  We consider the problem of a firm seeking to use personalized pricing to sell
an exogenously given stock of a product over a finite selling horizon to
different consumer types. We assume that the type of an arriving consumer can
be observed but the demand function associated with each type is initially
unknown. The firm sets personalized prices dynamically for each type and
attempts to maximize the revenue over the season. We provide a learning
algorithm that is near-optimal when the demand and capacity scale in
proportion. The algorithm utilizes the primal-dual formulation of the problem
and learns the dual optimal solution explicitly. It allows the algorithm to
overcome the curse of dimensionality (the rate of regret is independent of the
number of types) and sheds light on novel algorithmic designs for learning
problems with resource constraints.
