Competitive equilibrium always exists for combinatorial auctions with graphical pricing schemes

TL;DR

This paper proves that a competitive equilibrium always exists in a broad class of combinatorial auctions with graphical valuations, providing explicit algorithms and extending to multi-unit auctions, highlighting quadratic pricing as practical.

Contribution

It introduces a new class of valuations with guaranteed competitive equilibrium and provides constructive algorithms, extending the theory to multi-unit auctions.

Findings

Existence of competitive equilibrium for graphical valuations.

Explicit algorithms for competitive pricing vectors.

Extension to multi-unit combinatorial auctions.

Abstract

We show that a competitive equilibrium always exists in combinatorial auctions with anonymous graphical valuations and pricing, using discrete geometry. This is an intuitive and easy-to-construct class of valuations that can model both complementarity and substitutes, and to our knowledge, it is the first class besides gross substitutes that have guaranteed competitive equilibrium. We prove through counter-examples that our result is tight, and we give explicit algorithms for constructive competitive pricing vectors. We also give extensions to multi-unit combinatorial auctions (also known as product-mix auctions). Combined with theorems on graphical valuations and pricing equilibrium of Candogan, Ozdagar and Parrilo, our results indicate that quadratic pricing is a highly practical method to run combinatorial auctions.