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.

翻译：我们研究具有m个物品和n个多单位需求代理人（具有可加性、单位总和估值）的（非对称）单边分配问题中的公平机制。对称情形（m=n）即单边匹配问题，已针对单位需求代理人这一类别进行了广泛研究，特别关注随机优先机制（Random Priority）和由Bogomolnaia及Moulin引入的概率序列机制（Probabilistic Serial）。在单位总和估值函数的假设下，Christodoulou等人证明了在单边匹配问题中，随机优先机制和概率序列机制的无政府状态代价均为$\Theta(\sqrt{n})$。尽管随机优先机制和概率序列机制都是序数机制，但即使在更广泛的基数机制类别中，这些近似保证也是可能的最佳结果。要将这些结论推广至一般设定，存在两个技术障碍。其一，非对称性（$m\neq n$）会带来问题，尤其是当物品数量远大于代理人数量时。其二，需要研究多单位需求代理人而非仅单位需求代理人。我们的方法是研究概率序列机制的一种基数机制变体，我们称之为基数概率序列机制（Cardinal Probabilistic Serial）。我们为该机制建立了结构定理，并利用这些定理推导出无政府状态代价的界。我们的第一个主要结果是，对于具有多单位需求代理人的非对称单边分配问题，无政府状态代价的上界为$O(\sqrt{n}\cdot \log m)$。该上界同样适用于概率序列机制，且任何公平机制都存在一个互补的下界$\Omega(\sqrt{n})$。我们的第二个主要结果是，无政府状态代价随物品数量增加而恶化。具体而言，两种机制均需要对物品数量具有对数依赖性。