Dynamic Pricing in an Unknown and Sales-dependent Evolving Marketplace

We consider a firm who sells a single product with finite inventory over a finite horizon via dynamic pricing. The market size is a polynomial function of cumulative historic sales. The firm does not know the coefficients in the market size function before the start of the season and must learn it over time. The firm aims at finding a pricing policy that yields as much revenue as possible. We show that the firm's revenue is upper bounded by her optimal revenue in a setting that she perfectly knew all coefficients in the market size function ex ante and the system is deterministic (fluid model). For this fluid model, we show that by replacing prices with sales quantities as the decision variables, the problem becomes a convex program that can be efficiently solved. We propose a maximum likelihood estimate - reoptimized (MR) policy. Under this policy, in each period, the firm performs learning and optimization jobs. In the learning job, the firm uses the maximum likelihood estimate approach to form a point estimate of unknown coefficients. In the optimization job, the firm resolves the fluid model with updated information on remaining inventory, remaining horizon and the estimate of the unknown coefficients. We establish an upper bound of the regret of our policy for the regime that the initial inventory and the length of the horizon are proportionally scaled up.