Airlines’ competition in aircraft size and service frequency in duopoly markets

Transportation Research Part E 43 (2007) 409–424 www.elsevier.com/locate/tre

Airlines’ competition in aircraft size and service frequency in duopoly markets Wenbin Wei a

a,*

, Mark Hansen

b

Department of Aviation and Technology, San Jose State University, One Washington Square, San Jose, CA 95192-0061, United States b Institute of Transportation Studies, National Center of Excellence in Aviation Operations Research, University of California, Berkeley, Berkeley, CA 94720, United States Received 17 January 2005; received in revised form 30 November 2005; accepted 27 January 2006

Abstract We are interested in how airlines make decisions on aircraft size and service frequency in a competitive environment. We apply three game-theoretic models to analyze airlines’ choices in duopoly markets: one short-haul market and one long-haul market. We study how airlines’ choices in a competitive environment may vary with flight distance, and also do sensitivity analysis to explore how the equilibrium results may change when air travel demand is higher, as it may happen in the future. Our research considers the competition factor in airlines’ decisions on both aircraft size and service frequency, and the impact of these decisions on both the cost and demand sides of airlines’ business. Different from previous studies, our research is based on cost, market share and demand models derived from empirical studies.  2006 Elsevier Ltd. All rights reserved. Keywords: Aircraft size; Service frequency; Game-theoretic; Competition

1. Introduction The airline industry has been changed significantly after the September 11th of 2001. Due to the current economic uncertainties, security concerns and major carriers’ reconstruction and reorganization progress, it is still not clear how the airline industry will look like in the near future. But there is no doubt that air travel demand will keep growing and the constraint of airport capacity will still be a challenge to the aviation industry. Since the last decade of the 20th century, passengers, airlines and airports have all been complaining about congestion and delay at major airports in the US. As most airports are already saturated, it’s hard to imagine that the congestion problem could be resolved quickly and easily in the next one or two decades in the new millennium. *

Corresponding author. Tel.: +1 408 924 3206; fax: +1 408 924 3198. E-mail address: [email protected] (W. Wei).

1366-5545/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2006.01.002

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Most airlines and airports expect and demand that more new runways should be built to improve airport capacity and to reduce airport congestion and delay. But physical runway expansion is very expensive, difficult and sometimes infeasible due to land use, environmental and economic concerns. In the meantime, there is an alternative often overlooked by the airlines, airport and the policy makers: the airport throughput will be increased if airlines choose, or somehow are forced, to increase aircraft size, rather than increase service frequency, to accommodate the air travel growth. If airlines use larger aircraft, with the same number of service frequency, an airport can serve more passengers. Without any physical runway expansion, flight delay can be reduced through airlines’ adjustment of their choices in aircraft size and service frequency, i.e., larger aircraft and less frequency. But currently, the majority of total operations by all jet aircraft in some major airports are by aircraft with fewer than 150 seats, especially in the very high-density markets such as Los Angles–San Francisco. Boeing and Airbus, the two main aircraft providers, have two different perceptions of future aircraft size. In Boeing’s (2005) ‘‘Current Market Outlook’’, the company emphasizes that, to accommodate future air travel growth, airlines will offer more frequencies as a primary form of non-price competition. They forecasts that ‘‘the single-aisle airplanes dominate future deliveries’’, and ‘‘the share of 747 and larger airplanes will fall from 6% to 4%’’ at the end of 2024. At the same time, Airbus’s (2005) ‘‘Global Market Forecast 2004–2023’’ predicts that, at the end of 2023, ‘‘the very larger aircraft will account for 6% of the world passenger fleet, the same percentage as represented by 747s today’’, and the ‘‘twin-aisle and large aircraft will take a bigger role’’ in the future. And therefore, the ‘‘average seats per aircraft will increase 20% from 181 to 215’’. Airbus emphasizes that airlines could be forced to use larger aircraft if planned and required airport facilities cannot be completed on time to meet future demand. In practice, Airbus has just delivered their A380, the aircraft with more than 500 seats, while Boeing already gave up their plan of the 747 Jumbo Jet called 747X in year 2002, and is now focusing on its development of the 787, a cost-efficient but conventional jet with 200–300 seats. But we cannot find any description of research methodology that Boeing and Airbus apply in their forecasts of future aircraft size. In our research, we are interested in exploring how airlines make decisions on aircraft size and service frequency, especially in a competitive environment. The two bases of our research are a cost model and a market share model for airlines using different combinations of aircraft size and service frequency in their operations. Then, we apply game-theoretic models to investigate airlines’ choices of aircraft size and service frequency in competitive markets. Correspondingly, previous literatures on this subject can be classified in three categories: the cost side, the demand side and competition analysis. On the cost side, such literatures as Miller and Sawers (1970), Keeler (1972), Douglas and Miller (1974), Meyer and Oster (1984), Morrison (1984), Bailey et al. (1985), Morrison and Winston (1986), Kirby (1986), and Hansen and Kanafani (1989) study cost economics of aircraft size and focus on how airlines’ operation costs vary with different size of aircraft. More recently, based on a translog model, Wei and Hansen (2003) develop an econometric cost function for aircraft operating cost and find that ‘‘economies of aircraft size and stage length exist at the sample mean of their data set, and that for any given stage length there is an optimal size, which increases with stage length’’. The scale properties of the cost function are changed considerably if pilot unit cost is treated as endogenous, since it is correlated with size. And they conclude that ‘‘the cost-minimizing aircraft size is therefore considerably smaller, particularly at short stage lengths, when pilot cost is treated as endogenous’’. On the demand side, Eriksen (1977), Abrahams (1983), Viton (1986), Russon and Hollingshead (1989), and Coldren et al. (2003) study the role of aircraft size and service frequency in airlines’ demand and market share. More recently, Wei and Hansen (2005) build a nested logit model to study the roles of aircraft size, service frequency, seat availability and fare in airlines’ market share and total air travel demand in non-stop duopoly markets. They find that ‘‘airlines can obtain higher returns in market share from increasing service frequency than from increasing aircraft size’’. In general, if there is more than one airline in the market, one carrier’s market share and revenue depend not only on its own service but also on the services provided by all other airlines in the market. Thus we need to systematically study airlines’ choices of aircraft size in a competitive environment. Researchers started to use theoretical models to analyze airlines’ competition in 1970s. Douglas and Miller (1974) study market equilibrium and pinpoint the effect of regulated price on total capacity provided by airlines, and the relationship between price and market structure. Schmalensee (1977) explores further the basic properties of theoretical

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equilibrium for price-regulated non-cooperative oligopoly markets. Dorman (1983) builds an airline competition model for deregulated markets, where airlines compete in both price and frequency. But none of these theoretical models is applied to any real case. Later, Norman and Strandenes (1990) use game-theoretic Bertrand and Cournot models to study airlines’ price and frequency competitions in a deregulated market, and find that the Bertrand and Cournot models give identical results, since aircraft capacity constraint is binding in all equilibria. But they don’t take aircraft size as a decision variable in their demand analysis. Ghobrial (1983) studies airlines’ frequency competition in a hub-and-spoke system based on a statistical market share model, followed by Hansen (1990) which applies game-theoretic models to study airlines’ frequency competition in a hub-dominated environment. Hansen (1996) finds the Nash equilibrium for both service frequency and price in a hub-and-spoke network. But none of these studies takes aircraft size as a decision variable. In summary, there are extensive studies on the cost economics of aircraft size and on the role of aircraft size and service frequency in airlines’ demand and market share. The empirical cost function and market share model are developed in such literatures as Wei and Hansen (2003, 2005). The studies on airline competition start with theoretical models in analyzing the effects of regulation or deregulation on the structure of airline industry and the air travel market in terms of fare, service route and service frequency. Game-theoretic models have been the main tools to study airlines’ competition in service frequency or price in either a market level or a network (e.g. hub-and-spoke) level. But very few, if any, model takes aircraft size into consideration in the analysis of airlines’ competition strategies, either in theory or in practice. To fill this gap, our research seeks to build models for competition analysis that take both aircraft size and service frequency as airlines’ decision variables. Our research is different from previous studies and makes contribution to the literature in six important areas: (1) While previous literatures on aircraft size are either focused on cost side or demand side, our research considers both the cost and demand sides and assumes that airlines’ business goal is to maximize its profit, which is the difference between the revenue and the cost; (2) We consider the competition factor in influencing airlines’ choices of aircraft size and service frequency, and capture the fact that one airline’s behavior is also driven by the behavior of other airlines in the same market; (3) Both aircraft size and service frequency are regarded as decision variables in our competition analysis, which is rarely seen in any other competition model; (4) Different from pure theoretic studies, our studies are based on cost and demand models derived from empirical studies, and the parameters and coefficients applied in our game-theoretic models are selected to be as close to reality as possible; (5) We focus on duopoly markets, which allows us to apply the game-theoretic models based on cost and demand functions derived from the airlines’ actual operation data. Without losing generality, this simple setting will not only reveal the most fundamental relationships among various factors in the complicated airlines’ decision-making process, but also allow us to obtain the competition equilibrium, which is usually difficult to derive and rarely seen in empirical studies; (6) We consider three different game-theoretic models, corresponding to three practical cases of airlines’ competition in reality, so that we can explore the complexity of airlines’ competition behavior in depth, such as how the market equilibrium changes with different stage and demand level. At last, it is necessary to emphasize that our research problem, i.e., airlines’ decisions on aircraft size and service frequency in the strategic level, is different from airlines’ decisions on aircraft in the tactical or operational level, i.e., ‘‘fleeting planning’’ problem or ‘‘fleet assignment’’ problem in the airline industry. Usually, airlines’ strategic choices of aircraft are affected by airlines’ long-term business strategy, industry development, government regulation, airport policy, and aircraft manufacturers. Therefore, airlines’ strategic decision is a long-term decision, while the tactical and operational decisions are short term decisions. In the tactical level, airlines make decision on the actual acquisition of new aircraft through the company’s department or function of ‘‘fleet planning’’, which consider such factors as fleet commonality, purchasing price, and service routing structure (e.g. hub-and-spoke vs. point-to-point). In the operation level, airlines make decisions on what aircraft should fly in each segment on a daily base through the department or function of ‘‘fleet assignment’’, with coordination of other departments such as ‘‘scheduling’’ and ‘‘capacity planning’’. Most major airlines in the US apply some computer packages to complete the ‘‘fleet assignment’’ task. The core of these computer packages is a mathematical programming model, which consists of a profit-maximization objective function and a set of constraints for aircraft availability, scheduling feasibility and service covering. As the output of these computer models, what type of aircraft will actually fly on each route on the daily basis will be determined, which is constrained by the aircraft fleet available in the company and the pre-determined routing network.

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While airlines’ decisions on aircraft size and service frequency in both the tactical and operational levels are constrained by airlines’ choices of routing structure, airlines’ decisions on aircraft size in the strategic level, which is the subject in our research, are usually made at the same time, and to be consistent, with their strategic decisions on routing structure. For example, if capacities of major airports are not able to meet the increasing demand of aircraft operations, airlines can ‘‘strategically’’ change their operating network such as focusing more on hub-and-spoke service instead of direct service in the long run, and keep the size of aircraft the same in their future fleet. In this research, we start our analysis in the level of a market (with one origin and one destination), and assumes airlines’ strategic choices of aircraft size and frequency are consistent with airlines’ strategic choices of their routing network. In the future, our analysis can be extended to the network-level when we have a corresponding market share model which can captures the roles of both aircraft size and service frequency in airlines’ demand and market share in the network level. The next section of this paper introduces the two bases of our research: airline cost function and market share model, and then describes three different game-theoretic models for studying airlines’ competition in aircraft size and service frequency. We introduce a solution method, and discuss different properties of the derived equilibrium in these game theoretic models. The third section shows how these models can be applied in practice through a case study, and explains our sensitivity analysis of airlines’ strategic choices of aircraft size to accommodate higher level of demand in the future. At last, we will summarize our analysis, research results and implications, and direct further studies. 2. Model formulation We assume airlines’ objective in choosing different aircraft size and service frequency is to maximize their profit, the difference between the revenue and the cost incurred. Therefore, the two bases of our research are: (1) airline cost function and (2) airline demand and market share models, which capture the roles of aircraft size and service frequency in airlines’ cost, demand, market share and profit. Based on these functions, we build three game-theoretic models to analyze what aircraft size and service frequency airlines would choose to maximize their profit under three different competition scenarios. 2.1. Two bases: cost function and market share model In this research, we directly apply the cost function specified in Wei and Hansen (2003) and demand and market share models specified in Wei and Hansen (2005). On the cost side, Wei and Hansen (2003) study the relationship between aircraft cost and size for large commercial passenger jets. The general aircraft operating cost function is specified as follows, in which each data point is for aircraft type k, operated by airline i, during time period t. DC ikt ¼ f ðSeatikt ; ASLikt ; PFuelikt ; PPilotikt ; Ai Þ

ð1Þ

where DCikt is the average aircraft operating cost per flight for airline i, aircraft type k, in time period t Seatikt is the average seat capacity for aircraft type k, airline i, in time period t ASLikt is the average stage length for aircraft type k, airline i, in time period t PFuelikt is the unit fuel price per gallon for aircraft type k, airline i, in time period t PPilotikt is the unit pilot cost per block hour for aircraft type k, airline i, in time period t Ai is an airline-specific factor capturing characteristics of its individual production technology and efficiency Then a de-mean translog model is used to specify the cost function: N N X N h i h i X h ih i X lnðDC ikt Þ  lnðDCÞ ¼ Ai þ bj lnðX jikt Þ  lnðX j Þ þ kj‘ lnðX jikt Þ  lnðX j Þ lnðX ‘ikt Þ  lnðX ‘ Þ j¼1

j¼1

‘Pj

ð2Þ

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where Dikt lnðDCÞ N X jikt

is the same as defined in (1) above is the sample mean of the log of aircraft operating cost is the total number of independent variables in the model is the value of independent variable j for airline i, aircraft type k, in time period t (four independent variables are defined in the general cost function above) lnðX j Þ is the sample mean of the log of the independent variable j Ai, bj, kj‘ are coefficients to be estimated The model is calibrated using the data of the 10 largest US airlines in the US domestic market from the first quarter of 1987 to the fourth quarter of 1998. The main estimated coefficients are summarized in Table 1. Since the first and the second-order coefficients on aircraft size are 0.81 and 0.35, respectively, Wei and Hansen (2003) claim that, at the mean values of the data, there are economies of aircraft size, and that the elasticity of direct operating cost with respect to aircraft size increases with aircraft size. The second-order coefficient on the interaction between size and stage length is 0.18, which implies that ‘‘as stage length increases, cost elasticity decreases’’ and ‘‘there are stronger scale economies in aircraft operating cost at longer distances’’. Therefore, they argue that ‘‘for any given stage length there is an optimal (least-cost) aircraft size, which increases with stage length’’. On the demand side, we directly apply the market share model derived in Wei and Hansen (2005), which, based on a nested logit model, studies the roles of aircraft size, service frequency, and fare in airlines’ market share and total air travel demand in non-stop duopoly markets. From the database products of Onboard and O&D Plus, time series quarter data from the 1st quarter of 1989 to the 4th quarter of 1998 for six of ten largest airlines are used to estimate the model coefficients. The calibrated market share model is specified as !1:093  0:445  0:004 S im Freqim Seatim expðFareim Þ ¼ ð3Þ expðFarejm Þ S jm Freqjm Seatjm

Table 1 Statistical estimation results for aircraft operating cost function Explanatory variable

Estimated coefficients

T statistic

Seat ASL Pfuel Ppilot Seat * seat ASL * ASL Pfuel * Pfuel Ppilot * Ppilot Seat * ASL Seat * Pfuel Seat * Ppilot ASL * Pfuel AA dummy AS dummy CO dummy DL dummy HP dummy NW dummy TW dummy UA dummy US dummy WN dummy

0.813 0.748 0.238 0.424 0.346 0.320 0.123 0.137 0.184 0.665 0.160 0.504 0.100 0.145 0.054 0.200 0.177 0.120 0.132 0.121 0.154 0.545

24.51 34.52 7.18 18.06 15.16 10.57 1.96 13.88 3.16 6.14 3.23 6.27 5.48 5.12 2.37 12.92 6.25 5.43 5.55 6.65 8.19 13.28

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where: Sim, Sjm are the market share of airlinei and airline j, respectively, in market m Freqim, Freqjm are the service frequency provided by airline i and airline j, respectively, in market m Seatim, Seatjm are the aircraft size (number of seats per flight) used by airline i and airline j, respectively, in market m Fareim, Farejm are the average fare charged by airline i and airline j, respectively, in market m Since the coefficient for the ratio of service frequency is 1.093, which is greater than the coefficient for the ratio of aircraft size, 0.445, airlines can obtain higher returns in market share from increasing service frequency than from increasing aircraft size, and this also confirms an S-curve effect of service frequency on airlines’ market share. Therefore they conclude that ‘‘airlines have no economic incentives to use aircraft larger than the least-cost aircraft, since for the same capacity provided in the market, increase of frequency can attract more passengers than increase of aircraft size’’. We also use the total demand model derived in Wei and Hansen (2005) in our research. This total demand model specifies that the total number of passengers traveling in a market for all airlines is a function of the total population and average personal income in both origin and destination metropolitan areas, and of the service variables (including aircraft size, service frequency, and ticket price) of all airlines serving in this market. Based on these previously derived cost function, market share and total demand models, we can find directly the effects on airlines’ profit when airlines choose different combinations of aircraft size and service frequency. Therefore, we can build game-theoretic models to find airlines’ rational strategic choices of aircraft size and service frequency in a competition environment, under the ‘‘profit maximization’’ assumption. 2.2. Assumptions in game-theoretic models In this study, we apply non-cooperative non-zero sum game-theoretic models to study airlines’ strategic choices of aircraft size and service frequency. We concentrate on airlines’ long-term behavior on the level of market (a city-pair), in particular a duopoly market, where all the local passengers are served by direct non-stop flights. There are two players in the game, and we term them airline 1 and airline 2. Each player selects an aircraft size and a service frequency. Each airline’s objective is to maximize its profit. We assume that the fare charged to passengers by each airline is exogenous to their decisions on aircraft size and service frequency, i.e., fare is not a strategic decision variable in these games. On the one hand, despite the occasional ‘‘fare war’’, airlines are generally unwilling to use pricing as a competitive variable. On the other hand, the simple average fare variable in our applied demand and market share models (Wei and Hansen, 2005) cannot capture the complex airlines’ revenue management process, although it is not too hard to incorporate a fare variable in the formulated games. One of the most important concepts in game theory is the ‘‘strategy’’ of each player. A ‘‘strategy’’ is different from an ‘‘action’’ in depicting players’ behavior or choices in game-theoretic models. A strategy of a player is a full description of what he/she would do, i.e., choices of ‘‘action’’, in every feasible situation (or ‘‘contingency’’) in the game. Therefore, in multiple-stage games, a player’s ‘‘strategy’’ includes, in each stage, all this player’s possible choices of ‘‘action’’ conditioning on last stage’s results, while in one-stage game, a ‘‘strategy’’ is simply the player’s choice of ‘‘action’’. We concentrate on the pure strategy of the game, i.e., uncertainty is not taken into consideration here, and we assume that each airline will choose only one type of aircraft and one frequency. We also keep the assumption of ‘‘complete information’’ in our games, i.e., we assume that each airline knows the other airline’s payoffs, available actions and other information; moreover, each airline knows that the other airline has such complete information, and this ‘‘knowing’’ is also known to each other, and so on. Also all the games in our research are formulated for one period. A central concept in game theory is the equilibrium property of the game. The most fundamental equilibrium is called Nash equilibrium (NE), which is defined as a set of strategies chosen by each player in the game,

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where no player has the incentive to change their own strategy when given the other players’ strategy. The more strict equilibrium in multiple-stage games, called sub-game perfect Nash equilibrium (SPNE), is defined as a profile of strategies, of which the continuation strategies (parts of the strategies, which are applicable within each sub-game) constitute a Nash equilibrium in all sub-games. In our formulated multiple-stage games below, each stage of the game corresponds to a sub-game. The sub-game perfect Nash equilibrium (SPNE) strategy, which excludes all ‘‘incredible threat’’ strategies, is more powerful and reasonable than NE strategy in multiple stage games. Therefore we focus on SPNE strategy, rather than NE strategy, for the leader-andfollower game and the two-level hierarchical game. For interested readers, more theoretic properties for NE and SPNE and their relationships can be found in game-theory references such as Gibbons (1992) and Bierman and Fernandez (1993). The purpose of applying game-theoretic models is to help us understand and predict airlines’ strategic decisions, the processes leading to which are generally very diverse and complex. Since we are interested in airlines’ interactive behaviors and their long term strategies in the choices of aircraft size and service frequency, there are two basic questions that need to be answered in order to analyze airlines’ decisions in a duopoly market: (1) Does each airline make decisions on the choice of aircraft size and on the choice of service frequency sequentially or simultaneously? (2) Do the two airlines in the market make decisions at the same time, or does one airline have an advantage over the other and make decisions before the other? Based on the answers to these two questions, we use three different games to formulate the competition between the two airlines. These three games are: a one-shot simultaneous game, a leader-and-follower Stackelberg game, and a two-level hierarchical game. Next, for each of these three game-theoretic models, we first describe the competition process that the game assumes, and then formulate the game mathematically and find its equilibrium solutions. 2.3. Game 1: one-shot simultaneous game The first game is called a one-shot simultaneous game. In this type of game, the two airlines in the market choose aircraft size and service frequency at the same time, each assuming that the other airline will have a fixed choice once the choice is made. According to the ‘‘complete information’’ assumption mentioned above, both airlines know all the available choices of each other as well as the payoffs from each combination of choices. In this simultaneous game, when each airline makes a choice, they assume that the other airline will make an optimal choice for its own benefit, and each airline’s optimal choice depends on its competitor’s choice. Mathematically, in this case, there are two optimization problems facing the airlines, and aircraft size and service frequency are decision variables for each airline. We use the notations S1, F1, S2 and F2 as variables to represent the size and frequency used by the first and the second airlines. Profit 1(S1, F1, S2, F2) and Profit 2(S2, F2, S1, F1) denote the profit functions for airline 1 and airline 2, respectively, when airline 1 chooses S1 and F1, and airline 2 chooses S2 and F2. Finally, we use the symbol arg max Profit(Æ) to denote the argument (or arguments) that makes function Profit(Æ) achieve maximum. Then the two airlines’ decision problem, under the one-shot simultaneous game, can be formulated through the following two simultaneous optimization models. 8 Profit 1ðS 1 ; F 1 ; S 2 ; F 2 Þ < Maximize S 1 ;F 1 ð4Þ : Maximize Profit 2ðS 2 ; F 2 ; S 1 ; F 1 Þ S 2 ;F 2

The solutions to this problem, denoted as ½ðS 1 ; F 1 Þ; ðS 2 ; F 2 Þ, is the Nash equilibrium (NE) strategy in this game, in which airline 1’s optimal choice of aircraft size and frequency is ðS 1 ; F 1 Þ, given airline 2’s optimal choice of ðS 2 ; F 2 Þ, and vice versa. 2.4. Game 2: leader-and-follower Stackelberg game The second game is a leader-and-follower game, in which we assume that, due to specific contracts or historical reasons, one airline has some dominance in the market, so that it has the advantage of deciding size and

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frequency before the other, while the second airline makes its decisions after observing the choice made by the first airline. This is called a Stackelberg game in game theory. Mathematically, the first airline will optimize its decisions in terms of both aircraft size and frequency, recognizing the fact that the second airline will make its optimal choice based on what the first airline does. Using the same notation as in the one-shot simultaneous game, the airlines’ decision problem can be formulated as:   Maximize Profit 1 S 1 ; F 1 ; S 2 ; F 2 ð5Þ S 1 ;F 1

Subject to:



 S 2 ; F 2 ¼ argmax Profit 2ðS 1 ; F 1 ; S 2 ; F 2 Þ

ð6Þ

S 2 ;F 2

The solution process is the reverse of the sequence of airlines’ decision process. In the first step, we solve the sub-optimization problem (6) for airline 2, and find airline 2’s optimal choices conditioning on any possible decision made by airline 1. In the second step, we plug in the result (airline 2’s reaction) from the first step to the optimization problem of airline 1, and find the optimal choices for airline 1 by solving the unconstrained optimization problem (5). Corresponding to the two steps of optimization problems above, we can find the sub-game perfect Nash equilibrium strategy for the whole game, denoted as ½ðS 1 ; F 1 Þ; ðS 2 ðS 1 ; F 1 Þ; F 2 ðS 1 ; F 1 ÞÞ, in which airline 1 will choose ðS 1 ; F 1 Þ first; and after observing what airline 1 chooses, airline 2’s choice is ðS 2 ðS 1 ; F 1 Þ; F 2 ðS 1 ; F 1 ÞÞ, which depends on airline 1’s choices of (S1, F1). 2.5. Game 3: two-level hierarchical game The third game is called two-level hierarchical game. In this case, we assume that two airlines make decisions on aircraft size and service frequency in two stages: in the first stage, airline 1 and airline 2 make their decisions on aircraft size simultaneously, and then in the second stage, after they know each other’s decisions on aircraft size, the two airlines make decisions on service frequency simultaneously. Mathematically, the airlines’ decision problem can be formulated as a two-level optimization problem. In the first level or the first stage of the game, the competition is an aircraft size competition between two airlines; both airlines perceive the second level frequency competition, or are constrained by future frequency competition. The mathematical formulation, with objective functions and constraints, for this problem is shown as   8 Profit 1 S 1 ; F 1 ; S 2 ; F 2 < Maximize S1   ð7Þ : Maximize Profit 2 S 2 ; F 2 ; S 1 ; F 1 S2

Subject to: (    F 1 ¼ argmaxF 1 Profit 1 S 1 ; F 1 ; S 2 ; F 2   F 2 ¼ argmaxF 2 Profit 2 S 2 ; F 2 ; S 1 ; F 1

ð8Þ

Obviously, the problem can be solved in two steps. In the first step, we solve the low-level frequency competition problem with any given pair of aircraft size, which is expressed as a constraint (8) above. In the second step, we solve the high-level size competition problem (7) constrained by the reaction results from the frequency competition. Corresponding to the two steps of the optimization problem above, there are two sub-games in this game. The sub-game perfect Nash equilibrium strategy for the whole game consists of the Nash equilibrium in both sub-games, which means that, in the first stage, airline 1 and 2 choose their aircraft size S 1 and S 2 , respectively; in the second stage, both airlines will choose service frequency on the basis of their observation of the aircraft sizes (S1, S2) they choose in the first stage. In the next section, we apply the formulated models in a case study to see how the competition models can be used to analyze airlines’ strategic decisions on aircraft size and service frequency, and how these decisions may change with other factors such as flight distance and demand level in the market.

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3. Model application In airlines’ business, one market is more or less different from others, since every market has some specific features. Due to changes either on the supply side (such as the number of airlines in service, airlines’ service variables and network structure) or on the demand side (such as population and economy), the environment of each market keeps changing. Therefore, every market is unique and dynamic. The purpose of this paper is not to develop models that can fully capture the reality of any specific market, nor to generalize findings from one market to all other markets. On the contrary, we take a hypothetical and idealized, yet typical, duopoly market as an ‘‘example’’ and study how competition could influence airlines’ choices of aircraft size and service frequency, and how other factors such as flight distance and demand level in the market may change the competition equilibrium. Since the two bases of our research—airline cost function and market share model— are derived from empirical studies based on airlines’ actual operation data, we’d like to choose the same coefficients or parameters for the model of our case study as those in related literatures, i.e., Wei and Hansen (2003, 2005). The detailed market environment and airlines’ service coefficients for our case study are described next. 3.1. Case description (1) There are two hypothetical markets in our case study: a short-haul market with a 400-mile flight distance, and a long-haul market with a 2400-mile flight distance. Population and income data for the origin and destination that are needed for the demand model are based on the metropolitan areas of Albuquerque of New Mexico (ABQ) and Phoenix-Mesa of Arizona (PHX) in 1998. The ABQ–PHX market is one of these 13 markets, the data of which are used to derive the market share and demand model in Wei and Hansen (2005). The population is 680,439 and 2,933,542, and the average personal income (the total personal income divided by the total population) is $23,819 and $25,302 for ABQ and PHX, respectively, in 1998. (2) There are two identical airlines operating in the market. Their operating characteristics are based on an ‘‘average’’ Airline, based on the data used to calibrate the cost model in Wei and Hansen (2003). This airline’s unit fuel cost and unit pilot cost are $0.46 per gallon and $354 per block hour, whose logarithms are equal to the logarithms of the mean of all the data in the sample. For this airline, we take its dummy variable (the fixed effect in aircraft cost model) as 0.545, the value for Southwest Airlines derived in Wei and Hansen (2003), which is also listed in Table 1. (3) The fares charged by the two airlines are the same, fixed at 50 dollars (one-way) for the 400-mile market, and 300 dollars (one-way) for the 2400-mile market. (4) Both airlines will choose one of four available sizes of aircraft: 100-seat, 200-seat, 300-seat and 400-seat, and choose one of 10 service frequencies: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28 times per day. (5) The two airlines have the same quarterly number of connecting passengers in both the short-haul and long-haul markets, which is assumed to be the number of connecting passengers (43,601) that Southwest Airlines actually had in the Albuquerque-Phoenix market in the fourth quarter of 1998 according to Wei and Hansen (2003). The revenue generated from connecting passengers is based on the assumption of $30 per passenger in the short-haul market and $150 per person in the long-haul market. Following the notations of S1, F1, S2 and F2, which are introduced in the last section as variables to represent airline 1’s and airline 2’s general choices of aircraft size and service frequency, we use s1,i, f1,j, s2,k and f2,l to represent the specific choice of aircraft size i by airline 1, service frequency j by airline 1, aircraft size k by airline 2, and service frequency l by airline 2, respectively. Next, we apply the three game-theoretic models, together with previously developed cost function, market share and demand model, to the case described above, and derive equilibrium solutions as airlines’ strategic choices of aircraft size and service frequency under three different competition scenarios. 3.2. One-shot simultaneous game In the last section we denote the airlines’ general strategy for this game as [(S1, F1), (S2, F2)]; then a specific strategy in this section can be represented as [(s1,i, f1,j), (s2,k, f2,l)], which consists of both airlines’ choices of

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Table 2 Indices of airlines’ operation resulted from the NE and SPNE strategies in the one-shot simultaneous game in the case study Operation indices Aircraft size (seats) Service frequency (daily) Load factor (%) Demand (# of local passengers, quarterly) Cost ($, quarterly) Revenue ($, quarterly) Profit ($, quarterly)

400-mile market 100 13 68.5 9.1 · 104 28.68 · 105 35.80 · 105 7.12 · 105

2400-mile market 200 7 51.2 5.6 · 104 9.35 · 106 14.95 · 106 5.60 · 106

aircraft size and service frequency. Since there are 4 choices of aircraft size and 10 choices of service frequency by each airline, there are totally 4 · 10 · 4 · 10 = 1600 possible strategies available in this game. A Nash equilibrium strategy ½ðs1 ; f1 Þ; ðs2 ; f2 Þ in this one-shot simultaneous game means airline 1 chooses aircraft size and service frequency ðs1 ; f1 Þ and airline 2 chooses ðs2 ; f2 Þ, and neither airline has the incentive to change its choices when given the other’s choices. Mathematically, we want to find combinations of ðs1 ; f1 Þ and ðs2 ; f2 Þ from all 1,600 possible combinations that will solve the optimization problem (4), although there is no guarantee that there exists a solution or the solution is unique because of the non-zero-sum property of the game and the discreteness of choices available for each airline. In this case study, we find that there are unique Nash equilibrium solution strategies for the short-haul and long-haul markets. The solutions are ½ðs1 ¼ 100; f1 ¼ 13Þ; ðs2 ¼ 100; f2 ¼ 13Þ, and ½ðs1 ¼ 200; f1 ¼ 7Þ; ðs2 ¼ 200; f2 ¼ 7Þ, respectively. This means, under the Nash equilibrium condition, both airlines will choose aircraft of 100 seats and operate 13 times per day in the 400-mile market, and choose aircraft of 200 seats and operate 7 times per day in the 2400-mile market. Other operating indices for the airlines under these Nash equilibrium strategies are listed in Table 2. Comparing the airlines’ final actions resulted from the NE strategies in the short-haul and long-haul markets, we find that the size of aircraft used by the airlines in the short-haul market is smaller than in the longhaul market. This is not surprising in light of our applied cost function (Wei and Hansen, 2003), which shows that airlines’ economy of aircraft size depends on flight distance—the longer flight distance is, the larger the least-cost aircraft will be. We also find that the long-haul market has less demand and higher profit than the short-haul market. This is partly due to our specification of predetermined fares that are proportional to distance—$50 for the 400-mile short-haul market and $300 for the 2400-mile long-haul one. This implies a higher ‘‘mark-up’’ in the long-haul market because of substantial economies of flight distance. Due to the same reason, we also find that the frequency in the long-haul market is less than in the short-haul market, and the load factor in the long-haul market is lower than in the short-haul market. 3.3. Leader-and-follower Stackelberg game In the leader-and-follower game, we use a ‘‘conditioning’’ notation to indicate that airline 2’s discrete choices depend on airline 1’s choice, and a specific strategy in this game can be represented as [(s1,i, f1,j); (s2,k, f2,l)j(s1,1, f1,1), (s2,m, f2,n)j(s1,1, f1,2), . . . , (s2,x, f2,y)j(s1,4, f1,10)]. As in the one-shot simultaneous game, there are 4 · 10 = 40 possible combinations of choices of aircraft size and service frequency available to each airline, but the total number of possible strategies in this game is 40 · 4040, which is far larger than that in the first game. To find the sub-game perfect Nash equilibrium strategy in this game, which is mathematically expressed in (5) and (6), we can follow the two-step backward induction solution method—presumably we will find a unique solution, since it is a single airline’s optimization problem in each step. In the first step, we find airline 2’s optimal reaction to each of airline 1’s choices of aircraft size and service frequency. The result for the 2400mile market is shown in Table 3, in which each row of the first two columns specifies one of airline 1’s choices of aircraft size and service frequency; the third and fourth columns are airline 2’s best reactions to airline 1’s corresponding decisions; and the last two columns are the derived profits for each airline under their choices specified in each row. Based on the information in the first four columns, we can write airline 2’s NE strategy in this sub-game as [(100, 1)j(200, 4), . . . , (200, 7)j(200, 7), . . . , (400, 28)j(400, 1)]. In the second step, among all 40

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choices available to airline 1, we need to find the one that is best to airline 1’s own interest in terms of profit, under the condition that airline 1 perceives that airline 2 will respond in the way specified in its NE strategy. It turns out that airline 1’s best choice is [200, 7], i.e., airline 1 will choose aircraft of 200 seats and serve seven times per day. Finally we find the SPNE strategy for the whole game as [(200, 7); (100, 1)j(200, 4), . . . , (200, 7)j(200, 7), . . . , (400, 28)j(400, 1)], in which the first part indicates airline 1’s strategy, and the others are airline 2’s strategies. Based on this SPNE strategy, the final actions that the airlines will take in the 2400-mile market are: airline 1 uses aircraft with 200 seats and operates seven times per day; and after observing airline 1’s choice, airline 2 will use the same aircraft and operate with the same frequency. Surprisingly, we find that the final actions are the same as in the one-shot simultaneous game, although the ‘‘strategies’’ (the term in game theory) are different. We go through the same procedure for the 400-mile short-haul market, and find

Table 3 Airline 2’s NE strategies depending on airline 1’s choices in the leading-and-following game for the 2400-mile market in the case study If airline 1 chooses

Then airline 2 will choose

Aircraft size (seats)

Service frequency (per day)

Aircraft size (seats)

Service frequency (per day)

100 100 100 100 100 100 100 100 100 100 200 200 200 200 200 200 200 200 200 200 300 300 300 300 300 300 300 300 300 300 400 400 400 400 400 400 400 400 400 400

1 4 7 10 13 16 19 22 25 28 1 4 7 10 13 16 19 22 25 28 1 4 7 10 13 16 19 22 25 28 1 4 7 10 13 16 19 22 25 28

200 200 200 200 200 400 400 400 400 400 200 200 200 200 400 400 400 400 400 400 200 200 200 400 400 400 400 400 400 400 200 200 200 400 400 400 400 400 400 400

4 4 7 7 7 1 1 1 1 1 4 7 7 7 1 1 1 1 1 1 4 7 7 1 1 1 1 1 1 1 4 7 7 1 1 1 1 1 1 1

Profit of airline 1 ($ · 106)

Profit of airline 2 ($ · 106)

0.40 1.62 0.41 5.08 5.32 1.15 3.16 5.26 7.42 9.65 1.64 4.78 5.60 5.54 1.21 3.84 6.56 9.37 12.23 15.17 2.78 4.63 4.34 1.76 5.28 8.94 12.71 16.56 20.47 24.43 3.82 3.53 1.93 5.90 10.74 15.73 20.82 25.99 31.22 36.51

5.13 5.13 6.99 5.20 4.00 3.82 3.82 3.82 3.82 3.82 5.13 7.75 5.60 4.18 3.82 3.82 3.82 3.82 3.82 3.82 5.13 7.13 5.12 3.82 3.82 3.82 3.82 3.82 3.82 3.82 5.13 6.81 4.86 3.82 3.82 3.82 3.82 3.82 3.82 3.82

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that the final actions that airlines will take are also the same as in the one-shot simultaneous game for the 400mile market. Therefore, for both the short-haul and long-haul markets, we find a situation where there is neither advantage nor disadvantage for the leader in the Stackelberg game if we compare its results with those from the simultaneous game. This outcome results partly from airlines’ being given discrete choices in our case study. If the size and frequency variables were treated as continuous, the results for the two airlines in this game, and also the outcomes of the one-shot simultaneous game and the leader-and-follower game would almost certainly be different. Cases that demonstrate different equilibrium results from the one-shot simultaneous game and the leader-and-follower game can be seen in ‘‘sensitivity analysis’’, the last part of this section. 3.4. Two-level hierarchical game In the two-level hierarchical game, a specific strategy involving two airlines and two sub-games can be represented as [(s1,i, s2,j); (f1,s, f2,t)j(s1,1, s2,1), (f1,k,f2,l)j(s1,1, s2,2), . . . , (f1,m, f2,n)j(s1,4, s2,4)]. The total number of strategies in the game is 16 · 10016! As we describe in the leader-and-follower game, the Sub-game Perfect Nash equilibrium strategy in this two-level game can also be found in two steps, although there is no guarantee that there exists such a strategy or the strategy is unique. In the first step, for each possible result of the two airlines’ choices of aircraft size, we find the airlines’ corresponding Nash equilibrium strategy in frequency competition. Table 4 shows this result for the 2400-mile market. Each row of each of the first two columns is a possible choice of aircraft size for one of the two airlines; the third and the fourth columns are the airlines’ Nash equilibrium reactions in frequency competition corresponding to airlines’ choices of aircraft size; and the last two columns are the profits for the two airlines according to airlines’ choices of aircraft size and service frequency in each row. Based on the information in Table 4, we can write the Nash equilibrium strategy in the second level competition (second subgame) as [(13, 13)j(100, 100), (13, 7)j(100, 200), (13, 4)j(100, 300), . . . , (4, 4)j(400, 400)]. It should be noted that there is no NE strategy for some specific choices of aircraft size; we denote ‘‘NA’’ in the table under such circumstance. Afterwards, we can find the Nash equilibrium strategy for airlines competing in aircraft size, with the knowledge that both airlines will choose the service frequency specified in the NE strategy above for frequency competition. It turns out that the Nash equilibrium strategy in size competition (first sub-game) is

Table 4 Airlines’ Nash equilibrium strategies in frequency competition depending on the two airlines’ choices of aircraft size in the first level for the 2400-mile market in the case study For each competition result from 1st level

Results of frequency competition in 2nd level

Airline 1s aircraft size (seats)

Airline 2s aircraft size (seats)

Airline 1s service frequency (per day)

Airline 2s service frequency (per day)

100 100 100 100 200 200 200 200 300 300 300 300 400 400 400 400

100 200 300 400 100 200 300 400 100 200 300 400 100 200 300 400

13 13 13 NA 7 7 7 NA 4 4 4 4 NA NA 4 4

13 7 4 NA 13 7 4 NA 13 7 4 4 NA NA 4 4

Profit for airline 1 ($ · 106)

Profit for airline 2 ($ · 106)

3.50 5.32 6.81 NA 4.00 5.60 7.13 NA 3.48 4.63 5.95 5.65 NA NA 4.93 4.61

3.50 4.00 3.48 NA 5.32 5.60 4.63 NA 6.81 7.13 5.95 4.93 NA NA 5.65 4.61

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[200, 200], i.e., both airlines will choose aircraft with 200 seats. Then the unique sub-game perfect Nash equilibrium strategy for the whole game is [(200, 200); (13, 13)j(100, 100), (13, 7)j(100, 200), (13, 4)j(100, 300), . . . , (4, 4)j(400, 400)], in which the first term specifies airlines’ NE strategy in size competition, and the other terms specify the NE strategies in frequency competition. We go through the same procedure for the 400-mile shorthaul market. It turns out that the final actions that each airline will take are the same as in the one-shot simultaneous game for both short-haul and long-haul markets. Although we find that airlines will take the same actions in the two-level hierarchical game as in the oneshot simultaneous game, the competition environments and strategy formulations in these two games are quite different. We think that the assumptions in the two-level game are more reasonable and useful than in the oneshot simultaneous game, since airlines’ decisions on aircraft size and fleet are more long-term oriented than their decisions on frequency. Moreover, the solution procedure for the two-level hierarchical game provides more information than the one-shot simultaneous game. The Nash equilibrium strategy in the second level tells us the frequency competition results for each pair of aircraft size selected by the two airlines. Therefore, we can analyze cases where one or both airlines do not have choices of aircraft size and thus there is no competition in aircraft size at all. For example, in reality, Southwest Airline uses only Boeing 737 series aircraft in their fleet; we can analyze how this situation influences its operating strategy and efficiency in the market, where its competitor has the opportunity of changing airline size as a long-term strategy. If we assume that the size of aircraft operated by Southwest Airlines is fixed at 100-seat, while the other airline can choose one of the four sizes of aircraft in the 2400-mile market, then based on the information provided in Table 4, we find that the SPNE solution is: Southwest uses 100-seat aircraft and operate 13 time per day, and the other airline uses 200-seat aircraft and operates seven times per day. Their derived profits are $5.32 · 106 and $4.00 · 106, respectively, which are both less than those derived previously in the two-level hierarchical game. This implies that restriction in choice of aircraft size for one airline will not only reduce its own profit, but also make the whole market less efficient. 3.5. Sensitivity analysis After we have found the equilibrium solutions in the case study, we also do some sensitivity analysis to explore how these results may change with different demand levels in the future. Listed in Table 5 are the solution results for three scenarios of different demand levels in the 400-mile shorthaul market: the first scenario is the base scenario, which is described in our earlier case study; the second scenario assumes that the total air travel demand is uniformly 25% larger than that derived in the demand model for the base scenario; and the third scenario assumes that the demand is 50% higher than that derived in the demand model for the base scenario. As we have discussed in the base scenario above, in all the three games, each airline will choose aircraft size with 100 seats and operate 13 times per day in the 400-mile market. In the second scenario, the results from the first and third games are: each airline will use 100-seat aircraft and operate 19 times per day; but in the second (or leader-and-follower) game, the leader will operate 22 times per day, and the follower will operate 16 times per day. In the third scenario, the results from the first and third games are: each airline will use 100-seat

Table 5 Sensitivity studies under three scenarios of different demand levels, for the 400-mile short-haul market Scenario

Results from 1st and 3rd games

Results from 2nd game

(Size, frequency)

Load factor (%)

(Leader’s size, frequency) vs. (follower’s size, frequency)

Leader’s load factor (%) vs. follower’s load factor (%)

Base case

(100, 13)

68.5

25% linear increase in demand

(100, 19)

61.7

50% linear increase in demand

(100, 25)

59.6

(100, 13) (100, 13) (100, 22) (100, 16) (100, 31) (100, 22)

68.5 68.5 61.5 62.3 58.1 57.4

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aircraft and operate 25 times per day; but in the second (or leader-and-follower) game, the leader will operate 31 times per day, and the follower will operate 22 times per day. From these results, we first notice that there exists strong advantage for a leader in the leader-and-follower games in both the second and third scenarios, although neither advantage nor disadvantage is found in the base case. Secondly, airlines use 100-seat aircraft in all these scenarios, and only increase service frequency to accommodate increasing travel demand in this 400-mile market. Moreover, the load factors in the second and third scenarios are slightly lower than those in the base case, indicating that, when total air travel demand is higher in the future, airlines will still focus on increasing service frequency in market competition, and increasing aircraft size will gain little benefit. The sensitivity analysis and the prediction above remind us of the question of airlines’ choice of aircraft size in the future. The analysis shows that, at least for the idealized 400-mile short-haul market, it is service frequency rather than aircraft size that plays a more important role in airlines’ market share and competition; and thus airlines prefer to increase service frequency, rather than increase aircraft size, to accommodate higher demand and to compete with each other. But this is not a sufficient proof to support the claim that airlines will actually use small aircraft in the future, since a strong argument from the aircraft manufacturer Airbus, which is valid, is that airport capacity cannot accommodate constant increase of service frequency, and thus airlines will have to use larger aircraft in the future to meet higher demand. Other factors that may also influence airlines’ strategic choices of aircraft size in the future are discussed in the next section. 4. Summary and further studies In this research, we model airlines’ choices of both aircraft size and service frequency from profit-maximization perspective by taking competition into consideration. Our approach is different from previous studies, which usually consider only the cost side or the demand side of the question, and seldom take into account airlines’ competition in aircraft size. Our game-theoretic competition models are based on empirically derived cost function and market share model, which are calibrated through airlines’ actual cost and market data. Different from any pure theoretic model, which usually result in nice-looking equilibrium solutions, gametheoretic studies based on empirical cost and demand function are not guaranteed with any equilibrium solution at all. Therefore our approach and results are rarely seen in literature, but, comparing with those from pure theoretic studies, our findings are much more useful to practical decision-making and policy implementation. We apply three different game-theoretic models in the non-stop duopoly market, the simplest and most fundamental competitive market, and study how competition equilibrium may change under different competition scenarios, different stage lengths and demand levels, so that we can understand better how competition may influence airlines’ choices of aircraft size and service frequency in various markets. In the case study of two hypothetical and identical airlines in two idealized duopoly markets, we find that the size of aircraft used by airlines in the short-haul market is smaller than that used in the long-haul market, which is mainly due to the dependence of economy of aircraft size on flight distance. In the long-haul market, demand is less than in the short-haul market, but profit is much higher, which is due to the economy of stage length in airlines’ cost, and also due to our specification of a different predetermined fare for each market. Finally, in the sensitivity analysis for the short-haul (400-mile) market, we find that there exists strong advantage for a leader in the Stackelberg game by attracting higher demand. We also find that airlines always use the 100-seat—the smallest—aircraft in different games in all three scenarios of different demand levels, and only increase service frequency to accommodate increasing travel demand. These findings are not very surprising to us. According to Wei and Hansen (2005), airlines can gain higher market share by increasing service frequency than by increasing aircraft size at the same percentage. Therefore, airlines have no incentives to use aircraft larger than the minimal-cost ones. Wei and Hansen (2003) find that the sizes of minimal-cost aircraft range from 180 seats to 250 seats for stage lengths from 400 mile to 2400 mile. Due to the predominant role of service frequency in airlines’ competition in market share, airlines’ actual choice of aircraft size resulted from our models, i.e., 100 seats in 400 mile, is smaller than the minimalcost one. It is our belief that, in the short run, airlines will most likely continue to use the smallest, yet cost efficient, aircraft in the competitive market in the future; and if airport capacity allows, airlines will increase service

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frequency to accommodate increasing demand. Also, airlines’ daily use of aircraft is constrained by the fact that airlines tend to have large proportion of small aircraft in their fleet. Due to imperfect information of future demand, market competition (such as the trend of airlines’ merging and international competition and cooperation), and other uncertain factors, airlines having more small aircraft, rather than fewer large aircraft, can more easily and flexibly adapt their scheduling and routing planning to current and future uncertainties. In the long run, if aircraft manufacturers can make new, larger and more cost-efficient aircraft through technology development, especially to be used in short-haul markets, then airlines can have more incentives to use larger aircraft in their operations, since the aircraft cost function would be changed and the minimal-cost aircraft size would be larger. Therefore, in order to alleviate runway congestion and reduce flight delays, the government should not only focus on investments in increasing airport capacity, such as runway expansion and more advanced air traffic control technologies, but also provide more incentives to manufactures to build more cost-efficient larger aircraft (such as Boeing 787). On the other hand, when airport capacity is not able to meet the increasing demand of aircraft landings and departures, airlines can adjust their longterm business strategies, such as increase aircraft size, change their operating network (e.g. focusing more on hub-and-spoke service instead of direct service), or do both. This research shows a promising beginning towards understanding airlines’ choices of aircraft size and service frequency in a competitive environment. Two aspects of our studies deserve further work to improve our analysis. First, cost analysis and demand analysis are the two bases of our research, and we only have market-level cost function (Wei and Hansen, 2003) and market-level market share and demand models (Wei and Hansen, 2005) at present. In the future, it is necessary to develop and apply network-level models to capture the impact of operating network structure in airlines’ decisions on aircraft size and service frequency. Secondly, more sensitivity analysis and policy analysis can be done to understand airlines’ choice behavior. For example, we can study how airlines may adjust their choices when airport capacity is constrained; how public policy such as landing fees can influence airlines’ choices, which could possibly make airport operate more efficiently; how our equilibrium results may change with increase of factor prices such as fuel price; how the difference in labor cost influences airlines’ choices of aircraft size; how social welfare changes in different markets and policy scenarios; and how market features such as ground transportation and city properties could influence airlines’ decisions. Acknowledgements This research was partly funded by the Federal Aviation Administration (FAA) through the National Center of Excellence in Aviation Operations Research (NEXTOR) in the United States. References Abrahams, M., 1983. A service quality model of air travel demand: an empirical study. Transportation Research Part A 17A (5), 385–393. Airbus, 2005. Global Market Forecast 2004–2023. Bailey, E.E., Graham, D.R., Kaplan, D.P., 1985. Deregulating the Airlines. The MIT Press, Cambridge, Massachusetts. Bierman, H.S., Fernandez, L., 1993. Game Theory with Economic Applications, second ed. Addison-Wesley Publishing Co., New York. Boeing, 2005. Boeing Commercial Airplanes Current Market Outlook 2005. Coldren, G.M., Koppelman, F., Kasturirangan, K., Mukherjee, A., 2003. Air Travel Itinerary Share Prediction: Logit Model Development at Major US Airline. Transportation Research Board Annual Meeting CD ROM. Dorman, G.J., 1983. A model of unregulated airline markets. Research in Transportation Economics 1, 131–148. Douglas, G.W., Miller III, J.C., 1974. Economic Regulation of Domestic Air Transport: Theory and Policy. Eriksen, S.E., 1977. Policy Oriented Multi-equation Models of US Domestic Air Passenger Markets. Ph.D. dissertation, Massachusetts Institute of Technology. Ghobrial, A.A., 1983. Analysis of the Air Network Structure: The Hubbing Phenomenon. Ph.D. dissertation, Institute of Transportation Studies, University of California at Berkeley. Gibbons, R., 1992. Game Theory for Applied Economists. Princeton University Press. Hansen, M., 1990. Airline competition in a hub-dominated environment: an application of noncooperative game theory. Transportation Research 24B (1). Hansen, M., 1996. Airline Frequency and Fare Competition in a Hub-Dominated Environment. Unpublished paper. Hansen, M., Kanafani, A., 1989. Hubbing and airline costs. Journal of Transportation Engineering 116 (6), 581–590.

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Keeler, T., 1972. Airline regulation and market performance. The Bell Journal of Economic and Management Science 3 (2), 399–424. Kirby, M., 1986. Airline economies of ‘scale’ and Australian domestic air transport policy. Journal of Transport Economics and Policy 20 (3), 339–352. Meyer, J.R., Oster Jr., C.V., 1984. Deregulation and the New Airline Entrepreneurs. The MIT Press, Cambridge, Massachusetts. Miller, R.E., Sawers, D., 1970. The Technical Development of Modern Aviation. Praeger Publishers, New York. Morrison, S., 1984. An economic analysis of aircraft design. Journal of Transport Economics and Policy 18, 123–143. Morrison, S., Winston, C., 1986. The Economic Effects of Airline Deregulation. The Brookings Institute, Washington, DC. Norman, V., Strandenes, S., 1990. Deregulation of Scandinavian Airlines: A Case Study of the Oslo–Stockholm Route. CEPR Discussion Paper, no. 403. Centre for Economic Policy Research, London. Russon, M., Hollingshead, C.A., 1989. Aircraft size as a quality of service variable in a short-haul market. International Journal of Transportation Economics XVI (3), 297–311. Schmalensee, R., 1977. Comparative static properties of regulated airline oligopolies. The Bell Journal of Economics 8 (Autumn), 565– 576. Viton, P., 1986. Air deregulation revisited: choice of aircraft, load factors, and marginal-cost fares for domestic air travel. Transportation Research A 20A (5), 361–371. Wei, W., Hansen, M., 2003. Cost economics of aircraft size. Journal of Transport Economics and Policy 37 (part 2), 277–294. Wei, W., Hansen, M., 2005. Impact of aircraft size and seat availability on airlines’ demand and market share in duopoly markets. Journal of Transportation Research, Part E: Logistics and Transport Review 41, 315–327.