GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.56

Preemptive Investment Game with Alternative Projects

Michi Nishihara

June 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Preemptive Investment Game with Alternative Projects∗
Michi NISHIHARA†

Abstract
This paper derives a preemptive equilibrium in strategic investment in alternative
projects. The problem is formulated in a real options model with a multidimensional
state variable that represents project-speciﬁc uncertainty. The proposed method enables
us to evaluate the value of potential alternatives. The results not only extend previous
studies with a one-dimensional state variable but also reveal new ﬁndings. Preemptive
investment takes place earlier and the project value becomes lower if the numbers of bothﬁrms and projects increase by the same amount. Interestingly, a strong correlation among
proﬁts from projects, unlike in a monopoly, plays a positive role in moderating preemptivecompetition.

JEL Classiﬁcations Code: C73, G13, G31.
Keywords: strategic real options, preemption, alternative projects, stopping game.

∗†

This work was supported by KAKENHI 20710116.
Corresponding Author, Address: Graduate School of Economics, Osaka University, 1-7 Machikaneyama-cho,

Toyonaka, Osaka 560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp

1

Introduction

The global ﬁnancial crisis that began in 2007 has increased uncertainty about future
market demand in industries throughout the world. It is becoming increasingly importantfor ﬁrm project management to take into account uncertainty and ﬂexibility in the future.
The real options approach, in which option pricing theory is applied to capital budgeting
decisions, better enables us to ﬁnd an optimal investment strategy and project valuation
involving such uncertainty and ﬂexibility than the Net Present Value (NPV) method could(see [4]).
Although the early literature on real options investigated monopolists’ investment
decisions, recent studies have investigated the problem of several ﬁrms competing in thesame market from a game theoretic approach. Many studies, such as [6, 9, 16], analyze
the preemptive equilibrium in a duopoly investment game.1 Their main result, thatcompetition among ﬁrms accelerates investment in a project, has been supported byempirical papers such as [14].
Most studies of strategic real options assume one-dimensional Geometric Brownian
Motion (GBM) to be the stochastic process (called the state variable) representing thefuture cash ﬂow from a project. This is because explicit results are more appealing due tothe diﬃculty of model calibration in many real options models. Although such simpliﬁcation could be justiﬁed for a problem concerning a single investment project, a problem
involving several projects should be modelled by a multidimensional state variable insteadof a one-dimensional state variable. In fact, several papers have investigated a monopolist’s investment decision involving several projects in a model with a bidimensional statevariable. For example, [5] investigated land development timing with an alternative land
use choice and [11] investigated timing in switching methods of nuclear waste disposal.2
To the best of my knowledge, however, there are no papers that investigate preemptive investment involving several projects with a multidimensional state variable.3 The
contribution of the paper is to ﬁrst clarify the preemptive equilibrium in an investment
game by several ﬁrms with alternative projects, using a multidimensional state variable.
This paper shows several properties of the investment region and the option value in a1

In [7, 10] derived the equilibrium strategies in a Cournot–Nash framework instead of the preemption game.

The competitive equilibrium where the output price moves between upper and lower barriers has also been
investigated in [4, 17]. On the other hand, [8, 13, 15] investigated the agency problem in a single ﬁrm by themethod of mechanism design.
2
These studies apply the results of ﬁnancial options for multiple assets (see Chapter 6 in [3]) to capitalbudgeting. Although in several papers a problem with a bidimensional state variable is reduced to a onedimensional case by homogeneity, such cases are very restrictive.
3
One paper, [1], conducted a case study on the preemptive competition in the textile industry with three
types of uncertainty, but the preemptive game is essentially modelled on the one-dimensional state variable. So,
theoretically, their paper is no diﬀerent from the previous papers [6, 9, 16].

1

model where ﬁrms optimize both investment time and project choice among remaining
projects that have not been chosen by the leading competitors.
The use of this model is also motivated by the following practical issue. When weevaluate the value of a project by the real options method, we are often puzzled by thequestion of which value in monopolistic and strategic models is reliable. Indeed, thediﬀerence is likely to be quite large because the theoretical models with a one-dimensional
state variable calculate the extreme values. This paper provides us with a useful criterion
toward solving such a problem. That is, we should evaluate the value of considering a
potential alternative in a strategic model with a multidimensional state variable. I ﬁnd
that the strategic option values with a symmetric alternative are 40% ∼ 60% of monopoly
with two alternative projects, or equivalently, 70% ∼ 80% of monopoly with a single
project.
Furthermore, I show that preemptive investment takes place earlier and the option
value becomes lower if the numbers of both ﬁrms and projects increase by the sameamount. It is intuitively explained that in the preemptive equilibrium all the ﬁrms are
dragged into a scenario with the worst project. Taking into account the fact that the
number of competitors is likely to increase with the number of alternatives, the result
seems consistent with empirical studies on strategic real options such as [14].
Another new ﬁnding is that preemptive competition is moderated by the correlation
among proﬁts from projects. This contrasts with the monopoly situation where strong
correlation among cash ﬂows decreases the value of project choice. Thus, the sensitivity of the correlation with project value in an oligopoly depends on a trade-oﬀ between
moderation of the preemptive competition (positive eﬀect) and a decrease in the value of
project choice (negative eﬀect). In particular, when there are as many projects as ﬁrms,
the competition deprives ﬁrms of the value of project choice and hence a strong correlation
increases the option value.
Finally, let me mention several applications of the model in this paper. As mentioned
above, the model is suitable for strategic investment involving several alternatives. An
example is a war among ﬁrms opening new stores. A follower must open a store in adiﬀerent place or of a diﬀerent type from that of the leader. In the situation where big
ﬁrms ﬁght for market share in emerging countries, an alternative to preemptive entry into
the market in India might be preemption into the market in the Republic of South Africa.
The model also applies to M&A struggles. For instance, in the pharmaceutical industry large corporations strategically acquire venture businesses that develop new drugs.
Because many M&As take place by private negotiation rather than through a public bidding process, it is necessary for a ﬁrm to preempt the competitors. In the pharmaceuticalindustry numerous potential targets generate a low correlation in gains in takeovers, andthen severe preemptive competition occurs.

2The paper is organized as follows. Section 2 introduces the setup and the preliminary
results in three cases; a monopoly with a single project, a duopoly with a single project,
and a monopoly with two alternative projects. Section 3 describes the new results. In
particular I present the details in a duopoly with two projects, though the results can beextended to an oligopoly of n ﬁrms with m projects in Section 3.3. The results of theinvestment region and the project value contrasts a duopoly or oligopoly with a monopoly.
Section 4 concludes the paper.

2

Preliminaries

Consider a risk-neutral4 ﬁrm that has an option to invest in a project. Consider two
kinds of projects denoted by i = 1, 2. When a ﬁrm conducts project i at time t with sunkcost Ii (> 0), it receives a temporary proﬁt Xi (t).5 Assume that the proﬁt Xi (t) follows acontinuous diﬀusion process:dXi (t) = µi (Xi (t), t)dt + σi (Xi (t), t)dBi (t),(1)

where (B1 (t), B2 (t)) is a two-dimensional Brownian Motion (BM) with correlation coeﬃcient ρ. Mathematically, the model is built on the ﬁltered probability space (Ω, F, P ; Ft )generated by (B1 (t), B2 (t)) as usual. The set Ft means the available information set totime t, and a ﬁrm optimizes its investment strategy under this information. Let r(> 0)and T (> 0) denote the constant risk-free rate and maturity of the option throughout the
paper. We may take T = ∞ when we consider a perpetual option, as in many real options
models.

2.1

Monopoly with a single projectAs a benchmark, we consider a ﬁrm that has a monopolistic option to invest in a single
project, i. It is well known that the option value at time t(≤ T ) with the state variable
Xi (t) = xi is equal to the value function of the following optimal stopping problem:
Vi1 (xi , t) = sup Ext i [e−r(τ −t) (Xi (τ ) − Ii )1{τ ≤T } ],
τ ∈Tt

(2)

where Tt denotes the set of all stopping times τ satisfying τ ≥ t and Etxi [·] is the expectation conditional on Xi (t) = xi .6 Throughout the paper, the superscript and the subscript
on Vi1 represent the number of ﬁrms and available project(s), respectively; that is, Vi1 in(2) means the value function in a monopoly with a single project i.
Many diﬀusions satisfy the following properties.
4

Generally we can assume risk-adjusted proﬁt dynamics (1) rather than the risk-neutrality assumption.
The proﬁt can be interpreted as the discounted cash ﬂow during the lifetime of the project.
6
We do not consider 1{τ ≤∞} but 1{τ <∞} in the case of T = ∞ throughout the paper.
53Assumption (i) The value function Vi1 (·, t) is a (ﬁnite) continuous increasing function.
Assumption (ii) There exists a ﬁnite investment trigger x1i (t) such that the optimal
stopping time τi1 (t) of problem (2) is written as the threshold strategy:
τi1 (t) = inf{s ≥ t | Xi (s) ∈ Si1 (s) = [x1i (s), ∞)}.

(3)

We restrict our attention to a continuous diﬀusion X(t) satisfying the assumptions above.
In addition, as in the related papers, we assume nonnegativeness of X(t) as follows.
Assumption (iii) Xi (t) is nonnegative. If Xi (s) = 0 for any s, Xi (t) = 0 for all t ≥ s.
The assumptions are not restrictive. In fact, we can take a wide range of diﬀusions
including a GBM, i.e., (1) with µi (Xi (t), t) = µi Xi (t), σi (Xi (t), t) = σi Xi (t) where µi (< r)and σi (> 0) are constant, and a mean-reverting process (1) with µi (Xi (t), t) = η(X̄ −Xi (t)), σi (Xi (t), t) = σi Xi (t) where η, X̄ and σi are positive constants.
Note that for a GBM with T = ∞, Vi1 (xi , t) is explicitly derived independently fromtime t (see [4]). In fact, the option value Vi1 (xi ) is expressed as: µ ¶βi
 xi(x1i − Ii ) (0 ≤ xi < x1i )11xVi (xi ) =i x −I(xi ≥ x1i ).
i
i

(4)

Here, x1i is the constant investment trigger deﬁned by:
x1i =

βi
Ii ,
βi − 1(5)where βi is the positive characteristic root:
sµ
¶
1
µi
µi
1 2 2rβi = − 2 +−+ 2 (> 1).
2 σi
σi2 2
σi

2.2

Duopoly with a single projectThis subsection considers two symmetric ﬁrms that struggle to take a single project i.
The following outcome, called “preemptive investment”, is well known. For details, refer
to [6, 9, 16]. Assume that the initial value satisﬁes Xi (0) ≤ I.
We can solve the game between the ﬁrms backward. We begin by supposing that oneof the ﬁrms (called the leader) has ﬁrst invested at time t(≤ T ) with Xi (t) = xi , andwe ﬁnd the optimal decision of the other (called the follower). Because the follower’s
opportunity to invest is completely lost, the follower’s proﬁt is 0. On the other hand,
the leader’s proﬁt is xi − Ii . In the situation where neither ﬁrm has invested, each ﬁrm
attempts to preempt the other in order to obtain the leader’s payoﬀ if Xi (t) − Ii > 0.
As a result, in the preemptive equilibrium, both ﬁrms attempt to invest at the zero-NPV
time:
τi2 = inf{t ≥ 0 | Xi (t) − Ii = 0}4

(6)and gain no project value:
Vi2 (xi , t) = 0.

(7)

Recall that the superscript 2 and the subscript i represent duopoly with a single projecti.
Strictly speaking, both ﬁrms’ investment strategy at (6) proves to be a Nash equilibrium in the stopping game formulated under the appropriate assumption.7 The outcome
can be interpreted to mean that the leading ﬁrm invests at (6), but the follower cannot
conduct a project. The leader’s proﬁt is also zero because of investing too early. This isa well-known preemptive equilibrium in the strategic real options literature (refer to [9]).

2.3Monopoly with two alternative projects

This subsection considers a ﬁrm that has a monopolistic option to invest a single project
among projects 1, 2. The model applies to the situation where a ﬁrm cannot executeboth projects for a reason such as budget constraint. The problem has been essentiallyinvestigated in [5] and Section 6 in [3]. In contrast, [2] investigated investment with
diﬀerent scales under a one-dimensional state variable, i.e., the case where ρ = 1, X1 (0) 6=
X2 (0) and I1 6= I2 .
The option value at time t(≤ T ) with Xi (t) = xi is equal to the value function of the
optimal stopping problem as follows:1
V1,2(x, t) = sup Ext [e−r(τ −t) max(Xi (τ ) − Ii )1{τ ≤T } ].
i=1,2

τ ∈Tt

(8)

1 in (8) means the value function in monopoly with projects 1, 2.
Recall that V1,2
1 in problem (8) becomes:
The optimal stopping time τ1,21
1
τ1,2(t) = inf{s ≥ t | X(s) ∈ S1,2
(s)},

(9)

1 (s) is deﬁned by:where the stopping region S1,211
S1,2(s) = {x ∈ R2+ | V1,2
(x, s) = max(xi − Ii )}.
i=1,2

(10)

1 (t) proves to be the union of two disjoint convex sets corresponding
The stopping region S1,2

to the immediate investment region of each project when X(t) follows a GBM (refer to
Section 6 in [3] and Figure 3 in Section 3.2).
Let us now focus on two symmetric projects, i.e., µ1 = µ2 , σ1 = σ2 and I1 = I2 . In
this case, the larger the correlation coeﬃcient ρ, the more likely it is that proﬁts X1 (t)1 decreases and the stopping region
and X2 (t) take close values. Then the option value V1,2
1 (t) enlarges with the correlation. This can be explained in terms of a decrease of
S1,2
7This assumption is that if two ﬁrms choose the same timing, one of the ﬁrms is chosen as the leader withprobability 1/2. Most studies, including [6, 16], are built on this assumption.

5

diversiﬁcation eﬀects. In particular, in the case of the perfect correlation, i.e., ρ = 1,
1 and the investment time τ 1 for x = x , agree with those in athe option value V1,2
1
2
1,2

monopoly with a single project, i.e., Vi1 and τi1 , respectively. The eﬀect of a correlation
will be compared in detail with that in a duopoly with two projects in Section 3.
The next section is the main contribution of the paper. Although the results can beextended in the case of n ﬁrms with m projects in Section 3.3, I ﬁrst present the detailsof a duopoly with two projects in order to avoid unnecessary confusion.

3

Several ﬁrms with several alternative projects

3.1

Duopoly with two alternative projects

This subsection investigates two symmetric ﬁrms that compete for one of two projects
1, 2. Assume that the one that ﬁrst invests (the leader) can choose the better project
while the other (the follower) loses the opportunity to invest in that project. The leader’sadvantage of being able to choose the better project brings about preemptive competition
between the ﬁrms. As mentioned in Section 1, the model has a wide range of applications,such as preemption in the new market and M&A struggles. Relevant to this model,[12] investigated a duopoly with two projects following a one-dimensional state variable.
Assume Xi (0) ≤ Ii (i = 1, 2).
As in Section 2.2, the problem can be solved in a reverse manner. Suppose that theleader has ﬁrst invested in the better project i(t) at time t(≤ T ) with X(t) = x, where
the function i(t)8 is deﬁned by:i(t) = kif Xk (t) − Ik = max(Xi (t) − Ii ).
i=1,2

(11)

Under this assumption, we ﬁnd the optimal response of the follower. Because for i 6= i(t)the follower has the monopolistic option to invest in a single project i, the option value
and the optimal investment timing coincide with Vi1 , τi1 (see (2) and (3)), respectively.
On the other hand, the leader’s payoﬀ is equal to maxi=1,2 (Xi (t) − Ii ).
2F (t)Let us return to the situation where neither ﬁrm has invested. The region S1,2

where the leader’s proﬁt dominates that of the follower is:2F
S1,2(t) = {x1 − I1 ≥ V21 (x2 , t)} ∪ {x2 − I2 ≥ V11 (x1 , t)}.
2F (t). In addition, one
Each ﬁrm attempts to preempt the competitor as long as X(t) ∈ S1,2of the ﬁrms reluctantly invests X(t) ∈ S11 (t) ∪ S21 (t) if it knows that the other invests at
time:
2F
2F
τ1,2
= inf{t ≥ 0 | X(t) ∈ ∂S1,2(t)},
8

We do not have to be concerned about the value of i(t) when X1 (t) − I1 = X2 (t) − I2 .

6

(12)

2F (t) denotes the boundary of S 2F (t). This is because for X(t) ∈ S 1 (t) ∪ S 1 (t)where ∂S1,2
1,2
1
2
2F
immediate investment generates a higher proﬁt than the option value to wait until τ1,2

(this will be shown in the proof of Proposition 1). Therefore, the preemptive investment
2 (t) becomes:
region S1,22
S1,2(t) = {x1 − I1 ≥ V21 (x2 , t)} ∪ {x2 − I2 ≥ V11 (x1 , t)} ∪ S11 (t) ∪ S21 (t).

(13)

The preemptive investment takes place at:22
τ1,2= inf{t ≥ 0 | X(t) ∈ ∂S1,2
(t)},

(14)

2 (t) denotes the boundary of S 2 (t) which consists of three parts, i.e.:
where ∂S1,2
1,2
2
∂S1,2
(t) = {xi ≤ x1i0 (t) − Ii0 + Ii , xi − Ii = Vi10 (xi0 , t)}
|
{z
}
(a)

∪ {xi0 ≤ x1i0 (t), xi0 − Ii0 = Vi1 (xi , t)}
|

{z

}

(b)

∪ {xi0 = x1i0 (t), (Vi1 )−1 (x1i0 (t) − Ii0 ) ≤ xi ≤ x1i0 (t) − Ii0 + Ii },
|
{z
}

(15)

(c)

for i such that:
x1i (t) − Ii ≥ x1i0 (t) − Ii0 ,(16)

where i0 denotes project i0 6= i throughout the paper.
2 (t). The ﬁrst part (a)Figure 1 illustrates the preemptive investment boundary ∂S1,2

is the region where the leader’s investment in project i generates the same value as thefollower’s option value to invest in project i0 . In the second part (b), both ﬁrms are
indiﬀerent to being the leader with project i0 and the follower with project i. In the lastpart (c), both ﬁrms prefer to be the follower with project i to being the leader with project2F (t). However, one of the ﬁrms invests ﬁrst if it knows that the otheri0 due to X(t) ∈/ S1,22F (t). It must be noted that, unlike the monopolist investment
does not invest until τ1,2
2 (t) is independent of the correlation
region, the preemptive investment boundary ∂S1,2

coeﬃcient ρ.
2 )) with X(t) = x is written as:
The option value (of the leader) at time t(≤ min(T, τ1,2
2
2
V1,2
(x, t) = Ext [e−r(τ1,2 −t) max(Xi (τ1,2) − Ii )].
2

i=1,2

(17)

The leader’s advantage of choosing the better project is completely lost by its earlier
investment than the optimal timing. Furthermore, the leader’s proﬁt becomes less thanthat of the follower if and only if the process X(t) hits part (c).
Although so far we intuitively see the preemptive outcome, to do a more precise
derivation we formulate the following stopping game by two symmetric ﬁrms j = 1, 2.
Deﬁne the action space of both ﬁrms as follows:
A = {(τ, i) | τ ∈ T0 , i : Fτ measurable random variable taking values in {0, 1}}.

7

(c)
(b)
(a)

2
Figure 1: The preemptive investment boundary ∂S1,2(t)

Deﬁne the ﬁrm 1’s payoﬀ π1 as:
π1 (τ1 , i1 , τ2 , i2 ) = E[1{τ1 <τ2 } e−rτ1 (Xi1 (τ1 ) − Ii1 ) + 1{τ1 >τ2 } e−rτ2 Vi10 (Xi02 (τ2 ), τ2 )2e−rτ1
+1{τ1 =τ2 }(Xi1 (τ1 ) − Ii1 + Vi10 (Xi02 (τ2 ), τ2 ))],
2
2

(18)

where (τ1 , i1 ) and (τ2 , i2 ) in π1 (τ1 , i1 , τ2 , i2 ) denote the strategies of ﬁrm 1 and 2, respectively. The last term of (18) corresponds to the assumption in footnote 7. We also deﬁne
the payoﬀ of ﬁrm 2 as π2 symmetrically.
We wish to ﬁnd a Nash equilibrium in the stopping game, i.e., (τ˜1 , i˜1 , τ˜2 , i˜2 ) ∈ A × A
satisfying both:
π1 (τ˜1 , i˜1 , τ˜2 , i˜2 ) =max π1 (τ1 , i1 , τ˜2 , i˜2 ),(19)

max π2 (τ˜1 , i˜1 , τ2 , i2 ).

(20)(τ1 ,i1 )∈A

and
π2 (τ˜1 , i˜1 , τ˜2 , i˜2 ) =(τ2 ,i2 )∈A

2 (t) denote (14), replacing initial time 0 with t. We assume that the diﬀusion
Let τ1,2

process X(t) satisﬁes the following condition9 :Assumption (iv)22(t)) − Ii )] (x ∈/ S1,2(t)).
max(xi − Ii ) ≤ Ext [e−r(τ1,2 (t)−t) max(Xi (τ1,2
2

i=1,2

9

i=1,2

I do not know any proof, but the assumption is satisﬁed in many cases as far as I can judge from a wide

range of computations. Even if Assumption (iv) is not satisﬁed, the violation is so small that we can regard theoutcome as an approximate equilibrium.

8The next proposition shows that the intuitive equilibrium above is indeed a Nash equilibrium in the stopping game.
2 , i(τ 2 ), τ 2F , i(τ 2F )) is a Nash equilibrium in
Proposition 1 The pair of strategies (τ121212
12
2 , τ 2F are deﬁned by (14),(12), and thethe stopping game, where the stopping times τ12122 ), i(τ 2F ) are deﬁned by (11), respectively.
functions i(τ12122 , i˜ = i(τ 2 ), τ˜ = τ 2F , i˜ = i(τ 2F ). Recall
Proof To simplify the notations, let τ˜1 = τ12
1
2
2
12
12
12
0
0
˜
˜
˜
that i2 denotes project i2 6= i2 . Take an arbitrary (τ1 , i1 ) ∈ A. We calculate:

π1 (τ1 , i1 , τ˜2 , i˜2 )= E[1{τ1 <τ˜2 } e−rτ1 (Xi1 (τ1 ) − Ii1 ) + 1{τ1 >τ˜2 } e−rτ˜2 Vi˜1 0 (Xi˜2 0 (τ˜2 ), τ˜2 )2e−rτ1+1{τ1 =τ˜2 }(Xi1 (τ1 ) − Ii1 + Vi˜1 0 (Xi˜2 0 (τ˜2 ), τ˜2 ))]2
2−rτ1≤ E[1{τ1 <τ˜2 } emax(Xi (τ1 ) − Ii ) + 1{τ1 ≥τ˜2 } e−rτ˜2 max(Xi (τ˜2 ) − Ii )]
=

i=1,2
−rτ1
sup
E[e
max(Xi (τ1 ) − Ii )]i=1,2
τ1 ∈T0 ,τ1 ≤τ˜2

i=1,2

= E[e−rτ˜1 (Xi˜1 (τ˜1 ) − Ii˜1 )]

(21)

(22)

= π1 (τ˜1 , i˜1 , τ˜2 , i˜2 )where (21) results from V ˜1 0 (Xi˜2 0 (τ˜2 ), τ˜2 ) = maxi=1,2 (Xi (τ˜2 ) − Ii ) and (22) is proved asi2follows.
2 (t).
By Assumption (iv), immediate investment is not optimal for X(t) = x ∈/ S1,22 (t) \ S 2F (t)On the other hand, immediate investment is optimal for X(t) = x ∈ S1,2
1,2

(the triangle-like region in Figure 1). In fact, for any τ1 < τ˜2 , maxk=1,2 (Xk (τ1 ) − Ik ) ≤2F (τ ). Then, for i0 satisfying (16) we have:
Vi1 (Xi (τ1 ), τ1 ) (i = 1, 2) because of X(τ1 ) ∈/ S1,21supτ1 ∈Tt ,τ1 ≤τ˜2

Ext [e−r(τ1 −t) max (Xk (τ1 ) − Ik )] ≤ Ext [e−r(τ1 −t) Vi10 (Xi0 (τ1 ), τ1 )]k=1,2≤ Vi10 (x, t)(23)= xi0 − Ii0 ,where (23) follows from the supermartingale property of the discounted price processe−rt Vk1 (Xk (t), t). Thus, (22) (and hence (19)) has been proved. We can similarly show¤(20) under Assumption (iv).

Proposition 1 includes the results in a duopoly with a single project. In fact, if
Xi (0) = xi > Xi0 (0) = 0 the preemptive equilibrium in Proposition 1 agrees with thatin Section 2.2. For most of the diﬀusion process Xi (t), higher volatility σi brings aboutlater investment τi1 and higher option value Vi1 . In such a case, by (13) the preemptive
2 becomes smaller, which leads to later investment τ 2 and a higher
investment region S1,2
1,2
2 . That is to say, the eﬀects of volatility σ in a duopoly are inherited
option value V1,2
ifrom a monopoly.

9

If X(t) follows a GBM and T = ∞, we have an explicit form of the time homogeneous
2 by (4), (5) and (15) .
investment boundary ∂S1,2

Corollary 1 Assume that T = ∞, µi (Xi (t), t) = µi Xi (t), and σi (Xi (t), t) = σi Xi (t),where µi (< r) and σi (> 0) are constant for i = 1, 2. For all t > 0, the preemptive
2 is:investment boundary ∂S1,2

(
2
∂S1,2
=

xi ≤ x1i0 − Ii0 + Ii , xi − Ii =(µ

∪ xi0 ≤ x1i0 , xi0 − Ii0 =

xi
x1i

µ

xi0
x1i0

)

¶β1

(x1i0 − Ii0 )¶β1)(x1i − Ii )©
ª
∪ xi0 = x1i0 , (Vi1 )−1 (x1i0 − Ii0 ) ≤ xi ≤ x1i0 − Ii0 + Ii ,where i satisﬁes (16).
2 would be a big beneﬁt in
The explicit derivation of the investment boundary ∂S1,2
2 (see (17)) becomes the solution
applications of the model. Although the option value V1,2
2 instead of an explicitof the corresponding partial diﬀerential equation with boundary ∂S1,2

form, I would like to emphasize that the results are quite useful for applications.
For a general diﬀusion process X(t) we can show the following properties of the in2 , the timing τ 2 , and the option value V 2 .
vestment region S1,2
1,2
1,2

Proposition 2 The following relationships hold.
Investment Region
12S1,2(t) ⊂ Si1 (t) ⊂ S1,2
(t),

(24)

2
1
τ1,2
≤ τi1 ≤ τ1,2
,

(25)

2
1
0 = Vi2 (xi , t) ≤ V1,2
(x, t) ≤ Vi1 (xi , t) ≤ V1,2
(x, t).

(26)

Investment Timing

Option Valuefor all i = 1, 2.
2 (x, t) ≤ V 1 (x , t) (i = 1, 2) because the others are clear (from
Proof I prove only V1,2ii2 ) − I ) ≤ V 1 (X (τ 2 ), τ 2 ) (i = 1, 2), because of
Figure 1). We have maxk=1,2 (Xk (τ1,2
i 1,2
k
1,2
i
2 )∈
2F (τ 2 ) \ ∂S 2F (τ 2 ). Then we calculate (17):Xi (τ1,2/ S1,21,2
1,2 1,22
2
2), τ1,2)]
V1,2(x, t) ≤ Ext [e−r(τ1,2 −t) Vi1 (Xi (τ1,2
2≤ Vi1 (xi , t),(27)

where (27) follows from the supermartingale property of the discounted price processe−rt Vi1 (Xi (t), t).

¤

10

The point of Proposition 2 is that preemptive investment in a duopoly with two
projects is less eﬃcient than investment in a monopoly with a single project (needlessto say, than that in monopoly with two projects). In other words, the preemptive competition becomes more severe if the numbers of both ﬁrms and projects increase by thesame amount. This result is consistent with both the theoretical and empirical results inprevious studies (cf. [7, 14]). We can say that the result extends previous ﬁnding in thesense that the model considers the follower’s choice of an alternative project.
Let us consider two symmetric projects with the same initial value x1 = x2 . We focus
on the correlation coeﬃcient ρ. In the sensitivity analysis in the model, this correlation is
the most important because the previous strategic models with a one-dimensional statevariable cannot reveal its eﬀects. For example, what happens if the proﬁts X1 (t) andX2 (t) are perfectly correlated, i.e., ρ = 1? In that case, no preemption occurs because2
the two projects generate the same proﬁt. Indeed, the preemptive investment timing τ1,2
2 (see (14) and (17)) coincide with τ 1 and V 1 in monopoly withand the option value V1,2iia single project, respectively. Taking this and (26) into account, we can easily show thefollowing corollary.
Corollary 2 Consider the symmetric projects with x1 = x2 . The following equalities
hold for the correlation coeﬃcient ρ:2
1
max V1,2(x, t) = Vi1 (xi , t) = min V1,2
(x, t) (i = 1, 2),

ρ∈[−1,1](28)

ρ∈[−1,1]2 (x, t) and the minimum of V 1 (x, t).
where ρ = 1 gives the maximum of V1,2
1,2
2 (x, t), unlike the investment
It should be noted that in a duopoly the option value V1,2
2 (t) (see (15)), depends on the correlation coeﬃcient ρ. Recall that in aboundary ∂S1,2

monopoly a weaker correlation increases the option value by diversiﬁcation. In contrast,
in a duopoly a stronger correlation increases the strategic option value by moderationof the preemptive competition. The preemptive competition is moderated by a stronger
correlation because the leader’s advantage of project choice is reduced. This result is
consistent with frequent takeovers in the pharmaceutical industry where there are uncorrelated potential targets.

3.2

Numerical examples

This subsection presents numerical examples of the results. Assume that X(t) follows asymmetric GBM. I set the same base parameter values as [3]:r = 6%, µ1 = µ2 = 0%, σ1 = σ2 = 20%, I1 = I2 = 100,
which are also similar to those of [5]. All option values are computed for the initial point
x(t) = (100, 100).

11

2 (t), 6 months, 1 year, 5 years,
Figure 2 illustrates the investment boundaries ∂S1,2

and 10 years before maturity. The investment boundary is composed of two parts (a)and (b) with a vertex on (x11 (t), x12 (t)) which is a pair of the investment triggers in amonopoly with a single project.10 Needless to say, the investment region becomes larger
as time to maturity. This implies that the option value increases with time to maturity.
In fact, the option values 6 months, 1 year, 5 years, and 10 years before maturity are
2 = 3.72, 5.15, 9.83, and 12.16, respectively.
V1,2

Let us now examine the eﬀects of the correlation coeﬃcient ρ, which is the mostinteresting feature in the model. Fix time to maturity as 1 year. Figure 3 depicts the
2 (t) in a duopoly with those of a monopoly with two projects,
investment boundary ∂S1,2
1 (t). The investment boundary in a duopoly, unlike that of a monopoly, isi.e., ∂S1,2

independent of the correlation. We see from Figure 3 that the investment region in amonopoly becomes smaller with the correlation. In other words, the monopolistic option
value decreases with the diversiﬁcation eﬀects.
Table 1 presents the option values and percentages for a range of correlation coeﬃcients
2 in a duopoly increases to V 1 = 7.15 with ρ, while the option
ρ. The option value V1,2i
1 in a monopoly drops to V 1 = 7.15, as shown in the previous subsection. For avalue V1,2
ireasonable correlation ρ = −0.2 ∼ 0.8 the option value in a duopoly is 40% ∼ 60% of themonopolist with two projects, or equivalently 70% ∼ 80% of the monopolist with a single
project.
2 /V 1 , V 2 /V 1 are
It should be noted that the results concerning the percentages V1,2
1,2 1,2irobust for time to maturity T , drift µ, and volatility σ. For example, for ρ = 0, the option
2 = 12.16, which is more than twice that of Table
value 10 years before maturity is V1,2
2 /V 1 = 42.72%, V 2 /V 1 = 74.73%. The option value
1, while the percentages are V1,2
1,2
1,2
i
2 = 12.32 and V 2 /V 1 =and the percentages for volatility σ = 0.5 and ρ = 0 are V1,2
1,2
1,2
2 /V 1 = 69.99%, respectively.
38.87%, V1,2i

In a valuation of a project by a real options approach, it sometimes occurs that amonopolistic model and strategic model generate polar valuations, namely, the value inthe former is too high while that in the latter becomes too low. Then, a substantial
problem for a practitioner arises. How can we judge the gap and which value is reliable?
The model of the paper would provide us with a useful criterion in such a case. Thatis, we should evaluate the value of a project considering a potential alternative using themethodology of this paper.

10

All computations in the paper use a bivariate version of the lattice binomial method with 500 time steps,

and hence the discretization is rougher for longer times to maturity.

12ρ

Table 1: Option values.
2
1
2
1
2
V1,2
V1,2
V1,2
/V1,2
V1,2
/Vi1

−0.4

4.99 13.41

37.27%

69.85%−0.2

5.06 12.99

38.99%

70.78%

05.15 12.51

41.18%

72.01%

0.2

5.26 11.97

43.99%

73.6%

0.4

5.41 11.34

47.75%

75.7%

0.6

5.62 10.58

53.17%

78.64%

0.8

5.96

9.57

62.27%

83.33%

1

7.15

7.15

100%

100%

200
180
160

6 months
1 year
5 years
10 years

X2(t)

140
120
100
80
60
40
20
50

100
X1(t)

150

200

2
(t), 0.5, 1, 5, and 10 years before maturity.
Figure 2: The preemptive boundary ∂S1,2

13

300
duopoly
ρ=0
ρ=0.4
ρ=0.8

250

X2(t)

200
150
100
50
50

100

150
X1(t)

200

250

300

2
1
Figure 3: The investment boundary ∂S1,2(t) and ∂S1,2(t) for ρ = 0, 0.4, and 0.8.

3.3

n ﬁrms with m alternative projects

This subsection extends the results in Section 3.2 to an oligopoly of n ﬁrms with malternative projects. Assume that:Xi (0) ≤ Ii

(i = 1, 2, . . . , m).

(29)

As in Section 3.2, we can solve the problem backward. We restrict our attention to thecase of symmetric projects to avoid unnecessary confusion. For asymmetric projects thepreemptive investment regions include the region corresponding to part (c) in (15), but
the results below remain true.
Let us ﬁrst look at a case where n ≤ m. Consider the last two ﬁrms’ game amongm − n + 2 projects. Assume that at time t with X(t) = x, m − n + 2 projects in−1 , . . . , imremain. If one of the ﬁrms (the leader) invests in project ik at time t, the follower’s option
value becomes V 1(X(t), t), where iˇk denotes the exclusion of ik . Recall thatˇ
in−1 ,...,ik ,...,,imthe superscript and the subscript represent the number of ﬁrms and the available projects,
respectively.
Then, the region where the leader’s payoﬀ exceeds the follower’s, denoted by Si2n−1 ,...,im (t),is:Si2n−1 ,...,im (t) = ∪k=n−1,...,m {xik − Iik ≥ Vi1n−1 ,...,iˇ ,...,,im (x, t)}.
k

(30)

Under the assumption that:
X(t) = x ∈/ Si2n−1 ,...,im (t),

14

(31)

both ﬁrms attempt to invest at the preemptive time:11
τi2n−1 ,...,im (t) = inf{s ≥ t | X(s) ∈ ∂Si2n−1 ,...,im (s)}and gain the option value:−r(τi2Vi2n−1 ,...,im (x, t) = Ext [en−1 ,...,im(t)−t)

max

k=n−1,...,m(Xik (τi2n−1 ,...,im (t)) − Iik )].

It is readily veriﬁed from (30) that the investment region Si2n−1 ,...,im (t) has the relationship:
Si1n−1 ,...,im (t) ⊂ Si1n−1 ,...,iˇ ,...,im (t) ⊂ Si2n−1 ,...,im (t)kfor any k (cf. (24)). Thus we have:τi2n−1 ,...,im (t) ≤ τi1n−1 ,...,iˇ ,...,im (t) ≤ τi1n−1 ,...,im (t)kwith respect to the timing (cf. (25)) and:Vi2n−1 ,...,iˇ ,...,im (x, t) ≤ Vi2n−1 ,...,im (x, t) ≤ Vi1n−1 ,...,iˇ ,...,im (x, t) ≤ Vi1n−1 ,...,im (x, t)kkwith respect to the value (cf. (26)).
Next let us turn back to the 3 ﬁrm game among projects in−2 , . . . , im at time t with
X(t) = x. If one of the ﬁrms (the leader) invests in project ik at time t, the two followers’
option value becomes Vi2

n−2 ,...,iˇk ,...,im(X(t), t) derived under assumption (31) earlier.

The region where the leader has the advantage of preemption, denoted by Si3n−2 ,...,im (t),is:Si3n−2 ,...,im (t) = ∪k=n−2,...,m {xik − Iik ≥ Vi2n−2 ,...,iˇ ....,im (xik , t)}k(32)

Under the assumption that:
X(t) = x ∈/ Si3n−2 ,...,im (t),all the ﬁrms attempt to invest at the preemptive time:
τi3n−2 ,...,im (t) = inf{s ≥ t | X(s) ∈ ∂Si3n−2 ,...,im (s)}and gain the option value:
Vi3n−2 ,...,im (t) = Ext [e−r(τi3n−2 ,...,im(t)−t)

max

k=n−2,...,m(Xik (τi3n−2 ,...,im (t)) − Iik )].

Note that the state variable X(τi3n−2 ,...,im (t)) satisﬁes (31) necessary at the initial point in
the duopoly game.
It readily follows from (32) that the relationship:
Si2n−2 ,...,im (t) ⊂ Si2n−2 ,...,iˇ ,...,im (t) ⊂ Si3n−2 ,...,im (t)k11

To be precise, similar assumptions to footnote 7 and Assumption (iv) are necessary.

15

hold for any k. By this we have:τi3n−2 ,...,im (t) ≤ τi2n−2 ,...,iˇ ,...,im (t) ≤ τi2n−2 ,...,im (t)kandVi3n−2 ,...,iˇ ,...,im (x, t) ≤ Vi3n−2 ,...,im (x, t) ≤ Vi2n−2 ,...,iˇ ,...,im (x, t) ≤ Vi2n−2 ,...,im (x, t).
kk

By backward induction we can show the following results in n ﬁrms’ game at time 0nwith Xi (0) = xi < Ii (i = 1, 2, . . . , m). The preemptive investment region S1,...,m(t), thennpreemptive timing τ1,...,m, and the option value V1,...,m(x, t) are:n−1nS1,...,m(t) = ∪k=1,...,m {xik − Iik ≥ V1,...,(x , t)},ǩ,...,m knτ1,...,m= inf{t ≥ 0 | X(t) ∈ ∂Si31 ,...,m (t)},andnnV1,...,m(x, t) = Ext [e−r(τ1,...,m −t) max (Xk (τ1,...,m) − Ik )],nk=1,...,mrespectively. The following relationships and inequalities hold:
n−1
n−1
n
S1,...,m(t) ⊂ S1,...,(t) ⊂ S1,...,m(t),ǩ,...,mn−1n−1nτ1,...,m(t) ≤ τ1,...,(t) ≤ τ1,...,m(t),ǩ,...,m

and
n−1
n−1
n
n
V1,...,(x, t) ≤ V1,...,m(x, t) ≤ V1,...,(x, t) ≤ V1,...,m(x, t).
ǩ,...,mǩ,...,mThe results are a generalization of Proposition 2. Again, the point is that the preemptivecompetition intensiﬁes if the numbers of both ﬁrms and projects increase by the sameamount. It is intuitively explained that in the preemptive equilibrium all the ﬁrms are
dragged into a scenario with the worst project.
Let us focus on the symmetric projects with the same initial value. The perfectcorrelation gives V11 (x, t) in a monopoly with a single project. For n = m, as in Corollaryn2, V11 (x, t) agrees with the maximum of V1,...,n(x, t). On the other hand, it does notnecessarily hold for n < m. This is because the last ﬁrm’s monopolistic value Vi1n ,...,im (x, t)
decreases with the correlation. Generally, the sensitivity of the correlation in an oligopoly
depends on a trade-oﬀ between moderation of the preemptive competition (positive eﬀect)
and a decrease in a value of project choice (negative eﬀect).
In the case where the number of ﬁrms is larger than that of projects, i.e., n > m, it
can be easily shown that at each stage all the remaining ﬁrms attempt to invest with thezero-NPV timing and hence obtain nothing. The outcome is precisely the same as thatin Section 2.2.

16

4Conclusion

This paper has investigated the preemptive equilibrium in a real options model withthe multidimensional state variable, which represents potential alternative projects. The
results are summarized as follows.
First, preemptive investment takes place earlier and the option value becomes lower
if the numbers of both ﬁrms and projects increase by the same amount. The result canbe regarded as extension of the previous results with a one-dimensional state variable aswell as being consistent with empirical ﬁndings.
Second, the preemptive competition is moderated by the correlation among proﬁts
from projects. The eﬀect contrasts with that in a monopoly where a strong correlation
decreases the value of project choice. The sensitivity of the correlation to the project value
in an oligopoly depends on a trade-oﬀ between moderation of the preemptive competition
and a decrease in the value of project choice.
Third, the strategic option values with a symmetric alternative is 40% ∼ 60% of amonopoly with two alternative projects, or equivalently 70% ∼ 80% of a monopoly witha single project. This indicates the importance of the existence of a potential alternative.
Although monopolistic and strategic models with a one-dimensional state variable tend
to calculate extreme values, the method in this paper allows a reasonable valuation taking
account of the follower’s potential alternative investment.
Lastly, I should point out important but diﬃcult topics for future research. The paper
assumes that proﬁts from the projects are not sensitive to a competitor’s alternative
investment. However, the leader’s cash ﬂow could be aﬀected by the follower’s initiationof a project even if it is an alternative project that is diﬀerent from the leader’s project.
Also, the projects may have diﬀerent maturity in some cases.

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