Price Optimization for a Multi-Stage Choice Model

Considering the real-world situations where past purchases could influence future prices, this research examines the multi-product price optimization problem under a multi-stage choice model. Particularly, the seller commits to a multi-stage pricing policy and determines product prices based on the customer's purchase history, and the customer makes purchase decisions such that the expected utility is maximized. We show that the pricing problem has a unique optimal solution under some mild conditions and the optimal solution satisfies a modified equal adjusted markup property. Based on the property, the problem can be solved efficiently by reducing it to a single-dimensional search problem. Moreover, the optimal pricing policy has an important property, namely, the product with a higher adjusted markup in earlier stages should always lead to lower prices in subsequent stages. We also show that compared to customers that are myopic, the seller should offer higher first-stage prices and lower second-stage prices to forward-looking customers, which will lead to a higher profit. Numerical analyses are also conducted to demonstrate the above results