Computing Equilibria in Markets with Budget-Additive Utilities

TL;DR

This paper introduces a new combinatorial algorithm for computing market equilibria with budget-additive utilities, extending linear utility models and addressing multiple equilibria issues.

Contribution

It presents the first efficient algorithm for Fisher markets with budget-additive utilities and explores computational hardness for related equilibrium problems.

Findings

Developed a descending-price algorithm for Pareto-optimal equilibria.

Proved NP-hardness of welfare-maximizing equilibrium computation.

Established PPAD-hardness for equilibria with separate satiation points.

Abstract

We present the first analysis of Fisher markets with buyers that have budget-additive utility functions. Budget-additive utilities are elementary concave functions with numerous applications in online adword markets and revenue optimization problems. They extend the standard case of linear utilities and have been studied in a variety of other market models. In contrast to the frequently studied CES utilities, they have a global satiation point which can imply multiple market equilibria with quite different characteristics. Our main result is an efficient combinatorial algorithm to compute a market equilibrium with a Pareto-optimal allocation of goods. It relies on a new descending-price approach and, as a special case, also implies a novel combinatorial algorithm for computing a market equilibrium in linear Fisher markets. We complement these positive results with a number of hardness results for related computational questions. We prove that it is NP-hard to compute a market equilibrium that maximizes social welfare, and it is PPAD-hard to find any market equilibrium with utility functions with separate satiation points for each buyer and each good.