Stochastic Strategic Patient Buyers: Revenue maximization using posted prices

We consider a seller faced with buyers which have the ability to delay their decision, which we call patience. Each buyer's type is composed of value and patience, and it is sampled i.i.d. from a distribution. The seller, using posted prices, would like to maximize her revenue from selling to the buyer. Our main results are the following. $\bullet$ We formalize this setting and characterize the resulting Stackelberg equilibrium, where the seller first commits to her strategy and then the buyers best respond. $\bullet$ We show a separation between the best fixed price, the best pure strategy, which is a fixed sequence of prices, and the best mixed strategy, which is a distribution over price sequences. $\bullet$ We characterize both the optimal pure strategy of the seller and the buyer's best response strategy to any seller's mixed strategy. $\bullet$ We show how to compute efficiently the optimal pure strategy and give an algorithm for the optimal mixed strategy (which is exponential in the maximum patience). We then consider a learning setting, where the seller does not have access to the distribution over buyer's types. Our main results are the following. $\bullet$ We derive a sample complexity bound for the learning of an approximate optimal pure strategy, by computing the fat-shattering dimension of this setting. $\bullet$ We give a general sample complexity bound for the approximate optimal mixed strategy. $\bullet$ We consider an online setting and derive a vanishing regret bound with respect to both the optimal pure strategy and the optimal mixed strategy.