Online Market Making and the Value of Observing the Order Book

TL;DR

This paper investigates online market-making with action-dependent feedback, proposing algorithms that leverage order book observations to achieve improved regret bounds in stochastic and adversarial settings.

Contribution

It introduces new algorithms for online market making that utilize limited order book feedback, significantly enhancing regret guarantees over traditional bandit models.

Findings

Achieves $O(\sqrt T)$ regret in stochastic settings without smoothness assumptions.

Extends results to mean-reverting price processes with high-probability bounds.

Designs an explore-then-perturb algorithm with $O(T^{2/3})$ regret in adversarial settings.

Abstract

We study an online market-making problem in which a learner sequentially posts bid and ask prices for a single asset while interacting with traders holding private valuations. Unlike existing online learning formulations that assume fully censored feedback, we introduce an action-dependent feedback model inspired by real limit order books: when a trade occurs, the trader's valuation remains hidden, whereas when no trade occurs, informative feedback about supply and demand is revealed. We show that this additional information fundamentally changes the learnability of the problem. In the stochastic setting with i.i.d. market prices, we propose an elimination-based algorithm that achieves $O(\sqrt T)$ regret with high probability, without requiring any smoothness assumptions on the distribution of trader valuations. We then extend this result to a broad class of mean-reverting price processes by considering both local, autoregressive dynamics and a weaker global drift condition based on cumulative deviations from the mean. Under either assumption, we establish high-probability $O(\sqrt T)$ regret bounds, relying on a new concentration inequality of independent interest. Finally, in the adversarial setting with oblivious prices, we design an explore-then-perturb algorithm that guarantees $O(T^{2/3})$ regret in expectation. Our results quantify the value of observing the order book in online market making and demonstrate that even limited, action-dependent feedback can substantially improve regret guarantees compared to standard bandit feedback models.