We study the fundamental, classical mechanism design problem of single-buyer multi-item Bayesian revenue-maximizing auctions under the lens of communication complexity between the buyer and the seller. Specifically, we ask whether using quantum communication can be more efficient than classical communication. We have two sets of results, revealing a surprisingly rich landscape - which looks quite different from both quantum communication in non-strategic parties, and classical communication in mechanism design. We first study the expected communication complexity of approximately optimal auctions. We give quantum auction protocols for buyers with unit-demand or combinatorial valuations that obtain an arbitrarily good approximation of the optimal revenue while running in exponentially more efficient communication compared to classical approximately optimal auctions. However, these auctions come with the caveat that they may require the seller to charge exponentially large payments from a deviating buyer. We show that this caveat is necessary - we give an exponential lower bound on the product of the expected quantum communication and the maximum payment. We then study the worst-case communication complexity of exactly optimal auctions in an extremely simple setting: additive buyer's valuations over two items. We show the following separations: 1. There exists a prior where the optimal classical auction protocol requires infinitely many bits, but a one-way message of 1 qubit and 2 classical bits suffices. 2. There exists a prior where no finite one-way quantum auction protocol can obtain the optimal revenue. However, there is a barely-interactive revenue-optimal quantum auction protocol. 3. There exists a prior where no multi-round quantum auction protocol with a finite bound on communication complexity can obtain the optimal revenue.

翻译：我们研究在买方与卖方之间通信复杂度视角下的基础经典机制设计问题——单买家多物品贝叶斯收益最大化拍卖。具体而言，我们探讨量子通信是否比经典通信更高效。我们得到两组结果，揭示了一个令人惊讶的丰富图景——该图景既不同于非策略性参与方的量子通信，也不同于机制设计中的经典通信。首先研究近似最优拍卖的期望通信复杂度：针对具有单位需求或组合估值的买家，我们给出量子拍卖协议，可在获得任意逼近最优收益的同时，其通信效率相比经典近似最优拍卖呈指数级提升。然而这些协议存在一个缺陷——卖家可能需向偏离行为的买家收取指数级支付。我们证明该缺陷是必然的：给出了期望量子通信量与最大支付量乘积的指数下界。随后在极简设定（买家对两物品的加性估值）中研究精确最优拍卖的最坏情况通信复杂度。我们得到以下分离结果：1. 存在一种先验分布，其中最优经典拍卖协议需要无限比特，但单向消息仅需1量子比特和2经典比特即可达成。2. 存在另一种先验分布，使得任何有限单向量子拍卖协议都无法获得最优收益，但存在近乎交互的最优收益量子拍卖协议。3. 还存在一种先验分布，使得任何通信复杂度有界的多轮量子拍卖协议都无法获得最优收益。