Estimation of Choice-Based Models Using Sales Data from a Single Firm

Choice-based revenue management (RM) problems use discrete choice models to predict product-level demands. The estimation of choice-based RM models involves solving for choice parameters as well as an arrival rate. The latter represents a measure of unconstrained demand, i.e., the population of customers who arrive and purchase a product from our firm as well as those who arrive and either decide to purchase a product from a competitor or not to purchase at all. Talluri and van Ryzin were the first to propose a parameter estimation method for this problem in 2004. However, their method, which uses ExpectationMaximization (EM) routines to solve an expected log likelihood function, exhibits prohibitively long estimation times and often leads to counter-intuitive results. We reformulate the Talluri and van Ryzin parameter estimation method using marginal (versus expected) log likelihood functions. This enables us to eliminate the use of the EM algorithm, which results in solution times that are improved by orders of magnitude. The marginal log likelihood formulation allows us to decompose the problem into two steps: one that estimates the discrete choice model parameters, and the other that estimates the arrival rate and overall market share. We discuss theoretical properties of our marginal log likelihood formulation and prove that it converges to a local (and often global) optima. The proof is based on showing how the multidimensional non-concave nonlinear objective function can be reformulated as smaller globally concave problem with an easy to find and unique solution, plus a single-dimensional nonlinear optimization problem. Simulations based on industry hotel data clearly demonstrate the superior computational performance of our decomposition method over existing methods. Our simulations also reveal that incorporating continuous (versus discrete) arrival rates into the parameter estimation method is critical in order to obtain unbiased parameter estimates.