Robust Multi-Product Pricing

Abstract:     
This paper concerns dynamic pricing of multiple perishable products when there is model uncertainty, which we formulate as a worst-case stochastic intensity control problem where ambiguity is modeled using the notion of relative entropy. One feature of our formulation is that the demand models for different products can have different levels of ambiguity, a situation that arises (for instance) if a new product is being sold along side an established one. We show that this multiple-ambiguity multi-product robust pricing problem is equivalent to another (non-standard) risk-sensitive pricing problem, and show that it can be decentralized under additional assumptions on the demand rate model. The risk-sensitive problem has several unusual features: (i) the net income from sales of each product is valued by its certainty equivalent under an exponential utility function where the aversion parameter is determined by the level of ambiguity of its demand model, (ii) the overall goal is to maximize the sum of the certainty equivalents over all products, and (iii) products making sales are required to compensate other products for the use of common
resources according to a revenue sharing rule. We characterize the revenue sharing rule which leads to an equivalence between the risk-sensitive problem we have just described and the original robust pricing problem. This generalizes risk-sensitive/robust control duality to the case where different components of the model have different levels of model uncertainty. Finally, we show that the robust multi-product problem can be decentralized and solved in terms of modified robust/risk-sensitive single-product problems, if the demand rate functions satisfy certain independence assumptions. The modification of the single product problems involves the introduction of a cost to account for the value of inventory that is used at each sale. This cost is closely related to the revenue sharing rule associated with the robust/risk-sensitive control equivalence.

Keywords: multi-product pricing, dynamic pricing, revenue management, model ambiguity, model uncertainty, multiple levels of model uncertainty, relative entropy, revenue sharing, risk-sensitive control, robust control, intensity control, decentralization

JEL Classifications: C44, C61, C73, D81

Working Paper Series