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WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ENTRY AND EXIT:
SUBGAME PERFECT EQUILIBRIA IN
CONTINUOUS -TIME STOPPING GAMES
by
Chi-fu Huang
Massachusetts Institute of Technology
and
Lode Li
Yale School
WP//3269-91-EFA
April 1991
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
ENTRY AND EXIT:
SUBGAME PERFECT EQUILIBRIA IN
CONTINUOUS -TIME STOPPING GAMES
by
Chi-fu Huang
Massachusetts Institute of Technology
and
Lode Li
Yale School
WP//3269-91-EFA
April 1991
'»S/
Entry and Exit:
Subgame
Perfect Equilibria in
Continuous-Time Stopping Games*
Chi-fu Huang* and
Lode
Li*
April 1986
Last Revised.
March 1991
Abstract
We
study a class of continuous time entry-exit games in the extensive form, where the
is modelled by a Brownian motion. There may be multiple
The equilibrium strategies which represent the bounds of all possible strategies in a subgame perfect equilibrium are explicitly characterized. A necessary and
sufficient condition for the uniqueness of a subgame perfect equilibrium is also given.
stochastically changuig environment
subgame
perfect equilibria.
"We thank Jean- Francois Mertens for many insightful comments and suggestions and an anonymous referee
much shorter proof for a proposition. Barry NaJebuff pointed out an error in an earlier version.
'MIT Sloan Sloan School of Management and Yale School of Organization and Management.
providing a
Yale School of Organization and Management.
for
/
INTRODUCTION
1
Introduction
1
One
of the most important economic decisions faced by industrial firms
exit
from an industry. This decision
mode
is
of particular interest
and the economic
of competition
life
on the entry and exit decisions of a firm,
which Wilson (1990)
of the authors focus on the strategies firms can
for
and importance since
span of an industry. There
for
employ to gain
example, Fudenberg and Tirole (1986), Fudenberg, Gilbert,
Milgrom and Roberts (1982a. b). The others focus on the
is
it
determines the
a vast economic literature
Some
a good recent survey.
is
or protect
Stiglitz,
effect of the
entry-exit decisions; see, for example, Fine and Li (1989),
the decision to enter or to
is
monopoly power;
see,
and Tirole (1983), and
market environment on the
Ghemawat and Nalebuff
(1985), and
Londregan (1990).
Much
of the literature in entry-exit decisions of a firm, however, uses discrete-time models.
notable exceptions are Fudenberg and Tirole (1986) and
Ghemawat and Nalebuff
and Tirole considered a model under certainty when there there
exists
(
1985).
asymmetry
The
Fudenberg
of information
between the firms, and Ghemawat and Nalebuff focused on a model under certainty with complete
information.
As the game
theoretical extension of the optimal stopping theory, the literature of continuous
time stopping games focuses almost exclusively on the normal form games
and
Li (1990)
and the references therein). There
is
now an emerging
(see. for
example. Huang
literature on continuous time
extensive form games; see for example, Simon (1987) and Simon and Stinchcombe (1988). In these
papers, however, continuous time games are analyzed by taking limits of the outcomes of discrete
time games and there
The purpose
exit
is
no exogenous uncertainty.
of our paper
games done
is
twofold.
First,
we extend some
of the existing analyses of
enm
either in continuous time under certainty or in discrete time under uncertainty
to continuous time under uncertainty.
Second, in so doing, we also contribute to the continuous
time game theory by directly working with continuous time without taking limits of discrete time
outcomes.
The
rest of this
paper
is
organized as follows. Section 2 formulates the entry-exit problem in
continuous time for a single firm facing a stochastically changing demands modelled by a Brownian
motion. The unique optimal entry-exit decisions are characterized as barrier policies: enters when
the
demand
level.
Our
rises
above a
critical level
and exits when the demand decreases to another
analysis in this section overlaps
firm costly exits,
it
is
somewhat with
Dixit (1989).
We
critical
assume that once a
prohibitively costly to reenter; while Dixit allows a firm to reenter with a
finite cost after a costly exit.
In addition,
we allow the
profit rate of a firm to
be any bounded
l
:\GLE FIRM PROBLEMS
assumes that this function
increasing function of the Brownian motion, while Dixit
game
Section 3 analyzes an exit
The two
of a duopoly.
is
exponential.
firms in the industry, one strong
one weak in a sense to be formalized, are seeking the best time to
equilibrium
is
One subgame
exit.
demand
the one in which the strong firm does not exit unless the
is
and
perfect
such that
it
always exits before the strong
cannot sustain even as a monopoly and as a result the weak firm
We
firm does.
We
call this
equilibrium the ^natural equilibrium" for future reference.
subgame
also identify other candidates for a
perfect equilibria by characterizing the exact
upper bound and lower bound on each firm's subgame perfect equilibrium
phenomena occur
esting
example, either firm
in these equilibria. For
strategies.
may
exit
Some
inter-
when the demand
This happens for the strong firm, for example, because the
has been on the average increasing.
exits until the
weak firm plays tough and would not
demand
The strong
falls significantly.
firm
against the current
then trades off the potential of becoming a monopolist after the weak firm exits
duopoly
An
losses.
increasing
become a monopolist and
is
demand
increases the expected waiting time for the strong firm to
Thus
a bad news.
it
when
exits
demand
the
reaches a critical level
from below.
Finally
we
give a set of necessary
and
librium. In such case, the unique equilibrium
We
continue in Section 4 to consider a
exhibit a
entrant
subgame
makes the
the incumbent
unique subgame
sufficient conditions for a
is
the ^natural equilibrium'' discussed above.
game
of an incumbent versus a potential entrant.
life
span of the incumbent shorter than that when
may indeed remain
consequence the incumbent
is less
no potential entrant and thus
In addition, the entrant
as a
monopoly throughout
may
it
tolerant to the current
exits earlier
a
it is
its lifetime.
monopoly
than a monopoly
will
as a duopoly. This
able to enter after the incumbent exits.
And
the
demand
monopoly
is
low,
As
profits.
a
than a monopoly facing
even before the entrant enters.
so because
is
monopoly even though
When
losses
not enter the industry even though the
both firms can be profitable
may be
We
perfect equilibrium. In this equilibrium, as expected, the existence of a potential
the possibility of future duopoly competition limits the potential future
re
perf- ct equi-
demand
is
above the
level
by waiting longer, the entrant
the benefit from being a
monopoly
in the
future outweighs the losses in the current duopoly profit.
Section 5 contains
Single
2
We
;•
some concluding remarks and
all
the proofs are in the appendix.
Firm Problems
consider a single firm's entry and exit decisions in this section.
i firm in an industry facing a stochastic element, e.g.,
erratic
random
shocks. Forrnallv we model the uncertain
To begin, imagine that there
demand, that
is
demand by taking
subject to small and
the state space
fi
to
:
SISGLE FIRM PROBLE
2
be the space of continuo
3
-
-
-
complete des:-..
time
the state
t if
A
process
r
_
.
=
:
is
demand
the
_
_
_"
-
<
I
this information
<
s
measurable space
=
:
5
from observing
"infinity"
A"
'
.':
r
..
we
A"
:
f
ar.:
will also us
We
space
P
-
is
ar. i
r.d
from time
to
el
:
.
.
.
_•.
we
will ~cc:
_
r =
:
r
7
r P)
.-
.
1
.
ribec
{f :t €
t
~,^. r
l
:
.
[0,oc)}, the increasing
for
,
7
•
is
--
-
-elds
-
pr
.
--•'.-_*-
is
a
\se tha:
-
v.
ms
x
riven
:'rom ab:
increasing
5
tr
-
r
..
is
-
pi
-
-
-
-
:
.
t
-,
<
;.
-
mal time
a firm faces
:\rr
S
is
the industry, what
-
respeci
in:"
li
the firm
t).
-
is air-
Dterested
mei.-
1
t_
pi
.
Formallv.
P
-
-Tice and
that
I)
the firm
indu.-' ry if
in
7
1
a fan:.
A" :
with resp-
I
variables A*
spa
-.
further assume that ther-
of
random
the c;- pleted prol
below. In order to a
-
-
i-::.;
:.;:.:-continuous in that
When
fi
-
u
technical iifncultj even more,
VBrownian motion with
ai
neld on
riis information
5
demand
.
motion
.
.
We
.>
,
r
r
(/i.<r
=
-
hicr. A"
32, coroL:-.
.
r
;
frcr
ingea
will also enlarge
ith respect to
::
•
~
rical realization
..
demands
~.
to denote the
:
atei
To simplify the
with respect to
t;-
1
3. and
have used
represented
is
_
.
"
-
.
a*.
:e modelled by a c:
can
measura
index
.
md
contains the historical
t
-
.
e-deper
_.
The informa'
-lly.
1
vroblem the firm
time to
faces
er.'
rt
ndnsti
is
re
sets
sets are cieisi.-
-
the
2
SINGLE FIRM PROBLEM*
expected discounted future profits conditional on A'o
=
x:
rl
Tt
/ e- x(Xt )dt
Ex
sup
where
T
denotes the collection of
that, by the strong
Markov property
X,
of
e-
f
Tt
x(X
t
is
the riskless interest rate. Note
the objective function of the firm can be written as
r7
Ex
>
optional times and r
all
(1)
Jo
reT
=
)dt
/(i)
- Ex [e~ rl
/(At)],
Jo
where we understand that
E x [e- rT f(XT )]=
(
eJ{T«x>\
rTMf(X(u,T(u))P
x (du>),
and where
/(*)
Ex
and
f(x)
[-]
is
the expectation under
-*lf
P x By
.
rt
e-
the fact that
ir(X t )dt
tt
is
(2)
increasing and
X
is
a
Brownian motion.
continuous and increasing.
is
Given the discussion above.
(1)
is
equivalent to
mf £x [e- rT /(Xr)].
T
(3)
Putting
v(x)
= f(x)-M Ex [e- rT f(XT )],
(4)
£
provided that a solution exists for
to enter the industry
(3). the
knowing that
it
will
second problem the firm faces
is
to find the optimal time
behave optimally afterwards:
sup
Ex [e
r5
v(A's)],
(5)
seT
where we again used the strong Markov property of X.
We now show
and y e
.
below that these two problems have unique solutions: There are two barriers y"
The unique optimal
unique entry time
is
the
exit
first
The next theorem shows
Theorem
2.1
is
the
first
time that the demand
time that the demand
is
higher than y e
X
is
lower than y" and the
.
the existence of a solution to (3) and this solution
is
a barrier policy.
There exists a y' < y° so that
T' =
is a
time
inf{f
solution to (3) for all x £
J?.
>
:
v(X
t
)
=
0}
=
inf{f
>
:
X
t
<
y'}
(6)
2
SINGLE FIRM PROBLEMS
5
next theorem shows that the barrier policy characterized in
The
Theorem
2.1
the unique
is
optimal exit time.
Theorem
2.2
The T~ defined
unique solution to
in (6) is the
3
(3J.
Using identical arguments, one can show that (5) has a unique solution, which
terized by a barrier: enter the industry
that y e
>
if
demand
the
X
t
is
is
also charac-
c
greater than a given level y
y° as otherwise the firm will be better off staying outside.
We
.
It is
clear
leave the proofs to the
reader and record this result in the following theorem:
Theorem
2.3
There exists a unique solution to
r
some y e >
for
y°.
= mi{t >
:
D|
.V;
(5).
= v(X
I
t
)}
This unique solution
= mi{t >
:
X >y
t
is
e
).
where
v(x)
=
Ex {e~ rT v{Xt)].
sup
reT
Combining Theorem
the firm
is
Theorem
and
2.1
2.2. the
optimal entry time and exit time for the firm given that
currently outside the industry are recorded below:
2.4 Suppose that the firm
entry time for the firm
is
T
e
is
currently outside the industry.
and
defined in (7)
T tx =
inf{r
Then
the unique optimal
the unique optimal exit time for the firm is
r X
>
:
t
<
y'}.
Besides the qualitative results reported above, the optimal barriers y c and y' and the associated
expected discounted future profits can be calculated explicitly using Harrison (1985, chapter
These are recorded
Proposition 2.1
3).
in the following proposition.
y"
is
the
unique number satisfying
'
h(y')=
l°°
e-
a 2
w(z)dz
=
0,
(9)
=
0,
(10)
-V
and
e
t/
is
the unique
number
satisfying
h{y e )= -
We
^
e
a'z
*(z)dz
used a different argument in an earlier version to prove this theorem. The current proof
bv an anonymous
referee.
is
suggested to us
2
SINGLE FIRM PROBLEMS
where
Moreover, y e
>
y°
>
y*
.
a.
=
a- 2 [^ 2 +
+
p],
(11)
a*
=
a- 2 [^ 2 + 2a*r -
/,].
(12)
2<j2r
In addition,
-i
\
v(x)
=
Ej?\-rT')f
MAY T e)]M =
x [e
w(i)
v
;
=
J
e
/ v{y )0{x,y
v(x)
j
Ex [e- TT 'f(X T -)}=
f(x)-
)
ifx<t/e
iix>f
,
(13)
.
1
I
J
l
e
'
,,
,
f(x)-
1
„°
I
-
2/
?
m
^
ifx>y
,x
,
fl
f(y')d(x,y')
(14)
;
where
^'^-i
Now we
exp[-a*(2/-x)]
if
>
y
(lb}
x;
The optimal
have completely solved the optimal entry and exit problem of a firm.
policies are very simple.
The
firm should enter the industry
should exit afterwards when the demands
industry, the firm will not exit the
because there
is
first
time
falls
its
below
y"
.
if
the
rises
Note that since y" <
instantaneous profit
a strictly positive probability that the
demand
demand
above y e and
once in the
y°,
becomes negative. This
rat>-
will in the future
is
be strictly higher
firm chooses to remain in the industry in anticipation of the rise in
demand.
Similarly, the firm does not enter the industry the first time its instantaneous profit rate
becomes
than
y°.
Thus the
positive as there
profit negative.
The
is
a strictly positive probability that the
So the firm waits until the demand
explicit expressions for y'
and y
e
demand
is sufficient
soon decline to make the
will
high to enter.
also allow us to derive the following comparative statics
through direct computation:
Proposition 2.2 Let
ni
>
7r 2
and
let
these two profit functions, respectively.
increasing in r and y e
A
later.
is
j/j
and
Then
y\
<
However, the optimal entry time
it
y%.
anticipates that
it
all levels
for the firm
will in the future
corresponding optimal exit barriers for
Moreover, y"
increasing in o and decreasing in
firm with a uniformly higher profit rate for
later as
be the
y^
is
decreasing in
p.
and a and
r.
of
demand than another
firm will exit
with a uniformly higher profit rate
may
enter
be suffering from losses longer. Also, the higher
the expected increase in the demand, the later a firm will exit; while the higher the interest rate,
the higher the exit barrier and the earlier the firm will exit.
The former
is
obvious and the latter
follows since the firm does not exit immediately after the instantaneous profit
in the anticipation of future profits
and an increase
in the interest rate
becomes negative
makes future
profits less
2
SINGLE FIRM PROBLEMS
7
valuable. In addition, the larger the volatility of the
demand, the lower the
since the exit option of the firm limits the downside risk of the uncertain
added upside potential by an increase
The comparative
statics for the
a makes the firm more willing to
in
optimal entry time are
less intuitive
a has two
and
in the
effects:
the increase makes
meantime
firm will be
making
it
more
it
suffer current losses.
because a change of the
likely that the firm will suffer loss in the
a negative instantaneous profits.
the volatility of the demands,
may
may
or
not increase the
the riskless interest rate also has two effects. First,
it
increases the exit barrier and thus
suffer losses shorter.
The
makes the firm
latter
incentive to enter earlier.
The combined
the optimal exit time
also earlier,
longer. There
one hand,
it
is
is
it
it
increase
near future
On
and because of
So an increase
An
span of the firm.
more
to enter
afford to enter earlier
costly for the
the other hand,
it
will
and the former gives the firm
optimal entry time
in the optimal entry
in
increase in
makes the time span over which the firm
unclear whether the total
is
costly.
life
makes waiting
effects are that the
no clear direction of change
makes waiting more
In anticipation of the latter
entry barrier and enters the industry later.
its
Second,
An
depresses the exit barrier and thus increases the time span over which the
the former, the firm increases
firm.
so
is
demands and thus the
parameters also affects the optimal exit time on which the optimal entry time depends.
in
This
exit barrier.
life
is earlier.
But since
span of the firm
time when
[i
increases.
will
On
be
the
increases the time span over wliich
the firm will suffer losses by decreasing the exit barrier and thus creates an incentive for the firm
not to enter until the
We
demand
is
sufficiently high.
2.1
Lei
,,>
T
* (y)
_
J
and d = a/6. Solving equations
y'
=
y°-^ln(l +
e
=
y°
Suppose for
i
=
+
ify> y°\
ify<y°,
«.
-\-6,
an increasing step function with a,b >
y
(9)
and
(10),
we have
(16)
rf),
— ln{l + i[l a.
(1
+
d)— /-*]}.
(17)
a
1,2.
a.-
*i(y)
where a\ > a? > 0,6o >
is
effects.
conclude the section by looking at the following simple example.
Example
he
These are two opposing
&i
>
0,
=
and y® <
increasing in y° and decreasing in d.
ify>y°-
i
y®.
Then
tt\
> n2
e
In this example, y
higher profit rate enters the market earlier.
,
d\
is
> d 2 ami hence
,
decreasing in n,
y[
i.e.,
<
y% since y
e
the firm with
THE EXIT GAME
3
The
3
Now we
8
Game
Exit
Denote by n tJ (X
investigate the situation in which there are two firms in the industry.
the profit rate for firm
and j =
1,2.
These
of Section 2, that
n,j(y)
>
whole
at
for y
time
monopoly
As the
t is
if
t
.
and x tJ {y) <
The
depend on the
occurs. In this exit game,
strategies.
<
for y
profit for a firm in
exit decisions
we
will focus
that the
an duopoly situation
tt,i(j/)
>
Xi2(y) for all y.
it is
made by
,
where
demand
is
It
>
then follows that y^
As a
the other firm.
so that
j/°-
naturally less than that in a
its exit
y®
2
-
decision
a gaming situation
result,
specification of the
game.
It
We
assume that once a
will
follows that at any time
employed by a firm depending on whether
just has to specify the strategies
1,2
for the industry as a
a monopoly or a duopoly,
prohibitively costly to reenter.
it is
—
i
our attention on subgame perfect Nash equilibria in pure
Thus we need an extensive form
firm exits from the industry,
Assume
yf..
depends on whether
profit rate of a firm
will certainly
t
have the same characteristics as that in the single firm context
Thus we assume that
situation.
X
is
they are bounded, increasing, and nonconstant. And, there exist
yf.
X
demand
there are j firms in the market and the
profit functions
is,
>
i
t )
opponent
its
t,
one
in the
is
industry. Note that since a firm
becomes a monopoly once
afterwards should simply be
unique optimal exit time established in Section 2 in a single firm
its
game
context. Therefore, the
will
be completely specified
state w, the strategy a firm follows given that
Formally,
time with
1.
let T,
T,-(u>, t)
:
>
ft
t
x 9?+
P
x
i-»
-a.s.
K+
We
opponent
exits, its
we designate
if
opponent
be the strategy of firm
also
is still
i,
at
optimal strategy
any time
t
and
in
any
in the industry.
where
T,(-,r)
:
ft
—
>
5? + is
an optional
impose the regularity conditions:
the set
A=
is
2.
its
its
{(w,s) e
x
ft
ft +
:
T(u,s) =
s, 5
e 3?+}
(18)
progressively measurable; 4
Ti(u,t)
is
right-continuous in
t.
For brevity of notation, we will often use T,(5) to denote T,-(w,5(w)) as a
an optional time S. Our interpretation of T,
opponent are both
in the
industry and
its
is
as follows:
opponent
not exit immediately in the states where T,(S)
will
> S and
random
At any optional time 5,
if
variable for
firm
i
and
continue to be in the industry, firm
will exit
immediately
in the states
i
its
will
where
A process Y is progressively measurable if, as a mapping from Q x 5R + to 5?, its restriction to the time set [0, t] is
measurable with respect to the product sigma-field generated by T% and the Borel sigma-field of [0, t}. The progressive
sigma-field is the sigma-field on Q x 5? + generated by all the progressively measurable processes. A subset of fi x 9? +
is progressively measurable if it is an element of the progressive sigma-field. A good reference for these is Dellacherie
and Meyer (1978).
4
THE EXIT GAME
3
=
T,(5)
9
of the two regularity conditions will
The purpose
S.
T
about how T,(<) changes over time. Denote by
become
the space of
all
clear later. Roughly, both are
the mappings
T
:
ft
x
Ji +
>— 3?+
that satisfy these above conditions.
Given
T
G
T
and an optional time 5, we put
= ini{t> S
f(S)
f(S)
In words,
time of
exit
its
is
the
time after 5 that the strategy
first
is
That
able to
is,
T
=
(19)
t}.
instructs the firm to exit prior to the
opponent.
To make our interpretation
one
:T(w, t)
each time
tell at
we need
T(S) and T{S)
of
to
show that
whether one should continue or exit according to T(S) and T(S).
t
show T(S) and T(S) are optional times. This
to
we need
precisely correct, however,
is
among
the subjects of the
following proposition.
Proposition 3.1 Suppose
with
T(S) > S
T(f{S)) = f(S)
Note that the
is
and
a.s.,
This
is
if
T
G
and S
T
we define
is
an optional time. Then T(S)
according to (19), T(S)
is
is
an optional time
above proposition follows from the hypothesis that
last assertion of the
In words,
t.
it
says that a firm will indeed exit at
T(T(S))
if it still
related to the kind of intertemporal consistency discussed by Perry and
and Simon and Stinchcombe (1988). To understand the necessity of
following example.
should remain
in
Moreover,
an optional time.
a.s.
right-continuous in
at S.
T
that
=
Let T(t)
for all
1
/
the industry from time
specification implies that
T -
1/2, but
=
His strategy at that time, T(l/2)
1,
=
G [0.1/2] and T(t)
to time 1/2 but exit
T(T) =
instructs
1
.
The
after time 1/2 tell him, however, to exit immediately!
G
T
remains
Reny (1990)
suffices to consider the
it
for all
>
/
1/2.
That
is.
one
immediately after time 1/2. This
/ T At time
him to remain
t
this,
T
1/2, one
is
not sure what to do.
in the industry while his strategies
right-continuity of T(t) in
t
eliminates
this possibility.
We
will use T_, to
denote firm
opponent's strategy. Given T_,, firm
i's
i
solves the following
program:
sup
Ex
UT
where v t:
A
is
e-
r
'7r t2
(X
t
)<ft
+
l
{
/0
u
•J
defined as in (14) with
Nash equilibrium
that given
TAT.
rTAT_,
r
/
7r
replaced by
of the extensive form exit
T_,-, Ti solves
(20) for
i
=
1,2, for
all
f > f_ i} e"
r:r
-^i(^f_
(20)
)
i
7r,j.
game
i £ S.
is
a pair of strategies (jTi,T2 ) G
T
x
T
so
.
3
THE EXIT GAME
10
Note that the action taken by firm
filrf^f^ + ^'l^f^f.,}- Tnat
an optional time
it
on the
exits at T,; while
A
r
Ex
sup
£
is
tlie set i
firm -i exits
e-^- s ^ l2 (X
+ l {ns)> t. l( s)f~
)dt
t
perfect equilibrium (Ti,T2 )
equilibrium (T{,T2 ),
1
,
1
Px
probability.
for all optional
T (t)
lies in
t
(S)}
if
i
behaves
like
a monopoly.
any optional time S, Ti(S)
for
'
said to be unique
is
[)) MX tJS))\Ts
(21)
any other subgame perfect
if for
- 7 ,(5')l{7- '(s)<r
P1 -
a.s.
time S
it
differ
on a
set of (u>,t)
But they can imply the same
5> < j
of (20)
(S )-\
i
in equilibrium.
whose projection onto
exit times of the
=
two firms
T{(S)l,f, ,§*<,£,
in the
,
Proposition 3.2 For every r £ T,
there exists a
T
£
equivalent to a
T
so that r
=T
subgame
starting from
with probability one.
s^
is
There can be two
of a strictly positive
ft is
and (21) look formidable as they are looking
G T. The following proposition shows that (20)
The
gives the right sense of uniqueness.
the implied exit action taken by firm
and T[ which
The optimizations
,(S)}
|
f
T
T{
,
l
an optional time S as long as T,(5)lrj.
ping
firm
first,
exits first,
t"
conditional on Ts-
This definition of uniqueness seems rather weak. But
strategies T,
where firm
'
is
we have
7 i(£ )l{f (S)<t_
significance of
T ^ T-i)^
T(S)AT_,(S)
A subgame
and
on
a subgame perfect equilibrium
where f^H-^s] denotes the expectation under
for all x
is '
Nash equilibrium {Ti,T2 )
in a
i,
3?,
J/o
TeT
> T-,}, where
set {f,
Nash equilibrium (Ti,T2 )
solves, for all x
or the exit time of firm
i,
for a
much
a.s.
complicated map-
simpler optimization.
Thus, (20)
is
equivalent
to
sup
Ex
e-
/
rt
ir t2
(X
t
)dt
+
note that the optimization of (22)
T
rather than by indirectly looking for a
The question
addressed here.
time for
S £
Before
T
is
What
and
T
is
£
important
T
is
in order. Let y"
:
problem when the
an optional time r directly
is
is
a more
difficult
the fact that
one and
T(S)
is
is
not
an optional
as reported in Proposition 3.1.
attention to the existence of a
some notation
for
(22)
£ T.
our purpose, however,
for
rT
-i;t i(X _.)
f
performed by searching
of whether (21) can similarly be simplified
we turn our
equilibrium,
{r> f_ }ei
reT
We
l
profit function
is
7r,
:
.
are always j firms in the industry during
Then
Nash equilibrium and a subgame
perfect
be the unique optimal exit barrier in the single firm
y'
its life
is
the optimal exit barrier for firm
}
span. For example, y 22
is
i
when
there
the optimal exit barrier
3
THE EXIT GAME
when
for firm 2
firm
is.
is
1
y*,'s.
time 5. Also,
by
7r tJ
.
And
let
<
j/'j
^
>
ir
But
.
2j
l2
For convenience,
/,_,
this
we
is
use
We
.
assume further that
Note that these
2.
not necessary.
T *(5)
last
Figure
in (2)
and
7r tl
y'
n <
and
assumed
tt, 2
y 21 ai 'd
Vn <
J/22-
assumptions are insured by
depicts one possible relative
1
> S
to denote inf{<
t
and h tJ be the functions defined
:
X
t
<
J/,*,}
(9), respectively,
f° r
an >' optional
with n replaced
let
9 IJ (x,y)
The
y'
a stronger firm than firm
the hypothesis that
positions of
Given the ordering on
has no chance to be a monopoly.
Proposition 2.2 shows that
earlier,
That
it
11
=
following proposition shows that a
f, J
(x)-0(x,y)ftJ (y).
subgame
(23)
perfect equilibrium exists for the exit
game by
construction.
Proposition 3.3 (T\,T2 ) = {T\{t),T2 (t);t £
The expected discounted future
J?+)
is
a subgame perfect
Ti(t)
=
7\-,(0.
r2 (t)
=
T22 (t).
Nash
equilibrium, where
profits in the equilibrium for the two firms, or equilibrium payoffs,
are, respectively.
e*(
v1
_
x)
defined in
is
+H x ,yh)9u(yh,yu)
=
(J,)
i-
2)
behaves
like
one easily
One
exit
T,(S)
-
game.
We
will
perfect equilibria.
in
t
is
is
monopoly throughout
is
its life
time
a "stationary equilibrium" in
a "copy" of their strategies at time
a set of necessary
0.
and
In this
sufficient conditions for
Proposition 3.3 to be the unique subgame perfect equilibrium
go about accomplishing this by
The
24 \
T,(5) with probability one.
of the important results of this paper
the equilibrium identified
/
-
.
time than the "weaker" firm. This equilibrium
verifies that
22
a duopoly throughout. Also, the "strong" firm always
that the strategies for the two firms at any time
case,
y'
(25)
with x replaced by n tJ
and the "weaker" firm (firm
life
ifK
22 (x),
In this equilibrium, the '"stronger" firm (firm 1) acts like a
has a longer
'f2r>^22-
\ vn{x)
x
v\ {x)
where v tJ
9l2(x,V22)
J
first
in
the
identifying candidates of other
subgame
Then
sufficient
readers will find these candidate equilibria rather interesting.
conditions for there not to exist candidates other than the one in Proposition 3.3 will then be given.
The
following proposition records a useful restriction on
times of the two firms.
The
exit
time of firm
i
all
subgame
cannot be later than
its
perfect equilibrium exit
monopoly
exit
time and
,
THE EXIT GAME
3
12
cannot be earlier than
time when
its exit
it is
certain to remain a duopology throughout its
life
span.
T
Proposition 3.4 Let (T\,T2 ) €
time
T
X
subgame perfect equilibrium.
be a
Then for
all
optional
S
T;2 (S) < t,(5)l {ti(5) < t_ i(5)}
The
+^(5)l {ti(5)>t_ i(S)} <
corollary below gives a trivial sufficient condition for {Ti(t)
to be the unique
subgame
Corollary 3.1 Suppose
Px -
T&(S)
a.s.Vx e ».
= Txl {t),T2 (t) = T22 (t),
t
£
3ft
+)
perfect equilibrium.
<
that y'l2
y 21
.
Then (Tx (0 = T^(t),T2 (t) =
T22 (t),t
G 3?+)
is the
unique
subgame perfect equilibrium.
When
y\ 2
<
demand below which
tne l eve l of
J/21'
higher than that at which firm
earlier
than firm
in
1
firm 2 cannot survive as a monopolist
is
even
cannot survive as a duopolist. So naturally, firm 2 exits always
1
any subgame as firm 2
much
is
too
much weaker than
firm
1
and thus we
have a unique subgame perfect equilibrium.
For the rest of the analysis, we therefore assume that
2/21
<
I11 this case,
2/i2-
when demand
is
between y 2x and y\ 2 neither firm can survive as a duopolist and both can survive as a monopolist.
,
Thus there may be more than one equilibrium.
The
following proposition
instrumental for the main theorem of this section.
is
characterizes the unique optimal exit time of firm
1
when
It explicitly
firm 2 does not exit until the
demand
is
lower than y 2] in every subgame.
Proposition 3.5 Let
T2 (t) = T2\(t)
Ti(t)
Tit)
=
inf {s
M
=
[»},»}]
A°2
=
(-oo.i&J,
y\
=
y'n
1
_
f
1
I
m
T2
a solution to (21) given
is
t
(x,y)
=
>
t
:
X
s
=
VI £
r(t)l {x
$+
.
Then
^ )eAl}
+ T^t)l {Xr(t)eAoMl)
where
,
e A\ U A%},
U(-oo,yJJ,
inf {y
>
(27)
(28)
y'21
:
™i(jm6i) ^ °) >
y'n-
J\
if
inf {y
>
2/21
:
m
i(2/' 2/21)
<
°)
>
2/21
< vh\
otherwise,
00,
a
(26)
-
z
(l-e^ a
'
+a
-^-y^7r t2 (z)dz
+
J%
a
'\l-e-^' +a -^-y^7r tl (z)dz,
(29)
y
3
THE EXIT GAME
and where
and
a.
13
and
are defined in (11)
a"
Moreover,
(12), respectively.
T[ G
if
T
another
is
solution to (21), then
pI
Ti(S)l{f{(S)<f2 (S)) - ^ ^) 1 {f
1
Firm
behavior depicted
l's
any optional time
in the
first
and firm 2
will exit
and firm
it
-
when
will exit
1
A'< will
will follows its
words as follows. At
in
A' ( enters
reaches (-00,3/21]- Otherwise,
becomes a monopoly and
1
-
above proposition can be described
r, if firm 2 is still present, firm
from above or from below before
a s Vl G R.
(S)>T2 (S)}
1
the set [y\,
reach the set
u
(
]
cither
— 00,1/21]
unique optimal strategy
thereafter.
When
firm
1
finds itself playing a
demand
that firm 2 will not exit until the
demand
incur losses as the current
become
the opportunity to
1
duopoly game
is
below
falls
lower than y\ 2
a monopolist
if
the
at r
.
with a demand A' T G (j^l'^i)'
So staying
y^ v
But, remaining
in
demand
falls
means firm
for firm
1
as
may become
it
will suffer losses
1
is
a bad news.
suffer too
The
proof
is
if
much
XT
>
y\ 2 , firm
1
exits
when the demand
On
On
demand
1
turns out to be a
the other hand, a rising
demand
when the demand
rises to
balance,
below y'n
.
1
Continuing on, firm
balanced out by the prospect of becoming a monopoly
loss to be
in
exits.
1
will
the future.
proposition below records firm 2*s exit time as the unique best response to (26), whose
very similar to that for Proposition 3.5 and
Proposition 3.6 Suppose
that y\
<
{ A' r(oe
a solution to (21) given T\ of (26), where
r{t)
=
-1
2
_
inf{s
(
I
>
inf{</
t
:
>
^
A\
}
is
(30)
defined in Proposition 3.5,
X, g A\ U,4 2 }.
y'n
m 2(y, y'n)
<
if
0},
inf {y
>
y'
12
:
<
<
™2(y,
y'n)
2
'
e-l a +a 'H x z ))nn(z)d431)
0}
y 22
\
otherwise;
y\
=
inf{y G [y~2X -y\]
n,-(a!,y)
=
-
Jy
omitted.
+ T2l (t)l {Xrit) , AiMl)
00,
('
is
Then
oc.
W)= ^)l
is
falls
will
industry gives firm
reach y\, the prospect of the potential future monopoly profit becomes so dismal and firm
Similarly,
1
to reach y 2X an ^ firm 2 exits. So firm
a monopolist sooner.
longer and
the industry, firm
in the
trades off this potential future gains with the current losses. Falling
good news
knows
' l
e-*'*(\-
n 2 {y\,y) < 0},
t
-l*' +*-**-* ))x
l2 (
z
)dz+ [* e~ a
Jy'„
'
{\
-
3
THE EXIT GAME
where
m
T2 (t)
be
is
2
14
and
defined in (29),
and
a.
a' are defined in (11)
and
(12), respectively.
Moreover,
if
to (21) given T\ of (26), then
another solution
T2(S) l {f{{S)>T
2
(S)}
-
Px -
2n
2(^') 1
{fi(S)>f2(S)}
Vi 6 ».
a.s.
In response to (26), firm 2 plays a stationary two-barrier policy at an optional time r as a
duopoly: exits when demand reaches A\ before
falls
below y 2l
demand
is
The
.
in the future against the current
Note here that the
a duopoly,
is
As a consequence, the best response of firm
on
its
y\
>
subgame
?/2i-
3.4,
if
y\
=
in
in the industry, the earlier
we know
T21 (S)
that
is
the upper
perfect equilibrium starting at an optional time r.
1
characterized in Proposition 3.5
bound
perfect equilibrium exit times. This lower
Moreover,
the
monotone property studied
opponent stay
From Proposition
subgame
When
in its exit decision.
Li (1990a) in the sense that the longer its
of firm 2's exit time in any
similar to that for (26).
are considering satisfies the
a firm will exit as the best response.
bound
is
always trades off the potential of being a monopoly
it
duopoly losses
game we
exit
when the demand
reaches A\; otherwise, exits
interpretation of this two-barrier policy
below y 22 and firm 2
Huang and
it
oo, tuen firm l's exit time at
is
is
a lower bound
always tighter than Ti 2 (S) as
any subgame starting from r must be
greater than T^(S). This together with Proposition 3.4 implies that firm 2's equilibrium exit time
must be
T22 (S)
and we have a unique subgame perfect equilibrium.
Corollary 3.2 Let {T\,T2
Proposition 3.5
€
)
T
x
T
be a
equal to oo, then (Ti(t)
is
subgame perfect equilibrium and suppose that
— T{
x
{fy ,T2 (t)
= T22 (t);t £
9R+)
of
subgame
the unique
is
y\
perfect equilibrium.
To search
for the necessary
and
sufficient conditions for uniqueness,
a subgame perfect equilibrium whose equilibrium exit time for firm
and that
for firm 2
is
the least upper
bound
of
all
result together with Proposition 3.4 gives a set of
which subgame perfect equilibrium
singleton sets are given.
From Proposition
a subgame perfect equilibrium,
Our arguments
go as follows.
bound of firm
erty described in
it is
subgame
We
must
the largest lower
exit times for each of the
lie.
two
Finally, conditions for these
3.3, since (T(t)
identify
bound
perfect equilibrium exit times.
*
= T1 1 (t),T2(t) = T22
\
E
t
two
R+
This
between
firms,
bounds on the subgame perfect equilibrium
y\ of Proposition 3.5
is
not equal to
perfect equilibrium exit times at any
Huang and
equilibrium exit times.
two
is
first explicitly
sets to
)
is
be
always
then the unique subgame perfect equilibrium.
for deriving the tighter
Suppose that
l's
exit times
subgame
1
we
Li (1990), (30)
infinity.
Since (26)
exit times
is
a lower
subgame, by the monotone prop-
becomes a new upper bound of
its
subgame
perfect
repeat this procedure to generate tighter and tighter upper bounds on
THE EXIT GAME
3
15
firm 2's and tighter and tighter lower bounds on firm l's
If
is itself
upper bound, respectively,
and firm
for firm l's
3.1 Suppose that there exist y\, y 2
mi(yi,y2 ) =
0.
ri
2 (2/i,2/2)
>
inf{?/
{
=
yi
ni(y2 ,yi)
0,
:
m 2 (y,
2/1)
<
,
and where m, and
0,
and w(y u y 2 ,y 2 ) <
=
0},
profit for firm
1
if
replaced by y 2
j/ji
=
T2 (t)
if
2/2
with
>
inf{j/
and
S
y^ <
fa
Xs —
y\ replaced by y 2
2/i
<
2/i
<
<
y\ 2
2/2
such
where
0,
\
,
and w(yi,y 2 ,y 2 )
respectively,
with
<
y2
m2 (y,yi) < 0} < y22
:
2/1
when
it
is
the expected
exits at Tj*j(5)
and firm 2
Then
.
+ T^(t)l {Xrit)eAAAl}
=T2\(t)l {Xr{t)eAl} + T(t)l {XT(t)eAAAi}
Ti(t)
Moreover,
(31).
as a duopoly at an optional time
exits at (30) with
subgame
and
perfect equilibrium exit times.
an d
otherwise
are defined in (29)
n,
subgame
2's
Vi
00,
is a
3.3, this fixed point
a subgame perfect equilibrium and provides the exact largest lower bound and the exact
Theorem
that
perfect equilibrium exit times.
procedure has a fixed point other than the equilibrium of Proposition
this
least
subgame
r(t)l {XrU)eAi]
»+
tex+,
t
,
.
€
,
(32)
perfect equilibrium, where
(T[,T2
is
)
r(t)
=
mf{s>t:X eAi[JA 2 }.
M
m
=
[yuyi}\J(-°o,y"u },
=
s
y'
22 \
[2/2,
U (_00
'
y^-
another subgame perfect equilibrium, then
< M(0l{Tj'(0<f'(t)} + ^ri(0 1 {f '(()>T '(()} 2
2
Ti(t)
^1*1^)
1
and
T22 (t) <
? 2(0 1 {X|(t)<T '(t)}
1
P
T
-a.s.
for
all
x £
A'o
>
j/22'
Second,
if
if
fi
rm
A'o
We
of
exits
fi
[2/2.2/22]-
rm
is
2 exits
and firm
rises to reach y 2 ,
Thus firm 2
3.1
a "stationary equilibrium" in the sense that every
when the demand decreases
A'o € (2/1,2/2)- firm 2
demand
Theorem
can thus describe the equilibrium by looking at
^ exits
G
1
:
ft.
The equilibrium
"copy" of T,(0).
+ J 2l(0 1 {f|(t)>f '(t)} ^ ^(O
it is
and firm
1
a
1
to y\ 2
and firm
immediately and firm
1
is
1
a
7',(0).
We
7",(r) is
a
take cases. First,
if
continues to y'n as a monopoly.
monopoly throughout. Third,
both stay on with the former making a negative
bad news
for firm 2 as firm
1
will
profit.
If
the
be in the industry for a long time.
becomes a monopoly. On the other hand,
if
the
demand drops
to reach
3
THE EXIT GAME
j/i,
which
is
16
lower than
3/j 2
,
firm
knows that firm
1
y 2 from below or reaches y 2l from above, so firm
future
is
until the
gloomy and firm 2 becomes a monopoly. Fourth,
demand
demand
the
nrm
decrease to yjj,
The
demand
1
plays
relative positions of the barriers are depicted in Figure
There are two interesting features of
demand has been
when the demand
is
between
to reach y\ before
it
decreases to reach
when the demand has been
j/2
and
two firms' optimization p Dblems of
and
The
concludes that the lower bound on firm
starting from an optional time
As a consequence, there
S
l's
exists a unique
from the
£2 are
subgame
between
is
3/1
the
and y 2
,
down
to y\\ or
demand
goes up
reaches
exit as the
when
exit
is
first
between
and y 2
3/1
.
order conditions for the
perfect equilibrium in addition to the
— 00,3/^) and A? becomes — 00,3/22). Then one
(
(
subgame
T^(S) and
is
may
it
may
existence of a solution to these equations with the
Otherwise, A\ becomes
3.3.
is
monopoly strategy henceforth.
happens when the demand
desired order implies the existence of a stationary
one in Proposition
monopoly
continues as a monopoly.
1
When demand
strong firm (firm 1)
Vi,
3/2,
(21).
its
when
exits
1
Second, the strong firm exits before the weak firm does
3/2-
3/1,
h rm
(3/2,1/1),
up to reach y 2 before
drifts
y\, the
declining. This
The equations that determine
as firm 2 will continue
First, either firm
two occasions.
in
demand
the weak firm (firm 2) exits as the
e
in the
1.
this equilibrium.
happens
increasing. This
X
reaches y 2 and firm
firm 2 exits immediately and firm
2/2,
[3/1,3/1],
decreases to y 2 as the prospect of being a
it
gloomy. Otherwise, firm 2 exits when the
<
Xq G
exits immediately. Fifth, if
1
increases to reach y\ before
Finally, if A'o
if
reaches
monopoly
exits as the prospect of being a
1
demand
2 will not exit until either the
the upper
subgame
subgame
perfect equilibrium exit time at any
bound
time
for firm 2's exit
perfect equilibrium. This fact
is
is
T22 (S).
recorded in the
corollary below:
Corollary 3.3
if
and only
if
We now
(T\(t)
= T1*1 (i),T2(2) = T22 (t);t G
there do not exist
3/1,
3/1,
3/2,
and
3/2
3R+)
the unique
is
subgame
perfect equilibrium,
that satisfy the conditions of
Theorem
3.1.
demonstrate the possibility of a unique or multiple subgame perfect equilibria
in the
following example.
Example
3.1 Let
*» iy)
be increasing step functions with a,i
assume
We
1.
that
[i
<
a l2
ify<4,
-b ih
>
0, 6, 2
>
>
6,1
0,
and
3/^
<
yf2 for i,j
=
1,2.
Also
0.
choose the parameters, a tJ
Choose y"n
>
-{
y'
,
2l
y'
,
u
,
and
,
3/°,
6,_,
and
3/°,
so that 3/^
through the following steps:
<
y'
2l
<
y'
12
,
y 2l
<
3/^,
and
y'
12
-
3/^
<
y*
2l
-
y\ x
.
3
THE EXIT GAME
2.
17
Let
—e a
k(x)=
'
I
+ —e- a
It
can
shown
be
\i
<
(or a'
>
€
>
-
and
fc(y; 2
V\2
~
<
2/21
Choose
—
i/ii
>
)
—
J/11
2/"i
2/2i
~~
^(j/i2
2/n
2/21
)
>
//;a/ 6 12
> &n
_
£>y
<
2/2i
fe(y?2-
(33
»
34
>
<
~"
(2/i2
J/21))
- 20).
3/2*1
k(-(y'u -
fc(0)
^'(
and
)-fc(Q)
2/2-j))
-
c
<
(y[ 2
-
J/2i)/2-
r/ten, set
A-(t/'
^21
^22
a ' 50 noticing that
b,
:
so that
~
&21
>
36)
fc(0)'
'
622
("55^ ana" 6 2 2
,
f^^ anc (^.A
^2/
- ^(0)
2/ll +
2
0) - ^(0)'
2/21
^(-(yT 2
fc(y* 2 - 2/21 -0 - ^(Q)
2
(35)
1.
*(2/21
tn
r
It <*
> *(-(»ia -
<)
as determined in step
b 12
A oie
+
- 20 -
that satisfies (35), (36)
e
:
-0-*( Q)
-y2i
-
fc(y*2
fc(j/"i
i*:<o,
sufficiently small, the following will hold,
*(»5i
since
.
a.).
£«*>-«-*)>{
T/ius, /or
1
that
"<•>-<- -«-{<
and for
'
a.
a"
-^
1
(37)
(38)
-2e)-fc(0)'
&y 6?<?/ Furthermore,
^-(yr 2
-
i/21)
-
fc
(°)
nq)
[
621
*(-(vI 2 -v5i))-*(0)
'
by (36).
3.
Let
1/22
=
2/12
+
$>
where
>
f
is
so chosen that
h2
*(yr2
-y2i)-*(Q)
um
-
^(-(2/2*2-2/2 i))-fc(0)'
621
77ie existence 0}
^.
Finally,
such a
S
follows from (39) since
we choose a tJ and
y®-
to satisfy
2/u
where
d,j
=
a tJ /b
x:i
.
for i.j
=
lim^oy^ =
1,2.
=
2/"
-^ln(l+d
i;
)
J/*2-
GAME OF AN INCUMBENT VERSUS A POTENTIAL ENTRANT
4
Let
m
1
=
J/2
+
2/21
and
e
=
(y 1 ,y 2 )
2/i
= yh ~
T
-hi
e
a
Then
£
-^' +a
"(l - e
^ -^)dz - 6n
z
f°
Jyu*
Jy?
=
e
a,y2
18
"
(-MM-Ui
2/2))
-
*(0)j
-
e°'*(l
+ b n (k(y 2 - y'u )
e
-^ +a '^-^)dz
fc(0)))
=
by (31) and
=
n 2 (yi,2/ 2 )
I"'
622
e- a ' z
J y2
=
6y
for
-4/so,
(,?<?,).
m
2
e~
>
by (40). Therefore, y 2
much
It is
(b 22 (k( yi
e
a
-y 2 )-
=
and
oc
yi
=
y\ 2
.
and y?rm
i
-
fc(0))
mxCyJa, y 21
u»7i
k(0)))
=
(Ti,T2 )
M%?2 " »5i) - *(0))) >
+
in (32) with above
easier to construct the case in which there
-
doable since k(y 21
>
- b n (k( yi -y'21 )-
y' ))
2l
So,
1,
we simply
)
y{i)
=
e
fc
< hi <
611
mi(j, tfi)
fc(0))
'*(-b 22 (k(-(y'22 -
1
is
- e -^' +a '^-'))dz
Jyii*
example, after completion of step
This
a
e~ '*(l
y G (jfo, y 22 ],
all
(y. y'l2 )
a 'y>
T
{l-e-^ +a ^-^)dz-b 2l
B -*«(-6
and
y, is
an equilibrium.
a unique equilibrium.
In the above
set
(y2i
- yn)
^(-(2/12
> k(-(y[ 2 -
is
j/ t
0.
y 21
12 (*(-(tfa
)).
-
-fe(Q)
2/21))
-
^(0)
Then, for any y £
- &)) -
*(0))
+
[y 2 i,yl 2 ]
&u(*(»5i
noi exi7 before firm 2's longest possible exit time
T2 \.
- y'u )
fc(O)))
>
The uniqueness of
0,
the
equilibrium follows from Corollary 3.2.
Game
4
In this section,
of an
we
Incumbent Versus a Potential Entrant
investigate the situation where firm
option to exit, while firm 2
exit.
This
is
thus a
game
is
1 is
initially in the
market and has a single
not in the market at the beginning and has options to enter and then
of an incumbent versus a potential entrant, henceforth abbreviated as
simply the entry game. Unlike
in the previous section,
conditions for the uniqueness of a
subgame
our focus here
not to provide sufficient
perfect equilibrium, but to demonstrate that one such
equilibrium exists. Using the ideas similar to those of Section
for this equilibrium to be the
is
3,
one can identify
sufficient conditions
unique subgame perfect equilibrium. This procedure, however, being
tedious and highly computational, will not be repeated here and
is left
for the interested reader.
Before we proceed formally, we note the following. First, in any subgame perfect equilibrium,
once the two firms are
in
the industry in the
same time, therehence, they must be playing a subgame
GAME OF AX INCUMBENT VERSUS A POTENTIAL ENTRANT
4
19
perfect equilibrium in the exit
game
discussed in Section
enters, then afterwards, firm 2
must
follow its unique optimal single firm entry
characterized in Section
Given
game with
entry
By
f!
:
>
with T[(t)
x
subgame
a focus on
P
t
not exited, firm
be firm
all
if
1
>—
l's
for all x €
9?.
and only
if
firm
T2 (0 =
t.
1
1
Tf and T2
,
restrict
and
exit decisions
In words, if at time
exits,
and only
where
T2 (0
e
Tf(t)
if
an optional time
is
firm 2 has not entered and firm
t
=
t.
Similarly, let
T\
fi
:
>
an optional time with T£{t)
is
l's
exits.
1
strategy before firm 2 enters, where Tf(t)
if
the one
is
our attention on firm
entry decision before firm
2's
x
t
P
has
<— 9?+
Jt +
x
1
-a.s. for
has not exited and firm 2 has not entered, firm 2 enters immediately
We
assume that Tf and
measurability condition stipulated for T, in Section
possible
we can
perfect equilibria,
immediately
will exit
t,
exits before firm 2
1
and the two observations noted above, in analyzing the
and on firm
+ be firm
3i
-a.s.
time
If at
§?.
x
strategy before firm
2's
x £
R+
firm
the analysis of Section 3. this unique equilibrium
this hypothesis
exit decision before firm 2 enters
Let !{"
if
For simplicity, we assume that there exists a unique subgame perfect
2.
equilibrium for the exit game.
in Proposition 3.3.
Second,
3.
T2e
are right-continuous in
Denote by Ti and
3.
T2
and
t
satisfy the
the space of
all
the
respectively.
Define
f x (S) =
inf{*
> 5
=
t}
f2e (S) =
inf{f
> S :T2e (0 =
r}
:
Tf(t)
and
for all
e
Tf e Tj and T2 e
A subgame
Tf(S)
T2
.
rf(S)ATf(S)
Ex
TeT
where v\ x
sup
ET
where v\ x
The
is
is
js
r{t{S) - S
e-^- s ^ u (X,)ds +
r2 (5)
defined in (25) and
t'21
is
otherwise
e
- r(
it
^ (S) - 5, r
defined in (4) with
it
21
)
6 Ti X T2 so that
(A>, (S) )l {tiI(S) , f(S)} |7-s
replaced by - :i
tt
subgame
of this proposition, being similar to that of
exits
composed of (r r .T2
solves
h^(Xf{S) )l {ns)<fiZ{S)} +
firm 2 sets a critical level y\ such that
1
is
e-^^- S Kr(X n{S) n {n(S)<ns)} \^S
following proposition reports a stationary
The proof
firm
game
time 5,
defined in (24). and
[e-
Proposition 3.1, f{(S) and f{(S) are optional times.
perfect equilibrium of the entry
solves, for all optional
sup
By
enters the market the
3.1, is
first
1
exits.
game.
omitted. In this equilibrium
time demand
and then plays the unique subgame perfect equilibrium of the
behaves optimally as a monopolist after firm
,
.
perfect equilibrium for the entry
Theorem
]
The
exit
value y\
is
above
game
is
j/ 2
before
thereafter;
determined so
4
GAME OF AN INCUMBENT VERSUS A POTENTIAL ENTRANT
as to balance the marginal benefit of being a duopoly presently
a future monopoly. Firm
1
firm 2's entry; otherwise,
it
sets a critical level yf such that
plays the unique
subgame
it
20
and the marginal benefit of being
exits
demand
if
is
below yf before
perfect equilibrium of the exit
game
after
firm 2 enters.
We
will use
j/ji
an(^
IJ22
to denote the entry barrier for firm 2 as a
respectively, as described in
Theorem
Proposition 4.1 (Tf,T2e )
is
a
-
(-00,3/j
7
]
t
:
X, > y|J.
subgame perfect equilibrium for
!?(*)
=
r(t)l {Xr(t)€Al}
Ti(t)
=
It1 (T(t))l {Xrtt)eM}
and
+
the entry game, where
T^(T(t))l {XTlt)€A2}
+
(41)
,
r(t)l {Xr{t)€A2} ,
(42)
= mf{s > t:Xs G A t U A 2 }
r(t)
with Ai
= inf{s >
as a duopoly,
put
2.3. In addition,
T^{t)
monopoly and
A 2 = [y^+oc), and
where y\ £ (l/iujfo) an ^ V\ £
(2/221
+ 00
)
^e
are
unique solution to
^2(2/1,2/2)
=
0,
"1(2/1,2/2)
=
0,
with
e
w[ 1 . e -(..4«')(»-»i)] TaW(b _ e (..+.)»
=
w/iere
is
p,-j
T eAn
*
2
7r
n (z)[I - e-^+Ote-*)]^ +
existence of a potential entrant
"
e "°
VM +
2a 2 r
[gii(»a,y5a)
may
decreases to reach y* before
demand
This
to exit.
is
it
levels in the future,
Second, the entrant sets
3/22-
force the
incumbent to
its
increases to reach y\-
makes firm
due to the opportunity of
profit in the future.
1
exit earlier
its
The
(44)
1
t/jj.
Thus the
time.
is
no
This occurs when
becoming a duopolist
the incurrence of current losses.
exits higher
becoming a monopolist
its
life
>
than when there
potential of
less tolerant to
entry level y 2e before firm
Therefore the entrant sacrifices
monopoly
- Jia(lft,»Sa)],
First, in the equilibrium, y^
potential entrant and hence the incumbent has a shorter economic
at higher
(43)
7r 21
defined in (23).
Three interesting observations can be made.
demand
(z)(fz,
e
J V21
-2*2
ni(yi,y2 )
-° ,2
/
if it
than
its
duopoly entry
level
waits long enough for firm
1
current duopoly profits in exchange of the potential
CONCLUDING REMARKS
5
Finally, there are
two possible
By "peace", we mean
market by
different
21
results of the
incumbent versus entrant game: "peace" or "war".
that the entrant enters after the
time segment.
If
incumbent
and both firms share the
exits
the entrant enters before the incumbent exits, then they battle
as a duopoly in the market.
Concluding Remarks
5
We
have studied a class of entry-exit timing games
changing environment
is
in
continuous time where the stochastically
modelled by a Brownian motion.
In
doing so, we work directly in continu-
ous time with the extensive forms of the class of continuous-time entry-exit games. Given that the
stochastically changing environment
libria
without too much
is
/
to investigate the existence a
subgame
Huang and
all
identified
some subgame
perfect equi-
stationary equilibria in that the strategy
a copy of the strategy played at time
played by a firm at any time
the one of
we have
These equilibria are
difficulty.
is
stationary,
=
t
0.
It will
be interesting
perfect equilibrium in a general stochastic environment like
Li (1990).
There are many other economic applications of the continuous-time stochastic timing games
besides the entry and exit decisions of firms analyzed in this paper.
These include product and
process technology choices, multiproduct and multimarket competition, and multiplant global competition. Indeed,
any situation
in
which
all
players
players' well-beings are affected by time, by
choices
made by
some
make dichotomous
choices over time,
stochastically changing elements,
other players can be formulated as a stopping game.
technologies developed in the paper
may
and the
and by the
Then the methods and
apply.
References
6
1.
2.
C, and P. Meyer, Probabilities and Potential A: General Theory of Process,
North-Holland Publishing Company, New York, 1987.
Dellacherie,
Dixit, A.,
"Entrv and Exit Decisions under Uncertaintv," Journal of Political Economy
97:620-638, 1989!
Games and
3.
Fine, C. and L. Li, "Equilibrium Exit in Stochastically Declining Industries",
Economic behavior, 1:40-59, 1989.
4.
Fudenberg, D., R.J. Gilbert. J.E. Stiglitz and J. Tirole, "Preemption, Leapfrogging, and
Competition in Patent Races", European Economic Review, 24:3-31, 1983.
5.
Fudenberg, D. and
J.
Tirole,
"A Theorv
of Exit in Duopolv", Econometrica, 56:943-960.
1986.
6.
Ghemawat,
P.
and B. Nalebuff, "Exit", Rand Journal of Economics, 16:184-194, 1985.
PROOFS
A
7.
8.
9.
22
Harrison, J.M., Brown/an Motion and Buffered Flow, John Wiley
&
Sons,
new York,
1985.
Huang, C. and L. Li, "Continuous Time Stopping Games with Monotone Reward Structures",
Mathematics of Operations Research, 15:496-507, 1990.
Londregan,
J.,
"Entry and Exit over the Industry Life Cycle,"
RAND
Journal of Economics,
21:446-458, 1990.
10.
Mertens, J.F., "Strongly Supermedian Functions and Optimal Stopping,"
Z.
Wahrsciiein-
lichkeitstheorie view. Geb., 26:119-139, 1973
11.
Milgrom, P., and J. Roberts, "Limit Pricing and Entry under Incomplete Information:
Equilibrium Analysis," Econometrica 50:443-59, 1982a.
12.
Milgrom,
P..
and
An
Roberts, "Predation, Reputation, and Entry Deterrence," Journal of
J.
Economic Theory 27:280-312, 1982b.
M., and P. Reny, "Non-Cooperative Bargaining with (Virtually)
published manuscript, University of Western Ontario, 1990.
13. Perry,
No
Procedure," un-
"Reexamination of the Perfectness Concept for Equilibrium Point
Games", International Journal of Game Theory, 4:25-55, 1975.
14. Selten, R.,
15.
Simon,
ley,
16.
"Basic Timing Games," unpublished manuscript, University of California at Berke1987.
L.,
May
Simon,
in Extensive
L.,
and M. Stinchcombe, "Extensive Form Games
in
Continuous Time: Pure Strate(To appear in Economet-
gies," discussion paper, University of California at Berkeley, 1988.
rica.
17.
)
RAND
Whinston, M. D., "Exit with Multiplant Firms,"
Journal of Economics, 19:568-588,
1988.
18.
Wilson, R.,
"Two
Surveys: Auctions and Entry Deterrence," Technical Report No.
3,
Stanford
Institute for Theoretical Economics, 1990.
Appendix
A
Proofs
Proof of Theorem
2.1:
5
Proof. From the hypothesis that w is bounded, / is bounded. Theorem 1 of Huang and Li
(1990) shows that there exists a solution to (3) and this solution is characterized by the first time
that e~ Tt f(X ) is equal to the largest regular submartingale dominated by it. 6 We claim that this
largest submartingale is e~ Tt 4>{X t ) = e~ Tt (f(X t ) - v(X )).
t
t
6
We gratefully acknowledge that a referee suggested the current proofs for Theorems 2.1 and 2.2 to us.
A regular submartingale {Y ;t £ 5R + } is an optional process so that for any bounded optional time T, E[Yf] <
t
and
for all optional times
to the sigma-field on
fi
S > T,
.E^V'sI^t]
< Yt
x [0,oo) generated by
all
oo
where an optional process is a process measurable with respect
the processes adapted to F having right-continuous paths.
a.s.,
PROOFS
.4
23
_rt
/(A<). It is easy
we show that e _r '<j!>(A() is a regular submartingale dominated by e
rt
rt
for all x. Thus e~ 4>{X ) < e~ f(Xt) a.s. Note that for any optional times
that v{x) >
First
to see
t
S>T,
Ex [e- Tb ct>(Xs)\TT
)
=
Ex e-^\niE
Xs [e-^-^f(X T )\TT
r>S
>
e~
=
e~
rT
inj
:
T
£x [e-( T
-T
}
)/(A- T )|^T ]
a.s.
4>(Xt),
rt
where we have used the strong Markov property of X Thus e~ 4>(X t ) is a regular submartingale
Tt
dominated by e~ f(X t ).
rt
Second we want to show that e~ 4>(X ) is the largest regular submartingale dominated by
Tt
Tt
e~ f{X ). Suppose otherwise and let Y be a regular supermartingale dominated by e~ f(X )
and Y(u,t) > e~ Tt <f>{X{u,t)) on a set of (u;,r) whose projection on to Q is of a strictly positive
.
t
t
t
probability. Define
Sn =
It is
*oc}
>
0.
>
:
Y(t)
> c" r( (/)(A'
(
)
n
n is
>
Te i%_ Sn
=
However, V(5„) > e~ rS "Q(Xs r,) + ^ with a
cf)(A's n
Finally,
)
to
Y(Sn
)
£:[y(T)\rSn
is
)
a.s.
strictly positive probability.
with a strictly positive probability, which
we want
so that
t
e-^iXsJ^^i^A^nXr^sJ
_r5n
e
+ -}.
an optional time and by the hypothesis there exists an n >
S
Tt
Since that e- f(X ) > Y(t), we have
easily verified that
PT {Sn <
inf{/
Thus e~ rS "<p(Xsn ) >
a contradiction.
show that
T* =
inf{r
>
:
/(A,) -
<
<f>{Xt )
0}
a barrier policy, that is, T' is the first time that A' ( is less than a level y*. First we observe
that f(x) — 4>(x) = v(x) and it is obvious that v(x) is increasing by the fact that
is spatial
homogeneous and that ir is increasing and nontrivial. Thus there must exist a y' so that
is
A
T' =
inf{f
>
A, <
:
y'}.
The fact that y' < y° follows from the observation that when
industry as strictly positive profits are being generated.
I
Proof of Theorem
Proof.
S > T'
Assume
t
>
y°,
it
is
better to stay in the
2.2:
Proposition 2 of
a.s.
X
Huang and Li (1990) shows
PX {S > T") > 0.
that
if
5
is
another optimal exit time, then
therefore
Put
Tn =mf{t>0:X <
t
y'
- -}.
well-known that Tn J T" a.s. (Only the case A'o = y' is nontrivial. If T° = limn^ocTn ^ 0,
> A'o for < t < T° a.s.. Since Aj-o = y',
the zero-one law implies that T° >
a.s. and thus
t
starting again from T° by using the strong Markov property and repeating the above arguments.
It
is
X
PROOFS
A
24
T° = lim n _oo Tn < T° a.s. if T° = oc. Thus it must be that T° = oo a.s.
Xt > y" for all ( £ Sf + For a nontrivial a, this is clearly impossible.) Thus
= \J n {S > ^"}- ^ we can show that, for every n, a.s. on {S > Tn } we have
we conclude
that
This implies that
{5 >
X""}
.
E x [e- rS 4>{X s )\TTn >
)
e-
rT
"«A(AVJ,
then we are done as this will imply
E x [e- rS 4>{Xs)\TT '] >
e-
rT
>(X T .),
hence
Ex [e- rS f(X s > Ex [e- TS 4>(X s > Ex [e~ rT
)}
and 5
is
Now
?'
=
suboptimal.
t = in{{t >
let
)}
Tn Xt =
:
y") and set So
'
'
= Ex [e- rT /(A»] =
<f>(XT .)]
= Tn
,
S2
= S \JTn and
,
S\
=
S2
*(*),
A
7"-
Then
for
1,2.
e-
TS
'4>(X s J
<
Ex [e- rS-^<t>(Xsi+1 )\?Si
is strict on {Si > So}
is a submartingale. Inequality (45) for i =
<f>
negative profits are made between So and S\. To see this, we note that
since
e-
=
e-
TSo
4>(X So
)
°f(X So
)
rS
Ex
Ex [e- TSl <j>(XSl )\Fso]
- Ex [e-^f(XSl )\TSo
Tt
]
*(X
t
)dt\TSa
]-Ex
Js
=
E x [[
since only
roo
e-
[
= {S > Tn }
-
roc
=
(45)
),
e-
[
Tt
Tr(X t )dt\fs
)
JSi
'
e-
rt
ir(X )dt\fSo }<0,
t
where the first equality follows since on <j>(Xs,) = f(Xs,) for i = 0,1, and the inequality follows
since because 7r(A' ) <
for t G [5o,5i). The two inequalities in (45) (i =
and i = 1) imply that
on {S > Tn ],
(
Ex [e- rS 4>(X s )\TTn
}
Ex [e-^d>(X s,)\TSo
Ex [Ex [e- rS^(Xs )\TSl }\Ts
s
> Ex [e-' ><t>(X Sl )\Fs
=
=
}
2
}
}
>
which was to be proved.
Proof of Proposition
rT
e- »4>{XTn ),
I
3.1:
Proof. To prove that T(S) is an optional time it suffices to prove that T(S)At is ^-measurable,
T(S)At e Tu for all e 9?+; see Dellacherie and Meyer (1978, IV.49.3). First note that S At £ 7
as S is an optional time. By the right-continuity of T{u>,t) in t, we know T(t) is Borel measurable
in t. By the composition of two mappings, it follows that T(u,S(u) A t) A t 6 T
Next note that
or
*
t
t
T(u,S(u))At
=
=
[T(w,5(w)A0]l{S(u,)<t}(w)
+
.
[T(«,5(«)A0]l{s< w )>t}(w)
[T(u>,S(u)At)]l {S{ul) < t )(u)+tl {S(uj)>t] (L;)
a.s.
PROOFS
A
25
T
By the fact that {5(w) > t) G t as S
optional time. The assertion that T(S)
Next we want to show that T(S)
G
t
T
x
as
T(5)
is
an
an optional time. Observe that
is
f(S(^)) =
an optional time, we know T(S) A
a.s. is obvious.
is
> S
inf{/
>
:
An
(u,t) G
[S(w)oo)},
where ,4 is defined in (18). By the hypothesis that A is a progressive set, and the fact that the
stochastic interval [S, oo) is also a progressive set, then Dellacherie and Meyer (1982, IV. 50) shows
that
T(S)
is
an optional time.
from the hypotheses that T(t)
Finally, the last assertion follows
Proof of Proposition
Proof.
A=
known
=
{(w,t)
l [0 T t(;)](w,0 r (
(
,
T = S P r -a.s.
Proof of Proposition
First,
we want
A=
W) +
£flx» + :T(w,0 =
that the stochastic interval [r(w),oo)
also easily verified that
Proof.
t.
Let r G T. Define
Then
is
right-continuous in
3.2:
T(u,t)
It
is
is
f l(T( W ),oo)(^,0-
t, f
£+} =
G
[r(w),oo).
Thus
progressively measurable.
T
£ T.
It is
I
for all x.
3.3:
show that
to
{(«,*)
:
G T. By the definition of T,(t), we
T,
T,(w,5)
=
5
5,
G
£+} =
{(«,«)
:
know
that
X(w,«) < y£}.
(//,<j)-Brownian motion is a progressively measurable process. Thus A is a progressively
measurable subset and T, 6 T.
Second, we want show that (T\,T2 ) is a Nash equilibrium. It is obvious that (T\,T2 ) G T x T.
The
Next, by definition, f,
= T~
for
i
=
1,2.
Thus
it
show that T"
suffices to
t
rAT"
^
tGT
sup
" e~ Tt ir i2 (Xt )dt
I
+
l {r
> Tj'}e
solves, for all x G S,
TT
"v xi (X r;j
i ^ i_and i,? = 1,2.
- 1 and let r be a solution to the above program.
program We
Fix x G H. In
"n the above program, take j =
want to show that r = T"->
obvious that
We first claim that r < Txl Suppose
It la
AL
is uuvjuua
uiai t >
c
J 22
^_ T"
22 a.s.
otherwise. Then the expected discounted future profits for firm 2 is
whejre
1
Ex
.
'
'
.
/
"
f
Jo
e-
Tt
7r 22
(X )dt+
t
[
.
e-
k
rr;,A7
<
ET
=
Ex
=
ET
e-
I
r
Tn AT
t-
Tt
)dt
+£ r
[e-'
)dt
+Ex
[e-
ir 22
(X
t
i: 22
(X
t
ir 22
(Xt )dt
Tt
j
rT^Ar
/
Jo
Tt
* 21 (X t)dt
Jt.'.^t
e-
rt
r
T 'u.
TT
21
(A T .)l (:>r .)]
1
»V2i(yu)l { r > T;1 }\
PROOFS
A
26
where the strict inequality follows since at T^, Xt- < 2/n < 2/21 aim tne unique optimal exit time
x
time for firm 2 in the monopoly case calls for exiting immediately. But P {t > T^} > 0. Thus
>
t
<
Tj'j
a.s.
Given that r < T^ a.s., throughout firm 2's life span, it is a duopoly. Thus the unique optimal
exit time for it is T22 and r = T22
Similar arguments establish that T{x also solves the above program when j = 2. Since x is
arbitrary, (T\,T2 ) is a Nash equilibrium.
Third, we want to show that (Ti,T2 ) is subgame perfect. Let S be any optional time. It is
easily seen that fi(5) = T^(S) and f2 (S) = T;2 (S). It suffices to show, given fi(S), f2 (S) solves,
.
for all
i£jf,
rTAT,(S)
Ez
sup
reT
and
is
vice versa.
By
'
f
e-^- s K t2 (X
the strong
Ex
'
f
+
l
Markov property of
independent of the values of
sup
)dt
t
X
e-
{T> fl(5)}
^
r(tl(S) - 5)
^i(A'ti(S) )|^s
X
Xs, Tj^S)
and the fact that, conditional on
before 5, the above program is equivalent to
rt
Tr t2
(X
t
)dt
+
l
{T>T .i( s )} e-
rTn( 5
)t, tl
(A' T].
i(S)
)|A'(5)
T>S
£ r [-|^"s]
T
the expectation conditional on A's under P
used in showing that {T\,T2 ) is a Nash equilibrium show that
where
is
.
T^S)
program. The proof for
Given that (T\,T2 )
I
computation.
is
Proof of Proposition
Proof.
is
Then arguments
T22 (S)
identical to those
a solution to the above
is
identical.
Nash equilibrium, the
rest of the assertion
can be verified by direct
3.4:
clear that at S, firm t's strategy from then on must imply that it will stay on at
can not sustain as a duopoly when its opponent stays on forever. Similarly, firm z's
strategy from 5 on must not stay longer than it can sustain as a monopoly.
I
It is
least until
it
Proof of Corollary
Proof.
3.1:
for all x.
Proof.
,
t
I
Proof of Proposition
will
< y 2l it follows from Proposi< T{2 {S) < 7\(S) P x -a.s. for all x. So firm
x
It then follows that T,(S) - T' (S) P -a.s.
Let (T\,T2 ) be a subgame perfect equilibrium. Given y*12
tion 3.4 that for any optional time 5, T2 (S) < T^(S)
1 will always stay longer than firm 2 in any subgame.
We
first
3.5:
record in the following
lemma
the explicit expressions of
some functional that
be useful.
Lemma
A.l Suppose
z
< x < y,x,y,z £ U, and T =
N
inf{«
>
:
X
t
G {y,z}}- Define
_ 6(x,y)-6(x,z)9(z,y)
«'«'*)-
l-0(z,y)e(y,z)
(46)
A
PROOFS
27
Then
Ex [e~ TT :X T =
=
y)
v(x,y.z). and
Ex [e- Tl ;XT =
z]
=
v(x.z.y).
Furthermore.
a
-nx.y.r)
(a.
-=.y)
dy
+
a')d(y,z)
a. + a'6(y.z)9(z,y)
~
—v[i.z.yK
u w
dz
i
1
-6(y,z)0{z,y)
(a.
+
a')6(z,y)
v(x.y.z)
ij>(x,z,y),
\-9(y.z)6(z,y)
dz
We
v(j.y.z).
v(x.y.z).
V[x.:.y\
Proof.
a.6(z,y)6(y.z)
l-B{z,y)9(y,z)
d_
For the proof of the
direct computation.
+
\-9(z.y)6(y,z)
d_
a
m
part see Harrison (19S5). and the partial derivatives obtain from
first
I
proceed to prove Proposition 3.5.
we restrict our attention to finding the solution to
First
sup
reT
rrAT2\(S)
Ex
e-
Js
r
^- s K (X )dt
+
i2
t
l
{
r > T: (S) }
1
e-
^ S) -
T{T
S]
v,
l
(X T
.
i{S)
)\Ts
(47
T>S
Then arguments similar to those of Theorem 2.2 will show that
is a barrier policy.
any other solution to (47) will be equal to this barrier policy. This barrier policy is T\(S) in the
assertion. Then one easily verifies that (26) is an element of T using similar arguments used in
so that t
Proposition 3.3.
We begin with A' 5
(26) for
some
y
Ui(x,y,y2i)
by (21) and
= x
€
(
i/21
-
^12]
*
Given that firm 2 plays
^(5),
firm l's payoff by playing
is
=
Lemma
/j 2 (x)
m
- v(x.y,y 2l
)fn (y) - 0(x, y£ x y)fn(y'21) +
,
v(x, yj v y)t'n(j/3i).
(48)
A.l. Differentiate u^ with respect to y gives:
dui(x,y,y2i)
(a'
+
a.)v(x.
y. y'.\
)
(fi2(y)
- 0(y.y2i)fu{y2i))
i-*(y,^i)%2i.y)
n i2(y)
-0 - 0{y-y'2i)8{y2i-y))
y))hi
2 {y) +
a
Ae- 'y^(x,y.y-21
<<,
e (y>y'2i) v n(y'2i))
)
n»i(y,J/2i)>
1
-Oiy-yliWiyli-y)
A is strictly positive constant. Observe that the sign of the partial derivative depends
the sign of mi. Direct computation yields:
upon
where
drnijy-yh)
=
dy
e-X 12
(
y )(l- C
-K +O .)(^,) {>0
>^.>^
)
<y?2i
;
where we recall that t/°2 is such that "12(2/12) = 0.
Note that mi(i/2i J/21 = r i i( I/21 >
and that mi
y = 00 is the upper barrier independent of the initial
•
'
)
is
continuous in
state.
m^yf,. J/21 - 0> tnen
Suppose otherwise. mi(y° 2 -J'2i < 0y.
If
)
'
)
)
A
PROOFS
28
By the continuity of mi there exists a unique
y = y or y = +00 maximizes u\. Note that
0( 1,3/21)
=
y £
{y^vVn) sucn tnat
+
ip(x,y2i,y)
m i(2/,2/2i) =
0-
Now,
either
*l>(x,y,y2i)0(y,y2i),
or
Thus,
Ui(l,00,J/2l)
=
=
=
- Ul(z»y>02l)
V (^,j/,J/2l)/l2(y)-(^(a;,2/2l) -^(^,J/21 J/))/l2(y2l)
>
1
+
(^( I >I'2l)
~
- *(& y3iW(*i j/, !& )/i2(»5i) + *(y, ySiM*, y,
^(x,2/,t/2i)(/i2(j/) - *(»»»ai)/w(»3i).+ ^(2/,2/2i) u n(2/2i))
v>(x,
2/>
V2i)/u(y)
V>(«, 2/21.
2/))
Ull(»21
2/21)^11(2/21
the optimal upper barrier if y < y* 2 otherwise, the infinity is. Suppose y < y{ 2
For x 6 (2/1,2/12]' nrm 1 quits
j/i, firm 1 exits when 3/1 is reached before firm 2 exits.
immediately. To summarize, firm l's exit time is the entry time of A' in [3/1, t/j 2 before 2's exit and
I
is the entry time of A' in ( — 00,3/^] after the exit of firm 2.
Then, y
For x <
is
-
,
]
Proof of Theorem
3.1:
Proof.
Using arguments similar to those of Proposition 3.5, firm l's exit time as the unique
is a two-barrier policy characterized by y\ and y\. Like in Proposition 3.5,
=
=
if y\
T,*(/);z = 1,2, / 6 !ft + ) is the unique subgame perfect equilibrium. Otherwise,
00, (Xi(<)
we have y\ < y\ < t/* 2 firm 1 exits as a duopoly when the demand enters the set [2/1,2/1]- Then
firm 2's exit time as the unique best response is also characterized by a two-barrier policy like (30)
with y\ and y\ replaced by y\ and y\ with y^ < y\ < y\- Let 3/" and 3/™ be the optimal barriers
for firm i in the n-th iteration. Repeat the iteration as long as 3/™ < 00. By the monotonicity
of the game, 3/™ and y^ are increasing in n and £™ and y 2 are decreasing in n. Therefore, they
have a limit point. By the hypothesis of the theorem, this limit point is a fixed point of the above
iterative procedure and is a Nash equilibrium at any optional time r. The equilibrium strategies
are stationary strategies and thus thus are subgame perfect.
The second assertion follows from the fact that the fixed point has been reached by starting
from an upper bound of all of firm 2's subgame perfect equilibrium exit times and the monotonicity
of the game.
I
best response to (30)
:
Proof Corollary
3.3:
Proof.
We
1: suppose that there do not exist 3/1 and 3/2 that satisfy mi (3/1,3/2) =
with
the
desired order of the 3/, 's. This implies that a lower bound of firm l's
"2(3/1, 3/2)
subgame perfect equilibrium exit times is T^t) for all t. Proposition 3.4 then implies that the
unique subgame perfect equilibrium must be the one in Proposition 3.3.
Case 2: Suppose that there are 3/,'s and j/, 's satisfy the desired conditions except that 1^(3/1,3/2,3/2) >
0. Then again, a lower bound on firm l's perfect equilibrium exit times is T^(t) for all t. So we
have a unique subgame perfect equilibrium.
and
take cases. Case
=
A
PROOFS
Case
3:
29
Suppose that there
exist
j/,'s
and
y,'s
so that
7711(2/1,2/2)
=
0,
712(2/1,2/2)
=
0,
and
< 0, with t/21 < 2/2 < 2/1 < j/i2- Then the violation of the hypothesis of Theorem 3.1
Then it is easily seen that 2/1 = 2/12 an ^
the y,'s. Suppose first that j/2 = 00
come
from
must
=
the
conditions
of
Theorem
3.1
are
satisfied. Therefore, it must be that
0.
Hence
rai(oc,2/* 2 )
u '(y\-> ?V2»!/2)
•
There are several
1-
"1(2/2,1/1)
=
possibilities:
and y
times is T^(t) for
equilibrium.
2.
n\
(2/2, 2/1)
A'5
Xs
=
j/j,
J/i,
=
ar,
al
d
<
x
t
i/i
^j.
Then a lower bound
of firm l's
subgame
perfect equilibrium exit
and Proposition 3.4 implies that there exists a unique subgame perfect
=
Then
2/2-
at the
subgame
starting at the optional time S wit
is not a Nash equilibrium.
So this
both firms
fir
exit immediately, which clearly
tl
cannot be the
case
3.
There do not
exist
t/i
and
1/2
so that "1(2/2,2/1)
perfect equilibrium exit times
subgame perfect equilibrium.
The proof
is
T^(t)
for all
for necessity part uses similar
=
t.
Then a lower bound of firm l's subgame
Then by Proposition 3.4, there is a unique
0.
arguments.
Figure
1:
Relative Positions of the Barriers
Date Due
Lib-26-67
i
3
ii
hi
n
in mi mi inn
TOfiO
ii
OObbbSIE
a