  Posted price mechanisms (PPM) constitute one of the predominant practices to
price goods in online marketplaces and their revenue guarantees have been a
central object of study in the last decade. We consider a basic setting where
the buyers' valuations are independent and identically distributed and there is
a single unit on sale. It is well-known that this setting is equivalent to the
so-called i.i.d. prophet inequality, for which optimal guarantees are known and
evaluate to 0.745 in general (equivalent to a PPM with dynamic prices) and $1 -
1/e \approx 0.632$ in the fixed threshold case (equivalent to a fixed price
PPM). In this paper we consider an additional assumption, namely, that the
underlying market is very large. This is modeled by first fixing a valuation
distribution F and then making the number of buyers grow large, rather than
considering the worst distribution for each possible market size. In this
setting Kennedy and Kertz [Ann. Probab. 1991] breaks the 0.745 fraction
achievable in general with a dynamic threshold policy. We prove that this large
market benefit continue to hold when using fixed price PPMs, and show that the
guarantee of 0.632 actually improves to 0.712. We then move to study the case
of selling k identical units and we prove that the revenue gap of the fixed
price PPM approaches $1-1/\sqrt{ 2k\pi}$. As this bound is achievable without
the large market assumption, we obtain the somewhat surprising result that the
large market advantage vanishes as $k$ grows.
