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

2026-04-12

AI Summary

This work addresses the computation of buyer-optimal Walrasian equilibria in combinatorial markets with payment frictions—such as transaction taxes or commissions—where such equilibria were previously intractable. The authors introduce a unified ascending auction framework that integrates discrete convex analysis and L♮-convex optimization to handle heterogeneous markets characterized by strong substitutes and piecewise-linear payment functions. Their key contribution is a strongly polynomial-time algorithm that relies solely on a demand oracle, circumventing the need for exponential information to identify valid price update directions. This approach not only yields an efficient computational procedure but also provides a clear economic interpretation of the auction dynamics through a well-defined potential function. Furthermore, the method extends the unit-demand model with imperfectly transferable utility to general combinatorial settings.

Technology Category

• Game Theory and Economic Paradigms: Auctions and Market-Based Systems
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web measurements

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.

Problem

• combinatorial markets
• payment frictions
• Walrasian equilibrium
• strong substitutes
• buyer-optimal equilibrium

Innovation

• ascending auctions
• discrete convex analysis
• payment frictions
• Walrasian equilibrium
• L♮-convexity