Balanced Posted Prices in Stationary Two-Sided Markets

Abstract

We study an intermediary that buys nonperishing items from arriving sellers and sells them to arriving buyers, subject to limited storage. Sellers and buyers arrive according to independent Poisson processes. The intermediary pays and charges the same fixed item prices, aiming to maximize long-run gains from trade: buyer value minus seller cost. Our benchmark may choose allocations directly, has unlimited storage, and must purchase each unit before allocating it. Its optimal long-run gain rate equals the value of a linear program of average purchase and allocation rates, with or without knowledge of future arrivals. For buyers whose bundle values are maxima of nonnegative additive valuations (XOS), we compute static common prices that guarantee half of this benchmark for arbitrary finite item capacities, even one storage slot per item, regardless of how ties are resolved without using future arrivals. The prices are computable in polynomial time using demand and supporting-value queries. The construction equalizes itemwise lower bounds on buyer and seller contributions to gains from trade, accounting for procurement costs. We call this contribution balance. As a single-item foundation, we also equalize the rates of purchases and sales selected by the price, before checking inventory availability. We call this rate balance. For capacity $C$, it yields the tight guarantee $C/(C+1)$. The two principles parallel proofs of the classical prophet inequality based on half the expected benchmark and on a median threshold, adapting these balancing ideas to recurring two-sided markets with limited inventory.