Approximating Competitive Equilibrium by Nash Welfare

2024-02-15

ACM-SIAM Symposium on Discrete Algorithms

AI Summary

This paper investigates the relationship between competitive equilibrium (CE) and maximum Nash welfare (MNW) allocations under non-homogeneous concave utilities, aiming to bridge the theoretical gap between the PPAD-hardness of CE computation (e.g., for SPLC utilities) and the polynomial-time tractability of MNW via convex optimization. We introduce the novel concept of *Gale-substitutable utilities*, and prove that under such utilities, the MNW allocation is a strong approximate CE: each agent’s utility is at least half that in any exact CE, and the allocation satisfies approximate envy-freeness. This framework subsumes generalized network utilities—including SPLC and Leontief-free utilities—for the first time. Furthermore, we establish that for *any* concave utility, every CE achieves at least 0.69 times the maximum Nash welfare—a tight bound. Our approach integrates convex optimization, Gale’s demand system, generalized flow models, and approximate equilibrium theory.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Exploring competitive equilibrium via Nash welfare
• Approximating CE for Gale-substitutes utilities
• Maximizing Nash welfare for concave utilities

Innovation

• Gale-substitute utility functions
• approximate competitive equilibrium
• generalized network utilities