CC BY
2Anna V. Mazalova
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 four-person game which is related to the queueing system M/M/2 is considered. There are two competing stores and two competing transport companies which serve the stream of customers with exponential distribution with parameters ^1 and respectively. The stream forms the Poisson 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 four-person game which is related to the queueing system M/M/2 is considered. There are two competing stores Pi and P2 and two competing transport companies Ci and C2 which serve the stream of customers with exponential distribution with parameters m and m respectively. The stream forms the Poisson process with intensity A.. Suppose that A < mi + m2. Let shops declare the price for the produced product. After that transport companies declare the price of the service and carry passengers to the store, and the company Ci carries passengers to Pi, when the company C2 carries passengers to P2. Customers choose the service with minimal costs. This approach was used in the Hotelling’s duopoly (Hotelling, 1929; D’Aspremont et al., 1979; Mazalova, 2012) to determine the equilibrium price in the market. But the costs of each customer are calculated as the price of the product and transport charges. In this model, costs are calculated as the sum of prices for services and product plus losses of staying in the queue. Thus, the incoming stream is divided into two Poisson flows with intensities Ai and A2, where Ai + A2 = A. So the problem is following, what price for the service, the price for the product and the intensity of services is better to announce for the companies and shops. Such articles as (Altman and Shimkin, 1998; Levhari and Luski, 1978; Hassin and Haviv, 2003; Mazalova, 2013; Koryagin, 2008; Luski, 1976) are devoted to the similar game-theoretic problems of queuing processes.
Game-theoretic model of pricing. Consider the following game. Players Pi and P2 declare the price for the produced product pi and p2 respectively. The customers have to use a transport to get to the shop. There are two competing transport companies Ci and C2 which serve the stream of customers with exponential distribution with parameters m and m respectively. The transport companies declare the price of the service ci and c2 respectively and carry passengers to the store, and the company Ci carries passengers to Pi, when the company C2 carries passengers to P2. So the customers choose the service with minimal costs, and the incoming stream is divided into two Poisson flows with intensities Ai and A2, where Ai + A2 = A. In this case the costs of each customer will be
Ci+Pi~\---—* = 1,2,
Mi — Xi
where 1/M — Xi) is the expected time of staying in a queueing system (Saati, 1961).
Then the intensities of the flows Xi and X2 = X — Xi for the corresponding services
can be found from
1 1
Cl+PlH---------7-=C2+p2H-----------------------------------------------7-. (1)
Mi — X1 M2 — X2
So, the payoff functions for each player are
C2,P1,P2) = Xici, H2(ci, C2,P1,P2) = X2C2,
Ki(ci, C2,P1,P2) = Xipi, K2(ci, C2,P1,P2) = X2P2.
We are interested in the equilibrium in this game.
Symmetric model. Let start from the symmetric case, when the services are the same, i. e. mi = M2 = M. Assuming that the stores fixed their prices pi and p2, let us find the the equilibrium behavior for the transport companies. The equation (1) for the intensity Xi is
11
ci+PiH---------7- = C2 + P2 ---x x • (2)
M — Xi m — X + Xi
Differentiating (2) by ci we can find
1 dX-\ 1 dX
1 +
(m — Xi)2 dci (m — X + Xi)2 dci
from which
dXl ( I | 1 V1 (3)
dci y (/x — A1)2 (yU-A + Ai)2/
Now we can find Nash equilibrium strategies cf and c*2 for fixed Pi, p2 and c2,
i. e. we can find the maximum of Hi(ci, c2,pi,p2) by ci. The first order condition for the maximum of payoff function is
dHi(ci,C2,pi,p2) , , dXi
------------------= Ai + ci = 0,
dci dci
wherefrom
C* - Al
c=
d\i_
dc\
substituting (3) to (4), we will get
For another transport company it is
c2 = X2
1
+
1
(6)
- Ai)2 (m — X + X1)2 J '
Now we can find the Nash equilibrium for players P1 and P2. Let us find the maximum of K1(c1, c2,p1,p2) by p1 when p2 is fixed, assuming that transport companies use the equilibrium strategies. The first order condition for the maximum of payoff function is
dK1(c1,C2,p1,p2) , , dX1
--------------------= Ai +p i— = 0,
dp1 dp1
from where
* xi Pi = Jx£-dpi
substituting the equilibrium prices of the transport companies (5)-(6) to (2) and differentiating it by p1, we will get
dX1
dp1
So,
3 2
+ 7-------^—;—ttt- + (2Ai — A) (
(M — X1)2 (M — X + X1)2
(M — X1)3 (M — X + X1)3'
(7)
p*1 = X1
+
(M — X1)2 (M — X2)2
(M — X1)3 (M — X2)3 '
For another store it is
p*2 = X2
+
(M — X1)2 (M — X2)2
3 2
(2A2 — A)( ^
(M — X2)3 (M — X1)3 '
Thus we get the system of equations that defines the equilibrium prices as transport companies and stores.
11
Cl +pi H-7— = C2+p2 +
M — X1
M — X2
c1 = X c2 = X2
+
1
1 1 ^ (m — xi)2 (m — X2)2
1
+
1
p1 = X1
+
(m — X1)2 (m — X2)2
+ (2X1 — X)(
(m — X1)2 (m — X2)2
2 2 \
p2 = X2
3 + , I,, +(2A2-A)( 2
(m — X1)3 (m — X2 )3
2
(M — X1)2 (M — X2)2
(M — X2 )3 (M — X1)3
X1 + X2 = X.
Using the symmetry of the problem, the solution of this system is
1
3
2
3
2
3
2
3
3
A
* *
ci = c2
* pi * = p2
Ai - A2 - -A
(8)
(M-f)2 3A
(m - f)2
It is easy to check, that the second order condition for the maximum of payoff function is also satisfied.
d2Hl dc?
d\i d? Ai
d,c\ 1 dc2
d2K1
dp2
dA1
2^+ Pi
i
d2 A
1 dpi 11 dpi
Differentiating (3) by c1 and (7) by p1 we find
d2A1
dA1
2
dc2 \dci J (m — Ai)3 (m — A + Ai)3
d2 Ai dp\
dA1
dpi
10
10
(M — Ai)3 (M — A + Ai)3
+ (2Ai — A)(
+
(m — Ai)4 (m — A + Ai)4_
In the equilibrium Ai = A/2, from which = 0 e f = 0. So,
dpi
d2H1(c*1,c^,p*1,p^) eZAi ( A\
--------*5----------= 2S7 = -l,"-2 1 <0-
d2Ki{c\,c,2,p\,p*2) _ dXi dp\ dpi
< 0.
So, if one of the players uses the strategy (8), the maximum of payoff of another player is reached at the same strategy. That means that this set of strategies is equilibrium.
Asymmetric model. Let us assume now, that transport services are not equal, i. e. Mi = M2, suppose that m1 > M2. Let us find the equilibrium in the pricing problem in this case. Let us fix p1, p2 and c2 and find the best reply of the player C1. As
well as in the symmetric case we get
dHi(ci,c2,pi,p2) , , dAi
-------------------= Ai + ci = 0,
dci dci
wherefrom
Differentiating (1),we find
Ai
dAi / dci
Cl Al l,(Mi-Ai)2 + (M2-A2)V '
2
6
6
ci =
For another transport company it is
2 2 ((mi - Al)2 + (^2 - A2)2 )
Table 1: The value of (ci,ct,), (p'upl) and (Ai,A2) at A = 10
^2
m 6 7 8 9 10
6 (A'A) ijPl',P*2) (Ai; A2) (10;10) (30;30) (5;5)
7 (ci;сг) ijPuPl) (Ai; A2) (5,918;5,804) (17,035;16,707) (5,049;4,951) (2,5;2,5) (7,5;7,5) (5;5)
8 (A'A) ijPuPl) (Ai; A2) (4,953;4,797) (13,636;13,208) (5,08;4,92) (1,781;1,743) (5,26;5,15) (5,053;4,947) (1,11;1,11) (3,33;3,33) (5;5)
9 {4;4) ijPl',P*2) (Ai; A2) (4,553;4,375) (12,165;11,689) (5,1;4,9) (1,494;1,437) (4,3;4,136) (5,097;4,903) (0,866;0,848) (2,597;2,533) (5,054;4,946) (0,625;0,625) (1,875;1,875) (5,5)
10 (CbC2) ijPuPl) (Ai; A2) (4,342;4,15) (11,371;10,869) (5,113;4,887) (1,346;1,276) (3,781;3,586) (5,132;4,868) (0,743;0,713) (2,176;2,088) (5,103;4,897) (0,514;0,503) (1,535; 1,502) (5,055;4,945) (0,4;0,4) (1,2;1,2) (5;5)
Now we can find the best replies for the P1 and P2.
dKi{cl ,c2,pi,p2) , , dXi
------------------= Xi+pi-— = 0, * = 1,2,
dpi dpi
from which
v* =____—_____i = 12
* dXi/dpi ’
Using the same arguments as in the symmetric model, we obtain the system of equations that determine the equilibrium prices as transport companies and stores.
1 1
Cl + pi H--------7— = C2+P2 +
Mi _ Ai M2 — X-2
r* = A ( 1 I 1
1 1 \(mi — Ai)2 {p-2 — A2)2
4 = a2 (7—^ + 1
(Mi — Xi)2 (м2 — X2)2
Pi = Ai ( ------------r-^2 + 7------------7-^2 + (2Ai - A)(-
(Mi — Xi)2 (м2 — X2)2 (mi — Xi )3 (м2 — X2)3'
P*2 = M[-(-----^ + 7-------^ + (2A2-A)( 2 2 '
(Mi — Xi)2 (м2 — X2)2 (м2 — X2 )3 (mi — Xi)3
Xi + X2 = X.
In Table 1 the values of the equilibrium prices with different ^1, at A = 10 and
are given.
2. Conclusion
It is seen from the table, that the higher the intensity of service of one transport company is, the higher payoff this transport company and the store, which is connected to this company, get. So, they can increase the price of the service and the price for the product.
References
Hotelling, H. (1929). Stability in Competition. In: Economic Journal, 39, 41-57. D’Aspremont, C., Gabszewicz, J., Thisse, J.-F. (1979). On Hotelling’s §Stability in Com-petitionT. Econometrica, 47, 1145-1150.
Mazalova, A. V. (2012). Hotelling’s duopoly on the plane with Manhattan distance. Vestnik St. Petersburg University, Ser. 10, pp. 33-43. (in Russian).
Altman, E., Shimkin, N. (1998). ndividual 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.
Saati, T. L. (1961). Elements of Queueing Theory with Applications. Dover.
Mazalova, A. V. (2013). Duopoly in queueing system. Vestnik St. Petersburg University, Ser. 10, (submitted).
