We consider repeated multi-unit auctions with uniform pricing, which are widely used in practice for allocating goods such as carbon licenses. In each round, $K$ identical units of a good are sold to a group of buyers that have valuations with diminishing marginal returns. The buyers submit bids for the units, and then a price $p$ is set per unit so that all the units are sold. We consider two variants of the auction, where the price is set to the $K$-th highest bid and $(K+1)$-st highest bid, respectively. We analyze the properties of this auction in both the offline and online settings. In the offline setting, we consider the problem that one player $i$ is facing: given access to a data set that contains the bids submitted by competitors in past auctions, find a bid vector that maximizes player $i$'s cumulative utility on the data set. We design a polynomial time algorithm for this problem, by showing it is equivalent to finding a maximum-weight path on a carefully constructed directed acyclic graph. In the online setting, the players run learning algorithms to update their bids as they participate in the auction over time. Based on our offline algorithm, we design efficient online learning algorithms for bidding. The algorithms have sublinear regret, under both full information and bandit feedback structures. We complement our online learning algorithms with regret lower bounds. Finally, we analyze the quality of the equilibria in the worst case through the lens of the core solution concept in the game among the bidders. We show that the $(K+1)$-st price format is susceptible to collusion among the bidders; meanwhile, the $K$-th price format does not have this issue.

翻译：我们考虑了重复进行的统一定价多单元拍卖，此类拍卖广泛应用于碳许可证等商品的分配。每轮交易中，$K$个同质商品被出售给一组边际效用递减的买家。买家提交对各单元的出价，随后设定每单位价格$p$以售出所有商品。我们考察了两种拍卖变体：价格分别设定为第$K$高报价和第$(K+1)$高报价。我们从离线与在线两种场景分析该拍卖的性质。在离线场景中，考虑特定参与者$i$面临的问题：利用包含竞争对手历史报价的数据集，寻找使参与者$i$在该数据集上累积效用最大化的报价向量。通过将该问题等价转化为在精心构造的有向无环图中寻找最大权重路径，我们设计了一个多项式时间算法。在线场景中，参与者随时间推移使用学习算法更新报价。基于离线算法，我们设计了高效的在线竞价学习算法，在全信息与强盗反馈结构下均能达到次线性遗憾值，并给出了对应的遗憾下界。最后，我们通过核心解概念分析竞拍者博弈中最坏情况下的均衡质量，证明$(K+1)$价格式易导致竞拍者合谋，而$K$价格式则无此问题。