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Strategic Risk from Supply Chain Disruptions

Wallace J. HoppStephen M. Ross School of Business, University of Michigan, Ann Arbor, MI

Seyed M. R. Iravani and Zigeng LiuDepartment of Industrial Engineering and Management Science,

Northwestern University, Evanston, IL

Abstract

Supply chain disruptions can lead to both tactical (loss of short term sales) and strategic (loss of longterm market share) consequences. In this paper, we model the impact of regional supply disruptionson competing supply chains. We describe generic strategies that consist of two stages: (i) preparation,which involves investment prior to a disruption in dedicated backup capacity and/or measures thatfacilitate quick detection of a problem, and (ii) response, which involves post-disruption purchase ofshared (non-dedicated) backup capacity for a component whose availability has been compromised.Using expected loss of profit due to lack of preparedness as a measure of risk, we characterize theconditions that pose the greatest risk and suggest ways to reduce the risk exposure. This analysisreveals that a dominant firm in the 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 gainmarket share. We distinguish between preparedness activities related to monitoring and detectionas systemic preparation, which affects many components simultaneously, and preparedness activitiesrelated to securing extra dedicated backup capacity as targeted preparation, which affects only a specificcomponent. We characterize conditions that affect the optimal ratio of investment in these two typesof preparation and conclude that larger firms should use targeted preparation defensively, while smallerfirms should use it offensively. Finally, we discuss how firms can use both long term information (e.g.,estimates of the likelihood of disruptions) and real-time information (e.g., severity of disruption afterit occurs) as part of their preparedness strategy.

Keywords: supply chain disruption, risk, preparedness

———————————————————————————————————————————————

1 Introduction

High profile incidents of global terrorism have elevated concerns about risks in all aspects of private

and public life, including management of business operations. However, while terrorist activities are

particularly newsworthy, other sources of risk, such as natural disasters and political/economic shocks,

present a more widespread challenge to managers. Regardless of the source, it is clear that preparation

for and management of major disruptions is an important part of modern operations management.

In this paper, we focus on the risk of supply disruption, which is inherent in global supply chains.

We divide this risk into two categories: (1) tactical risk, which characterizes events that result in only a

short term loss of sales revenue, and (2) strategic risk, which is associated with events that result in a

long term loss of market share, in addition to a loss of sales.

An example of a strategic risk event occurred on March 17, 2000, when a ten-minute fire at a Royal

Philips Electronics semiconductor plant in Albuquerque, New Mexico, “touched off a corporate crisis

that shifted the balance of power between two of Europe’s biggest electronics companies...” (Wall Street

1

2Hopp et al. Strategic Risk from Supply Chain Disruptions

Article Submitted to Management Science

19981997 1999 2000 2001 2002 2003 2004

Market Share

10%

30%

20% MOTOROLA

SAMSUNG

NOKIA

End of OutageFire at Philips’ Plant

Year ERICSSON

Figure 1: Global Mobile Market Share from 1997 to 2004 (Gartner (2006)).

Journal, January 29, 2001). This occurred because, besides directly destroying several thousand chips for

mobile phones, the fire contaminated the clean room environment in the semiconductor plant, effectively

shutting it down for weeks. At the time, both Nokia and Ericsson were sourcing microchips from the

Philips plant. However, while Nokia was able to quickly shift production to other Philips plants and

some Japanese and American suppliers, Ericsson was trapped by its sole source dependence on the

Philips plant. Consequently, Ericsson had no way to make a rapid response to the disruption, and wound

up losing around $400 million in sales by the end of the first disruption-impacted quarter (Elgin (2003)).

Worse, six months after the fire, Ericsson’s market share of the global handset market had fallen by

3%, its stock price had decreased by 12%, and Ericsson’s mobile phone division reported a $2.34 billion

loss for 2000 (Sheffi (2005)). This setback almost certainly accelerated Ericsson’s decline in the handset

market (see Figure 1). In combination with a variety of other problems, this led Ericsson to merge its

mobile phone unit with Sony in 2001 (Kharif (2001), Sheffi (2005)).

Such dramatic consequences from a seemingly minor disruption seem to be the rule rather than

the exception. Hendricks and Singhal (2005) considered a sample of 827 disruptions (ranging from a

shipment delay to a key part shortage) announced between 1989 and 2000 and examined stock prices

over a three-year period beginning one-year prior to the announcement of a disruption and ending two

years after it. They found that firms who reported supply chain disruptions had stock returns over the

three-year period that were nearly 40 percent lower than comparable firms that did not report disruptions.

Evidently, supply chain disruptions are major business events that have lasting effects.

However, not all consequences of supply chain disruptions are negative. At the end of 2001, Nokia

and Samsung reported large increases in market share, while Motorola stopped a steady decline in market

Hopp et al. Strategic Risk from Supply Chain DisruptionsArticle Submitted to Management Science 3

share (see Figure 1). Apparently, Ericsson’s loss was these other firms’ gain, since some customers who

were unsupplied by Ericsson shifted their purchases to other brands. Evidently, supply chain disruptions

can pose opportunities for strategic gain, as well as loss.

To understand why the Philips disruption had such an unbalanced impact on Nokia and Ericsson, we

must go back to the year 1995. At that time, due to poor delivery performance and a relatively weak

product line, Nokia was experiencing mismatches between supply and demand. To remedy the situation,

Nokia installed a monitoring process, which enabled it to spot the disruption quickly (even before Philips

officially notified it of the problem) and to initiate a dialogue with Philips about alternate supplies. Once

Nokia’s component-purchasing manager knew about the fire, he quickly reported it to his upper-level

managers, in keeping with Nokia’s culture of encouraging dissemination of bad news. Nokia decided to

scrutinize the situation and call Philips daily to monitor the event. When Nokia realized the shortage due

to the fire would last for more than month, it assembled a task force to investigate all possible alternative

sources of supply (Sheffi (2005)). This included capacity of other Philips facilities, as well as that of other

suppliers. The ability to use non-Philips chips was facilitated by an earlier decision by Nokia to redesign

its phones to accommodate a wider range of chips that could be supplied by various suppliers in Japan

and the United States. In contrast, Ericsson did not realize the severity of the disruption until Nokia

had locked up all available supplies of the original chips and Ericsson did not have product flexibility to

allow use of other more widely available chips.

A major reason that a small-scale disruption resulted in a huge loss for Ericsson is that it resulted

in the loss of unsatisfied customers (Stauffer (2003)). In hindsight, it is clear that Ericsson grossly

underestimated the impact of what it thought would be a one-week delay of its chip supply. When

Philips called two weeks after the fire to explain that the delay would be much longer, Ericsson realized

too late that it would have thousands of disappointed customers, some of whom would not return. In

recognition of this type of customer dynamics, Zsidisin et al. (2000) concluded that firms implementing

high efficiency supply management techniques (e.g., single sourcing, just-in-time deliveries for production)

must be aware that such practices may increase their exposure to risk. Stauffer (2003) noted that it is

no longer feasible to try to cover losses from supply chain disruptions through insurance, since the 9/11

terrorist attack and other major events have altered the policies of insurance companies to limit coverage.

Although there has been a great deal of research devoted to the design, coordination and improvement

of supply chains (see, e.g., Fisher et al. (1997), Fine (2000), Lee (2003), Graves and Willems (2003), Gan

et al. (2005)), relatively little of this has focused on supply chain risks. Much of what has been done is

descriptive in nature. For example, Johnson (2001) divided supply chain risks into two categories: (a)



demand risks, including seasonal imbalances, fab volatility, and new product adoption, and (b) supply

risks, such as manufacturing and logistics capacity limitations, currency fluctuations, and supply disrup-

tions from political issues. Chopra and Sodhi (2004) further refined these into a taxonomy of risks faced

in supply chains and qualitatively discussed the different strategies for mitigating them. Kleindorfer and

Saad (2005) discussed the implications of designing supply chain systems to deal with disruption risks,

based on a conceptual framework, which they termed the “SAM-SAC” Framework, and generated several

empirical results from a data set of the U.S. Chemical Industry. Christopher and Peck (2004) also dis-

cussed how to design resilient supply chains in qualitative terms, with an emphasis on fully recognizing

the nature of supply chain risks. In this same vein, Christopher and Lee (2004) suggested that improved

end-to-end visibility is a key element for mitigating supply chain risk.

Some researchers have considered supply chain disruptions in multi-echelon settings. Snyder and

Shen (2006) investigated both supply uncertainty and demand uncertainty in simple multi-echelon supply

chains using simulation. They pointed out that these two types of uncertainties require opposite responses

and therefore firms should consider both types simultaneously to identify an optimal strategy. Hopp and

Liu (2006) developed an analytical model for striking a balance between the costs of inventory and/or

capacity protection and the costs of lost sales in an arborescent assembly network subject to disruption.

They showed that, under certain conditions, it is optimal locate inventory or capacity protection at no

more than one node along each path to the customer. In a related vein, Tomlin (2006) considered a

firm that has two suppliers: one inexpensive and unreliable and the other expensive and reliable with

volume flexibility. They described management strategies for using the two types of supplier to mitigate

disruptions under various environmental assumptions. Tomlin and Snyder (2006) further considered

disruption mitigation strategies for the situation in which disruptions are stochastic and the firm gets

some advance warning of the disruptions. Tomlin and Tang (2008) studied five stylized models (which

correspond to five types of flexible strategies) and suggested that flexibility can be used as a powerful

defensive protection mechanism to mitigate supply chain risks.

While the above research is beginning to reveal principles of supply chain risk mitigation, to our

knowledge, there is not yet a comprehensive modeling framework with which to evaluate both tactical

and strategic risks to a firm from supply disruptions. In this paper, we develop such a framework and use

it to analyze both how firms in competitive environments should prepare for potential disruptions and

how they should respond to disruptions once they occur. In some cases, this analysis gives quantitative

confirmation of intuitive behavior. In others, it reveals insights that are not otherwise apparent.

The remainder of this paper is structured as follows: In Section 2, we introduce a duopoly model


with a third party, in which two firms compete for a common backup supply after a disruption of the

primary supply of a key component, but the third party, which produces a competitive product, is not

affected by the disruption. Note that this matches the Philips-Nokia-Ericsson scenario, in which Nokia

and Ericsson are the duopoly and firms like Samsung and Motorola collectively constitute the outside

firm. Section 2.1 analyzes the Backup Capacity Competition (BC Competition) within this model in order

to characterize the optimal plan of action to be followed by firms after a disruption. In Section 2.2 we

study the Advanced Preparedness Competition (AP Competition), in which firms compete by investing

in strategies for mitigating supply chain risks in advance of a disruption. In Section 2.3 we perform a

sensitivity analysis of factors that influence the AP Competition. In Section 3 we investigate the factors

that influence the degree of strategic risk faced by firms and develop insights into preparation strategies.

Section 4 summarizes our conclusions and points to future research directions.

2 Model Formulation

To create a framework within which to evaluate the strategic impact of a supply disruption, we begin

by considering a market consisting of two firms, A and B, that sell competing products. Both products

require a key component that is purchased from a common supplier or region that is subject to disruption.

Because the key components is essential, a disruption in supply will prevent production. In addition to

the two firms affected by the disruption, there is a third firm C, which offers a substitute for the products

offered by firms A and B. However, because the third firm C’s product is sufficiently different from those

of firms A and B to allow it to rely on different components, it is not affected by the supply disruptions

under consideration. Because of this, a disruption presents an opportunity for Firm C to steal customers

from Firms A and B, but not for Firms A and B to steal customers from Firm C.1

We model the disruption as a random event that completely stops the supply of the disrupted com-

ponent and consists of two periods: (i) a fixed lead time T between the time of the disruption and the

time when the backup capacity is able to begin supplying the key component, and (ii) a random duration

of time after the lead time that extends to the end of the outage, which we assume follows a cumulative

probability distribution F (·) with an average of µ units of time (see Figure 2).

To create a model, we suppose Firm i (i ∈ A,B) has capacity to produce Ki products per unit

time and sells its products at a profit margin of ri per product. Each product requires one unit of the

key component. Before a disruption, demand for Firm i’s product is d0i

per unit time. To allow us to

1Note that if firms have independent suppliers, a disruption will affect only one of the competing firms. In our framework,this can be represented by a model with Firms A and C, but no Firm B. As such, it represents a simpler case than the onewe consider in this paper.



TimeOutage

EndsBackup CapacityComes On Line

Outage Occurs

(with mean )µ Remaining Outage Time

(T) Fixed Lead Time

Pre-Disruption Period Outage (Post-Disruption Period)

Figure 2: Time sequence of events after a disruption.

consider cases where the firm purchases more backup capacity for the key component than it actually

needs, we define the “holding cost” for one unit unused backup capacity for Firm i be hi per unit time.

We assume that, during the outage, customers of Firm i who are unable to purchase from Firm i will

buy the product from the other firm, Firm j, as long as Firm j can provide substitute products. If there

is no alternate supply with which to replace the disrupted parts, then neither firm will be able to meet

demand. We further assume that, the third party, Firm C, offers a less-than-perfect substitute for the

products offered by Firms A and B, which customers may turn to if neither Firm A nor Firm B have

supply available.

We focus primarily on the situation where a backup supplier exists that could provide an alternate

supply of the disrupted parts. However, if the backup capacity is limited, as we would expect it to be,

the question of how it is allocated is critical. Several factors could influence which firm (A or B) has an

advantage in securing the backup capacity. One firm might have a prior relationship with the backup

supplier that gives them an edge. Or a firm’s size or reputation might make them a more desirable

customer. Or, as seems to have been the case in the competition between Nokia and Ericsson, it might

simply be a matter of who monitors the situation more carefully and therefore asks first. Our framework

will accommodate any of these, but we will focus on the first-come-first-serve rule as the most interesting

scenario.

We assume a contractual business-to-business (OEM) relationship between the firms and their cus-

tomers. That is, the firms have contracts with their existing customers and are therefore obligated to fill

orders from these customers before filling orders from new customers. In contrast, in a retail business-to-

consumer environment, firms often cannot control which customers—existing or new—have their orders

filled first. (For example, Kellogg cannot ensure that their corn flakes go to loyal Kellogg customers, as

opposed to Post customers who are unable to buy their usual brand.) Under the business-to-business

conditions, a firm will supply its own customers first. But if the firm is able to secure a backup capacity of

the disrupted parts beyond those needed for its own customers (and it has ample supplies of other parts

and the necessary production capacity), then it could meet demand from the competitor’s customers. We


assume that the firm will take advantage of the situation to do this if the competitor firm is unable to

satisfy its own customers during the outage. If one firm makes sales to the other firm’s customers, then

the second firm will lose revenue in the short term. Furthermore, if some of these customers shift their

future sales to the new firm, then the second firm will also lose long term market share.

Our assumption that, during the outage, customers of a firm unable to meet demand are willing to

switch to another firm, is motivated by the fact that these customers are themselves firms. If the customer

firms make use of lean practices, they will have limited supplies of components on hand. To avoid shutting

down production, they must seek alternate supplies almost immediately in the face of a disruption. This is

what happened when cellular services companies unable to obtain Ericsson handsets were quick to satisfy

their customers with alternate brands. However, whether a customer will switch permanently depends

on his/her brand loyalty. We model brand loyalty of Firm i’s (i ∈ A,B) customer by the length of time

that customers of Firm i wait during the disruption before they permanently switch to the other firm’s

product. Specifically, we assume that a customer of Firm i who uses Firm j’s (j 6= i, j ∈ A,B, C)product during an outage, waits for an exponentially distributed amount of time with an average of γij

before permanently switching to Firm j’s product, and hence we refer to γij as Firm i’s customer loyalty

relative to Firm j. We define mij as the (finite) net present value of the long term market share that

Firm j gains from one customer who permanently switches from Firm i to Firm j, where j 6= i; define

mii as the (finite) net present value of the market share of Firm i to Firm i.

In situations where the first firm (either A or B) to detect the disruption and contact the backup

supplier will have the option to buy as much of the backup capacity as it chooses, there is incentive

to invest in early detection capabilities. Indeed, this is precisely what Nokia did when it installed its

new monitoring system. Hence, we include a preparedness period, in which firms invest in monitoring

technology, research potential backup suppliers and take other measures to enable them to be first to

secure the backup capacity in the event of a disruption. Specifically, we consider the following two periods:

• Pre-Disruption Period: Before a disruption occurs, both Firm A and Firm B serve their own

customers. Each firm estimates the likelihood of a disruption of the key component, which may differ

from the true likelihood of the disruption (p0). After a disruption occurs, firms will seek available

backup capacity for the key component. There are two types of backup capacity: (a) dedicated

capacity that is committed to a specific firm (e.g., due to a contractual or informal relationship),

and (b) shared capacity, for which the two firms compete, if the dedicated backup capacity is

not enough to cover the shortage. To model this competition, we denote the amount of Firm i’s

sales (i ∈ A,B) that cannot be covered by the dedicated capacity as d0i, and the shared backup



capacity by S. Both firms know that if the disruption occurs, the firm that detects the disruption

and moves first has the advantage of securing some or all of the total available backup capacity

for the key component. Therefore, each firm must decide how to invest in preparedness in order to

be able to detect the disruption first. This results in an Advanced Preparedness Competition (AP

Competition), in which the two firms invest in preparedness and thereby determine their respective

probabilities of being first to detect the problem in the event of a disruption.

• Post-Disruption Period: After the disruption occurs, the firm (either A or B) that detects the

disruption must decide how much of the total available backup capacity of S to buy in order to

maximize its short- and long term profits. This results in a Backup Capacity Competition (BC

Competition), in which one of the firms (i.e., the winner of the AP competition) is designated as the

“Leader” and gets first chance at purchasing the backup capacity. After the first firm has made its

purchase, the other firm, the “Follower”, then decides how much of the leftover backup capacity to

purchase. Once both firms have made their purchase amount decisions, there is a lead time period

required to bring the backup capacity on-line. During this period, both Firm A and Firm B lose

all of their short term sales, as well as some long term market share, to Firm C. The amount of the

market share of Firm i (i ∈ A,B) that is lost to Firm C during the lead time is d0i− di , where

di is the demand for Firm i’s product at the end of the lead time. Since the profit loss during

the lead time is unavoidable for both firms, we can treat the moment immediately after the lead

time as the beginning of the post-disruption period. Furthermore, we assume that, after the lead

time, if the available backup capacity is not enough to cover the total market, then all unsatisfied

customers will buy substitute products from Firm C during the outage. The number of customers

that permanently switch to Firm C, depends on the customer brand loyalty.

It is clear that if ample backup capacity of the key component existed (i.e., S is infinite), then there

would be no AP Competition or BC Competition between Firm A and Firm B, because there would be

no shortage to allow an opportunity to “poach” customers from the other firm. Hence, Firm i will only

purchase di units of the capacity per unit time to satisfy its own customers (i, j ∈ A,B). However,

when the backup capacity is limited, then both the AP Competition and the BC Competition have

an important impact on firms’ short- and long-term profits and market shares. We analyze these two

competitions (games) in reverse order, using profit maximization for the objective in the BC Competition

and a Nash equilibrium to define the outcome in the AP Competition.


2.1 Backup Capacity Competition (BC Competition)

In this section we consider the beginning of the post-disruption period when the winner of the AP

competition, which we call the Leader, has detected the disruption before the other firm, which we call

the Follower. The question that the Leader faces is how much of the shared backup capacity it should

buy, while the Follower can only purchase from whatever backup capacity is left after the Leader has

made its purchase. Both firms will seek to maximize profit from both short term sales and long term

sales. Note that it may be optimal for the Leader to purchase more capacity than it needs for its own

customers and to poach those of the Followers.

One factor that each firm must consider when making its purchase of backup capacity is its production

capacity, which corresponds to the maximum number of products it can produce given ample supplies of

raw materials. If, for instance, Firm A has only enough production capacity to meet demand from its

existing customers, then it cannot make sales to the customers of Firm B, regardless of how much backup

capacity it purchases.

We define Yi,k as the amount of backup capacity that Firm i (i ∈ A,B) buys, when Firm i is in

position k (k ∈ L,F, where L = Leader, F = Follower). Let ci (i ∈ A,B) be the premium cost of

one unit of the backup capacity for Firm i, and consider Πi,k (for i ∈ A,B and k ∈ L,F) as the sum

of the total expected profit of Firm i, given it is in position k during the outage. The total expected

profit includes both the short term sales profit during the outage and also the long term sales profit due

to market share gained from the competitor. Without loss of generality, suppose Firm A is the Leader

and Firm B is the Follower. Then we have:

ΠA,L(YA,L , YB,F ) = rA(Average number sold to A’s customers during the outage)

+rA(Average number sold to B’s customers during the outage)

+mBA(Average number of customers (i.e., market share) gained from B)

−mAB(Average number of customers (i.e., market share) lost to B)

−mAC (Average number of customers (i.e., market share) lost to C)

−cA(Average backup capacity bought during the outage)

−hA(Average “inventory” of unused backup capacity during the outage).

Since it is clear that the Follower will not purchase backup capacity in excess of its own production



capacity (because it cannot use it), we can express the expected profit of the Leader as:

ΠA,L(YA,L , YB,F ) = rA

(∫ ∞

0min

YA,L , dA

tdF0(t)

)

+rA

(∫ ∞

0min

[dB − YB,F ]+,min

[YA,L − dA ]+,KA − dA

tdF0(t)

)

+mBA

( ∫ ∞

0min

[dB − YB,F ]+, min

[YA,L − dA ]+,KA − dA

× (1− e− t

γBA )dF0(t)

)

−mAB

( ∫ ∞

0min

[YB,F − dB ]+, [dA − YA,L ]+

(1− e

− tγAB )dF0(t)

)

−mAC

(∫ ∞

0

[[dA − YA,L ]+ − [YB,F − dB ]+

]+(1− e

− tγAC )dF0(t)

)

−cA

(∫ ∞

0YA,LtdF0(t)

)

−hA

(∫ ∞

0

[YA,L −minYA,L , dA −min

[dB − YB,F ]+,

min[YA,L − dA ]+,KA − dA]+

tdF0(t)).

After some algebra, we can reduce this to:

ΠA,L(YA,L , YB,F ) = rA min

YA,L , dA

µ

+ψA min

[dB − YB,F ]+, min[YA,L − dA ]+, KA − dA

−mABξAB min

[YB,F − dB ]+, [dA − YA,L ]+

−mAC ξAC

[[dA − YA,L ]+ − [YB,F − dB ]+

]+

−cAYA,Lµ

−hA

[YA,L −minYA,L , dA −

min

[dB − YB,F ]+, min[YA,L − dA ]+,KA − dA

]+µ, (1)

where ψi = riµ + mjiξji, and

ξij =∫ ∞

0(1− e

− tγij )dFo(t) : j 6= i, i ∈ A,B, j ∈ A,B, C

is defined as the probability that a customer of Firm i permanently switches to Firm j after the outage.

Since Firm B (the Follower) knows its demand, and cannot obstruct Firm A, it has no incentive to

hold inventory. Therefore, we can write its profit function as:

ΠB,F (YA,L , YB,F ) = rB min

YB,F , dB

µ + ψB min

[dA − YA,L ]+, [YB,F − dB ]+

−mBAξBA min

min[YA,L − dA ]+,KA − dA

, [dB − YB,F ]+

−mBC ξBC

[[dB − YB,F ]+ −min

[YA,L − dA ]+, KA − dA

]+− cBYB,F µ. (2)


Hence, for a given YB,F , where 0 ≤ YB,F ≤ minS, KB, the problem of finding the optimal backup

capacity purchase for Firm A (the Leader) reduces to the following optimization problem:

Problem PA: max∀ Y

A,L

ΠA,L(YA,L , YB,F )

Subject to:

0 ≤ YA,L ≤ S

The optimal backup capacity purchase for the Follower (Firm B) can also be obtained for a given YA,L ,

where 0 ≤ YA,L ≤ minS, KA, from the optimization problem as follows:

Problem PB: max∀ Y

B,F

ΠB,F (YA,L , YB,F )

Subject to:

0 ≤ YB,F ≤ maxS − YA,L , 0

Proposition 1 shows that the optimal backup capacity purchases for both firms are restricted to a

small number of possible values, and shows how these values depend on the Leader’s production capacity

(KA) and the amount of available backup capacity (S). The proof of Proposition 1 and all other analytical

results can be found in the Online Appendix.

PROPOSITION 1:

(Large Capacity Leader:) If the Leader’s production capacity is greater than the combined market

share of both firms (i.e., KA ≥ dA + dB), and

(1) if the backup capacity is less then the combined market share of both firms (i.e., S ≤ dA + dB), then

the only possible choices for the optimal backup capacity purchases for the two firms are:

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB ),(0,minKB , S), (S − dB , 0), (S − dB , dB),

(dA , 0), (dA , S − dA), (S, 0)

;

(2) if the backup capacity is larger than the combined market share of both firms (i.e., S > dA + dB),

then the only possible choices for the optimal backup capacity purchases for the two firms are:

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB ),(0, minKB , dA + dB

), (dA , 0), (dA , dB ),

(dA + dB , 0), (S, 0)

.

(Small Capacity Leader:) If the Leader’s production capacity cannot cover the combined market share

of both firms (i.e., KA < dA + dB), and



(1) if the backup capacity is less than the Leader’s market share (i.e., S ≤ KA), then the only possible

choices for the optimal backup capacity purchases for the two firms are:

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB ), (0, minKB , S), (S − dB , 0), (S − dB , dB ),

(dA , 0), (dA , S − dA), (S, 0)

;

(2) if the backup capacity is larger than the Leader’s market share, but it is smaller than the combined

market share of both firms (i.e., KA < S ≤ dA + dB), then the only possible choices for the optimal

backup capacity purchases for the two firms are:

(Y ∗A,L

, Y ∗B,F

) ∈


(dA , 0), (dA , S − dA), (KA , 0), (KA , S −KA)

;

(3) if the backup capacity can cover the combined market share of both firms (i.e., S > dA + dB), then

the only possible choices for the optimal backup capacity purchases for the two firms are:

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB ), (0, minKB , dA + dB), (dA , 0), (dA , dB), (KA , 0),(S − (dA + dB ) + KA , 0

),

(S − (dA + dB) + KA , (dA + dB )−KA

).

Proposition 1 presents a set of solutions that dominates all feasible solutions of the BC Competition,

and therefore includes all candidates for the optimal solution. Using the results of this proposition, we

characterize the structure of the optimal solution of the BC Competition in Proposition 2.

PROPOSITION 2: The optimal policy of the Backup Capacity Competition has a five-region structure.

The thresholds that describe the five regions of the optimal policies for both the Leader and the

Follower are presented in Table 8 and Table 9 in Online Appendix B. Based on the system parameters,

the optimal policy for the BC Competition results in one of the following five scenarios: (i) Firms A and

B Forfeit, (ii) Firm A is aggressive, (iii) Firm A protects, (iv) Firm A forfeits to Firm B, and (v) Firm

B forfeits to Firm A.

To further describe the optimal policy, we present an example in Figure 3, in which maxdA , dB ≤S ≤ KA < dA + dB . As Figure 3 shows, the optimal solution results in five different scenarios:

(i) Firms A and B Forfeit: When rA ≤ cIA

and rB ≤ c1B, the optimal strategies for the Leader and

Follower are (Y ∗A,L

, Y ∗B,F

) = (0, 0).2 This scenario presents situations in which neither Firm A nor Firm B is

willing to buy any backup capacity due to its high premium cost. Consequently, all unsatisfied customers

2Note that the thresholds, cIA

and c1B

, do not depend on the result of the BC Competition, but only on the firms’ systemparameters. Furthermore, cI

Aand c1

Bare increasing functions of cA and cB , respectively, (see Table 9 in Online Appendix

B).


rAA

( 0 , minS, K )B

A Forefeits

A is Aggressive

( S , 0 )

( S , 0 )

B Forefeits

( S - d , 0 )B

A Protects

B B

( d , 0 )

( S - d , d ) ( d , S - d )AA

A

A & B Forefeit

( 0 , 0 )

B

B

c

c

cA

cIAc

rB

II

IV II

1

Figure 3: Five-region structure for optimal BC Competition strategy when (maxdA, d

B ≤)S ≤ K

A< d

A+d

B.

will buy from Firm C, and depending on their customer loyalty, some of them may permanently switch

to Firm C after the outage.

(ii) Firm A is Aggressive: When rA > cIIA

and rB > cIIB

, the optimal strategies for the Leader and

Follower are (Y ∗A,L

, Y ∗B,F

) =(minS, KA, [S −KA ]+

), where the Leader (Firm A) buys the minimum

of the entire backup capacity and its full production capacity. The reason is one of the following: (1)

it is profitable for Firm A to satisfy its own customers and Firm B’s customers using backup capacity,

or (2) although it is not profitable to satisfy Firm A’s customers due to the high premium cost, it is

still profitable for Firm A to satisfy Firm B’s customers. (Recall that, we are considering a business-to-

business environment, in which Firm A must satisfy all its own customers before it satisfies any of Firm

B’s customers.) Thus, under either of these two cases, Firm A buys either the entire backup capacity

or an amount equivalent to its full production capacity. Notice that, although there may be still some

remaining backup capacity for Firm B to purchase, it is guaranteed that the unsatisfied customers of

Firm B will be enough for Firm A to use up all of its production capacity.

(iii) Firm A Protects: When the premium cost of the backup capacity is high, it is not profitable

in the short term for Firm A to serve its customers using the backup capacity; however, the firm may

need to protect itself in the long term from losing customers to Firms B and C. There are two types of

protection strategies for Firm A (the Leader):

• (Y ∗A,L

= S − dB ), which occurs when Firm A mainly focuses on protecting itself from losing market

share to Firms B. To prevent Firm B from poaching Firm A’s customers, Firm A purchases an

amount that leaves only dB units of backup capacity for Firm B to satisfy its own customers. Note



that this case occurs when Firm A’s customer loyalty relative to Firm C is high, so that even if Firm

A cannot satisfy its customers during the outage, it will lose relatively few unsatisfied customers to

Firm C.

• (Y ∗A,L

= dA), which occurs when Firm A protects itself from losing market share to both Firms B

and Firm C. In contrast to the above case, this case occurs when Firm A’s customer loyalty relative

to Firm C is low. Thus, compared to the high premium cost of the backup capacity, it is economical

for Firm A to buy the backup capacity and prevent Firm C from poaching its customers.

(iv) Firm A Forfeits: In this case Y ∗A,F

= 0 (but Y ∗B,F

6= 0), which corresponds to the situation where the

premium cost of the backup capacity is too expensive for Firm A to purchase for any purpose; therefore,

Firm A does not buy any capacity.

(v) Firm B Forfeits: This corresponds to (Y ∗A,F

, Y ∗B,F

) =(minS, dA + dB , KA, 0

). In this case, the

premium cost of the backup capacity is too expensive for Firm B to use it to satisfy its own customers.

Therefore, Firm B does not buy any capacity.

2.2 Advanced Preparedness Competition (AP Competition)

In the previous section, we assumed the allocation of backup capacity is entirely determined by which firm

approaches the backup supplier first. But we recognize that supplier behavior may be influenced by other

factors, such as: (1) firm size: the largest firm may receive the highest priority; (2) business history: the

firm perceived by the supplier as a better customer may receive higher priority. (3) contractual agreements:

a firm may contract with a supplier to provide backup capacity if needed (effectively converting shared

backup capacity to dedicated backup capacity). In practice, the outcome may well be determined by a

combination of these factors, as well as the firms’ preparedness, which affects the speed with which they

recognize the situation and approach the shared backup supplier.

While a firm can take advantage of the first two factors; it cannot easily influence them. A firm

can invest in dedicated backup capacity to influence the third factor, so we will examine when this is

attractive later in the paper. Here we focus on the firms’ preparedness, which seems to have been the

dominant factor in the competition between Nokia and Ericsson. We consider a case in which the firm

can affect the outcome by making more effort to carefully monitor the situation and detect the disruption

first. This results in the Advanced Preparedness Competition (AP Competition).

Suppose xi (i ∈ A,B) is the effort that Firm i spends in preparedness activities such as monitoring

and detection. We assume that, πi, the probability that Firm i detects the disruption first (and therefore


becomes the Leader in the BC Competition) is proportional to the preparedness effort of Firm i as a

fraction of total preparedness effort. Specifically, we assume:

πi =xi

xi + xj: j 6= i, and i, j ∈ A,B,

and πj = 1− πi, i 6= j.

Recall that we defined p0 as the true likelihood of a disruption. However, it is often the case that

firms do not know p0 and so can only estimate it. Hence, we define:

• pi0 is Firm i’s estimate of the likelihood of a disruption;

• pj0,i is Firm i’s belief of Firm j’s estimate of the likelihood of a disruption.

In the following, we define the AP Competition for Firm i (i ∈ A,B) as the game in which Firm i

plays the AP Competition based on its estimate of the likelihood of a disruption (i.e., pi0), its belief of

Firm j’s estimate of the likelihood of a disruption (i.e., pj0,i), and its belief that Firm j knows Firm i’s

estimate of the disruption probability is pi0 .

We let Ce denote the cost per unit of effort spent on preparedness activity, and ΠAPi is defined as

Firm i’s total expected profit in the AP Competition for Firm i, which is given by:

ΠAPi = pi0

(πiΠi,L(Y ∗

i,L, Y ∗j,F )+πjΠi,F (Y ∗

i,F , Y ∗j,L)−miC(d0

i −di))

+(1−pi0)(rid

0i (µ+T )

)−Cexi +miid

0i ,

where rid0i (µ+T ) is the total regular sales profit of Firm i during the (µ+T )-day interval if no disruption

occurs and miid0i is Firm i’s discounted future sales profit.

At the same time, Firm i believes that Firm j uses pj0,i as its estimate for the likelihood of a disruption,

and hence, believes that Firm j is maximizing the following:

ΠAPj,i = pj0,i

(πjΠj,L(Y ∗

j,L, Y ∗i,F )+πiΠj,F (Y ∗

j,F , Y ∗i,L)−mjC(d0

j−dj))+(1−pj0,i)

(rjd

0j (µ+T )

)−Cexj+mjjd

0j .

Symmetrically, Firm j solves its own AP competition problem, in which it maximizes ΠAPj and believes

that Firm i maximizes ΠAPi,j , in which the probability of a disruption is thought to be pi0,j .

To simplify notation we use Π∗i,L and Π∗i,F instead of Πi,L(Y ∗i,L, Y ∗

j,F ) and Πi,F (Y ∗i,F , Y ∗

j,L), respectively.

Hence, the AP Competition for Firm i is to find xi that maximizes ΠAPi , the firm’s total expected profit:

maxxi

ΠAPi = pi0

(πiΠ∗i,L + πjΠ∗i,F −miC(d0

i − di))

+ (1− pi0)(rid

0i (µ + T )

)− Cexi + miid

0i j 6= i,

where πi = xi/(xi + xj) and πj = 1− πi.

Note that both firms would like to increase their chance (πi) to become the Leader in the BC Com-

petition. This can be achieved by increasing preparedness effort (i.e., xi). However, increasing xi does



not necessarily guarantee an increase in the probability of being first, since this probability also depends

on the amount of effort by the other firm. This results in a competition in preparedness efforts between

the two firms (Firm A and Firm B).

PROPOSITION 3: A unique Nash Equilibrium exists for each firm’s Advanced Preparedness Compe-

tition.

By Proposition 3, there is a unique Nash Equilibrium for the AP Competition for Firm i, i.e., (x∗i , x∗j ),

and there is also a unique Nash Equilibrium for the AP Competition of Firm j, i.e., (w∗j , w∗i ). That is, x∗i is

the actual preparedness effort Firm i spends given it believes Firm j’s preparedness effort x∗j , while, w∗j is

the actual preparedness effort Firm j spends given it believes Firm i’s preparedness effort w∗i . Therefore,

the preparedness investment by Firms A and B in the AP Competition is given by (x∗i , w∗j ), which is the

result of the Nash equilibrium of two games (AP Competition for Firm i and AP Competition for Firm

j). Consequently, Firm i’s probability of becoming the Leader in the BC Competition is,

π∗i =x∗i

x∗i + w∗j, π∗j = 1− π∗i .

Note that both Firms A and B find their optimal effort levels based on their estimates of the probability

of a disruption, which might be different from the actual probability of disruption. Considering the actual

probability of a disruption, p0 , the actual expected profit of Firm i, Πi under the effort levels (x∗i , w∗j ) is

Πi = p0

( x∗ix∗i + w∗j

Π∗i,L +w∗j

x∗i + w∗jΠ∗i,F −miC(d0

i − di))

+ (1− p0)(rid

0i (µ + T )

)− Cex

∗i + miid

0i .

2.3 Sensitivity Analysis

To better understand the AP Competition, we consider the case we call the symmetric information

case, in which both firms know the other firm’s estimate of the likelihood of a disruption (i.e., pi0,j =

pi0 , i 6= j, i, j ∈ A,B), but their estimate of the likelihood of a disruption may be accurate (i.e.,

pi0,j = pi0 = p0 , i 6= j, i, j ∈ A,B) or may be inaccurate (i.e., pi0,j = pi0 6= p0 , i 6= j, i, j ∈ A,B),where p0 is the actual likelihood of a disruption.

Under these conditions, Proposition 4 shows that we can clearly characterize the sensitivity of the

optimal effort level of the firms to cost structure and customer loyalty.

PROPOSITION 4: Under the symmetric information conditions, for a given level of Firm j’s pre-

paredness effect, the optimal preparedness effort of Firm i (i.e., x∗i ) is nondecreasing in mij, miC , mji,

and ri, and is nonincreasing in γij, γiC , γji, ci, and hi, for j 6= i, and i, j ∈ A,B.

Consider cases in which the system parameters of the problem (e.g., profit margin, production capacity,

etc.) are such that the optimal backup capacity purchasing policy corresponds to the “A is Aggressive”


area (see Figure 3). That is, if it is the Leader, Firm A will buy all available backup capacity in an

attempt to steal sales from Firm B. We define Firm A to be an “aggressive” firm if this occurs.

PROPOSITION 5: Under the symmetric information conditions, if both firms are “aggressive” and

are identical except for their size (i.e., dA 6= dB), then both firms spend the same amount of effort on

preparedness, and therefore will have the same chance (50%) to become the Leader in the BC Competition.

Intuitively, the larger firm faces a small market share gain (from the competitor) as the Leader, but

faces a large market share loss (to the competitor) as the Follower in the BC Competition. In contrast, the

smaller firm faces a large market share gain as the Leader, but a small market share loss as the Follower.

Since the two firms are identical except for their size, the opportunity cost (i.e., the gap between the

expected profit from being the Leader and that of being the Follower) is exactly the same for both firms.

Hence, both firms will invest the same amount in preparedness in the AP Competition.

3 Numerical Study

Characterizing the optimal policies in the BC Competition and the AP Competition gives us a basic

understanding of the behavior of competing firms in the face of supply chain disruptions. But from a

management perspective, the most valuable outcome of our framework is the insight it can provide into

the conditions that present the greatest risk from supply chain disruptions.

To qualitatively and quantitatively investigate the factors that influence the consequences of a dis-

ruption, we use our framework to examine the sensitivity of a specific risk measure to a variety of factors.

The measure of risk we use is Firm i’s loss due to lack of preparedness, which is defined as:

Ωi = Πi −(p0

(Π∗i,F −miC(d0

i − di))

+ (1− p0)(rid

0i (µ + T )

)+ miid

0i

). (3)

This measure represents the difference between the expected profit to Firm i if it made strategic prepa-

ration in the AP Competition and if it did not. We assume that in both cases, the rival firm prepares

optimally. Therefore, when Firm i does not prepare, it usually winds up being the Follower in the BC

Competition. Hence, Ωi characterizes risk as the opportunity cost of not preparing for a disruption. Note

that, it is possible for Ωi to be negative in extreme cases where Firm i would “over-prepare” because

its estimate on the likelihood of a disruption is much larger than the true likelihood. In such cases, not

preparing at all is economically preferred to such over-preparation. Without loss of generality, we focus

on the loss due to lack of preparation for Firm A, ΩA .

Although we have an analytic relation between Ωi and the various parameters of the model, it is so

complex that, at most, we can perform single parameter sensitivity analysis using it. The expression itself

cannot show us how the model factors interact and compare with each other. Therefore, we make use of



regression analysis to generate a statistical relation between loss due to lack of preparedness (ΩA) and

the various factors in the model. The results of this regression show us which factors are most important

in identifying high risk situations.

3.1 Design of Numerical Experiments

Our numerical study is based on a test suite that includes 322 × 2× 4 cases, created using values for the

32 model parameters that cover the range of scenarios we could observe in practice. For example, we

chose values of $60, $90, and $120, for the profit margin ri for Firm i. These numbers were chosen to

reflect the cell phone market with a 50% gross margin (Marsal (2007)). Another example is the set of

values for p0, which is 0.01, 0.05, 0.2, 0.4. Since p0 is the true likelihood that a disruption occurs this

year, p0 = 0.05 means that such a failure will happen on average once every 20 years. To motivate our

choices of p0, we note that the annual frequency of earthquakes in the U.S. with magnitude larger than

seven is 0.03 events per year (Mathewson (1999)), while the rate in Taiwan is five to ten times higher

than this. For details on the definitions and values for all model parameters see Online Appendix A.

From a managerial perspective, we are interested in identifying the most important factors that affect

the firm’s risk exposure from a given component, so that management can target their preparedness efforts

where they will have the greatest impact. Since it is not reasonable to expect a firm with thousands of

components to estimate all of the model parameters for every part in their portfolio, our hope is that a

simpler model consisting of only a few important factors can reliably identify the high risk components.

In our numerical study we look at the top five factors that affect risk. To find the top five factors that

have the most impact on the strategic risk for each firm, we performed a stepwise regression analysis in

which, Ωi, the strategic risk for firm i (i.e., loss due to lack of preparedness) was the dependent variable.

We considered a set of 547 independent variables corresponding to the 8 system parameters, 5 profit

factors, and 19 normalized factors. Specifically, we considered 32 linear terms, 19 squared terms, and 496

interaction terms (combinations of the 32 predictor variables). The value of α for entering or removing

a variable in our step-wise regression was 0.05. We performed the step-wise regression to find the best

regression model with 1, 2, . . . , and 5 independent variables that best explain the variation in Ωi (i.e.,

have the highest R-square).

To get a sense of how different competitive conditions affect the factors that characterize supply chain

risk, we consider both the ABC and AB environments in our numerical studies.


Number of Independent Variables in the Regression ModelVariables in the Model R2 p0LLC p0LLB

dA

S |pB0−p0p0

| KB−d

B

dA

1 42.1 +2 46.8 + +3 47.6 + + +4 48.7 + + + −5 49.2 + + + − +

Table 1: Regression models with the five most important factor(s) under the ABC Model when Firm A is largerthan Firm B. Sign + or − corresponds to the sign of coefficients in the regression model.

3.2 ABC Model

In this section, our regression studies are based on the ABC Model, which considers two competitors

(Firms A and B) subject to disruption and a third firm (Firm C) not vulnerable to disruption.

In Table 1 we display the best fitting regression models with one to five parameters under the condition

that Firm A is larger than Firm B. The results of Table 1 show that:

Top 1 Factor: If only one factor were included in the regression model, it would be p0LLC, Firm A’s

expected worst-case long term loss to the third party C, which is clearly essential in quantifying risk and

can be written as:

p0LLC =p0 ×mAC

γAC

× (T × d0

A+ µ× dA).

As expected, this term has a positive coefficient. While we have seen expected short term loss used as

a gauge of supply chain risk in industry, we are not aware of any firms that evaluate long term risk. So

this observation is of practical significance. It says that, under the range of conditions described by our

numerical examples, the expected long term loss to the third party C (not to the competitor, Firm B) is

the most important factor in determining the risk of a supply disruption.

Top 2 Factors: The best two-factor model adds p0LLB, Firm A’s expected worst-case long term loss

to Firm B, which can be written as:

p0LLB = p0 ×mAB ×µ

γAB

× dA ,

to Firm A’s expected worst-case long term loss to the third party C. Again this factor appears in the

model with a positive coefficient, as expected. Along with expected long term loss to Firm C, this factor

characterizes total expected long term loss.

Top 5 Factors: The regression model with five factors has an R2 of about 50%, and, in addition to the

two long term loss factors, includes



Number of Independent Variables in the Regression ModelVariables in the Model R2 p0LLC p0LG p0LLB

KA−d

A

dB

p0SG

1 27.2 +2 35.1 + +3 39.6 + + +4 41.1 + + + +5 42.1 + + + + +

Table 2: Regression models with the five most important factor(s) under the ABC Model when Firm A is smallerthan Firm B. Sign + or − corresponds to the sign of coefficients in the regression model.

1. Firm A’s sales relative to the total available backup capacity (dAS ), which implies that Firm A faces

greater risk when backup capacity is limited,

2. the absolute percent error in Firm B’s estimate of the likelihood of a disruption (|pB0−p0

p0|), which

has a negative coefficient and hence implies that more error (i.e., positive or negative) by Firm B

in estimating the likelihood of a disruption results in less risk to Firm A, and

3. the fraction of Firm A’s demand that Firm B has capacity to meet after satisfying its own customers

(KB−d

Bd

A), which we term Firm B’s “poaching potential”. This factor has a positive coefficient,

implying that risk to Firm A increases in Firm B’s poaching potential.

In Table 2 we display the best regression models with one to five factors under the condition that

Firm A is smaller than Firm B. From the results of Table 2, we can conclude the following:

Top 1 Factor: The best one factor model again uses p0LLC, Firm A’s expected worst-case long term

loss to the third party C.

Top 2 Factors: The best two-factor model adds p0LG, Firm A’s expected best-case long term gain,

which can be written as:

p0LG = p0 ×mBA ×µ

γBA

× dB

This is an interesting and potentially significant result, since it suggests that the smaller firm in a market

should consider the possibility of gaining market share during a supply chain disruption. Failure to do

this can substantially decrease expected profit.

Top 5 Factors: The regression model with five factors has an R2 of about 42%, and, in addition to

the two factors already mentioned, includes (1) Firm A’s expected worst-case long term loss to Firm B

(p0LLB), (2) Firm A’s “poaching potential” (KA−d

Ad

B), and (3) Firm A’s expected best-case short term


gain (p0SG), which can be written as:

p0SG = p0 × µ× rA × dB .

Thus, the most important factors for predicting risk when Firm A is smaller than Firm B are similar to

those for the case where Firm A is larger than Firm B, except that they also include some factors related

to potential market share gains during a disruption.

To gain more insight into these results, we classify factors related to a firm’s sales profit and/or market

share losses as loss factors and factors related to a firm’s capability to capture sales and/or market share

from the competitor firm as gain factors. Any factors that are in neither of these categories are classified

as neutral factors.3 For instance, in Table 1, p0LLC (Firm A’s expected worst-case long term loss to the

third party C) is a loss factor from Firm A’s perspective. Note that none of the factors listed in Table 1

are gain factors for Firm A. The factor p0LG (Firm A’s expected best-case long term gain) in Table 2 is

a gain factor from Firm A’s perspective. Based on the above results, we conclude that,

Observation 1:

• When Firm A is larger than Firm B, loss factors (e.g., expected long term loss, “poaching potential”of competition, ...) are the primary determinants of the level of strategic risk faced by Firm A.

• When Firm A is smaller than Firm B, gain factors (e.g., expected long term gain, Firm A’s “poach-ing potential”, ...) become important determinants of the level of strategic risk that faced by FirmA.

The structural reason behind the first conclusion of Observation 1 follows from the fact that for the

larger firm fewer outcomes lead to gains than to losses. For instance, when dAS is large (greater than one,

i.e., Firm A’s sales exceed the total available backup capacity), then even if Firm A is the Leader in the

BC Competition, it will still lose some customers to the third party C. On the other hand, when dAS is

quite small (i.e., total backup capacity is substantially larger than the total sales of Firms A and B),

then, although it is profitable for Firm A to purchase supply sufficient to cover its own customers, it is

too expensive for Firm A to corner the backup capacity market in order to steal customers from Firm

B. This is because, to do that, Firm A must buy all backup capacity S, even though some of them (i.e.,

S − dA − dB ) will not be used. Only when backup capacity is in a relatively narrow vicinity of dA + dB

does Firm A have a good opportunity to gain sales and market share from Firm B. The fact that “gain

scenarios” tend to require specialized conditions implies that Firm A will be more likely to find products

where the value of preparation is in the prevention of loss than in the generation of gain. This observation3We classify all 32 model parameters into these three categories (i.e., loss factors, gain factors, and neutral factors) in

Table 6 of Online Appendix A.



is consistent with the widely used financial investment strategy known as dollar-cost averaging (DCA),

which is based on the Sharpe ratio and is also known as the reward-to-variability ratio. This strategy

attempts to reduce the risk of investing too much at the “wrong” time and too little at the “right” time

by balancing the upside potential and the downside risk. The DCA strategy weights downside risk more

heavily than upside potential, even though this may result in investors losing opportunities to profit from

the upside potential. Despite some recent research questioning the use of the Sharpe ratio as the preferred

measure (e.g., Leggio and Lien (2003)), researchers still find that the DCA ranking based on different

performance measures (e.g., the Sortino ratio) tends to outperform alternative investment strategies, such

as lump-sum investing.

The intuition behind the second conclusion of Observation 1 is that when Firm A is the smaller player

in the market (i.e., dA < dB ), it has incentive to pay more attention to gain factors when evaluating

protection policies, because in this case there is more market share (up to the total demand of the

competitor, dB ) to poach than the case in which Firm A is the larger player. Hence, Firm A, the smaller

player, considers more “gain factors” in selecting components to focus on. Therefore, when a firm has

market sales smaller than those of the competitor, the firm should consider the important factors listed

in Table 2 in selecting components to prioritize in its preparation strategy.

Our numerical study confirms that a firm faces the greatest risk when (1) the likelihood of a disruption

is large, (2) its market share is valuable, (3) its customers loyalty (especially relative to the third party

unaffected by a disruption) is low, (4) its market sales exceed the available total backup capacity, and

(5) total backup capacity exceeds sales of the competition and the competitor firm has a high “poaching

potential”.

Although these risk factors are all qualitatively intuitive, they: (a) validate our model, (b) confirm

our intuition, and more importantly (c) provide quantitative comparison of the relative importance of

the individual factors, as they are ranked in Tables 1 and 2.

Since Firm B is in direct competition with Firm A, and since its product is a assumed to be a closer

substitute to the product of Firm A than is the product of Firm C, one would expect that Firm A

should be more concerned about its customers’ loyalty relative to Firm B than to Firm C. Observation

2, however, shows that this is not the case.

Observation 2: Under our assumptions that the third party: (1) is not affected by the disruption, (2)provides a less-than-perfect substitute for the products offered by Firms A and B, which customers onlyturn to if neither Firm A nor Firm B has product available, and (3) has enough production capacity, afirm’s customer loyalty relative to the third party is a more important factor than its customer loyaltyrelative to the competitor.


This observation comes from the fact that the third party is always ready to poach any unsatisfied

customers, regardless of whether the firm is the Leader or the Follower in the AB Competition. More

specifically,

• When the firm is the Follower, the competitor can poach the firm’s customer only after satisfying its

own customers. So, even when the (shared) backup capacity is limited (S ≤ the competitor’s sales)

and thus the firm faces no danger of losing customers to the competitor, it can still lose customers

to the third party.

• When the firm is the Leader, although the competition cannot poach the firm’s customer, the firm

can still lose customers to the third party (during the lead time T ).

An implication of Observation 2 is that the closer the substitute provided by the third party (Firm C),

the greater the risk to Firm A. The reason is that a more attractive product from Firm C will diminish

Firm A’s customer loyalty relative to Firm C.

As an example, consider three different firms, Huawei, Cisco and Juniper, that manufacture routers.

Although Huawei’s routers are not yet in the same class as the high-end models offered by Cisco and

Juniper, it represents a potential threat to the Cisco-Juniper duopoly because its cut-rate pricing has a

huge appeal to smaller-scale telecommunication firms and it has deep-pocketed support from the Chinese

government. Therefore, it is logical to assume that Huawei is waiting for chances to take business from

the Cisco-Juniper duopoly. Both Cisco and Juniper source microchips mainly from Taiwan, while Huawei

sources from both Taiwan and Shanghai, so if a disruption of Taiwan were to affect a Cisco product and

the competing Juniper product, then a less-than-perfect Huawei substitute might pick up demand because

it could remain available while Cisco and Juniper are unable to meet demand. As long as the Huawei

product is perceived as a sufficiently acceptable substitute by some customers, then the disruption delays

may drive some Cisco and Juniper customers to Huawei. As the quality of the Huawei offerings improve,

the risk to Cisco and Juniper will increase.

3.3 AB Model

The above results show that supply chain risk is influenced strongly by the presence of a third party

(Firm C), which is not subject to a supply chain disruption. This is not surprising, since, unless customer

loyalty is extremely high, a disruption event is an excellent opportunity for the third party to steal market

share from the disrupted firms.

To get a deeper understanding of the drivers of risk, we now consider the situation where there is no

third party. That is, the market consists of a duopoly in which both firms are subject to a supply chain



Number of Independent Variables in the Regression ModelVariables in the Model R2 p0LLB p0LG

dA

S |pB0−p0p0

| KA−d

A

dB

1 33.1 +2 40.8 + +3 43.1 + + −4 45.4 + + − −5 46.0 + + − − +

Table 3: Regression models (without the third party) with the top 1–5 most important factor(s) under the ABModel when Firm A is larger than Firm B.

Number of Independent Variables in the Regression ModelVariables in the Model R2 p0LG

dB

S p0SGK

A−d

A

dB

|pB0−p0p0

|1 31.7 +2 34.3 + −3 37.1 + − +4 38.8 + − + +5 39.9 + − + + −

Table 4: Regression models (without the third party) with the top 1–5 most important factor(s) under the ABModel when Firm A is smaller than Firm B.

disruption. More specifically, we assume that products by competitors who do not rely on the vulnerable

supplier are not close substitutes for the products produced by the duopoly firms.

We repeated our regression analysis under these conditions, which resulted in Tables 3 and 4, from

which we can draw the following conclusions:

Observation 3: Gain factors have more impact on Firm A’s strategic risk in the AB Model than in theABC Model.

As Table 3 shows, when Firm A is larger than Firm B, there is only one loss factor among the five,

namely Firm A’s expected worst-case long term loss (p0LLB). On the other hand, when Firm A is smaller

than Firm B, there is no loss factor among the five factors listed in Table 4. As stated in Observation 1,

loss factors are the primary determinants of the strategic risks.

Intuitively, the isolated market in the AB Model (involving only Firms A and B) makes it more

attractive to firms to pay attention to gain factors when evaluating protection policies, since they are

not at risk of losing customers to a third party. In a sense, this duopolistic market presents a zero sum

situation, in which Firms A and B compete for each others’ customers.

3.4 Predictive Power of the First Important Factor

As we noted previously, some firms in industry (including a firm with whom we have worked, which is

widely considered to be among the most sophisticated about managing supply chain risks) use expected

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short term loss as a measure of supply chain risk. We call this the expected short-term loss heuristic. Of

course, the five factor models summarized in Tables 1-4 provide a more accurate characterization of risk.

But they also require considerably more data. Therefore, to get a sense of how well a single factor risk

metric can work, we suppose that Firm A considers its specific environment (e.g., ABC or AB ; larger or

smaller than Firm B), and uses the single most important factor from the regression model to identify the

products in their portfolio for which they should invest in specific preparedness activities (e.g., setting up

tighter supply monitoring mechanisms, cultivating a closer relationship with the supplier, pre-qualifying

backup suppliers, modifying designs to make products more flexible with regard to input components,

etc.). Clearly, a single factor is not a perfect predictor of risk due to lack of preparedness (i.e., the

R2 of our single-factor regression model is not very high), but it does not need to be. The problem is

not to predict actual losses (or gains) with numerical precision, but rather to accurately identify those

components for which risk is greatest.

To test the utility of our model, we evaluate the ability of the most important factor in Tables 1, 2, 3,

and 4, which we call the Top 1 Factor heuristic, or T1F heuristic, against the expected short term loss

heuristic to choose top five riskiest components among 150 randomly picked items from our test suite.

We did this by computing the expected short term loss heuristic:

p0 × (T + µ)× rA × d0

A

for each component, and then performing the following simulation experiment:

Step 1: Randomly pick 150 components from the set of scenarios. The record for each component

contains (1) the values for all of the model factors from which we can calculate the exact value of

ΩA , and (2) the value of the most important factor (which depends on the specific environment)

and the value of the expected short term loss heuristic.

Step 2: Sequence (from large to small) the components according to the value of the most important

factor from 1 to 150, and calculate the total risk from the top five components as UFactor =∑5

i=1 Ω(i)A

.

Step 3: Sequence (from large to small) the components according to the value of the expected short term

loss heuristic, and calculate the total risk from the top five components as UHeuri =∑5

i=1 Ω(i)A

. (Note

that this value will differ from that in Step 2 if the indices are different.)

Step 4: Record the value Udiff

= (UFactor −UHeuri)/UHeuri, which represents the percent gain of profit

that Firm A experiences if the firm invests in preparedness for the five items determined by the



Difference in ABC Model AB ModelPerformance A is Larger A is Smaller A is Larger A is Smaller

Umax

diff207.5% 127.9% 229.4% 248.4%

Udiff

126.14% 71.98% 120.96% 104.77%

Table 5: Numerical study of the percent gain of Firm A’s expected profit due to using the most important factorin place of the expected short term loss heuristic.

value of the most important factor, rather than investing in the five components determined by the

expected short-term loss heuristic.

Step 5: Go to Step 1 until 50 replicates have been run (so that the obtained sample mean is close to the

true mean). Find the maximum and mean of Udiff

and denote them by Umax

diffand U

diff, respectively.

The results of this experiment are shown in Table 5.

Observation 4: On average, using the most important factor (i.e., T1F heuristic) to select the riskiestcomponents is much better than using the expected short term loss heuristic to rank components. Whencustomer loyalty differs across products, the T1F heuristic significantly outperforms the expected shortterm loss heuristic in most cases because it explicitly considers expected long term loss.

Observation 5: When customer loyalty is similar across products or when customer loyalty is lower foritems with higher short term loss of sales (i.e., so that short term loss is highly correlated with long termloss), then the expected short term loss heuristic is also a good predictor of total risk.

Note that in practice, the condition in Observation 5 that, “customer loyalty is lower for items with

higher short term sales loss”, does not always hold. In general, we would expect high demand products

to have high customer loyalty because of customers’ preference and due to network externality effects.

For example, comparing Apple’s notebook with Toshiba’s notebook, Apple’s notebook has high sales and

high customer loyalty because of network externalities. On the other hand, Toshiba’s notebook is not as

big a seller as Apple’s notebook in the U.S. market and does not have the same level of network driven

loyalty (Martellaro (2008) and Hruska (2008)). Hence, we would expect a disruption to cause a larger

short term loss of sales to Apple’s notebook but a larger long term loss of market share to Toshiba’s

notebook. Consequently, short term loss may not be a good predictor of long term loss, which implies

that firms using short term loss to allocate preparedness efforts may not be investing optimally. Indeed,

as we note in the following observation, the loss can be substantial.

Observation 6:

• In both ABC and AB models, the percent gain in profit a firm experiences from using the T1Fheuristic in place of the expected short term loss heuristic in evaluating Firm A’s risk due to lackof preparation is substantial.

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• For the smaller firm, the percent gain is higher in the AB model than the ABC Model.

As Table 5 shows, the average percent gain in profit in most cases is more than 100%. This implies

that firms can gain a significant increase in their profit, if they prepare for a supply possible supply chain

disruption based on the most important factor presented in this paper rather than the expected short

term loss factor.

Furthermore, as shown in Table 5, when Firm A is the smaller firm, the average and maximum of the

percent gain (104.77% and 248%) are higher in the AB model than in the ABC Model (71.98% and 127%).

The reason is as follows. Comparing Table 2 and Table 4, we see that the smaller firm’s most important

risk factor is a “loss factor” in the ABC model, but a “gain factor” in the AB model. In contrast, the

expected short-term loss heuristic uses a loss factor to evaluate the smaller firm’s risk. Knowing that the

smaller firm in the AB model should focus on the gain factor (which is what the T1F heuristic does),

we expect the performance of the T1F heuristic to be better than that of the expected short-term loss

heuristic in the AB model, but not so much better in the ABC model.

3.5 Impact of the Post-disruption Information

In our basic modeling assumptions, we assumed that (1) firms do not know the true likelihood of a

disruption and so can only estimate it, and (2) a random duration of outage time after the lead time

follows a cumulative probability distribution with an average of µ units of time, which represents cases

in practice where a firm has some knowledge about the ability of its key components suppliers to recover

from a disruptive event. For example, in the 1990’s, Lattice Semiconductor Corp. mainly sourced

semiconductor products from wafer fabs in Japan (Seiko Epson) instead of the more popular Taiwan

area. This decision was based on historical data for Seiko Epson that showed its average recovery period

from a disruptive event was less than two weeks. So, while Seiko Epson was vulnerable to earthquake as

were the Taiwanese suppliers, it was expected to be disrupted for less time by such an event. Long term

recovery data can help manufacturers reduce risk of supply chain disruptions.

In addition to long term data, firms can also make use of real time data about the severity of the

outage (i.e., the length of the mean outage time) immediately after disruption occurs. For instance,

knowing that the disruption is due to a fire, an earthquake, or other incident can help the firm refine its

estimate of the outage time. Gathering information about the incident, as Nokia did from Philips, can

help firms refine their estimate even further.

In this section, we examine (1) whether or not our previous insights (Observations 1 to 3) still hold

under the case in which the post-disruption information exists, and (2) which firm, the firm with a larger



market share or the firm with a smaller market share, benefits more from the post-disruption information.

To do these, we consider the following information scenarios:

• Pre-Disruption Information: Prior to a disruption, Firm i knows that, if a disruption occurs, it

could be one the following two types:

(a) with probability pS

i0the disruption will be a long disruption with Severe consequences, which

lasts an average of µS units of time with distribution FS (·), or

(b) with probability of pM

i0the disruption will be a short disruption with Minor consequences,

which lasts an average of µM units of time with distribution of FM (·).

Firms use this information to play the Advanced Preparedness Competition and decide how much

to invest in preparation. We refer to the disruption with severe consequences as a type S disruption,

and the one with minor consequences as a type M disruption.

• Post-Disruption Information: After the disruption, we use the conditional probability Pi(h|h),

i ∈ A,B, and h ∈ S, M, to denote the probability that Firm i is correct about the type of

disruption. This conditional probability represents the accuracy of the firm in its assessment of

the disruption type. If the post-disruption information is perfect, then Pi(S|S) = 1, Pi(S|M) = 0,

Pi(M |S) = 0, and Pi(M |M) = 1. Note that, we assume Pi(h|h) 6= Pj (h|h), because we want to

investigate the impact of one firm having more accurate post-disruption information than the other.

We incorporate these new information scenarios as follows:

Advanced Preparedness (AP) Competition: Firm i plays the AP Competition based on its esti-

mates of the likelihoods of each type of disruption, pS

i0and p

M

i0, and its belief of Firm j’s estimates

of these likelihoods, pS

j0,iand p

M

j0,i(i 6= j and i, j ∈ A,B).

Backup Capacity (BC) Competition: After a disruption occurs, firms play the BC Competition

based on the conditional probability Pi(h|h), i ∈ A,B, and h, h =∈ S,M. We define Πh|hi,L(·, ·)

to represent Firm i’s expected profit when it is the Leader in the BC Competition, and it believes

the disruption is of type h when the disruption is actually of type type h. Note that, since Firm

i believes the disruption is type h, it will use the distribution of Fh(·) with an average of µ

hto

optimize its expected profit.

We also introduce the following new factors to study the impact of the Post-disruption Information:

µS/µM : the ratio of the mean outage times of the two types of disruption.


pS

0/p

M

0: the ratio of the probabilities of the two types of disruption.

We repeated the regression analysis of Sections 3.2 and 3.3 under these conditions. We observed the

same top five factors that have the most impact on ΩA, Firm A’s loss due to lack of preparedness, and

hence Observations 1 to 3 hold under the case in which post-disruption information exists.

To answer the question of which firm benefits more from post-disruption information, we evaluated

the Value of Information, which measures the difference between the expected profit of Firm i in the

following two cases:

• Complete post-disruption information case in which Firm i knows for sure whether the disruption

is severe or minor, immediately after it occurs.

• Incomplete post-disruption information case in which Firm i knows only the estimated (i.e., condi-

tional) probabilities of the two types of disruption.

Note that, in both cases, the profit of Firm i depends on whether Firm j has the complete information or

not. To make a conservative (lower bound) assessment of the value of information for Firm i, we assume

that in the complete information case, Firm j also has complete information, and in the incomplete

information case, Firm j also uses its estimates of the probabilities of the two types of disruption.

Regression analysis on “Value of Information” reveals most of the same top factors we observed earlier

for the regressions on “Loss Due to Lack of Preparedness”. However, µS/µM , the ratio of mean outage

times of the two types of disruption, enters the top five factors correlated with the “Value of Information”

for the larger firm under both the ABC and AB Models and for the smaller firm under the ABC Model

(see Tables 11 to 14 in Online Appendix C for details). This variable also becomes the sixth factor for the

smaller firm under the AB Model. The ratio of the probabilities of the two types of disruption, pS

0/p

M

0, is

the seventh factor correlated with the “Value of Information” for the larger firm and the eleventh factor

for the smaller firm under both the ABC and AB Models. These results indicate that the longer and

more likely type S disruptions become, the greater the value of post-disruption information. However, as

we note below, the size of the firm affects the value of information.

Observation 7: The firm with the larger market share benefits from post-disruption information morethan the one with the smaller market share.

The intuition behind this observation is that the larger firm should pay more attention to loss factors

when evaluating protection policies than the smaller firm, and is therefore more inclined to pay for the

additional protection provided by better post-disruption information.



3.6 Dedicated Backup Capacity as a Protection Option

Up to this point, we have assumed that there are two types of backup capacity: (a) dedicated capacity

that is committed to a specific firm (e.g., due to a contractual or informal relationship), and (b) shared

capacity, for which the two firms compete. We have treated the dedicated backup capacity as fixed. But

in practice firms may be able to take steps (e.g., negotiate contracts) to acquire additional dedicated

backup capacity to use in case of a disruption. To examine the conditions under which investment in

such capacity is an attractive alternative to investment in monitoring and detection (i.e., xi, i ∈ A,B),we define θi, i ∈ A,B as the effort by Firm i devoted to acquiring extra dedicated backup capacity. We

define Cθ

e,i(i ∈ A,B) as the cost per unit of such effort. The effort level θi can also be considered as

the cost of purchasing capacity options from other suppliers, which will be exercised only if there is a

disruption of the regular supplier.

Once Firms A and B have made their investments in dedicated backup capacity (i.e., chosen θA

and θB), which will come on line when the outage occurs, the model becomes the same as that studied

above, so all results in Sections 3.2 – 3.4 hold. But we now face the new question of which factors affect

the investment ratio θ∗i /(θ∗i + x∗i ), the fraction of preparation investment that is devoted to acquiring

dedicated backup capacity if needed, as opposed to reacting quickly.

In Section 2.2, we defined ΠAPi as Firm i’s total expected profit in the AP Competition for Firm i with

two (decision) variables: xi (i ∈ A,B). In this Section ΠAPi becomes a function with four (decision)

variables: xi and θi (i ∈ A,B), which are simultaneously decided. Adding θi into the AP Competition

changes the game. See Online Appendix D for details of the game and corresponding analytical results.

To find the top factors that have the most impact on the preparation investment ratio, we performed

regression analysis of a numerical study like those presented earlier with θ∗i /(θ∗i + x∗i ) as the dependent

variable. Because the 4th

and 5th

factors have a very small effect on R2, and hence do not offer much

predictive value, we report only the top three factors. These are:

• If Firm A has the larger market share, then

1. the most important factor affecting the investment ratio for Firm A is p0LLC (Firm A’s

expected worst-case long term loss to Firm C) with a positive sign. So, the larger firm that

pays more attention to loss factors when evaluating protection policies, will invest more in

extra dedicated backup capacity as its expected worst-case long term loss to Firm C increases,

which implies the risk level Firm A faces gets higher.

2. the second most important factor isd0

AS (Firm A’s uncovered demand after using the original


dedicated backup capacity relative to the total available shared backup capacity), with a

positive sign. Intuitively, when the shared backup capacity is very limited, as the larger firm,

which pays more attention to loss factors when evaluating protection policies, it has more

incentive to protect against a loss by securing extra dedicated backup capacity.

3. the third most important factor is p0LLB (Firm A’s expected worst-case long term loss to

Firm B) with a positive sign. Intuitively, the larger Firm A’s expected worst-case long term

loss to Firm B, the higher level of risk it faces, and hence, as the firm paying more attention

to loss factors when evaluating protection policies, it will increase the preparation investment

in securing extra dedicated backup capacity.

• If Firm A has the smaller market share, then

1. the most important factor is p0LG (Firm A’s expected best-case long term gain from Firm

B) with a negative sign. Intuitively, when the opportunity for gain increases, the smaller

firm, which pays more attention to gain factors when evaluating protection policies, has more

incentive to try to win the AP Competition vis investment in monitoring and detection, rather

than avoiding it through investment in extra dedicated backup capacity. When the shared

backup capacity, S, is enough to satisfy Firm A’s customers plus some or all of the competitor’s

unsatisfied customers, Firm A prefers to invest more in monitoring and detection with an eye

toward acquiring the extra shared backup capacity and poaching the competitor’s unsatisfied

customers. However, when the shared backup capacity is limited (S < mindA , dB), then the

firm cannot acquire enough shared backup capacity to poach the competitor’s customers. So,

it may be worthwhile to invest more in extra dedicated backup capacity to be positioned to

take market share from the competitor.

2. the second and third most important factors affecting the investment balance in preparedness

activities of Firm A are p0LLC and p0LLB with positive signs, which implies that when Firm

A’s expected worst-case long term loss to Firm C/B increases, Firm A will invest more in

securing extra dedicated backup capacity.

Based on the above results, we conclude that,

Observation 8: Increased opportunity for gain motivates Firm i with the smaller market share toincrease the fraction of its preparedness effort in monitoring and detection rather than in securing extradedicated backup capacity than the firm with the larger market share (i.e., θ∗i /(θ∗i + x∗i ) is smaller forsmaller firm). This ratio for Firm i increases as the available shared backup capacity S decreases and thededicated backup capacity of Firm j decreases.



Observation 8 discusses situations in which firm should increase its investment in acquiring dedicated

backup capacity (i.e., conditions that cause θ∗i /(θ∗i + x∗i ) to increase). In the following, we examine the

conditions that cause firms to prefer to devote more effort to monitoring and detection than to securing

extra dedicated backup capacity (i.e., cause θ∗i /(θ∗i + x∗i ) < 0.5). To do this, we group the regression

results from the above numerical study into two groups: one with preparation investment ratios less than

0.5; the other with preparation investment ratios greater than or equal to 0.5. We compared the values

of independent variables in these two groups and observed:

Observation 9: The firm with the larger market share prefers investing in monitoring and detection when(1) its customer loyalty relative to the competitor and the third party are both high, (2) its uncovereddemand (before using the extra dedicated backup capacity) relative to the total available shared backup

capacity (d0

iS ) is low, and (3) its expected worst-case long term loss to the third party (p0LLC) is low.

Observation 10: The firm with the smaller market share prefers investing in monitoring and detectionwhen (1) its competitor’s customer loyalty relative to the firm (γji) is low, (2) its uncovered demand(before using the extra dedicated backup capacity) relative to the total available shared backup capacity

(d0

iS ) is low, and (3) its expected best-case long term gain from the competitor (p0LG) is high.

From Observations 9 and 10, we can see that the larger firm prefers to invest in monitoring and

detection when its customer loyalty is high, which makes the firm feel safe to take a relatively aggressive

action. In contrast, the smaller firm prefers to invest in monitoring and detection when its competitor’s

customer loyalty relative to the firm is low, which gives it a great opportunity for gain by poaching the

competitor’s unsatisfied customers.

From a managerial perspective, preparedness activities related to monitoring and detection can be

viewed as a systemic preparation, since they can protect majority or all components. The Nokia-Ericsson

story is an example of systematic preparedness. To appreciate Nokia’s response to the Philips fire in

March 2000, it is informative to look back to the year 1995. At that time, Nokia was experiencing poor

delivery performance and frequent imbalances between supply and demand. To fix these problems, Nokia

launched a collaborative planning process involving its key suppliers and installed a monitoring process.

This system was critical to the detection and evaluation of the situation in the wake of the Philip fire.

The seeds for Nokia’s fast response in 2000 had been sown years earlier. Moreover, the monitoring system

that proved so useful in the Philips fire would be equally useful for other types of disruptions of other

components.

In contrast, preparedness activities related to securing extra dedicated backup capacity can be viewed

as targeted preparation, since they are specific to a small number of components. A firm with thousands of

components probably cannot economically provide extra dedicated backup capacity for all its components.


Hence, such protection makes sense only for very critical or very risky components. An example of such

protection was Intel’s response to the 1999 Taiwan earthquake. At the time, Taiwan was the world’s third

largest supplier of semiconductor materials, including microprocessor chips. These chips were central to

the battle between the AMD Athlon and the Intel Pentium III, which was in full swing in 1999. Prior to

the earthquake, AMD believed it was poised to exploit surging demand with its powerful new Athalon

K7. Unfortunately, all Athlon facilities were located in Taiwan, while Intel had a Pentium III facility in

Korea (i.e., dedicated backup capacity), which sourced semiconductors from Korean suppliers (XbitLabs

(1999)). Because Intel was able to exploit this capacity during the Taiwan outage, it weathered the event

better than AMD and thus staved off a potential loss of market share.

3.7 Impact of Firms’ Risk Attitude

Note that, up to this point, we have assumed that all firms are risk neutral. In this section, we perform

a detailed numerical study to extend our results in Sections 3.2 and 3.3 to cases where firms are Risk

Averse (RA) or Risk Seeking (RS). Specifically, we consider: (1) a risk-averse (RA) exponential utility

function of the form u(x) = −e−ax

, and (2) a risk-seeking (RS) exponential utility function of the form

u(x) = eax

, where a is a positive constant. We test cases where a = 0.1, 1, 10, and 100 and compared the

results to the risk neutral case. Since there are three types of risk attitude (Risk Averse, Risk Seeking

and Risk Neutral) for each of the two firms, Firms A and B, we have nine main cases as shown in Table

15 in Online Appendix E. Comparing these revealed the following:

Observation 11: A risk averse firm tends to pay more attention to loss factors when evaluating protectionpolicies (i.e., with more loss factors in its top five risk factors), while a risk seeking firm tends pay moreattention to gain factors when evaluating protection policies (i.e., with more gain factors in its top fiverisk factors). Furthermore, the fact that the competitor is more risk averse motivates a firm to pay moreattention to gain factors when evaluating protection policies. In contrast, a more risk seeking competitormotivates a firm to pay more attention to loss factors when evaluating protection policies.

This observation complements our previous observations about firm size. Larger and/or risk averse

firms should pay attention to loss factors when evaluating protection policies, while smaller and/or risk

seeking firms should pay attention to gain factors when evaluating protection policies. Factors that make

the opponent pay more attention to loss (gain) factors when evaluating protection policies tend to make

a firm pay more attention to gain (loss) factors when evaluating protection policies. These results are

intuitive, but they confirm that our earlier insights about the conditions that magnify the risks from

supply chain disruptions are still valid when firms are not risk neutral.



4 Conclusions and Future Work

In this paper we have developed a modeling framework that captures both the tactical consequences

(short term sales loss) and the strategic consequences (long term market share shifts) of supply chain

disruptions in competitive environments. The resulting models help strike a balance between the costs

of preparedness and the risks of disruptions in such environments.

To gain structural insights into policies for protecting supply chains against disruptions, we separated

the overall process into two periods: (1) a “pre-disruption period” in which the Advanced Preparedness

(AP) Competition occurs (i.e., firms invest in dedicated backup capacity and/or monitoring and detection

to enhance their probabilities of being first to detect a disruption), and (2) a “post-disruption period”,

during which the Backup Capacity (BC) Competition occurs (i.e., firms purchase shared backup capacity

and use it to satisfy their own customers and possibly steal market share from the opponent).

For the BC Competition, we showed that the optimal backup capacity purchasing policy is a five-region

policy, which can be computed by comparing a small number of possible values. For the AP Competition,

we showed a unique Nash Equilibrium exists for each firm, which describes the preparedness investment

for the market.

From a managerial perspective, our numerical results suggest that a firm will face large tactical and

strategic risks when (1) the likelihood of a disruption is large, (2) the firm’s market share is highly

valuable, (3) its customers exhibit low loyalty (especially relative to a third party), (4) its market sales

exceed the available backup capacity, and (5) the backup capacity exceeds sales of the competitor firm

and the competitor has the potential to use this capacity to steal market share. To reduce the firm’s risk

due to lack of preparedness, it should (a) improve its speed of detecting and reacting to a disruption, (b)

cultivate greater customer loyalty, and (c) increase the amount of backup capacity (this includes both

developing additional dedicated backup capacity and making it more likely that the firm will get whatever

backup capacity is shared). On-Line Appendix F provides a list of policies firms can pursue to mitigate

the tactical and strategic consequences of supply chain disruptions.

Since collecting data is time-consuming and expensive, it is not practical for firms to carefully estimate

all of the parameters used in this paper. Fortunately, our results suggest that the single most important

factor (i.e., Top 1 Factor) can make reasonable predications of which components in the firm’s catalog

pose the highest risks. The data collection overhead of this single factor is similar to that for a single

factor method currently being used in industry, although it can predict risks much more accurately. From

a qualitative standpoint, our regression studies illustrate that large and/or risk averse firms should focus


on investing in preparedness to protect sales and market share from disruptions, while small and/or risk

seeking firms should invest in preparedness in hopes of taking advantage of a disruption to gain market

share.

We can distinguish between preparedness activities related to monitoring and detection as systemic

preparation, which affect many components simultaneously, and preparedness activities related to securing

extra dedicated backup capacity as targeted preparation, which affect only a few specific component.

Because of their limited range, targeted protection is only economical for vital or highly risky components.

Systemic protection can be achieved in a variety of ways, most of which revolve around acquiring, sharing,

and using information. This includes both long term information (e.g., estimates of the likelihood and

duration of disruptions) and real-time information (e.g., severity of disruption after it has begun). Our

results suggest that larger firms have more incentive to invest in both types of information because they

have more market share to lose than do smaller firms.

While our framework offers significant managerial insights into the science of supply chain risk man-

agement, there is still considerable need for further research. We describe two of these below.

First, in this paper we have studied the scenario in which a disruption affects a single component of

a single product of one firm and its competitor is affected in a symmetric fashion. However, a firm may

have many final products that rely on the disrupted key component. Alternatively, a disruption (e.g., an

earthquake) may affect an entire region and hence several components at once. To help guide real-world

decision making, the framework needs to be extended to cover these scenarios.

Second, we have considered only business-to-business (OEM) relationships between firms and their

customers. The business-to-consumer (retail) environment remains open. In a retail situation, firms have

no control over which customers – existing or new – will have their orders filled first. Hence, even if a firm

secures an amount of backup capacity of the disrupted component that is smaller than its sales, it may

still make some sales to unsatisfied customers of the competitor. This may mean that environments with

small amount of available backup capacity present greater risk than they do under the OEM assumption.

To characterize risk in this setting, we first need to better understand the customer brand loyalty, so we

can model the dynamics of how customers choose products and switch between brands.

Acknowledgement: This research was supported in part by the National Science Foundation under

grants DMI-0243377 and DMI-0423048.

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ONLINE APPENDIX A

Design of Numerical Experiments

We created a large set of scenarios (322 × 2× 4 cases in total) using values for the 32 model parameters

chosen to cover the majority of realistic scenarios we could observe in practice. We randomly picked onemillion scenarios from this large set and fit a regression model to this reduced sample.

Below we summarize the 8 parameters of the model, along with 5 profit factors, and 19 normalized factorsused in our numerical studies. We classify the factors as gain, loss, or neutral factors in Table 6.

A.1 Single Parameters:

We consider eight parameters in our model including p0, µ, CV (= σ/µ), T , γAB , γAC , γBA , and γBC .The values of these parameters are summarized in Table 7. Note that, given the distribution of the outagetime, the customer loyalty coefficients of 18, 42, 130, 260 correspond to cases in which (approximately)80%, 50%, 20%, 10% of customers who use the competitor’s (or the third party’s) products during theoutage will permanently switch to those firms. We assume that the remaining outage times (after leadtime T ) are normally distributed.

A.2 Profit Factors:

We include the following five variables corresponding to the profit loss or gain for each firm, i.e., SLTotal

,LLB, LLC, LG, and SG.

• Firm A’s worst-case short term loss:

SLTotal

= SLT + SLµ = T × rA × d0

A+ µ× rA × dA

which is the loss of profit during the lead time and outage that occurs, if Firm A cannot satisfy anyof its customers.

• Firm A’s worst-case long term loss to Firm B (i.e., LLB) and to Firm C (i.e., LLC):

LLB = mAB ×µ

γAB

× dA

LLC = LLCT + LLCµ = mAC ×T

γAC

× d0

A+ mAC ×

µ

γAC

× dA

These long terms losses correspond to the worst case in which Firm A cannot satisfy any of itsown customers. Note that, Firm A’s worst-case long term loss to Firm B is actually expressed asmAB × ξAB × dA , where

ξAB =∫ ∞

0(1− e

− tγAB )dFo(t),

and Fo is a distribution with mean µ and standard deviation σ. However when σ → 0, we have,

ξAB = 1− e− µ

γAB ' µ

γAB

.

Therefore, to simplify, we define,

LLB = mAB ×µ

γAB

× dA .

We use the same approximation to define LLC and LG.


Firm A’s Gain Factors Firm A’s Loss Factors Firm A’s Neutral FactorsSG SL

Totalp0

LG LLB, LLC µ,CV ,TγBA γAB ,γAC γBC

mAA365rA

, mBB365rB

, mBC365rA

mBA365rB

mAB365rA

, mAC365rA

crA

|pB0−p0

p0|

KA−dAdB

KB−dBdA

|pA0,B−p

A0

pA0

|rA

rB

|pA0−p0

p0| 365hA

rA,365hB

rBdAS ,dB

S

|pB0,A−p

B0

pB0

| Sd

A+d

B

Table 6: Firm A’s classified model factors.

• Firm A’s best-case short term gain (i.e., SG) and long term gain (i.e., LG):

SG = µ× rA × dB LG = mBA ×µ

γBA

× dB

which correspond to the case where Firm B cannot satisfy any of its own customers.

A.3 Normalized Factors:

We consider the following 19 normalized factors (whose ranges are determined by the values of associatedparameters and summarized in Table 7):

• rA

rB

, which is the relative profitability of the two firms;

• mij

365ri, which is the present-worth factor of one unit market share of Firm i for Firm j based on a 15%

interest factor for continuous compounding interest (Thuesen et al. 1977), i ∈ A,B, j ∈ A,B, C.So, for example, if mAB

365rA= 0.1 and rA = $120, then mAB = $4380;

• 365hiri

, which is the relative annual cost for Firm i to hold one unit of unused backup capacity,i ∈ A,B;

• crA

, which is the relative premium Firm A pays per unit of backup capacity;

• diS , which is Firm i’s sales relative to the total available backup capacity, i ∈ A, B;

• Ki−didj

, which is defined as the “poaching potential” of Firm i (notice that, the numerator is Firmi’s remaining production capacity after satisfying its own customers), i 6= j, i, j ∈ A,B;

• Sd

A+d

B, which is the fraction of total market sales represented by the backup capacity;

• |pi0−p0p0

|, which is the absolute percent error in Firm i’s estimate of the likelihood of a disruption,i ∈ A,B;

• |pj0,i−pj0

pj0|, which is the absolute percent error in Firm i’s belief of Firm j’s estimate of the likelihood

of a disruption, i 6= j, i, j ∈ A,B.



Unnormalized Parameters Normalized FactorsParameters Values Factors Rangesµ (day) 15, 30, 60CV (σ

µ) 16 , 1

3

T (day) 5, 10, 15γAB (day) 18, 42, 130γAC (day) γAB , γAB + 100, 260γBA (day) 18, 42, 130γBC (day) γBA , γBA + 100, 260rA ($/unit) 60, 90, 120rB ($/unit) 60, 90, 120 r

ArB

0.5 ∼ 2mAB ($/unit) 4380, 16425, 54750 m

AB365r

A0.1 ∼ 2.5

mAC ($/unit) 4380, 16425, 54750 mAC

365rA

0.1 ∼ 2.5mAA ($/unit) 4380, 16425, 54750 m

AA365r

A0.1 ∼ 2.5

mBA ($/unit) 4380, 16425, 54750 mBA

365rB

0.1 ∼ 2.5mBC ($/unit) 4380, 16425, 54750 m

BC365r

B0.1 ∼ 2.5

mBB ($/unit) 4380, 16425, 54750 mBB

365rB

0.1 ∼ 2.5

(365hA) ($/unit/year) 12, 22.5, 30 365hA

rA

0.1 ∼ 0.5

(365hB ) ($/unit/year) 12, 22.5, 30 365hB

rB

0.1 ∼ 0.5c ($/unit) 12, 90, 150 c

rA

0.1 ∼ 2.5

dA (unit) 1000, 3000, 5000 dAS 0.1 ∼ 10

dB (unit) 1000, 3000, 5000 dBS 0.1 ∼ 10

KA (unit) dA , dA + 0.5dB , dA + dB

KA−d

Ad

B0 ∼ 1

KB (unit) dB , dB + 0.5dA , dA + dB

KB−d

Bd

A0 ∼ 1

S (unit) 500, 5000, 10000 Sd

A+d

B0.05 ∼ 5

p0 0.01, 0.05, 0.2, 0.4pA0 when p0 = 0.01 0.005, 0.01, 0.05 |pA0

−p0p0

| 0 ∼ 4

when p0 = 0.05 0.025, 0.05, 0.1 |pA0−p0

p0| 0 ∼ 1

when p0 = 0.2 0.1, 0.2, 0.3 |pA0−p0

p0| 0 ∼ 0.5

when p0 = 0.4 0.2, 0.4, 0.6 |pA0−p0

p0| 0 ∼ 0.5

pB0 when p0 = 0.01 0.005, 0.01, 0.05 |pB0−p0

p0| 0 ∼ 4

when p0 = 0.05 0.025, 0.05, 0.1 |pB0−p0

p0| 0 ∼ 1

when p0 = 0.2 0.1, 0.2, 0.3 |pB0−p0

p0| 0 ∼ 0.5

when p0 = 0.4 0.2, 0.4, 0.6 |pB0−p0

p0| 0 ∼ 0.5

pB0,A when pB0 < 0.1 0.5pB0 , pB0 , 2pB0 |pB0,A−p

B0

pB0

| 0 ∼ 1

when pB0 ≥ 0.1 0.5pB0 , pB0 , 1.5pB0 |pB0,A−p

B0

pB0

| 0 ∼ 0.5

pA0,B when pA0 < 0.1 0.5pA0 , pA0 , 2pA0 |pA0,B−p

A0

pA0

| 0 ∼ 1

when pA0 ≥ 0.1 0.5pA0 , pA0 , 1.5pA0 |pA0,B−p

A0

pA0

| 0 ∼ 0.5

Table 7: Values of input parameters and ranges of factors for numerical studies.


References

Thuesen, H.G., Fabrycky, W.J., and G.J.Thuesen, 1977. Engineering Economy (5rd Ed.). Prentice-Hall, Inc.,New Jersey.



ONLINE APPENDIX B

Proofs of Analytical Results

PROPOSITION 1:

(Large Capacity Leader:) If the Leader’s production capacity is greater than the combined market share of bothfirms (i.e., K

A≥ d

A+ d

B), and

(1) if the backup capacity is less then the combined market share of both firms (i.e., S ≤ dA

+ dB), then the only

possible choices for the optimal backup capacity purchases for the two firms are:

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB ),(0, minKB , S), (S − dB , 0), (S − dB , dB ),

(dA, 0), (d

A, S − d

A), (S, 0)

;

(2) if the backup capacity is larger than the combined market share of both firms (i.e., S > dA

+ dB), then the

only possible choices for the optimal backup capacity purchases for the two firms are:

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB ),(0,minK

B, d

A+ d

B), (d

A, 0), (d

A, d

B),

(dA + dB , 0), (S, 0)

.

(Small Capacity Leader:) If the Leader’s production capacity cannot cover the combined market share of bothfirms (i.e., KA < dA + dB ), and

(1) if the backup capacity is less than the Leader’s market share (i.e., S ≤ KA), then the only possible choicesfor the optimal backup capacity purchases for the two firms are:

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB), (0, minK

B, S), (S − d

B, 0), (S − d

B, d

B),

(dA , 0), (dA , S − dA), (S, 0)

;

(2) if the backup capacity is larger than the Leader’s market share, but it is smaller than the combined marketshare of both firms (i.e., K

A< S ≤ d

A+ d

B), then the only possible choices for the optimal backup capacity

purchases for the two firms are:

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB), (0, minK

B, S), (S − d

B, 0), (S − d

B, d

B),

(dA , 0), (dA , S − dA), (KA , 0), (KA , S −KA)

;

(3) if the backup capacity can cover the combined market share of both firms (i.e., S > dA

+ dB), then the only

possible choices for the optimal backup capacity purchases for the two firms are:

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB), (0, minK

B, d

A+ d

B), (d

A, 0), (d

A, d

B), (K

A, 0),

(S − (d

A+ d

B) + K

A, 0

),

(S − (d

A+ d

B) + K

A, (d

A+ d

B)−K

A

).


PROOF OF PROPOSITION 1:

We only present the proof for the case of Large Capacity Leader, i.e., KA≥ d

A+ d

B. The proof for Small Capacity

Leader is similar, and is therefore omitted. To simplify notation, we let

φij

= mijξij: i 6= j, i ∈ A,B, j ∈ A, B,C

Since KA ≥ dA + dB , then we have KA − dA ≥ dB ≥ [dB − Y ∗B,F

(YA,L)]+, where Y ∗B,F

(YA,L) is Firm B’s optimalpurchase amount of backup capacity after Firm A purchases Y

A,L.

min

[dB− Y ∗

B,F(Y

A,L)]+, min

[Y

A,L− d

A]+,K

A− d

A

=

min

[dB− Y ∗

B,F(Y

A,L)]+, [Y

A,L− d

A]+,K

A− d

A

= min

[d

B− Y ∗

B,F(Y

A,L)]+, [Y

A,L− d

A]+

.

Therefore, equation (31) becomes:

ΠA,L

(Y

A,L, Y ∗

B,F(Y

A,L))

= rA

min

YA,L

, dA

µ

+ψA

min

[dB− Y ∗

B,F(Y

A,L)]+, [Y

A,L− d

A]+

−φAB

min

[Y ∗B,F

(YA,L

)− dB]+, [d

A− Y

A,L]+

−φAC

[[d

A− Y

A,L]+ − [Y ∗

B,F(Y

A,L)− d

B]+

]+

−cAYA,Lµ

−hA

[Y

A,L−minY

A,L, d

A −min

[d

B− Y ∗

B,F(Y

A,L)]+, [Y

A,L− d

A]+

]+

µ. (4)

It is easy to show that ΠA,L

is a piecewise linear function in YA,L

. It is linear because equation (4) is linear in YA,L

,and it is piecewise linear due to the existence of Min and Max functions.

To proof the results for Parts (1) and (2) of the case for “Large Capacity Leader” (i.e., KA ≥ dA +dB ), we considerthe following two corresponding cases:

• Case 1: When S ≤ dA + dB ;

• Case 2: When S > dA + dB .

CASE 1: If S ≤ dA + dB , then equation (4) becomes

ΠA,L

(Y

A,L, Y ∗

B,F(Y

A,L))

= rA

min

YA,L

, dA

µ + ψ

Amin

[d

B− Y ∗

B,F(Y

A,L)]+, [Y

A,L− d

A]+

− φAB

min

[Y ∗B,F

(YA,L

)− dB]+, [d

A− Y

A,L]+

− φAC

[[dA − YA,L ]+ − [Y ∗

B,F(YA,L)− dB ]+

]+

− cAYA,Lµ. (5)

For the piecewise linear functions, it is proven that the optimal solution is one of the break points (Hager andPark 2004). Hence, the optimal value of Y

A,Lis one of four break points of equation (5), namely 0, [S − d

B]+,

mindA, S, and S. Below we consider each of these three cases:

Case 1-i, YA,L

= 0: For this case, the objective function of Firm A is

ΠA,L

(0, Y ∗

B,F(YA,L)

)

= −φAB

min

[Y ∗B,F

(YA,L

= 0)− dB]+, d

A

− φ

AC

[d

A− [Y ∗

B,F(Y

A,L= 0)− d

B]+

]+

= −φAB

[Y ∗B,F

(YA,L

= 0)− dB]+ − φ

AC

(d

A− [Y ∗

B,F(Y

A,L= 0)− d

B]+

). (6)

Note that, since S ≤ dA

+ dB, we have [Y ∗

B,F(Y

A,L= 0)− d

B]+ ≤ S − d

B≤ d

A.

On the other hand, equation (32) is reduced to

ΠB,F (0, YB,F ) = rB min

YB,F , dB

µ + ψB min

dA , [YB,F − dB ]+

− φBC [dB − YB,F ]+ − cBYB,F µ

= rB

minYB,F

, dBµ + ψ

B[Y

B,F− d

B]+ − φ

BC[d

B− Y

B,F]+ − c

BY

B,Fµ, (7)



which is piecewise linear and has three break points: 0, mindB, S, and minK

B, S. So, the optimal value of

Y ∗B,F

∈

0, dB, minK

B, S

. Consequently, the possible combination of the optimal capacity purchase for firms

A and B in this case are

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB), (0,minK

B, S)

.

Case 1-ii, YA,L

= [S − dB]+: Notice that, if [S − d

B]+ = 0, i.e., S ≤ d

B, then it becomes similar to Case 1-i, so

here we are only interested in the case of [S − dB]+ = S − d

B.

Since S ≤ dA + dB , YA,L = S − dB ≤ dA and Y ∗B,F

(YA,L) ≤ dB must hold. Then the objective function of Firm Abecomes

ΠA,L

(S − dB , Y ∗

B,F(YA,L)

)= (rA − cA)(S − dB )µ− φAC

(dA − (S − dB )

). (8)

It is easy to see that the reason why YA,L

= S − dB

is that Firm A aims at protecting itself from loosing marketshare to Firm B (although sales profit is negative), i.e., if Y

A,L= 0, then Y ∗

B,F= S due to the linearity property of

the profit function. Based on Firm A’s protection strategy, the objective function of Firm B becomes

ΠB,F

(S − dB, Y

B,F) = (r

B− c

B)Y

B,Fµ− φ

BC(d

B− Y

B,F), (9)

which is a linear function with two end points: 0 and dB. Thus, the optimal capacity purchase Y ∗

B,F∈ 0, d

B.

Consequently, in this case the only possible combination of the optimal capacity purchase for firms A and B are

(Y ∗A,L

, Y ∗B,F

) ∈

(S − dB , 0), (S − dB , dB )

.

Case 1-iii, YA,L = mindA , S: For this case, the objective function of Firm A is

ΠA,L

(mind

A, S, Y ∗

B,F(Y

A,L))

= (rA− c

A)mind

A, Sµ− φ

AC

(d

A−mind

A, S)

= (rA − cA)mindA , Sµ− φAC [dA − S]+. (10)

On the other hand, since Firm A purchases capacity to only satisfy its own customers, i.e.,YA,L = mindA , S, theobjective function of Firm B, becomes

ΠB,F

(mindA, S, Y

B,F) = (r

B− c

B)Y

B,Fµ− φ

BC(d

B− Y

B,F), (11)

which is a linear function with two end points: 0 and [S − mindA, S]+ = [S − d

A]+. Thus, Y ∗

B,F∈ 0, [S −

mindA, S]+ = [S − d

A]+. Consequently, in this case the only possible combination of the optimal capacity

purchase for firms A and B are

(Y ∗A,L

, Y ∗B,F

) ∈

(mindA, S, 0), (mind

A, S, [S − d

A]+)

.

Case 1-iv, YA,L

= S: For this case, Y ∗B,F

(YA,L

) = 0 is the only possibility. Since S ≤ dA

+dB, then [S−d

A]+ ≤ d

B

must hold. Hence, the objective function of Firm A is

ΠA,L

(S, 0) = rA

minS, dAµ + ψ

A[S − d

A]+ − φ

AC[d

A− S]+ − c

ASµ, (12)

and the objective function of Firm B, becomes

ΠB,F

(S, 0) = −φBA

min[S − d

A]+, d

B

− φBC

(d

B− [S − d

A]+

).

The only possible combination of the optimal capacity purchase for this case is (Y ∗A,L

, Y ∗B,F

) ∈ (S, 0).

If we combine the results for Case 1-i to Case 1-iv, we get the following possible combinations of the optimalcapacity purchase for firms A and B, when S ≤ d

A+ d

B

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB ),(0,minKB , S), (S − dB , 0), (S − dB , dB ),

(mindA , S, 0)

,(mindA , S, [S − dA ]+

), (S, 0)

.


Note that, (1) the sixth possible combination(mind

A, S, 0)

becomes (dA, 0), when d

A≤ S, and reduces to the

last one, (S, 0), when dA

> S; (2) the seventh possible combination(mind

A, S, [S− d

A]+

)becomes (d

A, S− d

A),

when dA≤ S, and reduces to the last one, (S, 0), when d

A> S. Thus, the total possible combinations are:

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB),

(0, minK

B, S), (S − d

B, 0), (S − d

B, d

B

),

(dA, 0), (d

A, S − d

A), (S, 0)

.

This completes the proof for Part (1) of Proposition 1 for the case with “Large Capacity Leader”.

CASE 2: Similar to Case 1, it is easy to show that if S > dA

+dB, then equation (31) is a piecewise linear function

with four break points: 0, dA, d

A+ d

Band S. We now discuss each of the four cases.

Case 2-i, YA,L

= 0: For this case, the objective function of Firm A becomes

ΠA,L

(0, Y ∗

B,F(Y

A,L))

= −φAB

min[Y ∗

B,F(Y

A,L= 0)− d

B]+, d

A

− φAC

[d

A− [Y ∗

B,F(Y

A,L= 0)− d

B]+

]+. (13)

On the other hand, the objective function of Firm B is reduced to

ΠB,F (0, YB,F ) = rB min

YB,F , dB

µ + ψB min

dA , [YB,F − dB ]+

− φBC

[dB− Y

B,F]+ − c

BY

B,Fµ, (14)

which is a piecewise linear function with three break points: 0, dB , and minKB , dA + dB, which constitutes thepotential values for Y ∗

B,F. Consequently, the possible combinations for the optimal capacity purchase for Firms A

and B are

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB ),(0, minKB , dA + dB

).

Case 2-ii, YA,L = dA : For this case, the objective function of Firm A becomes

ΠA,L

(dA , Y ∗

B,F(YA,L)

)= (rA − cA)dAµ. (15)

Due to Firm A’s protection strategy, the objective function of Firm B becomes

ΠB,F

(dA, Y

B,F) = r

Bmin

Y

B,F, d

B

µ− φ

BC[d

B− Y

B,F]+ − c

BY

B,Fµ

= (rB− c

B)Y

B,Fµ− φ

BC[d

B− Y

B,F]+, (16)

which is a piecewise linear function and has two break points: 0 and dB. Consequently, the possible combinations

of the optimal capacity purchase for Firms A and B are

(Y ∗A,L

, Y ∗B,F

) ∈ (d

A, 0), (d

A, d

B).

Case 2-iii, YA,L = dA + dB : As shown in equation (1), the coefficient of the h is

[YA,L −minYA,L, dA −min

[dB − Y ∗

B,F (YA,L)]+,min[YA,L − dA ]+,KA − dA

]+

µ, (17)

Since in this case, YA,L = dA + dB , then we have

YA,L −minYA,L, dA = (d

A+ d

B)− d

A= d

B.

Hence, (17) becomes: [d

B−min

[d

B− Y ∗

B,F (YA,L)]+,mind

B,K

A− d

A

]+

µ. (18)

Note that, KA − dA ≥ dB holds, and hence (18) reduces to[dB −min

[dB − Y ∗

B,F (YA,L)]+, dB

]+

µ.



• If dB≥ Y ∗

B,F (YA,L), then the coefficient of the h is Y ∗B,F (YA,L)µ.

• If dB

< Y ∗B,F (YA,L), then the coefficient of the h is d

Bµ.

Therefore,

ΠA,L

(d

A+ d

B, Y ∗

B,F(Y

A,L))

= rAd

Aµ + ψ

A[d

B− Y ∗

B,F(Y

A,L)]+

− cA(d

A+ d

B)µ− h

Amin

d

B, Y ∗

B,F(Y

A,L= d

A+ d

B)µ. (19)

On the other hand, the objective function of Firm B is reduced to

ΠB,F

(dA

+ dB, Y

B,F) = (r

B− c

B)Y

B,Fµ− φ

BA[d

B− Y

B,F]+, (20)

which is a piecewise linear function and has two break points: 0 and minS − dA − dB , dB. Thus, Y ∗B,F

=0, minS − d

A− d

B, d

B.

However, the combination of(Y ∗

A,L= d

A+ d

B, Y ∗

B,F= minS − d

A− d

B, d

B) implies that only part of Y ∗

A,L,

which is the difference between dB

and Y ∗B,F

, will be used by Firm A to gain both sales profit and market sharefrom Firm B. But, due to the linearity property, it cannot be an optimal solution, because, in this case, if oneunit of firm B’s customer is profitable, then Firm A will definitely want to gain all possible units. So, if FirmA knows that B will buy all or some of the remaining backup capacity, i.e., minS − d

A− d

B, d

B, to protect

itself, the decision of dA

+ dB

will never be optimal for Firm A. Therefore, Y ∗A,L

= dA

+ dB

can be optimal onlyif Y ∗

B,F= 0, i.e., the only possible combinations of the optimal capacity purchase for firms A and B in this case is

(Y ∗A,L

, Y ∗B,F

) ∈ (dA

+ dB, 0).

Case 2-iv, YA,L

= S: In this case, Y ∗B,F

(YA,L

= S) = 0 is the only possibility. Thus, (Y ∗A,L

, Y ∗B,F

) ∈ (S, 0). Theobjective function of Firm A is

ΠA,L(S, 0) = rAdAµ + ψAdB − cASµ− hA(S − dA − dB )µ. (21)

If we combine the results for Case 2-i to Case 2-iv, we get the following possible combinations of the optimalcapacity purchase for firms A and B, when S > d

A+ d

B,

(Y ∗A,L

, Y ∗B,F

) ∈

(0, 0), (0, dB ),(0, minKB , dA + dB

), (dA , 0), (dA , dB ), (dA + dB , 0), (S, 0)

.

This completes the proof for Part (2) of Proposition 1 for the case of “Large Capacity Leader”. ¤

PROPOSITION 2: The optimal policy of the Backup Capacity Competition has a five-region structure.


We present the proof for the case of Large Capacity Leader, i.e., KA ≥ dA + dB . The proof for Small CapacityLeader is similar, and is therefore omitted.

PROOF for the Follower, Firm B:

We first determine the regions for the optimal capacity purchase for the Follower. We consider the following twocorresponding cases (same as the configuration of the proof for Proposition 1):

• Case 1: When S ≤ dA + dB . For this case, we introduce three thresholds on Firm B’s unit profit margin to

(i) separate the region in which Firm B forfeits and the region in which Firm B is aggressive,

(ii) separate the region in which Firm B forfeits and the region in which Firm B protects, and

(iii) separate the region in which Firm B protects and the region in which Firm B is aggressive.

• Case 2: When S > dA + dB . For this case, we introduce four thresholds on Firm B’s unit profit margin to

(i) separate the region in which Firm B forfeits and the region in which Firm B is aggressive,

(ii) separate the region in which Firm B forfeits and the region in which Firm B protects,


(iii) separate the region in which Firm B protects and the region in which Firm B is aggressive, and

(iv) separate the following two responses: Given Firm A purchases the amount of dA + dB , Firm B caneither buy the remaining to protect itself from losing customers to Firm A (this response implies thatd

A+ d

Bcannot be optimal for Firm A), or buy nothing (this response implies that d

A+ d

Bis optimal

for Firm A).

The summary of the thresholds introduced in this proof can be found in Table 8.

We present the proof based on the same cases that we presented in Proposition 1.

Case 1-i, YA,L = 0: Define

cIB

= cB− φ

AB

[minK

B, S − d

B

]+ + φBC

mind

B, minK

B, S

µ minKB, S .

We now show that when rB ≥ cIB, then ΠB,F

(0, minK

B, S) ≥ Π

B,F(0, 0). By equation(7) we have

ΠB,F

(0,minK

B, S)−Π

B,F(0, 0)

= rB

min

minKB, S, d

B

µ + ψ

B

[minK

B, S − d

B

]+

− φBC

[d

B−minK

B, S]+ − c

BminK

B, Sµ

−(rB min0, dBµ + ψB [0− dB ]+ − φBC [dB − 0]+ − cB (0)µ

),

= rB

min

minKB, S, d

B

µ + (r

Bµ + φ

AB)[minK

B, S − d

B

]+

− φBC

[d

B−minK

B, S]+ − c

BminK

B, Sµ

− (−φBC

dB),

= rB

minKB, Sµ + φ

AB

[minK

B, S − d

B

]+

+ φBC mindB ,minKB , S− cB minKB , Sµ,

= rB

minKB, Sµ−

(c

BminK

B, Sµ

− φAB

[minK

B, S − d

B

]+ − φBC

mind

B,minK

B, S

).

It is clear that if rB≥ cI

B, then the right-hand side is positive, which implies

ΠB,F

(0, minK

B, S) ≥ Π

B,F(0, 0).

Therefore, if rB≥ cI

B, then Y ∗

B,F(Y

A,L= 0) = minK

B, S; otherwise, Y ∗

B,F(Y

A,L= 0) = 0.

DefinecII

B= c

B− φ

BC

µ.

We now show that when rB≥ cII

B, then Π

B,F(0, d

B) ≥ Π

B,F(0, 0). By equation (7) we have

ΠB,F

(0, dB)−Π

B,F(0, 0) = r

Bd

Bµ− c

Bd

Bµ

−(r

Bmin0, d

Bµ + ψ

B[0− d

B]+ − φ

BC[d

B− 0]+ − c

B(0)µ

),

= (rB− c

B)d

Bµ− (−φ

BCd

B),

= rBd

Bµ− (c

Bd

Bµ− φ

BCd

B).

It is clear that if rB≥ cII


ΠB,F

(0, dB) ≥ Π

B,F(0, 0).

Therefore, if rB≥ cII

B, then Y ∗

B,F(Y

A,L= 0) = d

B; otherwise, Y ∗

B,F(Y

A,L= 0) = 0.

DefinecIII

B= c

B− φ

AB

µ.



We now show that when rB≥ cIII

B, then Π

B,F

(0, minK

B, S) ≥ Π

B,F(0, d

B). By equation (7) we have

ΠB,F

(0,minK

B, S)−Π

B,F(0, d

B)

= rB min

minKB , S, dB

µ + (rBµ + φAB )

[minKB , S − dB

]+

− φBC

[dB −minKB , S]+ − cB minKB , Sµ

−((rB − cB )dBµ

),

= (rB− c

B)minK

B, Sµ + φ

AB

(minK

B, S − d

B

)

− (rB− c

B)d

Bµ,

= rB

(minKB , S − dB

)µ−

(cB

(minKB , S − dB

)µ

− φAB

(minKB , S − dB

)).

It is clear that if rB ≥ cIIIB

, then the right-hand side is positive, which implies

ΠB,F

(0,minK

B, S) ≥ Π

B,F(0, d

B).

Therefore, if rB≥ cIII

B, then Y ∗

B,F(Y

A,L= 0) = minK

B, S; otherwise, Y ∗

B,F(Y

A,L= 0) = d

B.

Case 1-ii, YA,L

= S − dB: We now show that when r

B≥ cII

B, then Π

B,F(S − d

B, d

B) ≥ Π

B,F(S − d

B, 0). By

equation (9) we have

ΠB,F

(S − dB, d

B)−Π

B,F(S − d

B, 0) = (r

B− c

B)d

Bµ− φ

BC(d

B− d

B)

−((r

B− c

B)(0)µ− φ

BC(d

B− 0)

),

= (rB− c

B)d

Bµ + φ

BCd

B,

= rBd

Bµ− (c

Bd

Bµ− φ

BCd

B).

It is clear that if rB≥ cII


ΠB,F

(S − dB , dB

) ≥ ΠB,F (S − dB , 0).

Therefore, if rB ≥ cIIB

, then Y ∗B,F

(YA,L = S − dB ) = dB ; otherwise, Y ∗B,F

(YA,L = S − dB ) = 0.

Case 1-iii, YA,L

= dA: By equation (11), it is easy to see that, if r

B≥ cII

B, then Y ∗

B,F(Y

A,L= d

A) = S − d

A;

otherwise, Y ∗B,F

(YA,L

= dA) = 0.

Case 2-i, YA,L

= 0: Define

ciB

= cB −φAB

(minKB , dA + dB − dB

)+ φBC dB

µ minKB, d

A+ d

B .

By equation (14), we have

ΠB,F

(0, minKB , dA + dB

)−ΠB,F (0, 0)

= rB

min

minKB, d

A+ d

B, d

B

µ + ψ

B

[minK

B, d

A+ d

B − d

B]+

− φBC

[dB −minKB , dA + dB

]+ − cB minKB , dA + dBµ−

(r

Bmin

0, d

B

µ + ψ

Bmin

d

A, [0− d

B]+

− φBC

[dB− 0]+ − c

B(0)µ

),

= rB

min

minKB, d

A+ d

B, d

B

µ

+ (rBµ + φ

AB)[minK

B, d

A+ d

B − d

B

]+− c

BminK

B, d

A+ d

Bµ

− (−φBC

dB),

= rB minKB , dA + dBµ−(cB minKB , dA + dBµ

− φAB


)− φBC dB

).


It is clear that, if rB≥ ci

B, then the right-hand-side is positive, which implies that

ΠB,F

(0, minK

B, d

A+ d

B) ≥ Π

B,F(0, 0).

Therefore, if rB ≥ ciB

holds, then Y ∗B,F

(YA,L = 0) = minKB , dA + dB; otherwise, Y ∗B,F

(YA,L = 0) = 0.

Definecii

B= cB −

φBC

µ= cII

B.


ΠB,F

(0, dB)−Π

B,F(0, 0) = (r

B− c

B)d

Bµ− (−φ

BCd

B),

= rBd

Bµ− (c

Bd

Bµ− φ

BCd

B).

It is clear that, if rB≥ cII

B, then the right-hand-side is positive, which implies that

ΠB,F

(0, dB) ≥ Π

B,F(0, 0).



(YA,L = 0) = dB ; otherwise, Y ∗B,F

(YA,L = 0) = 0.

Defineciii

B= c

B− φ

AB

µ= cIII

B.


ΠB,F

(0,minKB , dA + dB

)−ΠB,F (0, dB )

= rB

min

minKB, d

A+ d

B, d

B

µ + ψ

B

[minK

B, d

A+ d

B − d

B]+

− φBC

[dB −minKB , dA + dB

]+ − cB minKB , dA + dBµ−

((r

B− c

B)d

Bµ),

= rB

min

minKB, d

A+ d

B, d

B

µ

+ (rBµ + φ

AB)[minK

B, d

A+ d

B − d

B

]+− c

BminK

B, d

A+ d

Bµ

− (rB− c

B)d

Bµ,

= rB

(minK

B, d

A+ d

B − d

B

)µ−

(c

B

(minK

B, d

A+ d

B − d

B

)µ

− φAB


)).

It is clear that, if rB ≥ cIIIB

, then the right-hand-side is positive, which implies that

ΠB,F

(0, minK

B, d

A+ d

B) ≥ Π

B,F(0, d

B).

Therefore, if rB ≥ cIIIB


(YA,L = 0) = minKB , dA + dB; otherwise, Y ∗B,F

(YA,L = 0) = dB .

Case 2-ii, YA,L = dA : By equation (16), we have

ΠB,F

(dA, d

B)−Π

B,F(d

A, 0) = r

Bd

Bµ− c

Bd

Bµ− (−φ

BCd

B)

= rBd

Bµ− (c

Bd

Bµ− φ

BCd

B).

It is clear that, if rB ≥ cIIB

, then the right-hand-side is positive, which implies that

ΠB,F

(dA, d

B) ≥ Π

B,F(d

A, 0).



(YA,L = dA) = dB ; otherwise, Y ∗B,F

(YA,L = dA) = 0.

Case 2-iii, YA,L = dA + dB : Define

civB

= cB− φ

BA

µ,



and notice that, cIIB≥ civ

B, since we have assumed that ξ

BA≥ ξ

BC. By equation (20)

ΠB,F (dA + dB ,minS − dA − dB , dB)−ΠB,F (dA + dB , 0)

= rB

min

minS − dA− d

B, d

B, d

B

µ

− φBA

[dB −minS − dA − dB , dB

]+− c

BminS − d

A− d

B, d

Bµ

− (−φBA

dB)

=(r

B− (c

B− φ

BA

µ))minS − d

A− d

B, d

Bµ.

Therefore, if rB≥ civ

Bholds, then Y ∗

B,F(Y

A,L= d

A+d

B) = minS−d

A−d

B, d

B; otherwise, Y ∗

B,F(Y

A,L= d

A+d

B) =

0.

In summary, we found thresholds on Firm B’s unit profit margin in two cases (same as the configuration of theproof for Proposition 1): (The summary of the thresholds can be found in Table 8.)

• Case 1: When S ≤ dA

+ dB. For this case, we introduce three thresholds on Firm B’s unit profit margin,

namely,

(i) cIB, separating the region in which Firm B forfeits and the region in which Firm B is aggressive,

(ii) cIIB

, separating the region in which Firm B forfeits and the region in which Firm B protects, and

(iii) cIIIB

, separating the region in which Firm B protects and the region in which Firm B is aggressive.

• Case 2: When S > dA + dB . For this case, we introduce four thresholds on Firm B’s unit profit margin,namely,

(i) ciB, separating the region in which Firm B forfeits and the region in which Firm B is aggressive,

(ii) ciiB(= cII

B), separating the region in which Firm B forfeits and the region in which Firm B protects,

(iii) ciiiB

(= cIIIB

), separating the region in which Firm B protects and the region in which Firm B is aggressive,and

(iv) civB

, which separates the following two responses: Given Firm A purchases the amount of dA

+dB, Firm

B can either buy the remaining to protect itself from losing customers to Firm A (this response impliesthat dA + dB cannot be optimal for Firm A), or buy nothing (this response implies that dA + dB isoptimal for Firm A).

PROOF for the Leader, Firm A:

We now present our proofs for the regions of the optimal capacity purchasing policy for the Leader. We considerthe following two corresponding cases:

• Case 1: When S ≤ dA + dB ;

• Case 2: When S > dA

+ dB.

CASE 1, S ≤ dA + dB : For this case, in the following we will construct the threshold structure for optimal backupcapacity competition strategy under condition KA ≥ dA +dB ≥ S. We consider the following four sub-cases, whichcan occur depending on the profit margin of Firm B:

1. As Firm A’s profit margin increases (i.e., we move from left to right along the x-axis of the five-regionstructure), the following three scenarios occur:

• Both Firm A and Firm B forfeit, i.e., (Y ∗A,L

, Y ∗B,F

) = (0, 0), which guarantees that Y ∗B,F

(∀ YA,L

) = 0.

• Firm A protects and Firm B forfeits, i.e., (Y ∗A,L

, Y ∗B,F

) = (S − dB , 0) and (Y ∗A,L

, Y ∗B,F

) = (dA , 0).

• Firm A buys all and Firm B forfeits, i.e., (Y ∗A,L

, Y ∗B,F

) = (S, 0).


In this sub-case, we want to construct the relationship among ΠA,L

(0, 0), ΠA,L

(S − dB, 0), Π

A,L(d

A, 0), and

ΠA,L

(S, 0). In other words, we want to find the thresholds among the “A and B Forfeit” region, the “AProtects” region, and the “B Forfeits” region.


• Firm A forfeits and Firm B protects, i.e., (Y ∗A,L

, Y ∗B,F

) = (0, dB). This guarantees that when the Leader,

Firm A, decides to protect itself against losing customers to Firm B (and Firm C), the Follower, FirmB, will purchase to satisfy as many of its own customers as possible to protect against losing customersto Firm C.

• Both Firm A and Firm B protect, i.e., (Y ∗A,L

, Y ∗B,F

) = (S − dB, d

B) and (Y ∗

A,L, Y ∗

B,F) = (d

A, S − d

A).

• Firm A is aggressive, which results in (Y ∗A,L

, Y ∗B,F

) = (S, 0).


(0, dB), Π

A,L(S−d

B, d

B), Π

A,L(d

A, S−d

A),

and ΠA,L

(S, 0). In other words, we want to find the thresholds among the “A Forfeits” region, the “AProtects” region, and the “A is Aggressive” region.


• Firm A forfeits and Firm B is aggressive, i.e., (Y ∗A,L

, Y ∗B,F

) = (0,minKB , S).• Firm A protects and Firm B forfeit, i.e., (Y ∗

A,L, Y ∗

B,F) = (S − d

B, 0) and (Y ∗

A,L, Y ∗

B,F) = (d

A, 0).


, Y ∗B,F

) = (S, 0).

In this sub-case, we want to construct the relationship among ΠA,L(0, minKB , S), ΠA,L(S − dB , 0),ΠA,L(dA , 0), and ΠA,L(S, 0). In other words, we want to find the thresholds among the the “A Forfeits”region, the “A Protects” region, and the “A is Aggressive” region.



, Y ∗B,F

) = (0,minKB , S).• Both Firm A and Firm B protect, i.e., (Y ∗

A,L, Y ∗

B,F) = (S − d

B, d

B) and (Y ∗

A,L, Y ∗

B,F) = (d

A, S − d

A).


, Y ∗B,F

) = (S, 0).

In this sub-case, we want to construct the relationship among ΠA,L(0, minKB , S), ΠA,L(S − dB , dB ),ΠA,L(dA , S − dA), and ΠA,L(S, 0). In other words, we want to find the thresholds among the “A Forfeits”region, the “A Protects” region, and the “A is Aggressive” region.

CASE 1 – Subcase 1: In this sub-case, by investigating the relationship among ΠA,L

(0, 0), ΠA,L

(S − dB, 0),

ΠA,L

(dA, 0), and Π

A,L(S, 0), we find the thresholds among the “A and B Forfeit” region, the “A Protects” region,

and the “B Forfeits” region.If rA ≥ cI

A, which is defined as

cIA

= cA− φ

AC

µ,

then by equation (6) and equation (8), we have

ΠA,L(S − dB , 0)−ΠA,L(0, 0) = (rA − cA)(S − dB )µ− φAC

(dA − (S − dB )

)− (−φAC dA)

=(r

A− (c

A− φ

AC

µ))(S − d

B)µ ≥ 0;

also, by equation (8) and equation (10), we have

ΠA,L

(dA, 0)−Π

A,L(S − d

B, 0) = (r

A− c

A)d

Aµ−

((r

A− c

A)(S − d

B)µ− φ

AC

(d

A− (S − d

B)))

=(rA − (cA −

φAC

µ))(

dA − (S − dB ))µ ≥ 0.

That is, cIA

is the threshold between the “A and B Forfeit” region and the “A Protects” region. Also, cIA

is thethreshold between the following two scenarios: the “A Protects against only B” scenario and the “A Protectsagainst both B and C” scenario.



• If rA≥ cI

A, then we have

ΠA,L

(dA, 0) ≥ Π

A,L(S − d

B, 0) ≥ Π

A,L(0, 0).

Now, to check the relationship between ΠA,L(S, 0) and ΠA,L(dA , 0), we define

cIIA

= cA− φBA

µ.

If rA≥ cII

A, then by equation (10) and equation (12), we have

ΠA,L

(S, 0)−ΠA,L

(dA, 0) = r

Ad

Aµ + ψ

A(S − d

A)− c

ASµ− (

rAd

Aµ− c

Ad

Aµ)

=((r

Aµ + φ

BA)− c

Aµ)(S − d

A)

=(r

A− (c

A− φ

BA

µ))(S − d

A)µ ≥ 0.

That is, cIIA

is the threshold between the “A Protects” region and the “B Forfeits” region.

• If rA

< cIA, then we have

ΠA,L

(0, 0) ≥ ΠA,L

(S − dB, 0) ≥ Π

A,L(d

A, 0).

Now, to check the relationship between ΠA,L

(S, 0) and ΠA,L

(0, 0), we define

cIIIA

= cA− φ

BA[S − d

A]+ + φ

ACmind

A, S

µS.

If rA ≥ cIIIA

, then by equation (6) and equation (12), we have

ΠA,L

(S, 0)−ΠA,L

(0, 0) = rA

minS, dAµ + ψ

A[S − d

A]+ − φ

AC[d

A− S]+ − c

ASµ− (−φ

ACd

A)

= rASµ− (

cASµ− φ

BA[S − d

A]+ − φ

ACmind

A, S) ≥ 0.

That is, cIIIA

is the threshold between the “A and B Forfeit” region and the “B Forfeits” region.


(0, dB), Π

A,L(S − d

B, d

B),

ΠA,L

(dA, S − d

A), and Π

A,L(S, 0), we find the thresholds among the “A Forfeits” region, the “A Protects” region,

and the “A is Aggressive” region.

By equation (6), equation (8), and equation (10), we know

• if rA ≥ cIA, then

ΠA,L(dA , S − dA) ≥ ΠA,L(S − dB , dB ) ≥ ΠA,L(0, dB ).

That is, cIA

is the threshold between the “A Forfeits” region and the “A Protects” region.By equation (10) and equation (12), we know, if r

A≥ cII

A, then

ΠA,L

(S, 0) ≥ ΠA,L

(dA, S − d

A).

That is, cIIA

is the threshold between the “A Protects” region and the “A is Aggressive” region.

• if rA

< cIA, then

ΠA,L(0, dB ) ≥ ΠA,L(S − dB , dB ) ≥ ΠA,L(dA , S − dA).

By equation (6) and equation (12), we know, if rA ≥ cIIIA

, then

ΠA,L(S, 0) ≥ ΠA,L(0, dB ).

That is, cIIIA

is the threshold between the “A Forfeits” region and the “A is Aggressive” region.

CASE 1 – Subcase 3: In this sub-case, by investigating the relationship among ΠA,L(0, minKB , S), ΠA,L(S−d

B, 0), Π

A,L(d

A, 0), and Π

A,L(S, 0), we find the thresholds among the “A Forfeits” region, the “A Protects” region,



If rA≥ cIV


cIVA

= cA−

φAB

(minK

B, S − d

B

)+ φ

AC

((S − d

B)− (

minKB, S − d

B

))

µ(S − dB)

= cA− φ

AC

µ− (φ

AB− φ

AC)(minK

B, S − d

B

)

µ(S − dB)

,


ΠA,L

(S − dB, 0)−Π

A,L

(0,minK

B, S)

= (rA− c

A)(S − d

B)µ− φ

AC

(d

A− (S − d

B))

−(− φ

AB[minK

B, S − d

B]+ − φ

AC

(d

A− [

minKB , S − dB

]+))

= rA(S − d

B)µ−

(c

A(S − d

B)µ

− φAB

(minK

B, S − d

B

)− φAC

((S − d

B)− (

minKB, S − d

B

))) ≥ 0.

That is, cIVA

is the threshold between the “A Forfeits” region and the “A Protects against only B” region.If r

A≥ cV


cVA

= cA −φ

AB

[minK

B, S − d

B

]+ + φAC

(d

A− [

minKB, S − d

B

]+)

µdA

= cA− φ

AC

µ− (φ

AB− φ

AC)[minK

B, S − d

B

]+µd

A

,

(Notice that dA ≥ [S − dB ]+, so cV

A≥ cIV

A.)


ΠA,L(dA , 0)−ΠA,L

(0,minKB , S)

= (rA − cA)dAµ

−(− φ

AB

[minK

B, S − d

B

]+ − φAC

(d

A− [

minKB, S − d

B

]+))

= rAd

Aµ−

(c

Ad

Aµ

− φAB

[minK

B, S − d

B

]+ − φAC

(d

A− [

minKB, S − d

B

]+))≥ 0.

That is, cVA

is the threshold between the “A Forfeits” region and the “A Protects against both B and C” region.

Recall, by equation (8) and equation (10), we have shown, if rA ≥ cIA, then

ΠA,L(dA , 0)−ΠA,L(S − dB , 0) = (rA − cIA)(dA − (S − dB )

)µ ≥ 0.

Therefore, (Note that cIA≥ cV

A≥ cIV

Aautomatically holds),

• if rA ≥ cIA, then

ΠA,L

(dA, 0) ≥ Π

A,L(S − d

B, 0) ≥ Π

A,L(0, minK

B, S).

That is, cIA

is the threshold between the “A Forfeits” region and the “A Protects” region.Now, we want to check the relationship between ΠA,L(S, 0) and ΠA,L(dA , 0). Note that, we have shown thatif rA ≥ cII

A, then

ΠA,L(S, 0) ≥ ΠA,L(dA , 0).

That is, cIIA

is the threshold between the “A Protects against both B and C” region and the “A is Aggressive”region.

• if rA ≤ cIA

but rA ≥ cVA

(so rA ≥ cIVA

must hold), then

ΠA,L

(S − dB, 0) ≥ Π

A,L(d

A, 0) ≥ Π

A,L(0, minK

B, S);



also if rA≤ cV

Abut r

A≥ cIV

A, then

ΠA,L

(S − dB, 0) ≥ Π

A,L(0, minK

B, S) ≥ Π

A,L(d

A, 0);


(S, 0) and ΠA,L

(S − dB, 0), we define

cV IA

= cA −φ

BA[S − d

A]+ + φ

AC

(mind

A, S − (S − d

B))

µdB

.

If rA≥ cV I


ΠA,L

(S, 0)−ΠA,L

(S − dB, 0) = r

AminS, d

Aµ + ψ

A[S − d

A]+ − φ

AC[d

A− S]+ − c

ASµ

−((r

A− c

A)(S − d

B)µ− φ

AC

(d

A− (S − d

B)))

= rAd

Bµ−

(c

Ad

Bµ

−φBA

[S − dA]+ − φ

AC

(mind

A, S − (S − d

B))) ≥ 0.

That is, cV IA

is the threshold between the “A Protects against only B” region and the “A is Aggressive”region.

• if rA≤ cIV

A, then

ΠA,L

(0, minKB, S) ≥ Π

A,L(S − d

B, 0) ≥ Π

A,L(d

A, 0).


(S, 0) and ΠA,L

(0,minKB, S), we define

cV IIA

= cA− φ

AB[minK

B, S − d

B]+ + φ

BA[S − d

A]+

µS

−φAC

(mindA , S − [

minKB , S − dB

]+)

µS.

If rA≥ cV II


ΠA,L

(S, 0)−ΠA,L

(0, minK

B, S)

= rA

minS, dAµ + ψ

A[S − d

A]+ − φ

AC[d

A− S]+ − c

ASµ

−(− φ

AB

[minK

B, S − d

B

]+ − φAC

(d

A− [

minKB, S − d

B

]+))

= rASµ−(cASµ− φAB

[minKB , S − dB

]+ − φBA [S − dA ]+

− φAC

(mindA , S − [

minKB , S − dB

]+))≥ 0.

That is, cV IIA



(0, minKB, S), Π

A,L(S−

dB , dB ), ΠA,L(dA , S − dA), and ΠA,L(S, 0), we find the thresholds among the “A Forfeits” region, the “A Protects”region, and the “A is Aggressive” region.

By equation (6) and equation (8), we know, if rA≥ cIV

A, then

ΠA,L

(S − dB, d

B) ≥ Π

A,L

(0,minK

B, S).

By equation (6) and equation (10), we know, if rA ≥ cVA

, then

ΠA,L(dA , S − dA) ≥ ΠA,L

(0, minKB , S).

By equation (8) and equation (10), we know, if rA ≥ cIA, then

ΠA,L

(dA, S − d

A) ≥ Π

A,L(S − d

B, d

B).

Therefore, (Not that cIA≥ cV

A≥ cIV

Aautomatically holds),


• if rA≥ cI

A, then

ΠA,L

(dA, S − d

A) ≥ Π

A,L(S − d

B, d

B) ≥ Π

A,L(0,minK

B, S).

That is, cIA

is the threshold between the “A Forfeits” region and the “A Protects” region.

By equation (10) and equation (12), we know, if rA≥ cII

A, then

ΠA,L

(S, 0) ≥ ΠA,L

(dA, S − d

A).

That is, cIIA

is the threshold between the “A Protects against both B and C” region and the “A is Aggressive”region.

• if rA≤ cI

Abut r

A≥ cV

A(so r

A≥ cIV

Amust hold), then

ΠA,L

(S − dB, d

B) ≥ Π

A,L(d

A, S − d

A) ≥ Π

A,L(0,minK

B, S),

and if rA≤ cV

Abut r

A≥ cIV

A, then

ΠA,L

(S − dB, d

B) ≥ Π

A,L(0, minK

B, S) ≥ Π

A,L(d

A, S − d

A).

By equation (8) and equation (12), we know, if rA≥ cV I

A, then

ΠA,L

(S, 0) ≥ ΠA,L

(S − dB, d

B).

That is, cV IA

is the threshold between the “A Protects against only B” region and the “A is Aggressive”region.

• if rA≤ cIV

A, then

ΠA,L

(0,minKB, S) ≥ Π

A,L(S − d

B, d

B) ≥ Π

A,L(d

A, S − d

A).

By equation (6) and equation (12), we know, if rA≥ cV II

A, then

ΠA,L(S, 0) ≥ ΠA,L

(0, minKB , S).

That is, cV IIA


CASE 2, S > dA + dB : For this case, in the following we will construct the threshold structure for the optimalbackup capacity competition strategy under condition KA ≥ dA + dB and S > dA + dB . We consider the followingfive sub-cases:


• Both Firm A and Firm B forfeit, i.e., (Y ∗A,L

, Y ∗B,F

) = (0, 0), which guarantees that Y ∗B,F

(∀ YA,L

) = 0.


, Y ∗B,F

) = (dA, 0).

• Firm A buys enough to satisfy the total market and Firm B forfeits, i.e., (Y ∗A,L

, Y ∗B,F

) = (dA

+ dB, 0).

In this sub-case, we want to construct the relationship among ΠA,L(0, 0), ΠA,L(dA , 0), and ΠA,L(dA + dB , 0).In other words, we want to find the thresholds among the “A and B Forfeit” region, the “A Protects” region,and the “B Forfeits” region.


• Firm A forfeits and Firm B protects, i.e., (Y ∗A,L

, Y ∗B,F

) = (0, dB). This guarantees that when the Leader,

Firm A, decides to protect itself against losing customers, the Follower, Firm B, will purchase to satisfyas many of its own customers as possible to protect against losing customers to Firm C.


, Y ∗B,F

) = (dA, d

B).


, Y ∗B,F

) = (S, 0).

In this sub-case, we want to construct the relationship among ΠA,L(0, dB ), ΠA,L(dA , dB ), and ΠA,L(S, 0). Inother words, we want to find the thresholds among the “A Forfeits” region, the “A Protects” region, and the“A is Aggressive” region.





, Y ∗B,F

) = (0,minKB, d

A+ d

B).


, Y ∗B,F

) = (dA , 0).

• Firm A is aggressive and Firm B forfeits. More specifically, The Follower, Firm B, decides to buynothing when the Leader, Firm A, purchases the amount of d

A+ d

Bto cover the total market, i.e.,

(Y ∗A,L

, Y ∗B,F

) = (dA + dB , 0).


(0,minKB, d

A+ d

B), Π

A,L(d

A, 0), and

ΠA,L

(dA

+ dB, 0). In other words, we want to find the thresholds among the “A Forfeits” region, the “A

Protects” region, and the “A is Aggressive” region.



, Y ∗B,F

) = (0,minKB , dA + dB).• Firm A protects and Firm B forfeit, i.e., (Y ∗

A,L, Y ∗

B,F) = (d

A, 0).


, Y ∗B,F

) = (S, 0). Note that, comparing with the thirdscenario in the above sub-case, here, instead of purchasing the amount of d

A+ d

B, Firm A has to buy

all available backup capacity to corner the whole market, because if Firm A purchases the amount ofdA + dB , Firm B will buy the remaining backup capacity to protect itself against losing customers toFirm A, and hence d

A+ d

Bis not optimal for Firm A.

In this sub-case, we want to construct the relationship among ΠA,L(0,minKB , dA + dB), ΠA,L(dA , 0), andΠ

A,L(S, 0). In other words, we want to find the thresholds among the “A Forfeits” region, the “A Protects”

region, and the “A is Aggressive” region.



, Y ∗B,F

) = (0,minKB, d

A+ d

B).


, Y ∗B,F

) = (dA , dB ).


, Y ∗B,F

) = (S, 0).

In this sub-case, we want to construct the relationship among ΠA,L(0, minKB , dA +dB), ΠA,L(dA , dB ), andΠA,L(S, 0). In other words, we want to find the thresholds among the “A Forfeits” region, the “A Protects”region, and the “A is Aggressive” region.


(0, 0), ΠA,L

(dA, 0), and

ΠA,L(dA + dB , 0), we find the thresholds among the “A and B Forfeit” region, the “A Protects” region, and the “BForfeits” region.

First of all, it is easy to see that ΠA,L

(YA,L

= dA

+ dB, Y

B,F= 0) ≥ Π

A,L(Y

A,L= S, Y

B,F= 0) is always satisfied

due to the saving of holding cost.

• If rA≥ ci


ciA

= cA −φ

AC

µ= cI

A,


ΠA,L

(dA, 0)−Π

A,L(0, 0) = (r

A− c

A)d

Aµ− (−φ

ACd

A)

=(r

A− (c

A− φ

AC

µ))d

Aµ ≥ 0.

That is, cIA

is the threshold between the “A and B Forfeit” region and the “A Protects” region.


• If rA≥ cii


ciiA

= cA− φ

BA

µ= cII

A,


ΠA,L

(dA

+ dB, 0)−Π

A,L(d

A, 0) = r

Ad

Aµ + ψ

Ad

B− c

A(d

A+ d

B)µ− (r

A− c

A)d

Aµ

=(r

A− (c

A− φ

BA

µ))d

Bµ ≥ 0.

That is, cIIA

is the threshold between the “A Protects” region and the “B Forfeits” region.

• If rA≥ ciii


ciiiA

= cA− φ

BAd

B+ φ

ACd

A

µ(dA

+ dB)

,


ΠA,L

(dA

+ dB, 0)−Π

A,L(0, 0) = r

Ad

Aµ + ψ

Ad

B− c

A(d

A+ d

B)µ− (−φ

ACd

A)

=(r

A(d

A+ d

B)µ− (

cA(d

A+ d

B)µ− φ

BAd

B− φ

ACd

A

)) ≥ 0.

That is, ciiiA

is the threshold between the “A and B Forfeit” region and the “B Forfeits” region.

CASE 2 – Subcase 2: In this sub-case, by investigating relationship among ΠA,L(0, dB ), ΠA,L(dA , dB ), andΠ

A,L(S, 0), we find the thresholds among the “A Forfeits” region, the “A Protects” region, and the “A is Aggressive”

region.

• By equation (13) and equation (15), we know, if rA≥ ci

A(i.e., cI

A), then

ΠA,L

(dA, d

B) ≥ Π

A,L(0, d

B).

That is, cIA


• If rA ≥ civA

, which is defined as

civA

= cA

S − dA

dB

+hA(S − d

A− d

B)

dB

− φBA

µ,


ΠA,L(S, 0)−ΠA,L(dA , dB ) = rAdAµ + ψAdB − cASµ− hA(S − dA − dB)µ− ((rA − cA)dAµ

)

= rAd

Bµ−

(c

A(S − d

A)µ + hA(S − d

A− d

B)µ− φ

BAd

B

)≥ 0.

That is, civA


• If rA≥ cv


cvA

= cA

S

dA + dB

+hA(S − dA − dB )

dA + dB

− φBAdB

µ(dA + dB )− φAC dA

µ(dA + dB ),


ΠA,L

(S, 0)−ΠA,L

(0, dB) = rAdAµ + ψAdB − cASµ− hA(S − dA − dB)µ− (−φ

ACd

A)

= rA(d

A+ d

B)µ−

(c

ASµ + hA(S − d

A− d

B)µ− φ

BAd

B− φ

ACd

A

)≥ 0.

That is, cvA


CASE 2 – Subcase 3: In this sub-case, by investigating the relationship among ΠA,L(0,minKB , dA + dB),Π

A,L(d

A, 0), and Π

A,L(d

A+ d

B, 0), we find the thresholds among the “A Forfeits” region, the “A Protects” region,




• If rA≥ cvi


cviA

= cA− φ

AB

(minK

B, d

A+ d

B − d

B

)

µdA

−φ

AC

(d

A− (

minKB, d

A+ d

B − d

B

))

µdA

,

= cA− φ

AC

µ− (φ

AB− φ

AC)(minK

B, d

A+ d

B − d

B

)

µdA

,


ΠA,L(dA , 0)−ΠA,L

(0,minKB , dA + dB

)

= (rA− c

A)d

Aµ

−(− φ

ABmin

minK

B, d

A+ d

B − d

B, d

A

− φ

AC

[d

A− (

minKB, d

A+ d

B − d

B

)]+)

= rAd

Aµ−

(c

Ad

Aµ

− φAB

(minK

B, d

A+ d

B − d

B

)− φAC

(d

A− (minK

B, d

A+ d

B − d

B))) ≥ 0.

That is, cviA


• We have shown that if rA≥ cII

A, then

ΠA,L

(dA

+ dB, 0) ≥ Π

A,L(d

A, 0).

That is, cIIA


• If rA≥ cvii


cviiA

= cA −φAB


)

µ(dA + dB )− φBAdB

µ(dA + dB )

−φ

AC

(d

A− (

minKB, d

A+ d

B − d

B

))

µ(dA

+ dB)

,


ΠA,L

(dA

+ dA, 0)−Π

A,L

(0, minK

B, d

A+ d

B)

= rAd

Aµ + ψ

Ad

B− c

A(d

A+ d

B)µ

−(− φ

ABmin

minK

B, d

A+ d

B − d

B, d

A

− φAC

[d

A− (

minKB, d

A+ d

B − d

B

)]+)

= rA(d

A+ d

B)µ−

(c

A(d

A+ d

B)µ

− φAB


)− φBAdB

− φAC

(d

A− (

minKB, d

A+ d

B − d

B

)))≥ 0.

That is, cviiA



(0,minKB, d

A+ d

B),

ΠA,L(dA , 0), and ΠA,L(S, 0), we find the thresholds among the “A Forfeits” region, the “A Protects” region, andthe “A is Aggressive” region.

• We have shown that if rA≥ cvi

A, then

ΠA,L

(dA, 0) ≥ Π

A,L

(0, minK

B, d

A+ d

B).

That is, cviA



• If rA≥ civ


ΠA,L

(S, 0)−ΠA,L

(dA, 0)

= rAdAµ + ψAdB − cASµ− hA(S − dA − dB)µ− (

(rA− c

A)d

Aµ)

= rAdBµ−(cA(S − dA)µ + hA(S − dA − dB )µ− φBAdB

)≥ 0.

That is, civA


• If rA≥ cviii


cviiiA

= cA

S

dA

+ dB

+hA(S − d

A− d

B)

dA

+ dB

− φAB

(minK

B, d

A+ d

B − d

B

)

µ(dA

+ dB)

− φBA

dB

µ(dA

+ dB)

−φ

AC

(d

A− (

minKB, d

A+ d

B − d

B

))

µ(dA

+ dB)

,


ΠA,L

(S, 0)−ΠA,L

(0, minKB, d

A+ d

B)

= rAdAµ + ψAdB − cASµ− hA(S − dA − dB)µ

−(− φAB min

minKB , dA + dB − dB , dA

− φAC

[dA −


)]+)

= rA(d

A+ d

B)µ−

(c

ASµ + hA(S − d

A− d

B)µ

− φAB

(minK

B, d

A+ d

B − d

B

)− φBA

dB

− φAC

(dA − (minKB , dA + dB − dB )

)) ≥ 0.

That is, cviiiA



(0,minKB, d

A+ d

B),

ΠA,L(dA , dB ), and ΠA,L(S, 0), we find the thresholds among the “A Forfeits” region, the “A Protects” region, andthe “A is Aggressive” region.

• By equation (13) and equation (15), we know, if rA ≥ cviA

, then

ΠA,L

(dA, d

B) ≥ Π

A,L

(0, minK

B, d

A+ d

B).

That is, cviA


• By equation (15) and equation (21), we know, if rA≥ civ

A, then

ΠA,L(S, 0) ≥ ΠA,L(dA , dB ).

That is, civA


• By equation (13) and equation (21), we know, if rA≥ cviii

A, then

ΠA,L(S, 0) ≥ ΠA,L(0,minKB , dA + dB).That is, cviii

Ais the threshold between the “A Forfeits” region and the “A is Aggressive” region.

This completes the proof of Proposition 2 for the “Large Capacity Leader”. ¤

PROPOSITION 3: A unique Nash Equilibrium exists for each firm’s Advanced Preparedness Competition.


To prove that a unique Nash Equilibrium exists for the AP Competition of Firm A, first we must show the following(Rudin 1976):



Thresholds Expression

When S ≤ dA + dB

cIA

cA −φ

ACµ

cIIA

cA −φ

BAµ

cIIIA

cA −φ

BA[S−d

A]+

µS − φAC

mindA

,SµS

cIVA

cA −φ

ACµ − (φ

AB−φ

AC)(

minKB

, S−dB

)µ(S−d

B)

cVA

cA −φ

ACµ −

(φAB

−φAC

)

[minK

B, S−d

B

]+

µdA

cV IA

cA −φ

BA[S−d

A]+

µdB

−φ

AC

(mind

A,S−(S−d

B)

)

µdB

cV IIA

cA −φ

AB

[minK

B, S−d

B

]+

µS − φBA

[S−dA

]+

µS −φ

AC

(mind

A, S−

[minK

B, S−d

B

]+)

µS

cIB

cB −φ

AB

[minK

B, S−d

B

]+

µ minKB

, S −φ

BCmin

d

B,minK

B, S

µ minKB

, S

cIIB

cB −φ

BCµ

cIIIB

cB −φ

ABµ

When S > dA + dB

cIA

cA −φ

ACµ

cIIA

cA −φ

BAµ

ciiiA

cA −φ

BAd

Bµ(d

A+d

B) −

φAC

dA

µ(dA

+dB

)

cviA

cA

S−dA

dB

+ hA(S−dA−d

B)

dB

− φBAµ

cvA

cAS

dA

+dB

+ hA(S−dA−d

B)

dA

+dB

− φBA

dB

µ(dA

+dB

) −φ

ACd

Aµ(d

A+d

B)

cviA

cA −φ

ACµ − (φ

AB−φ

AC)(

minKB

, dA

+dB−d

B

)µd

A

cviiA

cA −φ

AB

(minK

B, d

A+d

B−d

B

)µ(d

A+d

B) − φ

BAd

Bµ(d

A+d

B) −

φAC

(d

A−(

minKB

, dA

+dB−d

B

))

µ(dA

+dB

)

cviiiA

cAS

dA

+dB

+ hA(S−dA−d

B)

dA

+dB

− φAB

(minK

B, d

A+d

B−d

B

)µ(d

A+d

B)

− φBA

dB

µ(dA

+dB

) −φ

AC

(d

A−(

minKB

, dA

+dB−d

B

))

µ(dA

+dB

)

ciB

cB −φ

AB

(minK

B, d

A+d

B−d

B

)

µ minKB

, dA

+dB − φ

BCd

Bµ minK

B, d

A+d

B

cIIB

cB −φ

BCµ

cIIIB

cB −φ

ABµ

civB

cB −φ

BAµ

Table 8: Thresholds for the “Large Capacity Leader” case.


Thresholds Expression

cIA

cA −φ

ACµ

cIIA

cA −φ

BAµ

cIIIA

cA −φ

BA[S−d

A]+

µS − φAC

mindA

,SµS

cIVA

cA −φ

ACµ −

(φAB

−φAC

)

(minK

B, S−d

B

)

µ(S−dB

)

cVA

cA −φ

ACµ −

(φAB

−φAC

)

[minK

B, S−d

B

]+

µdA

cV IA

cA −φ

BA[S−d

A]+

µdB

−φ

AC

(mind

A,S−(S−d

B)

)

µdB

cV IIA

cA −φ

AB

[minK

B, S−d

B

]+

µS − φBA

[S−dA

]+

µS −φ

AC

(mind

A, S−

[minK

B, S−d

B

]+)

µS

c8A

cA −φ

BA(K

A−d

A)

µKA

− φAC

dA

µKA

c9A

cA −φ

BA(K

A−d

A)

µ

(K

A−(S−d

B)

) −φ

AC

(d

A−(S−d

B)

)

µ

(K

A−(S−d

B)

)

c10A

cA −φ

AB

[minK

B, S−d

B

]+

µKA

− φBA

(KA−d

A)

µKA

−φ

AC

(d

A−[

minKB

, S−dB

]+)

µKA

c11A

cA

S−(2dA

+dB

)+KA

µ(KA−d

A) + hA

S−(dA

+dB

)

µ(KA−d

A) −

φBAµ

c12A

cA

S−(dA

+dB

)+KA

KA

+ hAS−(d

A+d

B)

KA

− φBA

(KA−d

A)

µKA

− φAC

dA

µKA

c13A

cA −φ

AB

(minK

B, d

A+d

B−d

B

)µ(d

A+d

B) −

φAC

(d

A−(

minKB

, dA

+dB−d

B

))

µ(dA

+dB

)

c14A

cA −φ

AB

(minK

B, d

A+d

B−d

B

)µK

A− φ

BA(K

A−d

A)

µKA

−φ

AC

(d

A−(

minKB

, dA

+dB−d

B

))

µKA

c15A

cA

S−(dA

+dB

)+KA

µKA

+ hAS−(d

A+d

B)

µKA

− φAB

(minK

B, d

A+d

B−d

B

)µK

A

− φBA

(KA−d

A)

µKA

−φ

AC

(d

A−(

minKB

, dA

+dB−d

B

))

µKA

c1B

cB −φ

AB

[min

mind

A+d

B,S, K

B

−d

B

]+

µ min

mind

A+d

B,S, K

B

−φ

BCmin

d

B,min

mind

A+d

B,S, K

B

µ min

mind

A+d

B,S, K

B

cIIB

cB −φ

BCµ

cIIIB

cB −φ

ABµ

c4B

cB −φ

BCµ − (φ

BA−φ

BC)(S−d

A−d

B)

µ(S−KA

)

c5B

cB −φ

BCµ − (φ

BA−φ

BC)(K

A−d

A)

µdB

Table 9: Thresholds for the “Small Capacity Leader” case.



• Function ΠAPi does have a unique maximum, i ∈ A,B.

• Function ΠAPj,i does have a unique maximum, i 6= j and i, j ∈ A,B.

• Function xi(x

j) is continuous in x

jand strictly concave in x

j, i 6= j and i, j ∈ A,B.

Let Π∗i,L and Π∗i,F be the optimal expected profits for Firm i when it is the Leader and the Follower, respectively,in the BC Competition. It is easy to see that for Firm i, the profit of being the Leader in the BC Competitionis, at least, as much as being the Follower, and furthermore, if Π∗i,L = Π∗i,F , then Firm i will definitely have nointerest in the AP Competition, which is obviously not an interesting case. So here we assume that Π∗i,L > Π∗i,F ,i ∈ A,B. Also xi > 0, since xi represents Firm i’s preparedness effort, and that zero implies that Firm i has nointerest in the AP Competition.

To show that, for any given Firm B’s preparedness effort xoB, function ΠAP

i has a unique maximum, xA, consider

f(xA|xo

B) ≡ ΠAP

A = pA0

(π

AΠ∗

A,L+ π

BΠ∗

A,F−m

AC(d0

A− d

A))

+ (1− pA0)rA

d0A(µ + T )

−CexA+ m

AAd0

A,

= CAL

( xA

xA + xoB

)+ CA

F

( xoB

xA + xoB

)− CexA

+ constant, (22)

where CAL = p

A0Π∗A,L

, CAF = p

A0Π∗A,F

. Note that CAL , CA

F and Ce are independent of xi, and xi is positive. So,

f′(x

A|xo

B) =

d

d xA

f(xA|xo

B) =

(CAL − CA

F )xoB

(xA

+ xoB)2

− Ce.

By setting f′(x

A|xo

B) = 0, and solving for x

A, we get

xA

=

√(CA

L − CAF )

Cexo

B− xo

B,

=

√pA0(Π∗A,L

−Π∗A,F

)Ce

xoB− xo

B. (23)

Notice that, CAL − CA

F ≥ 0 always holds, since p0 is non-negative and we have already argued that Π∗A,L

> Π∗A,F

.It is clear that for any fixed xo

B, there exists a unique x

A, which is a non-negative real number, i.e., the optimal

solution of f(xA|x

B) does exist and it is unique.

Furthermore,

f′′(x

A|xo

B) =

−2(CAL − CA

F )xoB

(xA + xoB)3

< 0,

which implies that the optimal solution of f(xA |xB ) is a maximum.

Similarly, we can show that, for any given Firm A’s preparedness effort xoA, function ΠAP

B,Ahas a unique maximum,

xB. Consider,

f(xB |xoA) ≡ ΠAP

B,A= pB0,A

(πBΠ∗

B,L+ πAΠ∗

B,F−mBC (d0

B− dB )

)+ (1− pB0,A)rBd0

B(µ + T )

−CexB+ m

BBd0

B

= pB0,A

Π∗B,L

( xB

xoA

+ xB

)+ p

B0,AΠ∗

B,F

( xoA

xoA

+ xB

)− CexB

+ constant, (24)

and,

xB

=

√p

B0,A(Π∗

B,L−Π∗

B,F)

Cexo

A− xo

A. (25)

We now show that function xA(x

B) is continuous in x

Band strictly concave in x

B. Letting H =

√CA

L−CAF

Ce, which

is a non-negative constant, from equation (23) we obtain

xA(x

B) = H ×√x

B− x

B+ constant.


First of all, it is easy to see that the above expression of xA(x

B) is continuous by the form of its function. Secondly,

it is strictly concave becaused2

dx2B

xA(x

B) = −1

4H × (x

B)−3/2 < 0

is satisfied. Similarly, xB(x

A) is continuous and strictly concave.

Having the above results, we now show that Nash Equilibrium exists and is unique. Since we have already shownthat x

A(x

B) is strictly concave, to prove that the Nash Equilibrium exists, we need to show that at the very

beginning of the range of xB∈ (0,∞), we must have d

dxB

xA(x

B) > 1 (Rudin 1976). Intuitively, this condition

guarantees xA(xB ) will intersect with the line, xA = xB , and the similar condition of xB (xA) will guarantee itsintersection with x

A= x

B. Therefore, a unique Nash Equilibrium does exist.

We now check whether there exists a xB, which is within the very beginning of the range (0,∞), such that

H2 (xB )−1/2 − 1 > 1.

d

dxB

xA(x

B) =

H

2(x

B)−1/2 − 1 > 1

⇔ x1/2B

< H4

⇔ xB <CA

L−CAF

16Ce,

which obviously can be easily satisfied by some xB

within the very beginning of the range (0,∞), since the right-hand-side is just a positive constant. Similar arguments hold for xB (xA). This completes the proof of Proposition3. ¤

PROPOSITION 4: Under the symmetric information conditions, for a given level of Firm j’s preparedness effect,the optimal preparedness effort of Firm i (i.e., x∗i ) is nondecreasing in mij, miC , mji, and ri, and is nonincreasingin γij, γiC , γji, ci, and hi, for j 6= i, and i, j ∈ A, B.


We assumed that both firms know the other firm’s estimate of the likelihood of a disruption, i.e., pi0,j = pi0 , i 6=j, i, j ∈ A, B, and based on the firm’s knowledge on the actual likelihood of a disruption, we consider two mutuallyexclusive and collectively exhaustive sub-cases:

• Case 1: both firms know the actual likelihood of a disruption.

• Case 2: their estimate of the likelihood of a disruption may not be accurate.

Since the proof procedure is similar, we will just give one detailed example. In the following, we show that theproposition hold under Case 1.

In the proof of Proposition 3, we have already shown that, for any given Firm B’s preparedness effort xoB, Firm

A’s optimal preparedness effort is

xA

=

√CA

L − CAF

Cexo

B− xo

B.

Thus, to show that xA is nondecreasing (or nonincreasing) in say parameter α, we only need to show that CAL −CA

F

is nondecreasing (or nonincreasing) in α. According to equations (31) and (32), under complete information, wehave

CAL = po

(rA min

Y ∗

A,L, dA

µ + ψA min

[dB − Y ∗

B,F]+, min

[Y ∗

A,L− dA ]+,KA − dA

−mABξAB min

min[Y ∗

B,F− dB ]+,KB − dB

, [dA − Y ∗

A,L]+

−mAC ξAC

[[dA − YA,L ]+ −min

[Y ∗

B,F− dB ]+,KB − dB

]+

− cAY ∗A,L

µ− hA(some non-negative terms)),



and

CAF = po

(rA min

Y ∗

A,F, dA

µ + ψA min

[dB − Y ∗

B,L]+,min

[Y ∗

A,F− dA ]+,KA − dA

−mAB

ξAB

min

min[Y ∗

B,L− d

B]+,K

B− d

B

, [d

A− Y ∗

A,F]+

−mAC

ξAC

[[d

A− Y ∗

A,F]+ −min

[Y ∗

B,L− d

B]+, K

B− d

B

]+

− cAY ∗

A,Fµ).

Therefore,

CAL − CA

F = po

(r

A

(minY ∗

A,L, d

A −minY ∗

A,F, d

A)µ

− cA(Y ∗

A,L− Y ∗

A,F)µ− h

A(some non-negative terms)

+ ψA min

[dB − Y ∗B,F

]+, min[Y ∗

A,L− dA ]+,KA − dA

− ψA

min

[dB− Y ∗

B,L]+, min

[Y ∗

A,F− d

A]+,K

A− d

A

−mAB

ξAB

min

min[Y ∗

B,F− d

B]+,K

B− d

B

, [d

A− Y ∗

A,L]+

+ mABξAB min

min[Y ∗

B,L− dB ]+,KB − dB

, [dA − Y ∗

A,F]+

−mAC

ξAC

[[d

A− Y ∗

A,L]+ −min

[Y ∗

B,F− d

B]+,K

B− d

B

]+

+ mAC ξAC

[[dA − Y ∗

A,F]+ −min

[Y ∗


]+). (26)

It is easy to see that for Firm i, the purchasing amount when the firm is the Leader is at least as many as that whenthe firm is the Follower. The reason is the when the firm is the Leader it has access to a larger backup capacitythan that when it is the Follower. Thus, Y ∗

A,L≥ Y ∗

A,Fand Y ∗

B,L≥ Y ∗

B,F. We have

• With respect to the coefficient of ψA , i.e., the third and the fourth lines on the right-hand-side of equation(26), we have [dB − Y ∗

B,F]+ ≥ [dB − Y ∗

B,L]+ and [Y ∗

A,L− dA ]+ ≥ [Y ∗

A,F− dA ]+, so

min

[dB− Y ∗

B,F]+, min

[Y ∗

A,L− d

A]+,K

A− d

A

−min

[d

B− Y ∗

B,L]+, min

[Y ∗

A,F− d

A]+, K

A− d

A

≥ 0,

i.e., the coefficient of ψA

in equation (26) is non-negative. Furthermore, we know

ψA

= rAµ + m

BAξ

BA

= rAµ + mBA

∫ ∞

0

(1− e− t

γBA )dFo(t).

So, equation (26) (and therefore xA) is nondecreasing in mBA and nonincreasing in γBA . Furthermore, thesummation of the third and the fourth lines on the right-hand-side of equation (26) is also nondecreasing inr

A.

• With respect to the coefficient of rA

in the first line on the right-hand-side of equation (26), we have

minY ∗A,L

, dA −minY ∗

A,F, d

A ≥ 0

since Y ∗A,L

≥ Y ∗A,F

. Therefore, we can conclude that the coefficient of rA in the first line on the right-hand-sideof equation (26) is non-negative, and therefore equation (26) (and therefore xA) is nondecreasing in rA .

• With respect to the coefficient of mABξAB , i.e., the fifth and the sixth lines on the right-hand-side of equation(26), we know [Y ∗

B,L− d

B]+ ≥ [Y ∗

B,F− d

B]+ and [d

A−Y ∗

A,F]+ ≥ [d

A−Y ∗

A,L]+, so the coefficient of mABξAB in

equation (26) is non-negative. So, equation (26) (and therefore xA) is nonincreasing in γ

AB. Furthermore, the

coefficient of mAB

(i.e., the fifth and the sixth lines in on the right-hand-side of equation (26)) is non-negative.Therefore, that part of equation (26) is non-decreasing in mAB .

• With respect to the other coefficient of mAC ξAC , i.e., the last two lines on the right-hand-side of in equation(26), we will show that

[[d

A− Y ∗

A,L]+ −min

[Y ∗

B,F− d

B]+,K

B− d

B

]+

−[[dA − Y ∗

A,F]+ −min

[Y ∗


]+

≤ 0. (27)


It is easy to see that,

– if [dA−Y ∗

A,L]+ = 0, then equation (27) is smaller than or equal to zero. This is because δ ≡ min

[Y ∗

B,F−

dB]+,K

B−d

B

≥ 0 since both [Y ∗B,F

−dB]+ ≥ 0 and K

B−d

B≥ 0. Therefore, the first term of equation

(27) becomes [0− δ]+ = 0. Furthermore, the second term of equation (27) is non-negative.

– if dA − Y ∗A,L

> 0, then by Proposition 1, we know, Y ∗A,L

∈ 0, S − dB. Hence,

∗ if Y ∗A,L

= 0, then Y ∗A,F

= 0 and Y ∗B,L

= Y ∗B,F

must hold. This is because Y ∗A,F

≤ Y ∗A,L

always holds,so Y ∗

A,L= 0 implies that Y ∗

A,F= 0. Thus, Firm A always purchases nothing, i.e., Firm A’s position

in the BC Competition has no effect on Firm B’s purchase decision, and hence Y ∗B,L

= Y ∗B,F

. As aresult, equation (27) is equal to zero.

∗ if Y ∗A,L

= S − dB , then it is easy to see that, Y ∗B,L

= minKB , S and Y ∗B,F

∈ 0, dB must hold.Because Y ∗

A,L= S − d

Bmeans that to be the Leader Firm A protects, which by Proposition 2

implies that Firm B must be aggressive to steal Firm A’s customers. So Y ∗B,L

= minKB, S, and

Y ∗B,F

∈ 0, dB can be obtained directly from Proposition 1. As a result, equation (27) becomes

(dA − (S − dB )

)−(dA −minS − dB ,KB − dB

)

= minS − dB,K

B− d

B − (S − d

B) ≤ 0.

Therefore, equation (27) is non-positive, i.e., the coefficient of mAC

ξAC

in equation (26) is non-negative.Consequently, equation (26) (and therefore x

A) is nonincreasing in γ

AC. Furthermore, we can conclude that

equation (26) (and therefore xA) is nondecreasing in mAC .

Finally, it is clear that equation (26) (and therefore xA) is nonincreasing in cA and hA . This completes the prooffor Case 1. The proof of Case 2 is similar, and therefore omitted. ¤

PROPOSITION 5: Under the symmetric information conditions, if both firms are “aggressive” and are identicalexcept for their size (i.e., d

A6= d

B), then both firms spend the same amount of effort on preparedness, and therefore

will have the same chance (50%) to become the Leader in the BC Competition.


We know that the probability for Firm i to be the Leader in the BC Competition is,

π∗i =x∗i

x∗i + w∗j, π∗j = 1− π∗i ,

where x∗i is the preparedness effort Firm i spends in the AP Competition and w∗j is the preparedness effort Firm jspends in the AP Competition.

Based on the proof of Proposition 3, we have the expression of x∗A from equations (23) and (25). Recall that x∗i isFirm A’s actual preparedness effort Firm i given it believes Firm j’s preparedness effort is x∗j . Plugging the uniqueNash Equilibrium for the AP Competition for Firm i, i.e., (x∗A, x∗B), into equations (23) and (25), under interfirminformation, we have

x∗A =

√p

A0(Π∗A,L−Π∗

A,F)

Cex∗B − x∗B ,

and

x∗B =

√p

A0(Π∗B,L−Π∗

B,F)

Cex∗A − x∗A.

Note that interfirm information indicates pi0,j = pi0 , i, j ∈ A,B, and identical firms implies pA0 = p

B0 .

We can write out the preparedness investment by Firm A in the AP Competition, x∗A, as follows:

x∗A

=pA0

Ce

(∆Π∗A)2∆Π∗B(∆Π∗A + ∆Π∗B)2

, (28)

where ∆Π∗i = Π∗i,L −Π∗i,F , i ∈ A,B.



Similarly, we obtain w∗B

based on equations (23) and (25) as follows:

w∗B

=p

A0

Ce

(∆Π∗B)2∆Π∗A(∆Π∗B + ∆Π∗A)2

. (29)

Now, by the Equations (28) and (29), the probability for Firm i to be the Leader in the BC Competition becomes,

π∗i

=x∗i

x∗i + w∗j=

∆Π∗i∆Π∗i + ∆Π∗j

.

Therefore, to show π∗i

= 50%, we only need to show that ∆Π∗i = ∆Π∗j .

In the following, since two firms are identical except the firm size, we omit the subscript of all parameters exceptthe firm size that indicates Firm i or Firm j (e.g., K ≡ KA = KB ). Also, without loss of generality, we assumed

A> d

B.

When dA

> dB, based on the relationship among d

A, d

B, K, and d

A+d

Bwe have the following two different cases:

dB≤ d

A≤ K ≤ d

A+ d

Band d

B≤ d

A≤ d

A+ d

B≤ K. Furthermore, under each case, we also need to discuss the

relationship among the backup capacity S, firms’ sales, the total market sales, and their production capacities, sofor each case, we consider five mutually exclusive and collectively exhaustive sub-cases:

• Case 1: dB≤ d

A≤ K ≤ d

A+ d

B:

– S ≤ dB;

– dB ≤ S ≤ dA ;

– dA≤ S ≤ K;

– K ≤ S ≤ dA + dB ;

– dA

+ dB≤ S.

• Case 2: dB≤ d

A≤ d

A+ d

B≤ K:

– S ≤ dB;

– dB ≤ S ≤ dA ;

– dA≤ S ≤ d

A+ d

B;

– dA + dB ≤ S ≤ K;

– K ≤ S.

Since the proof procedure is similar, we will just give one detailed example. In the following, we show that theproposition hold under the situation where d

B≤ d

A≤ K ≤ S ≤ d

A+ d

B.

When K ≤ S ≤ dA + dB , based on Proposition 1, since both firms are “aggressive”, i.e., given the chance, a firmwill buy all available backup capacity in an attempt to steal sales from the competitor, it is easy to see that theonly two possible optimal purchases combinations are:

(Y ∗i,L, Y ∗

j,F ; Y ∗j,L, Y ∗

i,F ) ∈

(K, S −K;K,S −K), (K, 0; K, 0)

.

• If (Y ∗i,L, Y ∗

j,F ;Y ∗j,L, Y ∗

i,F ) = (K,S −K; K,S −K), then we have,

∆Π∗i = rµ(di − (S −K)

)+ (rµ + mξ)(K − di) + mξ(K − dj) + mξ

C

((di + dj)− S

)

−cµ(K − (S −K)

),

= (r − c)µ(2K − S) + mξ(2K − (di + dj)

)+ mξC

((di + dj)− S

),

which clearly depends on the total duopoly market size, di + dj .

• If (Y ∗i,L, Y ∗

j,F ;Y ∗j,L, Y ∗

i,F ) = (K, 0;K, 0), then we have,

∆Π∗i = (r − c)µK + mξ(2K − (di + dj)

)+ mξ

C

((di + dj)−K

),

which clearly depends on the total duopoly market size, di + dj .


So, when dB≤ d

A≤ K ≤ S ≤ d

A+ d

B, ∆Π∗i = ∆Π∗j holds, and hence each firm has a probability of 50% to win

the BC Competition. This completes the proof for this case. The proofs of other cases are similar, and thereforeomitted. ¤

References

Rudin, W., 1976. Principles of Mathematical Analysis (3rd Ed.). McGraw-Hill, New York.



ONLINE APPENDIX C

Numerical Study on Impact of the Post-disruption Information

In this appendix, we examine the impact of post-disruption information on the firm’s preparedness investment, aswell as on their capacity decisions after the disruption. To do this, we consider the following information scenarios:

• Pre-Disruption Information: Prior to a disruption, Firm i knows that, if a disruption occurs, it could be onethe following two types:

(a) with probability pS

i0the disruption will be a long disruption with Severe consequences, which lasts an

average of µS

units of time with distribution FS(·), or

(b) with probability of pM

i0the disruption will be a short disruption with Minor consequences, which lasts

an average of µM

units of time with distribution of FM

(·).Firms use this information to play the Advanced Preparedness Competition and decide how much to investin preparation. We refer to the disruption with severe consequences as a type S disruption, and the one withminor consequences as a type M disruption.

• Post-Disruption Information: After the disruption, we use the conditional probability Pi(h|h), i ∈ A,B,

and h ∈ S,M, to denote the probability that Firm i is correct about the type of disruption. This conditionalprobability represents the accuracy of the firm in its assessment of the disruption type after the disruptionoccurs. If the post-disruption information is perfect, then Pi(S|S) = 1, Pi(S|M) = 0, Pi(M |S) = 0, andPi(M |M) = 1. Note that, we assume Pi(h|h) 6= Pj (h|h), because we want to investigate the impact of onefirm having more accurate post-disruption information than the other.

We incorporate these new information scenarios as follows:

Advanced Preparedness (AP) Competition: Firm i plays the AP Competition based on its estimates of thelikelihoods of each type of disruption, p

S

i0and p

M

i0, and its belief of Firm j’s estimates of these likelihoods,

pS

j0,iand p

M

j0,i(i 6= j and i, j ∈ A,B).

So we have, ΠAPi , which defined as Firm i’s total expected profit in the AP Competition for Firm i as:

ΠAPi = p

S

i0

( xi

xi + xjΠ

S

i,L(Y ∗i,L, Y ∗

j,F ) +xj

xi + xjΠ

S

i,F (Y ∗i,F , Y ∗

j,L)−miC (d0i − di)

)

+pM

i0

( xi

xi + xjΠ

M

i,L(Y ∗i,L, Y ∗

j,F ) +xj

xi + xjΠ

M

i,F (Y ∗i,F , Y ∗

j,L)−miC (d0i − di)

)

+(1− pS

i0− p

M

i0)(rid

0i (µ + T )

)− Cexi + miid

0i ,

where Πh

i,k(Y ∗i,k, Y ∗

j,k) (i, j ∈ A,B and i 6= j, k, k ∈ L,F, and h ∈ S,M) is defined as the sum of the

total expected profit of Firm i, given it is in position k during a type h disruption.

At the same time, Firm i believes that Firm j uses pS

j0,iand p

M

j0,ias its estimates for the likelihood of a

disruption, and hence, believes that Firm j is maximizing the following:

ΠAPj,i = p

S

j0,i

( xj

xi + xjΠ

S

j,L(Y ∗j,L, Y ∗

i,F ) +xi

xi + xjΠ

S

j,F (Y ∗j,F , Y ∗

i,L)−mjC (d0j − dj)

)

+pM

j0,i

( xj

xi + xjΠ

M

j,L(Y ∗j,L, Y ∗

i,F ) +xi

xi + xjΠ

M

j,F (Y ∗j,F , Y ∗

i,L)−mjC (d0j − dj)

)

+(1− pS

j0,i− p

M

j0,i)(rjd

0j (µ + T )

)− Cexj + mjj d

0j .

Backup Capacity (BC) Competition: After a disruption occurs, firms play the BC Competition based on theconditional probability Pi(h|h), i ∈ A,B, and h, h =∈ S, M. We define Π

h|hi,k (Y ∗

i,k, Y ∗j,k

) (i, j ∈ A,Band i 6= j, k, k ∈ L,F, and h, h ∈ S, M) to represent Firm i’s expected profit when it is in position kin the BC Competition, and it believes the disruption is of type h when the disruption is actually of type h.Note that, since Firm i believes the disruption is type h, it will use the distribution of F

h(·) with an average

of µh

to optimize its expected profit in the BC Competition.


To answer the question of which firm benefits more from post-disruption information, we evaluated the Value ofInformation, which measures the difference between the expected profit of Firm i in the following two cases:

• Complete post-disruption information case in which, immediately after the disruption occurs, Firm i knowsfor sure whether the disruption is severe or minor.

• Incomplete post-disruption information case in which Firm i knows only the estimated (i.e., conditional)probabilities of the two types of disruption.

Note that the expected profit of Firm i in both cases depend on whether Firm j has the complete informationor not. To get a conservative (i.e., lower bound) on the value of information for Firm i, we assume that in thecomplete information case, Firm j also has complete information, and in the incomplete information case, Firm jdoes not have he complete information and uses its estimates of the probabilities of the two types of disruption.

For the case of incomplete post-disruption information, Firm i’s actual expected profit under effort levels (x∗i , w∗j )

is:

Πincmpl

i = pS

0Pi(S|S)


ΠS|Si,L (Y ∗

i,L, Y ∗j,F ) +

w∗jx∗i + w∗j

ΠS|Si,F (Y ∗

i,F , Y ∗j,L)−m

iC(d0

i − di))

+pS

0P

i(M |S)


ΠM|Si,L (Y ∗

i,L, Y ∗j,F ) +

w∗jx∗i + w∗j

ΠM|Si,F (Y ∗

i,F , Y ∗j,L)−m

iC(d0

i − di))

+pM

0Pi(M |M)


ΠM|Mi,L (Y ∗

i,L, Y ∗j,F ) +

w∗jx∗i + w∗j

ΠM|Mi,F (Y ∗

i,F , Y ∗j,L)−miC (d0

i − di))

+pM

0Pi(S|M)


ΠS|Mi,L (Y ∗

i,L, Y ∗j,F ) +

w∗jx∗i + w∗j

ΠS|Mi,F (Y ∗

i,F , Y ∗j,L)−miC (d0

i − di))

+(1− pS

0− p

M

0)(rid

0i (µ + T )

)− Cex

∗i + miid

0i ,

On the other hand, Firm i’s actual expected profit under the effort levels (x∗i , w∗j ) when both firms have complete

post-disruption information can be written as,

Πcmpl

i = pS

0


ΠS|Si,L (Y ∗

i,L, Y ∗j,F ) +

w∗jx∗i + w∗j

ΠS|Si,F (Y ∗

i,F , Y ∗j,L)−m

iC(d0

i − di))

+pM

0


ΠM|Mi,L (Y ∗

i,L, Y ∗j,F ) +

w∗jx∗i + w∗j

ΠM|Mi,F (Y ∗

i,F , Y ∗j,L)−m

iC(d0

i − di))

+(1− pS

0− p

M

0)(rid

0i (µ + T )

)− Cex

∗i + miid

0i ,

We define the difference between Πcmpl

i and Πincmpl

i as a conservative proxy (i.e., an underestimation) for the “Valueof Information for Firm A”.

We introduce the following new factors to study the impact of the Post-disruption Information:

µS/µ

M: the ratio of the mean outage times of the two types of disruption.

pS

0/p

M

0: the ratio of the probabilities of the two types of disruption.

The ranges of the new parameters and factors are summarized in Table 10 for (i, j ∈ A,B and i 6= j; q ∈ M,S.In Tables 11 to 14, we display the best regression models with one to five factors (corresponding to the“Value of Information for Firm A” as the dependent variable) under the conditions that Firm A is eitherlarger than Firm B or smaller than Firm B and under either the ABC model or AB model. Note that,

p0 = pM

0+ p

S

0, µ = p

M

0× µM + p

S

0× µS .



Un-normalized Parameters Normalized FactorsParameters Values Factors Rangesµ

M(day) 10, 20, 30

µS

(day) = µM× µ

S

µM

µS

µM

1.5, 3, 5

pM

00.01, 0.05, 0.1, 0.25

pS

00.01, 0.05, 0.1, 0.25

pS

0

pM

0

0.04 ∼ 25

pq

i0when p

q

0= 0.01 0.005, 0.01, 0.05 |p

q

i0−p

q

0p

q

0| 0 ∼ 4

when pq

0= 0.05 0.025, 0.05, 0.1 |p

q

i0−p

q

0p

q

0| 0 ∼ 1

when pq

0= 0.1 0.05, 0.1, 0.15 |p

q

i0−p

q

0p

q

0| 0 ∼ 0.5

when pq

0= 0.25 0.125, 0.25, 0.375 |p

q

i0−p

q

0p

q

0| 0 ∼ 0.5

pq

j0,iwhen p

q

j0< 0.05 0.5p

q

j0, p

q

j0, 2p

q

j0|p

q

j0,i−p

q

j0

pq

j0| 0 ∼ 1

when pq

j0≥ 0.05 0.5p

q

j0, p

q

j0, 2p

q

j0|p

q

j0,i−p

q

j0

pq

j0| 0 ∼ 1

Pi(S|S) 0, 0.25, 0.75, 1

Pi(M |S) = 1− P

i(S|S)

Pi(M |M) 0, 0.25, 0.75, 1Pi(S|M) = 1− Pi(M |M)

Table 10: Values of new input parameters and ranges of new factors for numerical studies.

Number of Independent Variables in the Regression ModelVariables in the Model R2 p0LLC p0LLB

µS

µM

dA

S |pB0−p0p0

|1 29.1 +2 36.8 + +3 40.1 + + +4 42.5 + + + +5 43.3 + + + + −

Table 11: Regression models with the top 5 most important factor(s) corresponding to the “Value of Informationfor Firm A” under the ABC Model when Firm A is larger than Firm B.


Number of Independent Variables in the Regression ModelVariables in the Model R2 p0LLC p0LG p0LLB

µS

µM

KA−d

A

dB

1 26.7 +2 33.2 + +3 38.1 + + +4 40.5 + + + +5 41.3 + + + + +

Table 12: Regression models with the top 5 most important factor(s) corresponding to the “Value of Informationfor Firm A” under the ABC Model when Firm A is smaller than Firm B.

Number of Independent Variables in the Regression ModelVariables in the Model R2 p0LLB p0LG

dA

S

µS

µM

KA−d

A

dB

1 31.9 +2 38.3 + +3 42.7 + + −4 46.2 + + − +5 47.8 + + − + +

Table 13: Regression models with the top 5 most important factor(s) corresponding to the “Value of Informationfor Firm A” under the AB Model when Firm A is larger than Firm B.

Number of Independent Variables in the Regression ModelVariables in the Model R2 p0LG

dB

S p0SGK

A−d

A

dB

|pB0−p0p0

|1 24.1 +2 32.2 + −3 36.0 + − +4 38.5 + − + +5 39.1 + − + + −

Table 14: Regression models with the top 5 most important factor(s) corresponding to the “Value of Informationfor Firm A” under the AB Model when Firm A is smaller than Firm B.



ONLINE APPENDIX D

Analytical Results for the Model with Systematic andTargeted Preparedness Investments

In this appendix, in addition to the effort that Firm i spends in preparedness activities related to moni-toring and detection (i.e., xi, i ∈ A, B), we also consider the effort that Firm i spends in preparednessactivities related to acquiring extra dedicated backup capacity (i.e., θi, i ∈ A,B). We considered C

θ

e,i

(i ∈ A,B) as the cost per unit of effort θi. The effort level θi can also be considered as the cost ofpurchasing capacity options from other suppliers, where the options will be exercised only if there is adisruption at the regular supplier. We define Θi (i ∈ A, B) as the upper limit on the available extradedicated backup capacity for Firm i.

We define ΠAPi (xi | xj) (i ∈ A,B) as Firm i’s total expected profit in the AP Competition for Firm

i (given Firm j’s preparedness effort level is xj) with one decision variables: xi. ΠAPi (xi, θi | xj , θj),

i ∈ A,B, Firm i’s total expected profit in the AP Competition for Firm i (given Firm j’s preparednesseffort levels are xj and θj), is a function with two decision variables: xi and θi, which are simultaneouslydecided by Firm i. Note that, we still consider two periods: Pre-Disruption Period and Post-DisruptionPeriod, but after adding θi into the AP Competition, we have:

• Pre-Disruption Period: Before a disruption occurs, both Firm A and Firm B serve their own cus-tomers. After a disruption occurs, firms will seek available backup capacity for the key component.There are three types of backup capacity: (a) dedicated capacity that is specific to each firm, (b)additional dedicated backup capacity that has already been contracted to be used in case of a dis-ruption, and (c) shared capacity, for which the two firms compete, if the dedicated backup capacityis not enough to cover the shortage. To model this competition with three possible backup capacitysources, we denote the amount of Firm i’s sales (i ∈ A,B) that cannot be covered by the dedi-cated capacity as d0

i, and the amount of Firm i’s sales (i ∈ A,B) that cannot be covered after

using both the dedicated capacity and the additional dedicated backup capacity is d0i− θi, where θi

is defined as the investment by Firm i (measured in unit capacity) devoted to acquiring additionaldedicated backup capacity. We denote the shared backup capacity by S. Under this setting, eachfirm must decide (1) how to invest in acquiring extra dedicated backup capacity, and (2) how toinvest in monitoring and detection in order to be able to detect the disruption first. This resultsin an Advanced Preparedness Competition for Firm i with two decision variables: xi and θi, whichare simultaneously decided.

• Post-Disruption Period: After the disruption occurs, the firm (either A or B) that detects thedisruption must decide how much of the total available backup capacity of S to buy in order tomaximize its total short- and long-term profits. This results in a Backup Capacity Competition(BC Competition). Once both firms have made their purchase decisions, there is a lead time periodrequired to bring the backup capacity (i.e., dedicated and shared) on line. During this period, bothFirm A and Firm B lose all of their short term sales, as well as some long term market share, toFirm C. The amount of the demand of Firm i (i ∈ A,B) that is lost to Firm C during the leadtime is (d0

i− θi)− di , where di is the demand for Firm i’s product at the end of the lead time.

All detailed analytical results are given in the following. Note that,

• adding θi (i ∈ A,B) into the AP Competition only changes the structure of the AP Competition:previously firm only makes one decision on how to invest in monitoring and detection, but nowfirm needs to make two simultaneous decisions of how to invest in acquiring extra dedicated backupcapacity and how to invest in monitoring and detection. Therefore, Proposition 3 needs to be provenin this new setting. On the other hand, since the relationship between the optimal preparednesseffort in monitoring and detection of Firm i (i.e., x∗i ) and the system parameters (e.g., mij , miC ,mji, ri, γij , γiC , γji, ci, and hi) will not be affected by adding θi into the AP Competition, it is easy


to see that Proposition 4 will still hold in the new setting. Also, the competition between two firmson the preparedness effort in monitoring and detection still exists and its investment structure willnot be affected by adding θi into the AP Competition. Thus, Proposition 5 will also hold in thenew setting.

• adding θi (i ∈ A,B) into the AP Competition does not change the structure of the BC Compe-tition: after the disruption occurs, same as previous setting, firms still competition on the shardbackup capacity. The only difference is that, previously before the disruption occurs, the amount ofunsatisfied customers of Firm i was d0

i; now, before the disruption occurs, that is d0

i−θi. Therefore,

it is clear that the structure of the Backup Capacity Competition will not be affected by addingθi into the AP Competition, and hence Proposition 1 and Proposition 2 will still hold in the newsetting.

Two new propositions are the main contents of this Online Appendix. Proposition 6 shows the possiblechoices for the optimal extra dedicated backup capacity purchases for both firms, and Proposition 7shows that a unique Nash Equilibrium exists for each firm’s Advanced Preparedness Competition withtwo kinds of preparedness activities under some situation.

PROPOSITION 6:

CASE 1: If ΘA < d0A, and

(1) if ΘB < d0B, then the only possible choices for the optimal extra dedicated backup capacity purchases

for both firms are:

(θ∗A, θ∗

B) ∈

(0, 0), (0, ΘB ), (ΘA, 0), (ΘA, ΘB)

;

(2) if d0B ≤ ΘB < d0

A +d0B, then the only possible choices for the optimal extra dedicated backup capacity

purchases for both firms are:

(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0, ΘB ), (ΘA, 0), (ΘA, d0B),

(ΘA, mind0

B + (d0A −ΘA),ΘB

);

(3) if ΘB ≥ d0A + d0

B, then the only possible choices for the optimal extra dedicated backup capacitypurchases for both firms are:

(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0, d0A + d0

B), (ΘA, 0), (ΘA, d0B),

(ΘA, d0

B + (d0A −ΘA)

);

CASE 2: If d0A ≤ ΘA < d0

A + d0B, and


for both firms are:

(θ∗A, θ∗

B) ∈

(0, 0), (0, ΘB ), (d0

A, 0), (d0A, ΘB ),

(mind0

A + (d0B −ΘB ), ΘA, ΘB

), (ΘA, 0)

;

(2) if d0B ≤ ΘB < d0



(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0, ΘB ), (d0A, 0), (d0

A, d0B), (ΘA, 0)

;



(3) if ΘB ≥ d0A + d0


(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0, d0A + d0

B), (d0A, 0), (d0

A, d0B), (ΘA, 0)

;

CASE 3: If ΘA ≥ d0A + d0

B, and


for both firms are:

(θ∗A, θ∗

B) ∈

(0, 0), (0,ΘB ), (d0

A, 0), (d0A,ΘB),

(d0

A + (d0B −ΘB ), ΘB

), (d0

A + d0B, 0)

;

(2) if d0B ≤ ΘB < d0



(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0,ΘB), (d0A, 0), (d0

A, d0B), (d0

A + d0B, 0)

;

(3) if ΘB ≥ d0A + d0


(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0, d0A + d0

B), (d0A, 0), (d0

A, d0B), (d0

A + d0B, 0)

;


Before we show that the optimal extra backup capacity purchases for both firms are restricted to a smallnumber of possible values, we need to show:

• Part I: ΠAPA (xA, θA | xB, θB), Firm i’s total expected profit in the AP Competition for Firm i

given Firm j’s preparedness effort levels are xj and θj , is linear in θA, and ΠAPA (xA, θA | xB, θB)

can be written as,

ΠAPA (xA, θA | xB, θB) = pA0

( xA

xA + xBΠA,L(Y ∗

A,L, Y ∗

B,F| θA, θB) +

xB

xA + xBΠA,F (Y ∗

B,L, Y ∗

A,F| θA, θB)

−mAC (d0A − θA)(1− e

− TγAC )

)+ (1− pA0)

(rA(d0

A − θA)(µ + T ))

−Cx

exA − C

θ

eθA + mAA(d0

A − θA), (30)

where Πi,k(Y ∗i,k

, Y ∗j,k| θi, θj) (i, j ∈ A,B and i 6= j, and k, k ∈ L,F) is the total expected profit

of Firm i when it is in position k in the BC Competition, given firms’ preparedness effort levels onacquiring extra dedicated backup capacity are θi and θj . For simplification, in the following, wewill write Πi,k(Y ∗

i,k, Y ∗

j,k| θi, θj) as Πi,k(θi, θj).

• Part II: ΠAPA (xA, θA | xB, θB) is piecewise linear in θA for given xj and θj .

Part I: In this part, we will show that ΠAPA (xA, θA | xB, θB) is linear in θA for given xj and θj .

First of all, it is easy to see that all terms in equation (30) except the first term,

pA0

xA

xA + xBΠA,L(θA, θB),


and the second term,

pA0

xB

xA + xBΠA,F (θA, θB),

are linear in θA, so we only need to show that both ΠA,L(θA, θB) and ΠA,F (θA, θB) are also linear in θA.

In the following, we will show that for any given θB, both Π∗A,L

(θA, θB) and Π∗B,F

(θA, θB) are linear inθA. Let’s look at the Backup Capacity Competition.

Part I – Case 1. d0i < θi ≤ Θi, i ∈ A,B:

It is easy to see that this case cannot be optimal. (All customers of Firm i can be satisfied by Firm i,so it cannot be optimal for the competitor of Firm i to buy more than what it needs to satisfy its owncustomers, i.e., the amount of d0

i .)

Part I – Case 2. θi = d0i , i ∈ A,B:

It is clear that in this case, no firm will play the Backup Capacity Competition.

Part I – Case 3. d0i < θi ≤ Θi, but θj ≤ mind0

j , Θj: For simplification, we call this as CONDITION3 .

Under CONDITION 3 , there are two possible cases:

• Subcase 1: If d0j ≤ Θj , then θj ≤ mind0

j , Θj becomes θj ≤ d0j . But, note that, it is easy to

show that the combination of d0i < θi and θj = d0

j cannot be optimal, so CONDITION 3 becomesd0

i < θi(≤ Θi), but θj < d0j (≤ Θj).

• Subcase 2: If d0j > Θj , then θj ≤ mind0

j , Θj becomes θj ≤ Θj . CONDITION 3 becomesd0

i < θi(≤ Θi), but θj ≤ Θj(< d0j ).

In both cases, Firm i not only can cover all its own customers, but also has an amount of θi − d0i extra

capacity, which can be used to steal the competitor’s unsatisfied customers. Therefore, the optimalamount of the shared backup capacity that Firm i will buy is the difference between the remaining extracapacity, θi − d0

i , and the number of competitor’s unsatisfied customers, dj (θj) − Yj,F , where dj (θj)(=

(d0j − θj)e

− TγjC ) is the uncovered demand of Firm j’s product at the end of the lead time, which is a

function of θj . Without loss of generality, let i = A and j = B, the optimal shared backup capacitypurchase amount of Firm A given it is the Leader in the BC competition is

Y ∗A,L

=[[dB (θB)− YB,F ]+ − (θA − d0

A)]+

,

where dB (θB) = (d0B − θB)e−

TγBC .

It is easy to see that, for any given θB, both ΠA,L(θA, θB) and ΠB,F (θA, θB) are linear in θA.



Part I – Case 4. For all other values of θi: We have,

ΠA,L(YA,L , YB,F | θA, θB) = rA min

YA,L , dA(θA)

µ

+ψA min

[dB (θB)− YB,F ]+,min[YA,L − dA(θA)]+,KA − dA(θA)

−mABξAB min

[YB,F − dB (θB)]+, [dA(θA)− YA,L ]+

−mAC ξAC

[[dA(θA)− YA,L ]+ − [YB,F − dB (θB)]+

]+

−cAYA,Lµ

−hA

[YA,L −minYA,L , dA(θA) −min

[dB (θB)− YB,F ]+,

min[YA,L − dA(θA)]+,KA − dA(θA)

]+µ, (31)

where di(θi) = (d0i − θi)e

− TγiC , which is linear in θi, i ∈ A,B, and

ΠB,F (YA,L , YB,F | θA, θB) = rB min

YB,F , dB (θB)

µ

+ψB min

[dA(θA)− YA,L ]+, [YB,F − dB (θB)]+

−mBAξBA min

min[YA,L − dA(θA)]+,KA − dA(θA)

,

[dB(θB)− YB,F ]+

−mBCξBC

[[dB (θB)− YB,F ]+

−min[YA,L − dA(θA)]+,KA − dA(θA)

]+

−cBYB,F µ. (32)

The forms of equations (31) and (32) are the same as equations (1) and (2), respectively. It is easy toshow that, for any given θB, both ΠA,L(θA, θB) and ΠB,F (θA, θB) are linear in θA.

Part II: In this part, we will show that ΠAPA (xA, θA | xB, θB) is piecewise linear. Actually, it is clear that

ΠAPA (xA, θA | xB, θB) is piecewise linear due to the existence of Min and Max functions in Πi,k(Y ∗

i,k, Y ∗

j,k|

θi, θj) for i, j ∈ A,B and i 6= j, and k, k ∈ L,F, as shown in equations (31) and (32).

Based on what we proved in Parts I and II, in the following we will show that the optimal extra backupcapacity purchases for both firms are restricted to a small number of possible values.

We examine the nine possible scenarios, i.e.,

• Scenario 1: ΘB < d0B and ΘA < d0

A.

• Scenario 2: ΘB < d0B and d0

A ≤ ΘA < d0A + d0

B.

• Scenario 3: ΘB < d0B and ΘA ≥ d0

A + d0B.

• Scenario 4: d0B ≤ ΘB < d0

A + d0B and ΘA < d0

A.


A + d0B and d0

A ≤ ΘA < d0A + d0

B.


A + d0B and ΘA ≥ d0

A + d0B.

• Scenario 7: ΘB ≥ d0A + d0

B and ΘA < d0A.



B and d0A ≤ ΘA < d0

A + d0B.


B and ΘA ≥ d0A + d0

B.

Here we only present the proof for the first scenario , The proofs of the rest are similar to that of thefirst scenario and are therefore omitted.

Scenario 1: For the piecewise linear functions, it is proven that the optimal solution is one of the breakpoints. Hence, in this case, for any given xB, θB and any xA, the optimal value of θA is one of two breakpoints of ΠAP

A (xA, θA | xB, θB), namely 0 or ΘA. Similarly, for Firm B, for any given x0A, θ0

A and any xB,the optimal value of θB is one of two break points of ΠBP

B,A(xB, θB | x0A, θ0

A), namely 0 or ΘB. Therefore,in this scenario , the only possible choices for the optimal extra backup capacity purchases for both firmsare:

(θ∗A, θ∗

B) ∈

(0, 0), (0, ΘB ), (ΘA, 0), (ΘA, ΘB)

.

In summary, the only possible choices for the optimal extra backup capacity purchases for both firms are:

• If ΘB < d0B, then,

– when ΘA < d0A, we have

(θ∗A, θ∗

B) ∈

(0, 0), (0, ΘB ), (ΘA, 0), (ΘA, ΘB)

;

– when d0A ≤ ΘA < d0

A + d0B, we have

(θ∗A, θ∗

B) ∈

(0, 0), (0,ΘB), (d0

A, 0), (d0A,ΘB),

(mind0

A + (d0B −ΘB ),ΘA,ΘB

), (ΘA, 0)

;

– when ΘA ≥ d0A + d0

B, we have

(θ∗A, θ∗

B) ∈

(0, 0), (0, ΘB ), (d0

A, 0), (d0A, ΘB ),

(d0

A + (d0B −ΘB ),ΘB

), (d0

A + d0B, 0)

;

• If d0B ≤ ΘB < d0

A + d0B, then,


(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0,ΘB ),

(ΘA, 0), (ΘA, d0B),

(ΘA, mind0

B + (d0A −ΘA), ΘB

);


A + d0B, we have

(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0, ΘB ), (d0A, 0), (d0

A, d0B), (ΘA, 0)

;


B, we have

(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0, ΘB ), (d0A, 0), (d0

A, d0B), (d0

A + d0B, 0)

;

• If ΘB ≥ d0A + d0

B, then,




(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0, d0A + d0

B),

(ΘA, 0), (ΘA, d0B),

(ΘA, d0

B + (d0A −ΘA)

);


A + d0B, we have

(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0, d0A + d0

B), (d0A, 0), (d0

A, d0B), (ΘA, 0)

;


B, we have

(θ∗A, θ∗

B) ∈

(0, 0), (0, d0

B), (0, d0A + d0

B), (d0A, 0), (d0

A, d0B), (d0

A + d0B, 0)

.¤

In all of our numerical study for cases with both targeted and systematic preparedness effort, we observedthat a unique Nash Equilibrium exists for each firm’s Advance Preparedness Competition. Although wewere not able to prove this for the general case, we prove this in Proposition 7 for a spacial case.

PROPOSITION 7: If both firms are “aggressive” and are identical except for their size (i.e., dA 6= dB),then a unique Nash Equilibrium exists for each firm’s Advanced Preparedness Competition with two kindsof preparedness activities.

PROOF OF PROPOSITION 7: For any given Firm B’s preparedness effort pair (xB , θB ), Firm A’stotal expected profit in the AP Competition for Firm A is

ΠAPA (xA, θA | xB, θB) = pA0

( xA

xA + xBΠA,L(Y ∗

A,L, Y ∗

B,F| θA, θB) +

xB

xA + xBΠA,F (Y ∗

B,L, Y ∗

A,F| θA, θB)


− TγAC )

)+ (1− pA0)

(rA(d0

A − θA)(µ + T ))

−Cx

exA − C

θ

eθA + mAA(d0

A − θA).

To simplify notation we use ΠA,L(θA, θB) and ΠA,F (θA, θB) instead of ΠA,L(Y ∗A,L

, Y ∗B,F

| θA, θB) andΠA,F (Y ∗

B,L, Y ∗

A,F| θA, θB), respectively, and then we have

RA ≡ RA(θA, θB) ≡ ΠAPA (xA, θA | xB, θB) = pA0

( xA

xA + xBΠA,L(θA, θB) +

xB

xA + xBΠA,F (θA, θB)


− TγAC )

)+ (1− pA0)

(rA(d0

A − θA)(µ + T ))

− Cx

exA − C

θ

eθA + mAA(d0

A − θA).

By setting R′A

= 0, and solving for xA , we get

xA =

√pA0

(ΠA,L(θA, θB)−ΠA,F (θA, θB)

)

Cx

exB

− xB . (33)

Similarly, for any given Firm A’s preparedness effort pair (xA , θA), we have

xB =

√pB0,A

(ΠB,L(θB, θA)−ΠB,F (θB, θA)

)

Cx

exA

− xA . (34)


By equations (33) and (34), we can write out x∗A

as follows

x∗A

=1

Cx

e

(pA0∆Π∗A)2(pB0,A∆Π∗B)(pA0∆Π∗A + pB0,A∆Π∗B)2

, (35)

where ∆Π∗i = Π∗i,L(θ∗i , θ∗j )−Π∗i,F (θ∗i , θ

∗j ), i = A,B, i 6= j. Similarly, we have

x∗B

=1

Cx

e

(pB0,A∆Π∗B)2(pA0∆Π∗A)(pA0∆Π∗A + pB0,A∆Π∗B)2

, (36)

After ignoring all constant terms, by equations (35) and (36), we have

RA(θ∗A, θ∗B) =p2

A0∆Π∗A

pA0∆Π∗A + pB0,A∆Π∗BΠ∗

A,L(θ∗A, θ∗B) +

pA0pB0,A∆Π∗BpA0∆Π∗A + pB0,A∆Π∗B

Π∗A,F

(θ∗A, θ∗B)

− (pA0∆Π∗A)2(pB0,A∆Π∗B)(pA0∆Π∗A + pB0,A∆Π∗B)2

−mAC (1− e− T

γAC )θ∗A − rA(1− pA0)(µ + T )θ∗A − Cθ

eθ∗A −mAAθ∗A.

In the proof of Proposition 5, we have shown that if both firms are “aggressive” and are identical exceptfor their size (i.e., dA 6= dB ), then we have ∆Π∗A = ∆Π∗B. Hence, we get

RA(θ∗A, θ∗B) =p2

A0

pA0 + pB0,A

Π∗A,L

(θ∗A, θ∗B) +pA0pB0,A

pA0 + pB0,A

Π∗A,F

(θ∗A, θ∗B)

− (pA0)2pB0,A

(pA0 + pB0,A)2(Π∗

A,L(θ∗A, θ∗B)−Π∗

A,F(θ∗A, θ∗B)

)

−mAC (1− e− T

γAC )θ∗A − rA(1− pA0)(µ + T )θ∗A − Cθ

eθ∗A −mAAθ∗A.

= ηA

1Π∗

A,L(θ∗A, θ∗B) + η

A

2Π∗

A,F(θ∗A, θ∗B)− η

A

3θ∗A + constant,

where

ηA

1=

p3A0

(pA0 + pB0,A)2, η

A

2=

2p2A0

pB0,A + pA0p2B0,A

(pA0 + pB0,A)2, and

ηA

3= mAC (1− e

− TγAC ) + rA(1− pA0)(µ + T ) + C

θ

e+ mAA .

Without loss of generality, let dA > dB . As we showed in the proof of Proposition 5, based on therelationship among dA , dB , K, and dA + dB we have the following two main cases, each including fivemutually exclusive and collectively exhaustive sub-cases:

• Case 1: dB ≤ dA ≤ K ≤ dA + dB :

– S ≤ dB ;

– dB ≤ S ≤ dA ;

– dA ≤ S ≤ K;

– K ≤ S ≤ dA + dB ;

– dA + dB ≤ S.

• Case 2: dB ≤ dA ≤ dA + dB ≤ K:

– S ≤ dB ;

– dB ≤ S ≤ dA ;



– dA ≤ S ≤ dA + dB ;

– dA + dB ≤ S ≤ K;

– K ≤ S.

Furthermore,

• based on the proof of Propositions 1 and 2, (Y ∗i,L, Y ∗

j,F ; Y ∗j,L, Y ∗

i,F ) can have different values. Forexample, given K ≤ S ≤ dA + dB , from Proposition 1, we know,

(Y ∗A,L

, Y ∗B,F

) ∈


(dA , 0), (dA , S − dA), (KA , 0), (KA , S −KA)

,

i.e., there are 9 possible optimal choices for the optimal shared backup capacity purchases whenK ≤ S ≤ dA + dB . Since we assume that, both firms are “aggressive”, i.e., given the chance, a firmwill buy all available backup capacity in an attempt to steal sales from the competitor, based on theproof of Proposition 2, it is easy to see that the only two possible optimal purchases combinationsare: (Y ∗

i,L, Y ∗j,F ; Y ∗

j,L, Y ∗i,F ) = (K,S −K;K,S −K) and (Y ∗

i,L, Y ∗j,F ; Y ∗

j,L, Y ∗i,F ) = (K, 0;K, 0).

• On the other hand, based on the proof of Proposition 6, there are a total of 9 possible combinationswith respect to the regions of ΘA and ΘB, i.e.,

– 1: ΘB < d0B and ΘA < d0

A.

– 2: ΘB < d0B and d0

A ≤ ΘA < d0A + d0

B.

– 3: ΘB < d0B and ΘA ≥ d0

A + d0B.

– 4: d0B ≤ ΘB < d0

A + d0B and ΘA < d0

A.

– 5: d0B ≤ ΘB < d0

A + d0B and d0

A ≤ ΘA < d0A + d0

B.

– 6: d0B ≤ ΘB < d0

A + d0B and ΘA ≥ d0

A + d0B.

– 7: ΘB ≥ d0A + d0

B and ΘA < d0A.

– 8: ΘB ≥ d0A + d0

B and d0A ≤ ΘA < d0

A + d0B.

– 9: ΘB ≥ d0A + d0

B and ΘA ≥ d0A + d0

B.

For example, given ΘA < d0A and ΘB < d0

B, from Proposition 6, the only possible choices for theoptimal extra dedicated backup capacity purchases for both firms are:

(θ∗A, θ∗

B) ∈

(0, 0), (0, ΘB ), (ΘA, 0), (ΘA, ΘB)

.

Since the proof procedure is similar, as an example we only present the proof for one scenario in which K ≤S ≤ dA+dB , (Y ∗

i,L, Y ∗j,F ; Y ∗

j,L, Y ∗i,F ) = (K,S−K;K, S−K), and (θ∗

A, θ∗

B) ∈

(0, 0), (0, ΘB ), (ΘA, 0), (ΘA, ΘB)

.

Now, we have

RA(θ∗A, θ∗B) = ηA

1

((r − c)µK + mξ(K − dA)

)+ η

A

2

((r − c)µ(S −K)−mξ(K − dB)

)

−ηA

3θ∗A + constant,

where di = (d0i − θ∗i )e

− TγiC . After some algebra, we get

RA(θ∗A, θ∗B) = (ηA

1mξe

− TγAC − η

A

3)θ∗A − η

A

2mξe

− TγBC θ∗B + constant.

It is easy to see that,


• if ηA

1mξe

− TγAC − η

A

3> 0, then Firm A prefers θ∗A = ΘA regardless of whether θ∗B = 0 or ΘB , i.e.,

ΘA is strictly dominated;

• if ηA

1mξe

− TγAC − η

A

3≤ 0, then Firm A prefers θ∗A = 0 regardless of wether θ∗B = 0 or ΘB , i.e., 0 is

(weakly) dominated.

That is, there is always a dominated strategy for Firm A.

For Firm B, we have the similar result,

• if ηB

1mξe

− TγBC − η

B

3> 0, then Firm B prefers θ∗B = ΘB regardless of whether θ∗A = 0 or ΘA , i.e.,

ΘB is strictly dominated;

• if ηB

1mξe

− TγBC − η

B

3≤ 0, then Firm B prefers θ∗B = 0 regardless of whether θ∗A = 0 or ΘA , i.e., 0 is

(weakly) dominated.

That is, there is always a dominated strategy for Firm B too.

We summarize the possible optimal extra dedicated backup capacity purchases combinations are:

• when ηA

1mξe

− TγAC − η

A

3> 0 and η

B

1mξe

− TγBC − η

B

3> 0, then (θ∗A, θ∗B) = (ΘA,ΘB);

• when ηA

1mξe

− TγAC − η

A

3> 0 and η

B

1mξthene

− TγBC − η

B

3≤ 0, (θ∗A, θ∗B) = (ΘA, 0);

• when ηA

1mξe

− TγAC − η

A

3≤ 0 and η

B

1mξe

− TγBC − η

B

3> 0, then (θ∗A, θ∗B) = (0,ΘB);

• when ηA

1mξe

− TγAC − η

A

3≤ 0 and η

B

1mξe

− TγBC − η

B

3≤ 0, then (θ∗A, θ∗B) = (0, 0).

So, we can claim that if both firms are “aggressive” and are identical except for their size (i.e., dA 6= dB ),then a unique Nash Equilibrium exists for each firm’s Advanced Preparedness Competition with twokinds of preparedness activities. ¤



ONLINE APPENDIX E

Numerical Study on Impact of Firms Risk Attitudes

In this appendix, we perform a numerical study which extends our results in Sections 3.2 and 3.3 forrisk-neutral firms to cases where firms are Risk Averse (RA) and/or Risk Seeking (RS). Specifically, weconsider: (1) a risk-averse (RA) exponential utility function of the form u(x) = −e

−ax, and (2) a risk-

seeking (RS) exponential utility function of the form u(x) = eax

, where a is a positive constant. We testcases where a = 0.1, 1, 10, and 100 and compare the results to the risk neutral cases.

Based on equation (3), as a RA firm, Firm i’s loss due to lack of preparedness is:

ΩRA

i = Πi −(p0

(− e−(Π∗i,F−miC(d0

i−di)+miid0i

))+ (1− p0)

(− e−(rid

0i (µ+T )+miid

0i

))), (37)

where Πi is the actual expected profit of Firm i, when the firm follows the optimal preparedness strategy:

Πi = p0

(− e

−(

x∗ix∗

i+w∗

j

(−e

−(Π∗i,L−miC (d0i−di)+miid0

i−Cex∗i ))+

w∗jx∗

i+w∗

j

(−e

−(Π∗i,F−miC (d0i−di)+miid0

i−Cex∗i ))))

+(1− p0)(− e−

(rid

0i (µ+T )+miid

0i−Cex∗i

)). (38)

The equation for the measure of risk for a RS firm can be obtained similarly.

Firm A (Larger) Firm B (Smaller)Case 1 Risk-averse Risk-averseCase 2 Risk-averse Risk-neutralCase 3 Risk-averse Risk-seekingCase 4 Risk-neutral Risk-averseCase 5 Risk-neutral Risk-neutralCase 6 Risk-neutral Risk-seekingCase 7 Risk-seeking Risk-averseCase 8 Risk-seeking Risk-neutralCase 9 Risk-seeking Risk-seeking

Table 15: Possible combinations of firms’ risk attitudes.

As shown in Table 15, since there are three types of risk attitude (Risk Averse, Risk Seeking and RiskNeutral) for each of the two firms, we will have 9 cases, each of which corresponds to a possible combinationof firms’ risk attitudes.

Tables 16 to 19 display the best regression models with five parameters under the ABC Model and ABModel with different combination of firms’ risk attitudes. Note that, the case in which both firms arerisk-neutral has already been studied in Sections 3.2 and 3.3, and is therefore omitted here.

As these tables show, when evaluating protection policies against disruption, a risk averse firm tends topay more attention to loss factors while a risk seeking firm tends to pay more attention to gain factors.Furthermore, we find that for cases where the competition is more risk averse, a firm should focus moreon gain factors. On the other hand, the competition being more risk seeking makes a firm paying moreattention to loss factors when evaluating protection policies.

This observation complements our previous observations about firm size. Larger and/or risk averse firmsshould pay attention to loss factors when evaluating protection policies, while smaller and/or risk seeking


firms should pay attention to gain factors when evaluating protection policies. Factors that make theopponent pay more attention to loss (gain) factors when evaluating protection policies tend to make a firmpay more attention to gain (loss) factors when evaluating protection policies. These results are intuitive,but they confirm that our earlier insights about the conditions that magnify the risks from supply chaindisruptions are still valid when firms are not risk neutral.

Firm A Firm B R2 Top 5 Factors

RA RA 57.1 p0LLC p0LLBK

B−d

B

dA

dA

S |pB0−p0p0

|RA RN 59.4 p0LLC p0LLB

KB−d

B

dA

dA

S p0SLtotal

RA RS 60.4 p0LLC p0LLBK

B−d

B

dA

p0SLtotal

dA

S

RN RA 45.2 p0LLC p0LLBd

A

S |pB0−p0p0

| KB−d

B

dA

RN RS 58.6 p0LLC p0LLBK

B−d

B

dA

dA

S |pB0−p0p0

|RS RA 50.3 p0LG p0LLC p0LLB

KA−d

A

dB

dA

S

RS RN 47.7 p0LGK

A−d

A

dB

p0LLCd

A

S p0LLB

RS RS 41.3 p0LG p0LLCK

A−d

A

dB

p0LLBd

A

S

Table 16: Regression models for the top five factors for Firm A in the ABC Model with different combination offirms’ risk attitudes, when Firm A is the larger firm.


RA RA 50.3 p0LLC p0LLBd

B

S p0LG |pB0−p0p0

|RA RN 52.8 p0LLC p0LLB

dB

S p0LG |pB0−p0p0

|RA RS 54.9 p0LLC p0LLB

dB

S

KB−d

B

dA

p0LG

RN RA 40.1 p0LLC p0LG p0LLBd

B

S

KA−d

A

dB

RN RS 52.7 p0LLC p0LLBd

B

S p0LGK

A−d

A

dB

RS RA 43.8 p0LGK

A−d

A

dB

p0LLC p0SG p0LLB

RS RN 41.5 p0LGK

A−d

A

dB

p0LLC p0SG p0LLB

RS RS 39.2 p0LGK

A−d

A

dB

p0SGd

B

S p0LLC

Table 17: Regression models for the top five factors for Firm A in the ABC Model with different combination offirms’ risk attitudes, when Firm A is the smaller firm.




RA RA 51.1 p0LLK

B−d

B

dA

dA

S

mAB

365rA

|pB0−p0p0

|RA RN 48.4 p0LL

KB−d

B

dA

dA

S

mAB

365rA

p0SLtotal

RA RS 55.6 p0LLK

B−d

B

dA

p0SLtotal

dA

S

mAB

365rA

RN RA 41.7 p0LL p0LGd

A

S |pB0−p0p0

| KA−d

A

dB

RN RS 54.4 p0LL p0LGd

A

S

KB−d

B

dA

|pB0−p0p0

|RS RA 44.3 p0LG p0LL

KA−d

A

dB

dA

S

KB−d

B

dA

RS RN 41.8 p0LGK

A−d

A

dB

p0LLd

A

S

KB−d

B

dA

RS RS 37.9 p0LG p0LLK

A−d

A

dB

KB−d

B

dA

dA

S

Table 18: Regression models for the top five factors for Firm A in the AB Model with different combination offirms’ risk attitudes, when Firm A is the larger firm.


RA RA 45.2 p0LL p0LGd

B

S |pB0−p0p0

| KB−d

B

dA

RA RN 41.1 p0LL p0LGd

B

S |pB0−p0p0

| KB−d

B

dA

RA RS 47.3 p0LLd

B

S

KB−d

B

dA

p0LG |pB0−p0p0

|RN RA 38.1 p0LG

dB

S p0SGK

A−d

A

dB

|pB0−p0p0

|RN RS 49.7 p0LG

dB

S p0LLK

A−d

A

dB

|pB0−p0p0

|RS RA 39.3 p0LG

KA−d

A

dB

p0SGd

A

S

mAB

365rA

RS RN 36.2 p0LGK

A−d

A

dB

p0SGd

A

S

mAB

365rA

RS RS 35.4 p0LGK

A−d

A

dB

p0SGd

B

S

dA

S

Table 19: Regression models for the top five factors for Firm A in the AB Model with different combination offirms’ risk attitudes, when Firm A is the smaller firm.


ONLINE APPENDIX F

Implications for Risk Management

We can boil down our insights from the our analytical and numerical analysis to produce the following list of policiesfirms can pursue to mitigate the tactical and strategic consequences of supply chain disruptions.

F.1. Speed Polices

In scenarios where a supply disruption affects multiple firms (i.e., the ABC and AB models), earlier detectionand quicker reaction make it more likely the firm will be able to secure the limited backup capacity before thecompetition. Therefore, we suggest firms (1) install monitoring processes on key components in order to spotdisruptions quickly and to better distinguish a true disruption from day-to-day variations, and (2) build a company-wide culture of awareness and communication so that relevant parties will be made aware of suspicious situationsquickly and will be able to work collaboratively to craft a response.

In addition to promoting detection and response speed from the within firms, firms can improve reaction speed toan external disruptive event by strengthening relationships with suppliers. Enhancing the firm’s reputation withsuppliers can make it more likely that these suppliers will share information with the firm and work with it toresolve a problem. For instance, the 1997 fire at Aisin (which made 99% of Toyota’s P-valves) provided an exampleof a successful business relationship between a manufacturer and a supplier, which led to a rapid collaborativeresponse to a disruptive situation (see Reitman (1997), Nishiguchi and Beaudet (1998) for discussions).

F.2. Customer Loyalty Polices

Under the ABC and AB models, our regression analysis indicates that Firm A faces lower risk when it has highercustomer loyalty (relative to either Firm B or Firm C). Indeed, high customer loyalty pays dividends not onlyduring disruptive events, but also during normal operations. It has been estimated that the cost of acquiring a newcustomer is six to ten times higher than the cost of maintaining a repeat customer (LeBoeuf (2000)). So a firm’smarketing dollars will go further if some of them are devoted to building, nurturing, and developing its currentcustomer relationships. Our results suggest that such efforts will also make the firm more resilient to supply chaindisruptions.

There are many ways a firm can cultivate greater customer loyalty, including:

• Discounts. During normal production, discounts for repeat customers can be an effective way to retaincustomers. In a similar fashion, firms can avoid permanently losing customers who cannot be satisfiedduring a supply outage by offering discounts or rebates to customers who are willing to return after theoutage. Similarly, if the firm has similar product(s) available as substitute(s) to the disrupted one(s), it candiscount these during the outage to encourage customers of the disrupted product(s) to buy them insteadof competitive products. Customer recover can substantially impact profitability. Studies indicate thatcustomer recovery investments yield returns of 30%-150% (Wreden (2009)).

• Exceed customer expectations. Delivering more than customers expect is one of the most powerful ways togain customer loyalty. For instance, Lexus had a recall early in its history in the United States. They calledcustomers, arranged to pick up their cars, and left identical cars as loaners for the day in their places. Whenthey returned the car that evening, it was fixed, cleaned, and had a full tank of gas and a coupon for a freeoil change. Lexus wisely used this disruption as an opportunity to impress its customers and hence increasecustomer loyalty (Sheffi (2005)).

• Pay great attention to unhappy customers. It is impossible for a firm to keep 100% of its customers happy, sostrategies are needed to identify unhappy customers who may be inclined to post negative comments (or evenvideos) on the web. A firm should consider giving unhappy customers an outlet for their comments, so thatit can monitor and respond to them. For example, in 1999, Intel was notified by a mathematics professorabout a floating-point-operation defect of its Pentium 486DX. When Intel failed to respond, the professorposted this problem on a mathematics web site, where it rapidly generated public attention. Intel initiallyresponded by offering to replace the 486DX chip for users who could prove a need for using floating-pointoperations. This served to further aggravate customer unhappiness and led to a demand for a complete recall.Intel finally gave in and recalled the 486DX, which cost more than four-hundred million dollars, and evenmore importantly, damaged its reputation and customer loyalty (Mueller and Soper (2001), Sheffi (2005)).An example of a successful response to a difficult situation occurred in 2001, after the 9/11 attack shut down



all U.S. air freight traffic, when Continental Teves (a supplier of automotive, industrial, and agriculturalproducts) saw an opportunity to impress customers affected by the disruption. On the afternoon of 9/11,Continental Teves picked out those shipments deemed most important to its customers, promised to reroutethem via ground transportation, and tried its best to deliver them on time. Although this action was costly,Continental Teves successfully impressed its customers and made them more loyal in the long term (Marthaand Vratimos (2002)).

F.3. Backup Capacity Polices

Our results show clearly that a firm faces the greatest risk when the shared backup supply for a component subjectto disruption is limited relative to sales. So, one way to reduce risk is to take steps to increase this potential supply.There are several ways this could be done, including:

• Increase the number of firms with capability to supply the needed component. This might be accomplishedby multi-sourcing a component or by otherwise allocating business to suppliers to help them develop specificcapabilities.

• Contractually obligate suppliers to be able to deliver more than the contracted amount within a specifiedamount of time. This would ensure that a secondary supplier could be turned into a primary supplier ifneeded. For research on capacity contracts, see Barnes-Schuster, Bassok, and Anupindi (2002), Kleindorferand Wu (2003), and Anupindi and Jiang (2008).

• Design products to be more flexible with regard to their constituent components. As we noted, Nokia didthis prior to the Philips fire and was able to purchase chips from suppliers who were not previously suitable.

• Cultivate process and organizational flexibility. For example, the previously cited case in which a host ofsuppliers were tapped by Toyota to produce p-valves to replace the supply disrupted by the fire at an Aisinplant is an illustration of the power of having an organizationally flexible supply chain. Working together intask force mode, the many firms in Toyota’s keiretsu quickly expanded p-valve capacity that did not existprior to the incident. For research on flexibility contracts, see Pasternack (1985), Tsay (1999), and Tsay andLovejoy (1999).

References

Anupindi, R. and L. Jiang, 2008. Capacity Investment Under Postponement Strategies, Market Competition, andDemand Uncertainty. Management Science, November 1, 54(11), 1876 - 1890.

Barnes-Schuster, D., Y. Bassok, and R. Anupindi, 2002. Coordination and Flexibility in Supply Contracts withOptions. Manufacturing and Service Operations Management, Vol. 4, No. 3, 171-207.

Kleindorfer, P. R. and D. J. Wu, 2003. Integrating Long- and Short-Term Contracting via Business-to-BusinessExchanges for Capital-Intensive Industries. Management Science, November 1, 49(11), 1597-1615.

LeBoeuf, M., 2000. How to Win Customers and Keep Them for Life. Berkley Trade.

Martha, J. and E. Vratimos, 2002. Creating a Just-in-Case Supply Chain for the Inevitable Disaster. MercerManagement Journal, No. 14.

Mueller, S. and M. E. Soper, 2001. Microprocessor Types and Specifications. June, 8, 2001. Available onhttp://www.informit.com/articles/article.aspx?p=130978&seqNum=28

Nishiguchi, T. and A. Beaudet, 1998. The Toyota group and the Aisin fire. MIT Sloan Management Review 40,No. 1, 49-59.

Pasternack, B., 1985. Optimal pricing and returns policies for perishable commodities. Marketing Science 4,166-176.

Reitman, V., 1997. To the Rescue: Toyota’s Fast Rebound After Fire at Supplier Shows Why It Is Tough. TheWall Street Journal, May 8th, page A1.

Sheffi, Y., 2005. The Resilient Enterprise: Overcoming Vulnerability for Competitive Advantage. MIT Press.Cambridge, Massachusetts.

Tsay, A. A., 1999. The Quantity Flexibility Contract and Supplier-Customer Incentives. Management Science,Vol. 45, No. 10, 1339-1358.


Tsay, A. A. and W. S. Lovejoy, 1999. Quantity Flexibility Contracts and Supply Chain Performance. Manufac-turing and Service Operations Management, Vol. 1, No. 2, 89-111.

Wreden, N., 2009. How to Recover Lost Customers. Available on: http://www.smartbiz.com/article/articleview/112/1/7/

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