Motivated by digital marketplaces such as Uber and Airbnb, we study how a two-sided platform designs a menu of qualities, consumer prices, and provider wages. Consumers and providers self-select according to their private values and costs, so the platform must design its menu to attract consumers while inducing enough supply to meet their demand. We study this coordination problem over a continuous quality space, requiring supply to cover demand at every quality threshold. We formulate menu design as an optimization over trading distributions, which specify how much trade occurs at or above each quality. Through convexification of the technological cost function, we characterize the profit- and welfare-optimal trading distributions. This characterization shows how production technology alone can reduce menu size without sacrificing optimal profit or welfare, regardless of the type distributions, and determines when optimal menus can be finite. We then show that profit maximization can be highly efficient: under monotone hazard rate conditions on both sides, a profit-optimal menu achieves at least $1/(e-1)$ of optimal social welfare. We next consider the practical setting in which the menu size is restricted to a prescribed number of offers. We establish how the restricted menu approximates the unrestricted optimum in profit. We develop an algorithm that computes a menu within this size limit in polynomial time. The computed menu simultaneously approximates the unrestricted optimal profit and social welfare with additive error.
We consider online learning in bilateral trade on platforms such as Uber, Airbnb, and eBay. In each round, a platform posts a payment to a seller and a price to a buyer, both with private valuations, and observes only whether trade occurs. Its objective is to maximize cumulative gains-from-trade, the sum of the differences between buyers' and sellers' valuations across completed trades. We measure the platform's performance by its regret against the best fixed common price. Financial self-sufficiency motivates global budget balance (Bernasconi et al., 2024), which allows profits from some trades to subsidize others, provided that the final balance is nonnegative. Such financial planning should also account for the time value of money: profits earned earlier can accumulate interest, providing additional funds for future subsidies. Motivated by this observation, we introduce global budget balance with interest, under which a platform starting with no funds can use both trading profits and accrued interest to finance subsidies. For a horizon of $T$ rounds and a fixed exponent $θ\ge0$, we consider the per-round interest rate $r_T = T^{-θ}$. We study two classic valuation models, adversarial and correlated i.i.d., and obtain tight bounds on regret against the best fixed common price in both. As the interest rate grows, optimal regret in each model drops from $\tildeΘ(T^{3/4})$ to $\tildeΘ(T^{5/7})$. The intermediate rates differ: $\tildeΘ(T^{3θ/4})$ for adversarial valuations when $20/21 < θ< 1$, and $\tildeΘ(T^{(2θ+3)/7})$ for correlated i.i.d. valuations when $1 < θ< 9/8$. Finally, as an extension, we study $α$-regret, with approximation factor $α\ge1$, for adversarial valuations against the best feasible distribution over price pairs, and show that the optimal competitive ratio rises from 2 to 3 as interest strengthens.