cs.LGOct 17, 2026

No-Regret Revenue Maximization in Sequential Market Making Under Inventory Constraints

Authors: Davide Maran, Marcello Restelli, Francesco Emanuele Stradi

Organizations: Politecnico di Milano

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Abstract

Market makers keep trading efficient by setting bid and ask prices for buyers and sellers. In sequential market making, trades happen independently based on price thresholds, requiring the market maker to maintain balanced, low-risk inventory over T rounds. This paper studies how to maximize revenue with unknown buyer and seller valuations using regret minimization. When the market maker only sees trade success or failure, an algorithm achieves regret and inventory bounds proportional to T to the power of 2/3. When full order book data is available for failed trades, regret and inventory limits improve significantly to the square root of T. We also establish a fundamental trade-off via a lower bound, by proving that tighter round-by-round inventory limits strictly force higher minimum regret.

Figures & tables

Figure 1: A favorable and arguably realistic regime. Left: the valuation distributions have little overlap, yielding a narrow optimal spread. Right: the corresponding quantities F(b∘)F(b^{\circ}) and S(a∘)S(a^{\circ}) are both small, namely of order η≪1\eta\ll 1 .
Protocol 1 Maker-Takers Interaction
Figure 2: Illustration of the discretization argument in Proposition 4.3 . The point (b0,a0)(b_{0},a_{0}) is an optimizer (or an ε\varepsilon -optimizer) of JJ over [0,1]2[0,1]^{2} , while (b0′,a0′)∈GN2(b_{0}^{\prime},a_{0}^{\prime})\in\mathcal{G}_{N}^{2} is the grid point selected in the proof, with b0≤b0′≤1/N+b0b_{0}\leq b_{0}^{\prime}\leq 1/N+b_{0} and a0−1/N≤a0′≤a0a_{0}-1/N\leq a_{0}^{\prime}\leq a_{0} .
Figure 3: Illustration of Double Double Pointer . Both scans start from (b0,aM−1)(b_{0},a_{M-1}) and move right or down, evaluating each visited feasible pair before moving. Left: the scan computes the benchmark value JlowJ^{\mathrm{low}} using the lower confidence bounds; the green box marks a maximizing pair. It continues until a pointer exits its grid or no feasible pair remains. Right: the algorithm restarts and scans using the upper confidence bounds, stopping at the first pair whose optimistic value reaches the lower benchmark.
Figure 4: This figure provides heuristic intuition for the lower-bound construction developed in Section 6.4 . Suppose that the agent aims to guarantee an imbalance bound of the form ∣It∣≤H=O(Tβ)|I_{t}|\leq H=\mathcal{O}(T^{\beta}) for some β<1/2\beta<1/2 . If the agent always selects the JJ -maximizing actions b∘b^{\circ} and a∘a^{\circ} , then, since F(b∘)=S(a∘)=\nicefrac12F(b^{\circ})=S(a^{\circ})=\nicefrac{{1}}{{2}} , the resulting imbalance process ItI_{t} evolves as a driftless random walk; this behavior is illustrated by the shaded curve. To prevent the walk from leaving the interval [−H,H][-H,H] (delimited by the dashed lines), the agent must occasionally intervene by selecting suboptimal actions, represented by the black points. As the figure suggests, the smaller HH is, the more frequently such corrective interventions are needed, leading to a fundamental trade-off between the strength of the imbalance constraint and the resulting cumulative regret. The core of the proof is to show that expected regret grows at least linearly with the expected total amount of correction, up to an additive O(H)\mathcal{O}(H) term. This correction is of order Ω(\nicefracTH)=Ω(T1−β)\Omega(\nicefrac{{T}}{{H}})=\Omega(T^{1-\beta}) .
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

SymbolMeaning
TTTime horizon.
Vtbuy,VtsellV^{\text{buy}}_{t},V^{\text{sell}}_{t}Buyer’s and seller’s private valuations at time tt .
Bt,AtB_{t},A_{t}Bid–ask pair selected at time tt .
ItI_{t}Inventory at time tt .
α\alphaPer-round inventory threshold.
OPT\mathrm{OPT}Optimal reward value.
Appendix
Table 6
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