Ascending Auctions for Combinatorial Markets with Frictions: A Unified Framework via Discrete Convex Analysis

TL;DR

This paper introduces a unified ascending-auction framework for combinatorial markets with payment frictions, extending existing models and providing a polynomial-time algorithm using demand-oracle queries.

Contribution

It generalizes ascending auctions to include payment frictions and computes the buyer-optimal equilibrium efficiently using discrete convex analysis.

Findings

Developed a characterization of valid price-update directions.

Designed a strongly polynomial-time algorithm using demand-oracle queries.

Provided economic interpretations of auction dynamics through convexity.

Abstract

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.