MODEL AND ALGORITHM OF COMPETITION BETWEEN
HIGH-SPEED RAILWAY AND AIR TRANSPORT
Game Theory Based
Jianan Zhang and Peng Zhao
School of Traffic and Transportation, Beijing Jiaotong University, Shangyuancun, Beijing, China
Keywords: High-speed railway, Game theory, Competitive strategy.
Abstract: This paper considered the access cost of passengers, and constructed a passenger’s total travel cost function.
We employed the concept of “linear city” to analyze the market share between high-speed rail (HSR) and
airline (AIR), and analyzed the relation between the two competitors with the non-cooperative game theory.
We took the fixed fare and variable fare rate as decision variables, and established an optimizing model to
calculate the fares of each mode, and then put forward a heuristic algorithm to solve the model. Taking the
Wuhan-Guangzhou transportation corridor as the background, the model and algorithm were used to
calculate the optimizing fares. And analyzed change of solutions with different value of time, then discussed
the competitive strategies of the HSR. The result demonstrated this model reflects facts reasonably and can
be used to generate better competitive strategies.
1 INTRODUCTION
The construction and development of high-speed rail
(HSR) brings people new choice of travel and the
profound impact on passenger transport market
structure. In the competition with airline (AIR), HSR
attracted a considerable part of AIR flow with its
own advantages. This is obviously in long-distance
travel. Such as Wuhan-Guangzhou HSR, which is a
recent opening line, makes flights between the
Wuhan to Guangzhou reduced. However, the HSR
development in China is still at the initial stage, air
transport, as the main mode has been developed in
China for many years, it has a relatively stable
market share and mature operating system, the
airlines carried out fully research and policy
adjustments face to the HSR opening and operation.
Compared with huge investment and operating costs
in HSR, AIR has flexible transport organization,
experienced management, higher pure running speed,
and other advantages, if HSR cannot adjust the
competitive strategies for more passengers, it will
affect the future development. Therefore, the study
of how should HSR dealing with air transport
competition is true important for its development.
The experts and scholars from home and abroad
has done many researches in the competition
strategies of transportation mode, especially the
passengers flow sharing and price stratigy in the
conditions of different transportation modes. In early
researches, McFadden referenced the utility theory
of economics and studied the issiue of market share
of transport modes (Mcfadden, 1989). Williams
proposed nested Logit Model to describe the
problem of flow sharing in different transport
modes(Williams, 1991). Recently, Yao used Nested
Structure Model to do demand forecast of various of
transport modes of inter-cities(Yao, 2005), and he
got the conclusion that, the amount of inter-city
travel come risen with the reduction of time and cost,
and the improvement of service frequency (Si,
2005).Roman, Espino had analyzed the competition
of HSR and air transport in Madrid-Barcelona
transport corridor in Spanish(Roman and Espino,
2007). They estimated the parameters from the
survey data, and found that HSR had been more
competitive in long-distance transport. Givoni
explored the issue that some air companies regard
the HSR network as the extension of their route and
analyzed the conditions and pattern of cooperation
between air transport companies and HSR (Givoni,
2007). Obviously, the analysis of the competitive
strategy should be connected with the forecasts of
the market sharing, and so far, the Logit Model
563
Zhang J. and Zhao P..
MODEL AND ALGORITHM OF COMPETITION BETWEEN HIGH-SPEED RAILWAY AND AIR TRANSPORT - Game Theory Based.
DOI: 10.5220/0003586105630568
In Proceedings of the 13th International Conference on Enterprise Information Systems (PMSS-2011), pages 563-568
ISBN: 978-989-8425-56-0
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

based on Utility Theory is the main way getting
passenger volume of modes.
Actually, the main competition between different
modes of transport is a game, while the use of game
theory is rare. This paper according to the cost that
passengers start from home to station or airport
(access-cost) and the cost from station to destination
(egress-cost) to constructed a cost function.
Analyzed the passenger volume of HSR and AIR
with the theory of “linear city”(Hsu C. W., 2009),
at the same time, analysis the competition of HSR
and AIR with non-cooperative game theory, on this
basis, established a optimization model and gave the
heuristic algorithm. Finally, we use an example to
analysis the competitive strategies of HSR.
2 BACKGROUND
The competitive strategy about passenger service
includes service planning, service level, and so on.
Obviously, ticket price is the main method to adjust
competitive strategy. It is also the main factor
influents passenger’s choice among different
transport modes. So, the main point of this paper is
study how HSR pricing.
What the operators care most is passenger’s
choice for mode. When passengers choose mode for
their travel, cost is the key factor. In this article, the
costs we talk about refer to currency expenses and
time expenses. Most of the early researches only
draw main attention to currency expenses which
passengers pay for the travel from origination station
to destination station, neglecting the access and
egress cost. For passengers themselves, however,
they will take access-cost and egress-cost into
consideration. Usually the access-cost and egress-
cost for HSR and AIR are different from each other.
As a result, operators should establish pricing
strategy according to passenger’s access (egress)-
cost. Recently, researchers gradually realized the
importance of access (egress)-cost. However,
specific calculation methods of access（egress）-
cost have not been mentioned yet. We brought
access (egress)-cost into the function of cost, and
then found that a passenger’s expenses consist of
fare, access (egress)-cost and time expenses. Since
there’s lots of access (egress) modes and they show
little influence on the whole travel cost, the time a
passenger spend on access (egress) becomes the
most effective factor for cost. In order to help
calculate, we only take the access (egress) time into
account, and the following is function of whole
travel cost:






k
e
mkk
a
mkk
k
m
k
mmm
k
m
vtvtvtdbpgpC 
(1)
We summarize our notation below:
Table 1: Symbols of travel cost function.
Symbol Meaning
k
m
C
The cost of passenger k choose the transport mode
m
K
Passengers set
M
Transport modes set, M={HSR, AIR}
m
g
p
The fixed fare of transport mode
m
m
bp
The variable fare rate of transport mode
m
k
m
d
The travel distance that passenger k choose
transport mode
m
k
m
t
The travel time that passenger k choose transport
mode
m
a
mk
t
The access time that passenger k choose transport
mode
m
e
mk
t
The egress time that passenger k choose transport
mode
m
k
v
The value of time of passenger k
This function divides the travel cost into two
parts. What is in the braces is the generally cost, and
this part entirely depends on the mode what
passengers choose. And the other part is the
connection cost. It depends on access-cost and
egress-cost.
3 THE MODEL
3.1 Linear City
Under the conditions of the determined fare and
access (egress) modes, travel cost can be calculated
depend on the passenger travel cost function. It is
possible to calculate the whole amount of passengers
take every transport mode when passengers choose
to minimize their total travel cost. This paper will
use the concept of linear city to predict passenger
flow of HSR and AIR. At first we can assume that
city is ribbon shape, with the ends of which are HSR
station and airport (Hotelling, 1929). To simplify the
problem, we assume that the sum of access cost to
the railway station and the airport are the length of
the linear city, and it can be marked by
D
. Then, set
the access charges of passenger
k
from home to high
speed rail station
k
x
, and set the access charges of
passenger from home to airport
k
D
x
(Figure 1).
Figure 1: Schematic diagram of linear city.
k
x
k
x
D

ICEIS 2011 - 13th International Conference on Enterprise Information Systems
564

Similarly, the assumption can be also used to the
egress cost of passenger
k
, then, each line represents
a linear city. Put the two lines on the same plane, we
can get a two-dimensional graph of passenger
k
from the departure city to the destination city
(Figure 2).
Figure 2: Schematic diagram of two-dimensional
connection charges.
Each point on Figure 2 represents a combination
of connection costs.
k
x
is the abscissa of the point
and it is the access cost of passenger
k
from home to
the HSR station, and
k
xD 
represents cost of
passenger to the airport. We can also achieve the
egress charges from the ordinate of Figure 2. The
two-dimensional graph can express visual difference
of connection costs among passengers. We can see,
when
e
D
is equal to
a
D
, the total connection costs of
passengers covered by diagonal
a choosing HSR is
equal to AIR. When the generally cost of HSR and
AIR during operation are equal, passengers below
the diagonal will preference for HSR, others will
preferred to fly. At that time, diagonal of the
rectangle will be the boundary of passengers to
choose HSR or airline. Assuming running time of
HSR and AIR is for certain, then
k
k
m
vt
in the formula
(1) is constant. When the price of mode
m change,
the diagonal
a will move up or down, and
determines passenger flow volume of the two
transport modes. For example, HSR reduce fares,
and
k
HSR
C
or
k
AIR
C
represent the generally cost of
passenger
k
takes HSR or AIR, then the minus of
k
HSR
C
and
k
AIR
C
can be expressed as:


k
e
AIRkk
a
AIRkk
k
AIR
k
AIRAIRAIR
k
e
HSRkk
a
HSRkk
k
HSR
k
HSRHSRHSR
k
AIR
k
HSR
vtvtvtdbpgp
vtvtvtdbpgp
CCC



(2)
For easy calculation, assuming the population of
linear cities is uniformly distribution，when the cost
of passengers take two modes are different, the line
a will shift up or down. Like Figure 3 shown，the
reason is cost of passengers who covered by straight
line
a should be the same whatever mode they take.
So, passenger flow volume of mode
m can be
calculated by the following formula:























 C
D
D
DC
D
D
D
DD
Q
q
e
a
a
a
e
e
ea
m
2
1
(3)
Figure 3: Schematic distribution of passenger flow.
3.2 Model
Price is the decision variable of the split line in the
two-dimensional graph, and it also determines the
passenger volume of the HSR and AIR. Goal of this
problem is to maximize profits of each enterprise, so
the problem is a constrained extremely problem.
Because the ticket price of HSR and AIR consist of
two parts, income of each mode is the product of
price and the number of passengers who choose each
mode, and profit for the enterprise is the difference
between revenue and cost. Guide price of fixed fare
and variable fare in the objective function
constrained by government. So the model can be
expressed as:






11
max
..
mm
qq
kk
mmm mm mmm
kk
lh
mm m
lh
mm m
kk a e
mmmmk mkkmkk
kk a e
mmmmk mkkmkk
zgpq bpd cdq
GP gp GP
BP bp BP
st
gp bp d t v t v t v
g
p bpd tv tv tv

  
 



 








(4)
k
x
ka
xD 
k
y
ke
yD 
a
e
D
High Speed
Rail
Station
Airport
Departure
City
Airport
Arrival
City
a
a
D
High Speed
Rail
Station
MODEL AND ALGORITHM OF COMPETITION BETWEEN HIGH-SPEED RAILWAY AND AIR TRANSPORT -
Game Theory Based
565

Table 2: Symbols of the model.
Symbol Meaning
m
z
Profits of mode
m
m
q
The number of passengers that select mode
m
l
m
GP
Lower limit of the fixed fare of mode
m
h
m
GP
Upper limit of the fixed fare of mode
m
l
m
BP
Lower limit of variable fare rate of mode
m
h
m
BP
Upper limit of variable fare rate of mode
m
The decision variables in the model are fixed fare
and variable fare rate of mode
m , which is HSR or
AIR. Fixed fare and variable fare rate are strategies
of HSR and AIR. Actual passenger flow volume of
the two sides in the game is directly related to the
two kinds of fares. By solving the above model,
HSR or AIR can calculates the best pricing
strategies when they face to competition from the
other.
4 HEURISTIC ALGORITHM
According to the above model, the competition
between HSR and AIR in middle distance passenger
transport market aspect is non-cooperative game;
non-cooperative game model’s solution is Nash
equilibrium. Nash equilibrium is a strategy
combination，and each participant's strategy is the
most superior one in the situation of other
participant strategy has been determined. Specific to
the problem in this paper, we can use the equation
below to express Nash equilibrium (Hsu C. W.,
2009):






 
**
****
,,,
,,,
mmmmm
mmmmm
bpgpbpgp
bpgpbpgp




(5)
That is to say, once the model achieved the Nash
equilibrium under the condition of the other
parameter has been determined, HSR and AIR
cannot get more profit however they adjust their
strategies. This model includes two decision
variables, namely fixed fare and variable fare rate,
because what this article studies is the game of HSR
and AIR, it needs to determine the strategy
combination which contains four variables, the best
strategy can be expressed by:






****
,,,
AIRAIRHSRHSR
bpgpbpgp
(6)
m
q
is the volume of transport mode m , whose
computational method has already been given in the
previous section. Because the model’s objective
function cannot differential everywhere, we cannot
solve the model directly. In order to obtain the
model’s Nash equilibrium, this article gives the
heuristic algorithm below.
First create an initial solution according to
relatively simple rule, and this solution will be the
initial ticket price of HSR and AIR. Afterward,
select one solution from the two modes as the known
condition, then use the (2) and (3) to calculate
m
q
,
subsequently uses (4) to calculate ticket price of
mode
m

as mode m

's second solution and this
is part of the first iteration. Similarly, we can get the
second solution of mode
m . After times of
iterations, when the solution meets the termination
condition, we obtain the solution of the whole
model.
This article uses the average guide ticket price of
HSR and AIR as the problem’s initial solution,
namely:






00 0 0
,, ,
,, ,
22 2 2
mm m m
lhlh l hl h
mmmm m mm m
gp bp gp bp
GP GP BP BP GP GP BP BP



  


 


 
 

(7)
And set the termination condition the two iterative
difference of solution is smaller than a small value

.
5 CASE STUDY
5.1 Data
In order to confirm the model and the algorithm’s
validity, we take the Wuhan to Guangzhou corridor
as an example, to determine the competition strategy
of HSR. The result indicates how the operation cost
and value of time will influence the strategy of HSR.
Because only the passengers between Wuhan to
Guangzhou can be fight for by HSR and AIR, this
case only consider the direct passengers between
Wuhan to Guangzhou. At present, Wuhan and
Guangzhou’s passenger already to be possible to
choose the HSR, they may also choose AIR, and the
relation of two transport modes meet the above
game model’s basic condition.
This case study involves the essential data of
passenger demand between Wuhan and Guangzhou,
such as the value of time, distance of each mode
between Wuhan and Guangzhou, operation cost of
HSR and AIR, and so on.
 Passenger Transportation Demand. At present,
there are 29 pairs of EMUs between Wuhan and
Guangzhou every day, each EMU can take 1200
passengers, then HSR can deliver 34800 passengers
ICEIS 2011 - 13th International Conference on Enterprise Information Systems
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one day in each direction; In the AIR aspect, there
are 10-11 flights every day between Wuhan and
Guangzhou, the plane seat capacity varies differently
based on the type, this article takes the average
number of one week as the capacity, and the number
is 2700 passengers per day. Multiplied the above
two way’s delivery capacity by the average booking
rate coefficient 0.7 as Wuhan and Guangzhou’s
passenger flow demand reference value.
 Value of Time. Residents’ average income is
25000 Yuan in Wuhan and the number is
40000Yuan in Guangzhou, then the averaging value
32500 Yuan. Considering that income of passengers
who take AIR or HSR is higher than the average
number, therefore we multiplied by 1.5 as the
passengers’ income in one year, and it is 48750
Yuan per year. Supposing that everyone works 365-
104-14=247 days, every day we work for 8 hours,
then one year everyone works 247*8=1976 hours.
Each hour's income is 24.67 Yuan approximately is
equal to 25 Yuan.
 The HSR running time is 3 hours, and AIR
running time is 1.5 hours. Guangzhou and Wuhan's
linear urban length is 2 hours.
 The travel distance of HSR is 1069km; with the
distance of AIR is 1000km.
 The fixed fare scope of HSR is 50 Yuan to 100
Yuan, and the Variable fare rate scope is 0.3 Yuan
per passenger-kilometre to 0.5 Yuan per passenger-
kilometre; the fixed fare scope of AIR is 70 Yuan to
100 Yuan, and variable fare rate scope is 0.4 Yuan
per passenger-kilometre to 0.8 Yuan per passenger-
kilometre.
 The HSR operation cost is 0.3 Yuan per
passenger-kilometre, while the cost of AIR is 0.4
Yuan per passenger-kilometre (Chang, 2004).
5.2 Result Analysis
After the running of computer programme, we got
the results of the problem shown by Table 3:
Table 3: Result of model.
Item HSR AIR
Fixed fare(Yuan) 54 70
Variable fare
rate(Yuan per
kilometre)
0.41 0.5
Passenger
volume(person)
19258 6992
Profits(Yuan) 3304480 1188640
It is easy to calculate the total ticket price of
HSR is 492 Yuan, and the total ticket price of AIR is
570 Yuan. At present, the price of HSR and AIR
between Wuhan to Guangzhou were 490 Yuan and
740 Yuan. Obviously, according to parameter in the
article, the fare of AIR is slightly high; this is also
one of the reasons that after the operation of HSR,
passengers who take AIR reduce rapidly. For better
show of how the value of time influences the
passengers’ choice, we simulated when the value of
time changes between 20 Yuan per hour to 100
Yuan per hour. Result as is shown in Table 4.
Table 4: Simulation results.
Value of
time
Passenger
volume(person)
Profits(Yuan)
HSR AIR HSR AIR
20 22053 4197 3699381 730320
40 15286 10964 2570327 1848410
60 12599 13651 2113980 2351950
80 11193 15057 1854550 2584850
100 10193 16057 1725674 2701130
Table 4 shown when the ticket price is stable and
value of time increase, the passengers who choose
HSR will reduce, and the profits of HSR and AIR
will change. That is to say, although the travel time
of each mode is short and almost equal, when
passengers’ value of time higher than the threshold,
travel time will be the key factor that determine
which mode to choose. Therefore, HSR operators
should analyze the influence of fare to the benefit,
and then create more scientific competition
strategies.
6 CONCLUSIONS
This paper constructed a game model of competitive
strategy optimization and gave the heuristic
algorithm. Finally, through a case study, compared
the changes of fares and revenue between the HSR
and AIR, and then analyzed what HSR operators
should adopt competitive strategies. In this passage,
we considered the access and egress cost, and this is
helpful to calculate the cost of travel more accuracy.
The travel cost function can reflect the total cost of
passenger travel; it is the basis to make scientific,
rational and competitive strategy. In addition, we
cited "linear city" theory to abstract processing the
urban passenger departure and arrival, and
predicting the passenger volume of various modes,
experiments show that this method is easy and with
rationality. Parameters in the model including access
and egress time, travel distance and value of time,
and so on, these parameters can be estimated
according to the actual situation, and then simulate
MODEL AND ALGORITHM OF COMPETITION BETWEEN HIGH-SPEED RAILWAY AND AIR TRANSPORT -
Game Theory Based
567

different scenarios and analysis by changing the
parameters’ values. From the results of Wuhan to
Guangzhou case study we can see, game model and
algorithm is effectively and we can calculate the
results within a reasonable time, and the solution is
realistic. The results of this study can also provide a
reasonable reference for HSR operators.
ACKNOWLEDGEMENTS
We want to acknowledge the financial support from
“the Fundamental Research Funds for the Central
Universities”.
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