  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.
