Bilateral Trade Under Heavy-Tailed Valuations: Minimax Regret without a Variance Bound

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 ∈ (1,2) or its scale σp, and -- in the nonparametric case -- none of the effective H"older smoothness β ∈ (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 ∈ (1,2), with possibly infinite variance. An epoch-based algorithm with truncated means achieves regret widetildeO(T(2-p)/p) in the parametric model and widetildeO(T1-2β(p-1)/(β p + d(p-1))) when the market value function is β-H"older, with matching Omega(·) 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 ∈ (0,1], interpolating between the classical nonparametric rate at p=2 and the trivial linear rate as p → 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 mathbbE[g(m,V,W) - g(π,V,W)] ≤ L|m-π|2 -- no second moment is needed.