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Maximin fairness with mixed divisible and indivisible goods

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Abstract

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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Notes

  1. Such a solution was credited to B. Knaster and S. Banach by Steinhaus [55].

  2. An MMS allocation always exists when there are two agents [20].

  3. An allocation is said to satisfy equitability if every agent gets the same utility.

  4. While an MMS allocation may not exist in general, such an allocation always exists in the case of two agents.

  5. The \(\gamma (I)\) is defined to be the maximum value of \(\alpha\) instead of the supremum. This is because the density functions are non-atomic and the maximum \(\alpha\) can always be achieved.

  6. In the indivisible setting, the corresponding result of adding an item may lower the best MMS guarantee for a problem instance is easy to get.

  7. We adopt the name “monotonicity property” from Amanatidis et al. [4].

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Authors and Affiliations

  1. School of Physical and Mathematical Sciences, Nanyang Technological University, Jurong West, Singapore

    Xiaohui Bei, Xinhang Lu & Hongao Wang

  2. School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen, China

    Shengxin Liu

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  1. Xiaohui Bei
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A preliminary version appeared in Proceedings of the 35th AAAI Conference on Artificial Intelligence (AAAI) [17]. This project/research is supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 2 (MOE2019-T2-1-045).

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Bei, X., Liu, S., Lu, X. et al. Maximin fairness with mixed divisible and indivisible goods. Auton Agent Multi-Agent Syst 35, 34 (2021). https://doi.org/10.1007/s10458-021-09517-7

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