Private Learning in Bilateral Trade

2026-06-01

AI Summary

This work investigates the learnability of ε-differentially private bilateral trade mechanisms from a finite number of valuation samples within the PAC learning framework, aiming to approximate either optimal profit or social welfare. We establish the first impossibility result showing that, under arbitrary distributions, differential privacy and near-optimal performance are fundamentally incompatible. However, under the assumption of σ-smooth distributions, we demonstrate that nearly tight sample complexities become achievable. Specifically, for profit maximization, we design an α-optimal private mechanism requiring Õ(1/(σεα²)) samples; for welfare maximization, only Õ(1/(εα) + 1/α²) samples suffice, a bound that is essentially tight in α.

Abstract

Bilateral trade models one of the most fundamental economic interactions: the intermediation between two strategic agents, a seller and a buyer, willing to trade a good. We consider the learning version of the problem, where the goal is to learn a mechanism from a sampled dataset of agents' valuations to maximize either profit or economic efficiency. While known learning algorithms are characterized by high sensitivity to the input dataset, we specifically study this problem through the lens of differential privacy, ensuring that each data point does not significantly affect the probability of learning any specific mechanism. For our results, we adopt the PAC-learning framework: with high probability, the learning algorithm should output a mechanism that is at most an additive $α$ away from optimal, in a $\varepsilon$-differentially private way. As a first result, we show that differential privacy and (near)-optimality are not achievable for general distributions. Surprisingly, assuming that the distribution underlying the agents' valuations is $σ$-smooth, we recover nearly optimal sample-complexity bounds for both economic efficiency and profit. For profit, we show how to construct in polynomial time an $α$-optimal and $\varepsilon$-differentially private mechanism using $\tildeΘ(\frac{1}{σ\varepsilonα^2})$ samples. For efficiency, measured by the gain from trade, we achieve the same result using $\tildeΘ(\frac{1}{\varepsilonα}+\frac{1}{α^2})$ samples. Notably, these bounds are essentially tight in the precision parameter $α$, since achieving $α$-optimality (ignoring differential privacy) requires at least $\frac{1}{α^2}$ samples.

Problem

• bilateral trade
• differential privacy
• mechanism learning
• economic efficiency
• profit maximization

Innovation

• differential privacy
• bilateral trade
• PAC learning
• smooth distributions
• sample complexity