Extend affine fee-setting results to multiple sellers

Extend the constant-approximation and optimality results established for one-seller one-buyer exchange to markets with multiple sellers. Specifically, determine whether an intermediary using an affine fee schedule w(P) = (1 − α)P + β—such as the prior-independent schedule w(P) = P − φ_B(P) under affine buyer virtual values, or the constant schedule w(P) = η_{v−c} under MHR assumptions—can achieve a constant-factor approximation to the intermediary’s optimal revenue (and, when pertinent, surplus) when there are multiple sellers with independent private costs drawn from known product distributions under analogous regularity or MHR conditions.

Background

The paper analyzes 1-seller 1-buyer exchange with an intermediary and shows that simple affine fee schedules can be optimal or near-optimal under various conditions (affine buyer virtual values, MHR assumptions). It further extends some results to multiple buyers via a reduction but does not address multiple sellers.

Introducing multiple sellers changes the strategic environment on the supply side (e.g., competition among sellers, multi-dimensional allocation decisions), and it is unclear whether the same affine schedules retain their approximation guarantees. Clarifying this would broaden the practical applicability of fee-setting mechanisms used by platforms like eBay and Amazon.