Repeated Bilateral Trade: The Quest for Fairness

2026-06-13

AI Summary

This work addresses the design of a unified pricing mechanism for repeated bilateral trade under unknown valuation distributions, aiming to balance fairness and efficiency. The authors introduce a one-parameter family of fairness objectives interpolating between Rawlsian and Nash welfare, formulated via non-positive Hölder means aggregating buyers’ and sellers’ net utilities. This formulation reveals an underlying two-dimensional singular integral kernel structure, which motivates a novel pure-exploration learning problem. To tackle it, they develop a threshold-based rectangular double-sum estimator that effectively handles row–column dependencies and singular weighting. By integrating nonparametric statistics with online learning techniques, they establish minimax-optimal convergence rates under arbitrary unknown marginal distributions. Matching fixed-confidence sample complexity and regret bounds—differing only by polylogarithmic factors—are derived across the entire fairness spectrum, fully characterizing the optimal learning rates.

Abstract

We study repeated bilateral trade from a fairness perspective. At each round, a fresh seller-buyer pair arrives, and the platform posts a price before observing the traders' valuations. Trade occurs only if both agents accept the price. Rather than maximizing only the gain from trade, we consider platforms that seek balanced divisions of the generated surplus. We show that natural fairness desiderata lead to a one-parameter Rawls-to-Nash family of fair-gain objectives, obtained by aggregating the seller's and buyer's net gains through nonpositive Hölder means. Unlike the standard gain-from-trade objective and the Rawlsian fair-gain objective studied in prior work, our proposed objectives induce a new statistical structure in which expected rewards are recovered from threshold feedback through a two-dimensional singular-kernel integral identity. This leads to a nonstandard pure-exploration problem whose natural estimators are rectangular double sums with row-column dependence and singular weights. Assuming independent i.i.d. seller and buyer valuation sequences with arbitrary unknown marginals, we characterize the optimal learning rates for the whole Rawls-to-Nash family of fair-gain objectives, giving matching fixed-confidence sample-complexity and regret bounds up to polylogarithmic factors.

Problem

• repeated bilateral trade
• fairness
• surplus division
• fair-gain objectives
• valuation sequences

Innovation

• fairness in bilateral trade
• Rawls-to-Nash objectives
• singular-kernel integral identity
• pure-exploration learning
• nonparametric estimation with threshold feedback