Optimal Pricing in Repeated Posted-Price Auctions

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.