On the use of buy up as a model of customer choice in revenue management

We consider settings in which a revenue manager controls bookings over a sequence of flights. The revenue manager uses a buy-up model to select booking limits and updates estimates of the model parameters as data are accumulated. The buy-up model we consider is based upon a simple model of customer choice, wherein each low-fare customer who is not able to purchase a low-fare ticket will, with a fixed probability, "buy up" to the high fare, independent of everything else. We analyze the evolution of the parameter estimates (e.g., the buy-up probability) and chosen booking limits in situations where the buy-up model is misspecified, that is, in situations where there is no setting of its parameters for which its objective function gives an accurate representation of expected revenue as a function of the booking limit. The analysis is motivated by the common situation in which a revenue manager does not know precisely how customers behave but nevertheless uses a parametric model to make decisions. Under some assumptions, we prove that the booking limits and parameter estimates converge and we compare the actual expected revenue at the limiting values with that associated with the booking limits that would be chosen if the revenue manager knew the actual behavior of customers. The analysis shows that the buy-up model often works reasonably well even when it is misspecified, and also reveals the importance of understanding how parameter estimates of misspecified models vary as functions of decisions.