Revenue Management Games

Transcription

1 Revenue Management Games Sergue Netessne and Robert A. Shumsky 2 Unversty of Rohester W. E. Smon Graduate Shool of Busness Admnstraton Rohester, NY 4627 Otober, 2000 netessnse@smon.rohester.edu 2 shumsky@smon.rohester.edu

2 Revenue Management Games Abstrat: A well-studed problem n the lterature on arlne revenue or yeld) management s the optmal alloaton of seat nventory among fare lasses gven a demand dstrbuton for eah lass. In the lterature thus far, passenger demand s an exogenous parameter. owever, the seat alloaton desons of one arlne affet the passenger demands for seats on other arlnes. In ths paper we examne the seat nventory ontrol problem wth two fare lasses and two arlnes under ompetton. Eah arlne hooses an optmal bookng lmt for the lower-fare lass whle takng nto aount the overflow of passengers from ts ompettor. We show that under ertan ondtons ths 'revenue management game' has a pure-strategy Nash equlbrum, and for speal ases we show that the equlbrum s unque. We also ompare the total number of seats avalable n eah fare lass wth, and wthout, ompetton. Analytal results for one speal ase as well as numeral examples demonstrate that, all else equal, under ompetton more seats are proteted for hgher-fare passengers than when a sngle arlne ats as a monopoly or when arlnes form an allane to maxmze overall profts.

3 . Introduton Consder the arlne flght shedules dsplayed n able. We see that WA and Delta shedule flghts between the same orgn and destnaton, at exatly the same tmes, usng the same equpment, and hargng nearly the same pre for advane-purhase tkets. hs paper examnes how suh dret ompetton for ustomers affets a fundamental revenue management deson, the alloaton of seat nventory among fare lasses. Arlne Flght Arraft Departure Pre advane purhase) WA 3832 Jetstream 4 8:35am $23.50 Delta 622 Jetstream 4 8:35am $ WA 3834 Jetstream 4 2:24pm $23.50 Delta 6204 Jetstream 4 2:24pm $ WA 3836 Jetstream 4 5:50pm $23.00 Delta 6206 Jetstream 4 5:50pm $ able. Flght shedule for WA and Delta ROC-JFK, Aprl 23, here s a substantal lterature analyzng arlne eonoms under ompetton as well as a reent stream of operatons researh lterature on the problem of optmal seat alloaton. owever, there are no publshed papers that plae the seat alloaton problem n a ompettve framework. As the example above llustrates, arlnes fae ntense ompetton, and the mpat of ompetton on seat nventory desons and arlne revenues s of nterest to arlne plannng and marketng managers, as well as government regulators. In ths paper we model ompetton between two arlnes offerng two flghts that serve as substtutes for ustomers. Eah arlne s faed wth an ntal demand from passengers who wsh to purhase tkets, but eah arlne may also sell tkets to passengers that were dened a reservaton on the ompetng arlne. ene, the optmal apaty lmts for eah lass the bookng lmts) on eah arlne are nterdependent. We ompare the optmal revenue management poles of two ompetng arlnes wth the poly of a monopolst who operates both flghts or, equvalently, the poles of two arlnes who form an allane to maxmze total profts. We show that under ertan ondtons a pure-strategy Nash equlbrum exsts for the ompettve ase, and we dentfy speal ases under whh the equlbrum s unque and stable. Gven the assumpton that low and hgh-fare tket pres reman onstant, we fnd that under

4 ompetton more seats are alloated for hgher-fare ustomers, and fewer seats are alloated for the lower-fare ustomers, than under entralzed ontrol. Readers famlar wth the arlne ndustry may fnd ths uxtaposton of ompettve analyss and yeld management unusual. In general, ompettve desons and seat alloaton desons are made by dfferent funtonal unts wthn arlnes, at dfferent tmes n the plannng proess, and over extremely dfferent tme horzons. he deson to enter markets, the assgnment of arraft to partular markets, and the reaton of a shedule takes plae over a tme horzon of years and months. he alloaton of seat nventory among ustomer lasses s an operatonal deson wth a tme horzon of weeks, days, or even mnutes see Jaobs, Ratlff and Smth [2000] for a general desrpton of ths plannng proess). owever, our smple model wll show that ompetton on a partular route at a partular tme an have a profound effet on yeld management desons. In general, arlne planners have reently shown an nreased nterest n the ntegraton of arlne funtons. For example, numeral experments n Jaobs, Ratlff and Smth [2000] demonstrate the value of smultaneous optmzaton of yeld management and shedulng desons. Yuen and Irrgang [998] emphasze the benefts of ntegratng sales, yeld management, prng and shedulng desons. Publatons that onsder the nteratons among eonom fores, strateg arlne market entry desons, and arlne shedules nlude the network desgn models of ederer and Nambmamdom [999], Dobson and ederer [994], and the empral work by Borensten and Rose [994]. Another body of researh fouses on the arlnes' shedulng desons under ompetton usng varants of the spatal model developed by otellng [929]. See, for example, the reent empral papers by Borensten and Netz [999] and Rhard [999]. hese papers fous on broad ompettve problems and gnore the spefs of seat nventory alloaton. In ths paper we wll not be onerned wth the reasons arlnes shedule ther flghts at the same tme or wth the prng deson for eah flght. Rather, we wll onentrate on the mplatons of ompettve shedulng on seat nventory ontrol. here are numerous papers n the area of revenue management that fous on arlne seat nventory ontrol, although to our knowledge only one addresses the ssues desrbed here. For fundamental results on the general subet of seat nventory ontrol see Belobaba [989], 2

5 Brumelle et.al. [990], and a useful lterature revew by MGll and van Ryzn [999]. Our paper s related to the work by and Oum [998], whh frst ntrodued the seat alloaton problem for two arlnes n ompetton. he model developed by and Oum has a relatvely restrtve assumpton about how demand s alloated among arlnes: total demand s splt aordng to the proporton of seats avalable on eah arraft n eah fare lass, and the overflow proess s not expltly modeled. In addton, the paper by and Oum dentfes one, symmetr equlbrum but does not determne whether the equlbrum s unque. Our approah s more general and the results more advaned; we wll plae no restrtons on how ntal demand s dstrbuted and wll show that for speal ases of the problem the equlbrum s unque. he lterature on nventory management has seen a stream of losely related papers devoted to ompetton among frms n whh the frms determne nventory levels and ustomers may swth among frms untl a sutable produt s found ths has been desrbed as a 'newsboy game'). Parlar [988] examnes the ompetton between two retalers fang ndependent demands. e establshes that a unque Nash equlbrum exsts. Karalanen [992] formulates the problem for an arbtrary number of produts wth ndependent demands. ppman and MCardle [997] examne both the two-frm game and a game wth an arbtrary number of players. In ther models, ntal ndustry demand s alloated among the players aordng to a pre-spefed 'splttng rule.' hs ntal alloaton may be ether determnst e.g., 40% of demand to player ) or stohast the rule tself depends on the outome of a random experment). For the two-frm game they establsh the exstene of a pure-strategy Nash equlbrum and show that the equlbrum s unque when the ntal alloaton s determnst and strtly nreasng n the total ndustry demand for eah player. Reent extensons of these models nlude Mahaan and van Ryzn [999] who model demand as a stohast sequene of utlty-maxmzng ustomers. For an arbtrary number of frms, they demonstrate that an equlbrum exsts and show that t s unque for a symmetr game. Rud and Netessne [2000] analyze a problem smlar to Parlar [988] but for an arbtrary number of produts. Gven mld parametr assumptons they establsh the exstene of, and haraterze, a unque, globally stable Nash equlbrum. Many of these papers ompare total nventory levels under frm ompetton wth nventory levels when frms ooperate. ppman and MCardle [997] show that ompetton 3

6 an lead to hgher nventores, and Mahaan and van Ryzn [999] derve smlar results gven ther dynam model of ustomer purhasng. On the other hand, wth the substtuton struture of the model of Rud and Netessne [2000], under ompetton some frms may stok less than under entralzaton. In ths paper we fnd that ompetton leads to an nrease n hgh-fare seat 'nventory,' a result smlar to earler fndngs. owever, our model dffers n many respets from the newsboy ompetton desrbed by ppman and MCardle. As n Mahaan and van Ryzn, the method of alloatng arrvng ustomers to frms s more natural than the stylzed splttng rules proposed by ppman and MCardle. In our model, demand for eah fare lass on eah arlne an follow an arbtrary dstrbuton, and we allow an arbtrary orrelaton struture among demands. Numeral experments wll demonstrate that the degree of orrelaton an have a sgnfant mpat on seat alloaton desons, and an even determne whether a pure-strategy equlbrum exsts. here s also a fundamental dfferene between the problem onsdered here and the problem onsdered n the nventory lterature. ere we onsder the alloaton of a fxed nventory pool between two produts, whle the nventory lterature assumes that the nventory of eah produt s a deson varable. In our problem, the effet of a hange n one arlne's bookng lmt s qute omplex. As the bookng lmt rses, demand by low-fare passengers to a ompettor delnes whle hgh-fare demand to the ompettor rses. In addton, we wll see that the bookng lmt of an arlne an affet the volume of ts own hgh-fare demand. In the next Seton we desrbe the revenue management game and provde examples of senaros n whh Nash equlbra do, and do not, exst. We examne one varaton of the game for whh we establsh the exstene of a pure-strategy Nash equlbrum. Seton 3 fouses on ompetton wth partal overflow, models n whh only low-fare or only hgh-fare passengers spll to a ompetng arlne. In Seton 4 we ompare analytally the behavor of a monopolst or allane between arlnes) wth the behavor of two arlnes under ompetton. Seton 5 desrbes numeral examples and ompares the serve levels perentage of ustomers who are able to purhase tkets) under the ompettve and ooperatve ases. In Seton 6 we dsuss the mplatons of our results on the prate of yeld management and desrbe areas for future researh. 4

7 2. he Full Revenue Management Game Suppose two arlnes offer dret flghts between the same orgn and destnaton, wth departures and arrvals at smlar tmes. We assume that other flghts on ths route are sheduled suffently far away n tme so that they an be gnored. For smplty, we assume that both flghts have the same seat apaty and that there are only two fare lasses avalable for passengers: a 'low-fare' and a 'hgh-fare.' A tket purhased at ether fare gves aess to the same produt: a oah-lass seat on one flght leg. As s tradtonal n the lterature on arlne seat nventory ontrol, we assume that demand for low-fare tkets ours before demand for hgh-fare tkets, as s the ase when advane-purhase requrements are used to dstngush between ustomers wth dfferent valuatons on pre and purhase onvenene. Customers who prefer a low fare and are wllng to aept the purhase restrtons wll be alled 'low-fare ustomers'. Customers who prefer to purhase later, at the hgher pre, are alled 'hgh-fare ustomers'. We also assume that there are no ustomer anellatons. o maxmze expeted profts, both arlnes establsh bookng lmts for low-fare seats. One ths bookng lmt s reahed, the low fare s losed. Sales of hgh-fare tkets are aepted untl ether the arplane s full or the flght departs. If ether type of ustomer s dened a tket at one arlne, the ustomer wll attempt to purhase a tket from the other we all these overflow passengers ). herefore, both arlnes are faed wth a random ntal demand for eah fare lass as well as demand from ustomers who are dened tkets by the other arlne. Passengers dened a reservaton by both arlnes are lost. Fgure shows both overflow proesses as well as the followng notaton:, = passenger lasses, for low-fare passengers and for hgh-fare passengers. C = apaty of the arraft. B = bookng lmt for low fare establshed by arlne =,2. D k = a random varable, demand for lass k tkets at arlne, k =, and =, 2. p k = revenue from lass k=, passengers less varable ost. 5

8 Arlne Arlne 2 D C gh fare lass hgh-fare overflow from to 2 hgh-fare overflow from 2 to gh fare lass C D2 B B2 D ow fare lass low-fare overflow from to 2 low-fare overflow from 2 to ow fare lass D2 Fgure. he Bas Compettve Model We assume that both arlne's pres are the same and that p < p. We also assume that the random varables D k have nonnegatve support. owever, to derve the followng results establshng the exstene of a pure-strategy equlbrum we do not assume that the umulatve dstrbutons are ontnuous we may have dsrete or ontnuous probablty dstrbutons), and there may be an arbtrary orrelaton struture among demands. In Seton 3, however, to establsh the exstene of a unque equlbrum we wll assume that fnte probablty denstes exst and that low and hgh-fare demands are ndependent. In ths paper we study ompetng arlnes engagng n a nonooperatve game wth omplete nformaton. Eah arlne attempts to maxmze ts profts by adustng ts bookng levels. In other words, the bookng level B [0,C] s the strategy spae of arlne for smplty, we assume that the bookng level may be any real number n ths range). Eah arlne knows the strategy spaes and demand dstrbutons of ts own flght as well as those of the ompetng arlne. An mportant assumpton of the model s that the ntal demands D k are exogenous; they are not affeted by the bookng lmts hosen by eah arlne. hs assumpton s onsstent wth the newsboy game models of Parlar, Karalanen, and ppman and MCardle. owever, one mght argue that the bookng lmts determne seat avalablty, and that n the long run ths aspet of serve qualty affets ntal demand. A more omplete model would norporate ths relatonshp between bookng lmts and demand, and the soluton would supply equlbrum demands as well as equlbrum bookng lmts. For our applaton, however, the relatonshp between bookng lmts and demand s weakened by marketng efforts suh as advertsng and frequent-flyer programs. In addton, the use of travel agents and on-lne reservaton tools 6

9 redues the margnal searh ost assoated wth makng eah bookng. Gven low searh osts, the deson as to whh arlne to query frst may depend on fators that domnate the lkelhood that the query wll result n a bookng. Our model smplfes other aspets of the atual envronment. For example, the model assumes that passengers dened a tket n one lass do not attempt to upgrade or downgrade to another lass. he model also assumes that a passenger, when frst dened a tket, wll not shft to a later or earler flght operated by the same arlne. owever, all results presented n ths paper also apply to a model n whh some fraton less than one) of passengers dened a tket on one arlne attempt to purhase a tket from the other arlne, whle some fraton greater than zero) are lost to both arlnes. o smplfy the model and mnmze the number of parameters, we assume that all passengers dened a tket from ther frst hoe overflow to ther seond-hoe arlne. he model ontans only two fare lasses, when n realty there may be many more see Belobaba [998] for an ntroduton to the omplextes of real-world yeld management systems). We also assume that the arlnes' bookng lmts are stat. hat s, the bookng lmt s set before demand s realzed and no adustments are made as low-fare demand s observed. As we wll see, even ths relatvely smple deson an be dffult to analyze n a ompettve game, and ths smple model allows us to fous on a few mportant questons. ow wll an optmal bookng lmt under ompetton dffer from a bookng lmt under a entralzed soluton, wth a sngle arlne or when two arlnes ooperate to maxmze total profts? ow does the exstene of 'spll' demand affet the alloaton of seat nventory? What s the effet of ompetton on profts, even when pres are held onstant? 2. ow-fare then gh-fare Spll hus far we have not desrbed the order of events n the game. We begn wth what may be the most natural order:. Arlnes establsh bookng lmts B and B ow-fare passengers arrve to ther frst-hoe arlnes and are aommodated up to the bookng lmts. 7

10 3. ow-fare passengers not aommodated on ther frst-hoe arlnes 'spll' to the alternate arlnes and are aommodated up to the bookng lmts. 4. gh-fare passengers arrve to ther frst-hoe arlnes and are aommodated wth any remanng seats, up to apaty C n eah arraft. 5. gh-fare passengers not aommodated on ther frst-hoe arlnes 'spll' to the alternate arlnes and are aommodated n any remanng seats, up to apaty C n eah arraft. o desrbe the problem n terms of ustomer demand and bookng lmts, defne: D = D D B ), total demand for low-fare tkets on arlne, =, =2 and =2, =. R = C mn D, B ), the number of seats avalable for hgh-fare passengers on arlne =,2. D = D D R ) he total revenue for arlne s, total demand for hgh-fare tkets, =, =2 and =2, =. [ p mn D, B ) p mn D, R )] π = E. ) Eah arlne wll maxmze ths expresson, gven the bookng lmt of ts ompettor. It wll be nstrutve to examne the frst dervatve of ths obetve funton. It s tedous to fnd the dervatve by the tradtonal methods e.g., applyng ebntz's rule). Instead, by applyng the tehnques desrbed n the Appendx of Rud and Netessne [2000], we fnd for =, =2 and =2, =, π B = p p Pr D Pr D > B ) p > B, D Pr D < B, D > C B, D > R, D > B ) < C B ). 2) Although ths s a omplex expresson, there s a straghtforward nterpretaton for eah term. An nremental nrease n the bookng lmt B by arlne has three effets on that arlne's total revenue. Frst, revenue from low-fare ustomers nreases wth probablty Pr D > B ). Seond, revenue from the hgh-fare ustomers dereases wth probablty Pr D > C B, D > B ). Whle these two effets are dret onsequenes of the hange n B, there s a thrd, ndret effet. Revenue from hgh-fare ustomers may derease beause ) an 8

11 nrease s B may redue the overflow of low-fare ustomers from to, ) a reduton n the number of low-fare ustomers at may nrease the number of seats avalable for hgh-fare ustomers at, ) ths may redue the overflow of hgh-fare ustomers from to and v) a delne n the overflow from may redue the number of hgh-fare ustomers aommodated at. he probablty of ths sequene of events s the thrd term on the rght-hand sde of equaton 2), whh mples that an nrease n the bookng lmt of arlne an result n a derease n hghfare demand to arlne. Beause the strategy spaes of the arlnes are ompat and the payoff funtons are ontnuous see Proposton, below), a Nash equlbrum n mxed strateges must exst. owever, a pure-strategy Nash equlbrum may, or may not, exst for arlnes playng ths game. Fgure 2 shows the best reply funtons, or reaton funtons r B ), of two arlnes, eah wth C =200 and multvarate normal demands the parameters for ths example wll be desrbed n detal n Seton 5). Fgure 3, showng a game wth multple equlbra, was also generated wth the multvarate normal dstrbuton agan, detals are gven n Seton 5). Fgure 4 dsplays two reaton funtons, eah wth two dsontnutes, produng a game wthout any pure-strategy equlbrum. An extremely unlkely demand pattern was used to produe ths outome. Fgure 4 was generated from: Bmodal demand dstrbutons for eah fare lass and arlne. he dstrbutons were reated by mxng two normal dstrbutons, one representng low-volume demand mean = 20 seats) and the other representng hgh-volume demand mean = 50 seats). Strong negatve orrelatons between low-fare and hgh-fare demands. When low-fare demand was hosen from the low-volume dstrbuton, hgh-fare demand was hosen from the hgh-volume dstrbuton, and ve-versa. As a result, ρ D, D ) = 0. 9 for =,2. 3 A large dfferene between hgh and low fares p / = 4 ). p 3 It s nterestng to note that n prate the strong negatve orrelaton would present an exellent opportunty for eah arlne to prate dynam yeld management, wth an adustable bookng lmt dependent on observed lowfare demand. Gven suh dynam deson-makng, there may well be a ompettve equlbrum. 9

12 B r2b) B r2b) rb2) 50 rb2) B Fgure 2: Unque Nash Equlbrum B Fgure 3: Multple Equlbra r 2B ) 50 r B 2) B Fgure 4: No Pure-Strategy Equlbrum B Whle we annot spefy analytally the general haratersts that would guarantee the exstene of an equlbrum, expresson 2) offers some nsght. For most reasonable probablty dstrbutons and for most values of B and B 2, the frst two terms domnate the thrd term, so that π B p Pr D > B ) p Pr D > C B, D > B ). 3) hs expresson s smlar to the frst-order ondtons for the standard two-fare seat alloaton problem of a stand-alone arlne, although here exogenous demands D k have been replaed by total demands D. Brumelle et.al [990] show that when the demands are monotonally k assoated, so that Pr D > C B D > B ) s nondereasng n B, then the obetve funton of arlne s quas-onave n B. Gven that the two players fae obetve funtons that are ontnuous and quas-onave n eah bookng lmt, there exsts a pure-strategy Nash equlbrum Mouln, 986). 0

13 hs reasonng does not provde us wth prese ondtons for the exstene of an equlbrum, but we have found ths analyss to be helpful when examnng the results of our numeral examples. When demands D and D and D are strongly negatvely orrelated then the total D are not monotonally assoated. In ths ase, the obetve funtons for eah arlne are not unmodal, produng the dsontnutes n the reaton funtons shown n Fgure 4. When orrelated, D and D are weakly negatvely orrelated, ndependent, or postvely D and D mantan the postve assoaton property and a pure-strategy equlbrum exsts. We wll see n Seton 5 that the latter ase apples for most reasonable problem parameters. Now we do dentfy two suffent ondtons for the exstene of a pure-strategy equlbrum. Frst, f low-fare demand s extremely hgh so that Pr D > C) = for =,2, then a pure-strategy equlbrum must exst and, under ertan ondtons, the equlbrum must be unque and stable. In ths ase, low-fare overflow s gnored by eah arlne beause there s already a surplus of low-fare ustomers, and arlnes only ompete for hgh-fare ustomers. Beause ths s a speal ase of the model presented n Seton 3., further dsusson and a proof wll be presented later see Proposton 3 and Corollary ). he seond ondton nvolves a revson of the tmng of the game. hs s presented n the next seton. 2.2 gh-fare then ow-fare Spll We wll now hange the order of events and assume that low-fare ustomers that overflow are aepted only after all other passengers have been aommodated. he order of events s as follows:. Arlnes establsh bookng lmts B and B ow-fare passengers arrve to ther frst-hoe arlnes and are aommodated up to the bookng lmts. 3. gh-fare passengers arrve to ther frst-hoe arlnes and are aommodated wth any remanng seats, up to apaty C n eah arraft. 4. gh-fare passengers not aommodated on ther frst-hoe arlnes 'spll' to the alternate arlnes and are aommodated n any remanng seats, up to apaty C n eah arraft.

14 5. ow-fare passengers not aommodated on ther frst-hoe arlnes 'spll' to the alternate arlnes and are aommodated up to the bookng lmts. o mantan the flavor of the tmng desrbed n Seton 2., n Step 5 we only book low-fare passengers up to the bookng lmt, even f addtonal seats are avalable. Note that ths game requres eah arlne to dstngush between low-fare passengers who hoose that arlne frst from those that ome to the arlne as a seond hoe. Whle ths may not always be possble, under ths re-orderng, t s possble to establsh the exstene of a pure-strategy Nash equlbrum beause an adustment n B does not affet the hgh-fare demand faed by arlne. Frst defne: mn D, B ), number of low-fare tkets sold n the frst round D = D D C mn B, D )), total demand for hgh-fare tkets at arlne D B ), overflow of low-fare passengers mn B, C D ) D ), number of seats avalable to the overflow low-fare passengers. he total revenue for arlne s: p mn D, B) π = E 4) p mn D B ), mn B, C D) D ) ) p mn C mn B, D ), D) Proposton. Gven the game orderng defned by steps -5 above, a pure-strategy Nash equlbrum n bookng lmts B, B 2 ) exsts. Proof: We wll show that the obetve funton for eah player s ontnuous and submodular n B, B 2 ). herefore, the obetve funton s ontnuous and supermodular n B, -B 2 ), whh are suffent ondtons for the exstene of a pure-strategy Nash equlbrum opks, 998). o see that the obetve funton s ontnuous, note that the strategy spae s fnte so that for any gven demand realzaton the obetve funton s bounded. In addton, the obetve funton s ontnuous n B, B 2 ) for any gven demand realzaton. herefore, by the bounded onvergene theorem, the expetaton 4) s ontnuous Bllngsley, 995). 2

15 o prove submodularty, note that the expetaton of a submodular funton s submodular, the sum of submodular funtons s a submodular funton, and a submodular funton multpled by a postve onstant s a submodular funton opks, 998, emma 2.6. and Corollary 2.6.2). herefore, we wll prove that for any gven demand realzaton, eah of the three terms n the sum 4) s submodular n B, B ). he frst term, mn D, B ), depends only on B, so t s submodular. For the last two terms we wll employ the followng two lemmas for the sake of readablty, n these lemmas and for the remander of the proof the term 'nreasng' mples nondereasng and the term 'dereasng' mples nonnreasng): emma Adopted from opks, 998, Example f).) If g B ) s nreasng and g B ) s dereasng then mn g B ), g B )) s a submodular funton n B, B ). emma 2 opks, 978, able ) Suppose g B, B ) s nreasng n both B and B and s a submodular supermodular) funton n B, B ). Also suppose that f z) s an nreasng onave onvex) funton. hen f g B, B )) s a submodular supermodular) funton n B, B ). mn We re-wrte the seond term of the obetve funton as D ) ) B ), mn B, C D ) D = D B ) mn 0, mn B, C D ) D ) D B ) ). 5) he term D B ) depends only on B and hene s submodular. o prove that the seond term n 5) s submodular, we wll employ emmas and 2. Sne f z) = mn 0, z) s a onave nreasng funton of z, t remans to show that mn B, C D ) D ) D B g B, B ) = ) 6) s an nreasng submodular funton. We frst show that t s an nreasng funton. It s obvous that ths funton s nreasng n B. Further, from the defnton of B, C D ) D ) D above, mn s ether lnearly dereasng n B for some demand realzatons) or does not hange as B hanges. In addton, ) D B s also ether lnearly nreasng or 3

16 nvarant n B. By examnng the two terms n 6), we see that when the seond term s lnearly nreasng n B then the frst term s ether lnearly dereasng or does not hange. When the frst term s lnearly dereasng n B then the seond term must be nreasng. herefore, the seond term domnates, and g B, B ) s nreasng n both B and B. We now show that g B, B ) s also submodular. Frst, mn B, C ) s nreasng D n B, dereasng n B, and by emma a submodular funton n B, B ). herefore, mn B, C D ) D s nreasng and supermodular n B, -B ). In addton, the funton f z) = z) = max 0, z) s onvex and nreasng n z, so that by emma 2 B, C D ) D ) mn s a supermodular funton n B, -B ) and therefore a submodular funton n B, B ). ene, g B, B ) s also a submodular funton. hs ompletes the proof for the seond term of 4). he thrd term of the obetve funton s C mn B, D ), D ) C mn B, D ) s dereasng n B, and C mn B, D ), D ) mn. Note that D s nreasng n B. By emma, mn s submodular. hs ompletes the proof. Whle we an be sure of a pure-strategy equlbrum n ths ase, we annot be sure that the equlbrum s unque. Condtons for unqueness wll be desrbed n the next seton. 3. Competton wth Partal Overflow In ths seton we onsder ompetng arlnes wth only hgh-fare passengers overflowng from one arlne to another Seton 3.) and wth only low-fare passengers overflowng Seton 3.2). For eah ase we wll fnd ondtons under whh a pure-strategy equlbrum exsts and s unque. It s, of ourse, reasonable to ask why we should be onerned wth these speal ases sne both hgh and low-fare ustomers are lkely to look for a seat on another arlne f one annot be found on the preferred arlne. In fat, these speal ases are good approxmatons of the general game desrbed n Seton 2., as long as the number of overflow ustomers from one of the two fare lasses s small. In addton, the model to be presented n Seton 3. 4

17 nludes the ase when hgh-fare passengers swth arlnes whle demand for low-fare tkets s suffently large to sell all avalable low-fare tkets. Moreover, analyss of these speal ases sheds some lght on the reasons why the full game presented n Seton 2 may fal to have a pure-strategy equlbrum. We wll see here that a game wth only hgh-fare overflow always has a pure-strategy equlbrum, whle a game wth only low-fare overflow may not. If only hgh-fare ustomers spll to a ompettor, then the arlnes are nvolved n a supermodular game smlar to the nventory game desrbed by Parlar and by ppman and MCardle. In terms of yeld management, an nrease n the bookng lmt by one arlne nreases demand by hgh-fare passengers to the ompettor, thus lowerng the ompettor's bookng lmt. Eah player's reaton funton s monoton n the other player's strategy, and an equlbrum must exst. owever, when both types of overflow our the response funtons need not be monoton, as n Fgure 4. Addtonal ondtons are needed to establsh the exstene of a pure-strategy equlbrum. 3. gh-fare Overflow Only We now assume that there s no overflow of the low-fare passengers and only hgh-fare passengers approah the other arlne when ther frst-hoe arlne s not avalable. Fgure 5 llustrates the flow of passengers. Note that the followng defntons dffer slghtly from the 'full-overflow' ase presented n Seton 2.. R = C mn D, B ), the number of seats avalable for hgh-fare passengers on arlne. D = D D R ), total demand for hgh-fare tkets on arlne, =, =2 and =2, =. 5

18 D Arlne Arlne 2 gh-fare lass D C D, ))) mn B D C D, ))) mn B gh-fare lass D2 B B2 mnb 2,D2) mnb,d) D ow-fare lass ow-fare lass D2 Fgure 5: gh-fare passengers overflow he total revenue for arlne s π = E [ p mn D, B ) p mn D, R )].. 7) he frst dervatve of the obetve funton wll be useful n the followng theorems. We fnd π B = p Pr D > B ) p Pr D > C B, D > B ). 8) he exstene of a pure-strategy Nash equlbrum, establshed n the followng proposton, follows from the supermodularty of the game. hs result holds for any demand dstrbuton, nludng dstrbutons wth orrelaton among arlnes and among fare lasses. As was the ase wth Proposton, the demand dstrbuton may be ontnuous or dsrete. Proposton 2. Gven overflow by hgh-fare ustomers only, a pure-strategy Nash equlbrum n bookng lmts B, B 2 ) exsts. Proof: By the reasonng presented n the proof of Proposton, the obetve funton 7) s ontnuous. We wll now show that both mn D, B ) and mn D, R ) are submodular, so that the obetve funton s submodular for any gven demand realzaton and therefore the expetaton 7) s submodular. hs s suffent to establsh the exstene of a pure-strategy Nash equlbrum opks, 999). 6

19 Observe that mn D, B ) depends only on B and hene s submodular. By the defntons above, R s dereasng n B and D s nreasng n B. By emma n the proof of Proposton, mn D, R ) s submodular. o show there s a sngle, unque equlbrum, we make the followng assumptons: Assumpton : here exsts, for eah random varable, a fnte probablty densty funton f τ ) = d Pr D < τ ) dτ. In addton, the densty funtons τ ) > 0 D k k / =,2. f for 0 τ C and Assumpton 2: Demands for low-fare and hgh-fare tkets are ndependent. More formally, let D = D, D 2) and D = D, D 2). We assume that D and D are mutually ndependent. Assumpton 3: PrD > C) > 0 for =,2. D Proposton 3. Gven overflow by hgh-fare ustomers only and Assumptons -3, there s a unque, globally stable Nash equlbrum n B, B 2 ). Proof: We wll haraterze the best reply funtons reaton funtons) of the players n the game and then wll show that the funtons are a ontraton on B, ). herefore, a sngle, unque equlbrum exsts and s stable. We wll frst show that eah funton π, wth B2 B held onstant, reahes ts maxmum at a unque pont B [ 0, C). Gven Assumpton 2, the frst dervatves of the obetve funtons may be wrtten as π B = Pr D > B ) p p Pr D > C B )) =,2. 9) From Assumpton 3, the frst dervatve s always less than zero at the upper boundary C: π B B= C = Pr D > C) p p Pr D > 0)) = Pr D > C) p p ) < 0. 0) Now onsder two ases. If Pr D > C) < p / p then 9) s postve when evaluated at the lower boundary: 7

20 π B B= 0 = Pr D > 0) p p Pr D > C)) = p p Pr D > C) > 0. ) By assumpton, Pr D > C B ) s strtly nreasng n B, and the obetve funton s strtly quas-onave n the nterval [0, C]. If there s an nteror soluton t s determned by the followng frst-order ondtons note that we have expanded the termd ): p Pr D D C mn D, B )) > C B ) =, =, = 2 and = 2, =. 2) p If Pr D > C) p / p, then the obetve funton s not nreasng at 0 and the slope does not hange sgn n the nterval [0, C]. herefore, the obetve s maxmzed at B = 0. Equaton 2) and the boundary ondton spefy reaton funtons r ) and r ) for B2 the two players. When the value of the reaton funton s n the nteror 0, C) then mplt dfferentaton of 2) fnds the magntudes of the slopes of the reaton funtons: 4 2 B r B ) f D = B D> C B C B f D ) Pr C B D ) > C B ) <. 3) If the value of the reaton funton s a boundary soluton, B = 0, then r B ) / B = 0 <. herefore, the reaton funtons B ), r )) r are a ontraton on B, ). 2 2 B B2 From Proposton 2, we know that at least one equlbrum pont exsts. From the proof of heorem 2.5 of Fredman [986], f there s at least one equlbrum pont and the reaton funton s a ontraton then the game has exatly one equlbrum pont. In addton, the expresson for the dervatve n 3) mples that r B B 2 2 ) r2 B) B < 4) so that the equlbrum s stable Mouln, 986). 4 In 3), the expresson τ ) f represents the densty funton of D gven event A. D A 8

21 hs result allows us to say somethng stronger about the full-overflow ase of Seton 2. when low-fare demand s suffent to fll both arraft. Corollary. Assume overflow by both low-fare and hgh-fare ustomers. Gven Assumptons and 2, and gven that low-fare demand s extremely large Pr D > C) = for =,2), there s a unque, globally stable Nash equlbrum n B, B 2 ). Proof: In ths ase, arlnes only ompete for hgh-fare ustomers and the overflow of low-fare ustomers an be gnored beause there are no extra seats to aommodate them. In the full model obetve funton ), we replae mn D, B ) wth B. hs s a speal ase of the model examned n Seton 3.. herefore, the unqueness and stablty results hold here. 3.2 ow-fare Overflow Only We wll now assume that hgh-fare passengers do not overflow and that only low-fare passengers swth arlnes f ther frst hoe s fully booked see Fgure 6). Arlne Arlne 2 D Full fare lass Full fare lass D2 B B2 MnB 2,D 2) MnB,D ) D Dsount fare lass D - B) D2 - B2) Dsount fare lass D2 Fgure 6. ow-fare passengers overflow Frst defne: D = D D B ), total demand for low-fare tkets on arlne, =, =2 and =2, =. R = C mn D, B ), the number of seats avalable for hgh-fare passengers on arlne. 9

22 he number of low-fare tkets sold s equal to mn D, B ) and the total revenue for arlne s [ p mn D, B ) p mn D, R )] π = E. 5) Surprsngly, a pure-strategy equlbrum need not exst for ths smple game. he obetve funton s not neessarly submodular or quas-onave. owever, under Assumptons -3 the equlbrum s unque and stable. Proposton 4. Gven overflow by low-fare ustomers only and Assumptons -3, there s a unque, globally stable Nash equlbrum n B, B 2 ). Proof: Gven ndependene between hgh and low-fare demands, the frst dervatve of the obetve funton 5) s π B = Pr D > B ) p p Pr D > C B )). 6) he obetve funton s quas-onave on [0,C] and t an be shown that the optmal soluton s always n the nteror, 0,C). he frst-order ondton p Pr D > C B ) = 7) p depends only on B and not on the ompettor's aton B. herefore, 7) defnes the unque optmal soluton for eah arlne and eah reaton funton has a slope of zero. he reaton funtons are a ontraton on B, ) and, followng the reasonng of the proof of Proposton B2 3, ths ontraton leads to a unque, globally stable equlbrum. hs soluton s dental to the soluton for a stand-alone arlne. When hgh-fare ustomers do not swth arlnes and hgh-fare and low-fare demands are ndependent, the optmal bookng lmts for both stand-alone arlnes and arlnes n ompetton are not nfluened by the demand dstrbutons of low-fare ustomers. 4. Comparng the Compettors and a Monopolst We wll now ompare the behavor of two arlnes n ompetton wth the behavor of a monopolst. Note that the term 'monopolst' does not neessarly mply that a sngle frm s the 20

23 only arrer on a partular route. he 'monopolst' may be two arlnes n an allane to oordnate yeld management desons. In addton, two arlnes may ompete on a partular route at ertan tmes of day, whle eah arlne may hold a vrtual monopoly at other tmes of day beause ts ompettor has not sheduled a ompetng flght at a pont lose n tme. For example, Unted Arlnes has the only dret flght from Rohester, NY to the Washngton DC area n the evenng, whle most of ts flghts durng the mornng and afternoon ompete dretly wth flghts by US Arways. In general, we wll fnd that the total bookng lmt for the monopolst s never less than the sum of the bookng lmts of two ompetng arlnes. In ths seton we provde a proof of ths result, gven a model wth hgh-fare overflow only the model presented n Seton 3.). In the followng seton we wll present numeral experments utlzng the full model of Seton 2.. o smplfy the omparson, we assume that the pre raton p /p and the dstrbutons of onsumer demands D k are equal under the ompettve and monopoly envronments. In Seton 6 we wll dsuss the mplatons of these assumptons. Our results are onsstent wth the fndngs of ppman and MCardle [997], who analyze ompetng newsvendors. hey fnd that ompetton never leads to a derease n total nventory. he 'nventory' of eah newsvendor s analogous to the stok of proteted hgh-fare seats, C B, and the demand for newspapers s analogous to demand by hgh-fare ustomers. owever, our problem norporates a sgnfant omplaton, the stohast demand by lowfare, as well as hgh-fare, ustomers. Frst we revew the ase wth no ompetton and only one arraft wth apaty C n the market for further detals, see Belobaba, 989, and Brumelle et.al., 990). Sne there s ust one arraft, we wll suppress the subsrpt =,2 whh denotes the arraft n the ompettve ase. After establshng a bookng lmt B, the arlne wll sell mn D, B) low-fare tkets and mn D, C mn D, B) ) he frst dervatve s hgh-fare tkets. herefore the total revenue s [ p mn D, B) p mn D, C mn D, B) )] π = E. 8) D F π B = p Pr D > B) p Pr D > C B, D > B). 9) 2

24 As mentoned n Seton 2, the frst-order ondtons are suffent for a soluton when Pr D > C B D B) s nondereasng n B [Brumelle et al. 990]. Note that ths ondton > s satsfed f D and D are ndependent. Gven ths property, a soluton B * wthn the nterval 0,C) satsfes 5 * * Pr D > C B D > B ) = p p. 20) Now onsder an arlne wth a monopoly or an allane between two arlnes) operatng two flghts. Passenger arrvals and overflows follow the order of events desrbed by Steps -5 at the begnnng of Seton 2.. Whle ths ase may seem to be more omplex than the sngleflght problem, t redues to the smpler problem desrbed above, sne the passenger overflow from one arraft s aptured by the same frm n the other arraft. We an wrte the obetve funton n ths two-arraft ase as [ p D D, B B ) p mn D D, 2C mn D D, B )] π = E mn B2 2) and the frst dervatve s smlar to 9) above, wth B = B B2 : π B = p Pr D D2 > B ) p Pr D D 2 > 2C B, D D2 > B ). 22) Now we onsder the stuaton ntrodued n Seton 3.. Assume that low-fare ustomers do not overflow to a seond-hoe flght whle hgh-fare passengers do overflow. he obetve funton for the monopoly arlne s p π = E mn D ) ), B) mn D2, B2 ) p mn D D 2,2C mn D, B) mn D2, B2 ) * * An nteror soluton B, ) satsfes the followng frst-order ondtons for =, =2 and =2, =: 6 B2. 23) 5 here s also a boundary ondton. If Pr D > C) p / p then B * = 0. 6 Agan, there are boundary ondtons. We present ondtons for 'extreme' solutons here. If * * Pr D D > 2C ) p / p then B, B ) = 0,0). If Pr D D > C mn D, C) p / p for =,2, then * * B, B ) = C, ). C 2 22

25 π B * B *, B ) = p Pr D > B * ) p Pr D D 2 > 2C B * mn D, B * ), D > B * ) = 0. 24) * * here may be multple values of B, ) that satsfy 24). B2 hs frst-order ondton and the frst-order ondtons 2) that unquely determne the ompettve equlbrum allow us to ompare, analytally, the entralzed and ompettve solutons. Proposton 5. Assume overflow by hgh-fare ustomers only and Assumptons -3. Also assume that the optmal soluton for the monopolst as well as the equlbrum under ompetton are n the nteror, e.g., B 0, C). hen the total number of proteted seats, B B 2, s lower under ompetton than under the entralzed soluton. Proof: Defne a B, =,2, as the optmal desons for the monopolst 'a' for allane) and defne B, =,2, as the equlbrum desons under ompetton. he allane soluton s determned by the frst-order ondtons, equatons 24). Gven Assumptons and 2, these frst-order ondtons may be re-wrtten as a a D > 2C B mn D, B )) Pr D 2 = p for =, =2 and =2, =. 25) p he ompettve optmalty ondtons 2) may be re-wrtten as: Pr D D > 2C B mn D, B )) 2 Pr D D < 2C B mn D, B ), D > C B ) 2 = p p 26) for =, =2 and =2, =. Note that 25) and 26) dffer by a sngle probablty term n the lefthand sde of 26). Sne ths extra term n nonnegatve and the rght-hand sdes are equal, the followng nequaltes hold smultaneously: a a D D > 2C B mn D, B )) Pr D D > 2C B mn D, B )) Pr a a D D > 2C B mn D, B )) Pr D D > 2C B mn D, B )) Pr Sne ths must be true for any value of C, these nequaltes defne stohast orders on two pars, 27). 28) 23

26 of sngle-valued funtons of random varables algebra manpulaton, 27) and 28) may be re-wrtten as where D and D. o make ths lear, after some a a a D D2 mn B D2, B B2 ) st D D 2 mn B D2, B B2 ), 29) a a a D D 2 mn B2 D, B B2 ) st D D2 mn B2 D, B B2 ), 30) X st Y ndates that X s smaller than Y n the usual stohast order. Beause of the ndependene between low-fare and hgh-fare demands Assumpton ) and the preservaton of stohast order under onvoluton Shaked and Shanthkumar, 994), a a a mn B D2, B B2 ) st mn B D2, B B2 ), 3) a a a mn B 2 D, B B2 ) st mn B2 D, B B2 ). 32) Fnally, by ontradton, assume that a a B B2 > B B2. hen for both nequaltes 3) and 32) to hold we would need smultaneously assumpton. ene, a a B B2 < B B2. a B B < and a B2 < B2, whh s nonsstent wth the Proposton 5 mples that, under ompetton, at least as many seats are held for hgh-fare ustomers as s optmal under ont proft maxmzaton. For the monopolst, every hgh-fare passenger who does not fnd a seat at arlne and turns to arlne s not 'lost' to the frm. Under ompetton, however, when arlne establshes a lower bookng lmt, arlne lowers ts bookng lmt as well as the two arlnes ompete for hgh-fare passengers. 5. Numeral Experments o determne whether the prevous seton's results apply to the full-fledged game desrbed n Seton 2., we alulate numerally both the ompettve equlbrum and the optmal monopoly soluton under a wde varety of parameter values. Our goal s to see whether the bookng lmt set by the monopoly, a a B B2, s onsstently greater than or equal to the total bookng lmt under ompetton, B B2. For eah senaro, demand s dstrbuted aordng to a multvarate normal dstrbuton and trunated at zero; any negatve demand s added to a mass pont at zero. Solutons are found by a 24

27 smple gradent algorthm and the gradents themselves, expressons 2) and 22), were evaluated by Monte Carlo ntegraton a smple searh proedure was also used f the obetve funton was not quas-onave). he senaros are reated by ombnng the followng parameters. - Rato of hgh fare to low fare: o over a range that nludes many atual pre ratos, we use the followng values: p / p = [.5, 2, 3, 4]. - Proporton of demand due to low-fare passengers: et µ µ ) be the average low-fare hgh-fare) demand for arlne, =,2. Beause n prate low-fare demand s often greater than hgh-fare demand, we assume that µ, and we use proportons µ µ ) µ / µ = [0.5, 0.75, 0.9]. Below we wll also dsuss experments n whh µ µ ) < 0.5. / µ - Proporton of demand due to arlne : et µ k µ k2) be the average demand for arlne 2), for demand lass k=,. Due to symmetry, we need only test senaros where µ k < µ k2. We use ratos µ k / µ k µ k 2) = [0., 0.25, 0.5]. - Varablty: o lmt the number of parameters, we assume that all four ustomer demand dstrbutons have the same oeffent of varaton, CV. We use values CV = [0.25, 0.5,,.5, 2]. Note that CV's hgher than rarely our n prate Jaobs, Ratlff and Smth [2000] desrbe 0.2 to 0.6 as a reasonable range for the CV). owever, we felt that there s some value n examnng envronments wth hghly varable demands. When we present the results below, we present both the aggregate results and the results for low CVs CV = 0.25 or 0.5). - Correlaton: Agan, to lmt the number of parameters, we assume that the orrelatons among all demands are equal. When four random varables are dstrbuted aordng to the multvarate normal dstrbuton, the lowest possble ommon orrelaton s /4 ) = 0.33; when the ommon orrelaton s lower than ths bound the ovarane matrx s not postve defnte ong, 980). For orrelaton, we use values ρ = [-0.3, 0.0, 0.5, 0.9]. When ombned, these parameters defne 4 *3*3*5*4 = 720senaros. 25

28 Before we examne aggregate statsts from the 720 senaros, let us fous on a sngle 'baselne' senaro. We hoose p / p = 2, CV = 0.5, and ρ = 0, set the mean low-fare demand to eah arlne at 50 passengers, and set the mean hgh-fare demand at 50 passengers so that µ / µ µ ) =0.75 and µ k / µ k µ k 2) = 0.5. Whle ertan parameter values nluded n the ranges above are unlkely to our n prate, ths senaro s relatvely plausble. Fgure 2 dsplays the reaton funtons of the arlnes, gven these parameter values. here s a unque equlbrum, resultng n B = B 44. herefore, the total bookng lmt s 2 = 288 and the arlnes reserve a total of 2 seats for hgh-fare ustomers. A monopolst, on the other hand, has an optmal total bookng lmt of B a B a 300 seats, wth 00 seats set asde 2 = for hgh-fare ustomers. If we defne the "serve level" as the probablty that a ustomer s able to purhase a seat on ether arraft, the dfferene n bookng lmts produes sgnfantly dfferent serve levels for eah ustomer lass. Under ompetton, 45% of low-fare ustomers found a seat on ether flght, whle under a monopoly the low-fare serve level rses to 50%. On the other hand, hgh-fare passengers beneft from ompetton. her serve level s 77% under ompetton, 70% under the monopoly. Whle ths partular example produed a unque equlbrum, n Seton 2 we saw that the full-overflow game may have multple equlbra or may not have any equlbra at all. Suh an outome would omplate the omparson between ompettve and monopoly bookng lmts. owever, by examnng the arlne response funtons for eah of the 720 senaros, we saw that n every ase an equlbrum exsts and was unque. All response funtons were ontnuous, and most produed a stable equlbrum, as n Fgure 2. As mentoned above, an extremely low negatve) orrelaton between hgh and low-fare demands an generate the outome shown n Fgure 4, n whh no pure-strategy equlbrum exsts. We have also found nstanes of multple equlbra when the rato µ µ ) s low e.g., 0.) and orrelaton / µ s negatve or zero. We wll dsuss these ases at the end of ths Seton. Frst we ompare the total bookng lmts n the ompettve and monopoly envronments for the orgnal 720 senaros. In every senaro, the bookng lmt for the monopoly s equal to, or greater than, the sum of the bookng lmts for the arlnes n ompetton. he mean a a dfferene B B ) B B ) aross all senaros s 5 seats, and the dfferene vares from

29 seats to 3 seats. When we examne only those senaros wth CV=0.25 or CV=0.5, the dfferenes are smaller. Under these senaros, the average dfferene s 9 seats wth a range from 0 to 03 seats. In general, the largest dfferenes our when orrelaton s low ρ = -0.3) and expeted demands are equally balaned among arlnes and lasses when µ µ ) =0.5 and / µ a a µ k / µ k µ k 2) =0.5). able 2 dsplays the dfferene B B2 ) B B2) for eah value of ρ. Eah olumn of able 2 represents an average over 80 senaros. As the orrelaton nreases, the dfferene between the monopoly and ompettve ases dereases. Avg. monopoly total bookng lmt Avg. ompettve total bookng lmt a a B B2 ρ = 0.3 ρ = 0.0 ρ = 0.5 ρ = B B a a Average B B ) B B ) ow-fare serve level monopoly-ompettve) 0.0% 4.%.3% 0.4% gh-fare serve level monopoly-ompettve) -0.0% -4.7% -.7% -0.4% able 2. Demand orrelaton and the effets of ompetton. he dfferenes n bookng lmts have a sgnfant effet on the serve levels offered to eah ustomer lass. Over all ases, the serve level offered to low-fare ustomers rose an average of 4% under the monopoly 39% to 43%), whle the serve level offered to hgh-fare ustomers delnes an average of 4% under the monopoly 75% to 7%). For senaros wth low CVs the average dfferenes were a bt smaller: 3.7% and 3.4%, respetvely. In addton, the range of results was extremely large. In fve senaros out of 720, monopoly low-fare serve levels were over 50% greater than the low-fare serve levels under ompetton. he dfferene n hgh-fare serve levels was as hgh as 3%. In general, the dfferene n total profts between the monopoly and ompettve ases was small. Averaged over all 720 senaros, profts to the monopoly are ust 0.3% hgher than the total profts under ompetton, wth a range from 0% to 5%. When restrted to senaros wth CV=0.25 or CV=0.5, the average dfferene n profts s 0.2% wth a range from 0% to 3.5%. he largest dfferenes n proft were seen when orrelaton s low, p / p s hgh, and expeted 27

30 demands are equally balaned among arlnes and lasses. hese small dfferenes n proft are not unexpeted sne n most ases the obetve funton s relatvely 'flat' near the optmum. It s more dffult to make these omparsons when the proporton of demand due to lowfare passengers s small µ µ ) < 0.5) beause senaros wth multple ompettve / µ equlbra begn to appear. For example, wth µ µ ) =0., we dentfed one senaro, / µ shown n Fgure 3, wth three equlbra: B = 6, B 36), B = 36, B 6), and 2 = 2 = 2 = B = 22, B 22). owever, under ths senaro the monopoly soluton s B a B a 93. As 2 = was true for the orgnal 720 senaros, at eah ompettve equlbrum the total bookng lmt s smaller than or equal to the bookng lmt hosen by a monopoly. hs was true for all examned senaros wth µ µ ) < 0.5. / µ 6. Observatons and Future Researh In ths paper we have examned how ompetton affets a fundamental deson n yeld management, the alloaton of seats among low and hgh-fare lasses. Besdes the tehnal results onernng the exstene and unqueness of ompettve equlbra and the analytal expressons for the frst-order ondtons, our prmary fndng s that the sum of the arlnes' bookng lmts under ompetton s no hgher than the total bookng lmt produed when total profts from both flghts are maxmzed as n a monopoly or when arlnes ooperate n settng bookng lmts). Under ompetton more hgh-fare tkets and fewer low-fare tkets may be sold than under a monopoly. hs s not an obvous result, for n many standard eonom models ompetton leads to a fall n pres e.g., a smple Bertrand model of pre ompetton). ere, we have held pres onstant, but ompetton leads to a realloaton of nventory among ustomer segments, produng a rse n the average pre pad for an arlne seat. Under the monopoly soluton, low-fare ustomers are more lkely to fnd a seat, and are more lkely to fnd a seat on a frst-hoe arlne, than under ompetton. Wth pres held onstant, a monopolst would mprove serve for the low-pre segment whle dmnshng serve for the hgh-pre segment. hs may be partularly nterestng n regulatory envronments n whh anttrust laws prohbt arlnes from olludng on pres but allow them to oordnate yeld management desons. 28