  In online sales, sellers usually offer each potential buyer a posted price in
a take-it-or-leave fashion. Buyers can sometimes see posted prices faced by
other buyers, and changing the price frequently could be considered unfair. The
literature on posted price mechanisms and prophet inequality problems has
studied the two extremes of pricing policies, the fixed price policy and fully
dynamic pricing. The former is suboptimal in revenue but is perceived as fairer
than the latter. This work examines the middle situation, where there are at
most $k$ distinct prices over the selling horizon. Using the framework of
prophet inequalities with independent and identically distributed random
variables, we propose a new prophet inequality for strategies that use at most
$k$ thresholds. We present asymptotic results in $k$ and results for small
values of $k$. For $k=2$ prices, we show an improvement of at least $11\%$ over
the best fixed-price solution. Moreover, $k=5$ prices suffice to guarantee
almost $99\%$ of the approximation factor obtained by a fully dynamic policy
that uses an arbitrary number of prices. From a technical standpoint, we use an
infinite-dimensional linear program in our analysis; this formulation could be
of independent interest to other online selection problems.
