We study the problem of regret minimization for a single bidder in a sequence of first-price auctions where the bidder discovers the item's value only if the auction is won. Our main contribution is a complete characterization, up to logarithmic factors, of the minimax regret in terms of the auction's \emph{transparency}, which controls the amount of information on competing bids disclosed by the auctioneer at the end of each auction. Our results hold under different assumptions (stochastic, adversarial, and their smoothed variants) on the environment generating the bidder's valuations and competing bids. These minimax rates reveal how the interplay between transparency and the nature of the environment affects how fast one can learn to bid optimally in first-price auctions.

翻译：我们研究单个竞标者在序列一价拍卖中最小化遗憾的问题，其中竞标者仅在中标时才获知物品价值。我们的主要贡献是完整刻画（在对数因子范围内）基于拍卖透明度的极小极大遗憾率，该透明度用于控制拍卖人在每次拍卖结束时披露竞争投标信息的数量。我们的结论适用于生成竞标者估值与竞争投标的不同环境假设（随机、对抗及其平滑变体）。这些极小极大速率揭示了透明度与环境性质之间的相互作用如何影响在一价拍卖中最优投标学习速度。