Relative Entropy, Exponential Utility, and Robust Dynamic Pricing

Abstract

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 purpose 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.

Risk management policies for dynamic capacity control

“…The works of Perakis and Roels [15], Thiele [16] and Lim and Shanthikumar [17] are exemplary for addressing the problem of uncertainty in the demand function by robust optimisation. Lim and Shanthikumar [17] show that the robust pricing problem is equivalent to a single-product revenue management problem with an exponential utility function without model uncertainty.…”

List pricing versus dynamic pricing: Impact on the revenue risk

“…Perakis and Roels (2007) analyze robust capacity controls for network revenue management using the maximin and minimax regret criteria. Lim et al (2008) extend their relative entropy approach from the single product, see Lim and Shanthikumar (2007), to multi products cases and a proposed a risk-sensitive pricing problem involving a risk sharing rule. Mitra and Wang (2005) discuss various risk indices for risk modeling of network revenue management problems.…”

Mean-Field Games and Ambiguity Aversion

“…Ambiguity aversion has been studied in many different context, and a limited list of contributions from a stochastic control perspective includes Uppal and Wang (2003), Maenhout (2004), Hansen et al (2006), Hansen and Sargent (2007), Lim and Shanthikumar (2007), Jaimungal and Sigloch (2012), Skiadas (2013), and . Thus far, however, there has been only limited study of model uncertainty in the stochastic game context, and in particular in mean-field games (MFGs).…”