European Journal of Operational Research 211 (2011) 263–273
Contents lists available at ScienceDirect
European Journal of Operational Research
journal homepage: www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
A game theoretic approach to coordinate pricing and vertical co-op advertising
in manufacturer–retailer supply chains
Mir Mehdi SeyedEsfahani a,⇑, Maryam Biazaran a, Mohsen Gharakhani b
a
b
Department of Industrial Engineering, AmirKabir University of Technology, Tehran 15875-4413, Iran
Department of Industrial Engineering, Iran University of Science and Technology, Tehran 16846-13114, Iran
a r t i c l e
i n f o
Article history:
Received 23 March 2010
Accepted 15 November 2010
Available online 23 November 2010
Keywords:
Marketing
Supply chain coordination
Pricing
Vertical cooperative advertising
Game theory
a b s t r a c t
Vertical cooperative (co-op) advertising is a marketing strategy in which the retailer runs local advertising and the manufacturer pays for a portion of its entire costs. This paper considers vertical co-op advertising along with pricing decisions in a supply chain; this consists of one manufacturer and one retailer
where demand is influenced by both price and advertisement. Four game-theoretic models are established in order to study the effect of supply chain power balance on the optimal decisions of supply chain
members. Comparisons and insights are developed. These embrace three non-cooperative games including Nash, Stackelberg-manufacturer and Stackelberg-retailer, and one cooperative game. In the latter
case, both the manufacturer and the retailer reach the highest profit level; subsequently, the feasibility
of bargaining game is discussed in a bid to determine a scheme to share the extra joint profit.
Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction
Supply chain coordination has been the focus of many research
studies without which channel members tend to maximize their
own profit. A prime example associated with such an uncoordinated system is ‘‘double marginalization’’, in which the retailer
makes arbitrary decisions without considering supplier’s profit
margin (Spengler, 1950). Another example is the ‘‘bullwhip effect’’
which occurs when supply chain members make decision ignoring
the others; this, in return, will lead to the spread of distorted demand information moving upstream (Lee et al., 1997).
Sahin and Robinson (2002) proposed two key drivers of supply
chain performance involving information sharing and coordination. Fugate et al. (2006) classified supply chain coordination
mechanisms into three categories: (1) price coordination, (2)
non-price coordination and (3) flow coordination. Based on this
classification, pricing and vertical cooperative advertising, two
business decisions discussed in this paper, are placed in the first
and second category, respectively.
A considerable amount of research has been conducted in recent years on different aspects of supply chain coordination including pricing, production, purchasing, inventory, etc. In this paper,
optimal pricing and vertical co-op advertising decision is discussed
in a single-manufacturer–single-retailer supply chain in which
⇑ Corresponding author. Tel.: +98 66466497; fax: +98 2188500995.
E-mail addresses: msesfahani@aut.ac.ir (M.M. SeyedEsfahani), iicty@aut.ac.ir (M.
Biazaran), gharakhani@iust.ac.ir (M. Gharakhani).
0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2010.11.014
consumer demand is influenced by both price and advertising
efforts.
Manufacturers and retailers use advertising programs to convince customers to purchase their products. Their efforts are different in the sense that, the aim of manufacturer’s national
advertising is to influence potential customers and raise brand
awareness, while the Retailer’s local advertising is intended to
bring potential customers to the point of desire and action (Huang
and Li, 2001). Vertical co-op advertising is an arrangement whereby a manufacturer agrees to pay for a portion or the entire costs of
local advertising undertaken by a retailer. The percentage of local
advertising cost that the manufacturer agrees to pay is called ‘‘participation rate’’ (Bergen and John, 1997). The main reason for the
manufacturer to use co-op advertising is to strengthen the brand
image and promote immediate sales at the retail level (Hutchins,
1953).
The vertical co-op advertising plays an important role in firms’
marketing programs. Total expenditures on co-op advertising in
the United States in 2000 were estimated at $15 billion; an approximately fourfold increase in real terms compared with $900 million
in 1970 (Nagler, 2006). Berger (1972) was the first to address the
vertical co-op advertising problem mathematically. Using a real
world application, he showed the proposed quantitative analysis
can be applied in determining the optimal decisions appropriately.
A common approach in the literature to analyze the role of coop advertising in supply chain coordination is to use game theoretical models; these exist in two categories: static and dynamic. In
static models, interactions among supply chain members are discussed in a single period. Examples in this category are Dant and
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Berger (1996), Bergen and John (1997), Kim and Staelin (1999),
Karray and Zaccour (2006, 2007), Huang and Li (2001), Huang
et al. (2002) and Li et al. (2002). In dynamic models, a goodwill
function is introduced to express the carry-over effect of advertising. Most of the studies in this category ignore the participation
rate despite its fundamental role. Reader is referred to Jørgensen
et al. (2001, 2000) and Jørgensen and Zaccour (2003b) for more
examples. On the other hand, when the retailer has perfect knowledge about manufacturer’s decision in advertising policy or in the
case it has already been announced, the required assumptions of
game-theoretic model do not hold. Berger et al. (2006) considered
such issue in co-op advertising problem. Determining retail/wholesale price has been the focus of many studies, as a fundamental
task in the supply chain management literature. Jeuland and
Shugan (1983, 1988), McGuire and Staelin (1983), Moorthy
(1988), Ingene and Parry (1995a,b, 1998, 2000) and Choi (1991,
1996) discuss channel coordination in the context of two-level
supply chain; they do this by adopting two common pricing mechanisms as well as two-part tariffs and quantity discounts. There are
a number of studies that consider pricing and advertising decisions
simultaneously in supply chain coordination. Jørgensen and
Zaccour (1999) proposed a differential game model in which they
consider pricing and advertising decisions in a two-level supply
chain under channel conflict and coordination. In their study, consumer demand is influenced by retail price and advertising goodwill. Jørgensen et al. (2001), considered the leadership role in a
single-manufacturer–single-retailer marketing channel; each
player controls her advertising and margin. In their model, consumer demand is influenced by both advertising goodwill and retail price. They proposed four game-theoretic models and
compared the results. Jørgensen and Zaccour (2003a) also modeled
consumer demand as the multiplicative product of retail price and
advertising goodwill in dynamic setting, and then compared results in coordinated strategies with all those of uncoordinated.
In the static framework, Yue et al. (2006) extended the model of
Huang et al. (2002); they did this by considering a price-sensitive
demand and studied the impact direct discount from manufacturer
to the costumer may have on the channel coordination. In his paper,
Zaccour (2008) attempted to study the conditions that may lead the
manufacturer to achieve the integrate channel solution by means of
a two-part tariff wholesale price. He further compared static and
dynamic models in which demand function is affected by price
and advertising. He et al. (2009) modeled a single-manufacturer–
single-retailer supply chain as a stochastic Stackelberg differential
game; in this game the demand is a function of both retailer’s price
and advertising. Szmerekovsky and Zhang (2009) considered pricing and advertising in a two-member supply chain; where costumer demand depends on both retail price and advertisement.
They obtained both the manufacturer and the retailer’s optimal
decisions by solving the Stackelberg-manufacturer. Xie and Neyret
(2009) and Xie and Wei (2009) followed a similar approach; they
compared the cooperative game optimal results with those of
non-cooperative. Xie and Neyret (2009) investigated four game
models, three of which were non-cooperative and one was cooperative; whereas, Xie and Wei (2009) only considered two game models including Stackelberg-manufacturer and cooperative game.
This paper is closely related to the last three studies just mentioned. According to Choi (1991), different demand-price functions
lead to considerably different results. Following Choi’s results, in
this paper, a relatively general demand function is proposed, compared to what Xie and Wei (2009) did in their model. In addition,
we investigate one cooperative and three non-cooperative gametheoretic models; in contrast to only two models discussed by
Xie and Wei. Major differences between this paper and three most
related studies mentioned above are summarized in Table 1.
To our best knowledge, most of the studies in the subject of
power balance have assumed a dominant manufacturer. This considers the manufacturer as leader and the retailer as follower
(Berger, 1972; Somers et al., 1990). Nowadays, this issue is the focus of many research studies (e.g. see Kumar, 1996; Kadiyali et al.,
2000; Geylani et al., 2007). There exist some different approaches
in the supply chain coordination; for instance, consider a powerful
manufacturer, such as P&G, who is able to order certain shelves in
her retailer’s stores, whereas a powerful retailer, such as Wal-Mart,
is able to limit manufacturer’s margin or demand extra requirements including RFID attachment, inventory management, quality
control, etc.
Keeping in mind both approaches, we propose four scenarios
including (1) equal power as in Nash game, (2) powerful manufacturer or Stackelberg-manufacturer game, (3) powerful retailer or
Stackelberg-retailer game and (4) the state of integration or cooperation game.
The remainder of the paper is organized as follows: in Section 2,
the model framework is presented. Four game-theoretic models
based on one cooperative and three non-cooperative games are
discussed in Section 3. Section 4 is dedicated to illustrate the results of four proposed models. The feasibility of cooperation and
solution of bargaining game is discussed in Section 5. Finally, the
conclusion including summary of the main results and some directions for future research is given in Section 6. Proofs of all propositions appear in the Appendix.
2. Model framework
Consider a supply chain that consists of a single manufacturer,
selling her products through a single retailer that, in turn, sells
the manufacturer’s product only. The manufacturer decides on
the wholesale price w, National advertising expenditures A, and
participation rate t. The retailer, on the other hand, decides on
the retail price p and local advertising costs a. Bearing in mind
the prevalent assumption in the literature (Jørgensen and Zaccour,
1999, 2003a; Yue et al., 2006; Szmerekovsky and Zhang, 2009; Xie
and Wei, 2009; Xie and Neyret, 2009), it can be assumed that the
consumer demand D(p, a, A) to have the following form:
Dðp; a; AÞ ¼ D0 gðpÞ hða; AÞ;
ð1Þ
Table 1
Comparing the current paper with three most related studies.
Demand function
Price effect
Szmerekovsky and Zhang (2009)
p
e
(e > 1)
Xie and Neyret (2009)
Xie and Wei (2009)
Proposed model
a1 b1p (a1, b1 > 0)
a1 b1p (a1, b1 > 0)
ða1 b1 pÞv ða1 ; b1 > 0Þ
pffiffiffi
pffiffiffi
k1 a þ k2 A ðk1 ; k2 > 0Þ
N
SM
SR
Co
Advertising effect
a2 b2acAd (a2, b2,c, d > 0)
a2 b2acAd (a2, b2,c, d > 0)
pffiffiffi
pffiffiffi
k1 a þ k2 A ðk1 ; k2 > 0Þ
Game structures
–
SM
–
–
N
SM
SR
Co
–
SM
–
Co
1
p, retail price; a, local advertising expenditures; A, national advertising expenditures; N, Nash game; SM, Stackelberg-manufacturer game; SR, Stackelberg-retailer game; Co,
cooperation game.
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where D0 is the base demand, g(p) and h(a, A) reflect the effect of retail price and advertising costs on demand, respectively. Based on
the concept of price elasticity, the demand to a product changes
when its price changes; in other words, an increase in the price results in a decrease in demand. Change in quantity demanded in response to price change becomes excessive for the product that has a
higher elasticity, and this product type is called highly elastic. Based
on the economic interpretation of the price elasticity of demand,
it
p
can be derived for proposed demand function as Ed ¼ ð1=v Þ 1p
.
There exist two kinds of demand curve involving straight line and
non-linear. The reader interested in price elasticity of demand is referred to Begg et al. (2002). In the literature, practitioners proposed
different demand functions of all possible curvatures including
straight line, convex and concave (Hsu, 2006; Batten, 1988; Choi,
1991). As proposed by Piana (2004), three types of society bring
about different shapes of demand curve. The linear demand curve
arises when the reserve prices follow a uniform distribution. While,
a concave demand curve arises when a wide number of consumers
have the same middle reserve prices and only few rich or poor exist.
By contrast, a polarized distribution of reserve price with most consumers having low reserve prices and only few are rich or middle
leads to a convex demand curve.
Unlike the common assumption of the linear relationship between demand and retail price, in this paper, g(p) is assumed to
have a relatively general form as:
1
gðpÞ ¼ ða bpÞv ;
ð2Þ
where a, b and v are positive constants. Values of v < 1, v = 1, and
v > 1 yields convex, linear and concave demand-price curves,
respectively.
The advertising effect h(a, A) is seen in a similar way as mentioned by Xie and Wei (2009):
pffiffiffi
pffiffiffi
hða; AÞ ¼ k1 a þ k2 A;
ð3Þ
where k1 and k2 are positive constants that, respectively, reflect the
effectiveness of local and national advertising in generating sales.
As it can be observed from Eq. (3), h(a, A) is an increasing concave
function of both a and A. Since the additional advertising generates
continuously diminishing returns, it would be consistent with the
‘‘advertising saturation effect’’. Such effect is concluded to characterize the shape of sales-advertising function (Simon and Arndt,
1980). Kim and Staelin (1999) and Karray and Zaccour (2006) used
a similar approach to link sales and advertising efforts. Combining
Eqs. (1)–(3), the demand function would be the following:
pffiffiffi
pffiffiffi
1
Dðp; a; AÞ ¼ D0 ða bpÞv ðk1 a þ k2 AÞ:
ð4Þ
In order to avoid negative demand, the following condition should
be verified:
Dðp; a; AÞ > 0 ) p <
a
b
ð5Þ
:
The profit functions of the two channel members and the system as
a whole are calculated as:
pffiffiffi
1
pffiffiffi
Pm ðw; A; tÞ ¼ D0 ðw cÞða bpÞv ðk1 a þ k2 AÞ A ta;
1
pffiffiffi
pffiffiffi
Pr ðp; aÞ ¼ D0 ðp w dÞða bpÞv ðk1 a þ k2 AÞ ð1 tÞa;
1
pffiffiffi
pffiffiffi
Pmþr ðp; a; AÞ ¼ D0 ðp c dÞða bpÞv ðk1 a þ k2 AÞ A a;
ð6Þ
the whole system. To avoid the backwash effect of the profit in
Eqs. (6)–(8), we have:
Pm > 0 ) w > c; Pr > 0 ) p > w þ d > w;
Pmþr > 0 ) p > c þ d:
Combining Eq. (5) and the last inequality, it can be concluded that
a b(c + d) > 0. To simplify the analysis process throughout this paper, we use an appropriate change of variables similar to the one
used by Xie and Neyret (2009):
a0 ¼ a bðc þ dÞ;
p0 ¼
b
a0
ðp ðc þ dÞÞ;
0
k1 ¼ D0
a0v1þ1
b
k1 ;
w0 ¼
0
k2 ¼ D0
b
a0
ðw cÞ;
a0v1þ1
b
k2 :
Based on the above equations, we have:
p<
a
b
() bp bðc þ dÞ < a bðc þ dÞ ()
bðp ðc þ dÞÞ
a bðc þ dÞ
< 1 () p0 < 1;
p > w þ d () p ðc þ dÞ > w c () p0 > w0 :
By applying the above changes to Eqs. (6)–(8), it can be rewritten as
follows:
1
pffiffiffi
pffiffiffi
P0m ðw0 ; A; tÞ ¼ w0 ð1 p0 Þv k01 a þ k02 A A ta;
pffiffiffi
pffiffiffi
0
0
P
¼ ðp w Þð1 k1 a þ k2 A ð1 tÞa;
pffiffiffi
pffiffiffi
1
P0mþr ðp0 ; a; AÞ ¼ p0 ð1 p0 Þv k01 a þ k02 A A a:
0
0
r ðp ; aÞ
0
0
1
p0 Þv
ð9Þ
ð10Þ
ð11Þ
For the sake of simplicity, we will remove the superscript (0 ) in the
sequel.
3. Four game models
In this section, four game-theoretic models based on three noncooperative games including Nash, Stackelberg-manufacturer and
Stackelberg-retailer (SM and SR in the sequel) with one cooperative is discussed.
3.1. Nash game
When the manufacturer and the retailer have the same decision
power, they determine their strategies independently and simultaneously. This situation is called a Nash game and the solution to
this structure is the Nash equilibrium. To determine the Nash equilibrium, manufacturer and retailer’s decision problems are solved
separately and these are as follows:
max
1
pffiffiffi
pffiffiffi
Pm ðw; A; tÞ ¼ wð1 pÞv ðk1 a þ k2 AÞ A ta
ð7Þ
max
0 6 A and 0 6 t 6 1;
pffiffiffi
pffiffiffi
1
Pr ðp; aÞ ¼ ðp wÞð1 pÞv ðk1 a þ k2 AÞ ð1 tÞa
ð12Þ
s:t: 0 6 w 6 1;
ð13Þ
s:t: w 6 p 6 1 and 0 6 a:
ð8Þ
where c and d are positive constants, respectively, denoting the
manufacturer’s unit production cost and the retailer’s unit handling
cost, in addition to purchasing cost w. Throughout this paper, subscripts m,r, and m + r represent the manufacturer, the retailer and
An obvious result is that the optimal value of t would be zero in the
manufacturer’s point of view, since it has a negative coefficient in
the objective function (12). In addition, Pm is increasing in line with
w, which means the optimal value for w is p. However, since
p > w, w cannot be equal to one, or otherwise there would be no
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profit for both sides. We apply a similar approach as proposed by
Jørgensen and Zaccour (1999) and Xie and Neyret (2009) to tackle
the problem; we assume that the retailer will not sell the product
if she does not get a minimum unit margin. We take manufacturer’s
unit margin as such minimum level and replace the wholesale price
constraint with:
lr > lm ) p w P w ) w 6 p=2;
where lr = p w and lm = w are retailer’s and manufacturer’s unit
margins, respectively. Hence, the optimal value of w is 2p.
t ¼ 0;
p
w¼ ;
2
2
1
k2 wð1 pÞ1=v :
A¼
2
ð16Þ
ð17Þ
ð18Þ
In order to determine the SR equilibrium, the retailer’s decision
problem is solved through incorporating optimal values of t, w, A
in her profit function.
Proposition 3. The SR game has the following unique equilibrium:
Proposition 1. The Nash game has the following unique equilibrium:
wSR ¼
t N ¼ 0;
v
2v
pN ¼
;
2v þ 1
2v þ 1
v2þ2
v2þ2
1 2
1
1 2
1
AN ¼ k2 v 2
a N ¼ k1 v 2
:
4
2v þ 1
4
2v þ 1
wN ¼
pSR ¼
In this section, the relationship between the manufacturer and
the retailer is modeled as a sequential non-cooperative game,
where the manufacturer is the leader and the retailer the follower.
The solution to this structure is called the Stackelberg equilibrium.
In order to obtain it, the best response of the follower and in the
second stage should be determined at first. The leader’s decision
problem is solved based on the follower’s response. Regarding
the retailer’s solution in Nash game, the retailer’s response is as
follows:
1þ1 !2
1
1w v
k1 v
;
2ð1 tÞ
v þ1
ð14Þ
v þw
p¼
:
v þ1
ð15Þ
Solving the manufacturer’s decision problem (12) by incorporating
optimal values of a and p, the SM equilibrium can be calculated.
Proposition 2. The SM game has the following unique equilibrium:
2
wSM ¼
k¼
2v ðv þ 1Þk þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2
4v 2 ðv þ 1Þ2 k ðk þ 1Þv 2 þ ðv þ 2Þ2
2
ðv þ 2Þ2 þ 4k ðv þ 1Þ2
aSM ¼
¼
t SM ¼
1 wSM
v þ1
!
1=v 2
wSM k2 1 wSM
;
2
v þ1
v ð1 wSM Þ
k1 v
wSM ðv þ 2Þ þ v
wSM ð3v þ 2Þ v
;
wSM ðv þ 2Þ þ v
pSM ¼
v
;
v þ1
v
v þ1
aSR ¼
;
k2 2
v
16
2
ASR ¼
;
k1 2
v
16
v
v2þ2
1
;
v þ1
2
v þ2
1
:
þ1
t SR ¼ 0;
The previous three subsections discussed three non-cooperative
games. Now, we model the manufacturer–retailer relationship as a
cooperative game in which both channel members agree to cooperate and maximize the profits of the whole system.
Proposition 4. The cooperation game has the following unique
solution:
A ¼
1
k2 v
2
pco ¼
v
:
v þ1
co
1
v þ1
v1þ1 !2
;
co
a ¼
1
k1 v
2
v
v1þ1 !2
1
;
þ1
The solution (pco, aco, Aco) gives the maximum profits for the whole
system. The wholesale price w and participation rate t, are open
to take any value between zero and 1. Despite this, it is obvious that
the individual profit of neither the manufacturer nor the retailer is
not independent of w and t. Both sides would participate in the
cooperation only if their individual profits are higher than those
of non-cooperative cases. Thus, we will discuss the feasibility of
the cooperative game in the next section. Table 2 summarizes the
optimal solution of four game models.
;
4. Discussion of the results
k2
;
k1
1þ1 !2
2
v
3.4. Cooperation
3.2. Stackelberg-manufacturer game
a¼
1
2
ASM
v þ wSM
:
v þ1
3.3. Stackelberg-retailer game
We now model the manufacturer–retailer relationship as a
sequential non-cooperative game in which the retailer has more
decision power, and is the leader. The solution to this structure is
called the SR equilibrium. The first step in determining it, similar
to the previous section, is to find the best response of the manufacturer. The manufacturer response as provided in Section 3.1 is:
In this section, we will discuss the comparisons among the optimal solution of the four games. All the presented comparisons are
given based on different values for the parameters k and v. The
parameter k, reflects the effectiveness of national advertising versus local advertising and is defined as k = k2/k1. The parameter v,
defines the shape of demand-price function. The demand-price
function would be linear, convex or concave on condition that v
is equal to, less than or greater than 1, respectively. Comparisons
are independent of values for k1 and k2.
Considering the optimal solution summarized in Table 2, all of
the decision variables are some function of the values k and v; this
means the optimum decisions of both sides can be calculated
through estimating these two parameters. This would make the issue of determining these parameters more challenging, since the
quality of the results depends greatly on the quality of the estimated parameters. Therefore, to determine the best policy, both
decision makers should estimate these parameters at first. To do
so, one should start a deep market research in order to specify
the behavior of the demand to advertisement, both national and local. The parameters k1 and k2 are positive constants which measure
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Table 2
Summary of the optimal solutions in four game models.
Nash game
SM game
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2 2 2
2
Wholesale price w
v
2v þ1
2v ðv þ1Þk þ
Retail price p
2v
2v þ1
v þwSM
v þ1
National advertising A
Local advertising a
1 2
4 k2
v2
1 2
4 k1
v2
v2þ2
1
2v þ1
2
v þ2
1
2v þ1
wSM k2
2
4v v þ1Þ k ðk þ1Þþv
2
ðv þ2Þ2 þ4k ðv þ1Þ2
Participation rate t
0
wSM ð3v þ2Þv
wSM ðv þ2Þþv
Price elasticity of demand Ed
2
1
k¼
v
v þwSM
1wSM
k1 v
SR game
Cooperation game
1 v
2 v þ1
–
v
v þ1
k22
16
1w
v þ1
v ð1wSM Þ
wSM ðv þ2Þþv
v þ2Þ
2ð
2
SM 1=v
2ð
1
þ1
1wSM v
v þ1
2
2
k1
16
v2
v2
1
v2þ2
v
v þ1
1
2 k2
v þ1
1
v2þ2
1
2 k1
v þ1
0
–
1
1
v
v
1
v1þ1 2
1
v1þ1 2
v þ1
v þ1
k2
.
k1
the potential increase in sales due to an increase in advertising
costs for both retailer and manufacturer, respectively. Since all
optimal solutions are some functions of v, it should be estimated
precisely. Necessity of a good and its corresponding parameter v
are inversely related; the higher the degree of necessity, the lower
the value of v (e.g. for luxury goods v seems to increase). Different
amounts for the shape parameter may be found in different industries depending on the nature of goods provided and the behavior
of their final consumers. In this paper, our distinct contribution is
to provide a flexible environment which enables the decision maker to recognize the shape of demand function and obtain the right
parameters of the identified model using market data. As mentioned before, a special form of this problem has been discussed
earlier in the literature where shape parameter assumed to be 1
(Xie and Wei, 2009). This paper supposes a more general and flexible model.
Fig. 1. Retail and wholesale prices.
4.1. Comparisons on prices
The summary of the results provided in Table 2 show that the
optimal retail prices in the SR and the cooperation game are the
same. It is obvious that the optimal retail price in the Nash game
is higher than those in SR and cooperation.
In the SM game, the retail price is higher than when the retailer
is the leader, because the manufacturer imposes a higher wholesale price, hence, in order to obtain a tolerable profit margin, the
retailer chooses a higher price. Fig. 1 illustrates the comparison between optimal values of wholesale price and retail price in four
game models. It can be observed from Fig. 1 that the resulting optimum solution divides the parameters space into two distinct regions. In region (I) the optimal price for the Nash game is higher
than that of MS, whereas in region (II) both retail and wholesale
prices for SM game are higher than that of Nash.
4.2. Comparisons on advertising expenditures
Fig. 2. National advertising expenditures.
Taking the summarized solutions in Table 2 into consideration,
the following simple outcome about the retailer’s advertising is
obtained:
8ðk; v Þ : aco > aSM > aSR > aN :
The highest local advertising expenditure is made in the cooperative game and the lowest occurs in the Nash game. When the manufacturer is the leader, the retailer spends more on advertising,
because the manufacturer participates in local advertising cost,
tSM > 0.
As illustrated in Fig. 2, the highest national advertising expenditure occurs in the cooperation, the same result as in local advertising case. Depending on the values of the parameters k and v, the
comparison results are different. However, one may believe that
the manufacturer spends more on national advertising when she
is the follower to the retailer rather than in the Nash Game in all
three regions.
4.3. Comparisons on participation rate
As shown in Table 2, the manufacturer’s participation rate in retailer’s local advertising expenditures equals zero in both Nash
game and SR game. Fig. 3 illustrates the behavior of the optimal
participation rate w.r.t. k and v in the SM game. When the effectiveness of local advertising is higher than that of national efforts,
the lower participation rate is preferred by the manufacturer. On
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M.M. SeyedEsfahani et al. / European Journal of Operational Research 211 (2011) 263–273
Fig. 3. Participation ratio tSM.
the other hand, lower values of m will result in lower participation
rate.
4.4. Comparisons on profits
Fig. 5. Retailer’s profits.
The profit is the most important performance measure in supply chain. Fig. 4 compares the optimal profit of the manufacturer,
obtained in three non-cooperative models.
It can be implied that the manufacturer prefers to play the Stackelberg with retailer rather than to be in conflict situation with the
retailer in the Nash game. Moreover, in region (I), the manufacturer
desirably prefers to be the retailer’s follower than to be the channel’s leader; while, in region (II), the manufacturer prefers to be
the leader.
As illustrated in Fig. 5, the retailer prefers to play the Stackelberg game in all three regions rather than to play separately as
in the Nash game. The retailer naturally prefers to be the leader
of the channel, whereas she agrees to be the follower in region (I).
Fig. 6 compares the whole system’s profits across four games. As
a popular result in the literature, the highest profit will be achieved
in the cooperation case; i.e. the channel members decide to make
decision cooperatively in order to maximize the whole system’s
profits.
In regions (I) and (II) total profit of the supply chain has a higher
level in Stackelberg game considering retailer as the leader. Even
though playing SM game will result in higher total profit as shown
in region (III). Furthermore, in regions (II) and (III) the Stackelberg
Fig. 6. System’s profits.
game has a higher profit than Nash game. Whereas, surprisingly total profit has a higher level in the Nash game compared with SM
game.
4.5. Feasibility of the cooperation game
Fig. 4. Manufacturer’s profits.
In Section 3.4, we reached the analytical solution to the cooperative game, and observed that the manufacturer and the retailer
will only agree to make joint decisions if their individual profit is
higher in the cooperative game than in the non-cooperative ones.
In this section, we use the comparison results in Section 4.4 to ver-
M.M. SeyedEsfahani et al. / European Journal of Operational Research 211 (2011) 263–273
269
ify the feasibility of this problem. In order to understand this, we
need to show that (pco, wco, aco, Aco, tco) exists:
co co
SM
SR
N
max
co
co
co
Pco
m ¼ Pm ðp ; w ; a ; A ; t Þ P max Pm ; Pm ; Pm ¼ Pm ;
ð19Þ
co co
SM
SR
N
co
co
co
Pco
:
¼ Pmax
r ¼ Pr ðp ; w ; a ; A ; t Þ P max Pr ; Pr ; Pr
r
ð20Þ
By integrating Eqs. (19) and (20), equivalently we have:
co
co
max
max
Pco
:
mþr ¼ Pm þ Pr P Pm þ Pr
ð21Þ
By combining Figs. 4 and 5, Fig. 7 is obtained, which has five regions. Table 3 determines the maximum profits of the manufacturer
and the retailer in each region.
The next step is to verify the condition in Eq. (21) for each region of Fig. 7 to understand if a feasible solution is achieved. In region (I), the maximum profit of the manufacturer corresponds to
the SR game, while the retailer’s is obtained in the SM game.
Fig. 8 illustrates the relative difference between cooperation and
non-cooperation.
D1 ¼
SR
SM
Pco
mþr Pm þ Pr
Pco
mþr
Fig. 8. Relative difference D1in regions (I).
100:
According to Fig. 8, the relative difference is positive, hence, the
condition in Eq. (21) holds true and the feasible solution is certain
to exist. The comparison results in regions (II) and (III) are identical
and imply that the maximum profit of both sides is obtained in the
SR game. As illustrated in Fig. 6, it is obvious that the condition in
Eq. (21) holds true in these two regions, as the cooperation case
yields the highest profits for the whole system. Thus, the cooperation game is feasible in these two regions. In regions (IV) and (V),
the maximum profits of the retailer and the manufacturer are ob-
Fig. 9. Relative difference D2in regions (IV) and (V).
tained, respectively in the SR and the SM. Similar to the approach
used in region (I), Fig. 9 illustrates the relative difference D2 between cooperation and non-cooperation; this displays that the condition in Eq. (21) holds true in these two regions and the feasibility
of the cooperation is obvious.
D2 ¼
Fig. 7. Five regions to discuss the feasibility of the cooperation game.
Table 3
Maximum profit of supply chain members in five regions of Fig. 7.
Pmax
m
Pmax
r
(I)
PSR
m
PSR
m
PSM
m
PSM
r
PSR
r
PSR
r
(IV) and (V)
Pco
mþr
100:
We showed that the cooperation game is feasible; therefore, this
resulted in the manufacturer and the retailer’s willingness to
cooperate. The next issue that is to be resolved is the sharing of
the extra gained profit. The profit-sharing problem is discussed
in Section 5.
5. Bargaining problem
Region
(II) and (III)
SR
SM
Pco
mþr Pm þ Pr
In this section, a feasible region for the variables w and t is presented. Finally, the Nash bargaining model will be used to solve the
profit-sharing problem in this region. We use a similar approach
employed by Xie and Wei (2009) and Xie and Neyret (2009) to
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M.M. SeyedEsfahani et al. / European Journal of Operational Research 211 (2011) 263–273
illustrate the feasible region for this problem. The extra profit of
the manufacturer and the retailer are as follows:
co
m
km
km
DP ¼
ðD CÞ;
km þ kr
km þ kr
kr
kr
DPr ðw ; t Þ ¼
DP ¼
ðD CÞ;
km þ kr
km þ kr
Ckm þ Dkr
) w B t aco ¼
:
km þ kr
DPm ðw ; t Þ ¼
max
m
DPm ¼ P P
pffiffiffiffiffiffi
pffiffiffiffiffiffiffi
¼ wð1 pco Þ1=v k1 aco þ k2 Aco Aco taco Pmax
m
¼ wB taco C > 0;
max
DPr ¼ Pco
r Pr
¼ ðp wÞð1 The solution to this problem is as follows:
1
pco Þv
ð22Þ
pffiffiffiffiffiffi
pffiffiffiffiffiffiffi
k1 aco þ k2 Aco ð1 tÞaco Pmax
r
¼ wB þ taco þ D > 0;
ð23Þ
where B, C and D are chosen as follows:
pffiffiffiffiffiffi
pffiffiffiffiffiffiffi
B ¼ ð1 pco Þ1=v k1 aco þ k2 Aco > 0;
> 0;
C ¼ Aco þ Pmax
m
D ¼ pco B aco Pmax
> 0:
r
Inequalities (22) and (23) specify region between two parallel
lines as illustrated in Fig. 10. Every pair (w, t) in this region presents a feasible solution to the bargaining problem. The closer this
point approaches the line, Pm ¼ Pmax
the lower the manufacturer’s
m
share and the higher the retailer’s. All the pairs (w, t) located on a
line parallel to Pm ¼ Pmax
lead to the same profit for
m
max
Pm ¼ Pmax
þ DPr for the
m þ DPm the manufacturer and Pr ¼ Pr
retailer.
According to Nash (1950), the bargaining outcome (w⁄, t⁄) is obtained by maximizing the product of individual utilities over the
feasible solution. Consider the following utility functions for the
manufacturer and the retailer:
um ðw; tÞ ¼ DPm ðw; tÞkm ;
ur ðw; tÞ ¼ DPr ðw; tÞkr :
To obtain the Nash’s solution, the following optimization needs to
be solved:
Max um ðw; tÞ:ur ðw; tÞ ¼ DPm ðw; tÞkm :DPr ðw; tÞkr :
ð24Þ
Eq. (24) presents a line parallel to Pm ¼ Pmax
m . If km > kr, this line
would be closer to Pr ¼ Pmax
, hence, the manufacturer’s share of
r
the extra profit will be greater than the retailer’s, and vice versa.
When km = kr, then the manufacturer and the retailer will split
the extra profit equally. In this section, we learn about the relationship (24) between the optimal values of t and w in the cooperation game. Without any further assumptions, determination of
individual values of w and t is impossible.
6. Conclusions
Optimal decisions of pricing and vertical co-op advertising in
four game-theoretic models are derived; these consist of one cooperative three non-cooperative games in a single-manufacturer–
single-retailer supply chain. At this point, the consumer demand
is influenced by both price and advertising expenditures. In the
proposed model, the relationship between price and demand has
a relatively general form compared with the classic linear relationship. Comparison results reflect the significant effect the shape of
demand-price function (i.e. linear, convex or concave) may have
on optimal values of decision variables and supply chain members’
profit. The practical aspects of our proposed model include
addressing the advertising saturation effect in demand function
modeling and different channel structures (cooperation, Nash
game and the case of dominant member). The results show that
the retail price is the lowest when two members of the supply
chain decide to cooperate, whereas the advertising expenditures
are higher in non-cooperative games. Also the highest amount of
profit for the whole system is achieved in cooperation case. It
has also been shown that the manufacturer prefers to be the retailer’s follower rather than to be in conflict situation (Nash game).
The feasibility of the cooperation game is verified, and the feasible
region for the bargaining problem is defined. This is the point when
supply chain members can share the extra-profit gained by the
cooperation. The Nash bargaining model is used to solve the bargaining problem. The optimal values for participation rate and
wholesale price for the cooperation game are not specified individually, however, a relationship between their optimal values is derived. There are several possible directions for the future studies.
First, one can involve more decision-makers to enrich the results
and add competitive characteristics to the model. Second, one
can adopt a different form of demand function and finally, other
bargaining schemes may be applied in order to achieve different
results.
Acknowledgment
The authors are grateful to the referees for their valuable comments and illustrative suggestions.
Appendix
Fig. 10. Feasible region of the bargaining problem.
Proof of Proposition 1. The second partial derivative of Pm in Eq.
(12) w.r.t. A is negative, hence, Pm is concave w.r.t. A and the
optimal value is achieved by solving the first order condition:
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M.M. SeyedEsfahani et al. / European Journal of Operational Research 211 (2011) 263–273
1
1
@ Pm 1
¼ k2 wð1 pÞv A2 1 ¼ 0 ) A ¼
2
@A
2
1
1
k2 wð1 pÞv :
2
ðA1Þ
To solve the retailer’s problem (13), we define variable x as
1
ðp wÞð1 pÞv . To determine the domain of x, we solve the first order equation below and compare the critical value with values of x
at p = 0 and p = w:
1
@x 1
v þw
¼ ð1 pÞv 1 ðv pðv þ 1Þ þ wÞ ¼ 0 ) p1 ¼
2 ðw; 1Þ;
@p v
v þ1
p ¼ w ) x ¼ 0;
p ¼ p1 ¼
v1þ1
v þw
1w
)x¼v
v þ1
v þ1
2
2
@ Pm
k1
1w v
ðv 2wðv þ 1ÞÞ
¼
@w
2ðv þ 1Þð1 tÞ v þ 1
pffiffiffi 11
k2 A
1w v
ðv wðv þ 1ÞÞ
þ
v ðv þ 1Þ v þ 1
2þ1
2
tk1 v
1w v
;
þ
2
2ð1 tÞ v þ 1
1
@ Pm wk2 1 w v
1;
¼ pffiffiffi
@A
2 A v þ1
2þ1
2
@ Pm
k1 v
1w v
ð2wðv þ 1Þ v ð1 wÞ
¼
@t
4ð1 tÞ3 ð1 þ v Þ v þ 1
> 0;
tð2wðv þ 1Þ þ v ð1 wÞÞÞ:
p ¼ 1 ) x ¼ 0:
Thus, the maximum of x is obtained while p = p1 and the minimum
value of x equals zero. Now we rewrite the retailer’s decision problem as follows:
max
pffiffiffi
pffiffiffi
Pr ðx; aÞ ¼ xðk1 a þ k2 AÞ ð1 tÞa
s:t: 0 6 x 6 v
1þ1
1w v
and 0 6 a:
v þ1
It is obvious that the retailer’s profit is increasing with x, thus, the
optimal value of x is:
x ¼ xmax ¼ v
1þ1
1w v
:
v þ1
ðA2Þ
It is also obvious that Pr is concave w.r.t. a; this is because the second partial derivative of Pr w.r.t. a is negative and therefore, the
optimal value of is obtained as follows:
@ Pr 1
1
¼ k1 xa2 ð1 tÞ ¼ 0 ) a ¼
@a
v
1
k1 x
2ð1 tÞ
2
:
ðA3Þ
Solving Eqs. (A1)–(A3) considering w ¼ 2p and t = 0, we achieve the
Nash equilibrium:
wN ¼
AN ¼
v
2v þ 1
;
pN ¼
1 2 2
1
k v
4 2
2v þ 1
2v
;
2v þ 1
v2þ2
;
0 6 t 6 1 t < 1;
0 6 w 6 1 w < 1;
0 < A 6 A 6 Amax ;
where t, w, A are positive constants and sufficiently close to zero.
Amax is a positive constant, which is assumed to be as great as we
desire, employed to restrict the solution area. The new solution area
is a closed and bounded set, and the objective function is defined
over it. The ‘‘extreme value theorem’’ a.k.a. ‘‘Weierstrass theorem’’,
states that if a real valued function is continuous over a closed and
bounded set, this function must attain its minimum and maximum
value, each at least once. Over the new set of constraints, Pm verifies the conditions of the ‘‘extreme value theorem’’. Thus, we can
ensure that there exists a solution to this problem. This solution
must fulfill the KKT first order necessary conditions. The KKT conditions for this problem are as follows:
0 @ ð P Þ 1
0 @g 1
m
i
@w
6
C X
C
B @ ð@w
B @g
B Pm Þ C þ
B i C ¼ 0;
u
i
@ @t A
@ @t A
@ ðPm Þ
@a
t N ¼ 0;
aN ¼
As the partial derivatives of Pm show, this function is not differentiable at t = 1, w = 1 (if v > 1) and A = 0. When participation rate t approaches 1, the profit function, in turn, approaches negative infinity.
When w is equal to 1, the profit function equals A < 0. We can assume a minimum level for the national advertising expenditures, to
ensure that the profit function is differentiable at this point. The
new set of constraints is as follows:
i¼1
g 1 ¼ w 6 0;
1 2 2
1
k v
4 1
2v þ 1
This completes the proof of Proposition 1.
v2þ2
g 3 ¼ t 6 0;
:
h
Proof of Proposition 2. Substituting Eqs. (14) and (15) into the
expression of Pm , the decision problem (12) becomes:
!
1
1þ1
2
pffiffiffi
1w v
k1 v
1w v
Max Pm ¼ w
þ k2 A A
v þ 1 2ð1 tÞ v þ 1
2þ2
2
tk1 v 2
1w v
4ð1 tÞ2 v þ 1
@g i
@a
g 2 ¼ w ð1 w Þ 6 0;
g 4 ¼ t ð1 t Þ 6 0;
g 5 ¼ A þ A 6 0;
g 6 ¼ A Amax 6 0;
ui g i ¼ 0 for i ¼ 1; . . . ; 6;
ui P 0 for i ¼ 1; . . . ; 6:
Table A1
Possible combinations of active constraints.
No constraint is active
Only one constraint is active
Only two constraints are
active
s:t: 0 6 w 6 1; 0 6 A; 0 6 t 6 1:
Before solving this non-linear programming problem, we revise the
constraints to make sure that the objective function is continuous
and differentiable in the solution area. Below comes the partial
derivatives of objective function (12) w.r.t. its corresponding
variables.
Three constraints are active
Possible combinations
of active constraints
combinations
(all ui = 0 )
g1,g2,g3,g4,g5,g6
(g1,g3), (g1,g4), (g1,g5),
(g1,g6),
(g2,g3), (g2,g4), (g2,g5),
(g2,g6),
(g3,g5), (g3,g6), (g4,g5),
(g4,g6),
(g1,g3,g5), (g1,g3,g6),
(g1,g4,g5),
(g1,g4,g6), (g2,g3,g5),
(g2,g3,g6),
(g2,g4,g5), (g2,g4,g6),
Total
1
6
12
8
27
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M.M. SeyedEsfahani et al. / European Journal of Operational Research 211 (2011) 263–273
All the possible combinations of active constraints are shown in
Table A1. The KKT necessary conditions need to be verified for each
combination in order to achieve all candidate local maximum
points.
Note that the constraints g1 and g2 are inconsistent; as a result,
no combination with both g1 and g2 active is possible. The same
result holds true for (g3, g4) and (g5, g6).
By verifying KKT first order necessary conditions in 27 combinations, only one feasible KKT point can be achieved and that is
where no constraint is active.
0 @ðP Þ 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8
2
v 2 ðv þ1Þ2 k2 ðk2 þ1Þþv 2 ðv þ2Þ2
>
;
w ¼ 2v ðv þ1Þk þ ðv4þ2Þ
>
2
>
þ4k2 ðv þ1Þ2
>
>
<
2
1w
v þ1
A¼
>
>
>
>
>
:
v þ2Þv
:
t ¼ wð3
wðv þ2Þþv
1=v
k ¼ kk21 ; ðA4Þ
¼
aSM ¼
ðA5Þ
;
¼
t SM ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2 2
2v ðv þ 1Þk þ 4v 2 ðv þ 1Þ2 k ðk þ 1Þ þ v 2 ðv þ 2Þ2
2
ðv þ 2Þ2 þ 4k ðv þ 1Þ2
wSM ð3v þ 2Þ v
;
wSM ðv þ 2Þ þ v
pSM ¼
v þ wSM
:
v þ1
1
0 6 a:
;
v þ1
2
We can conclude that Pr is an increasing function of y, therefore the
optimal value for y is:
y ¼ ymax ¼
v
1
2 v þ1
v1þ1
ðA7Þ
:
k
ASM
ðA8Þ
By solving Eqs. (16)–(18), (A7) and (A8), The SR equilibrium would
be:
v
;
v þ1
pSR ¼
v2þ2
wSR ¼
1
2
ASR ¼
k2 2
v
16
2
;
k2
;
k1
1þ1 !2
1 wSM v
;
v þ1
1=v !2
wSM k2 1 wSM
;
2
v þ1
v
s:t: 0 6 y 6
2
@ Pr 1
1
1
k1 y :
¼ k1 ya2 1 ¼ 0 ) a ¼
2
2
@a
ðA6Þ
v ð1 wSM Þ
k1 v
SM
w ðv þ 2Þ þ v
1
2
v1þ1
Therefore, the optimal value of is achieved by solving the following
first order equation:
Based on the result of ‘‘extreme value theorem’’, this KKT solution is
the only local maximum candidate; hence, we conclude that this
point is the global maximum of Pm . Therefore we obtain the SR
equilibrium solving Eqs. (14) and (15) and Eqs.(A4)–(A6) as shown
below:
wSM ¼
pffiffiffi
@ 2 Pr
1
3
¼ k1 ya2 < 0:
4
@a2
@ðPm Þ
@a
wk2
2
Pr ¼ y k1 a þ k22 y a
It is also obvious that Pr is concave w.r.t. a; because the second partial derivative of Pr w.r.t. a is negative as shown below:
m
C
B @ð@w
Pm Þ C
All ui ¼ 0 ) B
@ @t A ¼ 0 and All g i < 0
)
max
1
v þ1
v
;
v þ1
t SR ¼ 0;
2
;
aSR ¼
k1 2
v
16
1
v þ1
v2þ2
:
Proof of Proposition 4. To solve this problem, we define z as
p(1 p)1/v To determine the domain of z, we solve the first order
equation below and compare the value of at that point with values
of z while p = 0 and p = 1.
1
@z 1
v
¼ ð1 pÞv 1 ðv pðv þ 1ÞÞ ¼ 0 ) p3 ¼
;
@p v
v þ1
p ¼ w ) z ¼ 0;
v1þ1
v
1
p ¼ p3 ¼
> 0;
)z¼v
v þ1
v þ1
p ¼ 1 ) z ¼ 0:
Proof of Proposition 3. Substituting Eqs. (16)–(18) into the
expression of Pr , the decision problem (13) becomes:
max
pffiffiffi 1
p
Pr ðp; aÞ ¼ ð1 pÞ1=v k1 a þ k22 pð1 pÞ1=v a
2
4
s:t: 0 6 p 6 1 and 0 6 a:
To solve this problem, we define variable y as 2p ð1 pÞ1=v . To determine the domain of y, we solve the first order equation below and
compare the critical value of y at that specific point with values of y
at p = 0 and p = 1.
1
@y
1
v
¼
ð1 pÞv 1 ðv pðv þ 1ÞÞ ¼ 0 ) p2 ¼
;
@p 2v
v þ1
p ¼ w ) y ¼ 0;
1
v 1 v þ1
> 0;
p ¼ p2 ) y ¼
2 v þ1
Hence, the maximum of z is obtained while p = p3 and the minimum
value of z equals zero. Now we rewrite the decision problem as
follows:
max
pffiffiffi
pffiffiffi
Pmþr ¼ z k1 a þ k2 A a A
s:t: 0 6 z 6 v
1
v þ1
v1þ1
;
0 6 a and 0 6 A:
Now we derive the partial derivative of Pmþr w.r.t. z, that is:
pffiffiffi
pffiffiffi
@ Pmþr
¼ k1 a þ k2 A > 0:
@z
Thus, Pmþr is an increasing function of z, therefore, the optimal value of z is:
1
v þ1
v1þ1
p ¼ 1 ) y ¼ 0:
z ¼ zmax ¼ v
Hence, the maximum of y is achieved at p = p2 and the minimum value of y equals zero. Now we rewrite the retailer’s decision problem
as follows:
The optimal values of a and A can be derived from the first order
conditions below:
:
ðA9Þ
M.M. SeyedEsfahani et al. / European Journal of Operational Research 211 (2011) 263–273
2
1
k1 z ;
2
2
1
1
1
k2 z :
¼ k2 zA2 1 ¼ 0 ) A ¼
2
2
@ Pmþr 1
1
¼ k1 za2 1 ¼ 0 ) a ¼
2
@a
ðA10Þ
@ Pmþr
@A
ðA11Þ
The Hessian matrix is a negative definite matrix and fulfills the second-order condition for a maximum.
2
H¼
@ 2 Pmþr
4 @a2
@ 2 Pmþr
@A@a
@ 2 Pmþr
@a@A
@ 2 Pmþr
@A2
3
5¼
2
kp
1z
ffiffi
4 4a a
0
0
kp
2z
ffiffi
4A A
3
5:
Thus, the Eqs. (A9)–(A11) lead to the following solution for the
cooperative game:
A ¼
1
k2 v
2
pco ¼
v
:
v þ1
co
1
v þ1
v1þ1 !2
;
co
a ¼
1
k1 v
2
1
v þ1
v1þ1 !2
;
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