Batch auctions are a classical market microstructure, acclaimed for their fairness properties, and have received renewed interest in the context of blockchain-based financial systems. Constant function market makers (CFMMs) are another market design innovation praised for their computational simplicity and applicability to liquidity provision via smart contracts. Liquidity provision in batch exchanges is an important problem, and CFMMs have recently shown promise in being useful within batch exchanges. Different real-world implementations have used fundamentally different approaches towards integrating CFMMs in batch exchanges, and there is a lack of formal understanding of different design tradeoffs. We first provide a minimal set of axioms that are well-accepted rules of batch exchanges and CFMMs. These are asset conservation, uniform valuations, a best response for limit orders, and non-decreasing CFMM trading function. In general, many market solutions may satisfy all our axioms. We then describe several economically useful properties of market solutions. These include Pareto optimality for limit orders, price coherence of CFMMs (as a defence against cyclic arbitrage), joint price discovery for CFMMs (as a defence against parallel running), path independence for simple instances, and a locally computable response of the CFMMs in equilibrium (to provide them predictability on trade size given a market price). We show fundamental conflicts between some pairs of these properties. We then provide two ways of integrating CFMMs in batch exchanges, which attain different subsets of these properties. We further provide a convex program for computing Arrow-Debreu exchange market equilibria when all agents have weak gross substitute (WGS) demand functions on two assets -- this program extends the literature on Arrow-Debreu exchange markets and may be of independent interest.

翻译：批量拍卖是一种经典的市场微观结构，以其公平性而备受赞誉，并在基于区块链的金融系统背景下重新受到关注。恒定函数做市商（CFMMs）是另一项市场设计创新，因其计算简单性以及通过智能合约提供流动性的适用性而受到赞扬。批量交易所中的流动性提供是一个重要问题，而CFMMs最近显示出在批量交易所内具有应用潜力。不同的现实世界实现采用了根本不同的方法将CFMMs整合到批量交易所中，并且缺乏对不同设计权衡的正式理解。我们首先提出一组最小公理，这些是批量交易所和CFMMs公认的规则。这些公理包括资产守恒、统一估值、限价单的最优响应以及CFMM交易函数的非递减性。一般而言，许多市场解决方案可能满足我们所有的公理。然后，我们描述了市场解决方案的几个经济上有用的特性。这些特性包括限价单的帕累托最优性、CFMMs的价格一致性（作为对循环套利的防御）、CFMMs的联合价格发现（作为对并行运行的防御）、简单实例的路径独立性，以及均衡状态下CFMMs的局部可计算响应（为它们在给定市场价格下提供交易规模的可预测性）。我们展示了其中一些特性对之间的根本冲突。接着，我们提供了两种将CFMMs整合到批量交易所的方法，这些方法实现了这些特性的不同子集。此外，当所有代理对两种资产具有弱总替代（WGS）需求函数时，我们提出了一个用于计算阿罗-德布鲁交换市场均衡的凸规划——该规划扩展了关于阿罗-德布鲁交换市场的文献，并可能具有独立的研究价值。