Dynamic Revenue Management in Airline Alliances

Transcription

1 Dynami Revenue Management in Airline Allianes Christopher P. Wright, Harry Groenevelt Simon Graduate Shool of Business, University of Rohester, Rohester, New Yor Robert A. Shumsy Tu Shool of Business, Dartmouth College, Hanover, New Hampshire February, 2009 Major airlines are selling inreasing numbers of interline itineraries, in whih flights operated by two or more airlines are ombined and sold together. One reason for this inrease is the rapid growth of airline allianes, whih promote the purhase of interline itineraries and therefore virtually extend the reah of eah alliane member s networ. This pratie, however, reates a diffiult oordination problem: eah member of the alliane maes revenue management deisions to maximize its own revenue, and the resulting behavior may produe sub-optimal revenue for the alliane as a whole. Airline industry researhers and onsultants have proposed a variety of stati and dynami mehanisms to ontrol revenue management deisions aross allianes (a dynami mehanism adjusts its parameters as the number of available seats in the networ hanges). In this paper, we formulate a Marov-game model of a two-partner alliane that an be used to analyze the effets of these mehanisms on eah partner s behavior. We begin by showing that no Marovian transfer priing mehanism an oordinate an arbitrary alliane. Next, we examine three dynami shemes, as well three forms of the stati sheme widely used in pratie. We derive the equilibrium aeptane poliies under eah sheme and use analytial tehniques, as well a numerial analyses of sample allianes, to generate fundamental insights about partner behavior under eah sheme. The analysis and numerial examples also illustrate how ertain transfer prie shemes are liely to perform in networs with partiular harateristis. Key words: revenue management, yield management, airlines, Marov game, allianes 1

2 1. Introdution When one of the authors reently planned a trip from Boston to Barelona, British Airways offered a onvenient itinerary for $823. The itinerary began with a leg on British Airways from Boston to London, followed by a seond leg on another airline, Iberia, to Barelona. We will all suh an itinerary an interline itinerary, for it inludes servie on multiple airlines. The availability of this two-leg interline itinerary for this fare is ontingent on two deisions: (1) the airline who sells the tiet (the mareting airline, in this example British airways) must mae a seat available on one leg, and (2) the operator of the other leg (the operating airline, Iberia) must agree to aept the onneting passenger from the mareting airline. Beause both airlines pratie revenue management, these deisions depend upon the prie paid by the onsumer to the mareting airline for the tiet and the prie paid by the mareting airline to the operating airline for the use of a seat. This artile examines agreements among airlines that govern the latter prie, sometimes alled revenue sharing, transfer prie, or proration agreements. We show how these agreements have subtle and potentially signifiant effets on individual airline behavior as well as on the total revenue olleted by multiple airlines aross their ombined networs. It is beoming inreasingly important to understand the impat of proration agreements on airline revenue management beause sales of interline itineraries have been growing. This inrease is due in part to a type of mareting arrangement alled a ode-share agreement. Under this arrangement, the operating airline s flight is also listed as a flight with the mareting airline s name. In the example above, the seond leg had an Iberia flight number, but was also labeled British Airways Flight 7073 from London to Barelona. Analysis of data from the U.S. Department of Transportation (BTS, 2006) reveals that the fration of interline itineraries within the U.S. rose from 10% in 1998 to 20% in 2004, and most of those interline itineraries were mareted under ode-share agreements. Overall, 46% of revenues olleted from U.S. domesti flights in 2004 ame from interline itineraries. A seond fator driving up interline traffi is the growth of airline allianes. These allianes usually ombine ode-share agreements with other arrangements, suh as shedule oordination and the merger of frequent-flier programs. In Marh 2006, 59% of all worldwide ASMs (available seat miles, a measure of total apaity equal to the number of seats multiplied by the number 2

3 of miles flown, summed over all flights) were flown by airlines belonging to one of the three largest international allianes, Star, SyTeam or Oneworld (Lott, 2006). Both international and U.S. alliane ativity are expeted to grow (Lott, 2006; Belden, 2007; Shumsy, 2006). While these international allianes failitate formal mareting and operational arrangements among airlines, this paper uses the term alliane in a muh weaer sense: an alliane is formed by any two airlines that exhange interline passengers and that have a proration agreement for the revenue olleted from the sale of interline itineraries. In pratie, the rules for revenue sharing are usually laid out in speial prorate agreements (SPAs) that are negotiated by alliane partners. In the absene of an SPA, airlines follow the rules set out by the International Air Transport Assoiation (see IATA, 2007, for details of the rules to be implemented over the next few years). Whether they are enoded in an SPA or by the IATA, most rules inlude fixed transfer pries for partiular flight/fare-lass ombinations, or other simple alloation proedures suh as a split in revenue (a proration rate) based on relative mileage. For example, under suh a mileage proration sheme, British Airways would reeive the lion s share of the $823 in the example above for its Boston-to-London flight. Throughout this paper we all suh rules stati shemes, for they do not adjust proration rates as demand is realized and seats are sold. Although stati shemes are easy to manage, they an lead to suboptimal deision-maing by member airlines and lost revenue for the alliane as a whole. For our example, under mileage proration Iberia would reeive a relatively small share of the tiet revenue when flying interline passengers. Therefore, Iberia may hoose to fous on its own (non-interline) ustomers and not hold seats for British Airways ustomers, who may be more lurative for the alliane. The underlying flaw in any stati proration sheme is that the revenue-sharing proportions are not adjusted to reflet the atual value of seat inventory. Beause the revenue management system of eah airline in the alliane maximizes the revenue of that airline, an airline may rejet an itinerary if the transfer prie undervalues the real-time value of its seats, even if the total revenue from the itinerary is large. Given the defiienies of stati shemes, major airlines are onsidering dynami shemes, suh as the use of the real-time opportunity osts of seats, or bid pries, as transfer pries. In this paper we examine dynami shemes that have been desribed in the published literature (see 3

4 Vinod, 2005) and have been suggested to the authors by industry exeutives and revenue managers. In the industry there is interest in dynami shemes, but also muh unertainty. There are ertainly tehnial and legal barriers to implementation; for example, antitrust legislation in the U.S. prohibits the exhange of ertain types of information among airlines. But another signifiant barrier is unertainty over how revenue-maximizing airlines would respond to dynami shemes, and whether suh shemes would produe real benefits to the alliane. In fat, there is no published literature on this topi and, to our nowledge; there has been no rigorous analysis of these effets by researhers within the industry. This paper is a first attempt to fill that gap. Now we summarize the organization of the paper and its results. After reviewing the relevant literature in 2, in 3 we desribe our general model for a two-airline alliane. In 4 we desribe a entralized networ and determine the first-best poliies for a entralized yield management system. In 5 we desribe a two-airline alliane model and analyze stati and dynami transfer prie shemes. In this setion we first use a ounterexample to show that no dynami sheme is guaranteed to maximize alliane-wide revenue, unless the dynami sheme inludes revenue-sharing rules that depend upon the sample path of inventory sales (note that a sheme based on sample paths would be orders of magnitude more omplex than the stati and dynami shemes being onsidered by the airlines). We then derive equilibrium poliies for the alliane partners under ertain dynami shemes, and we use the analysis to highlight the strengths and weanesses of the shemes in terms of total alliane revenue. In 6 we desribe numerial experiments that support the insights from 5. The experiments also ompare the performane of stati and dynami shemes, given ertain networ parameters. We find that stati shemes an perform as well as dynami shemes for ertain networs, but that the performane of a stati sheme that is optimal for one networ an degrade quily as the networ parameters hange. Dynami shemes often perform better and are more robust. We find, however, that the performane of dynami shemes an be signifiantly redued if eah operating airline hooses a transfer prie to maximize its own revenue. Suh would be the ase, for example, if the partners initially agree to use bid pries as transfer pries, but then eah partner attempts to inrease its revenue by reporting inorret bid pries. Finally, in 7 we summarize our results and desribe future researh. 4

5 There are a few aveats for the results in this paper. First, we mae strong assumptions about the amount of information available to eah alliane partner. Speifially, we use a Marov game model to desribe the alliane, and we use the Nash equilibrium to desribe the airlines behavior under eah proration sheme. For legal and tehnial reasons the airlines annot oordinate their revenue management systems, so it is appropriate to use the tools of nonooperative game theory. To eep the problem tratable, however, we assume that the airlines share the same information about the state of the game and the distribution of future events, e.g., eah airline has perfet information about its partner s inventory level, and both have idential foreasts of future arrival probabilities and revenue distributions over the entire alliane networ (in tehnial terms, we define a game of omplete information). While this assumption is not realisti, we believe that the amount of transpareny in the industry is inreasing. For example, airlines regularly use the web to monitor the lowest available fare of their ompetitors, and hire maret intelligene servies suh as QL2 ( to gather information about ompetitor ations. Our model represents a logial extreme ase, and the full-information assumption allows us to generate fundamental insights on how ertain proration shemes behave. We provide additional details and disussion of our information-sharing assumptions in 3.2. In general, analysis of games with inomplete information will be an interesting area for further researh. A seond aveat is that that the numerial experiments desribed in 6 were onduted using small networs, in terms of the number of flights and the number of seats. Again, our purpose is to gain basi insights, i.e., to identify the fundamental advantages and disadvantages of eah transfer prie sheme. In addition, we demonstrate that many of these insights apply as the number of seats in the networ grows. An important area for additional researh, however, will be to examine alliane performane over networs of realisti size. Finally, at a higher level than our analysis, the alliane partners are engaged in a ooperative proess to determine whih routes should be available for interline traffi and what proration rules to use on those routes. In general, the partners see to inreases alliane-wide revenue and to alloate revenues so that all members are willing to partiipate in the alliane. We do not model this higher-level proess. Instead, our model provides information about how airlines behave, and how total revenues are affeted, given sets of interline routes and partiular proration rules. Models that fous on the higher-level problem have begun to appear in the literature, e.g., 5

6 Agarwal and Ergun (2007) examine the alloation mehanism design problem for argo shippers. For airline revenue management, suessful high-level negotiations depend upon information about the effets of partiular proration shemes on networ revenues. To our nowledge, the model presented here is the first to provide suh information. 2. Literature Review Revenue management (also referred to as yield management) and its appliation to the airline industry have reeived a great deal of attention sine the 1970s when Littlewood (1972) first desribed the basi problem. In that artile, Littlewood introdues the result (now referred to as Littlewood s rule) that a request for a seat should be fulfilled only if its revenue exeeds the expeted future value of the seat in question. This intuitive rule forms the basis of many ontrol poliies in both theory and pratie. Numerous authors have expanded on Littlewood s wor. See, for example, Belobaba (1989) who examines a problem with multiple fare-restrition ombinations, Glover et al. (1982) who loos at the passenger mix problem in a networ environment, You (1999) who examines a dynami priing model, and Talluri and van Ryzin (2004a) who utilize a disrete hoie model of demand. For a more thorough desription of the revenue management literature, see the survey by MGill and van Ryzin (1999) and the boo by Talluri and van Ryzin (2004b). The use of ompetitive game theory in revenue management has been limited. Vulano et al. (2002) examine a dynami game in whih a seller faes a sequene of ustomers who ompete with eah other in an aution for a fixed number of units. Netessine and Shumsy (2004) examine both horizontal and vertial ompetition between two airlines, where eah airline flies a single leg. Several aspets of airline allianes have been examined in the literature. Barron (1997) disusses many of the legal impliations of airline allianes, fousing on ode-sharing agreements used widely in the industry. Par (1997) and Bruener (2001) examine the eonomi effets of allianes on fares, traffi levels, profits and maret welfare. Bruener and Whalen (2000) provide an empirial analysis of the effets of international allianes on fares, showing that interline fares harged by allianes are approximately 25% lower than those harged by non-allied arriers. Ito and Lee (2006) examine the impat of domesti allianes on airfares. 6

7 Little attention, however, has been given to how revenue management should be implemented by an airline alliane. Wynn (1995) desribes simple transfer prie shemes based on the value of loal fares. Boyd (1998a) disusses the methodologial and tehnial hallenges of the alliane revenue management problem. He also refers to a more formal analysis in an unpublished woring paper (Boyd, 1998b) in whih he formulates a stati linear program to desribe the alliane revenue management problem. Boyd then derives onditions under whih the seat alloation between the two airlines maximizes alliane-wide revenue under this model. Vinod (2005) desribes many of the alliane oordination mehanisms now being onsidered by the airlines, but provides no formal analysis of their advantages and disadvantages. Some of the shemes analyzed in this paper orrespond to mehanisms desribed by Vinod. Shumsy (2006) argues that low-ost ompetitors are driving the networ airlines to rely on allianes for an inreasing proportion of their traffi. Both Shumsy (2006) and Fernandez de la Torre (1999) disuss the need for more researh on the effetiveness of alliane agreements, a need we attempt to fill here. In their paper on revenue management games, Netessine and Shumsy (2004) desribe and analyze a stati alliane revenue-sharing mehanism for a two-leg networ based on the expeted flow of passengers. In this paper we analyze the performane of dynami oordination mehanisms that are designed for arbitrary alliane networs and are similar to shemes that are proposed by, or atually used by, the airlines. Ongoing researh by Houghtalin et al. (2007) and Agarwal and Ergun (2007) loos at various aspets of allianes, fousing speifially on argo arriers. In addition to the inherent differene between the argo and passenger revenue management problems (see Kasilingam 1996), their analysis differs from ours in two fundamental ways: 1) they fous on the high-level alliane formation problem, with ooperative game theory as the appropriate method, while we formulate a nonooperative game, given an existing alliane and partiular revenue-sharing rules; and 2) they fous on a deterministi optimization problem in whih all demand for argo servie has been realized before routing deisions are made, while our passenger yield management problem is most appropriately desribed by a model in whih demand is unertain and arrives over time. Finally, this paper is related to the extensive literature on supply hain oordination. See Nagarajan and Sošić (2008) and Cahon and Netessine (2004) for overviews of the related literature that use, respetively, ooperative and ompetitive game theory. There are several attributes 7

8 of our problem, however, that distinguish it from this researh stream. First, the flow of produts in the traditional supply hain literature moves in one diretion, from raw materials to the onsumer. Therefore the unused produt does not move sideways within a level. Seond, in the supply hain literature, prodution of a produt begins at one level with one set of firms (suppliers) and demand is fulfilled at another level by another set of firms (retailers). Neither attribute holds for our problem. For a speifi ontrast, onsider the literature on assembly systems, for we an thin of a multi-leg itinerary as a final produt assembled from multiple omponents. In the traditional supply hain literature, an assembler reeives omponents from several suppliers that are ombined to reate a new produt to sell (e.g, Nagarajan and Basso, 2008 and Granot and Yin, 2008.) In our model either airline may serve as the mareting airline, the de fato assembler, and either airline may serve as the operating airline, the de fato supplier. In addition, the traditional researh on supply hain oordination fouses on either singleperiod newsvendor problems (e.g., Lariviere and Porteus, 2001) or repeated games in whih inventory is replenished between eah repetition of the game (e.g., Cahon and Zipin, 1999). The harateristis of suh problems are quite different from ours, a finite, multi-period problem with fixed apaity alloated to a stohasti arrival stream. Certain results from our paper may be similar in interpretation to results from the researh on the eonomis of supply hains. For example, the effet of the partner prie sheme in an be seen as a form of doublemarginalization (Spengler, 1950). In general, however, our problem ontext, model and ey results are quite different from those in the supply hain literature. 3. General Alliane Networ Model We onsider a dynami model of an alliane onsisting of two partner airlines (arriers), indexed by { 1, 2} (in a slight abuse of notation, we will denote the other airline by - instead of by 3 ). The model an be seen as an extension of the networ model desribed by Talluri and van Ryzin (1998) into a two-player game framewor. Eah flight leg in the networ is haraterized by an origin, destination, and departure time (for the remainder of this paper the terms flight and flight leg are used interhangeably). The number of flights operated by airline is denoted m m 1 + m is the total number of m and 2 flights offered by the alliane. The alliane offers n itineraries, and eah itinerary is either a sin- 8

9 gle flight or a series of onneting flights within one or both networs. The set of all itineraries is denoted N and has ardinality n. Within the alliane, these itineraries are divided into three subsets: those that involve only airline 1 s flights (N 1 ), those that involve only airline 2 s flights (N 2 ) and those that use flights from both airlines (N S ). Let n 1, n 2 and n S be the ardinality of eah subset, so that n = n1 + n2 + ns. We will refer to the sets N 1 and N 2 as intraline itineraries and the set N S as interline itineraries beause at least one leg on any itinerary in N S will not be operated by the airline that sold the tiet. We use the matrix A to speify the inventory requirements of the itineraries offered by the alliane. The matrix element therefore the olumn vetor A is the number of seats on flight i required for itinerary j, and ij j A speifies the total inventory required from the alliane networ to satisfy itinerary j. In disussions below, we will assume that eah request is for an individual passenger (i.e., A { 0,1} ij ), however group (multi-seat) requests ould be handled by reating additional olumns with eah positive element equaling the number of passengers in the group. For example, an itinerary from Rohester, NY to Denver, CO that passes through Chiago, IL would have 1 s in the rows for Rohester-Chiago and from Chiago-Denver. To handle a family of 4 looing to mae the same trip, A would need another olumn with 4 s in those same rows. For larity, A an be partitioned as follows, A = 1 m 1 m 1 +1 m N 1 N 2 N S 1 n 1 n 1 +1 n 1 + n 2 n 1 + n 2 +1 n A1 0 AS1 0 A2 AS2 suh that the first n 1 olumns have only positive elements in the first m 1 rows (airline 1 s networ), the next n 2 olumns have only positive elements in the last m 2 rows (airline 2 s networ) and the final n S olumns have positive elements in both sets of rows. While interline itineraries may be sold by either alliane partner (requests for itineraries in N S may be reeived by either airline 1 or 2), we assume that intraline itineraries are only sold by the airline that operates the flights (requests for itineraries in N are only reeived by airline ). In pratie, airlines do sell tiets for itineraries that are exlusively on another airline s networ. With some additional notation, this possibility an be inorporated into the model, and all of the 9

10 following results will ontinue to hold. To eep the exposition and notation simple we will assume that eah airline handles its own intraline requests. The number of remaining (unsold) seats for flight i is denoted x i. The m-dimensional vetor + x 1 m1 m1 1 m is the joint vetor of remaining inventory for the alliane: x { x,, x, x x } The Demand Proess We onsider a K-period booing horizon, with the urrent period, denoted, dereasing from K to 0. The probability that airline reeives a request for itinerary j in period is q 0. We assume that eah period is short enough suh that the probability that the alliane reeives more than one itinerary request in a given period is negligible. The probability that no request arrives is then: q = 1 q 0. (1) 0 {1,2} j N The revenue R assoiated with a request to airline for itinerary j in a given period, onditional on a request being made, is a nonnegative random variable with nown umulative distribution funtion (CDF) F ( r ). We assume that R has a finite expetation. The omplementary CDF is F ( r) = 1 F ( r ). Note that in R is the arrier that reeives the onsum- er s request (the mareting airline). We assume that F ( r) is differentiable with nown density funtion f ( r ). However, wherever we express our results in terms of f ( r ), similar results an be found for non-ontinuous distributions Assumptions about the Arrival Proess and Information-Sharing We assume that the distribution of eah R is independent of the realized revenue in preeding periods. Even simple (first-order) dependeny, while theoretially easy to handle with our model, would be notationally and omputationally umbersome. In addition, assume that eah player formulates an open-loop dynami program that does not utilize the realized arrival/revenue stream as feedba for its optimization problem. One ould imagine several losed-loop variations of our model. For example, demand intensity for a given itinerary ould be haraterized by an unnown parameter, whih would be updated as demand is realized. Suh models would be quite omplex and in pratie would liely be handled by up- 10

11 dating the inputs to the model over the horizon without expliitly aounting for the future effets of this updating proess when alulating the urrent value funtions. We also assume that there is independene between aeptane deisions in one period and the arrival proess in subsequent periods. Speifially, we assume that a ustomer, when denied a tiet, will not submit a new request to the alliane for the same or a similar itinerary. This assumption is onsistent with the assumptions that underlie many of the models in the revenue management literature. Inorporating multiple ustomer preferenes into the optimization problem is an area of ongoing researh (e.g., see Talluri and van Ryzin, 2004a). Within allianes, this behavior would add an interesting wrinle to our problem beause the revenue management deisions of eah airline ould, potentially, affet the arrival proess of its partners. As noted in 1, in our model the airlines share full information about their partner s inventory levels, foreasts of arrival proesses, and revenue distributions. This allows the airlines to alulate, in eah time period, a ommon expeted value for a seat on any flight in the networ. Using the terminology of game theory, we assume that eah airline nows the strategies and payoffs of its partner and therefore plays a game of omplete information. While this model is stylized, it allows us to generate fundamental insights into the advantages and disadvantages of stati and dynami transfer prie shemes. While we assume that eah airline nows the potential payoffs of its partners, we do not assume that eah airline immediately observes realized payoffs. Speifially, under the partner prie sheme of 5.4.3, the operating airline must post its transfer pries for interline inventory without nowing the realized revenue assoiated with an interline request in that period (of ourse, the mareting airline sees any realized revenue). Therefore, the partners are playing a game of imperfet information. This assumption reflets an important soure of information asymmetry found in the real world. For a given itinerary there exist numerous lasses and distribution hannels through whih the tiet an be sold; the range of pries aross these lasses and hannels reates the distribution F ( r ) of revenue for eah itinerary. Although the operating airline may now the distribution of revenue beause pries are publily posted, it annot now the speifi lass being sold or hannel being used at the moment the mareting airline reeives a speifi purhase request. 11

12 3.3. Assumptions about Revenue Sharing In general, the proration sheme used by the alliane will influene both the total revenue reeived by the alliane and the alloation of revenues to eah of the partners. We assume that the ultimate goal of eah partner is to maximize its own wealth (revenue). It is reasonable to assume, however, that by forming an alliane, the partners are seeing a sheme that inreases their joint profits, using some form of ex-ante revenue distribution (e.g., a partiipation fee) to ensure that all members of the alliane will ontinue to partiipate. In pratie, this problem is often solved by finding a set of interline routes on eah airline that leads to a rough balane in the revenue exhanged between the airlines (Ito and Lee, 2006). The hoie of mehanism for the distribution of total revenues is a bargaining problem that we do not examine here. We assume that some mehanism has already been hosen and that both airlines are willing to partiipate in the alliane. Therefore, our primary fous will be on examining how the various trading shemes affet total alliane revenue. 4. Centralized Control Here we desribe the optimal poliy for a single, entralized ontroller maing all deisions to maximize total alliane revenue. In general, members of an airline alliane annot adopt entralized revenue management ontrols (see the end of this setion for further disussion), but these results are useful as they lead to an upper-bound on the total revenue for the alliane. We will all this upper bound the first-best revenue Deision Proess The fundamental deision made by the entralized ontroller is whether to aept or rejet a request for an itinerary j, given the revenue offer R and the urrent state of the system: the re- x denote the total maining periods,, and the remaining inventory of the alliane, x. Let ( ) (urrent and future) expeted value for the alliane given inventory x with remaining periods j Δ J x, A be the opportunity ost to the alliane of the inventory required for itinerary j: and let ( ) For onveniene, let ( x) = J ΔJ j j ( x A ) J ( x) J ( x A ),. (2) whenever one of the omponents of x is negative. u j ; x, suh that A poliy for entralized ontrol onsists of a set of aeptane rules, ( ) J 12

13 u j 1 if, at time with remaining inventory x, the alliane is ; = willing to sell a tiet for itinerary j with revenue r, 0 otherwise. ( r x) j j We now define the joint arrival probability, q, and the orresponding onditional CDF, F, of the onditional revenue, R j () r, for a request made to the alliane (rather than to a partiular partner ) for itinerary j in period : q j = q 1 j + q 2 j F j 1 j 2 j q q = j j. q q 1 j 2 j () r F () r + F () r The Bellman equations for optimal entralized ontrol an then be written as: J J 0 0 j j j j j j j ( x) q J ( x) + q E[ R u ( R, x) + J ( x A u ( R, x )] = 1 j N ( x) = 0 x 0 j where u ( r, x) arg max { 0,1} 1 j { ru + J ( x A u) } = 1 u Given a request, the entralized ontroller either aepts the request, reeiving the assoiated revenue and reduing the inventory level, or denies the request, moving to the next period with the same inventory Optimal Poliies The deision faed by the entralized ontroller is idential to the deision faed by a single airline that maximizes the revenue generated by the ombined networ of the alliane. We an, therefore, apply results derived for a single airline networ. PROPOSITION 1. The optimal aeptane poliy for entralized ontrol is of the form: u j ( r; x) 1 = 0 if r ΔJ 1 otherwise. j ( x; A ) PROOF. See Talluri and van Ryzin 1998, Proposition 1. 13

14 Under the optimal poliy, the alliane aepts any request with assoiated revenue greater than or equal to the alliane s opportunity ost of the inventory used on that itinerary. Simply put, it aepts a request if it is benefiial (in expetation) to do so. Pratial limitations, however, prevent most allianes between large partners from eding ontrol of their revenue management systems to a entral ontroller and using an optimal poliy suh as the one desribed in Proposition 1. Barriers to oordination inlude tehnial inompatibilities among revenue management systems within an alliane, ompetitive onsiderations (alliane partners are often ompetitors on many routes and therefore do not want to merge revenue management systems), and antitrust laws. There are examples, however, of entralized ontrol in the airline industry. Regional airlines sometimes allow their national partners to ollet all revenues and mae all booing deisions, and revenue-sharing is aomplished with a fixed payment per flight to the regional partner (e.g., similar arrangements are used in the Continental/ExpressJet and United/Sywest allianes; see Shumsy, 2006). For the remainder of this paper we ompare this entralized poliy with the poliies followed by airlines when revenue management deisions are distributed among the partners in the alliane. That is, the following deentralized ontrol shemes have been used, or are intended for use, among major airlines suh as the primary members of the SyTeam, Star and OneWorld allianes. 5. Deentralized Control In this setion we examine airline behavior when revenue management deisions are deentralized among alliane partners. We assume that eah alliane partner is free to aept or rejet a request for an interline seat, as is true under the free sale system that is ommonly used by major airline allianes (Boyd, 1998a). In our model, interline sales follow the following steps. First, an airline (hereafter: the mareting airline) reeives a request for an interline itinerary. Next, a transfer prie is set for the seats on flights operated by its partner (the operating airline) that are needed to omplete the itinerary (there are a variety of methods for setting transfer pries, and we will desribe speifi shemes in 5.3 and 5.4). Next, the operating airline deides whether to mae its seat available, and then the mareting airline deides whether or not to sell the omplete itinerary. Finally, if the itinerary is sold the transfer prie is paid to the operating airline. 14

15 In 5.1, we will desribe our model for the alliane under deentralized ontrol. In 5.2, we will show by ounterexample that no transfer priing sheme an guarantee optimality under suh a system, and we gain insights into the pitfalls inherent in transfer priing shemes by examining the equilibrium behavior of the alliane partners under a generi deentralized sheme. In 5.3 and 5.4, we will desribe the equilibrium behavior of the partners under speifi transfer prie shemes. In 5.3, we examine stati proration, in whih revenue from all interline tiets is split aording to a fixed proportion. This sheme is urrently used within many allianes. In 5.4, we analyze three dynami shemes, whih are based on systems proposed by Vinod (2005) and on systems that are being onsidered in the industry. In 5.5, we disuss the benefits of allowing the operating airline to set the transfer prie and therefore share any surplus revenue reeived by the alliane for an interline request. In 5.6 we onsider how revenue is alloated between partners under eah sheme, and in 5.7 we disuss the omputational hallenges one faes when attempting to alulate the equilibrium behavior of partners in an alliane Deision Proess We model the set of dynami deisions for both airlines as a finite-horizon Marov game (Heyman and Sobel, 2004). While at the highest level the formation of the alliane an be viewed as a ooperative game, the ontratual revenue-sharing mehanism must be implemented within eah airline's revenue management system. These revenue management systems are inherently non-ooperative, optimizing eah airline s revenue without taing into aount eah deision s impat on the partner. Therefore we assume that, given the transfer-priing rules of the alliane, the revenue management systems of the airlines are loed in a non-ooperative game. The two alliane airlines are the players in the game, and in 3.2 we desribed the information available to eah player. The players possible ations are quite simple: whether to aept or rejet an itinerary request. In addition, under the partner prie sheme desribed in 5.4.3, the operating airline has one more ation, setting the transfer prie. Beause we use a Marov game, immediate payments and transition probabilities in eah state depend only on the ation in that state. Let ( x) J denote the total (urrent and future) expeted value for airline given inventory J x denoting, as before, the total value for the alliane, so 1 = J x + J x. As in definition (2), the opportunity ost of the inventory used by an x with remaining periods, with ( ) 2 that J ( x) ( ) ( ) 15

16 itinerary is denoted with a ΔJ term, though here we are onerned with eah airline s individual opportunity ost, ΔJ j j ( x A ) J ( x) J ( x A ),, in addition to the opportunity ost of the alliane as a whole, Δ J ( x, A j ) ontrol, we define J ( x) = A poliy for airline onsists of a set of aeptane rules, ( ; x) u ( r x) whenever a omponent of x is negative. u. As with entralized, suh that: 1 if, at time with remaining inventory x, airline is willing ; = to sell a tiet for itinerary j with net revenue r, 0 otherwise, Under the partner prie sheme, the poliy also inludes setting the internal transfer prie, for eah sub-itinerary., is a real number assoiated with eah airline, itinerary j, invento- is also a funtion of the revenue assoiated The transfer prie, p ( x) ry level x and period. For ertain shemes p ( x) p, with the request. To simplify the notation, however, we will not inlude R as an argument of x be p. Airline s partner must pay p ( x) to airline to sell the interline itinerary j. Let ( ) the n-vetor of all transfer pries in period. Note that we allow transfer pries to vary aross eah and every itinerary even if the subitinerary used on the operating airline is the same aross multiple itineraries. A speifi alliane arrangement may not allow for this level of detail. In partiular, the mareting airline may request a sub-itinerary from the operating airline without revealing the entire itinerary, and therefore within eah period the alliane will use a single transfer prie for eah sub-itinerary on the operating airline, regardless of the itinerary being sold by the mareting airline. While we do not examine the preise effets of this assumption, one would expet that a redution in the amount of shared information would redue the overall value of the alliane under deentralized ontrol. While the speifi form of the Bellman equations in the deentralized alliane will depend on the transfer prie sheme used, the general form an be written as, p 16

17 J J 0 ( x u, p ( x) ) = ( x) = 0 x 0, j N j N S j N S j N q q q { 0,1} q E E j [ R u ( R ) + J ( x A u ( R )] ~ ~ j ~ [ R ( x) u ( R ( x), x) + J ( x A u ( R ( x), x )] E E ~ j ~ [ p ( x) u ( R ( x), x) + J ( x A u ( R ( x), x )] j 0 1 [ J ( x A u ( R )] + q J ( x) 1 whereu ( r, x) = argmax 1 u j ~ { ru + J ( x A u) } and R ( x) = R p ( x). + +, + (3) The first summation orresponds to airline s intraline itinerary requests. As with the entralized model, airline must then deide whether to aept a request. The seond summation orresponds to airline s interline itinerary requests. Again, it must hoose to aept or deny the request, however, the revenue on whih this deision will be made is the revenue assoiated with the request less the transfer prie paid to the alliane partner. The remaining two summations orrespond to interline and intraline requests to airline s partner, while the final term orresponds to the no arrival ase. While airline reeives no revenue in the ases orresponding to the final summation, the hange in its partner s inventory does affet its future expeted value. Note that in (3) the aept/rejet ontrol variables u represent ations taen by the mareting airline, and the formulation does not expliitly allow the operating airline to rejet a request even though, under free sale, this ation is available to the operating airline. We will see that it will not be neessary to expliitly model the operating airline s aeptane poliy under any of the dynami shemes desribed in 5.4, for under all three shemes the transfer prie is always suffiiently large suh that the operating airline will hoose to aept the sale. Under the stati shemes of 5.3 the operating airline may hoose to rejet a sale, and in that Setion we will disuss a modifiation to (3) Non-Optimality of Marovian Transfer-Prie Shemes Before examining speifi transfer prie shemes in detail, we desribe a simple ounter- 17

18 example to demonstrate that no Marovian transfer sheme an guarantee networ optimality, as long as the transfer sheme is based solely on sales of interline itineraries. By Marovian we refer to shemes that are ompletely defined by the urrent state of the networ and do not depend upon past states. Non-Marovian shemes that depend on the partiular sample path (the history of whih airline sold eah seat, for how muh, and when) ould ahieve optimality in the following ounter-example. The omplexity of suh shemes, however, would mae them impossible to implement. Consider two airlines, 1 and 2, eah operating one flight; eah flight has one remaining seat. Table 1 shows the expeted demand over a two-period horizon. In the seond olumn, an itinerary (x,y) requires x seats on airline 1 and y seats on airline 2. In the seond to last period (period 2), eah airline is equally liely to reeive a request for its intraline itinerary with assoiated revenue of $250. In the final period, airline 1 reeives a request for an interline itinerary for $400 with probability one. Clearly, it would be best for the alliane if the airline reeiving the intraline request were to turn it down, leaving its inventory for the interline itinerary. Table 1 Data for ounter-example to transfer priing optimality Period, Itinerary, A j Mareting Airline Revenue Probability 2 (1, 0) Airline 1 $ (0, 1) Airline 2 $ (1, 1) Airline 1 $400 1 Let p be the transfer prie in the final period paid to airline 2 if there is suffiient inventory remaining. Therefore at the beginning of period 2 the opportunity osts of the intraline inventory for airline 1 and airline 2 are ($400 p) and p, respetively. Beause eah intraline request an be fulfilled without any of its partner s inventory, eah airline would maximize its own value by aepting an intraline request if its revenue exeeds its opportunity ost of its inventory (see Theorem 1 below for a formal proof of this behavior.) Thus, to prevent either airline from filling an intraline request, p must satisfy both p > $250 and $400 p > $250, or $250 < p < $150, whih is a ontradition. 18

19 Note that networ optimality ould be guaranteed if payments are made for intraline itineraries as well as interline itineraries. For example, assume that the airlines set p=$100. Then, in period 2, let airline 1 offer $151 to airline 2 if airline 2 agrees not to sell the intraline tiet if a request for that tiet arrives. Given suh a subsidy sheme, neither airline will aept an intraline request and the networ is optimized. Suh transfer payments for intraline tiets, however, are impossible to implement for a variety of tehnologial, ompetitive, and legal reasons. Although no realisti Marovian transfer-prie sheme is universally optimal, ertain shemes have intuitive appeal. For example, some pratitioners have suggested that a seat s opportunity ost (sometimes alled its bid prie) would be a logial transfer prie (Vinod, 2005). While we will analyze eah transfer prie sheme separately, there are some ommon results worth noting. These results hold for all the shemes (stati and dynami) analyzed below. The results will also provide us with more general insights into why any transfer prie sheme an fail to ahieve first-best. THEOREM 1. For the Marov game desribed in 5.1, there exists a unique, pure strategy Marov perfet equilibrium in whih the mareting airline adopts the poliy, u ( r; x) j A ) p ( x) 1 if r ΔJ 1 + = 0 otherwise. PROOF. See Appendix 1. Theorem 1 shows that the mareting airline will aept any request that provides it with net revenue that exeeds its opportunity ost of the inventory used in the itinerary. The net revenue is the revenue reeived from the external ustomer for the itinerary minus any transfer prie paid to the operating airline. The ounter-example presented above illustrates an adverse onsequene of the result in Theorem 1. Sine there is no transfer prie paid for the sale of an intraline itinerary, eah airline maes intraline deisions without onsidering that deision s effet on its partner s revenue. Therefore, even if a entralized ontroller were to mae all interline aeptane deisions (removing deision rights on interline itineraries from the mareting and operating arriers), the hoie of the revenue-sharing method for interline itineraries would still affet the purely intra- 19

20 line deisions of the partners. This point is emphasized in the following orollary and subsequent disussion. Corollary 1. The equilibrium ontrol for intraline requests is of the form: u ( r; x) 1 = 0 r ΔJ 1 otherwise. j A ) PROOF. Immediate from Theorem 1 and p ( x) = for j N. 0 The ritial revenue level for the Airline s intraline deision is its own opportunity ost of the inventory used for the itinerary, muh lie the optimal deision for a single airline. In this ase, however, the optimal (entralized) deision for the alliane shown in Proposition 1 is determined by the total opportunity ost of the itinerary of both partners. That is, the ritial value should be: 1 j j j ( x A ) = ΔJ A ) + ΔJ ( x A ) Δ J,, 1 1, the effet of the hange in one airline s (here, the mareting air- j We refer to Δ J 1 A ) j line s) inventory on its partner s value, as the seond-order effet. We refer to Δ J A ) effet on the airline s own value, as the first-order effet. Our intuition is that inventory has a positive value, however, a simple example demonstrates that seond-order effets whih orrespond to the partner s value of the inventory an be either positive or negative. Consider the alliane shown in Figure 1, in whih airline 1 operates flights A and B, while Airline 2 only operates flight C. Airline 1 offers itineraries AB and B (both intraline), and airline 2 offers AC (an interline itinerary). Figure 1 Sample alliane for ontrol omparisons 1, the Flight A Flight C Flight B Airline 1 Airline 2 20

21 Looing at the seond-order effet (that on airline 2) of the sale of airline 1 s itineraries, we expet oppositely signed values. The sale of a B itinerary frees up (in expetation) the A inventory needed by airline 2 to fill an AC request, so we expet a negative opportunity ost for Airline 2. That is, airline 2 is better off if airline 1 sells a B itinerary. Conversely, a sale of an AB itinerary uses up A inventory, so we expet a positive opportunity ost for airline 2; airline 2 is worse off if airline 1 sells an AB itinerary. Formally, we expet: ΔJ B 2 AB A ) < 0 and ΔJ 1 A ) > Therefore, Airline 1 will overvalue its B itineraries, not selling them when it would benefit the alliane to do so, and under-value its AB itineraries, selling them when it does not benefit the alliane. Figure 2 illustrates the effet of these deisions on expeted alliane revenue. The straight lines (horizontal and diagonal) are alliane values, given that the itinerary request is aepted ( Sale ) or rejeted ( No Sale ). The bold lines indiate the aeptane poliies in the deentralized alliane, and alliane losses are shown in gray. Figure 2 Loss of potential revenue from ineffiient intraline itinerary aeptane poliies J 1 Alliane Value () x Alliane-harming sales aepted by airline 1 Sale No Sale J 1 Alliane Value ( x) Alliane-benefiial sales rejeted by airline 1 Sale No Sale J 1 ( x A ) j J 1 ( x A ) j j j ( x A ) ΔJ ( x A ) 1, 1 1 ΔJ, Realization of R j 1 j 1 j A ) ΔJ 1( x A ) Δ J, Realization of R j 2 j ( a) ΔJ A ) 0 1 > 2 j ( b) ΔJ A ) 0 1 < While no transfer prie sheme an guarantee that the seond-order effet will be inorporated in eah airlines intraline deision-maing, we will show in 5.5 that ertain transfer-prie 21

22 shemes an indiretly redue the impat of ignoring seond-order effets, leading to more effiient intraline deisions Stati Proration In pratie, revenue sharing for interline sales is often governed by stati proration (SP) ontrats that prorate the revenue reeived from an aepted request aording to fixed proportions. (Suh ontrats are often enfored via relatively infrequent, ex-post sharing of revenue information, so that the model formulated here is onsistent with the information-sharing assumption desribed at the end of 3.2). One form of stati proration speifies how revenue should be split for eah and every itinerary. If airline is the operating airline and arries a ustomer who paid the mareting airline r for itinerary j, airline reeives α r as a transfer payment while the mareting airline retains (1 α )r in revenue. To simplify the notation we will assume that α - = (1 α ), so in this ase the mareting airline s share of the revenue is α - r for itinerary j. We will refer to this form of stati proration as Itinerary-Speifi SP. In pratie, airlines sometimes use a ommon proration rate for multiple itineraries. The most extreme version uses a ommon (or universal) proration rate α for all itineraries. We examine two types of universal shemes, one based on the identity of the mareting airline and another that fixes the proportion for eah airline and ignores whether an airline is the mareting or operating arrier. First, under Universal SP (Mareting), the mareting airline reeives (1 α ) r while the operating airline reeives α r. In the analysis below it will be useful to assoiate a proration rate with a partiular airline, so under Universal SP (Mareting), α =α if is the operating airline and α =(1 α ) if is the mareting airline. Seond, under Universal SP (Airline-speifi), we assume that airline 1 reeives α r and that airline 2 reeives (1 α ) r. Therefore, α =α if =1 and α =(1 α ) if =2. Table 2 summarizes the transfer pries paid to the operating airline under the stati proration shemes. The table also summarizes the dynami transfer prie shemes that will be desribed in detail in

23 Table 2 Summary of Transfer pries paid to operating airline Stati Proration (SP) Dynami Transfer Pries Itinerary-Speifi SP R Δ J 1 x, Universal SP (Mareting) α R Bid Prie Δ J 1 x, A (partial value) Universal SP R α to Airline 1 and ΔJ x, A (Airline-speifi) 1 ( 1 α )R to Airline 2 ΔJ x, A α Bid Prie (full value) ( A j ) ( ) Bid-prie Proration j ( ) j ( ) Partner Prie 1 R Chosen by the operating airline (see Theorem 4) Note that distintions among the three stati proration shemes will be relevant in the numerial examples desribed in 6. The results in this Setion, however, apply to all three stati shemes. For onveniene we use the proration term α throughout the following analysis, although for the universal shemes the j an be eliminated. THEOREM 2. For the mareting airline, the equilibrium interline aeptane poliy under a stati proration sheme is: u ( r; x) 1 = 0 if r ΔJ 1 otherwise. j A ) α PROOF. See Appendix 1. The interline aeptane poliy in Theorem 2 ensures that the mareting arrier earns at least its opportunity ost of the inventory used. Given stati proration and the model formulation in (3), however, an interline itinerary may be aepted when its total revenue is less than the operating airline s opportunity ost of its seats, and less than the full alliane s opportunity ost as well. An obvious modifiation would be to give the operating airline, as well as the mareting airline, a veto over the sale of interline itineraries. 23