In practice, most auction mechanisms are not strategy-proof, so equilibrium analysis is required to predict bidding behavior. In many auctions, though, an exact equilibrium is not known and one would like to understand whether -- manually or computationally generated -- bidding strategies constitute an approximate equilibrium. We develop a framework and methods for estimating the distance of a strategy profile from equilibrium, based on samples from the prior and either bidding strategies or sample bids. We estimate an agent's utility gain from deviating to strategies from a constructed finite subset of the strategy space. We use PAC-learning to give error bounds, both for independent and interdependent prior distributions. The primary challenge is that one may miss large utility gains by considering only a finite subset of the strategy space. Our work differs from prior research in two critical ways. First, we explore the impact of bidding strategies on altering opponents' perceived prior distributions -- instead of assuming the other agents to bid truthfully. Second, we delve into reasoning with interdependent priors, where the type of one agent may imply a distinct distribution for other agents. Our main contribution lies in establishing sufficient conditions for strategy profiles and a closeness criterion for conditional distributions to ensure that utility gains estimated through our finite subset closely approximate the maximum gains. To our knowledge, ours is the first method to verify approximate equilibrium in any auctions beyond single-item ones. Also, ours is the first sample-based method for approximate equilibrium verification.

翻译：在实践中，大多数拍卖机制并非策略证明的，因此需要均衡分析来预测投标行为。然而，在许多拍卖中，精确的均衡并不为人所知，人们希望了解——无论是手动还是计算生成的——投标策略是否构成近似均衡。我们开发了一个框架和方法，用于基于先验分布样本以及投标策略或样本投标，估计策略配置文件与均衡的距离。我们通过将代理的效用增益偏离到策略空间构造的有限子集中的策略来估计该增益。我们使用PAC学习给出误差界，适用于独立和相互依赖的先验分布。主要挑战在于，仅考虑策略空间的有限子集可能会遗漏较大的效用增益。我们的工作与先前研究在两个关键方面有所不同。首先，我们探讨了投标策略如何改变对手感知的先验分布——而不是假设其他代理真实投标。其次，我们深入研究了相互依赖先验下的推理，其中一个代理的类型可能暗示其他代理的独特分布。我们的主要贡献在于为策略配置文件建立充分条件，并为条件分布设定接近性标准，以确保通过我们的有限子集估计的效用增益紧密逼近最大增益。据我们所知，我们的方法是首个在单物品拍卖之外的任何拍卖中验证近似均衡的方法。此外，我们的方法是首个基于样本的近似均衡验证方法。