The role of a market maker is to simultaneously offer to buy and sell quantities of goods, often a financial asset such as a share, at specified prices. An automated market maker (AMM) is a mechanism that offers to trade according to some predetermined schedule; the best choice of this schedule depends on the market maker's goals. The literature on the design of AMMs has mainly focused on prediction markets with the goal of information elicitation. More recent work motivated by DeFi has focused instead on the goal of profit maximization, but considering only a single type of good (traded with a numeraire), including under adverse selection (Milionis et al. 2022). Optimal market making in the presence of multiple goods, including the possibility of complex bundling behavior, is not well understood. In this paper, we show that finding an optimal market maker is dual to an optimal transport problem, with specific geometric constraints on the transport plan in the dual. We show that optimal mechanisms for multiple goods and under adverse selection can take advantage of bundling, both improved prices for bundled purchases and sales as well as sometimes accepting payment "in kind." We present conjectures of optimal mechanisms in additional settings which show further complex behavior. From a methodological perspective, we make essential use of the tools of differentiable economics to generate conjectures of optimal mechanisms, and give a proof-of-concept for the use of such tools in guiding theoretical investigations.

翻译：做市商的作用是同时以指定价格提供买入和卖出一定数量商品（通常是金融资产，如股票）。自动做市商（AMM）是一种根据预定计划提供交易的机制；该计划的最佳选择取决于做市商的目标。关于AMM设计的文献主要集中于以信息获取为目标的预测市场。近期受DeFi启发的相关研究则聚焦于利润最大化的目标，但仅考虑单一类型商品（与基准货币交易），包括在逆向选择下的情形（Milionis et al. 2022）。在包含多种商品（可能存在复杂捆绑行为）情境下的最优做市机制尚未得到充分理解。本文证明，寻找最优做市商等价于一个最优传输问题，且对偶问题中的传输计划具有特定几何约束。我们表明，针对多种商品及逆向选择的最优机制可以利用捆绑策略——既能改善捆绑购买和销售的价格，有时也接受"实物"支付。我们提出了其他情境下的最优机制猜想，展示了进一步复杂的行为。从方法论角度，我们充分利用可微经济学工具生成最优机制猜想，并为这类工具指导理论探索提供了概念验证。