交易机制中的复杂性—近似权衡：AMM 与 LOB
Complexity-Approximation Trade-offs in Exchange Mechanisms: AMMs vs. LOBs
AI总结：
提出统一 CFMM 与 LOB 的交易机制框架，以基函数锥包定义复杂性，刻画近似能力与状态复杂度的权衡，并分析最优表达机制及 Uniswap v3。
摘要：
本文提出了一个用于两种资产之间交易机制设计与分析的通用框架，该框架统一了两种主导性交易范式——恒定函数做市商（CFMM）和限价订单簿（LOB）——并使二者能够相互比较。在我们的框架中，每个流动性提供者（LP）向交易所提交一条向下倾斜的需求曲线，指明其在每个价格下希望持有的风险资产数量；交易所买卖风险资产，以满足汇总后的提交需求。一般而言，这种机制具有预算平衡性，并能够实现价格发现。不同的交易机制对应于对可接受需求曲线集合的不同限制。本文的主要目标是形式化贯穿交易机制设计的近似—复杂性权衡。例如，CFMM 为了简洁性而放弃表达能力：LP 的总需求曲线可以用恒定空间描述，但大多数需求曲线无法由相应一维函数族中的任何函数良好近似。直观上，LOB 作出相反的权衡：任何向下倾斜的需求曲线都可以由一组限价订单良好近似，但描述 LOB 状态所需的空间可能很大。本文引入了一种通用的“交易复杂性”度量，其定义为通过锥包生成交易所允许的全部需求函数所需的最小基函数集合。在建立该复杂性度量后，我们研究“最优表达能力”交易机制的设计，即在允许任意向下倾斜需求曲线得到良好近似的前提下，复杂性最低的机制。作为案例研究，我们通过该框架的视角解读广泛使用的 Uniswap v3 AMM 中的复杂性—近似权衡。
英文摘要：
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.
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