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.
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Ordinal Maximin Share Approximation for Chores
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.
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- PAR ID:
- 10353560
- Date Published:
- Journal Name:
- Proceedings of the 21st International Conference on Autonomous Agents and Multiagent Systems
- Page Range / eLocation ID:
- 597–605
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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