Repeated Bilateral Trade: The Quest for Fairness

We address the problem of maximizing Gain from Trade (GFT) in repeated buyer-seller exchanges subject to global budget balance constraints. While this problem is well-understood in purely adversarial and stochastic settings, these environments exhibit a sharp dichotomy: adversarial environments allow for no-regret learning against the best fixed-price mechanism, whereas stochastic environments allow for no-regret learning against the best distribution over prices that is budget balanced in expectation. This gap is significant, as policies balanced in expectation can increase the GFT by a multiplicative factor of two. In this work, we bridge these extremes by studying perturbed markets, where an underlying stochastic distribution is subject to an adversarial corruption $C$. We design an algorithm that adaptively scales with the level of corruption, achieving an $\tilde{\mathcal{O}}(T^{3/4}) + \mathcal{O}(C\log(T))$ regret bound against the best budget-balanced distribution over prices. Simultaneously, our algorithm maintains the worst-case $\tilde{\mathcal{O}}(T^{3/4})$ regret bound relative to a per-round budget-balanced baseline, ensuring optimality even in fully adversarial environments.

In contextual bilateral trade under full feedback, the posted price does not affect which valuations are observed. We show that in this model such action-independent feedback removes the polynomial adaptation penalty familiar from heavy-tailed bandits: fully parameter-free algorithms attain the oracle minimax $T$-exponents up to logarithmic factors, with no knowledge of the moment order $p \in (1,2)$ or its scale $σ_p$, and -- in the nonparametric case -- none of the effective Hölder smoothness $β\in (0,1]$. The statistic that makes model selection possible is a paired squared-loss difference, whose noise-square term cancels exactly, leaving noise damped by the candidate gap. The resulting bilateral-trade regret rates are new. Trader valuations have bounded conditional densities and heavy tails -- finite $p$-th moments for some $p \in (1,2)$, with possibly infinite variance. An epoch-based algorithm with truncated means achieves regret $\widetilde{O}(T^{(2-p)/p})$ in the parametric model and $\widetilde{O}(T^{1-2β(p-1)/(βp + d(p-1))})$ when the market value function is $β$-Hölder, with matching $Ω(\cdot)$ lower bounds -- under a mild nondegeneracy condition -- via Assouad's method and a fixed-support mixture construction -- characterizing the minimax rate in $T$ up to logarithmic factors over the effective smoothness range $β\in (0,1]$, interpolating between the classical nonparametric rate at $p{=}2$ and the trivial linear rate as $p \to 1^+$. The enabling structural step extends the self-bounding property of Bachoc et al. (ICML 2025) from bounded to real-valued valuations: within our conditionally independent, conditionally centered noise model, bounded conditional densities and finite first moments suffice for the expected regret of any price $π$ to satisfy $\mathbb{E}[g(m,V,W) - g(π,V,W)] \le L|m-π|^2$ -- no second moment is needed.

Bilateral trade models the task of intermediating between two strategic agents, a seller and a buyer, who wish to trade a good. We study this problem from the perspective of a profit-maximizing broker within an online learning framework, where the agents' valuations are generated by a smooth adversary. We devise a learning algorithm that guarantees a $\tilde{O}(\sqrt{T})$ regret bound, which is tight in the time horizon $T$ up to poly-logarithmic factors. This matches the minimax rate for the stochastic i.i.d. case, and is also well separated from the adversarial setting, where sublinear-regret is unattainable. By extending the strong regret guarantees from the i.i.d. case to the smooth adversary, we significantly broaden the scope of settings where such fast rate is achievable, while closing an important gap in the regret landscape of this fundamental economic problem. To overcome the challenges posed by this adversary, we leverage a continuity property of smooth instances and combines this with a hierarchical net-construction of the broker's action space, which is analyzed via algorithmic chaining. We showcase the applicability of these techniques by deriving a similarly tight $\tilde{O}(\sqrt{T})$ regret bound for a related mechanism design model: the joint ads problem.