Approximating Competitive Equilibrium by Nash Welfare
Abstract
We explore the relationship between two popular concepts on allocating 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 EisenbergGale convex program that maximizes Nash welfare over feasible allocations yields a competitive equilibrium. However, these two concepts diverge for nonhomogeneous utilities. From a computational perspective, maximizing Nash welfare amounts to solving a convex program for any concave utility functions, computing CE becomes PPADhard already for separable piecewise linear concave (SPLC) utilities. We introduce the concept of Galesubstitute utility functions, an analogue of the weak gross substitutes (WGS) property for the socalled Gale demand system. For Galesubstitutes utilities, we show that any allocation maximizing Nash welfare provides an approximateCE with surprisingly strong guarantees, where every agent gets at least half the maximum utility they can get at any CE, and is approximately envyfree. Galesubstitutes include examples of utilities where computing CE is PPAD hard: in particular, all separable concave utilities, and the previously studied nonseparable class of Leontieffree utilities. We introduce a new, general class of utility functions called generalized network utilities based on the generalized flow model; this class includes SPLC and Leontieffree utilities. We show that all such utilities are Galesubstitutes. 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.09994
 arXiv:
 arXiv:2402.09994
 Bibcode:
 2024arXiv240209994G
 Keywords:

 Computer Science  Computer Science and Game Theory;
 Mathematics  Optimization and Control