Tight Regret Bounds for Bilateral Trade under Semi Feedback
Yaonan Jin
TL;DR
The paper studies fixed-price bilateral trade under semi-feedback with adversarial values, focusing on regret minimization against the best fixed-price benchmark under Global Budget Balance. It introduces a two-phase mechanism, GBB-Semi, combining ProfitMax (to ensure nonnegative profit) with a discretized Exp3-LS20-based update that uses semi-feedback signals and surrogate Gains from Trade ${\widetilde{\text{GFT}}_k^t}$. By adopting a $K = \widetilde{\Theta}(T^{1/3})$ grid and near-diagonal actions, it achieves a regret of $\widetilde{O}(T^{2/3})$, matching the known $\Omega(T^{2/3})$ lower bound up to polylog factors. The analysis establishes GBB feasibility with high probability, provides unbiased estimators for the surrogate gains, and combines phase-wise bounds to close the semi-feedback regret landscape for adversarial values, extending the understanding between full and partial feedback regimes.
Abstract
The study of \textit{regret minimization in fixed-price bilateral trade} has received considerable attention in recent research. Previous works [CCC+24a, CCC+24b, AFF24, BCCF24, CJLZ25, LCM25a, GDFS25] have acquired a thorough understanding of the problem, except for determining the tight regret bound for GBB semi-feedback fixed-price mechanisms under adversarial values. In this paper, we resolve this open question by devising an $\widetilde{O}(T^{2 / 3})$-regret mechanism, matching the $Ω(T^{2 / 3})$ lower bound from [CJLZ25] up to polylogarithmic factors.