  We study anonymous posted price mechanisms for combinatorial auctions in a
Bayesian framework. In a posted price mechanism, item prices are posted, then
the consumers approach the seller sequentially in an arbitrary order, each
purchasing her favorite bundle from among the unsold items at the posted
prices. These mechanisms are simple, transparent and trivially dominant
strategy incentive compatible (DSIC).
  We show that when agent preferences are fractionally subadditive (which
includes all submodular functions), there always exist prices that, in
expectation, obtain at least half of the optimal welfare. Our result is
constructive: given black-box access to a combinatorial auction algorithm A,
sample access to the prior distribution, and appropriate query access to the
sampled valuations, one can compute, in polytime, prices that guarantee at
least half of the expected welfare of A. As a corollary, we obtain the first
polytime (in n and m) constant-factor DSIC mechanism for Bayesian submodular
combinatorial auctions, given access to demand query oracles. Our results also
extend to valuations with complements, where the approximation factor degrades
linearly with the level of complementarity.
