The Price of Anarchy of the Asymmetric One-Sided Allocation Problem

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 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. Under the assumption of unit-sum valuation functions, Christodoulou et al. proved that the price of anarchy is $\Theta(\sqrt{n})$ 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 there are two technical obstacles. One, asymmetry ($m\neq n$) is problematic especially when the number of items is much greater than the number of items. Two, it is necessary to study multi-unit demand agents rather than simply unit demand agents. Our approach is to study a cardinal mechanism variant of Probabilistic Serial, which we call Cardinal Probabilistic Serial. We present structural theorems for this mechanism and use them to obtain bounds on the price of anarchy. 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. This upper bound applies to Probabilistic Serial as well and there is a complementary lower bound of $\Omega(\sqrt{n})$ for any fair mechanism. Our second main result is that the price of anarchy degrades with the number of items. Specifically, a logarithmic dependence on the number of items is necessary for both mechanisms.