We consider the problem of a designer who wants to allocate resources to representatives, that then distribute the resources they receive among the individuals they represent. Motivated by the way Feeding America, one of the largest U.S. charities, allocates donations to food banks, which then further distribute the donations to food-insecure individuals, we focus on mechanisms that use artificial currencies. We compare auctions through the lens of the Price of Anarchy, with respect to three canonical welfare objectives: utilitarian social welfare (sum of individuals' utilities), Nash social welfare (product of individuals' utilities), and egalitarian social welfare (minimum of individuals' utilities). We prove strong lower bounds on the Price of Anarchy of all auctions that allocate each item to the highest bidder, subject to a mild technical constraint; this includes Feeding America's current auction, the First-Price auction. In sharp contrast, our main result shows that adapting the classic Trading Post mechanism of Shapley and Shubik to this setting, and coupled with Feeding America's choice of budget rule (each representative gets an amount of artificial currency equal to the number of individuals it represents), achieves a small Price of Anarchy for all generalized $p$-mean objectives simultaneously. Our bound on the Price of Anarchy of the Trading Post mechanism depends on $\ell$: the product of the rank and the ``incoherence'' of the underlying valuation matrix, which together capture a notion of how ``spread out'' the values of a matrix are. This notion has been extremely influential in the matrix completion literature, and, to the best of our knowledge, has never been used in auction theory prior to our work. Perhaps surprisingly, we prove that the dependence on $\ell$ is necessary: the Price of Anarchy of the Trading Post mechanism is $\Omega(\sqrt{\ell})$.

翻译：我们研究一个设计者如何将资源分配给代表，再由代表将所得资源分配给其所代表的个体的问题。受美国最大慈善机构之一Feeding America向食品银行分配捐赠物资（食品银行随后将捐赠进一步分发给粮食不安全个体）的方式启发，我们重点关注使用人工货币的机制。我们通过无政府价格（Price of Anarchy）的视角比较拍卖机制，并针对三种经典福利目标：功利主义社会福利（个体效用之和）、纳什社会福利（个体效用之积）和平等主义社会福利（个体效用最小值）进行分析。我们证明，在温和的技术约束下，所有将每件物品分配给最高出价者的拍卖（包括Feeding America当前使用的拍卖和第一价格拍卖）的无政府价格都存在很强的下界。与此形成鲜明对比的是，我们的主要结果表明，将Shapley和Shubik的经典交易站机制（Trading Post mechanism）适配于此场景，并结合Feeding America的预算规则（每位代表获得的人工货币数量等于其代表的个体数量），能够同时对所有广义$p$-均值目标实现较小的无政府价格。我们对交易站机制无政府价格的界依赖于$\ell$：即底层估值矩阵的秩与其“非相干性”的乘积，二者共同刻画了矩阵值“分散程度”的概念。这一概念在矩阵补全文献中极具影响力，且据我们所知，在我们的工作之前从未在拍卖理论中使用过。可能令人惊讶的是，我们证明了对$\ell$的依赖是必要的：交易站机制的无政府价格为$\Omega(\sqrt{\ell})$。