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OPERATIONS RESEARCH
Vol. 00, No. 0, Xxxxx 0000, pp. 000–000
issn 0030-364X |eissn 1526-5463 |00 |0000 |0001
INFORMS
doi 10.1287/xxxx.0000.0000
c
0000 INFORMS
Hidden-City Ticketing: the Cause and Impact
Zizhuo Wang
Department of Management Science and Engineering, Stanford University, zzwang@stanford.edu
Yinyu Ye
Department of Management Science and Engineering, yinyu-ye@stanford.edu
Hidden-city ticket is an interesting airline ticket pricing phenomenon. It occurs when an itinerary connecting
at an intermediate city is less expensive than a ticket from the origin to the intermediate city. In such a
case, passengers traveling to the intermediate city will have the incentive to pretend to be traveling to the
final destination, deplane at the connection point and forgo the unused portion of the ticket. Hidden-city
opportunities are not uncommon nowadays.
In this paper, we establish a mathematical model to analyze the cause of hidden-city ticketing and its
impact on both airlines’ revenues and consumers’ welfare. We consider a flight network revenue management
model and show that the hidden-city opportunity may arise when there is a large difference in price elasticity
of demand on related itineraries. We show that when the passengers take advantages of such opportunities
whenever possible, the airlines had better to react, and the optimal reaction will no longer contain any
hidden-city opportunities. However, the airline’s revenue always decreases from that when passengers do
not practice such ticketing strategies. We show that the decrease could be as much as half of the original
optimal revenue, but it cannot be more if the airline takes a hub-and-spoke network. Meanwhile, as a result
of airline’s reaction, the fares to the final destination of a hidden-city itinerary will rise, which eventually
will hurt the passengers. We take an exogenous competition model in our analysis and validate it through a
game theoretic approach. Numerical results are presented to illustrate our results.
1. Introduction
Since the deregulation of the airline industry in 1970s, airlines have employed more and more
sophisticated pricing strategies to strive for more revenues from passengers. Many interesting pric-
ing phenomena are frequently observed. Among these is the “hidden-city” ticketing opportunity
in which the price quoted by an airline for an itinerary from city A to city B is more expensive
than the price quoted by the same airline for an itinerary from city A to city C with a connection
at city B. When such a situation happens, travelers from A to B will have an incentive to pretend
to be traveling from A to C, deplane at city B and forgo the unused portion of the ticket. In
this case, we call city B the “hidden-city”. An example of a hidden-city ticketing opportunity is
illustrated as follows1:
A one-way flight from San Francisco (SFO) to Dallas/Fort Worth (DFW) on Feb. 25th, 2011
costs $597 for American Airlines. However, if one searches for a one-way flight from SFO to
Tempa (TPA) for the same day, the fare is only $229 for American Airlines with a connection at
DFW (we call DFW the hidden-city in this case).
This pricing may not make sense to many travelers. How can it be that the entire flight is less
expensive than a shorter segment of the flight? However, this phenomenon prevails in current air-
lines’ pricing, especially in the modern hub-and-spoke airline networks (A hub-and-spoke network
consists hub cities and spoke cities. Itineraries between spoke cities are connected at a certain
hub. This flight network is most adopted by major airlines to provide more services and increase
the transportation efficiency, we will formally define this term in Definition 4). According to an
investigation paper GAO (2001), about 17% of the markets among major US airlines (in all fare
1
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classes) exhibit hidden-city ticketing opportunities (which they defined in GAO (2001) as 100 dol-
lars’ difference in high fare class and 50 dollars’ difference in low fare class). Since such a ticketing
strategy hurts airlines’ revenues and competitive positions, most airlines explicitly prohibit the
use of hidden-city ticketing in the terms and conditions of a ticket. However, travelers are seldom
adversely sanctioned for such practice because it is difficult and costly for airlines to pursue an
individual passenger. Moreover, the usage of such practice is not yet sufficiently widespread to
justify expensive controls.
Although it is debatable whether the hidden-city ticketing practice is legal or moral, it is very
important to understand why such seemingly unreasonable pricing policies are established and
persist; the cost to the airlines if hidden-city ticketing is completely admitted and used by passen-
gers whenever possible; and the long term impact it may have on both the airlines and travelers.
Understanding these issues is fundamental for decision makers involved, such as law makers, airline
executives as well as travelers.
For this purpose, we consider a multi-period flight network revenue management model. At each
period, the airline decides which itinerary (connection city) and price to offer for each origin-
destination (O-D) pair. We show that the hidden-city ticketing opportunity could exist in this
model when there is a significant difference in price elasticity of demand on related itineraries. We
also quantify the magnitude of the difference for this phenomenon to happen. Our finding, being
consistent with discussions in previous literatures, indicates that different level of competition on
different routes is the main factor that causes this phenomenon.
After understanding the cause of hidden-city opportunity, we study the cost of such ticketing
practice for the airlines if every passenger takes advantage of it whenever possible. First we show
that if the airlines do not react to this practice, their revenues could be severely hurt. Then we
establish a modified dynamic programming decision model for the airlines in which every passenger
takes advantage of all possible hidden-city ticketing opportunities. We show that in this model, the
optimal strategies of the airlines will no longer contain any hidden-city opportunities, but even the
optimal strategies cannot fully mitigate the negative revenue effect of such ticketing policies. We
show that an upper bound on the revenue losses can be established at half of its original optimal
revenue if the airline uses a hub-and-spoke network. We also show that this upper bound is actually
tight. Therefore, admitting hidden-city ticketing will be quite detrimental to airlines’ revenues and
thus from the airlines’ point of view, they do have a strong incentive to legally or contractually
forbid such ticketing practice.
On the other hand, although the airlines have a strong incentive to prohibit hidden-city ticketing,
passengers may find it attractive since it instantly saves their money. However, in a long run,
using the hidden-city ticketing may also hurt the passengers through the externalities that the
behavior causes. For a hub-and-spoke flight network, if hidden-city ticketing is fully admitted in
certain period and all passengers take advantage of such opportunities, the optimal fares to the
spoke cities will increase in that period. The rises in those fares not only immediately hurt those
who travel to the spoke cities, but also significantly reduce the profitability of airlines for serving
those spoke cities which in turn may result in a reduction or suspension in service towards those
cities. Therefore, our result suggests that in the long run, the use of hidden-city ticketing may also
decrease travelers’ benefits, creating a lose-lose situation.
Our work is based on a dynamic programming framework for network revenue management
problems. For the most part of this paper, we consider an exogenous competition environment.
We justify this model by also considering an endogenous competition model and argue that our
model is a special case when all other competitors use their equilibrium strategies. Therefore, this
approach, although not entirely perfect, is a good approximation of the pricing game occurred in
real practice.
There have been extensive researches on the flight revenue management problems in the past
twenty years, see Elmaghraby and Keskinocak (2003), Talluri and van Ryzin (2005), Bitran and
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Caldentey (2003) and references therein for a comprehensive review on this subject. Among pre-
vious researches, Gallego and van Ryzin (1997) laid a foundation for the flight network revenue
management. They formulated the flight network revenue management problem as a dynamic pro-
gram with remaining time and inventory as the state variables. This idea, taken by Adelman (2007),
Cooper (2002), Talluri and van Ryzin (1998, 1999), Topaloglu (2009) and others, is also adopted
in this paper.
In most earlier models, people assume that the demand over different itineraries are independent.
However, in practice, passengers choose from all itinerary/price combinations available to them
according to certain choice behaviors, and the demand functions over different itineraries might
be highly correlated. Talluri and van Ryzin (2004) proposed a choice-based revenue management
model and proposed the idea of offering a “product set” at each time period. Several properties
of optimal solutions were examined in their work. Later, Liu and van Ryzin (2008) extended this
model to network case and propose a decomposition method to solve it. Several more researches
worked on how to obtain a good solution for this model, see, e.g., Miranda Bront et al. (2009),
Kunnumkal and Topaloglu (2010), van Ryzin and Vulcano (2008), Zhang and Cooper (2005). In
some sense, our work can be viewed as a special case of the model in Liu and van Ryzin (2008),
where the “product set” contains a certain connecting flight for each O-D pair at a certain price,
and in addition to the traditional customer choice behavior, we add “hidden-city” ticketing as one
choice of the passenger. Although sharing the same framework, this distinguishing consideration
make our analysis more focused and many interesting results can be drawn.
Recently, many works on revenue management consider the competition between different play-
ers. This is very realistic given the fact that internet search engines have enabled people to compare
prices offered by different airlines very conveniently. Gallego and Hu (2000) proposed a game theo-
retic model for a single product revenue management problem. They considered a continuous-time
fluid model and showed that Nash equilibrium exists for both an open-loop and a close-loop case.
In another paper, Lin and Sibdari (2009) considered a discrete-time model and showed similar
results. In our paper, we first prove that in general, a pure-strategy Nash equilibrium exists for the
network revenue management game, and show that our exogenous competitive model considered
throughout the paper can be viewed as a special case when all other players are playing their
equilibrium strategies. This result justifies our model to draw our above findings and may be of
independent interest for further studies of competitive flight networks.
The remainder of the paper is organized as follows: In Section 2, we introduce our model for
the flight network revenue management and set up a dynamic programming formulation for the
model. In Section 3, we analyze the optimal pricing strategy and show that the hidden-city ticketing
opportunity can arise in this model. Then in Section 4, we construct the airlines’ optimization
problem when hidden-city ticketing is admitted and passengers take advantage of it whenever
possible. In Section 5, we study the effect on airlines’ revenues due to the hidden-city practice.
We show that the optimal revenue must decrease and it could decrease by a half but no more.
In Section 6, we study the impact on the fares when the airlines optimally react to hidden-city
ticketing and discuss the long term impact of such ticketing practice. In Section 7, we use a game
theoretic approach to justify our model. We present some numerical results to illustrate our study
in Section 8 and we conclude in Section 9.
2. Network Revenue Management Model
We consider an airline network consisting of Ncities (the set of city is denoted by N) and define
Eto be all the flight legs this airline serves and Oto be all the O-D pairs served using these legs.
The airline sells tickets on this network and the goal is to maximize its revenue over the whole
flight network.
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Assume there is a fixed inventory on each flight leg, which we use a vector cto denote. We
consider a discrete time model with Tperiods indexed backwards. That is, we start from time T
and all flights depart at time 0. At each time t, the airline has to make the following decisions for
each O-D pair it serves (when it is searched by a certain customer):
1. Which route to provide, i.e., which connection node (or a direct flight) to provide;
2. What price to offer for that itinerary;
In our study, for the first question, we allow the airline to provide this passenger any possible
1-stop itineraries or a direct flight. Specifically, we don’t consider itineraries with two or more
stops (this simplifying assumption is reasonable since an itinerary with 2 or more stops are rarely
attractive). For the second question, we assume that the airline offers a price pt
ij . In the following,
we consider an exogenous competition environment, that is, when pt
ij is offered, a customer with
this request will choose this product with probability λt
ij (pt
ij )2. This function can be quite general,
for example, it could be a choice function based on the competitor’s price (e.g., a multinomial logit
demand function). However, we assume that λt
ij is known in advance. Given the historical data and
future forecasts an airline typically has, this assumption may not be too unrealistic. In Section 7,
we justify this assumption through a study of this problem from a game theoretical point of view.
Now we formulate this problem into a dynamic programming problem. As mentioned above,
we consider a discrete time model with Tperiod. We use Vt(x) to denote the optimal expected
revenue when there are tperiods left and the inventory level is x. We assume the arrival rate for
the O-D pair i→jat time tis ηt
ij , and that tis defined finely enough such that there is at most
one arrival during each time period. At each period, the decision of the airline is a mapping from
all O-D pairs in Oto a pair (k∈ N , p), where a mapping from (i, j)∈ O to (kt
ij , pt
ij ) means that
the airline will serve the O-D pair (i, j) through a connection at kt
ij , and asking for a price pt
ij . We
use kt
ij = 0 to denote a direct flight.
First, we consider the case where every passenger’s purchase choice is only dependent by the
prices of the O-D pair he travels, and is independent of the prices for other O-D pairs. This is the
case when hidden-city ticketing is legally or contractually prohibited or the passengers voluntarily
don’t use such ticketing strategy (e.g., passengers are not aware of such strategy). In this case, a
dynamic programming model for the optimal decisions for the airlines can be formulated as follows:
Vt(x) = max
kt
ij ,pt
ij
X
(i,j)∈O
ηt
ij λt
ij (pt
ij )pt
ij +Vt−1(x−Akt
ij
ij )+ (1 −X
(i,j)∈O
ηt
ij λt
ij (pt
ij ))Vt−1(x)
=Vt−1(x) + max
kt
ij ,pt
ij
X
(i,j)∈O
ηt
ij λt
ij (pt
ij )pt
ij +Vt−1(x−Akt
ij
ij )−Vt−1(x)
(1)
with boundary conditions
V0(x) = 0 ∀x,
Vt(0) = 0 ∀t,
and
Vt(x) = −∞ if xcontains negative entry.
Here we use Ak
ij to denote a vector that has 1 on ikth and kjth element (recall that xis the
capacity for each leg, so Ak
ij is a leg consumption vector where each entry corresponds to a leg).
When k= 0, Ak
ij denotes the vector that has 1 only in ijth element. It is not hard to notice that
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
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the optimization over kt
ij and pt
ij are separated. Given the values of Vt−1(·)’s, for each iand j, we
choose kt
ij such that
Vt−1(x−Akt
ij
ij ) (2)
is maximized. Denote the optimal kt
ij by ˆ
kt
ij . Then we choose pt
ij such that each
λt
ij (pt
ij )pt
ij +Vt−1(x−Aˆ
kt
ij
ij )−Vt−1(x)(3)
is maximized.
However, even with such a structure, to solve the exact dynamic programming (1) is usually
computationally intractable, since the state space is huge. Several heuristics have been proposed to
approximately solve this problem, see, e.g., Adelman (2007), Zhang and Adelman (2009). In those
approaches, people approximate the value functions by assuming some parametric structures of
them, solve the dynamic programming using this structure and use the approximate value functions
to make the current pricing decision. We refer the readers to those references for the details of
those algorithms.
Now we make some assumptions on the demand functions. We will use these assumptions fre-
quently in the later discussions.
Assumption 1. For any (i, j)∈ O and any t,
1. λt
ij (p)is non-increasing in p.
2. λt
ij (p)is continuous and differentiable.
3. For any c > 0,λ(p)(p−c)is quasiconcave in pand there exists a unique maximizer for
λ(p)(p−c)on (c, ∞).
Note that these assumptions are quite mild. The first two assumptions are some basic regularity
conditions. The last one guarantees that there is always a unique optimal price for any specific
itinerary. These assumptions are also made in previous literatures, e.g., Gallego and Hu (2000).
Specifically, we show that the most commonly-used demand function based on multinomial logit
(MNL) choice model satisfies Assumption 1.
Proposition 1. If all λt
ij (p)comes from the multinomial logit (MNL) choice model, i.e., λt
ij (p) =
exp(−βt
ij p)
αt
ij +exp(−βt
ij p)for some αt
ij , βt
ij >0, then Assumption 1 is satisfied.
Proof. Consider λ(p) = exp(−βp)
α+exp(−βp). It is easy to see that it satisfies the first two requirements of
Assumption 1. For the last requirement, we have
(λ(p)(p−c))0=λ0(p)(p−c) + λ(p)
=−αβ exp(−βp)
(α+ exp(−βp))2(p−c) + exp(−βp)
α+ exp(−βp)
=exp(−βp)
(α+ exp(−βp))2(−αβ(p−c) + α+ exp(−βp)).
Note that −αβ(p−c) + α+ exp(−βp) is decreasing in pand it has a unique root on (c, ∞).
Therefore, λ(p)(p−c) is quasiconcave in pand there is a unique maximizer on (c, ∞).
In the following section, we show that the hidden-city opportunity can exist in the above model
if there is a significant difference in price elasticity of demand on two itineraries. This finding gives
a quantitative explanation of this phenomenon.
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3. Cause of Hidden-City Phenomenon
In this section, we study the cause of hidden-city opportunity. Assume the airline prices its products
according to model (1). As has been shown in Section 2, at each time period, the decision maker
chooses kt
ij and pt
ij according to (2) and (3). Fix ˆ
kt
ij , we denote Vt−1(x)−Vt−1(x−Aˆ
kt
ij
ij ) by ct
ij .
Then the optimal price for the O-D pair i→jat time tis determined by:
pt
ij = arg max λt
ij (pt
ij )(pt
ij −ct
ij ).
In the following, we omit the subscripts/superscripts for the simplicity of notation. Also we assume
that Assumption 1 holds. By the optimality condition, at optimal ˆp, we have:
ˆp−c=−λ(ˆp)
λ0(ˆp).
Divide ˆpon both side, we have
1−c
ˆp=−(E(ˆp))−1,
where E(p) is the price elasticity of demand of this O-D pair at price p. Suppose hidden-city
opportunity exists on the route i→j→k, i.e., for the O-D pair i→j, the airline offers a flight with
price pij (either direct or 1-stop), and for another O-D pair i→k, the airline offers a connecting
flight with connection at j, however, the prices satisfy pik < pij. In the following, we use subscript 1
to denote the corresponding parameters for the O-D pair i→jand 2 to denote the corresponding
parameters for the O-D pair i→k. By the above argument, ˆp1and ˆp2satisfy:
1−c1
ˆp1
=−(E1(ˆp1))−1; (4)
1−c2
ˆp2
=−(E2(ˆp2))−1,(5)
where c1=Vt−1(x)−Vt−1(x−Akt
ij
ij ) and c2=Vt−1(x)−Vt−1(x−Aj
ik). However, we have
Vt−1(x−Akt
ij
ij )≥Vt−1(x−A0
ij )≥Vt−1(x−Aj
ik),
where the first inequality is because the choice of kt
ij over a direct flight in the offer for the O-D
pair i→j(or equal if kt
ij = 0) and the second inequality is because the monotonicity of Vt(x) in x
(which is not hard to show and the reader is referred to Gallego and van Ryzin (1997) and Lemma
1 in Section 4 for the detail). Therefore, we have c1≤c2. Then combined with the assumption that
ˆp1>ˆp2, we obtain that E1(ˆp1)> E2( ˆp2) (and since the price elasticity are all negative, we have
|E1(ˆp1)|<|E2(ˆp2)|. In what follows, we ignore the signs in elasticities and use the word “larger” to
mean larger in absolute value). We have the following proposition.
Proposition 2. If j∗is a hidden-city such that a direct flight from ito j∗is more expensive than
a connecting flight from ito another city kwith connection at j∗. Then we must have the price
elasticity of demand on the O-D pair i→kis greater that the price elasticity of demand on the
O-D pair i→j∗.
The result states that the difference in price elasticity of demand is a necessary condition for the
hidden-city phenomenon. The converse is not true. The occurrence of a hidden-city opportunity
also depends on the magnitude of this difference and the value of the extra leg. It might also vary
with time.
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Figure 1 Illustration of Example 1.
Note that this result is consistent with the discussions in previous literatures. For example, in
GAO (2001), the authors write “hidden-city opportunities may arise when a greater amount of
competition exists for travel between spoke communities than on routes to and from hub com-
munities”. When competition is greater, as classical economic theory explains, the price elasticity
of demand will be larger (at optimal price), which will result in a relatively lower optimal price
for that product. Although simple, this economic theory gives a condition for the occurrence of
hidden-city opportunity and illustrates this somewhat strange phenomenon.
In the following sections, we study how the airlines will react if all the passengers take advantage
of this pricing structure and what impact the hidden-city ticketing practice would have on the
revenues of the airlines.
4. Optimal Reaction when Hidden-City Ticketing are Used
In this section, we assume all the passengers are network strategic, that is, they exploit all the
hidden-city opportunities in ticketing. More precisely, when purchasing airline tickets for O-D pair
i→j, the passenger searches for all O-D pairs i→lwith kil =j3. Then he compares the cheapest
price to the price offered to the O-D pair i→j. If the former one is cheaper, the passenger will
purchase this hidden-city ticket without letting the airline know his true intention. We first show
that when all passengers are network strategic, the airline’s revenue could be severely affected if
the airline keeps its original pricing. Then in the rest of this section, we develop models for airlines
to optimally react to hidden-city ticketing practices. We start from the following example:
Example 1. (If airlines don’t react) Consider a very simple two-city network as shown in Figure
1. The only two O-D pairs the airline serves are from A to B and A to C with connection at B, and
there is no demand from B to C. We assume there is only 1 period left before departure and there
is only 1 inventory on both A→Band B→Cleg. In the following, we first consider a series of
cases where the demand functions are not continuous (a piecewise constant), however, we can use
continuous functions to approximate them with arbitrary degree of accuracy. Therefore the results
will also hold for continuous demand functions. Let the demand functions be:
λAC (p) = c, if p≤100
0,if p > 100 (6)
and
λAB (p) = 10/x, if p≤x
0,if p > x, (7)
where c < 0.1, x100 are constants chosen beforehand. We assume ηAB =ηAC = 1 in this period.
Then we have the optimal pricing without consideration of hidden-city ticketing will be to offer
100 for O-D pair A→Cand xfor O-D pair A→Band the optimal expected revenue will be
10 + 100c. (8)
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However, if hidden-city ticketing is used by the passengers, then the airline’s revenue is
1000
x+ 100c. (9)
By choosing xsufficiently large and csufficiently small (and use a continuous function to approxi-
mate λAB and λAC ), the ratio between (9) and (8) can be made arbitrarily close to 0. Therefore,
if the airlines don’t react to the hidden-city ticketing, the revenue can be severely hurt.
In the rest of this section, we study how an airline should react to hidden-city ticketing. We
establish another dynamic programming decision model for the network strategic passengers. This
model will be the foundation for our analysis in the following sections.
We use similar notations as in (1) and use ¯
Vto denote the value function when hidden-city
ticketing is used. In this case, the airline’s optimization problem when facing inventory xat time
tbecomes:
¯
Vt(x) = ¯
Vt−1(x) + max
pt
ij ,kt
ij X
(i,j)∈O
ηt
ij λt
ij (¯pt
ij )¯pt
ij +¯
Vt−1(x−¯
At
ij )−¯
Vt−1(x),(10)
where
¯pt
ij = min{pt
ij ,min
l:kt
il=j
pt
il},(11)
¯
kt
ij =arg minl:kt
il=jpt
il,if minl:kt
il=jpt
il < pt
ij
0,if minl:kt
il=jpt
il ≥pt
ij
(12)
and
¯
At
ij =Akt
ij
ij ,if ¯
kt
ij = 0
Aj
i¯
kt
ij ,if ¯
kt
ij 6= 0 .(13)
The boundary conditions are the same as before. Here ¯
At
ij is the actual consumption inventory
when O-D pair i→jis requested and ¯pt
ij is the actual revenue obtained for that O-D pair. If ¯
kt
ij
is non-zero, then a hidden-city opportunity exists and the cheapest possible fare is obtained by
pretending to travel from ito ¯
kt
ij ; if ¯
kt
ij is zero, then no hidden-city opportunity exists in this O-D
pair. Note that (10) is no longer separable for each O-D pair because the price in one O-D pair
may affect the price/demand for other O-D pairs as well as the itinerary actually consumed. This
makes it even harder to find the optimal policy for this problem. Before discussing how to solve
(10), we show some properties about the solution in this case. We start from a lemma about the
monotonicity of the value function ¯
V.
Lemma 1. For any t, any vector x≥y(we define all vector inequality as componentwise), we have
¯
Vt(x)≥¯
Vt(y).
We omit the proof of this lemma and go to our theorem.
Theorem 1. There exists a solution to (10) that does not contain any hidden-city opportunity,
i.e., when all passengers take advantage of such opportunities, the best response of the airlines will
not contain any such opportunity.
Proof. We prove by contradiction. Assume the optimal pricing policy for (10) still contains hidden-
city opportunities. Then there exists city iand j(hidden-city) such that
pt
ij >min
l:kt
il=j
pt
il.(14)
Next we discuss four cases and show that we can always modify the current pricing policy to obtain
one with at least the same revenue and the hidden-city opportunity is eliminated.
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Figure 2 An illustration of the proof of Theorem 1.
1. kt
ij = 0. In this case, we just lower the price pt
ij to ¯pt
ij . Then for the right hand side of (10),
the only thing changed is ¯
At
ij and it is decreased. Therefore, by Lemma 1, the expected revenue
weakly increases.
2. kt
ij 6= 0 but kt
ij is not a hidden-city in the solution to (10). In this case, instead of offering
the route i→kt
ij →jfor the O-D pair ito j, we offer a direct flight from ito jwith price
˜pt
ij = minl:kt
il=jpt
il. Comparing this new pricing policy to the original one, the only term in (10) that
changed is for the O-D pair ito j. However, the value ¯
Vt−1(x−¯
At
ij )−¯
Vt−1(x) weakly increases
in the new pricing policy since the new policy only consumes one unit of leg ij while the old one
consumes one unit of leg ij and j¯
kt
ij . Meanwhile, the effective price for the O-D pair i→jremains
the same. Therefore, the new pricing policy achieves a weakly higher revenue than the old one.
3. kt
ij is also a hidden-city in the solution to (10) but the cheapest route for traveling from ito kt
ij
is not i→kt
ij →j(or precisely, ¯
kikt
ij 6=j). Then again, instead of offering the route i→kt
ij →j, we
offer a direct flight i→jwith price ˜pt
ij = minl:kt
il=jpt
il. Then by the same argument as the previous
case, we claim that this modification achieves weakly higher revenue.
4. kt
ij is also a hidden-city and the route i→kt
ij →jis the cheapest route to travel from ito kt
ij
(or more precisely, pt
ij = min{pikt
ij ,minl:kt
il=kt
ij pt
il}). We draw this case in Figure 2.
In this case, we offer a direct flight for O-D pair i→jwith price ˜pt
ij = minl:kt
il=jpt
il and a direct
flight for O-D pair i→kt
ij with price ˜pikt
ij =pt
ij . Now we consider the terms in (10):
•For (i, j), the effective price is the same as the original pricing policy ( ¯pt
ij ), but the consumption
of the inventory is lower (changed from i→j→¯
kt
ij to i→j), thus according to Lemma 1 there is
an weakly increase in revenue from that term;
•For (i, kt
ij ), the effective price is the same as the original policy (pt
ij ), but the consumption of
the inventory is lower (changed from i→kt
ij →jto i→kt
ij ), thus there is an weakly increase in
revenue from that term;
•For all O-D pairs, the price and inventory consumption are the same.
To summarize, if there is a hidden-city opportunity in the network, there is always a modification
to the prices such that the revenue can be weakly improved. Also note that the above modification
can be terminated in finite steps, since at each step, the modification only modifies some prices
in the network to a lower price also existed in the network. And when this modification termi-
nates, there must be no hidden-city opportunities in the network and the revenue has been weakly
improved. Thus the theorem is proved. .
Although Theorem 1 is intuitive, it is an important result for our further study. A direct result
will be a simplification of the dynamic program in (10)-(13). According to Theorem 1, the dynamic
program for ¯
Vt(x) can be equivalently written as:
¯
Vt(x) = maxpt
ij ,kt
ij
X
(i,j)∈O
ηt
ij λt
ij (pt
ij )pt
ij +¯
Vt−1(x−Akt
ij
ij )+ (1 −X
(i,j)∈O
ηt
ij λt
ij (pt
ij )) ¯
Vt−1(x)
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subject to pt
ij ≤pt
il,∀kt
il =j, ∀i, j.
(15)
Comparing to the dynamic program in (1), the only difference in (15) is that it has some additional
constraints. Obviously, (15) is very hard to solve and one would still need to resort to approximation
schemes to solve it. One possible method is to use the approximate technique proposed in Zhang and
Adelman (2009) to approximate the value functions. Then, in order to find out the optimal pt
ij and
kt
ij , one can search among all connecting possibilities (polynomial time), solve the corresponding
constrained optimization (using the approximate value function ¯
Vt−1(·) and compare the optimal
values. Since our main emphasis is to explore interesting properties from the solution, the detail
solving method for (15) is beyond our discussion. It will be one direction of future research to
obtain good way to solve this problem.
In the next section, we will use the model proposed above to derive important bounds on the
airlines’ revenues when hidden-city opportunities are taken by every passenger.
5. Bounds on Revenues
In this section, we compare the airlines’ optimal revenue when hidden-city ticketing is not used to
that if hidden-city ticketing is fully admitted and every passenger takes advantage of it whenever
possible, i.e., the optimal value of (1) to the optimal value of (10). Intuitively, the revenue will
decrease because when hidden-city ticketing is used, the passengers are essentially playing against
the airlines. We will first formalize this intuition. Then we show that the loss is bounded up to
half of its original revenue when the airline uses a hub-and-spoke network structure. Although this
bound is loose, it is so far the first explicit bound for the revenue loss due to this ticketing practice.
Moreover, we show that this bound is actually tight, i.e., there exists a case (a hub-and-spoke
network) such that the loss equals to half of the original revenue. These results provide a detailed
analysis on the loss to the airlines when hidden-city opportunities are fully taken by the passengers.
We first prove that the revenue always decreases when hidden-city ticketing is used. We prove
the following theorem:
Theorem 2. Let Vt(x)and ¯
Vt(x)be defined in (1) and (10) respectively. Then Vt(x)≥¯
Vt(x).
Proof. We prove by induction on t. The result is obvious for t= 0 for all x. Assume it is true for
all s=t−1 for all x, then for time t, we have by (1) and (15):
Vt(x) = max
kt
ij ,pt
ij
X
(i,j)∈O
ηt
ij λt
ij (pt
ij )pt
ij +Vt−1(x−Akt
ij
ij )+ (1 −X
(i,j)∈O
ηt
ij λt
ij (pt
ij ))Vt−1(x)
,
(16)
¯
Vt(x) = maxkt
ij ,pt
ij
X
(i,j)∈O
ηt
ij λt
ij (pt
ij )pt
ij +¯
Vt−1(x−Akt
ij
ij )+ (1 −X
(i,j)∈O
ηt
ij λt
ij (pt
ij )) ¯
Vt−1(x)
.
subject to pt
ij ≤pt
il,∀kt
il =j, ∀i, j.
(17)
By induction assumption, the right hand side of (17) is less than that if ¯
Vt−1(x) is replaced by
Vt−1(x). And it is also with more constraints. Thus the theorem holds.
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The above results also follow from an alternative formulation of this problem. The alternative
formulation, although not directly useful in computing the optimal strategy, will provide an insight
of the relationship between (1) and (10).
We start from writing Vt(x) in (1) as the optimal value of the following optimization problem:
max
ps
ij (y),qs
ijk (y),V s(y)X
(i,j)∈O
ηt
ij λt
ij (pt
ij (x)) pt
ij (x) + X
k
qt
ijk (x)Vt−1(x−Ak
ij )−Vt−1(x)!+Vt−1(x)
subject to Vs(y)≤X
(i,j)∈O
ηs
ij λs
ij (ps
ij (y)) ps
ij (y) + X
k
qs
ijk (y)Vs−1(y−Ak
ij )−Vs−1(y)!+Vs−1(y)
∀s≤t, y ≤x
X
k
qs
ijk (y) = 1,∀(i, j )∈E, s ≤t, y ≤x
ps
ij (y), qs
ijk (y)≥0.
(18)
Here qt
ijk (x) is the “indicator variable” of which route to provide. It is easy to see that at optimality,
qt
ijk (x) will be all either 0 or 1. Similarly, we could also write ¯
Vt(x) in (15) as the optimal value of
the following optimization problem:
max
ps
ij (y),qs
ijk (y),¯
Vs(y)X
(i,j)∈O
ηt
ij λt
ij (pt
ij (x)) pt
ij (x) + X
k
qt
ijk (x)¯
Vt−1(x−Ak
ij )−¯
Vt−1(x)!+¯
Vt−1(x)
subject to ¯
Vs(y)≤X
(i,j)∈O
ηs
ij λs
ij (ps
ij (y)) ps
ij (y) + X
k
qs
ijk (y)¯
Vs−1(y−Ak
ij )−¯
Vs−1(y)!+¯
Vs−1(y),
∀s≤t, y ≤x
(ps
ij (y)−ps
il(y)) ·qt
ilj (y)≤0,∀s≤t, y ≤x, and (i, j),(l , j),(i, l)∈E
X
k
qs
ijk (y) = 1,∀(i, j )∈E, s ≤t, y ≤x
ps
ij (y), qs
ijk (y)≥0.
(19)
In this case, we can also easily argue that at optimality, qt
ijk (x) is all either 0 or 1. Therefore, (19)
is equivalent to (15). However, the only difference between (18) and (19) is that (19) has more
constraints (ps
ij (y)−ps
il(y)) ·qt
ilj (y)≤0. Therefore, it is clear that Theorem 2 holds.
Theorem 2 states that by admitting hidden-city ticketing, the airlines will indeed incur losses.
In the following, we study the magnitude of loss that can be incurred. We first define some terms
which will be useful in our later discussions.
Definition 1. (Pricing Network): At any time t, we define the current pricing network P N to be
the set of (pt
ij , kt
ij ), for all (i, j)∈ O.
Definition 2. (Hidden-City Pair): For any pricing network P N , we call an ordered pair of cities
(i, j) to be a hidden-city pair if and only if
1. (i, j)∈ O
2. There exists another city lsuch that kt
il =jand pt
ij > pt
il
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We also call ja hidden-city in this pricing network P N . We denote all the Hidden-City Pair in a
network by H.
Definition 3. (Hidden-City Branch): We define a hidden-city branch in a pricing network P N
to be a Hidden-City Pair (i, j) along with the set of all cities ls,s= 1, ..., n such that kt
ils=jand
pt
ij > pt
ils. We denote all the hidden-city branches in a network by B. And we call all lsin this case
the final destinations of the hidden-city pair (i, j).
Now we make the following assumption of our pricing network at any time.
Assumption 2. (Destination of Hidden-City is not a Hub): The final destinations of hidden-city
pairs are not used as connection at any time. To be more precise, assume the pricing network P N
is the pricing network solved from (1) at some time. If lsis a final destination of a hidden-city pair
of P N , then lsis not used as a connection point in P N.
The assumption that the destinations of hidden-city pairs are not connection points seems quite
strong, however it is quite reasonable in practice. In practice, the hidden-city is usually a hub
and the destinations of hidden-city pairs are usually non-hub cities. Therefore, the destinations of
hidden-city pairs will usually not be connection points since in such network, nearly all connections
take place in the hub (in fact, there may not be outbound flights from the destinations of hidden-
city pairs other than towards hub cities). Next we show that the most-common used hub-and-spoke
network satisfies Assumption 2.
Definition 4. (Hub-and-Spoke Network) We call a flight network a hub-and-spoke network if the
cities in this network are divided into two classes: “hubs” and ”non-hubs” (spokes). The flights
between spokes and hubs are direct and flights between spoke cities are connected at one of the
hubs.
As many studies suggest, most major airlines in the United States (as well as many small airlines)
adopt a hub-and-spoke network strategy nowadays, because of its efficiency and capability to serve
more O-D pairs. We have:
Proposition 3. Any hub-and-spoke network satisfies Assumption 2.
Therefore, Assumption 2 is quite mild in practice. With this assumption, we prove the following
theorem showing that the optimal revenue when hidden-city ticketing is used is within a factor of
2 to the optimal revenue when it is not used.
Theorem 3. Let Assumption 1 and 2 hold. Then ¯
Vt(x)≥1
2Vt(x), for any tand x.
Proof. Our proof is by induction. At each time t, we construct a modification to the optimal
prices of (1) such that it satisfies the constraints in (15) and the revenue is at least a half. We refer
to Appendix A for the detail proof.
Theorem 3 shows that by allowing hidden-city ticketing, the revenue can be reduced by at most
a half (under Assumption 1 and 2). The following counterexample shows that this bound is tight,
that is, in certain cases, allowing hidden-city ticket will indeed reduce the optimal revenue by half.
Example 2. (The revenue can be reduced by half) We consider the same example as in Example
1 with c= 0.1. Then we have the optimal pricing without hidden-city ticketing will be pricing 100
for O-D pair A→Cand xfor O-D pair A→Band the optimal expected revenue will be 20.
However, if hidden-city ticketing is permitted, it can be shown that the optimal pricing strategy
will be to price 100 on both O-D pairs and the expected revenue is 10 + 1000/x. By choosing x
sufficiently large (And use a continuous function to approximate λAB and λAC), the ratio can be
made arbitrarily close to 1/2.
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In this section, we showed that when hidden-city ticketing is admitted and used by all passengers,
the revenues of the airlines always decrease, comparing to the optimal revenue when hidden-city
ticketing is not used. We also showed that the decrement can be as much as half of the original
optimal revenue, but it can not be more if the airline takes a hub-and-spoke network. In the next
section, we discuss another important aspects, i.e., how the prices will change when airlines react
to hidden-city ticketing. We will still use model (15) to conduct our analysis.
6. Resulting Price Changes and the Long Term Effects
In the previous sections, we studied the effect on the airlines’ revenue when hidden-city ticketing is
used by the passengers. We showed that the airlines always suffer a loss even if they optimally react
(by changing the prices) to such a ticketing practice. Therefore, the use of hidden-city opportunities,
if not controlled, does hurt the airlines. Thus the airlines do have the incentive to forbid such
ticketing tricks. On the other hand, given such opportunities, the travelers have the incentive to
take advantage of them since it instantly saves their money. But this may not be true for the
travelers in the long run. American Airlines hints in one of its letter to their customers4that “If
American Airlines continues to lose revenue as a result of hidden-city transactions, the fares we
charge must inevitably rise”. Similar statements are also made by other airlines as well as in some
literatures studying this phenomenon. In this section, we will study how the optimal prices change
when hidden-city ticketing is taken into account in pricing. Specifically, will the prices increase?
Answering this question will be instrumental to study the long term impact of this practice to the
passengers and to help decision makers to make wise decisions on this controversial issue.
We start by considering a single period case. i.e., we consider model (1) but allow hidden-city
ticketing in the first time period (time period t). We study how the prices change in that first
period in the optimal reaction of the airlines.
We use pt
ij to denote the optimal prices for (1) and ˜pt
ij to denote the optimal prices when
hidden-city ticketing is used in the first time period. We prove the following theorem.
Theorem 4. Let Assumption 1 and 2 hold. Consider any hidden-city pair (i, j)defined in Defini-
tion 2, and any destination lof this hidden-city pair. Then
1. ˜pt
il ≥pt
il
2. ˜pt
ij ≤pt
ij
Remark 1. Theorem 4 states that the fares between the hidden-city pair will decrease and the
fares from the origin to the destinations of the hidden-city pair will increase if hidden-city ticketing
is to be used by the passengers. The latter part of this finding, although only considers a one-period
model, is consistent with the empirical discussions in the previous literatures. For example, in GAO
(2001), the authors conclude “... if legislation required airlines to permit hidden-city ticketing,
airfares in certain markets (i.e., for travel between certain spoke communities connecting over a
hub) could increase immediately... ”. And as a result, the demand to the spoke cities will drop and
“airlines would consider reducing or eliminating service on these markets”. When this happens,
the benefits gained in the second half of Theorem 4 will disappear and all prices will be above the
original prices (or the service is halted). Clearly, this whole sequence of effect is both adverse to
the airlines as well as to the passengers.
Before we prove Theorem 4, we prove the following technical lemma.
Lemma 2. Let Assumption 1 holds. Then for any c2≥c1,p1=arg maxpλ(p)(p−c1)and p2=
arg maxpλ(p)(p−c2), we have p2≥p1.
Proof. By the optimality condition, we have
λ0(p1)(p1−c1) + λ(p1) = 0.
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Therefore, since λ0(p)≤0, we have
λ0(p1)(p1−c2) + λ(p1)≥0.
Then by Assumption 1 that λ(p)(p−c2) is quasiconcave and there exists a unique optimal solution,
we must have p2≥p1.
Proof of Theorem 4. We first consider the first part. We prove by contradiction. Assume ˜pt
il < pt
il.
We will show contradiction in two cases.
Case 1: ˜
kt
il still equals to j. In this case, we show that by raising ˜pt
il to pt
il while maintaining
all other prices, the revenue will increase. First, we argue that doing this won’t create any new
hidden-city opportunity for the pricing network (˜p, ˜
k). This is true because by Theorem 1, we know
that the pricing network (˜p, ˜
k) doesn’t contain any hidden-city opportunity. And by Assumption 2,
lcan not be used as a connection point. Therefore, increase the fare to lwon’t create any hidden-
city opportunity in this network. Then we show that the revenue would increase. This is because
pt
il =arg maxpλt
il(p)(p+Vt−1(x−Aj
il)−Vt−1(x)). Therefore, choosing pt
il will increase the revenue
over ˜pt
il.
Case 2: j0=˜
kt
il no longer equals to j. In this case, we want to claim that ˜pt
il must be no less than
pt
il. By definition, we have
pt
il =arg max λt
ij (p)p+Vt−1(x−Aj
il)−Vt−1(x).
Define
¯pt
il =arg max λt
il(p)p+Vt−1(x−Aj0
il )−Vt−1(x).(20)
Since kt
il =j, we must have Vt−1(x−Aj
il)≥Vt−1(x−Aj0
il ). Therefore, by Lemma 2, we have ¯pt
il ≥pt
il.
Now we claim that ˜pt
il ≥¯pt
il. If not, we increase ˜pt
il to ¯pt
il. First, by Assumption 2, this will not
create any new hidden-city opportunity for the pricing network (˜p, ˜
k) since lis not used as any
connection point. And also, since ¯pt
il is optimal to (20), the revenue is increased. Therefore, we
reach a contradiction in this case too. Therefore, the first part of Theorem 4 holds.
Next we consider the second part. Suppose ˜pt
ij > pt
ij . Then we modify the price network by
lowering the price ˜pt
ij . First we claim that this will not create any new hidden-city opportunities
because as stated in Assumption 2, jcan not the destination of any hidden-city pair. Also kt
ij must
remain the same, because Vt−1(x−Akt
ij
ij ) is the maximum one among all possible connections from
ito j. Using the fact that pt
ij is the optimal solution to
ηt
ij λt
ij (pt
ij )pt
ij +Vt−1(x−Akt
ij
ij )−Vt−1(x),
we know that the revenue would increase if we lower ˜pt
ij in this case.
Now we have shown the way prices will change in the single-period model. It is tempting to
extend this result to the full multi-period model, i.e., comparing pt
ij in (1) and (10). Intuitively,
prices to the hidden-city destination should still increase and the prices between the hidden-city
pair should decrease. However, as the following example shows, this is not always the case, even
the pair of city is a hidden-city pair for all the periods forward.
Example 3. (Theorem 4 may not hold for multi-period model) Consider a case with 2 periods
and 3 cities as shown in Figure 3 (the graph is the same as in Example 1).
At the beginning of period 2, the inventory level is x= (1,1) where the first entry denotes the
inventory on the the flight leg A→Band the second entry denotes the inventory on the flight leg
B→C. There are only demands for the itinerary A→Band A→Cin both period.
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
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Figure 3 Illustration of Example 2
Now we consider the following parameters: in time period t= 1 (the last time period), η1
AB =
η1
AC = 0.2, λ1
AB (p) = 1 −p/200 and λ1
AC (p) = 1 −p/100 (The demand functions are only defined on
the range where it is positive, otherwise it is zero. It is easy to verify that it satisfies Assumption
1 with a slight modification to guarantee differentiability, which doesn’t affect our analysis). With
these parameters, for the case when hidden-city ticketing is not used, we have the optimal price
p1
AB = 100 and p1
AC = 50, and we have the value function V1((1,1))= 0.2·0.5·100+ 0.2·0.5·50 = 15.
In the case when the hidden-city opportunity is taken by all passengers (solve (15)), the optimal
price ¯p1
AB and ¯p1
AC can be computed by solving:
maximize¯p1
AB ,¯p1
AC 0.2(1 −¯p1
AB
200 )¯p1
AB + 0.2(1 −¯p1
AC
100 )¯p1
AC
subject to ¯p1
AB ≤¯p1
AC
(21)
and this yields the optimal price ¯p1
AB = ¯p1
AC = 200/3 and V2((1,1))= 40/3. Note that these prices
satisfy the result in Theorem 4 since it is the last period. Now we consider the following parameters
for the second to last period (t= 2). We define η2
AB = 0.01, η2
AC = 0.5, λ2
AB (p)=1−p/200 and
λ2
AC (p) = 1−p/100 (the demand function is the same as before, only the arrival rates have changed).
Now for the case when hidden-city ticketing is not considered, according to (1), we solve the
following optimization:
maximizep2
AB ,p2
AC 0.01(1 −p2
AB
200 )(p2
AB −15) + 0.5(1 −p2
AC
100 )(p2
AC −15) (22)
and get the optimal prices p2
AB = 107.5 and p2
AC = 57.5 and the optimal expected value is
V2((1,1))= 24.4591. For the case when the hidden-city opportunity is taken by the passengers, we
solve the following optimization problem
maximize¯p2
AB ,¯p2
AC 0.01(1 −¯p2
AB
200 )(¯p2
AB −40
3) + 0.5(1 −¯p2
AC
100 )(¯p2
AC −40
3)
subject to ¯p2
AB ≤¯p2
AC
(23)
and get ¯p2
AB = ¯p2
AC = 57.1617 and the optimal expected value is ¯
V2((1,1))= 23.0269 (this also
verifies the relationship between Vand ¯
Vas stated in Theorem 2 and 3). Note that the optimal
prices on both legs decrease in this case. This is because that the optimal value of one period also
changes with the value functions of future periods. And when hidden-city opportunity is taken
by the passengers, the value functions of future periods may change in a way that the resulting
changes in price is in a different direction as Theorem 4 predicts.
However, as one can see, the above example is not really well-behaved because of the big difference
in ηwe designed to make our example work. In practice, one may expect that when hidden-city
opportunity is taken by the passengers, the optimal fares towards the destination of hidden-city pair
will increase, and the consequences in Remark 1 will take place. In the long run, it is detrimental
to both the airlines and the consumers, creating a lose-lose situation.
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7. A Game-Theoretic Approach
In the previous sections, we have considered an exogenous competition model in which we assume
there is a known demand function λt
ij (pt
ij ) when an airline posts a price pt
ij for the O-D pair
i→j. In reality, there are endogenous competitions between airlines and the customer’s choice is
a function of all the prices posted by different airlines. This is especially true since the emergence
of internet travel search engines which enable consumers to compare prices of difference airlines
conveniently. In this section, we consider a game theory model where the demand function is jointly
determined by all the prices posted by different airlines. We show that there exists a pure-strategy
Nash equilibrium for this game under a strong information structure. We then relate this model
to our previous discussions and show that our previous discussions can be viewed as a simplified
case when all the other players play their equilibrium strategies. In this section, we assume that no
hidden-city ticketing is admitted, therefore, the customer’s choice for one O-D pair only depends
on all prices offered by different airlines on this O-D pair.
We start from defining the information structure in the game. A similar definition can be found
in Gallego and Hu (2000) and Lin and Sibdari (2009) for the single product case.
Definition 5. (Strong Information Structure) All airlines have perfect knowledge about each
other’s inventory levels of each flight leg at any time period.
This strong information structure, although appears strong, is attainable in the current industry
practice. Nowadays, many travel search websites offer features of previewing seat availability maps
for each flight leg so that the real-time inventory can be deemed as public information. Under this
information structure, at the beginning of each time period, each airline first observes the inventory
levels of all other competitors, and makes the decision of its offering price.
Before discussing the Nash equilibrium, we need to define more notations. In the following, we
use letter rin parenthesis to denote the rth player, and “−r” to denote all the other players.
Specifically, we denote the price the rth airline chooses at time tfor O-D pair i→jby pt(r)
ij ,
the connection city this airline chooses by kt(r)
ij . Also in this case, the demand function will be
dependent on all other airlines’ prices on this O-D pair, which we denote by λt(r)
ij (pt(r)
ij , pt(−r)
ij ). Then
the value function V(r)
t(x(r), x(−r)) for player rat time twith inventory x(r)and other airlines’
inventory x(−r)has the following dynamic programming representation (other notations remain
the same as in (1)):
V(r)
t(x(r), x(−r)) = max
pt(r)
ij ,kt(r)
ij
X
(i,j)∈O
ηt
ij λt(r)
ij (pt(r)
ij , pt(−r)
ij )pt(r)
ij +V(r)
t−1(x(r)−Akt(r)
ij
ij , x(−r))
+X
q6=rX
(i,j)∈O
ηt
ij λt(q)
ij (pt(r)
ij , pt(−r)
ij )V(r)
t−1(x(r), x(q)−Akt(q)
ij
ij , x(−q,−r))
+(1 −X
(i,j)∈O X
q
ηt
ij λt(q)
ij (pt(r)
ij , pt(−r)
ij ))V(r)
t−1(x(r), x(−r))
=V(r)
t−1(x(r), x(−r))+
max
pt(r)
ij ,kt(r)
ij
X
(i,j)∈O
ηt
ij λt(r)
ij (pt(r)
ij , pt(−r)
ij )pt(r)
ij +V(r)
t−1(x(r)−Akt(r)
ij
ij , x(−r))−V(r)
t−1(x(r), x(−r))
+X
q6=rX
(i,j)∈O
ηt
ij λt(q)
ij (pt(r)
ij , pt(−r)
ij )V(r)
t−1(x(r), x(q)−Akt(q)
ij
ij , x(−q,−r))−V(r)
t−1(x(r), x(−r))
.
(24)
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
Operations Research 00(0), pp. 000–000, c
0000 INFORMS 17
At each time period, given all other firms’ choices (and observe the inventory of all airlines), the
airline picks pt(r)
ij and kt(r)
ij to maximize V(r)
t(x(r), x(−r)).
To prove that there exists a pure-strategy Nash equilibrium for this multi-period game, we
use an induction method (which is also used in Lin and Sibdari (2009) to prove the existence of
a pure-strategy Nash equilibrium in the single product pricing game). First, at time 0, all the
value functions are equal to 0, thus any pure policy will be a pure-strategy Nash equilibrium, and
V(r)
0(x) = 0 are the expected revenues under the equilibrium policy, for all rand x. Now consider
any time period t, and denote the current inventory vector by x. By induction assumption, a pure-
strategy Nash equilibrium exists for all periods 1,2, ..., t −1, which means that for any inventory
level yand s≤t−1, there exists an equilibrium policy (p∗, k∗) (not necessarily unique, we pick
arbitrary one in the case when there are multiple equilibriums). We denote the expected revenue
obtained by equilibrium policy at time s≤t−1 and inventory level yby ˆ
Vs(y) = ( ˆ
V(r)
s(y))m
r=1. Note
then each airline’s optimization problem at period tcan be represented as (24) with the value
functions on the right hand side replaced by the expected revenue ˆ
V’s. Then it is easy to see that
for every O-D pair i→j, the optimal connection decision is to offer the route kt(r)
ij with the highest
ˆ
V(r)
t−1(x(r)−Akt(r)
ij
ij , x(−r)); and to offer pt(r)
ij ’s to maximize
Φr(pt(r)) = X
(i,j)∈O
ηt
ij λt(r)
ij (pt(r)
ij , pt(−r)
ij )(pt(r)
ij −C1
ij )−X
q6=rX
(i,j)∈O
C2
ijq λt(q)
ij (pt(r)
ij , pt(−r)
ij ),(25)
where C1
ij =ˆ
V(r)
t−1(x(r), x(−r))−ˆ
V(r)
t−1(x(r)−Akt(r)
ij
ij , x(−r))>0, C2
ijq =ηt
ij (ˆ
V(r)
t−1(x(r), x(−r))−
ˆ
V(r)
t−1(x(r), x(q)−Akt(q)
ij
ij , x(−q,−r))) >0 are independent of the decision pt(r)
ij .
In order to prove the existence of Nash equilibrium, we make the following assumptions to the
demand functions:
Assumption 3. For each r,tand (i, j)∈ O,Φr
ij (pt(r)
ij ) = ηt
ij λt(r)
ij (pt(r)
ij , pt(−r)
ij )(pt(r)
ij −C1
ij )−
Pq6=rC2
ijq λt(q)
ij (pt(r)
ij , pt(−r)
ij )is quasi-concave in pt(r)
ij . And the maximizer of Φr
ij (pt(r)
ij )is uniformly
bounded for all pt(−r)
ij .
Proposition 4. If the customer’s choice satisfies the multinomial logit (MNL) model, that is, the
choice function πi(p) = exp(αi−βpi)
1+Pjexp (αj−βpj). Then it satisfies Assumption 3.
Proof. The proof is similar to the proof of Lemma 3.1 in Lin and Sibdari (2009). We take derivative
to Φr
ij (p(r)). We have (we omit the subscript i, j in pfor convenience)
∂Φr
ij (p(r))
∂p(r)=−βexp (αr−βp(r))(1 + Pq6=rexp (αq−βp(q)))
(1 + Pqexp (αq−βp(q)))2(p(r)−C1
ij ) + exp (ar−βp(r))
(1 + Pqexp (αq−βp(q)))
−exp (ar−βp(r))Pq6=rβexp (aq−βp(q))C2
ijq
(1 + Pqexp (αq−βp(q)))2
=exp (αr−βp(r))
(1 + Pqexp (αq−βp(q)))2·"−β(1 + X
q6=r
exp (αq−βp(q)))(p(r)−C1
ij )+
(1 + X
q
exp (αq−βp(q))) −βX
q6=r
exp (aq−βp(q))C2
ijq #.
(26)
Then note that the expression in the bracket is monotonically decreasing in p(r). Therefore, the
original function could have at most one point at which the derivative is zero, therefore, Φr
ij (p(r))
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
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must be quasi-concave in p(r). And for the upper bound of the maximizer, it is not hard to see
from (26) that any stationary point of Φr
ij (p(r)) (necessary condition for a maximizer) must be less
than C1
ij + (1 + Pqexp (αq))/β. Therefore, the theorem holds.
We will use the following theorem from Vives (1999) and Debreu (1952) to prove the existence
of a pure-strategy Nash equilibrium in our model.
Theorem 5. Consider a game. If the strategy sets are nonempty convex and compact subsets of
Euclidean space and the payoff to player iis continuous in the actions for all firms and quasiconcave
in its own action, then there is a pure-strategy Nash equilibrium.
With Theorem 5 and the discussions above, we have
Theorem 6. If Assumption 3 holds, then there exists at least one pure-strategy Nash equilibrium
in each period in this multi-period game defined in (24).
Corollary 1. Under a multinomial logit (MNL) choice model for the consumers, there exists at
least one pure-strategy Nash equilibrium for the airlines in this multi-period game defined in (24).
In the following discussion, we assume that Assumption 3 holds, therefore, there exists a pure-
strategy Nash equilibrium in this game. We argue that our original model of exogenous competition
can be viewed as a simplified case when every player plays its equilibrium strategy.
The relation is established by comparing the dynamic program in (1) and that in (24). To equate
those two, first we have to assume that all other players play their equilibrium prices, which removes
the dependence of λ(·) on the prices of other airlines. Then we need to make a modification to the
value functions by redefining Vt(x) by:
V(r)
t(x(r), x(−r)),
Vt−1(x−Akt
ij
ij ) by:
V(r)
t−1(x(r)−Akt(r)
ij
ij , x(−r)),
and Vt−1(x) by
V(r)
t−1(x(r), x(−r))+ Pq6=rP(i,j)∈O ηt
ij λt(q)
ij (pt(r)
ij , pt(−r)
ij )V(r)
t−1(x(r), x(q)−Akt(q)
ij
ij , x(−q,−r))−V(r)
t−1(x(r), x(−r))
1−P(i,j)∈O ηt
ij λt(r)
ij (pt(r)
ij , pt(−r)
ij ).
In the above transformations, the first two are natural. The last one, although looks complicated, is
just a modification to the original function with consideration of the changes in other competitors’
inventory, under the expectation that all other competitors use the equilibrium pricing strategy. The
equilibrium analysis justifies our exogenous optimization model, a modification of the competitive
market model, used to establish all the results in the preceding sections.
8. Numerical Results
In this section, we perform numerical experiments to illustrate our previous studies. We consider a
small hub-and-spoke network with four cities and three flight legs as shown in Figure 4. We assume
there are only demands from Ato B,Ato Cand Ato D. And flights from Ato Cand Dare
connected at B. We assume the capacity on the flight legs A→B,B→C,B→Dare 70, 20 and
25 respectively and there are totally 400 time periods before departure.
In our numerical study, we assume that the arrival rates for each O-D pair is 0.25 and doesn’t
change with time, that is ηt
AB =ηt
AC =ηt
AD = 0.25. And we define the demand function λusing the
mulitnomial logit (MNL) model as follows:
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
Operations Research 00(0), pp. 000–000, c
0000 INFORMS 19
Figure 4 A Small Flight Network.
λt
AB (pAB ) = exp (−β1pAB )
α1+ exp (−β1pAB )(27)
λt
AC (pAC ) = exp (−β2pAC )
α2+ exp (−β2pAC )(28)
λt
AD(pAD ) = (exp −β3pAD )
α3+ exp (−β3pAD )(29)
In our experiments, we define α2= 1, α3= 1.5, β1= 0.01, β2= 0.01 and β3= 0.008. Then we
choose different values of α1to see the effect of different market environments on the hidden-city
opportunities on this network. Note that a smaller α1means that the traveler has a higher chance
to choose this flight when it is being requested, and the market competition on the O-D pair A→B
is relatively low. The opposite is true when α1is large. In our experiments, we choose α1to be
0.05, 0.1, 0.25, 0.35, 0.5 and 1. The computation results are shown in Table 1.
In Table 1, V∗is the optimal revenue when hidden-city ticketing is not used, i.e., the optimal
value of the dynamic program (1). In the third and fourth row, ˜
Vis the revenue obtained by
using the policy solved by (1), however, all passengers take advantage of hidden-city opportunities
whenever possible. ¯
Vis the revenue obtained by the optimal response of the airlines when hidden-
city ticketing is admitted and used by all passengers, i.e., the optimal value of the dynamic program
(15). In the next six rows, p∗shows the optimal prices for the three O-D pairs at the first time
period (t= 400) when hidden-city ticketing is not used; and ¯pshows the corresponding optimal
prices when passengers use hidden-city opportunities. The letter “H” in the parenthesis identifies
hidden-city opportunities on the price network. And since the size of this problem is relatively small,
the numbers in Table 1 are all solved from the exact dynamic program, not from an approximation.
In Table 1, we can see that when competition between origin Aand hub Bis relatively low
comparing to that between origin Aand spokes Cand D(i.e., when α1is small), there could be
hidden-city opportunities existing in this network. And the revenue of the airline will be reduced
when passengers take advantage of such opportunities. When airlines optimally react to this tick-
eting method, the loss can be reduced but the revenue is still less than that when hidden-city
ticketing is not used. These results are consistent with Theorem 2 and 3, however, it hints that
in practice, the decrease of revenue may be much smaller than the worst case scenario indicated
in Theorem 3. With the competition on the O-D pair A→Bincreases (i.e., when α1increases),
the hidden-city opportunity becomes less significant, and the revenue difference becomes smaller.
We plot this change in Figure 5. When certain level of competition is reached on leg A→B, the
hidden-city opportunity will partially (α1= 0.35) or completely (α1= 0.5) disappear. This is also
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
20 Operations Research 00(0), pp. 000–000, c
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α1= 0.05 0.1 0.25 0.35 0.5 1
V∗19003 15772 12058 10868 9718 7908
˜
V17849(−6.07%) 15116(−4.16%) 11905(−1.27%) 10818(−0.46%) 9718(−0.00%) 7908(−0.00%)
¯
V18626(−1.98%) 15443(−2.09%) 12001(−0.47%) 10845(−0.21%) 9718(−0.00%) 7908(−0.00%)
p∗
AB 302.83 254.23 193.28 172.18 152.92 128.32
p∗
AC 215.07(H) 190.14(H) 165.58(H) 161.07(H) 156.75 154.35
p∗
AD 240.87(H) 214.51(H) 182.11(H) 173.53 165.72 161.08
¯pAB 277.11 232.05 183.27 167.54 152.92 128.32
¯pAC 277.11 232.05 183.27 167.54 156.75 154.35
¯pAD 277.11 232.05 183.27 173.53 165.72 161.08
Table 1 Revenues and optimal prices with/without hidden-city ticketing
Figure 5 Relationship between α1and the revenue loss
in accordance with our discussion in Section 3 in which we claim that the price elasticity of demand
is the main reason to cause hidden-city opportunity.
Lastly, we look at the optimal pricing reactions of the airlines when hidden-city ticketing is
admitted and all passengers take advantage of it. First, we see from Table 1 that the optimal
reactions of the airlines will not contain any hidden-city opportunities, which is consistent with
Theorem 1. Also, as stated in Theorem 4, the fares towards the destination of hidden-city pairs
would increase and the fares towards the hub would decrease. This is also verified in our numerical
results. Besides, our results provide an intuition of how to modify the optimal prices of the original
problems to obtain the optimal prices when hidden-city ticketing is used.
9. Conclusion
In this paper, we build, for the first time, a mathematical model to study the hidden-city ticketing
phenomenon in airline ticket pricing. We consider a dynamic flight network revenue management
model under which the hidden-city opportunities could arise when there is a large difference in the
price elasticity in different O-D pairs. We show that if passengers take advantage of the hidden-city
opportunity whenever possible, the airlines had better to react in price, otherwise their revenues
could be severely reduced. Then we study the optimal reaction of the airlines towards this ticketing
practice. We first show that the optimal reaction of the airlines will no longer contain any hidden-
city opportunity. However, even with the optimal reaction, the airline’s revenue will still be reduced,
and the reduction could be as much as half of the original optimal revenue but not more. We also
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
Operations Research 00(0), pp. 000–000, c
0000 INFORMS 21
proved that in the optimal reaction, the fares to the spoke cities would increase and therefore,
practicing hidden-city ticketing would also hurt the passengers in the long run.
Our results suggest that hidden-city ticketing needs to be dealt with, since it could be detrimental
to both parties. Ideally, passengers could be aware of the long term effects of such practice, and
therefore voluntarily give up such opportunities. However, for a single passenger, his interest may
be quite limited to a cheaper one-time travel and thus will not consider the long term consequences
at all. Another possibility is to use contracts/warnings to bind the behaviors of the passengers,
which is mainly carried out in current practice. But the enforcement of those contracts/warnings is
doubtful. If all those measures fail to prevent passengers from taking such opportunities, the airlines
may need to change its current pricing structures. In practice, a mixed strategy of contractually
prohibiting and optimal responding may be the most appropriate, and the mixture depends on
each specific route and how many passengers will choose to conduct hidden-city transactions. As
we have shown in our discussions, these are all very important issues to the airlines as well as to
the whole industry. We hope that this paper could provide a starting point for further studies and
complete solution of this problem.
Appendix A: Proof of Theorem 3
We prove by induction on t. It is obvious that for t= 0, the claim holds. We assume that the claim
holds for t−1 for all feasible xand now we consider ¯
Vt(x).
In the following, we will construct a pricing network at time tfor (15) and show that this pricing
scheme combined with the optimal pricing in the later periods will achieve ¯
Vt(x)≥1
2Vt(x).
We start from the optimal solution for Vt(x). We denote the optimal solution for (1) to be pt
ij
and kt
ij . Now we construct a feasible solution (¯pt
ij ,¯
kt
ij ) for (15), and we want to show that
¯
Vt(x)≥X
(i,j)∈O
ηt
ij λt
ij (¯pt
ij )¯pt
ij +¯
Vt−1(x−A¯
kt
ij
ij )+ (1 −X
(i,j)∈O
ηt
ij λt
ij (¯pt
ij )) ¯
Vt−1(x)
≥1
2
X
(i,j)∈O
ηt
ij λt
ij (pt
ij )pt
ij +Vt−1(x−Akt
ij
ij )+ (1 −X
(i,j)∈O
ηt
ij λt
ij (pt
ij ))Vt−1(x)
=1
2Vt(x).(30)
First, we maintain all the connection nodes, i.e., we let ¯
kt
ij =kt
ij for all (i, j)∈ O. Now we need
to make some adjustments to the prices pt
ij such that the constraints in (15) is met. We do this
by redefining prices in each hidden-city branch in the pricing network (pt
ij , kt
ij ). For those O-D
pairs that is not covered in any of the hidden-city branches, we just keep the original price pt
ij .
And by the implication of Assumption 2, no two hidden-city branches contain the same O-D
pair. This allows us to define prices separately in different branches. Consider a single hidden-city
branch ((i, j),{l1, l2, ...ln}) in this network. By the definition of hidden-city branch, we know that
kt
ils=jand pt
ij > pt
ilsfor all s= 1, ..., n. In the following, without loss of generality, we assume that
pt
ij > pt
il1≥... ≥pt
iln. We consider two cases:
Case 1:
ηt
ij λt
ij (pt
ij )pt
ij ≥
n
X
s=1
ηt
ilsλt
ils(pt
ils)pt
ils.(31)
This is the case when the revenue generated from the O-D pair (i, j) dominates the revenue gen-
erated from all other O-D pairs in this hidden-city branch. In this case, we define ¯pt
ils= ¯pt
ij =pt
ij
for all s= 1, ..., n. We have
ηt
ij λt
ij (¯pt
ij )¯pt
ij +¯
Vt−1(x−Akt
ij
ij )−¯
Vt−1(x)+
n
X
s=1
ηt
ilsλt
ils(¯pt
ils)¯pt
ils+¯
Vt−1(x−Aj
ils)−¯
Vt−1(x)
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
22 Operations Research 00(0), pp. 000–000, c
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≥ηt
ij λt
ij (pt
ij )pt
ij +ηt
ij λt
ij (pt
ij )¯
Vt−1(x−Akt
ij
ij )−¯
Vt−1(x)+
n
X
s=1
ηt
ilsλt
ils(pt
ij )¯
Vt−1(x−Aj
ils)−¯
Vt−1(x)
≥1
2(ηt
ij λt
ij (pt
ij )pt
ij +
n
X
s=1
ηt
ilsλt
ils(pt
ils)pt
ils)
+ηt
ij λt
ij (pt
ij )¯
Vt−1(x−Akt
ij
ij )−¯
Vt−1(x)+
n
X
s=1
ηt
ilsλt
ils(pt
ils)¯
Vt−1(x−Aj
ils)−¯
Vt−1(x)
≥1
2 ηt
ij λt
ij (pt
ij )pt
ij +Vt−1(x−Akt
ij
ij )−Vt−1(x)+
n
X
s=1
ηt
ilsλt
ils(pt
ils)pt
ils+Vt−1(x−Aj
ils)−Vt−1(x)!
+ (ηt
ij λt
ij (pt
ij ) +
n
X
s=1
ηt
ilsλt
ils(pt
ils)) 1
2Vt−1(x)−¯
Vt−1(x),
(32)
where the second inequality is because the case assumption (31) and that λt
ij (·)’s are decreasing
functions and ¯
Vt(x−Ak
ij )−¯
Vt(x)’s are always negative. And the last inequality follows from the
induction assumption and reorganizing terms. Now we give some hints on the resulting formula. In
the resulting formula, the first term stands for the half of the revenue generated by this branch in
the case without hidden-city ticketing, therefore when combined, will stands for half of the revenue
in total. The second term is the residual difference. This will be combined in the end and we can
show that it can be eliminated. The detailed discussion will be given later in the proof.
Case 2:
ηt
ij λt
ij (pt
ij )pt
ij <
n
X
s=1
ηt
ilsλt
ils(pt
ils)pt
ils.(33)
In this case, we choose price pthat satisfies the following equation:
ηt
ij λt
ij (p) +
n
X
s=1
ηt
ilsλt
ils(pt
ils∨p) = ηt
ij λt
ij (pt
ij ) +
n
X
s=1
ηt
ilsλt
ils(pt
ils),
(34)
where a∨b= max(a, b). To illustrate, we choose psuch that the arrival rate is the same as before.
Then, for O-D pair from ito ls, if the original price is less than p, we use price p, otherwise we use
the original price. Also we use price pfor the O-D pair ito j.
First we claim that such palways exists. This follows from Rolle’s Mean Value Theorem. Note
that when ptakes value of pt
ij , the left hand side of (34) is smaller than the right hand side. And
when ptakes value of pt
iln, the left hand side of (34) is larger. Also by Assumption 1, we know that
the left hand side is a continuous function of p. Therefore there must exist psuch that (34) holds.
Now we bound the revenue for this p. We consider two further cases:
Case 2.1: p≥pt
il1. In this case, we have
ηt
ij λt
ij (p)p+¯
Vt−1(x−Akt
ij
ij )−¯
Vt−1(x)+
n
X
s=1
ηt
ilsλt
ils(p)p+¯
Vt−1(x−Aj
ils)−¯
Vt−1(x)
= (ηt
ij λt
ij (pt
ij ) +
n
X
s=1
ηt
ilsλt
ils(pt
ils)) ·p
+ηt
ij λt
ij (p)¯
Vt−1(x−Akt
ij
ij )−¯
Vt−1(x)+
n
X
s=1
ηt
ilsλt
ils(p)¯
Vt−1(x−Aj
ils)−¯
Vt−1(x)
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
Operations Research 00(0), pp. 000–000, c
0000 INFORMS 23
≥
n
X
s=1
ηt
ilsλt
ils(pt
ils)pt
ils+1
2ηt
ij λt
ij (p)Vt−1(x−Akt
ij
ij )−Vt−1(x)
+1
2
n
X
s=1
ηt
ilsλt
ils(p)Vt−1(x−Aj
ils)−Vt−1(x)+ (ηt
ij λt
ij (pt
ij ) +
n
X
s=1
ηt
ilsλt
ils(pt
ils)) 1
2Vt−1(x)−¯
Vt−1(x)
≥1
2(ηt
ij λt
ij (pt
ij )pt
ij +
n
X
s=1
ηt
ilsλt
ils(pt
ils)pt
ils+ηt
ij λt
ij (pt
ij )Vt−1(x−Akt
ij
ij )−Vt−1(x)
+
n
X
s=1
ηt
ilsλt
ils(pt
ils)Vt−1(x−Aj
ils)−Vt−1(x))+ (ηt
ij λt
ij (pt
ij ) +
n
X
s=1
ηt
ilsλt
ils(pt
ils)) 1
2Vt−1(x)−¯
Vt−1(x),
(35)
where the first equality is by grouping the terms and applying equation (34). The second inequality
is because the induction assumption and reorganizing terms. The third inequality is because the
case assumption (33) and that
n
X
s=1
(ηt
ilsλt
ils(p)−ηt
ilsλt
ils(pt
ils)) Vt−1(x−Aj
ils)−Vt−1(x)
≥
n
X
s=1
(ηt
ilsλt
ils(p)−ηt
ilsλt
ils(pt
ils)) Vt−1(x−Akt
ij
ij )−Vt−1(x)
= (ηt
ij λt
ij (pt
ij )−ηt
ij λt
ij (p)) Vt−1(x−Akt
ij
ij )−Vt−1(x).
(36)
Note that in (35), except the last term, it is the value that appears in (1), the last term will be
eliminated when we add all the O-D pairs together and apply the induction assumption, which we
will show later. Now we consider the other case:
Case 2.2: pilk> p ≥pilk+1 . In this case, we have
ηt
ij λt
ij (p)p+¯
Vt−1(x−Akt
ij
ij )−¯
Vt−1(x)+
k
X
s=1
ηt
ilsλt
ils(pt
ils)pt
ils+¯
Vt−1(x−Aj
ils)−¯
Vt−1(x)+
n
X
s=k+1
ηt
ilsλt
ils(p)p+¯
Vt−1(x−Aj
ils)−¯
Vt−1(x)
= (ηt
ij λt
ij (pt
ij ) +
n
X
s=k+1
ηt
ilsλt
ils(pt
ils)) ·p+
k
X
s=1
ηt
ilsλt
ils(pt
ils)pt
ils+ηt
ij λt
ij (p)¯
Vt−1(x−Akt
ij
ij )−¯
Vt−1(x)
+
k
X
s=1
ηt
ilsλt
ils(pt
ils)¯
Vt−1(x−Aj
ils)−¯
Vt−1(x)+
n
X
s=k+1
ηt
ilsλt
ils(p)¯
Vt−1(x−Aj
ils)−¯
Vt−1(x)
≥
n
X
s=1
ηt
ilsλt
ils(pt
ils)pt
ils+1
2ηt
ij λt
ij (p)Vt−1(x−Akt
ij
ij )−Vt−1(x)
+1
2
k
X
s=1
ηt
ilsλt
ils(pt
ils)Vt−1(x−Aj
ils)−Vt−1(x)+1
2
n
X
s=k+1
ηt
ilsλt
ils(p)Vt−1(x−Aj
ils)−Vt−1(x)
+ (ηt
ij λt
ij (p) +
k
X
s=1
ηt
ilsλt
ils(pt
ils) +
n
X
s=k+1
ηt
ilsλt
ils(p)) 1
2Vt−1(x)−¯
Vt−1(x)
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
24 Operations Research 00(0), pp. 000–000, c
0000 INFORMS
≥1
2 ηt
ij λt
ij (pt
ij )pt
ij +
n
X
s=1
ηt
ilsλt
ils(pt
ils)pt
ils+ηt
ij λt
ij (pt
ij )Vt−1(x−Akt
ij
ij )−Vt−1(x)
+
n
X
s=1
ηt
ilsλt
ils(pt
ils)Vt−1(x−Aj
ils)−Vt−1(x)!+ (ηt
ij λt
ij (pt
ij ) +
n
X
s=1
ηt
ilsλt
ils(pt
ils)) 1
2Vt−1(x)−¯
Vt−1(x),
(37)
where again the first equality is by grouping the terms and applying equation (34). The first
inequality is because the induction assumption. The last inequality is because the case assumption
(33) and that
n
X
s=k+1
(ηt
ilsλt
ils(p)−ηt
ilsλt
ils(pt
ils)) Vt−1(x−Aj
ils)−Vt−1(x)
≥
n
X
s=k+1
(ηt
ilsλt
ils(p)−ηt
ilsλt
ils(pt
ils)) Vt−1(x−Akt
ij
ij )−Vt−1(x)
= (ηt
ij λt
ij (pt
ij )−ηt
ij λt
ij (p)) Vt−1(x−Akt
ij
ij )−Vt−1(x).
(38)
Now we have proved that for each hidden-city branch, we can modify the prices such that we can
achieve half of the revenue locally. Now we consider all the O-D pairs in (15). We group them
into each hidden-city branches (for which the prices are modified) and those O-D pairs that don’t
appear in any hidden-city branch (for which the prices are not modified). We have:
¯
Vt(x)≥¯
Vt−1(x) + X
(i,j)∈O
ηt
ij λt
ij (¯pt
ij )¯pt
ij +¯
Vt−1(x−A¯
kt
ij
ij )−¯
Vt−1(x)
≥1
2
Vt−1(x) + X
(i,j)∈O
ηt
ij λt
ij (pt
ij )pt
ij +Vt−1(x−Akt
ij
ij )−Vt−1(x)
+ (1 −X
(i,j)∈O
ηt
ij λt
ij (pt
ij )) ¯
Vt−1(x)−1
2Vt−1(x)
≥1
2Vt(x) + (1 −X
(i,j)∈O
ηt
ij λt
ij (pt
ij )) ¯
Vt−1(x)−1
2Vt−1(x)
≥1
2Vt(x).
(39)
In (39), the first inequality is because of equation (15), the second inequality is because the above
discussion on each term in the hidden-city branch combined with all those terms that don’t appear
in the hidden-city branch. The third inequality is because we assumed that pt
ij and kt
ij are the
optimal solutions to (1). And the last inequality is because we assume the time window is small
enough such that only one arrival may appear in each period.
Theorem 3 then follows.
Wang and Ye: Hidden-City Ticketing: the Cause and Impact
Operations Research 00(0), pp. 000–000, c
0000 INFORMS 25
Endnotes
1. Prices are found on American Airlines Website (www.aa.com) on Feb. 23rd, 2011
2. In this paper, we assume that the choice function is affected only by the prices, i.e., the flight
route does not affect the choice. Although this assumption is obviously not true in practice (e.g.,
a direct flight is obviously more attractive to most passengers than a 1-stop flight), given the hub-
and-spoke network adopted by most airlines, whether a direct flight or connecting flight is provided
is usually pre-determined. Furthermore, as will be shown later, the decision on the route and price
can be separated in this model, therefore this assumption is reasonable
3. We only allow passenger to forgo the latter half of a ticket, forgoing the first half of a ticket can
be easily discovered and airlines will almost surely not allow that passenger to be on board on the
second flight
4. See http://www.aa.com/i18n/agency/Booking_Ticketing/Ticketing/hidden_city_ltr.
jsp&locale=de_DE
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