Approximating Competitive Equilibrium by Nash Welfare

TL;DR

This work investigates the relationship between competitive equilibrium (CE) and maximum Nash welfare (NSW) in Fisher markets with general concave utilities. It shows that NSW can be efficiently computed for any concave utilities and provides a bridge to CE via Gale-demand duality, introducing the Gale-substitutes property to identify broad utility classes where NSW allocations approximate CE well. The authors prove that for Σ-Gale-substitutes utilities, NSW yields at least half the CE utility for every agent, and separable utilities (a key non-homogeneous class) fall into this category; generalized network utilities are also encompassed, with thrifty equilibria extending results to satiable cases. They establish a tight price-of-anarchy bound of $ (1/ ext{e})^{1/ ext{e}} $ between CE and NSW and discuss the computational hardness of CE versus the tractability of NSW. Overall, the paper provides a unifying, algorithmically favorable framework for approximating competitive equilibria across a broad spectrum of utility functions, with implications for fair division and market design.

Abstract

We explore the relationship between two popular concepts in the allocation of divisible items: competitive equilibrium (CE) and allocations with maximum Nash welfare, i.e., allocations where the weighted geometric mean of the utilities is maximal. When agents have homogeneous concave utility functions, these two concepts coincide: the classical Eisenberg-Gale convex program that maximizes Nash welfare over feasible allocations yields a competitive equilibrium. However, these two concepts diverge for non-homogeneous utilities. From a computational perspective, maximizing Nash welfare amounts to solving a convex program for any concave utility functions, computing CE becomes PPAD-hard already for separable piecewise linear concave (SPLC) utilities. We introduce the concept of Gale-substitute utility functions, which is an analogue of the weak gross substitutes (WGS) property for the so-called Gale demand system. For Gale-substitutes utilities, we show that any allocation maximizing Nash welfare provides an approximate-CE with surprisingly strong guarantees, where every agent gets at least half the maximum utility they can get at any CE, and is approximately envy-free. Gale-substitutes include utility functions where computing CE is PPAD hard, such as all separable concave utilities and the previously studied non-separable class of Leontief-free utilities. We introduce a broad new class of utility functions called generalized network utilities based on the generalized flow model. This class includes SPLC and Leontief-free utilities, and we show that all such utilities are Gale-substitutes. Conversely, although some agents may get much higher utility at a Nash welfare maximizing allocation than at a CE, we show a price of anarchy type result: for general concave utilities, every CE achieves at least $(1/e)^{1/e} > 0.69$ fraction of the maximum Nash welfare, and this factor is tight.

Figures (1)

• Figure 1: The graph illustrates the containment relationships among different utility classes, with the classes where the Fisher market equilibrium can be computed efficiently highlighted in grey.

Theorems & Definitions (80)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Definition 2.1
• Definition 2.2
• Definition 2.3