APPLICATIONS OF.A GENERALIZATION OF A SET
INTERSECTION THEOREM OF VON NEUMANN
by
Robert M. Freund
Sloan W.P. No. 1528-84
February 1984
1
Abstract
In his paper on the expanding economic model, von Neumann developed a
set intersection theorem that asserts that the intersection of two special
sets in the cross-product of two convex sets cannot be empty.
This theorem
is generalized in two ways, first by expanding the number of sets involved
beyond two, and then by relaxing the restriction of cross-products.
The
generalization of the theorem yields a direct proof of the existence of
anequilibrium in an n-person noncooperative game, and a direct proof of the
existence of a +-flexible solution to a system of linear inequalities.
Keywords
set intersection, fixed point, game theory, flexible solutions,
noncooperative game.
I.
Introduction
A cell is defined to be a nonempty closed and bounded convex set.
S e Rm l and
of R such that for each
for each
T, {x
y
Rml
x
Let
R = S x T.
T E em 2 be cells and let
U
nd V be subsets
is a cell, and
U(x) = {y E Rm21 (x, y) E U
S,
(x, y) E V
is a cell.
Let
von Neumann's set
intersection theorem [2] states that under the above conditions, U n V
0.
This theorem was used to prove the existence of equilibria in von Neumann's
expanding economic model [2], and can be used to demonstrate the existence
of an equilibrium in a two-person noncooperative game.
In this note,
von Neumann's theorem is generalized in two ways, first by expanding the
number of sets involved beyond two, and then by relaxing the condition that
This generalization yields a direct proof
be a cross-product of cells.
R
of the existence of an equilibrium in an n-person noncooperative game.
It
also yields a proof of the existenceof a 4f-flexible solutions to systems of
linear inequalities [4].
II.
The Generalized Set Intersection Theorem
Let
pk
R eR m l xRm2 x ... xRmn
denote the projection of
m l
x ... xm
k = {(yl, ,
k
k-
closed subset of
k,
y e Ok
k = 1,..., n let
onto the coordinate space
l x Rmkk+l x ... x]Rmn , i.e.
yk-1 yn
k l
k
m
k-l
k+..
yn). there exists x em
k-l
(Y
R
For each
be a cell.
xk
k+l
y )
E R.
Let
S1
..., S
R, with the property that for each
k
k
k-1
the
= {xk
(yl theorem
he setset sk(y)
intersection
..., yk-1
The generalized set intersection theorem is:
-1-
k
k+1
x k y k+l
with
with
each be a
k = 1,..., n, and each
n)
k
S k }isacell.
i
11
II
nsk
n Sk
Under the above conditions,
Theorem
.
k=l
Note that von Neumann's theorem is a special case of the above, where
n
T c mR
Rm l,
S
2,
R = S x T, U = S2 , and V = S1
2
Proof of the Theorem: The proof follows from Kakutani's fixed point theorem,
Qk to be the projection of R
k = 1,..., n, define
For each
see [1].
onto the coordinate space IRm k, i.e. Qk = {k
y x., k+l
k
with (yl,
ERmk
yn) E R).
.,
For each x
(x1,
=
x
Rmk) that contains R.
n
: Q + Q*, where Q* is the set of cells contained
define F(x) = F(x), where
., xn )
k+
Sk(x,+ ..
Fk(x)
pk
..., xn ) E Q , define the point-to-set mapping
,
..., Fn(x))
F(x) = (Fl(x),
in Q, by
k
... ,
E
Q = Q 1 x Q2 x ... x Qn,
Let
is the smallest cross-product (of sets in
i.e. Q
y
I there exists
if x
R.
If
R,
x
x e R is the unique point in R which minimizes
the Euclidean distance II - xll 2 over all points x in R. Because each set
S k is closed, and Q is bounded, F(.) is upper semi-continuous and maps
elements of Q to cells that are subsets of Q.
point theorem, there exists
~
x E F(x) means x
i.e. I
E S
show that
for
x
Suppose
~k S 1
E S(x, ...,
n) for all k = 1, .. , n,
It only remains to
.k
xk
1
-k-1
,x
'x
112 to x is minimized over all points in R.
By a separating hyperplane thoerem, x(x - x)
x E R, and
-k-1
x
, ,
x e R, then
x is not an element of R. Let x be the unique point in R
for any
.
k+1
x
If
must be an element of R.
x E F(x) = F(x).
(I
1
(x
.k-1
x
x E F(1).
k = 1, ..., n, proving the theorem.
whose distance i x -
k
4ES(xk
x e Q such that
Thus, by Kakutani's fixed-
x(x - x) > x(x - x).
-k+l
x,
-k, -k+l
...
Since
which means that
E S
k c R, for all k =
-2-
x(3x - x)
I E F(x),
xn),
Then
1,..., n.
Therefore,
_k
k
(i1
-k
k- -k
k) <
xk(xk
k(xk - xk)
-x)
k = 1, ..., n.
for all
x(
x), whereby
-
Summing this last
x( - x) < x(x - x), wb4-h is a contradiction.
k, we obtain
inequality over
, xn )(
xk
... k+
-k-l
x e R, and so the theorem is proved.
Therefore
III. Applications of the Generalized Set Intersection Theorem
A.
N-Person Noncooperative Game Theory
n
An n-person noncooperative game is characterized by
of which has
players, each
mk pure strategies, k = 1, ..., n, and a loss function
a(j; il, ... , in)
defined as the loss incurred by player
if the k th
j
player chooses pure strategy ik , where ik e {1, ..., mk}, k = 1, ..., n.
A mixed strategy for player
k
T
k is a vector x
k - (xk 6I~m
mkl
{x
which is the standard simplex in mR k , k = 1, ..., n.
and for any strategy vector
mi
m2
x
mn
n
z a(k; i l ,
il=l i2 =1
A strategy
x e T, the loss to player k is given by the function
.
fk( )
k
xj = 1, x
. ,i X
..
-
in=l
i
1
T is said to constitute a (Nash) equilibrium if
..
xk+, x
x
,
..., xn)
for any
Tk , for
x
k = 1, ..., n.
Most proofs of the existence of an equilibrium rely directly on Brouwer's or
Kakutani's fixed-point theorem.
Below, the result is derived as a consequence
of the generalized set intersection theorem, rendering more of a geometric
interpretation to the existence of an equilibrium.
Let
Proof of the Existence of an Equilibrium.
Sk ={
-
-
R
-
fk(x)< f k(x,
R is a cell, and
.. x-k-1
Sk is closed, and
x
Sk(x
-3-
k
R = T, and let
-k+l
xn)
..., xk-1
anyx
xk+
0
Tl x Tm2 x ... x Tmk,
T
Let
mk
k
Tm k
xn) is a
III
subsimplex of
Thus the hypotheses of the set
Tk, and so is a cell.
~x
intersection theorem are satisfied, whereby there exists
V S k
Sk for
x
k = 1, ..., n.
f
,
But then
x k-l,,
.
x, x
,
...
for any
n)
Tk
x
,
Flexible Solutions to Linear Inequalities
B.
R = {x E En
Let
Ax
for a given matrix and vector (A, b), and
b}
R is bounded, whereby R is a cell.
assume that
For a given
x E R, define
k = 1, ..., n,
for
=
mki
aik>
to the boundary of
R
-
i
is said to be
,
min
aik
-aik
k(x) and sk(x) are the distances from
along the k t h coordinate in the positive and
9k-flexible if
k
sk)S+(x) =
(1 -
< (1, .... 1),
with 0
For a given vector
negative directions, respectively.
R
sk(x) =
and
)
0
where Ai. is the i th row of A.
x
,
is an equilibrium.
and so
x
R such that
k
k sk(x), for
k = 1, ...,
s+(x)
k
This corresponds to
see van der Vet [4].
k
s+(x) +sk(x)
s+(x) + s(x) # 0.
where
k = 1, ..., n,
x
If
x
is said to be a
is simultaneously
in thecase
=
k
k-flexible for all
-flexible solution.
In [4], van der Vet uses Brouwer's fixed-point theorem to prove that for
any
with
0
~ • (1, ..., 1), a
-flexible solution exists.
Herein, this
existence result is proved as a consequence of the generalized set intersection
theorem.
Proof of the Existence of a
-Flexible Solution:
Let
R be given as above,
k
and let
m 1 = m2 = ... = mn = 1.
For a given
y E k
,
Define
define
-4-
P
as in section II
for k = 1, ..., n.
min {xk
I
(Y'
--
+ k max {Xk
I
(Yl'
-'' Yk-l' Xk' Yk+l'
(Y1 ,
-,
f (y) = (1 -)
fk(y) is the value of xk such that
is
Ok-flexible.
Define
Yk-l' Xk' Yk+l'
Yk-l' fk(Y)
Sk= {(Yl ',
k
S
c R, and
k
S (y) is a single point for
k
y e pk
... ,
for
with
x E S,
k = 1, ..., n.
k = 1, ..., n , whereby
x is a
Yn)
n.
Yn)
....
Y
pk},
Because each
1
n
R and S
satisfy the hypotheses of the set intersection theorem.
x e R
-'' Yn) E R}
Yk+l'
fk()Yk+l
i.e., Sk is the graph of the function fk(.), k = 1,
yn)e RI
'
S
Thus there exists
This in turn means that
x
is
k-flexible
-flexible solution.
Remark
The geometry of the sets Sk in the flexible solution problem is similar
to that of von Neumann's set intersection theorem for the case when
n
2.
It was this similarity that motivated the development of a generalization of
von Neumann's theorem applicable to the problem of
-flexible solutions.
References
[1]
Kakutani, S., "A Generalization of Brouwer's Fixed Point Theorem,"
Duke Math. J.,
[2]
von Neumann, John,- "Uber Ein okonomisches Gleichungssystem une eine
Verallgemeinerung des Brouwerschen Fixpunktsatzes," Ergeb. Math.
Kolloquiems, 8, 1937.
[3]
[4]
8 (1941), 457-459.
, translation of [2] in Review Econ. Studies.
van der Vet, R. P., "Flexible Solutions to Systems of Linear Inequalities,"
European Journal of Operational Research, 1 4, July 1977, 247-254.
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I
1945-1946.
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