LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
EQUILIBRIUM EXISTENCE RESULTS FOR
SIMPLE ECONOMIES AND DYNAMIC ^AMES
By
3ul R.
^o^
WtinJc
tAurcJr Rao(
Sevk\
^leindorfer
Murat^
Sertel
?a,o\
December 1972
MASSACHUSETTS
TECHNOLOGY
50 MEMORIAL DRIVE
IBRIDGE. MASSACHUSFTTR ni
INSTITUTE OF
MASS. iH3T. TECH.
JArn4l973
DEV/tY LIBRARY
EQUILIBRIUM EXISTENCE RESULTS FOR
SIMPLE ECnNOMIES AND DYNAMIC OAMES
By
Paul R. J^leindorfer^«^-\
December 1972
^^^
WonUe..
631-72
This paper was prepared upon the kind invitation of the IEEE Conference
on Decision and Control in New Orleans, 1972, where it was delivered, December
13.
It carries Preprint 1/72-22 of the International Institute of Management,
D-1000 Berlin 33, Griegstrasse 5.
00. ^5/- 73
RECEIVED
JAN
M.
I.
T.
18
1973
LIBRARIES
INTRODUCTION AND SUMMARY
O.
The purpose of this paper is to bring
some siirplicity
and crenerality to the investigation of
equilibriuir existence
in certain "simple" dynamic games
""
The way we deal with
these objects is, essentially, by reducing
them to what we
term "simple economies", these latter being,
from the view.
point of equilibrium existence results, at
once more general
and less cumbersome. In §1, we exhibit
some topological facts
about a certain rather general class of
simple economies, includinc, the fact
(1.7, Main Theorem) that they each possess
non-empty and compact set of eauilibria.
Then, in §2, we
define the simple dynamic games of interest,
immediately reducing them (2.5, Reduction Lemma) to
the simple economies
studied. In §3, we show (3.0) that the
simple dynamic games
in question have non-empty and compact
sets of equilibria,
using the fact (1.7) that the simple
economies to which they
reduce have such sets of equilibria. We then
illustrate by
example (3.3) that a general class of
discrete-time, deterministic games with convex performance criteria
is covered
a
by the equilibrium existence results
just described. This
class includes dynamic games for which
certain non-linearities
in the next-state map are allowed,
and for which controls
are
restricted to compact reaions, these regions
themselves
varying as a function of state. Of course,
we do not intend
1
)
An excellent bibliography of previous
work on dynamic games
can be compiled throucrh th- references
cited within
[Kuhn and Szego,
1971^
and from^the survey paper,
[Ho,
197oJ.
that results particularized to this quite
special example
be understood as the main thrust of this paper.
Historically, the study of the existence of
(competitive)
general equilibrium in economies exhibits quite
a long-standing
and extensive literature. A crucial turning
point in that
literature is afforded by the Arrow & Debreu
fl954j study,
benefiting from Debreu' s [1952] earlier investigation
and
using a FPT (fixed point theorem) of Eilenberg
& Montgomery
[1946[. This work is generalized in [Sertel,
of the more powerful FPT
'
s
of [Prakash
&
by use
1971]
Sertel, 1971].
Essentially, our present results can easily be
demonstrated
as corollaries to this last mentioned,
but owing to the relative inaccessibility of both this work and
the fixed point
theory it employs, we restrict ourselves here
to what can be
done by using the relatively well-known FPT
of Fan [1952]
which [Prakash
&
Sertel,
1970]
generalizes. Even with these
handicaps, our main theorem generalizes the Arrow
[1954
1
&
Debreu
equilibrium existence result.
We work in locally convex spaces. Such
spaces include
normed spaces and the conjugate space of the
Banach space
of all real-valued continuous functions on
a given compact
space. This conjugate space, in turn, is
the natural habitat
of probability measures under the weak
(or w*) topology
(see Parthasarathy
[1967|). Our working in locally convex
spaces is motivated by the hope and conjecture
that equilibrium existence results for stochastic
dynamic games may also
be obtained by reducing them to the
simple economies intro-
duced here.
0.0.
Notation and Conventions
we denote
^
N = {1,2,
m
We fix
N^ = N
,n},
\J
denote the Cartesian product
n
and denote
W
e
Given any set
{O}.
we consider the family
W,
e
For the set of natural numbers
:
= {1,2,...}.
=
{"'"Z
^Z =
TI
Z
-
i
|
"""Z
x...x
and any
Z
1,... ,m}
"^z
,
i=l
j^Z
„z
generic elements being in lower case;
,
When a given
Z.
e
=
z
(
z
,
.
.
.
,^z)
"^z
e
and
bv
^Z
is understood,
we denote projections and components according to
V
,
V^
= k^
^""^
N'^Z)
= ^z, k =
U^)
^
m.
Every Cartesian product of topological spaces is
understood to carry the product topology.
Let
be a set. Then
Z
non-empty subsets of
topology,
sets of
When
denotes the set of
Z
is equipped with a
denotes the set of non-empty closed sub-
C(Z)
Z.
Z.
[z]
When
Z
lies in a real vector space,
denotes the set of non-empty convex subsets of
abbreviate
C{Z)r\
Finally,
P
0,(2)
to
C{^(Z)
We
.
denotes the set of reals; whenever
considered as a topological space,
topology.
Z.
Q.(Z)
R
carries the usual
SIMPLE ECONOMIES AND THETR EQUILIBRIA
The main task of this section is to introduce the
notion of a simple economy, define equilibria of such,
and then establish some fundamental results
includ-
ing our main theorem giving sufficient conditions under
which simple economies possess non-empty and compact sets
of equilibria. Apart from the interest of these notions
and results in their own right, altogether they provide
the main framework and the tools for our analysis of the
equilibrium existence question for the dynamic games formulated in the next section.
1.0.
1
.0.0.
Definition
S
A simple economy is an ordered quadruplet
:
< W, U, T, A >
=
,
where
1.0.1.
W
=
{X^
a
?^
I
e
A}
i^
is a non-empty family of non-empty sets
generic elements
X = nx
A "
X° =
,
x
X„,
A\{a}
^
for generic elements
1.0.2.
U
=
{u
is an
u
on
:
X
-t-
X
g
n
Rjcx
(a
E
,
from which we define
denoting
e
with
X
x
e
X
and
x'
A);
A}
associated family of real-valued functions
X
T
1.0.3.
(t^
=
X
:
-*
r^^J'"^
^^
^
is an associated family of maps
non-empty subset
1.0.4.
{a
=
7^
X°
:
-^
t
&a-
'
°
assignino a
t^
to each
X
(x) cT
x
e
X;
^^
^
is a self-indexed family of maps defined, at each
x"
by
X",
e
oi(x")
=
where
{x
t
x"
:
Pemark
the maps
x°
e
e
X
,
x°)
> ^Sup u
(.,
x")
}
o
iz
e
a
t
a
(z
a
,
x")}.
It is evident that in a simple economy wherein
;
cisely
a
(X
is defined by
Px 1
->
= {z
t(x°')
1.1.
(x°)|u
t
e
are each independent of
t
t
x",
(x
,
= t
x")
i.e.,
t
treated by Arrow
&
a
for all
(x")
= t OTT^a
X
a
(a
e
^).
x
x
we have pre-
,
e
X
and
The "economies"
Debreu Il954| are all of this sort.
For a discussion of some major deficiencies of such a
notion of an economy, most of which are present also in
that of a "simple economy"
•simple'!
— we
—
hence, the oualifier
refer the reader to [Sertel, 1971; pp.
18-22]
Remark
1.2.
To check that
:
1
self-contradiction,
.O is not a
i.e., that simple economies are well-defined, we need
only indicate that an object
satisfyina 1.0.0.
S
will permit 1.0.4. so long as, for each index
the maps
t(.,x):X
a
e
-
1.0.3,
A,
have fixed points
-*-[x|
ich
u
a
x")
(.,
attains a supremum on each such set of fixed
points. All this will obtain, e.a., when
compact spaces, each
= t
t
(x"
and
oifyCt,
X", a
e
E
u
t
(.,
x
consist of
W
is upper semi-continuous,
)
consists of compact spaces
(X)
A)
To aid in verbalizing thoughts, we borrow the follow-
ing nomenclature of [Sertel, 197f].
1.3.
1.3.1.
Nomenclature:
X
Let
be as in 1.0.
S
of
is called the behavior space
*
a
called a behavior of
1.3.2.
x"
is called the
x'^
being called an
x"
e
X";
a
iff
x
a
e
«,
X
x
a
being
;
ot
g-exclusive behavior space
a-exclusive behavior
(of S),
(of S)
iff
1.3.3.
is called the collactive behavior space
X
being called a collective behavior (of
X
X
u
is called the utility function of
1.3.5.
t
(resp., t
mation (resp.
of
a,
of
effective feasibility trans forr^ation
,
t^(X) = {t^(x)|x
iff
ct
d
t
e
a
and
a,
a
X}
e
d
iff
S)
)
being called the feasi -
being called
feasibility
a
(X);
is called the personnel
behavor (of
a;
is called the feasibility transfor -
)
bility space of
A
iff
S)
X;
e
1.3.4.
1.3.6.
(of S)
a
(of S)
beina called a
a
,
A.
e
Regarding the following useful tool dealing with continuity properties of set-valued mappings, the definitions
and facts in the appendix may be consulted.
1.4.
Theorem
a
s:
:
Let
u
:
P
x
Q
->
p
be a continuous function on
non-empty compact Hausdorff space
O
-»•
C(P)
P
x
fi,
be continuous, and define a map
let
a
on
by
= {p
a(q)
s(q)|u(p,
e
Sup u(., q)
>
q)
(q
}
0).
e
s(q)
Then
maps
a
Proof
upper semi-continuously into
Q
Given any
;
q
-
closed in the compact
s(q)
P
hence,
pact, so that the continuous
it. This shows that
is compact, being
0,
;
x
s{q)
for each
q
establish the upper semi-continuity of
peatedly use the closed graph theorem
the graph
>*
=
r
{
(q
,
p)|q
e
for each
q
e
a(q)
Q,
hence, closed in
p
Q,
it will follow that
P,
is com-
{q}
attains a supremum on
u
a{q) ^
C(P).
we will re-
a,
and show that
A. 2
a(q)}
e
of
is compact,
r^
= Trp(({q}
^({q}
x
X
C
P
P)
To
Q.
e
is
a
so that,
r^|
H ^^
n
is compact,
P.
We now show that
r
^^
><
is closed. As
s
is continuous, it is upper semi-continuous, so that,
by A.
its graph
2,
is closed. As
u(P
X
Q)
^R
r^ =
e
by
0,
graph
r^ =
C
u(P
X
{
(q
,
Q)
By continuity of
(p,
q)
e
P
X
O, p
s (q)
e
}
C.
P
P
>*
x
0,
is compact and, of course, Hausdorff.
v(q)
X
e
is continuous on the compact
u
By A. 4, the function
q
{(q, p)|q
Q,
=
v:
-^
Sup u
s(q)x{q}
r) |q
e
0,
is closed.
u,
P
defined, at each
is continuous,
r = v(q)
}
Hence,
r^
CZ Q
so is the graph
r = u(p,
q)}ClP>'
x
p
=
r
x
^
u
(P
so that its
v(0)
ig closed.
{(p, q,
x
Q)
r)
of
u
closed. Thus,
(r^
>«
Therefore, its projection
hence closed. Now
r
is closed, hence compact
r^
P)
into
G
P
is compact,
Q
x
is nothing but
G
which,
O^g'
clearly, is closed. This completes the proof.
*
1.5.
Corollary
In the last theorem, assume also that
:
convex in a real vector space, that
on
X
P
Then
Proof
for each
{q}
maps
a
Let
:
a (q)
with
s(q)
{p
C(2(P).
C(2.(P)
All we need to show is that
Q.
e
is convex. Now
of
s(0)C
and that
Q,
e
e
is
P
is quasi-concave
upper semi- con tinuously int^
O
q
q
u
.
a (q)
is nothing but the intersection
the former of
P]u(p, q) > v(q)},
which is convex by assumption, and the latter of which
is convex by quasi-concavity of
shows that
P
x
{a}.
This
completing the proof.
is convex,
a(q)
on
u
#
1.6.
Definition
S
= <W,
each
=
E (x)
n
A
A
of
point
S
X
iff
E
:
X
[xj
-*
defined, at
by
X,
E
is a map
T, A>
U,
x
The evolution of a simple economy
:
X
(x)
(tt
X*^
)
=
n
a (x")
.
A
is called an equilibrium or contract
X
e
a
e
E(x).
The set
is called the contractual set of
C
of contracts of
S.
This section culminates with the following:
S
10
1.7.
Main Theorem
Let
:
= <W,
S
be an ordered
U, T, A>
quadruplet such that, using the notation of
1.7.1.
W
1.0.1. with each
is as in
X
1.0.,
compact and con-
a
vex in a locally convex Hausdorff topological vector
space;
1.7.2.
is as in
r
1.0.2.
continuous on
1.7.3.
T
is as in
T
=
{
(x
d
T
a
t
1.7.4.
for each
a
1.0.3.
and,
a
'
,
a
for each
x")
e
o
x'
X,
a
X
t
(x")
= {(x
x")
:
aa
x')
,
X
-*-
[X
X
e
X
a
X
Ix'
o'a
c
t
where
= {x
t
e
t
(x
aa
convex for each
1
is a self-indexed family of maps
(x")
the graph
(x
a
,
x°)}
ot
(x")|u
(x
x")
,
>
a
^Sup
x°)}
,
x"
e
u
(.,
is as in 1.0.4.
for each
a
e
A,
continuously into
t
maps
C(liX
)
;
X
of
x";
and
such that
Then
(a)
is
a
{x°J
x
a
A,
e
e
u
X
(.,
A
P,
e
and quasi-concave on
X
x", x')|(x
,
a
a
and,
upper semi-
x")},
11
(b)
for each
(c)
S
^
e
maps
a
,
into
X
C{^(X
and
);
is a simple economy.
furthermore,
If,
for each
1.7.5.
a
a
e
t
/^ ,
is lower semi-continuous,
a
then also
(d)
for each
(e)
the contractual set
a
P,
e
is upper semi-continuous;
a
of
C
and
is non-empty and
S
compact.
T
a
X
T
(x
t
a
x
(.,
of
)
t
to
a
is convex by assumption and it is closed since
{x"}
X
a
of the restriction
)
is closed. Thus,
t
a
( .
x°
,
is into
)
CQ^iX
a
by A.
);
2,
it is also upper semi-continuous. Hence, by Fan's FPT
(;v.5),
of
X
its graph
X
a
X
.
a
T
a
intersects
(x")
the diagonal
A
o
Furthermore, this intersection is one of
two closed and convex sets, and so is closed and convex.
Being closed in the compact
X
Moreover, its projection into
showina that
^^^Ply
\«.X
t(x°)
e
CO(X
^""-^^^^cx'
a
).
^"'
X
it is compact,
,
is nothing but
X
Now the graph of
^a^
e
X^
X
x°
X
t
t
Xjx^
(x
)
is
a
= y^}).
12
which is compact, hence closed. Thus,
is upper
t
semi-continuous. So much proves (a).
hence compact, so that the continuous restriction of
to
X
X
subset of
X
X
a
t
)
Clearly, this subset is nothing but
.
since
{x"},
proves
t
a
(x°)
[x ]
proving
(c)
(
(e)
ad (d)):
Applying
Being upper semi-continuous by
1.7.5,
(e)
From
x"
upper semi-continuously into
E
of
S
maps
so that, by Fan's
E
and
(b)
point. In other words,
Actually, since
and
(a)
is proved.
(ad
:
1.2)
(Cf.
is now continuous.
t
(d)
)
we see that
(b)
is a simple economy
S
1.4.,
evolution
,
from
),
now assume 1.7.5.
,
lower semi-continuous by
CCIKY.)
is convex. This
(a),
]OC(l(X
[x
i.e., that
,
and
(d)
As
(c)):
a(X°')C
maps
by
,
on
u^
(b)
(ad
For
(x
and it is convex by quasi-concavity of
a{x°'),
u
attains a supremum on a (non-empty) closed
(x"}
S
X
FPT
(d)
,
for each
Q.^{y.
a
).
A,
e
a
Thus, the
upper semi-continuously into
[1952],
E
has a fixed
has an eauilibrium and
C
?^
is upper semi-continuous, its graph
0.
13
r
=
{
(x,
x'
)
e
X
X
x|x'
pact in the compact space
where
h
e
E{x)}
X
is the diagonal of
is closed, hence com-
x.
x
X
x
compact. This completes the proof.
C =
As
x,
C
tt
(Fp
O
A),
is actually
14
SIMPLE DYNAMIC GAMES
The purpose of this section is to define simple dynamic
games and their equilibria. The existence of these equilibria
will be studied in the next section by reference to simple
economies derived from simple dynamic qames
The present sec-
.
tion gives a constructive procedure for obtaining a unique
derived simple economy from any given simple dynamic aame.
2.O.
Definition
A simple dynamic game
:
(
s
.d.g.
)
is an ordered
octuple.
2.0.0.
S
=
< W,
=
{X.
Y, Y,
6,
0),
U,
A}
3^
A >
T,
where
2.0.1.
W
j^
I
non-empty sets
d
e
X.
a
is a non-empty family of
with generic elements
from which we define
X = nX.
"
A
denoting
(a
2.0.2.
Y
2.0.3.
Ye
e
x
e
x
and
A);
is a non-empty set;
[y]
;
x
e
x
and
x" =
X.
x.
e
n
X.,
A\{a}
6
a
a
for aeneric elements
15
2.0.4.
6:XxY-*-Y
from which
is a function,
define the derived functions
^i
:
^X
x
vre
Y
->
inductively
Y
and
^6{^x, i) = 6(^x, ^ ^5(^x, i))
and
7
Mj^x,
Y.)
=
X
Y
<5(^x,
(
y;) '•
••/
6
(„x, Y_))J
< Z
e
Y,
Ik
e
N,
where
j^X
:
<5
is a map defined by
Y
-*
°6(^x, i) = i
2.0.5.
ti)
2.0.6.
U
=
{d).
Y
:
(^x
[x.]|a
--
an XxYx
=
{u.
j^X,
e
:
n
is a family of maps;
A}
e
Y);
e
i;
Y-vPliEA} isa
family of
'
real-valued utility functions.
2.0.7.
T
=
{t.
X
:
Y
X
whence, for each
derived maps
t.
K a
.
:
„X
n
X
Y
—
^t
-»-
->-
a
.
:
Hx.]
'-K
a-'
[x.]
|
i
X
x
A}
is a family of maps
we inductively define the
A,
e
e
y
-^
[x.]
as follows:
(footnote 2) see next page)
and
16
=
""^^(n^' y)
t,('^x,^-lj(^x, y))
and
k-1
t.(^x,
=
n
1 ^
I^^J
=
i^(^)
Y.)
^t.(^x, i)
where
^a
'•
n^
""
°t^(„x, i)
2.0.8.
A
a map
(^x
n^5<n^
^X, i
e
(footnote of page 15)
x", y)
(
'
e
Y);
e
,y)
^^^^
e
x"
X
Y.
X*
a
x
:
x"
x
y
-^
Lx
J
y
^>
The reader may find it convenient
to assume on first reading that
(x
defined by
is a self-indexed family of mans
such that, for any
2)
i-s
t.(x°',y)
= x.
for all
This assumption implies, in particular,
k^i^n^'i) = k^i
for all
as will become clear below,
(^x,i)
e
^X
X
Y,
which,
corresponds to the case where
the feasible control region is fixed.
17
where
n^
•
n*^a
and
'^
X
"^
(^x°, ^)
G
is defined for each
[n^^J
^x"x Y
a
e
A
by
In order to facilitate discussion we
introduce the
follov/inrr
2.1.
terminology.
Terminology;
From here on, whenever we speak of
game", we will always mean a
The elements of
2.0.
2.1.0,
2.1.1.
Let
x^
will be called the
(^x^)
(a-plan space),
d-control
'
X
^
(
and
g-plan
x^
iff
)
x.
t
X"
(^x"
scace
(U lan
plan
iff
)
E
X
J").
space ),
X
e
X
(
X
x
e
x.
e
x"
(^x")
a-exclusive plan
X
)
will be called
)
iff
will be called the control
(^X)
and
(
g-exclusive control space
and
(
.
anana
X.
e
a
a-exclu5ive control
^^
be as in
S
will be called an
will be called the
(g-exclusive plan space).
x''
"dynamic
g-control space
(^x^)
'
(^X")
a
will be referred to as follows.
S
will be called the planning horizon
n
(
s.d.g..
(^x)
X)
will be called a control
2.1.2.
called the state space
V7ill be
Y
a state iff
2.1.3.
y
e
s pace
of initial states
will be called an initial state iff
2.1.4.
the k-extended next state map
k-string next-state map
2.1.5.
Y
will be called the next-state map
6
will be called
y
Y.
will he called the
Y
and
,
.
^
k
.
,
and
y.
X*
•
6
will be called
will be called the
,6
.
will be called the initial control feasibility speci -
u)
fication
,
u).
a
will be called the initial a-control
•
feasibility specification
,
and
d.
e
will be
[x.^
called an initial a-control feasibility for
e
i
2.1.6,
d.
= <;.(i).
d
will be called the control feasibility specification
T
will be called the
t.
_t.
11
iff
will be called the utility
function of
^
u.
a
2.1.7.
Y
a
(„t.)
n ct
d-control feasibility specification
will be called the
bility specification
.
and
^d.
o-plan
'
(j^d.)
(effective)
feasi—
will be called an
19
(effective
a-r>lan
=
d.
n a
2,1.8.
h
(^x,
n
t.
n a
feasi hllity for a plan
)
o-exclusive plan
(^d. =
t.
n a
n a
y)
i-
(
n
x"*,
will be called the personnel of
will be called
a
x
(an
and an initial state
^x"")
player
y)
•'^
S
,
)
^
iff
.
while each
a
e
^
.
The connotation of the above denotation will becoire
clear as we proceed; however,
few points can be irade at
a
this tir^e. The key to understandina the above formulation of
dynamic names is 2.0.8, v;hich
Given any initial state
^x
,
each player
computes the set
now elucidate briefly.
y.
^^'^
a-exclusive plan
^^Y
is assumed to develop an
a
In doing so, player
2.0.8.
v;e
^(^x",
y;)
a-plan obeying
takes the given
a
(
x°',
and
y^)
of solutions to the following
optimal control problem:
Maximize
„X
n
X
{
a
n
x.
ana
u.
y}
x"",
X
n
*-
(
,
n
x", y,
*-
y)
n-"
Y
subject to:
/k.
k. o k-1
6(x.,ic,
y),
k.
;
y =
X.
where
.
E
t.
n
a
n^ =
^n^^a'
(
n
X.
a
,
n'^"^
.
keN,
x,
,
n
x", y;)
^
=
<^^
/
k
+
"^^
o.
y=Y,
1
e
N
^'^'^
J
=
^^
"v^
20
are the sequences of controls and states over the planning
horizon, left superscript beinq the "time" index. Note
that, by 2.0.7,
x.
°t.(^x, ^) =
e
a-initial control feasibility, while, for
we have
x.
t.( x.
e
k
control recrion for
x"
,
•
y)
,
,
qiven
a
(I).(y),
k =
1
,
.
.
.
-
,n
1,
so that the feasible
is determined by the initial state
x.
and the controls precedino the
k
We postpone a discussion of the conditions under
which
s.d.q.
a
is well-defined until we have demon-
S
strated that any s.d.q. may be reduced to a simple econofny.
Sufficient conditions for the existence of
will then
S
be clear from similar conditions already stated
2.2.
Pemark and Notation
Let
:
=<W,
When
S
that
S'
Y.
Y',
and
S'
6,
u,
r,
Then clearly
^ 0.
T,
A
is also a dynamic game.
>
are related in this fashion, we say
is a restriction of
S.
will be called the restriction of
When
y.
S
is understood,
v;e
When
S
denote
Y'
=
(i)
C X'
Definition
S
E
= < W,
:
n
:
Y,
S'
in this case by
The evolution of a simple dynamic game
Y,
5,
w,
XxY-v
rXxYl
-J
—
Ln
r,
T,
h
>
S'
to the initial state
S(Y).
2.3.
and
be a s.d.g. specified as
S
Yl^Y'
in 2.0, and suppose
S'
(1.2.
for simple economies.
1.7(c))
is a map
defined, at each
21
(„x, i)
E
^X
X
by^
i,
TV
A point
iff (^x, y)
e
called an equillbriuin of
vrill be
^^
(f,^'
^^n^' ^^
"^^^
•
^^^ °^ equilibria of
called the contractual set of
Re mark
2.4.
a
Sup
'^^^
u.
(
a
X
r
n
x",y}
-i
k.
; /k.
y = 6( x.
a
^''^'^x.
a
3)
Note that
i;
satisfy
A,
e
'^a^n^a'n^ 'n« ^n^& 'n^"
„X.
n ct
S
S.
It may be verified that the contractual set of
:
which, for each
^'
S
£
^t.
a
F
(
X
n
X
X. ,„x°,y
.
'-^
n a n
y)
n-'
Y
k-1..
ktot
,
^
,
X. , k^ ,y)
'^
n a n
y)
,
'
k
Y
^
.,
,
,
E
k+1
N,
E
= Yr
^'
N.
is actually a inap into
"[n^a^'^'XlC l^^"!'
22
We now show that every simple dynamic qame can be
reduced to
simple economy whose evolution and, hence,
a
whose contractual set are identical to those of the given
simple dynamic game.
2.5.
reduction Lemma:
Let
S
= < W, Y,y
there is a simple economy
(a)
derivable from
uniquelv
i
-
S
S,
,
6 ,aj
be a s.d.a..
,n,T, A >
= < W,i\T,7^ >,
with
X =
n
x
X
Y,
—
and for which:
= E(x)
(b)
E(x)
(c)
the contractual sets for
Proof
:
(ad
(a)):
be aiven. Denote by
X
(x
Let a
^
y)
;
and
S
s.d.g.
the set
and
coincide.
S
and an object
.<^
^^(a).
a ^ A
Consider the
follov7ing sequence of definitions which specify the ele-
ments of the simple economy
S
to be derived from
S.
2.5.1
U
A \{a}
Xt
^
(a
e
PJ
""
;
23
Define the family of functions
2.5.2.
u.(x)
= u,(.^.(i),
u. (x)
= /2,
{u.
X
->
X
-<-
P a
]
e
A^}
by
^Mx))|
X
X;
e
J
Define the family of functions
2.5.3.
:
(t.
:
rx.'Ma
e
A.)
by
t. (x)
=
t. (X)
=
t. (x)
,
n a
i
A
e
X
Define the family of maps
2.5.4.
X;
e
(x)
ir^
{a
x'''
:
[x.!|o
-»
e
A_|_}
by
=
afx")
(x.
t. (x") |u. (x. ,x
a
a
a
e
a
)
Sup
=
u.{.,x")},
.a
t,(x'»)
where
t.
:
= {z.
t. (x")
x"
X. iz.
e
is defined by
fx.!
-•
We now denote
e
t.
(z.,
yJ^
A =
(a)
x")
and,
.
for
a
e
A, we denote
aeA_|_
X
a
= X.
a
,
u
a
= u.
a
,
t
a
= t..
a
These identifications yield
24
(from 1.0.1
S
-
1.0.4
.-\nd
2.5.1
-
2.5.4) a unique economy
= < W,U,T,A >.
(
and
S
ad
(b)
and
(c)
)
Clearly, if the evolutions of
(see 1.6 and 2.3). This leaves only
that is direct from definitions
1.6
2.6.
(b)
to be shown, and
(compare 1.0.4
(2.^.8)
and
(2.3) with 2.5.4)
Definition
:
derived from
2.5
S
are identical, their contractual sets will coincide
(a)
Let
S
S
be a
s.d.n.
The simple economy
S
via the procedure niven in the proof of
will be called the derived
(simple) economy of
F.
25
Equilibrium Existenc e Results for Dynaniic
GamPs
^-
.
This section applies the equilibrium
existence results
obtained in section
for economies to the case of
1
dynamic
names. Given the reduction procedure
of 2.5, the only concern is to demonstrate conditions
on the elements of a given
dynamic name S under which the
elements of the derived
economy of S satisfy the hypotheses
of Theorem 1.7. This
we now do.
3.0.
Proposition:
Let
S
= < W,Y,Y,«,:,tT,T,^ > be
specified as
in 2.0, where
3.0.1.
X.
is a compact and convex subset
of a locally
5^
convex Hausdorff topological vector
space
[so that the product
X,
(a
h
e
too, is compact and convex
in a locally convex Hausdorff
topoloaical vector space]
3.0.2.
Y
is closed and convex subset
of a locally
7^
Hausdorff topolooical vector space;
3.O.3.
3.0.4.
Y
e
6
:
is compact and convex;
[y]
X
X
Y
linear on
-^
Y
X.
is continuous and,
x
{x°'}
x
Y;
for each
x°
t
x'
26
3.0.5.
For each
with
3.0.6.
^,
e
a
z
l" ,
for each
(
For each
w.
Y
:
a
a
'
'
^
y.
X.'
—
an yxyx
u.
n
:
x", v)
e
n
•'-
A,
e
is a continuous man
[x.]
-v
—
a
convex for each
(y)
For each
and,
3.n.7.
0).
a
t.
X
:
map for v;hich the section
Y-»-P
|x
-+
is continuous
on
quasi-concave
^
y
—
x
X*^
y
x
n
is a continuous
I
r.(x°)
=
{
(x. ,x° ,y ,xl
)
}
X.
{x"}
X
a
each
x"
D
X
x"
e
x
X
a
Ixl
a
'
and each
t
e
k
(x.
a
a
x" ,y
where
N,
e
,
.
)
a
a
ct
e
is convex for
is the
"D.ID Y
closed convex hull of the set
^D = {y
e
for some
y|
^
(n^'i^
1/
n^
~^i(^>^>y)
Y =
Then
is a siirple dynamic aame;
(a)
S
(b)
for each
a
e
/,
maps
a
continuously into
(c)
CQ.(j^X^)
the contractual set of
striction of
S
^^x"
y
upper semi-
and
;
-
S
x
in fact, of everv re-
to a compact, convex
Y
'
C
Y -
is non-empty and compact.
Proof
:
(ad
(a)
and
(b)
)
be a well-defined s.d.g.
:
By Proposition 2.5,
and
a
e
A
S
will have the
asserted continuity and convexity properties if the
will
27
derived economy of
of Theorem 1.7.
nomy of
S
fulfills the hvpotheses (1.7.1-1.7.5)
<W,U,T,A>
S =
Let
be the derived eco-
We verify each of the hypothesis 1.7.1-5 sepa-
S.
rately.
(ad
1.7.1):
(ad
1.7.2):
1.7.1 is fulfilled in view of 3.0.1 and 3.O.3.
1.7.2 follows for
tinuity of
for each
n
x"
a
c
n
x"
on
6
tinuity and linearity properties of
= a,
a
'^n^"-^
^
con-
,
—
For
(3.0.4).
6
.
For
1.7.3):
For
»
1.7.
map (2.5.2)
(ad
X.
follows from tne con-
which, in turn,
,
from 3.0.6.
A\{a}
e
and linearitv of
6,
a
1.7,3 reauires for each
A\{a},
£
1.7.3 follows trivially from 3.O.3.
= a,
a
a
that the
A
e
graph
n
T.
=
{(
a
X., „x^,
nan
kl
na
of
t«
e
xJ)
nana
Let
Y
n
'
n
x", i'
y)
X., „x", y)}d„X
nana'n-^
n
^t.
e
n
x
(
X.
(^x^,
a
and
e
y.)
e
^^^
""
t
t.
n a
„X
Y
—
x
„X.
(
X.,
n a'
y,
—
^
n
na
T*
x
(
,
^x", —
v)
n
v)
=
be convex
}
^"
be given and note that, by comoactness of
A
X»
x.l xl
n a'n a
X
a
e
'
be closed and that the graph
{(„x.,
for each
„xJ)|( X,,
n a
a
y,
-^
,
T«
is closed
if each of its components
t«
iff
is so,
t»
k +
is upper semi-
1
e
N.
In fact,
28
since
t.
is a composition
uous by 3.0.7), projection
by 2.0.4, 3.0.4),
continuous,
3.0.5.
k
Also,
„t
t.
of
{contin-
t.
(continuous
^~'^i
°t.
is continuous by 2.0.7 and
is upper semi-continuous and
u
ii
and
is continuous, hence upper semi-
t-
N.
E
Thus,
(see 2.0.7)
tt
is
T.
n a
closed.
We turn now to the convexity of the sections
n^S
Fix
i^-
(n^"'
(^x\
that the sections
Where
^
^al^J
^5
^
^n^
of
(^x\
T,
'
=
y,
,
,
let
So,
it is
k
^n^'a'
"^a
^n^
^}
n^"'
'
(ad 1.7.4):
f^'x^ v}
^^
^T.
X
'Cy,
kp
(^.^^
^T.
linear on
is
u
i.^
^
'^t.
nH
X.
x
{
a
^n
^^}
y
allows the
is convex,
oarUcular,
tt,
roting,
)
^X
(in 2.0.7) of
is
'^
t.
a
is convex, observe that
(^x.,
^i^J
'
n^""
y,
'
is convex for each
1.7.4
(2.0.7)
Using
1}.
n
^^^^ 3.0.7)
(^:^^ ^)
(in
(^x^, ^)
V
That
n -
{l
e
linear and recalling the definition
that
^
X
,
]<•_
of course, that projection
/\
X..
^
N)
straightforward to verify that
reader to compute that
(,x.,
X
^X-
e
is convex is direct from the definition
which is into
see
.
1
i^
„x", v)}<r:^X
Now the convexity of
lb
xl)
+
(k
(n^-,
2.0.4 and 3.0.4.
k-1 •
5,
We first verify
y.
are convex
^^
((^^,,
x
%
and 3.0.5.
t.
(^i", ^)
""t.
i)
^^"
e
Y_)
I^
i)
^"^
k +
1
"^^^
e
the fact that
N.
follows directly from 2.5.4
y
29
(ad
1.7.5)
The map
:
t^,
2.5.3), is certainly
a
e
A\{a}.
belncr a constant map
semi-continuous. Now, fix
lov.'er
To establish the lower sem.i-continulty of
"
t
=
n a
n
logy of
^x^.
is open.
This beinq trivially so when
It suffices to show that
henceforth that
V
o|
t
for some
^i.
(2.0.8) of
e
V,
k
each
a
^^"^
n^a
keN,
'^P
contained in
^
k
+
1
e
^""^
n^k'
= x.
,
k_
"p
aeneric elements
each
^'1^
-
«
and the map
n
4)
X^
t
define
writina'^
k.
^
and
--
"P
^k^a'
r
°r
^P
k>'
^'""t^f x.,g)}.
:
j^P
x
p
-v
V
{^L}
[^"^""p]
and,
for
""px...:^'
x
e
^
.P
(k
k^
^y
N
e
:
)
Fo]
.
y.^ x ^ X
k
n
(n^i'k^i^
a
a
^
n^a
"
a
bv
-^
k^i'
by
The reader will, please, excuse our momentary departure,
in
defining
j^P,
from our usual notational convention as
announced in 0.0.
y
—
as
We will con-
U.
e
°P =
^«)'
^i
x
V.
define the function
N,
^x
n
contains a nbd of each
V
n
'^
=
and complete our proof by showing that
U
is a nbd of
we assume
to rewrite
4^
N =>
e
We now proceed to show that
l.,
n a
of its points. Toward that, suppose
struct a set
V = 0,
it is useful to define
0.
j^
and, using the definition
V = {a
a
Now let
t..
^^ ^"^ basic open set in the product topo-
^^"^a
k^N
t
^
we first note that by 2.5.3,
^'
(see I.0.4,
30
^T(^p,q) = ''tj(''x(,^r),q)
Froir
2.0,7
(^p,a)
we see that, for any
H,(^x,,a)
=
k +
N
e
1
^P
c
Since
q
^(^p,q)
for soire
r
e
for every
3
,
k
p,
(
k
n,..., p)e^^,keN =>
(
N.
g
tinuous, since
.
.
.
a
N,
k +
1
e
]^
k +
*
k
set
0,
V
n
=
k
V
(k
(following the order
N
N,
e
1
^ (i._-iP/f5
e
is con-
x
—
verification ad
is obviously so. Thus,
x
inductively define open sets
for
k
N)
e
and,
,
i
W,
-^V,
(
j e {
1
,
.
.
for
n-2,...,0),
k = n-1,
.
,k
}
)
and
,
obeying:
n,
'"•W
p
v;ith
T(i,_^Pr"))
e
is certainly lov/er seir.i-continuous
t
To construct
each
p
(as seen in the
is so
t.
each
e
V
c
for each
;^lso,
and since
1
6)
,
as
U
•
1.7.3, above)
k +
0.
and for any
,
From this we are able once more to rewrite
U = {nEO|
X
=
nV
^Prn)\
(
n
''"'^V.)
j=k+1
n
''T(,p,q)
Ji
0}
J
with
k
(^P,...,''p,q)
e
(
n
j=l
^v.)
^
X
^nc'^w,
''uCo.
^P,
^v,Cr
*"
Fto check that this is possible, we use the followina facts
inductively (in the order
k = n-1,
n-2,...,0)
:
(i)
x
is
31
A
lov/er Kerri-continuous and
V.
-i=k+1
is open;
''^V
(ii)
''^^',
n
r
j=k+1
(""p,
.
n
1
.
.
,'^f ,'^)
rC
X
^Y
Nv;
G
w
and
,
'"''V
so
''iC-f,^),
'^
e
-
usina
(iii)
W
open, so
i.'=;
-"'
and
(i)
(ii)
,
is chosen as a basic open set in the product
=1
1
topolocry of
A
U =
Set
PX...X k P
X
0.
construction, it is clear that
Fror^ its
U.
V
k=o
is open and that
Cr.
V
For this, we take any
constructinq
for every
n
point
a
k +
1
(
Since
N.
e
q
e
p,...,"f>)
q
V.)
=1
V
e
p
T(.p,q)
e
we have
vCZ°\<l,
^
n
t( p,q)
G
(
k
r)
V
j=k
Then,
f^
Choose
0.
as any point of this
f>
^p
n
)
•
k-1
-
T
~
k +
e
1
N,
is chosen for every
(i^_^P»<l)
k
e
( 1
,
.
.
^
e
A
j=k
V.
(k
e
{ 1
,
.
.
.
,k
} )
and
o
clear, it follows that
(
^n,
.
.
.
,'^n,q)
e(
n
j
^W ^ f,
so that
(
and affords an element
f)
k+1
-
p.
exists, so that
satisfies 1.7.5,
U
C ^'
being
=1
-
.
^V
^
^"'^r.) (\ ''\(rP,n)
j=k+1
Pf---/"p)
e
^
k
(
by
i^
e
^
"
Now
a
such that
rC
e
intersection. Now assume that, for
kp
and show
I''
-
(0
j
q
It remains to show only that
V.
c
)
x
is non-emnty
^
This shows that the desired
V^V,
establishina that
and completina the verification of the
hypotheses of theorem 1.7.
~
'^ncz'^W.
S
.
,k}
32
(
That
ad c)
has a non-empty and compact contractual
S
set equilibrium is now a direct consequence of theorem 1.7
and proposition 2.5
(b)
That any restriction of
.
(non-empty) compact and convex set
to a
S
^^^ ^ non-empty
X'CI X
and compact contractual set follows from 2.2 and a simple
check that the restriction
3.
O.I
-
S'
of
S
so obtained satisfies
(where only 3.0.3 is involved). This completes
3.0.7
the proof.
3.1.
Remark
If,
:
in 3.0, one takes the restriction of
initial state
i
e
Y,
then 3.0
an equilibrium for all
exists
X
E
X
^
e
Y,
im.plies that
(c)
i.e.,
satisfying 2.4.1 with
for each
Y.
~
S
Z
to the
S
^
has
(y.)
there
X.'
Thus, under
Y.-
the hypotheses of the last proposition, there is an equilibrium
plan for each feasible initial state. Proposition 3.0 also
shows that there exist well-defined simple dynamic games, so
that 2.0 is not a self-contradiction. Of course, this V7ell-
definedness holds under much weaker conditions than 3.0.1-7.
Clearly,
^'^n^a^
^t*
"""^
^a'
will be a continuous map of
^a'
^"^
^
^x"'
x
any additional convexity or linearitv assumptions.
and recalling 1.2 and 2.5, the derived economy
hence,
S
S
Piven this,
of
(and,
S
itself) will be well-defined if 3.0.1-3 hold,
is continuous,
continuous
into
y
^^^ only assumed continuous without
and if,
for each
a
e
A,
w.,
a
t*,
a
and
if
u.
a
*
are
33
3.2.
Pemark
In the sinple dynainic qaires we have considered so
:
feasible control rerrion to vrhich the
far, the
k+1
a-control
•
is constrained to belonn depends on the previous
X.
a
k
control
k-1
and state
X
•
(See the discussion iinmediately
y.
preceding 2.2). The reader will note, however, that the case
k
where the control region in question also depends on
naturally under the case that
v/e
illustrate this by considerina
k
on
•
a
falls
consider generally. Pelow we
dependence of these reaions on
k
and, while this dependence is on
y,
•
y
•
alone, the
y
reader will easily be able to extend the siinple idea involved
k
to the general case where the dependence on
k
is not through
Suppose
3.2.1.
:
Y
E
for each
a
A,
e
vex
(k
E
N)
u.
=
and
"f.
a
t
a
'^r{y.)
=
a
{(x.,y)
a
where
,
'^D
'V .
X.
e
(y)
CO(X.)
e
^dIx.
x
a
'
a
e
I*,
a
(y)
a-controls
k+1
.
a
o
for all
}
con-
are con-
to their respective control reaions is,
S
hence, as follows:
£
>f
•
x.
a
'^'''i.
=
is as in 3.0.7.
The fashion in which the
strained in
.
a
[x.] satisfies
->
with
Y,
satisfies 3.0.1-4
W,Y,Y,(5,w,t\T,A >
<
is a continuous map for v/hich
y^
y
^F .
•
y
alone.
0.6 and that,
and 3.0.6
v/here
k-1
and
•
y
=
S
•
x
>f
•('^y)
,
k +
1
E
N,
a
e
/.
6
34
Moreover, it is straiahtforward to check that
satis-
S
fies 3.0.5 and 3.0.7; so that the conclusions of Proposition
3.0 are valid.
The followina exanple illustrates the application of 3.0
to a salient class of deterministic dynamic games.
3.3.
Example
{I.
Let
:
N|a
A}
of a s.d.cr.
S
3.3.2.
Y =
3.3.3.
Ye
3.3.4.
6
3.3.5.
w.
m
be a finite set, and let
A
e
t
and
m
e
= < W,Y,Y,
6
{m.
e
N|a
,(!>
,U,T,A > as follows:
R;
is compact and convex;
[y]
satisfies 3.0.4;
:
where
Y
->
4- .
is defined by
|X.]
:
Y
-»
C (X.
)
e
A),
be niven. We specify the elements
N
w. C^)
is defined by
=
"^
^^^^
4'.(y)
'
=
^
^
^'
35
{x^
< O)
X^l^li^ik^rY)
e
is a given continuous
3.3.6.
t^:XxY->
is defined
|_x.]
t.(x,y) = H'.(6(x,y)),
(^x,y,^y)
"f. ("y)
+
k=o
{'^f.
X
:
are each
a
Y
X
a
Y
X
X
(x,y)
^y)
,
N,
a
\li
(a
^
e
*
X
e
"^
**
^i
^
"*
ji
a
A);
for each
i
by
A
e
Y;
X
u(^x,i,^y) =
by
^^
^f.i^-'^'^k,
E
"
3.3.8.
^X
e
in which
,
function,
°y
where
e
and where
Y,
"
->
P|k +
e
1
and {"f.
p)
E
Y
:
->
P|a
e
M
family of continuous and concave functions;
satisfies 2.0.8
(a
e
P)
Suppose, in addition, that the follov;ina condition holds.
3.3.9.
For all
a
e
A,
conditions on
if
(ii)
ip .
a
4'.
v/ill be
"¥.
a
satisfies:
for each
y
satisfies 3.2.1.
e
(i)
ij;.
Y,
\p .
a
[Note that these
satisfied if
:
X.
a
x
Y
is
\h .
^a
P
-»
linear or
is continuous;
f.
a
is convex on
X.
x
{y};
36
(iii)
for all
y
there exists
Y,
e
x.
X.
e
^P^(k^,Y)
See, e.n. Hogan
(iv)
3. o.
3.1,
for each
a
y
S,
x,y,
u.
"^
a n
A,
e
(
"^
n
5(
^f.
n
n
x
e
i^^^k.,^^^k\^y))
n
k*a
;/k6
(
ijj.c'^x.,
X.
,
^
""y)
X
k-1-.
,
< O,
y)
^x.
,
•
oy = X'
ex.,
n
x,y)) =
Subiect to
ky =
'^D,
as specified above,
there exists an
Y,
—
e
•^
("f.("y) +
Maxii^un.
ft
x
k
e
N.
satis-
1-3.0.7, Therefore, by 3.0, and recallino 2.4 and
fies
n
X.
[1971^, Theoreins 10,12.3
It is easily verified that
for all
is convex on
ij/^
v;ith
a
ct
and
O;
<
.,
,
k
e
N;
k
e
N,
X
satisfyina,
'
'
37
4
.
Llirltations and Extensions
Our main results are the eauilibriuri existence result
for siirple econoiries
(1.7), the lemira
(2.5)
reducinc sirrnle
dynamic qames to simple economies, and the eauilibrium
existence result (3.0) for simple dynamic cames
The reader may have been struck by our somewhat curious
obstinacy in always referrina to the objects we treat as
"simple". Here we try to indicate the deficiencies in the
above formulation which gave rise to this standing oualifi^
cation.
For simple economies, these deficiencies amount to our
havinn irinored the followina:
1.
informational and/or perceptional imperfections
on the part of behavors;
2.
the usefulness of specifying "effective" preferences
(utility functions) as dependent upon
a
desian pa-
rameter which may be called an "incentive scheme";
3.
the fact that, in practice, the feasibility trans-
formations depend not only on the behavior chosen
from
a
given feasibility, but also on the aiven
feasibility itself
-
the resources one has tomorrow
depend not only on what one does today, but also on
the resources available today.
38
From the viewpoint of eauilibrium existence, these
oininissions turn out not to be crucial, since it can be shdwn,
under fairly weak assumptions, that generalized economies
(called "social systems" in
[Sertel,
1971^), including the
above features, retain the property of having non-empty, compact sets of equilibria
(see
[[Sertel,
197l]]).
Nonetheless,
from the viewpoint of social system design {e.q., selecting
an appropriate incentive scheme)
,
these characteristics are
clearly essential elements of the problem.
T^nother element of simplicity in our work and the pre-
vious literature is the assumption that the collective feasi-
bility is box-shaped, i.e., that what is currently feasible for
a.
is independent of the current
a-exclusive behavior.
One
would, of course, like to remove this restriction, as many
of the problems of central planning (and control theory)
are
not completely decomposable on the constraint side, even by
"dual" m.ethods
(incentive schemes)
.
Turning now to dynamic games, those treated here suffer
from the same restrictions discussed above for simple economies. Moreover, there appear to be sionificant difficulties
in interpreting the informational and behavioral connotations
implied by the evolutions
(and their equilibria)
of these
simple dynamic games. In fact, in the current formulation,
each nlayer
a
e
A
computes his
a-plan given an
a-exclusive
plan. This can be likened to the situation in an auction hall,
where each player calls out his projected plan, and recomputes
39
his plan based on the projected
(i.e.
called out) plans of
all other players. The equilibrium existence question posed,
then, is sirply whether there exists a plan which is repeatable
in the sense of beino called out twice in succession. We find
it difficult, however, to interpret the equilibrium of this
"auction" process, unless
n =
in the case where plans
1,
are exhibited only as thev are enacted;
i.e.,
where
equilibria are to be interpreted as repeatable in the sense
of enactable twice in a row.
Finally, as may be seen from 2.4, the equilibria whose
existence is established here correspond to open loop equilibria of dynamic names as they are usually defined
However, the existence of closed loop equilibria
1970|).
|Ho,
(see, e.a.,
can also be studied by similar methods. Essentially, one starts
with
oiven simple dynamic name
a
dynamic name
of
and derives from it
S
by replacina the
d-control spaces
by the function space of stratecry maps
S
laws)
S'
to
X..
a
a
•
Y.
(or control
from the observed state and control history
tine k)
k
(until
The (non-trivial) issue to be resolved is
the determination of conditions under which the dynamic qame
S'
resultina from this transformation will inherit from
the convexity,
linearity, continuity, and compactness assump-
tions required by Proposition 3.0
5)
For
a
P
-
or, more aenerally.
discussion of some aspects of the meaninrrfulness of
prrviilibrium in dynamic oames
,
see
IPtarr
&
Ho,
1969a,
1969bl
40
throuah an appropriate reduction, the assumptions
required
by Theorem 1.7.
In nrincirle,
the same apt-roach links the
above results to stochastic dynamic names, thouoh
clearly
much work remains to be done in specifying the transfor-
mations reauired for reducina these to the simple
economies
and dynamic frames introduced here.
41
^
•
Appendi x
In the interest of accessibility,
here we collect a
minirral amount of topoloaical information concerning certain
maps whose values or arcruments are non-empty sets. In what
follov/s,
are topolocrical spaces. Our first de-
and
P
finition is extracted from [[Michael, 1951^, of course.
(
N.B
see
A.I.
.
For Isf, usf and finite topolorries on a hyperspace,
195lQ.)
[Michael,
Definition
Let
:
F
:
P
->
is lov;er semi-continuous
(use)
J
[o]
(Isc)
be a map. We say that
iff it is continuous when
semi finite
(Isf)
[Equivalently
[[resp.
,
is aiven the low er
[o]
upper semi-finite
we say that
,
F
is Isc
F
upper semi-contin u ous
[[resp.,
(usf)^]
(resp., use)
topolooy,
iff the
set
{p
is open
(resp.
,
(resp., closed)
continuous when
valently,
Isc
c
P
F
I
closed)
.3
n V}
in
P
We say that
[o]
whenever
F
VC
is open
is continuous iff it is
is qiven the finite topology.
say that
V7e
(p)
F
[Equi-
is continuous iff it is both
and use.
The proof of the following "closed araph theorem" may
be obtained,
3.3 in
for example, by combining propositions 3.2 and
[Prakash
of [Fan,
19 52]
:
&
Sertel,
1970],
where 3.3 is also I.emma
2
42
Proposition
P. 2.
Assume that
:
Hausdorff, and let
f
P
:
Denote the oraphs of
f
and
P
->
0,
are both compact
O
F
P
:
and
F
by
-*
a
be two maf
C (0)
and
G,
s
respec-
tively:
q
=
<^
=
It is iff
{
(
{p,Ci)
1
(P/q)
1
p
c
P,
q
=
f (p) },
P
e
P,
q
E
F(p)
(resp., G)
q
CP
}.
is closed that
x
f
is
continuous (resp., F is use).
The followinq fact, which is a part of corollary 3.1.4.3
in
[sertel,
turns out to be
1971^,
useful tool basic to
a
optimization.
A. 3.
Proposition
w
:
C(P)
w(d,q)
Equip
P
=
x
be non-emptv and compact Hausdorff,
n
x
u:Pxq->r
and let
Define
Let
:
be a continuous real-valued function.
O - P
by
Sup u(p,q)
ped
(deC(P), QcQ)
with the finite topoloqy. Then
C(P)
w
is continuous
We have the followinn obvious
A. 4.
Corollary
:
Let all be as in A.
continuous. Defining
v(q)
v
=
w(s
(rr)
v
,
is then continuous.
:
q)
3,
O - P
(a
and let
by
e
O)
,
s
:
->
C(P)
be
43
A. 5.
Fixed Point Theorem
Theorem
1;
Fan IS 52
Given
:
locally convex Hausdorff topoloqical vector space
if
X^
is compact and convex and if
L
is an
use
point
x
e
transformation,
X
such that
x
F
:
X
-^
a
L,
CQ^iX)
then there exists a (fixed)
e
F(x)
.
44
REFERENCES
Arrow, K.J. and G. Debreu,
Competitive Economy", Econometrica
a
G.
,
vol.
22,
"A Social Equilibrium Existence Theorem",
,
3,
in
Proceedings of the National Academy of Sciences
(U.S.A.), vol.
Eilenberg,
No.
265-290, 1954.
pp.
Debreu,
"Existence of an Equilibrium for
S.
38,
pp.
,
886-893, 1952.
and D. Montgomery,
"Fixed Point Theorems for
Multi-Valued Transformations", American Journal of
Mathematics
Fan, K.,
,
vol.
68,
dd.
214-222, 1946.
"Fixed-point and Minimax Theorems in Locally Convex
Topological Linear Spaces", Proceedings of the National
Academy of Sciences
,
(U.S.A.), vol.
38,
pp.
121-126,
1952.
Ho.
Y.C.
,
"Differential Games, Dynamic Optimization, and
Generalized Control Theory", Journal of Optimization
Theory and Applications, vol.
Hogan, W.F.,
6,
No.
3,
Sept.
1970.
"Point to Set Maos in Mathematical Programming",
Working Paper No. 170, Western Management Science
Institute, UCLA, Feb., 1971.
45
Kuhn, r. and C. Szego,
Gaires and
^''ichael,
North Holland, 1971.
,
"Topoloaies on Spaces of Subsets",
E.
Transactions of the
vol.
The Theory of Differential
(ed.)
Pelated Topics
71,
Parthasarathy
152-182,
pp.
K.P.,
,
7^ir,erican
Society
,
1951.
Probability Measures on Metric Sp aces,
/Academic Press,
1967.
Prakash, P. and M.^. Sertel,
Theoreir.s
r''atheriatical
"^^ixed Point and Minirax
in Semi-Linear Spaces", Workincr paper,
Sloan School of Manaaement, Mgc^ciachusetts
(484-70)
Institute of Technolony, Sent., 1970.
Sertel,
"Flerients of Eouilihrium Methods for Social
^''.P.
/Analysis",
TM.T.
,
Jan.
tinpublished Ph.D. Dissertation,
,
1971
Starr, A.W. and Y.C. Ho,
.
"Nonzero-Sun Differential Haires"
Journal of Optimization Theory and Applications
Starr,
vol.
3,
7\.W.
and Y.C. Ho,
No.
3,
March,
"Further Properties of Nonzero-Sum
Differential Games",
and Applications
,
,
1969.
vol.
Journal of Optimization Theory
3,
No.
4,
April,
1969.
y^
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