Managing Supply Uncertainty under Chain-to-Chain Competitionpersonal.cb.cityu.edu.hk/biyishou/Disruption_competition.pdf · Managing Supply Uncertainty under Chain-to-Chain Competition

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Managing Supply Uncertainty under Chain-to-ChainCompetition

Biying ShouDepartment of Management Sciences, City University of Hong Kong, [email protected]

Erick Zhaolin LiFaculty of Economics and Business, The University of Sydney, Australia, [email protected]

We study the competition between two supply chains that are subject to supply uncertainty. Each supplychain consists of a supplier and a retailer. The retailers engage in a Cournot competition by determiningorder quantities from their exclusive suppliers. The yields of the suppliers are random. We examine thesuppliers’ and the retailers’ decisions at three different levels. At the operational level, we demonstrate that aretailer should order more if its competing retailer has a less reliable supply, and it should order less if its ownsupply becomes less reliable. At the design level, we characterize the optimal parameters for a wholesale pricecontract with linear penalty for any given supply uncertainty. At the strategic level, we show that supplychain centralization is a dominant strategy, and it always makes the customers better off. Nevertheless, if thesupply risk is low, centralization could actually decrease the the supply chain profit, compared with the casewhere both chains do not choose centralization. This results in a prisoner’s dilemma. However, if the supplyrisk is high, centralization always increases supply chain profit. The benefit of supply chain centralization isstrengthened by high supply uncertainty.

Key words : Supply chain risk management, supply uncertainty, supply chain competition, game theoryHistory :

1. IntroductionWith advances in information technology and the increasingly tense global competition, firms arecooperating more closely than ever to optimize the performance of their supply chains. As a result,the traditional model of firm-to-firm competition is giving way to a new paradigm of chain-to-chaincompetition (Barnes (2006), Ha and Tong (2008)). Moreover, supply uncertainty is becoming amajor concern for global supply chain management. An industrial survey conducted by Protivitiand APICS (American Production and Inventory Control Society) showed that 66% of respondentsconsidered supply interruption one of their most significant concerns among all supply chain relatedrisks (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 chain-to-chain competition. In particular, we study theinteraction 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 tofulfill the retailer’s order due to external (e.g., snowstorm, customs delays) or internal causes (e.g.,labor strike, machine breakdown).

We can find many real-life examples where our model of chain-to-chain competition with supplyuncertainty is applicable. For instance, in the automobile industry, competing car manufactur-ers often sell through independent and exclusive dealerships. Events such as labor strikes, partsshortages, earthquakes, and fires may disrupt supply. For example, due to the recent earthquakein Japan, Toyota was forced to cut production because of compulsory power cuts and problemsin getting the parts to factories. Akio Toyoda, president of Toyota, announced in May 2011: “InJapan, we are currently operating at about 50%. We will mark that as the low point from whichwe intend to increase production. As for production outside Japan, we are currently producing at

1

Author: Article Short Title2 Article submitted to Production and Operations Management; manuscript no.

about 40%.” The supply shortage of Japanese cars have seen its impact on the car dealerships.The BBC news reported (May 2011), “As car demand in the US, China and Asia continues torecover, an increasing number of customers are stopping by at dealer showrooms only to find thereare not enough Japanese cars. Consequently, the Japanese car makers are losing market shares toAmerican, European and South Korean car companies.”

Another example to consider is from the pharmaceutical industry. Two Chinese companiesaccount for approximately 70% of the market for manufacturing Heparin Sodium, an anti-coagulantextracted from healthy pig intestines and widely used in open-heart surgery. Currently, both com-panies source from independent suppliers. Because quality problems with pig intestines often causedisruption to their normal supply, one of the companies is considering vertical integration. Simi-larly, some diary product manufacturers in China have their own farms while others procure fromindependent farmers. Different supply risks and supply chain structures are likely result in differentcompetitive strategies.

We can also find chain-to-chain competition in many other contexts. For instance, Wu and Chen(2005) discussed the chain-to-chain competition between the Chinese wireless service providers whoare served by exclusive equipment suppliers. McGuire and Staelin (1983) discussed competitionbetween fast-food chains. These supply chains also face challenges managing supply disruptionsdue to equipment failure and quality problems.

In this paper, we aim at addressing the following research questions:• How does supply uncertainty and chain-to-chain competition affect the ordering decisions?• How to determine the wholesale price and shortage penalty in a decentralized supply chain?• Does centralization provide a competitive advantage when dealing with competition 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 asDecentralized Competition). We derive the equilibrium decisions for each game in terms of theretailer ordering quantity and (for a decentralized chain) the contract terms. By comparing thethree equilibria, we examine the impact of supply uncertainty and chain-to-chain competition onthe value of centralization and customer service.

The key results of this paper are summarized as follows:• Operational level: 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. Retailer’s order sizes can be largeror smaller compared to the benchmark setting where supply uncertainty is absent.• Design level: At the design level, we characterize the optimal contract terms for wholesale

price with linear penalty for supply shortage.• Strategic level: We show that supply chain centralization is the dominant strategy under supply

uncertainty and chain-to-chain competition. Moreover, in terms of expected market supply andprice, customers are always better off with centralization than without. Nevertheless, supply chaincentralization may not always result in positive gains for the supply chain itself. In fact, we showthat if supply risk is low, centralization could actually decrease supply chain profit comparing tothe case that both chains choose to be decentralized, which results in a prisoner’s dilemma. On theother hand, if supply risk is high, centralization always increases the supply chain profit. In otherwords, the benefit of supply chain centralization is strengthened by high supply uncertainty.

The remainder of this paper is organized as follows. Section 2 reviews the related literature.Section 3 describes the model and introduces the notation. Section 4 studies the operational anddesign level decisions under three different game settings: the centralized game, the hybrid game,and the decentralized game. Section 5 compares the three equilibria from Section 4 and derivesstrategic insights on supply chain profits and customer benefits. Finally, we conclude in Section 6.

Author: Article Short TitleArticle submitted to Production and Operations Management; manuscript no. 3

2. LiteratureOur study is closely related to three areas: supply uncertainty, supply chain contracting, and supplychain 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 functionof 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)), (ii) the stochastic lead-timemodel, 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 supplieras one of two states: “up” or “down” (e.g., Arreola-Risa and DeCroix (1998), Gupta (1996), Meyeret al. (1979), Parlar and Berkin (1991), Song and Zipkin (1996), Tomlin (2006), Snyder and Shen(2006b), Snyder and Shen (2006a), Yang et al. (2009)). Specifically, for the supply disruption model,the orders are fulfilled on time and in full when the supplier is “up,” and no order can be fulfilledwhen 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 theseconflicts. The key task in supply contract design is to align the objectives of the various firms withthe 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 andcompetition 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 structureswith deterministic demand. Boyaci and Gallego (2004) studied two competing supply chains wherethe manufacturers and the retailers could act to affect service quality, and examined the value ofcentralization given full information. They showed that centralization is the dominant strategy, butit reduces supply chain profit, presenting a prisoner’s dilemma. Ha and Tong (2008) investigated twocompeting 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.

Very few papers have focused on the effects of supply uncertainty in a competitive environment.Motivated by the case of the U.S. influenza vaccine market, Deo and Corbett (2009) examinedthe interaction between supply uncertainty and a firm’s strategic decision regarding market entry.They found that, in an equilibrium state, supply uncertainty can lead to a high number of firmsconcentrating in an industry as well as a reduction in the industry output and the expectedconsumer surplus. Babich et al. (2007) studied a supply chain where one retailer does businesswith competing high-risk suppliers that may default during their production lead times. They showthat low supplier default correlations dampen competition between the suppliers and increase theequilibrium wholesale prices. Therefore, the retailer prefers suppliers with highly correlated defaultevents, despite consequently losing the benefits of diversification. In contrast, the suppliers andthe channel prefer negatively correlated defaults. Hopp et al. (2008) investigated the impact ofregional supply disruption on competing supply chains where the firms’ strategies consist of twostages: (i) preparation, investing in measures that facilitate quick detection of a problem priorto a disruption, and (ii) response, post-disruption purchasing of backup capacity for componentswith compromised availability. As a measure of risk, they used the expected loss of profit due tolack of preparedness, and they found that the products that pose the greatest risk are those withvaluable 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 marketshare, while a weaker firm should focus on being ready to take advantage of a supply disruptionto gain market share.


Our paper differs from the above-mentioned papers in that we focus on the ordering decisions,contract design, and centralization choices given supply uncertainty and chain-to-chain competi-tion. We contribute to the literature by offering a systematic examination of how to design andoperate supply chains to effectively deal with supply risks and competition.

3. The ModelTo investigate the impact of supply uncertainty and chain-to-chain competition, we consider themodel presented in Figure 1.

Supplier 1

Supplier 2

Retailer 1

Retailer 2

Market

Figure 1 Model of Supply Chain Competition and Supply Uncertainty

• There are two competing supply chains that offer the same product in the market. Eachchain consists of one retailer and one 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 onquantity.• A supplier provides the product to its retailer exclusively, and it may be subject to supply

uncertainty. If retailer i places an order of Qi, he receives αiQi, where αi is a random variablebetween 0 and 1 (inclusive). The mean and standard deviation of αi is µi and σi. We use coefficientof variation (CV), δi = σi/µi, to measure level of 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, and thus the two supply chains only interact once. The

market price of the product is determined by a function p(x) =A−x, where x is the total amountof inventory available in the market (i.e., the sum of the actual deliveries received by the tworetailers) and the positive constant A characterizes the maximum market demand. This pricingfunction is widely used in quantity competition papers (e.g., in Ha and Tong (2008)). The functionp(x) =A−x reflects the fact that a larger supply leads to a lower market price.• We include two marginal costs for each supply chain i (similar as in Deo and Corbett (2009)):

(i) C1,i per unit of production quantity, and (ii) C2,i per unit of successful delivery. If supplier iproduces an order of Qi and the yield is αiQi, the expected total cost is E[C1,iQi +C2,iαiQi] =C1,iQi +C2,iµiQi. To simplify the notation, we can further define Ci =C1,i/µi +C2,i, and thus theexpected total cost is CiQi. To exclude the non-interesting case where neither party sells to themarket, we assume A>Ci. We consider the general case where the two chains may have differentcosts.• 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 theentire market.

The sequence of events is as follows, and both chains (i= 1,2) act simultaneously at each stage.1. Contract decision: supplier i offers a contract to retailer i.2. Order decision: retailer i decides on the order quantity Qi from supplier i.3. Uncertain supply : retailer i receives αiQi from supplier i .


4. Quantity competition: the market is cleared based on the total realized products from bothsupply chains and each party receives the payoff.

4. Design and Operation DecisionsIn 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 centralizedchains. Second, we derive the equilibrium for the hybrid competition game between a centralizedchain and a decentralized chain. Finally, we derive the equilibrium for the decentralized competitiongame between two decentralized chains. Comparisons between the three settings and insights intothe strategic decisions are provided in Section 5.

4.1 Centralized Competition GameWe first consider the case in which both supply chains are centralized, i.e., retailers and suppliersare fully aligned to achieve the supply chain’s optimal performance. We first consider each chain asa central planner (instead of considering the supplier and the retailer as two decision makers), andanalyze their quantity competition under supply disruptions. We seek to derive the order quantity,Qi and Qj, for each supply chain at the market equilibrium.

The goal of the central planner i (i.e., the person who dictates supply chain i) is to maximizethe expected profit for supply chain i, which can be expressed as follows:

E(πi) =E(pαiQi−C1,iQi−C2,iαiQi) = (A−Ci)µiQi−Q2i (µ

2i +σ2

i )−QiQjµiµj (1)

The following theorem characterizes the unique pure strategy equilibrium and the correspondingplayers’ payoffs for the Centralized Competition game. For notation convenience, we denote Ki =A−Ci, Kj =A−Cj, and λ=Ki/Kj.

Theorem 1. The unique pure strategy Nash equilibrium of the Centralized Competition gameis (Q∗i ,Q

∗j ), where the optimal order quantities of supply chains are

(i) when 12(1+δ2j )

<λ< 2(1 + δ2i ),

Q∗i =2Ki−Kj + 2δ2jKi

µi(4δ2i + 4δ2j + 4δ2i δ2j + 3)

, (2)

Q∗j =2Kj −Ki + 2δ2iKj

µj(4δ2i + 4δ2j + 4δ2i δ2j + 3)

, (3)

(ii) when λ≤ 12(1+δ2j )

,

Q∗i = 0, (4)

Q∗j =Kj

2µj(1 + δ2j ), (5)

(iii) when λ≥ 2(1 + δ2i ),

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.


The proof is provided in the Appendix. Now consider the special case where there is no supplyuncertainty (µi = µj = 1, δi = δj = 0). First, if the costs are the same (Ci =Cj =C), the CentralizedCompetition game degenerates into a standard Cournot duopoly game, and the optimal orderquantities become Q∗i =Q∗j = (A−C)/3 and the expected profits are equal to (A−C)2/9. Second,if the difference between the two supply chain’s costs is significant, one chain simply does not enterthe market and the other chain becomes the market monopoly. For example, if Ki ≥ 2Kj, chain ibecomes the monopoly player and its optimal order quantity becomes (A−Ci)/2.

Next, we examine the impact of supply uncertainty. In the special case when Ci =Cj and δi = δj,Theorem 1 echoes the findings in Lemma 1 of Deo and Corbett (2008) when there are two firms.The order quantities of Chain i and j are identical and decrease in the yield uncertainty δ. Nowconsider the more general setting where the costs and yield uncertainties of two chains are different,we find: first, when the market has only one monopoly player (cases (ii) and (iii) in Theorem1), the optimal order quantity and the corresponding expected profit decrease when the supplyuncertainty increases; second, when both chains sell to the market (case (i) in Theorem 1), weshow that the following is true.

Corollary 1. when both chains sell to the market (i.e., when 12(1+δ2j )

< λ < 2(1 + δ2i )), the

equilibrium order quantity and expected profit for supply chain i increase in δj and decrease in δi.That is, supply chain i’s order quantity and expected profit increase when the supply uncertainty ofthe competing chain increases, and decrease when its own supply uncertainty increases.

6.50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

0.6

0

0.1

0.2

0.3

0.4

0.5

Supply Uncertainty Level of Chain j (δj)

Expe

cted

Sup

ply

of C

hain

i

δi=0

δi=0.5

δi=1

δi=2

Figure 2 Expected supply

6.50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

0.3

0.05

0.1

0.15

0.2

0.25


Expe

cted

Pro

fit o

f Cha

in i

δi=0

δi=0.5

δi=1

δi=2

Figure 3 Expected profit

Corollary 1 can be proven easily by examining the first order derivative. Figures 2 and 3 presentsome numerical examples. For the purpose of illustration, we let Ci =Cj =C and normalize (A−C)to 1. The horizontal axis in both figures is the uncertainty level of supply chain j, δj. In Figure 2 thevertical axis is the expected order quantity of supply chain i, E(Qi) = µiQi, whereas in Figure 3 thevertical axis is the expected profit of supply chain i. The four curves represent different uncertaintylevels for supply chain i, with δi = 0,0.5,1,2, respectively. The 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. For example, if supply chaini has perfect supply (i.e., δi = 0), while the uncertainty of supply chain j (δj) increases from 1 to2, supply chain i’s expected supply increases by 10% and the expected profit increases by 22%.


• Given a fixed uncertainty level for the competing supply chain, the expected order quantityand profit of supply chain i decreases as its own uncertainty level increases. For example, assumingthat the uncertainty level of supply chain j is fixed at δj = 1, while the uncertainty of supply chaini (δi) increases from 0.5 to 1, supply chain i’s expected supply decreases by 40% and the expectedprofit 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.• 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 theprofits for the standard monopoly game without supply uncertainty.

The next corollary shows the impact of supply uncertainty on the expected total market supply.

Corollary 2. When both chains sell to the market, the expected total market supply at theNash equilibrium for the centralized Competition game is

E(αiQi +αjQj) = µiQ∗i +µjQ

∗j =

Ki +Kj + 2Kiδ2j + 2Kjδ

2i

4δ2i + 4δ2j + 4δ2i δ2j + 3

(8)

The expected total market supply decreases in δi and δj.

4.2 Hybrid Competition GameIn this section, we study the case where one supply chain is centralized while the other is not.We are interested in discovering how the equilibrium order quantities and the profits change whencompared to the centralized game. In particular, does the centralized chain have a competitiveedge over the decentralized chain in the hybrid scenario?

Without loss of generality, we assume that supply chain i is centralized and that supply chainj is decentralized. We use superscript “h” to denote the hybrid competition game. The expectedprofit of supply chain i is

E(πhi ) =E(pαiQhi −C1,iQ

hi −C2,iαiQ

hi ) = (A−Ci)µiQh

i −Qh2i (µ2

i +σ2i )−Qh

iQhjµiµj. (9)

For supply chain j, we assume that the supplier charges wholesale price ωj per unit successfuldelivery, and pays a penalty sj per unit of shortage. This is a linear contract that is easy toimplement and is widely adopted in practice. The expected profit of retailer j is

E(πhj,r) = E[(p−wj)αjQhj + sj(1−αj)Qh

j ]= (A−wj)µjQh

j −Qh2j (µ2

j +σ2j )−Qh

iQhjµiµj + sj(1−µj)Qh

j . (10)

Define m=wjµj−sj(1−µj) as the supplier j’s expected profit margin for each unit ordered by theretailer, i.e., with probability µj the supplier receives ωj for each unit of successful delivery to theretailer and with probability 1− µj the supplier pays sj for each unit of shortage. The expectedprofit of supplier j becomes

E(πhj,s) = (m−Cjµj)Qhj . (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 Qh

i and Qhj 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.

When a different contract term m is offered by supplier j in the first stage, central planner i andretailer j play a different subgame at the second stage. For an extensive game, the most commonlyused solution concept is a subgame perfect Nash equilibrium (SPNE), which is a refinement of theconcept of a Nash equilibrium. An equilibrium is an SPNE if the players’ strategies constitute aNash equilibrium in every subgame.

The SPNE of the Hybrid Competition game can be characterized as follows.Definition 2. A pure strategy SPNE (Qh

i (m),Qhj (m),m∗) of the Hybrid Competition game

satisfies1. 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,r(Q

hi ,Q

hj (m))),

Qh∗j (m)∈ arg max

Qhj≥0E(πhj,r(Q

hi (m),Qh

j ,m)).

2. Supplier j sets the contract term 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 ofcontract term m. Similar as before, we let Ki =A−Ci, Kj =A−Cj, and λ=Ki/Kj.

Lemma 1. At a pure strategy SPNE of the Hybrid Competition game,1. if Aµj − 2

(δ2j + 1

)µjKi <m<Aµj −

µjKi

2δ2i+2,

Qhi (m) =

m−Aµj + 2Kiµj + 2Kiµjδ2j

µiµj(4δ2i + 4δ2j + 4δ2i δ2j + 3)

, (12)

Qhj (m) =

2Aµj − 2m−Kiµj − 2mδ2i + 2Aδ2i µjµ2j(4δ

2i + 4δ2j + 4δ2i δ

2j + 3)

. (13)

2. if m≤Aµj − 2(δ2j + 1

)µjKi,

Qhi (m) = 0, (14)

Qhj (m) =

Aµj −m2(δ2j + 1

)µ2j

. (15)

3. if Aµj −µjKi

2δ2i+2≤m≤Aµj,

Qhi (m) =

Ki

2µi (δ2i + 1), (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> 2δ2i + 2, m∗ can be any value in [Cjµj,Aµj],

Qh∗i =

Ki

2µi(1 + δ2i ),

Qh∗j = 0.


ii) if2+2δ2i

8δ2i+8δ2j+8δ2i δ2j+7

<λ≤ 2δ2i + 2,

m∗ = Aµj −(Ki + 2Kj + 2Kjδ

2i )µj

4 (δ2i + 1),

Qh∗i =

7Ki− 2Kj + 8Kiδ2i + 8Kiδ

2j − 2Kjδ

2i + 8Kiδ

2i δ

2j

4(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)(δ2i + 1)µi

,

Qh∗j =

2Kj + 2δ2iKj −Ki

2µj(4δ2i + 4δ2j + 4δ2i δ2j + 3)

.

iii) if 14δ2j+4

<λ≤ 2+2δ2i8δ2i+8δ2j+8δ2i δ

2j+7

,

m∗ = Aµj − 2(δ2j + 1

)µjKi,

Qh∗i = 0,

Qh∗j =

Ki

µj.

iv) if λ≤ 14δ2j+4

,

m∗ = Aµj −Kjµj

2,

Qh∗i = 0,

Qh∗j =

Kj

4(δ2j + 1

)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 3. The equilibrium order quantity and expected profit for supply chain i increasein δj and decrease in δi. That is, one supply chain’s order quantity and expected profit increaseas its competing supply chain’s uncertainty increases and decrease as its own supply uncertaintyincreases.

Corollary 3 can be easily proven by checking the first order derivative. Figures 4 and 5 presentsome numerical examples. For illustration purposes, we let Ci = Cj = C and normalize (A− C)to 1. The horizontal axis in both figures represents the uncertainty level of supply chainj, δj. InFigure 4 the vertical axis is the expected order quantity of supply chain i, whereas in Figure 5 thevertical axis is the expected order quantity of supply chain j. The four curves represent differentuncertainty levels of supply chain i, with δi = 0,0.5,1,2, respectively.

Corollary 4. At the SPNE of the Hybrid Competition game, the expected market supplydecreases in δi and δj. As a result, the expected market price increases in δi and δj.

Next we compare the equilibrium order quantities under the Hybrid Competition Game withthose in the Centralized Competition Game. We find that Qh∗

i >Q∗i and Qh∗j <Q∗j . This indicates

that under the same uncertainty level, in the Hybrid Competition game the order quantity forthe centralized supply chain i increases while that for the decentralized supply chain j decreases,when comparing with the Centralized Competition game. Figure 6 provides a numerical example.Here we let Ci = Cj = C and δi = δj = δ and normalize (A−C) to 1. The horizontal axis is theuncertainty level of supply chains δ, and the vertical axis is the expected order quantity of eachsupply chain. The middle curve represents the order quantity of chain i and j in the CentralizedCompetition game, whereas the top and bottom curves represent the order quantity of chain i(centralized) and chain j (decentralized) in the Hybrid Competition game.


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0.6

0

0.1

0.2

0.3

0.4

0.5


Expe

cted

Sup

ply

of C

hain

i

δi=0

δi=0.5

δi=1

δi=2

Figure 4 Expected supply of chain i

6.50 1 2 3 4 5

0.25

0

0.05

0.1

0.15

0.2


Expe

cted

Sup

ply

of C

hain

j

δi=0

δi=0.5

δi=1

δi=2

Figure 5 Expected supply of chain j

3.50 0.5 1 1.5 2 2.5 3

0.45

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Level of Supply Uncertainty

Expe

cted

Sup

ply

Hybrid Competition, Chain i

Centralized Competition

Hybrid Competition, Chain j

Figure 6 Expected supply of chain i

We have similar results for the expected profits of supply chains. More detailed discussions onthe comparison of expected market supplies and profits of different games are provided in Section5.

4.3 Decentralized Competition GameAssume that supply chain i∈ {1,2} adopts a linear contract with a wholesale price ωi and a penaltysi for failed delivery. We use superscript “u” to indicate that both chains are decentralized. Definemi = µiωi− (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 Qu

i and Quj 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).


The SPNE of the Decentralized Competition game can be characterized as follows.Definition 4. A pure strategy SPNE (Qu

i (mi,mj),Quj (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 thechoices 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 =Aµi−mi and Bj =Aµj −mj.

Lemma 2. At a pure strategy SPNE of the Hybrid Competition game,

1. ifBiµiµj

2(µ2i+σ2i )<Bj <

2Bi(µ2j+σ

2j )

µiµj,

Qui =

2Biσ2j −Bjµiµj + 2Biµ

2j

4σ2i σ

2j + 4σ2

i µ2j + 4σ2

jµ2i + 3µ2

iµ2j

(18)

Quj =

2Bjσ2i −Biµiµj + 2Bjµ

2i

4σ2i σ

2j + 4σ2

i µ2j + 4σ2

jµ2i + 3µ2

iµ2j

(19)

2. if Bj ≥2Bi(µ

2j+σ

2j )

µiµj,

Qui = 0 (20)

Quj =

Bj2(µ2

j +σ2j )

(21)

3. if Bj ≤Biµiµj

2(µ2i+σ2i )

,

Qui =

Bi2(µ2

i +σ2i )

(22)

Quj = 0 (23)

Similar as before, we define Ki =A−Ci, Kj =A−Cj, and λ=Ki/Kj.

Theorem 3. At the SPNE of the Decentralized Competition game, supplier i (i= 1,2) choosesthe contract terms m∗i (equivalently B∗i ) and order quantity Qu∗

i as follows:

i) if2+2δ2i


<λ<8δ2i+8δ2j+8δ2i δ

2j+7

2δ2j+2,

B∗i =8Kiµi + 2Kjµi + 8Kiµiδ

2i + 8Kiµiδ

2j + 2Kjµiδ

2i + 8Kiµiδ

2i δ

2j

16δ2i + 16δ2j + 16δ2i δ2j + 15

,

B∗j =2Kiµj + 8Kjµj + 2Kiµjδ

2j + 8Kjδ

2i µj + 8Kjµjδ

2j + 8Kjδ

2i µjδ

2j

16δ2i + 16δ2j + 16δ2i δ2j + 15

,

Qu∗i =

2(7Ki− 2Kj + 8Kiδ

2i + 8Kiδ

2j − 2Kjδ

2i + 8Kiδ

2i δ

2j

) (δ2j + 1

)(16δ2i + 16δ2j + 16δ2i δ

2j + 15

) (4δ2i + 4δ2j + 4δ2i δ

2j + 3

)µi

,

Qu∗j =

2(7Kj − 2Ki− 2Kiδ

2j + 8Kjδ

2i + 8Kjδ

2j + 8Kjδ

2i δ

2j

)(δ2i + 1)(

16δ2i + 16δ2j + 16δ2i δ2j + 15

) (4δ2i + 4δ2j + 4δ2i δ

2j + 3

)µj

.


ii) if λ< 14(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 14(1+δ2j )

≤ λ≤ 2δ2i+2


,

B∗i = Kiµi,B∗j = 2Kiµj + 2Kiµjδ

2j ,

Qu∗i = 0,

Qu∗j =

Ki

µj.

iv) if8δ2i+8δ2j+8δ2i δ

2j+7

2+2δ2j≤ λ≤ 4(1 + δ2i ),

B∗i = 2Kjµi + 2Kjµiδ2i ,

B∗j = Kjµj,

Qu∗i =

Kj

µi,

Qu∗j = 0.

v) if λ> 4(1 + δ2i ),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 followingcomparative statics.

Corollary 5. At the SPNE of the decentralized Competition game when both retailers sell tothe market, the equilibrium order quantity and expected profit for supply chain i increase with δjand decreases with δi. That is, one supply chain’s order quantity and expected profit increase as itscompeting supply chain’s uncertainty increases, and its order quantity and expected profit decreaseas its own supply uncertainty increases.

Corollary 6. At the SPNE of the Decentralized Competition game when both retailers sell tothe market,, the expected market supply increases in δi and δj. As a result, the expected marketprice decreases in δi and δj.

5. Strategic DecisionsIn the previous section, we derive the optimal ordering decision and linear contract design (fordecentralized chain) for the Centralized, Hybrid and Decentralized Competition games. In thissection we consider the strategic-level question: with supply uncertainty and chain-to-chain com-petition, is there any incentive for a supply chain to become centralized? What is the impact ofsupply uncertainty on the benefit/loss from supply chain centralization?


First, we examine whether a supply chain has an incentive to become centralized when bothchains sell to the market. Based on the equilibrium order quantities we have derived under eachgame when both chains sell to the market, we can calculate the corresponding expected pay-offsas follows.

Under the Centralized Competition game:

E(π∗i ) =(δ2i + 1)

(Kj − 2Ki− 2Kiδ

2j

)2(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 (24)

E(π∗j ) =

(δ2j + 1

)(Ki− 2Kj − 2Kjδ

2i )

2(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 (25)

Under the Decentralized Competition game:

E(πui ) =2(δ2j + 1

) (6δ2i + 6δ2j + 6δ2i δ

2j + 5

) (2Kj − 7Ki− 8Kiδ

2i − 8Kiδ

2j + 2Kjδ

2i − 8Kiδ

2i δ

2j

)2(16δ2i + 16δ2j + 16δ2i δ

2j + 15

)2 (4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 (26)

E(πuj ) =2(δ2i + 1)

(6δ2i + 6δ2j + 6δ2i δ

2j + 5

) (2Ki− 7Kj − 8Kjδ

2j − 8Kjδ

2i + 2Kiδ

2j − 8Kjδ

2i δ

2j

)2(16δ2i + 16δ2j + 16δ2i δ

2j + 15

)2 (4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 (27)

Under the Hybrid Competition game where chain i is centralized and chain j is decentralized:

E(πhi ) =

(2Kj − 7Ki− 8Kiδ

2i − 8Kiδ

2j + 2Kjδ

2i − 8Kiδ

2i δ

2j

)216

(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2(δ2i + 1)

(28)

E(πhj ) =

(6δ2i + 6δ2j + 6δ2i δ

2j + 5

)(2Kj −Ki + 2Kjδ

2i )

2

8(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2(δ2i + 1)

(29)

Under the Hybrid Competition game where chain j is centralized and chain i is decentralized:

E(πh′

i ) =

(6δ2i + 6δ2j + 6δ2i δ

2j + 5

) (2Ki−Kj + 2Kiδ

2j

)28(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 (δ2j + 1

) (30)

E(πh′

j ) =

(2Ki− 7Kj − 8Kjδ

2j − 8Kjδ

2i + 2Kiδ

2j − 8Kjδ

2i δ

2j

)216

(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 (δ2j + 1

) (31)

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 )

Table 1 summarizes these results. We compare these expected profits under different competitiongames and obtain the following theorem.

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 maynot result in positive gains for each supply chain, as the payoff difference between the centralizedand 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 integration (i.e., central-ization) as a strategic level game, Theorem 4 implies that integration would be a unique NashEquilibrium. However, a prisoners’ dilemma may occur.

To further understand the implications of Theorem 4, let us consider the special case withsymmetric supply chains, i.e., δi = δj = δ and Ci =Cj =C. Let K =A−C. It can be verified thatthe expected profits for supply chain i and j under the Centralized Competition game become

E(π∗i ) =E(π∗j ) =(δ2 + 1)K2

(2δ2 + 3)2 ,R.

To simplify exposition, we define it as “R”. The expected supply chain profits under the Decen-tralized Competition game become

E(πui ) =E(πuj ) =2K2 (12δ2 + 6δ4 + 5) (δ2 + 1)

(4δ2 + 3)2(2δ2 + 3)

2 =2(12δ2 + 6δ4 + 5)

(4δ2 + 3)2 R.

The expected supply chain profits under the Hybrid Competition game are as follows.

E(πhi ) =E(πh′

j ) =(4δ2 + 5)

2K2

16 (2δ2 + 3)2(δ2 + 1)

=(4δ2 + 5)

2

16 (δ2 + 1)2R,

E(πhj ) =E(πh′

i ) =(12δ2 + 6δ4 + 5)K2

8 (2δ2 + 3)2(δ2 + 1)

=(12δ2 + 6δ4 + 5)

8 (δ2 + 1)2 R.

Here “h” denotes the case when chain i is centralized and chain j is not, whereas “h′” denotes thecase when chain j is centralized and chain i is not.

These results are summarized in Table 2. Figure 7 presents some numerical examples, where thex axis is level of supply uncertainty δ, the y axis is expected supply chain profit, and the threecurves represent the Centralized, Hybrid and Decentralized Competition, respectively. Note thatchain i is the centralized one in the Hybrid Competition game.

We can see that a decentralized supply chain i is better off by moving toward centralization

if supply chain j is centralized, since (12δ2+6δ4+5)

8(δ2+1)2< 1. Furthermore, a decentralized supply chain

i is better off by moving toward centralization if the supply chain j remains decentralized, since4(δ2+5)2

16(δ2+1)2>

2(12δ2+6δ4+5)(4δ2+3)

2 . Hence, centralization is the dominant strategy for supply chain i. A similar

argument is true for supply chain j, in other words, centralization is the dominant strategy forboth supply chains.

Table 2 Payoff Table for Strategic Choice of Integration with Symmetric Chains

Supply Chain jcentralized decentralized

Supply centralized R,R(4δ2+5)

2

16(δ2+1)2R,

(12δ2+6δ4+5)8(δ2+1)

2 R

Chain i decentralized(12δ2+6δ4+5)

8(δ2+1)2 R,

(4δ2+5)2

16(δ2+1)2R

2(12δ2+6δ4+5)(4δ2+3)

2 R,2(12δ2+6δ4+5)

(4δ2+3)2 R

Furthermore, we can show that when δ <√22

a chain receives less profit in the centralized gamecompared with the decentralized game. This is the case of prisoner’s dilemma, i.e., both supplychains are worse off under the centralized scenario although centralization is the dominated strat-egy. 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 generalize their find-ings and show that it still exists with low levels of supply risk. However, when δ >

√22

, we have2(12δ2+6δ4+5)

(4δ2+3)2 < 1, i.e., the profit for each chain in the centralized competition game is higher than

that in the decentralized competition game. Hence, if the level of supply uncertainty is high, itis better for the supply chain to centralize since it aligns the intra-chain objectives and resultsin higher gain. In other words, the benefit of supply chain centralization is strengthened by highsupply risks.

3.50 0.5 1 1.5 2 2.5 3

0.18

0

0.04

0.08

0.12

0.16


Expe

cted

Pro

fit

Hybrid Competition, Chain i


Hybrid Competition, Chain j

Decentralized Competition

Figure 7 Comparison of the expected supply chain profits

Now a related question is: do individual retailer and supplier in a decentralized chain have theright 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 deliveredorder, and a penalty for each unit that the supplier fails to deliver. Let φ ∈ [0,1] be the fractionof supply chain revenue the retailer keeps, so the supplier earns the remaining (1−φ). The profitfunctions 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)]},E(π) =E(πr) +E(πs) =E(pQα).

Here α is a random variable that represents yield uncertainty. Assume that α is generally distributedin (0,1] with average value of µ. When ωµ− (1−µ)S = 0, we have

E(πr) =E[φpQI] = φE(π),

i.e., the profit of the retailer is an affine function of the central planner. Hence, such a revenuesharing contract coordinates the supply chain. Based on this finding, we can show that the followingis true.

Lemma 3. For any given linear price and penalty contract, there always exists a coordinatingrevenue-sharing contract that makes both the supplier and retailer better off.


The proof for Lemma 3 is provided in the Appendix. Next, we examine customer service in termsof expected total market supply and price under the Centralized, the Hybrid, and the DecentralizedCompetition games. From the customers’ point of view, a higher total expected market supplymeans a lower expected market price, and thus a better customer service.

Theorem 5. When both chains sell to the market, the centralized competition game providesthe best service to customers in terms of total expected market supply, the hybrid competition gameprovides the second, and the decentralized competition game provides the worst service, i.e.,

µiQ∗i +µjQ

∗j >µiQ

hi +µjQ

hj >µiQ

uj +µjQ

uj . (32)

The proof is provided in the Appendix. Theorem 5 indicates that customers always benefit fromsupply chain centralization with more supplies and a lower price. Figure 8 provides a numericalexample, where we assume that the two supply chains have the same costs and the same levels ofsupply uncertainty. The horizontal axis is the level of supply uncertainty. The three curves representthe expected total market supply under the Centralized, Hybrid, and Decentralized Competitiongames, respectively.

3.50 0.5 1 1.5 2 2.5 3

0.7

0

0.1

0.2

0.3

0.4

0.5

0.6


Expe

cted

Tot

al M

arke

t Sup

ply


Hybrid Competition

Decentralized Competition

Figure 8 Comparison of the expected total market supplies

6. ConclusionIn this paper, we provide a systematic examination of how to design and operate supply chains toeffectively deal with supply uncertainty and competition for short life-cycle products. We derivethe equilibrium decisions for ordering decisions and contract terms for the Centralized, Hybrid,and Decentralized competition games. By comparing these three scenarios, we obtain the followingmanagerial insights on supply chain management with the presence of competition and supplyuncertainty.

First, at the operational level we show that a retailer should order more if its competing retailer’ssupply becomes less reliable, and order less if its own supply becomes less reliable. Retailer’s ordersizes can be larger or smaller compared to the case where supply uncertainty is absent.

Second, at the design level we characterize the optimal terms for a wholesale price contractwith linear penalty for any given level of supply uncertainty. We also show that, for such a linear


contract there exists a coordinating revenue management contract which can make both supplierand retailer better off.

Third, at the strategic level we show that supply chain centralization is always the dominantstrategy under supply uncertainty and chain-to-chain competition. Nevertheless, depending on thelevel of supply uncertainty, centralization efforts may or may not result in positive gains for eachsupply chain. For example, given competition between two symmetric supply chains, centralizationefforts guarantee a positive gain for each supply chain when the supply is highly uncertain. However,when the supply is sufficiently reliable, centralization reduces the profit for each supply chain,which results in a prisoner’s dilemma. In other words, the benefit of supply chain centralizationis strengthened by high supply uncertainty. Furthermore, centralization always benefits customersby giving them a lower expected market price.

The above findings are derived through closed-form analysis, facilitated by several simplifyingassumptions. We can extend our analysis in several directions. First, we can consider contingentsourcing, i.e., after a retailer observes his supplier’s status, he may purchase from another reliablesource or from the spot market. Second, we may explore the effect of correlated supply disruptions,i.e., a severe crisis could disrupt both chains simultaneously. Third, suppliers and retailers mayhave private information about demand and reliability, and level of supply uncertainty could bean endogenous decision. Incorporating these factors will add extra complexity to the analysis andmake the closed-form results very tedious and difficult to obtain. Finally, we may study differenttypes of games (e.g., Bertrand competition) and different type of contracts (e.g. with nonlinearprice and penalty cost).

AppendixProof of Theorem 1. Under the Centralized Competition Game, the expected profits for chain i (i= 1,2;

j 6= i) are:

E(πi) =E(pαiQi−C1iQi−C2iαiQi) = (A−Ci)µiQi−Q2i (µ

2i +σ2

i )−QiQjµiµj ,

which is a concave function in Qi. The first-order conditions for each chain are:

(A−Ci)µi− 2Qi(µ2i +σ2

i )−Qjµiµj = 0(A−Cj)µj − 2Qj(µ

2j +σ2

j )−Qiµiµj = 0

Solving these equations, we obtain the equilibrium production quantities as shown in Theorem 1.Proof of Lemma 1. Under the Hybrid Competition game, the expected profit for chain i (centralized)

becomes:E(πhi ) =AµiQ

hi −Qh2

i (µ2i +σ2

i )−QhiQ

hjµiµj −CiQh

i .

The expected profits for retailer and supplier in chain j (decentralized) are respectively:

E(πhj,r) =AµjQhj −Qh2

j (µ2j +σ2

j )−QhiQ

hjµiµj −mQh

j ,

E(πj,s) = (m−Cjµj)Qhj ,

Now consider the ordering decisions for any given m. The first-order conditions are:

Aµi− 2Qhi (µ2

i +σ2i )−Qh

jµiµj −Ciµi = 0Aµj − 2Qh

j (µ2j +σ2

j )−Qhi µiµj −m= 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) = (m − Cjµj)Qj for

Cjµj ≤m ≤ Aµj . Let Ki = A − Ci,Kj = A − Cj , λ = Ki/Kj ,Gi = Aµi − Ciµi,Gj = Aµj −m. Because Gj

and m are one-to-one matching, we seek the optimal Gj in [0,Kjµj ] that maximizes the supplier’s profitE(πj,s) = (Aµj −Gj −Cjµj)Qj .

Based on the results of Lemma 1, we discuss the following three cases to derive supplier j’s optimal decision,


Case A: if Kjµj ≤Giµiµj

2(σ2i+µ2

i), i.e., λ≥ 2δ2i +2, we only need to consider scenario 3) in the first stage (Lemma

1). Gj can be any value in (0,Kjµj ],Qi = Gi

2(µ2i+σ2

i),Qj = 0.

Case B: ifGiµiµj

2(σ2i+µ2

i)≤Kjµj <

2Gi(µ2j+σ

2j )

µiµj,i.e., 1

2δ2j+2≤ λ< 2δ2i + 2, we need to compare supplier’s expected

profit under scenarios 1) and 3) in the first stage. We first consider scenario 1) whereGiµiµj

2(σ2i+µ2

i)<Gj <Kjµj .

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 =(Ki + 2Kj + 2Kjδ

2i )µj

4(δ2i + 1).

When λ< 2δ2i + 2, it satisfiesGiµiµj

2(σ2i+µ2

i)<Gj <Kjµj . Acoordingly,

Qj =2Kj + 2δ2iKj −Ki

2µj(4δ2i + 4δ2j + 4δ2i δ2j + 3)

,

E(πj,s) =1

8

(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)−1 (δ2i + 1

)−1 (2Kj −Ki + 2Kjδ

2i

)2.

Because the expected profit in scenario 3) is 0, obviously scenario 1) is a better choice. This means that if1

2δ2j+2≤ λ< 2δ2i + 2,Gj = 1

4(δ2i + 1)

−1(Ki + 2Kj + 2Kjδ

2i )µj .

Case C: Kjµj ≥2Gi(µ

2j+σ

2j )

µiµj, i.e., λ≤ 1

2δ2j+2, we need to compare supplier’s expected profit under scenarios

1), 2) and 3) in the first stage. First consider case 2) where2Gi(µ

2j+σ

2j )

µiµj≤Gj ≤Kjµj ,Qi = 0,Qj =

Gj

2(µ2j+σ2

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≤ 1

4δ2j+4, then

(A−Cj)µj

2≥ 2Gi(µ

2j+σ

2j )

µiµj. Hence Gj =

(Kjµj

2,Qj =

Aµj−Cjµj

4σ2j+4µ2

j,E(πj,s) =

K2j

8δ2j+8.

• If 14δ2

j+4

< λ ≤ 12δ2

j+2

then(A−Cj)µj

2<

2Gi(µ2j+σ

2j )

µiµj. Hence Gj =

2Gi(µ2j+σ

2j )

µiµj,Qj = Gi

µiµj, E(πj,s) =(

Kj − 2Ki− 2Kiδ2j

)Ki.

Next consider scenario 1). Based on the first-order condition, we have

Gj =1

4

(δ2i + 1

)−1 (Ki + 2Kj + 2Kjδ

2i

)µj

We need to check whether the solution falls into the feasible regionGiµiµj

2(σ2i+µ2

i)≤Gj ≤

2Gi(µ2j+σ

2j )

µiµj. There are

two cases to discuss:• if

2+2δ2i8δ2

i+8δ2

j+8δ2

iδ2j+7≤ λ ≤ 1

2δ2j+2, then the solution falls into the feasible region, hence Gj =

14

(δ2i + 1)−1

(Ki + 2Kj + 2Kjδ2i )µj ,Qj = 1

2µj

−Ki+2Kj+2δ2iKj

4δ2i+4δ2

j+4δ2

iδ2j+3

, E(πj,s) = 18

(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)−1(δ2i + 1)

−1

(2Kj −Ki + 2Kjδ2i )

2.

• if λ≤ 2+2δ2i8δ2

i+8δ2

j+8δ2

iδ2j+7, then the solution is smaller than the lower bound of the feasible region, hence

Gj =2Gi(µ

2j+σ

2j )

µiµj,Qj = Gi

µiµj,E(πj,s) =


2j

)Ki.

Because the expected profit in scenario 3) is 0, we only need to compare scenarios 1) and 2). We find• When λ≤ 1

4δ2j+4,the expected profits under scenario 2 is larger than that under scenario 1. Thus Gj =

(A−Cj)µj

2,Qj =

Aµj−Cjµj

4σ2j+4µ2

j.

• When 14δ2

j+4≤ λ ≤ 2+2δ2i

8δ2i+8δ2

j+8δ2

iδ2j+7,the optimal solution under scenarios 1 and 2 are the same, thus

Gj =2Gi(µ

2j+σ

2j )

µiµj,Qj = Gi

µiµj.

• When2+2δ2i

8δ2i+8δ2

j+8δ2

iδ2j+7

< λ≤ 12δ2

j+2, the expected profit in scenario 1 is larger than that in scenario 2.

Thus,

Gj =1

4

(δ2i + 1

)−1 (Ki + 2Kj + 2Kjδ

2i

)µj ,


Qi =7Ki− 2Kj + 8δ2iKi− 2δ2iKj + 8δ2jKi + 8δ2i δ

2jKi

16µiδ4i δ2j + 16µiδ4i + 32µiδ2i δ

2j + 28µiδ2i + 16µiδ2j + 12µi

,

Qj =2Kj + 2δ2iKj −Ki

2µj(4δ2i + 4δ2j + 4δ2i δ2j + 3)

.

Summarizing cases A, B and C, we derive the SPNE of Gj ,Qi and Qj , as shown in Thoerem 2.Proof of Lemma 2. Under the Decentralized Competition game, the expected profit of retailer i(i= 1,2)

is:E(πi,r) =AµiQ

ui −Qu2

i (µ2i +σ2

i )−QuiQ

uj µiµj −miQ

ui .

For any given mi, retailer i wants to find the optimal ordering quantity to maximize its expected profit. It’seasy to show that the expected profit function is concave in Qi. The first-order condition leads to:

Aµi− 2Qui (µ2

i +σ2i )−Qu

j µiµj −mi = 0.

For exposition simplicity, we define Bi =Aµi−mi, where i= 1,2. Thus,

Bi− 2Qui (µ2

i +σ2i )−Qu

j µiµj = 0.

Solving the two first-order conditions, we obtain the equilibrium ordering quantities Quj and Qu

j for any givenBj and Bj , as shown in Lemma 2.

Proof of Theorem 3. Consider the suppliers’ decision for decentralized game. Supplier i’s optimal choiceof mi to maximize E(πi,s) = (mi −Ciµi)Qi, or equivalently, to maximize E(πi,s) = (Aµi −Bi −Ciµi)Qi for0≤Bi ≤ (A−Ci)µi. Based on the results of Lemma 2, we charaterize the best-response functions of Bj andBj 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 ≤

Biµiµj

2(σ2i+µ2

i)

=µjBi

2µi(δ2i +1), we only need to consider scenario 3) in the first stage. Bj can be

any value in (0,Kjµj ],Qi = Bi

2(µ2i+σ2

i),Qj = 0.

Case B: ifBiµiµj

2(σ2i+µ2

i)≤ Kjµj ≤

2Bi(µ2j+σ

2j )

µiµj, i.e.,

µjBi

2µi(δ2i +1)≤ Kjµj ≤

2(δ2j+1)µjBi

µi, we need to compare the

expected profits under scenarios 1) and 3) in the first stage to decide the the optimal choice of Bj . Wefirst consider scenario 1) in the first stage where

µjBi

2µi(δ2i +1)<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:

(Aµj −Bj −Cjµj)(2σ2

i + 2µ2i

4σ2i σ

2j + 4σ2

i µ2j + 4σ2

jµ2i + 3µ2

iµ2j

)− 2Bjσ2i −Biµiµj + 2Bjµ

2i

4σ2i σ

2j + 4σ2

i µ2j + 4σ2

jµ2i + 3µ2

iµ2j

= 0.

The solution is

Bj =(Bi + 2Kjµi + 2Kjµiδ

2i )µj

4 (δ2i + 1)µi. (33)

We can show that this solution satisfiesµjBi

2µi(δ2i +1)<Bj <Kjµj . Accordingly,

Qj =1

2

(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)−1 (2Kjµi−Bi + 2Kjµiδ

2i

)µ−1i µ−1j ,

E(πj,s) =1

8

(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)−1 (δ2i + 1


2i

)2µ−2i .

Because the expected profit for supplier j in case 3) is 0, obviously case 1) is better, which means that ifµjBi

2µi(δ2i +1)≤Kjµj ≤

2(δ2j+1)µjBi

µi,Bj =

(Bi+2Kjµi+2Kjµiδ2i )µj

4(δ2i +1)µi.

Case C: Kjµj ≥2Bi(µ

2j+σ

2j )

µiµj=

2(δ2j+1)µjBi

µi, 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) where2(δ2j+1)µjBi

µi≤Bj ≤Kjµj ,Qi = 0,Qj =

Bj

2(µ2j+σ2

j). It’s straightforward to show that E(πj,s) is a concave function

w.r.t Bj . Let ddBj

E(πj,s) = 0. We get Bj =Kjµj

2. We need to consider two cases:


• IfKjµj

2≥ 2Bi(µ

2j+σ

2j )

µiµj, then Bj =

Kjµj

2,Qj =

Kjµj

4σ2j+4µ2

j,E(πj,s) =

K2j

8δ2j+8.

• IfKjµj

2<

2Bi(µ2j+σ

2j )

µiµj,then Bj =

2(δ2j+1)µjBi

µi,Qj = Bi

µiµj,E(πj,s) = µ−2i

(Kjµi− 2Bi− 2Biδ

2j

)Bi.

Next we consider scenario 1) in the first stage. We need to check whether the FOC solution expressed by

Eq.(33), Bj =(Bi+2Kjµi+2Kjµiδ

2i )µj

4(δ2i +1)µi, satisfies

Biµiµj

2(µ2i+σ2

i)<Bj <

2Bi(µ2j+σ

2j )

µiµj. Because

Biµiµj

2(µ2i+σ2

i)<Kjµj , we find

1

4σ2i + 4µ2

i

(Biµiµj + 2Kjσ

2i µj + 2Kjµ

2iµj

)− Biµiµj

2(µ2i +σ2

i )=

1

4

(σ2i +µ2

i

)−1 (2Kjσ

2i −Biµi + 2Kjµ

2i

)µj > 0.

However, to have1

4σ2i + 4µ2

i

(Biµiµj + 2Kjσ

2i µj + 2Kjµ

2iµj

)−

2Bi(µ2j +σ2

j )

µiµj> 0,

it should satisfy

Bi >2Kjµ

3iµ

2j + 2Kjσ

2i µiµ

2j

8σ2i σ

2j + 8σ2

i µ2j + 8σ2

jµ2i + 7µ2

iµ2j

.

Hence we need to consider two cases:• If

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j<Bi <

Kjµiµ2j

2(µ2j+σ2

j), thenBj = 1

4σ2i+4µ2

i(Biµiµj + 2Kjσ

2i µj + 2Kjµ

2iµj) ,E(πj,s) =

18

(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)−1(δ2i + 1)

−1(2Kjµi−Bi + 2Kjµiδ

2i )

2µ−2i .

• If Bi <2Kjµ

3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j, then Bj =

2Bi(µ2j+σ

2j )

µiµj,Qj = Bi



2j

)Bi.

Because the expected profit of supplier j under scenario 3) is always 0 and that under scenarios 1 and 2are positive, we only need to compare scenarios 1 and 2. We find:

• when(A−Cj)µj

2≥ 2Bi(µ

2j+σ

2j )

µiµj, i.e., Bi ≤

Kjµiµ2j

4(µ2j+σ2

j), the expected profit of supplier j under scenario 2) is

larger than that under scenario 1), therefore the optimal decision is: Bj =Kjµj

2,Qj =

Kjµj

4σ2j+4µ2

j,E(πj,s) =

K2j

8δ2j+8.

• whenKjµiµ

2j

4(µ2j+σ2

j)≤Bi <

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j, the expected profit and choice of Bj under scenarios 1)

and 2) are the same, thus Bj =2Bi(µ

2j+σ

2j )

µiµj,Qj = Bi



2j

)Bi.

• When2Kjµ

3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j≤Bi ≤

2(σ2i +µ

2i )Kj

µi, the expected profit in scenario 1 is larger than that

in scenario 2. Thus,

Bj =1

4

(δ2i + 1

)−1 (Bi + 2Kjµi + 2Kjµiδ

2i

)µ−1i µj .

Summarizing cases A, B and C, the best response of Bj for any given Bi is:

i) if Bi >2(σ2

i +µ2i )Kj

µi,Bj can be any value in (0,Kjµj ],Qi = Bi

2(µ2i+σ2

i),Qj = 0,E(πj,s) = 0.

ii) if2Kjµ

3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j<Bi ≤

2(σ2i +µ

2i )Kj

µi,

Bj =1

4

(δ2i + 1


2i

)µ−1i µj

Qi =

(1

4

)(δ2i + 1

)−1 (4δ2i + 4δ2j + 4δ2i δ

2j + 3

)−1µ−2i

(7Bi− 2Kjµi + 8Biδ

2i + 8Biδ

2j − 2Kjµiδ

2i + 8Biδ

2i δ

2j

)Qj =

1

2

(4δ2i + 4δ2j + 4δ2i δ

2j + 3


2i

)µ−1i µ−1j .

iii) ifKjµiµ

2j

4(µ2j+σ2

j)<Bi ≤

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j,Bj =

2Bi(µ2j+σ

2j )

µiµj,Qj = Bi



2j

)Bi.

iv) if Bi ≤Kjµiµ

2j

4(µ2j+σ2

j),Bj =

Kjµj

2,Qi = 0,Qj =

Kjµj

4σ2j+4µ2

j,E(πj,s) =

K2j

8δ2j+8.

Best-response function of Bi :Similarily, we can obtain the best-response function of Bi for any given Bj as follows.

i) if Bj >2(σ2

j+µ2j )Ki

µj,Bi can be any value in (0,Kiµi],Qj =

Bj

2(µ2j+σ2

j),Qi = 0,E(πi,s) = 0.

ii) if2Kiµ

3jµ

2i+2Kiσ

2jµjµ

2i

8σ2jσ2i+8σ2

jµ2i+8σ2

iµ2j+7µ2

jµ2i<Bj ≤

2(σ2j+µ

2j )Ki

µj,

Bi =1

4

(δ2j + 1

)−1 (Bj + 2Kiµj + 2Kiµjδ

2j

)µ−1j µi


Qj =

(1

4

)(δ2j + 1

)−1 (4δ2j + 4δ2i + 4δ2j δ

2i + 3

)−1µ−2j

(7Bj − 2Kiµj + 8Bjδ

2j + 8Bjδ

2i − 2Kiµjδ

2j + 8Bjδ

2j δ

2i

)Qi =

1

2

(4δ2j + 4δ2i + 4δ2j δ

2i + 3

)−1 (2Kiµj −Bj + 2Kiµjδ

2j

)µ−1j µ−1i .

iii) ifKiµjµ

2i

4(µ2i+σ2

i)<Bj ≤

2Kiµ3jµ

2i+2Kiσ

2jµjµ

2i

8σ2jσ2i+8σ2

jµ2i+8σ2

iµ2j+7µ2

jµ2i,Bi =

2Bj(µ2i+σ

2i )

µjµi,Qi =

Bj

µjµi,E(πi,s) = µ−2j (Kiµj − 2Bj − 2Bjδ

2i )Bj .

iv) if Bj ≤Kiµjµ

2i

4(µ2i+σ2

i),Bi = Kiµi

2,Qj = 0,Qi = Kiµi

4σ2i+4µ2

i,E(πi,s) =

K2i

8δ2i+8.

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 >2(σ2

j+µ2j )Ki

µjand Bi >

2(σ2i +µ

2i )Kj

µi, Bi can be any value in (0,Kiµi] and Bj can be any value

in (0,Kjµj ]. Because Bj >2(σ2

j+µ2j )Ki

µjand Bj ≤ Kjµj , Kjµj >

2(σ2j+µ

2j )Ki

µj, which leads to λ <

µ2j

2(σ2j+µ2

j)

=

12(δ2

j+1)≤ 1

2. However, because Bi >

2(σ2i +µ

2i )Kj

µiand Bi ≤ Kiµi, Kiµi >

2(σ2i +µ

2i )Kj

µi, which leads to λ >

2(σ2i +µ

2i )

µ2i

= 2(δ2i + 1)≥ 2. Contradiction.

2. if2Kiµ

3jµ

2i+2Kiσ

2jµjµ

2i

8σ2jσ2i+8σ2

jµ2i+8σ2

iµ2j+7µ2

jµ2i< Bj ≤

2(σ2j+µ

2j )Ki

µj,

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j< Bi ≤

2(σ2i +µ

2i )Kj

µi, that is,

2+2δ2j8δ2

i+8δ2

j+8δ2

iδ2j+7Kiµj <Bj ≤ 2µjKi

(δ2j + 1

),

2+2δ2i8δ2

i+8δ2

j+8δ2

iδ2j+7µiKj <Bi ≤ 2µiKj (δ2i + 1) , we have

Bi =1

4

(δ2j + 1


2j

)µ−1j µi

Bj =1

4

(δ2i + 1


2i

)µ−1i µj .

Solving these two equations, we obtain

Bi =8Kiµi + 2Kjµi + 8Kiµiδ

2i + 8Kiµiδ

2j + 2Kjµiδ

2i + 8Kiµiδ

2i δ

2j

16δ2i + 16δ2j + 16δ2i δ2j + 15

,

Bj =2Kiµj + 8Kjµj + 2Kiµjδ

2j + 8Kjδ

2i µj + 8Kjµjδ

2j + 8Kjδ

2i µjδ

2j

16δ2i + 16δ2j + 16δ2i δ2j + 15

.

To satisfy2Kiµ

3jµ

2i+2Kiσ

2jµjµ

2i

8σ2jσ2i+8σ2

jµ2i+8σ2

iµ2j+7µ2

jµ2i<Bj ≤

2(σ2j+µ

2j )Ki

µj,

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j<Bi ≤

2(σ2i +µ

2i )Kj

µi, it’s easy to

show that we need2+2δ2i

8δ2i+8δ2

j+8δ2

iδ2j+7

<λ<8δ2i +8δ2j+8δ2i δ

2j+7

2δ2j+2

.

3. ifKiµjµ

2i

4(µ2i+σ2

i)< Bj ≤

2Kiµ3jµ

2i+2Kiσ

2jµjµ

2i

8σ2jσ2i+8σ2

jµ2i+8σ2

iµ2j+7µ2

jµ2i,Kjµiµ

2j

4(µ2j+σ2

j)< Bi ≤

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j,we have Bi =

2Bj(µ2i+σ

2i )

µjµiand Bj =

2Bi(µ2j+σ

2j )

µiµj,which meansBi

Bj=

2(µ2i+σ

2i )

µjµiand Bi

Bj=

µiµj

2(µ2j+σ2

j). However, since

2(µ2i+σ

2i )

µjµi−

µiµj

2(µ2j+σ2

j)

= 12

(σ2j +µ2

j

)−1µ−1i µ−1j

(4σ2

i σ2j + 4σ2

i µ2j + 4σ2

jµ2i + 3µ2

iµ2j

)> 0. This leads to contradiction.

4. if Bj ≤Kiµjµ

2i

4(µ2i+σ2

i),Bi = Kiµi

2,Bi ≤

Kjµiµ2j

4(µ2j+σ2

j),Bj =

Kjµj

2, we have Kiµi

2≤ Kjµiµ

2j

4(µ2j+σ2

j)

andKjµj

2≤ Kiµjµ

2i

4(µ2i+σ2

i), that

is, λ≤ 12(δ2

j+1)

and λ≥ 2(δ2i + 1).Contradiction.

5. if Bj >2(σ2

j+µ2j )Ki

µj= 2(δ2j + 1)Kiµj ,Bi can be any value in (0,Kiµi] and

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j<

Bi ≤2(σ2

i +µ2i )Kj

µi(

2+2δ2i8δ2

i+8δ2

j+8δ2

iδ2j+7Kjµi <Bi ≤ 2µiKj (δ2i + 1)).Since Bj = 1

4(δ2i + 1)

−1(Bi + 2Kjµi + 2Kjµiδ

2i )

µ−1i µj > 2(δ2j + 1)Kiµj ≥ 2(δ2j + 1)Biµj/µi, we get Bi ≥2Kjµi+2Kjµiδ

2i

8δ2i+8δ2

j+8δ2

iδ2j+7

which contradicts with

2+2δ2i8δ2

i+8δ2

j+8δ2

iδ2j+7Kjµi <Bi ≤ 2µiKj (δ2i + 1) .Similarly, we can derive contradiction when Bi > 2(δ2i + 1)Kjµi

and2+2δ2j

8δ2i+8δ2

j+8δ2

iδ2j+7Kiµj <Bj ≤ 2µjKi

(δ2j + 1

).

6. if Bj >2(σ2

j+µ2j )Ki

µjand

Kjµiµ2j

4(µ2j+σ2

j)< Bi ≤

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j, Bi can be any value in (0,Kiµi] and

Bj =2Bi(µ

2j+σ

2j )

µiµj. Since Bj =

2Bi(µ2j+σ

2j )

µiµj>

2(σ2j+µ

2j )Ki

µj,which means Bi > Kiµi. Contradiction. Similarly, we

can derive contradiction when Bi >2(σ2

i +µ2i )Kj

µiand

Kiµjµ2i

4(µ2i+σ2

i)<Bj ≤

2Kiµ3jµ

2i+2Kiσ

2jµjµ

2i

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j.


7. if Bj >2(σ2

j+µ2j )Ki

µjand Bi ≤

Kjµiµ2j

4(µ2j+σ2

j), Bi can be any value in (0,Kiµi] and Bj =

Kjµj

2. To satisfy

Bj >2(σ2

j+µ2j )Ki

µjand Bi ≤

Kjµiµ2j

4(µ2j+σ2

j), we need

Kjµj

2>

2(σ2j+µ

2j )Ki

µj,i.e., λ < 1

4(1+δ2j). Similarly, we can show that

if λ> 4(1 + δ2i ),there is an equilibrium where Bj can be any value in (0,Kjµj ] and Bi = Kiµi

2.

8. if2Kiµ

3jµ

2i+2Kiσ

2jµjµ

2i

8σ2jσ2i+8σ2

jµ2i+8σ2

iµ2j+7µ2

jµ2i<Bj ≤

2(σ2j+µ

2j )Ki

µj,Kjµiµ

2j

4(µ2j+σ2

j)<Bi ≤

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j,

Bi =1

4

(δ2j + 1


2j

)µ−1j µi

Bj =2Biµj(1 + δ2j )

µi

Solving these equations, we obtain Bi = Kiµi,Bj = 2Kiµj + 2Kiµjδ2j . In order to satisfy

2Kiµ3jµ

2i+2Kiσ

2jµjµ

2i

8σ2jσ2i+8σ2

jµ2i+8σ2

iµ2j+7µ2

jµ2i<Bj ≤

2(σ2j+µ

2j )Ki

µj,Kjµiµ

2j

4(µ2j+σ2

j)<Bi ≤

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2j, we need

2+2δ2j8δ2

i+8δ2

j+8δ2

iδ2j+7Kiµj <

2Kiµj + 2Kiµjδ2j and

Kjµi

4(1+δ2j)< Kiµi ≤ 2+2δ2i

8δ2i+8δ2

j+8δ2

iδ2j+7Kjµi. It’s easy to show that the former is satisfied

and the latter is eqivalent to 14(1+δ2

j)<λ≤ 2+2δ2i

8δ2i+8δ2

j+8δ2

iδ2j+7. Similarly, we can show that if

8δ2i +8δ2j+8δ2i δ2j+7

2+2δ2j

≤λ< 4(1 + δ2i ), there is an equilibrium where Bj =Kjµj ,Bi = 2Kjµi + 2Kjµiδ

2i .

9. if2Kiµ

3jµ

2i+2Kiσ

2jµjµ

2i

8σ2jσ2i+8σ2

jµ2i+8σ2

iµ2j+7µ2

jµ2i<Bj ≤

2(σ2j+µ

2j )Ki

µj,Bi ≤

Kjµiµ2j

4(µ2j+σ2

j), we have

Bi =1

4

(δ2j + 1


2j

)µ−1j µi

Bj =Kjµj

2.

Solving these equations, we obtain Bi = 18δ2

j+8

(4Kiµi +Kjµi + 4Kiµiδ

2j

),Bj = 1

2Kjµj . In order to

satisfy2Kiµ

3jµ

2i+2Kiσ

2jµjµ

2i

8σ2jσ2i+8σ2

jµ2i+8σ2

iµ2j+7µ2

jµ2i< Bj ≤

2(σ2j+µ

2j )Ki

µj, we need

2+2δ2j8δ2

i+8δ2

j+8δ2

iδ2j+7Kiµj <

12Kjµj ≤ 2Kiµj +

2Kiµjδ2j , which equals to 1

4(1+δ2j)≤ λ <

8δ2i +8δ2j+8δ2i δ2j+7

4+4δ2i

. In order to satisfy Bi ≤Kjµiµ

2j

4(µ2j+σ2

j), we need

18δ2

j+8


2j

)≤ Kjµi

4(1+δ2j), which is equal to λ≤ 1

4(1+δ2j). Thus, we need λ= 1

4(1+δ2j). Note

that when λ = 14(1+δ2

j),Bi = 1

8δ2j+8


2j

)= Kiµi,Bj =

Kjµj

2= 2Kiµj + 2Kiµjδ

2j . Simi-

larly, we can show that if λ= 4(1 + δ2i ),Bj =Kjµj ,Bi = 2Kjµi(1 + δ2i ).

10. ifKiµjµ

2i

4(µ2i+σ2

i)< Bj ≤

2Kiµ3jµ

2i+2Kiσ

2jµjµ

2i

8σ2jσ2i+8σ2

jµ2i+8σ2

iµ2j+7µ2

jµ2i

and Bi ≤Kjµiµ

2j

4(µ2j+σ2

j),Bi =

2Bj(µ2i+σ

2i )

µjµi,Bj =

Kjµj

2. Because

Bi =2Bj(µ

2i+σ

2i )

µjµi=

2(µ2i+σ

2i )

µjµi

Kjµj

2=Kjµi(1 + δ2i )≥Kjµi, this contradicts with Bi ≤

Kjµi

4(1+δ2j)≤ Kjµi

4. Similarly,

we can derive contradiction ifKjµiµ

2j

4(µ2j+σ2

j)<Bi ≤

2Kjµ3iµ

2j+2Kjσ

2i µiµ

2j

8σ2iσ2j+8σ2

iµ2j+8σ2

jµ2i+7µ2

iµ2jand Bj ≤

Kiµjµ2i

4(µ2i+σ2

i).

Summarizing all the cases discussed above, we derive the Subgame Perfect Nash Equilibriums as shownin Theorem 3.Proof of Theorem 4.

E(πhi )−E(πui ) =


2i − 8Kiδ

2j + 2Kjδ

2i − 8Kiδ

2i δ

2j

)216

(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2(δ2i + 1)

−2(δ2j + 1

) (6δ2i + 6δ2j + 6δ2i δ

2j + 5

) (2Kj − 7Ki− 8Kiδ

2i − 8Kiδ

2j + 2Kjδ

2i − 8Kiδ

2i δ

2j

)2(16δ2i + 16δ2j + 16δ2i δ

2j + 15

)2 (4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2=


2i − 8Kiδ

2j + 2Kjδ

2i − 8Kiδ

2i δ

2j

)2 (128δ2i + 128δ2j + 64δ4i + 64δ4j + 256δ2i δ

2j + 128δ2i δ

4j + 128δ4i δ

2j + 64δ4i δ

4j + 65

)16

(16δ2i + 16δ2j + 16δ2i δ

2j + 15

)2 (4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2(δ2i + 1)

> 0

E(π∗j )−E(πhj ) =

(δ2j + 1

) (Ki− 2Kj − 2Kjδ

2i

)2(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 −(6δ2i + 6δ2j + 6δ2i δ

2j + 5

) (2Kj −Ki + 2Kjδ

2i

)28(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2(δ2i + 1)

=

(2Kj −Ki + 2Kjδ

2i

)2 (2δ2i + 2δ2j + 2δ2i δ

2j + 3

)8(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2(δ2i + 1)

> 0


E(π∗i )−E(πh′i ) =

(δ2i + 1

) (Kj − 2Ki− 2Kiδ

2j

)2(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 −(6δ2i + 6δ2j + 6δ2i δ

2j + 5

) (2Ki−Kj + 2Kiδ

2j

)28(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 (δ2j + 1

)=


2j

)2 (2δ2i + 2δ2j + 2δ2i δ

2j + 3

)8(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 (δ2j + 1

)> 0

E(πh′j )−E(πuj ) =

(2Ki− 7Kj − 8Kjδ

2j − 8Kjδ

2i + 2Kiδ

2j − 8Kjδ

2i δ

2j

)216

(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 (δ2j + 1

)−

2(δ2i + 1

) (6δ2i + 6δ2j + 6δ2i δ

2j + 5

) (2Ki− 7Kj − 8Kjδ

2j − 8Kjδ

2i + 2Kiδ

2j − 8Kjδ

2i δ

2j

)2(16δ2i + 16δ2j + 16δ2i δ

2j + 15

)2 (4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2=


2j + 8Kjδ

2i + 8Kjδ

2j + 8Kjδ

2i δ

2j

)2 (128δ2i + 128δ2j + 64δ4i + 64δ4j + 256δ2i δ

2j + 128δ2i δ

4j + 128δ4i δ

2j + 64δ4i δ

4j + 65

)16

(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)2 (16δ2i + 16δ2j + 16δ2i δ

2j + 15

)2 (δ2j + 1

)> 0.

This implies that a decentralized supply chain is better off by centralization regardless whether the othersupply chain is centralized or not. Hence, centralization is the dominant strategy.

Proof of Lemma 3. We examine the centralization decision for a supply chain in two cases: (i) when theother supply chain is centralized; and (ii) when the other supply chain is decentralized.

Case (i) when the other supply chain is centralized: Without loss of generality, we assume that supplychain j uses a linear contract and supply chain i is centralized. As Theorem 4 suggested, E(πhj ) < E(π∗j ),meaning that the entire supply chain j would be better off if it centralizes. Suppose that under a given linearcontract, retailer j earns a profit of E(πhj,r) and supplier j earns a profit of E(πhj,s). Consider when supplierj and retailer j engage in a revenue-sharing contract with profit split ratio φi = E(πhj,r)/E(πhj ). It is clearthat retailer j is better off since

φiE(π∗j )>φiE(πhj ) =E(πhj,r)

E(πhj )E(πhj ) =E(πhj,r).

and so is supplier j since

(1−φi)E(π∗j )> (1−φi)E(πhj ) =E(πhj,s).

Case (ii) when the other supply chain is decentralized: Without loss of generality, we assume that supplychain j uses a linear contract and supply chain i can choose either to coordinate or stick with a given linearcontract. As Theorem 4 suggested, E(πui ) < E(πhi ). We define φi = E(πui,r)/E(πui ) and let φi be the profitsplit ratio of a revenue-sharing contract, we see that both the retailer and supplier are better off under thiscontract.

In conclusion, there exists a coordinating contract that achieves Pareto improvement in a supply chainregardless whether the other supply chain is centralized or not.Proof of Theorem 5. Based on the results in Theorems 1, 2 and 3, we find

µiQ∗i +µjQ

∗j =

2Ki−Kj + 2Kiδ2j

(4δ2i + 4δ2j + 4δ2i δ2j + 3)

+2Kj −Ki + 2Kjδ

2i

(4δ2i + 4δ2j + 4δ2i δ2j + 3)

=Ki +Kj + 2Kiδ

2j + 2Kjδ

2i

4δ2i + 4δ2j + 4δ2i δ2j + 3

,

µiQh∗i +µjQ

h∗j =

7Ki− 2Kj + 8Kiδ2i + 8Kiδ

2j − 2Kjδ

2i + 8Kiδ

2i δ

2j

4(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)(δ2i + 1)

+2Kj −Ki + 2Kjδ

2i

2(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)=

5Ki + 2Kj + 6Kiδ2i + 8Kiδ

2j + 6Kjδ

2i + 4Kjδ

4i + 8Kiδ

2i δ

2j

4(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)(δ2i + 1)

,

µiQu∗i +µjQ

u∗j =

2(δ2j + 1

) (7Ki− 2Kj + 8Kiδ

2i + 8Kiδ

2j − 2Kjδ

2i + 8Kiδ

2i δ

2j

)(16δ2i + 16δ2j + 16δ2i δ

2j + 15

) (4δ2i + 4δ2j + 4δ2i δ

2j + 3

)+

2 (δ2i + 1)(7Kj − 2Ki− 2Kiδ

2j + 8Kjδ

2i + 8Kjδ

2j + 8Kjδ

2i δ

2j

)(16δ2i + 16δ2j + 16δ2i δ

2j + 15

) (4δ2i + 4δ2j + 4δ2i δ

2j + 3

) .


It can be verified that

µiQ∗i +µjQ

∗j − (µiQ

h∗i +µjQ

h∗j ) =

(2Kj −Ki + 2Kjδ2i ) (2δ2i + 1)

4(4δ2i + 4δ2j + 4δ2i δ

2j + 3

)(δ2i + 1)

> 0,

µiQh∗i +µjQ

h∗j −(µiQ

u∗i +µjQ

u∗j ) =

(7Ki− 2Kj + 8Kiδ2i + 8Kiδ

2j − 2Kjδ

2i + 8Kiδ

2i δ

2j )

(6δ2i + 8δ2j + 8δ2i δ

2j + 5

)4(16δ2i + 16δ2j + 16δ2i δ

2j + 15

) (4δ2i + 4δ2j + 4δ2i δ

2j + 3

)(δ2i + 1)

> 0

Hence, µiQ∗i +µjQ

∗j >µiQ

h∗i +µjQ

h∗j >µiQ

u∗j +µjQ

u∗j .

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