论文标题
重新审视拍卖中的简单性:原始复杂性
Simplicity in Auctions Revisited: The Primitive Complexity
论文作者
论文摘要
在本文中,我们重新审视了机制简单性的概念。我们认为$ M $物品的卖家面临估值$ v $的单个买家。我们观察到,以前的定义复杂性度量的尝试通常无法分类,这些机制被直观地认为是简单的(例如,“单独销售”机制)。如果可以通过进行一些被认为简单的原始操作来找到最大化买家利润的捆绑包,我们建议将菜单视为简单。 \ emph {菜单的原始复杂性}是(自适应地)在菜单中找到利润 - 最大化条目所需的原始操作数量。在本文中,我们研究的原始操作本质上是计算“单独销售”机制的结果。 原始复杂性是否捕获了直觉上简单的其他拍卖的简单性?我们考虑\ emph {捆绑尺寸定价},这是一种常见的定价方法,其中捆绑包的价格仅取决于其大小。我们的主要技术贡献是确定在各种环境中捆绑尺寸定价菜单的原始复杂性。我们表明,对于任何分配$ \ cal d $,加权矩阵排名估值,甚至在其价值之间具有任意相关性的分布,总有一个捆绑尺寸的定价菜单,具有低原始复杂性,与最佳捆绑捆绑式定价菜单的收入几乎相同。作为此证明的一部分,我们提供了一种随机算法,对于任何加权的Matroid等级评估$ V $和Integer $ k $,找到最有价值的尺寸$ K $的集合,只有一个poly-logarithmic的需求和价值查询。我们表明,这个结果在几个方面基本紧密。例如,如果估值$ V $是子模型,那么找到最有价值的$ k $的集合需要许多查询(这解决了Badanidiyuru等人的开放问题[EC'12])。
In this paper we revisit the notion of simplicity in mechanisms. We consider a seller of $m$ items, facing a single buyer with valuation $v$. We observe that previous attempts to define complexity measures often fail to classify mechanisms that are intuitively considered simple (e.g., the "selling separately" mechanism) as such. We suggest to view a menu as simple if a bundle that maximizes the buyer's profit can be found by conducting a few primitive operations that are considered simple. The \emph{primitive complexity of a menu} is the number of primitive operations needed to (adaptively) find a profit-maximizing entry in the menu. In this paper, the primitive operation that we study is essentially computing the outcome of the "selling separately" mechanism. Does the primitive complexity capture the simplicity of other auctions that are intuitively simple? We consider \emph{bundle-size pricing}, a common pricing method in which the price of a bundle depends only on its size. Our main technical contribution is determining the primitive complexity of bundle-size pricing menus in various settings. We show that for any distribution $\cal D$ over weighted matroid rank valuations, even distributions with arbitrary correlation among their values, there is always a bundle-size pricing menu with low primitive complexity that achieves almost the same revenue as the optimal bundle-size pricing menu. As part of this proof we provide a randomized algorithm that for any weighted matroid rank valuation $v$ and integer $k$, finds the most valuable set of size $k$ with only a poly-logarithmic number of demand and value queries. We show that this result is essentially tight in several aspects. For example, if the valuation $v$ is submodular, then finding the most valuable set of size $k$ requires exponentially many queries (this solves an open question of Badanidiyuru et al. [EC'12]).