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http://www.archive.org/details/finiteplayerapprOOfude
working paper
department
of economics
FINITE PLAYER APPROXIMATIONS TO A
CONTINUUM OF PLAYERS
Drew Fudenberg
David K. Levine
Number 455
January 1987
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
FINITE PLAYER APPROXIMATIONS TO A
CONTINUUM OF PLAYERS
Drew Fudenberg
David K. Levine
Number 455
January 1987
FINITE PLAYER APPROXIMATIONS TO A CONTINUUM OF PLAYERS
Drew Fudenberg
David
K.
Levine
August 1986
Revised:
January 1987
*
We are grateful to Andreu Mas-Colell for helpful conversations, and
NSF Grants SES 85-09484 and SES 85-09697 for financial support.
Department of Economics, University of California, Berkeley, and
University of California, Los Angeles, respectively.
Dewey
:i/^» s ,W|P
MAY 23
If
-
N0V1
-^
t^
r
f
ABSTRACT
In this paper we are interested in the lower hemi-continuity of the
Nash equilibrium correspondence with respect to the number of players.
Specifically, given an equilibrium in a game with a continuum of players,
and a finite player game that approximates the continuum game, is there an
equilibrium of the finite player game that is "close" to the equilibrium
of the continuum game?
200SS31
1
.
Introduction
In this paper we are interested in the lower hemi-continuity of the
Nash equilibrium correspondence with respect to the number of players.
Specifically, given an equilibrium in a game with a continuum of players,
and a finite player game that approximates the continuum game,
is
there an
equilibrium of the finite player game that is "close" to the equilibrium of
the continuum game?
We show that lower hemi-continuity obtains if the players' payoffs are
differentiable
,
strictly concave in their own actions, and moreover have the
property that the indirect effect that players have on each other is small
in the appropriate sense.
This bound on the cross -player effects, when
carefully specified, implies that the operator representing the cross partial derivatives of players payoffs with respect to each other's actions is
compact.
It implies also that the continuum game cannot exhibit a robust
indeterminacy, and is connected to similar conditions used to establish
determinacy in continuum economies in Kehoe
[1986],
and Kehoe, Levine and Romer [1987],
,
Levine
,
Mas-Colell and Zame
A similar condition is used by
Araujo and Scheinkman [1977] in studying the value function in dynamic
programming.
The upper hemi-continuity of the Nash correspondence has been studied
extensively by Green [1984], and by Fudenberg and Levine [1986].
paper studied lower-hemicontinuity as well, but it considered
e
Our [1986]
-equilibria,
providing conditions for an exact equilibrium of a given game to be an
equilibrium of games that are nearby.
However,
the literature on limits of
monopolistically competitive equilibrium, including the papers of Green
[1980], Mas-Colell
e-
[1982], Novshek and Sonnenschein [1980],
and Roberts
[1980], have examined lower-hemi-continuity of exact "equilibria.
These
results depend heavily on there being a fixed finite number of types
(although not all of the above papers impose strict concavity in own
action).
With a finite number of types, the cross partial operator has
Our formulation is more general,
finite rank, and so is certainly compact.
allowing situations where payoffs of individuals depend on their own actions
in idiosyncratic ways, and on countably many weighted averages of the
actions of others.
The model's increased generality may help provide a
better understanding of the essence of the lower-hemicontinuity results.
Section
2
and considers the
of the paper sets up the model,
mathematical issue of when
a
strategy for finitely many players is "like" a
strategy for infinitely many of them.
Section
develops a version of the
3
implicit function theorem that applies when changes in a parameter change
This is needed since,
the domain on which the function is defined.
depending on the number of players, strategy profiles lie in different
spaces.
Section 4 extends our topology on finite- and infinite-player
to consider what it means for a finite dimensional linear operator to
games,
It concludes with the sufficient
be "like" an infinite dimensional one.
conditions for lower-hemi-continuity
Finally, Section
.
5
shows that these
conditions are satisfied when players affect each other only through
weighted averages of all players actions.
2.
The Model
The space
A
of players actions is a compact convex subset of a fixed
finite dimensional vector space
that
A
is the closure of its
Hausdorff topological space
with a fixed norm.
A*
The space of players is a compact
interior.
P,
distinct closed nested subsets of
while
P.
We shall assume
P.
,
P.
,
.
.
.
P
are a sequence of
We also adopt 'the convention that
- P.
P
as a subset of
P
(or
P
We generally think of
(P)
that is
;
A-
think of
example
players, but this is not essential.
is called an atom of
€ P)
p
n
as containing a countable dense set,
P
is an isolated point.
p
P-
[0,1],
and
[0,1]
has no atoms and
P
from players to actions.
continuous mapr
x:
in place of
x
X
x(p)
of continuous mappings from
P
denote the identity)
-A
P
:
The space. of all
.
A*
in the sup
Every
interior.
Consequently, there are natural restriction mappings
r - r
x
-+
has a continuous restriction
A*
-*
In this
This is a bounded convex subset
.
is equal to the closure of its
P
n-l/n,l}.
game is a continuous map
We write
X*
X
Moreover,
For concreteness it is helpful to
- (0,l/n,2/n
P
such maps (in the sup norm) is denoted
of the Banach space
connected component of
is a
p
p S P
has only atoms.
P
A strategy profile of the n
norm.
if
(P)
P
A player
and
r
x
n
P
:
- A*.
n
X* * X*
:
(we let
These are obviously continuous linear
.
operators
If
for
x
n
x e X*
€ X*,
n
then
that
n
x
-*
n
"converges" to
€ X*
n
r x
x
provided
v
|x
'
n
-r xl
n
More generally, we say
x.
-»
This makes precise
the
r
0.
'
idea that a profile in a finite player game converges to a profile in the
continuum game.
U C X
Let
be open in
Corresponding to
X*.
U
Notice that •by the Tietze extension theorem, for every
J
x
n
€ X*
with
linear map
r
r x
n n
- x
n
and
is onto all of
|x
'
n
I
- |x
1
'
X*,
n
I
1
.
are sets
x
€ X*
n
n
- r U.
U
there is
Consequently
J the continuous
^
and, by the open mapping principle,
U
is an open set.
The payoff
function of the n
J
We write
depends on
7r
n
(a,x)
p's
in place
of
r
n
game
is a function
°
n
(p,a,x).
n
n
:
P
n
x A x X
Note that the payoff
-
7r
*
n
P
n
action both directly, and indirectly through the effect
IR.
that it has on the joint distribution of players and actions.
atom and
a
x\a
the continuous function derived by replacing
is
then it is sometimes convenient to define
.
'
be the payoff in which
If
?r[a
and
= x,
x\a
Definition
all
p e P
;r
[a
,
^
x]
with
x
= n (a ,x\a
n
,x]
to
)
so we define
7r
P (a
n
,
x).
x e X
game
°
of the n
n
satisfies for
xP G A
and
n
n
is an
p
captures both the direct and indirect effect.
a
A Nash equilibrium
:
n
there should be no indirect effect,
is not atomic,
p
If
P [x P
,x]
P
>
n
,r
n
[x
P ,x].
We shall always assume
Assumption
1
For each
:
x
7r
n
as a function of
[a,x]
differentiable and strictly
concave; both
J
respect to
Let
so
6
tj>
n
n
- D w [x
(x)
,x]
an
e X
equilibrium
n
x
.
n
.
Under Assumption
r
1
continuously
is
and its derivative with
n
are jointly continuous with respect to
a
(x)
it
a
d>
n
a.
continuous in
is
(x)
and
p
p,
Moreover, it is necessary
and sufficient for a Nash
J
e U
n
that
d>
n
(x
n
)
=
0.
We assume that we are given an equilibrium
goal is to give sufficient conditions relating the
guarantee that there are solutions
x
n
G U
with
n
to
4>
<f>
n
(x
<Hx) - 0.
with
x £ U
n
)
<f>
—
Our
which
and
x
-*
n
x.
In other words, we will provide conditions for the Nash equilibrium
correspondence to be lower hemi- continuous in the continuum of players
limit.
3
An Implicit Function Theorem
.
Suppose that instead of the discrete parameter
X
,
games are indexed by
A
and for all
A
n
and separate spaces
the game is played in
X.
If
<^(A,x) -
and
U
in
is the equilibrium condition,
that for
and
close enough to
X
x(A)
x
-+
the implicit function theorem implies
is non-singular,
D <f>(X,x)
as
A
X
-*
continuously dif ferentiable
is
<f>
there are solutions
X
x(A)
to
<f>(X
,x(X)) -
Our goal is to make sense of how an argument
.
like this might work when games are parameterized by a discrete parameter
n,
and the spaces
on
n.
on which the equilibrium condition is defined depend
X
Now let us consider
V A,x € U,
then
|D rf<A',x')
Definition
d>
is
n
D
-
x
>
e
x
|x'-x| < 6;
35, N s.t.
<KV,x)| <
11
Assumption
n > N
2
:
implies
The family
differentiable, and
<f>
n
implies
is uniformly differentiable if each
<f>
continuously
J differentiable, and
|x'-x| < 6,
|A'-A| < 1/N
c.
The discrete family
:
If this is continuously differentiable
$(A,x).
I
Vj>
n
(r x')
V x € U,
-
n
(including
<j>
-
(r x)
D<£
n
(r x)
n
n -
«>)
and
Y
>0,
£
I
<
35,
Ns.t.
e.
'
is
uniformly
<i(x).
n
Our basic tool is
The Inverse Function Theorem
A:
X
-+
Y
r?(x)
< b(|A
- 0.
X
{x e
and suppose
|Ax+r7(x)|
Ax +
X
are Banach spaces and
is a non-singular continuous linear operator.
-- U-b > 0,
Suppose
:
|
r?
|
|x-x|
U * Y
:
-k)
is
*i
_ 1ax-t?(x)
Ia"
< b)
Lipshitz with constant
then there is a unique point
>
Moreover
i"
Let
1 !" 1
1
-^
k.
If
x e U
with
Proof
This is an implication of the contraction mapping fixed point
:
theorem.
Observe that
U
Obviously
is a
Ax +
x - -A
if and only if
-
rj(x)
complete metric space; we must show that
r?(x).
-A
is a
rj
contraction of U.
First we show that
-A
maps
r)
U
into
This follows from the
U.
sequence of inequalities
|-A'
1
»
7
< l-A'Scx)-*! + |-A"
(x)-x|
X
<
|A"
< Ia"
1
r
7
(x)+A" r?(x)|
(x)+Ax| + |rj(x)-r7(x)|)
|
(|r?
!
(bdA"
1
1
1 !" 1
-^)
+
/cb)
- b.
(We obtain the last inequality by substituting the bounds on
|Ax
-
r7
(x)
|
.
|a
1
r7
-
|
This shows that
x
(x)+A"
k
>
1
r
7
Ix-xl
-
I
|
|
r,
(x)
-r,
and so
k < 1
|
(x'
)
-A
|
< \a'
r\
1
k
\
|x-x'
|
,
is a contraction.
<
|
|x-x|
-A"
r?(x)
-A"
r?(x)-x|
1
!
< Ia"
I
xl
-
+
|
r?(x)+A"
-A"
1
U(x)-^(x)
h(x)-Axl +
Ia"
U(x)-Ax| +
Ia"
h(x)-Axi _
hw-^i
!
I
r?(x|
k
Ix-xl
we find
1
,;.;,
_1
exists and is unique.
< Ia"
|x-x|
< |a
(x')|
|a
0,
It remains to estimate
Solving for
and
Moreover
)
|-A"
and since
|x-x|
s Ia'
!
1
I
A
|k
I
A
I
QED
-k
We now apply the inverse function theorem to the family
<f>
The Implicit Function Theorem
that
family,
J
and
Proof
—
x
b
|x' -x|
< b
Moreover,
.
A
Take
:
Hi (r x)
so small and
(3.2)
IVV
(3.3)
l^
n
(x
This implies
1 '" 1
x
(
n n
we need only
J show
k
n
on
n
'
< a/2.
'
n
"
Df(rx)|
n n
A X
n n
(r x)
r\
|A
n
n
k
-
I
< b.
]
> a/4
n
Since
|a
< b;
|x-x|
where
n
< {a/k)
+ (a/4)
(x
n
(x
n
)
-
n
;
> 0.
Now
'
< b,
|x-x|
)
-
Dtf
^(x^
+
-
4>
n
(^
n)
constant
is the Lipshitz
r
n
<b,
'
nn
|W n (x n
+
k.
> 3a/4,
I
'
|x'-x|
nn
nn
n
|A
and
A^
we need only
show
J
x
'
^ l^(x')
<f>
n
From the inverse function theorem
.
'
-
d>
< a/4
< ba/4
I
'
We have for
U n (x')
n
n > N
Consequently,
•
'
'
1
such that
.
n
Then there
n
'
'
nn
where
(r x) - 0.
a - lim inf
so large that for
-
'n
|x -r xl
x
n
" ha/U
I
n>
'
r\
unique
a
4>
" 3a/4
+
|a r x
1
}
n
By assumption
n
^n
=
n>
n
and that
x.
-»
n
.
N
(rx')
|Di
»?
x
nnn
(3.1)
Define
n
and for large
enough
&
b
e r U'
n
n
choose
for
n
uniformly dif ferentiable
is a
<j>
> 0,
(r x)t"
|D<£~
'
C U,
U'
is
inf
lira
Suppose
:
n
-
rf
)
-
|x;-x
|x;-x
(X„)
nn
-
n
-rx,
n
x'
n
- r x'
n
nn
nnn'
D4 (x)(x'-x
nn nn
)|
Df(rx)](x'-x)|
n
n n
n
n
n
|
|
< (a/2)
nnn
X;-X|
11
(x )(x'-x )|
and the Fundamental Theorem of Calculus.
|
< a/4
|x'-x]
'
follows from (3.1)
To see this, write
K<o
'nn
0.5)
+j*j
nn
•
D
[D4
|jj
u
Mnnn
J(
x
nn
(x n
U'nn
From (3.1), and
(3.6)
4.
|x-x|,
/J
|D^
n
+
< b,
|x'-x|
(x
nn
+ t(x'-x ))
(x
< [ji|Di
J
t(x'-x))
nn
-
nn
nn
Di (x )](x'-x )dt|
-
(xj|dt] |x'-x
M nn
'nn'
|.
it follows that
+ t(x;-x ))
n
n
-
x :- x )l
-
D^(x
n )|
< a/4.
Q.E.D.
Convergence of Operators
We wish to apply the inverse function theorem to conclude that a Nash
equilibrium
<^(x)
of the continuum game can be approximated by Nash
-
equilibria of games with finitely many players.
that
D<^>
is non-singular for sufficiently large
(r x)
were indexed by a continuous parameter
D^
then
sufficiently near to
\
non-singular
D$(x)
D
(A,x)
<f>
X,
is non-
In the case at hand, however,
A.
not sufficient to ensure that
is
If the games
n.
and played on a common space
n,
non- singular would imply that
(X,x)
singular for
To do so we must ensure
D<f>
(r x)
is non-
singular as well.
Suppose
F
nnn
:
We will say
that
J
X* * X*
F
(4.1).
x
-
X*
are continuous linear operators,
x
n
€ X*
n
with
|x|—l
n
1
'
such that
1
'
'nnn
|x
-+
|x|-l
n
e X*
n
X*
F:
if for every
sequence
~i
j
F
-*•
n
there is a sequence
and
r x
exists by
the Tietze
(such a sequence
J
n
-*
I
1
extension theorem)
(4.2)
if
(4.3)
|F x
'
-r zl
'mm'
|x
n n
-
r
n
Fx
-»
I
n
for some subsequence
m
then
x
-*
m
z
- 0.
1
A simpler and more obvious definition of convergence would be to
require that
F r x - Fx
n n
for every
J
x € X*.
If we add the requirement that
uniformly bounded, this can be shown equivalent to our definition
is
|F
I
of convergence holding for convergent sequences only.
We refer to this as
To show the extra bite from requiring that
weak convergence.
and
F
F
are similar for non-convergent sequences consider the following example.
P -
Set
[0,1]
- (0,1/n
P
,
1)
A - [0,1]
and
,
Let
.
be the
F
matrix in which every entry in the first column is one, every entry in the
second column minus one, and all remaining entries zero, i.e.,
F
F = 0.
while
For any fixed
Consequently
n
J
continuous.
In other words
all
x
n
-»
F
If r
n n
'
|r
F
then
n
x(0)
r x(l/n)
-
n
1
-f
However,
|f|
x
|F
In this case
> lim sup|F
since
-
n
I
-
r
x
is
x(l/n)
n
0.
-»
I
'
so the norm is not
2,
1
not necessarily convergent,
,
in our sense.
0,
0,
'
'
'
n
-»
I
x-r Fxl - If r xl - |r x(0)
n
n n
n
weakly.
J
-»
n
- (1,0,0,...).
Thus,
x.
F
if
F
x,
If we allow sequences
continuous.
can choose
1-10
1-10
-
F x
n n
In fact,
- 1,
while
then we
Fx -
for
be shown that
it can easily
J
|.
while our definition is a fairly restrictive notion of convergence, it
does not follow that if
eventually non- singular
A - [0,1]
and
F
-*
n
F
and
F
is non-singular
then
°
A counterexample with
.
F
n
maps
s r FR
n n
(0,1/n
1}
>
x(p/2 + 1/2)
n
P -
is
< p < 1/3
x(2p)
R
[0,1]
n
is the operator
(Fx)(p) -
If
P -
F
X*
n
to its linear extrapolation in
obviously
to
J converge
b
non-singular and
1/3 < p < 1.
F
F.
the operators
Moreover it can be shown that
are always singular.
is that while it is non-singular,
X*,
F
is
The problem with the operator
its restriction to the invariant subspace
F
10
x(p) -
of functions with
< p < 1/3
for
one-to-one
is
but not onto.
No non-singular finite dimensional operator can be singular on an invariant
subspace
F
is a
it is unreasonable to think that an operator like
In other words,
.
reasonable limit of finite dimensional operators.
One important class of operators for which the finite and infinite
dimensional versions are similar is the diagonal operators.
the linear operators from
is
representable by a continuous map
- f (p)x(p)
(F x)(p)
multiplication with
F:
X*
-+
X*.
that is,
,
f
operator
A diagonal
°
to itself.
F
f
Let
:
- L(A)
P
,
F
L(A)
n
be
X* - X*
n
n
:
such that
is the operator induced by pointwise
F
Similarly we can define a diagonal operator
.
These operator arise naturally as the derivative of
respect to the players' own actions:
4>
with
the diagonal is simply the second
derivative of players payoffs with respect to their own actions.
It is easy to show that for a diagonal operator
- sup
F
p6P
|f;
x
If
- (su P
|
(p)
,
I
and
n
_1
1
lf"
(
P)
)
|
.
peP
Moreover, for diagonal operators
lim inf If"
|"
> 0.
Lemma 4.1
If
F
sup
PS?
Proof
|f
:
:
* F
and
F
non-singular implies
F
if and only
if
J
This follows from
and
n
F
F
are diagonal
b
F
-»
n
(p)-f(p)| - 0.
That sup
peP
show the converse.
|f
n
(p)-f(p)| -
implies
F
-*
n
F
is straightforward.
Suppose there is a subsequence with
W€
11
|f
Choose
and
x
(p
IF x
-
n n
1
n
< e/2.
l|
-
)
f(p
-
)
x (p ) n n
so that
n
|x
|F|
n (p n
r Fx
n
|f
n (P n )
-
|f
n (P n
-
)
)|
>e.
We may
assume
J
1.
x
n
has
-r Fx
'nnnn'
|F x
-»
I
Then
I
>
n'
f(P )x(P )| ^
n
n
f(P
n )|
|f(P
-
n
)d-x n (p n ))|
^ e/2.
This contradiction establishes the Lemma.
Q.E.D,
Notice that as far as diagonal operators are concerned, we could have used
weak convergence, rather than convergence.
We now characterize a broad class of cases for which
and
F
-*
n
F
imply
that
J
lim inf If
'
n
>
I
0.
An operator
non-singular
F
on a Banach
C
'
space is compact if the closure of the image of an arbitrary bounded set is
compact.
A broad class of compact operators can be found by taking strong
limits of operators with a finite dimensional range; in some Banach spaces,
all compact operators have this form.
As a result,
of the derivative of a players incentives
<f>
it is natural to think
with respect to what everyone
else does as compact provided that there are only finitely many important
channels of interaction.
next section.
A formal model of what this means is found in the
From a mathematical point of view, compact operators are
important, because,
like diagonal operators, they can be approximated by
finite dimensional operators.
Proposition L.2
diagonal;
F,B
;
Suppose
F
n
-F;F n
non-singular and
C
- B
n
+C;F-B+C;
n
compact.
First we prove a preliminary lemma.
Then
B
n
lim inf |F
- B
|
are
>
0.
12
Lemma 4,3
Proof
Under the hypotheses
of Proposition 4.2
J
:
Let
:
|x
'
n
-
I
Since
1.
+ C x
IB x
B
|x
'
nn
r Bx
-
nn
n n
1
-+
n
there is a sequence
F,
•*
n
C
n
C.
I
n
-
1
X
in
1
In other words
satisfying (4.1) to (4.3).
(4.4)
F
1
B
r
-
n
Cx
-
I
0.
n'
We will show
|b"
(4.5)
1
n
1
By Lemma 4.1 and
C x
r
-
n n
B^Cx n'
non- singular
B
- 0.
I
n
|B
eventually bounded above.
is
|
Consequently from (4.4)
|b"||Bx
+Cx
n
n n
nn
'
'
-
r
'
Bx
nn
r
-
-
Cii
nn'
I
implying
|x
n
-1
_1
x
nnn
+ B
C
B
-
nnn
r Bx
B
-
nnn
Cx
r
I
-
.
1
This expression is at least as great as
x
'nnn
Ib"
C
-lrx
'nn
Since by (4.1)
n
-»
I
term goes to zero.
n
I
-
x
'nnn'
|x -r
1
I
n
nnn +rB*Cx
r Bx
B'
•
-r x
'nnn'
|x
Cx
r B
-
Since
n
nnn'
I
it suffices to show that the final negative
0,
|x
B" r Cx
-
n
- 1,
I
it is bounded above in absolute value
by
(iVn
1
1
Since
|B
'
and
B
n
+ |C|)
'
|r B*
'
is bounded,
t
'
_1
1
n
-
B
r
nn'
|.
we need only
show
J
'
|r B
-
n
'
B
n
r
I
*
n'
.
Since
B
(and their inverse) are diagonal
°
n
|r B"
1
Since
!
'
B
-»
B,
n
_1
1
-
B
n
r
n
I
1
- sup
„
peP
|B"
'
1
(p)
r
-
B"
n
1
(p)|.
r
'
n
the final expression goes to zero by Lemma 4.1.
Q.E.D.
13
Proof of Proposition 4,2
x
n
e X
with
n
|x
'
implies
r
|B
'
F x
n n
n
|x
(4.6)'
v
'
-
I
n
(4.7)
IB
'
and
1
I
1
'
•*
I
F x
-»
I
n n
Since
.
|b"
'
bounded,
is
I
n
1
this
'
Consequently
.
1
C x
B"
-
n
n
Lemma 4.3, choose
Bv
'
v
If the proposition is not true we can find
:
x
n n
1
-
n n
|x|-l,
n
with
€ X
n
C x
n
- 0.
I
r B"
n
Cx
n
I
|x -r
n
x
n n
I
and
-*
- 0.
1
Notice that this step is where we need strong rather than weak convergence:
there is no reason
(4.8)
|x
1
Since
+ r
n
is compact,
r
C
chosen subsequence.
'mm
implies
r
|x
+ r
m
x
Since
B
Cx
contradicts
m
zl
n
B^Cx
and
-»
0.
I
-
From (4.6)
0.
1
n
is bounded,
r B
m
Cx
B
Cx
-+
r z
m
m
-»
m
for a properly
J
z
Since
.
I
E
'
n
I
(4.8)
-*
,
1
Then from (4.2),
•* -z:
,
we conclude
+ B" C - B*
I
x
n
This implies
r
1
n
z
-»
m
should have a convergent subsequence.
x
x +B
Cx
(B+C) = B" F
Now we provide conditions on
Proposition 4.2.
'mm'
I
D4 (x)
n
I
-*
0.
Since
I
'
x
m
1—1
this
1
non-singular.
Q.E.D.
that permit us to appeal to
Recall that in the payoff function
7r
P (a,x),
player p's
own action affects his payoff both directly and through its "indirect
effect" on the vector
x.
We have already
J assumed that
»r
n
which
[a,x],
incorporates the indirect effects into the first argument, is concave in
x.
We will now also assume that the direct effects themselves are concave:
Definition
:
B
n
(x)
whose entries are
is the diagonal
operator
tb
D
aa
7r
n
(a,x)
14
Assumption
B
n
3
For all
:
B
x,
(x)
strictly negative definite, and
is
(r x) - B(x).
n
From Proposition 4.2 we see that we should also assume
Assumption
4
D^
:
(r x)
D^(x)
The condition
-+
D</>(x);
is compact.
B(x)
•
compact is a restriction on the indirect effect
B(x)
-
D<f>(x)
that players can have on each other.
In the next section we show that it is
satisfied in the case where
depends on
average value of
(a,x)
it
only through the
x
x.
Under these assumptions we can use Proposition 4.2 to show that the
Nash correspondence is lower hemi- continuous at the continuum of players
limit.
Theorem (Lower Hemicontinuitv')
Assumptions
sequence
x
D^(x)
non-singular, and
is
through 4 are satisfied, then there exists an
1
n
$(x) - 0,
If
:
such that
n > N,
,
Finally,
J
<f>
*n
(x
n
-
)
D4>
n
(r x)
B
-
n
n
x
to show
let us note a useful fact:
suffices to show that
and
-»
n
D<?S
n
x.
(r x)
n
D^(x)
-*
n
converges to
(r x)
and a
N
D^(x)
-
B(x)
it
This
.
follows from
Lemma 4.4
B
n
- B
Proof
:
;
Suppose
and
C
x
the fact that
that
IB x
'nn
- C.
n
Given
C
r
F
n
n
C.
-»
Bit
nn'
I
•
|x
'
+
n
Then
€ X*,
n
n
= B
F
n
I
n
=
C
n
;
B
n
and
B
are diagonal
°
- F.
1,
find
x e X*,
|x
1
'
Then since
0.
F = B + C,
B
and
n
Consequently
^
^
B
|F x
'
n n
I
>=
n'
1,
are diagonal
r Fx
nn'
I
-
guaranteed b^
as °
.
it follows
as required,
Q E D
.
.
.
15
5
.
Compactness With Weighted Averages
This section shows that the sufficient conditions of Section 4 are
satisfied by games in which each player cares only about a weighted average
of his opponents' actions, and not about the play of each individual opponWe will begin by considering a single weighted average.
ent.
This is, of
the case in Cournot competition with a single homogeneous good,
course,
which is the game in which lower hemi-continuity has been most extensively
Below, we extend these results to the case in which payoffs depend
studied.
on a finite number of weighted averages or even infinitely many.
A-
We now take
P -
[0,1],
and
[0,1]
P
- (0 1/n, 2/n,
,
.
.
.
,n-l/n,
1
}
We suppose that
jr£(p,a,x
h(q)xVn)
- g(p,a, 2
)
^Pn
7r(p,a,x) - g(p ,a,J"h(q)x dq)
and
is
C
1
and
P x
g:
A x A
-+
where the weighting function
,
is
IR
2
C
n < »
Assumption
1,
P
-*
IR
and concave in its second argument.
We will show that in this case Assumptions
For
h:
1
through 4 are satisfied.
concavity and continuity, follows from the
fact that
g(p,a,
2
is
C
jointly in
in
a.
For
n -
p
2
h(q)xVn
and
q,
Assumption
«>
and,
1
+ h(p)a/n)
for large enough
x s Jh(q)x dq.
theory implies
x
Notice that if
-*
x.
strictly concave
follows immediately from the properties of
For notational convenience define the averages
and
n,
x
-*
x
h(q)x /n
n
n
then ordinary integration
We now verify Assumptions
x
2
n
= 2
qeP
through
4.
We begin b}
16
computing
<j>
n
and
(x)
(x)
4>
:
*£(x) - D g(p,xP,x
(5.1)
2
^
P (x)
- D g(p,x
P
n
+ h(p)D g(p xP,x )/n
3
n
)
>
,x).
2
Now we will differentiate (5.1), and show that
as
can be decomposed
r
B
satisfies Assumptions
3
entry,
b
(p)
e
C
n
h
n
C
= c h
n n
n
D<i(x)
respectively,
in a way
that
J
J
B + C,
be the diagonal matrix whose p
B
n
- (h(p)/n) D
(p)
entry is
g(p,xP,X
n
32
).
- D
n (p)
g(p,xP,x
n
23
+ (h(p)/n)D
)
33
g(p,x£,x),
n
- (h(0)/n,h(l/n)/n,...,h(l)/n).
.
containing the
D„„g
- D
c
h
and
-D 22 g(p,xP,x n ),
(p)
In the continuum limit,
and
Let
4.
n
be the corresponding column vector, and set
c
Then
and
n
(r x)
set
,
c
Let
and
+ C
n
be the diagonal matrix whose p
E
To define
+ E
n
is
,
b
and let
n
D<4
is
Then
terms,
is as before,
B
C - ch,
and
where
g(p,xP,x)
23
the linear functional defined by
D(f>
n
=B n +E n +C,Diji>-B
n
is strictly negative definite,
Next observe that
D<ji(x)
-
Clearly
+ C.
Assumption
B - ch
hx - J"h(q)x(q)dq.
3
is
E
n
+ C
-*
n
C.
Since
E
n
B
n
-
B,
and since
B
n
satisfied.
has rank one,
To satisfy Assumption 4, we must also show that
equivalent to
the diagonal matrix
and is thus compact.
D$ (r x)
-»
converges
clearly
to
b
J
D$(x)
,
which
and is
is
17
it suffices to show
diagonal,
b
C
C
-»
n
To do this, we assume that we are given a sequence
We must construct a continuous approximation
|x
(1)
v
'
(2)
x
(3)
v
'
|C
n
-r x
x
n n
n
Recall that
C
to choose
x
n
x
- c h
n n
n
C - ch
and
Since
.
c
'
be continuous.
= x.
it will be sufficient
.
h x
n n
-
as
h
x
does with
n
has a (continuous) limit
x
x
at
).
(p-,p.,...,p
x
n
hx
-
1/n
1/n;
-»
n
n
that is,
,
x
will have
n
Along the k
n
must
n
x
n
.
To do
constant segments
centered
°
will do.
x (p) - x
segment, we will set
the area in which
,
1
2
x
then we can simply
x,
x
* x
n
n
Thus as
increasingly steeply sloped continuous function.
n n
That is,
.
Moreover,
.
then connect these segments by linear interpolation.
h x
-»
n'
to be a continuous approximation to the step-function
x
associated with
1
hx
However, we must also worry
&
J about non- convergent
this, we set
is an
such that
must yield essentially the same average with the
x
If
c
-+
n
so that the scalar difference
n
00
n
- 1,
and
does,
n
limiting weight- functional
x
n'
1.
|
- 0.
|
1
the approximating
take
x
'
|x
- 0,
|
r Cx
-
with
n
-
with
1
converges if
n
'
n n
x
x
(p,
)
We
.
grows,
x
To ensure that
must &go to zero faster than
n
[The computations are available from the authors upon
request.
Finally, we verify Assumption
|F -F'l
1
Now
c
1
h
n n
n
n
-
c'h
< sup
nn
f,
'
P eP „
1
+
(p)-b'(p)|
n K
n r
c
l
Lc°
It is clear that for
le
'
is less than
measured in the sup or
norm.
lb
1
Set
2.
norm,
'
n
and
-c'
'
|h
,
I,
n 1
'
e
(p)-e'(p)| +
n
nn ll h n'l
'
-*
- D$ (r x'),
F'
,
is
Evidently
Ic h
n n
'
where
c
'
n
-c'h I.
n n
1
is
-c'
n
1
measured in the dual or
L,
1
sup
|b
(p)-b^(p)
sup
p€P
and similarly
\c
< sup
peP
|
"
peP^
|e
(p).-e' (p)
<
|
|b(p)-b'(p)| +
e
,
e
-c'|
< |c-c'| +
-F'|
< sup |b(p)-b'(p)| + |c-c'|
ptP
respectively.
e
Consequently, we have
the uniform bound
|F
and the first half of Assumption
are continuous in
from (5.1)
+ 3e
follows from the fact that
2
The last half,
x.
Ih^
that
d>
n
(r x) - $(x)
n
and
b
is
c
immediate
.
In this case we write
Next we consider several weighted averages.
g(p,a,x
x
where
)
x
- J h (q)x^dq
similarly in the finite case.
is the i
weighted average, and
In this case
k
? - D.gCp.xP.x 1
4>
n
2& r
n n
,
.
.
.
k
,x ) +
n
1
1
)
£_
.gCp.xP.x
h (p)D.
r
l+i b r n n
x
k
n)
i-1
while only the arguments of
and
b
<$P
in (5.1) change.
The definition of
b
are unchanged, we have
e>)
-
(h^/n)
D
g(p,xP,x^,...,x
.
(1+ )2
k
)
and
k
J-l
while
C
We write
D<6
'
n
= B
(p)
n
"
+ I
2(2+i) SC-P.x
-
k
L
k
1
+ Z . C
.
E
i=l n
i=l
n
x
,
.
•
and
•
,x
Dd>
).
= B + 2. . C 1
1=1
.
The proof
19
is as in the case of a single average, with one exception:
i
tha t
c
*
c
n
Define
and
i
In
x
h) - supi
r
n
c
1-1n
• |c -r c
,c)
l-ln
n
to be the "sup" distance between
I
be the extrapolation of
n
|h x -hx
n'
x-l'nn
.n
x
c
n
described above, and set
n
to be the "linear functional"
I
,
i
.
1
C
Z.
we must show
i
distance.
1
.
'
Since
.
„k
* Z.
i
mn'nnn'"
d (c
'
d (h
.
implies that
let
c
,
.
Jc
and
- c h
n n
.
- c h
C
.
we have
,
k
k
k
1
/^nn/^nnn
)
C x
-
i-1
We showed above
C x
y
/
<
n
,
c
(that is the °
gap between
n'l «
(c
1
n
^
1
+ |c
)
l
1
i
|d.(h
'
'
n
1
,h
:
i-1
-+
)
IV^Ld
'
,
i-1
d (c
«
)
r
,
h x
n n
and it can be shown that
and
hx
n
is
uniform in
in ,h
d. (h
x
n
)
-
and
),
This reasoning may also be extended to a function of infinitely many
averages
g(p,x ,x ,x ,...)
converge absolutely.
provided that when
In other words,
[3/4,1]
tions in
h
2
is
uniform on
and so forth.
x
[1/2,1],
In this case
above,
the sums
the later averages in the sequences
have relatively little impact on utility.
is uniform;
k - «
g
An important case is where
h
3
on
h
[1/4,1/2],
depends on all of
x,
4
h
on
but varia-
over very small intervals do not make very much difference.
20
References
Scheinkman [1977], "Smoothness, Comparative Dynamics and
Araujo, A., and J.
the Turnpike Property," Econometrica
Fudenberg, D., and
Levine [1986],
D.
.
45
(April),
601-20.
"Open-Loop and Closed-Loop Equilibria
in Dynamic Games with Many Players," forthcoming, Journal of Economic
Theory
Green,
.
[1980],
E.
"Noncooperative Price Taking in Large Dynamic Markets,"
Journal of Economic Theory
Kehoe, T.
,
.
52,
975-94.
Levine, A. Mas-Colell and W. Zame [1986],
D.
"Determinacy of
Equilibrium in Large Square Economies," mimeo, MSRI
Kehoe, T.
,
Levine, and
D.
P.
Romer [1987],
"Steady States and Determinacy
with Infinitely-Lived Consumers," mimeo, Rochester.
Mas-Colell, A.
.
Novshek, W.
"The Cournotian Foundations of Walrasian Equilibrium
An Exposition of Recent Theory," Ch.
Theory:
Theory
[1982],
W.
,
Hildenbrand (ed.), New York:
and H. Sonenschein [1980],
6
in Advances in Economic
Cambridge University Press.
"Small Efficient Scale as a
Foundation for Walrasian Equilibrium," Journal of Economic Theory
.
22,
243-56.
Roberts, K.
"The Limit Points of Monopolistic Competition," Journal
[1980],
of Economic Theory
361
i
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22,
256-78.
OQk
Date Due
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