Market Making without Regret

Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni, Luigi Foscari, Vinayak Pathak
Proceedings of Thirty Eighth Conference on Learning Theory, PMLR 291:799-837, 2025.

Abstract

We consider a sequential decision-making setting where, at every round $t$, the learner (a market maker) posts a bid price $B_t$ and an ask price $A_t$ to an incoming trader (the taker) with a private valuation for some asset. If the trader’s valuation is lower than the bid price, or higher than the ask price, then a trade (sell or buy) occurs. Letting $M_t$ be the market price (observed only at the end of round $t$), the maker’s utility is $M_t-B_t$ if the maker bought the asset, it is $A_t-M_t$ if they sold it, and it is $0$ if no trade occurred. We characterize the maker’s regret with respect to the best fixed choice of bid and ask pairs under a variety of assumptions (adversarial, i.i.d., and their variants) on the sequence of market prices and valuations. Our upper bound analysis unveils an intriguing connection relating market making to first-price auctions and dynamic pricing. Our main technical contribution is a lower bound for the i.i.d. case with Lipschitz distributions and independence between market prices and takers’ valuations. The difficulty in the analysis stems from a unique relationship between the reward and feedback functions that allows learning algorithms to trade off reward for information in a continuous way.

Cite this Paper

BibTeX
@InProceedings{pmlr-v291-cesa-bianchi25a, title = {Market Making without Regret}, author = {Cesa-Bianchi, Nicol\`{o} and Cesari, Tommaso and Colomboni, Roberto and Foscari, Luigi and Pathak, Vinayak}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {799--837}, year = {2025}, editor = {Haghtalab, Nika and Moitra, Ankur}, volume = {291}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--04 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v291/main/assets/cesa-bianchi25a/cesa-bianchi25a.pdf}, url = {https://proceedings.mlr.press/v291/cesa-bianchi25a.html}, abstract = {We consider a sequential decision-making setting where, at every round $t$, the learner (a market maker) posts a bid price $B_t$ and an ask price $A_t$ to an incoming trader (the taker) with a private valuation for some asset. If the trader’s valuation is lower than the bid price, or higher than the ask price, then a trade (sell or buy) occurs. Letting $M_t$ be the market price (observed only at the end of round $t$), the maker’s utility is $M_t-B_t$ if the maker bought the asset, it is $A_t-M_t$ if they sold it, and it is $0$ if no trade occurred. We characterize the maker’s regret with respect to the best fixed choice of bid and ask pairs under a variety of assumptions (adversarial, i.i.d., and their variants) on the sequence of market prices and valuations. Our upper bound analysis unveils an intriguing connection relating market making to first-price auctions and dynamic pricing. Our main technical contribution is a lower bound for the i.i.d. case with Lipschitz distributions and independence between market prices and takers’ valuations. The difficulty in the analysis stems from a unique relationship between the reward and feedback functions that allows learning algorithms to trade off reward for information in a continuous way.} }
Endnote
%0 Conference Paper %T Market Making without Regret %A Nicolò Cesa-Bianchi %A Tommaso Cesari %A Roberto Colomboni %A Luigi Foscari %A Vinayak Pathak %B Proceedings of Thirty Eighth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Nika Haghtalab %E Ankur Moitra %F pmlr-v291-cesa-bianchi25a %I PMLR %P 799--837 %U https://proceedings.mlr.press/v291/cesa-bianchi25a.html %V 291 %X We consider a sequential decision-making setting where, at every round $t$, the learner (a market maker) posts a bid price $B_t$ and an ask price $A_t$ to an incoming trader (the taker) with a private valuation for some asset. If the trader’s valuation is lower than the bid price, or higher than the ask price, then a trade (sell or buy) occurs. Letting $M_t$ be the market price (observed only at the end of round $t$), the maker’s utility is $M_t-B_t$ if the maker bought the asset, it is $A_t-M_t$ if they sold it, and it is $0$ if no trade occurred. We characterize the maker’s regret with respect to the best fixed choice of bid and ask pairs under a variety of assumptions (adversarial, i.i.d., and their variants) on the sequence of market prices and valuations. Our upper bound analysis unveils an intriguing connection relating market making to first-price auctions and dynamic pricing. Our main technical contribution is a lower bound for the i.i.d. case with Lipschitz distributions and independence between market prices and takers’ valuations. The difficulty in the analysis stems from a unique relationship between the reward and feedback functions that allows learning algorithms to trade off reward for information in a continuous way.
APA
Cesa-Bianchi, N., Cesari, T., Colomboni, R., Foscari, L. & Pathak, V.. (2025). Market Making without Regret. Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:799-837 Available from https://proceedings.mlr.press/v291/cesa-bianchi25a.html.

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