Shilling is the use of artificial bids to make competition appear stronger and push prices upward. We study repeated first-price auctions in which shilling affects feedback but not allocation: the learner wins or loses against the real competing bid, but after a loss observes the maximum of the real bid and an independent shill bid. Thus the manipulation changes what the learner observes and hence how it learns to bid, without changing the outcome of the current auction. We analyze regret with respect to the best bid benchmark, assuming that the shill-bid distribution is known. Even then, shilling can mask the real bid, while useful side information appears only through intermittent low-shill events. Our algorithm combines a robust interval-elimination branch, which ignores the shilled report and achieves the dynamic-pricing rate $\tilde{\mathcal{O}}(T^{2/3})$, with an optimistic branch that debiases losing-side reports and exploits the resulting suffix information when it is reliable and achieves the first-price auctions rate $\tilde{\mathcal{O}}(\sqrt{T})$. A validation and racing procedure lets the algorithm use these optimistic updates without knowing the right scale or feedback geometry in advance. We complement the upper bounds with a matching lower bound, up to logarithmic factors, in the single-active-region case. Overall, the results show that even feedback-only shilling can sharply alter the statistical difficulty of repeated bidding.

翻译：虚假投标是指使用人为出价使竞争看似更激烈并推高价格。本文研究重复第一价格拍卖，其中虚假投标影响反馈但不影响分配：学习者赢或输给真实竞争出价，但在输掉后观察真实出价与独立虚假投标中的最大值。因此，操纵改变了学习者的观测内容，进而改变其学习出价的方式，但不改变当前拍卖的结果。本文假设虚假投标分布已知，并基于最优出价基准分析遗憾。即使如此，虚假投标能掩盖真实出价，而有用的辅助信息仅通过间歇性的低虚假事件出现。我们的算法结合了一个鲁棒的区间消除分支（忽略虚假报告，实现动态定价率$\tilde{\mathcal{O}}(T^{2/3})$）和一个乐观分支（去偏输方报告，并在可靠时利用由此产生的后缀信息，实现第一价格拍卖率$\tilde{\mathcal{O}}(\sqrt{T})$）。通过验证和竞速过程，算法无需预先知道正确尺度或反馈几何结构即可使用这些乐观更新。在单活动区域情况下，我们证明了上界与下界匹配（相差仅对数因子）。总体而言，结果表明即使仅反馈层面的虚假投标也能显著改变重复竞价问题的统计难度。