Dynamic Pricing with Multiple Products and Partially Specified Demand Distribution

We study a dynamic pricing problem with multiple products and infinite inventories. The demand for these products depends on the selling prices and on parameters unknown to the seller. Their value can be learned from accumulating sales data using statistical estimation techniques. The quality of the parameter estimates is influenced by the amount of price dispersion; however, a large amount of variation in the selling prices can be costly since it means that suboptimal prices are used. The seller thus needs to balance optimizing the quality of the parameter estimates and optimizing instant revenue, i.e., exploitation and exploration. In this study we propose a pricing policy for this dynamic pricing problem. The key idea is to use at each time period the price that is optimal with respect to current parameter estimates, with an additional constraint that ensures sufficient price dispersion. We measure the price dispersion by the smallest eigenvalue of the design matrix and show how a desired growth rate of this eigenvalue can be achieved by a simple quadratic constraint in the price-optimization problem. We study the performance of our pricing policy by providing bounds on the regret, which measures the expected revenue loss caused by using suboptimal prices.