Statistical Methods For Optimal Pricing Under Random Demand With Unknown Distribution

In this thesis, we study a dynamic learning and pricing problem faced by a decision maker who observes an uncertain demand with unknown distribution. The objective of the decision maker is to find the optimal price that maximizes the revenue rate with lowest cost and minimal time interval. At each price point, the decision maker collects some amount of sales data and compares it with historical data using statistical methods, and then calculates the next price on the fly. The dynamic pricing algorithms we developed aims to find simple and efficient price adjustment rules so that decision makers can find the optimal price quickly and accurately in the process of learning the market response.We develop two dynamic pricing algorithms for the pricing problem with unknown demand distribution - Hill-Climbing Algorithm and Bi-section Methods. Both are based on the theory of hypothesis testing in statistics. We use T-test on random samples collected under different prices to decide which price is better and then calculate the next price that may be closer to the optimal price based on current data.We also propose a stop condition which allows the decision-maker to control how close to the optimal solution the terminating result is within certain accuracy. Specifically, the stop condition guarantees that the revenue rate at the stopping price will be within a user-defined range around the true optimal revenue rate with a pre-specified statistical confidence. To the best of our knowledge, the proposal algorithms and stopping condition for the first time provide decision makers a statistical control on the quality of the terminating solution, which is generally lacking in other dynamic pricing algorithms such as stochastic approximations (gradient-based approach or Newton-Raphson approach) or the simple learning algorithm proposed by Besbes and Zeevi(2009) .We conduct numerical analysis for comparing our algorithms with aforementioned existing algorithms. The main indicators for measuring algorithm performance are the probability of finding the true optimal price and the revenue rate, etc. Based on our comparison, the Bi-section Method is better than other algorithms in these indicators.