Fairness in Repeated Bilateral Trade Explored with Rawls-to-Nash Objectives

This paper delves into the dynamics of repeated bilateral trade, focusing on the concept of fairness rather than solely maximizing profit. In this model, a new seller-buyer pair emerges in each round, and a platform sets a price without prior knowledge of the traders' valuations. The core idea is to study platforms that prioritize an equitable distribution of the generated surplus. The researchers propose a "Rawls-to-Nash" family of fair-gain objectives. These objectives are derived from natural fairness desiderata and aggregate the net gains of both seller and buyer using nonpositive Hölder means. This approach differs significantly from standard gain-from-trade maximization and previous Rawlsian fair-gain objectives, introducing a new statistical structure where expected rewards are recovered via a two-dimensional singular-kernel integral identity from threshold feedback. This new structure gives rise to a nonstandard pure-exploration problem, characterized by estimators that are rectangular double sums with row-column dependence and singular weights. Assuming independent and identically distributed valuation sequences for sellers and buyers with unknown marginals, the study precisely characterizes the optimal learning rates for the entire Rawls-to-Nash family of fair-gain objectives, providing matching fixed-confidence sample-complexity and regret bounds.