Complexity of Efficiency-Revenue Trade-offs in Bayesian Auctions

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, auctions that optimize revenue without losing too much welfare, say. 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 ask the question of 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 independent valuation distributions and three or more bidders.