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Journal of Industrial and Management Optimization
doi:10.3934/jimo.2022102
PRODUCING TWO SUBSTITUTABLE PRODUCTS UNDER A
SUPPLY CHAIN INCLUDING TWO MANUFACTURERS AND
ONE RETAILER: A GAME-THEORETIC APPROACH
Hamed Jafari∗
Assistant Professor, Industrial Engineering Group, Golpayegan College of Engineering
Isfahan University of Technology, Golpayegan 87717-67498, Iran
Soroush Safarzadeh
Assistant Professor, Department of Industrial Engineering
Quchan University of Technology, Quchan, Iran
(Communicated by Gerhard-Wilhelm Weber)
Abstract. In the current study, a supply chain structure is considered con-
sisting of duopolistic manufacturers and monopolistic retailer. In the con-
sidered structure, the manufacturers product two substitutable products and
deliver them to end customers. One of the manufacturers sells his/her prod-
ucts only through the common retailer whereas another one sells directly to
customers as well as through the common retailer. Obviously, a competitive
behavior is established in order to price the products under the considered
supply chain. In this point of view, a game-theoretic approach is applied in-
cluding Nash, Manufacturers-Stackelberg, and Retailer-Stackelberg games to
analyze the pricing decisions. Then, the equilibrium strategies obtained from
the investigated games are discussed and some insights are presented. The
results indicate that Manufacturers-Stackelberg and Nash games respectively
lead to the highest and the lowest profits for the manufacturers and the whole
system. Furthermore, the highest profit for the common retailer is obtained
from Retailer-Stackelberg game.
1. Introduction. Nowadays, logistics and supply chain management are known
as critical business concerns [36]. These two concepts play an important role to
achieve competitive advantages [30,16].
Christopher [5] presented a definition for logistics as “Logistics is the process of
strategically managing the procurement, movement, and storage of materials, parts,
and finished inventory through the organization and its marketing channels in such
a way that current and future profitability are maximized through the cost effective
fulfillment of orders.” Also, supply chain management has attracted considerable
attentions in recent years [2,15,3]. As defined by Christopher [5], “Supply chain
management is the management of upstream and downstream relationships with
suppliers and customers in order to deliver superior customer value at less cost to
the supply chain as a whole.”
2020 Mathematics Subject Classification. 91A80, 90C20, 90C90.
Key words and phrases. Supply chain management, pricing, two substitutable products, game
theory, Nash equilibrium, Stackelberg model.
∗Corresponding author: Hamed Jafari.
1
2 HAMED JAFARI AND SOROUSH SAFARZADEH
Price is considered as a significant competitive factor in each market [18,32,17,
14,6,7]. In this point of view, to specify the prices under the various supply chains
can lead to a strategic decision for decision makers [21].
Game theory is known as a mathematical approach to analyse the complicated
situations established among the rational players [48,37,34,12,46]. Recently,
it has found a wide range of applications in different supply chain structures to
make decisions under the various competitive and cooperative behaviors (e.g., see:
[35,28,47,4,13,24,11]).
The aim of the current study is to apply a game-theoretic framework in order to
price two substitutable products under a supply chain structure including duopolis-
tic manufacturers and monopolistic retailer. In this setting, attempts are made to
address some of the researches that have applied a game-theoretic approach to price
two substitutable products under the various structures as follows.
Xiao and Choi [42] proposed a dynamic game theoretical framework to price two
substitutable products in a two-echelon supply chain consisting of two manufactur-
ers and two retailers.
Xiao and Yang [43] and Ai et al. [1] investigated a pricing competition on two
competing supply chains under the demand uncertainty, while Xie et al. [44] con-
sidered the quality improvement concept in addition to the pricing strategies under
a similar supply chain. Furthermore, Wu [41] applied a Nash bargaining scheme
to share the profit obtained from the pricing integration among the members in a
similar structure.
Some studies have investigated the pricing decisions on a supply chain consisting
of two manufacturers and a common retailer. Edirisinghe et al. [8], Sinha and
Sarmah [33], Wei et al. [38], and Wu [40] analyzed the effects of different channel
power structures, whereas Li et al. [23] considered this structure to set prices under a
supply disruption. Zhao et al. [50] applied a fuzzy modeling approach to investigate
the effects of uncertainty on the pricing decisions in which the production costs and
demands are as the fuzzy variables. Pan et al. [31] and Li et al. [22] established
this structure in which the manufacturers can select the wholesale price and the
revenue sharing contracts to sell their products to the retailer. Moreover, Lu et
al. [26] investigated the effects of the services provided by the manufacturers on
the pricing competition established among two competing manufacturers and their
common retailer.
Xu et al. [45] considered a supply chain including two suppliers and one man-
ufacturer where a supplier is a big company abroad while another one is a small
local company.
In this research, a game-theoretic approach is applied to set prices under a supply
chain structure including two competing manufacturers and one common retailer.
In this structure, the manufacturers procure two substitutable products to sell them
to customers. One of the manufacturers sells his/her products only via the common
retailer, while another one sells his/her products directly to customers in addition
to via the common retailer. In this setting, the pricing competition is investigated
under the various interactions established among the members, i.e., if the members
have a similar power structure, then they can make the decisions under the Nash
game and as they have a different power structure, the Manufacturers-Stackelberg
and the Retailer-Stackelberg games are established among the members.
The rest of the study is organized as follows: In Section 2, the considered prob-
lem is described in details. The game-theoretic framework including the Nash,
GAME THEORY FOR PRODUCING TWO SUBSTITUTABLE PRODUCTS 3
Manufacturers-Stackelberg, and Retailer-Stackelberg games are investigated in Sec-
tions 3-5, respectively. In Section 6, the obtained results are provided. Moreover,
the conclusions are presented in Section 7.
2. Model description. In this research, a supply chain structure is established
including duopolistic manufacturers and monopolistic retailer. The manufacturers
separately product two substitutable products and sell them to customers. The
first manufacturer sells his/her products directly to customers as well as through
the common retailer, whereas the second one sells his/her products only through
the common retailer.
The first manufacturer sets the price related to the first product in the direct
channel in addition to the wholesale price of this product to the common retailer.
The second manufacturer specifies the wholesale price related to the second prod-
uct to the retailer. Furthermore, the retailer charges the prices concerning the
substitutable products to end customers. The notations are defined as follows:
KiManufacturing cost per unit for the manufacturer i(i= 1, 2)
φMarket scale for the first product in the direct channel
ϑiMarket scale for the product iin the retail channel
λSelf-price sensitivity of the demands
ζCross-price sensitivity of the demands
WiWholesale price set by the manufacturer ito the common retailer
PPrice set by the first manufacturer to customers in the direct channel
PiPrice set by the common retailer to customers for the product i
QDemand for the first product in the direct channel
QiDemand for the product iin the retail channel
MiProfit value of the manufacturer i
RProfit value of the common retailer
SC Profit value of the whole system
The considered assumptions are as follows:
(1) The manufacturers have enough production capacities to meet their demands.
(2) The demands are specified based on the prices charged to customers.
(3) The demand in each channel is more sensitive to the price set in this channel
than to the sum of the prices set in other channels, i.e., λ > 2ζ.
(4) As the manufacturers sell their products through the common retailer, the
retailer’s profit margin is not less than the manufacturers’ profit margins, i.e., Pi−
Wi≥Wi−Ki(i= 1,2).
The demands are incorporated into the problem as follows:
Q=φ−λP +ζ(P1+P2) (1)
Q1=ϑ1−λP1+ζ(P+P2) (2)
Q2=ϑ2−λP2+ζ(P+P1) (3)
Below, the profits are also formulated:
M1= (P−K1)Q+ (W1−K1)Q1(4)
M2= (W2−K2)Q2(5)
R= (P1−W1)Q1+ (P2−W2)Q2(6)
SC =M1+M2+R(7)
4 HAMED JAFARI AND SOROUSH SAFARZADEH
The following constraints should be considered into the problem to ensure that
the profit margins and demands in the channels are nonnegative:
P≥K1, W1≥K1, W2≥K2, P1≥W1, P2≥W2(8)
Q, Q1, Q2≥0 (9)
In the next sections, the game-theoretic framework is applied to make the pric-
ing decisions under the considered supply chain. In this point of view, the game-
theoretic models are developed including the Nash, Manufacturers-Stackelberg, and
Retailer-Stackelberg to find the equilibrium prices under the various behaviours es-
tablished among the members.
Note the proofs of all theorems presented in the next sections are provided in
Appendix A.
3. Nash game. Under the Nash game, the members have a similar power struc-
ture. Thus, they attempt to make their decisions by maximizing their profits in-
dependently and simultaneously (i.e., see: [29,27,25,19]). The Nash game is
formulated as follows:
max
P, W1
M1= (P−K1)Q+ (W1−K1)Q1s.t. P ≥K1, W1≥K1
max
W2
M2= (W2−K2)Q2s.t. W2≥K2
max
P1, P2
R= (P1−W1)Q1+ (P2−W2)Q2s.t. P1≥W1, P2≥W2
(10)
Now, the attempts are made to find the equilibrium prices under the Nash game.
Theorem 3.1. The profit functions M1and M2are linearly increasing with respect
to W1and W2, respectively.
From assumption (4), we have: Wi≤(Pi+Ki)/2 (i= 1,2). Thus, regard-
ing Theorem 3.1, the optimal values of the wholesale prices are equal to Wi=
(Pi+Ki)/2 (i= 1,2) from the manufacturers’ point of view.
By substituting Wi= (Pi+Ki)/2 (i= 1,2) into the profit functions M1,M2,
and R, the Nash game is reformulated as follows:
max
PM′
1= (P−K1)Q+(P1−K1)
2Q1s.t. P ≥K1
max
P1, P2
R′=(P1−K1)
2Q1+(P2−K2)
2Q2s.t. P1≥K1, P2≥K2
(11)
Obviously, the second manufacturer sets only the wholesale price W2. Therefore,
the decision variable of this manufacturer is fixed by setting W2= (P2+K2)/2.
Theorem 3.2. M′
1is concave with Pand R′is jointly concave with P1and P2.
As a result, if the prices calculated from solving the first order deviations of M′
1
in Pand R′in P1and P2meet the relations 8and 9, then they can be considered
as the equilibrium prices under the Nash game.
Theorem 3.3. Under the Nash game, the equilibrium prices are as follows:
GAME THEORY FOR PRODUCING TWO SUBSTITUTABLE PRODUCTS 5
W1N=A1K1+A2K2+A3φ+A4ϑ1+A5ϑ2
A6
W2N=A7K1+A8K2+A9φ+A10ϑ1+A11 ϑ2
A6
PN=A12K1+A13 K2+A14φ+A15 ϑ1+A16ϑ2
A6
P1N=A17K1+A2K2+A3φ+A4ϑ1+A5ϑ2
A6
P2N=A7K1+A18K2+A9φ+A10 ϑ1+A11ϑ2
A6
(12)
where, we have:
A1= 12λ3+ 2λ2ζ−17λζ2−7ζ3
A2=ζ2(λ+ζ)
A3= 2ζ(λ+ζ)
A4= (2λ+ζ)(2λ−ζ)
A5=ζ(4λ+ζ)
A6= 8λ3−13λζ2+ 5ζ3
A7= 4λ2ζ+ 5λζ2+ζ3
A8= 24λ3−37λζ2−13ζ3
A9= 4ζ(λ+ζ)
A10 =ζ(8λ+ 3ζ)
A11 = 8λ2−3ζ2
A12 = 4λ3+λ2ζ−4λζ2−ζ3
A13 = 2ζ(λ+ζ)(λ−ζ)
A14 = 4(λ+ζ)(λ−ζ)
A15 =ζ(3λ+ 2ζ)
A16 =ζ(2λ+ 3ζ)
A17 = 4λ3+ 2λ2ζ−4λζ2−2ζ3
A18 = 8λ3−11λζ2−3ζ3
In the next section, the equilibrium prices are given under the Manufacturers-
Stackelberg game.
6 HAMED JAFARI AND SOROUSH SAFARZADEH
4. Manufacturers-Stackelberg game. In the Manufacturers-Stackelberg game,
the power structure of the manufacturers is considered to be higher than of the
common retailer. In this setting, the decisions are specified at two levels. The
manufacturers make their decisions firstly and then the common retailer sets his/her
prices (i.e., see: [39,9,10,49,20]). The Manufacturers-Stackelberg game is modelled
as follows:
Level1 : max
P, W1
M1= (P−K1)Q+ (W1−K1)Q1s.t. P ≥K1, W1≥K1
max
W2
M2= (W2−K2)Q2s.t. W2≥K2
Level2 : max
P1, P2
R= (P1−W1)Q1+ (P2−W2)Q2s.t. P1≥W1, P2≥W2
(13)
Theorem 4.1. The profit function Ris jointly concave with P1and P2.
By solving the first order deviations of Rin P1and P2, the prices set by the
retailer are determined with respect to the manufacturers’ prices as follows:
Theorem 4.2. Given the manufacturers’ decisions, the optimal prices set by the
retailer are:
Pi(W1, W2, P ) = ζ(λ+ζ)P+ (λ+ζ) (λ−ζ)Wi+λϑi+ζ ϑj
2(λ+ζ)(λ−ζ)
(14)
By substituting P1(W1, W2, P ) and P2(W1, W2, P ) into the manufacturers’ profit
functions, M′
1and M′
2are calculated. Now, the manufacturers set their decisions
by maximizing their own profits, simultaneously.
Theorem 4.3. M′
1is jointly concave with W1and P. Furthermore, M′
2is concave
with W2.
Theorem 4.4. In the Manufacturers-Stackelberg game, the equilibrium prices are
as follows:
W1MS =B1K1+B2K2+B3φ+B4ϑ1+B5ϑ2
B6
W2MS =B7K1+B8K2+B9φ+B9ϑ1+B10 ϑ2
B11
PMS =B12 K1+B13K2+B14φ+B15 ϑ1+B16ϑ2
B17
P1MS =B18 K1+B19K2+B20φ+B21 ϑ1+B22ϑ2
B23
P2MS =B24 K1+B25K2+B26φ+B22 ϑ1+B27ϑ2
B23
(15)
where, we have:
B1= 8λ4−8λ3ζ−11λ2ζ2+ 3λζ3−2ζ4
B2= 4λ3ζ−2λ2ζ2−6λζ3
B3= 8λ2ζ−6λζ2−2ζ3
B4= 8λ3−8λ2ζ−5λζ2+ 2ζ3
GAME THEORY FOR PRODUCING TWO SUBSTITUTABLE PRODUCTS 7
B5= 4λ2ζ+ 2λζ2−5ζ3
B6= 16λ4−8λ3ζ−15λ2ζ2+ 5λζ3+ 4ζ4
B7= 4λ2ζ−8λζ2+ 2ζ3
B8= 4λ3−8λ2ζ+ 2λζ2
B9= 2λζ −2ζ2
B10 = 4λ2−8λζ + 3ζ2
B11 = 8λ3−16λ2ζ+λζ2+ 4ζ3
B12 = 8λ4−8λ3ζ−13λ2ζ2+ 5λζ3+ 2ζ4
B13 = 2λ3ζ−2λζ3
B14 = 8λ3−8λ2ζ−2λζ2+ 2ζ3
B15 = 8λ2ζ−3λζ2−2ζ3
B16 = 6λ2ζ−3ζ3
B17 = 2(λ+ζ)(8λ3−16λ2ζ+λζ2+ 4ζ3)
B18 = 8λ5−8λ4ζ−11λ3ζ2+λ2ζ3+ 4ζ5
B19 = 4λ4ζ−4λ3ζ2−4λ2ζ3+ 4λζ4
B20 = 16λ3ζ−22λ2ζ2+ 2λζ3+ 4ζ4
B21 = 24λ4−48λ3ζ+ 13λ2ζ2+ 12λζ3−4ζ4
B22 = 20λ3ζ−28λ2ζ2−5λζ3+ 10ζ4
B23 = 4(λ+ζ)(λ−ζ)(8λ3−16λ2ζ+λζ2+ 4ζ3)
B24 = 16λ4ζ−24λ3ζ2−17λ2ζ3+ 21λζ4−2ζ5
B25 = 8λ5−16λ4ζ−2λ3ζ2+ 16λ2ζ3−6λζ4
B26 = 12λ3ζ−12λ2ζ2−6λζ3+ 6ζ4
B27 = 24λ4−48λ3ζ+ 6λ2ζ2+ 24λζ3−9ζ4
Now, in the following section, the equilibrium decisions in the Retailer-Stackelberg
game are obtained.
8 HAMED JAFARI AND SOROUSH SAFARZADEH
5. Retailer-Stackelberg game. In this section, the Retailer-Stackelberg game is
analysed in which the power structure of the common retailer is more than of the
manufacturers. At the first level, the common retailer sets his/her prices, whereas
the prices related to the manufacturers are determined at the second level. The
Retailer-Stackelberg game is formulated as follows:
Level1 : max
P, W1
M1= (P−K1)Q+ (W1−K1)Q1s.t. P ≥K1, W1≥K1
Level2 :max
W2
M2= (W2−K2)Q2s.t. W2≥K2
max
P1, P2
R= (P1−W1)Q1+ (P2−W2)Q2s.t. P1≥W1, P2≥W2
(16)
As stated in Theorem 3.1, the profit functions M1and M2are linearly increasing
with W1and W2, respectively. So, from the manufacturers’ point of view, the
optimal values of the wholesale prices are Wi= (Pi+Ki)/2 (i= 1,2). In this
setting, the game is reformulated as follows:
Level1 : max
P1,P2
R′=(P1−K1)
2Q1+(P2−K2)
2Q2s.t. P1≥K1, P2≥K2
Level2 :max
PM′
1= (P −K1) Q + (P1−K1)
2Q1s.t. P ≥K1
(17)
Regarding Theorem 3.2,M′
1is concave with P. Thus, given the retailer’s prices,
the optimal value of Pis obtained by solving the first order deviation of M′
1in P
as follows:
Theorem 5.1. Knowing the prices specified by the common retailer, the optimal
price set by the first manufacturer in the direct channel is as follow:
P(P1, P2) = 3ζP1+ 2ζP2+ 2φ+ (2λ−ζ)K1
4λ
(18)
Substituting P(P1, P2) into R′, the new profit function R′′ is calculated.
Theorem 5.2. R′′ is jointly concave with respect to P1and P2.
By maximizing the profit function R′′, the optimal values of P1and P2are
determined. Now, the equilibrium prices under the Retailer-Stackelberg game are
obtained by substituting the optimal values of P1and P2into relation 18 and
Wi= (Pi+Ki)/2 (i= 1,2).
Theorem 5.3. Under the Retailer-Stackelberg game, the equilibrium prices are:
W1RS =C1K1+C2K2+C3φ+C4ϑ1+C5ϑ2
C6
W2RS =C7K1+C8K2+C9φ+C5ϑ1+C10ϑ2
C6
PRS =C11K1+C12 K2+C13φ+C14 ϑ1+C15ϑ2
C6
P1RS =C16K1+C17 K2+C18φ+C19 ϑ1+C20ϑ2
C6
P2RS =C21K1+C22 K2+C23φ+C20 ϑ1+C24ϑ2
C6
(19)
GAME THEORY FOR PRODUCING TWO SUBSTITUTABLE PRODUCTS 9
where, we have:
C1= 48λ4+ 8λ3ζ−104λ2ζ2−61λζ3
C2=ζ4−2λ2ζ2
C3= 8λ2ζ+ 8λζ2+ζ3
C4= 16λ3−8λζ2
C5= 16λ2ζ+ 10λζ2
C6= 64λ4−144λ2ζ2−80λζ3−ζ4
C7= 8λ3ζ+ 6λ2ζ2−5λζ3−ζ4
C8= 48λ4−108λ2ζ2−62λζ3−2ζ4
C9= 8λ2ζ+ 8λζ2−ζ3
C10 = 16λ3−12λζ2
C11 = 32λ4+ 8λ3ζ−52λ2ζ2−46λζ3−17ζ4
C12 = 16λ3ζ−39λζ3−22ζ4
C13 = 32λ3−52λζ2−20ζ3
C14 = 24λ2ζ+ 16λζ2−2ζ3
C15 = 16λ2ζ+ 24λζ2+ 3ζ3
C16 = 32λ4+ 16λ3ζ−64λ2ζ2−42λζ3+ζ4
C17 = 2ζ4−4λ2ζ2
C18 = 16λ2ζ+ 16λζ2+ 2ζ3
C19 = 32λ3−16λζ2
C20 = 32λ2ζ+ 20λζ2
C21 = 16λ3ζ+ 12λ2ζ2−10λζ3−2ζ4
C22 = 32λ4−72λ2ζ2−44λζ3−3ζ4
C23 = 16λ2ζ+ 16λζ2−2ζ3
C24 = 32λ3−24λζ2
6. Discussions and results. In this section, the results obtained from the inves-
tigated models are revealed.
10 HAMED JAFARI AND SOROUSH SAFARZADEH
6.1. Comparisons of the equilibrium decisions under the various games.
In this subsection, the equilibrium prices, demands, and profits are compared under
the investigated games. The following results can be derived through a simple
analytical comparison between each two games.
Corollary 6.1. Comparisons of the prices given by the models are presented as
follows:
WiMS ≥WiRS ≥WiN
PMS ≥PRS ≥PN
PiMS ≥PiRS ≥PiN
(20)
Regarding Corollary 6.1, the highest and the lowest prices are obtained from the
Manufacturers-Stackelberg and Nash games, respectively. Due to more competition
established among the members under the Nash game, they set lower prices to do
not lose their customers. Moreover, the decision powers of the manufacturers are
higher than of the retailer. In this view, the prices made under the Manufacturers-
Stackelberg game are higher than under the other games.
Corollary 6.2. Comparisons related to the demands are provided as follows:
QMS ≥QRS ≥QN
QiN≥QiRS ≥QiMS (21)
The Manufacturers-Stackelberg and Nash games respectively lead to the largest
and the shortest values for the demand of the first product in the direct channel.
This inference is reversed for the demands of the products in the retail channel.
Customers are price sensitive and hence higher prices lead to lower demands. Con-
sequently, the highest and the lowest retail demands are given under the Nash and
Manufacturers-Stackelberg games, respectively.
Corollary 6.3. The fol lowing relations hold among the profits obtained from the
developed games:
M1MS ≥M1RS ≥M1N
M2MS ≥M2RS ≥M2Nif D1≥0, D2≥0, and D3≥0
M2RS > M2MS ≥M2Nif D1<0, D2≥0, and D3≥0
M2RS ≥M2N> M2M S if D1<0, D2<0, and D3≥0
RRS ≥RN≥RMS
SC M S ≥SCRS ≥S CN
(22)
where, D1,D2, and D3are defined in Appendix A.
The highest and the lowest profits for the first manufacturer and the whole system
are obtained from the Manufacturers-Stackelberg and Nash games, respectively.
From the common retailer’s point of view, the Retailer-Stackelberg game leads to
the highest profit. Moreover, the highest profit of the second manufacturer is given
by the games under some threshold conditions.
GAME THEORY FOR PRODUCING TWO SUBSTITUTABLE PRODUCTS 11
6.2. Numerical example. In this subsection, a numerical example is provided
to well illustrate the research problem. The values of the considered parameters
have been presented in Table 1. Without loss of generality, it is assumed that
the manufacturing costs as well as the market scales in the retail channel for both
products are the same, i.e., K1=K2and ϑ1=ϑ2. Applying the investigated
game theoretical models, the equilibrium strategies related to the prices, demands,
and profits are obtained under the various power structures. The results have been
summarized in Table 2.
Table 1. The values of the parameters in the numerical example
Parameters K1=K2φ ϑ1=ϑ2λ ζ
Values 100 200 300 3.0 1.0
Table 2. The results obtained from the investigated models for
the numerical example
Games Prices
W1W2P P1P2
Nash 130.23 130.23 141.86 160.47 160.47
Manufacturers-
Stackelberg
161.39 153.16 157.59 195.09 190.98
Retailer-
Stackelberg
134.58 133.86 145.24 169.15 167.73
Demands
Q Q1Q2
Nash 95.35 120.93 120.93
Manufacturers-
Stackelberg
113.29 63.29 79.75
Retailer-
Stackelberg
101.15 105.51 111.21
Profits
M1M2R SC
Nash 7647.38 3656.03 7312.06 18615.47
Manufacturers-
Stackelberg
9534.33 3627.72 6637.06 19799.11
Retailer-
Stackelberg
8224.57 3766.11 7414.31 19405.00
6.3. Sensitivity analysis. Now, the effects of the considered parameters are in-
vestigated on the members’ profits and the profit of the while system obtained from
the models. For this reason, the parameters are changed in the numerical example
presented in the previous subsection. The changes of the profits with respect to the
manufacturing costs (i.e., K1and K2), the market scale of the first product in the
direct channel (i.e., φ), the market scales of the products in the retail channel (i.e.,
ϑ1and ϑ2), the self-price sensitivity of the demands (i.e., λ), and the cross-price
sensitivity of the demands (i.e., ζ) have been exhibited in Figures 1-5, respectively.
12 HAMED JAFARI AND SOROUSH SAFARZADEH
Figure 1. Changes of the profits in the manufacturing costs
Corollary 6.4. When the manufacturing costs of the products increase, the mem-
bers’ profits and the profit of the whole system decrease under the investigated games.
Figure 2. Changes of the profits in the market scale for the first
product in direct channel
Corollary 6.5. Higher the market scale of the first product in the direct channel
leads to higher profits under the developed games.
Corollary 6.6. If the market scales of the products in the retail channel increase,
then the profits are raised in all games.
Corollary 6.7. By increasing the self-price sensitivity of the demands, the mem-
bers’ profits and the profit of the whole system are reduced under the game models.
Corollary 6.8. More the cross-price sensitivity of the demands leads to more profits
for the members and the whole system.
GAME THEORY FOR PRODUCING TWO SUBSTITUTABLE PRODUCTS 13
Figure 3. Changes of the profits in the market scales for the prod-
ucts in the retail channel
Figure 4. Changes of the profits in the self-price sensitivity of the demands
6.4. Effects of opening the direct channel. In this section, the profit values of
both manufacturers are compared under the same conditions in order to recognize
the effects of establishing the direct channel on their profits. For this reason, it
is assumed that the production costs for both manufacturers and the total market
scales for both products are the same (i.e., K1=K2=Kand ϑ1+φ=ϑ2=ϑ).
By substituting relations K1=K2=K,ϑ1=ϑ−φ, and ϑ2=ϑinto the profit
values of the manufacturers 1 and 2, the new profits MA1and MA2are obtained,
respectively. Now, we compare the profit functions M A1and M A2under the
developed game models.
14 HAMED JAFARI AND SOROUSH SAFARZADEH
Figure 5. Changes of the profits in the cross-price sensitivity of
the demands
Corollary 6.9. One can derive that:
MA1N≥M A2Nif D4≥0
MA1M S ≥MA2M S if D5≥0
MA1RS ≥M A2RS if D6≥0
(23)
where, D4,D5, and D6are defined in Appendix A.
Regarding Corollary 6.9, the profits received by the first manufacturer under
different game models are higher than by the second manufacturer, if some threshold
conditions are met. As a matter of fact, improving the manufacturer’s profit by
opening a direct channel depends on the values of the considered parameters.
7. Conclusions. In this research, a game theoretical framework was developed
to specify the prices of two substitutable products under a supply chain consist-
ing of duopolistic manufacturers and monopolistic retailer. The first manufacturer
sells his/her products directly to customers in addition to via the common retailer
whereas the second manufacturer sells only through the common retailer. In this
point of view, the pricing decisions were analyzed under different competitions es-
tablished among the members. As the members had a similar power structure, they
set the decisions under the Nash game. If the power structure of the manufacturers
was more than of the retailer, then the Manufacturers-Stackelberg game was de-
veloped. Moreover, the Retailer-Stackelberg game was established when the power
structure of the common retailer was higher than of the manufacturers.
Then, the results given by the investigated games were revealed. It was found
that the highest and the lowest prices are specified under the Manufacturers-
Stackelberg and Nash games, respectively. The Manufacturers-Stackelberg and
Nash games lead to the largest values respectively for the demand of the first prod-
uct in the direct channel and the demands of the products in the retail channel. The
highest and the lowest profits for the first manufacturer and the whole system are
given by the Manufacturers-Stackelberg and Nash games, respectively. The highest
GAME THEORY FOR PRODUCING TWO SUBSTITUTABLE PRODUCTS 15
profit of the second manufacturer is obtained under a threshold condition. Fur-
thermore, the Retailer-Stackelberg game leads to the highest profit for the common
retailer.
Finally, a sensitivity analysis was implemented on the considered parameters. We
derived that by increasing the manufacturing costs, the members’ profits and the
profit of the whole system decrease. As the market scales of the products increase,
the profits increase in all games. When the self-price sensitivity of the demands
increases, the profit values of the members and the whole system are raised, while
this inference is reversed by increasing the cross-price sensitivity of the demands.
There are some directions for future researches. In the current study, production
costs for the first manufacturer to sell his/her products through the direct and retail
channels were similar. Considering different production costs faced by the first
manufacturer under these two channels would be an interesting idea. Furthermore,
different contract mechanisms (like discount, profit/revenue sharing, etc.) can be
considered in future studies.
Appendix A. Notations and proofs.
Proof of Theorem 3.1.The first and second order deviations of M1and M2respec-
tively to W1and W2are:
∂M1
∂W1
=Q1=ϑ1−λP1+ζ(P+P2)≥0,∂2M1
∂W12= 0
dM2
dW2
=Q2=ϑ2−λP2+ζ(P+P1)≥0,d2M2
dW22= 0
Clearly, from the above relations, M1and M2are linearly increasing with W1and
W2, respectively.
Proof of Theorem 3.2.We have:
dM′
1
dP =φ−2λP +ζ(P1+P2) + λK1+ζP1−K1
2,d2M′
1
dP 2=−2λ < 0
∂R′
∂P1
=1
2[ϑ1−2λP1+ζ(P+P2) + λK1+ζ(P2−K2)]
∂R′
∂P2
=1
2[ϑ2−2λP2+ζ(P+P1) + λK2+ζ(P1−K1)]
H="∂2R′
∂P1
2
∂2R′
∂P2∂ P1
∂2R′
∂P1∂ P2
∂2R′
∂P2
2#=−λ ζ
ζ−λ
Considering assumption (3), the Hessian matrix His negative definite. Thus, from
the above relations, M′
1is concave and Ris jointly concave with their own prices.
Proof of Theorem 3.3.Solving the first order deviations of M′
1in Pand R′in P1
and P2and substituting them into relation Wi= (Pi+Ki)/2 (i= 1,2), the prices
presented in relation 12 can be calculated. By some algebraic manipulations that
are not presented here for brevity, it is proved that these prices meet constraints 8
and 9. Therefore, they are feasible and can be considered as the equilibrium prices
under the Nash game.
16 HAMED JAFARI AND SOROUSH SAFARZADEH
Proof of Theorem 4.1.We have:
∂R
∂P1
=ϑ1−2λP1+ζ(P+ 2P2) + λW1−ζW2
∂R
∂P2
=ϑ2−2λP2+ζ(P+ 2P1) + λW2−ζW1
H="∂2R
∂P1
2
∂2R
∂P2∂ P1
∂2R
∂P1∂ P2
∂2R
∂P2
2#=−2λ2ζ
2ζ−2λ
From assumption (3), His negative definite and thus Ris jointly concave in P1and
P2.
Proof of Theorem 4.2.Solving the first order deviations of Rin P1and P2, relation
14 can be given.
Proof of Theorem 4.4.By calculating the first and second order deviations, we have:
∂M ′
1
∂W1
=1
2(ϑ1−2λW1+ζW2+ 2ζ P + (λ−ζ)K1)
∂M ′
1
∂P =ζ ϑ1+ζϑ2+ 2 (λ−ζ)φ+ 2ζ(λ−ζ)W1+ζ(λ−ζ)W2
2(λ−ζ)
−4λ2−λζ −ζ2P−(2λ2−3λζ −ζ2)K1
2(λ−ζ)
H="∂2M′
1
∂W1
2
∂2M′
1
∂P ∂ W1
∂2M′
1
∂W1∂ P
∂2M′
1
∂P 2#="−λ ζ
ζ−2(λ2−λζ−ζ2)
(λ−ζ)#
dM′
2
dW2
=1
2[ϑ2−2λW2+ζW1+ζ P +λK2],d2M′
2
dW22=−λ < 0
Regarding assumption (3), His negative definite. Therefore, M′
1is jointly concave
in W1and P. Moreover, M′
2is concave with W2.
Proof of Theorem 5.1.Relation 18 is obtained by solving the calculated first order
deviation in P.
Proof of Theorem 5.2.We have:
∂R′′
∂P1
=−24λ2−ζ2P1+8λζ + 5ζ2P2+ 2 2λ2+λζ −2ζ2K1
8λ
−4λζ + 3ζ2K2−2ζφ + 4λϑ1
8λ
∂R′′
∂P2
=8λζ + 5ζ2P1−42λ2−ζ2P2−2λζ + 3ζ2K1
8λ
+2 2λ2−ζ2K2+ 2ζφ + 4λϑ2
8λ
H="∂2R′′
∂P1
2
∂2R′′
∂P2∂ P1
∂2R′′
∂P1∂ P2
∂2R′′
∂P2
2#="−4λ2−ζ2
4λ
8λζ+5ζ2
8λ
8λζ+5ζ2
8λ−2λ2−ζ2
2λ#
The hessian matrix His negative definite by considering assumption (3). So, R′′ is
jointly concave in P1and P2.
GAME THEORY FOR PRODUCING TWO SUBSTITUTABLE PRODUCTS 17
Proof of Theorem 5.3.The prices presented in relation 19 are determined by solving
the above first order deviations in P1and P2and substituting them into relations
18 and Wi= (Pi+Ki)/2 (i= 1,2). Some algebraic manipulations show the
given prices are feasible and therefore they are as the equilibrium prices under the
Retailer-Stackelberg game.
Notations.
D1=F1F2
F3
+λF4F5
F6
where, we have:
F1= 32 λ3ϑ2−2K1ζ4−2K2ζ4−32 K2λ4−2φ ζ 3+ 12 K1λ2ζ2+
72 K2λ2ζ2−10 K1λ ζ3+ 16 K1λ3ζ+ 36 K2λ ζ 3+ 16 λ φ ζ2+
16 λ2φ ζ + 20 λ ϑ1ζ2+ 32 λ2ϑ1ζ−24 λ ϑ2ζ2
F2= 16 K2λ5+ 8 K1ζ5+ 10 K2ζ5−16 λ4ϑ2 + 9 φ ζ4+ϑ1ζ4−ϑ2ζ4
+ 53 K1λ2ζ3−6K1λ3ζ2−20 K2λ2ζ3−44 K2λ3ζ2−2λ2ϑ1ζ2+ 36 λ2ϑ2ζ2
+ 43 K1λ ζ4−24 K1λ4ζ+ 18 K2λ ζ 4+ 17 λ φ ζ3−8λ3φ ζ + 18 λ ϑ2ζ3
F3=−64 λ4+ 144 λ2ζ2+ 80 λ ζ 3+ζ42
F4= 2 K1ζ3−4K2λ3−4K2ζ3+ 4 λ2ϑ2−2φ ζ2−2ϑ1ζ2+ 3 ϑ2ζ2
+ 2 λφζ + 2 λ ϑ1ζ−8λ ϑ2ζ−8K1λ ζ2+ 4 K1λ2ζ+K2λ ζ 2+ 8 K2λ2ζ
F5= 24 K2λ5+ 2 K1ζ5−16 K2ζ5−24 λ4ϑ2−6φ ζ 4−10 ϑ1ζ4+ 9 ϑ2ζ4
+ 17 K1λ2ζ3+ 24 K1λ3ζ2+ 64 K2λ2ζ3−26 K2λ3ζ2+ 12 λ2φ ζ 2+ 28 λ2ϑ1ζ2
−6λ2ϑ2ζ2−21 K1λ ζ4−16 K1λ4ζ+ 2 K2λ ζ 4−48 K2λ4ζ+ 6 λφζ3
−12 λ3φ ζ + 5 λ ϑ1ζ3−20 λ3ϑ1ζ−24 λ ϑ2ζ3+ 48 λ3ϑ2ζ
F6= 16 8λ3−16 λ2ζ+λ ζ2+ 4 ζ3
−8λ5+ 16 λ4ζ+ 7 λ3ζ2−20 λ2ζ3+λ ζ 4+ 4 ζ5
D2=λF7F8
F9
+F10F11
F12
where, we have:
F7= 2 K1ζ3−4K2λ3−4K2ζ3+ 4 λ2ϑ2−2φ ζ2−2ϑ1ζ2+ 3 ϑ2ζ2
+ 2 λφζ + 2 λ ϑ1ζ−8λ ϑ2ζ−8K1λ ζ2+ 4 K1λ2ζ+K2λ ζ 2+ 8 K2λ2ζ
F8= 24 K2λ5+ 2 K1ζ5−16 K2ζ5−24 λ4ϑ2−6φ ζ 4−10 ϑ1ζ4+ 9 ϑ2ζ4
+ 17 K1λ2ζ3+ 24 K1λ3ζ2+ 64 K2λ2ζ3−26 K2λ3ζ2+ 12 λ2φ ζ 2+ 28 λ2ϑ1ζ2
−6λ2ϑ2ζ2−21 K1λ ζ4−16 K1λ4ζ+ 2 K2λ ζ 4−48 K2λ4ζ+ 6 λφζ3
−12 λ3φ ζ + 5 λ ϑ1ζ3−20 λ3ϑ1ζ−24 λ ϑ2ζ3+ 48 λ3ϑ2ζ
F9= 16 8λ3−16 λ2ζ+λ ζ2+ 4 ζ3
18 HAMED JAFARI AND SOROUSH SAFARZADEH
−8λ5+ 16 λ4ζ+ 7 λ3ζ2−20 λ2ζ3+λ ζ 4+ 4 ζ5
F10 =K1ζ3−8K2λ3+ 7 K2ζ3+ 8 λ2ϑ2+4φ ζ2+ 3 ϑ1ζ2−3ϑ2ζ2+ 4 λφζ
+ 8 λ ϑ1ζ+ 5 K1λ ζ2+ 4 K1λ2ζ+ 15 K2λ ζ 2
F11 = 8 λ3ϑ2−6K1ζ4−2K2ζ4−8K2λ4−4φ ζ3+ 2 ϑ1ζ3−2ϑ2ζ3+K1λ2ζ2
+ 15 K2λ2ζ2−17 K1λ ζ3+ 12 K1λ3ζ+ 5 K2λ ζ3+ 4 λ2φ ζ + 3 λ ϑ1ζ2−11 λ ϑ2ζ2
F12 =−16 λ3+ 26 λ ζ2+ 10 ζ3−32 λ3+ 52 λ ζ2+ 20 ζ3
D3=−F13F14
F3
−F15F16
F12
where, we have:
F13 = 32 λ3ϑ2−2K1ζ4−2K2ζ4−32 K2λ4−2φ ζ3+ 12 K1λ2ζ2+ 72 K2λ2ζ2
−10 K1λ ζ3+ 16 K1λ3ζ+ 36 K2λ ζ 3+ 16 λφζ2+ 16 λ2φ ζ + 20 λ ϑ1ζ2
+ 32 λ2ϑ1ζ−24 λ ϑ2ζ2
F14 = 16 K2λ5+ 8 K1ζ5+ 10 K2ζ5−16 λ4ϑ2 + 9 φ ζ4+ϑ1ζ4−ϑ2ζ4
+ 53 K1λ2ζ3−6K1λ3ζ2−20 K2λ2ζ3−44 K2λ3ζ2−2λ2ϑ1ζ2+ 36 λ2ϑ2ζ2
+ 43 K1λ ζ4−24 K1λ4ζ+ 18 K2λ ζ 4+ 17 λ φ ζ3−8λ3φ ζ + 18 λ ϑ2ζ3
F15 =K1ζ3−8K2λ3+ 7 K2ζ3+ 8 λ2ϑ2+4φ ζ2+ 3 ϑ1ζ2−3ϑ2ζ2+ 4 λφζ
+ 8 λ ϑ1ζ+ 5 K1λ ζ2+ 4 K1λ2ζ+ 15 K2λ ζ 2
F16 = 8 λ3ϑ2−6K1ζ4−2K2ζ4−8K2λ4−4φ ζ3+ 2 ϑ1ζ3−2ϑ2ζ3+K1λ2ζ2
+ 15 K2λ2ζ2−17 K1λ ζ3+ 12 K1λ3ζ+ 5 K2λ ζ 3+ 4 λ2φ ζ + 3 λ ϑ1ζ2−11 λ ϑ2ζ2
D4= 128K2λ7−256K2λ6ζ−424K2λ5ζ2+ 584K2λ4ζ3+ 648K2λ3ζ4
−136K2λ2ζ5−192K2λζ6−32K2ζ7−256K ϑλ5ζ+ 16K ϑλ4ζ2+ 808Kϑλ3ζ3
+464Kϑλ2ζ4−152K ϑλζ 5−80Kϑζ6−128Kφλ6+ 192Kφλ5ζ+ 480K φλ4ζ2
−424Kφλ3ζ3−604K φλ2ζ4+ 52K φλζ5+ 72Kφζ6+ 120ϑ2λ3ζ2+ 240ϑ2λ2ζ3
+120ϑ2λζ4−128ϑφλ5+ 256ϑφλ4ζ+ 352ϑφλ3ζ2−456ϑφλ2ζ3−484ϑφλζ 4
−60ϑφζ5+ 192φ2λ5−192φ2λ4ζ−488φ2λ3ζ2+ 224φ2λ2ζ3+ 407φ2λζ 4+ 72φ2ζ5
D5= 512K2λ9−2624K2λ8ζ+ 3136K2λ7ζ2+ 3352K2λ6ζ3−7253K2λ5ζ4
−248K2λ4ζ5+ 4175K2λ3ζ6−618K2λ2ζ7−636K2λζ8+ 72K2ζ9−1472K ϑλ7ζ
+5664Kϑλ6ζ2−4224K ϑλ5ζ3−5010K ϑλ4ζ4+ 5004Kϑλ3ζ5+ 798Kϑλ2ζ6
−952Kϑλζ 7−72K ϑζ8−640Kφλ8+ 3264Kφλ7ζ−4480Kφλ6ζ2
−1592Kφλ5ζ3+ 6282K φλ4ζ4−1090K φλ3ζ5−2272Kφλ2ζ6+ 360Kφλζ7
+288Kφζ 8+ 1008ϑ2λ5ζ2−2456ϑ2λ4ζ3+ 771ϑ2λ3ζ4+ 916ϑ2λ2ζ5
GAME THEORY FOR PRODUCING TWO SUBSTITUTABLE PRODUCTS 19
−261ϑ2λζ6−110ϑ2ζ7−384ϑφλ7+ 3008ϑφλ6ζ−7456ϑφλ5ζ2
+5960ϑφλ4ζ3+ 1458ϑφλ3ζ4−2934ϑφλ2ζ5+ 100ϑφλζ6+ 368ϑφζ 7
+704φ2λ7−3712φ2λ6ζ+ 6576φ2λ5ζ2−3456φ2λ4ζ3
−1845φ2λ3ζ4+ 1866φ2λ2ζ5+ 64φ2λζ6−224φ2ζ7
D6= 1024K2λ9−2048K2λ8ζ−4672K2λ7ζ2+ 6976K2λ6ζ3+ 9888K2λ5ζ4
−6176K2λ4ζ5−8661K2λ3ζ6+ 191K2λ2ζ7+ 1280K2λζ8−84K2ζ9−2048K ϑλ7ζ
+128Kϑλ6ζ2+ 8256K ϑλ5ζ3+ 3808K ϑλ4ζ4−6480Kϑλ3ζ5−4622Kϑλ2ζ6
−419Kϑλζ 7−2K ϑζ8−1024Kφλ8+ 1536Kφλ7ζ+ 5120Kφλ6ζ2−4288K φλ5ζ3
−9664Kφλ4ζ4+ 912K φλ3ζ5+ 5014K φλ2ζ6+ 1210Kφλζ7−76Kφζ8+ 960ϑ2λ5ζ2
+1664ϑ2λ4ζ3+ 528ϑ2λ3ζ4−176ϑ2λ2ζ5−ϑ2λζ6−1024ϑφλ7+ 2048ϑφλ6ζ
+4096ϑφλ5ζ2−4032ϑφλ4ζ3−7712ϑφλ3ζ4−2864ϑφλ2ζ5−206ϑφλζ6−ϑφζ 7
+1536φ2λ7−1536φ2λ6ζ−5824φ2λ5ζ2+ 1408φ2λ4ζ3+ 7352φ2λ3ζ4
+3520φ2λ2ζ5+ 316φ2λζ6−17φ2ζ7
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Received January 2022; revised April 2022; early access June 2022.
E-mail address:hamed.jafari@iut.ac.ir
E-mail address:s.safarzadeh@qiet.ac.ir
