Repeated Bilateral Trade Against a Smoothed Adversary
This work addresses algorithmic pricing in repeated trade settings, providing theoretical guarantees for regret against smoothed adversaries, which is incremental but offers new insights into optimal rates.
The paper tackles the problem of repeated bilateral trade with an adaptive adversary, characterizing regret regimes for fixed-price mechanisms under different feedback models. It shows minimax regret of order √T with full feedback, linear regret with partial feedback when posting the same price, and optimal T^{3/4} regret when posting different prices, with a lower bound proof as the main technical result.
We study repeated bilateral trade where an adaptive $σ$-smooth adversary generates the valuations of sellers and buyers. We provide a complete characterization of the regret regimes for fixed-price mechanisms under different feedback models in the two cases where the learner can post either the same or different prices to buyers and sellers. We begin by showing that the minimax regret after $T$ rounds is of order $\sqrt{T}$ in the full-feedback scenario. Under partial feedback, any algorithm that has to post the same price to buyers and sellers suffers worst-case linear regret. However, when the learner can post two different prices at each round, we design an algorithm enjoying regret of order $T^{3/4}$ ignoring log factors. We prove that this rate is optimal by presenting a surprising $T^{3/4}$ lower bound, which is the main technical contribution of the paper.