loss under these fairness constraints and establish their prices of fairness.
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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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Fair allocation of conflicting items

2021-12-23
Halvard Hummel , Magnus Lie Hetland
Abstract We study fair allocation of indivisible items, where the items are furnished with a set of conflicts , and agents are not permitted to receive conflicting items. This kind … Abstract We study fair allocation of indivisible items, where the items are furnished with a set of conflicts , and agents are not permitted to receive conflicting items. This kind of constraint captures, for example, participating in events that overlap in time, or taking on roles in the presence of conflicting interests. We demonstrate, both theoretically and experimentally, that fairness characterizations such as EF1, MMS and MNW still are applicable and useful under item conflicts. Among other existence, non-existence and computability results, we show that a $$1/\Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>Δ</mml:mi></mml:mrow></mml:math> -approximate MMS allocation for n agents may be found in polynomial time when $$n&gt;\Delta &gt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>&gt;</mml:mo><mml:mi>Δ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> , for any conflict graph with maximum degree $$\Delta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi></mml:math> , and that, if $$n &gt; \Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>&gt;</mml:mo><mml:mi>Δ</mml:mi></mml:mrow></mml:math> , a 1/3-approximate MMS allocation always exists.
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Truthful and Fair Mechanisms for Matroid-Rank Valuations

2022-06-28
Siddharth Barman , Paritosh Verma
We study the problem of allocating indivisible goods among strategic agents. We focus on settings wherein monetary transfers are not available and each agent's private valuation is a submodular function … We study the problem of allocating indivisible goods among strategic agents. We focus on settings wherein monetary transfers are not available and each agent's private valuation is a submodular function with binary marginals, i.e., the agents' valuations are matroid-rank functions. In this setup, we establish a notable dichotomy between two of the most well-studied fairness notions in discrete fair division; specifically, between envy-freeness up to one good (EF1) and maximin shares (MMS). First, we show that a known Pareto-efficient mechanism is group strategy-proof for finding EF1 allocations, under matroid-rank valuations. The group strategy-proofness guarantee strengthens an existing result that establishes truthfulness (individually for each agent) in the same context. Our result also generalizes prior work from binary additive valuations to the matroid-rank case. Next, we establish that an analogous positive result cannot be achieved for MMS, even when considering truthfulness on an individual level. Specifically, we prove that, for matroid-rank valuations, there does not exist a truthful mechanism that is index oblivious, Pareto efficient, and maximin fair. For establishing our results, we develop a characterization of truthful mechanisms for matroid-rank functions. This characterization in fact holds for a broader class of valuations (specifically, holds for binary XOS functions) and might be of independent interest.
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Two Birds With One Stone: Fairness and Welfare via Transfers

2021-06-01
Vishnu V. Narayan , Mashbat Suzuki , Adrian Vetta
We study the question of dividing a collection of indivisible goods amongst a set of agents. The main objective of research in the area is to achieve one of two … We study the question of dividing a collection of indivisible goods amongst a set of agents. The main objective of research in the area is to achieve one of two goals: fairness or efficiency. On the fairness side, envy-freeness is the central fairness criterion in economics, but envy-free allocations typically do not exist when the goods are indivisible. A recent line of research shows that envy-freeness can be achieved if a small quantity of a homogeneous divisible good (money) is introduced into the system, or equivalently, if transfer payments are allowed between the agents. A natural question to explore, then, is whether transfer payments can be used to provide high welfare in addition to envy-freeness, and if so, how much money is needed to be transferred. We show that for general monotone valuations, there always exists an allocation with transfers that is envy-free and whose Nash social welfare (NSW) is at least an $e^{-1/e}$-fraction of the optimal Nash social welfare. Additionally, when the agents have additive valuations, an envy-free allocation with negligible transfers and whose NSW is within a constant factor of optimal can be found in polynomial time. Consequently, we demonstrate that the seemingly incompatible objectives of fairness and high welfare can be achieved simultaneously via transfer payments, even for general valuations, when the welfare objective is NSW. On the other hand, we show that a similar result is impossible for utilitarian social welfare: any envy-freeable allocation that achieves a constant fraction of the optimal welfare requires non-negligible transfers. To complement this result we present algorithms that compute an envy-free allocation with a given target welfare and with bounded transfers.

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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An improved approximation algorithm for maximin shares

2021-06-08
Jugal Garg , Setareh Taki
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Maximin Fairness with Mixed Divisible and Indivisible Goods

2021-05-18
Xiaohui Bei , Shengxin Liu , Xinhang Lu +1 more
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 … 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 ratios of the instances. On the algorithmic front, we propose a constructive algorithm that will always produce an \alpha-MMS allocation for any number of agents, where \alpha takes values between 1/2 and 1 and is a monotonically increasing function determined by how agents value the divisible goods relative to their MMS values.
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Extension of Additive Valuations to General Valuations on the Existence of EFX

2023-08-01
Ryoga Mahara
Envy freeness is one of the most widely studied notions in fair division. Because envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among … Envy freeness is one of the most widely studied notions in fair division. Because envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling notion is envy freeness up to any item (EFX). Informally speaking, EFX requires that no agent i envies another agent j after the removal of any item in j’s bundle. The existence of EFX allocations is a major open problem. We study the existence of EFX allocations when agents have general valuations. 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 an EFX allocation always exists when one can leave at most n − 1 items unallocated. We develop new techniques and extend some results of additive valuations to general valuations on the existence of EFX allocations. We show that an EFX allocation always exists (i) when all agents have one of two general valuations or (ii) when the number of items is at most n + 3. We also show that an EFX allocation always exists when one can leave at most n − 2 items unallocated. In addition to the positive results, we construct an instance with n = 3, in which an existing approach does not work. Funding: This work was partially supported by Kyoto University and Toyota Motor Corporation [Joint Project “Advanced Mathematical Science for Mobility Society”].
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Indivisible Mixed Manna: On the Computability of MMS+PO Allocations

2021-07-18
Rucha Kulkarni , Ruta Mehta , Setareh Taki
No abstract available. No abstract available.
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On Achieving Fairness and Stability in Many-to-One Matchings

2020-09-12
Shivika Narang , Arpita Biswas , Y. Narahari
The past few years have seen a surge of work on fairness in social choice literature. This paper initiates the study of finding a stable many-to-one matching, under cardinal valuations, … The past few years have seen a surge of work on fairness in social choice literature. This paper initiates the study of finding a stable many-to-one matching, under cardinal valuations, while satisfying fairness among the agents on either side. Specifically, motivated by several real-world settings, we focus on leximin optimal fairness and seek leximin optimality over many-to-one stable matchings. We first consider the special case of ranked valuations where all agents on each side have the same preference orders or rankings over the agents on the other side (but not necessarily the same valuations). For this special case, we provide a complete characterisation of the space of stable matchings. This leads to FaSt, a novel and efficient algorithm to compute a leximin optimal stable matching under ranked isometric valuations (where, for each pair of agents, the valuation of one agent for the other is the same). The running time of FaSt is linear in the number of edges. Building upon FaSt, we present an efficient algorithm, FaSt-Gen, that finds the leximin optimal stable matching for ranked but otherwise unconstrained valuations. The running time of FaSt-Gen is quadratic in the number of edges. We next establish that, in the absence of rankings, finding a leximin optimal stable matching is NP-Hard, even under isometric valuations. In fact, when additivity and non-negativity are the only assumptions on the valuations, we show that, unless P=NP, no efficient polynomial factor approximation is possible. When additivity is relaxed to submodularity, we find that not even an exponential approximation is possible.

Indivisible Mixed Manna: On the Computability of MMS + PO Allocations

2020-01-01
Rucha Kulkarni , Ruta Mehta , Setareh Taki
In this paper we initiate the study of finding fair and efficient allocations of an indivisible mixed manna: Divide m indivisible items among n agents under the fairness notion of … In this paper we initiate the study of finding fair and efficient allocations of an indivisible mixed manna: Divide m indivisible items among n agents under the fairness notion of maximin share (MMS) and the efficiency notion of Pareto optimality (PO). A mixed manna allows an item to be a good for some agents and a chore for others. The problem of finding $\alpha$-MMS allocation for the (near) best $\alpha\in(0,1]$ for which it exists, remains unresolved even for a goods manna with constantly many agents, while the problem of finding $\alpha$-MMS+PO allocation is unexplored for any $\alpha\in(0,1]$. We make significant progress on the above questions for a mixed manna. First, we show that for any $\alpha>0$, an $\alpha$-MMS allocation may not always exist, thus ruling out solving the problem for a fixed $\alpha$. Second, towards computing $\alpha$-MMS+PO allocation for the best possible $\alpha$, we obtain a dichotomous result: We derive two conditions and show that the problem is tractable under these two conditions, while dropping either renders the problem intractable. The two conditions are: (i) number of agents is a constant, and (ii) for every agent, her absolute value for all the items is at least a constant factor of her total (absolute) value for all the goods or all the chores. In particular, first, for instances satisfying (i) and (ii) we design a PTAS - an efficient algorithm to find an $(\alpha-\epsilon)$-MMS and $\gamma$-PO allocation when given $\epsilon,\gamma>0$, for the highest possible $\alpha\in(0,1]$. Second, we show that if either condition is not satisfied then finding an $\alpha$-MMS allocation for any $\alpha\in(0,1]$ is NP-hard, even when a solution exists for $\alpha=1$. To the best of our knowledge, ours is the first algorithm that ensures both approximate MMS and PO guarantees.
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On Lower Bounds for Maximin Share Guarantees

2023-08-01
Halvard Hummel
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 … 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 ≥ 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 &gt; 0, there exists a number of agents n_c such that an MMS allocation exists for any instance with n ≥ n_c agents and at most n + c items, where n_c ≤ ⌊0.6597^c · c!⌋ for allocation of goods and n_c ≤ ⌊0.7838^c · c!⌋ for chores. Furthermore, we show that for n ≠ 3 agents, all instances with n + 6 goods have an MMS allocation.
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Fairly Allocating Goods and (Terrible) Chores

2023-08-01
Seyed Hossein Hosseini , Aghaheybat Mammadov , Tomasz Wąs
We study the fair allocation of mixture of indivisible goods and chores under lexicographic preferences---a subdomain of additive preferences. A prominent fairness notion for allocating indivisible items is envy-freeness up … We study the fair allocation of mixture of indivisible goods and chores under lexicographic preferences---a subdomain of additive preferences. A prominent fairness notion for allocating indivisible items is envy-freeness up to any item (EFX). Yet, its existence and computation has remained a notable open problem. By identifying a class of instances with "terrible chores", we show that determining the existence of an EFX allocation is NP-complete. This result immediately implies the intractability of EFX under additive preferences. Nonetheless, we propose a natural subclass of lexicographic preferences for which an EFX and Pareto optimal (PO) allocation is guaranteed to exist and can be computed efficiently for any mixed instance. Focusing on two weaker fairness notions, we investigate finding EF1 and Pareto optimal allocations for special instances with terrible chores, and show that MMS and PO allocations can be computed efficiently for any mixed instance with lexicographic preferences.
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Simplification and Improvement of MMS Approximation

2023-08-01
Hannaneh Akrami , Jugal Garg , Eklavya Sharma +1 more
We consider the problem of fairly allocating a set of indivisible goods among n agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations … We consider the problem of fairly allocating a set of indivisible goods among n agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The Garg-Taki algorithm gives the current best approximation factor of (3/4 + 1/12n). Most of these results are based on complicated analyses, especially those providing better than 2/3 factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of (3/4 + min(1/36, 3/(16n-4))). For small n, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques.
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Rainbow Cycle Number and EFX Allocations: (Almost) Closing the Gap

2023-08-01
Shayan Chashm Jahan , Masoud Seddighin , Seyed-Mohammad Seyed-Javadi +1 more
Recently, some studies on the fair allocation of indivisible goods notice a connection between a purely combinatorial problem called the Rainbow Cycle problem and a fairness notion known as EFX: … Recently, some studies on the fair allocation of indivisible goods notice a connection between a purely combinatorial problem called the Rainbow Cycle problem and a fairness notion known as EFX: assuming that the rainbow cycle number for parameter d (i.e. R(d)) is O(d^β .log(d)^γ), we can find a (1 − ϵ)-EFX allocation with O_ϵ(n^(β/β+1) .log(n)^(γ/β+1)) number of discarded goods. The best upper bound on R(d) is improved in a series of works to O(d^4), O(d^(2+o(1))), and finally to O(d^2). Also, via a simple observation, we have R(d) ∈ Ω(d). In this paper, we introduce another problem in extremal combinatorics. For a parameter l, we define the rainbow path degree and denote it by H(l). We show that any lower bound on H(l) yields an upper bound on R(d). Next, we prove that H(l) ∈ Ω(l^2 / log(l)) which yields an almost tight upper bound of R(d) ∈ Ω(d.log(d)). This, in turn, proves the existence of (1−ϵ)-EFX allocation with O_ϵ(√n .log(n)) number of discarded goods. In addition, for the special case of the Rainbow Cycle problem that the edges in each part form a permutation, we improve the upper bound to R(d) ≤ 2d−4. We leverage H(l) to achieve this bound. Our conjecture is that the exact value of H(l) is ⌊l^2/2⌋ − 1. We provide some experiments that support this conjecture. Assuming this conjecture is correct, we have R(d) ∈ θ(d).
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Maximin fairness with mixed divisible and indivisible goods

2021-06-30
Xiaohui Bei , Shengxin Liu , Xinhang Lu +1 more
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On picking sequences for chores

2023-07-07
Uriel Feige , Xin Huang
We consider the problem of allocating m indivisible chores to n agents with additive disvaluation (cost) functions. It is easy to show that there are picking sequences that give every … We consider the problem of allocating m indivisible chores to n agents with additive disvaluation (cost) functions. It is easy to show that there are picking sequences that give every agent (that uses the greedy picking strategy) a bundle of chores of disvalue at most twice her share value (maximin share, MMS, for agents of equal entitlement, and anyprice share, APS, for agents of arbitrary entitlement). Aziz, Li and Wu (2022) designed picking sequences that improve this ratio to 5/3 for the case of equal entitlement. We design picking sequences that improve the ratio to 1.733 for the case of arbitrary entitlement, and to 8/5 for the case of equal entitlement. (In fact, computer assisted analysis suggests that the ratio is smaller than 1.543 in the equal entitlement case.) We also prove a lower bound of 3/2 on the obtainable ratio when n is sufficiently large.
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Guaranteeing Maximin Shares: Some Agents Left Behind

2021-08-01
Hadi Hosseini , Andrew Searns
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 … 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 approximate MMS guarantees. We then prove the existence of allocations that guarantee MMS for 2/3 of agents, and devise a polynomial time algorithm that achieves this bound for up to nine agents. A key implication of our result is the existence of allocations that guarantee the value that an agent receives by partitioning the goods into 3n/2 bundles, improving the best known guarantee when goods are partitioned into 2n-2 bundles. Finally, we provide empirical experiments using synthetic data.
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Fair Allocation of Conflicting Items.

2021-04-13
Halvard Hummel , Magnus Lie Hetland
We study fair allocation of indivisible items, where the items are furnished with a set of conflicts, and agents are not permitted to receive conflicting items. This kind of constraint … We study fair allocation of indivisible items, where the items are furnished with a set of conflicts, and agents are not permitted to receive conflicting items. This kind of constraint captures, for example, participating in events that overlap in time, or taking on roles in the presence of conflicting interests. We demonstrate, both theoretically and experimentally, that fairness characterizations such as EF1, MMS and MNW still are applicable and useful under item conflicts. Among other existence, non-existence and computability results, we show that a $1/\Delta$-approximate MMS allocation for $n$ agents may be found in polynomial time, for any conflict graph with maximum degree $n>\Delta>2$, and that, if $n > \Delta$, a 1/3-approximate MMS allocation always exists.

An Algorithmic Framework for Approximating Maximin Share Allocation of Chores

2019-01-01
Xin Huang , Pinyan Lu
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 … 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.
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The Fairness Fair: Bringing Human Perception into Collective Decision-Making

2024-03-24
Hadi Hosseini
Fairness is one of the most desirable societal principles in collective decision-making. It has been extensively studied in the past decades for its axiomatic properties and has received substantial attention … Fairness is one of the most desirable societal principles in collective decision-making. It has been extensively studied in the past decades for its axiomatic properties and has received substantial attention from the multiagent systems community in recent years for its theoretical and computational aspects in algorithmic decision-making. However, these studies are often not sufficiently rich to capture the intricacies of human perception of fairness in the ambivalent nature of the real-world problems. We argue that not only fair solutions should be deemed desirable by social planners (designers), but they should be governed by human and societal cognition, consider perceived outcomes based on human judgement, and be verifiable. We discuss how achieving this goal requires a broad transdisciplinary approach ranging from computing and AI to behavioral economics and human-AI interaction. In doing so, we identify shortcomings and long-term challenges of the current literature of fair division, describe recent efforts in addressing them, and more importantly, highlight a series of open research directions.
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1/2-Approximate MMS Allocation for Separable Piecewise Linear Concave Valuations

2024-03-24
Chandra Chekuri , Pooja Kulkarni , Rucha Kulkarni +1 more
We study fair distribution of a collection of m indivisible goods among a group of n agents, using the widely recognized fairness principles of Maximin Share (MMS) and Any Price … We study fair distribution of a collection of m indivisible goods among a group of n agents, using the widely recognized fairness principles of Maximin Share (MMS) and Any Price Share (APS). These principles have undergone thorough investigation within the context of additive valuations. We explore these notions for valuations that extend beyond additivity. First, we study approximate MMS under the separable (piecewise-linear) concave (SPLC) valuations, an important class generalizing additive, where the best known factor was 1/3-MMS. We show that 1/2-MMS allocation exists and can be computed in polynomial time, significantly improving the state-of-the-art. We note that SPLC valuations introduce an elevated level of intricacy in contrast to additive. For instance, the MMS value of an agent can be as high as her value for the entire set of items. We use a relax-and-round paradigm that goes through competitive equilibrium and LP relaxation. Our result extends to give (symmetric) 1/2-APS, a stronger guarantee than MMS. APS is a stronger notion that generalizes MMS by allowing agents with arbitrary entitlements. We study the approximation of APS under submodular valuation functions. We design and analyze a simple greedy algorithm using concave extensions of submodular functions. We prove that the algorithm gives a 1/3-APS allocation which matches the best-known factor. Concave extensions are hard to compute in polynomial time and are, therefore, generally not used in approximation algorithms. Our approach shows a way to utilize it within analysis (while bypassing its computation), and hence might be of independent interest.
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An Algorithmic Framework for Approximating Maximin Share Allocation of Chores

2021-07-18
Xin Huang , Pinyan Lu
We consider the problem of fairly dividing m indivisible chores among n agents. The fairness measure we consider here is the maximin share. The previous best known result is that … We consider the problem of fairly dividing m indivisible chores among n agents. The fairness measure we consider here is the maximin share. The previous best known result is that there always exists a 4/3-approximation maximin share allocation[3]. With our algorithm, we can always find a 11/9-approximation maximin share allocation for any instance. We also discuss how to improve the efficiency of the algorithm and its connection to the job scheduling problem. The full paper can be found at https://arxiv.org/abs/1907.04505.
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Fair Allocation with Interval Scheduling Constraints

2021-01-01
Bo Li , Minming Li , Ruilong Zhang
We study a fair resource scheduling problem, where a set of interval jobs are to be allocated to heterogeneous machines controlled by agents. Each job is associated with release time, … We study a fair resource scheduling problem, where a set of interval jobs are to be allocated to heterogeneous machines controlled by agents. Each job is associated with release time, deadline, and processing time such that it can be processed if its complete processing period is between its release time and deadline. The machines gain possibly different utilities by processing different jobs, and all jobs assigned to the same machine should be processed without overlap. We consider two widely studied solution concepts, namely, maximin share fairness and envy-freeness. For both criteria, we discuss the extent to which fair allocations exist and present constant approximation algorithms for various settings.
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Approximation Algorithms for Computing Maximin Share Allocations

2017-10-31
Georgios Amanatidis , Evangelos Markakis , Afshin Nikzad +1 more
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 … 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 main result is a $2/3$-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang, which also produces a $2/3$-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of their algorithm. Furthermore, motivated by the apparent difficulty, both theoretically and experimentally, in finding lower bounds on the existence of approximate solutions, we undertake a probabilistic analysis. We prove that in randomly generated instances, with high probability there exists a maximin share allocation. This can be seen as a justification of the experimental evidence reported in relevant works. Finally, we provide further positive results for two special cases that arise from previous works. The first one is the intriguing case of $3$ agents, for which it is already known that exact maximin share allocations do not always exist (contrary to the case of $2$ agents). We provide a $7/8$-approximation algorithm, improving the previously known result of $3/4$. The second case is when all item values belong to $\{0, 1, 2\}$, extending the $\{0, 1\}$ setting studied in Bouveret and Lema\^itre. We obtain an exact algorithm for any number of agents in this case.
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Approximation Algorithms for Maximin Fair Division

2017-06-20
Siddharth Barman , Sanath Kumar Krishna Murthy
We consider the problem of dividing indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin … We consider the problem of dividing 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 focussed on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee.
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