Garg, Jugal, Tao, Yixin and Végh, László A. 
ORCID: 0000-0003-1152-200X
(2025)
Approximating competitive equilibrium by Nash welfare.
In: Azar, Yossi and Panigrahi, Debmalya, (eds.)
Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA).
Society for Industrial and Applied Mathematics Publications, pp. 2538-2559.
ISBN 978-1-61197-832-2
Abstract
We explore the relationship between two popular concepts in the allocation of divisible items: competitive equilibrium(CE) and allocations that maximize 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, whereas 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.
| Item Type: | Book Section |
|---|---|
| Additional Information: | © 2025 The Author(s) |
| Divisions: | Mathematics |
| Subjects: | Q Science > QA Mathematics |
| Date Deposited: | 12 Feb 2025 11:48 |
| Last Modified: | 19 Feb 2025 11:30 |
| URI: | http://eprints.lse.ac.uk/id/eprint/127291 |
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