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# Models of the Spiral-Down Effect in Revenue Management

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OPERATIONS RESEARCHVol. 00, No. 0, Xxxxx 0000, pp. 000–000issn 0030-364X |eissn 1526-5463 | 00 | 0000 | 0001INFORMSdoi 10.1287/xxxx.0000.0000c°0000 INFORMSModels of the Spiral-Down Effectin Revenue ManagementWilliam L. CooperDepartment of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, [email protected] Homem-de-MelloDepartment of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL 60208,[email protected] J. KleywegtSchool of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205,[email protected] spiral-down effect occurs when incorrect assumptions about customer behavior cause high-fare ticketsales, protection levels, and revenues to systematically decrease over time. If an airline decides how manyseats to protect for sale at a high fare based on past high-fare sales, while neglecting to account for the factthat availability of low-fare tickets will reduce high-fare sales, then high-fare sales will decrease, resultingin lower future estimates of high-fare demand. This subsequently yields lower protection levels for high-faretickets, greater availability of low-fare tickets, and even lower high-fare ticket sales. The pattern continues,resulting in a so-called spiral down. We develop a mathematical framework to analyze the process by whichairlines forecast demand and optimize booking controls over a sequence of flights. Within the framework, wegive conditions under which spiral down occurs.Subject classifications : Pricing: revenue management. Forecasting: estimation and control. Probability:applicationsArea of review : Manufacturing, Service, and Supply Chain OperationsHistory :1. IntroductionRevenue management involves the application of quantitative techniques to improve profits by con-trolling the prices and availabilities of various products that are produced with scarce resources.The best known revenue management application occurs in the airline industry, where the prod-ucts are tickets (for itineraries) and the resources are seats on flights. Over the past decade bothpractitioners and academics have helped to develop a considerable and rapidly growing literatureon revenue management. Much of this work is reviewed in Talluri and van Ryzin (2004b), Bitranand Caldentey (2003), and Boyd and Bilegan (2003).In almost every instance of published work, the starting point of the analysis is some setof assumptions regarding an underlying stochastic or deterministic demand process. With theseassumptions in hand (and assumed to be correct), most papers proceed to analyze the model andderive policies that are good or optimal for the formulated model. In the airline context, such apolicy usually prescribes which types of tickets should be available at which times, and under whichcircumstances.However, the situation faced by revenue managers in practice is different in at least two keyregards: assumptions may be incorrect, and model parameters are not known. There are a varietyof reasons why a revenue manager may use a model with incorrect assumptions. Among theseare (a) availability of intuitively-pleasing decision rules — such as the Littlewood rule consideredherein, (b) simplification for analytical tractability, (c) availability of forecasting and optimization1Cooper, Homem-de-Mello, and Kleywegt: Spiral-Down Effect in Revenue Management2 Operations Research 00(0), pp. 000–000,c°0000 INFORMSsoftware, and (d) lack of understanding of the problem. Moreover, a revenue manager may be awareof a modeling error, but may not fully comprehend its consequences. We are specifically interestedin the consequences of using incorrect models, especially if the parameters of such models areestimated with available data. Even if the data are good (say correctly untruncated demand data)and a good forecasting method is used, the problem remains that parameters are being estimatedfor an inappropriate model, and consequently there often do not exist parameter values that willmake the revenue manager’s model correct.In revenue management practice, there is a pro cess whereby controls (e.g., protection levels) areenacted, sales occur, flights depart, new data are observed, and parameter estimates are updated.The updated estimates are then used to choose new controls for the next set of flights, and so on.An important question is what can happen in such a forecasting and optimization process if therevenue manager uses a good forecasting method, but the chosen controls are based on erroneousassumptions.As an example, suppose that there are two classes of tickets and that customers are flexible, thatis, they are willing to buy either low-fare or high-fare tickets, but they will buy the low-fare ticketsif both are available. Suppose also that the airline chooses how many seats to reserve for high-faretickets (i.e., the protection level) based on past sales of high-fare tickets, while neglecting to accountfor the fact that availability of low-fare tickets will reduce sales for high-fare tickets. Then, if morelow-fare tickets are made available, low-fare sales will increase and high-fare sales will decrease,resulting in lower future estimates of high-fare demand, and subsequently lower protection levelsfor high-fare tickets and greater availability of low-fare tickets. The pattern continues, resultingin a downward spiral of high-fare sales, protection levels, and revenues. It is of concern that theflawed model produces suboptimal controls (which is no surprise), but of even greater concern is thephenomenon that the controls can become systematically worse as the forecasting and optimizationprocess continues. Boyd et al. (2001) have used simulation to demonstrate this spiral-down effect,which is known to some practitioners. However, to our knowledge, this phenomenon has not beenstudied in the literature, although Kuhlmann (2004) alluded to the underlying issue with hisremark that “although airlines had spent considerable sums making forecasting, allocation, andother elements of revenue management more precise, they failed to deal with some of the inherentflawed assumptions of revenue management. For instance, if a carrier sold 50 B-class passengerson any given day, that was then established as the historical demand for B class, ignoring the factthat the absence of availability of other classes might have skewed the result.”In this paper we introduce a generic framework for the study

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