  We study revenue optimization pricing algorithms for repeated posted-price
auctions where a seller interacts with a single strategic buyer that holds a
fixed private valuation. We show that, in the case when both the seller and the
buyer have the same discounting in their cumulative utilities (revenue and
surplus), there exist two optimal algorithms. The first one constantly offers
the Myerson price, while the second pricing proposes a "big deal": pay for all
goods in advance (at the first round) or get nothing. However, when there is an
imbalance between the seller and the buyer in the patience to wait for utility,
we find that the constant pricing, surprisingly, is no longer optimal. First,
it is outperformed by the pricing algorithm "big deal", when the seller's
discount rate is lower than the one of the buyer. Second, in the inverse case
of a less patient buyer, we reduce the problem of finding an optimal algorithm
to a multidimensional optimization problem (a multivariate analogue of the
functional used to determine Myerson's price) that does not admit a closed form
solution in general, but can be solved by numerical optimization techniques
(e.g., gradient ones). We provide extensive analysis of numerically found
optimal algorithms to demonstrate that they are non-trivial, may be
non-consistent, and generate larger expected revenue than the constant pricing
with the Myerson price.
