CC BY
3Anna V. Melnik
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes, Universitetskii pr. 35, St.Petersburg, 198504, Russia E-mail: annamazalova@yandex. ru
Abstract A non-cooperative m-person game which is related to the queueing system M/M/m is considered. There are n competing transport companies which serve the stream of customers with exponential distribution with parameters ^i, i = 1, 2 ,...,m respectively. The stream forms the Pois-son process with intensity A. The problem of pricing and determining the optimal intensity for each player in the competition is solved.
Keywords: Duopoly, equilibrium prices, queueing system.
1. Introduction
A non-cooperative n-person game which is related to the queueing system M/M/m is considered. There are n competing transport companies, which serve the stream of customers with exponential distribution with parameters Mi, i = 1,2, ...,m respectively. The stream forms the Poisson process with intensity A. Suppose that
m
A < M*. Let companies declare the price for the service. Customers choose the
i= 1
service with minimal costs. This approach was used in the Hotelling’s duopoly (Hotelling, 1929; D’Aspremont, Gabszewicz, Thisse, 1979; Mazalova, 2012) to determine the equilibrium price in the market. But the costs of each customer are calculated as the price for the service and expected time in queue. Thus, the incoming stream is divided into m Poisson flows with intensities Ai, i = 1, 2, ...,m,
m
where Ai = A. So the problem is following, what price for the service and
i= 1
the intensity for the service is better to announce for the companies. Such articles as (Altman, Shimkin, 1998; Levhari, Luski, 1978; Hassin, Haviv, 2003), and (Mazalova, 2013; Koryagin 2008; Luski, 1976) are devoted to the similar game-theoretic problems of queuing processes.
2. The competition of two players
Consider the following game. There are two competitive transport companies which serve the stream of customers with exponential distribution with parameters m1 and M2 respectively. The transport companies declare the price of the service c1 and c2 respectively. So the customers choose the service with minimal costs, and the incoming stream is divided into two Poisson flows with intensities A1 and A2, where A1 + A2 = A. In this case the costs of each customer will be
c* H---, ^ v * = 1,2,
Mi(Mi — Ai)
* This work was supported by the St. Petersburg State University under grants No. 9.38.245.2014
where Aj/^j(^j — Aj) is the expected time of staying in a queue (Taha, 2011). So, the balance equations for the customers for choosing the service are
i Al . A2 ci H------------- —r = c2 +
Ml(Ml — A1) M2(M2 — A2 )
So, the payoff functions for each player are
Hi(ci,c2) = A1C1, H2(ci, C2) = A2C2,
We are interested in the equilibrium in this game.
Nash equilibrium. For the fixed c2 the Lagrange function for finding the best reply of the first player is defined by
L\ = A1C1 + k f c\ -\----- --—— — c2------------ -------------- - J + 7(ai + A2 — A). (1)
V Mi(Mi — Ai) M2(M2 — a2 ) /
For finding the local maxima by differentiating (1) we get
^ = \1+k = 0 dci
dLx _ k k\\ _
9Ai A*i (a*i — Ai) — A1)2 ^
5Li k k\2
+ 7 = 0
dA2 ^2(^2 — A2) ^2(^2 — A2)2
from which
Mi(Mi — Ai) ^2(^2 — A2) Mi(Mi — Ai)2 ^2(^2 — A2)2
Symmetric model. Start from the symmetric case, when the services are the same, i. e. ^i = ^2 = ^. It is obvious from the symmetry of the problem, that in equilibrium cf = c| = c* and Ai = A2 = So
2 |) /x(/x - f)2)
So, if one of the players uses the strategy (2), the maximum of payoff of another player is reached at the same strategy. That means that this set of strategies is equilibrium.
Asymmetric model. Assume now, that transport services are not equal, i. e. Mi = M2, suppose that ^i > ^2. Find the equilibrium in the pricing problem in this case. The system of equations that determine the equilibrium prices of transport companies is
Ai A2
C1 + —(-----rr =c2 + —7--------TT
Mi(Mi — Ai) M2(M2 — A2 )
1.1 Ai A2
C1 — A1 ----------------------7--------------H--------------------7--------------T T H-----------------7-----------, NO +
Mi(Mi — Ai) ^2(^2 — A2) Mi(Mi — Ai)2 ^2(^2 — A2)2
* — ' I 1 i 1 i Al i
c2 A2 I --------7-------7 r H-------7--------7 r H------7--------T~v7 +
Ml(Ml — Al) M2(M2 — A2) Ml(Ml — Al)2 ^2(^2 — A2)2
Al + A2 = A.
In Table 1 the values of the equilibrium prices with different m^ M2 at A = 10 and are given.
Table 1: The value of (c1,c2), (pi,p2) and (Ai, A2) at A = 10
M2
Ml 6 7 8 9 10
7 (ci;c2) (Ai;A2) (5,41 ;5,1) (5,15;4,85) (2,5;2,5) (5;5)
8 (ci;c2) (Ai;A2) (4,04;3,64) (5,25;4,75) (1,75; 1,65) (5,14;4,86) (1,11;1,11) (5;5)
9 (ci;c2) (Ai;A2) (3,4;2,98) (5,33;4,67) (1,4;1,26) (5,27;4,73) (0,87;0,82) (5,14;4,86) (0,625;0,625) (5;5)
10 (ci;c2) (Ai;A2) (3,06;2,62) (5,39;4,61) (1,21;1,04) (5,36;4,64) (0,73;0,66) (5.26;4,74) (0,52;0,59) (5,13;4,87) (0,4;0,4) (5;5)
3. The competition of m players.
Let us increase the number of players. There are m competitive transport companies which serve the stream of customers with exponential distribution with parameters M*, i = 1, 2,..., m respectively. The transport companies declare the price of the service Cj, i = 1, 2,..., m and the customers choose the service with minimal costs. The incoming stream is divided into n Poisson flows with intensities A*, i = 1, 2,..., m,
m
where A* = A. Thus, the balance equations for the customers for choosing the
j=l
service are
Al Ai
Cl H----- -—- = Cj H---------- -----—, * = 1, •••, m.
Mi(Mi — Ai) Mi(Mi — Ai)
The payoff functions for each player are
Hj(ci,..., Cj) = AjCj, i = 1,..., m.
Find the equilibrium in this game.For the fixed c*, i = 2,..., m the Lagrange function for finding the best reply of the first player is defined by
m / A A \ m
Li = ciAi + ^2 ki ( ci H - Cj------ —* . ) + 7(^3 A* - A). (3)
j=2 ^ Mi(Mi — Ai) Mi(Mi — Ai) / j=l
Differentiating (3),we find
m
P = Ai+5> = 0,
dci j=2
m m
or E EMi
0-^1 _ . i=2_________i = 2________ „
9Ai Cl M^i-Ai) ^1(^1 - Ai)2 7 ’
dLi ki ki A* .
+ 7 = 0, * = z,m.
dAi Mi(Mi — Ai) Mi(Mi — Ai)2
from which
f 1 , 1
''n I / a \o “r
1^0,7 = (MJ — Aj )2 (m — Ai)2
j = 0,j=ilMj 7
* 1 Ai * 1 Ai+l n 1 //1 \
c* ^ ? \~T — c*+1 ^ ? \ V’ 1 ~ ’m ~ ( }
Mi(Mi — AiJ Mi+l(Mi+l — Ai+l)
i=1
4. The competition of 2 players on graph.
Fig. 1: Competition of 2 players on graph Gi
Consider competition on the graph Gl, which is equivalent to a part of the Helsinki Metro. Let’s define the game as r =< I, II, Gl, Zl, Z2, Hl, H2 >, where I, II are 2 competitive transport companies which serve the stream of customers with exponential distribution with parameters m*, i = 1, 2 on graph Gl =< V, E >. V = {1, 2, 3, 4} is the set of vertices, E = {el2, e23, e24} - the set of edges. Z* = {R, R}
is the set of routes of player i. Each rout is a sequence of vertices. So there are two routs R = {1, 2,3} and R = {1, 2,4}, i = 1,2. The stream of passengers forms the Poisson process with intensity A, where
A-
0 Ai2 Ai3 Ai4
0 0 A23 A24
0 0 0 0
0 0 0 0
The transport companies declare the price of the service ck-, i = 1, 2, k =1,2, j = 2, 3, 4, j = k and the customers choose the service with minimal costs. The incoming stream A is divided into two Poisson flows with intensities Ak- = A*- + A|-, k = 1, 2, j = 2, 3,4, j = k. We are interested in equilibrium in this game.
The balance equations are
where
c12 + a1 = c12 + a2,
c23 + a2 = c23 + a2,
c24 + a3 = c24 + a2,
c13 + a1 + a2 = 4, + a1 + a2,
1 112 2 2 C]_4 + + «3 = Cj_4 + ai + a3,
Afcj = A- + A|j, k = 1, 2, j =2, 3,4, j = k,
A12 + A13 + A
14
- A12 - A13 - A14)
A13 + A:
23
LLL \i \i V
2 I 2 A13 A23J
A14 + A:
24
mVm* \i \i \ 2 I 2 14 A24J
The payoff functions for each player are
i = 1, 2.
Hi((cR 1 ,CR2 ,CRi ,CR2 ) = ^ 53 Akj Ckj,
k=1 j=2,j=fc
The Lagrange function for finding the best reply of the first player is defined by
L
1 = 53 53 Akjckj + k1 (c12 + a1 - c12 - a2) + k2 (4 + a2 - 4 - a2) +
k=1 j=2 = k
a
1
a=
2
a=
3
+k3 (c24 + a1. — i
24
c13 + a1 + a2 — c23 — a2 — a2) +
+k5 (014 + a1 + a3 — — a2 — a|) .
Differentiating this equation we find
dLl - A}2 + h = 0, |^ = A23 + k2= 0,
dc12
dc23
dLl - A^4 + *3 = 0, = Aj3 + k4 = 0,
dc24
dc13
^L = X\4 + k5=0.
dC14
Since Akj = Afcj — A2-, k =1, 2, j = 2, 3,4, j = k, we get dL1 _ 1
dA12
1 \ A da1 da2 N
Cl‘2 + ikl + h + h)[dxr + dxrJ
dL1 1 / da2 da2 \
dL1 1 / dag da3 \
WrC2, + {3+ ’■)\Mt + WJ'
dLi
= c13 + (kl + k4 + ks)
da1
+
da2
dA13 dA13 /
) + (k2 + k4)
da1
____2_ , ga2
^A13 5A13
8L\
<9A}4
c14 + (kl + k4 + fe) (^3 ^ ^5)
<9a2 da\
dA14
dA24
Symmetric model. Consider symmetric case, when the services are the same, i. e. M1 = M2 = M. It is obvious from the symmetry of the problem, that in equilibrium
ckj = ckj = clj and xlj = xlj = ^r,k=l,2,j = 2, 3,4, j + k. So
A12 + A13 + A14
c12
A12+A13 + A14 '
2 -
+
(Ai2 + A13 + A14)2 2/x (m - Al2+A^+Al4;
c23
c24
A23 + A13 a (a ^23+^13
2 V2 2
A24 + A14
M (B. _
2 V 2
2
A12 + A13 + A14
+
(A23 + A13)2
) M (f -
A23 + A13 \ 2 2 )
+
(A24 + A14)2 ..( I1 ^24 + ^14 A ^
^ I 2--------------2-J
c13
m (m —
A12+A13 + A14 '
2 -
(A12 + A13 + A14)2 ' , . . . . . , 0 '
2m (m —
2
2
3
2
Л23 + А13 (A23 + A13)2
P(f-^)2 * _ A12 + Ліз + A14 (A12 + A13 + A14)2
CU ~ М(М~ Лі2+Л^+Лі4) 2M (M- ^+ліз+л14)2
^24 + ^14 (A24+A14)2
-Л^) ^ _ A^)2
In Table 2 the values of the equilibrium prices with different at A12 = 1,
A23 = 1, A24 = 2, A13 = 3, A14 = 1 are given.
Table 2: The value of equilibrium prices at Л12 = 1, Л23 = 1, Л24 = 2, A13 = 3, Л14 = 1
prices 10 11 12 13 14 15
cl 2 0,089 0,069 0,055 0,045 0,038 0,032
C23 0,44 0,327 0,25 0,198 0,16 0,13
c24 0,24 0,188 0,15 0,12 0,089 0,083
сїз 0,53 0,396 0,305 0,243 0,199 0,16
C14 0,33 0,258 0,204 0,165 0,137 0,115
5. Conclusion
It is seen from the table, that the higher the intensity of service is, the lower price
this transport company declare. But the prices c23 and c24, that correspond to the
edges, where the pass is divided on two roads, are greater, that ci2, because after
this division the intensity of service is divided too.
References
Hotelling, H. (1929). Stability in Competition. Economic Journal, 39, 41-57.
D’Aspremont, C., Gabszewicz, J., Thisse, J.-F. (1979). On Hotelling’s “Stability in Competition”. Econometrica, 47, 1145-1150.
Mazalova, A. V. (2012). Hotelling’s duopoly on the plane with Manhattan distance. Vestnik St. Petersburg University, 10(2), 33-43. (in Russian).
Altman, E., Shimkin, N. (1998). Individual equilibrium and learning in processor sharing systems. Operations Research, 46, 776-784.
Levhari, D., Luski, I. (1978). Duopoly pricing and waiting lines. European Economic Review, 11, 17-35.
Hassin, R., Haviv, M. (2003). To Queue or Not to Queue / Equilibrium Behavior in Queueing Systems, Springer.
Luski, I. (1976). On partial equilibrium in a queueing system with two services. The Review of Economic Studies, 43, 519-525.
Koryagin, M.E. (1986). Competition of public transport flows. Autom. Remote Control, 69:8, 1380-1389.
Taha, H. A. (2011). Operations Research: An Introduction, ; 9th. Edition, Prentice Hall.
Mazalova, A. V. (2013). Duopoly in queueing system. In: Vestnik St. Petersburg University, 10(4), 32-41. (in Russian).
