An official website of the United States government
In fair division of indivisible goods, ℓ-out-of-d maximin share (MMS) is the value that an agent can guarantee by partitioning the goods into d bundles and choosing the ℓ 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' ordinal rankings of bundles. We prove the existence of ℓ-out-of-⌊(ℓ + 1/2)n⌋ MMS allocations of goods for any integer ℓ ≥ 1, and present a polynomial-time algorithm that finds a 1-out-of-⌈3n/2⌉ MMS allocation when ℓ=1. We further develop an algorithm that provides a weaker ordinal approximation to MMS for any ℓ > 1.
more »
« less
This content will become publicly available on April 11, 2026
Improved Maximin Share Approximations for Chores by Bin Packing
We study fair division of indivisible chores among n agents with additive cost functions using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist for more than two agents, the goal has been to improve its approximations and identify interesting special cases where MMS allocations exist. We show the existence of· 1-out-of-9n/11 MMS allocations, which improves the state-of-the-art factor of 1-out-of-3n/4.· MMS allocations for factored instances, which resolves an open question posed by Ebadian et al. (2021).· 15/13-MMS allocations for personalized bivalued instances, improving the state-of-the-art factor of 13/11.We achieve these results by leveraging the HFFD algorithm of Huang and Lu (2021). Our approach also provides polynomial-time algorithms for computing an MMS allocation for factored instances and a 15/13-MMS allocation for personalized bivalued instances.
more »
« less
- Award ID(s):
- 1942321
- PAR ID:
- 10592418
- Publisher / Repository:
- {AAAI} Press
- Date Published:
- Journal Name:
- Proceedings of the AAAI Conference on Artificial Intelligence
- Volume:
- 39
- Issue:
- 13
- ISSN:
- 2159-5399
- Page Range / eLocation ID:
- 13881 to 13888
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
In fair division of indivisible goods, l-out-of-d maximin share (MMS) is the value that an agent can guarantee by partitioning the goods into d bundles and choosing the l 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' ordinal rankings of bundles. We prove the existence of l-out-of-floor((l+1/2)n) MMS allocations of goods for any integer l greater than or equal to 1, and present a polynomial-time algorithm that finds a 1-out-of-ceiling(3n/2) MMS allocation when l = 1. We further develop an algorithm that provides a weaker ordinal approximation to MMS for any l > 1.more » « less
-
null (Ed.)Fair division is a subfield of multiagent systems that is concerned with object distribution. When objects are indivisible, the Maximin Share Guarantee (MMS) is a desirable fairness notion; however, it is not guaranteed to exist. While MMS allocations may not always exist, a relaxation of MMS is guaranteed to exist. We show that there exists a family of instances for which this relaxation fails to guarantee the MMS value for all but a small constant number of agents.more » « less
-
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.more » « less
-
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 dłeq n for which 1-out-of-d MMS allocation exists. Our main contribution is a polynomial-time algorithm for 1-out-of-ł 2n/3 MMS allocation, and a proof of existence of 1-out-of-łfloor 3n/4 MMS allocation of chores. Furthermore, we show how to use recently-developed algorithms for bin-packing to approximate the latter bound up to a logarithmic factor in polynomial time.more » « less
