We develop a unified ascending-auction framework for computing Walrasian equilibria in combinatorial markets with strong substitutes valuations and piecewise-linear payment functions. Our auction extends the celebrated ascending auctions of Gul and Stacchetti (2000) and Ausubel (2006) to accommodate payment frictions (e.g., transaction taxes or commission fees). This is achieved by incorporating directional price updates that reflect heterogeneous payment structures. Our framework also generalizes the unit-demand imperfectly transferable utility models of Alkan (1989, 1992) to a fully combinatorial setting, thereby unifying these paradigms. Furthermore, this is the first study to compute the minimum -- also known as the buyer-optimal -- equilibrium in combinatorial markets with such frictions. Our analysis builds upon discrete convex analysis. Our main technical contribution is a characterization of valid price-update directions, together with a strongly polynomial-time algorithm for computing them. Notably, the algorithm uses only demand-oracle queries and never requires handling information of exponential size. To compute such a direction, we formulate a lexicographic extension of the polymatroid sum problem and characterize its dual solution via a reduction to a convex flow problem. Exploiting the $\text{L}^\natural$-convexity of the dual objective, we show that the desired direction can be constructed from the minimal dual solution. This convexity also yields transparent economic and potential-based interpretations of the auction dynamics, strengthening the connection between ascending auctions and discrete optimization.

翻译：我们提出了一个统一的上升拍卖框架，用于计算具有强替代性估值和分段线性支付函数的组合市场中的瓦尔拉斯均衡。我们的拍卖将Gul和Stacchetti (2000)与Ausubel (2006)的经典上升拍卖扩展到容纳支付摩擦（例如交易税或佣金费用）。这是通过引入反映异质支付结构的方向性价格更新来实现的。该框架还将Alkan (1989, 1992)的单位需求不完全可转移效用模型推广至完全组合环境，从而统一了这些范式。此外，这是首次在带有此类摩擦的组合市场中计算最小（即买方最优）均衡的研究。我们的分析基于离散凸分析。主要技术贡献在于刻画了有效的价格更新方向，并提出了一个强多项式时间算法来计算这些方向。值得注意的是，该算法仅使用需求预言查询，且无需处理指数规模的信息。为计算此类方向，我们构造了多拟阵和问题的字典序扩展，并通过归约为凸流问题来刻画其对偶解。利用对偶目标的$\text{L}^\natural$-凸性，我们证明了所需方向可由最小对偶解构造得出。这一凸性还为拍卖动力学提供了透明的经济解释与基于势函数的解读，强化了上升拍卖与离散优化之间的关联。