0.
^.s
^
^1
HD28
.M414
.rn:^?
'^^
amiotm
^.
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Continuous Time Stopping Gaines with
Monotone Reward Structures
Chi-fu Kuang and Lode Li
To appear in "Mathematics of Operations Research'
\-rP
#2109-89-EFA
Revised
March 1989
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Continuous Time Stopping Games with
Monotone Reward Structures
Chi-fu Kuang and Lode Li
To appear in "Mathematics of Operations Research"
IVP
#2109-89-EFA
Revised
March 1989
Continuous Time Stopping Games with Monotone Reward
Structures*
Chi-fu
Huang and Lode Li
Management
School of
Massachusetts Institute of Technology
Cambridge,
02139
MA
July 1987
Revised,
March 1989
ABSTRACT
We
prove the existence of a Nash equilibrium of a class of continuous time stopping games when
certain monotonicity conditions are satisfied.
'This is an extensively revised version of the first half of a paper entitled "Continuous Time Stopping Games"
by the authors. They would like to thank conversions with Michael Harrison and Andreu Mas Colell. Three
anonymous referees and seminar participants at Princeton University' and Yale Universit>' provided useful com-
ments. Remaining errors are of course the authors'.
Introduction
1
1
Introduction
1
The game
was
theoretic extension of the optimal stopping theory in the discrete time framewori<
initiated by
The extensions
Ohtsubo
Chaput
Dynkin
two-person zero-sum stopping games.
[1969] in an analysis of a class of
of Dynkin's
work include Chaput
[1986] in discrete time
[1974], Kiefer [1971],
Elbakidze [1976], Neveu [1975], and
[1974],
and Bensoussan and FViedman
[1974],
Bismut
[1977, 1979],
Krylov [1971], Lepeltier and Maingueneau [1984], and Stettner
continuous time.
[1982] in
Mamer
Non-zero-sum stopping games have been studied by
discrete time and by Bensoussan and
Nakoulima
Friedman
[1977],
Morimoto
and Ohtsubo [1986]
[1986],
Nagai
in
and
[1987],
[1981] in continuous time.
The widespread and
games can be found
potential use of stopping
and management science. Examples include the entry and
optimal investment
in
in
economics, finance,
exit decisions of firms, job search,
research and development, and the technology transition in industries.
(See Fine and Li [1986],
Mamer and McCardle
dynamic games
stochastic
[1986]
in
[1987],
Reinganum
which each player's strategy
is
[1982] etc.)
many
In fact,
a single dichotomous decision at
each time can be formulated as a stopping time problem.
The
existing literature on continuous time non-zero-sum stopping
games mentioned above,
with the exception of Morimoto [1986], uses stochastic environments that have the Markov
Morimoto
property.
games. The purpose of this paper
[1986] considers cyclic stopping
is
to
provide an existence theorem for Nash equilibria for a class of non-zero-sum non-cyclic stopping
games
non-Markov environment.
a
in
We
basically extend the discrete time analysis of
Some
[1987] to a continuous time setting.
Mamer
properties of a symmetric Nash equilibrium are also
characterized.
In Section 2
we formulate an A'-person
continuous time non-zero-sum stopping game.
Reward processes
are optional processes that
may
at
The
in
rest of this
paper
is
organized as follows.
be unbounded and can take the value
-oo
t
=
+co.
A
martingale approach
is
adopted
Section 3 to show the existence of optimal stopping policies of players under fairly general
conditions.
proved
same
in
The
existence of a Nash equilibrium in
Section 4 by using Tarski's lattice theoretic fixed point theorem.
We show
in
is
the
section that, for a symmetric stopping game, there always exists a symmetric equilibrium
when the reward processes
when
games with monotone payoff structures
it
exists,
satisfy a
monotone condition. Moreover, a symmetric equilibrium,
must be unique when reward processes
are separable in a sense to be
made
precise.
We
satisfy another
monotone condition and
discuss two duopolislic exit
games
in
Section 5
The formulation
2
2
to demonstrate the two kinds of monotonicity conditions posited in Section
4.
T}ic reward
processes of these two exit games also satisfy the separability condition mentioned above.
The formulation
2
Let (n, 7, P) be a complete probability space equipped with an increasing family of sub-sigma-
{7t]
7
of
field
t
€
or a filtration,
F=
{Jj;
G
t
5?+}.
F
5R+} by Joo arid assume that
=
1.
right continuity: 7t
2.
complete: 7q contains
We
F
,
interpret each
We
shall denote the smallest sigma-field containing
usual conditions:
satisfies the
Clg^^t ^>' ^^'^
all
w € n
the P-null sets.
to be a complete description of the state of the world.
models the way information about the true state of the world
is
The
filtration
revealed over time.
In a
discrete time finite state setting, a filtration can be thought of as an event tree.
Let 5R+ denote the extended positive real
An
line.
T
optional time
is
a function from
n
into
=
1.
An
3i+ such that
{ujeQ:T{u>) <t} e
where
!R+ denotes the positive real line.
optionaJ time
T
is
bounded
said to be
if
An
yte^+,
7t
optional time
T
there exists a constant
is
finite
K
G
P{T <
if
9?+ such that
oo}
P{T < K) =
1.
Let
of
all
T
be an optional time.
events
AE
7oo
The
sigma-field 7t, the collection of events prior to T, consists
such that
yte^+.
Af]{T<t}e7t
A
process
X
in this
paper
is
a
mapping
X
:
fi
x 3?+
h->
SR
measurable with respect to
7®B{yi+), the product sigma-field generated by 7 and the Borel sigma-field
X
is
said to be adapted,
if
sigma-field, denoted by 0,
is
right-continuous paths (see,
X{t)
is
measurable with respect to
the sigma-field on
n
7(
V^
G
of !R+
5R+.
.
A
The
process
optional
x 5R+ generated by adapted processes having
Chung and Williams
A
process
optional
if it
is
mecisurable with respect to 0. Naturally, any adapted process with right-continuous paths
is
optional.
[1983]).
is
It is
also
For any process
clear that X{t)
random
known
=
e.g.,
that any optional process
is
[1983]).
adapted
(see, e.g.,
is
Chung and Williams
X, we write X+{t) = max[X(0,0] and A'"(0 = max[-A'(0,0].
X'^{t)
- X~{t). For an optional time T, we
variable X{uj,T{(jj)).
will use
X{T)
it
to denote the
The formulation
2
We
consider an A' player s£o;)pinff yame. Players are indexed by
game
of this
3
»
=
1,2,..., A'.
The
payoffs
are described by a family of optional reward processes:
z,(u;,<;T_.);<e5R+,t= 1,2, ...,7V,
where r_, runs through
payoff that player
strategies
i
- l)-tuple of optional times. Interpret
(A^
all
receives in state
w when
employed by players other than
Note that the value
his strategy
is
T,
z,{uj,t;T_,) to be the
and T,{u)
=
t,
if
T-, are the
t.
a reward process "at infinity" specifies the payoff to a player
ol
if
never stops. There are at least two possibilities depending on the nature of the game. First,
a player does not receive any reward
Second,
if
from time
the value at time
to time
t
if
t
when he never
stops,
we simply
set 2,(w,
+oc; T_,)
=
he
if
0.
of reward process represents accumulated payoffs a player receives
he does not stop until time
t,
then a natural selection of the value at
infinity is
2,(w,+oo;T_,)
=
limsup,_+^2,(a;, t;r_,).
For an example of the second case, see Section
In the
5.
above two cases,
it is
easily verified that
the reward processes as defined on 5R+ are optional and their values at infinity are measurable
with respect to
The
Tea-
following assumptions on reward processes will be
Assumption
1 For any
[N -
l)-tupl£ optional times r_,
we have E[2,{T,;T_,)] > -co. Moreover, there
"ilt^.O
>
z,{u,t-T-,) for
Assumption
all {oj,t)
e
fi
x
this paper:
and any bounded optional time
m
martingale^
is a
=
T,
{m{t);t £ 9?+} such that
^+.
The reward process 2,(w,/;T_,)
2
made throughout
is
upper-semi-continuous on the right and
quasi-upper-semi-continuous on the left}
Assumption
1
ensures that
-oo < sup E[z,{T;T.,)] < +oo
reT
t
€
"The process m = {m{t);t G »+}
S-)-, and £[m(t)| J,] = m(s) a.s. for
it is
is
all
a martingale
t
uniformly integrable. Thus E[m(T)|77.1
An
optional process x
we have lim8up„_„2(T
-f-
=
{z[t);t
l/n)
<
6 S?+}
>
s.
= m{T)
is
=
1,2,...,} increasing to
t,
it
adapted to {I,;i £ !ft+}, £||m(/)|J < oo for all
is defined on the extended positive real line,
optional times t >T.
is
a.s. for
m
upper-semi-continuous from the right
x[t) on the set {r
said to be quasi-upper-semi-continuous on the left
{Tn;ri
if
Note that since
if
<
for
limsup„_^i(T„) <
if
for every optional time t
= {x{t);t 6 3? + } is
and any sequence of optional times
oo}. Similarly, an optional process z
each optional time
i(t).
t
3
Existence of best responses
for all T_,,
which we
optional times.
The
— oo
at
=
t
Theorem
will formally state in
usefulness of
process satisfying Assumption
the value
4
1
Assumption
1,
where
become
2 will
T
denotes the collection of
clear later.
Note that
all
reward
a
can be unbounded from above and from below, and can take
+oo.
Given the strategy of
his
T, that solves the following
opponents T_,, the objective of player
i
is
to find an optional time
program:
supE[r.(T;T_,)!.
(1)
TeT
If
such an optional time
A Nash
T, exists,
we
equilibrium of the stopping
such that T, solves
(1) for
i
=
In this section
game
we
is
a best response to T_,-.
an TV-tuple of optional times
show that there always
will
Thompson
(T,;
«'
=
1, 2,
.
Thompson's analysis
some
Our
exists a solution to (1).
,
.
.
analysis closely follows that of
This solulion
Thompson
[1971], players are allowed to use only the finitely-valued optional times.
however, the admissible strategy space
A')
is
composed
of
all
the optional times.
Our
quite straightforward. For completeness of this paper
is
will
[1971
.
Here,
generalization
we
shall provide
details of this generalization.
Note that Fakeev
[1970]
and Maingueneau [1976/77] also analyzed optimal stopping prob-
lems by allowing non-finite optional times
£'[sup( 2~(i;r_,)]
dass D, that
1
is
T,
1,2,..., A^.
also be said to be a "best response."
of
sometimes say
Existence of best responses
3
In
will
is,
in
a non-Markov setting. Fakeev [1970] required that
< oo and Maingueneau [1976/77] worked with reward processes that
random
variables {z,{T\T-i);T
and 2 admit processes that
lie
G T}
are uniformly integrable.
are of
Assumptions
outside the framework of Fakeev [1970] and Maingueneau
[1976/77].
We
proceed
first
with some definitions.
Definition 1 A regular supermartingale {X{t);t G
!R+
}
is
an optional process so that for any
bounded optional time T,
E[X-{T)]<^
and for
all
optional times
S >T, E[X{S)]
is
(2)
well-defined and
E[X{S)\7t] < X{T)
a.s.
(3)
3
Existence of best responses
Remark
1
Out
5
definition of a regular supermartingale
Mertens /1969, Theorem
include values at infinity.
of a regular supermartingale have right
The regular supermartingale defined
right.
An
Y{u),
and
Y
optional process
> W{uj,t)
t)
is
for all (w,
said to
t)
left
generalization of Merten [1969J to
is a
shown
l] has
limits
and
that almost all of the paths
are upper semi-continuous
W
above the optional process
lie
same
in Definition 1 also has the
outside a subset of
nx
9?+
the
characteristics.
y >
denoted by
,
from
whose projection on n
is
of
H',
if
P-measure
zero.
A
Y
regular supermartingale
W
an optional process
Y >
i(
W
is
minimum
the
and
ii
X
regular supermartingale
(MRS)
lying above
>W for any other regular supermartingale
then
A',
X >Y.
The
exists a
following proposition shows that, for any
MRS
G ^+}
t
G
9f+,
and
<
6
9?+}-
\)-tuple of optional times T_,, there exists a
G
lying above the reward process {z,{t;T-,);t
,
right-continuous on the set {[iu,t)
and
- l)-tuple of optional times T^,, there
lying above the reward process {2,(<; T_,);
Proposition 1 For any [N —
{Y,{t;T-,);t
(A'^
G Q X
3?+
:
Y,{uj,t;T-t)
>
2,(0;,
t;
^-,
}
.
MRS,
The
MRS
denoted by
Y,{T-,)
T-,)}. Moreover, fix
f
is
>
let
7,
Then P{t, < 00}
=
1
=
>
inf{s
t
:
2.(s;T_,)
> y;(s;T_.) -
e}.
and
E\Y,{u-T.,)\Tt] =y.(t;T_.)
a.s.
Proof Put
z^{t-T.,)
The
by
process z^[T-,)
M.
is
Dellacherie and
Meyer
[1982,
Now
y.(<;T_.)
We
<
V<
G ^+.
bounded above by a martingale by Assumption
Y-^{T-i) lying above 2,^(T_,).
for all
= m^x[z,{t-T.,),-M\
Appendix
I,
Theorem
22]
1
and
is
bounded below
shows that there exists a
MRS
define
= essinf{y,^(t;T_.);M=l,2,...}
G ^+
claim that Yi{T-i)
optional. Second,
Zi{T-i), for any
is
the
MRS
lying above z,[T-,). First,
we show that Yf{T-,)
is
it is
easy to see that }\(T-,)
a regular supermartingale. Since Y,{T^,)
bounded optional time T, E[|y,(T;T-,)|] < 00 by Assumption
1.
lies
Next
is
above
let
T
Existence of best responses
3
6
and 5 be two optional times with
T <
By Assumption
S.
£'[7,(5; T_,)]
1,
is
strictly less
than
+00. Moreover,
< E{Y,'^{S-T_,)\Tt]
E[Y,{S-T_,)\Tt]
< yA^(T;T_.)
where the
from the
as. for
fact that
Y/^{T-,)
M
Now
is
7.(7; T_,)
a.s.
from similar arguments of Lemmas
2.2
and 4.2-4.5 of
I
[1971].
define an optional time
r.(T_,)
The
=
a regular supermartingale.
rest of the cissertion follows
Thompson
1,2,...
a regular supermartingale. Relation (4) implies that
is
E[Yi{S;T-i)\TT] < essinf r,^(T;T_.)
The
N=
inequality follows from the definition of y,(T_,) and the second inequality follows
first
Thus Y,{T-i)
all
following
Theorem
is
=
inf{i
e
5R+
:
Y,{t;T.,)
=
z,{t;T^,)}.
i.
Moreover,
our main theorem of this section.
1 T,(T_,) is a best response to T_, for player
£[y.(0;T_,)]
= sup
E{z,{T-T^,)]
TeT
and E[\Y,{0;T_i)\] <
Proof
Fix n >
oo.
and
let
T„
=
inf{s
>
:
y.(s;r_.)
<
2.(s;T_.)
+ -}.
n
By Proposition
1, ?„ is finite a.s.
and
E[y,(r„;T_.)l
By the hypothesis that
that y,(r_,)
is
z,{T_,)
is
=
^[r.(0;T_.)].
(5)
upper-semi-continuous on the right (Assumption
right-continuous on the set {Yi[s;T-,) > 2,(s;T_,)} (Proposition
have
^[^.(r„;T_.)]
>
E[Y,(t„;T-,)]
1
-
n
E{Y,{0-T-,)] -
1
n
>
1
sup E\z,[T-T^,)]
TeT
"
1),
2), the fact
and
(5)
we
Existence of best responses
3
Letting n —> oc and noting that
sup E[z,{T-T^,)] > E[z,{r„-T_,)]
Vn,
TeT
we
get
sup E[z,{T-T-,)]
=
ElY,{0-T-,)].
(6)
TeT
Moreover, Assumption
<
implies that £[|yj(0;T_,)|]
1
oo.
These are the second and the third
assertions.
Next put
r'
<
t'
=
We
T,{T_,).
limnTn.
The
limit
is
well-defined since t„
increasing in n.
is
>
lim E[^.(r„;T_.)]
n—OO
=
£;[y,(0;T_.)]
>
E[r.(7';T_.)],
where the second inequality follows from the hypothesis that
continuous on the
left.
It
=
Recall from above that E{\Yi{0;T-,)\]
<
r'
=
assertion.
The
in
T,{T^i).
By
(6),
it
E[Y,{t';T-,)]
=
Therefore
oo.
follows that T,{T-,)
(T_,)
is
quasi-upper-semi-
is
it
E[Y,{0-T-,)\.
must be that
z,(r';T_,)
=
a best response to T^,. This
Y,{t';T-,)
is
the
first
I
following propositions give
Section
2,
then follows that
E[z,{r';T-,)]
and
clear that
have
E[Y,{T';T^,)]>E{z,{r'-T-,)]
a.s.
It is
some properties
of a best response
which we
will find useful
4.
Proposition 2 Fix T_,. Let
above £,(T_,).
t,
be a best
response for player
t
and
let
Yi{T-,) be the
MRS
lying
Then
y.(7.;T_.)
=
z.(r.;T_,)
a.s.,
(7)
and hence
T,{T_,)
Proof By
lies
above
the fact that F,(T_,)
2,(T'_,),
is
we have, almost
<
T,
a.s.
a regular supermartingale,
(8)
r, is
a best response,
surely,
£ly.(r.;T_,)] < E[r.(0;T_.)]
=
E{z,{t,;T_,)]
<
E\Y,{t,;T_,)],
and
V, (7"_,)
3
Existence of best responses
8
where the equality follows from the second assertion of Theorem
E[y.(0;T_.)]
It
then follows from Y,{T-i) > z,{T-i)
=
E[y.(r.;T_.)l
and the
a.s.
y,(7-,;T_,)
This
=
=
1
.
Hence
E[z,{t,-T.,)].
last assertion of
2,(r,;T_,)
Theorem
1
that
a.s.
is (7).
By
the definition oi T,{T-i),
we then have T,(T_i) <
a.s.,
t,
which
is
(8).
I
Note that
(7) says that
Yi{T-t) meets 2,(r_,).
2,(T'_,),
This
Ti{T_i)
is
the
any best response
to T-,
r
Since T,{T^i) by definition
minimum
is
must be the random times
the
best response for player
i,
time that Y,{T_,)
which
equal to
is
given that his opponents play T.,.
is (8).
The
following proposition shows that a best response
t,
chosen by player
to be optimal at
any optional time S as long as at S player
Proposition 3
Let
{'«
first
at
> S)
r,
,
is a
r,
be a best
response to T_, and
let
S
at time
continues
has not stopped according to
»
be
»
r,
an optional time. Then on the set
solution to
sup
E{z,{r;T_,)\rs].
t€T,t>5
Proof
If
the event
Suppose therefore that
B =
B
is
0,
is
nothing to prove.
there exists an optional time ?
> S as. such that
is
of zero probability, there
of a strictly positive probability.
Suppose that the assertion
P[B) >
> 5} € Ts
{r,
is false.
Then
where
B = {<^eB:
E{z,{t-T_,)\Ts]
>
E\z,{r,;T_,)\rs].
Define
^
It is eaisily
verified that t'
is
T,{u)
Uujen\B;
f[u)
liujeB.
I
^'^^
\
an optional time and
E{z,{t'-T^,)]> E[z,{t,-T.,)],
a contradiction to the hypothesis that
r,
is
a best response to
T_,-.
Hence the assertion
follows.
Naish equilibria
4
Nash
4
when
have a monotone structure
z,'s
equilibria
we
In this section
will
players satisfy certain
when
show that there
monotone
2;,'s
9
have a monotone structure
Nash equilibrium when the reward processes
exists a
of
structures.
Consider the following eissumptions:
Assumption
Zi{ijj,t;T-i)
—
There exists
3
—
There exists
-
There exists
5
r,(w,f';T_,)
Assumption
Q e T
and hotels gather
higher
demand
game. This
Denoting by
is
T^
1, "^ is
be referred to
a
3:
if
for
if r,
=
is
eis
>
t'
1
and for u e Q,
t,
t'
G ^+ and
t
>
t'
1
and for
t,
t'
E
and
t
>
t'
oj
in the
e
fl,
!R+
game, the higher the "incremen-
Assumption
Assumption
satisfy
for, say, oligopolists facing a fixed
5
a strengthening of Assumption
is
Assumption
4 is the
$
well-defined. Since
=
demand
func-
Examples
4.
3 or 4 can be found in Section 5.
the collection of A''-tuple of optional times,
we
define
<I>
:
T'
>—»
T'
as
(T.(T_.)).'li.
specifies best responses for players,
it
will
sometimes
a reaction mapping.
5, a.s. for all
said to be
$(r) and S'
The
t
Being clustered together, they can generate a
more natural assumption
i'
=
1,2,
.
.
.
,
= $(5)
—
t
A'^.
any two A^-vectors of Markov times
mapping $
t'
>
e ^+ and
a player's incremental reward decreases the longer his opponent stay
For two A'^-vectors of optional times
> 5
t'
such as convention business for hotels.
$(T,,...,r;v)
r
t,
u;
game. For example, we often observe departnient
in the
games having reward processes that
By Theorem
e h,
opponents stay
nearby locations.
in
tion for the industry as a whole.
of
and for
decreasing in T-i.
is strictly
for their services
converse of Assumption
P{Q) =
with
3 says that the longer his
stores
in the
P{h) -
with
Q e 7
reward a player gets by staying
tal"
1
z,[oj,t']T^t) is nonincreasing in T-i.
Assumption
Zi{ijj,t]T-,)
P(n) =
with
Zi[oj,t';T-,) is nondecreasing in T-,.
Assumption 4
Zi[LL),t,T-i)
Q e T
t
t'
<
.
.
,Tfi)
and S
The mapping $
> S,
monotone decreasing
implies
[ri,.
if
r'
for
=
is
=
(S,,
.
.
said to be
$(r) and 5'
,5^-),
.
we denote by
monotone increasing
= ^{S)
imply
t'
> S'
any two A^-vectors of Markov times
.
r
The
> S
S'
following proposition shows that
monotone decreasing under Assumption
$
4.
is
monotone increasing under Assumption
3
and
is
Nash
4
when
equilibria
Proposition 4 $
if
Assumption
>
S. Let
t'
=
monotone increasing
is
Assumption S
if
that Assumption 3
and 5'
$(7-)
=
[^.•(5,';r_.)
-
some
{.;
where we have used the
<
z.(r.';r_.)]l^
>
on a
S'-
S,'
E[z,{Sl-S_,) - r,(r,';5_,)l.';.]U >
A G
(see, e.g., Dellacherie
Tj'
r' a.s.
It is
=
=
where we have used
the
minimum
(9)
and where
rest of the assertion follows
The
following
2
We
on the
is
claim that
E[z,{tI-
is
S_,)U +
z.{Sl\
E[z,{S[S-,)\A
+
and Meyer [1978, Theorem
set A.
(9)
an optional time and
a, is a best
cr,
<
S-,
response to 5_,. To
the
first
5-,)^^]
z,[SlS^,)\A^Jr'
E\z,{S[-S_i)\,
A'^
denotes
H
\
>4.
This
best response to 5_,; see Proposition
The
Theorem
a.s.
0,
Thus
3.
easily checked that a,
= E
is
z,(t,';5_,)
measure.
5_.)]U.
that
E[z,{a,- 5_.)]
S[
=
surely,
on both sides of the above relation gives
T^'
>
fact that
A
z.(r,';
£;[2.(5,';r_.)-2.(r,';r_.)|^r'llA
set of strictly positive
we note
see this,
=
ct,
-
>
£[2,(5,-; S_,)|.%.')
^
monotone decreasing
we know, almost
3
[z.(S:;5_.)
IV. 56]), where the inequalities follows from Proposition
Oi
is
5.'}
By Assumption
i.
Taking conditional expectations with respect to
define
and
$(5). Suppose the set
of strictly positive mear-jre for
Now
valid
is
Choose two A^-vectors of optional times
satisfied.
is
A=
is
10
is valid.
4
Proof Suppose
T
have a monotone structure
z,'s
2.
from similar arguments.
main theorem
is
a contradiction to the fact that
Thus A must be
of
measure
I
of this section:
There exists a Nash equilibrium of the stopping game under Assumption
S.
zero.
Nash
4
Proof Suppose
complete
have a monotone structure
z,'s
that Assumption 3
T^
mapping from
ing
when
equilibria
lattice. It
to
is
satisfied.
11
From Proposition
4,
<C>
is
monotone
a
Proposition VI.1.1 of Neveu [1975] implies that (T'^',>)
itself.
then follows from Tarski's fixed point theorem (see Tarski [1955])
exists a fixed point for
<^.
easily verified that the fixed point
It is
increas-
is
a
tliat
is
a
there
Nash equilibrium.
I
A
stopping game
we have
2,(u;,t;T_,)
measure
zero.
=
A Nash
Under Assumptions
exists a
is
3,
symmetric
said to be
if
any
for
(A^
2j(w,t;T_,) except on a subset of
equilibrium {Ti)f_^
is
fi
- l)-tuple of optional limes T_,
x !R+ whose projection to
a symmetric equilibrium
if
T,
—
T, a.s. for
H
is
of
all i,j.
the following theorem shows that, in a symmetric game, there always
symmetric equilibrium.
Theorem
3 Suppose that Assumptions S are satisfied.
Then
there exists a
symmetric Nash
=
$;(T__,) a.s. for
equilibrium in a symmetric stopping game.
PROOF
Let
D=
and
all
F
let
where
t,,;',
^
be the restriction of
<I>,
denotes the
eT^
{T
to D.
theorem.
D
is
a
complete
The
lattice.
for each
T e
of $. Therefore,
show that F
similar to the proof of Proposition 4
verified that
Note that
component
t'-th
-.T,^ T, a.s.Vi,;}
cissertion
is
D, $,(T_,)
F maps D
monotone
Arguments
into D.
increasing.
It
is
also easily
then follows from the Tarski's fixed point
I
In the next
theorem we specialize our model to the case where the number of players
equal to two. In this case, there exists a Nash equilibrium under Assumption
Theorem
4 Suppose that
N =
Then
2.
4.
Nash equilibrium under Assumption
there exists a
is
^
for the stopping game.
Proof From
the
Proposition
component
t'-th
4,
of ^{T).
composite mapping $i o $2
we know $
It is
:
T
:
T'^ i—
T^
is
monotone decreasing. Let
easily seen that $,'s are
•—>
It
is
monotone increasing
G
T
such that ^i{^2{Ti))
Even
metric
in the
two player
=
Tj.
Hence
case, there
game under Assumption
4; for
since
then follows from the Tarski's fixed point theorem (see
Tarski [1955]) that there exists a fixed point of the composite mapping, that
Ti
(r_,) be
monotone decreasing. Consider the
T. This composite mapping
$1 and $2 are monotone decreasing.
<I>,
may
(Tj, <f>2(7'i))
is
a
Nash equilibrium.
is,
there exists
I
not exist a symmetric Nash equilibrium for a sym-
an example of nonexistence see Huang and
Li
[1987].
4
Ncish equilibria
when
have a monotone structure
z,'s
Note that we only consider pure strategy equilibria
symmetric equilibrium
strategies are allowed.
Nash equilibrium
following theorem, however, shows that
for a general
is
4
if
that a
mixed
there exists a symmetric
if
game under Assumption
player symmetric
A'^
Our conjecture
paper.
in this
symmetric stopping game under Assumption
exists for a
The
12
5
and
if
the reward
processes satisfy a separability property, then this equilibrium must be the unique symmetric
Nash equilibrium.
Theorem
5 Suppose that Assumption 5
is
equilibrium in a symmetric stopping game.
and
satisfied
Then
this
that there exists a
Nash equilibrium
is
symmetTic Nash
the unique
symmetric
equilibrium provided that reward processes satisfy the following condition of separability:
T,S G
T^~^ and
t
=S
B€
on a set
a.s.
Zi{u,,t;T)
Proof
Let
to an element of
T
By Theorem
.
1
,
almost surely
= z,{oj,t;S)yte^+yoje
T^
be a mapping from
"I*
', then
to
this
all
r'
<
S'
To
.
see this,
we suppose
of strictly positive probability for
(^.(r;;r_.)
-
some
2,(5,'; r_.))
i.
U
{rl
k(r:;r-.) -
where we have used the
The
is
left side
Now
let
fact that
of the relation
strictly positive
and
T' ,T' e
D
2.(5,';
is
is
r_.)l.vl Ia
A £
> S
a.s.
>
and
r'
claim that
£
<
<
{z.irl^S-,)
5,
monotone
- z,iS';S.,))
surely,
1^.
7^' gives
E
Uirl-S.,) -
Ts' (see Dellacherie
a contradiction to Proposition
#
is
and S' G '^{S) we
'^{t)
we know almost
^.(5.';
5_.)j.vl
U,
and Meyer [1978, Theorem IV.56J).
nonnegative almost surely by Proposition
and T*
^
5.'}
By Assumption
Taking conditional expectations with respect to
E
t
We
the best responses
all
that the set
A =
is
well-defined.
is
decreasing in the sense that for two optional times
have
B.
the subsets of T'^ that gives
mapping
If
3.
Hence the
right side
3.
T' be two symmetric Nash equilibria.
We
will
show that
this
leads to contradiction by considering three cases.
Case
T'
1:
T* < T'
a.s.
Since T' and T' are Nash equilibria,
e *(r'). The monotonicity
of
*
implies that T'
> T'
we must have T' e *(T") and
a.s.,
hence T'
=
T'
a.s..
This a
contradiction.
Case
2:
T'
previous case.
< T'
.
Using the monotonicity of
'i'
,
we have
a contradiction similar to the
4
Nash
equilibria
Case
3:
fe
when
have a monotone structure
z,'s
P{T' < T'} e
(0,1).
*(T). By the monotonicity of
* and
T > T
Let
B=
{T'
T =
Define
and
a.s.
< T'}. By Dellacherie and Meyer
T,' is
3,
equality.
T'_^
That
and since
T >
T'
T
a lattice,
E *(T") and
T'
E D.
Let
E *{T'), we know
a.s.
Theorem
IV. 56],
=
lB^[^.(7;';r_.)|.Vl
<
lBE[z,{f,;T^,)\7r]
=
lBElz,(f,;ri.)17j-;],
BE
We
It--
have
,
Proposition 3 implies that the above inequality
=
lBE[z,{f,-T'_,)\Tr] a.s.
(10)
define
^ ^'^^-\
where
[1978,
is
is
lBE\z,{T:-T'_,)\7r]
Now we
T > T*
D
and the second equality follows again from the separability assumption. Since
a best response to
must be an
Since
.
equality follows from the separability assumption, the inequality follows from
first
Proposition
^T'
the fact that T'
lB^[^.(T;;T^)|/rl
where the
T'
13
B'^
= Q\B.
Dellacherie and
Since
Meyer
BE
[1978,
Tt-
"^
^f,'^^
Theorem
f(a;)
's
ifuEB^,
easily verified that
IV. 53]).
We
T°
is
a optional time (see,
claim that T° E *(T). To see
e.g.,
we note
this,
that
E[z,{T:;T^,)]
=
E\lBZi{T°;T.,)
=
E[lBZ,iTl;T^,)+lB'Z,{f,-T_,)]
=
E[1b2.(T.;T_.)
=
^[r.(T.;T_.)],
where the third equality follows from
implies that T°
> T'
a,s.,
1.
We
+
B must
thus conclude that T'
= T'
lB'Z,{T°;T^,)\
1b^^.(T'.;T_,)!
Hence we proved our claim that T° E *(T). This
by the monotonicity of *.
contradiction. Therefore the set
Case
(10).
+
But on the
set
have a zero probability and T' >
as. and there
is
B, T°
=
T\T' ^
T'
T'
a unique Nash equilibrium.
.
< T',
a
This
is
I
5
Two
Two
5
we
symmetric, there
Assumption
games
two continuous time duopolistic stochastic
will consider
Assumption
satisfies
14
duopolistic exit
In this section
is
games
duopolistic exit
and thus has a Nash equilibrium by Theorem
3
Nash equilibrium by Theorem
a symmetric
is
Let the probability space and filtration be as specified
X=
I
and
=
The
1,2.
In the first
{W{t);t G
^+} adapted
game,
to F.
there
is
that the
|7r,j|
demand
is
natural to require that t,2(V^(<))
example we gave
for firm
is
>
We
bounded.
higher
W
when
firm's
in
the market
and are indexed
problem
is
to find an
profits.
for firm
t
for firm
t
:
when
it
t
when
there are two
is
interpret A' to be the industry
to be that
when
the only
demand
there are two firms
Thus
there are two firms in the market.
jr,i(X(<)) for
t
=
the
in
it
The reader can think about the
1,2.
Section 4 on department stores and hotels clustering together to generate
in
demand. Given that
a higher
expected
be the profit rate at time
only one firm in the industry and
Assume
industry.
let 7r,2(^V(<))
risk neutral
A
r.
satisfies
There are two processes
2.
There are two firms
a constant denoted by
its
The second game
Section
be the profit rate at time
let ;r,i(A'(<))
Suppose that
firms in the market.
is
maximize the present value of
firm in the market and
is
=
riskless interest rate
exit time in order to
when
W
game
this
if
first
4.
beginning of both games. These two firms are sissumed to be
at the
by
e^+}
{X{t)]t
in
games. The
In addition,
2.
3.
and therefore has a Nash equilibrium by Theorem
5
exit
its
competitor exits at the optional time T, the reward process
is
t
=
z,{io,t-T)
/o'^^'^'e—V,2(Vy(a;,s))ds+/'
c-";r„(X(u;,5))cf5,
te^^,
(11)
= limsup(_+^
Zi[u},+oo\T)
This reward process
It satisfies
7rjy(-)
by
=
when
absolutely continuous on [0,+oc) and adapted and therefore optional.
Assumptions
n2]{') for
Theorem
Now
is
j
=
z,{uj,t\T).
1, 2,
and
1,2, this
is
3,
and there exists a Nash equilibrium by Theorem
a symmetric
consider the second game.
that
game, the
there
is
a
When
symmetric Nash equilibrium
3.
Let ;r,j(A(<)) be the profit rate firm
there are j firms in the market and
Assume
game and
2.
|7r,j(-)|
total
is
demand
bounded.
here
natural that a firm's profit
require ;r,i(A'(<))
>
is
is
firm
«'
is still
in
Also interpret X[t) to be the demand.
when
it is
a duopoly than
receives at time
when
it
in
is
Unlike
the market.
in
t
=
1,2.
the
first
the market, where j
independent of the number of firms
lower
n^2{X{t)).
when
i
Thus
it
a monopoly. Hence
is
we
5
Two
duopolistic exit
Given that
its
games
15
competitor exits at the optional time T, the reward process of firm
i
is
(12)
= limsup,^^^
Zi{u>,+oo;T)
This reward process can be
z,{uj,t;T).
verified to
be optional and satisfies Assumptions
1, 2,
and
5.
ll
then follows from Theorem 4 that there exists a Nash equilibrium of this duopolistic exit game.
Readers interested
equilibria
when
Finally,
X
in
is
detailed analysis of this
game and
a Diurtiiian motion are referred to
we remark that the reward processes defined
condition of
Theorem
5.
discussion on refinements of
Huang and
in (11)
Nash
Li [1987].
and (12)
satisfy the separability
Bibliography
6
16
Bibliography
6
1.
A. Bensoussan and A. Friedman, Nonlinear variational inequalities and differential games
with stopping times. Journal of Functional Analysis 16 (1974), 305-352.
2.
A. Bensoussan and A. Friedman, Nonzero-sum stochastic differential games with stopping
times and free bomidary problem. Transactions of American Mathematical Society 231
(1977), 275-327.
3.
J.-M. Bismut, Sur un probleme de Dynkin. Z. Wabrscb. Verw. Gebiete 39 (1977), 31-53.
4.
J.-M. Bismut, Control de processus alternants et applications. Z. Wabrscb. Verw. Gebiete
47 (1979), 241-288.
5.
H. Chaput,
Markov games, Technical Report No. 33, Department
of Operations Research,
Stanford University, 1974.
6.
K.
Chung and
Boston
7.
R. Williams, 1983,
An
Inc.
C. Dellacherie and
P.
Meyer, 1978, Probabilities and Potential A: General Theory of
Process, North-Holland Publishing
8.
Company, New York.
C. Dellacherie and P. Meyer, 1982, Probabilities and Potential B: Theory of Martingales,
North-Holland Publishing Company,
9.
Introduction to Stochastic Integration, Birkhauser
E. Dynkin,
Game
New
York.
variant of a problem on optimal stopping, Soviet
Math.
Dokl.
10
(1969), 270-274.
10.
N. Elbakidze, Construction of the cost and optimal policies in a
a
11.
Markov
process.
game problem
of stopping
Theory Probab. Appl. 21 (1976), 163-168.
A. Fakeev, Optimal stopping rules for stochastic processes with continuous parameter.
Th. Prob. Appl. 15 (1970), 324-331.
12.
C. Fine and L. Li, Stopping time equilibria in games with stochcistically
monotone
payoffs.
Working paper #158-87, Operations Research Center, MIT, 1987.
13.
C.
Huang and
L. Li,
Continuous Time Stopping Games, Working Paper #1796-86, Sloan
School of Management, 1986.
6
Bibliography
14. C.
15.
17
Huang and
J. Kiefer,
L. Li,
Brownian motion
Optimal stopping
in
exit
games,
MIT mimeo,
1987.
games with continuous time, Prob. Th. Appl. 18
(1971),
556-562.
16.
N. Krylov, Control of
Markov Processes and W-spaces.
Math.
USSR-Izv 5
(1971),
233-266.
17. J. Lepeltier
and M. Maingueneau, Le jeu de Dynkin en theorie general sans rhypolhese
de Mokobodski. Stochastics 13 (1984), 25-44.
18.
M. Maingueneau, Temp
d'arret
optmaux
et theorie generale,
Seminarie de Probabilites
(1976/1977), 457-467.
19.
J.
20. J.
Mamer, Monotone stopping games,
Mamer and
ogy,
21. J.
22.
J.
Appl. Prob. 24 (1987), 386-401.
K. McCardle, Uncertainty, competition, and the adoption of new technol-
Afanagement Science 33 (1987), 161-177.
Mertens, Sur
la theorie
des martingales, C.r. Acad. Sc. Paris
268
(1969), 552-554.
H. Morimoto, Non-zero-sum discrete parameter stochcistic games with stopping times.
Probab. Th. Rel. Fields 12 (1986), 155-160.
23.
H. Nagai, Nonzero-sum stopping games of symmetric
Markov
processes, Prob.
Th. Rel.
Fields 75 (1987), 487-497.
24. O.
Nakoulima, Inequations variationnelles
des problemes de jeux stochastiques a
Bordeaux
25. J.
I,
et quasi-variationnelles bilaterales associees a
somme
nulle
ou non
nulle.
These, Universite de
no. 674, 1981.
Neveu, Discrete Parameter Martingales North-Holland Publishing Company, Amster,
dam, 1975.
26.
Y. Ohtsubo, Optimal stopping in sequential games with or without a constraint of always
terminating. Math. Oper. Res. 11 (1986), 591-607.
27. Y.
Ohtsubo,
A
nonzero-sum extension of Dynkin's stopping problem. Math. Opcr. Res.
12 (1987), 277-296.
28. J.
Reinganum, Strategic search theory. International Economic Review 23 (1982), 1-17.
^8^7 U58
6
Bibliography
jg
29. L. Stettner.
Zero-sum Markov games with stopping and impulsive
strategies.
Appl.
Math. Optim. 9 (1982), 1-24.
30. A, Tarski,
A
lattice theoretic fixed
point theorem, Pacific Journal of Mathematics^ 5
(1955), 285-309.
31.
M. Thompson, Continuous parameter optimal stopping problem,
theorie verw. Geb.
19 (1971), 302-318.
Z.
Wahrscheinlichkeits-
i
(•
Date Due
H012
'90
Lib-26-67
mil
3
TDfiO
III
mi mil
00Sb72fi3
III
M