'J^::^'.
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HD28
.M414
DfcVVEV
no.
EQUILIBRIUM EXIT IN STOCHASTICALLY
DECLINING INDUSTRIES
by
Charles H. Fine
Lode Li
WP 1804-86
June 1986
Sloan School of Management
Massachusetts Institute of Technology
Cambridge, Massachusetts, 02139
Preliminary draft.
Comments are welcome.
The authors gratefully acknowledge comments of John Cox, Chi-fu Huang, and
Jean Tirole.
The paper is a revised version of an earlier Sloan School Working Paper
(//1755-86) entitled "A Stochastic Theory of Exit and Stopping Time Equilibria.
EQUILIBRIUM EXIT IN STOCHASTICALLY DECLINING INDUSTRIES
Charles H. Fine
Lode Li
ABSTRACT
We study a complete information model of exit in which the stage payoffs
are governed by a nonstationary Markov process that reflects the stochastic
decline of the Industry.
problem.
For a monopolist, our model is an optimal stopping
For duopolists, we analyze the (perfect) stopping time equilibria of
the exit game.
There are multiple perfect equilibria of our exit game, in
contrast to several papers in the literature.
We explain how relaxing an
arguably unrealistic assumption in those models will give multiple equilibria
in those models also.
Finally, we show an equivalence between stopping time
equilibria, and perfect equilibria of complete-information, stochastic exit
games.
m^
1.
INTRODUCTION
The decline of products and industries is a natural by-product of the
growth and evolution of industrial economies.
Economic decline puts pressure
on firms' profits and on their ability to remain in the business.
eventually depart if the decline is severe.
Most firms
However, those who manage to stay
in after a shakeout can often earn significant profits.
In this paper, we analyze firms' decisions to exit an Industry that is
suffering from economic decline.
In contrast to the literature on entry, the
study of exit decisions by firms in oligopolistic industries is not
extensive.
Ghemawat and Nalebuff (1985) analyze a continuous time, perfect
information model for an asymmetric duopoly in a deterministically declining
industry, where the firms differ by their capacities and related fixed costs
per period.
Unless the high-fixed-cost firm has significantly lower operating
costs than the low-fixed-cost firm, the unique perfect Nash equilibrium in
their model is for the firm with the larger capacity and fixed costs to exit
at the first time that its duopoly profits are nonpositive.
The smaller firm
then remains until its monopoly profits become negative.
Fudenberg and Tirole (1986) also analyze an asymmetric duopoly exit game
in a continuous time model with deterministic demand paths.
However, they
assume that each firm does not know the fixed operating cost per period of its
rival.
Thus their model is a war of attrition where each firm continually
revises downward its estimate of the other's costs, as long as the contest for
the market lasts.
With an assumption that each firm assesses positive
probability that its rival will never find it optimal to exit, the model
-2-
yields a unique perfect Nash equilibrium where the higher cost firm exits the
market before the firm with lower costs.
We also model an industry that begins as a duopoly.
Nalebuff and Section
5
Like Ghemawat-
of Fudenberg-Tlrole, we assume that the Industry
declines over time so that the profits of the two firms shrink until one of
them exits, leaving a monopoly for the other.
In contrast to the results of
Ghemawat and Nalebuff, and Fudenberg and Tirole (each of whom obtains a unique
perfect equilibrium in a continuous time, deterministic model), there are
multiple exit-time equilibria in our stochastic, discrete time framework.
As
a consequence of the multiplicity of the equilibria, the stronger firm, which
has a stage-payoff advantage, may not always exit last.
The multiplicity of equilibria in our model is a consequence of the fact
that the industry demand process can jump from a point where both firms are
viable as duopolists to a point where neither firm is viable
but each is viable as a monopolist.
a.s
a
duopolist
At such a point in time, exit by either
could constitute an equilibrium in stopping times.
Both Ghemawat and Nalebuff
and Fudenberg and Tirole avoid this problem by assuming that demand falls
continuously (and detenninistically) over time.
However, their models would
also exhibit multiple equilibria as ours does if their demand process were
allowed to have jumps.
Thus, an assumption in the direction of more realism
(a demand process with discrete jumps) destroys the equilibrium uniqueness in
each of their models.
We characterize the generic form of any equilibrium in our model.
These
equilibria are obtained by solving a time-indexed sequence of fixed point
-3-
problems.
Among the equilibria, two are particularly Interesting because they
provide the upper and lower bounds for any equilibrium exit times and because
each player prefers one of the two equilibria to all other equilibria.
In
particular, each firm favors the equilibrium which makes his active period the
longest.
We provide a necessary and sufficient condition for the "natural"
equilibrium (in which the stronger firm always outlasts its rival) to be the
unique perfect equilibrium.
To model the exit game, we adopt a natural multiperson extension of the
optimal stopping time concept (see, e.g., Breiman (1964), Shiryayev (1978),
Dynkln (1969), and references in Monahan (1980)) which is the concept of
stopping time equilibrium, a Nash equilibrium in stopping time strategies.
(Our concept of stopping time equilibrium is not unlike that of Ghemawat and
Nalebuff or Fudenberg and Tirole, except that in their models the single-firm
optimal stopping problem is of little intrinsic interest because their models
have deterministic demand paths.)
In general, any stochastic dynamic game in
which each player's strategy is a single dichotomous decision at each stage
can be formulated as a stopping time problem.
Further, in such games, the
notion of equilibrium stopping time can be applied.
For this type of game, we
show that the stopping time equilibria correspond to the subgame perfect
equilibria in the natural extensive form game.
One characteristic of the firms' optimal policies in our model is that one
or both firms may earn negative profits in some period(s), but remain in the
market.
This can occur for either of two reasons.
First, industry demand can
suffer a stochastic negative shock, such that, demand will increase in the
next period, in expectation.
In such cases, fiirms will absorb the loss in
-4-
that period, knowing that demand is likely to recover.
with observed practice.
This result accords
Most industries experience stochastic downturns that
do not lead to mass exit from the industry.
A firm may also choose to absorb
losses and remain in the market in the expectation that its rival will exit
the industry first, leaving a more profitable, monopolistic industry for the
remaining firm.
Although the former effect is unique to our model, the latter
effect is also present in the Fudenberg-Tirole formulation.
For both the single-firm and two-firm cases, we solve examples with linear
demand and linear costs.
For the single-firm example, the optimal exit time
is decreasing in the firm's unit costs, the slope of the demand curve, and the
per unit fixed cost of being in the market.
in the firm's discount factor.
The optimal exit time increases
In the two-firm example with Cournot
competition, these results also hold for each firm.
In addition, the high
cost firm's optimal exit time increases in the low cost firm's unit costs.
The remainder of the paper is organized as follows:
In Section 2, the
single-firm exit problem is formulated as a stopping time problem and several
basic results are established.
In Section
3
the notion of stopping time
equilibrium is presented, and the main results regarding equilibrium exit
behavior in a duopoly are proved and discussed.
possible extensions follow in Section
4.
Concluding remarks and
^° this
aecti,.
"-Lace the
sloxle-f-f
^^°PP^ng .,,,
•
' °^''^ ^°- comparative
^
statl
^- assu^ne that at
ti
'
profitable. It
„.
may exit
»
the
m
^^« market.
costly.
'''^^
"nalnia. i.
°
^
'
°° '^^ optlmal
..
'^^s
market
.
'"' ^«
°o longer
Reentry I3
^
assumed to h*.
''^
prohibitively
,
'^
dynamf^
^
^-
^-^ustry demand
be 1,^
^''' '^
^-^ period t
^ «-rW se.ue
"=•
the.demand
t^e
«e<J"ence .(a
}.
a
p
take,
''^^"^ i° *'
Probabliitv
Z c fi
'
^L-i-cy m
.
g. and
measure p /
ha<j
^^^ ^ conditional
t^*'Vl>onZ.
late .
F„
^^^er
°" ^°°venlence,
discussion,
i, ,
;,e a,
-^n
order tn
assuoe 7 « d
=,
1
/
<;
"- •" -..;.:;;;- ir- -• - .....:r
--^astlcali, doml
dominates
^
'' ^^^^
^i-
t.
the
^-"^ ^hat
an ,>,/
''' ^^
P^^^HW
arm
.or
/
«
<
—
"cn,
,.„,„^^^
'
'^^
;^
t)
7T(a
We .
^^^"»e that
^ pre
present profi^<,
'^'
)
"
,
^
"^^o^ >
, ,
'
--«
°
flra's «^
"rategy
'°^ ^"^
1=
'''°
Period
.
^""^
^n'-'^)foraa
''''«
.
"^-tMe>
"^^^
t
or
^
^
^eclsio
ecision
'"'' ^°^ ^^^
for earh
», .. ,,
''
t
>o.
J3 ^
"^O
be stopping
^,
—
'=—
2-
•
"'^ '°"^"-ce
.
'"'"'^
-^out Whether
^^^
''^^ t^° «^ay
la the
market a^°^
^ero profit
set
^t th.
thereafter.
"^
z
'•••--
Formally,
-6-
let F^ " a(a
P
8<t), the o-field generated by (a ;s£t}.
;
as the information available at time
random time T:
available at time
In other words, information F
teM.
obtained by observing
t
N is said to be a stopping time of (P
-^
i^
We may think of
t
if
)
{a
;
s
{T£t}eP
<^
t}.
A
for every
is sufficient to tell
whether the event {T<t} has occurred.
Let
t
be the first time that demand has declined to the point where the
That is,
industry is no longer profitable.
assume that t*
t
" Inf
{t:fi(a
to avoid a trivial case and that t* <
>
°°
)
< 0,
a.s.}.
We
to guarantee that the
firm will exit in finite time.
Let
firm's discount factor.
6 be the
The firm's problem is to choose
stopping time T so as to maximize the expected discounted profit
T-1
E
[
I
7i(a.)
e'^
Proposition
problem.
1
a.]
I
^
t=0
^
(1)
.
below characterizes the optimal stopping times for this
It defines time Indexed sets B
policy is to exit at the first time
t
such that the firm's optimal
such that a
e B
.
The sets
B^ are defined by the recursively defined profit-to-go functions
u
(a
)
and take the form B
the u (•) functions.
=
{a
£
h
}
where the h 's are defined in terms of
Proposition 1 is preceded by a technical lemma that orders
the expectations of the positive parts of monotone ordered functions of random
variables that are ordered by stochastic dominance.
-7-
Lenuna 1
.
for zeZ.
\i
,
y„,
Suppose f,(*), f2(*) are increasing functions and
fJ,z),
>
And suppose z,, z~ are random variables with probability measures
Moreover,
respectively.
E(f ^(Zj^)l(Q oo)(fi(zi)]
IE
holds if E[1(Q „jfj^(z^)]
Proof ;
(z)
f
Since
f.^(z) >
\i.
stochastically dominates M„.
Then
inequality
[f 2(22^^(0, <=)^^2^^2^^^' ^°*^ strict
> 0.
£2(2) implies {f^(z) > 0}
_
{f2(z)
^t^l^^l^\0,«)(^l^h>)^-
^{f^(z)>0}^^^^^V^^
^
^{f2(z)>0}^/^^%^^^
^^{f2(z)>0}^(^^^^l^^^
>
0}
and y
>
y
,
^ ^{f2(z)>0}^2^^^<^V^^
= E[f2(z2) l^Q „)(f2(z2))].
Q.E.D.
Proposition 1;
(1) A unique minimal optimal stopping time for the problem described in (1)
can be characterized by T = inf{t:a
u^(a^) = TT(a^) +
'^Wj.^j^Ca^),
e B
)
=
where B
{u (a
)
£0},
and
^t+l^^t^ ° ^f"t+l^^t+l^ ^(0.<»)^"t+l^^t+l^^'^t^'
with w (•) =
(ii)
For
h
B
t
= sup{z
B
for
1
t*.
and zeZ, w (z)
>^
:
t
u (z)
for
t
<
>_
0}.
0.
>_
Then h
w ,,(2) and u (z)
>^
u
is increasing in t, B
(z).
=
(a
Let
£h
}
and
-8-
Part (1) follows from the optlmality principle of dynamic progranmiing
Proof
cf. Shlryayev (1978, Chapter 2).
To prove (11), note that
° E[n(a^^_^)l(Q^„j(TT(aj.^_^))lz] >
Wt*-l<^^^
Suppose w
- w
.
£
t
Lemma 1 implies w^(z) ^v^_^_^iz) since w^(
Finally, we need to show that u
that w (•) are monotone for
increasing, then u (•) "
tt
E[u (a )1.Q aj-)^"t^^t^^
as {a^
1
with
hj.}
Note that
Note that u (•) is Increasing since
t*-2.
Furthermore, u (z) - u
is increasing.
7t(»)
£
for
^
2
h^.
Uj-Csj.)
=
'
^
t
(
•)
•)
(z) = S(w
are monotone for
^(•)
sup{z:u^(z)
for
< 0}
Hence,
,
TT(a
)
>
(TT(a
)
or equivalently
is
•)
(
B
can be written
Therefore, the optimal policy is
)
£
For
(0).
< 0).
t
^
{a
},
0.
This cutoff
values that dictate optimal
and the set of a
and the optimal stopping time T.
it
*
Q.E.D.
falls so low that u (a
illustrates a sample path of
z "
*^
>.0.
t
exit, that is, the set of points less than or equal to h
profit line
'
^0.
is just the expected discounted profit from time t
is denoted by h
1
.,(z))
is increasing and w (•)
^^ increasing by Lemma 1.
to stop the first time that a
Figure
^
t
Again, by induction, if w
0.
(•) + 6w
*^
,,(z) - w
= E[u (a )1^q
oo^^^t^^t^^
until exit under the optimal stopping policy.
point for a
= Wj.^(2).
h
are denoted by B
,
.
as a function of t,
The dashed horizontal line represents the zero
0,
if a
falls above (below) this line, then
The cutoff points h
always lie below this line.
Thus,
the firm is sometimes willing to sustain a certain level of current loss in the
hope of receiving future profits.
E[Ti(a
.)la
]
< 0,
TT(a
)
and
<
the firm's optimal policy may be to remain in the market.
To see this, suppose
TT(aj.)
In fact, even if
TT(a
)
= -e <
and E[TT(a
+ e Et^^^t+i^lfo ")*'^^^t+l^^ '^t^
^ ^'
)la
]
<
0,
but
^^^° ^^^ ^^^^ ^^^ ^^^
^
^° period
-9-
for the option of staying in the market in the hope that demand will be higher
t
Because the firm can choose to exit at no cost after
than expected next period.
^
seeing a
the option of being able to observe
,
positive value.
The variable
tolerate in period
t
and profit from it has
aj^^,
represents how heavy a loss the firm will
h-^
in the hope of future profits.
The next proposition provides general comparative static results for the
It says that the optimal stopping time is increasing in the
model.
one-stage payoffs and in the discount factor.
Proposition
2
;
If ti'(z)
±
ti"(z)
and T'
±
T" almost surely.
for
1
or S'
e Z
z
Q"
then w
,
<
w
TT"(a
)
1
u
,
u
£
0, a.s.},
,
I
^
h
for
h
Proof
^
t
Let t' = inf{t
;
Assume t"
<
TT'(a^)
:
Obviously
°°.
£
t'
w
h^.
^
w
=
£
=0.
w
O
Ti"(z)
B
Assume that
This follows from the fact
.
11
(z) = u (z) and an application
+ Q w
is obvious by noticing that h
h
implies B
h^.
,
"
^
Then h
t"
I.
.(z)
'
of Lemma 1.
t
w
t
that u (z) = TT'(z) + Sw
In turn,
£
We want to show w
.
t
^
t
:
•
I
w
<^
For
t".
••
f
a.s.} and t" = inf{t
0,
<_
and B
B
S.
,
for
t
^
= u
-1
(0).
Thus,
0.
{T*
t} =
>
t
(n
B'}
3=0
^
Q (Ob"}-
The results for
Proposition
2
(T" > t}
for every
t
>
0.
So T'
<
T" almost surely.
^
8-0
Q'
±
S"
Q.E.D.
follow from a similar argument.
provides a very convenient tool for dealing with the
comparative statics analysis of the optimal stopping time.
It says that
looking at the stage payoff is sufficient.
For example, consider a monopolist who faces a stochastic linear demand
function and produces a homogeneous good at constant cost c.
The random
-10-
variable a^ is the intercept of the inverse demand and the constant
The opportunity cost of staying in the market is k.
slope.
b is the
(Alternately, the
firm has to pay a fixed fee, k, each period in order to participate in the
Then, at each period t, the maximum profit that the
production activity.)
monopolist can obtain is
(a -c)^
^^V 'I
-^
^
if a^ > c
- k
V
otherwise.
Clearly n(») is an increasing function.
or k increases.
Also
TT(a
)
= TT(a ,b,c,k)
decreases as
This fact, together with an application of Proposition
c,
b,
2,
implies that the optimal exit time T decreases almost surely as c,b,k
increases or
6
decreases.
That is, the monopolist optimally exits earlier
when the marginal cost of production is higher, a better alternative
opportunity exits (a higher k)
,
the price is less sensitive to the quantity
change (a higher b), or the firm is less patient (a smaller S).
3.
THE DUOPOLY MODEL - EXIT AS EQUILIBRIUM STOPPING TIME
We next analyze optimal exit in a duopoly.
characterize the optimal stopping equilibria.
the payoff to firm
demand is a
.
i
We formulate the exit game and
For l,j = 1,2, denote by
in period t, given there are
j
''^^/a
firms in the market and the
We assume that either firm will prefer high demand to low
demand and will prefer having the market to itself to sharing the market.
is,
"^/a
and zeZ.
)
)
are increasing in a
for i,j = 1,2, and
''^^o^^^
—
That
^11^^^ ^°^ i~l>2
At each time t, the active firms observe (a :s<t)
— and then simultaneously
3
decide whether to stay in the market and earn
ff.
,(a
)
(depending on the number
of firms who stay in) or to exit and get zero profit thereafter.
Firm i's
-11-
strategy is a stopping time T,: ^
i,
i
completely determined by observing the history
j, at time t is
^^
The payoff (as a function of T.) to firm
N.
-^
(a ;s<t),
i.e.,
—
s
.[ (a^.Tj)
Let
^2^a^)l(x^>,} ^
.
£
(0 < e.
e.
V^^^djlt}
be the discount factor for firm i.
1)
Then firm
i's problem is an optimal stopping problem if its opponent's exit time, T
is also a stopping time.
Given T
i
,
,
¥ i, a stopping time, let
T-1
^u
vj:
for
i
(aJ
- sup E{
T
t'O-'
I
ej
TT^
(a.,T.)
I
a.},
" 1,2, where the supremum is over all possible stopping times T.
Definition
1
;
(T,
,
T-) is a stopping time equilibrium if T,, T- are
stopping times and for i=l,2, and
2;
Firm
1
i,
'
t=0
Definition
?•
j
is stronger than Firm
2
if ^-.Az) > ^21^^^
for j=l,2 and every z.
To aid the game-theoretic analysis that follows, we first solve four
single-firm problems, two for each firm.
That is, by applying the results in
the previous section, we derive the optimal exit time for each firm
there were
j
firms in the market throughout i's stay in the market.
single-firm problem is indexed by
payoff.
i
and
j
i _a3
J^
Each
and takes ^jA*) as the stage
We emphasize that the functions calculated below and used in Facts
1-4 _do not represent equilibrium behavior.
Rather, they provide useful
manipulations of the two-firm profit function data,
^/j(*)» that will
-12-
Following the notation In
be used In the equilibrium analysis to follow.
Section
with w
ii
-^
2,
define recursively for i"l,2, j=l,2,
*
-
for
Propositions
Fact 1 ;
T,
.
1
t
It,
and
*
where
r
= inf {t:TT
t
(a
)
0,
_<
a.s.
By
}.
we have the following results:
2
inf {t
i
±
a
h j^} is the optimal stopping time where
h^J - sup{z:u^J(z) < 0}.
Fact 2
;
w
Fact 3
;
For i=l,2, and for all
t
Fact 4
are decreasing in
^
0, and T
^^
.
and h
t
a^.,
are increasing in
w^^^Ca^) Iw^^j^Ca^.),
for all i,j.
t
<
h^"""
h^^ for
almost surely.
If Firm 1 is stronger than Firm 2, then for j=l,2 and for all
;
^ii^s^ ^ ^ii^s^'
^t^
^
^t^
''''
'
^
^^^
°'
^j
'-
a^^,
hi
almost surely.
Figure
2
illustrates h
stopping times, T.
For i=l,2, and
,
*
1 j
,
t.., a realization of the {a
process, and
}
for the case when Firm 1 is stronger than Firm 2.
j;*i,
we use the following notation to denote the equilibrium
functions analogous to the single-firm functions described above.
is a stopping-time equilibrium.
For 1=1,2, and for each
subset of Z that represents the exit set for firm
and let B
denote the complement of this set.
1
t,
let B
Suppose (T ^,T ^^
denote a
in the equilibrium (T^,T-)
That is, T
=
inf{t:a
£
B
}
-13-
and B.
- Z\B.
=
is always an laterval of the form B
Whereas B
.
the single-firm problem, B.
{a
<_
h
}
in
in general, not an interval in the multiperson
is,
exit game.
*
For
t
^
t ..,
*
1
define w (•)
Then define recursively for
0.
=•
t
<_
t
-1
and
\
(Vi^
^\^\^ ^(o,-)"t^S^'Vi^-
- ^
" sup {z: 11.2(2) + S.w
Finally, define h
,(z)
The h
< O}.
variable plays
a somewhat different role in the multiperson game from the role of h
=
single-firm model where we had B
£
{a
describe the equilibrium strategy of firm
for the exit set
does not completely
H^^^* ^
^t^*
in the
Rather, it provides an upper bound
i.
of firm i.
B,
Our first result is that the optimal single-firm stopping times, T.- and T
impose upper and lower bounds for Firm i's equilibrium stopping time.
for any equilibrium, (T ,T ), T
1
-
intuitive argument goes as follows:
to exit as long as a
is above h
12
.
<
T
almost surely, for i"l,2.
T
.
That is,
The
At any time t, it is not i's best response
This follows because w
12
.
is the expected
profit from t+1 on obtained by acting optimally in the situation when other firm
12
is the minimum
can guarantee for itself.
If w ^, is the
will be in the market throughout the game.
possible future gain that firm
i
Therefore, w
i
expected equilibrium profit from t+1 on, then w ,,(a
TT^
(a^) + B^wJ+3^(a^)
t
"*
"i2^^t^
will stay active at time
a
falls below h
^
^w 12,,(a^)
° ^°^ j=1.2.
By a similar argument. Firm
t.
since w
^i^t+l^^t^
)
,
i
and
Hence, Firm i
will exit whenever
is the highest possible expected future gain
-14-
from t+1 on in the most optimistic situation, i.e., when he will be in the market
This completes the arguments required to prove
alone from t+1 on.
Proposition
3.
T^ -inf{t
a
:
1
T
„£T <T
Suppose
tit
e
,
B,
(T-,
,12) is an exit time equilibrium and
Then
}.
il
B
t—
<1
CB t12
it—
B,
for
>
t
—
and
almost surely, for i=l,2.
We now turn to an asymmetric situation in which Firm
Fi rm 2, that is,
^
^^ .(*)
11
situation has B,^ "
Bj^
and
is stronger than
The natural equilibrium for this
(•).
tt
1
" B 22 ; the weak firm exits the
Bjj.
first time that its expected present-plus-future duopoly profits are negative,
and the strong firm then remains as a monopolist until its expected monopoly
profits turn negative.
Although this is the unique euqilibrium that obtains
in the models of Fudenberg-Tirole and Ghemawat-Nalebuf f
,
it is only one of
many equilibria here.
To built intuition for understanding the complete characterization of the
equilibria in our model, consider the situation where a £(h
21
,h
12
].
In this case, either firm will find it profitable to stay in the market if
his opponent exits, but both firms have negative expected profits if neither
exits.
Exit by either the strong firm or the weak firm could constitute an
equilibrium.
In fact, any equilibrium in this game can be characterized by h
a set A
C
and a eB,
t
It
21 1
(h
,h
=
21
]
C.(h
(- °o,h
]
t
both firms are in at
12
,h
U A
,
1
,
h
2
,
If both firms are in the market at time
]:
then Firm 1 exits at
t
and Firm
2
stays.
If
t
t
and a eB
2
=
(-°°,
h
-
2
]
f) A
,
then only Firm
Proposition 4 formally states and proves this result.
2
exits.
and
t
-15-
Suppose Firm
Proposltlon 4
stronger than Firm
1 is
equilibrium (T ,T.) can be characterized as T
The exit sets
B
B
,
i=l,2,
(•-
"'
^t
_
.
,
" ^~
^t' ^2t
^
inf{t:a
_
°°'
,
"t+l^^t^
for
" 0,
<^
t
<
t >
t
for i=-l,2.
2,
21
h
,
1
],
A
= Z\A
,
t^^, and
w^^^(a^) = E[uJ^^(a^^^) l(o.=o)("t+l^^+i))
for
}
^t^ "^^t*
is a Borel set contained in the interval (h
A
efl
are defined recursively as follows:
t _^
11,
It "
=*
Any exit time
2.
^^
I
,
where
uj(a^) > (.^^(a^) + 3^w^^^^(a^)l5^^(a^)
+ (TT^l(a^) + e/^J:^(a^))l3_^^(a^).
Proof
;
For
^
i,
Backward induction.
t
> t*j^.
FiHj^j^Caj.)
> 0} -
= w
=
(a
j
)
P^^i/^t^
> 0} = 0,
and Pfa^. e
for 1=1,2.
B^^.}
i.j = 1,2.
= 1 for 1=1,2.
So P{a^
£
B^^} <
Also w^^Ca^.) = wjca^.)
The proposition is trivially true.
Now assume that it is true for
s
^
\^\^' vi^%^ ^ vi^^) i \li^\^
t+1.
Then by the definitions of h
12
^""p^^- "^
'-
^i
-
<
and
-16-
Suppose both firms are in the market at time
for i-1,2.
^^2t' ^2t^
adopted by Firm
Given any strategy
t.
Firm 1 will be a monopolist from
2,
and he will be involved in a game which gives expected payoff w
and equilibrium strategies follow from t+1 on.
on if a^
t
,
B^^,
e
e B„
,if a
Remaining active at t, Firm 1
expects to get
u'^(a^) - (TT^^^a^) +
S^wJ^^ (a^)) 1-^^ (a^) + (n^^Ca^) + S^wJ^^ (a^)
Firm I's best response then is to exit if u (a
t
Let
B.
and a
=
It
<
{uVa^) ~
t
t
< h
and B. = {u^(a )
It
t
t
0}
or if Firm
^It " ^^2t
^
stays and a
2
^~ "'
^t^^^
'-'
>
< h
^^2t
0}.
-
'
Firm
t
^^^^(^^
> 0.
)
t
will exit if Firm
1
2
That is,
.
^
and stay if u (a
<
)
t
)
^" "'
^^^^
''t^^-
Similarly,
®lt
" iB2^ri
(hj\ »)} U (B2^r\(hJ,
Given the strategy of Firm 1, if Firm
-.
(^a^)
-H
2
(3b)
")}.
does not exit at
t,
he will expect
S^w^^^ (a^)) l^^^Ca^).
and 2's best response should satisfy
^2t" ^^it^^-"
^2t
'
h^^]}
^^It*^ ^''t^'"^^
U
U
(4a)
{B^^flC-", h^]}
{B^t^(h^, »)}
(4b)
.
But, using (3a) and (3b) to expand the sets in (4b), we get
Bitn(hf.
») =
=
{{B2,n
B2^n
(-», hj^]}
(h^^, hj]
u
{B2t
since h^^
n(-», hj]}}n(h^\ «>
£
h^^^.
exits
-17-
Simllarly,
= .(hj,
\
00)
(B^j.
n
since h^^
(hj, h[]}
h^^ <
<
h^.
Using (4) and the above characterizations,
^2t' ^B2^n(h^\
'
Thus
B-
n
{B2t
hj]} U((h^.
")\{B2^n(h2,
U
since hj^
(h^^, h^]}
(hj, ")
is part of the equilibrium if and only if
^2t " ^^2t'^ ^^V'' ^t^^
^
^^t'
B^j^
<
hj ]}}
h^
.
satisfies
"^^
The only possible solution is of the form
—
®2t
"
^t
Furthermore, B„
^2t
'
2
^
^^t'
°°^
^*^^^^
'
Borel subset of (h
^t ^^ ^
is just the complement of B„
^t*^
^^'
21
_
If f2t^^^ = is
r> (hj.
Borel set in Z, then B„
"^^"
^t^'
1
,
h^]}
Aj.
= Z
\
,
1
,
h ].
i.e.
A^.
2
(J
(h ,"), a mapping from a Borel set in Z to a
is an equilibrium strategy if and only if B
fixed point of f2t^*^' ^'^•'
^
^2t''^2t^
^2t*
By (3), our characterizations of B-
^It ° ^^' ^\^^^
^It " ^^t^'
21
\'
"^'^^
•
^""^
and B„
give
is a
-18-
Finally, note that since
£
(a ), which implies w
u
-
^^^^.^^t^
(a
)
*'t+l^^t^
£w
(a
)
-
'
^'t+l^^t^
£ w
(a
)
"^t
*"^t^
-
"t^^t^
by Lemma 1.
We conclude the induction.
Q.E.D,
The importance of Proposition
4
lies in the fact that it characterizes all
In particular, it points out that we can obtain all
possible equilibria.
equilibria by varying A
21
1
(h , h ], for every
t.
Borel subset contained in the interval
a
,
Among these are two extreme equilibria which give
upper and lower bounds on any equilibrium exit times (T
of these, (T
= (-
B'
(T
,
T
h
°°,
T
,
)
11
is obtained by letting A
= (-
B*
],
°°,
h
22
],
=
,
T„
for all t;
)
.
The first
then
almost surely for every t.
In fact,
= ^-^ii* ^29^' 3°3logO"3 to the equilibrium of Fudenberg-Tirole and
)
Ghemawat-Nalebuff
.
This equilibrium is the one most preferred by Finn 1,
the stronger firm, because, as will be shown in Proposition 5, it gives Firm
1 the latest possible exit time.
The second extreme equilibrium, (T^^,T^) gives Firm 2, the weaker firm, the
and then defining
^' recursively
^ for
^t+l^^t^' -t^^t^' "t'-^t^
and B"
=
gives Firm
(h
2
,h
]
(J
(h
^^
,'")
*
1
12
It is obtained by setting h^_i = h ^^ for
latest possible exit time.
^°-
.
t
<
—
*
t,
il
oil
=
-1, A
,h
(h
t
Proposition 4, where
t
—t
B^^.
=
],
- t
,
l—OI
w ,,Ca^),
and h ,h
(- »,
t
—t
h^"'"
,
t
]
(J
— 1+1
t
(h^^.hj]
The next proposition shows that this equilibrium
the longest possible time in the market.
-19-
Let (T^, T2) be an equilibrium.
Proposltloa 5;
1
and T2
Proof;
Let B,
w (a
,
1
T-
Then T^ 1 T^
T2, almost surely.
^°*^
u^^^t^' "t^^t^' ^t' ^t'
),
f^
<
^t
^^t
'
^t^
correspond
to the equilibrium (T,, T-) in the usual way.
*
Note that for
11
Assume w^^^
Note that
and
Bj^^
Also,
C
t^^, w
^1
(-»,
(^,
h^l]
h^"^
C
82^^ (hj\
11
(a
(-»,
h^"^
C
^
A^
<VS>
C
11
u (a
-^
= w
1
w^^^
A^.
U
)
^
u
22
-
-,,
,
Then, h
w^^^.
i
- B^^.
11 ^
<
j.
a
h^
^
<
^,1
h^.
,
(h^^ hj]
- b^
.
(hj, ") - 82^, and
(h^l, h^]
^ u^(^^) ^^^
)
-2
2
22
= 0.
and w = w = w
"
2
i
>
U
]
h^] (J(h^, ")
)
1
(->, h^l] (j
C
(hj, «)
We just show Uj^(a
*
" w
^
and
w^^^
(h^^, h^]
B^^ = (h^^, «)
It follows that u
C
]
11
-2
1
^
>
w^.^^ > w^.^^
B^j. -
for every a^.
^
t
(3
)
U
(hj, ») = B^^.
-2
"^ (^r^
2
i
",.(^^)
as an example.
Vtti's" h:2t ^\^
-4
<S>
•
—
21
" ^
^*^^
-20-
Since
w^^j^Caj.)
we have
>
—
w^'^'Ca^)
t
t
1^1 1
t
and 12
Tj_
~> —w^(a^)
t
t
w^(a^)
t
1
T^
T^
1
and
Applying Lemma
^2f^ ^"if
w (a
tt
2 ^ir- ^W
As a result, B"
la completed.
ll
w^+i^^j.), ^21^^ ^2t ^^'^
i
)
^
—
^°*^
tt~t
w (a^)
^2t
>
w
again,
The Induction
(a.).
t
— ^2t—
1
^"P^^ ^^^'
^2t
Q.E.D.
.
Since Firm 1 is assumed to be stronger than Firm 2, the first extreme case
(T,,, T22) iSj in some sense, the more appealing equilibrium.
This
equilibrium is also simple in structure, i.e., a cut-off equilibrium.
Ghemawat
and Nalebuff (1985) show that this is the unique subgame-perfect equilibrium
in their deterministic, continuous time model.
Their result depends on their
An analogous
assumption that industry demand declines continuously over time.
assumption, for our model, would be that, with probability one, a
1-2
into the interval (J}^>h
21
before falling into (h
]
equilibrium has the weaker firm exiting first.
1
>J},.)«
falls
In this case, the unique
(This is stated and proved
Conversely, if Ghemawat and Nalebuff (or
formally in Proposition 6.)
Fudenberg and Tirole) were to allow jumps of sufficient magnitude into their
declining demand paths, they would lose uniqueness as our model does.
Proposition
6
;
Suppose Firm 1 is stronger than Firm
2.
Then (T.., T22)
is the unique equilibrium if and only if P {a
t
(n\
h^]}''
t
for
Proof:
For
t =
1 <
~
t
—
t2-l, P{a^
This implies that
B'
= B"
t- -1 where t„ = inf {t: P{
<
I
e
82^.}
I
= 0,
or P{a^ e
A
{a
(h^-"",
a.s. and consequently B'
h^]
= B"
e
s
^_,
3=0
U
.
B" }} = O}.
^t
(h^, ")} = 0.
Assume
-21-
B*
,
= B"
,
It then follows that
a. 8.
,
condition ?{a^
e
h^]} =
(h^""-,
=
h^.
imply that
h^.
=
B^^.
which together with the
a.s. for i'l.Z.
B'^^.
The
sufficiency follows.
To prove the necessary conditions, let
t
= sup
1 < s
(s:
t„-l, P{a
<
—
—
Z
(h^^,
e
3
S
0}.
>
h"*"]}
—
Then,
'"^
P(T2 > 12)
i
(a^ c B'-g}
n
P{
21
{a^ e
1
(h^\ hj]}}
s"0
= P{a^ e (h^^^,
t
since
t
h^]
—t
t-1
t-1
_
n
3=0
B'
C
28
a
I
_
B"
2s
r\
e
.
t-1
B"
J
2T
and
•
2t-l
(h^"",
t
P{
(a
1
h;^]
-t
e B':
23
s
3=0rt
= B" fl B'
2t
2t
}}
>
Q.E.D,
.
The following example illustrates some comparative static results that can
A Coumot duopoly faces a
be obtained with linear cost and demand functions.
stochastic linear inverse demand, p^ = a
and decreasing in
= 1,2.
t,
- bQ, where a
are independent
and produces a homogeneous good at constant costs
Assume that c,
^
c~ so that Firm
1 is
stronger than Firm
Firm i's opportunity cost of staying in the market is
k,
.
c.
,
2.
If both firms are
in the market, the equilibrium stage payoffs are
(a
- 2c
9b
5B
- k,
+
c
)2
—
- k
'^1
-
h
- c
> 2c
a
^^2
^1
^t
if
^
"
=1
i
-
^t
otherwise.
^=2 " =1
-22-
(a^ - 2C2 + c,)'
- k«
9b
if a
2
t
2c„ -
>
-
c.
1
2
TT22(aj.)
otherwise.
^
If one finn is in the market alone, the monopoly stage profits are as derived in
Example
1,
for
i
" 1,2,
(a
- c
),2
-4b
''
^i
\'-
^1
^1^^^
-
otherwise.
k.
It is easy to see that ^ji^a^^
k, " kj or c, = c-. k,
j
" 1,2 and every a
£
—
^I2^*t^ ^°^
^
" ^'^ ^^'^ every a
k-, then Firm 1 is stronger, i.e.,
""iJ-^J^
t.
£
^21^^t^
c-,
^°^
.
From the general results obtained earlier, we know that
exit equilibrium.
and if c,
The functions h
11
22
,
h^
(T^
-i
.
^22) is an
can be calculated as follows:
b/^1^
^1 ^ -/^bCk^ - Q^.[l^)
if k^ >
c.
otherwise.
^t
2C2 - c^ + /9b(k2 - B^w^^^)
if k^
i 32V1'
^t
2c„ 2
And h
11
b,
1
increases as c,, b, k^ increases or S. decreases, while h
as c-j b, k_ increases or
Ci,
otherwise.
c,
k,
c
,
I3„
decreases.
Thus T
22
increases
increases almost surely as
decreases or S, increases and T22 increases almost surely as C2, b,
kj decreases or c,, B~ increases.
-23-
We conclude this section by pointing out that the stopping time equilibria
and the subgame perfect equilibria of the exit game are equivalent.
By way of
constructing a stopping time equilibrium in Proposition 4, it is clear that the
stopping time T. = inf {s:
s
^
t,
a
e
B.
constitute an equilibrium for the
}
remaining subgame, given the firms surviving until
t.
Conversely, if we analyze
the exit problem in an extensive form game assuming players have perfect recall
of the historic information but not the future, etc., then any subgame perfect
equilibrium is a stopping time equilibrium.
Proposition
4.
7
.
The stopping time equilibria are subgame perfect.
CONCLUSION
This paper illustrates how a stochastic dynamic game like exit can be
formulated as a stopping time problem, how a stopping time equilibrium can be
found by solving a sequence of fixed point problems, and how to derive the
properties that the stopping time equilibria possess.
Furthermore it shows
how demand processes that are not continuous can give rise to multiple exit
time equilibria.
There are several interesting extensions or variations that might be
A straightforward, but
explored in the continuation of this work.
notationally burdensome, generalization is the exit game for an n-firm
oligopoly.
The equilibrium in which the stronger firms exit later always
exists if the firms are ordered by strength.
generalizing the model is to incorporate
A second direction for
entry decisions into the model to
study the firm's behavior over the life cycle of an Industry.
Though we only
-24-
discuas the exit problem for oligopoly here, the approach employed can be
applied to entry problems as well.
Finally, other interesting extensions may
include stopping time equilibria in stochastic exit games with incomplete
information (as in Fudenberg and Tirole) and stochastic exit games in
contlnous time.
-25-
Figure
1.
Single-firm stopping problem in the monopoly model
22
t
12
t
11
Figure
2.
Four single-firm stopping problems in the duopoly model.
-26-
REFERENCES
Brelman, L. , "Stopping-Rule Problems," Applied Combinatorial Mathematics
Wiley and Sons, 1964, pp. 284-319.
,
John
Dynkln, E.G., "Game Variant of a Problem of Optimal Stopping," Soviet Math
Dokl. , Vol. 10, 1969, pp. 270-274.
Fudenberg, D. and J. Tirole, "A Theory of Exit In Duopoly," Econometrlca , To
appear in 1986.
Ghemawat, P. and B. Nalebuff, "Exit," Rand Journal of Economics
2, 1985, pp. 184-194,
,
Vol. 16, No.
Hadar, J. and W.R. Russell, "Rules for Ordering Uncertain Prospects," American
Economic Review March, 1969, pp. 25-34.
,
Monahan, G.E., "Optimal Stopping in a Partially Observable Markov Process with
Costly Information," Operations Research , Vol. 28, No. 6, Nov. -Dec. 1980,
pp. 1319-1334.
Shiryayev, A.N., Optimal Stopping Rules
,
Springer-Verlag, 1978.
V07
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