Competitive Equilibria for Non-quasilinear Bidders in Combinatorial Auctions

TL;DR

This paper investigates Walrasian equilibria in combinatorial auctions without assuming quasilinearity, demonstrating the existence of fractional equilibria and relating integral solutions to configuration LPs, thus broadening the theoretical understanding.

Contribution

It extends the theory of Walrasian equilibria to non-quasilinear settings, showing existence results and linking integral equilibria to linear programming solutions.

Findings

Fractional Walrasian equilibrium always exists.

Existence of equilibrium reduces to an Arrow-Debreu style market.

Integral equilibria relate to solutions of a configuration LP.

Abstract

Quasiliearity is a ubiquitous and questionable assumption in the standard study of Walrasian equilibria. Quasilinearity implies that a buyer's value for goods purchased in a Walrasian equilibrium is always additive with goods purchased with unspent money. It is a particularly suspect assumption in combinatorial auctions, where buyers' complex preferences over goods would naturally extend beyond the items obtained in the Walrasian equilibrium. We study Walrasian equilibria in combinatorial auctions when quasilinearity is not assumed. We show that existence can be reduced to an Arrow-Debreu style market with one divisible good and many indivisible goods, and that a "fractional" Walrasian equilibrium always exists. We also show that standard integral Walrasian equilibria are related to integral solutions of an induced configuration LP associated with a fractional Walrasian equilibrium, generalizing known results for both quasilinear and non-quasilnear settings.