Network Capacity Management Under Competition

Transcription

1 Network Capacity Management Under Competition Houyuan Jiang 1 and Zhan Pang 2 1 Judge Business School, University of Cambridge, Trumpington Street, Cambridge CB2 1AG, UK. h.jiang@jbs.cam.ac.uk 2 Management School, Lancaster University, Lancaster LA1 4YX, UK. z.pang@lancaster.ac.uk This paper is dedicated to Professor Liqun Qi on the occasion of his 65th birthday Abstract We consider capacity management games between airlines who transport passengers over a joint airline network. Passengers are likely to purchase alternative tickets of the same class from competing airlines if they do not get tickets from their preferred airlines. We propose a Nash and a generalized Nash game model to address the competitive network revenue management problem. These two models are based on well-known deterministic linear programming and probabilistic nonlinear programming approximations for the non-competitive network capacity management problem. We prove the existence of a Nash equilibrium for both games and investigate the uniqueness of a Nash equilibrium for the Nash game. We provide some further uniqueness and comparative statics analysis when the network is reduced to a single-leg flight structure with two products. The comparative statics analysis reveals some useful insights on how Nash equilibrium booking limits change monotonically in the prices of products. Our numerical results indicate that airlines can generate higher and more stable revenues from a booking scheme that is based on the combination of the partitioned booking-limit policy and the generalized Nash game model. The results also show that this booking scheme is robust irrespective of which booking scheme the competitor takes. Keywords: Revenue management, capacity control, generalized Nash games, existence, uniqueness. 1 Introduction Revenue management has become a prevailing notion in airline and hospitality industries which provide perishable services or products. While many important components such as pricing, capacity management, overbooking, and choice modelling have been extensively studied, competition has not received enough attention although research on the topic is under way. For comprehensive reviews of revenue management theory and practice, see two recent excellent books [43, 50]. Most airlines in the world operate and compete fiercely over networks. Because of competition, most airlines operate with very small profit margins. Competition between airlines is complicated and multi dimensional. Possible competitive elements are network settings, frequencies, 1

2 timetabling, aircraft capacities, product offerings, pricing and capacity management among others. In this paper, we focus on capacity availability. It is well known in [50] that the capacity management problem can be satisfactorily modeled using dynamic programming. The resulting dynamic programs are computationally intractable due to the curse of dimensionality and are often approximated by various simpler mathematical programs such as deterministic linear programs, probabilistic nonlinear programs, and randomized linear programs, see for example [4, 9, 11, 49, 52]. Capacity management becomes even more complicated when several airlines compete over a network. Conceptually, capacity management under competition can be formulated by dynamic programming. But we are content to model it using simplified models such as deterministic linear programs in order to avoid the curse of dimensionality. There is no doubt that approximate models can only give airlines sub-optimal solutions. Nevertheless, Dockner et al. [12] argue that in regard to the assumption of payoff-maximizing behaviour, some researchers have suggested that firms are only bounded rational: they satisfice rather than maximize. Satisficing behaviour means that a firm is content with obtaining a certain level of payoff, not necessarily a maximal one. Chapter 8 of the book [50] is devoted to revenue management competition. Cachon and Netessine [5] present an excellent review on game theory in the area of supply chain management. In particular, they review techniques for proving the existence and uniqueness of a Nash equilibrium for static, dynamic and cooperative games. There are two notable features of revenue management competition that are different from traditional oligopolistic games. First, multiple products sold by the same airline share the same capacity. Second, unsatisfied demand for a particular product from one airline can be satisfied by other airlines; i.e., demand overflow is allowed because passengers are willing to substitute another airline. The first oligopolistic competition model with demand overflow between two firms is presented in [41] in a supply chain management setting where each firm produces a single product that can substitute similar products provided by competing firms. This work is extended in [28, 31, 33, 38]. Either deterministic or probabilistic rules to account for demand overflow are considered in [31, 38] while in [33], a choice model based on user s utility maximization is used for customers to decide from which firms they will purchase products. Explicit duopoly capacity management competition in a single-leg setting is considered in [29, 39, 48] where multiple products sold by each firm share the same capacity. In all these papers, the existence of a Nash equilibrium is investigated. Talluri [48] gives some weak and obvious conditions about the uniqueness of a Nash equilibrium, Netessine and Shumsky [39] establish a uniqueness result for a duopoly game with two classes of products when there is demand overflow of either the low- or high-fare class but not both. Li et al. [29] do not consider the uniqueness of a Nash equilibrium. Revenue management competition in both inventory and pricing is also studied by other researchers, see [1], [2], [10], [15], [16], [20], [21], [18], [37], [42], where demand overflow is not considered. Rather they assume that demand for each product for each firm can be determined by prices whether demand is deterministic or stochastic. We study capacity management games between several airlines in a network setting. To our 2

3 knowledge, this has not been considered in the literature. Our work is a natural extension of both network capacity management from a monopolistic setting to a oligopolistic setting, and capacity management competition from a single-leg setting to a network setting. Each airline sells multiple products as in traditional network revenue management problems, see [50]. Demand overflow is taken into account in a deterministic way as in [5, 38], i.e., a proportion of the passengers, who do not get the tickets they want from their preferred airline, approach other airlines for similar products. We develop one Nash and one generalized Nash game model to represent capacity management games. We prove the existence of a Nash equilibrium for both games. For a general network, we identify a sufficient condition to ensure the uniqueness of a Nash equilibrium for the perturbed (approximate) PNLP game. This uniqueness result in a general network setting follows from a recent result in [14]. In order to get more insights, we consider games over some simple networks. In particular, in the single-leg setting, we prove the uniqueness of a Nash equilibrium for the PNLP game under some suitable conditions. Note that Netessine and Shumsky [39] consider a Nash game in a single-leg setting with two classes of products and they only discuss the uniqueness of a Nash equilibrium in the interior region of the feasible set for a special case where there is no low-fare or highfare overflow. We derive some monotone comparative statics results. The comparative statics analysis for the duopoly PNLP game reveals that the smallest and largest undominated Nash equilibria are monotone nondecreasing in a firm s low-fare and in the competitor s high-fare, and are monotone nonincreasing in a firm s high fare and in the competitor s low fare. Further comparative statics results are derived for a symmetric PNLP game. In particular, the unique symmetric Nash equilibrium (the booking limit for the low fare) is monotone nondecreasing in the low fare, monotone nonincreasing in the high fare, and monotone nondecreasing in capacity. Furthermore, we give examples to show that the unique symmetric Nash equilibrium can be increasing or decreasing in number of firms in the competition. An iterative algorithm is proposed for finding Nash equilibria of both game models. We conduct numerical experiments. Our numerical results indicate that airlines can generate higher and more stable revenues from a booking scheme that is based on the combination of the partitioned booking-limit policy and the generalized Nash game model. The results also reveals that this booking scheme is robust irrespective of which booking scheme the competitor takes. The rest of the paper is organized as follows. In the next section, we propose game-theoretic models based on a deterministic linear programming model and a probabilistic nonlinear programming model to represent the response problem for an airline. We also illustrate both booking-limit and bid-price policies. In Section 3, we study the existence and uniqueness of Nash and generalized Nash equilibria. In Section 4, we focus on analysis for a single-leg network with two classes of products. Particularly, we prove the uniqueness of a Nash equilibrium under mild conditions and derive comparative statics results. In Section 5, we conduct numerical experiments on the performance of several booking schemes derived from both models in conjunction with either the partitioned booking-limit policy or the bid-price policy. In the last section, we make some concluding remarks and point out some possible further research topics. 3

4 2 Game-Theoretic Models and Booking Policies 2.1 Basic notation and assumptions I airlines, indexed by i = 1,, I, compete to provide services transporting passengers over a joint airline network where each airline may provide services in part of the network. Suppose I airlines sell K classes of products, indexed by k, over the network for a particular future departure date. Each product is a combination of a fare class and an itinerary over the network. Let r i be the unit price vector of dimension K for airline i. Since airlines do not necessarily provide identical services, some products may not be provided by all airlines. Without loss of generality, we may assume that rk i > 0 if product k is provided by airline i, and ri k = 0 otherwise. Let C i be the capacity vector of dimension M for airline i, where M is the number of legs over the network. Here each leg is a flight from one airport to another at a particular departure time. That is, two flights between two directly-connected airports are treated as two different legs. Let A i be the leg-product incidence matrix for airline i, i.e., A i mk = 1 if and only if airline i sells product k and product k covers leg m. Clearly A i is a matrix with M rows and K columns. If airline i does not sell product k, then column k of A i contains all zero elements and rk i = 0. If airline i does not fly on leg m, then Cm i = 0. The primary demand for product k of airline i is denoted by d i k, which is assumed to be a continuous random variable with a support [0, αk i ], where αi k is a sufficiently large positive constant. The primary demands for airline i are denoted by a vector d i = (d i 1,..., di K ) and the total demands in the network are represented by d = (d 1,..., d I ). The primary demand d i k is exogenous and is not affected by the booking limits of any airline. The demands for different products of the same airline can be correlated. Let f d ( ) and F d be the joint density function and cumulative distribution function of d, respectively. In our overflow setting, a rejected customer from airline i may make another booking request for the same product from other airlines. Suppose o ij k 0 denotes the overflow rate of product k from airline i to airline j. That is, if a customer, who prefers airline i, is rejected for a booking request for product k, by airline i, then they would make a booking request of product k from airline j with a probability o ij k. Clearly, j i oij k 1 for any i and k. We further assume that each customer has a unique airline for their initial preference. We make the following blanket assumptions for the rest of the paper: (A1) The prices of all products are endogenous and fixed for all airlines. (A2) Airline capacities as well as capacity allocation are treated as a vector of non-negative real numbers. (A3) d i k is a continuous random variable for all i and k with a sufficiently large support [0, αi k ]. (A4) Each customer is only interested in one particular product. Each customer makes a booking request from their preferred airline and with a certain probability, makes another booking request of the same product from another airline if their first booking request is rejected. If their second booking request is also rejected, then they become a lost customer to all airlines for this particular departure date. Some comments about the above assumptions are in order. Assumptions (A1) and (A2) are fairly standard in the literature of capacity management. Assumption (A2) is linked with so-called 4

5 fluid models for airline revenue management because it is often too difficult to solve optimization problems with integer variables. In practice, d i k is often a discrete random variable. But it is not unusual to approximate d i k by a continuous random variable as stated in Assumption (A3). Assumption (A4) is more restrictive and is assumed for tractability of our analysis. In the real world, passengers may have preference lists of service providers and products and they use preferences sequentially to guide them to search for products. This is explicitly modelled in [6]. Our model is a simplified version of the model of [6] by assuming that each preference list has only two elements. This simplification is also often employed in the literature, see [41], [28], [31], [38], [39]. 2.2 The probabilistic nonlinear programming formulation Let x i IR K, denoting the partitioned booking limits, be decision variables for airline i. The total effective demand for airline i is d i + j i oji [d j x j ] +, where z + = max{0, z} and max is taken componentwise for a vector z. This demand overflow approach has been used in [28], [31], [38], [41], where only a single product is offered by each of all firms/airlines, and in [39], where two products are offered by each of two airlines. The total effective demand for airline i is made up from its own primary demand d i and the overflow demand from other airlines, which is equal to j i oji [d j x j ] +. Assuming that partitioned booking limits x i for all other airlines other than i are given, airline i aims to determine its optimal partitioned booking limits by solving the following probabilistic nonlinear program (PNLP): PNLP i max x i IE[(r i ) T min(x i, d i + o ji [d j x j ] + )] j i s.t. A i x i C i, x i 0. The expectation is taken over the joint demand probability space of all airlines for all products. In PNLP i, the objective is that airline i maximizes its total expected revenue, where the sales for product k is equal to the smaller value of booking limit x i k and the effective demand d i k + j i oji k [dj k xj k ]+. The first constraint states that the capacity on each leg must not be violated. The last constraint simply shows that the booking limits are nonnegative. One simple observation is that the sizes of PNLPs for all airlines can be reduced by suitably removing the products/columns/variables that are not sold by a particular airline and the legs/rows that have zero capacities or that are not covered by a particular airline. This is particularly useful in reducing computational time when solving PNLPs. However, to keep notation simple, we shall not remove those columns or rows for the purpose of analysis because they do not affect our future analysis. 2.3 The deterministic linear programming formulation In the literature of revenue management, another popular approximate model is the so-called deterministic linear program (DLP), which is a further simplification of PNLP, where the stochastic nature of demand is entirely ignored. Let D i represent the expectation of demand random variable d i. 5

6 Assuming that partitioned booking limits x i for all other airlines other than i are given, airline i aims to determine its optimal partitioned booking limits by solving the following DLP: DLP i max (r i ) T x i x i s.t. A i x i C i, x i D i + o ji [D j x j ] +, j i x i 0. The first constraint states that the capacity on each leg must not be violated. The second constraint specifies that the total allocation for each product must not exceed the effective demand for this product. The last constraint simply shows that the booking limits are nonnegative. Based on an exact penalty method, we can reformulate DLP i into an equivalent nonlinear and nonsmooth program, whose feasible set depends only on the partitioned booking limit x i of airline i: max (r i ) T min(x i, D i + o ji [D j x j ] + ) x i j i s.t. A i x i C i, x i 0. Clearly, (1) is a special case of PNLP i when random variable d i k takes a single value of Di k for all i and k. We can define another variation of (1). Let ρ i > r i be a constant vector for each i. (1) max x i (r i ) T x i + (ρ i ) T min(0, D i + j i s.t. A i x i C i, x i 0. o ji [D j x j ] + x i ) (2) The statement below shows that DLP i, (1) and (2) are equivalent. Proposition 2.1 Let x i airline i. 0 be partitioned booking limits for all other airlines except for (a) If x i is an optimal solution to DLP i, then x i is also an optimal solution to (1). Conversely, if x i is an optimal solution to (1), then min(x i, D i + j i oji [D j x j ] + ) is an optimal solution to both DLP i and (1). (b) x i is an optimal solution to DLP i if and only if x i is an optimal solution to (2). Proof. (a) Suppose x i is an optimal solution to DLP i. Then x i is a feasible solution to (1). Therefore the optimal objective function value of DLP i is not greater than that of (1). Conversely, suppose x i is an optimal solution to (1). Then min(x i, D i + j i oji [D j x j ] + ) is a feasible solution to both DLP i and (1), and min(x i, D i + j i oji [D j x j ] + ) is still an optimal solution to (1). Hence the optimal objective function value of (1) is not greater than that of DLP i. Collectively, we have shown that DLP i and (1) have the same optimal objective function value. Then it is obvious that any optimal solution to DLP i is also an optimal solution to (1). Moreover, if x i is an optimal solution to (1), then min(x i, D i + j i oji [D j x j ] + ) is an optimal solution 6

7 to (1) because both x i and min(x i, D i + j i oji [D j x j ] + ) are feasible to (1) and the objective function of (1) has the same value at both feasible solutions, and min(x i, D i + j i oji [D j x j ] + ) is an optimal solution to DLP i. (b) We only need to prove that any optimal solution to (2) must satisfy D i + j i oji [D j x j ] + x i 0, i.e., any optimal solution to (2) must be a feasible solution to DLP i. By contradiction, assume that for an optimal solution x i of (2), it holds that for some k, k Dk i + j i oji k [Dj k xj k ]+ x i k < 0. Define yi such that yk i = xi k + k and yl i = xi l for any l k. Then x i y i 0 because D i + j i oji [D j x j ] + 0. This shows that y i is still a feasible solution to (2). By simple calculations, we can prove that y i is a better solution for (2) than x i, which is a contradiction. 2.4 Booking policies Both booking limits and dual prices or Lagrangian multipliers can be obtained from solving either DLP i or PNLP i. These solutions are used to form booking policies such as partitioned booking-limit and bid-price controls, see [50]. In a partitioned booking-limit policy, a fixed amount of capacity of each resource is allocated to every product offered. The demand for each product has access only to its allocated capacity and no other product may use this capacity. In both DLP i and PNLP i, these booking limits are set to be the respective optimal solutions x. A bid-price control policy sets a threshold price or bid price for each resource in the network. Roughly this bid price is an estimate of the marginal cost of consuming the next incremental unit of the resource s capacity. When a booking request for a product arrives, the revenue from the request is compared to the sum of the bid prices of all the resources required by the product. If the revenue exceeds the sum of the bid prices, the request is accepted, provided all the resources associated with the requested product are still available; if not, the request is rejected. In DLP i, the optimal solution of dual variables associated with the capacity constraints, which are of the format Ax C, are used as bid prices. Similarly the bid prices for PNLP i are the Lagrangian multipliers associated with the capacity constraints at their optimal solution. 3 Existence and Uniqueness of Nash Equilibrium In this section, we first formally define Nash and generalized Nash games based on the optimization models defined in the last section. We then define Nash and generalized Nash equilibrium points. In the end, we study the existence and uniqueness of (generalized) Nash equilibrium points. We follow [40] to define generalized Nash games and generalized Nash equilibrium points. Since generalized Nash games have found many other applications in addition to revenue management competition considered in this paper, we shall use firms and airlines interchangeably in a general context. Let x i be the strategy variable for firm i, x i the joint strategy variable for all other firms, and x = (x i, x i ) the joint strategy variable for all firms. Let π i (x i, x i ) be the payoff function for 7

8 firm i when the joint strategy is x. Assume that for any given strategies x i for all other firms, firm i must choose their strategy from their feasible set K i (x i ) = {x i : h i (x i ) 0, g i (x i, x i ) 0}, where h i : IR K IR N i and g i : IR K I IR M i. Here h i (x i ) 0 is a set of constraints that do not involve x i, and g i (x) 0 is a set of constraints that involve all strategy variables, which are usually called the joint/coupled constraints in the literature, see [45]. For any given strategy x i for other firms, firm i finds their best strategy x i by solving the following maximization problem: max π i (x i, x i ) x i (3) s.t. x i K i (x i ). Definition 3.1 A generalized Nash game defined by (3) for firm i is to find x, called a generalized Nash equilibrium, such that x i K i ( x i ) is an optimal solution for firm i when the strategies for all other firms are fixed to be x i. If there is no joint constraint in the game, i.e., g i (x) 0 diminishes for all i, then a generalized Nash game and a generalized Nash equilibrium reduce to a traditional Nash game and a Nash equilibrium respectively. The key difference between generalized Nash games and traditional Nash games is that the strategy space for a firm may depend on other firms strategies in the former, but not in the latter, although the payoff functions in both types of games are allowed to be functions of other firms strategies. The best response function is an important concept in game theory. For any given x i, the set of optimal solutions for (3) is called the response function for firm i and is denoted by BR i (x i ). To have a meaningful game, we often assume that BR i (x i ) is singleton for any given x i and for all i. Consequently, x is a generalized Nash equilibrium if and only if x i = BR i ( x i ) for all i. Intuitively, this optimality condition says that a firm is not better off if they unilaterally change their strategy. One can observe that the game based on DLP i for all airlines results in a generalized Nash game, while the game based on PNLP i results in a Nash game. The resulting two games are called DLP and PNLP games, respectively. By Proposition 2.1, the DLP game is equivalent to a traditional Nash game with nonsmooth and nonlinear payoff functions. Generalized Nash games have been used for modelling competitive pricing in [42] where there is no network and demand does not overflow between firms once prices are fixed. In [16], traditional Nash games have been used for proposing a unified game model combining pricing, capacity management, overbooking in a stochastic and dynamic framework for network revenue management. However, demand is not shared among competitors but the demand for individual airlines is determined by the prices of the same products offered by all airlines. Therefore, the game model in [16] is not an example of generalized Nash games. For other applications of generalized Nash games, see [13], [25], [40], [46]. 3.1 Existence of Nash equilibrium The existence of a Nash equilibrium (or generalized Nash equilibrium) for Nash games (or generalized Nash games) is an important topic in game theory. Without an equilibrium in a 8

9 game, firms do not know what strategy they should take. In many occasions, an equilibrium does exist. In this subsection, we provide conditions to ensure the existence of a Nash or a generalized Nash equilibrium for the two games proposed in Section 2. The following results are well known in the literature by noting that concavity is preserved under the min-operator, limits, addition and hence under expectation and integration signs; see for example [5], and that the integral is continuously differentiable if the integrand is globally Lipschiz continuous and continuously differentiable almost everywhere; see for example [23, 26]. Lemma 3.1 (a) For each airline i, the objective functions of DLP i, (1) and (2) are all concave with respect to x i. (b) For each airline i, the objective function of PNLP i is concave with respect to x i. (c) For any i, the objective function of PNLP i is continuously differentiable with respect to x. Proposition 2.1 proves an equivalence between DLP i, (1) and (2). The lemma below shows equivalences between different generalized Nash and Nash games defined by those optimization problems. All results can be easily proved by the definitions of the generalized Nash equilibrium and the Nash equilibrium, and Proposition 2.1. Lemma 3.2 (a) If x is a generalized Nash equilibrium for the DLP game, then x is a Nash equilibrium for the Nash game defined by (1). (b) x is a generalized Nash equilibrium for the DLP game if and only if x is a Nash equilibrium for the Nash game defined by (2). We are now ready to present existence results of a (generalized) Nash equilibrium for both DLP and PNLP games. Theorem 3.1 (a) There exists a generalized Nash equilibrium for the DLP game. (b) There exists a Nash equilibrium for the PNLP game. Proof. The results follow from Theorem 1 of [45], which states that a Nash equilibrium exists for a Nash game if the objective function for each firm is concave with respect to their own strategy and continuous with respect to the strategies of all firms and the strategy space for each firm is convex and compact. In light of Lemma 3.1 (b) and (c), the result in (b) follows from Theorem 1 of [45] directly. It is easy to verify that the conditions in Theorem 1 of [45] are satisfied for the Nash games defined by (2). Therefore the existence of a generalized Nash equilibrium of the DLP game follows from Lemma 3.2(b). 9

10 3.2 Uniqueness of Nash equilibrium The uniqueness of a Nash equilibrium is another important topic in game theory. If there is a unique equilibrium, firms can choose their strategies without vagueness. The importance of a unique equilibrium is highlighted in a statement of [5]: The obvious problem with multiple equilibria is that the firms may not know which equilibrium will prevail. Hence, it is entirely possible that a non-equilibrium outcome results because one firm plays one equilibrium strategy while a second firm chooses a strategy associated with another equilibrium. Let K = {x i : A i x i C i, x i 0}, K = K K... K I, and Φ(x) = ( x 1π 1 (x),, x iπ i (x),, x I π I (x)) T be a column vector. Define a variational inequality (VI) problem, which is to find a vector x K such that (y x) T Φ( x) 0, y K. It follows from page 24 of [14] that x is a Nash equilibrium for the PNLP game if and only if x is a solution of the above VI problem. Recall a unique equilibrium result for the variational inequality problem. Lemma 3.3 (Proposition of [14]) Let K = K K... K I be a Cartesian product. If Φ is a P function on K, then the VI defined by K and Φ has at most one solution. In order to obtain the uniqueness of a Nash equilibrium for the Nash or generalized Nash game based on Lemma 3.3, it is often required that the payoff function for firm i is continuously differentiable with respect to the strategy variables of firm i and the feasible set for each firm does not depend on the strategies of other firms. Since not both the above conditions are satisfied at the same time for the DLP game defined by either DLP i or (1) or (2), we are not able to establish the uniqueness of a Nash equilibrium for the DLP game. In [45], a uniqueness result of a normalized Nash equilibrium is proved for a generalized game in which all coupled constraints must be shared by all firms. However, firms in the generalized Nash game defined by (2.3) do not share coupled constraints. Therefore we shall focus on the uniqueness of a Nash equilibrium for the PNLP game. We now evaluate the Jacobian of the payoff functions of the PNLP game. Netessine and Rudi [38] state that since the function under the expectation is integrable and has a bounded derivative, it satisfies the Lipschitz condition of order one, and hence the expectation and the derivative can be interchanged. This argument follows from a result in [23]. Recall that the payoff function for airline i is K π i (x i, x i ) = IE[ k=1 r i k min(x i k, d i k + j i The first order derivatives of the payoff functions are o ji k [dj k xj k ]+ ]. π i (x i, x i ) x i k = r i k(1 F d i k + j i oji k [dj k xj k ]+ (x i k)), i = 1,, I, k = 1,, K, where F d i k + j i oji k [dj k xj k ]+ is the cumulative probability distribution of random variable d i k + j i oji k [dj k xj k ]+. The second order derivatives of the payoff functions are 10

11 2 π i (x i, x i ) ( x i = rkf i k )2 d i (x i k + k), j i oji k [dj k xj k ]+ 2 π i (x i, x i ) x i k xi l = 0, 2 π i (x i, x i ) = rko i ji x i k xj k f d i + k m i omi k k 2 π i (x i, x i ) = 0, x i k xj l [dm k xm k ]+ d j k >xj k i = 1,, I, k = 1,, K, i = 1,, I, k = 1,, K, l = 1,, K, l k, (x i k) Pr(d j k > xj k ), i = 1,, I, j = 1,, I, k = 1,, K, j i, i = 1,, I, j = 1,, I, k = 1,, K, l = 1,, K, j i, l k. The calculation of 2 π i (x i,x i ) is slightly involved, but it can be done through the derivative x i k xj k definition. The Jacobian Φ(x) is a sparse matrix of I K columns and I K rows. We can rewrite Φ(x) as N 11 N 1I... N I1 N II where N ij is a diagonal matrix in IR K K whose diagonal elements are ( 2 π i (x i, x i ),, 2 π i (x i, x i ) ). x i 1 xj 1 x i K xj K Having calculated the second-order derivatives of π i, it is easy to see that the Hessian matrix for the payoff function of firm i is N ii, where N ii is a diagonal matrix with non-negative diagonal entries rk i f d i + (x i k j i oji k [dj k xj k ]+ k ). If Nii has positive diagonal entries, then the response problem for firm i has a unique solution BR i (x i ) for given x i. In order to prove that the PNLP game has a unique solution, by Lemma 3.3, it suffices to prove that Jacobian Φ(x) is a P function on K. By Proposition in [14], if Jacobian Φ(x) is a P matrix for all x K, then Φ is a P function on K. By [17], any row diagonally dominant matrix is also a P -matrix. This shows that it is sufficient to prove that Φ(x) is row diagonally dominant for any x K. More precisely we need to prove the following: rkf i d i (x i k + k) > j i oji k [dj k xj k ]+ rko i ji k f d i + (x i k) Pr(d j k m i omi k > xj k ), j i k [dm k xm k ]+ d j k >xj k which holds if the following conditions are satisfied: (A5) For any i and k, j i oji k < 1; (A6) For any feasible point x K, the probability density function for d i k + j i oji k [dj k xj k ]+ has a positive value at any x i k such that xi K i. 11

12 Assumption (A5) is relatively weak and can be satisfied if overflow rate o ij k is small for all i, j and k. However, there is no obvious condition to ensure Assumption (A6) except for a special case where o ij k = 0 for all i, j and k. In this case, there is no demand overflow and the existence of a unique Nash equilibrium is equivalent to a unique optimal solution for airline i s maximization problem. It is obvious that if the payoff function of airline i is strictly concave (equivalently matrix N ii is positive definite) over its feasible region, then its optimal solution is unique. Furthermore, matrix N ii is positive definite over its feasible region if probability distribution f d i (x) > 0 for any x [0, α i k k ], see Assumption (A3). Assumption (A3) implies that U = d i k + V is also continuous random variable, where V = j i oji k [dj k xj k ]+. It is easy to check that for any x i k 0, we P r(u x i k) = x i k x i k v 0 0 f V,d i k (v, ξ)dξdv, where f V,d i is the joint probability density function of random variable (V, d i k k ) at (v, ξ). Then the probability density function of U is x i k 0 f V,d i k (v, x i k v)dv. One can see that for any x i k > 0, f U(x i k ) > 0 should hold under Assumption (A3). However, the probability density function for U has a value of zero at x i k = 0. This shows that in general, Assumption (A6) is not easy to satisfy. We propose two heuristic approaches to get around Assumption (A6). The first is the so-called regularization approach, in which we add a quadratic term in the objective function so that the response problem becomes a strictly concave problem. In particular, firm i considers the following optimization problem PNLP R i max x i π i R(x i, x i ) = IE[(r i ) T min(x i, d i + j i s.t. A i x i C i, x i 0. o ji [d j x j ] + )] ε k (x i k) 2 Here ε is a small positive constant. We now provide conditions to ensure the uniqueness of a Nash equilibrium for the PNLP R game. Theorem 3.2 For any ε > 0, Assumption (A5) is a sufficient condition for the PNLP R game to have a unique Nash equilibrium. Furthermore, when ε 0, any accumulative point of the sequence of the Nash equilibrium for the PNLP R game is a Nash equilibrium for the PNLP game. Proof. The existence of a Nash equilibrium for the PNLP R game can be proved similar to Theorem 3.1. Therefore, we only need to prove that the PNLP R game has at most one solution. It is straightforward to show that Φ R is a P -function over K. Therefore, the uniqueness result for the PNLP R game follows from Lemma 3.3 and Assumption (A5). By Theorem 1 of [24], when ε 0, any accumulative point of the sequence of the Nash equilibrium for the PNLP R game is a Nash equilibrium for the PNLP game. 12

13 The second approach uses some approximation in the feasible set for the response problem for all firms. Especially, in PNLP i, we introduce a small lower bound for each of all products: PNLP i F max x i π i R(x i, x i ) = IE[(r i ) T min(x i, d i + j i s.t. A i x i C i, x i ε. o ji [d j x j ] + )] In this case, we define a new joint strategy space K F such that K F = (K 1 F, K2 F,..., KI F ) and K i F = {xi : A i x i C i, x i ε}. It is straightforward to show that Φ F is a P -function over K F. Therefore, the uniqueness result for the PNLP R game follows from Lemma 3.3 and Assumption (A5). Unlike the PNLP R game, we are not able to prove that when ε 0, any accumulative point of the sequence of the Nash equilibrium for the PNLP F game is a Nash equilibrium for the PNLP game. However, in the next section, for a PNLP game over a single-leg network with two classes of products, we prove that for a sufficiently positive ε and under suitable conditions, any Nash equilibrium for the PNLP game is also a Nash equilibrium for the PNLP F game and vice versa, see Proposition 4.1. The following counter-example shows that the DLP game can have multiple Nash equilibrium points while its PNLP counterpart has a unique solution. Example 3.1 Consider a symmetric duopoly game over a simple star network consisting of three legs: A B, B C, and B D, which are indexed as 1, 2 and 3, respectively. By symmetry, we mean that all firms have the same setting with respect to all possible parameters. Assume the capacities on all three legs are 200 each. Suppose there are five itineraries: A B, B C, B D, A B C, and A B D. For each itinerary, there is exactly one product. This implies that there are five products, which are indexed by 1, 2, 3, 4, and 5, respectively. Assume the random demand for each of five products is uniformly distributed. The demand support for them are [0, 400], [0, 400], [0, 400], [0, 200], and [0, 200], respectively. Hence, the average demands for five products are D1 i = Di 2 = Di 3 = 200 and Di 4 = Di 5 = 100. Assume the unit revenues for five products are r1 i = ri 2 = ri 3 = 100 and ri 4 = ri 5 = 200. For simplicity, assume overflow rate o ji k = 0 for any i, j and k. By the above setting, the response problem based on the DLP for firm i is the following: max x i 100x i xi xi xi xi 5 s.t. x i 1 + x i 4 + x i 5 200, x i 2 + x i 4 200, x i 3 + x i 5 200, 0 x i 1, xi 2, xi 3 200, 0 x i 4, xi After some simple calculations, we obtain the response problem based on PNLP i : max x i ((x i 1 )2 + (x i 2 )2 + (x i 3 )2 + 2(x i 4 )2 + 2(x i 5 )2 )/800 + x i 1 + xi 2 + xi 3 + xi 4 + xi 5 s.t. x i 1 + x i 4 + x i 5 200, x i 2 + x i 4 200, x i 3 + x i 5 200, 0 x i 1, xi 2, xi 3, xi 4, xi 5. 13

14 It is easy to check that both (0, 100, 100, 100, 100) and (200, 200, 200, 0, 0) are optimal solutions for DLP i, with the optimal objective function value of 60, 000. On the other hand, PNLP i has a unique solution (85.714, , , , ) with the optimal objective function value of 50, Because all overflow rates are assumed to be zero, we can conclude that there are multiple Nash equilibria for the DLP game and a unique Nash equilibrium for the PNLP game. The next example demonstrates that the DLP game may have a unique Nash game. Example 3.2 Suppose both C i and C j are large positive numbers and demand D i and D j are small positive numbers. For simplicity, let us assume that there is exactly one product from each airline. Note that the capacity constraints A i x i C i and A j x j C j are inactive at any (generalized) Nash equilibrium for DLP i and DLP j. On the other hand, the demand constraints are active at any (generalized) Nash equilibrium for DLP i and DLP j. Then the response functions for firm i and j are: { D BR i (x j i + D j x j if x j D j ) = D i otherwise, { D BR j (x i i + D j x i if x i D i ) = D j otherwise. It is easy to see that (D i, D j ) is the unique generalized Nash equilibrium for the DLP game. 4 Analysis for Simple Networks In the previous section, we have analyzed the existence and uniqueness of Nash equilibrium in a general network setting. However, it is very difficult to derive the structural properties for these general models. To gain some insights from the capacity competition, we consider a network consisting of a single leg with two classes of products, which are denoted by 1 and 2. For any i, we assume that the unit prices for low and high-fare products are r1 i and ri 2, respectively, which satisfy condition r1 i < ri 2. Note that our model is different from the model of Netessine and Shumsky [39] who assume that low-fare passengers arrive earlier than high-fare passengers. We adopt the standard setting in network revenue management literature by assuming that the demands arrive simultaneously, see Chapter 3 of [50]. The pay off function in PNLP i can be rewritten as follow: π i (x i, x i ) = r i 1IE[min(d i 1 + j i o ji 1 (dj 1 xj 1 )+, x i 1)] + r2ie[min(d i i 2 + o ji 2 (dj 2 xj 2 )+, x i 2)]. (4) j i The feasible region for firm i is {x i : x i 1 + xi 2 Ci, x i 0}. 4.1 Space dimension reduction Our first preliminary result is to simplify the feasible region. 14

15 Lemma 4.1 For any Nash equilibrium of the PNLP game, there exists a Nash equilibrium such that the capacity constraint x i 1 + xi 2 Ci is binding for all i. Proof. Suppose x is a Nash equilibrium for the PNLP game and there exists i such that x i 1 + xi 2 < Ci. Construct ˆx as follows: ˆx i 1 xi 1, ˆxi 2 xi 2, ˆxi 1 + ˆxi 2 = Ci, ˆx j 1 = xj 1, and ˆxj 2 = xj 2 for any j i. We prove that ˆx is also a Nash equilibrium. First, (ˆx i 1, ˆxi 2 ) is still a feasible solution of the response problem for firm i. Second, (ˆx i 1, ˆxi 2 ) is still an optimal solution of the response problem for firm i because (x i 1, xi 2 ) is an optimal solution and the objective function value at (ˆx i 1, ˆxi 2 ) is not smaller than the objective function value at (xi 1, xi 2 ). Third, the probability such that the effect demand for product 1 (or 2) from firm i exceeds x i 1 (or xi 2 ) is zero since otherwise, (ˆx i 1, ˆxi 2 ) gives a strictly better solution than (xi 1, xi 2 ) for the response problem of firm i. Fourth, the probability of the primary demand of both products 1 and 2 from firm i overflows to any other firm j is zero when firm i chooses (x i 1, xi 2 ) as their strategy. Fifth, the probability of the primary demand of both products 1 and 2 from firm i overflows to any other firm j is again zero when firm i chooses (ˆx i 1, ˆxi 2 ) as their strategy. Sixth, the probability of overflow demand from firm i to any other firm j in zero when firm i takes strategy (x i 1, xi 2 ) or (ˆxi 1, ˆxi 2 ). Seventh, (x j 1, xj 2 ) is still an optimal solution for the response problem of firm j if all other firms take ˆx j as their joint strategy. This shows that ˆx is also a Nash equilibrium. Repeating the above process, we obtain a new Nash equilibrium such that the capacity constraints are all binding for all firms. By Lemma 4.1, finding a feasible solution for the response problem for firm i can be done by solving the following optimization problem with a single decision variable x i 1 (without confusion, x i 1 is replaced by xi in the sequel): max π i (x i, x i ) = r1 i IE[min(di 1 + j i oji 1 (dj 1 xj ) +, x i )] (5) +r2 i IE[min(di 2 + j i oji 2 (dj 2 (Cj x j )) +, C i x i )] s.t. 0 x i C i. 4.2 Uniqueness Taking the first and second order derivatives of π i with respect to x i, we have π i x i = ri 1P r(d i 1 + o ji 1 (dj 1 xj ) + x i ) r2p i r(d i 2 + o ji 2 (dj 2 (Cj x j )) + C i x i ), (6) j i j i and 2 π i x i x i = ri 1f d i 1 + j i oji 1 (dj 1 xj ) + (x i ) r i 2f C i (d i 2 + j i oji 2 (dj 2 (Cj x j )) + ) (xi ). (7) The next lemma shows that under some suitable condition, the response problem (5) has a unique solution. Lemma 4.2 (a) The objective function of the response problem (5) is a strictly concave function over [0, C i ]. 15

16 (b) The response problem (5) has a unique solution. Proof. (a) It follows from (7) that the objective function of the response problem (5) is a concave function over [0, C i ]. Assumption (A3) shows that the second order derive of π i is strictly negative over (0, C i ]. Therefore, the objective function of the response problem is strictly concave over [0, C i ]. (b) The result follows from (a) directly. Given other airlines booking limits x i, it follows from the assumption r1 i < ri 2 that π i x i = r1p i r(d i 1 + o ji 1 (dj 1 xj ) + C i ) r2 i < 0, x i =C i j i which shows that the optimality of the response problem of firm i cannot be obtained at C i. On the other hand, we can also ensure that under suitable assumptions, the optimality of the response problem of firm i cannot be obtained at x i = 0. A simple and intuitive assumption is r i 1 > r i 2P r(d i 2 + j i o ji 2 dj 2 Ci ), (8) which states that the maximum effective demand of high-fare passengers for firm i is small enough so that the expected profit of accepting the C i -th high-fare customer is lower than the unit price for a low-fare customer. A weaker condition than (8) is the following inequality: r i 1 r i 2P r(d i 2 C i ). (9) Lemma 4.3 (a) The optimal booking limit for the response problem (5) of firm i, x i, must be positive if assumption (8) is satisfied. (b) x i = 0 is the unique optimal booking limit for the response problem (5) of firm i if (9) is not satisfied. Proof. (a) Suppose (8) holds. By (6), for any x i [0, C i ], we have π i x i ri 1P r(d i 1 + o ji 1 (dj 1 xj ) + x i ) r2p i r(d i 2 + o ji 2 dj 2 Ci x i ). j i j i When x i = 0, P r(d i πi 1 0) = 1 and it follows from (8) that r i x i 1 ri 2 P r(di 2 + j i oji 2 dj 2 C i ) > 0, which shows that π i is strictly increasing at x i = 0. Therefore, x i = 0 must not be optimal. (b) Suppose (9) does not hold. Then at x i = 0, π i x i = ri 1 r2p i r(d i 2 + o ji 2 (dj 2 (Cj x j )) + C i ) r1 i r2p i r(d i 2 C i ) < 0, j i which shows that π i is strictly decreasing at x i = 0. Therefore, x i = 0 is an optimal solution by Lemma

17 Proposition 4.1 For the PNLP game, there exists a unique Nash equilibrium if condition (8) and the following condition are satisfied: o ji < 1, i. (10) j i Proof. By Lemma 4.3, x i > 0 for all i (the booking limit for the low-fare class). It is now true that each of all the firms in the new PNLP game has a non-zero booking limit in any Nash equilibrium. We can make some modifications about the game by replacing the feasible region for firm i by [ε, C i ]. Here ε is a fixed and sufficiently small positive constant. We have proved that Φ is a P -function on K, where K = ( K 1, K 2,..., K I ) and K i = [ε, C i ] for a fixed ε. Therefore, the PNLP game has a unique Nash equilibrium on K. So far, we have proved the uniqueness of a Nash equilibrium for the game defined by (5), in which we impose the condition that x i 1 + xi 2 = Ci, i.e., the capacity constraint is binding. We now prove that the Nash equilibrium for the PNLP game is unique. Otherwise, there exists a Nash equilibrium for the PNLP game which satisfies x i 1 + xi 2 < Ci for some i. The proof of Lemma 4.1 shows that we can construct another Nash equilibrium from this existing equilibrium by pushing either x i 1 or xi 2 up such that xi 1 + xi 2 = Ci. Clearly, in such a way, we can obtain more than one Nash equilibrium such that the capacity constraint for firm i becomes binding. Repeating this process, we can obtain multiple Nash equilibria for the PNLP game such that the capacity constraints for all firms are binding, which contradicts to the uniqueness of a Nash equilibrium with binding capacity constraints that we have proved above. Therefore, we have proved that the PNLP game has a unique Nash equilibrium. The above proposition presents sufficient conditions to ensure the uniqueness of a Nash equilibrium in a simple network setting, while, in the general network setting, we can only provide sufficient conditions for the uniqueness of a Nash equilibrium for the perturbed PNLP game. Condition (8) implies that the equilibrium booking limits must be interior points of the feasible region. Hence the results in Lemma 3.3 can be applied directly. In [39], they assume that equilibrium booking limits are interior points rather than give a sufficient condition to ensure equilibrium booking limits are interior points. 4.3 Comparative statics Comparative statics analysis provides a way of deriving managerial insights and results such as monotonicity of the optimal decisions with respect to changes of some parameters of response problems and Nash games. Next we study monotone comparative statics of booking limits. Proposition 4.2 Consider the PNLP game. (a) The optimal solution x i for the response problem of firm i is monotone nondecreasing in r1 i for any k. (b) The optimal solution x i for the response problem of firm i is monotone nonincreasing in r i 2 for any k. 17

18 (c) The optimal solution x i for the response problem of firm i is monotone nonincreasing in x j k for any j i and k. (d) The optimal solution x i for the response problem of firm i is monotone nondecreasing in C i. Proof. (a) By Theorem 4 of [36], we only need to prove that π i is supermodular in x i and has increasing differences in (x i, r1 i ), and the feasible region is a sublattice. It is trivial that πi is supermodular in x i because x i is the only decision variable for response problem of firm i. By Theorem 6 of [36], the fact that π i has increasing differences in (x i, r1 i ) follows from the fact that 2 π i / x i r1 i 0. The latter can be easily verified from the formula for the first order derivative of π i. The feasible region for the response problem is a simple interval, which is a sublattice. (b) The proof is analogue to that for (a) with only one exception which is to define a new variable s i 2 = ri 2. (c) The proof is analogue to that for (a) with only one exception which is to define a new variable y j k = xj k for any j i and any k. Then we can easily prove that πi is supermodular in x i and has increasing differences in (x i, yk i ). Therefore, Theorem 4 of [36] shows that the optimal solution x i for the response problem of firm i is monotone nondecreasing in y j k for any j i and k, which implies the desired result. (d) In contrast to Theorem 4 of [36] for a comparative statics result with regard to parameters in the payoff function, Theorem 4 of [36] is a comparative statics result with regard to parameters in the constraints and possibly in the payoff function. However, we are not able to prove our result in this proposition based on Theorem 4 of [36] because we are not able to prove that the payoff function has strictly increasing differences in (x i, C i ). Therefore, we employ a direct proof approach. From the first order condition (6), it can be seen that the optimal solution x i for the response problem (5) is obtained at the joint point of two curves y = r i 1P r(d i 1 + j i o ji 1 (dj 1 xj ) + x i ), and y = r i 2P r(d i 2 + j i o ji 2 (dj 2 (Cj x j )) + C i x i ). The first curve is monotone nonincreasing in x i and the second curve is monotone nondecreasing in x i. When C i is increased, the first curve remains unchanged and the second curve moves towards right. This shows that the joint point of the two curves moves towards the south-east direction, which implies that x i is not decreased when C i is increased. This completes the proof. Theorem 4.1 Suppose x is a Nash equilibrium of the PNLP duopoly game defined by (5) between firms i and j. Then (a) The smallest and largest serially undominated Nash strategies (x i, x j ) are monotone nondecreasing in r i 1 and rj 2. 18