This paper presents a general framework for the design and analysis of exchange mechanisms between two assets that unifies and enables comparisons between the two dominant paradigms for exchange, constant function market markers (CFMMs) and limit order books (LOBs). In our framework, each liquidity provider (LP) submits to the exchange a downward-sloping demand curve, specifying the quantity of the risky asset it wishes to hold at each price; the exchange buys and sells the risky asset so as to satisfy the aggregate submitted demand. In general, such a mechanism is budget-balanced and enables price discovery. Different exchange mechanisms correspond to different restrictions on the set of acceptable demand curves. The primary goal of this paper is to formalize an approximation-complexity trade-off that pervades the design of exchange mechanisms. For example, CFMMs give up expressiveness in favor of simplicity: the aggregate demand curve of the LPs can be described using constant space, but most demand curves cannot be well approximated by any function in the corresponding single-dimensional family. LOBs, intuitively, make the opposite trade-off: any downward-slowing demand curve can be well approximated by a collection of limit orders, but the space needed to describe the state of a LOB can be large. This paper introduces a general measure of {\em exchange complexity}, defined by the minimal set of basis functions that generate, through their conical hull, all of the demand functions allowed by an exchange. With this complexity measure in place, we investigate the design of {\em optimally expressive} exchange mechanisms, meaning the lowest complexity mechanisms that allow for arbitrary downward-sloping demand curves to be well approximated. As a case study, we interpret the complexity-approximation trade-offs in the widely-used Uniswap v3 AMM through the lens of our framework.

翻译：本文提出一个用于设计与分析两种资产间交换机制的通用框架，该框架统一了两种主导性交换范式——恒定函数做市商（CFMMs）与限价订单簿（LOBs），并实现了它们之间的比较。在此框架下，每位流动性提供者（LP）向交易所提交一条向下倾斜的需求曲线，指定其在各价格水平上希望持有的风险资产数量；交易所通过买卖风险资产以满足提交的总体需求。一般而言，此类机制能够实现预算平衡并促进价格发现。不同交换机制对应于对可接受需求曲线集合施加的不同约束。本文的主要目标是形式化贯穿交换机制设计的近似-复杂度权衡。例如，CFMMs 以牺牲表达力为代价换取简洁性：LP 的总体需求曲线可用常数空间描述，但大多数需求曲线无法被相应单维函数族中的任何函数良好近似。直观而言，LOBs 做出了相反的权衡：任何向下倾斜的需求曲线均可通过限价订单集合良好近似，但描述 LOB 状态所需的空间可能很大。本文引入一种通用的“交换复杂度”度量，其定义为能够通过锥形包络生成交易所允许的所有需求函数的最小基函数集合。基于此复杂度度量，我们探究了“最优表达力”交换机制的设计——即允许任意向下倾斜需求曲线被良好近似的最低复杂度机制。作为案例研究，我们通过该框架解读了广泛使用的 Uniswap v3 AMM 中的复杂度-近似权衡。