Sequential Posted Pricing and Multi-parameter Mechanism Design

We consider the classical mathematical economics problem of {\em Bayesian optimal mechanism design} where a principal aims to optimize expected revenue when allocating resources to self-interested agents with preferences drawn from a known distribution. In single-parameter settings (i.e., where each agent's preference is given by a single private value for being served and zero for not being served) this problem is solved [Myerson '81]. Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed [Ausubel and Milgrom '06], and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multi-dimensional setting (i.e., where each agent's preference is given by multiple values for each of the multiple services available) [Manelli and Vincent '07]. In contrast to the theory of optimal mechanisms we develop a theory of sequential posted price mechanisms, where agents in sequence are offered take-it-or-leave-it prices. These mechanisms are approximately optimal in single-dimensional settings, and avoid many of the properties that make optimal mechanisms impractical. Furthermore, these mechanisms generalize naturally to give the first known approximations to the elusive optimal multi-dimensional mechanism design problem. In particular, we solve multi-dimensional multi-unit auction problems and generalizations to matroid feasibility constraints. The constant approximations we obtain range from 1.5 to 8. For all but one case, our posted price sequences can be computed in polynomial time.