Non-cooperative competition among revenue maximizing service providers with demand learning Changhyun Kwon a , Terry L. Friesz a, * , Reetabrata Mookherjee b , Tao Yao a , Baichun Feng a a Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, University Park, PA 16802, USA b Zilliant Inc., 3815 S Capital of Texas Hwy, Suite 300, Austin, TX 78704, USA Received 11 March 2006; accepted 8 December 2007 Abstract This paper recognizes that in many decision environments in which revenue optimization is attempted, an actual demand curve and its parameters are generally unobservable. Herein, we describe the dynamics of demand as a continuous time differential equation based on an evolutionary game theory perspective. We then observe realized sales data to obtain estimates of parameters that govern the evolution of demand; these are refined on a discrete time scale. The resulting model takes the form of a differential variational inequality. We pres- ent an algorithm based on a gap function for the differential variational inequality and report its numerical performance for an example revenue optimization problem. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Revenue management; Pricing; Demand learning; Differential games; Kalman filters 1. Introduction In the rapidly growing literature on revenue manage- ment – see McGill and van Ryzin (1999) and Talluri and van Ryzin (2004) for comprehensive studies and a survey – one of the most important issues is how to model service provider demand learning. Demand is usually represented as a function of price, explicitly and/or implicitly, and the root tactic upon which revenue management is based is to change prices dynamically to maximize immediate or short-run revenue. In this sense, the more accurate the model of demand employed in revenue optimization, the more revenue we can generate. Although demand may be viewed theoretically as the result of utility maximization, an actual demand curve and its parameters are generally unobservable in most markets. In this paper, we first describe the dynamics of demand as a differential equation based on an evolutionary game theory perspective and then observe the actual sales data to obtain estimates of param- eters that govern the evolution of demand. A dynamic non-zero sum evolutionary game among ser- vice providers is expressed, in this paper, as a differential variational inequality. The providers also have fixed upper bounds on output as each faces capacity constraints on available resources. We intend to provide a numerical example of the revenue management model we will intro- duce later in this paper along with an efficient algorithm. 1.1. Revenue management model The service providers of interest are in oligopolistic game-theoretic competition according to a learning process that is similar to evolutionary game-theoretic dynamics and for which price changes are proportional to their signed excursion from a market clearing price. We stress that in this model firms are setting prices for their services while simultaneously determining the levels of demand they will serve. This is unusual in that, typically, firms in 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.12.041 * Corresponding author. Tel.: +1 814 863 2445; fax: +1 814 863 4745. E-mail addresses: chkwon@psu.edu (C. Kwon), tfriesz@psu.edu (T.L. Friesz), Reetabrata.Mookherjee@zilliant.com (R. Mookherjee), taoyao@psu.edu (T. Yao), buf118@psu.edu (B. Feng). www.elsevier.com/locate/ejor Available online at www.sciencedirect.com European Journal of Operational Research xxx (2008) xxx–xxx ARTICLE IN PRESS Please cite this article in press as: Kwon, C. et al., Non-cooperative competition among revenue maximizing ..., European Journal of Operational Research (2008), doi:10.1016/j.ejor.2007.12.041