  We study the revenue performance of sequential posted price mechanisms and
some natural extensions, for a general setting where the valuations of the
buyers are drawn from a correlated distribution. Sequential posted price
mechanisms are conceptually simple mechanisms that work by proposing a
take-it-or-leave-it offer to each buyer. We apply sequential posted price
mechanisms to single-parameter multi-unit settings in which each buyer demands
only one item and the mechanism can assign the service to at most k of the
buyers. For standard sequential posted price mechanisms, we prove that with the
valuation distribution having finite support, no sequential posted price
mechanism can extract a constant fraction of the optimal expected revenue, even
with unlimited supply. We extend this result to the the case of a continuous
valuation distribution when various standard assumptions hold simultaneously.
In fact, it turns out that the best fraction of the optimal revenue that is
extractable by a sequential posted price mechanism is proportional to ratio of
the highest and lowest possible valuation. We prove that for two simple
generalizations of these mechanisms, a better revenue performance can be
achieved: if the sequential posted price mechanism has for each buyer the
option of either proposing an offer or asking the buyer for its valuation, then
a Omega(1/max{1,d}) fraction of the optimal revenue can be extracted, where d
denotes the degree of dependence of the valuations, ranging from complete
independence (d=0) to arbitrary dependence (d=n-1). Moreover, when we
generalize the sequential posted price mechanisms further, such that the
mechanism has the ability to make a take-it-or-leave-it offer to the i-th buyer
that depends on the valuations of all buyers except i's, we prove that a
constant fraction (2-sqrt{e})/4~0.088 of the optimal revenue can be always be
extracted.
