Revenue Management Games: Horizontal and Vertical Competition

Transcription

1 Revenue Mngement Gmes: Horzontl nd Vertl Competton Sergue Netessne he Whrton Shool Unversty of Pennsylvn Phldelph, PA Robert A. Shumsky Smon Shool of usness Admnstrton Unversty of Rohester Rohester, NY Forthomng n Mngement Sene November 2001, Revsed June 2003, August 2004, November 2004 Abstrt: A well-studed problem n the lterture on rlne revenue (or yeld) mngement s the optml lloton of set nventory mong fre lsses gven demnd dstrbuton for eh lss. In prte, the set lloton desons of one rlne ffet the pssenger demnds for sets on other rlnes. In ths pper we exmne the set nventory ontrol problem under both horzontl ompetton (two rlnes ompete for pssengers on the sme flght leg) nd vertl ompetton (dfferent rlnes fly dfferent legs on mult-leg tnerry). Suh vertl ompetton n be the outome of ode shrng greement between rlnes, for eh rlne sells sets on the prtner rlnes flghts but the rlnes re unwllng, or unble, to oordnte yeld mngement desons. We provde qute generl suffent ondton under whh pure-strtegy Nsh equlbrum exsts n these 'revenue mngement gmes', nd we lso ompre the totl number of sets vlble n eh fre lss wth, nd wthout, ompetton. Anlytl results s well s numerl exmples demonstrte tht under horzontl ompetton more sets re proteted for hgher-fre pssengers thn when sngle rlne ts s monopoly, whle under vertl ompetton the bookng lmt my be hgher, or lower, thn the monopoly level, dependng upon the demnd for onnetng flghts n eh fre lss. Fnlly, we dsuss revenue-shrng ontrts tht oordnte the tons of both rlnes.

2 1. Introduton Consder n rlne ustomer lookng for n erly-mornng flght from Rohester, NY to Chgo n My he trveler n hoose between two rlnes, Amern nd Unted, who offer flghts t nerly dentl tmes (6:00 m nd 6:10 m, respetvely) t dentl pres (both hrge $266 for 14-dy nd $315 for 7-dy dvne purhse round-trp tkets). Now suppose tht our ustomer wshes to purhse Amern s 14-dy dvne tket. If the sets lloted to the 14-dy fre lss hve sold out, t s lkely tht the ustomer wll ttempt to purhse tket n the sme fre lss on the Unted flght tht deprts 10 mnutes lter. In generl, the lloton of set nventory mong fre lsses by one rlne ffets the quntty of ustomer demnd, nd optml set lloton, of the other rlne. If the rlnes re ompettors nd do not ollborte on set lloton desons, then the desons tht rse out of the resultng gme n dffer sgnfntly from the set llotons tht would be optml for sngle deson-mker wth ontrol over both rlnes. Here we use the term horzontl ompetton when desrbng rlnes tht ompete for ustomers on sngle flght-leg. We wll use the term vertl ompetton to refer to dfferent rlnes tht fly dfferent legs on mult-leg tnerry. For exmple, trveler who wshes to fly from Clevelnd to okyo n purhse onvenent two-leg tnerry wth the frst leg operted by Contnentl Arlnes from Clevelnd to Los Angeles, nd the seond leg operted by Northwest Arlnes from Los Angeles to okyo. (We ll ths ustomer onnetng pssenger. ) euse these two rlnes hve formed n llne tht llows eh rlne to sell tkets for flghts operted by the llne prtner (n rrngement lled ode shrng), the Clevelnd-Los Angeles-okyo tket n be purhsed from ether rlne. In ths se, the term vertl ompetton my seem to be msnomer, for rlnes n suh n llne ooperte by ode shrng. However, tehnl nd legl brrers often prevent the rlnes from expltly oordntng both prng nd set nventory desons. 1 As ws true for horzontl ompetton, the set nventory desons of one rlne ffet ustomer demnd nd nventory desons of the other, nd the bsene of perfet oordnton n led to nventory lloton desons tht dffer sgnfntly from the system-optml soluton. In ths pper we exmne how both horzontl nd vertl ompetton ffet rlne set nventory desons, nd we onsder how rlnes n n llne my oordnte these desons by greements 1 oyd (1998) desrbes how oordntng the revenue mngement systems of multple rlnes n be logstl nghtmre, nd he explns tht n rlne my beleve tht ts own propretry system provdes ompettve dvntge nd therefore the rlne my resst mergng systems wth n lly. In ddton, mny llnes re not mmunzed from nt-trust regultons, nd rlne revenue mngers for these rlnes re onerned tht the Federl Government my see overt oordnton of set nventory desons s volton of nttrust lw (Zukermn, 2002). 1

3 smlr to revenue-shrng ontrts. here s evdene tht over the pst ten yers llnes hve beome more numerous nd nresngly mportnt for the rlne ndustry. he U.S. rlnes hve reently formed the two lrgest llnes n hstory (Unted Arlnes nd US Arwys n 2002, nd elt Arlnes oned Northwest nd Contnentl n 2003). he frst mor nterntonl llnes were formed n the erly nd md-1990 s, nd nterntonl pssenger trff between hubs of ode-shrng prtners nresed by 43% between 1992 nd 1997, whle nterntonl trff between other hubs nresed by ust 7% (U. S. eprtment of rnsportton, 2000). here re mny dvntges to suh llnes. hey llow rlnes to expnd nto new mrkets nd to nrese the frequeny of flghts offered wthn mrkets served by both rlnes. If gven nttrust mmunty, llne prtners n nrese profts by oordntng tvtes suh s mrketng, purhsng, luggge hndlng nd flght shedulng. hese prtners my lso pply revenue mngement tehnques to optmze ont revenues (for thorough summry of the benefts nd osts of rlne llnes, see Fernndez de l orre, 1999). However, s we mentoned bove, there n be sgnfnt brrers to suh oordnton, nd the rlnes struggle to fnd deentrlzed mehnsms to ntegrte revenue mngement desons. here s substntl eonoms lterture nlyzng rlne behvor under ompetton nd the mpt of rlne llnes. here s lso reent strem of opertons reserh lterture on the problem of optml set lloton. However, to our knowledge there re no publshed ppers tht ple the set lloton problem n ompettve frmework, exmne the mpt of rlne llnes on these desons, or spefy ontrts tht enble rlnes to oordnte desons. We ddress these ssues here, for n understndng of how ompetton ffets revenue mngement desons would be helpful for rlne mngers who negotte ontrts wth potentl llne prtners or desgn systems to mplement llnes, s well s for government regultors who must evlute the mpt of rlne llnes. After revew of the lterture n Seton 2, n Seton 3 we desrbe the trdtonl sngle-rlne, sngle-leg, two-lss yeld mngement problem, nd we ntrodue ondton on the demnd dstrbuton tht wll be useful n lter setons. We dsuss horzontl ompetton n Seton 4. In ths se, eh rlne s fed wth n ntl demnd from pssengers who wsh to purhse tkets, but eh rlne my lso sell tkets to pssengers tht were dened reservton on the ompetng rlne. In Seton 5 we onsder vertl ompetton: eh rlne hs both lol nd onnetng demnd, but the number of onnetng pssengers booked on one rlne s lmted by the number booked by n llne prtner. In both ses the optml pty lmts for eh lss (the bookng lmts) on eh rlne re nterdependent. We ompre the revenue mngement poles of ompetng rlnes wth the poly of monopolst who opertes ll flghts or, equvlently, the poles of rlnes tht ooperte to mxmze totl profts. Anlytl results for spel ses s well s numerl exmples demonstrte tht under 2

4 horzontl ompetton more sets re proteted for hgher-fre pssengers thn when sngle rlne ts s monopoly, whle under vertl ompetton the bookng lmt my be hgher, or lower, thn the monopoly level, dependng upon the quntty of onnetng demnd n eh fre lss. In Seton 6 we present revenue-shrng ontrts tht oordnte the tons of both rlnes, nd n Seton 7 we dsuss mngerl nsghts nd onsder dretons for further work. 2. Relted Lterture Publtons tht onsder the ntertons mong eonom fores, strteg rlne mrket entry desons, nd rlne shedules nlude the network desgn models of Lederer nd Nmbmmdom (1999), obson nd Lederer (1994), nd the emprl work by orensten nd Rose (1994). Another body of reserh fouses on the rlnes' shedulng desons under ompetton usng vrnts of the sptl model developed by Hotellng (1929). See, for exmple, the reent emprl ppers by orensten nd Netz (1999) nd Rhrd (2003). hese ppers fous on brod ompettve problems nd gnore the spefs of set nventory lloton. In ths pper we wll not be onerned wth the resons rlnes shedule ther flghts t the sme tme or wth the prng deson for eh flght. Rther, we wll onentrte on the mpltons of ompettve shedulng on set nventory ontrol. here re numerous ppers n the re of revenue mngement tht fous on rlne yeld mngement. For fundmentl results on the generl subet of set nventory ontrol see elobb (1989), rumelle et l. (1990), nd useful lterture revew by MGll nd vn Ryzn (1999). Some of our results for horzontl ompetton re relted to the unpublshed work by L nd Oum (1998), whh desrbes set lloton problem for two rlnes n ompetton but uses reltvely restrtve ssumpton bout how demnd s lloted mong rlnes nd dentfes one, symmetr equlbrum. Zho nd Atkns (2002) desrbe model wth two rlnes ompetng for pssengers n one demnd lss. elobb nd Wlson (1997) desrbe smulton model tht s used to evlute the benefts of yeld mngement systems for rlnes under ompetton. he lterture on nventory mngement hs seen strem of losely relted ppers devoted to ompetton mong frms n whh the frms determne nventory levels nd ustomers my swth mong frms untl sutble produt s found. hs hs been desrbed s 'newsvendor gme' or s nventory ompetton nd s relted to our onept of horzontl ompetton. Prlr (1988) exmnes the ompetton between two retlers fng ndependent demnds. Lppmn nd MCrdle (1997) exmne both the two-frm gme nd gme wth n rbtrry number of plyers. In ther models, ntl ndustry demnd s lloted mong the plyers ordng to pre-spefed 'splttng rule.' hey estblsh the exstene of pure strtegy Nsh equlbrum nd show tht the equlbrum s unque when the ntl 3

5 lloton s determnst nd strtly nresng n the totl ndustry demnd for eh plyer. Reent extensons of these models nlude Mhn nd vn Ryzn (1999) who model demnd s stohst sequene of utlty-mxmzng ustomers. For n rbtrry number of frms, they demonstrte tht n equlbrum exsts nd show tht t s unque for symmetr gme. Netessne nd Rud (2003) nlyze problem smlr to Prlr (1988) but for n rbtrry number of produts. Gven mld prmetr ssumptons they estblsh the exstene of, nd hrterze, unque, globlly stble Nsh equlbrum. Mny of these ppers ompre totl nventory levels under ompetton wth nventory levels when frms ooperte. Lppmn nd MCrdle (1997) show tht ompetton n led to hgher nventores, nd Mhn nd vn Ryzn (1999) derve smlr results gven ther dynm model of ustomer purhsng. On the other hnd, wth the substtuton struture of the model of Netessne nd Rud (2003), under ompetton some frms my stok less thn under entrlzton. It s not mmedtely ler how these results n be extended to our model of horzontl ompetton, for the problem desrbed here nvolves the lloton of fxed nventory between two ustomer lsses. herefore, overstokng of nventory n one lss would mply understokng n the other lss. We wll lso see tht n our problem, n rlne s demnd depends upon ts own bookng-lmt deson, omplton not seen n ny newsvendor gme. In ddton, our demnd model for eh fre lss s more generl thn those tht hve been used n the lterture. Lppmn nd MCrdle s stylzed splttng rules generte demnds tht re ether ndependent or perfetly orrelted, whle Prlr only onsders ndependent demnds. In our model, ustomer demnd my follow n rbtrry dstrbuton. esdes beng more nturl desrpton of relworld demnd, llowng generl demnd dstrbuton leds to ddtonl nsghts. For exmple, the trdtonl newsvendor gme lwys hs pure strtegy equlbrum, but ours my not. We wll see tht the orrelton struture of demnd between produts hs n mportnt mpt on the dynms of the gme, nd our model llows us to fnd generl ondtons under whh the gme hs n equlbrum. Anlyss of vertl ompetton dtes bk to Spengler (1950) who demonstrted n smple supply hn tht the effet of double-mrgnlzton leds to suboptml system performne. Sne then there hve been numerous studes of vertl ompetton n the supply hn settng but, to the best of our knowledge, no nlyss n the revenue mngement settng. Whle there s both theoretl nd emprl reserh nvestgtng the mpt of ode shrng greements on tket pres (see, for exmple, ruekner nd Whlen 2000, nd ruekner 2003), to the best of our knowledge there s no publshed work nvestgtng the neffenes tht rse durng dy-to-dy yeld mngement desons. he spef terms of ode shrng greements re usully not vlble to publ. However, 4

6 Wynne (1995) nd oyd (1998) desrbe some of the oordnton mehnsms tht re used n prte. esdes short ounts n llur nd vn Ryzn (2004) nd oyd (1998b), there hs been no publshed nlytl nlyss of ontrts mong rlne llne prtners. In generl, there s muh onfuson, nd very lttle theoretl support, for llne prtners when they ttempt to desgn ontrts. he ontrts we propose re relted to work of Chon nd Lrvere (2000) on revenue shrng n supply hn mngement. See lso Chon (2004) for the survey of ontrtng lterture n opertons mngement. 3. he Sngle-Arlne, Sngle-Leg Problem efore dsussng the effets of ompetton on yeld mngement prtes, we revew the trdtonl stnd-lone yeld mngement problem, ntrodue some notton, nd derve result tht wll be used throughout the pper. Let p k be the net revenue from pssenger n lss k=l,h (low-fre nd hgh-fre), whh tkes nto ount the vrble ost of flyng tht pssenger. emnd for tkets t these pres s represented by the rndom vrbles k, k = L, H. A tket purhsed for ether fre provdes ess to the sme produt: oh-lss set on one flght leg. We ssume tht demnd for low-fre tkets ours before demnd for hgh-fre tkets, whh s the se when dvne-purhse requrements re used to dstngush between ustomers wth dfferent vlutons on pre nd flexblty. Customers who prefer low fre nd re wllng to ept the purhse restrtons wll be lled 'low-fre ustomers'. Customers who prefer to purhse lter, t the hgher pre, re lled 'hgh-fre ustomers'. We ssume tht there re no ustomer nelltons. o mxmze expeted profts, the rlne estblshes bookng lmt for low-fre sets. Note tht the estblshment of bookng lmt s n optml poly for eh rlne see rumelle et l. (1990). One the bookng lmt s rehed, the low fre s losed. Sles of hgh-fre tkets re epted untl ether the rplne s full or the flght deprts. hs model ontns only two fre lsses, when n relty there my be mny more (see elobb 1998 for n ntroduton to the omplextes of rel-world yeld mngement systems). We lso ssume tht the rlnes' bookng lmts re stt. ht s, the bookng lmt s set before demnd s relzed nd no dustments re mde s low-fre demnd s observed. hs model smplfes other spets of the tul envronment. For exmple, we ssume tht pssenger dened low-fre bookng does not ttempt to upgrde to the hgh-fre lss nd tht pssenger, when frst dened tket, wll not shft to lter or erler flght operted by the sme rlne. hese ssumptons pper often n the lterture on nonompettve revenue mngement, nd, gven the dded omplexty of our problem nvolvng ompetton, we dopt them here s well. 5

7 Gven tht the rrft hs pty C, the rlne s optmzton problem s ( ) ( ) ( ) mx π = E pl mn L, ph mn H, C mn L, +. (2.1) he frst-order ondton s, π = pl Pr L > ph mn H > C, L > ( ) ( ) ( ) ( ) p p ( C ) = Pr > mn > > = 0. L L H H L (2.2) If, re ndependent, t s esy to see tht there s unque soluton. o ddress dependent L H demnds, rumelle et l. (1990) ntrodued the monoton ssoton property. Gven tht ( C ) Pr > > s monotonlly nresng n ( monoton ssoton ), there s unque H L soluton to (2.2). Here we ntrodue relted property, tht 6 L nd H re totlly postve of order 2 (P 2 ). P 2 mples tht relztons of the rndom vrbles re more lkely to be hgh together, or low together, thn to be mxed hgh nd low (see Joe 1997 for thorough dsusson of P 2 nd ts propertes). Proposton 1. Suppose, re P 2 (otlly Postve of Order 2). hen there s unque soluton to L H the optmlty ondton (2.2) nd the obetve funton (2.1) s qus-onve. Proof: From Proposton 2.3 of Joe (1997), P2 mples tht nd Pr( > C > ) s nresng n. he result follows. H L L H re rght-tl nresng,.e., In the proof, the suffent ondton on, s tully weker thn the P 2 ondton: the two L H rndom vrbles must be rght-tl nresng (RI), property smlr to (but stronger thn) monoton ssoton. However, workng wth the P 2 property s onvenent, for we know tht there re mny useful bvrte dstrbutons tht re P 2. hese nlude ny set of ndependent rndom vrbles, the multvrte logst, Gmm nd F dstrbutons, nd the bvrte norml dstrbuton wth postve orrelton (Krln nd Rnott, 1980, nd ong, 1990). he P 2 property n lso be extended to the multvrte P 2 property (MP 2 ). oth P 2 nd MP 2 wll be useful n the next two setons. 4. Horzontl Competton Suppose two rlnes offer dret flghts between the sme orgn nd destnton, wth deprtures nd rrvls t smlr tmes. We use subsrpts,=1,2 to dstngush two ompetng rlnes. Flghts hve the set pty C nd there re two fre lsses vlble for pssengers: 'low-fre' nd 'hgh-fre.' If ether type of ustomer s dened tket t one rlne, the ustomer wll ttempt to purhse tket from the

8 other rlne nd we ll these overflow pssengers (see Fgure 1). herefore, both rlnes re fed wth rndom ntl demnd for eh fre lss s well s demnd from ustomers who re dened tkets by the other rlne. Pssengers dened reservtons by both rlnes re lost. Fgure 1 shows the overflow proesses. We ssume tht both rlnes pres stsfy pl < ph. We lso ssume tht the rndom vrbles k hve nonnegtve support nd tht the umultve dstrbutons re dfferentble. Arlne 1 Arlne 2 H1 C 1 C 2 hgh-fre overflow from 1 to 2 Hgh fre Hgh fre lss lss hgh-fre overflow from 2 to 1 H2 1 2 L1 Low fre lss low-fre overflow from 1 to 2 low-fre overflow from 2 to 1 Low fre lss L2 Fgure 1: Horzontl Competton Note tht ll results presented n ths seton lso pply to model n whh some frton (less thn one) of pssengers dened tket on one rlne ttempt to purhse tket from the other rlne, whle some frton (greter thn zero) re lost to both rlnes. o smplfy the model nd mnmze the number of prmeters, we ssume tht ll pssengers dened tket from ther frst hoe overflow to ther seond-hoe rlne. he model desrbes two rlnes enggng n non-oopertve gme wth omplete nformton. Eh rlne ttempts to mxmze ts profts by dustng ts bookng levels. In other words, the bookng level [0,C ] s the strtegy spe of rlne (for smplty, we ssume tht the bookng level my be ny rel number n ths rnge). Eh rlne knows the strtegy spes nd demnd dstrbutons of ts own flght s well s those of the ompetng rlne. hroughout the pper we wll use equlbrum bookng lmts set by ompetng rlnes, nd we wll be omprng these wth to denote, the optml bookng lmts estblshed by n llne of two rlnes tht oordnte yeld mngement desons (or, proft-mxmzng monopoly). An mportnt ssumpton of the model s tht the ntl demnds k re exogenous; they re not ffeted by the bookng lmts hosen by eh rlne. hs ssumpton s onsstent wth the newsvendor gmes of Prlr (1988) nd Lppmn nd MCrdle (1997). However, one mght rgue tht the bookng lmts determne set vlblty, nd tht n the long run ths spet of serve qulty ffets ntl demnd. A more omplete model would norporte ths reltonshp between bookng lmts nd 7

9 demnd, nd the soluton would supply equlbrum demnds s well s equlbrum bookng lmts. For our pplton, however, the reltonshp between bookng lmts nd demnd s wekened by mrketng efforts suh s dvertsng nd frequent-flyer progrms. In ddton, the use of trvel gents nd on-lne reservton tools redues the mrgnl serh ost ssoted wth mkng eh bookng. Gven low serh osts, the deson s to whh rlne to query frst my depend on ftors tht domnte the lkelhood tht the query wll result n bookng. he order of events n the gme s onsstent wth the trdtonl yeld mngement problem: 1. Arlnes estblsh bookng lmts 1 nd Low-fre pssengers rrve to ther frst-hoe rlnes nd re ommodted up to the bookng lmts. Low-fre pssengers not ommodted on ther frst-hoe rlnes 'spll' to the lternte rlnes nd re ommodted up to the bookng lmts. 3. Hgh-fre pssengers rrve to ther frst-hoe rlnes nd re ommodted wth ny remnng sets, up to pty C n eh rrft. Hgh-fre pssengers not ommodted on ther frst-hoe rlnes 'spll' to the lternte rlnes nd re ommodted n ny remnng sets, up to pty C n eh rrft. Note tht for n rlne to enfore bookng-lmt poly t s not neessry to dstngush between orgnl nd overflow pssengers. One demnd s relzed, eh rlne smply observes the rrvl of low-fre pssengers nd then the rrvl of hgh-fre pssengers. Eh group of ustomers ontns mxture of frst-hoe nd overflow pssengers. o desrbe the problem n terms of ustomer demnd nd bookng lmts, defne: L = + ) L ( L +, totl demnd for low-fre tkets on rlne, =1, =2 nd =2, =1. R = C mn(, ), the number of sets vlble for hgh-fre pssengers on rlne = 1,2. L H = + R ) H ( H +, totl demnd for hgh-fre tkets, =1, =2 nd =2, =1. he totl revenue for rlne s π = E pl mn( L, ) + ph mn( H, R ). (3.1) Eh rlne wll mxmze ths expresson, gven the bookng lmt of ts ompettor. It wll be nstrutve to exmne the frst dervtve of ths obetve funton. It s tedous to fnd the dervtve by the trdtonl methods (e.g., pplyng Lebntz's rule). Insted, by pplyng the tehnques desrbed n Rud (2000), we fnd for =1, =2 nd =2, =1, 8

10 π = p Pr( > ) p Pr( > C, > ) L L H H L p Pr( >, <, > R, < C ). H L L H H (3.2) Although ths s omplex expresson, there s strghtforwrd nterpretton for eh term. An nrementl nrese n the bookng lmt by rlne hs three effets on tht rlne's totl revenue. Frst, revenue from low-fre ustomers nreses wth probblty Pr( > ). Seond, revenue from the hgh-fre ustomers dereses wth probblty Pr( > C, > ). Whle these two effets L H L re dret onsequenes of the hnge n, there s thrd, ndret effet. Revenue from hgh-fre ustomers my derese due to the followng sequene of events: () n nrese s my redue the overflow of low-fre ustomers from to, () reduton n the number of low-fre ustomers t my nrese the number of sets vlble for hgh-fre ustomers t, () ths my redue the overflow of hgh-fre ustomers from to nd (v) delne n the overflow from my redue the number of hghfre ustomers ommodted t. he probblty of ths sequene of events s the thrd term on the rght-hnd sde of equton (3.2), whh mples tht n nrese n the bookng lmt of rlne n result n derese n hgh-fre demnd to rlne. Hene, even though there s no explt dependene between rlne s bookng lmt nd ts demnd, suh dependene s ntrodued through the gme. euse the strtegy spes of the rlnes re ompt nd the pyoff funtons re ontnuous, Nsh equlbrum n mxed strteges must exst. However, pure-strtegy Nsh equlbrum my or my not exst for rlnes plyng ths gme. 4.1 Nsh Equlbrum Condtons In the followng proposton we show tht f L nd H re P 2 then the gme hs pure strtegy Nsh equlbrum. In orollry we show tht these ondtons lso hold f (, ) = s MP 2. L1, L2 H1, H 2 hs result ndtes tht the exstene of pure strtegy equlbrum depends upon the orrelton of demnds mong fre lsses wthn eh rlne, nd does not dretly depend upon the demnd orrelton between rlnes (lter we wll desrbe numerl exmple tht ompres these two ses). hs result helps to dstngush our nlyss from the nlyss of Prlr (1988) nd of Lppmn nd MCrdle (1997). In the lss newsvendor gme there s only one type of demnd fed by eh frm, Nsh equlbrum lwys exsts nd nrese n nventory by one frm leds to derese n nventory by the ompettors ( gme of substtutes). In our model, the ddton of nother demnd lss ompetng for the sme resoure not only ompltes the nlyss but lso rses the possblty tht there wll not be ont pure strtegy t ll, nd/or tht n nrese n the bookng lmt of one rlne my led to n nrese 9

11 n the bookng lmt of the other rlne. However, we show tht under mny resonble demnd dstrbutons, suh ounterntutve behvor does not pper. Proposton 2. Suppose tht L nd H, =1,2, re P 2. hen pure strtegy Nsh equlbrum exsts. In ddton, the best response funtons re deresng (possbly wth umps down): f one plyer nreses the bookng lmt, t s optml for the other plyer to derese the bookng lmt. Proof: See the Appendx. It s worth notng tht the proof of Proposton 2 does not rely on showng tht the gme s submodulr (ths hs been shown for newsvendor gmes, see Lppmn nd MCrdle 1997), for the gme under horzontl ompetton s not submodulr. Insted, we utlze rsky s fxed pont theorem nd explore the struturl propertes of the best response funtons. Note lso tht proposton 2 spefes ondtons on nd L H, lthough relztons of H my depend upon ll four demnd relztons. he followng Corollry desrbes pproprte ondtons on these underlyng demnd dstrbutons. Corollry 1. If (, ) re multvrte P 2 (MP 2 ) n ther densty funtons, then the L1, L2 H1, H 2 results of Proposton 2 lso hold. Proof: From heorem 2.2 n Joe (1997), f (, ) ( ( ) f ( ), f ( ) f ( ) ) f1 2 3, 4 = re MP 2, then so re L1, L2 H1, H 2, where f re nresng funtons. euse (, ) L1, L2 H1, H 2, L nd H s n nresng funton of H re P 2 s well. he results of Proposton 2 follow. As we mentoned n Seton 3, there re mny useful dstrbutons tht stsfy the MP 2 property. Multvrte norml dstrbutons of three or more rndom vrbles re MP 2 f the nverse of the ovrne mtrx stsfes ertn propertes (speflly, t must be n M-mtrx, see Krln nd Rnott, 1980). On the other hnd, f multvrte norml dstrbuton s MP 2, then neessry ondton s tht ts orrelton oeffents re non-negtve. When the demnd dstrbutons for hgh nd low-fre pssengers ( nd L H ) re strongly negtvely orrelted, then L nd the Proposton 2 suggests tht we my not see well-behved gme. However, when demnd dstrbutons between rlnes (e.g., L nd L ) re negtvely ssoted, L nd the P 2 ondton. In tht se, even though s not P2, Proposton 2 my stll hold. H re not P 2, nd H my stll stsfy Fgures 2 nd 3, for exmple, show the best response funtons r( ) of two rlnes wth C = C =200 nd demnds tht re hrterzed by lrge, negtve orrelton. o generte Fgure 2, we reted demnd dstrbuton wth orrelton of -0.9 between hgh nd low-fre demnds wthn eh rlne, 10

12 but wth ndependent demnds between rlnes. 2 For Fgure 3, hgh nd low-fre demnds re ndependent, but the between-rlne orrelton s As suggested by the Proposton 2, the results re very dfferent. Fgure 2 shows gme wthout ny pure-strtegy equlbrum, nd ontns ump n whh rse n the bookng lmt of one rlne leds to shrp rse n the bookng lmt of the other rlne. he gme n Fgure 3, on the other hnd, s well-behved, wth sngle equlbrum t bookng lmt of 50 sets. In the generl horzontl ompettve gme desrbed bove, t s lso possble to hve more thn one equlbrum. 200 r2(1) r1(2) r 1 ( 2 ) r 2 ( 1 ) Fgure 2: est response funtons wth Fgure 3: est response funtons wth or(, ) = or(, ) = 0.9, or(, ) = or(, ) = 0, L H L H L H L H or(, ) = or(, ) = 0 or(, ) = or(, ) = 0.9 L L H H L L H H 4.2 Comprng the Compettors nd Monopolst We wll now ompre the behvor of two rlnes n ompetton wth the behvor of monopolst. Note tht the term 'monopolst' does not neessrly mply tht sngle frm s the only rlne on prtulr route. he 'monopolst' my be two rlnes n n llne to oordnte yeld mngement desons. In ddton, two rlnes my ompete on prtulr route t ertn tmes of dy, whle eh rlne my hold vrtul monopoly t other tmes of dy beuse ts ompettor hs not sheduled ompetng flght t pont lose n tme. In generl, we wll fnd tht under horzontl ompetton the totl bookng lmt for the monopolst s never less thn the sum of the bookng lmts of two ompetng rlnes. o help us understnd the dynms of the gme, we frst exmne smplfed model wth hgh-fre pssenger overflow only, nd then we dsuss numerl experments wth the full model. o enble proper 2 Intutvely, umps n the best responses our beuse, under negtve orrelton, eh rlne fes two entrely opposte sttes of the world: () hevy low-fre demnd nd no hgh-fre demnd nd () hevy hgh-fre demnd nd no low-fre demnd. herefore, eh rlne estblshes ether lrge bookng lmt, essentlly bettng on the frst stte of the world, or t n estblsh low bookng lmt whle hopng for the seond stte. Under ertn problem prmeters the two desons produe the sme expeted proft so tht the obetve funton s bmodl nd the best response s to ump from one mode to the other when the ompettor s deson hnges (see Appendx, Fgure 5). However, suh n extreme negtve orrelton s unlkely to our n prte. emnd forest errors for dfferent fre lsses re more lkely to be ndependent or wekly postvely orrelted (see Seton 4.3). herefore, the outome n Fgure 3 s more plusble. 11

13 omprson, we ssume tht the pres p k nd the dstrbuton of onsumer demnds the ompettve nd monopoly envronments. k re equl under Consder the stuton when low-fre ustomers do not overflow to seond-hoe flght whle hgh-fre pssengers do overflow. hs pples n the (unlkely) se tht low-fre ustomers re wllng to fly on only one rlne. A smlr, even smpler model pples to the se where low-fre demnd s suffently lrge so tht the bookng lmts of both rlnes re lwys rehed (e.g., t s hgh enough to fll n entre rrft) nd therefore low-fre overflow s rrelevnt. Wth no low-fre overflow, obetve funton (3.1) s repled by nd the optmlty ondton π / = 0 t π = E pl mn( L, ) + ph mn( H, R), (3.3) n be wrtten s, pl Pr( H > C L > ) =. (3.4) p It n be shown tht there s unque equlbrum n ths gme (see Netessne nd Shumsky, 2004, Proposton S1 for detls). In the followng proposton we wll ompre bookng lmts under ompetton wth system-optml bookng lmts. Note tht n ths proposton nd n ll subsequent propostons we ssume tht the optml solutons for the monopolst nd the equlbrum under ompetton re n the nteror,.e.,, (0, C ). Proposton 3. Assume tht eh rlne mxmzes obetve funton (3.3), so tht there s no low-fre overflow, nd tht H H nd L re P 2. hen the bookng lmts re lower under ompetton thn the entrlzed soluton: nd. Proof: he obetve funton of the llne s the sum of the two rlnes obetve funtons, π = π + π, nd the entrlzed optmlty ondton ( π + π ) / = 0 n be wrtten s, p π L 1 Pr( H > C L > ) = +. (3.5) ph p Pr( ) (, ) H L > Clerly, π / > 0 beuse n nrese n the bookng lmt by one rlne results n more hgh-fre pssengers for the other rlne wthout ny effet on demnd by low-fre pssengers. y omprng (3.4) nd (3.5) we fnd, 12

14 Pr( > C > ) < Pr( > C > ). (3.6) Now onsder the followng four ses: H L H L 1., > >. Gven the P 2 ssumpton, the probblty term n (3.6) s nresng n both nd, so ths se s mpossble. 2. >, <. Consder the hnge from to probblty term whle n nrese n. A derese n leds to derese n the leds to n nrese n the probblty term. For nequlty (3.6) to hold, the seond effet must domnte. ut for ths to hppen we need < where refers to n bsolute hnge n the bookng lmt. However, nlyss of the seond optmlty ondton π / leds to the opposte requrement > whh s ontrdton. 3. <, >. hs leds to nother ontrdton, by the sme resonng s the prevous se. he only remnng opton s tht nd. Proposton 3 mples tht, under ompetton, t lest s mny sets re held for hgh-fre ustomers s s optml under ont proft mxmzton. For the monopolst, every hgh-fre pssenger who does not fnd set t rlne nd turns to rlne s not 'lost' to the frm. Under ompetton, however, when rlne estblshes lower bookng lmt, rlne lowers ts bookng lmt s well s the two rlnes ompete for hgh-fre pssengers. A smlr proof demonstrtes tht when there s only low-fre overflow, nd no hgh-fre overflow, then nd. We do not beleve tht ths result hs prtl sgnfne, for t s dffult to mgne stuton n whh hgh-fre ustomers do not overflow (e.g., f there re few hghfre ustomers, then the bookng lmt s dusted ordngly so the overflow s lkely to our). However, ths result, when ombned wth Proposton 3, shows tht the outome of the full gme desrbed n Seton 4.1 s not ler, for the full gme hs both types of overflow. 4.3 Numerl nlyss o determne whether the prevous seton's results pply to the full-fledged gme, we lulte numerlly both the ompettve equlbrum nd the optml monopoly soluton n gme wth both low-fre nd hgh-fre overflow. We perform these numerl experments over rnge of prmeter vlues tht s suffently wde so tht we replte most possble rel-world senros. Our gol s to see whether the bookng lmt set by the monopoly, 1 2 +, s onsstently greter thn or 13

15 equl to the totl bookng lmt under ompetton, o fnd the pproprte rnge of prmeters we exmned vrety of soures, nludng publshed ppers, unpublshed Ph theses, nd dtbses olleted by the rlne ndustry nd the U.S. eprtment of rnsportton (O). Our most sgnfnt prmry soure s the O s qurterly Pssenger Orgn nd estnton Survey (O, 2004). hs dtbse, known s ether t se 1 or the O& Survey, ontns 10% smple of ll rlne tkets sold for flghts orgntng durng 3-month perod. Eh reord of eh tket nludes the tes vsted by the pssenger (the tnerry), the pre of the tket, nd fre ode ndtng whether the tket ws restrted or unrestrted. In generl, unrestrted, or full-fre, tkets re not subet to lmttons suh s dvne purhse requrements, mnmum/mxmum stys, or refund penltes. o estmte prmeters of our model tht re relted to fre lsses, suh s the rto H L p / p, we used the dtbse s fre ode s proxy for our hgh nd lowfre tegores. Gven tht our model llows for only two types of fre lsses, we beleve tht dvdng fres nto restrted nd unrestrted tegores s resonble pproxmton. In totl, the O& Survey provded us wth nformton on 471,000 tkets sold n 1500 dfferent mrkets (orgn/destnton prs). he followng lst summrzes the dt nlyss nd the prmeter vlues. For more detls on the O& Survey, our nlyss of these dt, nd the followng numerl experments, see Netessne nd Shumsky (2004). - Cpty (C 1 nd C 2 ): Fleet dt from the Federl Avton Admnstrton (FAA, 2004) ndtes tht the verge rplne of mor rlne hs 180 oh sets, vryng from pproxmtely 50 for regonl et to over 400 for oeng 747. Gven these dt, we desgn two sets of experments: symmetr se wth C 1 = C 2 = 200, nd n symmetr se wth C 1 = 200 nd C 2 = Rto of hgh fre to low fre ( p H / pl ): In our dt set derved from the O& Survey, the medn fre rto mong ll 1500 mrkets s p / p = 2.6. For over 90% of ll mrkets, the rto fell wthn H L rnge from 1.3 to 4. herefore, we defne three senros, p / p = [1.3, 2.6, 4], for both the symmetr nd symmetr ses. - Proporton of demnd due to low-fre pssengers: Let µ L ( µ H ) be the verge low-fre (hgh-fre) demnd for rlne, =1,2. he proporton of low-fre demnd s µ µ + ). In the dt from H L L /( L µ H the O& Survey, the medn vlue mong ll mrkets s 0.74, wth 90% of ll mrkets fllng between 0.5 nd 0.9. herefore, we set µ µ + ) = [0.5, 0.74, 0.9] for both the symmetr nd symmetr ses. L /( L µ H 14

16 - otl demnd nd demnd fed by eh rlne: Aordng to the rlne ndustry trde group the Ar rnsport Assoton (AA), the ndustry lod ftor (the utlzton of rplne sets) hs hovered ner 70% for the lst dede (AA, 2004). However, there s substntl flght-to-flght vrton round ths rnge, nd the pplton of revenue mngement tehnques genertes lod ftors tht re lower thn exogenous demnd. Here we ssume tht totl demnd s equl to totl rlne pty. In our experments, ths led to relzed lod ftors between 70% nd 98%, dependng upon the vlues of the other prmeters. We must lso llote demnd between the rlnes. It s most onvenent to desrbe ths lloton n terms of the lod pled on eh rlne, where the lod for rlne = ( µ + µ ) / C. For the symmetr se, we vry rlne 1 s lod through three prmeters, [0.5, 0.75, 1], whh mples tht rlne 2 s omplementry lod s [1.5, 1.25, 1]. For the symmetr se, rlne 1 s lod = [0.5, 1, 1.25] nd rlne 2 s lod=[2, 1, 0.5] (rell tht n the symmetr se, rlne 1 hs 200 sets nd rlne 2 hs 100). - Vrblty: o lmt the number of prmeters, we ssume tht ll four ustomer demnd dstrbutons hve the sme oeffent of vrton, CV. sed on dt nlyses n elobb (1987) nd n Jobs, Rtlff nd Smth (2000), resonble vlues for the CV = [0.2, 0.33, 0.6]. - Correlton: Gven the dt nlyss desrbed n elobb (1987) nd our own dsussons wth mngers who work wth yeld mngement systems n the rlne ndustry, orrelton n the forest error mong ustomer lsses s usully smll. When orrelton s sgnfnt, postve orrelton s probbly more prevlent thn negtve orrelton. herefore, we ssume tht orrelton ρ = [-0.3, 0.0, 0.3, 0.6]. o lmt the number of prmeters, we ssume tht the orreltons mong ll demnds re equl. - Probblty densty: elobb (1987) uses ndustry studes nd hs own dt nlyss to show tht the Norml dstrbuton s resonble model of demnd wthn eh fre lss. For eh of our senros, we ssume tht demnd s dstrbuted ordng to multvrte Norml dstrbuton nd s trunted t zero; ny negtve demnd s dded to mss pont t zero. When ombned, these prmeters defne 2 *3*3*3*3* 4 = 648 senros. Solutons were found by smple grdent lgorthm nd the grdents themselves were evluted by Monte Crlo ntegrton. For every senro, we found tht the bookng lmt set by the monopoly, 1 + 2, ws greter thn or equl to the totl bookng lmt under ompetton, he men dfferene + ) ( + ) ross ll senros s 9.3 sets, nd the dfferene vres from 0 sets to 111 sets. ( L H

17 In generl, the lrgest dfferenes our when orrelton s low nd when both pty nd demnd re blned mong rlnes nd lsses. For more detled desrpton of these results, see Netessne nd Shumsky (2004). If we defne the serve level of ustomer lss s the probblty tht ll ustomers of prtulr lss re ble to purhse set on ether rrft (smlr to the no-stokout probblty of nventory theory), these dfferenes n bookng lmts n produe sgnfntly dfferent serve levels. Over ll 648 senros, the serve level for low-fre ustomers rose n verge of 5.4 perentge ponts under the monopoly (43% to 48%), whle the serve level for hgh-fre ustomers delned n verge of 5.3 perentge ponts under the monopoly (76% to 71%). he rnge of results ws extremely lrge. he dfferene n low-fre serve levels ws s hgh s 62%, whle the dfferene n hgh-fre serve levels rehed 30%. o summrze, n ths seton we demonstrted tht the horzontl gme my not be wellbehved: best responses my exhbt umps nd, qute ounter-ntutvely, t my be optml for n rlne to nrese ts bookng lmt n response to the nrese of bookng lmt by the ompettor. We provded regulrty ondtons tht ensure exstene of ompettve equlbrum nd gurntee tht the best response funtons re deresng. hrough ombnton of nlytl results nd numerl experments, we found tht ompetng rlnes tend to reserve too mny sets for hgh-fre pssengers. In the onluson, Seton 8, we wll dsuss ddtonl mngerl nsghts from ths seton s nlyss, s well s from the followng nlyss of vertl ompetton. 5. Vertl Competton Now onsder two or more rlnes tht do not ompete horzontlly beuse ll of the rlnes operte flghts n non-overlppng mrkets: f rlne 1 opertes flght from A to, then no other rlne opertes flghts between those tes. However, the rlnes do ntert beuse they my exhnge onnetng pssengers. For exmple, f rlne 2 fles from to C, then pssengers my trvel from A to C by flyng on both rlne 1 nd rlne 2. he presene of both lol pssengers flyng one leg (from A to or from to C), nd onnetng pssengers flyng both legs leds to mportnt questons of oordnton between the rlnes. How mny hgh-fre sets should eh rlne set sde for onnetng pssengers? For lol pssengers? Wht hppens f the rlnes do not ollborte n these desons? Our nlyss of the O s O& Survey found mny exmples of flghts by the mor rlnes wth hgh proporton of nter-rlne trnsfer pssengers. For exmple, on flghts operted by Contnentl between etrot nd Clevelnd, we estmte tht 65% of pssengers trnsferred to or from other rlnes. Most of these pssengers flew on Northwest on the prevous leg nd trnsferred from 16

18 Northwest to Contnentl t Northwest s hub n etrot. In ddton, the dt show tht regonl rlnes often rry n even hgher proporton of pssengers who trnsfer to or from other rlnes. low-fre pssengers hgh-fre pssengers ll pssengers Mor U.S. Arlnes Amern Contnentl elt Northwest Unted USArwys Independent regonl rlnes Alsk Hwn Horzon Mes Skykng Skywest Regonl rlnes ontrolled by mor rlnes Amern Egle Comr (elt) elt Conneton Atlnt Southest Mesb/Northwest Arlnk Unted Express Atlnt ble 1: Frton of U.S. pssengers who re onnetng to or from nother rlne (soure: O s O& Survey) ble 1 shows the estmted frton of nter-rlne onnetng pssengers on flghts n the U.S. for mny of the lrgest rlnes lsted n the O s O& Survey dt from the 4 th qurter of 1999 (O 2004). o lulte these numbers we frst exmned eh flght leg operted by eh rlne nd found the number of pssengers whose tket showed ether trnsfer from dfferent rlne on the prevous leg or trnsfer to dfferent rlne on the next leg. he tul numbers n ble 1 were produed by dvdng the number of these nter-rlne onnetng pssengers by the totl number of pssengers. In ddton, the O& Survey lssfes eh tket s restrted or unrestrted, nd we used ths lssfton s proxy for our low-fre nd hgh-fre tegores (for more on ths lssfton sheme, see Seton 4.3). From the ble we see tht pproxmtely 15-20% of the pssengers on the mor rlnes re onnetng pssengers. Regonl rlnes ontrolled by the mor rlnes suh s Amern Egle nd Comr (whh s wholly owned subsdry of elt) tend to hve hgh proporton of onnetons to nd from the ontrollng prtner, round 70% - 90%. For these prtners, revenue mngement desons re oordnted so tht there s no vertl ompetton between the mor rlne nd the subsdry. On 17

19 the other hnd, onsder the ndependent regonl rlnes, whh lso hve extremely hgh proportons of onnetng pssengers but do not oordnte prng nd set lloton desons wth ther prtners. hese rlnes provde us wth exmples of vertl ompetton. For exmple, Skywest feeds pssengers to flghts operted by Contnentl, elt nd Unted, nd over 90% of ts pssengers re onnetng pssengers. Geogrphlly onentrted rlnes, suh s Alsk Arlnes nd Horzon Ar lso sw lrge number of trnsfers from other rlnes, 34% nd 44%, respetvely. he nlyss tht led to ble 1 nluded only flghts operted wthn the U.S. Most nterntonl flghts operted by the mor rlnes lso hve hgh frton of pssengers onnetng wth other rlnes (Fernndez de l orre, 1999). In ths seton we onsder the mpt of these onnetng pssengers on yeld mngement poles. hroughout ths Seton we ssume tht none of the rlnes ompete horzontlly, so tht eh opertes legs n seprte mrkets. Seond, we ssume tht the totl revenue generted by ny tnerry s the sum of the revenues tht would hve been generted by lol pssenger on eh leg of the route. For exmple, suppose rlne 1 fles on route from ty A to ty, whle rlne 2 fles from to C. Lol pssengers trvelng from A to generte revenue p k1, k = L, H to rlne 1 nd re worth p k 2 to rlne 2. Connetng pssengers trvelng from A to C re worth pk1+ pk2, k = L, H. We must lso mke ssumptons bout how the revenue s splt between the rlnes. In our model, the rlnes ode shrng greements llow eh rlne to book onnetng pssenger on ny other rlne, nd f n rlne mkes onnetng bookng t keeps the lol fre nd dstrbutes the remnder to the pproprte prtners. Note tht the tul lol fres my be negotted mong the rlnes n dvne. For exmple, they my be bsed on the length of the two legs, system lled 'mlege prorton' (oyd, 1998). However, we ssume tht the revenue ontrbuted by onnetng pssenger to eh rlne s equl to the revenue tht eh rlne would obtn from lol ustomer flyng sngle leg n the sme fre lss. hs smplfes the problem beuse onnetng nd lol demnds then fll nturlly nto the sme bookng-lmt lsses (or bukets) on eh rlne. Gven ths prtton of pty, eh rlne uses free sle mehnsm to llote pty to the other rlne. Speflly, eh rlne provdes rel-tme set vlblty nformton to ts prtner. If n rlne wnts to book low-fre set on the other rlne, t n, s long s the pproprte bookng lss s open. hs mehnsm s sometmes lled utomted odeshrng, nd t s n nresngly populr mehnsm for llotng pty mong rlne llne prtners (Fernndez de l orre, 1999 nd llur nd vn Ryzn, 2004). Frst, we wsh to determne whether the rlnes, eh optmzng ths obetve funton over ts own bookng lmts, reh ompettve equlbrum. In ddton, we wnt to ompre the behvor of these unoordnted rlnes wth the behvor of sngle, entrlzed rlne tht optmzes over the entre 18

20 network. In ft, there re vrety of methods tht entrl uthorty my use to llote sets mong the vrous types of demnd on eh flght leg (rell tht eh leg hs low nd hgh-fre lol demnds s well s low nd hgh-fre onnetng demnds long ny number of routes). llur nd vn Ryzn (2004) provde lud ntroduton to ths top. hey desrbe () 'prttoned bookng lmts' so tht eh type of demnd hs ts own bookng lss, () 'vrtul nestng ontrols' n whh flghts wth smlr proft mplton re pled n the sme vrtul lss, nd () bd-pre ontrols n whh eh bookng request s evluted n rel-tme gnst the totl expeted mrgnl ost of stsfyng tht request. We wll ssume tht our entrlzed rlne uses vrtul nestng ontrol system nd tht lol nd ll onnetng pssengers of eh fre lss ontnue to o-exst n the sme vrtul lss. In other words, the entrlzed rlne mntns the sme two-lss bookng lmt struture desrbed bove; the only dfferene s tht the rlne determnes dfferent, 'optml' bookng lmt for eh flght leg tht my tke nto ount the lrger revenue tht onnetng ustomers n provde. For mny prmeters ths s resonble method for llotng pty. For exmple, the expeted net mrgnl network proft generted by low-fre onnetng ustomer s often loser to the lol low-fre revenue thn the lol hgh-fre revenue (lthough the onnetng pssenger who trvels on more thn one leg genertes more revenue, tht pssenger my dsple lol ustomer on eh leg). Suh vrtul nestng systems re qute populr n prte beuse of ther smplty nd ther onformne wth exstng rlne reservton systems. Mntnng the sme bookng-lmt struture lso llows us to ompre dretly the ompettve equlbrum nd entrlzed ses. o gn some ntuton bout wht hppens under ompetton, we wll now onsder few extreme ses. In Seton 5.1 we exmne generl network topology wth ny number of rlnes mkng ny number of onnetons, but we lmt the lloton of onnetng ustomers to ether the hghfre or low-fre tegores, but not both. 3 We wll see tht n the frst senro ompetng rlnes set bookng lmts tht re lower thn s entrlly optml nd n the seond senro ompetng rlnes set bookng lmts tht re hgher thn entrlly optml. In Seton 5.2 we onstrut model of two rlnes exhngng pssengers ross two legs, nd we exmne numerlly the mpt of hvng pssenger onnetons n both fre tegores Generl networks wth onnetng ustomers n ether the low-fre or the hgh-fre buket 3 One mght lso onsder nother spel se wth both types of onnetng demnd but no lol demnd. Whle ths my pper nterestng, t turns out tht lol demnd plys n mportnt stblzng role n the gme nd tht n nlyss of ths spel se s unnformtve. For exmple, onsder ny number of rlnes tht operte sequene of legs onnetng tes A to to C. It n be shown tht when there s no lol demnd, ny soluton =,, suh tht 0 mn ( C), s Nsh equlbrum. 19

21 Consder n M-leg network between n rbtrry number of tes. he network my be seres of tes, A to to C, or t my be more omplex network nvolvng multple rlnes feedng pssengers to multple hubs. Eh leg on ths network s ontrolled by one of N rlnes, N M. Legs re ndexed by =1,,M nd rlnes re ndexed by k=1,,n. Cpty offered on eh leg s C. Pssengers trvel on ndvdul legs s well s on ny ombntons of suessve legs. Pssengers trvelng on sngle leg py p k, k = L, H nd pssengers trvelng on route tht nludes set Ω of severl legs py pk, k = L, H. he rlnes ode-shrng greement llows eh rlne to book onnetng pssenger on the other rlne, nd f n rlne ontrollng legs n Ω mkes onnetng bookng n lss k, t keeps Ω pk nd pys Ω pk to the rlnes ontrollng other legs. o desrbe the totl demnd fed by Ω / Ω eh rlne, we defne s vetor of bookng lmts on ll M legs nd on M-1 legs exludng leg. Furthermore, ( ) = + ( ) nd L L L ( ) ( ) = +, where H H H, = lol low-fre nd hgh-fre demnd spef to the leg, L H ( ) L, H ( ) s vetor of bookng lmts = onnetng low-fre nd hgh-fre demnd on the leg, gven the desons mde on the other legs. We ssume tht the onnetng demnds hve the followng propertes: L ( ), nd H ( ) s nresng n s deresng n. For the nlytl results n ths seton we wll not spefy funtonl form for ( ) or (, ), for these frst-order ssumptons re suffent. hese L H ssumptons re lso qute ntutve: hgher bookng lmt on ny one leg opens more sets for onnetng low-fre pssengers for other rlnes on the network, whle lower bookng lmt opens more sets for onnetng hgh-fre pssengers. In Seton 5.2 we wll desrbe one spef funtonl form for the onnetng demnds tht stsfes these ssumptons. he proft generted for n rlne from prtulr leg s, ( ( )) ( ( ) ) ( ) ( ) = E p mn +, + p mn +, C mn +, π L L L H H H L L 20

22 he totl proft of rlne k ontrollng set of legs Ω k s Π k = Ωk π. We wll ompre the bookng lmts resultng from suh vertl ompetton wth the bookng lmts tht monopolst rlne would set. For sngle monopolst rlne set Ω smply nludes ll M legs. We wll onsder two extreme stutons: n the frst, ll onnetng pssengers wll fly n the low fre only nd n the seond ll onnetng pssengers wll fly n the hgh fre only. efore we do so, we wll prove one theorem tht wll be nstrumentl lter on. Proposton 4. Consder n N-plyers non-oopertve gme wth eh plyer k=1,,n endowed wth vetor of strteges (,,...,,...) nd let k = k 1 k 2 nd pyoff Π ( ) k k denote the llne solutons. hen, f eh k ( ). Let denote equlbr of ths gme Π s supermodulr n nd moreover, f Π / 0, k, (orrespondngly, Π / < 0, k, ) then k k ( ). Proof: Consder fttous gme n whh every plyer s stll endowed wth vetor of strteges (,,...,,...) = but the pyoff now s k k 1 k 2 k k ( α) k ( ) α m( ) Note tht for α = 0 the equlbrum of ths gme s Π, =Π + Π, k. m k nd for α = 1 the equlbrum of ths gme s beuse eh plyer optmzes the sum of ll plyers pyoffs. Hene, n order to ompre the ompettve soluton wth the llne soluton t suffes to show tht the equlbrum of the gme defned by (5.1) s monotone n α. o show tht ths s the se, observe tht k (, α ) sum of supermodulr funtons. Further, k (, α ) Π α 2 Π ( ) = Π, k, m k k (, α ) Πm ( ) α Hene, f / 0, k, m =, k,,. m k 2 Πk, we hve tht k ( α) (5.1) Π s supermodulr n s Π, / α 0 so tht the gme s supermodulr n ll plyers strteges s well s n the prmeter α. From opks (1998), t follows tht the set of equlbr of ths gme s monotonlly nresng n α nd hene shown nlogously.. he seond prt s At ths pont, t s worthwhle pontng out tht n set-orderng sense. ht s, the lrgest nd 21

23 the smllest equlbr stsfy ths nequlty but not neessrly every soluton does. opks (1998, pg. 32) lls ths the ndued set orderng. Of ourse, f the equlbr re unque then onventonl wy. n the We now return to the rlne problem. Frst, onsder the problem wth onnetng pssengers n the low fre only. In ths se revenues on gven leg re ( ) ( ) ( ) ( ) ( ( ) ) = E p mn +, + p mn, C mn +,. π L L L H H L L Proposton 5. Suppose the dstrbuton of (, ) fre. hen. Proof: Note frst tht ll ( ) s P 2 nd ll onnetng pssengers re n the low L H π re lerly nresng n beuse n nrese n mkes more onnetng low-fre demnd vlble for the leg wthout ffetng hgh-fre demnd. Further, the frst dervtve s π Pr ( ( ) ) Pr (, ( ) ) ( ( ) ) ( L L pl ph H C L L( ) ) = p + > p > C + > L L L H H L L ( ) = Pr + > mn > + > Under the P 2 ssumpton, π / s nresng n Π k ( ). Hene, ondtons of Proposton 4 re stsfed nd so tht every ( ) π s supermodulr, nd so s We now turn to the se wth ll onnetng pssengers n the hgh fre. Rell tht, generlly spekng, H ( ) s funton of s well s.. he dependene on s due to the ft tht ffets demnd for low-fre onnetng pssengers on legs other thn whh, n ts turn, ffets the number of onnetng hgh fre pssengers tht n be ommodted whh ffets H ( ) f onnetng pssengers re only n the hgh fre, the dependene of H ( ) on n wrte H ( ). Revenues on gven leg re ( ) ( ) ( ) ( ) π = E pl mn L, + ph mn H + H, C mn L, Proposton 6. Suppose tht ll onnetng pssengers re n the hgh fre. hen Proof: Note frst tht ll ( ) π re deresng n beuse n nrese n.. However, dsppers nd we. mkes less onnetng 22

24 hgh-fre demnd vlble for the leg wthout ffetng low-fre demnd. Further, the frst dervtve s π herefore, π / s nresng n ( ) so tht every π ( ) s supermodulr, nd so s Π k ( ) ( ) ( ) = p Pr > p Pr + > C, >. L L H H H L. Hene, ondtons of Proposton 4 re stsfed nd. he results of Propostons 5 nd 6 re ntutve: ompetng rlnes undervlue onnetng ustomers beuse eh rlne only reeves frton of the totl revenue from those ustomers. When onnetng trff trvels only n the low-fre buket, ompetng rlnes estblsh lower bookng lmts nd sell fewer onnetng tkets thn s optml for the network. he opposte hppens when onnetng pssengers trvel only n the hgh-fre buket. hese results lso suggest tht when onnetng pssengers re lloted to both bukets, the omprson of bookng lmts under ompetton nd entrlzton my depend upon the proporton of onnetng pssengers n the two bukets, the oneture tht we verfy n the next seton Connetng pssengers n both bukets. In order to verfy our oneture n the prevous seton, we fous on smpler network, wth ust two rlnes (rlne 1 nd rlne 2) opertng two legs (legs 1 nd 2, respetvely). We lso ssgn prtulr funtonl form to ( ) nd (, ). When onsderng how these demnds re L H generted, the followng ssue rses: f both lol nd onnetng pssengers re ommodted n buket, how mny sets re sold to lol pssengers nd how mny to onnetng pssengers? For ths exmple, ssume tht both lol nd onnetng pssengers rrve unformly over the reservton perod. Frst onsder low-fre onnetng pssengers. If the bookng lmt of rlne s not onstrnt, then rlne n book ll vlble onnetng pssengers, nd these pssengers re vlble for rlne : ( ) =. However, f the bookng lmt of rlne s onstrnt, the number of lol nd L L onnetng tkets sold s proportonl to ther relzed demnds: L ( ) ( L /( L L )) 2 resonng pples for hgh-fre demnd. herefore, = +. Smlr L ( ) = mn,1 L, L + L (5.2) 23

25 ( ) C mn, L + L( ) (, ) = mn,1 H + H H H 24. (5.3) As requred, L ( ) s nresng n nd H(, ) s deresng n nd. Agn, defntons (5.2) nd (5.3) re useful exmples nd wll be used n numerl experments lter n ths seton, but they re not neessry for the nlytl results n Seton 5.1. Arlne s obetve funton s, ( ( )) ( ( ) ) ( ) ( ) π mn, mn,, mn, = E pl L + L + ph H + H C L + L In ble 1, we sw tht rlnes must often mke yeld mngement desons wth sgnfnt number of nter-rlne onnetons n ll fre lsses. o exmne ths generl se, we onduted numerl experments to fnd the dfferene between the ompettve equlbrum bookng lmts,, nd the system-optml bookng lmts, for vrety of problem prmeters. Here we wll present the results of one set of experments wth two dentl rlnes, eh wth 200 sets nd eh fng totl expeted demnd of 200 pssengers, nludng both lol nd onnetng pssengers n both fre lsses. (After the pplton of yeld mngement ontrols, ths produes lod ftors between 85% nd 95% n the followng set of experment.) We lso performed n ddtonl set of experments wth symmetr rlnes: one regonl rlne wth 100-set plne feedng mor rlne wth 200-set plne. he results of those experments were smlr to the results shown here nd hene re not reported. We gn used the O& Survey nd other soures of dt to determne the prmeters for the experments. sed on the nlyss desrbed n Seton 4.3, we set p / p to the bselne vlue of 2.6. All demnds were dstrbuted s ndependent norml rndom vrbles, trunted t 0, wth oeffents of vrton equl to 0.33 (elobb, 1987). Gven the results from the O& Survey, we lloted 74% of lol demnd to low-fre pssengers. o determne the proporton of hgh-fre nd low-fre onnetng trff, we gn exmned eh of the mrkets n the O& survey. We found tht mong nter-rlne onnetng pssengers, the frton of hgh-fre tkets vred from 0 to 0.9. herefore, we vred the frton E ( ) / E ( + ) ross ths rnge. Fnlly, let f represent the frton of demnd due to H L H pssengers onnetng from other rlnes, number tht s roughly equvlent to the sttst shown n the thrd olumn of ble 1. We set f =[0.25, 0.75], rnge tht nludes most of the mrkets n the O& Survey wth sgnfnt number of onnetng trvelers. In ll experments, we ssumed tht onnetng demnds were generted ordng to the H L

26 unformty ssumpton tht led to equtons (5.2) nd (5.3). We used Monte Crlo smulton to evlute the obetve funton when fndng the optml bookng lmts for eh rlne, nd vrblty n the generted demnd ounts for the rndom vrton n the trend lne. Fgure 4 shows the results of the experments. ompettve equl. bookng lmt - entrlzed bookng lmt, frton of onnetng demnd n hgh-fre bn, E ( )/ E ( + ) H L H f = 0.75 f = 0.25 Fgure 4: Comprng ompettve nd entrlzed bookng lmts he left-hnd sde of the fgure, wth E ( ) / E ( + ) = 0, s equvlent to the senro wth H L H ll onnetng pssengers n the low-fre buket (Seton 5.1), nd we see tht equlbrum bookng lmts re slghtly lower under ompetton whh s onsstent wth our results. When E ( ) / E ( + ) = 0.9, most onnetng pssengers re n the hgh-fre buket nd, s we found n H L H Seton 5.1, the bookng lmts re hgher under ompetton. he ntermedte ses show grdul trnston between the two extremes, nd t one pont the ompetng rlnes estblsh the entrlly optml bookng lmts. However, n the morty of exmples we exmned, the mpt of onnetng pssengers n the hgh-fre buket domnted, so tht bookng lmts tend to be hgher under ompetton. From Fgure 4 we see tht hvng proportonlly more onnetng demnd (hgher f ) tends to nrese the dfferene between the ompettve nd entrlzed bookng lmts. In ths set of experments, the dfferene n network profts between the optml soluton nd the ompettve equlbrum vred from 0, where the solutons re equl, to 0.6% when 75% of ll demnd s onnetng demnd, nd most of tht onnetng demnd s n the hgh-fre bn. Hgher proft dfferenes were seen wth even hgher levels of onnetng demnd nd stronger negtve orreltons mong demnds. For omprson, llur nd Vn Ryzn (2002) menton tht optml network ontrols n led to "mprovements on the order of 0.5% wth gns n be s hgh s 2% or more under hgh lod 25

27 ftors." Note, however, tht we re not omprng the ompettve equlbrum wth the truly optml ontrol sheme; our entrlzed soluton s the optml bookng lmt, gven the sub-optml vrtul nestng sheme tht ples lol nd onnetng demnds n the sme fre lss. 6. Revenue-shrng Agreements We hve shown tht both vertl nd horzontl ompetton on rlne routes n led to sub-optml lloton of sets mong fre lsses. Here we exmne how ontrtul rrngements n oordnte the tons of the two rlnes. hs s tmely subet, gven the prolferton of rlne llnes tht re presumbly tryng to do ust tht. Other work on ths subet nludes Wynne (1995), who bses trnsfer pres between rlnes on the vlue of lol fres hrged n eh mrket trveled by the onnetng pssengers, oyd (1998b) who fnds pres from the mrgnl vlue of eh set on eh rlne, nd Feng nd Gllego (2003), who exmne ontrts n whh frm pys fxed fre to n rlne, whle the rlne grees to ept bookngs s long s sets re vlble. In prte, the rlne ndustry hs been strugglng to develop effetve oordnton shemes for ts vrous llnes (Fernndez de l orre, 1999). here hs been welth of reserh on ontrtng n the re of supply hn mngement (see Chon, 2004, for n overvew). Most of these ontrts ombne wholesle pre wth nother lever, suh s buybk greements, revenue-shrng greements, or optons. In our settng there s no equvlent of the wholesle pre between suppler nd retler, nd therefore we wll spefy dfferent type of ontrt bsed on trnsfer pres. However, the method used to derve our ontrt s smlr to methods used to determne mny supply hn ontrts: we reformulte eh plyer s obetve funton so tht t beomes lner trnsformton of the entrlzed obetve funton. hs s smlr n sprt to the ostbsed revenue shrng desrbed by Fernndez de l orre (1999), who notes tht suh shemes [rse] the dffult queston of how to llote revenue generted by onnetng trff to the leg tht s odeshred. (pg. 155) ht s the problem we ddress here. Note tht ths tehnque s pproprte not only when rlnes ompete on bookng lmts, but lso when they ompete on pres. Here we wll derve sutble ontrt for rlnes under vertl ompetton (.e., rlnes n ode-shrng llne). Smlr tehnques dentfy ontrts tht oordnte two rlnes under horzontl ompetton. Eh rlne s obetve funton s ( ) ( ) ( ) π = E pl mn L, + ph mn H, C mn L,. where eh demnd s funton of pres s well s bookng lmts. urng negotton phse, the rlnes gree to splt profts n proportons nd 1-. One type of ontrt found n the supply-hn 26

28 lterture would smply llote totl revenues to the two rlnes ordng to proportons nd 1-. However, suh ontrts sk the rlnes to shre revenues from both onnetng nd lol trff, nd ths my be mpossble for tehnologl, ompettve, nd/or regultory resons. Insted we wll ssume tht the terms of the ontrt re smlr to the sheme proposed by oyd (1998b) nd desrbed n llur nd vn Ryzn (2004, Seton 6.2): when n rlne sells onnetng tket, t pys trnsfer pre to the other rlne for the set. However, our method for fndng the trnsfer pyment s dfferent from the dulty-bsed method proposed by oyd. Here we wll demonstrte ths tehnque for sles n the low-fre lss; the dervton for the hgh-fre ommsson s smlr. When rlne sells onnetng tket, t ollets revenue pl 1+ pl2 nd pys rlne trnsfer pre δ. he method for lultng the trnsfer pre δ s negotted n dvne, nd nnot be hnged one pyments begn. As n the prevous seton, lol low-fre pssenger on rlne s worth revenue. p L n In our formulton the trnsfer pre wll depend upon the expeted flow of trff, speflly, the expeted number of onnetng pssengers nd totl pssengers, gven eh rlne s bookng lmt. Let be the number of onnetng tkets sold n the low-fre lss nd let be the number of onnetng tkets sold by rlne (so tht 1 2 from lol pssengers, ( δ ) + = ). Gven ths notton, rlne 1 erns L1 mn ( L1, 1) 27 ( ) p pl 1+ pl2 2 1 from onnetng tkets sold by rlne 1 nd δ1( 1) from onnetng tkets sold by the prtner. o fnd the oordntng ontrt, we hoose δ 1 nd δ 2 so tht the expeted totl revenue from low-fre ustomers for rlne 1 s equl to proporton of the entrlzed revenue from low-fre ustomers: ( mn (, ) ) + ( + δ ) + δ ( ) L1mn ( L1, 1) L2mn ( L2, 2) E p p p = E ( p + p ) L1 L1 1 L1 L (5.4) Now let β be the rto of the expeted number of onnetng pssengers nd the expeted number β. Also let λ be the rto of the expeted number of onnetng tkets sold by rlne 1 nd the totl number of onnetng tkets sold: of pssengers ommodted by rlne : [ ]/ mn ( = E E L, ) λ = E [ 1]/ E[ ]. vdng (5.4) throughout by E mn ( 1, 1) L, p p p p p 1 ( 1 β ) + ( + δ ) λβ + δ ( β λβ ) = + L1 1 L1 L L1 L2 β. β2

29 Rerrngng terms nd dvdng through by β 1, = p p (5.5) L ( λ)( δ p ) λ( δ p ) ( ) 2 L1 1 1 L1 2 L2 1. β2 β1 Equton (5.5) shows how the trnsfer pres depend upon the trff flows. Gven vlue of δ 2 > p L 2, (5.5) ndtes tht the trnsfer pre δ 1 () nreses s λ nreses () nreses s β 1 nreses nd () dereses s β 2 nreses. Pont () s ntutve: s rlne 1 sells lrger proporton of onnetng trff, t reeves fewer trnsfer pyments from rlne 2, nd lrger trnsfer pre s needed to lgn ts behvor. Ponts () nd () re more subtle. For pont (), rse n β 1 ndtes tht onnetng tkets represent hgher proporton of rlne 1 s sles. hs mples tht rlne 1 s pyng rlne 2 trnsfer pre on hgher proporton of ts own tket sles beuse of derese n rlne 1 s lol demnd. euse rlne 1 ts n ts own self-nterest, fll n lol demnd would push rlne 1 to lower ts bookng lmt t the expense of onnetng trff on rlne 2. herefore, the nentve for rlne 1 to preserve room for onnetng trff must rse, nd δ 1 rses s β 1 rses. he reson for pont () s smlr. 7. Observtons nd Future Reserh In ths pper we hve exmned how bookng-lmt desons re ffeted by both horzontl ompetton (wth pssenger overflow) nd vertl ompetton (wth onnetng pssengers). We hve shown tht smple ondton on the demnd dstrbuton, totl postvty of order 2, s suffent to ensure pure strtegy Nsh equlbrum under horzontl ompetton nd under vertl ompetton when onnetng demnd s restrted to the hgh or low-fre lsses. In generl, we fnd tht the equlbrum behvor of ompetng rlnes n be very dfferent from the behvor of ompettors n newsvendor gme. In ths smpler gme, eh ompettor s obetve funton s onve nd the gme s submodulr. However, n our settng nether property holds, nd ths leds to best-response funtons tht n be dsontnuous, nd n be ether deresng or nresng. he reson for ths dsprty s tht n the newsvendor gme the deson s how muh nventory/pty to proure whle n our model the deson s to llote fxed pty mong two ustomer lsses. he problem of llotng fxed pty s not lwys wellbehved even n the bsene of ompetton,.e. the obetve funton my not be unmodl, resultng n the possble lk of ompettve equlbrum when ompetton s ntrodued. Our model demonstrtes tht omputerzed revenue mngement systems my generte pty ontrols tht exhbt non-ntutve behvor, e.g., my suggest sgnfnt nrese n bookng lmt wthout ny sgnfnt hnge n demnd, lthough suh behvor s unlkely. An understndng of the underlyng uses of these effets 28

30 n be useful for mngers. We hve lso found tht bookng lmts re lower under horzontl ompetton thn the bookng lmts found by entrl proft-mxmzer. Compettve equlbrum bookng lmts my be hgher, or lower, under vertl ompetton, lthough n 2-rlne, 2-leg se we found tht they tend to be hgher. he dfferenes n bookng lmts n led to sgnfnt dfferenes n the level of serve mong ustomer lsses. hese results n be useful for mngers who re plnnng expnson nto new mrkets or fng n entry by rvl. Although rlnes lose revenue beuse of ompetton, our results ndte tht some groups of ustomers (hgh-fre ustomers under horzontl ompetton nd low-fre ustomers under vertl ompetton) gn beuse ther serve s hgher under ompetton thn under monopoly. he net vlue to onsumer welfre of these losses nd gns re, however, not ler, for t depends upon the omprtve vlue those ustomers ple on the blty to obtn tkets n ther hosen fre lsses. We hve demonstrted nlytlly nd evluted numerlly both the dreton nd mgntude of revenue losses for the two rlnes due to ompetton. Effetve oordntng ontrts n redue these losses, nd whle there hs been muh reserh tvty n the re of supply hn ontrtng, the sme hs not ourred n the yeld mngement re. Here we hve desrbed set of revenue-shrng ontrts to oordnte the bookng-lmt desons of two rlnes wth onnetng pssengers. Our work s n ntl ttempt n ths dreton. Gven the prolferton of rlne llnes nd prevlene of ompetton on rlne routes, ddtonl efforts re needed. Our models of full ompetton nluded only two rlnes wth two pssenger lsses eh. In prte, multple rlnes n be n ompetton nd typlly more thn two fr lsses re offered by eh. Relxng eh of these ssumptons presents ertn hllenges. For exmple, model wth more thn two rlnes ompetng horzontlly s hrd to nlyze beuse one hs to spefy the order n whh pssengers overflow when one rlne runs out of sets (e.g., from rlne 1 to rlne 2 nd then to rlne 3 or frst to rlne 3 nd then rlne 2). Consderton of more thn two pssenger lsses ompltes the nlyss s well beuse the number of deson vrbles (bookng lmts) nd hene the number of optmlty ondtons would grow ordngly. euse dret nlyss s dffult, smulton hs been used to exmne the mpt of ompetton (elobb nd Wlson, 1997). On the other hnd, n Seton 5.1 we sw tht the nlyss of vertl ompetton on multple legs mong multple rlnes does not pose sgnfnt dffultes, s long s onnetng pssengers belong to only one fre lss. However, when there re multple fre lsses for onnetng pssengers, the smple supermodulr struture of the gme s not preserved. In ths se, the mpt of ompetton depends upon the proporton of ustomers n eh lss. Wllmson (1992) nd Fernndez de l orre (1999, pg. 185) use smulton to exmne the mpt of vrous optmzton tehnques nd oordnton mehnsms n rlne networks wth 29

31 lrger number of fre lsses. Another sgnfnt onern wth the nlyss s tht when omprng ompettve nd oopertve bookng lmts we ssume tht both pres nd exogenous demnd re onstnt. For some omprsons ths ssumpton s resonble. Under horzontl ompetton, two ompetng rlnes often hrge the sme pres throughout the dy for trvel on prtulr route, nd some hours n the dy re 'ompettve' whle others re monopolzed by sngle rlne. Pres re unform over ll flghts, but the tmng of flghts throughout the dy ffets the yeld mngement desons of both rlnes. Lkewse, n vertl llne, pres my be negotted fr n dvne, whle the yeld mngement desons hppen n reltme, ordng to rules tht my be qute smlr to those desrbed here. Fnlly, n prte rlnes my be ompetng on the sme leg (horzontlly) n ddton to ompetng on dfferent legs (vertlly), nd mny llnes nvolve both horzontl nd vertl oordnton. In our dt nlyss we dd dentfy exmples of horzontl ompetton wth reltvely lttle vertl ompetton (e.g., shuttle flghts between oston nd New York) nd vertl ompetton wthout sgnfnt horzontl ompetton (e.g., regonl rlnes wth monopoly n smll mrket tht s feedng the hub of mor rlne). However, the nlyss of smultneous horzontl nd vertl ompetton s n nterestng re for ddtonl reserh. Appendx: Proof of Proposton 2 o prove exstene, we wll employ rsky s fxed pont theorem whh sttes tht suffent ondtons for the exstene of pure strtegy equlbrum re tht the strtegy spe s losed nd bounded nd tht the best-response funtons re non-deresng (Vves, 2000). Note tht rsky s theorem does not requre tht the best responses re ontnuous: dsontnutes (or umps ) re llowed, but only umps up. Evdently, the strtegy spe [ 0, C ] [ 0, C ] s losed nd bounded. Insted of showng tht best responses re non-deresng, we wll show tht the best-response funtons re non-nresng beuse fter smple re-defnton, =, rsky s result pples to ths new gme (Vves, 2000, pge 33, Remrk 13). Consequently, umps down n the best-response funtons re llowed, but not umps up. We wll frst use the Implt Funton heorem (IF) to show tht best responses re deresng whenever they re dfferentble (Prt I). hs, however, does not elmnte the exstene of umps up beuse the IF n only hrterze the best response t ponts where t s dfferentble. In Prt II we wll demonstrte tht the best-response funtons n only hve umps down. Prt I. y the IF, 30

32 π π 2 2 = (5.6) 2 At plyer s best response, π < 0 nd we wll show tht π < 0 whenever the frstorder ondtons hold. Equvlently, we wnt to show tht the frst dervtve (3.2) s deresng n. Frst note tht, gven prtulr relzton ( L, L, H, H ) =, nresng n. he frst two probblty terms n be rewrtten s, L s deresng n nd H s euse pl Pr( L > ) ph Pr( H > C, L > ) = Pr( L > ) pl ph Pr( H > C L > ( L ) + ). L s deresng n, the frst probblty term Pr ( L ) (5.7) > s deresng n. From heorem 2.3 of Joe (1997), P 2 mples tht nd L H re rght-tl nresng,.e., Pr( > C > ( ) + ) s nresng n. Fnlly, sne the frst-order ondtons hold H L L t the equlbrum, we re ssured tht the expresson n squre brkets s postve nd therefore (5.7) s deresng n. Now we show tht the lst probblty term n (3.2) s nresng n nd, beuse ths term s multpled by ph, the dervtve must be deresng n. he fnl term n be rewrtten, Pr( L >, L <, H > R, H < C ) = Pr( >, + < +, + + > C +, < C + C ) L L L L L H L L H H he bookng lmt only ppers one n ths expresson, n the event L + L < +, nd the probblty of ths event nreses s nreses. herefore, the lst probblty term 2 nreses, π <, nd best responses re deresng whenever they re dfferentble Corollry to Prt I (wll be used n Prt II): Suppose ( ) s some pont on plyer s best response * * funton. hen, 0, 0,. 2 * * π < 0 holds for ny ( ) Proof: We only need to show tht p > p Pr( > C > ( ) + ) (5.8) everywhere on ( ) * * π s deresng n L H H L L, 0, 0,, nd then the resonng of Prt I pples to show tht * * over the pproprte rnge. We demonstrted tht (5.8) s true t ( ). 31

33 We know tht the probblty term on the rght s nresng n f P 2 holds so the result holds for ny. Smlrly, we n show tht ths term s nresng n f P 2 holds. Expnd ths term s, * + + ( ( ( ) )) ( ) Pr + + C + mn +, > C > H H L L L L here re three n ths expresson. An nrese n the frst or thrd vlues of lerly leds to n nrese n the probblty expresson, but the mpt of the seond s the opposte. However, the frst domntes the seond; the expresson ( mn ( ( ), )) C H H L L + s nresng n. herefore, f the nequlty (5.8) holds t ( ), t wll lso hold t ny * * < nd * <. * Prt II: We now elmnte the possblty of umps up n the best responses. Jumps our when the obetve funton s b-modl (or mult-modl). We wll only onsder the b-modl se (the multmodl se s nlogous). he proof s by ontrdton. Suppose tht there s ump up n the best response ( ) tht ours t 1 * 2 * π (, ) (, π ) 1 * 2 * ump up, π (, ε) π (, ε) * (see Fgure 5). here re two globl mxm 1 ( * ) 2 ( * ) <, =, nd the frst-order neessry ondton holds t eh of them. euse there s + < +. ht s, s we nrese mxmum beomes the unque globl mxmum. hs mples, dπ d < dπ d, 1 * 2 * (, ) (, ) * nfntesmlly, the seond beuse the obetve funton of plyer rses fster (or delnes more slowly) t the seond lol mxmum 2 thn t the frst lol mxmum 1. Usng the envelope theorem, π < π. (5.9) 1 * 2 * (, ) (, ) 2 A neessry ondton for (5.9) s tht π / > 0 somewhere between the two best response ponts 32

34 π * (, ) π * (, + ε ) ( ) 2 * ( ) 1 * 1 * 2 * ( ) ( ) 2 * ( + ) ( + ε ) ε 1 * * obetve funton t * obetve funton t * + ε plyer s best-response funton ( ) ( ) 1 * 2 * Fgure 5: A Jump Up 2 nd. However, from the Corollry to Prt I we know tht π / x x 0 for every ( ) ( ) 2 * *, 0, 0,, nd ths s suffent to show ontrdton. herefore, ump up s mpossble, nd the best-response funton s deresng. Aknowledgements We would lke to thnk Andy oyd, Zhhng Ch, Gerrd Chon, Mrshll Fremer, Mrtn Lrvere, Phllp Lederer, ll Loveoy, Mhel Rth, two nonymous referees nd n ssote edtor for helpful omments. We lso thnk Lur X.L. Lu for her help wth the dt nlyss. Referenes Ar rnsport Assoton (AA) Annul rff nd Cpty U.S. Arlnes Sheduled Serve. Avlble t elobb, P.P Ar trvel demnd nd rlne set nventory mngement, Ph.. hess, Msshusetts Insttute of ehnology. elobb, P Applton of probblst deson model to rlne set nventory ontrol. Opertons Reserh, vol. 37, no. 2, elobb, P he Evoluton of Arlne Yeld Mngement: Fre Clss to Orgn estnton Set Inventory Control. In Hndbook of Arlne Mrketng, utler, G. F. nd M. R. Keller, eds., MGrw-Hll. pp elobb, P. nd J. L. Wlson Impts of yeld mngement n ompettve rlne mrkets. Journl of Ar rnsportton Mngement, vol. 3, no. 1, 3-9. orensten, S. nd J. Netz Why do ll the flghts leve t 8 m?: Competton nd deprture-tme dfferentton n rlne mrkets. Interntonl Journl of Industrl Orgnzton, vol. 17, orensten, S. nd N. L. Rose Competton nd pre dsperson n the US rlne ndustry. Journl of Poltl Eonomy, vol. 102, no. 4,

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