Optimal Automated Market Makers: Differentiable Economics and Strong Duality

TL;DR

This work studies profit-maximizing market making across multiple goods under adverse selection, showing that the design problem is dual to an optimal transport problem with cost $\|x - y\|_1$ and a geometric constraint. It develops a duality framework, proving strong duality and demonstrating that optimal mechanisms can involve bundling and in-kind payments, not just separate per-good pricing. Differentiable economics (e.g., RochetNet) is used to explore the mechanism space and conjecture new optimal structures, which are then validated via explicit dual certificates in both 1D and multi-parameter settings. The results highlight that mixed bundling and in-kind transactions can yield substantial profit gains and suggest practical deployment implications for DeFi and prediction markets, while offering a general, data-driven approach to multi-parameter mechanism design.

Abstract

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.

Figures (17)

• Figure 1: Left: A visualization of the single-dimensional signed measure under adverse selection, for a uniform valuation distribution, $\lambda=\frac{1}{3}$ and initial valuation $c=\frac{1}{2}$. Right: A visualization of the two-dimensional signed measure under adverse selection, for uniform valuation distribution and initial valuation $c=(\frac{1}{2}, \frac{1}{2})$. There is positive mass $\frac{\lambda}{2}$ distributed along the boundaries and a point mass at the point $c$ and $-(2\lambda + 1)$ negative mass spread uniformly through the space.
• Figure 2: Learned allocation and payment rules for two kinds of goods and $c=(\frac{1}{2}, \frac{1}{2})$, where trader valuations are distributed uniformly. Top to bottom, $\lambda = 0$ (no adverse selection), $\lambda=\frac{1}{2}$, and $\lambda=1$ (full adverse selection, so no trade is desirable, hence the blank plot). The axes of the plot are the trader's valuation for either good. Each distinct region is associated with a specific menu item; these menu items are marked on the payment rule plots.
• Figure 3: Left: Partition of the unit square that corresponds to the optimal transport solution to the dual problem in the two-item case, for the specific values $\lambda=1$ (pure noise trading) and $c=(\frac{1}{2}, \frac{1}{2})$. The red edges are of the same length, denoted by $a_1$, while the blue edges are of the same length as well, denoted by $b_1$. The dark green edges share the same length $1 - 2a_1$. Right: A visualization of the optimal transport solution for the partitioned pentagon-shaped and rectangle regions in the two-item case, for the specific values $\lambda=1$ (pure noise trading) and $c=(\frac{1}{2}, \frac{1}{2})$.
• Figure 4: A plot of the allocation and payment rules for the true optimal mechanism under the noise trading ($\lambda=1$) model ($\pi(c, x) = c$) for $c=(\frac{1}{2}, \frac{1}{2})$. Each distinct region is associated with a menu item; these are marked on the payment plot. Compare to the learned mechanism in \ref{['fig:noiserochet']}, top.
• Figure 5: Learned allocation and payment rules for $c=(\frac{1}{3}, \frac{1}{3})$ for a uniform distribution with no adverse selection ($\lambda=1$). Each distinct region is associated with a menu item; these are marked on the payment plot. A conjectured optimal menu is in \ref{['tab:noisetradingonethird']}.

Theorems & Definitions (8)

• Lemma 3.1
• Theorem 3.2: Strong duality, following Villani2003TopicsDaskalakis2017StrongKash2016Optimal
• Lemma 3.3
• Lemma 3.4
• proof
• proof : Proof of \ref{['thm:strongduality']}
• proof : Proof of \ref{['lemma:byparts']}
• proof : Proof of \ref{['lemma:twofunctions']}