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On a cooperative advertising model for a supply chain with one manufacturer and one retailer
European Journal of Operational Research 219 (2012) 458–466
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On a cooperative advertising model for a supply chain with one manufacturer and one retailer Amir Ahmadi-Javid ⇑, Pooya Hoseinpour Department of Industrial Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
a r t i c l e
i n f o
Article history: Received 24 September 2010 Accepted 21 June 2011 Available online 22 July 2011 Keywords: Supply chain management Cooperative advertising Coordination Game theory
a b s t r a c t Huang and Li (2001), Huang et al. (2002), Li et al. (2002), Xie and Ai (2006) and Yue et al. (2006) recently studied a game-theoretic model for cooperative advertising in a supply chain consisting of one manufacturer and one retailer. However, the sales-volume (demand) function considered in this model can become negative for some values of the decision variables, and in fact, this does happen for the proposed Stackelberg and Nash equilibrium solutions. Yue et al. (2006) acknowledge the negativity problem and suggest two constraints to fix it; however, they do not incorporate these constraints into their mathematical analysis. In this paper, we show that the results obtained by analyzing the advertising model under the constraints suggested by Yue et al. can differ significantly from those obtained in the previous papers. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Cooperative advertising is a coordination mechanism for advertising activities in a supply chain. In cooperative advertising, the manufacturers may contribute part of the advertising costs which are paid by retailers (see Hutchins, 1953; Bergen and John, 1997). Some studies consider a static, single-period relationship to explore the detailed interactions among the factors involved in cooperative advertising (see e.g., Huang and Li, 2001; Huang et al., 2002; Yue et al., 2006; Karray and Zaccour, 2006, 2007). Others focus on long-term, multi-period cooperative advertising relationship between manufacturers and retailers and use dynamic models and differential games to analyze that relationship (see e.g., Jorgensen et al., 2000; Jorgensen et al., 2001; Jorgensen and Zaccour, 2003a; Jorgensen et al., 2003; He et al., 2011). A large body of literature is also devoted to the simultaneous study of pricing and cooperative advertising. For static models we refer the reader to Szmerekovsky and Zhang (2009), Xie and Neyret (2009), Xie and Wei (2009) and Yan (2009). For dynamic models, see He et al. (2009), Jorgensen et al. (2001), Jorgensen and Zaccour (1999) and Jorgensen and Zaccour (2003b). This paper analyzes a static, single-period model that has been studied by several authors in the past ten years. Huang and Li (2001), Huang et al. (2002), Li et al. (2002), Xie and Ai (2006), and Yue et al. (2006) study a static game-theoretic model for cooperative advertising in a supply chain with one manufac-
⇑ Corresponding author. Tel.: +98 2166413034. E-mail address: [email protected] (A. Ahmadi-Javid). 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.06.032
turer and one retailer. However, the sales-volume function considered in their model has the unfortunate property of becoming negative for some values of the decision variables. Here we correct the sales function and develop the same game-theoretic analysis for the modified sales function. The results we obtain from this analysis differ significantly from the original. The paper is organized as follows. The notation and model are presented in Section 2. Sections 3 and 4 respectively characterize the model’s Stackelberg and Nash equilibria. In Section 5, the model’s Pareto-efficient points are studied. In Section 6, we present some managerial implications arising out of the model; and, lastly, conclusion is given in Section 7. The proofs are given in Appendix A. 2. Modeling Consider a supply chain with one retailer and one manufacturer. We now define the notation to be used in the rest of the paper. a is the retailer’s local advertising expenditure; q is the manufacturer’s investment in the national brand name, including national advertising costs; t is the fraction of the total local-advertising expenditure that the manufacturer agrees to share with the retailer (0 6 t 6 1); qm is the manufacturer’s marginal profit for each product unit; qr is the retailer’s marginal profit for each product unit; c is the cost for each unit of q; pm is the manufacturer’s gross profit; pr is the retailer’s gross profit.
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The gross profit of the retailer and the manufacturer, respectively, are as follows:
pr ða; q; tÞ :¼ qr Sða; qÞ ð1 tÞa; pm ða; q; tÞ :¼ qm Sða; qÞ ta cq; where S(a, q) is the one-period sales function. Huang and Li (2001), Huang et al. (2002), Li et al. (2002), Xie and Ai (2006) and Yue et al. (2006) consider the following sales function:
Sða; qÞ :¼ a bac qd ; where the parameters a, c, d and b are positive constants. An important problem associated with the above sales function is that it is negative for some values of a > 0, q > 0. Yue et al. (2006) recognize this problem and propose two constraints to ensure that the national and local investments do not go below certain appropriate levels: i.e., a P a0 and q P q0, for given positive constants a0 and q0. However, they do not consider the two constraints in their analysis. Here, we show that the results obtained by analyzing the above advertising model when accounting for the constraints a P a0 and q P q0 can differ significantly from the original results. It should be noted that this negativity problem can be overcome using an entirely different approach, which is investigated by Ahmadi-Javid and Hoseinpour (2011). One might believe that these results are of little interest from a managerial viewpoint. In other words, it may seem that the problem of having a negative sales function is not a managerial concern. However, this argument can be countered using the following numerical example. For a supply-chain channel, suppose that the sales function is
Before proceeding, it should be noted that the parameters a0 and q0 are in fact related. Indeed, by defining a¯(q):= [b /(aqd)]1/c we notice the following equivalence:
ðqÞ; Sða; qÞ P 0 () a P a so that if q P q0 for a positive constant q0, then one must have a P a¯(q0). This means that when q0 is a given positive constant, a0 must satisfy a0 P a¯(q0) to guarantee that S(a, q) is nonnegative for a P a0 and q P q0. Thus, in the sequel, we assume that q0 is a given positive constant and that a0 ¼ a¯(q0). 3. Stackelberg equilibrium point Inthis section, we determine the Stackelberg equilibrium point aS ; qS ; t S of the model described in the previous section when imposing the two constraints a P a0 and q P q0. Let us define ar ðq; tÞ ¼ arg maxaPa0 ðpr ða; q; tÞÞ, and suppose that (q⁄, t⁄) is the solution to the following optimization problem:
max pm ar ðq; tÞ; q; t q;t
ðProblem SÞ
s:t: q P q0 0 6 t 6 1:
Then, point aS ; qS ; tS is defined as the Stackelberg equilibrium aS ; qS ; t S ¼ ar ðq ; t Þ; q ; t . Since pr(a, q, t) ¼ qrS(a, q) (1 t)a ¼ qr(a b acqd) (1 t)a is concave for a P a0, we have:
( ar ðq; tÞ
ðq0 Þ; ¼ max a
Sða; qÞ :¼ 100 220a0:1 q0:2 ; with qm ¼ 0.8, qr ¼ 0.7, c ¼ 1. By using the existing results, the Stackelberg equilibrium point aS ; qS ; t S for the model becomes
aS ¼ 7:61;
qS ¼ 15:21;
tS ¼ 0:04:
However, by substituting the investment values aS ; qS in the sales function, we obtain
S aS ; qS ¼ 4:20 < 0: It is clear that no manager would accept these results. She or he would immediately want to know what caused the negativity problem and how it can be overcome. In Section 6, we will also discuss the impacts of this problem on our understanding of cooperative advertising. The model under consideration in this paper is the same as the one proposed by Huang and Li (2001), but with two constraints a P a0 and q P q0, which were proposed by Yue et al. (2006). The following are the differences that exist between Huang and Li’s model and the other models derived from it: In Huang et al. (2002), c ¼ b ¼ 1; In Li et al. (2002) and Xie and Ai (2006), c ¼ b ¼ a ¼ 1; In Yue et al. (2006), c ¼ 1, and the sales function is
Sða; qÞ :¼ ða bac qd Þð1 eÞe ; where e is the price-deduction percentage offered by the manufacturer, and e is the price elasticity (both e and e are positive constants). Also, because the manufacturer offers an e pricereduction percentage directly to the customers, the manufacturer’s marginal profit becomes qm eP0 where P0 is the full price to the customers (see Yue et al., 2006, for more details). In fact, from a mathematical standpoint, the model presented in Yue et al. (2006) could be recast as the model considered in this paper through the following simple reparameterization:
a :¼ að1 eÞe ; b :¼ bð1 eÞe ; qm :¼ qm eP0 :
cqr b ð1 tÞq
) 1 cþ1 : d
From this formula, the Stackelberg equilibrium point can be obtained by solving the following two smooth optimization problems:
max qm a qm bqd q;t
cqr b ð1tÞqd
c 1þ c
t
ðq0 Þ 6 s:t: q P q0 ; 0 6 t < 1; a
cqr b ð1tÞqd
cqr b
1þ1 c
cq ðProblem S1Þ
1þ1 c
ð1tÞqd
ðq0 ÞÞc t a ðq0 Þ cq max qm a qm bqd ða q;t
ðq0 Þ P s:t: q P q0 ; 0 6 t 6 1; a
cqr b
1þ1 c
ð1tÞqd
ðProblem S2Þ :
The constraint t 6 1 in Problem S is replaced with the constraint t < 1 in Problem S1 because its objective function tend to 1 as t ? 1. We now introduce the following parameters, which are used in the rest of the paper for notational simplicity: cþ1 E :¼ qdþ 0
ccþ1 d
b1 cc
G :¼ qd0ac 1
c H :¼ ðcc bað1þcÞ qd 0 Þ
dc ðdþcþ1Þ 1c F :¼ b1þd dc ccð1þdÞ aqd0 : Furthermore, suppose that q1 ; t 1 and q2 ; t2 are the optimal solutions of Problem S1 and Problem S2, respectively, and that a1 and a2 are calculated as follows:
ai
9 8 1 !cþ1 = < cq b r ðq Þ; ; ¼ max a d ; : 0 1 t i qi
i ¼ 1; 2:
Taking into account the above notation, the following theorem characterizes the model’s Stackelberg equilibrium point.
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Theorem 1. The Stackelberg equilibrium point aS ; qS ; tS exists. To obtain qS ; t S , the optimal solutions of Problem S1 and Problem S2 must first be determined; subsequently, the optimal solution with greater objective value is qS ; tS . Tables 1 and 2 present the optimal solution of Problem S1 and Problem S2, respectively. The calculation of the value of aS is based on qS ; t S as follows:
Remark 1. The Stackelberg equilibrium point proposed by Huang and Li (2001), Huang et al. (2002), Li et al. (2002), and Yue et al. (2006) is the following:
qS ¼
h d cþ1 (
c
bcc ðqm cqr Þ
1 q qrcq m r
idþc1þ1
;
qm =qr P c þ 1; qm =qr < c þ 1;
8 9 1 !cþ1 < = cq b r ðq Þ; aS ¼ max a : d : 0 ; 1 t S qS
t S ¼
Tables 1 and 2 also present a1 and a2 to simplify the computation of aS .
This solution was recently modified as follows, in the note by Xie and Ai (2006) for the case qm/qr < c þ 1:
aS ¼
Note that, due to the complexity of the resulting expressions, explicitly comparing the optimal values for Problem S1 and Problem S2 is impractical, but by using Tables 1 and 2 it is possible to pinpoint the Stackelberg equilibrium point according to the known values of the parameters.
Set of conditions
Optimal solution of Problem S1
S1-a
S1-a1: qm (1 + c)qr > 0
q1 ¼ q0 ; t 1 ¼ 1 q rcq m r 1þ1 c a1 ¼ cbðqmqdcqr Þ
S1-a2: qm cqr > H S1-a3: qm cqr 6 E S1-b
cþ1 qm qr > E S1-b1 : q ð1þ cÞ r d q ð1þcÞ S1-b2 : r q qr > F m
q
S1-c1: qm cqr > G d q S1-c2 : qr q rcq
r
S1-c3:qm cqr 6 F
idþc1þ1 bcðdþ1Þ ðqm cqr Þ :
"
qm d cð1 þ cÞ
tS ¼ 0;
aS ¼
q1 ¼
cþ1
dqm cðcþ1Þ
t1 ¼ 0; a1 ¼
ðcqr Þc b d
cð1þcÞ dqm
dþc1þ1
ðcqr Þ1þd b
1 q1 ¼ daqd0 ðqm cqr Þ=c 1þd d dþ1 dþcþ1 1 t1 ¼ 1 cqr bc aqd0 dc dðq ccq Þ m r 1c a1 ¼ abqd q1 ¼ q0 ; t 1 ¼ 0; a1 ¼
S1-d1: qr > H
qd0
S1-e2: qm cqr 6 H
q1 ¼ q0 ; t 1 ¼ 1 1c a1 ¼ abqd
ðcqr Þc að1þcÞ qd0 b
In this section, we find the Nash equilibrium point qN ; aN ; tN for the model given in Section 2 when considering the two constraints a P a0 and q P q0. By defining
ar ðq; tÞ ¼ arg maxðpr ða; q; tÞÞ aPa0 qm ðaÞ; t m ðaÞ ¼ arg max ðpm ðq; a; tÞÞ; qPq0 ;06t61
dþ1 d
S1-f 2 : qr
1 d
P F ðqm cqr Þ
cÞ d S1-f 3 : qr qr ð1þ 6F qm S1-g
S1-g1: qm (c + 1)qr > 0 S1-g2: qm cqr > E S1-g3: qm cqr > F
c
ðcqr Þ
1þc
ðaqd0 Þ
a P a0 ; d1c
S2-a
S2-a1: qm > G ðdþ1Þ
S2-a2 : qr S2-b
1c t1 ¼ 0; a1 ¼ abqd 0
Theorem 2. The Nash equilibrium point exists and is unique, as presented in Table 3. 1
q1 ¼ ½ðd=cÞcþ1 bcc ðqm cqr Þdþcþ1 t1 ¼ 1 q qrcq m
r
Table 3 Nash equilibrium point.
1
a1 ¼ ½ðc=dÞd bcðdþ1Þ ðqm cqr Þdþcþ1
Case
Set of conditions
N-a
N-a1: qr 6 H
Optimal solution of Problem S2 1 1c dqm aqd0 1þd ; t 2 ¼ 0; a2 ¼ abqd q2 ¼ c
S2-c1: qr P H ðdþ1Þ
S2-c2 : qr
qd m P F
1þd q N-b1 : qm q r 6F m N-b2: qm > G
0
N-c q2
¼
S2-b2: qr 6 H S2-c
Nash equilibrium point 1c qN ¼ q0 ; tN ¼ 0; aN ¼ abqd 0
N-a2: qm 6 G
qd m < F
S2-b1: qm 6 G
0 6 t 6 1:
The next theorem presents the model’s Nash equilibrium point.
N-b Set of conditions
q P q0 ;
b
Table 2 Optimal solution of Problem S2. Case
ðSystem NÞ
t ¼ tm ðaÞ
0
q1 ¼
S1-f1: qr P H
:
q ¼ qm ðaÞ
1c
S1-e3: qm cqr 6 G S1-f
#dþc1þ1 ðcqr Þdþ1 b
a ¼ ar ðq; tÞ
r
S1-e1: qr 6 H
d
qm d cð1 þ cÞ
;
the Nash equilibrium point qN ; aN ; tN is the solution of the following system:
1þ1 c
cqr b
S1-d2: qm (c + 1)qr 6 0 1þc qm S1-d3 : qr q ð1þ 6E cÞ S1-e
"
#dþc1þ1 ðcqr Þc b
4. Nash equilibrium point dþ1cþ1
0
S1-d
cþ1
0
S1-b3: qm cqr 6 0 S1-c
d
It is easy to check that the function S(a, q) is not always positive for the above proposed Stackelberg equilibrium point. Theorem 1 shows that if we incorporate the two constraints a P a0 and q P q0 into the analysis, the Stackelberg equilibrium point differs from the above.
Table 1 Optimal solution of Problem S1. Case
qS ¼
0 h c d
q2 t 2
q0 ; t 2
¼
0; a2
¼ ¼
ðcqr Þc ðaqd0 Þ b
0; a2
¼
1þc
b aqd0
¼
d1c
1c
b aqd0
1c
N-c1: qr > H 1þc N-c2 : qr qqm 6E
qN ¼
qm daqd0
1 1þd
c
1c t N ¼ 0; aN ¼ abqd 0 1þ1 c qN ¼ q0 ; tN ¼ 0; aN ¼ cqqdr b 0
r
N-d
1þd >F N-d1 : qm qqr m 1þc qm N-d2 : qr q >E r
1
qN ¼ ½ðdqm =cÞ1þc bðcqr Þc cþdþ1 t N ¼ 0 1
aN ¼ ½ðc=ðdqm ÞÞd bðcqr Þ1þd cþdþ1
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Remark 2. The Nash equilibrium point proposed by Huang and Li (2001), Huang et al. (2002), Li et al. (2002), Xie and Ai (2006), and Yue et al. (2006) is as follows: 1
qN ¼ ½ðdqm =cÞ1þc bðcqr Þc cþdþ1 tN ¼ 0 1
aN ¼ ½ðc=ðdqm ÞÞd bðcqr Þ1þd cþdþ1 : It is easy to check that the function S(a, q) is not always positive for the above Nash equilibrium point. From Theorem 2, it can be seen that, if we consider the two constraints a P a0 and q P q0, the Nash equilibrium point is different from the above except under the two parametric conditions N-d1 and N-d2, which are given in Table 3. 5. Pareto-efficient points we obtain the Pareto-efficient points In this section, aPEl ; qPEl ; t PEl by solving the following optimization problem for different values of l P 0.
max pt ¼ pr ða; q; tÞ þ lpm ða; q; tÞ a;q;t
ðq0 Þ; s:t: a P a
ðProblem PE l Þ
q P q0 ;
6. Managerial implications In Sections 3–5, we derived the closed-form expressions for the Nash and Stackelberg equilibriums of the model proposed by Huang and Li (2001) under the two constraints suggested by Yue et al. (2006). Despite the fact that these expressions are much more complex than the expressions presented for Huang and Li’s model, we are still able to propose some interesting managerial implications. In this section, we give an overview of a few qualitative properties and compare them to those obtained in the other papers that use the same sales function. Note that in all the figures presented in this section, dash curves correspond to Huang and Li’s original model and solid curves to the modified model considered in this paper, i.e., Huang and Li’s model with Yue et al.’s constraints. Moreover, inequalities involving partial derivatives hold almost everywhere, because the associated partial derivatives exist almost everywhere. Hence, if we state that a quantity, e.g., the retailer’s or the manufacturer’s investment values, is increasing or decreasing with respect to a specific parameter, e.g., the retailer’s or the manufacturer’s marginal profits, then this monotonicity property holds in the range of the parameter where the partial derivative of the quantity with respect to the parameter exists. 6.1. Cooperation rate
0 6 t 6 1: Theorem 3. The Pareto-efficient points determined by Problem PEl for l P 0 are given in Table 4.
Remark 3. Huang and Li (2001), and Li et al. (2002) characterize the Pareto-efficient points aPEl ; qPEl ; tPEl for all values of l P 0 as follows:
aPEl ¼ qPEl ¼ 06
h c d d
c1þd bðqr þ qm Þ
h d cþ1
t PEl
c
idþc1þ1
cc bðqr þ qm Þ
idþc1þ1
6 1:
In fact, they state that the Pareto-efficient points for all l P0 are the same as the non-dominant point obtained for l ¼ 1. However, we can see from Table 4 that this statement is not true when we consider the two constraints a P a0 and q P q0.
Property 1. If the manufacturer and the retailer act simultaneously, the manufacturer will not share in the local-advertising costs, i.e., tN ¼ 0. However, if they move sequentially, the manufacturer may in some cases share the local-advertising costs. Moreover, the cooperation rate depends non-negatively on the manufacturer’s marginal profit and non-positively on the retailer’s marginal profit, i.e.,
@tS P 0; @ qm
@t S 6 0: @ qr
This property shows that the cooperative-advertising mechanism is not useful when the manufacturer and the retailer move simultaneously, but that, if they move sequentially, the manufacturer will sometimes be interested in cooperating and sharing local-advertising costs. The level of this cooperation monotonically depends on the marginal profits. If the retailer increases her marginal profit, then the manufacturer will not increase her cooperation level. If the manufacturer increases her marginal profit, she will not decrease her share in local efforts.
Table 4 Pareto-efficient points obtained from Problem PEl for l P 0 under four sets of parametric conditions. Case
Set of conditions
Optimal solution of Problem PEla
PE-a
PE-a1: max{1, lc}l1(qr + lqm) 6 E
aPEl ¼
PE-a2: max{1, l}l PE-b
1
(qr + lqm) > H
PE-b1: max{1, ld + 1}l1 (qr + lqm) 6 F PE-b2: l1(qr + lqm) > G
PE-c
PE-c1: l1(qr + lqm)6G PE-c2: max{1, l}l1(qr + lqm) 6 H
PE-d
PE-d1 max{1, ld + 1}l1 (qr + lqm) > F
I½0;1Þ ðlÞ 6
1þ1 c
cbðqr þlqm Þ minf1;lgqd0
tPEl
; qPEl ¼ q0
6 I½0;1 ðlÞ
1c
1 dðqr þlqm Þaqd0 1þd aPEl ¼ abqd ; qPEl ¼ lc 0 I½0;1Þ ðlÞ 6 tPEl 6 I½0;1 ðlÞ 1c aPEl ¼ abqd ; qPEl ¼ q0 0 I½0;1Þ ðlÞ 6 tPEl 6 I½0;1 ðlÞ dþ1cþ1 d c 1þd aPEl ¼ ldc bðqr þ lqm Þ minf1;lg
PE-d2: max{1, lc}l1(qr + lqm) > E qPEl ¼
cþ1 d lc
c
c
minf1;lg
I½0;1Þ ðlÞ 6 tPEl 6 I½0;1 ðlÞ a
In this table we use the indicator function defined as IA ðxÞ ¼
1 0
x2A for any A R. xRA
dþc1þ1 bðqr þ lqm Þ
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6.1.1. Comparison We know that, for the original model, the cooperation rate or its derivative with respect to marginal profits becomes zero if and only if qm/qr < c þ 1. However, for the modified model, we lose this important property. In Fig. 1, we depicted the participation rate versus the manufacturer’s marginal profit for two examples. The figure exhibits the following points: If we use the original model, the resulting cooperation rate differs considerably from the one obtained for the modified model. As we can see from the right plot in Fig. 1, it is possible for the original model’s cooperation rate to tend to 1, while the corresponding rate for the modified model is still zero. This shows that the minor error of ignoring the non-negativity of the sales function may result in the significant error of incorrectly determining the appropriate cooperation policy, that is, choosing full cooperation for local advertising instead of full non-cooperation. For the modified model, the condition qm/qr < c þ 1 is not required for the cooperation rate to be zero. 6.2. Investments in the Nash game Property 2. If the manufacturer and the retailer move simultaneously, each of their respective advertising investment depends nonnegatively on her own marginal profit and non-positively on her counterpart’s marginal profit, i.e.,
@aN @ qr
P 0;
@aN @ qm
6 0;
@qN @ qr
6 0;
@qN @ qm
P 0:
Property 2 shows the impact of marginal profits on the advertising efforts. When the members of the supply chain move simultaneously, their advertising efforts respond monotonically to the marginal profits.
However, for the modified model, depending on the parameter values, increasing or decreasing the marginal profits may have no effect on the local or national advertising values because the inequalities given in Property 2 are not strict. Fig. 2 compares the original and modified investment values versus the marginal profits for four examples. We can see the followings from Fig. 2: If we use the original model for the Nash game, the investment values are significantly different from the ones obtained for the modified model. For example, for the case plotted in the southwest section of Fig. 2, we see that the national investment for the modified model is constant while, for the original model, it rapidly tends to zero as the retailer’s marginal profit decreases. In many cases, increasing the marginal profits will not impact on investment values. This shows that what was previously understood, i.e., that the investment in advertising strictly depends on marginal profits, is incorrect for the modified model. Therefore, for example, we cannot always use an increase in the retailer’s marginal profit as an incentive to encourage the retailer to more promote local advertising.
6.3. Investments in the Stackelberg game Property 3. If the manufacturer and the retailer move sequentially, then the manufacturer’s investment is non-negatively dependent on her own marginal profit, i.e., @qS =@ qm P 0. Property 3 shows that, in the Stackelberg game, the only thing we can guarantee is that an increase in the manufacturer’s marginal profit will not decrease her investment in advertising. Our numerical study shows that we have
6.2.1. Comparison For the original model we have
@aS P 0; @ qr
@aN > 0; @ qr
however, due to the complexity of the results, we have no mathematical proof on hand.
@aN < 0; @ qm
@qN < 0; @ qr
@qN > 0: @ qm
@aS P 0; @ qm
@qS 6 0; @ qr
Fig. 1. The participation rate versus the manufacturer’s marginal profit in the Stackelberg game.
A. Ahmadi-Javid, P. Hoseinpour / European Journal of Operational Research 219 (2012) 458–466
463
Fig. 2. The advertising values versus marginal profits in the Nash game.
6.3.1. Comparison In Fig. 3, we plot four comparisons of the original and modified investment values for the Stackelberg game. The following points can be understood from Fig. 3: If we use the original model, the investment values deviate markedly from the correct ones. Consider for example the southwestern graph in Fig. 3. It can be seen that as the retailer’ marginal profit approaches 1, the national investment tends to zero for the modified model, but tends to 2.5 for the original model. In many cases, the previously held interpretation that advertising investments strictly depend on marginal profits, does not hold for the modified model. This means that the retailer’s marginal profit, for example, can no longer be used as an incentive that is certain to promote local advertising efforts or as a disin-
centive to force the manufacturer to increase her contribution to national advertising. For example, assume that
a ¼ 100; c ¼ 1; qr ¼ 0:1; qm ¼ 0:3; c ¼ 0:6; d ¼ 0:1; b ¼ 450; q0 ¼ 0:1: For this example, the level of national advertising does not vary, as long as the retailer’s marginal profit increases to 0.4 or decreases to 0 (see southwestern plot in Fig. 3). For the modified model, the investment values monotonically vary with respect to the marginal profits. However, for the original model, investment values are non-monotone functions of the marginal profits, which is very problematic since a slight change in each marginal-profit value may yield a imperceptible change in the level of advertising. For the example considered in the previous point, if we increase the retailer’s marginal profit from 0.1 to 0.5, because the discontinuity point 0.2 is placed
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Fig. 3. The advertising investment values versus marginal profits in the Stackelberg game.
between the numbers 0.1 to 0.5 (see the southwestern plot in Fig. 3), we have an unexpected increment in the nationaladverting effort. However, for the modified model, we have a decrease, which is more logical. Moreover, Yue et al. (2006) showed that, for the original model, the investments are linearly dependent, i.e., aS =qS ¼ cc=d. However, this property does not hold for the modified model. Fig. 4 clearly shows that investment values obtained from the modified model do not linearly depend on each other.
7. Conclusion This paper addresses a static, single-period cooperative advertising model introduced and studied by Huang and Li (2001), and
then by several other authors, using game theory. Although the model is interesting and useful, it lacks constraints to guarantee that the sales function will be non-negative for all values of the decision variables. Yue et al. (2006) proposed two constraints to solve this issue, but they did not investigate them mathematically. Here we analyzed the model, accounting for Yue et al.’s non-negativity constraints and find that almost all new quantitative and qualitative results are different from the previous ones. We show that our previous understanding of cooperative advertising, which stated that any change in the manufacturer’s or the retailer’s marginal profits impacts on the advertising investments, is not true for the modified model. This means that advertising efforts may remain constant even if one of the marginal-profit values increases or decreases. Moreover, where the manufacturer is the leader and the retailer is the follower, we demonstrate that, while the previous equilib-
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where
f ðq; tÞ ¼ qm a þ qm bqd
cqr b ð1tÞqd
c 1þ c
þt
cqr b ð1tÞqd
1þ1 c
þ cq;
g 1 ðq; tÞ ¼ q0 q; g 2 ðq; tÞ ¼ t; g 3 ðq; tÞ ¼ ðð1 tÞqd Þc
ðcqr Þc ðaqd0 Þ b
1þc
;
g 4 ðq; tÞ ¼ t 1: It can be seen that as t ? 1, the objective function tends to 1, so the fourth constraint g4(q, t) < 0 needs not to be considered here; however, we must consider it while checking the feasibility of the KKT points. For the above reformulated problem, without considering the fourth constraint, the associated Lagrangian function is written as follows:
lðq; t; uÞ ¼ f ðq; tÞ þ
3 X
ui g i ðq; tÞ with u ¼ ðu1 ; u2 ; u3 Þ:
i¼1
A KKT point (q, t) for Problem S1 satisfies the following system:
i : ui g i ðq; tÞ ¼ 0; ii : ui P 0; Fig. 4. The ratio of local investment to national investment versus the manufacturer’s marginal profit in the Stackelberg game.
rium advertising investments depend non-monotonically on the marginal profits, the new ones are monotone with respect to the marginal profits. We also present an example of a situation where, for the modified model, the manufacturer does not share in the local advertising, whereas the results from the original model predict that the manufacturer would pay almost all the local advertising costs at equilibrium. These observations show the importance of considering a nonnegative sales function for all values of the control variables. Unfortunately, this is sometimes overlooked in applications, especially in the cooperative advertising context. Acknowledgements We would like to thank editor-in-chief Professor Dyson and the three referees for their comments. We are also grateful to Professor Georges Zaccour and Professor Roland Malhame for their discussions that significantly improved an earlier version of this paper.
i ¼ 1; 2; 3
i ¼ 1; 2; 3
iii : rq;t lðq; t; uÞ ¼ 0: To find all feasible KKT points, we need first to solve the above system to determine KKT points, and then we have to check the feasibility of them. Solving the above system requires a case-by-case examination. Here, we describe the method used for the case S1-a of Table 1. Assume that the first constraint is active at the solution (q, t) of the above system, i.e., g1(q, t) ¼ 0, and the other two constraints are inactive, i.e., g2(q, t) < 0 and g3(q, t) < 0, which implies u2 ¼ u3 ¼ 0 by considering Condition (i). Next, we must obtain q, t, u1. First by Condition (iii) and u2 ¼ u3 ¼ 0, we have:
rq;t lðq; t; uÞ ¼ 0 )
@f qr ¼0)t ¼1 : @t qm cqr
By this expression and considering our assumption that g1(q, t) ¼ 0, we have q ¼ q0. Finally, by Condition (iii), we obtain u1 as the following:
u1 ¼
1 d q ð1 tÞ cqr b 1þc 1 m t q þ c: 1þc ð1 tÞqd cqr
The point t ¼ 1 q
qr
m cqr
; q ¼ q0 is a feasible KKT point if and only
if it satisfies the following conditions: Appendix A. Proofs A.1. Proof of Theorem 1
i:t<1 ii : 0 < t iii : ðð1 tÞqd Þc <
The objective function of Problem S is continuous, and tends to 1 as q ? þ 1 or t ? 1, so the problem has an optimal solution. Next we can find the optimal solution by finding the feasible KKT points of Problem S1 and Problem S2 (note that both Problem S1 and Problem S2 have optimal solutions and all points in their feasible solution sets satisfy the linear independence constraint qualification, so we can use the first-order KKT necessary condition; for more details see Bazaraa et al., 2006). In the following, for the sake of brevity, we only solve Problem S1. The method for solving Problem S2 is similar. First we rewrite Problem S1 in the following standard form:
min f ðq; tÞ q;t
s:t: g 1 ðq; tÞ; g 2 ðq; tÞ; g 3 ðq; tÞ 6 0; g 4 ðq; tÞ < 0;
ðcqr Þc ðaqd0 Þ
1þc
b
iv : u1 P 0: These conditions are equivalent to the following three parametric conditions (see case S1-a in Table 1): S1-a1: qm (1 + c)qr > 0 S1-a2: qm cqr > H S1-a3: qm cqr 6 E. All the feasible KKT points and their associated parametric conditions can be found along the same line described above. After finding all the feasible KKT points, the optimal solution of Problem S1 can be determined by directly comparing the objective values of these points, which is a rather complex procedure. Fortunately, for the above optimization problem we have a better alternative to determine the optimal solution. Indeed, it can be shown that the sets of parametric conditions of feasible KKT points completely
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partition the set of all possible values for two parameters qm, qr, i.e., {qm, qr : qm, qr > 0}. This means that for each set of parametric conditions we have only one feasible KKT point, which is exactly the optimal solution of Problem S1 under that set of parametric conditions because we know that Problem S1 has a solution. A.2. Proof of Theorem 2 To solve System N we need only to solve the problem maxqPq0 ;06t61 ðpm ðq; a; tÞÞ to find qm ðaÞ; tm ðaÞ . The procedure for solving this problem is the same as the procedure described for solving Problem S1 in the proof of Theorem 1. A.3. Proof of Theorem 3 To prove this theorem, Problem PEl needs to be solved. The procedure for solving Problem PEl is exactly the same as the procedure described for solving Problem S1 in the proof of Theorem 1. A.4. Proof of Properties 1–3 These properties can be proven easily by checking all of the solutions obtained for the Stackelberg and Nash equilibria in Theorems 1 and 2. References Ahmadi-Javid, A., Hoseinpour, P., 2011. A game-theoretic analysis for coordinating cooperative advertising in a supply chain. Journal of Optimization Theory and Applications 149, 138–150. Bazaraa, M.S., Sherali, H.D., Shetty, C.M., 2006. Nonlinear Programming Theory and Application. Jon Wily and Sons, Inc. Bergen, M., John, G., 1997. Understanding cooperative advertising participation rates in conventional channels. Journal of Marketing Research 35, 357–369. He, X., Krishnamoorthy, A., Prasad, A., Sethi, S.P., 2011. Retail competition and cooperative advertising. Operation research letters 39 (1), 11–16. He, X., Prasad, A., Sethi, S., 2009. Cooperative advertising and pricing in a dynamic stochastic supply chain: Feedback Stackelberg strategies. Production and Operations Management 18 (1), 78–94.
Jorgensen, S., Sigue, S.P., Zaccour, G., 2000. Dynamic cooperative advertising in a channel. Journal of Retailing 76, 71–92. Jorgensen, S., Sigue, S., Zaccour, G., 2001. Stackelberg leadership in a marketing channel. International Game Theory Review 3 (1), 13–26. Jorgensen, S., Taboubi, S., Zaccour, G., 2001. Cooperative advertising in a marketing channel. Journal of Optimization Theory and Application 110, 145–158. Jorgensen, S., Taboubi, S., Zaccour, G., 2003. Retail promotions with negative brand image effects: Is cooperation possible? European Journal of Operational Research 150, 395–405. Jorgensen, S., Zaccour, G., 1999. Equilibrium pricing and advertising strategies in a marketing channel. Journal of Optimization Theory and Applications 102 (1), 111–125. Jorgensen, S., Zaccour, G., 2003a. Channel coordination over time: Incentive equilibria and credibility. Journal of Economic Dynamics and Control 27 (5), 801–822. Jorgensen, S., Zaccour, G., 2003b. A differential game of retailer promotions. Automatica 39 (7), 1145–1155. Huang, Z., Li, S.X., 2001. Co-op advertising models in manufacturer–retailer supply chains: A game theory approach. European Journal of Operational Research 135, 527–544. Huang, Z., Li, S.X., Mahajan, V., 2002. An analysis of manufacturer–retailer supply chain coordination in cooperative advertising. Decision Sciences 33, 469–494. Hutchins, M.S., 1953. Cooperative Advertising. Roland Press, New York. Karray, S., Zaccour, G., 2006. Could co-op advertising be a manufacturer’s counterstrategy to store brands? Journal of Business Research 58, 1008–1015. Karray, S., Zaccour, G., 2007. Effectiveness of coop advertising programs in competitive distribution channels. International Game Theory Review 9, 151– 168. Li, S.X., Huang, Z., Zhu, J., Chau, P.Y.K., 2002. Cooperative advertising, game theory and manufacturer–retailer supply chains. Omega 30, 347–358. Szmerekovsky, J., Zhang, J., 2009. Pricing and two-tier advertising with one manufacturer and one retailer. European Journal of Operational Research 192 (3), 904–917. Xie, J., Ai, S., 2006. A note on cooperative advertising, game theory and manufacturer–retailer supply chains. Omega 34, 501–504. Xie, J., Neyret, A., 2009. Co-op advertising and pricing models in manufacturer– retailer supply chains. Computers and Industrial Engineering 56 (4), 1375– 1385. Xie, J., Wei, J., 2009. Coordinating advertising and pricing in a manufacturer–retailer channel. European Journal of Operational Research 197 (2), 785–791. Yan, R., 2009. Cooperative advertising, pricing strategy and firm performance in the e-marketing age. Journal of the Academy of Marketing Science 38, 510–519. Yue, J., Austin, J., Wang, M.C., Huang, Z., 2006. Coordination of cooperative advertising in a two-level supply chain when manufacturer offers discount. European Journal of Operational Research 168, 65–85.
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Interfaces with Other Disciplines
On a cooperative advertising model for a supply chain with one manufacturer and one retailer Amir Ahmadi-Javid ⇑, Pooya Hoseinpour Department of Industrial Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
a r t i c l e
i n f o
Article history: Received 24 September 2010 Accepted 21 June 2011 Available online 22 July 2011 Keywords: Supply chain management Cooperative advertising Coordination Game theory
a b s t r a c t Huang and Li (2001), Huang et al. (2002), Li et al. (2002), Xie and Ai (2006) and Yue et al. (2006) recently studied a game-theoretic model for cooperative advertising in a supply chain consisting of one manufacturer and one retailer. However, the sales-volume (demand) function considered in this model can become negative for some values of the decision variables, and in fact, this does happen for the proposed Stackelberg and Nash equilibrium solutions. Yue et al. (2006) acknowledge the negativity problem and suggest two constraints to fix it; however, they do not incorporate these constraints into their mathematical analysis. In this paper, we show that the results obtained by analyzing the advertising model under the constraints suggested by Yue et al. can differ significantly from those obtained in the previous papers. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Cooperative advertising is a coordination mechanism for advertising activities in a supply chain. In cooperative advertising, the manufacturers may contribute part of the advertising costs which are paid by retailers (see Hutchins, 1953; Bergen and John, 1997). Some studies consider a static, single-period relationship to explore the detailed interactions among the factors involved in cooperative advertising (see e.g., Huang and Li, 2001; Huang et al., 2002; Yue et al., 2006; Karray and Zaccour, 2006, 2007). Others focus on long-term, multi-period cooperative advertising relationship between manufacturers and retailers and use dynamic models and differential games to analyze that relationship (see e.g., Jorgensen et al., 2000; Jorgensen et al., 2001; Jorgensen and Zaccour, 2003a; Jorgensen et al., 2003; He et al., 2011). A large body of literature is also devoted to the simultaneous study of pricing and cooperative advertising. For static models we refer the reader to Szmerekovsky and Zhang (2009), Xie and Neyret (2009), Xie and Wei (2009) and Yan (2009). For dynamic models, see He et al. (2009), Jorgensen et al. (2001), Jorgensen and Zaccour (1999) and Jorgensen and Zaccour (2003b). This paper analyzes a static, single-period model that has been studied by several authors in the past ten years. Huang and Li (2001), Huang et al. (2002), Li et al. (2002), Xie and Ai (2006), and Yue et al. (2006) study a static game-theoretic model for cooperative advertising in a supply chain with one manufac-
⇑ Corresponding author. Tel.: +98 2166413034. E-mail address: [email protected] (A. Ahmadi-Javid). 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.06.032
turer and one retailer. However, the sales-volume function considered in their model has the unfortunate property of becoming negative for some values of the decision variables. Here we correct the sales function and develop the same game-theoretic analysis for the modified sales function. The results we obtain from this analysis differ significantly from the original. The paper is organized as follows. The notation and model are presented in Section 2. Sections 3 and 4 respectively characterize the model’s Stackelberg and Nash equilibria. In Section 5, the model’s Pareto-efficient points are studied. In Section 6, we present some managerial implications arising out of the model; and, lastly, conclusion is given in Section 7. The proofs are given in Appendix A. 2. Modeling Consider a supply chain with one retailer and one manufacturer. We now define the notation to be used in the rest of the paper. a is the retailer’s local advertising expenditure; q is the manufacturer’s investment in the national brand name, including national advertising costs; t is the fraction of the total local-advertising expenditure that the manufacturer agrees to share with the retailer (0 6 t 6 1); qm is the manufacturer’s marginal profit for each product unit; qr is the retailer’s marginal profit for each product unit; c is the cost for each unit of q; pm is the manufacturer’s gross profit; pr is the retailer’s gross profit.
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The gross profit of the retailer and the manufacturer, respectively, are as follows:
pr ða; q; tÞ :¼ qr Sða; qÞ ð1 tÞa; pm ða; q; tÞ :¼ qm Sða; qÞ ta cq; where S(a, q) is the one-period sales function. Huang and Li (2001), Huang et al. (2002), Li et al. (2002), Xie and Ai (2006) and Yue et al. (2006) consider the following sales function:
Sða; qÞ :¼ a bac qd ; where the parameters a, c, d and b are positive constants. An important problem associated with the above sales function is that it is negative for some values of a > 0, q > 0. Yue et al. (2006) recognize this problem and propose two constraints to ensure that the national and local investments do not go below certain appropriate levels: i.e., a P a0 and q P q0, for given positive constants a0 and q0. However, they do not consider the two constraints in their analysis. Here, we show that the results obtained by analyzing the above advertising model when accounting for the constraints a P a0 and q P q0 can differ significantly from the original results. It should be noted that this negativity problem can be overcome using an entirely different approach, which is investigated by Ahmadi-Javid and Hoseinpour (2011). One might believe that these results are of little interest from a managerial viewpoint. In other words, it may seem that the problem of having a negative sales function is not a managerial concern. However, this argument can be countered using the following numerical example. For a supply-chain channel, suppose that the sales function is
Before proceeding, it should be noted that the parameters a0 and q0 are in fact related. Indeed, by defining a¯(q):= [b /(aqd)]1/c we notice the following equivalence:
ðqÞ; Sða; qÞ P 0 () a P a so that if q P q0 for a positive constant q0, then one must have a P a¯(q0). This means that when q0 is a given positive constant, a0 must satisfy a0 P a¯(q0) to guarantee that S(a, q) is nonnegative for a P a0 and q P q0. Thus, in the sequel, we assume that q0 is a given positive constant and that a0 ¼ a¯(q0). 3. Stackelberg equilibrium point Inthis section, we determine the Stackelberg equilibrium point aS ; qS ; t S of the model described in the previous section when imposing the two constraints a P a0 and q P q0. Let us define ar ðq; tÞ ¼ arg maxaPa0 ðpr ða; q; tÞÞ, and suppose that (q⁄, t⁄) is the solution to the following optimization problem:
max pm ar ðq; tÞ; q; t q;t
ðProblem SÞ
s:t: q P q0 0 6 t 6 1:
Then, point aS ; qS ; tS is defined as the Stackelberg equilibrium aS ; qS ; t S ¼ ar ðq ; t Þ; q ; t . Since pr(a, q, t) ¼ qrS(a, q) (1 t)a ¼ qr(a b acqd) (1 t)a is concave for a P a0, we have:
( ar ðq; tÞ
ðq0 Þ; ¼ max a
Sða; qÞ :¼ 100 220a0:1 q0:2 ; with qm ¼ 0.8, qr ¼ 0.7, c ¼ 1. By using the existing results, the Stackelberg equilibrium point aS ; qS ; t S for the model becomes
aS ¼ 7:61;
qS ¼ 15:21;
tS ¼ 0:04:
However, by substituting the investment values aS ; qS in the sales function, we obtain
S aS ; qS ¼ 4:20 < 0: It is clear that no manager would accept these results. She or he would immediately want to know what caused the negativity problem and how it can be overcome. In Section 6, we will also discuss the impacts of this problem on our understanding of cooperative advertising. The model under consideration in this paper is the same as the one proposed by Huang and Li (2001), but with two constraints a P a0 and q P q0, which were proposed by Yue et al. (2006). The following are the differences that exist between Huang and Li’s model and the other models derived from it: In Huang et al. (2002), c ¼ b ¼ 1; In Li et al. (2002) and Xie and Ai (2006), c ¼ b ¼ a ¼ 1; In Yue et al. (2006), c ¼ 1, and the sales function is
Sða; qÞ :¼ ða bac qd Þð1 eÞe ; where e is the price-deduction percentage offered by the manufacturer, and e is the price elasticity (both e and e are positive constants). Also, because the manufacturer offers an e pricereduction percentage directly to the customers, the manufacturer’s marginal profit becomes qm eP0 where P0 is the full price to the customers (see Yue et al., 2006, for more details). In fact, from a mathematical standpoint, the model presented in Yue et al. (2006) could be recast as the model considered in this paper through the following simple reparameterization:
a :¼ að1 eÞe ; b :¼ bð1 eÞe ; qm :¼ qm eP0 :
cqr b ð1 tÞq
) 1 cþ1 : d
From this formula, the Stackelberg equilibrium point can be obtained by solving the following two smooth optimization problems:
max qm a qm bqd q;t
cqr b ð1tÞqd
c 1þ c
t
ðq0 Þ 6 s:t: q P q0 ; 0 6 t < 1; a
cqr b ð1tÞqd
cqr b
1þ1 c
cq ðProblem S1Þ
1þ1 c
ð1tÞqd
ðq0 ÞÞc t a ðq0 Þ cq max qm a qm bqd ða q;t
ðq0 Þ P s:t: q P q0 ; 0 6 t 6 1; a
cqr b
1þ1 c
ð1tÞqd
ðProblem S2Þ :
The constraint t 6 1 in Problem S is replaced with the constraint t < 1 in Problem S1 because its objective function tend to 1 as t ? 1. We now introduce the following parameters, which are used in the rest of the paper for notational simplicity: cþ1 E :¼ qdþ 0
ccþ1 d
b1 cc
G :¼ qd0ac 1
c H :¼ ðcc bað1þcÞ qd 0 Þ
dc ðdþcþ1Þ 1c F :¼ b1þd dc ccð1þdÞ aqd0 : Furthermore, suppose that q1 ; t 1 and q2 ; t2 are the optimal solutions of Problem S1 and Problem S2, respectively, and that a1 and a2 are calculated as follows:
ai
9 8 1 !cþ1 = < cq b r ðq Þ; ; ¼ max a d ; : 0 1 t i qi
i ¼ 1; 2:
Taking into account the above notation, the following theorem characterizes the model’s Stackelberg equilibrium point.
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Theorem 1. The Stackelberg equilibrium point aS ; qS ; tS exists. To obtain qS ; t S , the optimal solutions of Problem S1 and Problem S2 must first be determined; subsequently, the optimal solution with greater objective value is qS ; tS . Tables 1 and 2 present the optimal solution of Problem S1 and Problem S2, respectively. The calculation of the value of aS is based on qS ; t S as follows:
Remark 1. The Stackelberg equilibrium point proposed by Huang and Li (2001), Huang et al. (2002), Li et al. (2002), and Yue et al. (2006) is the following:
qS ¼
h d cþ1 (
c
bcc ðqm cqr Þ
1 q qrcq m r
idþc1þ1
;
qm =qr P c þ 1; qm =qr < c þ 1;
8 9 1 !cþ1 < = cq b r ðq Þ; aS ¼ max a : d : 0 ; 1 t S qS
t S ¼
Tables 1 and 2 also present a1 and a2 to simplify the computation of aS .
This solution was recently modified as follows, in the note by Xie and Ai (2006) for the case qm/qr < c þ 1:
aS ¼
Note that, due to the complexity of the resulting expressions, explicitly comparing the optimal values for Problem S1 and Problem S2 is impractical, but by using Tables 1 and 2 it is possible to pinpoint the Stackelberg equilibrium point according to the known values of the parameters.
Set of conditions
Optimal solution of Problem S1
S1-a
S1-a1: qm (1 + c)qr > 0
q1 ¼ q0 ; t 1 ¼ 1 q rcq m r 1þ1 c a1 ¼ cbðqmqdcqr Þ
S1-a2: qm cqr > H S1-a3: qm cqr 6 E S1-b
cþ1 qm qr > E S1-b1 : q ð1þ cÞ r d q ð1þcÞ S1-b2 : r q qr > F m
q
S1-c1: qm cqr > G d q S1-c2 : qr q rcq
r
S1-c3:qm cqr 6 F
idþc1þ1 bcðdþ1Þ ðqm cqr Þ :
"
qm d cð1 þ cÞ
tS ¼ 0;
aS ¼
q1 ¼
cþ1
dqm cðcþ1Þ
t1 ¼ 0; a1 ¼
ðcqr Þc b d
cð1þcÞ dqm
dþc1þ1
ðcqr Þ1þd b
1 q1 ¼ daqd0 ðqm cqr Þ=c 1þd d dþ1 dþcþ1 1 t1 ¼ 1 cqr bc aqd0 dc dðq ccq Þ m r 1c a1 ¼ abqd q1 ¼ q0 ; t 1 ¼ 0; a1 ¼
S1-d1: qr > H
qd0
S1-e2: qm cqr 6 H
q1 ¼ q0 ; t 1 ¼ 1 1c a1 ¼ abqd
ðcqr Þc að1þcÞ qd0 b
In this section, we find the Nash equilibrium point qN ; aN ; tN for the model given in Section 2 when considering the two constraints a P a0 and q P q0. By defining
ar ðq; tÞ ¼ arg maxðpr ða; q; tÞÞ aPa0 qm ðaÞ; t m ðaÞ ¼ arg max ðpm ðq; a; tÞÞ; qPq0 ;06t61
dþ1 d
S1-f 2 : qr
1 d
P F ðqm cqr Þ
cÞ d S1-f 3 : qr qr ð1þ 6F qm S1-g
S1-g1: qm (c + 1)qr > 0 S1-g2: qm cqr > E S1-g3: qm cqr > F
c
ðcqr Þ
1þc
ðaqd0 Þ
a P a0 ; d1c
S2-a
S2-a1: qm > G ðdþ1Þ
S2-a2 : qr S2-b
1c t1 ¼ 0; a1 ¼ abqd 0
Theorem 2. The Nash equilibrium point exists and is unique, as presented in Table 3. 1
q1 ¼ ½ðd=cÞcþ1 bcc ðqm cqr Þdþcþ1 t1 ¼ 1 q qrcq m
r
Table 3 Nash equilibrium point.
1
a1 ¼ ½ðc=dÞd bcðdþ1Þ ðqm cqr Þdþcþ1
Case
Set of conditions
N-a
N-a1: qr 6 H
Optimal solution of Problem S2 1 1c dqm aqd0 1þd ; t 2 ¼ 0; a2 ¼ abqd q2 ¼ c
S2-c1: qr P H ðdþ1Þ
S2-c2 : qr
qd m P F
1þd q N-b1 : qm q r 6F m N-b2: qm > G
0
N-c q2
¼
S2-b2: qr 6 H S2-c
Nash equilibrium point 1c qN ¼ q0 ; tN ¼ 0; aN ¼ abqd 0
N-a2: qm 6 G
qd m < F
S2-b1: qm 6 G
0 6 t 6 1:
The next theorem presents the model’s Nash equilibrium point.
N-b Set of conditions
q P q0 ;
b
Table 2 Optimal solution of Problem S2. Case
ðSystem NÞ
t ¼ tm ðaÞ
0
q1 ¼
S1-f1: qr P H
:
q ¼ qm ðaÞ
1c
S1-e3: qm cqr 6 G S1-f
#dþc1þ1 ðcqr Þdþ1 b
a ¼ ar ðq; tÞ
r
S1-e1: qr 6 H
d
qm d cð1 þ cÞ
;
the Nash equilibrium point qN ; aN ; tN is the solution of the following system:
1þ1 c
cqr b
S1-d2: qm (c + 1)qr 6 0 1þc qm S1-d3 : qr q ð1þ 6E cÞ S1-e
"
#dþc1þ1 ðcqr Þc b
4. Nash equilibrium point dþ1cþ1
0
S1-d
cþ1
0
S1-b3: qm cqr 6 0 S1-c
d
It is easy to check that the function S(a, q) is not always positive for the above proposed Stackelberg equilibrium point. Theorem 1 shows that if we incorporate the two constraints a P a0 and q P q0 into the analysis, the Stackelberg equilibrium point differs from the above.
Table 1 Optimal solution of Problem S1. Case
qS ¼
0 h c d
q2 t 2
q0 ; t 2
¼
0; a2
¼ ¼
ðcqr Þc ðaqd0 Þ b
0; a2
¼
1þc
b aqd0
¼
d1c
1c
b aqd0
1c
N-c1: qr > H 1þc N-c2 : qr qqm 6E
qN ¼
qm daqd0
1 1þd
c
1c t N ¼ 0; aN ¼ abqd 0 1þ1 c qN ¼ q0 ; tN ¼ 0; aN ¼ cqqdr b 0
r
N-d
1þd >F N-d1 : qm qqr m 1þc qm N-d2 : qr q >E r
1
qN ¼ ½ðdqm =cÞ1þc bðcqr Þc cþdþ1 t N ¼ 0 1
aN ¼ ½ðc=ðdqm ÞÞd bðcqr Þ1þd cþdþ1
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Remark 2. The Nash equilibrium point proposed by Huang and Li (2001), Huang et al. (2002), Li et al. (2002), Xie and Ai (2006), and Yue et al. (2006) is as follows: 1
qN ¼ ½ðdqm =cÞ1þc bðcqr Þc cþdþ1 tN ¼ 0 1
aN ¼ ½ðc=ðdqm ÞÞd bðcqr Þ1þd cþdþ1 : It is easy to check that the function S(a, q) is not always positive for the above Nash equilibrium point. From Theorem 2, it can be seen that, if we consider the two constraints a P a0 and q P q0, the Nash equilibrium point is different from the above except under the two parametric conditions N-d1 and N-d2, which are given in Table 3. 5. Pareto-efficient points we obtain the Pareto-efficient points In this section, aPEl ; qPEl ; t PEl by solving the following optimization problem for different values of l P 0.
max pt ¼ pr ða; q; tÞ þ lpm ða; q; tÞ a;q;t
ðq0 Þ; s:t: a P a
ðProblem PE l Þ
q P q0 ;
6. Managerial implications In Sections 3–5, we derived the closed-form expressions for the Nash and Stackelberg equilibriums of the model proposed by Huang and Li (2001) under the two constraints suggested by Yue et al. (2006). Despite the fact that these expressions are much more complex than the expressions presented for Huang and Li’s model, we are still able to propose some interesting managerial implications. In this section, we give an overview of a few qualitative properties and compare them to those obtained in the other papers that use the same sales function. Note that in all the figures presented in this section, dash curves correspond to Huang and Li’s original model and solid curves to the modified model considered in this paper, i.e., Huang and Li’s model with Yue et al.’s constraints. Moreover, inequalities involving partial derivatives hold almost everywhere, because the associated partial derivatives exist almost everywhere. Hence, if we state that a quantity, e.g., the retailer’s or the manufacturer’s investment values, is increasing or decreasing with respect to a specific parameter, e.g., the retailer’s or the manufacturer’s marginal profits, then this monotonicity property holds in the range of the parameter where the partial derivative of the quantity with respect to the parameter exists. 6.1. Cooperation rate
0 6 t 6 1: Theorem 3. The Pareto-efficient points determined by Problem PEl for l P 0 are given in Table 4.
Remark 3. Huang and Li (2001), and Li et al. (2002) characterize the Pareto-efficient points aPEl ; qPEl ; tPEl for all values of l P 0 as follows:
aPEl ¼ qPEl ¼ 06
h c d d
c1þd bðqr þ qm Þ
h d cþ1
t PEl
c
idþc1þ1
cc bðqr þ qm Þ
idþc1þ1
6 1:
In fact, they state that the Pareto-efficient points for all l P0 are the same as the non-dominant point obtained for l ¼ 1. However, we can see from Table 4 that this statement is not true when we consider the two constraints a P a0 and q P q0.
Property 1. If the manufacturer and the retailer act simultaneously, the manufacturer will not share in the local-advertising costs, i.e., tN ¼ 0. However, if they move sequentially, the manufacturer may in some cases share the local-advertising costs. Moreover, the cooperation rate depends non-negatively on the manufacturer’s marginal profit and non-positively on the retailer’s marginal profit, i.e.,
@tS P 0; @ qm
@t S 6 0: @ qr
This property shows that the cooperative-advertising mechanism is not useful when the manufacturer and the retailer move simultaneously, but that, if they move sequentially, the manufacturer will sometimes be interested in cooperating and sharing local-advertising costs. The level of this cooperation monotonically depends on the marginal profits. If the retailer increases her marginal profit, then the manufacturer will not increase her cooperation level. If the manufacturer increases her marginal profit, she will not decrease her share in local efforts.
Table 4 Pareto-efficient points obtained from Problem PEl for l P 0 under four sets of parametric conditions. Case
Set of conditions
Optimal solution of Problem PEla
PE-a
PE-a1: max{1, lc}l1(qr + lqm) 6 E
aPEl ¼
PE-a2: max{1, l}l PE-b
1
(qr + lqm) > H
PE-b1: max{1, ld + 1}l1 (qr + lqm) 6 F PE-b2: l1(qr + lqm) > G
PE-c
PE-c1: l1(qr + lqm)6G PE-c2: max{1, l}l1(qr + lqm) 6 H
PE-d
PE-d1 max{1, ld + 1}l1 (qr + lqm) > F
I½0;1Þ ðlÞ 6
1þ1 c
cbðqr þlqm Þ minf1;lgqd0
tPEl
; qPEl ¼ q0
6 I½0;1 ðlÞ
1c
1 dðqr þlqm Þaqd0 1þd aPEl ¼ abqd ; qPEl ¼ lc 0 I½0;1Þ ðlÞ 6 tPEl 6 I½0;1 ðlÞ 1c aPEl ¼ abqd ; qPEl ¼ q0 0 I½0;1Þ ðlÞ 6 tPEl 6 I½0;1 ðlÞ dþ1cþ1 d c 1þd aPEl ¼ ldc bðqr þ lqm Þ minf1;lg
PE-d2: max{1, lc}l1(qr + lqm) > E qPEl ¼
cþ1 d lc
c
c
minf1;lg
I½0;1Þ ðlÞ 6 tPEl 6 I½0;1 ðlÞ a
In this table we use the indicator function defined as IA ðxÞ ¼
1 0
x2A for any A R. xRA
dþc1þ1 bðqr þ lqm Þ
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6.1.1. Comparison We know that, for the original model, the cooperation rate or its derivative with respect to marginal profits becomes zero if and only if qm/qr < c þ 1. However, for the modified model, we lose this important property. In Fig. 1, we depicted the participation rate versus the manufacturer’s marginal profit for two examples. The figure exhibits the following points: If we use the original model, the resulting cooperation rate differs considerably from the one obtained for the modified model. As we can see from the right plot in Fig. 1, it is possible for the original model’s cooperation rate to tend to 1, while the corresponding rate for the modified model is still zero. This shows that the minor error of ignoring the non-negativity of the sales function may result in the significant error of incorrectly determining the appropriate cooperation policy, that is, choosing full cooperation for local advertising instead of full non-cooperation. For the modified model, the condition qm/qr < c þ 1 is not required for the cooperation rate to be zero. 6.2. Investments in the Nash game Property 2. If the manufacturer and the retailer move simultaneously, each of their respective advertising investment depends nonnegatively on her own marginal profit and non-positively on her counterpart’s marginal profit, i.e.,
@aN @ qr
P 0;
@aN @ qm
6 0;
@qN @ qr
6 0;
@qN @ qm
P 0:
Property 2 shows the impact of marginal profits on the advertising efforts. When the members of the supply chain move simultaneously, their advertising efforts respond monotonically to the marginal profits.
However, for the modified model, depending on the parameter values, increasing or decreasing the marginal profits may have no effect on the local or national advertising values because the inequalities given in Property 2 are not strict. Fig. 2 compares the original and modified investment values versus the marginal profits for four examples. We can see the followings from Fig. 2: If we use the original model for the Nash game, the investment values are significantly different from the ones obtained for the modified model. For example, for the case plotted in the southwest section of Fig. 2, we see that the national investment for the modified model is constant while, for the original model, it rapidly tends to zero as the retailer’s marginal profit decreases. In many cases, increasing the marginal profits will not impact on investment values. This shows that what was previously understood, i.e., that the investment in advertising strictly depends on marginal profits, is incorrect for the modified model. Therefore, for example, we cannot always use an increase in the retailer’s marginal profit as an incentive to encourage the retailer to more promote local advertising.
6.3. Investments in the Stackelberg game Property 3. If the manufacturer and the retailer move sequentially, then the manufacturer’s investment is non-negatively dependent on her own marginal profit, i.e., @qS =@ qm P 0. Property 3 shows that, in the Stackelberg game, the only thing we can guarantee is that an increase in the manufacturer’s marginal profit will not decrease her investment in advertising. Our numerical study shows that we have
6.2.1. Comparison For the original model we have
@aS P 0; @ qr
@aN > 0; @ qr
however, due to the complexity of the results, we have no mathematical proof on hand.
@aN < 0; @ qm
@qN < 0; @ qr
@qN > 0: @ qm
@aS P 0; @ qm
@qS 6 0; @ qr
Fig. 1. The participation rate versus the manufacturer’s marginal profit in the Stackelberg game.
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463
Fig. 2. The advertising values versus marginal profits in the Nash game.
6.3.1. Comparison In Fig. 3, we plot four comparisons of the original and modified investment values for the Stackelberg game. The following points can be understood from Fig. 3: If we use the original model, the investment values deviate markedly from the correct ones. Consider for example the southwestern graph in Fig. 3. It can be seen that as the retailer’ marginal profit approaches 1, the national investment tends to zero for the modified model, but tends to 2.5 for the original model. In many cases, the previously held interpretation that advertising investments strictly depend on marginal profits, does not hold for the modified model. This means that the retailer’s marginal profit, for example, can no longer be used as an incentive that is certain to promote local advertising efforts or as a disin-
centive to force the manufacturer to increase her contribution to national advertising. For example, assume that
a ¼ 100; c ¼ 1; qr ¼ 0:1; qm ¼ 0:3; c ¼ 0:6; d ¼ 0:1; b ¼ 450; q0 ¼ 0:1: For this example, the level of national advertising does not vary, as long as the retailer’s marginal profit increases to 0.4 or decreases to 0 (see southwestern plot in Fig. 3). For the modified model, the investment values monotonically vary with respect to the marginal profits. However, for the original model, investment values are non-monotone functions of the marginal profits, which is very problematic since a slight change in each marginal-profit value may yield a imperceptible change in the level of advertising. For the example considered in the previous point, if we increase the retailer’s marginal profit from 0.1 to 0.5, because the discontinuity point 0.2 is placed
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Fig. 3. The advertising investment values versus marginal profits in the Stackelberg game.
between the numbers 0.1 to 0.5 (see the southwestern plot in Fig. 3), we have an unexpected increment in the nationaladverting effort. However, for the modified model, we have a decrease, which is more logical. Moreover, Yue et al. (2006) showed that, for the original model, the investments are linearly dependent, i.e., aS =qS ¼ cc=d. However, this property does not hold for the modified model. Fig. 4 clearly shows that investment values obtained from the modified model do not linearly depend on each other.
7. Conclusion This paper addresses a static, single-period cooperative advertising model introduced and studied by Huang and Li (2001), and
then by several other authors, using game theory. Although the model is interesting and useful, it lacks constraints to guarantee that the sales function will be non-negative for all values of the decision variables. Yue et al. (2006) proposed two constraints to solve this issue, but they did not investigate them mathematically. Here we analyzed the model, accounting for Yue et al.’s non-negativity constraints and find that almost all new quantitative and qualitative results are different from the previous ones. We show that our previous understanding of cooperative advertising, which stated that any change in the manufacturer’s or the retailer’s marginal profits impacts on the advertising investments, is not true for the modified model. This means that advertising efforts may remain constant even if one of the marginal-profit values increases or decreases. Moreover, where the manufacturer is the leader and the retailer is the follower, we demonstrate that, while the previous equilib-
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where
f ðq; tÞ ¼ qm a þ qm bqd
cqr b ð1tÞqd
c 1þ c
þt
cqr b ð1tÞqd
1þ1 c
þ cq;
g 1 ðq; tÞ ¼ q0 q; g 2 ðq; tÞ ¼ t; g 3 ðq; tÞ ¼ ðð1 tÞqd Þc
ðcqr Þc ðaqd0 Þ b
1þc
;
g 4 ðq; tÞ ¼ t 1: It can be seen that as t ? 1, the objective function tends to 1, so the fourth constraint g4(q, t) < 0 needs not to be considered here; however, we must consider it while checking the feasibility of the KKT points. For the above reformulated problem, without considering the fourth constraint, the associated Lagrangian function is written as follows:
lðq; t; uÞ ¼ f ðq; tÞ þ
3 X
ui g i ðq; tÞ with u ¼ ðu1 ; u2 ; u3 Þ:
i¼1
A KKT point (q, t) for Problem S1 satisfies the following system:
i : ui g i ðq; tÞ ¼ 0; ii : ui P 0; Fig. 4. The ratio of local investment to national investment versus the manufacturer’s marginal profit in the Stackelberg game.
rium advertising investments depend non-monotonically on the marginal profits, the new ones are monotone with respect to the marginal profits. We also present an example of a situation where, for the modified model, the manufacturer does not share in the local advertising, whereas the results from the original model predict that the manufacturer would pay almost all the local advertising costs at equilibrium. These observations show the importance of considering a nonnegative sales function for all values of the control variables. Unfortunately, this is sometimes overlooked in applications, especially in the cooperative advertising context. Acknowledgements We would like to thank editor-in-chief Professor Dyson and the three referees for their comments. We are also grateful to Professor Georges Zaccour and Professor Roland Malhame for their discussions that significantly improved an earlier version of this paper.
i ¼ 1; 2; 3
i ¼ 1; 2; 3
iii : rq;t lðq; t; uÞ ¼ 0: To find all feasible KKT points, we need first to solve the above system to determine KKT points, and then we have to check the feasibility of them. Solving the above system requires a case-by-case examination. Here, we describe the method used for the case S1-a of Table 1. Assume that the first constraint is active at the solution (q, t) of the above system, i.e., g1(q, t) ¼ 0, and the other two constraints are inactive, i.e., g2(q, t) < 0 and g3(q, t) < 0, which implies u2 ¼ u3 ¼ 0 by considering Condition (i). Next, we must obtain q, t, u1. First by Condition (iii) and u2 ¼ u3 ¼ 0, we have:
rq;t lðq; t; uÞ ¼ 0 )
@f qr ¼0)t ¼1 : @t qm cqr
By this expression and considering our assumption that g1(q, t) ¼ 0, we have q ¼ q0. Finally, by Condition (iii), we obtain u1 as the following:
u1 ¼
1 d q ð1 tÞ cqr b 1þc 1 m t q þ c: 1þc ð1 tÞqd cqr
The point t ¼ 1 q
qr
m cqr
; q ¼ q0 is a feasible KKT point if and only
if it satisfies the following conditions: Appendix A. Proofs A.1. Proof of Theorem 1
i:t<1 ii : 0 < t iii : ðð1 tÞqd Þc <
The objective function of Problem S is continuous, and tends to 1 as q ? þ 1 or t ? 1, so the problem has an optimal solution. Next we can find the optimal solution by finding the feasible KKT points of Problem S1 and Problem S2 (note that both Problem S1 and Problem S2 have optimal solutions and all points in their feasible solution sets satisfy the linear independence constraint qualification, so we can use the first-order KKT necessary condition; for more details see Bazaraa et al., 2006). In the following, for the sake of brevity, we only solve Problem S1. The method for solving Problem S2 is similar. First we rewrite Problem S1 in the following standard form:
min f ðq; tÞ q;t
s:t: g 1 ðq; tÞ; g 2 ðq; tÞ; g 3 ðq; tÞ 6 0; g 4 ðq; tÞ < 0;
ðcqr Þc ðaqd0 Þ
1þc
b
iv : u1 P 0: These conditions are equivalent to the following three parametric conditions (see case S1-a in Table 1): S1-a1: qm (1 + c)qr > 0 S1-a2: qm cqr > H S1-a3: qm cqr 6 E. All the feasible KKT points and their associated parametric conditions can be found along the same line described above. After finding all the feasible KKT points, the optimal solution of Problem S1 can be determined by directly comparing the objective values of these points, which is a rather complex procedure. Fortunately, for the above optimization problem we have a better alternative to determine the optimal solution. Indeed, it can be shown that the sets of parametric conditions of feasible KKT points completely
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partition the set of all possible values for two parameters qm, qr, i.e., {qm, qr : qm, qr > 0}. This means that for each set of parametric conditions we have only one feasible KKT point, which is exactly the optimal solution of Problem S1 under that set of parametric conditions because we know that Problem S1 has a solution. A.2. Proof of Theorem 2 To solve System N we need only to solve the problem maxqPq0 ;06t61 ðpm ðq; a; tÞÞ to find qm ðaÞ; tm ðaÞ . The procedure for solving this problem is the same as the procedure described for solving Problem S1 in the proof of Theorem 1. A.3. Proof of Theorem 3 To prove this theorem, Problem PEl needs to be solved. The procedure for solving Problem PEl is exactly the same as the procedure described for solving Problem S1 in the proof of Theorem 1. A.4. Proof of Properties 1–3 These properties can be proven easily by checking all of the solutions obtained for the Stackelberg and Nash equilibria in Theorems 1 and 2. References Ahmadi-Javid, A., Hoseinpour, P., 2011. A game-theoretic analysis for coordinating cooperative advertising in a supply chain. Journal of Optimization Theory and Applications 149, 138–150. Bazaraa, M.S., Sherali, H.D., Shetty, C.M., 2006. Nonlinear Programming Theory and Application. Jon Wily and Sons, Inc. Bergen, M., John, G., 1997. Understanding cooperative advertising participation rates in conventional channels. Journal of Marketing Research 35, 357–369. He, X., Krishnamoorthy, A., Prasad, A., Sethi, S.P., 2011. Retail competition and cooperative advertising. Operation research letters 39 (1), 11–16. He, X., Prasad, A., Sethi, S., 2009. Cooperative advertising and pricing in a dynamic stochastic supply chain: Feedback Stackelberg strategies. Production and Operations Management 18 (1), 78–94.
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