In this paper, we investigate the computation of second-price pacing equilibria (SPPEs), a foundational model in online advertising auctions. We present a polynomial-time algorithm for computing exact SPPEs in instances with a constant number of goods. Our core technique maps buyers' pacing multipliers to the highest bids on each good, effectively partitioning the parameter space into a set of distinct geometric cells. By enumerating these cells, we fix the relative ordering of the bids and reduce the problem of equilibrium computation to a linear feasibility program. Finally, we demonstrate that this tractability extends to large-scale markets with an arbitrary number of goods, provided the goods can be aggregated into a constant number of valuation types.

翻译：本文研究二价拍卖中定价均衡的计算问题，该模型是在线广告拍卖领域的基础模型。我们提出了一种多项式时间算法，用于在商品数量恒定的情境下精确计算二价拍卖定价均衡。核心技术通过将买方的定价乘子映射为每种商品的最高出价，从而将参数空间划分为一组不同的几何单元。通过枚举这些单元，我们固定了出价的相对顺序，并将均衡计算问题简化为线性可行性规划。最后，我们证明，当商品可被聚合为恒定数量的估值类型时，这种可计算性可推广至任意数量商品的大规模市场。