Airport competitions on Airline networks: from the viewpoint of network theory

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1 Arport compettons on Arlne networks: from the vewpont of network theory Katsuhko Yonezak (Doshsha Unversty) and Haruo Ima (Kyoto Unversty) 1. Introducton Competton among arports to attract more flghts s very severe lately n the eastern Asa. Growng economes of Chna and Korea together wth multple arports n Japan whch contend to be a hub-arport made the Japanese arports the losers of ths competton. Insde Japan, we see the tendency of declnng flghts across regonal arports. Inspred by ths competton, we try to model competton among arports along wth competton among arlnes, based on network lnk theory a la Jackson and Wolnsky (1996). Among the theoretcal analyss on the structures of arlne network, promnent are the studes of effcences (Pels, Njkamp and Retveld (2000) 1 ) and the studes on the process of network formatons (Hendrcks, Pccone and Tan (1999), Hu 1 They studed the effcency of the arlne network, under assumptons of lnear margnal cost functon: MC 1 Q and the symmetrc demand functon: P Q/2, where α s a measure of the extent of the market and θs a parameter representng densty. In the precedng works, the optmalty of the Hub-and-Spoke network was not dscussed. In ths paper, we attempt to nvestgate ths ssue va comparng t wth the full-connect network. Dependng on the parameters (α,θ), there s a case where nono-optmalty s observed. However, as te fxed cost of lnk formaton becomes greater, then the optmalty of the Hub-and-Spoke network s restored. 1

2 (2005) and also (Brueckner, Dyer, and Spller (1992), Brueckner (2004), Hascm and Oz Shy (2004)). These studes focused on whether the patterns of the optmal network for arlne companes are Hub-and-Spoke or Pont-to-Pont, or f under the presence of competton among arlne companes, network formed by arlne companes are of Hub-and-Spoke type. Also recently, attentons are pad to the vertcal relatonshp between arlne companes and arports, as they have several dfferent nteractons. For nstance, arports can offer an opportunty for the arlne and ts alles to economze costs for ther busness operaton as well as for ts customers (Zhang,Fu and Yang(2010)), and there s a rsk sharng aspects (Hhara(2011)), too. Studes n these focus upon the relatonshps between arlne companes sufferng from volatle economes and arports whch also face compettons due to open-sky agreement etc. We lke to approach ths problem from slghtly dfferent aspects. One of the motvatons for dong ths s that the competton among arports as well as arlnes through decson on establshng flght networks, the result would be effcent. We consder a smple model wth 3 nodes. There s a beneft accrung to the arlne whch operates flght connectng two nodes, for each par of nodes. Each arport charges landng fee to each flght but n ths model, we assumes that fee s fxed and not charged strategcally. Also followng tradton of network theory, we assume that there s a dscountng of benefts, when two nodes are connected through two flghts rather than through a drect flght. Frst we start wth the case of a sngle arlne. In ths case, naturally the arlne operates optmally, accordng to the crteron gven by benefts mnus costs, where costs are just landng fees, and complete network, sngle hub network, and the empty network appear as the chosen and effcent pattern of flghts,. Next, we ntroduce another arlne. Agan by abstractng away of the prce competton, we assume that the beneft s splt between two arlnes when they serve the same route. Even wth ths smplfcaton, we show that the resultng pattern dffers from the prevous case. Not only that the pattern dverges from the effcent one 2

3 under some occason, we show that there emerges a case where only two nodes are lnked. Ths shows that there s a tendency for fewer servces due to competton among arlnes. Also the possblty of doubly served lnks would appear. Under the capacty constrant, agan the concluson remans the same qualtatvely. There s a choce of the assumpton on benefts f a connecton of nodes va two dfferent arlnes s allowed or not. Naturally, under the stuaton where ths s allowed, the effcency loss s mproved to some extent. Then we brefly overvew the result when there s prce competton and based on these observatons we comment on the mplcaton on the decrease n flghts of domestc ar servces. Below, we gve a descrpton of the model n Secton 2., that of results n Secton 3., and concludes n Secton Model Network and Notatons 2 In ths game, there are two arlne careers, ( A, B. ), and n ctes whch are also dentfed wth arports, N {1,2,..., n} ( n3) 3. A network s represented by a graph, G. A graph s comprsed of a par of two unordered (and non-empty) lsts of nodes (or vertexes) and an unordered lst of edges (or lnks). In the context of our model, a node represents a cty, and a lnk{α,β} s a subset of N contanng ctes α and β. Each lnk represents a drect flght servce between those ctes by the career, and one drect lnk mples possblty of round trp travel. G mples that n the network G, there s a lnk αβ, and smlarly, G mples that ths lnk s absent n the network G. Thus G s a lst of lnks. E.g., a star-shape network g* consstng of four nodes s represented by g ={12,13,14} where 1 s the central node. Let L ( g) be the set of lnks for node I n the network g. 2 As a general reference for the network theory, see Jackson and Wolnsky (19969 and Hendrcks, Pccone, and Tang (1996). 3 Greek letters appearng as a suffx represent ctes. 3

4 For nstance, the abovementoned star-shape network g* consstng of four nodes can be represented by L1( g ) {12,13,14}, L2( g ) {12}. A (complete graph: g N s a famly of all subsets of sze 2 n N. Fg. Several types of graphs (four nodes). ( a) complete (b) star (c) cycle (d) lnear In other words, t s a network correspondng to the set of all feasble lnks. Let N G g g g be the set of all possble networks on N. Well know types of graphs are, star graph, cycle graph, lne graph, and null graph. A complete graph can be rephrased as a graph where there s a lnk for all par of nodes (Fg. (a)). In a star graph, from one node, all other nodes are lnked (Fg. (b)9). A cycle graph s a regular and connected graph wth a degree 2, whch s also known as wa wheel graph Fg. (c9). A lne graph has a one-to-one correspondence between lnks and nodes and neghborng nodes are connected everywhere (Fg. d)). A null graph contans no lnk.. The network of a career I s represented by a functon G : NN {0,1} and each network must satsfy the followng condtons: (1) G (, ) G (, ) 1 f there s a drect connecton between and. (2) G (, ) 0 otherwse The sze of s gven by m (1 / 2), N G (, ), and t stands for the total number of lnks. An null network s denoted byφ. NG ( ) s.t. G} s the set of nodes whch possesses at least 1 lnk n G for any Gg. A path between α and β s a set ( 1, 2,..., k ) N( G) of dstnct nodes satsfyng 12, 23,, k 1k G, 1, k, or the sequence satsfyng 4

5 1. G ( n, n 1) 1 for t 1,..., z. t 2. t t1 s s1 t ( n, n ) ( n, n ) for t s. ( n, n ) ( n, n ) for any t and s. 3. t t1 s1 s The condton 1 states that for each par of adjacent ctes, there s a drect connecton, and 2 and 3 express the condton requrng that they can go through a drect lnk only once. Let z be the length of the path, whch counts the number of drect lnk n the path. We say that a two dstnct ctes α, β s connected n G f there s a path { n1, n2,..., nz 1} such that n1, nz 1. N( G) 1 s the set of ctes connected to the cty and n ( G) Stands for the cardnalty of N ( G). Gven a network g, we say ts subset h g a component f for any Nh ( ) and Nh ( ) wth then there s a path connectng α and β, and also f Nh ( ) and k N( G), then k G mples k h. We wrte C(G) for the set of all the components of G. Wth ths defnton, we can wrte that g h C( g) h. C C Especally, let be the maxmal connected sub-network n G. Formally, satsfes the followng condtons: 1. If C (, ) 1 then G (, ) 1 holds. 2. If NG ( ) s.t. G} holds at C (, ), then α and β are connected n C. 3. If G holds. C spans α and not β, tehn (, ) 0 We say a network that does not contan a cycle a forest, and all pars of ctes are connected by a sngle path, then we say that network a tree (or a complete forest). c A network G (wth a sze m 0 ) s called a Hub-and-Spoke network, f there s a cty βsatsfyng G N (, ) m Structure of the game Fxed costs necessary to operate a network depends upon the sze of the network. Also 5

6 costs c s subdvded nto L necessary to operate that route and landng costs f (denoted fx by for a cty x). Between each par of ctes, there s a demand consstng of people and goods whch s desred to be transported. Denote by n(, ) ths demand and we assume that the benefts for a arlne company s proportonal to ths quantty. Specfcally, beneft between operatng route α-β s represented by the beneft functon B(, ). Also ths demand s dscounted by a dscount factor (δ) as the route goes through more and more ctes. The model conssts of three stages. Competton among arports wth the landng fee as the strategc varable s ther frst stage, whle n the second stage, arlnes form networks and n the thrd stage, arlnes compete for customers.,. In the frst stage, each arport tres to maxmze the number of flghts to the arport through the adjustment of landng fees, (.e. max n ( G). In the second stage, arlnes smultaneously determne respectve network (correspond to the so-called Nash network). For the thrd stage competton among arlnes, we adopt the Bertrand prce competton. Wrtng the total demand by D( p ), demand for the arlne for the duopoly case s gven by D( p) for p pj D( pa, pb) D( p) / 2 for p pj 0 for p p j The soluton concept adopted s the subgame perfect equlbrum. In ths paper, we focus upon the formaton of network pattern, and evaluate each network by a effcency crteron, whch we defne next: Defnton: strongly effcent A strongly effcent network s defned by G * = arg max G n 1 n (, ) 3. Results 6

7 In th s secton, we consder the determnaton of netwrok structure by arlne careers for the case of three ctes ctes (x,y,z) and 1 arlne To start wth, let us nvestgate the case where there s only one arlne company. As a smplfyng hypothess, we assume that everythng s symmetrc,.e. Bxy (, ) Byz (, ) Bzx (, ) 1, f x f y f z f, and l x l y l z l. Also snce there s only one company, there s no second stage prce competton. Further, there could be several routes through whch benefts could be generated, but we assume away that beneft accrues only from the shortest path. Proposton 1: Strongly effcent arlne network s gven by 1. c :complete 2. 2 /2c :star 3. c 2 /2:empty Proof: The condtons compare the benefts generated by the demand between ctes and costs of creatng lnk. In 1.,benefts are generated from each lnk and so lnk s formed between all the ctes. Under 2, benefts are ncreased f each cty s lked by fewer lnks and wth less dstance. In the case of 3., benefts never exceeds the costs regardless of the way lnk s formed. Proposton 2: Formed arlne network s gven by 1. c :complete 2. 2 /2c :star 3. c 2 /2:empty Proof: The arlne company would form a lnk f the beneft generated by formng a lnk exceeds the costs of makng that connecton. Hence under 1., lnks are formed 7

8 between all the ctes. In 2., no postve beneft s generated by a sngle connecton alone, but desgnng lnks so that costs are mnmzed, then postve benefts are generated. In the case of 3., lnks are not formed because no matter how network s formed, postve benefts are generated. In ths case, effcency of networks for the socety and the proft maxmzaton for the arlne concde, the formed network s strongly effcent rd and 2 nd stage Next, we consder the case where there are two arlnes. Assumptons employed are smlar to the prevous secton. However, snce we have two arlnes, we must consder the second stage prce competton. Frst, we assume that consumers have no bas toward a partcular arlne n ther decson crtera and they choose flght only through the comparson of the servces offered. In ths case, the competton becomes that of Bertrand type wthout product dfferentaton. Then we take up the case where consumers care for the dentty of arlne. We n partcular assume that when two arlnes offer flghts for the same route, then they splt the benefts evenly Bertrand competton and Monopoly Case In ths case, we assume that there s no product dfferentaton (vertcally nor horzontally) n the ar flght market. Proposton 3 (1) If c 2 /2 s satsfed, then the arlne network formed s G( G A, G B ) {(, ),( G A, ),(, G B )}, where and are as follows: 1. c :complete 2. 2 /2c :star (2) When c 2 /2, arlne network s the empty network. 8

9 Under the Bertrand competton, t s possble to have a strongly effcent network n every occason Bertrand competton and duopoly case We consder the case where two arlne offer flghts for the same route, then the profts are splt equally. The crteron to form a lnk s ts proftablty. Lemma 1: Network for each career s gven by : 1. 1 c :complete c :star 3 3. c :1lnk 4. c :empty 4 Combnng the behavors of 2 companes, we get a pcture of the entre arlne networks, and we obtan: Proposton 3: Formed arlne networks are as follows: 1. 3 c 4 :complete 3 2. c :star 3. c :empty 4 Form ths proposton, we see that there s a case where effcency s not guaranteed by the formed network Arport compettons on Arlne networks Here, we dscuss the effect of arport competton on arlne networks based on the observatons made n the prevous secton. The objectve of the arlne company s to maxmze the number of flghts. We focus on the landng fee as the strategc varable to represent the competton among arports. 9

10 3.3.1 Symmetrc Case Suppose all the arport are dentcal. Arport competton by means of landng fees as the strategc varables. Proposton 4: Each arport lowers the fee to the level where ther benefts are essentally nl. But there shall be no effect on the lnk formatons by arlne companes Asymmetrc Case Now suppose that one arport has an advantage n the cost reducton compared to other arport. Proposton 5: In ths case, by lowerng ts landng fee much lower than 2 ( 2 )/4, t could nduce more flghts nto ts port. 4. Concludng remarks In ths paper, we analysed the relatonshp between network formaton by arlne companes and competton among arports form theoretcal vewpont, especally drawng heavly from the network formaton theory. We derve some condtons for formaton of specfc network types. In realty, advances n technology lke the nformaton technology whch allows more economzaton n flght operaton and also ntroducton of new arcraft wth new materals, declne n the transportaton costs could be dramatc. Theoretcally speakng, ths suggests that the network would contan more drect lnks (complete: pont-to-pont). However there s another forces comng from the rapd growth of demand for ar traffc due to globalzaton and hgh growth n the emergng economes. The competton n the Eastern Asa resembles the observaton made n 4.2.Campred to the Chna (Shangha Pudong arport) and Korea (Inchon arport), whch nvested enormously as part of natonal polcy to obtan the poston of an nternatonal hub, coupled wth the overwhelmng cost advantages, Japan (Narta arport) whch trals the growng 10

11 demand for ar traffc, lost compettve power, and ts passengers and freght have been surpassed by these rvals after early 90 s. Ths s also due to the lack of consstency n the Japanese arport polces. 6. Reference Brueckner, J.K., Dyer, N.J., P. Spller (1992) Fare determnaton n arlne hub-and-spoke networks, Rand Journal of Economcs 23, Brueckner, J.K., (2004)"Network Structure and Arlne Schedulng," The Journal of Industral Economcs 52, Hassn, O., and Oz Shy (2004) "Code sharng Agreements and Interconnectons n Markets for Internatonal Flghts," Revew of Internatonal Economcs 12, Hendrcks, Pccone and Tan (1999) Equlbra n Network, Econometrca vol.67. no.6 pp Hu, Qaoha (2005) Are hub-and-spoke networks better than pont-to-pont networks?, Case Western Reserve Unversty Techncal Memorandum Number 800 Jakson, M.O. and A. Wolnsky (1996) A Strategc Model of Socal and Economc Networks, Journal of Economc Theory 71, Pels, E., Njkamp, P. and P. Retveld (2000) A Note on the Optmalty of Arlne Network, Economc Letters Zhang, A., Fu, X., and H. Yang (2010) Revenue sharng wth multple arlnes and Arports,. Transportaton Research Part B, 44, pp

Air Transport Demand. Ta-Hui Yang Associate Professor Department of Logistics Management National Kaohsiung First Univ. of Sci. & Tech.

Air Transport Demand. Ta-Hui Yang Associate Professor Department of Logistics Management National Kaohsiung First Univ. of Sci. & Tech. Ar Transport Demand Ta-Hu Yang Assocate Professor Department of Logstcs Management Natonal Kaohsung Frst Unv. of Sc. & Tech. 1 Ar Transport Demand Demand for ar transport between two ctes or two regons