We study the revenue guarantees and approximability of item pricing. Recent work shows that with $n$ heterogeneous items, item-pricing guarantees an $O(\log n)$ approximation to the optimal revenue achievable by any (buy-many) mechanism, even when buyers have arbitrarily combinatorial valuations. However, finding good item prices is challenging -- it is known that even under unit-demand valuations, it is NP-hard to find item prices that approximate the revenue of the optimal item pricing better than $O(\sqrt{n})$. Our work provides a more fine-grained analysis of the revenue guarantees and computational complexity in terms of the number of item ``categories'' which may be significantly fewer than $n$. We assume the items are partitioned in $k$ categories so that items within a category are totally-ordered and a buyer's value for a bundle depends only on the best item contained from every category. We show that item-pricing guarantees an $O(\log k)$ approximation to the optimal (buy-many) revenue and provide a PTAS for computing the optimal item-pricing when $k$ is constant. We also provide a matching lower bound showing that the problem is (strongly) NP-hard even when $k=1$. Our results naturally extend to the case where items are only partially ordered, in which case the revenue guarantees and computational complexity depend on the width of the partial ordering, i.e. the largest set for which no two items are comparable.
翻译:我们研究的是收入保障和物品定价的近似性。最近的工作显示,用美元的各种项目,项目定价保证了与任何(购买-许多)机制所能达到的最佳收入相比的O(log n)美元近似值,即使买主任意组合估价。然而,发现良好的物品价格是具有挑战性的 -- 众所周知,即使根据单位-需求估值,也很难找到比美元(sqrt{n})更接近最佳物品价格的物品价格。我们的工作提供了对收入保障和计算复杂性的更精细分析,就任何(购买-许多)机制所能达到的最佳收入而言,可能大大少于美元。我们假设这些物品按美元类别进行分配,因此一个类别内的物品是完全订购的,而一个包的买主价值仅取决于每个类别中的最佳物品。我们的项目主要保证美元与最接近(购买-many)收入的准确性(即使是购买-manify)部分地计算收入,并且当我们最稳定地计算最稳定的物品时,PTAS的固定性项目也是固定的。