Bilateral Trade Under Heavy-Tailed Valuations: Minimax Regret with Infinite Variance

TL;DR

This paper investigates bilateral trade with heavy-tailed valuations, establishing minimax regret bounds and extending previous properties to infinite variance scenarios, advancing understanding of trading under complex valuation distributions.

Contribution

It extends the self-bounding property to real-valued valuations with infinite variance and characterizes the minimax regret rate for heavy-tailed valuation distributions.

Findings

Proves regret bounds for heavy-tailed valuations with finite p-th moments.

Establishes a matching lower bound using Assouad's method.

Interpolates between classical nonparametric and trivial linear rates.

Abstract

We study contextual bilateral trade under full feedback when trader valuations have bounded density but infinite variance. We first extend the self-bounding property of Bachoc et al. (ICML 2025) from bounded to real-valued valuations, showing that the expected regret of any price $\pi$ satisfies $\mathbb{E}[g(m,V,W) - g(\pi,V,W)] \le L|m-\pi|^2$ under bounded density alone. Combining this with truncated-mean estimation, we prove that an epoch-based algorithm achieves regret $\widetilde{O}(T^{1-2\beta(p-1)/(\beta p + d(p-1))})$ when the noise has finite $p$-th moment for $p \in (1,2)$ and the market value function is $\beta$-H\"older, and we establish a matching $\Omega(\cdot)$ lower bound via Assouad's method with a smoothed moment-matching construction. Our results characterize the exact minimax rate for this problem, interpolating between the classical nonparametric rate at $p=2$ and the trivial linear rate as $p \to 1^+$.