Duopoly game of callable products in airline revenue management European Journal of Operational Research

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European Journal of Operational Research

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Decision Support

Duopoly game of callable products in airline revenue management

Tingting Li

^a

, Jinxing Xie

^b,∗

, Shengmin Lu

^b

, Jiafu Tang

^a

aSchool of Management Science and Engineering, Dongbei University of Finance and Economics, Dalian 116025, China

bDepartment of Mathematical Sciences, Tsinghua University, Beijing 10 0 084, China

a rt i c l e i n f o

Article history:

Received 13 February 2015 Accepted 25 April 2016 Available online 3 May 2016 Keywords:

Revenue management Allocation game Demand uncertainty Callable products Duopoly

a b s t r a c t

Thispaperstudiesthecapacityallocationgamebetweenduopolisticairlineswhichcouldoffercallable products.Previousliteraturehasshownthatcallableproductsprovidearisklesssourceofadditionalrev- enueforamonopolisticairline.Weexaminetheimpactoftheintroductionofcallableproductsonthe revenuesand thebooking limits ofduopolisticairlines. Theanalyticalresults demonstratethat,when thereisnospilloflow-farecustomers,offeringcallableproductsisadominantstrategyofbothairlines and providesPareto gainstobothairlines.Whencustomersofbothfareclassesspill, offeringcallable productsisnolongeradominant strategyand mayharm the revenuesoftheairlines. Numericalex- amplesdemonstratethatwhetherthetwoairlinesoffercallableproductsandwhetherofferingcallable productsisbeneficialtothetwoairlinesmainlydependsontheirloadsandcapacities.Specifically,when thedifferencebetweentheloadsoftheairlinesislarge,theloadsoftheairlinesplaythemostimportant role.Whenthedifferencebetweentheloadsoftheairlinesissmall,thecapacitiesoftheairlinesplaythe mostimportantrole.Moreover,numericalexamplesshowthatthebookinglimitsofthetwoairlinesin thecasewithcallableproductsarealwayshigherthanthoseinthecasewithoutcallableproducts.

© 2016ElsevierB.V.Allrightsreserved.

1. Introduction

During the last three decades, the technology of revenue management has been used more and more widely around the world, and has playeda significant role in improvingthe profits ofcorporations.Asaresult,thisfieldhasattractedmuchattention from scholars (e.g., Steinhardt & Gönsch, 2012; Hu, Caldentey, &

Vulcano, 2013; Otero& Akhavan-Tabatabaei, 2015). In the airline industryinwhichthetechnologyofrevenuemanagementismost widelyused,airlinesusuallyhavetofacetwotypesofcustomers:

low-valuationcustomerswho acceptlow-fare ticketsonlybutare willing to book in advance and high-valuation customers who are willing to buy expensive tickets but arrive just before the plane takes off.Usually, the airlines cannot forecast the demand from high-valuation customers with certainty or convince them to book earlier than the low-valuation customers. Thus, “despite heavy investmentin sophisticated revenue management systems, airlines losemillions ofdollars a yearin potential revenue; both when low-fare bookings displace higher than expected high-fare bookings(‘cannibalization’)andwhenairlinesflyemptyseatspro- tected forhigh-fare bookingsthat do not materialize(‘spoilage’)”

∗ Corresponding author. Tel.: +86 10 6278 7812; fax: +86 10 6277 3400.

E-mail address: xiejinxing@tsinghua.org.cn , jxie@math.tsinghua.edu.cn (J. Xie).

(Gallego, Kou, & Phillips, 2008). Many kinds of mechanisms are proposed to hedge against demand uncertainty from high- valuation customers, e.g., overbooking (Aydın, Birbil, Frenk, &

Noyan, 2012; Karaesmen & Van Ryzin, 2004), last-minute dis- counts (Ovchinnikov & Milner, 2012), flexible products (Gallego

& Phillips, 2004), etc. All these mechanisms have shortcomings:

overbooking adds operational complexity to management; last- minutediscounts may induce the customers to wait rather than to book early; flexible products require that the customers are indifferentamongthealternativeflights.Toavoidtheaboveshort- comings, Gallego et al. (2008) proposed the concept of “callable products”, which refers to units of capacity sold to self-selected low-fare customers who willingly grant the airline the option to

“call” the capacity ata pre-specified recall price. The concept of callableproductsdoesnotaddoperationalcomplexityandcanbe usedtogetherwithothermechanisms.

Gallego etal. (2008) showed that callable products providea riskless source of additional revenue for a monopolistic airline.

In practice, airlines usually have to face other competitors. Seat allocation among different fare classes by one airline affects the demand andthe optimal seat allocation of other airlines. There- fore,thereareseveralquestionstobeaddressed.Inacompetitive environment, does offering callable products still provide a risk- lesssourceofadditional revenues?Howdoesthe introductionof callableproductsaffectthecapacityallocationdecisionsoftheair- lines? What isthe orderrelationship betweenthebooking limits http://dx.doi.org/10.1016/j.ejor.2016.04.054

0377-2217/© 2016 Elsevier B.V. All rights reserved.

undercompetitionandthoseundermonopoly?Thispaperaimsto answerthesequestions.

This paper studies the capacity allocation game between duopolisticairlineswhichcouldoffercallableproducts.Weexam- inehow theintroductionof callableproducts affects thebooking limits and the revenues of duopolistic airlines. It is shown that whenthe low-fare customersdo not spill,offering callable prod- uctsis a dominant strategy of both airlinesand provides Pareto gainstoboth(In thispaper,theword“spill” meansthat ifeither typeofcustomercannot besatisfiedbyoneairline,thecustomers gototheotherairlineandcanberecapturedbytheotherairline).

Whencustomers ofboth fareclasses spill,offering callable prod- uctsisnolongeradominantstrategyandmayharmtherevenues oftheairlines.Numericalexamplesdemonstratethatwhetherthe two airlinesoffer callableproducts andwhetheroffering callable productsisbeneficialtothetwo airlinesmainlydependsontheir loadsandcapacities(the loadofan airlineis theratioofthe av- eragetotal demand of the airline to the capacity of the airline).

Specifically,whenthedifferencebetweentheloadsoftheairlines is large, the loads of the airlines play the most important role.

Whenthedifferencebetweentheloadsoftheairlinesissmall,the capacitiesofthe airlinesplay themostimportant role.Moreover, numerical examples demonstrate that the booking limits of the twoairlines inthe casewithcallable products are always higher thanthoseinthecasewithoutcallableproducts.

The rest of the paper is organized as follows. Section 2 re- viewstheliteratureoncallableproducts andonrevenuemanage- ment game. Section 3 describes the key elements of the model.

Section 4 presents a comprehensive model analysis. Specifically, Section4.1givessufficientconditionsfortheexistenceandunique- nessoftheNashequilibrium;Section 4.2examinestheimpact of callableproducts ontherevenuesoftheairlineswhenthereisno low-farespill;Section4.3comparesthebookinglimitsundercom- petitionwiththoseundermonopoly;Section4.4conductsasensi- tivityanalysisofthebooking limitswithrespectto thepricepa- rameters.InSection5,werunnumericalexamplestoexaminethe impactofofferingcallableproductsontherevenuesandthebook- inglimitsof thetwo airlines, whereboth thelow-fare andhigh- farecustomersspill.Section6concludesthepaperandpointsout directionsforfutureresearch.

2. Literaturereview

Two streams of literature are related to our study: one is callableproducts,andtheotherisrevenuemanagementgame.

Manyformsofcallableproductshavebeenusedinvariousin- dustries.Some companies usean optionnamed “callback” to re- callpreviouslycommittedadvertisementtime bypayinga prede- terminedamount.Thecallableconcept isalsousedby Caterpillar toreduce theinventory riskof itsdealers(Sheffi,2005,pp. 229–

231).Biyalogorsky,Carmon, Fruchter,andGerstner(1999)showed thattheuseofoversellingwithopportunisticcancellationscanin- creaseexpectedprofitsinanairlinecontext.BiyalogorskyandGer- stner(2004)demonstratedthatcontingentpricingcanbeusedfor sellers in response to demand uncertainty. In contingent pricing arrangements,priceis contingent onwhetherthe sellersucceeds inobtaining ahigher pricewithin aspecified period.Itis shown thatcontingentpricingisprofitableregardlessofbuyers’riskatti- tudes,andthatcontingent pricingbenefits buyersaswell assell- ers.Gallegoetal.(2008)differedfromandextendedBiyalogorsky andGerstner(2004)inthefollowingways.First,Biyalogorskyand Gerstner(2004) considered salesofa singleunit of capacityand Gallego et al. (2008) extended the analysis to sales of multiple units.Second,BiyalogorskyandGerstner(2004)assumedcommon willingness-to-payamongbuyers,whereasGallegoetal.(2008)as- sumedthatdemandforcallableproductsisuncertainanddepends

ontherecallprice.Gallegoetal.(2008)showedthatcallableprod- uctsprovidearisklesssourceofadditionalrevenuetoamonopo- listicairline. Biyalogorsky(2009) considereda modelwithstrate- gic consumers who can decide when to show up in the market andinvestigatedwhether,inthefaceofstrategicbehaviorbycon- sumers, it can be profitable forsellers to use contingent pricing toinducethelow-higharrival patterntypicalintheairlineindus- try.Elmaghraby,Lippman,Tang,andYin(2009)examinedasitua- tionin whichthefirm offers bothcallable andnon-callableunits atdifferentpricesatanypointintime.Theyshowedthatstrategic customer behavior can renderthe customer to be worse off and the retailer to be better off. Therefore, more purchasing options donot necessarily benefitcustomers.Aydın,Birbil,andTopalo˘glu (2016)developedsingle-legrevenuemanagementmodelsthatcon- sider contingent commitment decisions, where commitment op- tion allows passengers to reserve a seat fora fixed duration be- fore making a final purchase decision. We introduce the concept ofcallable products intoa capacityallocationgame betweentwo airlinesandexamineits impactonthe revenuesandthe booking limitsofthetwoairlines.

The second stream of literature related to our study is rev- enue management game. Lederer and Nambimadom (1998) dis- cussedhow theentireairline networkdetermines theroutesand frequencies of flights when multiple airlines interact with each other.UsingdataonU.S.airlinedeparture timesfrom1975,when fares were regulated, and 1986, when fares were not regulated, Borenstein and Netz (1999) empirically estimated the effect of competition on product differentiation. Richard (2003) analyzed thewelfareconsequencesofairlinemergersintermsofticketprice andflightfrequency.Theaboveresearchconsidersthecompetition betweenprice,flightfrequencyanddeparturetime,whichisdiffer- entfromseatallocationcompetitionasconsideredinthispaper.

Netessine and Shumsky (2005) was the first published paper thatplacestheseatallocationprobleminacompetitiveframework and examines the seat inventory control problem. The analytical results demonstrated that more seats are protected forhigh-fare passengers under horizontalcompetition than when a singleair- lineactsasamonopoly.Li,Oum,andAnderson(2007)showedthe existenceofanequilibriumbookingstrategysuchthatbothairlines protectthesamenumberofseatsforthehighfareandthattheto- talnumberofseatsavailableforthelowfareundercompetitionis smallerthan thetotal numberofseatsthat wouldbe availableif thetwo airlinesweretocollude. Li,Zhang,andZhang(2008)ex- tendedLietal.(2007)byincorporatingthecostasymmetryofdif- ferentairlines.WhileNetessineandShumsky(2005)tookthedif- ferentiationapproachby assumingseparate demandforeach fare classofferedbyanairline,Lietal.(2007)andLietal.(2008)chose thehomogeneousmarketapproach,i.e.,twoairlinesfacecommon marketdemandandthedemandissplitbetweenthetwoairlines.

The splitting rules of the demand in Li et al. (2007, 2008) are analogous to Rule 3 (Incremental Random Splitting) in Lippman and McCardle (1997) and generate demand that is independent or perfectly correlated, whereas the demand form in Netessine andShumsky (2005)is moregeneralasdemandof differentfare classes and different airlines can be partially correlated. We in- corporatethe concept ofcallable products into theframework of NetessineandShumsky(2005)andexamineitsimpactontherev- enuesandthebookinglimitsofthetwoairlines.

Furthermore,Songand Parlar(2012) alsostudied thecapacity allocationgame betweentwo airlines,where thedemandformis similar to that in Netessine and Shumsky (2005).Song and Par- lar (2012) took into account the penalty cost for each reserva- tionofthe transfercustomersrejectedby anairline. Theyused a nonnestedmodeltoapproximatetheoriginalnestedbookinglimit modelandshowedtheexistenceofaunique Nashequilibriumin the noncooperative situation. Zhao, Atkins, (2002) madea major

attempttoaddressthejointpricingandallocation problemwhen two airlines compete for passengers in one demand class. Lim (2009) examinedthe practiceof oversellingin aduopoly context where late-arriving consumers value the good higherthan early- arrivingonesbuttheformer’sarrivalisuncertain.

There are some papers studying the airline alliances, includ- ingcapacitycontroloftheairlines,revenuesharingoftheairlines, etc. (e.g.,Graf & Kimms,2011; Kimms& Çetiner, 2012;Hu etal., 2013; Graf & Kimms, 2013). Particularly, Kimms and Grauberger (2016) investigatedtheproblemoftwo airlinesinwhichthey co- operatewithinanallianceononehand,butontheotherhandthey continuetocompete forcustomerswithinarevenuemanagement setting.Itisthefirstpapertoconsidersimultaneoushorizontaland vertical competition within allianceson a network withmultiple classes.

3. Problemdescription

Our model is similar to that of Netessine and Shumsky (2005)andthe differenceis that theairlines inour modelcould of- fercallableproducts.Supposetherearetwoairlinesofferingflights betweenthesameoriginandthesamedestination.Usesubscripts i=1,2todistinguishthetwoairlines.Bothairlinesfacetwotypes of customers: low-fare customers and high-fare customers. The low-fare customersacceptlow-fare ticketsonlybutare willingto book inadvance;thehigh-valuationcustomersare willingtobuy expensiveticketsbutarrivejustbeforetheplane takesoff.There- fore,wecanassumethatthelow-farecustomersarriveinthefirst period and the high-fare customers arrive in the second period.

Suppose that the capacity of Airline i is C_i, and that the prices of the low-fare tickets and high-fare tickets of Airline i are p_Li and p_Hi, respectively. The initial low-fare demand and the initial high-faredemandofAirlineiarerepresentedbytherandomvari- ablesD_LiandD_Hi,respectively.AssumethatthesupportsetsofD_Li andD_Hiarenonnegativeandthatthecumulativedistributionfunc- tions of D_Li and D_Hi are differentiable. If eithertype of customer cannot be satisfied by one airline, they spill to the other airline andcan be recaptured bythe other airline. We callthesespilled customers.

Thelow-fare ticketsareeithercallableproducts ornoncallable products: if an airline offers callable products, the low-fare tick- ets are callable products, which means that the low-fare tickets can berecalledata pre-specified price;ifanairline doesnot of- fercallableproducts,thelow-fareticketsarenoncallableproducts, whichmeansthatthelow-fareticketsareregularticketsandcan- notberecalled. Ifanairlineofferscallableproducts,whentheca- pacityavailableforthehigh-farecustomerscouldnotsatisfy their demand, the airline can recall some orall ofthe callable tickets (i.e.,low-faretickets).SupposetherecallpriceofAirlineiisp_i(p_Li

<p_i<p_Hi).DenotethebookinglimitofAirlineiasB_i.Eachairline needs to determine thebooking limit for thelow-fare customers atthebeginningofthefirstperiod.

Unless particularly specified, we analyze the casewhere both airlines offercallable products. The expectedrevenue ofAirline i is:

π

i=E

pLimin

(

^DTLi,Bi

)

+pHimin

D^T_Hi,Ci−min

(

^DTLi,Bi

)

+

(

^pHi−pi

)

^min

(

^min

(

^DTLi,Bi

)

+D^T_Hi−Ci

)

⁺,min

(

^DTLi,Bi

)

, (1)

whereD^T_Li=D_Li+(^DL j−B_j)^{+ isthetotallow-faredemandforAir-} lineiandD^T_Hi=D_Hi+(^DH j−C_j)^{+isthetotalhigh-faredemandfor} Airlinei (i,j=1,2andi =j).Themeaningsofthethreeitemsin Eq.(1)areasfollows:thefirstitemistherevenuefromsellingthe

Table 1

Notation description.

Notation Meaning

C i Capacity of Airline i

p Hi Price of the high-fare tickets of Airline i p Li Price of the low-fare tickets of Airline i p i Recall price of Airline i

D Hi Initial high-fare demand of Airline i D Li Initial low-fare demand of Airline i D ^T_Hi Total high-fare demand of Airline i D ^T_Li Total low-fare demand of Airline i B i Booking limit of Airline i πi Expected revenue of Airline i

low-fare tickets,the second item isthe revenue from selling the restoftheticketstothehigh-farecustomersandthethirditemis theadditionalrevenue broughtby recallingsome low-faretickets.

Themarginalrevenueofoneunitofcallableproductisp_Hi−p_i. SimilartoNetessineandShumsky(2005),weapplythemethod describedinRudi(2001,pp.27–31)toobtainthederivativeofAir- linei’sexpectedrevenue:

∂π

i

∂

^Bi =p_LiPr

(

^DTLi>B_i

)

−p_HiPr

(

^DTLi>B_i,D^T_Hi>C_i−B_i

)

+

(

^pHi−pi

)

^Pr

(

^DTLi>Bi,D^T_Hi>Ci−Bi

)

=Pr

(

^DTLi>Bi

)

pLi−piPr

(

^DTHi>Ci−Bi

|

^DTLi>Bi

)

.

(2)

For ease of exposition, Table 1 summarizes the notation we used,wherei=1,2.

Notethatthebooking limitB_i istheonlydecisionofAirline i. The pricesp_Li, p_Hi andp_i arenot decision variablesand they are assumedtobe determinedsomehowinadvance.Weconsiderthe effectsofdifferentpriceswithnumericalstudiesinSection5.Itis obviousthat,forAirlineitoobtainhigherprofit,itshouldbebet- tertodeterminethepricesp_Li,p_Hi andp_iandthebookinglimitB_i simultaneously. However, due to the complexityof computations andanalyses,thisproblemisrarelyconsidered inthecompetitive revenue management situation (Zhao & Atkins, (2002). Further- more,the decisions of pricing andbooking limit are at different decisionlevelsin reality,asstatedby Petrick,Gönsch, Steinhardt, andKlein(2010),“Revenuemanagementisessentiallyachievedby theapplicationoftwoinstruments:Inafirststep,onarathertac- ticalplanninglevel,pricedifferentiationisperformed,leadingtoa variety ofdifferently pricedproducts defined on thesame set of resources.Inasecond step,ontheoperationallevel,theavailabil- ityoftheproductsis permanentlyadjustedby meansofcapacity control,accordingtothecurrentforecastregardingfuturedemand within theselling horizon”.In addition,the focusofthe paperis onexaminingtheimpactofcallableproductsontherevenuesand thebooking limitsofthe airlines. Therefore,we assume that the pricesp_Li,p_Hi andp_iareexogenouslygiven.

4. Modelanalysis

This section provides a comprehensive model analysis.

Section 4.1 presents the conditions under which a unique Nash equilibrium exists. Section 4.2 examines the impact of callable productsontherevenuesoftheairlineswhenthereisnolow-fare spill. Section 4.3compares the booking limitsunder competition with those under monopoly. Section 4.4 conducts a sensitivity analysisto investigate the impact of priceparameters, i.e., ticket prices and recall prices, on the booking limits. Note that only Section 4.2 considers the situation where there is no spill of low-farecustomerswhileSections4.1, 4.3and4.4allconsiderthe situationwherebothlow-fareandhigh-farecustomersspill.

4.1.Nashequilibriumconditions

This subsectionpresentsthe sufficientconditionsforthe exis- tenceanduniquenessofapure-strategyNashequilibrium.Weuti- lizetwoproperties:totallypositiveoforder2(TP₂)andmultivari- atetotallypositiveoforder2(MTP₂)(forathoroughdiscussionof TP₂ andMTP₂,seeJoe,1997).

Definition1. (Joe,1997,p.23). Anon-negativefunction b on A², whereA⊂R,isTP₂ifforallx₁<y₁,x₂<y₂,withx₁,x₂,y₁,y₂ ∈ A,

b

(

^x1,x2

)

^b

(

^y1,y2

)

≥b

(

^x1,y2

)

^b

(

^y1,x2

)

. ⁽³⁾ Definition2. (Joe,1997,p.24).RandomvariablesXandYareTP₂ ifthejointprobabilitydensityfunctionofXandYisTP₂.

Definition3. (Joe,1997,p.24).LetX bearandomm-vectorwith densityf.Xoffismultivariatetotallypositiveoforder2(MTP₂)if

f

(

^x∨y

)

^f

(

^x∧y

)

≥f

(

^x

)

^f

(

^y

)

, ⁽⁴⁾ forallx,y∈R^m,where

x∨y=

(

^max

{

^x1,y₁

}

^,max

{

^x2,y₂

}

^,...,max

{

^xm,ym

} )

^,

x∧y=

(

^min

{

^x1,y₁

}

,min

{

^x2,y₂

}

,...,min

{

^xm,ym

} )

. ⁽⁵⁾ The definitionofTP₂ indicates that itis morepossible forthe realizationsof the two random variables tobe both low orboth high,thantobemixedlowandhigh.Manyusefulbivariatedistri- butionsareTP₂,suchasanysetofindependentrandomvariables, GammaandFdistributions,themultivariatelogisticandthebivari- atenormaldistributionswithpositivecorrelation(Kalin&Rinotta, 1980; Netessine & Shumsky, 2005). The TP₂ property can be ex- tendedtotheMTP₂.

Lemma1. (Theorem2.1in Vives,2001).Consideragamewithn(n

≥2)players. Ifthestrategy setsarenonempty convex andcompact subsetsof Euclideanspace andthe payoff tofirmi is continuous in theactionsofallfirmsandquasiconcaveinitsownaction, thereisa Nashequilibrium.

Proposition1. Whenbothairlinesoffercallable products,ifD_Li and D^T_Hi (i=1,2)areTP₂,thereexistsauniquepure-strategyNashequi- librium. Furthermore,the best responsefunctions of theairlines are decreasing.

Proposition 1 demonstrates that the best response functions aredecreasing, i.e., ifone airline increasesthe booking limit,the other airline will decrease the booking limit, which is the same asthatinNetessineandShumsky(2005).Recallthat inNetessine and Shumsky (2005) the Nash equilibrium exists but may not be unique.When the low-fare tickets of theairlines are callable, Proposition1indicatesthattheNashequilibriumisunique,which facilitatesthesubsequentanalysis.NotethatProposition1requires D_LiandD^T_Hi tobeTP₂ whiletherealizationofD^T_Hi dependsonthe realizationsofD_HiandD_Hj.Corollary1describesconditionsonthe fourunderlyingdemanddistributions.

Corollary1. Whenbothairlinesoffer callableproducts, if(D_L1,D_L2, D_H1,D_H2)areMTP₂intheirdensityfunctions,theresultsinProposi- tion1hold.

Inthepreviousanalysis,weassumethat,ifthelow-faretickets arecallable, all thelow-fare customersgrant the airlinesthecall option.Ifonlya fractionofthe low-farecustomers grantthe air- linesthecalloption,Proposition2givesconditionsunderwhicha pure-strategyNashequilibriumexists.

Proposition2. Suppose only

α

⁽⁰≤

α

≤1)proportionof thelow- fare customers grant the airlines the call option when the low-fare

ticketsarecallable.IfD_LiandD^T_Hi (i=1,2)areTP₂ or(D_L1,D_L2,D_H1, D_H2)areMTP₂,a pure-strategy Nashequilibrium exists.Inaddition, thebest-responsefunctionsoftheairlinesaredecreasing.

Formathematicaltractability,inthefollowinganalysis,westill discussthecasewhereallthelow-farecustomersgranttheairlines thecalloptionifthelow-fareticketsarecallable.Inthissituation, theuniqueness of theNash equilibrium(Proposition 1) facilitates theanalysis.Ifonlyafractionofthelow-farecustomersgrantthe airlinesthecalloption,weconjecturethattheinsightsofthecur- rentpaperstillhold.

4.2. Impactofcallableproductsontherevenuesoftheairlines

In this subsection, we consider the case where the low-fare customers do not spill to the other airline while the high-fare customers do spill. This applies to the situation where low-fare demand is so high that the booking limits of the two airlines can always be reached, and therefore low-fare spill is irrelevant (Netessine & Shumsky, 2005). The casewhere both the low-fare customersandthehigh-farecustomersspill isexamined numeri- callyinSection5.

Toinvestigatetheimpact ofcallableproducts ontherevenues of theairlines, we consider three other scenarios:neither of the twoairlinesofferscallableproducts;onlyAirlinei (i=1,2) offers callableproducts.

(1) If neither of the two airlines offers callable products, the expectedrevenueofAirlineiis:

π

i=E

p_Limin

(

^DLi,B_i

)

+p_Himin

_˜

D^T_Hi,C_i−min

(

^DLi,B_i

)

, (6)

whereD˜^T_Hi=D_Hi+

D_{H j}−C_j+min(^DL j,B_j)

+

isthetotalhigh-fare demandforAirlinei(i,j=1,2andi=j).Thefirstordercondition forthemaximizationof(6)is:

∂π

i

∂

^Bi =pLiPr

(

^DLi>Bi

)

−pHiPr

(

^DLi>Bi,D˜^T_Hi>Ci−Bi

)

=Pr

(

^DLi>B_i

)

p_Li−p_HiPr

(

^D˜THi>C_i−B_i

|

^DLi>B_i

)

=0.

(7)

Thatis,

Pr

(

^D˜THi>C_i−B_i

|

^DLi>B_i

)

⁼ _p^pLi

Hi

. (8)

(2)If onlyAirline i offers callable products, theexpected rev- enuesofAirlineiandjare

π

i=E

pLimin

(

^DLi,Bi

)

+pHimin

(

^D˜THi,Ci−min

(

^DLi,Bi

))

+

(

^pHi−pi

)

^min

((

^min

(

^DLi,Bi

)

+D˜^T_Hi−Ci

)

⁺,min

(

^DLi,Bi

))

, (9) and

π

^{j= E}

pL jmin

(

^DL j,Bj

)

+pH jmin

(

^DTH j,Cj−min

(

^DL j,Bj

))

, (10)

respectively. The first order conditions for the maximization of (9)and(10)are:

∂π

i

∂

^Bi =pLiPr

(

^DLi>Bi

)

−piPr

(

^DLi>Bi,D˜^T_Hi>Ci−Bi

)

=Pr

(

^DLi>B_i

)

p_Li−p_iPr

(

^D˜THi>C_i−B_i

|

^DLi>B_i

)

=0,

(11)

and

∂π

^j

∂

^Bj =p_{L j}Pr

(

^DL j>B_j

)

−p_{H j}Pr

(

^DL j>B_j,D^T_{H j}>C_j−B_j

)

=Pr

(

^DL j>B_j

)

p_{L j}−p_{H j}Pr

(

^DTH j>C_j−B_j

|

^DL j>B_j

)

=0,

(12)

respectively.Eqs.(11)and(12)canbesimplifiedas:

Pr

(

^D˜THi>Ci−Bi

|

^DLi>Bi

)

= p_Li

pi, ⁽¹³⁾

and

Pr

(

^DTH j>C_j−B_j

|

^DL j>B_j

)

= pL j

p_{H j}. ⁽¹⁴⁾

When D_Li and D^T_Hi are TP₂,applying a similar analysis asthe proofofProposition1,itcanbeshownthatthereisauniqueNash equilibriumforeach game corresponding tothe abovethree sce- narios.Let

π

_i^{k1k2 (k}1,k₂=0,1,i=1,2) denotetherevenueofAir- line i in different scenarios. The superscripts k₁ and k₂ indicate whetherthe low-faretickets ofthe twoairlinesare callable: 1is callable while 0is not. Forexample,

π

1¹⁰ denotes therevenue of Airline1when thelow-fareticketsofAirline 1arecallablewhile those ofAirline 2are not. Similarly, B^k_i^1k2 is the booking limit of Airlineiindifferentscenarios.

Recall that Gallego etal. (2008)showed that offering callable productsbrings arisklessrevenueimprovementtoamonopolistic airline, which is intuitive asan airline offers callable products if andonlyifitisbeneficialto itself.Proposition3indicatesthat, if there is nospill of low-fare customers,offering callable products isa dominantstrategyofboth airlinesandbrings Paretogainsto both.

Proposition3. SupposethatD_Li andD^T_Hi (ⁱ=1,2) ^areTP2.Ifthere isnospilloflow-farecustomers,thefollowinginequalitieshold:

(1)

π

1⁰⁰<

π

1¹⁰,

π

1⁰¹<

π

1¹¹; (2)

π

2⁰⁰<

π

2⁰¹,

π

2¹⁰<

π

2¹¹; (3)

π

_i⁰⁰<

π

_i¹¹,i=1,2.

Parts(1)and(2)ofProposition3indicatethatofferingcallable products improves an airline’s revenue no matter whether the other airline offerscallableproducts. Thus, offeringcallableprod- uctsisa dominantstrategyofboth airlines.Parts (1)and(2)can be interpretedasfollows.IfAirlinej doesnotoffercallable prod- ucts, Airline i’s (i,j=1,2 and i = j) revenue when Airline i of- fers callable products is no lower than that when Airline i does notoffercallableproducts.IfAirlinejofferscallableproducts,the high-farecustomersofAirlinejspilltoAirlineionlywhenthere- alizedhigh-faredemandofAirlinejishigherthanthecapacityof Airlinej.Giventhissituation,Airlinei’srevenuewhenAirlineiof- ferscallableproductsisnolowerthanthatwhenAirlineidoesnot offercallable productseither. Thus,offering callableproducts isa dominantstrategyofbothairlines.

Part (3) of Proposition 3 implies that offering callable prod- ucts provides Pareto gains to both airlines, which can be in- terpreted intuitively. Note that, when one airline does not offer callableproducts,someofitshigh-farecustomersmayspilltothe other airline. However, if one airline offers callable products, its high-fare customers spill to the other airline only when the re- alized high-fare demand is higher than the capacity of the air- line.Thus,offeringcallableproductsprovidesParetogainstoboth airlines. In other words, when there is no spill of low-fare cus- tomers, Proposition3demonstratesthat offeringcallableproducts isa dominantstrategyofboth airlinesandbrings Paretogainsto both.Whencustomersofbothfareclassesspill,Section5numeri- callyshowsthatofferingcallableproductsisnolongeradominant strategyandmayharmtherevenuesoftheairlines.

4.3. Comparingthecompetitorswithamonopolist

Thissubsectioncomparesthebehavior oftwoairlinesincom- petition with that of a monopolist. The monopolist maybe two airlinesinanalliancetocoordinatetherevenuemanagementdeci-

sions,i.e., booking limits(Graf & Kimms,2011; Kimms& Çetiner, 2012). Denote the optimal booking limits of the monopolist to be B^C_i (i=1,2) and the booking limits in equilibrium to be B^∗_i (i=1,2),respectively. Assume that B^∗_i andB^C_i are interior points, i.e.,B^∗_i,B^C_i ∈(⁰,C_i)^.

Proposition4. Whenbothairlinesoffercallableproducts,ifD_Li and D^T_HiareTP₂,andthetwoairlinesaresymmetric(i.e.,p_L1=p_L2,p_H1= p_H2, p₁=p₂,C₁=C₂,(D_L1,D_H1)and(D_L2,D_H2)areidenticallydis- tributed),thebookinglimitsundercompetitionarehigherthanthose undermonopoly:B^∗_i ≥B^C_i,wherei=1,2.

Proposition 4showsthat, when the two airlinesare symmet- ric,theairlinesalwaysprovidemoreticketstolow-farecustomers inthe decentralized situation compared to the centralized situa- tion, which impliesthat the airlinescompete more intensely for low-farecustomers.Thisisdifferentfromtheresultwhenthetwo airlinesdo not offer callable products:if only the high-fare cus- tomersspill,theairlinessetlowerbookinglimitsinthedecentral- izedsituation;ifonlythelow-farecustomersspill,theairlinesset higherbooking limitsin the decentralizedsituation (Netessine &

Shumsky,2005).Thereasonbehindthedifferenceisthatwhenthe low-fareticketsarecallable,theairlinescanavoidcannibalization fromthe low-fare customers by recalling the low-fare tickets, so bothairlinespaymoreattentionto thescrambleforthelow-fare customers.

4.4.Impactofpriceparametersonthebookinglimits

Themaindifferencebetweencallableproducts andtheregular productsisthatcallableproductscanberecalledatapre-specified recallprice.Obviously,recallpricehasagreatimpactonthebook- ing limits ofthe airlines. Although recall priceis not taken as a decisionvariable,wecanexamineitsimpactonthebookinglimits throughcomparativestaticanalysis.

Proposition5. IfD_LiandD^T_Hi (ⁱ=1,2)^areTP2,

(1) B^∗_i decreaseswithp_iwhileincreaseswithp_j(i,j=1,2,i=j);

(2) B^∗_i increaseswithp_Liwhiledecreaseswithp_Lj(i,j=1,2,i=j).

Part(1)of Proposition5showsthat asthe recallpriceofAir- linei increases, Airline i lowers its booking limit while Airline j raises its booking limit. This resultis intuitive asthe higher the recall priceofAirline i, the higherthe cost forAirline i to recall thelow-faretickets;atthesametime,Airlinejhasacomparative advantagein recalling thelow-fare tickets.This isalsoconsistent withProposition 1 whichstates that the bestresponse functions oftheairlinesaredecreasing.TheinterpretationofPart(2)issim- ilartothat ofPart (1).Notethat theimpactofp_Li onB_iandB_jis oppositetothat ofp_i,astheairlinesweighthedeterministic rev- enuep_Li againstthepotentialrecall costp_i whendeterminingthe bookinglimits.

Inaddition,we findthatthe bookinglimitsofthe airlinesare notaffectedbythepricesofthehigh-faretickets.Forthehigh-fare customers, the airline could avoid cannibalizationfrom the low- fare customers by recalling the low-fare tickets. Therefore, when setting the booking limits, the airlinesfocus on the tradeoff be- tweentheimmediaterevenuep_Liandthepotentialrecallcostp_i.

Proposition6examines thecaseinwhich thetwo airlinesare symmetric,i.e., p_L1=p_L2=p_L, p_H1=p_H2=p_H, p₁=p₂=p,C₁= C2=C,(DL1,DH1)and(DL2,DH2)areidenticallydistributed.

Proposition6. When thetwo airlinesaresymmetric,ifD_LiandD^T_Hi (ⁱ=1,2)^areTP2,B^∗₁isequaltoB^∗₂ and

(1) bothofthemdecreasewithp; (2) bothofthemincreasewithp_L.

Whentherecallpriceincreases,theairlinescostmoretorecall thelow-faretickets,sotheywilllowerthebooking limits.Wecan considerthefollowingtwoextremecases.First,ifthe recallprice isequal to the price of the low-fare tickets, the airlines will set thebookinglimit tobe thecapacity;second,iftherecall priceis equaltothe priceofthehigh-faretickets,thegame betweenthe twoairlinesdegeneratestothatwhenthetwoairlinesdonotoffer callableproducts.Part(1)ofProposition6impliesthat,whenthe twoairlinesaresymmetric,thebooking limitsofthetwo airlines whentheyoffercallableproductsarehigherthanthosewhenthey do not offer callable products. When the two airlines are asym- metric,numericalexamplesinSection5showsthattheresultstill holds.

5. Numericalexamples

In thissection, we runnumerical examplesto examine,when boththelow-fare andhigh-farecustomersspill,whetherthetwo airlinesoffer callable products andhow callable products impact the booking limits and the revenues of the airlines in equilib- rium.ReferringtotheparametervaluesinNetessineandShumsky (2005),theparametervaluesareasfollows.

• Capacity(C₁ andC₂): Weruntwo setsofexperiments:a sym- metric casewithC₁=C₂=200,andan asymmetric casewith C₁=200andC₂=100.

• Ratio of high fare to low fare (p_H/p_L): We define three scenar- ios, p_H/p_L=[1.3,2.6,4],forboththesymmetricandasymmet- riccases.

• Recallprice(p₁andp₂):Therecallpricesofthetwoairlinesare assumedtobeequal.Weset p₁=p₂=p_L+

α

(^pH−p_L),where

α

=[0.4,0.6,0.8].

• Proportion of demand due to low-fare passengers: Let

μ

Li and

μ

Hibe theaveragelow-faredemandandtheaveragehigh-fare demand ofAirline i, respectively. Forboth thesymmetric and asymmetriccases,set

γ

=

μ

Li/(

μ

Li+

μ

Hi)=[0.5,0.74,0.9].

• Total demand and demand faced by each airline: We consider two scenarios: first,the average total demand is equal tothe total airline capacity,i.e.,

μ

L1+

μ

H1+

μ

L2+

μ

H2=C₁+C₂, re- ferred to as TD=TC case; second, the average total demand is slightly larger than the total airline capacity, where

μ

L1+

μ

^H1+

μ

^L2+

μ

^H2=1.1(^C1+C2),referredtoasTD=1.1TCcase.

Describetheallocationofdemandbetweenthetwoairlinesin termsofload,wheretheloadforAirlineiequals(

μ

Li+

μ

Hi)/C_i. Foreaseofexposition,Table2presentstheloadsofthetwoair- linesindifferentcases.

• Variability:NetessineandShumsky (2005)supposed thecoeffi- cient ofvariation (CV) ofthe fourdemand distributions tobe the same andCV=[0.2, 0.33, 0.6].To limit thenumber ofpa- rameters,weassumeCVofthefourdemanddistributionstobe thesameandCV=[0.2,0.6].

• Correlation:NetessineandShumsky (2005)assumedthecorre- lations among all demand are equal and the correlation

ρ

= [−0.3,0.0,0.3,0.6].AsstatedbyNetessineandShumsky(2005), correlation amongairline demandclassesis usually smalland when correlationis significant,it ismore likelytobe positive thannegative.Thus,tolimitthenumberofparameters,weas-

Table 2

Loads of the two airlines in different cases.

Cases l 1 l 2

Symmetric, TD = TC [0.5, 0.75, 1] [1.5, 1.25, 1]

Symmetric, TD = 1.1TC [0.5, 0.8, 1.1] [1.7, 1.4, 1.1]

Asymmetric, TD = TC [0.5, 1, 1.25] [2, 1, 0.5]

Asymmetric, TD = 1.1TC [0.5, 1.1, 1.4] [2.3, 1.1, 0.5]

sumethatthecorrelationsamongalldemandareequalandthe correlation

ρ

=[0.2,0.5].

• Probability density: Foreach of the scenarios, assume that the demanddistribution isa multivariate Normal distributionand is truncated at zero. The negative values of the demand are addedtoamasspointatzero.

When combined, the above parameters define 2^∗3^∗3^∗3^∗6^∗ 2^∗2=1296 scenarios. There are 648 scenarios in Netessine and Shumsky (2005). The parameter values in the current paper are thesameasthoseinNetessineandShumsky(2005)exceptCVand

ρ

^{.In addition,there isno recallpricein Netessine andShumsky}

(2005).We usea simple gradientalgorithm to findthe solutions andevaluatethegradientsbyMonteCarlointegration.

Proposition3showsthat,whenthereisnospilloflow-farecus- tomers,offering callableproductsisadominantstrategyandpro- videsParetogainstobothairlines.Whenlow-farecustomersspill, numericalexamplesdemonstratethatwhetherthetwoairlinesof- fercallableproductsandwhetherofferingcallableproductsisben- eficialtothemmainlydependsontheloadsandthecapacitiesof thetwoairlines.Foreaseofexposition,werefertothecasewhere thetwoairlinesdonotoffercallableproductsasthecasewithout callable products andthe casewhere both airlinescould choose whetherornottooffercallableproductsasthecasewithcallable products.Wepresentthenumericalresultsinthesymmetriccase andtheasymmetriccase,respectively.

In the symmetric case, we obtain the following three results.

(1)Whentheloadsofthetwoairlinesarenotequal,andtheratio ofhighfare tolow fare(p_H/p_L)is loworthe recallprice(p₁ and p₂) is high, the airline witha lower loaddoes not offer callable productswhiletheairlinewithahigherloadofferscallableprod- uctsinequilibrium.Specifically, iftheparametervaluesbelong to Table3, Airline1doesnot offercallable productswhile Airline2 offerscallableproductsinequilibrium.Underotherparameterval- ues,bothairlinesoffercallableproductsinequilibrium.Thisresult isintuitive andwecould interpretitasfollows.Comparedtothe airlinewithahigherload,theairlinewithalowerloadisatadis- advantage.Theratioofhighfaretolowfarebeinglowortherecall pricebeinghighindicatesthatthemarketispessimistictotheair- lines.Thisresultimpliesthatwhenthemarketispessimistictothe airlines,theairlineatadisadvantagedoesnotoffercallableprod- uctsinequilibrium.(2)Whentheloadsofthetwoairlinesarenot equal,the revenueof theairline witha higher(resp.lower)load inthecasewithcallableproductsishigher(resp.lower)thanthat inthecasewithoutcallableproducts.Thatis,theairlineatanad- vantagealways benefits fromofferingcallable products whilethe airlineatadisadvantageisalwaysworseoff nomatterwhetheror not it offers callable products in thecase withcallable products.

Specifically, whentheloads ofAirlines1 and2are different,Air- line2’s (resp.Airline1’s) revenuein thecasewithcallableprod- uctsishigher(resp.lower)thanthat inthecasewithoutcallable products. (3) When theloads of the two airlinesare equal, both airlinesoffercallable products inequilibriumandthey are better off orworseoff byofferingcallableproductssimultaneously.Ifthe

Table 3

Parameter values in the symmetric case under which Airline 1 does not offer callable products while Airline 2 offers callable products.

Cases l 1 l 2 p H/ p L α TD = TC 0.5 1.5 130 0.4, 0.6, 0.8

260 0.8 0.75 1.25 130 0.4, 0.6, 0.8

260 0.8 TD = 1.1TC 0.5 1.7 130 0.4, 0.6, 0.8

0.8 1.4 130 0.4, 0.6, 0.8

Table 4

Parameter values in the symmetric case under which both airlines are better off by offering callable products.

Cases l 1 l 2 p H/ p L α TD = TC 1 1 260, 400 0.4 TD = 1.1TC 1.1 1.1 260, 400 0.4

Table 5

Parameter values in the asymmetric case under which Air- line 1 does not offer callable products while Airline 2 offers callable products.

Cases l 1 l 2 p H/ p L α

TD = TC 0.5 2 130 0.4, 0.6

130, 260, 400 0.8 TD = 1.1TC 0.5 2.3 130 0.4, 0.6, 0.8

ratioofhighfaretolowfare(p_H/p_L)ishighandtherecallprice(p₁ andp2)islow,bothairlinesarebetteroff whiletheyareworseoff underotherparametervalues.Specifically,bothairlinesarebetter off by offeringcallable products when theparameter values belong to Table4.Underother parametervalues,both airlinesare worse off by offering callableproducts.Therefore,callable productsmay bringParetogainsorPrisoner’sDilemmatothetwoairlines.

Inthe asymmetriccase, we obtainthe followingthree results.

(1)When the difference betweentheloads ofthetwo airlinesis large, andtheratioof highfareto lowfare (p_H/p_L) is lowor the recallprice(p₁ andp₂)ishigh,theairlinewithalowerloaddoes notoffercallableproductswhiletheairlinewithahigherloadof- fers callableproductsinequilibrium. Specifically,iftheparameter valuesbelongtoTable5,Airline1doesnotoffercallableproducts whileAirline2offerscallableproductsinequilibrium.Underother parameter values,both airlines offer callableproducts inequilib- rium.Theinterpretationoftheresultisasfollows.Therevenuesof theairlinesdependon theproportionofhigh-faredemandtoca- pacity(i.e.,

μ

H_i/C_i)toa largedegree.Thus,comparedtotheother airline, whetheran airlineis atan advantagedependson thera- tiooftheir proportionofhigh-faredemand tocapacity.Notethat

μ

H_i/C_i=(¹−

γ

)^li, where i=1,2. Obviously, the higher l_i is, the higher

μ

H_i/C_i is.Therefore,an airline witha higherloadis atan advantagenomatter whetherits capacityislargerthan theother airline.Thisresultimpliesthat,whenthemarketispessimistic,the airlineatadisadvantagedoesnotoffercallableproductsinequilib- rium.(2)Whentheloadsofthetwoairlinesarenotequal,therev- enueoftheairlinewithahigher(resp.lower)loadinthecasewith callableproductsishigher(resp.lower)thanthatinthecasewith- out callableproducts.Specifically,iftheloadofAirlinei ishigher, Airlinei’s(resp.Airlinej’s)revenueinthecasewithcallableprod- uctsishigher(resp.lower)thanthat inthecasewithoutcallable products, wherei,j=1,2 andi =j.Thisresultisintuitive andis similar to that inthesymmetric case. (3)When theloads ofthe twoairlinesareequal,bothairlinesoffercallableproductsinequi- librium.Ifthe ratioofhighfare tolow fare(p_H/p_L)is lowor the recall price(p₁ andp₂) ishigh, bothairlinesareworse off by of- feringcallableproducts.Specifically,iftheparametervaluesbelong toTable6,bothairlinesareworseoff byofferingcallableproducts.

Under other parameter values, theairline witha higher capacity is betteroff while the one withalower capacityis worse off by offering callableproducts. Theresultis intuitive.Note that,when theratioofhighfaretolowfareislowortherecallpriceishigh, themarketispessimistic.Sobothairlinesareworseoff byoffering callableproducts.Moreover,whentheloadsofthetwoairlinesare equal,theairlinewithahighercapacityisatanadvantage.

Table 6

Parameter values in the asymmetric case under which both airlines are worse off by offering callable products.

Cases l 1 l 2 p H/ p L α

TD = TC 1 1 130 0.4, 0.6

130, 260, 400 0.8 TD = 1.1TC 1.1 1.1 130 0.6 130, 260, 400 0.8

Ingeneral,whenthe low-farecustomersspill,offering callable products is no longer a dominant strategy and may harm the revenues of the airlines. Numerical examples demonstrate that whetherthe two airlinesoffer callableproducts and whetherof- fering callable products is beneficial to them mainly depends on theloads andthe capacitiesof thetwo airlines. Especially, when thedifferencebetweentheloadsoftheairlinesislarge,theloads oftheairlinesplaythemostimportantrole.Weobtainthefollow- ingtwoinsights: (1)iftheratioofhighfaretolowfareislowor therecallpriceishigh,theairlinewithalowerloaddoesnotoffer callableproductsinequilibrium;(2)therevenueoftheairlinewith ahigher(resp.lower)loadishigher(resp.lower)inthecasewith callableproducts than that inthecasewithout callableproducts.

Whenthedifferencebetweentheloadsoftheairlinesissmall,the capacitiesof theairlinesplay themostimportantrole. Ifthedif- ferencebetweenthecapacitiesoftheairlinesisalsosmall,thetwo airlinesareworseoff orbetteroff byofferingcallableproductssi- multaneously. Ifthe difference betweenthe capacities ofthe air- linesislarge,theairlinewithalowercapacityisalwaysworseoff byofferingcallableproductswhiletheonewithahighercapacity maybeworseoff orbetteroff whichdependsontheratioofhigh faretolowfareandtherecallprice.

Furthermore,Proposition6impliesthat, whenthetwoairlines are symmetric,the booking limits ofthe two airlineswhen they offercallableproductsarehigherthanthosewhentheydonotof- fercallableproducts. Numericalexamples showthat, thebooking limits of the two airlinesin the case withcallable products are higherthan those in thecase withoutcallable products. This re- sultisintuitive.Inthecasewithcallableproducts,oneairlinemay ormaynotoffercallableproductsinequilibrium.Iftheairlineof- ferscallable products,it canrecallsome ofthelow-fare ticketsif needed, so it will set a higher booking limit. If the airline does not offer callable products, its opponent will offer callable prod- uctsandsetahigherbookinglimit.Tocopewiththecompetition fromitsopponent,thisairlinewillsetahigherbookinglimitthan thatinthecasewithoutcallableproducts.

6. Conclusion

Thispaperintroducestheconceptofcallableproductsintothe capacityallocationgamebetweenduopolisticairlines.Weexamine theimpactoftheintroductionofcallableproductsontherevenues andthe booking limits ofthe two airlines. The analytical results demonstratethat,ifthereisnospilloflow-farecustomers,offering callableproducts isadominantstrategy ofbothairlinesandpro- videsParetogainstobothairlines.Ifcustomersofbothfareclasses spill,numerical examples demonstratethat whetherthe two air- linesoffercallableproductsandwhetherofferingcallableproducts isbeneficial tothetwo airlinesmainly dependson theloadsand thecapacitiesofthem.Moreover,numericalexamplesdemonstrate thatthebookinglimitsofthetwoairlinesinthecasewithcallable productsarealwayshigherthanthoseinthecasewithoutcallable products.

Thereare some limitationsinourstudy.First, tofocuson air- linecompetitionwithregardtobooking limits,wetaketheprices asexogenouslygiven, includingtheprices oflow-faretickets, the