Optimal Auctions with Correlated Bidders are Easy

TL;DR

This paper presents polynomial-time algorithms for designing near-optimal revenue mechanisms in auctions with correlated bidders, demonstrating small gaps between different types of truthful mechanisms and extending results to multi-item settings.

Contribution

It introduces efficient algorithms for optimal auction design with correlated bidders and compares the performance of truthful-in-expectation and deterministic mechanisms.

Findings

Polynomial-time algorithm for optimal randomized mechanism with correlated bidders.

Deterministic truthful mechanisms achieve a 5/3 approximation to optimal revenue.

Small performance gap between truthful-in-expectation and deterministic mechanisms.

Abstract

We consider the problem of designing a revenue-maximizing auction for a single item, when the values of the bidders are drawn from a correlated distribution. We observe that there exists an algorithm that finds the optimal randomized mechanism that runs in time polynomial in the size of the support. We leverage this result to show that in the oracle model introduced by Ronen and Saberi [FOCS'02], there exists a polynomial time truthful in expectation mechanism that provides a $(\frac 3 2+\epsilon)$-approximation to the revenue achievable by an optimal truthful-in-expectation mechanism, and a polynomial time deterministic truthful mechanism that guarantees $\frac 5 3$ approximation to the revenue achievable by an optimal deterministic truthful mechanism. We show that the $\frac 5 3$-approximation mechanism provides the same approximation ratio also with respect to the optimal truthful-in-expectation mechanism. This shows that the performance gap between truthful-in-expectation and deterministic mechanisms is relatively small. En route, we solve an open question of Mehta and Vazirani [EC'04]. Finally, we extend some of our results to the multi-item case, and show how to compute the optimal truthful-in-expectation mechanisms for bidders with more complex valuations.