Optimal Pricing in Repeated Posted-Price Auctions with Different Patience of the Seller and the Buyer

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. When the participants non-equally discount their cumulative utilities, we show that the optimal constant pricing (which offers the Myerson price) is no longer optimal. In the case of more patient seller, we propose a novel multidimensional optimization functional --- a generalization of the one used to determine Myerson's price. This functional allows to find the optimal algorithm and to boost revenue of the optimal static pricing by an efficient low-dimensional approximation. Numerical experiments are provided to support our results.