Oe\Nev
HD28
.M414
•
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
CONTINUOUS TIME STOPPING GAMES
by
Chi-fu Huang
Lode Li
WP 1796-86
July 1986
MASSACHUSETTS
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
INSTITUTE OF
CONTINUOUS TIME STOPPING GAMES
by
Chi-fu Huang
Lode Li
WP 1796-86
Sloan School of Management
Massachusetts Institute of Technology
Cambridge, Massachusetts, 02139
Preliminary Draft and Incomplete.
July 1986
CONTINUOUS TIME STOPPING GAMES
Chi-fu Huang
Lode Li
ABSTRACT
We prove
unique
of a Nash equilibrium of a class of continuous time
the existence
stopping games.
Under
symmetric
a
strict
equilibrium
for
monotonicity
a
condition,
there
exists a
stopping
game.
Any Nash
symmetric
equilibrium of a stopping game is subgame perfect.
in the
analyses is
the General Theory of Process.
will provide conditions under which there is a
The
machinery employed
In the next version, we
unique Nash
equilibrium for
general stopping games and will give examples of our general theory.
LIBRARIES
APR 1 6 1937
M, IT.
|
Prilimary Draft and Inronipletr
Continuous Time Stopping Games
Chi-fn HtiaiiR and Lodo Li
Sloan School of
Management
MasFachusetts Institutes of T(ThnnloRy
July, 1986
Abstract
We prove
Under a
the existence of a Nash equililninm of a class of continuous time stopping games.
strict
monotonicity condition, there exists a unique symmetric
symmetric stopping game. Any Nash erjuilibrium of a stopping game
The machinery employed
version,
we
will
in the analyses
is
the (^rnrrnl Throvy of
provide conditions imder which there
is
a unique
is
eeiuilibrivim for a
suhgame
Prnre.s.se.s
.
perfect.
hi the next
Nash equilibrium
general stopping games and will give examples of our general theory.
for
Table of Contents
1
Iiitrodnrtion
1
2
The fonmilatinn
2
3 Existeiifp of reaction funrHons
when
4
Nash
5
Couchiding remarks
equilibria
2,"s
have a monotone strurtun^
3
7
10
1.
Introduction
This jiappr presents the theoretic findings on the rontinnnns time version of the general stopping
games. The game theoretic extension of the optimal stopping theory in the discrete time framework
was initiated by Dynkin
[1969] in analysis of a class of
two i)erson zero sum stopping games. The
sum stopping games and Mamer
modifirations and extensions include C'hapnt [1974] in zero
in
[198G|
two jierson nonzero sum stojjping monotone games.
The widespread use
management
investment
in
Examples include the entry and
science.
in research
lie
is
equilibria
where
each
is
a consequence of the fact that the industry
demand
the stopping
natural extensive form game.
in the
is
the multiplicity of
game. Fine and Li [1986] pointed out that the
is
nniltiplicity of
jump from
process can
where neither firm
lioth firms are viable as duopolists to a i>oint
is
game
tyjx^ of
However, one of the major weakness of the discrete time stoj)ping games
hi an oligopolistic exit
dynamic
a single dichotomf)us flecision at each time can be formulated
time cfjuilibria corresjiond to the subgame perfect etjuilibria
equilibria,
(See Fine ancl
any stochastic
In fact,
Fine and Li [1986] also show that for this
as a stopping time problem.
job search, optimal
ti-aiisitinn in influstries.
[1985], Reingainini [1982] etc)
which each palyer's strategy
found in economics, finance, and
exit decisions of firms,
and develojmient, and the technology
Mamer and McCardle
Li [1986],
game
games can
or potential use of stopping
a point
vialiN^ as a duopolists,
but
viable as a monopolist. This jiroblem might be avoided by assuming that the information
evolves continuously, which could be valid only in a continuous time framework. This
one of the
is
reasons that motivate this study.
The only work we
proposed
Our
l)y
Dynkin
sttuly
is
game
are aware of dealing with
optimal stopping problems
is
Chai)ut [1974].
of
two
He
theoretic extension of continuous time
analyzcxl a two jjerson zero
jiarts.
hi this jiart,
continuous time stop|>ing game assuming the
we
of the reaction correspondences of play<'rs
Nash
game
equilibriinn in
lattice theoretic fixed j)oint
games with a monotone
theorem.
We
also
further structures are imposed. Moreover, any
A
are oi)tional jirocesses.
Markov
under
setting)
is
is
martingale
adoj)ted to
fairly general conditions.
i)ayotf structures follows
show that there
Nash
an A^-person
start with a general setting of
payc:)ff processi^s
ai)proacli (in contrast to the excessive function ajiprfjacli in the
existence of a
stopj)ing
[1969].
composed
show the existence
sum
a uniqu(^
The
from Tarskis
symmetric equilibrium
(-(luilibrium of th(^ stojiping
game
is
a
if
subgame
perfect equilibrium as in the discrete time case.
This version
is
a prelimilary and incomplete draft.
the uniqueness of an cfjuilibrium
useful
examples
in
when information
is
The continuation
of this
reveaiecl continuously.
economics, finance, and management science.
We
work
anticijiates
will also
construct
Tho formulation
2.
J P)
Lit (W,
.
1)C
F =
of /, or a fi/fmfjnii.
firl.l
To roiitaius
1.
roiniilrtr:
2.
right -rout iininiia.
Wr
Wp
a roiiiplotr pmhnliility span-
also assuiiio Miat Jn
.^r •!"'
shall (Iciiotc V/g^j
Wr
i^
w E W
iiitrri)rrt oarli
F
finite state setting,
G
V<
^.
the
tliat satisHi-s
family of
iiiriTHsiiiK
suli-pifriiia-
tisiia/ ntuilitirtiis:
3R4
trivial in that
nhiinfit
si^iiia-firlfl
?iii;tll<''^t
roiitaiiiiu^: {7,:f
tli<'
true statr of tho world
tlir
W ami
by
<?:ciicrat((l
is
it
to he a coinplctr drsniiitioii of
models tho way information ahout
time
3?+},
with an
r-iiull sets;
all tlir
- n,>/
7,
^
{Ti-f
ofjuiiiix-fi
^
9?
l>y
f }
the r-imll sets.
all
^oc-
state of the world.
The
filtration
nvcalcfl ov(-r time. In a discrete
is
a filtration ran he tlionRlit of as an event tree.
Before we j)roreed. some definitions are in order.
A
X
i)rofess
is
a
mapping
A'
U X
:
the prndurt sigma-field generated
he ndaptrd.
to
Hira.siira/>/e
if
X(t)
if
V«
G
is
9?+, the
is
3? that
is
The optiouH} sigmH-Rrh].
7,
(g)
denoti-f]
nn-asnraMe with
and the Borel sisma-fieM of
mapping (w.x)
5(10,
>-»
vY(a;..«)
where
<)),
by 0.
is
V^
J,
G
9?^.
A
5R+.
U x
of
as usual 5([n,f])
and Williams
A
is
U X
J®
is
is
S(3?+),
X
i>roress
process
into 3?
[(). f]
the sigma-field on
resi)ert to
is
said
pro^re.s.sive/y
measurable with
the Borel sigma-field of
5R+ generated by adai>ted
A
process
ojitional
if
measurable with respect to 0. Naturally, any adapted jirocess with right-continuo<is paths
is
processes having right-continuous
it
T
l>y
->
measurable with respect to
respect to the product sigma-field
((),<].
3?^.
optional.
Let
It
is
^+
al.so
known
i)atlis;
('Innig
cf.
that any optifmal process
denote the extended
jiositive real line.
is
ada])ted;
A Markov
cf.
tiiiir
[1983).
is
Chung and Williams
T
is
a fiuiction from
[1083].
U
into 3?+
8nch that
{w G
A Markov
if
time
T
is
all
T
events
KG
<t}e7,
T{uj)
if
P{T <
sij^niia-field
7t
X
be an
ojitional jirocess.
follows from Dellaclierie
random
ea.sy to
I.
A
st(.].])iii'';
tiui<-
T
is
bounded
i.
7oc ""fl* *hfi\
variable.
This
is
V/.
g3?+.
Putting
A'(+oo)
it
00} =
collection of <vents jninr N. T. consists of
^li"
Af]{T<t}e7,
Let
yt G3i+.
r{T < K) ^
3?+ P'lrh that
be a Markov time. The
A ^
:
said to be a st'tjijiinK timr
there exists a constant
Let
12
=
limsup,X(0.
and Meyer [1978. Theorem IV
90] that A'(
+ Oo)
is
an .-xtended real-valued
the convention we will use for any ojitional jirocess to aiijiear.
check that the process {X{l):
/
G ^+
} is
ojitiouMl
I)
tli.Mi f'.jlows
from
sinnl.ir
It
is
also
arguments
in
Thenrnn
and Moyor
IV. G4 of Dellarhrrie
[1978] that X{uj.T{uj))
is
an cxtcnrlrd roal-vahicd random
variable measurable with respect to Tj.
An
optional process
X satisfies
condition
A~
if
^[siip^"(()] < oo,
where E\] denotes the expectation with
resjiect to P.
<
E{sx\pX'^{t)]
Coming back
indexed by
i
=
we consider an
to economics,
1,2,
.
.
.
,
N. The payoff
of this
z,{uJ,t:T_,(w));t
where
runs through
r_,-
that player
i
(A^
all
receives in state
employed by players other than
—
We
i.
is
^4"*"
his strategy
if
OO.
game.
Players are
described by a family of adapted processes
r=
9?+.
Markov
l)-tuple of
w when
satisfies conflition
A^ i)layer dynnitiir stopping
game
e
It
1,2
A^.
times. Interpret 2,(w./;r_i) to be the payoff
is
T,
and
T,(u;)
=
t.
if
r_, are the strategies
assume that payoff processes are optional
jirocesses satisfying
A~
condition
(liven the strategy of his opponents T_,. the objective of jihyer
»'
is
Markov time
to find a
T,-
that solves the following program:
snp E{z,{T,:T^i)].
T,eT
where
A
T
is
the collection of
Nnfili eq\ii]ihniim of
all
Markov
A Nash
if
given a
times.
the stojiping
such that Ti solves (2.1) for alU'
=
r,
game
3.
set
an
S, putting A;
=
Aj for
all
Markov times
r,-
Markov times
A^-tujile of
?'
(T,;
=
N)
1,2
N.
2
{T,-
>
E{z,{T^;T_i)\rs]
on the
is
equilibrium of the stoj)ping ga.me {T,;?'
Markov time
(2.1)
>
such that
=
1,2, ... .A^}
is
said to
Ix'
suJigamr perfect
S}. we have
E[z,{t,;T.,]\Ts]
r,-
> S
on the
set
"...
A,, for
all i
=
1,2
A^.
Existence of reaction functions
In this section
we
shall
show that given the strategy
of his oj)j)onents, a jiiayer's best response
always exists.
Some
A
definitions are first given.
sujiormnrtingHlr
X
is
an optional process such that
for
all
bounded
stojijung times
T
and
S with T < S both
E{X-{S)] < oo
(3.1)
aud
E\XIS)\Tt\
wlirrr
fire satisfirrl,
if Iifitli
X
—X
and
Remark
X~{t)
=
3.1. Traflitionally. a ^rwrnlizri]
jiatlis is
Mcrtcns [19G9. TlK^orrni
and
riRJit
A
limits
left
and arr
snprrmartinfjalo
£^(A'(.'>)) exists
X
n])i>ci-
Lemma
tiinrs.
a iiuirtingah
an
a'la])tr(l jirnross.
not nrrcs-
of the ])atlis of a snpcrniaitinf^alr
all
si'mi-rontinuons fiom the
if
is
havo
lifrlit.
for every pair of stopi)in<.': times
.sii/)e;-
T < S
the expertation
following terhniral
Lrt {X{t)}
< X{T)
rrgnlnr
Markov times T < S
ev<-ry j)air of
foi'
if
a...
the
and
lemmas
}>r a
<
.sii/)er
a...
A'iT-)
are keys to the
main
(3.3)
results in this sertion.
irgulnr sujicniiHitingnlr nniJ
T nnd S
Irt
}<r
two Mnikov
Thru
< X{T)
E\X{S)\Tt]
ProKf
is
and
£^[-Y(.9)] exists
3.1.
prorrss
sii])iMiiiartin^alc in tlic traditional sonsc
K''ii<'rali7.('(l
^iA'(.S')|77-l
The
s\iiniiii:titiii'^:ilf is
sliown that almost
vrgnliiv
is
Any
2)
(-T
1) lias
supermartingale {X(/)}
expertation
A
nf X{t].
a sn]icrniartinKalc rirfincfl al)ov('
^[A'(.S-)| Jrl
A
i);irt
aro siiprrinartiiipalrs.
rontinnons
riglit
(3.2)
......
u\nx\-' X(t).()]. the iirRiitivc
sarily njititmal. satisfying (3 1) anii
with
< XIT)
Let
r(A') >
A = {S > T]
and A'
C A
Sni)pose
tin-
assertion
is
srt
{S
not trm-
> T)
Then
there exists A'
Markov times
-
T{u})
= +00
S'(i^)
on
-;
,"
if
w e
th.' set
A'.
A'.
otherwise:
=
S(uj)
—
-foo otherwise.
if
w e
A'.
Throroni IV. 53 of D<dlaelierie and Meyer [1978] ensures that T' ami S'
rf)nstrurtion, almost surely, S'
is
w-ith
as follows:
T'{w)
Since ,Y
^ Tt
snrh that
E\X{S)\7t] > X{T)
Define two
on the
'i..«.
>
T',
S <
.*?',
and
a sujx-r rej^ular supermartinRale,
^|A-(.S-')|/rl
T <
T'
.
Ihnce
we must have
< X(T')
4
......
/j-'
C
ar<'
7$'.
Markov
times.
By
Ndw
taking conditional expectations
vvitli
rcsjioct to
\a-E\X{S')\7t]
E Jt and
Since A'
<
E\X[S)\7t\
\a'X[T')
< X(T)
on
n.s.
on Ixith
nniltijilyinp; l)y 1^-
Ia'E[X{T')\7t]
^ \a'X[S) and
since \a<X.{S')
Jj
=
sirlrs
Rives
a..<,.
l,vA^(T),
we have
A\
tlie set
a clear contradiction.
I
Lemma
Lef
3.2.
From
Proof.
ho a
.sti/)eniia;f;;ipa/e .satisfying^ roiuJifinn
we know Hni/^-^
the hyjjothesis,
where A(oo)
is
A^
condition
X
the hniit;
l>e
A'(<) exists
and
Thm X
.
G
9?+}
is
T <
let
.'»
he two Markov times. For any constant
A^
the process
a supermartingale. Using the arguments in the iirevif)us i);).ragraph,
conditions A'^ ans
satsifies
Also, since
.
E\X'(S)\Tt]<X'(T)
lemma
c.
minimum
Then
Fatou's
If
oo,
X
also satisfies
of
Appendix
Remark 5a
A*"
A'*"
<
Z?[A''~(oo)]
4].
I],
a snpermartinp^ale.
For the general case,
is
su[>cr regular.
is
and
finite,
is
Mertens [19C9, Theorem
cf.
the assertion follows from Dellacherie and Mt^yer [1982.
,
since {X{f),t
understood to
A~
we jmt
is
X'^
X /\c.
=
a concave function,
we know
a.s.
implies that
^flim
Lr— oo
X''(S)\Tt]
<
J
liiii
r— oo
X'iT)
«....
Hence
E\X(S)\Tt]<X(T)
That
A
is,
is
a.s.
super regular.
I
The optional process
>
Y(ujj)
W{uj,t) for
all
Y
is
said to
lie
above the
(w,t) outside a subset of
ojitional ])i(icess
n X
W. denoted by Y >
di^ whose i)rojection on
U
is
of
W
,
if
P-measure
7,ero.
A
regular supermartingale
process
W
The
if
Y >
W
and
if
Y
is
A>F
the niinimnm vrgiihv su])rnn:utiitgnlr
for
following jnoposition follows directly from Mertens [19G9,
5
(MBS) above an
A
>
IV.
Theorem
7]
and
any other regular su])ermartingale
optional
Lemma
3.2.
rrnpositioii 3.1. Civr,, T^,.
y,{t:T^,)
rit^lil'nitliniKiiis.
j.s
tli'ir rxi.^ts
rif^ltt
MPS VAT,)
a
inntiiniKH^
=
y,{t:Y_,)
liiii
lyim: Hl«'vr s,(T_,).
If z,(l:T^,)
Fnitliriinitrc. }',(7'_,) is sii/ir; I't^iilnr
;s
ami
liIllsuI),_,.^(n
ttnd
= sup
r_,)
}',(();
£;[r,(r,;r_,i
T,eT
Pri'nf.
Till-
Meyer
assertion follows from tlm fart that z,(r_,) satisfies rfjiidition
first
(19G9, Tlionrrm
7).
The
of Ajipenriix
I
/I
and Mcrtrns
of Dellarlierie
and
(1982],
Y(oo)
i."*
finite a..t. sinre
Y
Lemma
from
Tlie rest of the assertions follows
that
Remark 23
soroiul assertion follows from
A~
satisfies
3.2,
Mertens (19G9, Theorem
Theorem
(see again Merten.s [19G9.
7),
and the
fart
4]).
I
We
shall heneefoith nse the notation
The
following result
Proposition
3.2.
F<>r
is
introduced
Proposition 3
in
1
also useful:
rvriy
M^rknv
=
y,(.9;r_,)
Prnnf See Theorem 22 of Ai)i)endix
tiinr
I
S wr
l]^vr
;)/i(josf sr;rr/y
e...ssup,ex,r>5^[-.(^;7'_,)|75l.
of Dellarherie anrl
Meyer
(1982).
I
Now
define a
Markov time
T,{T-i):
r,(7'_,)
Here
is
inf{<
G
3?+
our main theorem of this sertion
?
TheorPtTi 3.1.
.S'u/i/iMse
mnliniinii^ funu
It
¥(()
:
=-
?(^;T_,)},
shows that under some conditions
ther(>
always
a best resjionse to other players" strategies
exists for ])layer
Pronf. Using
=
tlir Irft
Lemma
,
thnt z^T^,)
nml
.saf/.s/irs
is
u]>jiri-srini-r<<iitiniii>iis fioin
ffinihtinn
A^
.
Th'ii T,{T
3.2, the assertion follows fr<uii similar
,)
is
arguments
thr
iif:lit.
(/u;iM-u/)pe/-.sr;jii-
thr }>rst ;rs/)onsr to T^,.
of
Theorem
7.3 of Tliomi)son
[1071]
I
From now on we
sliall
assmne
tjial
riMiditions of
Theorem
3
1
on the reward jiroresses are
satisfied.
The
following |)roj)ositions give
some
jjroperties of an optimal leaction
Proposition 3.3. Fix
innjoiiiig z,(T_,).
r_,-.
Let
n host rcfijinnsr fny jtlnyrr
}>r
r,
nnd
i
lot
Yj{T.i)
l)c
the
MBS
Then
=
Y,{Ti-T^i)
r,(r_,)
Zi{T,:T_i)
<
(3.4)
n..«..
a.s.,
r,-
(3.5)
and
E[Y,{T,-T.,)] -y,(n;r_,).
Proof.
Dy
the fart that y,(T_,)
that y,(r_,) hes ahove z,(r_,),
pnjxu' regular (cf.
is
Lniiiiia 3.2), that
r,
is
a heat icspoiise,
and
wo have
e[y,(t,: r_,)i
<
y,(n; r_,)
whrio the equality follows from Projiosition
=
3.2.
E[Y.(r,:T_,)\
It
(3.G)
>
then follows from the fart that Y,{T^,)
i?[2,(r,: r_,))
<
e[yAt,- r_,)).
(3.7)
Hrnrc
=
E[z,iT,:T.,)].
2,(r_,)
aii<l th(-
r/,..«.
fact that 2,(r_,) satisfies A'^
and
Ay,(r,;r_,)
This
is
By
=
2,(r,;T_,)
„....
(3.4)
the definition of r,(r_,).
Argnments used
we then have r,(r_,)
<
to prove (3.7) proves (3.G), sinee T,
r,
is
a..<i.,
an
whirh
is
(3.5).
response.
oi)tiiiial
I
Note that
(3.5) implies that r,(r_,)
that his opponents play
the nni<ine
is
minimum
best response for player
t,
given
T_,-.
Proposition 3.4. Lot
l>c
r,
an optimal
E\z,(r,-T_i)\Ts]
j-e.spon.se tn
>
z,(-'^-T-i)
Proof. Proposition 3.2 anci arguments similar to
E\z,ir,:T_i)\rs]
=
T^j
a.n<l Irt
S
1)r
on the set {S
Lemma
Y,{S-T^,)
>
<
n.
Marknv
Then
r,}.
3.1 imjily that,
Zi{S-T.i)
tiiiic
on the
set
{r,
>
S},
«....,
whirh was to he proved.
I
4.
Nash
equilibria
hi this section
jilayers satisfy a
when
we
will
monotone
2,'s
have a monotone structure
show that there
exists a
structure.
7
Nash
('(piililnium wIkmi the
reward processes of
Tlir
he made
a.s^uinptidii will
fi)lli>wiiip;
Assumption
For
4.1.
€
«'
t.
an./
5R,
t'
scrtinn.
tliis
tliri'iifilKiiit
>
I
a/jimsf siiiWy. c,(w.
.
/:
-
r_,(w))
z,(w,
«';
r_,(w))
is
noiiiitrrrnsiiig in r_,(u')
DrnotiiiK by
tlw
T'*^'
r.ill(-rtif)ii
i'(r,
By
Tli<^or<Mii
3
1.
^
i?
Markov
of yV-tui)li- of
=
rA.)
a rrartiou fuiichon for
we
tiiiics.
di-finc 4>
:
T''^'
-»
T''^'
.-v;
(r,(T„,))f:,.
playis
By
Tin'ornii 3.1,
tlic
A'
for
any two Maikov times r
tin-
iiia])])iiif^
is
wril-drfiiird.
^
is
said to hr tu'ninUiur
*(.<?) ii„],lirs r'
<
.<?'.
Tlir iiiai>piiiK
.«?'
=
The
following j)ioj)opitioii siiows that
Proposition 4.1.
Prnfif.
is
('honso two
is
<I'
S. t
=
and
4>(r)
luonotonc.
is
<I>
>
/iioijofone.
Maikov
of strirtly i)ositive
if
tiiiirs r
measure
>
S. Let
for sonie
t.
=
r'
^(r) and
By Assnm])tion
.S''
=
Snpjjose
4>(.9).
we know almost
4.1
tiie set
surely.
\z,{t':t_,]-z,(S]:t_,)\\,^
<\z,(T',:S_i)-Zi(S'.:S_i)\i^.
Taking conditional expectations with respect to 7s' on hotli sides of the above relation gives
where we have used the
where
-
<E\z,(T,:S^i)
-
tiiat
inecjuality follows
tlie first
Proposition
fart
<E\zi{T\:T_,)
3.2.
A G
Ts'
z,(S',:t_,)\7s']\a
z,(S',:S_,)\Js'\\a
Delhu
(<f
from Proposition 3
define
rr,
=
S^
A
a.K.
T-
on a
set of stiictly jiositive
note
tliat
It
is
measure.
ElzAry^.T-i)]
=
W<' claim that n,
El.-,(.S;;r„,)l,4
=
t'-
=
we have
<[>(r_,) is
tised (4.1)
the unique
and where
minimum
on the
.i..«
easily elierked that
= E
wliere
tlic tliird ini(|u:ility
follows from
Thus
E\z,{r',:r.,)\Tsi]^ zAS',:t_,)
Now
0,
and Meyer [1978. Theorem IV. 5C]).
h(-ri<'
and whric
4.
<
+
is
r,(
[e\z,(S',:t.,]1.^
rr,
+
:\
r,';
is
a
Ix-st
set
A
Markov
(4.1)
tim<'
and
resixinse to r_,
rr,
To
<
r,
.
rr,
see this,
following
is
the
first
r,-
we
r.,)rj
c,(r;;r_,)l..i, |J5;l]
E\z,(T::r.,)].
A"^
denotes
U
\
>1.
best resjxmse to r„,
This
is
Thus A
a contradiction to the fact that
nuist
be of measure
z.ero
I
The
^
main theorem
of this section:
8
Theorem
rrnof.
From rroposition
Nrveu
[1975] implies that
(cf.
Nash
Tlirrc exists a
4.1.
$
4.1
T^
rquijiliriiun
<<f
hmn
a monotoiir maiiiiiiiK
is
a complete lattice.
is
the sfn/i/tnip pajjir.
It
Tarski [1955]) that there exists a fixpoint for ^.
T'
in itself.
Proixisition VI. 1.1 of
then follows from Tarski's fixpoint theorem
It is
easily verified that the fixpoint
Nash
a
is
eqnilibrinm.
I
The
following theorem shows that
of
Markov times T we have
n
is
of
measure
asRiiniption
is
Assumption
z,(w,
—
T)
t;
we have
if
zj{uj,
a
symmetric game, that
T) except on a subset
t;
of
is,
UX
for
any (A^
—
l)-tuple
whose projection
3?_|.
to
symmetric etinilihrinm. provided that the following
zero, then there exists a imi<ine
satisfied.
For
4.2.
f.
G
t'
3?.^
mid
t
>
t'
nlnmst suirly. 2,(w.
.
<;
r_,(w))
-
2,(w,
/.';
r_,(w))
is
strirfly (irrrrnsing in T_,(w).
A
the
definition
l)ost
>
.9
needed. Let
a.s..
The
and
a mapi)ing fmni T'
'P l)r
responses to an element of
raiTcsjinndrurc.
7-
is
T
By Theorem
.
The reaction corresi)ondence
r'
G
'l'(r),
S'
G
is
we have
*(.9),
following jiroposition shows that
^
to
3.1, '^
all
the subsets of T'
well-defined.
is
said to he monofojie
r'
<
S'
if
call
'I'
all
the rcartion
any two Markov times
,i..9.
.
monotone, whose proof
is
for
We
that gives
is
similar to that of Propo-
sition 4.1.
Proposition 4.2. ^
rrnnf.
is
monotono
Choose two Markov times
if
r
Assuinptimi 4.2
>
S. Let
r'
=
is sntisfirt]
^(r) and
.*>'
=
^>(.'>).
Suppose the
set
A={r',>S:}
is
of strictly i)ositive
measure
for
some
i.
By
we know almost
Assuinjition 4.1
surely,
[^,(r;;r_,)-^,(5;:^_,)]l.,
<[z,(r:-S_,)-z,{S',:S_,)]l,,.
Taking conrlitional exi)ectations with
resjjcct to Ts' f'H
both
sid(~s
of the above relation gives
E[z,(r',:T_,)-z,{S',:r_,)\rs;]lA
<E{z,(r::S_,)-z,(S::S.,)\rs;]lA.
where we have used the
fact that
left-liand-sirle of the relation
hand-side
is
strictly jiositive
Here
is
theorem:
is
and
AG
Ts'- ^^
Dellach(-ri(> aufl
Meyer
[1978,
nonnegative almost surely by Proposition
is
a contradiction to Proposition 3.2.
Theorem
3.4.
Thus A
IV. 5G].
Hence the
nnist
The
right-
be of measure
Thporem
Siippn^^r
4.2.
f/j,if
thr f^nuir
is syiiintrtiji-
tlt'Tf rxisfs a iiniiiuf f^yiiunrtrii' N:ish 'ipuljlirinin
thnt A^^iiinpti'ni 4 2
nii'l
{"V
tli>- sliij)])iut^ t^mii'-.
f'"
''"
if:
sntiafjcd
Tlirn
Proof. Lft
F(r) -
aiul lot
Note
f}l^t
Dfl'I'lT')
for o^rli
roiiipoiiriit of 4>(T).
fixrd
jxiiiit
aiifl
Lot T'
1'c
Sinro r*
r^
F'(r)
=ii>fl
T e D.
So FfT)
is
= Dn*(7')
iionciiipty
thrrr exists a .syminrtiir
a fixrd point of
=
4>,(r_,)
F
^j{T_j)
and
T'
D
F
TG
T'^'.
for all i.j. wlicic 1>,(r) <l<liotcp thr j-tll
>—*
D
iiionntoiic sinrc
i^
is.
<I>
F
Tims
has a
fjuilihriuin
(
and tlirrofmc
e *(r*). by mouotonirity
a.f
<
T'
oni- for F'
is
a.fl.
for all T'
("lioosc
.
T ^ D
with
G F'(r) C *(r) So
T'
T >
< T
T'
n.s.
ti.-t.
and
*(r).
Similarly.
T <
T' «
.'
implies
T ^
Tlicrrfoic,
'I'fT')
T* must
tlie
])r
nniqni- symnictrir
Ofpiililiritim.
I
Fnially,
Theorprn
we have
Any
4.3.
iVa.s/i
r(jiii]i}<iiuiii
nf
thr- .sfo/i/u'/ij^ i^ainr /s su/'i^a/de jirrff-t.
Proof. Tliis follow? from ar^imrnts similar to
Lonima
3.1
and Proposition
3.2.
Concluding remarks
5.
As wo
iiavo
mentioned
in the introfhirtion. this version
demonstrated the existence of a Nasli
We
are
still
ecjiiilibriiuii
in
working on showing the unifjueness of an
is
])reliminary and inromjilete.
a rlass of ((iiitinuous
i'(jnilil)iinni
time
stojjjjinp:
We
Rame.s.
and on constrnetinR example.s of
general interest.
ReferpncGs
1
H
(Miai.ut,
Maikov games.
Tr(
hnir,-il
Report
No
H.^,
r~).i);ut iiirni
.
>f (
^ix'rat jcus
Research. Stan-
fonl University. 1974.
2
K. (^Inuip; and
I^
Williams. 1083.
An
Iiiinuhirtioii to Stnrhnslir Intrt^i-ttion.
Birkhanser Doston
Inr.
3.
C. Dellarherie and P. Meyer. 1978.
North-Holland
Pii])lishin(T
Pin},:i}>ilitir^
CVimi)any.
New
York.
10
and
r<>t<'nti:tl
A
(.Vnera/ Tijenry of Prorcss,
4.
C. Dcllachrrie and P. Mryrr,
1982,
North- Holland PublishinR Company,
5.
E.
Dynkin, Cajur variant of a
PinhHfnlitirf;
Now
proliloiii
nwl PdfrutinI D: Thmvy of Aiarf iupa/rs
York.
on optimal ptojipinp, Soviet Mntli.
Dnki 10
(19G9),
270 274.
G.
C. Fine and L. Li,
A
storhastir theory of exit and stojijiin^ time cquiHliria,
Working Pajjor
^1755-86, Sloan School of Managomont, MIT, 198G.
7. J.
Mamcr, Monotone stoppin ^ames, working
School of Management, IK^LA,
jiaper, (Jradnate
198G.
8.
.7.
Mamer and
K. McCardl(\ Uncertainty, competition, and the adoption of
working paper, Graduate School of Management,
des martingales, Cr.
UCLA,
Arnd
new technology,
1985.
268
9.
.1.
Mortens, Snr
ID.
.1.
Novell, Discrete Parainefrr Mniiiii^nlrs, North-Holland Piihlishing f'omiiany, Am.'^terdam,
la thoorio
Sr.
Pari.s
(19G9), 552 554.
1975.
11. J.
Reingannm, Strategic search theory, Intrrnationnl Eronmnir Review 23 (1982), 1-17.
12. A. Shiryayov, Optima.}
13.
A. Tarski,
A
Stopping Rules, Springer Vorlag,
lattice theoretic fixed jioint
New
York, 1978.
theorem, Rnrifir .Journal of Mathematics 5 (1955),
285 309.
14.
M. Thomi)son, Contimions jiarameter
oi)timaI stopjiing prohiem, Z. Wahisrhrinliclikeitstlieorie
vcrw. Geh. 19 (1971), 302 318.
11
99
7\V
3
TOAD 004 Obi 153