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微观市场结构，因其公平性而备受推崇，并在基于区块链的金融系统中重新引起关注。常数函数做市商（CFMM）是另一种市场设计创新，以其计算简单性和通过智能合约提供流动性的适用性而闻名。在批量交易所中提供流动性是一个重要问题，而CFMM最近在批量交易所中的实用性展现出潜力。不同实际实现采用了根本不同的方法将CFMM集成到批量交易所中，但对于不同设计权衡缺乏正式理解。我们首先提供一组最小公理，这些公理是批量交易所和CFMM公认的规则，包括资产守恒、统一估值、限价单的最佳反应以及非递减的CFMM交易函数。一般来说，许多市场解决方案可能满足我们所有公理。然后，我们描述了市场解决方案的几个具有经济实用性的属性，包括限价单的帕累托最优性、CFMM的价格一致性（作为对抗循环套利的防御）、CFMM的联合价格发现（作为对抗并行运行的防御）、简单实例的路径独立性以及均衡中CFMM的局部可计算响应（以便在给定市场价格时预测交易规模）。我们揭示了这些属性中某些配对之间的根本冲突。然后，我们提供了两种将CFMM集成到批量交易所的方法，它们实现了这些属性的不同子集。我们还提供了一个凸规划，用于计算所有代理对两种资产具有弱总替代（WGS）需求函数时的Arrow-Debreu交换市场均衡——该规划扩展了关于Arrow-Debreu交换市场的文献，可能具有独立的意义。