Robust Multi-product Pricing under General Extreme Value Models

We study robust versions of pricing problems where customers choose products according to a general extreme value (GEV) choice model, and the choice parameters are not given exactly but lie in an uncertainty set. We show that, when the robust problem is unconstrained and the price sensitivity parameters are homogeneous, the robust optimal prices have a constant markup over products and we provide formulas that allow to compute this constant markup by binary search. We also show that, in the case that the price sensitivity parameters are only homogeneous in each subset of the products and the uncertainty set is rectangular, the robust problem can be converted into a deterministic pricing problem and the robust optimal prices have a constant markup in each subset, and we also provide explicit formulas to compute them. For constrained pricing problems, we propose a formulation where, instead of requiring that the expected sale constraints be satisfied, we add a penalty cost to the objective function for violated constraints. We then show that the robust pricing problem with over-expected-sale penalties can be reformulated as a convex optimization program where the purchase probabilities are the decision variables. We provide numerical results for the logit and nested logit model to illustrate the advantages of our approach. Our results generally hold for any arbitrary GEV model, including the multinomial logit, nested or cross-nested logit.