Abstract
We study “fair mechanisms” for the (asymmetric) one-sided allocation problem with m items and n multi-unit demand agents with additive, unit-sum valuations. The symmetric case (\(m=n\)), the one-sided matching problem, has been studied extensively for the special class of unit demand agents, in particular with respect to the folklore Random Priority mechanism and the Probabilistic Serial mechanism, introduced by Bogomolnaia and Moulin [6]. These are both fair mechanisms and attention has focused on their structural properties, incentives, and performance with respect to social welfare. Under the standard assumption of unit-sum valuation functions, Christodoulou et al. [10] proved that the price of anarchy is \(\varTheta (\sqrt{n})\) in the one-sided matching problem for both the Random Priority and Probabilistic Serial mechanisms. Whilst both Random Priority and Probabilistic Serial are ordinal mechanisms, these approximation guarantees are the best possible even for the broader class of cardinal mechanisms.
To extend these results to the general setting of the one-sided allocation problems there are two technical obstacles. One, asymmetry (\(m\ne n\))is problematic especially when the number of items is much greater than the number of agents, \(m\gg n\). Two, it is necessary to study multi-unit demand agents rather than simply unit demand agents. For this paper, our focus is on Probabilistic Serial. Our first main result is an upper bound of \(O(\sqrt{n}\cdot \log m)\) on the price of anarchy for the asymmetric one-sided allocation problem with multi-unit demand agents. We then present a complementary lower bound of \(\varOmega (\sqrt{n})\) for any fair mechanism. That lower bound is unsurprising. More intriguing is our second main result: the price of anarchy of Probabilistic Serial degrades with the number of items. Specifically, a logarithmic dependence on the number of items is necessary as we show a lower bound of \(\varOmega (\min \{n\, , \, \log m\})\).
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Notes
- 1.
To avoid confusion, we emphasize that symmetry in this paper refers to an equal number of agents and items. This is a standard definition in one-sided markets. (The term symmetric does have a second meaning in the literature: all the agents have identical valuation functions).
- 2.
We remark that the proof of [10] for Probabilistic Serial does not apply with multi-unit demand agents, even in the simple symmetric (\(m=n\)) setting.
- 3.
In contrast, for a unit demand agent i, we have \(v'_i(S)=\max _{j\in S} v'_i(j)\).
- 4.
At time \(\frac{m}{n}\) all the items have been consumed because there are m units and each of the n agents consumes at rate 1.
- 5.
While we do not show the existence of pure strategy Nash equilibria of probabilistic serial in the multi-unit demand setting, the proof by Aziz et al. for the unit demand case [4] also applies to the multi-unit demand setting.
- 6.
We remark that this condition on the consumption time is not present in the original statement of the lemma as it is not needed when there are only n items.
- 7.
We remark that Lemma 4.2 implies that \(|X|\le n\).
- 8.
While we do not show the existence of pure strategy Nash equilibria of probabilistic serial in the multi-unit demand setting, the proof by Aziz et al. for the unit demand case [4] also applies to the multi-unit demand setting.
- 9.
To see this, take a single agent with value 1 for every item and let other agents having value 1 for the first item and \(\varepsilon \) for the remaining items.
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The authors are grateful to Hervé Moulin for comments and advice. We also thank the referees for suggestions improving the paper.
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Jiang, S., Ndiaye, N., Vetta, A., Wu, E. (2024). The Price of Anarchy of Probabilistic Serial in One-Sided Allocation Problems. In: Garg, J., Klimm, M., Kong, Y. (eds) Web and Internet Economics. WINE 2023. Lecture Notes in Computer Science, vol 14413. Springer, Cham. https://doi.org/10.1007/978-3-031-48974-7_24
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