This paper investigates the impacts of competition and market uncertainty on airlines network structures and capacity

Transcription

1 PRODUCTION AND OPERATIONS MANAGEMENT Vol. 9, No., January February 200, pp ISSN EISSN POMS DOI 0.340/poms r 2009 Production and Operations Management Society Network Game and Capacity Investment Under Market Uncertainty Qiaohai (Joice) Hu Krannert School of Management, Purdue University, West Lafayette, Indiana , hu23@purdue.edu This paper investigates the impacts of competition and market uncertainty on airlines network structures and capacity investment. The airlines choose their network structures and construct capacities while demands are unknown. After uncertainty is resolved, they determine the total number of seats to offer in each leg constrained by their capacities built earlier. We conclude that market uncertainty is the driving force of hub-and-spoke networks, whereas the market mean is the driving force of point-to-point networks. Which of the two countervailing forces dominates determines the equilibrium network structures. Moreover, we find that the airlines total expected profits in the mixed equilibrium in which the airlines employ different networks are larger than in the pure hub-and-spoke network equilibrium in which each airline employs the hub-and-spoke network. However, the mixed equilibrium does not necessarily yield larger profits than the pure pointto-point equilibrium in which each airline employs the point-to-point network. Key words: hub-and-spoke; point-to-point; cost advantage; flexibility; uncertainty History: Received: September 2007; Accepted: April 2009, after 2 revisions.. Introduction During the first 0 years of US airline deregulation (in the 980s), major airlines (e.g., Northwest) shifted dramatically from point-to-point networks to hub-andspoke networks. By flying passengers from different cities to and from a few hub cities, hub-and-spoke networks not only create economies of scale and density but also yield higher flight frequency and broader geographic coverage. However, some airlines moved back to point-topoint networks recently (e.g., Southwest). Growing level of congestion at major hub airports in the 980s created opportunities for low-fare, no-frills, and pointto-point services exemplified by Southwest airline. Shunning congested airports and direct competition with major airlines, low-cost carriers carved out a thriving market niche by reviving point-to-point services. In response, several major airlines (Continental, Delta, United, and US Airways) created subsidiaries offering similar services. So apparently competition plays an important role in airlines network selection. The roles of competition and uncertainty in shaping airlines network structures have been surprisingly neglected in the literature. This paper strives to provide some insights into how these factors affect airlines network selection, while suppressing the economies of scale and traffic density, which have been identified as the major factors that explain the adoption of hub-and-spoke networks after deregulation (Bailey et al. 985, Brueckner and Spiller 99, 98 Brueckner et al. 992, Caves et al. 984, Hendricks et al. 995a). A hub-and-spoke network pools passengers from several markets into the same leg. If demand on one market turns out to be lower, the airline could increase sales in other markets and thereby avoid or lessen the pressure to lower prices on the market with excess capacity. If demand in one market is high, the airline can ration low-value passengers on several other markets first before rationing passengers with higher willingness to pay on the market with excess demand. A point-to-point network, in contrast, does not possess this flexibility. Namely, a point-to-point airline has to commit to the capacity it has invested and dedicated to each market before demands are known and it cannot easily adjust capacities among different markets. If demand in one market is low, the point-topoint airline is forced to lower price. Consequently, if demand in one market is high, yielding excess demand, it has to raise price, thus rationing out passengers that would be retained if a hub-and-spoke network were employed. Based on above arguments, market uncertainty supports a hub-and-spoke network in a monopoly setting. However, it is not clear whether market uncertainty still favors hub-and-spoke network in the presence of competition. Will the airlines adopt the same network structures or diversify their network structures? To answer this question, we employ a duopoly model in which two competing airlines have to decide their

2 Production and Operations Management 9(), pp. 98 0, r 2009 Production and Operations Management Society 99 network structures and capacities before demands are known. A network requires at least three cities. To simplify the analysis, we let the two airlines compete on a three-city network that includes one hub city and two side cities. While demand is still unknown, the airlines determine their network structures, either a hub-and-spoke network or a point-to-point network, then construct capacity for each leg of their networks. After uncertainty resolves, they engage in Cournot competition. We conclude that market uncertainty is the driving force of hub-and-spoke networks, while market mean is the driving force of point-to-point networks. For a large coefficient of demand variation, only one equilibrium can arise: both airlines adopt hub-and-spoke networks. For a medium value of coefficient of variation, two equilibria may arise: both airlines employ hub-and-spoke networks, or one airline adopts the hub-and-spoke network structure while the other adopts the point-to-point network structure. For a low coefficient of demand variation, all three possible structure combinations can arise in equilibrium: either both airlines adopt hub-and-spoke network or point-topoint networks, or they adopt different network structures. Capacity costs fine-tune which of these equilibria will occur. Moreover, we find that the airlines total expected profits in the mixed equilibrium are greater than in the pure hub-and-spoke network equilibrium as one would expect because in the mixed equilibrium the hubbing airline enjoys a monopolistic flexibility value and the nonhubbing airline enjoys a cost advantage, which we will elaborate further in 3.2.2, in serving the connect market. However, the mixed equilibrium does not necessarily yield the greatest total profits. If the connect market is large enough relative to the capacity costs, the airlines total profits are the greatest in the pure point-to-point equilibrium in which both airlines employ point-to-point networks because the cost advantage dominates the flexibility value. A stream of analytical papers in the economics literature study the impacts of competition on network morphology in the US airline industry. Oum et al. (995) focus on the economies of scale and density of hub-and-spoke networks through a duopoly game on a three-city network. They conclude that the airlines have a tendency towards hub-and-spoke networks due to economies of density, but this tendency causes the prisoner s dilemma: both airlines adopt hub-and-spoke networks, but the competitive advantage cancels out; consequently, both might be worse off. Barla and Constantatos (999) study the impacts of demand uncertainty on a monopolistic airline s network structure in a three-city network. Using a similar framework, Barla and Constantatos (2005) study impacts of demand uncertainty and competition on airlines network structure selection. They assume that the airlines have separate hubs but common non-hub cities. Thus, their network consists of four cities, two hub cities and two non-hub cities. Each airline operates in a three-city network. They compete only for passengers who fly between the non-hub cities where uncertainty resides, and they are monopolists in other legs where demands are deterministic. The authors conclude that a point-topoint structure provides an airline a committed position in the market between the non-hub cities, while a huband-spoke network provides the airlines the flexibility of allocating capacities between the duopolistic market and their respective monopolist markets after uncertainty is resolved. The best response network depends on which of the two countervailing values dominates, the commitment value or the flexibility strategic values. Different from Barla and Constantatos (2005) whose airlines have different hub cities, our airlines compete on a common uncertain three-city network and share a hub. Consequently, their commitment vs. flexibility argument no longer holds for our model as the airlines in our model compete on more than one market. Hendricks et al. (995b) identify conditions under which an equilibrium point with competing hub-and-spoke networks exists for an n-city network with constant demands. Another stream of literature about competition in revenue management setting is also related. Netessine and Shumsky (2005) study a duopoly inventory allocating games between two classes of customers on one leg, and Lan et al. (2007) examine revenue management under limited demand information. The insights obtained from this paper are not limited to airline industry but are also applicable to other industries in a similar context. One such good case is whether to adopt flexible or dedicated technology when firms compete on multi-markets in the presence of uncertainty. A recent Wall Street Journal headline story features Honda s flexible plant where the same assemble line can make both Civic and the CR-V. The flexibility technology provides Honda a competitive edge against its rivals who are stuck with excess capacities for large sport utility vehicles and pick-up trucks because of high demands for small cars as gasoline price soars (Linebaugh 2008). The rest of the paper is organized as follows. Section 2 presents the model, Section 3 analyzes the network game by focusing on the impacts of respective demand uncertainty and market size, Section 4 discusses limitations of a few key assumptions, and Section 5 concludes. 2. Model and Problem Formulation 2.. Model Two airlines indexed by i (i 5, 2) compete across a collection of three cities: a hub city O and two side

3 00 Production and Operations Management 9(), pp. 98 0, r 2009 Production and Operations Management Society cities, A and B (Figure ). We call AO, OB, or AB as legs. When flying through a hub-and-spoke network, passengers who need to fly through only one leg to complete a trip are addressed as local markets, while passengers who need to fly both legs, AO and OB, to complete a trip are addressed as connecting market. Let y represent these two markets (y 5, 2; refers to the connecting market and 2 the local markets). Flying through a point-to-point network, connect passengers take only one flight traveling from A to B. Thus, it takes a hub-and-spoke network two flights (seats) to transfer a connect passenger, but it takes only one flight (seat) for a point-to-point network to do so. Moreover, as indicated by Figure, a hub-and-spoke network pools passengers from the local and connect markets to the same legs, but a point-to-point network does not. Henceforth, we address airlines who employ hub-and-spoke networks as hubbing airlines and those who employ point-to-point networks as nonhubbing airlines. To simplify the analysis, let the local markets be symmetric, e.g., markets AO and BO have a common demand function, and all three markets AO, BO, and AB are bidirectionally symmetric, e.g., the traffic from A to B has the same characteristics as the traffic from B to A. Relaxing the symmetric assumptions will increase the number of markets and affect the specific conditions for different equilibria but will not change the qualitative insights obtained in the ensuing analysis. At the first stage, each airline chooses a network structure, either a hub-and-spoke network (H) or a point-to-point network (P). Three combinations of network structures are possible: m refers to a mixed subgame in which one airline employs a hub-andspoke network and the other a point-to-point network (also referred to as (H, P) or (P, H) subgame); p, or (P, P), refers to a point-to-point subgame in which both airlines use point-to-point networks; and h, or (H, H), refers to a hub-and-spoke subgame in which both airlines use hub-and-spoke networks. Network adjustment is infrequent in airline industry, especially, when one of the end point airports is congested, and gates and landing slots are hard to obtain. Hence, network structure is a strategic long-term decision. Figure A Three-City Network At the second stage, while demands are still unknown, the airlines construct capacities. Capacity investment is also a long-term decision and is determined by the number of aircrafts that the airlines own and the number of available gates and landing slots on the airports. If having chosen a hub-and-spoke network at the first stage, the airlines decide capacities for AO or OB which serves both local and connecting passengers; if having chosen a point-to-point network, they determine capacities for the local and connecting markets, respectively. The term capacity refers to the maximum weekly or monthly number of flights that each airline can offer on each of its legs. Let K i represent hubbing airline i s capacity on each leg, and K yi be nonhubbing airline i s capacity on market y. Let c h and c p represent the unit capacity cost of the respective hub-and-spoke network and point-to-point network. Because a hub-and-spoke network requires more investment in the hub and more complex scheduling issues, we allow capacity costs to depend on the network structures. Airline i s expected profit at the second-stage capacity game is represented by P i. In the economics literature on airline competition, capacity costs are sometimes assumed to be concave in size in order to examine the influence of economies of scale on network structure. Hub-and-spoke network pools passengers into the hub, enabling airlines to use larger aircrafts and increase flight frequency and thereby yielding the economies of scale and density. These effects of hubbing are well-studied and known to support a hub-and-spoke network as mentioned earlier in the introduction. The linearity assumption and excluding aircraft size from consideration simplify the analysis and help eliminate the economies of scale and density, which have been well-studied and allow us to distill the impact of demand uncertainty and competition on the equilibrium networks. At the third stage, after demands are known, the airlines engage in the Cournot game, given their network structures and capacities that have been determined at the first two stages. Quantity means the actual weekly or monthly numbers of flights/seats offered on each leg by the airlines constrained by their capacities. Quantity decision is more flexible and can be adjusted more easily than capacity and network structures. Airlines can reduce number of flights by lowering the usage of aircrafts or putting some aircrafts into storage if demand is too low. However, if demand is high, it may not be easy for airlines to expand their capacities in a short run as building capacity requires acquiring additional airplanes and landing slots. The third-stage game is a two-market Cournot competition in which the airlines compete for connecting and local passengers. First, Cournot competition is typical for the study of network morphology. Second,

4 Production and Operations Management 9(), pp. 98 0, r 2009 Production and Operations Management Society 0 whether Cournot or Bertrand competition is more appropriate depends on the timing of the decisions. Because, once invested, capacity is very costly to change, while price is easily adjusted, quantity thus would be the strategic decision. On the other hand, markets characterized by production to order will fit the Bertrand model better: prices are set first, orders by customers are taken, and production follows. In addition, for markets with competition, airlines often match their prices. For example, a recent search for flights between Indianapolis, IN, and Washington, DC, from showed that all three non-stop flights offered by different airlines were priced at $265, and all four one-stop flights offered by different major airlines were priced at $46. Apparently, prices depend on the number of stops that the flights require rather than on who offers the services. Let q yi be the number of seats that airline i offers in market y, and let p y be the price at market y. We employ a linear inverse demand function on market y: p y ¼ M y ðq y þ q y2 Þ (y 5, 2). This function can be modified to reflect a direct flight value to the connecting passengers. For example, a nonhubbing airline could charge p v for each connect passenger, where v40 and represents direct flight value to connecting passengers, while a hubbing airline can only charge p. One can expect that this modification will increase the likelihood for the point-to-point network to be adopted by the airlines. In addition, qualitative results remain valid if a more general demand function were used as long as revenue functions are concave on their domains. Nonnegative random variable M y follows a bivariate probability distribution F(, ). The mean and the variance of its marginal distributions are denoted, respectively, by m y and s 2 y, and the correlation coefficient between the local and connecting markets by r. Since local traffic is usually larger than connect traffic due to the nature of a hub city, let m 2 m. For expository simplicity, define s 2 T ¼ 4s 2 2 þ s2 4rs s 2 and address it as market uncertainty throughout the paper. The market uncertainty increases as the demand variances increase, or as the coefficient of the correlation between the local and the connect markets decreases. s 2 T is the largest at r 5. That is, the local and connect markets are perfectly negatively correlated. Let p i denote airline i s profit at the third-stage quantity game. After uncertainty is resolved, airline i s problem at the third stage is to choose quantity q yt (y 5, 2) to maximize p i, given its own and competitor s capacities and network structures. Variable costs of serving a passenger are normalized to zero because airlines operating costs are mostly associated with offering a seat rather than serving a passenger. In addition, to ensure the airlines nonnegative profits on the markets, let c h m h /2 and c p m. Figure 2 Sequence of Events We assume that the unit capacity costs and demand distribution are common knowledge. At each of the three stages, the airlines engage in a simultaneousmove noncooperative game. The chronology can be further illustrated by Figure 2. After the airlines have chosen their network structures and capacities, at the third stage Cournot game, the airlines network structure and capacities at each leg also become common knowledge Problem Formulation Each airline has two network strategies (P or H). The pure strategy Nash equilibrium of this 22 noncooperative game (shown in Table ) could be identified by examining the airline s best response network structure given its competitor s. If airline i employs a hub-andspoke network, then its problems at the second and third stages can be formulated as follows, respectively. capacity game : P i ¼ maxfep i 2c h K i g K i X 2 quantity game : p i ¼ max yq yi ½M y ðq yi þ q y;j ÞŠ q i ;q 2i y¼ i; j ¼ ; 2; i 6¼ j s:t: q i þ q 2i K i ; q yi 0; y ¼ ; 2: If airline i employs a point-to-point network, then its problem at the second and third stages are as follows, respectively. X 2 capacity game : P i ¼ maxfep i c p yk yi g K i ;K 2i y¼ X 2 quantity game : p i ¼ max yq yi ½M y ðq yi þ q y;j ÞŠ q i ;q 2i y¼ i; j ¼ ; 2; i 6¼ j s:t: 0 q yi K yi ; y ¼ ; 2: Until Section 3, we assume that the airlines may hold back excess capacities if demands turn out to be Table 22 Network Game H H P, P 2 P, P 2 P P, P 2 P, P 2 P

5 02 Production and Operations Management 9(), pp. 98 0, r 2009 Production and Operations Management Society low. That is, they provide a fewer number of flights if realization of the markets is low. Many cases may arise in the quantity subgame depending on the airlines network structures and on whether the capacity constraints are binding or not. In the supporting information Appendix S, working backwards, we first analyze the capacity and quantity games for all combinations and then prove the existence of subgame perfect Nash equilibrium in the Cournot and capacity games and the uniqueness of the equilibrium point for the symmetric capacity subgames, and finally obtain closed-form solutions for each case of the Cournot game. However, capacities at equilibrium satisfy rather complicated equations; no explicit expressions can be derived. 3. Network Structure Game The analysis of the network game requires comparison of the airlines expected profits at the capacity subgame. Without equilibrium capacities at the second stage, it is not possible to obtain the airlines profits. Even a specific probability distribution for the demand intercepts (e.g., uniformly distributed between 0 and ) does not yield tractable expressions for the capacities and expected profits in the capacity subgames. In order to gain insights into the network game, we impose two additional assumptions to facilitate the analysis and later discuss their limitations and restrict domains of the intercepts so that these two assumptions hold with probability (refer to the supporting information Appendix S). ASSUMPTION. The hubbing airline always serves the connect market. This assumption excludes the extreme cases in which the local markets are so much larger than the connect market that the hubbing airline serves only the former. We have pointed out in Section 2 that the key characteristics of a hub-and-spoke network is pooling passengers from the local and connect markets into the same legs. If a hubbing airline serves only the local market, the pooling effect vanishes. Therefore, it operates as a nonhubbing airline. Consequently, without this assumption, the first-stage network game degenerates. ASSUMPTION 2. Each airline depletes its capacities: for a hubbing airline, q i þ q 2i ¼ K i, and for a nonhubbing airline, q yi ¼ K yi. Assumption 2 forces the airlines to deplete their capacities for any realization of the markets although depletion might induce negative prices when demands are very low. While this assumption is problematic in the long term, in a shorter term it is justified as pointed out by Carey and McCartney (2004): After September, airlines reduced the number of seats available for each kilometer they were flying by more than 20 percent in the United States. However, more than three-quarters of these reductions came from using active aircraft less, with the remaining cuts coming from putting aircraft into storage. Both measures allow airlines to lower variable costs, notably fuel and maintenance, by dropping unprofitable flights, but fixed costs such as lease and debt payments remain. What is more, because the excess capacity is readily available, it can come back quickly. In early 2002, airlines were already bringing back capacity into their networks rather than lose market share on key routes. This is particularly important, since available excess capacity in the present downturn exceeds that at similar points in the past two cycles. The problem here is that, given near-term fixed costs, airlines must sacrifice yield to keep seats filled.... This phenomenon exists in a wide range of industries. For example, according to Mackintosh (2003), automobile makers have been forced to slash prices below costs to keep lines running to maintain high capacity utilization rather than reduce production as models fall out of favor with the public. Assumption 2 eliminates the complexity caused by different demand realizations. A nonhubbing airline s third-stage decision is equivalent to its second-stage decision; however, after demands are known, at the third stage, a hubbing airline could still allocate its capacity between the local and connecting markets. Therefore, whenever there is a hubbing airline in the duopoly game, the capacity and quantity games still need to be analyzed separately. Thus the game still has three stages. Only if both airlines employ point-topoint networks, the three-stage game can be reduced to a two-stage game because capacity and quantity become the same. As indicated in the supporting information Appendix S, what obscures the analysis of the capacity game is that quantity games have too many cases that hinge on the realizations of the markets. Imposing Assumptions and 2 is equivalent to focusing on the analysis of the cases in which both airlines are capacity-constrained, thus depleting their capacities at the third stage. We henceforth study symmetric games, thereby suppressing airline identity index i. Proofs of all propositions are attached in the Appendix. 3.. Quantity and Capacity Games 3... Monopoly. To distill the effect of competition on network structures, we first study a monopoly case. Let P M p and PM h be the monopolistic airline s

6 Production and Operations Management 9(), pp. 98 0, r 2009 Production and Operations Management Society 03 respective expected profit at the second stage when it uses a point-to-point network and a hub-and-spoke network. For the remainder of this paper, we use increasing and decreasing for nondecreasing and nonincreasing, respectively. PROPOSITION. (i) A monopolist nonhubbing and hubbing airline s expected optimal profit at the second stage, respectively, are P M p ¼ X 2 yðm y¼ y c p Þ 2 =4 P M h ¼ 6 ðm þ m 2 3c h Þ 2 þ 2 ½ð2m 2 m Þ 2 þ s 2 T Š: (ii) The monopolist employs a hub-and-spoke network if qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s T 4 ðm þ m 2 3c p Þ 2 þðm þ m 2 Þ 2 6c p m 2 : ðþ Otherwise, it employs a hub-and-spoke network if and only if c h c M ¼ m pffiffi þ m 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 6 ðm þ m 2 3c p Þ 2 þðm þ m 2 Þ 2 6c p m 2 s 2 T: ð2þ M =@s 2 T c M =@ðs 2 T Þ2 0 Part (i) of Proposition implies that a hubbing monopolist s profit increases in the market uncertainty, while a nonhubbing monopolist s depends on its unit capacity cost and the demand intercept means but not on the market uncertainty. It can be shown that a hubbing monopolist s profit at the second stage is a convex function of M y. Thus, the demand intercept depends on the second moment of demand probability distribution, s 2 T, which increases as the demand variances increase or as the coefficient of correlation decreases. A hub-and-spoke network pools local and connecting passengers on the same legs, enabling a hubbing airline to allocate capacities between the local and connect markets accordingly after uncertainty is resolved: a larger quantity for the larger market. The flexibility of deferring allocating capacities till uncertainty is resolved offers a hubbing airline a flexibility value, which increases as the demand variances increase or the coefficient of correlation decreases. For brevity, in the remainder of this paper, we simply state, the flexibility value increases as market uncertainty increases. In contrast, a point-to-point network transports passengers through three separate routes. So it has to determine dedicated capacity for each market before uncertainty is resolved. Consequently, the expected profit of the nonhubbing airline at the second stage is a linear function of M y, depending only on the first moment of the demand intercept but not on the second moment, s 2 T. Part (ii) of Proposition shows that a monopolist employs a hub-and-spoke network if the market uncertainty is so large that () holds; otherwise, it does so only if the unit capacity cost of a hub-and-spoke network is below the threshold c M, which is defined in (2) and is convex increasing in s 2 T Duopoly. We now examine subgame perfect equilibrium under Assumptions and 2. Nash equilibrium points of the three capacity subgames, h, p, and m are summarized as follows. Since all the solutions are unique, all the subgames have a unique equilibrium point. PROPOSITION 2. Equilibrium capacities, prices, and profits of the three network subgames are characterized, respectively, in Tables 2 to 4. In Tables 2 or 4, in the symmetric subgame h or p, the airlines have the same capacity in each leg, so only one entry is entered. For the asymmetric subgame, the first entry is for airline and the second entry is for airline 2. Using Table 3, one can check that because m 2 4m by assumption, in each subgame a connecting passenger is better off buying one ticket that transfers him or her from one side city to the other rather than buying two separate tickets, i.e., 2p 2 4p. Consequently, the connect passengers do not engage in price arbitrage. Let p g y represent the equilibrium prices of market y (y 5, 2) at equilibrium of the subgame g where g 5 p, h, m. The following results can be derived easily from Table 2. COROLLARY. If c h c p, then p h 2 pm 2 pp 2 and p h 4pm 4pp. Corollary implies that the pure hub-and-spoke subgame results in the highest prices for both local and connecting passengers, while the pure point-topoint subgame results in the lowest ones. This ordering is due to the fact that hubbing airlines invest smaller capacities because of its flexibility of allocating capacity between the local and connect markets after uncertainty is resolved. Consequently, smaller capacity leads to higher prices. We note that when c h 5 c p, prices for the local markets are the same for Table 2 Equilibrium Capacities H P H 3 ðm þ m 2 3c h Þ 3 ðm þ m 2 þ 2c p 6c h Þ 3 ½m y þð3 yþc h 2c p Šðy ¼ ; 2Þ P 3 ½m y þð3 yþc h 2c p Šðy ¼ ; 2Þ 3 ðm c p Þðy ¼ ; 2Þ 3 ðm þ m 2 þ 2c p 6c h Þ

7 04 Production and Operations Management 9(), pp. 98 0, r 2009 Production and Operations Management Society Table 3 Equilibrium Prices p h m Local 3 ðm 2 þ 2c p Þ 3 ðm 2 þ 2c h Þ 3 ðm 2 þ c h þ c p Þ Connecting 3 ðm þ 2c p Þ 3 ðm þ 4c h Þ 3 ðm þ 2c h þ c p Þ all subgames, whereas the ordering of prices for the connecting market remains valid. At the same price a connect passenger would prefer a non-stop flight with a nonhubbing airline than a one-stop flight with a hubbing airline. Corollary implies that even the value of a non-stop flight is suppressed, the emergence of nonhubbing airlines benefits connecting passengers the most. By Assumption 2, the airlines deplete their capacities, so the ordering of the quantities offered to the market is converse to the ordering of the prices. That is, the total number of seats offered to each market is the largest in subgame p and the least in subgame h. As shown by Table 2, because a hubbing airline pools local and connect passengers on the same leg, the equilibrium capacity of a hubbing airline in subgame h or m increases as either market intercept mean increases. In contrast, the equilibrium capacity of a nonhubbing airline on a particular market, either in subgame p or m, depends only on the intercept mean of that market because it does not pool passengers from different markets onto the same legs. In subgames p or h, the equilibrium capacity decreases as the corresponding unit capacity cost increases. However, in subgame m, the airlines capacity depends on both c h and c p : the nonhubbing airline s capacity decreases as either c h or c p decreases, the hubbing airline s capacity decreases as c h increases or as c p increases. From Table 4, the first term of the expected profit in subgame h represents the uncertain nature of the markets: the greater s 2 T, the larger the expected profit. Hence, the flexibility value of a hub-and-spoke network persists under competition and is shared by the hubbing airlines. By Proposition, a monopolist enjoys a flexibility value of s 2 T =2. Because 2s 2 T =27os2 T =2, the hubbing airlines total flexibility value in subgame h is smaller than in a monopoly. Hence, competition weakens the flexibility value if both airlines employ hub-and-spoke networks. From Table 4, the terms in the square brackets of the expected profit in subgame h are deterministic. Moreover, it increases as either demand intercept mean increases, or as the unit capacity cost decreases. In subgame m, the hubbing airline enjoys the monopolistic flexibility value, s 2 T =2, while the nonhubbing airline s profit does not depend on the market uncertainty for the same argument as in Section 3... In subgame p, neither airline s expected profit depends on the market uncertainty Network Structure Game: Best Response We now characterize the airline s best response network, given its rival s network structure. Using Proposition 2 leads to the following results. PROPOSITION 3. (i) If the competing airline employs a hub-and-spoke network, the best response network is a hub-andspoke network if c h oc h ¼ c pðm þ 2m 2 Þ 3c 2 p þ s2 T =2 : 2ðm þ m 2 2c p Þ (ii) If the competing airline employs a point-to-point network, the best response is hub-and-spoke if c h oc h ¼ m þ m 2 þ 2c p sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 6 ðm þ m 2 þ 2c p Þ 2 6c p ðm þ 2m 2 Þ 9s2 T 8 The threshold costs c M (defined in (2)), c h,orc h depend on m i (i 5, 2), c p, c h,ands 2 T in rather complex expressions. In the subsequent analysis, we distill the impacts of market uncertainty, demand intercept means, and unit capacity costs, respectively, on the airlines network game. Subsequently, the threshold costs in Propositions and 3 can be further simplified and the equilibrium of the network game can be obtained Uncertainty Effect. To distill the impacts of uncertainty on airlines network structures, let : Table 4 Equilibrium Profits H H P s 2 T 27 þ 9 ½m2 þ 2m2 2 4c hðm þ m 2 Þþ6ch 2Š s 2 T 2 þ 9 ½m2 þ 2m2 2 þ 24c2 h þ 3c2 p 8c hð2c p þ m þ m 2 Þþ2c p ðm 2m 2 ÞŠ ðm þ 2c h 2c p Þ 2 þ 2ðm 2 þ c h 2c p Þ 2 P ðm þ 2c h 2c p Þ 2 þ 2ðm 2 þ c h 2c p Þ 2 9 ½ðm c p Þ 2 þ 2ðm 2 c p Þ 2 Š s 2 T 2 þ 9 ½m2 þ 2m2 2 þ 24c2 h þ 3c2 p 8c hð2c p þ m þ m 2 Þþ2c p ðm þ 2m 2 ÞŠ

8 Production and Operations Management 9(), pp. 98 0, r 2009 Production and Operations Management Society 05 c p 5 c h 5 c and m 5 m 2 5 m. The following results are immediate from Proposition. COROLLARY 2. A monopolist adopts a hub-and-spoke network if s T /m ; otherwise, it employs a hub-and-spoke network if c 2½0; c M Š[½c M ; 2u=3Š and a point-to-point network if c 2ðc M ; c M Þ where pffiffiffiffiffiffiffi p (ii) if 2=3 ost =mo ffiffi 3, then it has an (H, H) equilibrium if c 2ð0; c 2 Þ and an (H, P) equilibrium if c 2ðc 2 ; p m=2þ; (iii) if s T =mo ffiffiffiffiffiffiffi 2=3, then it has an (H, H) equilibrium if c 2ð0; c 2 Þ, an (H, P) equilibrium if c 2½c 2 ; c Þ[½c ; m=2þ, and a (P, P) equilibrium if c 2ðc ; c Þ. c M ¼ m 3 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 3 2 s 2 T; c M ¼ m 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 3 2 s 2 T: ð3þ Similarly, for the duopoly, the following results are immediate from Proposition 3, COROLLARY 3. (i) If the competing airline employs a point-to-point network, the best presponse is a hub-and-spoke network if s T =m4 ffiffiffiffiffiffiffi 2=3 and otherwise, a hub-andspoke if c 2ð0; c Þ[½c ; u=2š and a point-to-point network if c 2ðc ; c Þ. (ii) If the competing airline employs a hub-and-spoke network, the best presponse is a hub-and-spoke network if s T =m ffiffiffi 3 and otherwise, a hub-andspoke network if c 2ð0; c 2 Þ and a point-to-point network if c 2ðc 2 ; m=2þ where c ¼ m pffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 2m 8 2 3s 2 T; c ¼ m pffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 þ 2 2m 8 2 3s 2 T; ð4þ c 2 ¼ m pffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 3m 6 2 s 2 T: From (3) and (4), it can be derived that c M c c c 2 c 2. So a large coefficient of variation favors a hub-and-spoke response regardless of the competing airline s network structure. However, the monopolist s threshold cost above which it selects a hub-and-spoke network is smaller than a duopolist s who competes with a hubbing rival but is larger than a duopolist s who competes with a nonhubbing rival. Hence, competition raises the threshold coefficient of variation for a hub-andspoke to be the best response if the competing airline already employs a hub-and-spoke network. The equilibrium network structure is as follows based on the analysis of the best response network from Corollary 3. Proposition 4 is immediate from Corollary 3. Thus we omit its proof. The equilibrium network structures p are demonstrated by Figure 3. If s T =m4 ffiffi 3, a hub-and-spoke network is the best response regardless of its rival s network choice. Consequently, pffiffiffiffiffiffiffi the network p game has an (H, H) equilibrium. If 2=3 ost =mo ffiffiffi 3, the flexibility value of a hub-and-spoke response is smaller but is still large enough for a hub-and-spoke network to be the best response if the competing airline employs a point-topoint network, but not necessarily so if the competitor already uses a hub-and-spoke. Consequently, in addition to a hub-and-spoke equilibrium, a mixed equilibrium might arise. Specifically, a high unit capacity cost induces an (H, P) equilibrium, while a low unit capacity p cost induces an (H, H) equilibrium. If s T =mo ffiffiffiffiffiffiffi 2=3, the flexibility value of a hub-and-spoke network is so small that the airlines might be better off both employing point-to-point networks. Hence, a point-to-point equilibrium is possible. In addition, as indicated in part (iii) of Figure 3, capacity cost fine-tunes the resulting equilibrium. A low cost results in an (H, H) equilibrium, a high cost results in an (H, P) equilibrium, and a medium cost induces an (H, P) or a (P, P) equilibrium. The following results are derived from (3) and (4). COROLLARY 4. (i) If s T ¼ 0, c ¼ c 2 ¼ c M ¼ 0; (ii) 2 =@s 2 cm =@s 2 =@s 2 =@s 2 T cm =@s 2 T 0. Figure 3 Equilibrium Network Structures PROPOSITION 4. p (i) If s T =m4 ffiffi 3, then the network game has an (H, H) equilibrium;

9 06 Production and Operations Management 9(), pp. 98 0, r 2009 Production and Operations Management Society Figure 4 Equilibrium Network Structures The market size effect. To distill the impacts of demand intercept means on the airlines best response, let c p 5 c h 5 c but keep the market identity index y. Corresponding to Proposition, the following holds for a monopolist. COROLLARY 5. The monopolist chooses the hub-and-spoke network if m om M ¼ 3c=2 þ s2 T =ð6cþ and otherwise chooses the point-to-point network. Similarly, corresponding to Proposition 3, the following holds for the duopoly network game. COROLLARY 6. 2 c =@ðs 2 T Þ2 2 c 2 =@ðs 2 T Þ2 2 c M =@ðs 2 T Þ2 2 c =@ðs 2 T Þ2 2 c M =@ðs 2 T Þ2 0. The network game can be further illustrated by the two-dimensional Figure 4 based on Proposition 4 and Corollary 3. As demonstrated by Figure 4, the duopoly game has a large p region in between c and c,anhregion below c 2 and c 5 m/2, and an m region in the interior formed by the lines c, c, c 5 m/2, and c 2. For a very smaller s 2 T, p region is the largest followed by m region, with a negligible h region. However, as s 2 T increases, p region shrinks quickly, m region expands quickly, and h region expands slowly. At s 2 T ¼ 2m2 =3, p region disappears, m region starts to shrink, but h region continues to expand. Finally, once s 2 T hits 3m3, m region disappears and h region takes over the whole area to the right of s 2 T ¼ 3m2 and below c 5 m/2 (c m/2 is necessary and sufficient to guarantee nonnegative profits for the duopolists). The monopolist s two thresholds, c M and c M,are indicated by the dashed lines in Figure 4. When s 2 T om2, a monopolist has a p region in between c M and c M and an h region between c 5 2m/3 and c M or between c ¼ c M and c 5 0. When s T 2m 2, a monopolist chooses a hub-and-spoke network for c 0. By definition, c is the lower cost threshold for an airline to respond with a hub-and-spoke network when faced with a nonhubbing airline, c 2 is the lower threshold cost for an airline to respond with a hub-and-spoke network when faced with a hubbing rival, and c M the lower threshold for a monopolist to choose a hub-and-spoke network. Because c 4 c M 4c 2, as illustrated by Figure 4, for a fixed s 2 T, a monopolist s h region is smaller than a duopolist s who competes with a nonhubbing rival but is larger than a duopolist s who competes with a hubbing rival. Hence, the magnitude of the impacts of market uncertainty on the airline s network structure depends on whether competition is present and if so on the rival s network structure. (i) If the competing airline employs a hub-and-spoke network, the best response is a point-to-point network if and only if m 4m. (ii) If the competing airline employs a point-to-point network, the best response is a point-to-point network if and only if m 4m where m ¼ c þ s2 T 2c ; m ¼ 2c þ 3s2 T 6c : Interestingly, Corollaries 5 and 6 imply that network selection is independent of the local market mean whether there is competition or not. This irrelevance relies on the assumption that the two local markets are symmetric. Without this assumption, the local market mean will affect the thresholds for the best response network. Moreover, m om M om. Thus the threshold connect market mean for a monopolist to select the point-to-point network is larger than that for a duopoly competing with a hubbing rival, but it is smaller than that for a duopolist competing with a nonhubbing rival to select the point-to-point network. Hence, the threshold connect market mean for a point-to-point to be the best response depends on the presence of competition and the rival s network structure. Using Corollary 5, the equilibrium of the network game is determined as follows. PROPOSITION 5. (i) If m 4m, the network structure game has a (P, P) equilibrium. (ii) if m om om, it has an (H, P) or (P, H) equilibrium. (iii) If m om, it has an (H, H) equilibrium. The equilibrium network structure can be illustrated by Figure 2 T M 2 T 2 T ¼ 3 6c ;

10 Production and Operations Management 9(), pp. 98 0, r 2009 Production and Operations Management Society 07 Figure 5 Equilibrium Network Structures as a Function of c and r 2 T Table 5 Numerical Examples of Network Equilibrium m c Equilibrium Largest profit? h No 0.3 m Yes 0.7 p No h No 0.7 m Yes 2 0oco h No the duopoly game has an m equilibrium in between the lines m and m,apequilibrium above m, and an h equilibrium below m. Interestingly, the monopolist s threshold, m M, cuts the m region exactly in two halves. Above m M the monopolist selects the pointto-point network, and below it the monopolist selects the hub-and-spoke network. Moreover, the three lines never cross each other. When competing with a hubbing rival, knowing that the rival s cost of serving the connect market is twice as expensive as serving the local market, a point-to-point response enables the airline to enjoy the cost advantage in the connect market and thereby forces the hubbing rival to lower its capacity allocation on the more expensive connect market till its marginal profits in the local and connect markets equal. In contrast, a hub-and-spoke response forgoes this cost advantage but entitles the airline to a share of the flexibility value. Similarly, if the rival airline already employs a point-to-point network, a huband-spoke response makes the connect market expensive to serve but grants the airline the flexibility value which increases as market uncertainty increases. A point-to-point response enables the airline to enjoy the cost advantage, but because the rival already uses the same network, the cost advantage is weakened. Given the competing airline s network structure, the best response network, therefore, depends on whether the flexibility value or the cost advantage dominates. We now use numerical examples to illustrate how the equilibrium network structures change with respect to m and c for a giving s 2 T. Let s 2 T ¼ 6. Table 5 summarizes the resulting equilibrium for different values of c and m by Proposition 4 or 5. For m 5 8, c 5 0.5, the network game results in the h equilibrium. Keeping m unchanged, if c 5 0.3, the game yields the m equilibrium, which results in the largest total expected profit. If m 5 2, the game has the h equilibrium for any ca(0, ). One would guess that because symmetric network structure dampens either the flexibility value or the cost advantage, the airlines total expected profits will be most likely the highest in the mixed subgame, where the hubbing airline monopolizes the flexibility strategic value, while the nonhubbing airline monopolizes the cost advantage. However, in Section we show that this conjecture does not necessarily hold. The mixed equilibrium is superior to the pure hubbing equilibrium. If the connect market is large enough, the airlines total profits are the greatest in the pure point-to-point equilibrium not in the mixed equilibrium Comparison of Equilibria. Let P h and P p represent each airline s expected profits in the capacity subgames h and p, respectively. Let P m p and P m h represent the respective nonhubbing and hubbing airlines expected profit in subgame m. It would be interesting to compare the airlines total expected profits in all subgames. The following results are derived directly from Proposition 2. COROLLARY 7. (i) P m h þ Pm p 42Ph ; (ii) if m =c4y, then 2P p 4P m h þ Pm p 42Ph ; (iii) if bom =coy, then P m h þ Pm p 42Pp 42P h ; (iv) if m =cob, then P m h þ Pm p 42Ph 42P p where y ¼ 5 4 þ 9 8 ðs T c Þ 2 and b ¼ þ 2 ðs T c Þ 2. The airlines total expected profits are larger in subgame m than in subgame h. If the connecting market is very large, the total profits are the greatest in subgame p because the cost advantage of point-topoint network, which increases in m /c dominates the flexibility value of a hub-and-spoke network that increases in the market uncertainty. 4. Generalization and Discussion of Assumptions Assumption forces a hubbing airline to serve the connect market even if it turns out to be very small

11 08 Production and Operations Management 9(), pp. 98 0, r 2009 Production and Operations Management Society relative to the local markets. Without Assumption, a hubbing airline can supply an infinitesimal quantity in the undesirable market under extreme market realizations. So the impact of this assumption on the hubbing airline s profit is minimal. Consequently, the overall impact on the network game is not significant. Assumption 2 forces the airlines to deplete their capacities for any demand realization. This assumption affects a nonhubbing airline more than a hubbing airline because the latter can allocate its capacity between the markets after demands realize, while the former cannot. Consequently, this assumption inflates a hub-and-spoke network s flexibility value. Chod and Rudi (2005) study a capacity allocation problem to two markets. Like our model, after uncertainty is known, price is set and the firm allocates capacity between the markets. Hence, the capacity allocating problem is exactly like the hubbing airline s problem in our model except that their firm does not always deplete capacity for every demand realization. As demonstrated numerically by Chod and Rudi (2005) (page 7, 2nd paragraph), this inflation induced by Assumption 2 is not significant. In the supporting information Appendix S, we derive bounds on demand mean intercepts so that Assumptions and 2 hold with probability. To streamline and simplify the analysis we have assumed that the two local markets are symmetric. Consequently, the local market mean does not affect the equilibrium network structure as implied by Proposition 5. However, if the symmetric assumption is relaxed, then the local market means will also affect the equilibrium network. In addition, one also has to consider the correlation between the two local markets and the correlation between the connect market and each local market. The qualitative insights would remain consistent with Propositions 4 and 5. Large coefficient of variation tends to drive the airlines to adopt hub-and-spoke networks, while large market means relative to the capacity costs drives them to adopt point-to-point networks. Our analysis assumes that, at the third stage, the hubbing airline, if any, allocates its capacity between the local and connecting markets depending on the market realization. It is worth pointing out that in practice although hubbing airlines do pools passengers from different market into the same leg, they do not allocate seats among different markets but rather employ nesting capacity controls, bid prices, or dynamic pricing to management pricing and available seats for different classes of passengers. Literature along this line includes Lan et al. (2007) and Wright et al. (2009). Although the capacity allocation assumption is a limitation, since the main purpose of this paper is to examine the influence of competition and demand uncertainty on airlines network structure, and a hub-and-spoke network pools local and connect passengers onto the same legs, omitting the details of capacity control for a hubbing airline does not affect the key insights derived thus far: market uncertainty favors hub-and-spoke networks in the presence of competition. Furthermore, our model can be easily generalized to include more than two airlines. The key insights will remain valid but the expressions of the thresholds for the network equilibrium will change. 5. Concluding Remarks Network structure selection is more complicated in the presence of uncertainty and competition. Market uncertainty supports a hub-and-spoke network whether there is competition or not due to the flexibility value of a hub-and-spoke network. The flexibility value in the duopoly game is the same as in the monopoly if the airlines employ different network structures but is smaller than the monopolist s if both airlines employ hub-and-spoke networks. Hence, competition dampens the flexibility value of a huband-spoke network. A point-to-point network in the duopoly games enjoys a cost advantage that increases in the connect market mean. Similarly, the cost advantage is reduced if both airlines employ pointto-point networks. However, this cost advantage is amplified if the competing airline employs the huband-spoke network. Equilibrium network structures depend on which of the two countervailing effects, flexibility value and cost advantage, dominates. For a large coefficient of demand variation, the flexibility value is fairly high, regardless of the competing airline s network structure, the airline s best response is a hub-and-spoke network, thus resulting in the (H, H) equilibrium. However, for an intermediate coefficient of variation, the flexibility value decreases, making the (H, P) equilibrium possible in addition to the (H, H) equilibrium. For a small coefficient of variation, the flexibility value of hub-andspoke network is insignificant so that the airlines may be better off both adopting point-to-point networks, thus a (P, P) equilibrium is possible in addition to the other equilibria. Capacity cost fine-tunes which of the equilibria occurs. A large connect market supports a (P, P) equilibrium, an intermediate connect market results in an (H, P) equilibria, and a small connect market supports an (H, H) equilibrium. The local market mean, however, does not affect the network game because the costs of serving the local market are the same for both networks and the local markets are symmetric. Furthermore, we find that the (H, P) equilibrium is superior to the (H, H) equilibrium. However, it is not necessarily superior to the (P, P) equilibrium as