We study the problem of regret minimization for a single bidder in a sequence of first-price auctions where the bidder knows 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 transparency, which regulates 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：我们研究了单一投标人在一系列第一价格拍卖中最小化遗憾的问题，其中投标人仅在中标时知晓物品的价值。我们的主要贡献是，在考虑拍卖透明性的情况下，对极小化极大遗憾进行了完整刻画（精确到对数因子）。透明性调控拍卖人在每轮拍卖结束时披露竞争投标信息量的程度。我们的结果基于投标人估值和竞争投标所生成环境的不同假设（随机性、对抗性及其平滑变体）。这些极小化极大速率揭示了透明性与环境本质之间的相互作用如何影响在第一价格拍卖中学习最优投标策略的速度。