A fundamental assumption in classical mechanism design is that buyers are perfect optimizers. However, in practice, buyers may be limited by their computational capabilities or a lack of information, and may not be able to perfectly optimize. This has motivated the introduction of approximate incentive compatibility (IC) as an appealing solution concept for practical mechanism design. While most of the literature focuses on the analysis of particular approximate IC mechanisms, this paper is the first to study the design of optimal mechanisms in the space of approximate IC mechanisms and to explore how much revenue can be garnered by moving from exact to approximate incentive constraints. We study the problem of a seller facing one buyer with private values and analyze optimal selling mechanisms under $\varepsilon$-incentive compatibility. We establish that the gains that can be garnered depend on the local curvature of the seller's revenue function around the optimal posted price when the buyer is a perfect optimizer. If the revenue function behaves locally like an $α$-power for $α\in (1,\infty)$, then no mechanism can garner gains higher than order $\varepsilon^{α/(2α-1)}$. This improves upon state-of-the-art results which imply maximum gains of $\varepsilon^{1/2}$ by providing the first parametric bounds that capture the impact of revenue function's curvature on revenue gains. Furthermore, we establish that an optimal mechanism needs to randomize as soon as $\varepsilon>0$ and construct a randomized mechanism that is guaranteed to achieve order $\varepsilon^{α/(2α-1)}$ additional revenues, leading to a tight characterization of the revenue implications of approximate IC constraints. Our work brings forward the need to optimize not only over allocations and payments but also over best responses, and we develop a new framework to address this challenge.
A Game-Theoretic Analysis of the Empirical Revenue Maximization Algorithm with Endogenous Sampling
Peking University（北京大学）；Technion - Israel Institute of Technology（以色列理工学院）；Hong Kong University of Science and Technology（香港科技大学）；Alibaba Group（阿里巴巴集团）；Shanghai Jiao Tong University（上海交通大学）
经验收益最大化（ERM）是拍卖设计中最重要的价格学习算法之一：文献表明，它可以在重复拍卖和统一价格拍卖中学习到收益最大化拍卖者的近似最优保留价。然而，在这些应用程序中，提供输入的机构有动机操纵输入，以降低输出的价格。我们概括了Lavi et al（2019）提出的激励意识度量的定义，以量化由于$N$输入样本中的$m\ge 1$的变化而导致的ERM输出价格的降低，并提供了该度量到零的特定收敛率，因为对于不同类型的输入分布，$N$趋于无穷大。通过采用这一措施，我们构建了一个有效的，近似激励兼容的，和收入最优的学习算法使用ERM在重复拍卖对非近视投标人，并显示近似组激励兼容统一价格拍卖。
