Constant-function market makers (CFMMs), such as Uniswap, are automated exchanges offering trades among a set of assets. We study their technical relationship to another class of automated market makers, cost-function prediction markets. We first introduce axioms for market makers and show that CFMMs with concave potential functions characterize "good" market makers according to these axioms. We then show that every such CFMM on $n$ assets is equivalent to a cost-function prediction market for events with $n$ outcomes. Our construction directly converts a CFMM into a prediction market and vice versa. Conceptually, our results show that desirable market-making axioms are equivalent to desirable information-elicitation axioms, i.e., markets are good at facilitating trade if and only if they are good at revealing beliefs. For example, we show that every CFMM implicitly defines a proper scoring rule for eliciting beliefs; the scoring rule for Uniswap is unusual, but known. From a technical standpoint, our results show how tools for prediction markets and CFMMs can interoperate. We illustrate this interoperability by showing how liquidity strategies from both literatures transfer to the other, yielding new market designs.

翻译：常数函数做市商（CFMM），如Uniswap，是一种支持多种资产间交易的自动化交易所。我们研究了它们与另一类自动化做市商——成本函数预测市场——的技术关系。首先，我们为做市商引入公理，并证明具有凹势函数的CFMM根据这些公理刻画了“优质”做市商。随后，我们证明每个定义在n种资产上的此类CFMM均等价于一个针对n个结果事件的成本函数预测市场。我们的构造方法可直接将CFMM转换为预测市场，反之亦然。从概念上看，我们的结果表明，理想的做市公理与理想的信息激励公理等价，即市场善于促进交易当且仅当其善于揭示信念。例如，我们证明每个CFMM都隐含地定义了一个用于激励信念的正确评分规则；Uniswap的评分规则虽不常见，但已为学界所知。从技术角度看，我们的结果展示了预测市场与CFMM工具的可互操作性。通过展示来自两个领域的流动性策略如何迁移至对方领域并产生新的市场设计，我们阐释了这种互操作性。