We study the relationship between two central concepts in the allocation of divisible goods: 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 concepts coincide: the classical Eisenberg-Gale convex program that maximizes Nash welfare over feasible allocations yields a competitive equilibrium. However, they 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, 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.

翻译：本文研究了可分商品分配中两个核心概念之间的关系：竞争均衡（CE）与最大化纳什福利的分配方案，即加权效用几何平均最大化的分配方案。当参与者具有齐次凹效用函数时，这两个概念完全一致：经典的Eisenberg-Gale凸规划通过在可行分配中最大化纳什福利，即可得到竞争均衡。然而，对于非齐次效用函数，二者会出现分歧。从计算视角看，对于任意凹效用函数，最大化纳什福利等价于求解凸规划问题；而对于可分离分段线性凹（SPLC）效用函数，计算竞争均衡则属于PPAD难问题。本文提出了Gale替代效用函数的概念，这是针对所谓Gale需求系统的弱总替代（WGS）性质的类比。对于Gale替代效用函数，我们证明任何最大化纳什福利的分配都能提供具有惊人强保证性的近似竞争均衡：每个参与者至少获得其在任意竞争均衡中可能获得的最大效用的一半，并且分配方案近似无嫉妒。Gale替代效用函数涵盖了那些计算竞争均衡属于PPAD难问题的效用函数类别，包括所有可分离凹效用函数以及先前研究的非可分离类Leontief-free效用函数。基于广义流模型，我们引入了一类广泛的新型效用函数——广义网络效用函数。该类函数包含SPLC和Leontief-free效用函数，并且我们证明所有此类函数均属于Gale替代效用函数。反观之，尽管某些参与者在纳什福利最大化分配中可能获得远高于竞争均衡的效用，我们证明了一个“无政府代价”类型的结果：对于一般凹效用函数，每个竞争均衡至少能达到最大纳什福利的$(1/e)^{1/e} > 0.69$比例，且该比例因子是紧的。