  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.
