Relative Entropy, Exponential Utility, and Robust Dynamic Pricing

In the area of dynamic revenue management, optimal pricing policies are typically computed on the basis of an underlying demand rate model. From the perspective of applications, this approach implicitly assumes that the model is an accurate representation of the real world demand process and that the parameters characterizing this model can be accurately calibrated using data. In many situations, neither of these conditions are satisfied. Indeed, models are usually simplified for the purposes of tractability and may be difficult to calibrate because of a lack of data. Moreover, pricing policies that are computed under the assumption that the model is correct may perform badly when this is not the case. This paper presents an approach to single product dynamic revenue management that accounts for errors in the underlying model at the optimization stage. Uncertainty in the demand rate model is represented using the notion of relative entropy and a tractable reformulation of the ‘robust pricing problem’ is obtained using results concerning the change of probability measure for point processes. The optimal pricing policy is obtained through a version of the so-called Isaacs’ equation for stochastic differential games, and the structural properties of the optimal solution are obtained through an analysis of this equation. In particular, (i) closed form solutions for the special case of an exponential nominal demand rate model, (ii) general conditions for the exchange of the ‘max’ and the ‘min’ in the differential game, and (iii) the equivalence between the robust pricing problem and that of single product revenue management with an exponential utility function without model uncertainty, are established through the analysis of this equation. Key words– robust control, model uncertainty, ambiguity, rate control, revenue management, dynamic pricing, Girsanov Theorem, stochastic differential games, relative entropy, exponential utility, risk-sensitive control.