Managing supply uncertainty under supply chain Cournot competition
39
Transcript of Managing supply uncertainty under supply chain Cournot competition
Production, Manufacturing and Logistics
Accepted Manuscript
Managing Supply Uncertainty under Supply Chain CournotCompetition
Yaner Fang, Biying Shou
PII: S0377-2217(14)00961-8DOI: 10.1016/j.ejor.2014.11.038Reference: EOR 12656
To appear in: European Journal of Operational Research
Received date: 29 April 2013Accepted date: 25 November 2014
Please cite this article as: Yaner Fang, Biying Shou, Managing Supply Uncertainty underSupply Chain Cournot Competition, European Journal of Operational Research (2014), doi:10.1016/j.ejor.2014.11.038
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Highlights
• We study how to manage supply uncertainty under supply chain Cournot competition.
• We characterize the equilibrium ordering decisions and contract terms.
• We show that supply chain centralization is a dominant strategy.
• Centralization may decrease supply chain profit, resulting in prisoner’s dilemma.
• Centralization is more desirable under high supply risk and low competition.
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Managing Supply Uncertainty under Supply ChainCournot Competition
Yaner FangThe School of Management, University of Science and Technology of China, [email protected]
Biying ShouDepartment of Management Sciences, City University of Hong Kong, [email protected]
We study the Cournot competition between two supply chains that are subject to supply uncertainty. Each
supply chain consists of a retailer and an exclusive supplier which has random yield. We examine how the
levels of supply uncertainty and competition intensity affect the equilibrium decisions of ordering quantity,
contract offering, and centralization choice. We show that a retailer should order more if its competing
retailer’s supply becomes less reliable or if its own supply becomes more reliable. A supply chain with reliable
supply can take great advantage of the high supply risk of its competing chain. Furthermore, for decentralized
chains we characterize the optimal wholesale price contracts with linear penalty, under different supply risks
and competition scenarios. Finally, we show that supply chain centralization is a dominant strategy, and it
always makes the customers better off. Nevertheless, if the supply risk is low and the chain competition is
intensive, centralization could actually decrease the supply chain profit, compared with the case where both
chains do not choose centralization. This results in a prisoner’s dilemma. On the other hand, if the supply
risk is high and/or the competition level is low, centralization always increases the supply chain profit. The
desirability of supply chain centralization is enhanced by high supply uncertainty or low chain competition.
Key words : Supply uncertainty, supply chain competition, game theory.
1. Introduction
Due to fast development of information technology and more intense global competition, firms are
cooperating more closely than ever to optimize their supply chains. As a result, the traditional
model of firm to firm competition is giving way to a new paradigm of supply chain to supply chain
competition in the market place (Barnes 2006, Ha and Tong 2008).
Meanwhile, supply uncertainty has become a major concern for global supply chain manage-
ment. Supply uncertainty may be caused by various reasons, such as natural disasters, labor strikes,
custom delays, terrorist attacks, changes in government regulation, etc. An industrial survey con-
ducted by Protiviti and APICS (American Production and Inventory Control Society) showed that
66% of respondents considered supply interruption one of their most significant concerns among
all supply chain related risks (O’Keeffe 2006).
This paper offers a systematic examination of how to design and operate supply chains to effec-
tively deal with supply uncertainty and supply chain competition. In particular, we study the
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interaction between two competing supply chains. Each chain consists of one retailer and its exclu-
sive supplier. Both chains are subject to supply uncertainty, i.e., the supplier may not be able to
fulfill the retailer’s order due to exogenous causes.
There are many real-life examples of supply chain to supply chain competition, where supply
risk is also an important issue. Take the dairy industry in China as an example. The industry has
experienced rapid growth in recent years. The market was worth $40.6 billion in 2013, compared
with only $20.7 billion in 2008. The two largest national players, Yili Group and China Mengniu
Dairy Company Ltd., had a combined market share of 41% in 2012, leaving the other players far
behind. The uncertainty of high-quality milk supply poses great challenges to their dairy supply
chain management (e.g., the notorious 2008 milk contamination scandal). Both companies source
from independent and exclusive farms, but they are investing more heavily on establishing their
own farms domestically and internationally in recent years (Sharma and Zhang 2014, Zhang 2014).
Retailers often sell substitutable products in environments with end-market competition, where
the products market prices depend primarily on their output quantities in the market. In such com-
peting environments, supply uncertainty may exacerbate the competition between supply chains.
For example, Canon and Nikon are the two largest players in the digital single lens reflex (SLR)
camera market. In 2011, a large earthquake in Japan forced some Japan-based semiconductor man-
ufacturers to halt production as a result of rolling blackouts, impassable roads, and limited fuel
supplies. It led to the shortage of sensor, flash memories and so on, which affected Canon and
Nikon’s total production output and the market prices for the competing end-products (Miserere
2011, Lucero 2011, Nikon 2011, Canon 2011).
In our paper, we focus on the Cournot supply chain competition where the market price is
determined by the total quantity in the market, and investigate how supply risks, competition
intensity, and supply chain structures may affect firms’ operational strategies. Specifically, we aim
at addressing the following research questions:
• How does supply uncertainty and supply chain competition affect the ordering decisions?
• How to determine the wholesale price and shortage penalty in a decentralized supply chain?
• Does supply chain centralization provide a competitive advantage when dealing with compe-
tition and supply uncertainty?
We consider three different market structures: i) both supply chains are centralized (referred to
as Centralized Competition); ii) one supply chain is centralized while the other is decentralized
(referred to as Hybrid Competition); and iii) both supply chains are decentralized (referred to as
Decentralized Competition). We derive the equilibrium decisions for each game in terms of the
retailer ordering quantity and (for a decentralized chain) the contract terms. By comparing the
three equilibria, we examine the value of centralization.
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For the optimal ordering decisions, we show that a retailer should order more if its competing
retailer’s supply becomes less reliable or if its own supply becomes more reliable. That is, a retailer
with reliable supply can take great opportunity of it’s competing chain’s high supply risk. We also
show how the total market supply and price are affected by supply uncertainty and the intensity
of chain competition. In addition, we show how to optimally design the wholesale-price contract
with shortage penalty in a decentralized supply chain, under different supply risks and competition
scenarios.
Furthermore, we show that supply chain centralization is the dominant strategy under supply
uncertainty and chain-to-chain competition. Nevertheless, supply chain centralization may not
always result in positive gains for the supply chain itself. In fact, we show that if supply risk is
low and competition intensity is high, centralization could actually decrease supply chain profit,
comparing with the case where both chains are decentralized, which results in a prisoner’s dilemma.
On the other hand, if supply risk is high or competition level is small, centralization always increases
the supply chain profit. In other words, the desirability of supply chain centralization is enhanced
by high supply uncertainty and/or low supply chain competition.
The remainder of this paper is organized as follows. Section 2 reviews the related literature. Sec-
tion 3 describes the model. Section 4 investigates the ordering and contract design decisions under
three different game settings: the Centralized Competition Game, the Hybrid Competition Game,
and the Decentralized Competition Game. Section 5 compares the three equilibria from Section 4
and examines the strategic choice of supply chain centralization. Finally, Section 6 concludes the
paper.
2. Literature
Our study is closely related to three areas: supply uncertainty, supply chain contracting, and supply
chain competition.
The existing work on supply uncertainty can be divided into three categories: (i) the random-
yield model, which models the uncertainty by assuming that the supply level is a random function
of the input level (e.g., Yano and Lee 1995, Gerchak and Parlar 1990, Parlar and Wang 1993,
Swaminathan and Shanthikumar 1999, Kazaz 2004, Babich et al. 2007, Federgruen and Yang 2008,
Wang et al. 2008, Kazaz 2008, Deo and Corbett 2009, Gao et al. 2014), (ii) the stochastic lead-time
model, which models the lead-time as a random variable (see a comprehensive review in Zipkin
2000), and (iii) the supply disruption model, which typically models the uncertainty of a supplier
as one of two states: “up” or “down” (e.g., Arreola-Risa and DeCroix 1998, Gupta 1996, Meyer
et al. 1979, Parlar and Berkin 1991, Song and Zipkin 1996, Tomlin 2006, Snyder and Shen 2006a,b,
Yang et al. 2009, Shou et al. 2013). Specifically, for the supply disruption model, the orders are
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fulfilled on time and in full when the supplier is “up,” and no order can be fulfilled when the
supplier is “down”. Our proposed research builds upon the random yield model.
Conflict of interest is ubiquitous in a supply chain, and contracts are widely used to resolve these
conflicts. The key task in supply contract design is to align the objectives of the various firms with
the supply chain’s overall objective, and thereby reduce the inefficiency of the supply chain. Cachon
(2003) provides an excellent review of this field. There are several papers addressing contracting and
competition simultaneously, e.g., Parlar (1988), Cachon (2001), Corbett and Karmarkar (2001),
Carr and Karmarkar (2005), Boyaci and Gallego (2004), and Ha and Tong (2008). In particular,
Carr and Karmarkar (2005) examined quantity competition under different supply chain structures
with deterministic demand. Boyaci and Gallego (2004) studied two competing supply chains where
the manufacturers and the retailers could act to affect service quality, and examined the value of
centralization given full information. They showed that centralization is the dominant strategy, but
it reduces supply chain profit, presenting a prisoner’s dilemma. Ha and Tong (2008) investigated two
competing supply chains where only the retailers could act to directly affect market competition.
They studied the value of information sharing and the corresponding contracting choices. However,
none of these papers considered supply risks in a competitive environment.
A good number of researcher have investigated supply chain structures in a competitive environ-
ment, e.g. McGuire and Staelin (1983), Moorthy (1988), Anderson and Bao (2010). McGuire and
Staelin (1983) was the first one to study various supply chain structures, including a mixed struc-
ture with one centralized channel and one decentralized channel. They examined Nash equilibrium
and dominant structures in different situations, and showed that in the highly competitive market,
the manufacturers prefer to use decentralized system. This was a very interesting result, since in
a single supply chain a centralized system usually performers better than a decentralized system
because of “double marginalization” (Spengler 1950, Cachon 2003, Ru and Wang 2010). Based on
the model of McGuire and Staelin (1983), Moorthy (1988) provided the reason for decentralization
to be a Nash equilibrium strategy as “the nature of the coupling between demand dependence
and strategic dependence”. Anderson and Bao (2010) derived that whether decentralized system
outperforms centralized system depends on the coefficient of variation of these market shares. Wu
et al. (2009) considered three strategies, Vertical Integration, Manufacturer’s Stackelberg, and Bar-
gaining on the Wholesale price, in two supply chain competition with demand uncertainty. They
demonstrated that (1) in one period model, Vertical Integration is the unique Nash Equilibrium for
both supply chains; and (2) over infinitely many periods, Manufacturer’s Stackelberg or Bargaining
on the Wholesale price may become the Nash Equilibrium.
There are a few papers in operations management that study chain-to-chain competition. Demi-
rag et al. (2011) studied the performance of customer rebate and retailer incentive promotions
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under chain-to-chain competition, and showed that customer rebates can be more profitable for
the manufacturers under some cases in the presence of supply chain competition, which is different
from the monopoly case where the manufacturers are always better off with retailer incentives. Ha
et al. (2011) analyzed Cournot and Bertrand competition of two supply chains each consisting of
one manufacturer selling to one exclusive retailer, and investigated the impact of production disec-
onomy, information (signal) accuracy, competition intensity, and competition type between supply
chains on the demand information sharing incentive. Fang et al. (2013) focused on the equilibrium
contracting strategies between consignment contract and wholesale-price contract in supply chain
to supply chain competition under price-sensitive stochastic demand. However, very few papers
have considered the effects of supply uncertainty in a competitive environment.
Motivated by the case of the U.S. influenza vaccine market, Deo and Corbett (2009), which is
closely related to our study, examined the interaction between supply uncertainty and a firm’s
strategic decision regarding market entry. They found that, in an equilibrium state, supply uncer-
tainty can lead to a high number of firms concentrating in an industry as well as a reduction in
the industry output and the expected consumer surplus. Deo and Corbett (2009) assumed same
supply uncertainty levels and perfect competition between firms, while in this paper we consider
a more general model with different supply risks, different competition intensities, and different
supply chains structures. Furthermore, they focused on the question of market entry, which is very
different from our study. Babich et al. (2007) studied a supply chain where one retailer does busi-
ness with competing high-risk suppliers that may default during their production lead times. They
show that low supplier default correlations dampen competition between the suppliers and increase
the equilibrium wholesale prices. Therefore, the retailer prefers suppliers with highly correlated
default events, despite consequently losing the benefits of diversification. In contrast, the suppliers
and the channel prefer negatively correlated defaults. Hopp et al. (2008) investigated the impact
of regional supply disruption on competing supply chains where the firms’ strategies consist of
two stages: (i) preparation, investing in measures that facilitate quick detection of a problem prior
to a disruption, and (ii) response, post-disruption purchasing of backup capacity for components
with compromised availability. As a measure of risk, they used the expected loss of profit due to
lack of preparedness, and they found that the products that pose the greatest risk are those with
valuable market share, low customer loyalty, and relatively limited backup capacity. Furthermore,
they showed that a dominant firm in such a market should focus primarily on protecting its market
share, while a weaker firm should focus on being ready to take advantage of a supply disruption
to gain market share.
Our paper differs from the above-mentioned papers (i.e., Deo and Corbett 2009, Babich et al.
2007, Hopp et al. 2008) in that we focus on the ordering decisions, contract design, and centraliza-
tion choices given supply uncertainty and supply chain competition. We contribute to the literature
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by offering a systematic examination of how to design and operate supply chains to effectively deal
with supply risks and competition.
3. The Model
To investigate the impact of supply uncertainty and supply chain competition, we consider the
model presented in Figure 1.
Supplier 1
Supplier 2
Retailer 1
Retailer 2
Market
Figure 1 Model
There are two competing supply chains that offer the same product in the market. Each chain
consists of a retailer and an exclusive supplier. All parties are risk neutral. The two supply chains,
suppliers, and retailers are indexed by i and j, where i, j ∈ {1,2}, i 6= j. The retailers compete on
quantity.
The suppliers are subject to supply uncertainty. The uncertainty may be caused by natural
disasters, labor strikes, transportation disruption, etc., which are considered to be out of control
by the suppliers and the retailers. If retailer i places an order of Qi, he would receive αiQi where
αi is a random variable between 0 and 1 (inclusive). The mean and standard deviation of αi is µi
and σi. We use coefficient of variation (CV), δi = σi/µi, to measure the level of supply uncertainty:
the larger δi, the more uncertainty. We consider the general case where the two suppliers may have
different uncertainty levels.
The product has a short life cycle, thus both supply chains have only one ordering opportunity.
The market price of the product is determined as pi(Qi,Qj) =A−αiQi−γαjQj, where the positive
constant A represents the market size and γ ∈ [0,1] measures the competitive intensity. In the
extreme cases, when γ = 1 the two products are perfectly substitutable; when γ = 0 the two products
are completely different. This pricing function is widely used in quantity competition papers (e.g.,
Ha and Tong 2008).
We consider two marginal costs for each supply chain i: (i) C1i per unit of production quantity,
and (ii) C2i per unit of successful delivery. If supplier i produces an order of Qi and the yield is
αiQi, the expected total cost is E(C1iQi +C2iαiQi) = C1iQi +C2iµiQi. To simplify the notation,
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we define Ci =C1i/µi +C2i. Hence, the expected total cost becomes CiµiQi. To exclude the non-
interesting case where neither party sells to the market, we assume A>Ci. We consider the general
case where the two chains may have different costs.
We assume that the retailers have no option for backup (contingent) sourcing, that the supply
uncertainty are independent of each other, and that all parties have full information about the
entire market.
4. Design and Operation Decisions
In this section we examine the optimal ordering and contract design decisions under three settings.
First, we derive the equilibrium for the centralized competition game between two centralized
chains. Second, we derive the equilibrium for the hybrid competition game between a centralized
chain and a decentralized chain. Finally, we derive the equilibrium for the decentralized competition
game between two decentralized chains. Comparisons between the three settings and insights into
the strategic decisions are provided in Section 5.
4.1 Centralized Competition Game
We first consider the case in which both supply chains are centralized, i.e., the retailer and the
supplier in each chain are fully aligned (integrated) to achieve the whole supply chain’s optimal
performance. We use a central planner i (i= 1,2) to represent the supplier and the retailer in each
chain. We seek to derive the equilibrium order quantity, Q∗i and Q∗j , of each central planner and
examine how they are affected by supply uncertainty and supply chain competition.
The sequence of events is as follows:
1. Ordering decision: central planner i and j decide the order quantity Qi and Qj simultaneously.
2. Uncertain supply : central planner i receives αiQi and central planner j receives αjQj.
3. Quantity competition: the market is cleared based on the total realized products from both
supply chains and each party receives the payoff.
The goal of the central planner i is to maximize the expected profit for supply chain i, which
can be expressed as follows:
E(πi) = E(piαiQi−C1iQi−C2iαiQi)
= E(piαiQi)−CiµiQi
= E[(A−αiQi− γαjQj)αiQi]−CiµiQi
= (A−Ci)µiQi− (µ2i +σ2
i )Q2i − γµiµjQiQj, (1)
where E[αi] = µi, V ar[αi] = σ2i ,E[α2
i ] = µ2i +σ2
i and Ci =C1i/µi +C2i.
The following theorem characterizes the unique pure strategy equilibrium and the corresponding
players’ payoffs for the Centralized Competition Game. For notation convenience, we denote Ki =
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A−Ci, Kj =A−Cj, and λ=Ki/Kj for the rest of the paper. λ implies the cost differences between
the two chains. λ< 1 means that the cost of supply chain i is lower than that of j.
Theorem 1. The unique pure strategy Nash equilibrium of the Centralized Competition Game
is (Q∗i ,Q∗j ), where the optimal order quantities of supply chains are
(i) when γ
2(1+δ2j )<λ<
2(1+δ2i )
γ,
Q∗i =2(1 + δ2j )Ki− γKj
µi(4(1 + δ2i )(1 + δ2j )− γ2), (2)
Q∗j =2(1 + δ2i )Kj − γKi
µj(4(1 + δ2i )(1 + δ2j )− γ2), (3)
(ii) when λ≤ γ
2(1+δ2j ),
Q∗i = 0, (4)
Q∗j =Kj
2µj(1 + δ2j ), (5)
(iii) when λ≥ 2(1+δ2i )
γ,
Q∗i =Ki
2µi(1 + δ2i ), (6)
Q∗j = 0. (7)
Note that as we assume Ki > 0 and Kj > 0, we exclude the trivial case of Q∗i =Q∗j = 0.
All proofs are provided in the Appendix. Let us first look at the special case where there is no
supply uncertainty (µi = µj = 1, δi = δj = 0), two products are perfect substitutable (γ = 1) and the
costs are the same (Ci =Cj =C). In this case, the Centralized Competition Game degenerates into
a standard Cournot duopoly game, and the optimal order quantities become Q∗i =Q∗j = (A−C)/3
and the expected profits become (A−C)2/9, which are consistent with the classical results.
Now we examine the impact of supply uncertainties on the order quantities. Based on Theorem
1, we show that the following is true.
Corollary 1. When λ is small or large (i.e., λ≤ γ
2(1+δ2j )or λ≥ 2(1+δ2i )
γ), the market has only
one monopoly chain, whose optimal order quantity and expected profit decrease when its supply
uncertainty increases. When λ is medium (i.e., γ
2(1+δ2j )<λ<
2(1+δ2i )
γ), both chains sell to the market
and the equilibrium order quantity and expected profit for supply chain i increase in δj and decrease
in δi.
Corollary 1 says that one supply chain’s order quantity and expected profit increase when the
supply uncertainty of the competing chain increases, and decrease when its own supply uncertainty
increases. Note that in the special case when Ci = Cj, γ = 1 and δi = δj, Corollary 1 echoes the
finding in Lemma 1 of Deo and Corbett (2009) when there are two firms: the order quantities of
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0 2 4 60
0.1
0.2
0.3
0.4
0.5δi = 0
δi = 0.5
δi = 1
δi = 2
Supply Uncertainty Level of Chain j (δj)
Exp
ecte
d S
uppl
y of
Cha
in i
Figure 2 Expected supply
0 2 4 60
0.05
0.1
0.15
0.2
0.25δi = 0
δi = 0.5
δi = 1
δi = 2
Supply Uncertainty Level of Chain j (δj)
Exp
ecte
d P
rofit
of C
hain
i
Figure 3 Expected profit
Chain i and j are identical and decrease in the yield uncertainty δ. Here we provide more general
results for cases where the costs and yield uncertainties of two chains are different.
Figures 2 and 3 present some numerical examples to illustrate how a supply chain’s expected
supply and profit change with respect to its own supply uncertainty level and that of the competing
supply chain. For the purpose of illustration, we let Ci =Cj =C, γ = 1. We also normalize (A−C)
to 1, without loss of generality. The horizontal axis in both figures is the uncertainty level of
supply chain j, δj. In Figure 2 the vertical axis is the expected order quantity of supply chain
i, E(αiQi) = µiQi, whereas in Figure 3 the vertical axis is the expected profit of supply chain i.
The four curves represent different uncertainty levels for supply chain i, with δi = 0,0.5,1,2, which
represents cases when the supply risk is absent, low, medium, and high, respectively. The main
observations are as follows:
• For a given uncertainty level δi, the expected order quantity and profit of supply chain i
increase as its competing supply chain’s uncertainty level δj increases. That is, a supply chain can
take great opportunity of it’s competing chain’s supply risk. For example, if supply chain i has
perfect supply (i.e., δi = 0), when the uncertainty of supply chain j (δj) increases from 0 to 2,
supply chain i’s expected supply increases by 38% and the expected profit increases by 96%.
• Given a fixed uncertainty level for the competing supply chain, the expected order quantity
and profit of supply chain i decreases as its own uncertainty level increases. For example, assuming
that the uncertainty level of supply chain j is fixed at δj = 1, while the uncertainty of supply chain
i (δi) increases from 0.5 to 1, supply chain i’s expected supply decreases by 40% and the expected
profit decreases by 42%.
• As a supply chain’s uncertainty level becomes very large, its expected supply becomes very
small, and the competing supply chain behaves more like a monopoly player. In fact, we can
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analytically prove that Q∗i ≤ (A − C)/2 = 0.5 and E(π∗i ) ≤ (A − C)2/4 = 0.25, i.e., the optimal
order quantities and the expected profits with supply uncertainty are upper-bounded by the order
quantities and the profits for the standard monopoly game without supply uncertainty.
Next, we examine the impact of competitive intensity γ on the optimal order quantities and
expected profits. We have the following corollary:
Corollary 2. When the two competing supply chains have same costs and both sell to the
market (i.e., λ= 1 and γ
2(1+δ2j )< 1<
2(1+δ2i )
γ), the equilibrium order quantity and expected profit for
supply chain i decrease in γ. That is, higher competition leads to lower equilibrium order quantities
and expected profits for both supply chains.
The proof of the corollary is by looking at the first-order derivatives of Eqs. (2) and (3). Details are
shown in the Appendix. The next corollary shows the impact of supply uncertainty and competitive
intensity on the expected total market supply, denoted as T =E(αiQi +αjQj).
Corollary 3. When both chains sell to the market, the expected total market supply at the
Nash equilibrium for the Centralized Competition Game is
T ∗ = µiQ∗i +µjQ
∗j =
(2(1 + δ2j )− γ)Ki + (2(1 + δ2i )− γ)Kj
4δ2i + 4δ2j + 4δ2i δ2j + 4− γ2
(8)
The expected total market supply decreases in δi, δj, and γ.
4.2 Hybrid Competition Game
In this section, we study the case where one supply chain is centralized while the other is not.
We show how to design the optimal wholesale-price contract for the decentralized chain. We seek
to understand how the equilibrium order quantities and the profits change, comparing with the
Centralized Competition Game. In addition, we are also interested in discovering whether the
centralized chain has a competitive edge over the decentralized chain in this Hybrid Competition
Game.
Without loss of generality, we assume that supply chain i is centralized and that supply chain j
is decentralized. We use superscript “h” to denote the Hybrid Competition Game. The expected
profit of supply chain i is
E(πhi ) =E(piαiQhi −C1iQ
hi −C2iαiQ
hi ) = (A−Ci)µiQh
i −Qh2i (µ2
i +σ2i )− γQh
iQhjµiµj. (9)
For supply chain j, we assume that the supplier charges the retailer a wholesale price ωj per
unit of successful delivery, and pays the retailer a penalty sj per unit of unfilled order. This is a
linear contract that is easy to implement and is widely adopted in practice. The expected profit of
retailer j is:
E(πhj,r) = E[(pj −wj)αjQhj + sj(1−αj)Qh
j ]
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= (A−wj)µjQhj −Qh2
j (µ2j +σ2
j )− γQhiQ
hjµiµj + sj(1−µj)Qh
j . (10)
Define m = (wj − Cj)µj − sj(1 − µj) as the supplier j’s expected profit margin for each unit
ordered by the retailer, i.e., with probability µj the supplier receives ωj for each unit of successful
delivery with cost Cj, and with probability 1− µj the supplier pays sj for each unit of shortage.
The expected profit of supplier j becomes:
E(πhj,s) =mQhj . (11)
The setup of the Hybrid Competition Game is as follows.
Definition 1 (Hybrid Competition Game). A Hybrid Competition Game is defined as
• Players: a set of three players: central planner i, retailer j, and supplier j,
• There are two stages in the game
— Stage 1: supplier j announces the contract term m to retailer j.
— Stage 2: central planner i and retailer j choose production quantities Qhi and Qh
j simulta-
neously.
• Payoff: the expected profits for the three players are E(πhi,r(Qhi ,Q
hj )), E
(πhj,r(Q
hi ,Q
hj ,m)
), and
E(πhj,s(Q
hj ,m)
), respectively.
The goal of the central planner i is to maximize the expected total profit of chain i, whereas
retailer j and supplier j want to maximize their individual expected profits. The Hybrid Compe-
tition is a dynamic game. For such game, Subgame Perfect Nash equilibrium (SPNE) is the most
commonly used equilibrium concept (Osborne 2004). A strategy profile is a SPNE if the players’
strategies constitute a Nash equilibrium in every subgame.
Definition 2. A pure strategy SPNE (Qhi (m),Qh
j (m),m∗) of the Hybrid Competition Game
satisfies
1. Retailers compete with each other in quantities to maximize their profits for any given contract
m (not necessary optimal), i.e.,
Qh∗i (m)∈ arg max
Qhi ≥0E(πhi (Qh
i ,Qhj (m))),
Qh∗j (m)∈ arg max
Qhj≥0E(πhj,r(Q
hi (m),Qh
j ,m)).
2. Supplier j sets the contract term m to maximize its expected profit given the choices of
retailers, i.e.,
m∗ ∈ arg maxm≥0
E(πhj,s(Q
hj (m),m)
).
The following lemma characterizes the equilibrium strategies for both retailers as a function of
contract term m.
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Lemma 1. At a pure strategy SPNE of the Hybrid Competition Game,
1. if µjKj −2(1+δ2j )µjKi
γ<m<µjKj − γµjKi
2(1+δ2i ),
Qhi (m) =
2(1 + δ2j )µjKi− (Kjµj −m)γ
µiµj(4(1 + δ2i )(1 + δ2j )− γ2), (12)
Qhj (m) =
2(1 + δ2i )(µjKj −m)− γµjKi
µ2j(4(1 + δ2i )(1 + δ2j )− γ2)
. (13)
2. if m≤ µjKj −2(1+δ2j )µjKi
γ,
Qhi (m) = 0, (14)
Qhj (m) =
µjKj −m2µ2
j(1 + δ2j ). (15)
3. if µjKj − γµjKi
2(1+δ2i )≤m≤ µjKj,
Qhi (m) =
Ki
2µi(1 + δ2i ), (16)
Qhj (m) = 0. (17)
The following theorem characterizes the equilibrium contract terms m∗ and order quantities.
Theorem 2. At an SPNE of the Hybrid Competition Game, supplier j’s contract term m∗,
supply chain i’s order quantity Qh∗i , and retailer j’s order quantity Qh∗
j satisfy
i) if λ≥ 2(1+δ2i )
γ, m∗ can be any value in [0,Kjµj],
Qh∗i =
Ki
2µi(1 + δ2i ),
Qh∗j = 0.
ii) if2γ(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2
<λ<2(1+δ2i )
γ,
m∗ = µj(Kj
2− γKi
4(1 + δ2i )),
Qh∗i =
Ki
(8(1 + δ2i )(1 + δ2j )− γ2
)− 2γKj(1 + δ2i )
4µi(4(1 + δ2i )(1 + δ2j )− γ2)(1 + δ2i ),
Qh∗j =
2Kj(1 + δ2i )− γKi
2µj(4(1 + δ2i )(1 + δ2j )− γ2).
iii) if γ
4(1+δ2j )<λ≤ 2γ(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2
,
m∗ = Kjµj −2(1 + δ2j )µjKi
γ,
Qh∗i = 0,
Qh∗j =
Ki
γµj.
iv) if λ≤ γ
4(1+δ2j ),
m∗ =Kjµj
2,
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Qh∗i = 0,
Qh∗j =
Kj
4(1 + δ2j
)µj.
Note that as we assume Ki > 0 and Kj > 0 we exclude the case where Qh∗i =Qh∗
j = 0.
Similar to the Centralized Competition Game, we obtain the following comparative statics.
Corollary 4. When both chains sell to the market and the cost structures are symmetric (i.e.,
Ci = Cj), the equilibrium order quantity and expected profit for supply chain i increase in δj and
decrease in δi. That is, one supply chain’s order quantity and expected profit increase as its com-
peting supply chain’s uncertainty increases and decrease as its own supply uncertainty increases.
Figures 4 and 5 present some numerical examples of Corollary 4. For illustration purposes, we let
Ci =Cj =C,γ = 1 and normalize (A−C) to 1. The horizontal axis in both figures represents the
uncertainty level of supply chain j, δj. In Figure 4 the vertical axis is the expected order quantity of
supply chain i, whereas in Figure 5 the vertical axis is the expected order quantity of supply chain
j. The four curves represent different uncertainty levels of supply chain i, with δi = 0,0.5,1,2, each
representing the case with no, small, medium, and high levels of supply uncertainty, respectively.
0 2 4 60
0.1
0.2
0.3
0.4
0.5δi = 0
δi = 0.5
δi = 1
δi = 2
Supply Uncertainty Level of Chain j (δj)
Exp
ecte
d S
uppl
y of
Cha
in i
Figure 4 Expected supply of chain i
0 2 4 60
0.05
0.1
0.15
0.2
0.25
Supply Uncertainty Level of Chain j (δj)
Exp
ecte
d S
uppl
y of
Cha
in j
δi = 0δi = 0.5δi = 1δi = 2
Figure 5 Expected supply of chain j
Corollary 5. When both chains sell to the market and have the same cost (i.e., Ci =Cj), the
equilibrium order quantity and expected profit for supply chain i decrease in γ.
Figures 6 and 7 provide some numerical examples to illustrate Corollary 5. As the chain com-
petition intensity (γ) increases, the expected supplies of both chains (µiQh∗i and µjQ
h∗j ) and their
expected profits (E(πh∗i ) and E(πh∗j )) both decrease. The next corollary shows the impact of supply
uncertainty and competitive intensity on the expected total market supply and price.
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0 0.2 0.4 0.6 0.8 1
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Competitive Intensity γ
Exp
ecte
d S
uppl
y
µiQh∗i
µjQh∗j
Figure 6 Expected supply changes with γ
0 0.2 0.4 0.6 0.8 10.05
0.1
0.15
0.2
0.25
Competitive Intensity γ
Exp
ecte
d P
rofit
E(πh∗i )
E(πh∗j )
Figure 7 Expected profit changes with γ
Corollary 6. At the SPNE of the Hybrid Competition Game, when the two supply chains have
the same costs (i.e., Ci = Cj), the expected market supply decreases in δi and δj and γ; whereas,
the expected market price increases in δi and δj, but decreases in γ.
Next we compare the equilibrium order quantities under the Hybrid Competition Game with
those in the Centralized Competition Game. We obtain the following result:
Corollary 7. In the Hybrid Competition Game, the equilibrium order quantity for chain i is
greater than that in the Centralized Competition Game, i.e., Qh∗i >Q∗i ; while the order quantity of
chain j′s is smaller, i.e., Qh∗j <Q∗j .
Corollary 7 says that for the same uncertainty level and competition level, the order quantity for
the centralized supply chain i increases while that for the decentralized supply chain j decreases,
when comparing the Hybrid Competition Game with the Centralized Competition Game. Figure
8 provides a numerical example. Here we let Ci =Cj =C,δi = δj = δ, γ = 1 and normalize (A−C)
to 1. The horizontal axis is the uncertainty level of supply chains δ, varying from 0 to 3, and the
vertical axis is the expected order quantity of each supply chain. The middle solid curve represents
the order quantity of chain i and j in the Centralized Competition Game, whereas the top and
bottom dash curves represent the order quantity of chain i (centralized) and chain j (decentralized)
in the Hybrid Competition Game, respectively.
We have similar results for the expected profits of supply chains. Further discussions are provided
in Section 5.
4.3 Decentralized Competition Game
Now we study the case where both supply chains are decentralized and adopt wholesale price
contracts with linear penalty. Each supplier i ∈ {1,2} charges its retailer a wholesale price ωi per
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0 0.5 1 1.5 2 2.5 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Level of Supply Uncertainty δ
Exp
ecte
d S
uppl
y
µiQ
h∗i
µiQ∗i , µjQ
∗j
µjQh∗j
Figure 8 Comparison of expected supplies
unit of successful delivery and pays the retailer a penalty si per unit of unfilled order. We use
superscript “u” to indicate that both chains are decentralized. Define mi = µi(ωi−Ci)− (1−µi)si.Definition 3 (Decentralized Competition Game). A Decentralized Competition Game
is defined as
• Players: a set of four players: retailer i, retailer j, supplier i, and supplier j.
• There are two stages in the game
— Stage 1: two suppliers announce the contract term mi and mj simultaneously.
— Stage 2: two retailers choose production quantity Qui and Qu
j simultaneously.
• Payoff: the expected revenues for the four players are E(πui,r(Qui ,Q
uj ,mi)), E
(πuj,r(Q
ui ,Q
uj ,mj)
),
E(πui,s(Q
ui ,mi)
), and E
(πuj,s(Q
uj ,mj)
), respectively. The payoff functions are defined similarly as
in Eqs. (10) and (11).
Definition 4. A pure strategy SPNE (Qui (mi,mj),Q
uj (mi,mj),m
∗i ,m
∗j ) of the game between
two decentralized supply chains satisfies:
1. Retailers compete with each other in quantities to maximize their profits for any given con-
tracts (mi,mj) (not necessary optimal), i.e.,
Qui (mi,mj)∈ arg max
Qui ≥0E(πi,r
(Qui ,Q
uj (mi,mj),mi
)),
2. Suppliers compete with each other in contract terms to maximize their profits given the
choices of retailers, i.e.,
m∗i ∈ arg maxmi≥0
miQui (mi,m
∗j ),
The following results characterize the equilibrium strategies of the retailers and the suppliers.
For expositional simplicity, we let Bi = µiKi−mi and Bj = µjKj −mj.
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Lemma 2. At a pure strategy SPNE of the Hybrid Competition Game,
1. ifγµjBi
2µi(1+δ2i )<Bj <
2µjBi(1+δ2j )
γµi,
Qui =
2µj(1 + δ2j )Bi− γµiBjµ2iµj(4δ
2i δ
2j + 4δ2i + 4δ2j + 4− γ2)
(18)
Quj =
2µi(1 + δ2i )Bj − γµjBiµiµ2
j(4δ2i δ
2j + 4δ2i + 4δ2j + 4− γ2)
(19)
2. if Bj ≥2µjBi(1+δ
2j )
γµi,
Qui = 0 (20)
Quj =
Bj2µ2
j(1 + δ2j )(21)
3. if Bj ≤ γµjBi
2µi(1+δ2i )
,
Qui =
Bi2µ2
i (1 + δ2i )(22)
Quj = 0 (23)
Theorem 3. At the SPNE of the Decentralized Competition Game, supplier i (i= 1,2) chooses
the contract terms m∗i (equivalently B∗i ) and order quantity Qu∗i as follows:
i) if2γ(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2<λ<
8(1+δ2i )(1+δ2j )−γ2
2γ(1+δ2j )
B∗i =2µi (1 + δ2i )
(4Ki(1 + δ2j ) + γKj
)
16 (1 + δ2i ) (1 + δ2j )− γ2,
B∗j =2µj(1 + δ2j ) (4Kj (1 + δ2i ) + γKi)
16 (1 + δ2i ) (1 + δ2j )− γ2,
Qu∗i =
2(1 + δ2j
) ((8 (1 + δ2i ) (1 + δ2j )− γ2)Ki− 2γKj(1 + δ2i )
)
µi(16 (1 + δ2i ) (1 + δ2j )− γ2
) (4 (1 + δ2i ) (1 + δ2j )− γ2
) ,
Qu∗j =
2(1 + δ2i )((8 (1 + δ2i ) (1 + δ2j )− γ2)Kj − 2γKi(1 + δ2j )
)
µj(16 (1 + δ2i ) (1 + δ2j )− γ2
) (4 (1 + δ2i ) (1 + δ2j )− γ2
) .
ii) if λ< γ
4(1+δ2j ),B∗i can be any value in (0,Kiµi],
B∗j =Kjµj
2,
Qu∗i = 0,
Qu∗j =
Kj
4µj(1 + δ2j ).
iii) if γ
4(1+δ2j )≤ λ≤ 2γ(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2,
B∗i = Kiµi,
B∗j =2Kiµj(1 + δ2j )
γ,
Qu∗i = 0,
Qu∗j =
Ki
γµj.
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iv) if8(1+δ2i )(1+δ2j )−γ2
2γ(1+δ2j )≤ λ≤ 4(1+δ2i )
γ,
B∗i =2Kjµi (1 + δ2i )
γ,
B∗j = Kjµj,
Qu∗i =
Kj
γµi,
Qu∗j = 0.
v) if λ>4(1+δ2i )
γ,B∗j can be any value in (0,Kjµj],
B∗i =Kiµi
2,
Qu∗i =
Ki
4µi(1 + δ2i ),
Qu∗j = 0.
Note that as we assume Ki > 0 and Kj > 0 we exclude the case where Qu∗i =Qu∗
j = 0.
Similar to the Centralized Competition and Hybrid Competition games, we obtain the following
comparative statics.
Corollary 8. At the SPNE of the Decentralized Competition Game when both retailers sell to
the market and cost structures are symmetric (i.e., Ci = Cj), the equilibrium order quantity and
expected profit for supply chain i increase with δj and decrease with δi.
Corollary 9. At the SPNE of the Decentralized Competition Game with symmetric cost struc-
tures and supply uncertainties (i.e., Ci = Cj and δi = δj), if both retailers sell to the market, the
equilibrium order quantity and expected profit for supply chain i decrease with the competitive
intensity γ.
The above two corollaries present the impact of supply uncertainty and supply chain competition
on the equilibrium decisions and performances. On the one hand, one supply chain’s order quantity
and expected profit increase as its competing supply chain’s uncertainty increases, while decrease
as its own supply uncertainty increases. On the other hand, one supply chain’s order quantity and
expected profit decrease as the competition from the other supply chain intensifies.
Corollary 10. At the SPNE of the Decentralized Competition Game when both retailers sell to
the market and cost structures are symmetric (i.e., Ci =Cj), the expected market supply decreases
in δi and δj. As a result, the expected market price increases in δi and δj.
Corollary 11. At the SPNE of the Decentralized Competition Game with the symmetric cost
structures and supply uncertainties (i.e., Ci = Cj and δi = δj), if both retailers sell to the market
and cost structures, the expected market supply and market price both decrease in γ.
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5. Strategic Decisions
In the previous section, we derive the optimal ordering decision and contract design for the Central-
ized, Hybrid and Decentralized Competition games. In this section we consider the strategic-level
question: Does supply chain centralization provide a competitive advantage when dealing with
competition and supply uncertainty?
To answer this question, we first calculate the expected payoffs in the the Centralized, Hybrid
and Decentralized Competition games, based on the equilibrium order quantities we have derived
in the previous section (assuming that both chains sell to the market).
Under the Centralized Competition Game:
E(π∗i ) =(1 + δ2i )
(2(1 + δ2j )Ki− γKj
)2(4 (1 + δ2i )
(1 + δ2j
)− γ2
)2 (24)
E(π∗j ) =
(1 + δ2j
)(2(1 + δ2i )Kj − γKi)
2
(4 (1 + δ2i )
(1 + δ2j
)− γ2
)2 (25)
Under the Decentralized Competition Game:
E(πui ) =2(1 + δ2j
) (6 (1 + δ2i )
(1 + δ2j
)− γ2
) (2γ(1 + δ2i )Kj − (8 (1 + δ2i )
(1 + δ2j
)− γ2)Ki
)2(4 (1 + δ2i )
(1 + δ2j
)− γ2
)2 (16 (1 + δ2i )
(1 + δ2j
)− γ2
)2 , (26)
E(πuj ) =2(1 + δ2i )
(6 (1 + δ2i )
(1 + δ2j
)− γ2
) (2γ(1 + δ2j )Ki− (8 (1 + δ2i )
(1 + δ2j
)− γ2)Kj
)2(4 (1 + δ2i )
(1 + δ2j
)− γ2
)2 (16 (1 + δ2i )
(1 + δ2j
)− γ2
)2 . (27)
Under the Hybrid Competition Game where chain i is centralized and chain j is decentralized:
E(πhi ) =
(2γKj(1 + δ2i )−Ki(8 (1 + δ2i )
(1 + δ2j
)− γ2)
)2
16 (1 + δ2i )(4 (1 + δ2i )
(1 + δ2j
)− γ2
)2 , (28)
E(πhj ) =(2Kj(1 + δ2i )− γKi)
2 (6 (1 + δ2i )
(1 + δ2j
)− γ2
)
8 (1 + δ2i )(4 (1 + δ2i )
(1 + δ2j
)− γ2
)2 . (29)
Under the Hybrid Competition Game where chain j is centralized and chain i is decentralized:
E(πh′i ) =
(2Ki(1 + δ2j )− γKj
)2 (6 (1 + δ2i )
(1 + δ2j
)− γ2
)
8(1 + δ2j
) (4 (1 + δ2i )
(1 + δ2j
)− γ2
)2 , (30)
E(πh′j ) =
(2γKi(1 + δ2j )−Kj(8 (1 + δ2i )
(1 + δ2j
)− γ2)
)2
16(1 + δ2j
) (4 (1 + δ2i )
(1 + δ2j
)− γ2
)2 . (31)
Here “h” denotes the case when chain i is centralized and chain j is not, whereas “h′” denotes
the case when chain j is centralized and chain i is not.
Table 1 summarizes these results. We compare these expected profits under different competition
games and obtain the following theorem.
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Table 1 Payoff Table for Strategic Choice of Integration
Supply Chain jcentralized decentralized
Supply centralized E(π∗i ),E(π∗j ) E(πhi ),E(πhj )
Chain i decentralized E(πh′i ),E(πh
′j ) E(πui ),E(πuj )
Theorem 4. When both chains sell to the market, we find E(πhi ) > E(πui ), E(π∗i ) > E(πh′i ),
E(πh′j ) > E(πuj ), and E(π∗j ) > E(πhj ). Hence, given the competing supply chain’s fixed strategic
decision (to centralize or not), a supply chain is always better off by choosing to centralize. That is,
centralization is the dominant strategy for both supply chains. However, centralization may or may
not result in positive gains for each supply chain, as the payoff difference between the centralized
and decentralized games, E(π∗i )−E(πui ), can be either negative or positive.
The proof is provided in the Appendix. If we consider the choice of centralization as a strategic-
level game, Theorem 4 implies that centralization would be a unique Nash Equilibrium. However,
centralization may lead to lower profit, which means that a prisoners’ dilemma may occur.
To further understand the implications of Theorem 4, let us consider the special case of two
supply chains with same costs and supply risks, i.e., δi = δj = δ and Ci =Cj =C. Let K =A−C. It
can be verified that the expected profits for supply chain i and j under the Centralized Competition
Game become
E(π∗i ) =E(π∗j ) =(1 + δ2)K2
(γ+ 2δ2 + 2)2 ,R.
To simplify exposition, we let “R” represent E(π∗i ) and E(π∗j ). The expected supply chain profits
under the Decentralized Competition Game become
E(πui ) =E(πuj ) =2K2(1 + δ2)(6 (1 + δ2)
2− γ2)
(γ+ 2δ2 + 2)2(4δ2 + 4− γ)
2 =2(6 (1 + δ2)
2− γ2)
(4δ2 + 4− γ)2 R.
The expected supply chain profits under the Hybrid Competition Game are as follows,
E(πhi ) =E(πh′j ) =
K2 (γ+ 4δ2 + 4)2
16 (1 + δ2) (γ+ 2δ2 + 2)2 =
(γ+ 4δ2 + 4)2
16 (1 + δ2)2 R,
E(πhj ) =E(πh′i ) =
K2(6 (1 + δ2)2− γ2)
8 (1 + δ2) (γ+ 2δ2 + 2)2 =
(6(1 + δ2)2− γ2)
8 (1 + δ2)2 R.
These results are summarized in Table 2. It is easy to verify that6(1+δ2)
2−γ2
8(1+δ2)2 < 1 and
(γ+4δ2+4)2
16(1+δ2)2 >
2(6(1+δ2)2−γ2)(4δ2+4−γ)
2 . This means that centralization is better off for each chain, no matter whether the
other chain is centralized or not — in other words, centralization is the dominant strategy for
both supply chains. However, as shown in Figure 9, we can also see that when γ > 2(1+δ2)
3, we have
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2(6(δ2+1)2−γ2)(4δ2+4−γ)
2 > 1, which means that a chain receives less profit in the centralized game than in
the decentralized game. This leads to a “prisoner’s dilemma”, i.e., both supply chains are worse
off under the centralized scenario although centralization is the dominant strategy. On the other
hand, when γ < 2(1+δ2)
3, the profit for each chain in the Centralized Competition Game is higher
than that in the Decentralized Competition Game.
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Lev
el o
f Sup
ply
Unc
erta
inty
Level of Competitive Intensity
γ =2(1 + δ2)
3
E(π∗i ) > E(πu
i )
E(π∗i ) < E(πu
i )
Figure 9 Comparison of expected supply chain profits
Note that Boyaci and Gallego (2004) first discovered the phenomenon of the prisoner’s dilemma
in supply chain competition, assuming that there is no supply uncertainty. Here we show that their
finding is still true when supply risk is low and supply chain competition intensity is relatively
high, but it no longer holds if supply chain competition intensity is low and/or the supply risk
becomes more substantial.
Table 2 Payoff Table for Strategic Choice of Integration with Symmetric Chains
Supply Chain jcentralized decentralized
Supply centralized R,R(γ+4δ2+4)
2
16(δ2+1)2 R,
(6(δ2+1)2−γ2)8(δ2+1)
2 R
Chain i decentralized(6(δ2+1)2−γ2)
8(δ2+1)2 R,
(γ+4δ2+4)2
16(δ2+1)2 R
2(6(δ2+1)2−γ2)(4δ2+4−γ)
2 R,2(6(δ2+1)2−γ2)
(4δ2+4−γ)2 R
Figure 10 presents some numerical examples, where the x-axis is level of supply uncertainty δ, the
y-axis is expected supply chain profit. We let γ = 1 and normalize A−C to 1. There are four curves:
the black solid curve represents the expected profit of chain i (i= 1,2) in Centralized Competition
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0 0.5 1 1.5 2 2.5 30
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
√1/2
Level of Supply Uncertainty δ
Exp
ecte
d P
rofit
λ = 1, γ = 1
0 0.5 1 1.5 2 2.5 30
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Level of Supply Uncertainty δ
Exp
ecte
d P
rofit
λ = 1, γ = 1
Hybrid Competition, Chain i
Decentralized Competition
Centralized Competition
Hybrid Competition, Chain j
Figure 10 Comparison of the expected supply chain profits
Game; the blue dash curve represents the expected profit of chain i (i = 1,2) in Decentralized
Competition Game (Note that because the two chains are completely symmetric in both games,
chain 1’s profits are the same as those of chain 2); the purple dash curve represents the expected
profit of chain i, while the red dash curve represents the expected profit of chain j in the Hybrid
Competition Game, where we assume that chain i is centralized and chain j is decentralized.
We can see that a decentralized supply chain is better off by moving toward centralization if the
other supply chain is centralized, since the black solid curve is always above the red dash curve.
Furthermore, a decentralized supply chain is better off by moving toward centralization if the other
supply chain is decentralized, since the purple dash curve is always above the blue dash curve.
Hence, centralization is the dominant strategy for both supply chains.
When supply risk is high (δ ≥√
1/2), centralization leads to higher profits, since the blue dash
curve is above the black solid curve; however, when supply risk is low, it is the opposite. This means
that the desirability of centralization is increased by the presence of supply risk. This finding is
consistent with some real-life observations, e.g., as we discussed in Section 1, the two largest dairy
companies in China, Yili and Mengniu, are both increasingly vertically integrated, due to the high
risk of quality milk supply.
Now a related question is: do individual retailer and supplier in a decentralized chain have the
right incentives to integrate?
We may consider a revenue sharing contract where the supplier’s income consists of three parts:
ω for each unit ordered by the retailer, a percentage of retail revenue for each successfully delivered
order, and a penalty for each unit that the supplier fails to deliver. Let φ ∈ [0,1] be the fraction
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of supply chain revenue the retailer keeps, so the supplier earns the remaining (1−φ).1 The profit
functions for the retailer, the supplier, and the central planner are
E(πr) =E[(φp−ω)Qα+SQ(1−α)],
E(πs) =E{[(1−φ)p+ω]Qα−SQ(1−α)−CQα]},E(π) =E(πr) +E(πs) =E(pQα).
Here α is a random variable that represents yield uncertainty. Assume that α is generally distributed
in (0,1] with average value of µ. When ωµ− (1−µ)S−Cµ= 0, we have
E(πr) =E[φpQα] = φE(π),
i.e., the profit of the retailer is an affine function of the central planner. Hence, such a revenue
sharing contract coordinates the supply chain. Based on this finding, we can show that the following
is true.
Lemma 3. For any given linear price and penalty contract, there always exists a coordinating
revenue-sharing contract that makes both the supplier and retailer better off.
To prove Lemma 3, we examine the centralization decision for a supply chain (say chain j) in
two cases: (i) when the other supply chain is centralized; and (ii) when the other supply chain is
decentralized.
Case (i) when the other supply chain (chain i) is centralized: We compare the expected profits for
retailer j and supplier j in centralized and decentralized scenarios, to see whether they are better
off under centralization or not. As Theorem 4 suggests, E(πhj )< E(π∗j ), meaning that the entire
supply chain j would be better off if it centralizes. Suppose that under a given linear contract,
retailer j earns a profit of E(πhj,r) and supplier j earns a profit of E(πhj,s). Consider when supplier
j and retailer j engage in a revenue-sharing contract with profit split ratio φj =E(πhj,r)/E(πhj ). It
is clear that retailer j is better off since
φjE(π∗j )>φjE(πhj ) =E(πhj,r)
E(πhj )E(πhj ) =E(πhj,r).
and so is supplier j since
(1−φj)E(π∗j )> (1−φj)E(πhj ) =E(πhj,s).
1 Here we choose revenue sharing contract as an example of supply chain coordination tool, as it is a commonly-used coordinating contract in supply chain literature and practice (Cachon and Lariviere 2005). We do acknowledgethat there are many other types of supply chain contracts, such as buy-back contracts (Pasternack 1985), quantity-flexibility contracts (Tsay 1999), sales-rebate contracts (Taylor 2002), etc. We leave the comprehensive examinationof all types of supply chain contracts that can coordinate under supply uncertainty to future research.
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Case (ii) when the other supply chain (chain i) is decentralized: Without loss of generality,
we assume that supply chain j uses a linear contract and supply chain i can choose either to
coordinate or stick with a given linear contract. As Theorem 4 suggests, E(πui )<E(πhi ). We define
φi = E(πui,r)/E(πui ) and let φi be the profit split ratio of a revenue-sharing contract, we see that
both the retailer and supplier are better off under this contract.
Hence, we can conclude that there exists a coordinating contract that achieves Pareto improve-
ment in a supply chain regardless whether the other supply chain is centralized or not.
Note that in the above discussion, we just provide an example of how profit may be split between
the retailer and the supplier. Several existing papers have provided analysis and discussion about
how to derive the optimal revenue share φ in supply chain coordination (e.g., Wang et al. 2004,
Dana Jr and Spier 2001, Cachon and Lariviere 2005, etc.). The same logic can be applied here.
Hence, we omit the detailed discussions here.
Next, we examine customer service in terms of expected total market supply and price under
the Centralized, the Hybrid, and the Decentralized Competition games. A better customer service
means higher total expected market supply and lower expected market price.
0 1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Level of Supply Uncertainty δ
Exp
ecte
d T
otal
Mar
ket S
uppl
y
T ∗
T h∗T u∗
Figure 11 Expected market supply changes with δ
0 0.2 0.4 0.6 0.8 10.4
0.5
0.6
0.7
0.8
0.9
1
Competitive Intensity γ
Exp
ecte
d T
otal
Mar
ket S
uppl
y
T ∗
T h∗T u∗
Figure 12 Expected market supply changes with γ
Theorem 5. When both chains sell to the market, the Centralized Competition Game provides
the best service to customers in terms of total expected market supply, the Hybrid Competition
Game provides the second, and the Decentralized Competition Game provides the worst service,
i.e., T ∗ >T h >T u.
Theorem 5 says that customers always benefit from supply chain centralization with more sup-
plies and a lower price. Figures 11 − 14 provide numerical examples, where we assume that the
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two supply chains have the same costs and the same levels of supply uncertainty. We let γ = 0.5
in Figures 11 and 13 and δ= 0 in Figures 12 and 14. Figures 11 and 12 compare the total market
supplies in different competition games, and show that centralized competition game always leads
to the largest expected total market supplies. Figures 13 and 14 compare the retail prices in differ-
ent competition games and show that centralization always leads to lower retail prices. Therefore,
in terms of expected market supply and price, customers are always better off when both supply
chains are centralized.
0 1 2 31.6
1.7
1.8
1.9
2
2.1
Level of Supply Uncertainty δ
Ret
ail P
rice
p∗i , p∗j
ph∗i
ph∗j
pu∗i , pu∗j
Figure 13 Retail price changes with δ
0 0.2 0.4 0.6 0.8 11.5
2
2.5
Competitive Intensity γ
Ret
ail P
rice
p∗i , p∗j
ph∗i
ph∗j
pu∗i , pu∗j
Figure 14 Retail price changes with γ
6. Conclusion
In this paper, we provide a systematic examination of how to design and operate supply chains to
effectively deal with supply uncertainty and competition for short life-cycle products. We derive
the equilibrium decisions for ordering decisions and contract terms for the Centralized, Hybrid,
and Decentralized Competition games. By comparing these three scenarios, we examine the value
of supply chain centralization.
First, for the ordering decisions, we show that a supply chain should order more if its competing
chain’s supply becomes less reliable or if its own supply becomes more reliable. A chain with reliable
supply can take great advantage of the high supply risk of its competing chain. We also show that
the order sizes are upper-bounded by those in the standard monopoly game without uncertainty.
Second, we characterize the optimal terms for a wholesale price contract with linear penalty
for any given level of supply uncertainty and competition intensity. We also show that, for such a
linear contract there always exists a coordinating revenue-sharing contract which can make both
supplier and retailer better off.
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Finally, we show that supply chain centralization is always the dominant strategy under sup-
ply uncertainty and chain competition. Furthermore, centralization always benefits customers by
providing more market supply and lower price. Nevertheless, depending on the levels of supply
uncertainty and competition intensity, centralization may or may not result in positive gains for
each supply chain. Specifically, when the supply risk is low and chain competition is very intense,
centralization reduces the profit for each supply chain, resulting in a prisoner’s dilemma. On the
other hand, when supply risk becomes very substantial or the competition becomes less intense,
centralization leads to positive gain for each supply chain. However, In other words, the benefit of
supply chain centralization is strengthened by high supply uncertainty or low supply chain com-
petition. The desirability of supply chain centralization is enhanced by high supply uncertainty or
low chain competition.
There are several interesting directions for future research. First, we may consider the possibility
of contingent sourcing, i.e., after a retailer observes his supplier’s status, he may purchase from
another reliable source or from the spot market. Second, we may explore the impact of correlated
supply risks among competing supply chains, e.g., a severe crisis could poses supply risks to multiple
chains simultaneously. Third, suppliers and retailers may have private information about demand
and reliability, and level of supply uncertainty could be an endogenous decision. Finally, we may
study different types of games (e.g., Bertrand competition) and different type of contracts (e.g.
buy-back contracts).
Acknowledgement
The authors thank the Editor, Prof. Robert Dyson, and the referees for their constructive com-
ments. We also thank Dr. Erick Zhaolin Li and Dr. Jianwei Huang for their valuable suggestions.
This research is supported by the Hong Kong General Research Fund (Project No. CityU 149812)
and grant from City University of Hong Kong (Project No. 7004136).
Appendix.
Proof of Theorem 1. Under the Centralized Competition Game, the expected profits for chain i (i= 1,2;
j 6= i) are:
E(πi) = E(piαiQi−C1iQi−C2iαiQi)
= E((A−αiQi− γαjQj)αiQi−C1iQi−C2iαiQi)
= (A−Ci)µiQi−Q2i (µ
2i +σ2
i )− γQiQjµiµj ,
which is a concave function in Qi. The first-order conditions for each chain are:
(A−Ci)µi− 2Qi(µ2i +σ2
i )− γQjµiµj = 0
(A−Cj)µj − 2Qj(µ2j +σ2
j )− γQiµiµj = 0
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Solving these equations, we obtain the equilibrium production quantities as shown in Theorem 1.
Proof of Corollary 1. When both chains sell to the market (i.e., when γ
2(1+δ2j)< λ <
2(1+δ2i )
γ),
∂Q∗i∂δi
=
− 8δi(δ2j+1)(2Ki(1+δ2j )−γKj)
µi(4δ2i +4δ2j+4δ2
iδ2j+4−γ2)
2 < 0 since γ
2(1+δ2j)< Ki
Kj;∂Q∗i∂δj
=4γδj(2Kj(1+δ
2i )−γKi)
µ2i (4δ2i +4δ2
j+4δ2
iδ2j+4−γ2)
> 0 since Ki
Kj<
2(1+δ2i )
γ. Fur-
thermore,∂E(π∗i )∂δi
=−2µ2iQ
2i δi−γQiµiµj
∂Q∗j∂δi
< 0 and∂E(π∗i )∂δj
=−γQiµiµj∂Q∗j∂δj
> 0 since∂Q∗j∂δj
< 0 and∂Q∗j∂δi
> 0.
Proof of Corollary 2. In the case that both chains sell to the market (i.e., when γ
2(1+δ2j)<
λ <2(1+δ2i )
γ),∂Q∗i∂γ
= − (4(1+δ2i )(1+δ2j )+γ2)Kj−4γKi(δ2j+1)µi(4(1+δ2i )(1+δ2j )−γ2)
2 and∂E(π∗i )∂γ
= −Q∗iµiµj(Q∗j + γdQ∗jdγ
), where Q∗j +
γdQ∗jdγ
= 2 (δ2i + 1)(4(1+δ2i )(1+δ2j )+γ2)Kj−4γKi(1+δ
2j )
µj(4(1+δ2i )(1+δ2j )−γ2)2 . Then, when Ci = Cj , i.e., Ki = Kj = K,
∂Q∗i∂γ
=
− (4(1+δ2i )(1+δ2j )+γ2)−4γ(δ2j+1)µi(4(1+δ2i )(1+δ2j )−γ2)
2 K < 0, and∂E(π∗i )∂γ
< 0.
Proof of Corollary 3. When both chains sell to the market (i.e., when γ
2(1+δ2j)<λ<
2(1+δ2i )
γ),
∂E(αiQ∗i +αjQ
∗j )
∂γ=−
(4(δ2j + 1
)(δ2i + 1− γ) + γ2
)Ki +
(4 (δ2i + 1)
(δ2j + 1− γ
)+ γ2
)Kj(
4(1 + δ2i )(1 + δ2j )− γ2)2 < 0,
∂E(αiQ∗i +αjQ
∗j )
∂δi=−4δi
(2(1 + δ2j )Ki− γKj
)(2(1 + δ2j )− γ)
(4(1 + δ2i )(1 + δ2j )− γ2
)2 < 0,
∂E(αiQ∗i +αjQ
∗j )
∂δj=−4δj (2(1 + δ2i )Kj − γKi) (2(1 + δ2i )− γ)
(4(1 + δ2i )(1 + δ2j )− γ2
)2 < 0,
Proof of Lemma 1. Under the Hybrid Competition game, the expected profit for chain i (centralized)
becomes:
E(πhi ) = (A−Ci)µiQhi −Qh2
i (µ2i +σ2
i )− γQhiQ
hjµiµj .
The expected profits for retailer and supplier in chain j (decentralized) are respectively:
E(πhj,r) = E[(pj −wj)αjQhj + sj(1−αj)Qh
j ]
= E[(A−αjQhj − γαiQh
i −wj)αjQhj + sj(1−αj)Qh
j ]
= (µjKj −m)Qhj −Qh2
j (µ2j +σ2
j )− γQhiQ
hjµiµj ,
E(πhj,s) = E[wjαjQhj − sj(1−αj)Qh
j −C1jQhj −αjC2jQ
hj ]
= [wjµj − sj(1−µj)−µjCj ]Qhj =mQh
j ,
Now consider the ordering decisions for any given m. The first-order conditions are:
µiKi− 2Qhi (µ2
i +σ2i )− γQh
jµiµj = 0
µjKj −m− 2Qhj (µ2
j +σ2j )− γQh
i µiµj = 0
µiKi− 2Qhi (µ2
i +σ2i )− γQh
jµiµj = 0
Gj − 2Qhj (µ2
j +σ2j )− γQh
i µiµj = 0
Solving these equations, we find the equilibrium order quantities for any given m, as shown in Lemma 1.
Proof of Theorem 2. Supplier j’s optimal choice of m is to maximize E(πj,s) =mQhj for 0≤m≤Kjµj .
Let Ki =A−Ci,Kj =A−Cj , λ=Ki/Kj ,Gj = µjKj −m. Because Gj and m are one-to-one matching, we
seek the optimal Gj in [0,Kjµj ] that maximizes the supplier’s profit E(πj,s) = (Kjµj −Gj)Qhj .
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Based on the results of Lemma 1, we discuss the following three cases to derive supplier j’s optimal decision,
Case A: if µjKj ≤ γµjKi
2(1+δ2i), i.e., λ≥ 2(1+δ2i )
γ, we only need to consider scenario 3) in the first stage (Lemma
1). Gj can be any value in (0,Kjµj ],Qhi = Ki
2µi(1+δ2i),Qh
j = 0.
Case B: ifγµjKi
2(1+δ2i)≤ µjKj <
2(1+δ2j )µjKi
γ,i.e., γ
2(1+δ2j)≤ λ< 2(1+δ2i )
γ, we need to compare supplier’s expected
profit under scenarios 1) and 3) in the first stage. We first consider scenario 1) whereγµjKi
2(1+δ2i)<Gj <µjKj .
It is easy to show that E(πj,s) is a concave function w.r.t Gj . Solving the first order condition, we obtain
Gj =(2(1 + δ2i )Kj + γKi)µj
4(1 + δ2i ).
When λ<2(1+δ2i )
γ, it satisfies
γµjKi
2(1+δ2i)<Gj <µjKj . Accordingly,
Qhj =
2Kj(1 + δ2i )− γKi
2µj(4δ2i + 4δ2j + 4δ2i δ2j + 4− γ2)
,
E(πhj,s) =(2Kj(1 + δ2i )− γKi)
2
8(1 + δ2i )(4δ2i + 4δ2j + 4δ2i δ2j + 4− γ2)
.
Because the expected profit in scenario 3) is 0, obviously scenario 1) is a better choice. This means that if
γ
2(1+δ2j)≤ λ< 2(1+δ2i )
γ,Gj = 1
4(1 + δ2i )
−1(2(1 + δ2i )Kj + γKi)µj .
Case C: Kjµj ≥ 2(1+δ2j )µjKi
γ, i.e., λ≤ γ
2(1+δ2j), we need to compare supplier’s expected profit under scenarios
1), 2) and 3) in the first stage. First consider case 2) where2(1+δ2j )µjKi
γ≤Gj ≤Kjµj ,Q
hi = 0,Qh
j =Gj
2µ2j(1+δ2
j)
.It’s straightforward to show that E(πj,s) is a concave function w.r.t Gj . Solving the first order condition,
we get
Gj =Kjµj
2.
There are two cases to discuss:
• If λ≤ γ
4(1+δ2j), then
Kjµj
2≥ 2(1+δ2j )µjKi
γ. Hence Gj =
Kjµj
2,Qh
j =Kj
4µj(1+δ2j),E(πhj,s) =
K2j
8(1+δ2j).
• If γ
4(1+δ2j)< λ ≤ γ
2(1+δ2j)
thenKjµj
2<
2(1+δ2j )µjKi
γ. Hence Gj =
2(1+δ2j )µjKi
γ,Qh
j = Ki
γµj, E(πhj,s) =
(γKj−2(1+δ2j )Ki)Ki
γ2.
Next consider scenario 1). Based on the first-order condition, we have
Gj =(2(1 + δ2i )Kj + γKi)µj
4(1 + δ2i )
We need to check whether the solution falls into the feasible regionγµjKi
2(1+δ2i)≤Gj ≤ 2(1+δ2j )µjKi
γ. There are
two cases to discuss:
• if2γ(1+δ2i )
8δ2i+8δ2
j+8δ2
iδ2j+8−γ2 ≤ λ ≤ γ
2(1+δ2j), then the solution falls into the feasible region, hence Gj =
(2(1+δ2i )Kj+γKi)µj
4(1+δ2i)
,Qhj =
2(1+δ2i )Kj−γKi
2µj(4δ2i+4δ2
j+4δ2
iδ2j+4−γ2) , E(πhj,s) =
(2(1+δ2i )Kj−γKi)2
8(1+δ2i)(4δ2
i+4δ2
j+4δ2
iδ2j+4−γ2) .
• if λ≤ 2γ(1+δ2i )
8δ2i+8δ2
j+8δ2
iδ2j+8−γ2 , then the solution is greater than the upper bound of the feasible region, hence
Gj =2(1+δ2j )µjKi
γ,Qh
j = Ki
γµj,E(πhj,s) =
(γKj − 2(1 + δ2j )Ki
)Ki/γ
2.
Because the expected profit in scenario 3) is 0, we only need to compare scenarios 1) and 2). We find
• When λ ≤ γ
4(1+δ2j),the expected profits under scenario 2 is larger than that under scenario 1. Thus
Gj =Kjµj
2,Qh
j =Kj
4µj(1+δ2j).
• When γ
4(1+δ2j)≤ λ≤ 2γ(1+δ2i )
8δ2i+8δ2
j+8δ2
iδ2j+8−γ2 ,the optimal solution under scenarios 1 and 2 are the same, thus
Gj =2(1+δ2j )µjKi
γ,Qh
j = Ki
γµj.
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• When2γ(1+δ2i )
8δ2i+8δ2
j+8δ2
iδ2j+8−γ2 <λ≤ γ
2(1+δ2j), the expected profit in scenario 1 is larger than that in scenario
2. Thus,
Gj =(2(1 + δ2i )Kj + γKi)µj
4(1 + δ2i ),
Qhi =
(8(1 + δ2i )(1 + δ2j )− γ2)Ki− 2γ(1 + δ2i )Kj
4(1 + δ2i )µi(4δ2i + 4δ2j + 4δ2i δ2j + 4− γ2)
,
Qhj =
2(1 + δ2i )Kj − γKi
2µj(4δ2i + 4δ2j + 4δ2i δ2j + 4− γ2)
.
Summarizing cases A, B and C, we derive the SPNE of Gj ,Qhi and Qh
j , as shown in Theorem 2.
Proof of Corollary 4. In the case that both chains sell to the market and Ki = Kj =
K,∂Qh∗
i∂δi
= − δiK2
8(δ2j+1)(δ2i +1)(4(δ2j+1)(δ2i +1)−γ2−γ(δ2i +1))+γ4
µi(δ2i +1)2(4(1+δ2i )(1+δ2j )−γ2)2< 0,
∂Qh∗i
∂δj=
2γδj(2Kj(1+δ2i )−γKi)
µi(4(1+δ2i )(1+δ2j )−γ2)2> 0,
∂Qh∗j
∂δj=
− 4δj(δ2i +1)(2Kj(1+δ2i )−γKi)
µj(4(1+δ2i )(1+δ2j )−γ2)2< 0,
∂Qh∗j
∂δi=
2γδi(2Ki(1+δ2j )−γKj)
µj(4(1+δ2i )(1+δ2j )−γ2)2> 0 and ∂m
∗∂δi
=γδiµjKi
2(δ2i +1)2> 0.
Furthermore, for the centralized supply chain i,∂E(πh∗
i )
∂δi= −2µ2
iQ2i δi − γQiµiµj
∂Qh∗j
∂δi< 0 and
∂E(πh∗i )
∂δj=
−γQiµiµj ∂Qh∗j
∂δj> 0 since
∂Qh∗j
∂δj< 0 and
∂Qh∗j
∂δi> 0; for the decentralized supply chain j, E(πh∗j ) =
(2Kj(1+δ2i )−γKi)
2(6(1+δ2i )(1+δ2j )−γ2)8(1+δ2i )(4(1+δ2i )(1+δ2j )−γ2)2
,
∂E(πh∗j )
∂δi=
γδiK2(2(1 + δ2i )− γ
)( 48(δ2j + 1
)2 (δ2i + 1
)2 − 16γ(δ2i + 1
)2 (δ2j + 1
)
−12γ2(δ2j + 1
) (δ2i + 1
)+ γ3
(γ+ 2δ2i + 2
))
4 (δ2i + 1)2(4 (1 + δ2i )
(1 + δ2j
)− γ2
)3 > 0
and∂E(πh∗j )
∂δj= −δjK
2(2(1 + δ2i )− γ
)2 (12(1 + δ2i
) (1 + δ2j
)− γ2
)
2(4 (1 + δ2i )
(1 + δ2j
)− γ2
)3 < 0
when Ki =Kj =K.
Proof of Corollary 5. In the case that both chains sell to the market (i.e., when
2γ(1+δ2i )
8(1+δ2i)(1+δ2
j)−γ2 < λ <
2(1+δ2i )
γ), ∂m∗
∂γ= − µjKi
4(1+δ2i)< 0,
∂Qh∗i
∂γ=
4γKi(δ2j+1)−Kj(4(1+δ2i )(1+δ2j )+γ2)2µi(4(1+δ2i )(1+δ2j )−γ2)
2 and
∂Qh∗j
∂γ=
4γKj(δ2i +1)−Ki(4(1+δ2i )(1+δ2j )+γ2)2µj(4(1+δ2i )(1+δ2j )−γ2)
2 . When Ci = Cj , we have∂Qh∗
i
∂γ< 0 and
∂Qh∗j
∂γ< 0;
∂E(πhi )
∂γ=
−Qh∗i µiµj(Q
h∗j + γ
∂Qh∗j
∂γ) < 0 since Qh∗
j + γ∂Qh∗
j
∂γ= (δ2i + 1)K
4(δ2j+1)(δ2i +1−γ)+γ2
µj(4(1+δ2i )(1+δ2j )−γ2)2 > 0; and
∂E(πh∗j )
∂γ= −
K2(2(1+δ2i )−γ)(12(δ2j+1)
2(δ2i +1)−8γ(δ2j+1)(δ2i +1)+γ2(γ−δ2j−1)
)
2(4(1+δ2i )(1+δ2j )−γ2)3 < 0 when Ki =Kj =K.
Proof of Corollary 6. In the case that both chains sell to the market and Ki =Kj =K,let Th∗ =E(αiQh∗i +
αjQh∗j ), we have
∂Th∗
∂δi= −δiK
32(δ2j + 1
)2 (δ2i + 1
)2+ γ4 + 4γ2(1 + δ2i )2 − 8γ2
(δ2i + 1
)(1 + δ2j )− 16γ
(δ2i + 1
)2 (δ2j + 1
)
2 (δ2i + 1)2(4(1 + δ2i )(1 + δ2j )− γ2
)2 < 0,
∂Th∗
∂δj= − 2δjK
(2(1 + δ2i )− γ
)2(4(1 + δ2i )(1 + δ2j )− γ2
)2 < 0,
and∂Th∗
∂γ= −K
(2(1 + δ2j )− γ
) (2(1 + δ2i )− γ
)(4(1 + δ2i )(1 + δ2j )− γ2
)2 < 0.
Since phi =A−µiQhi − γµjQ
hj and phj =A−µjQ
hj − γµiQ
hi , we obtain that
∂phi∂δi
> 0,∂phi∂δj
=2γδj(2δ2i +1)K(2(1+δ2i )−γ)
(4(1+δ2i )(1+δ2j )−γ2)2>
0,∂phj∂δi
> 0,∂phj∂δj
=2δjK(2(1+δ2i )−γ2)(2(1+δ2i )−γ)
(4(1+δ2i )(1+δ2j )−γ2)2> 0,
∂phi∂γ
= −(2δ2i + 1
)Kγ2 + 4δ2i + 4δ2j + 4δ2i δ
2j − 4γδ2j − 4γ+ 4
2(γ2 − 4δ2i − 4δ2j − 4δ2i δ
2j − 4
)2 < 0,
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and∂phj∂γ
= −K 8(4δ2j + 3
) (δ2j + 1
) (δ2i + 1
)2+ γ4 − 2γ2
(2δ2j + 3
) (δ2i + 1
)− 8γ
(δ2i + 1
)2 (2δ2j + 1
)
4 (δ2i + 1)(γ2 − 4δ2i − 4δ2j − 4δ2i δ
2j − 4
)2 < 0.
Proof of Corollary 7. In the case that both chains sell to the market where λ<2(1+δ2i )
γ,
Qh∗i −Q∗i =−1
4γ
γKi− 2Kj(1 + δ2i )
µi (δ2i + 1)(4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
) > 0,
and
Qh∗j −Q∗j =
1
2
γKi− 2(1 + δ2i )Kj
µj(4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
) < 0.
Proof of Lemma 2. Under the Decentralized Competition game, the expected profit of retailer i(i= 1,2)
is:
E(πi,r) = (µiKi−mi)Qui −Qu2
i (µ2i +σ2
i )− γQuiQ
uj µiµj .
For any given mi, retailer i wants to find the optimal ordering quantity to maximize its expected profit.
It’s easy to show that the expected profit function is concave in Qi. The first-order condition leads to:
(µiKi−mi)− 2Qui (µ2
i +σ2i )− γQu
j µiµj = 0.
For exposition simplicity, we define Bi = µiKi−mi, where i= 1,2. Thus,
Bi− 2Qui (µ2
i +σ2i )− γQu
j µiµj = 0.
Solving the two first-order conditions, we obtain the equilibrium ordering quantities Quj and Qu
j for any
given Bj and Bj , as shown in Lemma 2.
Proof of Theorem 3. Consider the suppliers’ decision for decentralized game. Supplier i’s optimal choice
of mi to maximize E(πi,s) =miQi, or equivalently, to maximize E(πi,s) = (Aµi−Ciµi−Bi)Qi for 0≤Bi ≤(A−Ci)µi. Based on the results of Lemma 2, we characterize the best-response functions of Bj and Bj and
then compute SPNE.
Best-response function of Bj :
we first derive the best-response function of Bj for any given Bi. There are three possible cases.
Case A: if Kjµj ≤ γµjBi
2µi(1+δ2i),i.e., Bi ≥ 2µi(1+δ
2i )Kj
γ, we only need to consider scenario 3) in the first stage.
Bj can be any value in (0,Kjµj ],Qi = Bi
2µ2i(1+δ2
i),Qj = 0.
Case B: ifγµjBi
2µi(1+δ2i)<Kjµj ≤ 2µjBi(1+δ
2j )
γµi,i.e.,
γKjµi
2(1+δ2j)≤Bi < 2µi(1+δ
2i )Kj
γ, we need to compare the expected
profits under scenarios 1) and 3) in the first stage to decide the the optimal choice of Bj . We first consider
scenario 1) in the first stage whereγµjBi
2µi(1+δ2i)<Bj <Kjµj . It is easy to show that E(πj,s) is a concave function
w.r.t Bj . The first-order condition leads to:
(Kjµj −Bj)2µi(1 + δ2i )
µiµ2j (4 (1 + δ2i ) (1 + δ2j )− γ2)
− 2µi(1 + δ2i )Bj − γµjBiµiµ2
j (4 (1 + δ2i ) (1 + δ2j )− γ2)= 0.
The solution is
Bj =γµjBi
4µi(1 + δ2i )+Kjµj
2. (32)
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We can show that this solution satisfiesγµjBi
2µi(1+δ2i)<Bj <Kjµj . Accordingly,
Qj =2µiKj(1 + δ2i )− γBi
2µjµi(4 (1 + δ2i ) (1 + δ2j )− γ2
) ,
E(πj,s) =(2µiKj(1 + δ2i )− γBi)2
8µ2i
(4 (1 + δ2i ) (1 + δ2j )− γ2
)(1 + δ2i )
.
Because the expected profit for supplier j in scenario 3) is 0, obviously scenario 1) is better, which means
that ifγµjBi
2µi(1+δ2i)≤Kjµj ≤ 2µjBi(1+δ
2j )
γµi(i.e.,
γµiKj
2(1+δ2j)≤Bi ≤ 2µi(1+δ
2i )Kj
γ), Bj =
γµjBi
4µi(1+δ2i)
+Kjµi
2.
Kjµj ≤ 2µjBi(1+δ2j )
γµi=>
γµiKj
2(1+δ2j)≤Bi
Case C: Kjµj >2µjBi(1+δ
2j )
γµi, i.e., Bi <
γKjµi
2(1+δ2j), we need to compare the expected profits of supplier j under
scenarios 1), 2) and 3) in the first stage to decide the optimal choice of Bj . First consider scenario 2)
where2µjBi(1+δ
2j )
γµi≤Bj ≤Kjµj ,Qi = 0,Qj =
Bj
2µ2j(1+δ2
j). It’s straightforward to show that E(πj,s) is a concave
function w.r.t Bj . LetdE(πj,s)
dBj= 0. We get Bj =
Kjµj
2. We need to consider two cases:
• IfKjµj
2≥ 2µjBi(1+δ
2j )
γµi, then Bj =
Kjµj
2,Qj =
Kjµj
4µ2j(1+δ2
j),E(πj,s) =
K2j
8(1+δ2j).
• IfKjµj
2<
2µjBi(1+δ2j )
γµi,then Bj =
2µjBi(1+δ2j )
γµi,Qj = Bi
γµiµj,E(πj,s) =
Bi(γµiKj−2Bi(1+δ2j ))
γ2µ2i
.
Next we consider scenario 1) in the first stage. We need to check whether the FOC solution expressed
by Eq.(32), Bj =γµjBi
4µi(1+δ2i)
+Kjµj
2, satisfies
γµjBi
2µi(1+δ2i)<Bj <
2µjBi(1+δ2j )
γµi. Because
γµjBi
2µi(1+δ2i)<Kjµj (i.e., Bi <
2µi(1+δ2i )Kj
γ), we find
γµjBi4µi(1 + δ2i )
+Kjµj
2− γµjBi
2µi(1 + δ2i )=µj (2µiKj(1 + δ2i )− γBi)
4µi (δ2i + 1)> 0.
However, to have2µjBi(1 + δ2j )
γµi−(
γµjBi4µi(1 + δ2i )
+Kjµj
2
)> 0,
it should satisfy
Bi >2γµiKj(1 + δ2i )
8 (1 + δ2i ) (1 + δ2j )− γ2.
Hence we need to consider two cases:
• If2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2<Bi <
γKjµi
2(1+δ2j), then Bj =
γµjBi
4µi(1+δ2i)
+Kjµj
2,E(πj,s) =
(2µiKj(1+δ2i )−γBi)
2
8µ2i (4(1+δ2i )(1+δ2j )−γ2)(1+δ2i )
.
• If Bi <2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2, then Bj =
2µjBi(1+δ2j )
γµi, Qj = Bi
γµiµj, E(πj,s) =
Bi(γµiKj−2Bi(1+δ2j ))
γ2µ2i
.
Because the expected profit of supplier j under scenario 3) is always 0 and that under scenarios 1 and 2
are positive, we only need to compare scenarios 1 and 2. We find:
• whenKjµj
2≥ 2µjBi(1+δ
2j )
γµi, i.e., Bi ≤ γµiKj
4(1+δ2j), the expected profit of supplier j under scenario 2) is larger
than that under scenario 1) (i.e.,K2
j
8(1+δ2j)− Bi(γµiKj−2Bi(1+δ
2j ))
γ2µ2i
= 18
(4Bi(1+δ2j )−γµiKj)
2
γ2µ2i (δ2j+1)
> 0), therefore the
optimal decision is: Bj =Kjµj
2,Qj =
Kjµj
4µ2j(1+δ2
j),E(πj,s) =
K2j
8(1+δ2j).
• whenγµiKj
4(1+δ2j)≤Bi < 2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2, the expected profit and choice of Bj under scenarios 1) and 2) are
the same, thus Bj =2µjBi(1+δ
2j )
γµi,Qj = Bi
γµiµj,E(πj,s) =
Bi(γµiKj−2Bi(1+δ2j ))
γ2µ2i
.
• When2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2≤Bi < γKjµi
2(1+δ2j), the expected profit in scenario 1 is larger than that in scenario
2. Thus,
Bj =γµjBi
4µi(1 + δ2i )+Kjµj
2.
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Summarizing cases A, B and C, the best response of Bj for any given Bi is:
i) if Bi >2µi(1+δ
2i )Kj
γ,Bj can be any value in (0,Kjµj ],Qi = Bi
2µ2i(1+δ2
i),Qj = 0,and E(πj,s) = 0.
ii) if2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2<Bi ≤ 2µi(1+δ
2i )Kj
γ,
Bj =γµjBi
4µi(1 + δ2i )+Kjµj
2,
Qi =Bi(8 (1 + δ2i ) (1 + δ2j )− γ2
)− 2γµiKj (1 + δ2i )
4µ2i
(4 (1 + δ2i ) (1 + δ2j )− γ2
)(1 + δ2i )
,
and
Qj =2µiKj(1 + δ2i )− γBi
2µjµi(4 (1 + δ2i ) (1 + δ2j )− γ2
) .
iii) ifγµiKj
4(1+δ2j)<Bi ≤ 2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2,Bj =
2µjBi(1+δ2j )
γµi,Qj = Bi
γµiµj,and E(πj,s) =
Bi(γµiKj−2Bi(1+δ2j ))
γ2µ2i
.
iv) if Bi ≤ γµiKj
4(1+δ2j),Bj =
Kjµj
2,Qi = 0,Qj =
Kjµj
4µ2j(1+δ2
j),and E(πj,s) =
K2j
8(1+δ2j).
Best-response function of Bi :
Similarly, we can obtain the best-response function of Bi for any given Bj as follows.
i) if Bj >2µj(1+δ
2j )Ki
γ,Bi can be any value in (0,Kiµi],Qj =
Bj
2µ2j(1+δ2
j),Qi = 0,and E(πi,s) = 0.
ii) if2γµjKi(1+δ
2j )
8(1+δ2j )(1+δ2i )−γ2<Bj ≤ 2µj(1+δ
2j )Ki
γ,
Bi =γµiBj
4µj(1 + δ2j )+Kiµi
2,
Qj =Bj(8(1 + δ2j
)(1 + δ2i )− γ2
)− 2γµjKi
(1 + δ2j
)
4µ2j
(4(1 + δ2j
)(1 + δ2i )− γ2
) (1 + δ2j
) ,
and
Qi =2µjKi(1 + δ2j )− γBj
2µiµj(4(1 + δ2j
)(1 + δ2i )− γ2
) .
iii) ifγµjKi
4(1+δ2i)<Bj ≤ 2γµjKi(1+δ
2j )
8(1+δ2j )(1+δ2i )−γ2,Bi =
2µiBj(1+δ2i )
γµj,Qi =
Bj
γµjµi,and E(πi,s) =
Bj(γµjKi−2Bj(1+δ2i ))
γ2µ2j
.
iv) if Bj ≤ γµjKi
4(1+δ2i),Bi = Kiµi
2,Qj = 0,Qi = Kiµi
4µ2i(1+δ2
i),and E(πi,s) =
K2i
8(1+δ2i).
Subgame Perfect Nash Equilibriums:
Based on the best response functions, we analyze the following possible scenarios and compute the Subgame
Perfect Nash Equilibriums.
1. if Bj >2µj(1+δ
2j )Ki
γand Bi >
2µi(1+δ2i )Kj
γ, Bi can be any value in (0,Kiµi] and Bj can be any value
in (0,Kjµj ]. Because Bj >2µj(1+δ
2j )Ki
γand Bj ≤ Kjµj , Kjµj >
2µj(1+δ2j )Ki
γ, which leads to λ < γ
2(1+δ2j)≤
12. However, because Bi >
2µi(1+δ2i )Kj
γand Bi ≤ Kiµi, Kiµi >
2µi(1+δ2i )Kj
γ, which leads to λ >
2(1+δ2i )
γ≥ 2.
Contradiction.
2. if2γµjKi(1+δ
2j )
8(1+δ2j )(1+δ2i )−γ2<Bj ≤ 2µj(1+δ
2j )Ki
γ,
2γµiKj(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2<Bi ≤ 2µi(1+δ
2i )Kj
γ, we have
Bi =γµiBj
4µj(1 + δ2j )+Kiµi
2
Bj =γµjBi
4µi(1 + δ2i )+Kjµj
2.
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Solving these two equations, we obtain
Bi =8µiKi (1 + δ2i ) (1 + δ2j ) + 2γµiKj(1 + δ2i )
16 (1 + δ2i ) (1 + δ2j )− γ2,
Bj =8µjKj (1 + δ2i ) (1 + δ2j ) + 2γµjKi(1 + δ2j )
16 (1 + δ2i ) (1 + δ2j )− γ2.
To satisfy2γµjKi(1+δ
2j )
8(1+δ2j )(1+δ2i )−γ2<Bj ≤ 2µj(1+δ
2j )Ki
γ,
2γµiKj(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2<Bi ≤ 2µi(1+δ
2i )Kj
γ, it’s easy to show that we
need2γ(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2<λ<
8(1+δ2i )(1+δ2j )−γ2
2γ(1+δ2j)
.
3. ifγµjKi
4(1+δ2i)< Bj ≤ 2γµjKi(1+δ
2j )
8(1+δ2j )(1+δ2i )−γ2,γµiKj
4(1+δ2j)< Bi ≤ 2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2,we have Bi =
2µiBj(1+δ2i )
γµjand
Bj =2µjBi(1+δ
2j )
γµi,which meansBi
Bj=
2µi(1+δ2i )
γµjand Bi
Bj= γµi
2µj(1+δ2j). However, since
2µi(1+δ2i )
γµj− γµi
2µj(1+δ2j)
=
µi(4(1+δ2i )(1+δ2j )−γ2)2γµj(1+δ2j )
> 0 This leads to contradiction.
4. if Bj ≤ γµjKi
4(1+δ2i),Bi = Kiµi
2,Bi ≤ γµiKj
4(1+δ2j),Bj =
Kjµj
2, we have Kiµi
2≤ γµiKj
4(1+δ2j)
andKjµj
2≤ γµjKi
4(1+δ2i), that is,
λ≤ γ
2(δ2j+1)
and λ≥ 2(δ2i +1)
γ.Contradiction.
5. if Bj >2µj(1+δ
2j )Ki
γ,Bi can be any value in (0,Kiµi] and
2γµiKj(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2< Bi ≤ 2µi(1+δ
2i )Kj
γ.Since
Bj =γµjBi
4µi(1+δ2i)
+Kjµj
2>
2µj(1+δ2j )Ki
γ≥ 2µj(1+δ
2j )Bi
γµi, we get Bi <
2γµiKj(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2which contradicts with
2γµiKj(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2< Bi ≤ 2µi(1+δ
2i )Kj
γ.Similarly, we can derive contradiction when Bi >
2µi(1+δ2i )Kj
γand
2γµjKi(1+δ2j )
8(1+δ2j )(1+δ2i )−γ2<Bj ≤ 2µj(1+δ
2j )Ki
γ.
6. if Bj >2µj(1+δ
2j )Ki
γand
γµiKj
4(1+δ2j)< Bi ≤ 2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2, Bi can be any value in (0,Kiµi] and Bj =
2µjBi(1+δ2j )
γµi. Since Bj =
2µjBi(1+δ2j )
γµi>
2µj(1+δ2j )Ki
γ,which means Bi >Kiµi. Contradiction. Similarly, we can
derive contradiction when Bi >2µi(1+δ
2i )Kj
γand
γµjKi
4(1+δ2i)<Bj ≤ 2γµjKi(1+δ
2j )
8(1+δ2j )(1+δ2i )−γ2.
7. if Bj >2µj(1+δ
2j )Ki
γand Bj ≤ γµjKi
4(1+δ2i), Bi can be any value in (0,Kiµi] and Bj =
Kjµj
2. To satisfy Bj >
2µj(1+δ2j )Ki
γand Bi ≤ γµjKi
4(1+δ2i), we need
Kjµj
2>
2µj(1+δ2j )Ki
γ,i.e., λ < γ
4(1+δ2j). Similarly, we can show that if
λ>4(1+δ2i )
γ,there is an equilibrium where Bj can be any value in (0,Kjµj ] and Bi = Kiµi
2.
8. if2γµjKi(1+δ
2j )
8(1+δ2j )(1+δ2i )−γ2<Bj ≤ 2µj(1+δ
2j )Ki
γ,γµiKj
4(1+δ2j)<Bi ≤ 2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2,
Bi =γµiBj
4µj(1 + δ2j )+Kiµi
2,
Bj =2µjBi(1 + δ2j )
γµi.
Solving these equations, we obtain Bi =Kiµi,Bj =2Kiµj(1+δ
2j )
γ. In order to satisfy
2γµjKi(1+δ2j )
8(1+δ2j )(1+δ2i )−γ2<Bj ≤
2µjKi(1+δ2j )
γ,γµiKj
4(1+δ2j)< Bi ≤ 2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2, we need
2γµjKi(1+δ2j )
8(1+δ2j )(1+δ2i )−γ2<
2µjKi(1+δ2j )
γand
γµiKj
4(1+δ2j)< Kiµi ≤
2γµiKj(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2. It’s easy to show that the former is satisfied and the latter is eqivalent to γ
4(1+δ2j)< λ ≤
2γ(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2. Similarly, we can show that if
8(1+δ2i )(1+δ2j )−γ2
2γ(1+δ2j)
≤ λ< 4(1+δ2i )
γ, there is an equilibrium where
Bj =Kjµj ,Bi =2Kjµi(1+δ
2i )
γ.
9. if2γµjKi(1+δ
2j )
8(1+δ2j )(1+δ2i )−γ2<Bj ≤ 2µj(1+δ
2j )Ki
γ,Bi ≤ γµiKj
4(1+δ2j), we have
Bi =γµiBj
4µj(1 + δ2j )+Kiµi
2
Bj =Kjµj
2.
Solving these equations, we obtain Bi =µi(4Ki(1+δ
2j )+γKj)
8(1+δ2j)
,Bj =Kjµj
2. In order to satisfy
2γµjKi(1+δ2j )
8(1+δ2j )(1+δ2i )−γ2<
Bj ≤ 2µj(1+δ2j )Ki
γ, we need
2γµjKi(1+δ2j )
8(1+δ2j )(1+δ2i )−γ2<
Kjµj
2≤ 2µj(1+δ
2j )Ki
γ, which equals to γ
4(1+δ2j)≤ λ <
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8(1+δ2j )(1+δ2i )−γ2
4γ(1+δ2j)
. In order to satisfy Bi ≤ γµiKj
4(1+δ2j), we need
µi(4Ki(1+δ2j )+γKj)
8(1+δ2j)
≤ γµiKj
4(1+δ2j), which is equal to λ≤
γ
4(1+δ2j). Thus, we need λ= γ
4(1+δ2j). Note that when λ= γ
4(1+δ2j),Bi =
µi(4Ki(1+δ2j )+γKj)
8(1+δ2j)
=Kiµi,Bj =Kjµj
2=
2Kiµj(1+δ2j )
γ. Similarly, we can show that if λ=
4(1+δ2i )
γ,Bj =Kjµj ,Bi = Kiµi
2=
2Kjµi(1+δ2i )
γ.
10. ifγµjKi
4(1+δ2i)< Bj ≤ 2γµjKi(1+δ
2j )
8(1+δ2j )(1+δ2i )−γ2and Bi ≤ γµiKj
4(1+δ2j),Bi =
2µiBj(1+δ2i )
γµj,Bj =
Kjµj
2. Because Bi =
2µiBj(1+δ2i )
γµj=
2µi(1+δ2i )
γµj
Kjµj
2=
Kjµi(1+δ2i )
γ≥Kjµi, this contradicts with Bi ≤ γµiKj
4(1+δ2j)≤ Kjµi
4. Similarly, we can
derive contradiction ifγµiKj
4(1+δ2j)<Bi ≤ 2γµiKj(1+δ
2i )
8(1+δ2i )(1+δ2j )−γ2and Bj ≤ γµjKi
4(1+δ2i).
Summarizing all the cases discussed above, we derive the Subgame Perfect Nash Equilibriums as shown
in Theorem 3.
Proof of Corollary 8. When λ= 1, i.e., Ki =Kj =K, then, we have
∂Qu∗i∂δi
=
8K(1 + δ2j )δi
(64γ
(δ2j + 1
)2 (δ2i + 1
)2+ 64γ2
(δ2j + 1
)2 (δ2i + 1
)
−γ2(6γ2
(1 + δ2j
)+ γ3
)− 256
(δ2j + 1
)3 (δ2i + 1
)2)
µi(4(1 + δ2j
)(1 + δ2i )− γ2
)2 (16(1 + δ2j
)(1 + δ2i )− γ2
)2 < 0,
∂Qu∗i∂δj
=
4γδjK
(128
(δ2i + 1
)3 (δ2j + 1
)2 − 96γ(δ2j + 1
)2 (δ2i + 1
)2−γ3
(2γ(1 + δ2i ) + γ2
)+ 16γ3
(δ2i + 1
) (δ2j + 1
))
µi(4(1 + δ2j
)(1 + δ2i )− γ2
)2 (16(1 + δ2j
)(1 + δ2i )− γ2
)2 > 0,
∂E(πu∗i )
∂δi
= −(
1536(δ2j + 1
)4 (δ2i + 1
)3 − 384γ(δ2j + 1
)3 (δ2i + 1
)3 − 608γ2(δ2j + 1
)3 (δ2i + 1
)2
+8γ3(δ2j + 1
)2 (δ2i + 1
)2+ 18γ5
(δ2j + 1
) (δ2i + 1
)+ 116γ4
(δ2j + 1
)2 (δ2i + 1
)− 9γ6
(1 + δ2j
)− 2γ7
)
8δi(δ2j + 1
)K2(8(1 + δ2j
) (1 + δ2i
)− 2γ
(1 + δ2i
)− γ2
)(16(1 + δ2j
)(1 + δ2i )− γ2
)3 (4(1 + δ2j
)(1 + δ2i )− γ2
)3 < 0,
and
∂E(πu∗i )
∂δj
=
(1536
(δ2i + 1
)4 (δ2j + 1
)3+ γ7 − 640γ
(δ2j + 1
)3 (δ2i + 1
)3 − 384γ2(δ2i + 1
)3 (δ2j + 1
)2
−16γ5(δ2j + 1
) (δ2i + 1
)+ 160γ3
(δ2j + 1
)2 (δ2i + 1
)2+ 16γ4
(δ2i + 1
)2 (δ2j + 1
)+ 2γ6
(δ2i + 1
))
4γδjK2(8(1 + δ2i
) (1 + δ2j
)− (2γ+ γ2 + 2γδ2i )
)(4 (1 + δ2i )
(1 + δ2j
)− γ2
)3 (16 (1 + δ2i )
(1 + δ2j
)− γ2
)3 > 0.
Proof of Corollary 9. When Ki =Kj =K and δi = δj = δ, then, we have
∂Qu∗i∂γ
= − 4K(δ2 + 1
) (1 + δ2 − γ
) (γ+ 4(1 + δ2)
)2 (γ− 2(1 + δ2)
)2
µi (γ+ 4(1 + δ2))2 (γ− 2(1 + δ2))2 (γ+ 2(1 + δ2))2 (γ− 4(1 + δ2))2< 0,
and∂E(πi)
∂γ= −4
(δ2 + 1
)K2 4(1− γ) + (9− 2γ)4δ2 + (9− γ)4δ4 + 12δ6 + 8 + γ3
(2(1 + δ2) + γ)3 (4(1 + δ2)− γ)3< 0.
Proof of Corollary 10. Let Tu∗ = µiQu∗i +µjQ
u∗j . When Ki =Kj =K, we have
∂Tu∗
∂δi= −4δiK
(512
(δ2j + 1
)4 (δ2i + 1
)2 − 256γ(δ2j + 1
)3 (δ2i + 1
)2 − 128γ2(δ2j + 1
)3 (δ2i + 1
)
−16γ4(δ2j + 1
) (1 + δ2i
)+ 96γ2
(δ2j + 1
)2 (δ2i + 1
)2+ 12γ4
(δ2j + 1
) (δ2j + 1
)+ γ6 + 4γ5(1 + δ2j )
)
(γ2 − 16
(δ2j + 1
)(δ2i + 1)
)2 (γ2 − 4
(δ2j + 1
)(δ2i + 1)
)2 < 0,
and
∂Tu∗
∂δj= −4δjK
(512
(δ2i + 1
)4 (δ2j + 1
)2 − 256γ(δ2i + 1
)3 (δ2j + 1
)2 − 128γ2(δ2i + 1
)3 (δ2j + 1
)
−16γ4(δ2i + 1
) (δ2j + 1
)+ 96γ2
(δ2i + 1
)2 (δ2j + 1
)2+ 12γ4
(δ2i + 1
)2+ 4γ5(1 + δ2i ) + γ6
)
(γ2 − 16
(δ2j + 1
)(δ2i + 1)
)2 (γ2 − 4
(δ2j + 1
)(δ2i + 1)
)2 < 0.
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Let pu∗i =A−µiQu∗i − γµjQ
u∗j . When Ki =Kj =K, we have
∂pu∗i∂δi
=4δiK
(G1 + γ7 + 2γ6(1 + δ2j ) + 96γ3
(δ2j + 1
)2 (δ2i + 1
)2+ 12γ4
(δ2j + 1
)2+ 2γ5
(δ2j + 1
))
(γ2 − 4δ2i − 4δ2j − 4δ2i δ
2j − 4
)2 (γ2 − 16δ2i − 16δ2j − 16δ2i δ
2j − 16
)2 > 0,
since G1 = 512(δ2j + 1
)4 (δ2i + 1
)2 − 128γ(δ2j + 1
)3 (δ2i + 1
)2 − 128γ2(δ2j + 1
)3 (δ2i + 1
)2 − 128γ2(δ2j + 1
)3 (δ2i + 1
)−
16γ5(δ2j + 1
) (δ2i + 1
)> 0; and
∂pu∗i∂δj
=4γδjK
(G2 + 96γ
(δ2i + 1
)2 (δ2j + 1
)2+ 2γ4
(6δ2i + 7
) (δ2i + 1
)+ γ5
(2δ2i + 3
))
(γ2 − 16δ2i − 16δ2j − 16δ2i δ
2j − 16
)2 (γ2 − 4δ2i − 4δ2j − 4δ2i δ
2j − 4
)2 > 0,
since G2 = 512(δ2i + 1
)4 (δ2j + 1
)2 − 128(δ2i + 1
)3 (δ2j + 1
)2 − 16γ3(δ2j + 1
) (δ2i + 1
)− 128γ2
(δ2i + 1
)3 (δ2j + 1
)−
128γ(δ2i + 1
)3 (δ2j + 1
)2> 0.
Proof of Corollary 11. When Ki =Kj =K and δi = δj = δ, we have ∂Tu∗
∂γ=− 8K(1+δ2)(1+δ2−γ)
(4(1+δ2)−γ)2(2(1+δ2)+γ)2 < 0,
and∂pu∗i∂γ
=− 2K(δ2+1)(γ2+2γ+8δ4+14δ2+6)(2γ(δ2+1)+8(δ2+1)2−γ2)
2 < 0.
Proof of Theorem 4.
E(πhi )−E(πui ) =
(2γKj(1 + δ2i )−Ki(8δ
2i + 8δ2j + 8δ2i δ
2j + 8− γ2)
)2
16 (1 + δ2i )(4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
)2
− 2(1 + δ2j
) (6(1 + δ2i + δ2j + δ2i δ
2j )− γ2
) (2γ(1 + δ2i )Kj − (8(1 + δ2i + δ2j + δ2i δ
2j )− γ2)Ki
)2(4(1 + δ2i + δ2j + δ2i δ
2j )− γ2
)2 (16(1 + δ2i + δ2j + δ2i δ
2j )− γ2
)2
=
(γ4 + 64
(δ2j + 1
)2 (δ2i + 1
)2)(2γKj(1 + δ2i )−Ki(8δ
2i + 8δ2j + 8δ2i δ
2j + 8− γ2)
)2
16 (δ2i + 1)(4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
)2 (16(1 + δ2i + δ2j + δ2i δ
2j )− γ2
)2 > 0,
E(π∗j )−E(πhj ) =
(1 + δ2j
) (2(1 + δ2i )Kj − γKi
)2(4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
)2 −(2Kj(1 + δ2i )− γKi
)2 (6δ2i + 6δ2j + 6δ2i δ
2j + 6− γ2
)
8 (δ2i + 1)(4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
)2
=
(2(1 + δ2i )Kj − γKi
)2 (γ2 + 2(1 + δ2i + δ2j + δ2i δ
2j ))
8 (δ2i + 1)(4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
)2 > 0,
E(π∗i )−E(πh′i ) =
(1 + δ2i
) (2(1 + δ2j )Ki− γKj
)2(4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
)2 −(2Ki(1 + δ2j )− γKj
)2 (6δ2i + 6δ2j + 6δ2i δ
2j + 6− γ2
)
8(1 + δ2j
) (4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
)2
=
(2(1 + δ2j )Ki− γKj
)2 (γ2 + 2(1 + δ2i + δ2j + δ2i δ
2j ))
8(δ2j + 1
) (4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
)2 > 0,
E(πh′j )−E(πuj ) =
(2γKi(1 + δ2j )−Kj(8δ
2i + 8δ2j + 8δ2i δ
2j + 8− γ2)
)2
16(1 + δ2j
) (4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
)2
− 2(1 + δ2i
) (6(1 + δ2i + δ2j + δ2i δ
2j )− γ2
) (2γ(1 + δ2j )Ki− (8(1 + δ2i + δ2j + δ2i δ
2j )− γ2)Kj
)2(4(1 + δ2i + δ2j + δ2i δ
2j )− γ2
)2 (16(1 + δ2i + δ2j + δ2i δ
2j )− γ2
)2
=
(γ4 + 64
(δ2j + 1
)2 (δ2i + 1
)2)(2γKi(1 + δ2j )−Kj(8δ
2i + 8δ2j + 8δ2i δ
2j + 8− γ2)
)2
16(δ2j + 1
) (4δ2i + 4δ2j + 4δ2i δ
2j + 4− γ2
)2 (16(1 + δ2i + δ2j + δ2i δ
2j )− γ2
)2 > 0.
This implies that a decentralized supply chain is better off by centralization regardless whether the other supply
chain is centralized or not. Hence, centralization is the dominant strategy.
Proof of Theorem 5. Based on the results in Theorems 1, 2 and 3, we find
T ∗ =(2(1 + δ2j )− γ)Ki + (2(1 + δ2i )− γ)Kj
4 (1 + δ2i ) (1 + δ2j )− γ2,
Th∗ =2Kj
(δ2i + 1
) (2(1 + δ2i )− γ
)+Ki
(8(1 + δ2i
)(1 + δ2j )− γ
(γ+ 2δ2i + 2
))
4(4 (1 + δ2i ) (1 + δ2j )− γ2)(1 + δ2i ),
Tu∗ =2(1 + δ2j
) ((8(1 + δ2i
)(1 + δ2j )− γ2)Ki− 2γKj(1 + δ2i )
)(16 (1 + δ2i ) (1 + δ2j )− γ2
) (4 (1 + δ2i ) (1 + δ2j )− γ2
)
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+2(1 + δ2i
) ((8(1 + δ2i
)(1 + δ2j )− γ2)Kj − 2γKi(1 + δ2j )
)(16 (1 + δ2i ) (1 + δ2j )− γ2
) (4 (1 + δ2i ) (1 + δ2j )− γ2
) .
Since2γ(1+δ2i )
8(1+δ2i )(1+δ2j )−γ2<λ<
2(1+δ2i )γ
, we can verify that
T ∗−Th∗ =
(2(1 + δ2i )− γ
)(2Kj(1 + δ2i )− γKi)
4(4 (1 + δ2i ) (1 + δ2j )− γ2)(1 + δ2i )> 0,
Th∗−Tu∗ =
(8(1 + δ2i
)(1 + δ2j )− (γ2 + 2γ(1 + δ2i )
) (Ki
(8(1 + δ2i
)(1 + δ2j )− γ2
)− 2γKj
(1 + δ2i
))
4 (1 + δ2i )(16 (1 + δ2i ) (1 + δ2j )− γ2
) (4 (1 + δ2i ) (1 + δ2j )− γ2
) > 0.
Hence, T ∗>T h∗ >Tu∗.
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