Duopoly Game Of Callable Products In Revenue Management

During the last three decades, the technology of revenue management has beenused more and more widely around the world, and played a significant role inimproving the profits of these corporations. As a result, much attention of scholars hasbeen attracted into this field. In the airline industry in which the technology of revenuemanagement is most widely used, airlines usually have to face two types of customers:low-valuation customers who accept low-fare tickets only but are willing to book inadvance, and high-valuation customers who are willing to buy expensive tickets butarrive just before the plane takes off. The airlines do not know the demand ofhigh-valuation customers in the future, which makes the airlines face a dilemma—leaving high-valuation customers unsatisfied when facing large number of arrivals ifprotecting too few tickets for them,or the plane taking off with empty seats otherwise.The concept of callableproductsis one of therecently emerging methodsin revenuemanagement to respond to demand uncertainty of the customers, which means units ofcapacity sold to self-selected low-fare customers who willingly grant theairline theoption to “call” the capacity at a pre-specified recall price. Thereby we can avoid theabove problems by using callable products.The problem with relation to a single airline monopolistic in the market hasbeenwell studied in the literature.In practice, however,the airlinesusually have to faceother competitors. In this article,the concept of callable products is introduced into themodel of a duopoly game to explore the problem of capacity allocation in a competitiveenvironment. We provide a sufficient condition which is more general than those in theliterature about callable products, to prove theexistence and uniqueness of thepure-strategy Nash equilibrium. And the analysis on the equilibrium shows that: Anincrease inthe compensation (recall price) of one airline leads to a decrease inits ownbooking limit and an increase in its competitor’s.But if both the recall pricesincrease,their booking limits reduce at the same time. The ticket price of low-fare classhas theopposite impact, and the presence of the callable products makes the ticket price ofhigh-fareclass have little impact on the airlines’ decisions.Same as the situationwithoutcallable products, competition makes the overall revenue of the system decreasewhen selling callable products. The airlines selling callable products under competition willalways choose higher booking limitsthan under monopoly, unlike the situation withoutcallable products under which the booking limits may be higheror lower. It means amore intensive competitionfor the low-fare class and an improvement in the welfare ofthe consumers. We also show that selling callable products will improve one airline’srevenue.Thus the airline will choose this mode no matter whether its competitor sells ornot.And compared with the situation that neither of them sells callable products, it willimprove the revenue of both of the airlines whenthey both choose this mode.The major contributions of this paper are as follows:(1) Weextend the callablemechanism to the competitive environment, and prove the existence and uniqueness ofthe pure-strategy Nash equilibriumunder a condition that allowsthe demands aredependent.(2) We discuss the properties of the callable products under competition.Westudy the impact of recall price on the decision of both sides and compare the revenue inthe situations with and without callable mechanism.(3) With callable mechanism, wecomparethe booking limits with and without competition.