Pricing under the Generalized Extreme Value Models with Homogeneous Price Sensitivity Parameters

We consider pricing problems when customers choose according to the generalized extreme value (GEV) models and the products have the same price sensitivity parameter. First, we consider the static pricing problem, where we maximize the expected profit obtained from each customer. We show that the optimal prices of the different products have a constant markup over their unit costs. We provide an explicit formula for the optimal markup in terms of the Lambert-W function. This result holds for any arbitrary GEV model. Second, we consider single-resource dynamic pricing problems. We show that as we have more resource inventory or as we have fewer time periods left until the end of the selling horizon, the prices charged by the optimal policy decrease. Third, we consider dynamic pricing problems over a network of resources. We focus on a price-based deterministic approximation with prices as the decision variables, but this deterministic approximation fails to be a convex program. We transform the price-based deterministic approximation to an equivalent market-share-based deterministic approximation, with purchase probabilities as the decision variables. Surprisingly, the transformed problem is a convex program, and the gradient of its objective function can be computed efficiently. Computational experiments show that the market-share-based formulation provides substantial advantages over the original price-based formulation.