Batch trading systems and constant function market makers (CFMMs) are two distinct market design innovations that have recently come to prominence as ways to address some of the shortcomings of decentralized trading systems. However, different deployments have chosen substantially different methods for integrating the two innovations. We show here from a minimal set of axioms describing the beneficial properties of each innovation that there is in fact only one, unique method for integrating CFMMs into batch trading schemes that preserves all the beneficial properties of both. Deployment of a batch trading schemes trading many assets simultaneously requires a reliable algorithm for approximating equilibria in Arrow-Debreu exchange markets. We study this problem when batches contain limit orders and CFMMs. Specifically, we find that CFMM design affects the asymptotic complexity of the problem, give an easily-checkable criterion to validate that a user-submitted CFMM is computationally tractable in a batch, and give a convex program that computes equilibria on batches of limit orders and CFMMs. Equivalently, this convex program computes equilibria of Arrow-Debreu exchange markets when every agent's demand response satisfies weak gross substitutability and every agent has utility for only two types of assets. This convex program has rational solutions when run on many (but not all) natural classes of widely-deployed CFMMs.

翻译：批处理交易系统和恒定函数做市商（CFMMs）是两项截然不同的市场设计创新，近年来作为解决去中心化交易系统某些缺陷的方式而备受关注。然而，不同的实际部署采用了截然不同的方法将这两项创新相结合。本文基于描述每项创新有益特性的最小公理集证明，事实上存在唯一一种方法能同时保留两者的所有有益特性，即在批处理交易方案中整合CFMMs。同时交易多种资产的批处理交易方案的部署，需要一种可靠算法来逼近Arrow-Debreu交换市场的均衡。我们研究了包含限价订单和CFMMs的批处理场景下的这一问题。具体而言，我们发现CFMM设计会影响该问题的渐近复杂度，给出了一个易于验证的准则来检验用户提交的CFMM在批处理中是否具有计算可行性，并提出了一个可计算限价订单与CFMMs批处理均衡的凸规划方法。等价地，当每个代理的需求响应满足弱总替代性且每个代理仅对两类资产具有效用时，该凸规划可计算Arrow-Debreu交换市场的均衡。当在广泛部署的CFMMs的多个（但非全部）自然类别上运行时，该凸规划具有有理数解。