Efficiency-Revenue Trade-offs in Auctions

TL;DR

This paper explores the computational complexity of designing deterministic auctions that balance revenue and social welfare, showing NP-hardness in general but providing approximation algorithms for two bidders.

Contribution

It proves NP-hardness of deterministic mechanisms achieving optimal trade-offs and offers polynomial-time approximation algorithms for two bidders with correlated valuations.

Findings

NP-hardness of deterministic trade-off mechanisms for multiple bidders.

Polynomial-time approximation algorithms for two bidders with arbitrary correlations.

Optimal trade-offs can be approximated arbitrarily closely in polynomial time for two bidders.

Abstract

When agents with independent priors bid for a single item, Myerson's optimal auction maximizes expected revenue, whereas Vickrey's second-price auction optimizes social welfare. We address the natural question of trade-offs between the two criteria, that is, auctions that optimize, say, revenue under the constraint that the welfare is above a given level. If one allows for randomized mechanisms, it is easy to see that there are polynomial-time mechanisms that achieve any point in the trade-off (the Pareto curve) between revenue and welfare. We investigate whether one can achieve the same guarantees using deterministic mechanisms. We provide a negative answer to this question by showing that this is a (weakly) NP-hard problem. On the positive side, we provide polynomial-time deterministic mechanisms that approximate with arbitrary precision any point of the trade-off between these two fundamental objectives for the case of two bidders, even when the valuations are correlated arbitrarily. The major problem left open by our work is whether there is such an algorithm for three or more bidders with independent valuation distributions.