Pricing Problems under the Markov Chain Choice Model

Abstract

We consider pricing problems when customers choose under the Markov chain choice model. In this choice model, a customer arriving into the system is interested in a certain product with a certain probability. Depending on the price charged for this product, the customer decides whether to purchase the product. If the customer purchases the product, then she leaves the system. Otherwise, the customer transitions to another product or to the no purchase option with certain transition probabilities. In this way, the customer transitions between the products until she purchases a product or reaches the no purchase option. We study three fundamental pricing problems under this choice model. First, for the monopolistic pricing problem, we show how to compute the optimal prices efficiently. Second, for the competitive pricing problem, we show that a Nash equilibrium exists, prove that Nash equilibrium prices are no larger than the prices computed by a central planner controlling all prices and characterize a Nash equilibrium that Pareto dominates all other Nash equilibria. Third, for the dynamic pricing problem with a single resource, we show that the optimal prices decrease as we have more resource capacity or as we get closer to the end of the selling horizon. We also consider a deterministic approximation formulated under the assumption that the demand for each product takes on its expected value. Although the objective function and constraints in this approximation do not have explicit expressions, we develop an equivalent reformulation with explicit expressions for the objective function and constraints.