T: X .... Y will be called closed if its graph B(T) == (x, y) Ix E X, Y E T(x») is a closed subset of X X Y â¢. Let X be a normal topological space. If for every normal ...
PURE STRATEGY NASH EQUILIBRIUM POINTS AND THE LEFSCHETZ FIXED POINT THEOREM by Leigh Tesfatsion Discussion Paper No. 75-60, September 1975
Center for Economic Research Department of Economics University of Minnesota Minneapolis, Minnesota 55455
ABSTRAcr
A pure strategy Nash equilibrium point existence theorem is established for a class of
n-person
disconnected) strategy spaces.
games with possibly nonacyclic (e.g., The principal tool used in the proof
is a Lefschetz fixed point theorem for multivalued maps which extends the well known "Eilenberg-Montgomery fixed point theorem" to nonacyclic spaces.
Special cases of the existence theorem are also discussed.
PURE STRATEGY NASH EQUILIBRIUM POINTS
AND
THE LEFSCHETZ FIXED POINT
THEoREM*
by Leigh Tesfatsion
1.
INTRODUCTION
A pure strategy Nash equilibrium point existence theorem is estab1ished for a class of
n-person
disconnected) strategy sets.
games with possibly nonacyc1ic (e.g.,
The principal tool used in the proof is
a Lefschetz fixed point theorem for mu1tiva1ued maps, due to Ei1enberg and MOntgomery, which extends their better known "Ei1enberg-Montgomery fixed point theorem" (EM!) [9, Theorem 1, page 215J to nonacyclic spaces. Special cases of the existence theorem are also discussed. A number of economists and game theorists investigating pure strategy solution concepts have used the EMT [see 1, 6, 7, 17J.
On the other hand,
Lefschetz fixed point theorems do not seem to have appeared in either the economic or game theory literature. The Lefschetz approach to fixed point theorems may ultimately prove to be of particular importance in economic and game theory for two reasons:
generality of spaces which can be
considered; interesting related questions which can be investigated. Concerning the first reason, the objective of many fixed point theorems (e.g., the EMT) has been to establish that certain spaces have the fixed point property with respect to a class of maps which includes all continuous maps.
Since disconnected spaces do not have the fixed
point property with respect to all continuous maps, the hypotheses in
*Research underlying this paper was supported by National Science Foundation Grant GS-31276X.
2
these theorems generally include an acyclicity restriction (e.g., convexity) which implies connectedness.
In contrast, the Lefschetz fixed
point theorem is basically a statement about homotopy classes of maps rather than the class of all continuous maps; hence more general spaced (e.g., disconnected) can be considered. Concerning the second reason, the Lefschetz approach to fixed point theorems leads naturally to the concept of the "Nielsen number" of a map
f: Y .... Y ,
of fixed points of
a homotopy-invariant lower bound for the "number" f
[see 11, section 3; and 5, Chapters VI-VlI].
The Nielsen number provides a lower bound for the number of Nash equilibrium points in certain
n-person
games (see the proof of 2.8, below,
where a one-to-one correspondence is established between the pure strategy Nash equilibrium points of an map).
n-person
game and the fixed points of a
Secondly, for sufficiently restricted spaces and maps, a converse
to the Lefschetz fixed point theorem can be obtained [see 11, section 3; and 5, Chapter VllI J. 2.
THE EXISTENCE THEOREM
A number of needed intermediary results will be established prior to the existence theorem (2.8).
The following definitions and conventions
will be used. 2.1
DEFINITIONS AND CONVENTIONS.
A standard
normal form is given by a vector
(D.~En*@., D.~En*u.: D.~ En'*@~..... ~ ~
Rn)
n-person
game in
3
where 8
i
n* == [1" ,n}
[9 i , ••• }
=
is the
ith
(e{, ••• ,
is the
ith
E
I1 8 i
i
E
n* ,
Ui : IT *8 - R
player's strategy set and
player's objective function.
9~
if for each
is the player set, and, for each
n
i
A joint strategy
is a pure strategy Nash equilibrium point
i e n'l( ,
U.;(9 ', ••• , e.' l' 9., e~+l' ••• , 9') ::;; U.(9 ', ••• , 9~, ••• , 9~ , ... 1 ~~ ~ n ~ 1 ~ n
y
Cech homology based on all open coverings with coefficients in a fixed field
F
[see 10, Chapter IX] will be used throughout this paper,
except where otherwise specified. is acyclic with respect to the Cech homology for
~
Let
Y
Cech homology over
F-vector spaces
y
H. (Y ; F) ;; 0
A nonempty compact Hausdorff space
i
=I: 0
F
(C
F
- acyclic)
[Hi (Y; F)} satisfy
Ho (Y ; F) ;;; F
f: Y - Y y
denote the
provided
For each integer
n,
F-homomorphism induced by
let f
be a v
(f*)n: Hn(Y ; F) - Hn(Y ; F)
[see 10, Chapter IX, Section
y
If
Hn (Y ; F)
is finitely generated, define
trace of the matrix associated with of
Y.
(f)
*
Trace (f*)n
to be the
with respect to any basis n
H (Y; F) •
n
Let
T: Y ..... Y
Let
B(T) == [(x, y)
be a nru1tiva1ued map on a compact Hausdorff space
Define continuous maps
and
•
be a compact Hausdorff space and let
continuous map.
Y
I xeY,
yeT(x)}
r: B(T) - Y
r(x, y)
y
and
denote the graph of t: B(T) ..... Y
t(x, y)
=x
by
T.
4J.
4
Then the Lefschetz number the field
F
L(T; F)
of the map
T with respect to
is defined to be
L.
(_l)i Trace (r
1.
*
whenever the sum exists and is finite. Let
X and
Y be compact Hausdorff spaces.
T: X .... Y will be called closed if its graph is a closed subset of Let Y,
B(T) == (x, y) Ix
E
X, Y
E
X XY •
X be a normal topological space. If for every normal space
every closed subset
exists an extension of
A of f
Y,
and every map
to a neighborhood of
is called an absolute neighborhood retract Remark. example,
A multi valued map
f: A ..... X there
A in
Y,
then
X
(ANR).
This definition of ANR is not completely standard.
For
Hu [14 J uses "ANR" for spaces having the (above) extension
property with respect to metrizab1e rather than normal spaces Nevertheless, for compact metrizable (hence normal) spaces
Y.
X,
Hu has
shown [14, 5.2 - 5.3, pages 93 - 94] that his definition and the above definition are equivalent. Conventions.
The word "group" will always mean "abelian group."
The term "compactum" (plural "compacta") will be used for "compact metrizab1e space."
All products of topological spaces are assumed to
carry the product topology.
References to the coefficient field
F
will hereafter be suppressed. v
Many of the sources referred to below do not use Cech homology; for example, Eilenberg and Montgomery [9J use Vietoris homology,
T(x»)
5
Begle [2, 3J uses generalized Vietoris homology, and Dold [8J uses singular homology.
The following proposition clarifies the relation-
ship among these homology theories and justifies references to these various sources in later parts of the paper. 2.2
PROPOSITION. y
(i)
Vietoris and Cech homology groups are isomorphic y
over compacta, and generalized Vietoris and Cech homology groups are isomorphic over compact Hausdorff spaces, with respect to any coefficient group; (ii)
" Cech homology over compact Hausdorff spaces
satisfies the seven axioms of the EilenbergSteenrod system [see 10J with respect to any coefficient field (considered as a vector space over itself); y
(iii)
Cech and singular homology groups over ANR compacta are isomorphic with respect to any coefficient field.
Proof.
The first part of statement (i) follows from Theorem 26.1
[15, page 273 J and the second part is proved in Ref. 3 [pages 536 - 537J. Statement (ii) is proved by Eilenberg and Steenrod [10, Chapter IXJ. For any ANR compactlUD. M,
if
H'
and
H' ,
are two homology
theories satisfying the seven Ei1enberg and Steenrod axioms with isomorphic coefficient groups to
H~'(M;
G")
G'
and
for all integers
G"
n
,
then
H ' (M ; G ') n
is isomorphic
[24, Theorem 7.3, page 14lJ.
6
Thus statement (iii) follows from statement (ii) and Chapter VII in Ref. [10J, where it is proved that singular homology satisfies the seven Ei1enberg and Steenrod axioms for arbitrary coefficient group.
Q.E.D.
2.3 LEMMA.
Let
Y be a compact Hausdorff space such that the
homology vector spaces i
v
are finitely generated for each integer
H. (Y) ~
and vanish for sufficiently large
mu1tiva1ued map such that
~ech
T(y)
is
i.
Let
T: Y
Y be a closed
-+
C-acyclic for each
y
Y.
E
Then
exists and is finite.
L(T)
Proof.
Let
= [(x,
B
and let continuous maps
([x} X (T(x))
~
=y
T(x)
T(x)
E
Y, y
E
r: B .... Y and
r(x, y) Since by assumption
y)lx
is
is
T(x)}
denote the graph of
t: B .... Y be defined as above by
=x
t(x, y)
C-acyclic for each
C-acyclic for each
x
E
x E Y.
Y,
t
-1
-1
hence
(t,.() i
H(B)
By the Vietoris-
v
H. (Y) ~
into itself for each integer v
Since by assumption
v
isomorphically onto
is a well defined single-valued map and
a homomorphism of
=
(x)
v
Beg1e Mapping Theorem [3J,
T
(r
0
,~
i
t
H(Y);
-1
*
is
).
~
•
has a finite basis for each integer
H. (Y) ~
y
i
and
H. (Y) = 0 ~
for sufficiently large
well defined for each L(T)
i
Trace (r
*
0
t -1) .
*
and vanishes for suffiCiently large
= L. (- l)i Trace (r* t- 1). ~
i,
0
*
~
~
is
i.
exists and is finite.
Q.E.D.
Thus
7 2.4
COROLLARY.
be as in 2.3.
Let
Then
be an
M
L(T)
ANR
compactum and let
T: M .... M
exists and is finite.
By Corollary 7.2 [14, page 141J and 2.2 (ii) , these exists
Proof.
v
m
an integer and
v
=
H (M) n
such that for
0
H (M) n
is finitely generated for every
The claim n9w follows from
n>m.
n~m
2.3. Q.E.D.
2.5 and let
COROLLARY. T: Y .... Y
Let
Y
be a
be as in 2.3.
C-acyc1ic compact Hausdorff space, Then
L(T) = 1 •
v
Proof.
Ii
(Y)
By
C-acyclicity of
has only one generator.
o is well defined. zero map for map.
and
(r*
H (Y) n
By 2.3,
In particular,
n > 0
Y,
0
(r *
-1 t*)
0
=0
L(T) = t
-1 *
)
v
for
L
n > 0
(_l)i Trace (r
~
*
y
: H (Y) .... H (Y) n n n
v
: H (Y) ....
Ii
000
and
(Y)
0
1 t- )
*
is the
is the identity
Hence
L(T) = Trace (r*
0
-1 t* )0 = 1 Q.E.D.
2.6 and let
THEOREM [9, Theorem 5, page 217J.
M be an
T: M .... M be a closed mu1tiva1ued map such that
C-acyclic for each such that
Let
x
Remarks.
E
x EM.
If
L(T) 1= 0,
ANR
T(x)
then there exists
compactum is x EM
T (x) • Beg1e [2J has generalized 2.6 to compact homologically
locally connected spaces;
R. B. Thompson [see llJ has generalized 2.6
8 (for single-valued maps
T) to compact spaces which admit a "weak
semicomp1ex structure;" and A. Granas and L. Gorniewicz [12 J have generalized 2.6 to topologically complete
ANR's.
See also [16J
and [18J for additional generalizations.
2.7
COROLLARY (EMT)
C-acyc1ic
ANR
map such that x e M
[9, Theorem 1, page 215J.
compactum and let T(x)
is
M be a
Let
T: M .... M be a closed nru1tiva1ued x eM.
C-acyclic for each
Then there exists
x e T(x) •
such that
A C-acyc1ic, compact Hausdorff space is connected [15,
Remark.
11.18, page 261J.
Proof.
By 2.5,
L(T) = 1.
Thus 2.7 follows from 2.6.
Q.E.D •
2.8
(II
-
II
@. ,
n*
U. : 1.
n*
1.
II
@.
n*
.... Rn)
1.
be an
If
n-person game. 1)
r
Let
THEOREM.
II
n*
@.
ANR
is an
compactum;
1.
2)
u.1. :
nn-l, @.
3)
T(e)
is
.... R
is continuous for each
II
Ti (8 ')
=
[8*i
E
@.
1.
max U (8{, i 8. e@. 1.
for each
i
.....
E
II
... , e: l' 8.* , 8i,+ 1 , ... , 8~ ... , 8~ l' 8i , 8;+1' ... , 8n')}
1.
e n"
n* ;
1.
@.IU. (8{, 1.
e
E
@. where n* 1. is given in terms of
C-acyc1ic for each
T. : II @..... II n* 1. n* 1. n* its coordinate maps by
T ==
i
1.
for each
1.-
1.-
1.
9
4) then
r
L(T)
0,
where
T
is as in condition 3) ,
has at least one pure strategy Nash equilibrium point.
Proof. map.
'1=
Let
It will first be shown that Y == 8
and
X ==
82
T 8 • n
X ••• X
for each
Y X X
Since by assumption
U 1
Let
T*: X
is continuous over
(T ) 1
= Y X (graph
(T*)) ,T
Similarly,
T , ••• , Tn 2
T == IT ~T.
is well defined and closed.
n~
1.
-+
Y
and
Y
1
are
X
is a compact subset
be given by
Y X X ==
IT 8., T* is n*
1.
T* is closed.
By the Maximum Theorem [4, page 116J,
well defined. Since graph
x EX.
Then
Y X [x}
compact Hausdorff spaces; in particular, of
is a well defined closed
is well defined and closed.
are well defined and closed.
Hence the product
Combining conditions 1), 3), 4), and the above paragraph, it follows from 2.6 that there exists definition of
T,
e' e IT
8 n* i this implies that
such that
r
e'
E
T (e ')
•
has at least one pure strategy
Nash equilibrium point.
2.9
COROLLARY.
n-person game.
then
By
Q.E.D. Let
r
==
(n''n* 8.1. ,IIn* U.: IIn* 8.1. 1.
-+
Rn)
be an
If
1')
IIn'>'( 8.1. is a C-acyc1ic ANR compactum;
2')
Conditions 2) and 3) in 2.8 are satisfied,
r has at least one pure .strategy Nash equilibrium point. Proof.
By 2.5,
L(T)
= 1. The claim follows from 2.8 • Q.E.D.
10
3.
SPECIAL CASES
In this section special cases of the existence theorem 2.8 will be discussed. 3.1 of spaces
The following definitions will be used.
DEFINITIONS. A
~
X,
continuous map The space
If
then
r: X
~
i: A
~
A is a retract of
A such that roi
hood retract
X ~ Rn
(some
(ENR)
n)
who~ch
X.
A space
p E X if every open set V containing
point in
U.
p
° ~s
The space
A space
A has a
if
Y is called
h omeomorp hi ct oY. i: X
~
A
X is homotopic
X is locally contractible at a
U containing
such that
X)
A.
if there exists a neighbor-
X is contractible if the identity map
to a constant map on
set
is the identity map on
X of which it is a retract.
a euclidean neighborhood retract
point
X if there exists a
A is called a neighborhood retract (in
neighborhood in
space
X is the inclusion map for a pair
p
contains an open
V is contractible over
U to a
X is locally contractible if it is locally
contractible at every point. ANR
compacta appear in the hypotheses of both the existence
theorem 2.8 and its corollary 2.9. in economic and game theory are
Most of the spaces commonly used
ANR
convex subsets of Banach spaces are
compacta. ANR
For example, compact
compacta [14, 6.4 and 6.5,
page 96J; finite dimensional locally contractible compacta (e.g., finite discrete spaces) are
ANR
compacta [14, 7.1, page l68J; and
locally euclidean compacta (e.g., compact compacta [14, 8.3, page 98J.
n-manifo1ds) are
ANR
By a theorem of Haver [13, page 281J,
11
locally contractible compacta that are a countable union of finite dimensional compacta are
ANR
compacta.
As these examples demonstrate,
the hypotheses of 2.9 are significantly more restrictive than those of
2.8; for the
C-acyclicity restriction on the product of strategy
spaces in 2.9 implies a strong global type of connectedness, whereas the property of being an
ANR
compactum is a local property.
The following proposition clarifies the relationship between the Lefschetz number,
C-acyclicity, and contractibility.
This latter
restriction is used by Debreu (7J.
3.2
PROPOSITION.
Let
Y be a compact Hausdorff space.
Then
Y contractible; Y C-acyclic; L(f) ~ 0 for all continuous maps f: Y .... Y
Corollary 2.5 implies that the right-hand side forward
Remark.
implication can be strengthened to: closed maps
T: Y .... Y such that
Y C-acyclic
T(y)
is
~
L(T)
~
0
for all
C-acyclic for each
y
E
Y
Whether or not the converse of this stronger implication holds is apparently not known.
Proof.
By Theorem 3.4 [10, page 238J and Theorem 5.1 [10, page
240J, one point spaces are
C-acyclic and homotopic maps on a compact y
Hausdorff space induce the same homomorphism on the Cech homology groups of that space. C-acyclic.
By 2.5,
continuous map
Thus contractible compact Hausdorff spaces are C-acyclicity of
f: Y .... Y •
Y implies
L(f)
=1
for every
12 As for the converse implications, for every positive integer real projective
2n-space
RP
2n
n,
is a finite polyhedron which is
Q,
acyclic with respect to singular homology over the rational field but not contractible; and complex projective
2n-space
cp2n
is a
finite polyhedron which is not acyclic with respect to singular homology over the field
Z2
for every continuous map finite polyhedra are
of integers
f: cp2n
ANR
-+
mod 2 ,
yet
L(f; Z2)
cp2n [5, pages 31 - 32J.
=1=
0
Since
compacta, it follows by 2.2 (iii) that the
proof of 3.2 is complete.
Q.E.D.
The following proposition demonstrates that conditions 1) a.nd 1') in 2.8 and 2.9 may be replaced by restrictions on the individual strategy sets. 3.3
PROPOSITION.
M X N is an
ANR
ANRs
M and
N be
ANR
compactum; and if in addition
C-acyc1ic, then so is
Proof.
Let
compacta. M and
Then N are
M XN •
The topological product of a finite number of metrizab1e
is a metrizable
ANR
[14, 7.6, page 97J; compactness of
M XN
follows by Tychonoff's theorem. Suppose
M and
N are C-acyc1ic.
2.2 (iii), for each integer y
By 2.12 [8, page l8lJ and
n, y
y
Hn (M X N) ;; ~p+q=n Hp (M) ® Hq (N) ,
(1)
13
where
® denotes tensorproduct (with respect to
of tensorproduct, C-acyclicity of
i.e.,
0 ® F ;; 0 M and
M X N is
and
F ® F ;; F.
F) •
By definition
Hence by (1) and
N,
if
n
if
n = 0
1= 0
C-acyclic.
Q.E.D.
3.4
COROLLARY.
Condition 1) in 2.8 may be replaced by i e n*
For each
1°)
,
is an
Gl.
l.
ANR
compactum; and condition
in 2.9 may be replaced by
1 ')
For each
1*)
i
E
n*, ®
is a C-acyclic
i
compactum.
ANR
The restriction "T(e)
is
condition 3) of 2.8 may be replaced by for each
e
Proof.
e
II
®.,
n*
i
e
C-acyclic for each
E
e
II
n*
is a
"T. (e) l.
Gl."
C-acyclic
ANR
n* ."
l.
The first statement is immediate from 3.3.
in the proof of
in
l.
2.8, the coordinate maps
In particular, the image sets compact subsets of
Gl. , l.
e
for each
image sets is clear, since each
i
Gl.,
T., l.
E
i
II * Gl., n
E
n*
i
E n~'
D.
p
of
is obvious. q>
Define
q>:
is in the starkerne1 of Finally,
is a deformation of
i d
-0
q>
p
Let
in the
tE[O, 1J
is well defined.
0) - id(·)
into
D.
D by
deD,
D,
q> (.
into a point
D
D X [0, 1 J
(d, t) = [1 - tJ d + tp,
Since
Hence
id: D -0 D be the identi ty map on
be the constant map taking
starkerne1 of
q>
Every starconvex set is contractible.
g;
and i.e.,
q> (. ,
Continuity 1) == g(.)
D is contractible
(to p) Q.E.D.
3.7
LEMMA.
Every
ENR
is an
ANR.
15
Proof.
Without loss of generality, assume
retract in
for some
D is a neighborhood
By Theorem 7.1 [14, page 168J, every
n.
metrizab1e, finite dimensional, locally contractible space is an To prove the lemma, it thus remains to show that
ANR.
D is locally
contractible. Every open Rn
• open set ~n
subset of
Rn)
n-ba11
in
n R
is contractible (3.6).
contains an open
n-ba1l,
is locally contractible.
retract of an open set in
Rn.
Since every
(and hence each open By assumption,
D is a
Since retracts preserve local con-
tractibi1ity [14, 9.2, page 26J,
D is locally contractible. Q.E.D.
3.8
Let
n D~ R
be a compact starconvex set whose
contains an
n-ba1l
B.
PROPOSITION.
starkernel
D*
Then
D is a
C-acyc1ic
ANR
comp ac tum.
Proof.
After parallel translation and multiplication by some
positive constant, it can be assumed that the standard ball distance.
n B == [x
E
n R III x II
Then for every
in the interior
DO
of
d
D ,.
by projecting the interior Bn ~ D*, from 0
this cone lies in
E
s;
1} , where
D and for
rd
(Bn) 0
of
D.
o s;
n-ball
II • II
r < 1 ,
B ~ D*
is the
denotes euclidean the point
rd
lies
lies in the open cone obtained n B
from
d •
By compactness of
Since by assumption
D,
every ray
must therefore contain exactly one point on the boundary
CD
of
D.
16 Since
n 0 E B ~ D~( ~ D,
is compact as a closed subset of (x
E
n R III x
II
=
1}
is bounded away from
cD
D•
The map
f :
(j) ....
o ,. Sn-1
and
aD
-
given by F(d)
= d/1ldll
is therefore a well defined homeomorphism.
By radial extension, one
then obtains a homeomorphism (3)
given by F(rd) = rdl
I d II '
Every space homeomorphic with Rn,
hence an
3.7 that
ENR
D is an
dEaD,Os:rs:1 Bn
is a neighborhood retract of
[8, 8.5, page 81]. ANR.
By 3.6 and 3.2,
It thus follows from (3) and D is
C-acyc1ic. Q.E.D.
17 REFERENCES 1.
Arrow, K. J., and G. Debreu, 1954, Existence of an Equilibrium for a Competitive Economy, Econometrica 22, No.3, 265-290.
2.
Begle, E. G., 1950, A Fixed Point Theorem, Annals of Mathematics 51, No.3, 544-550.
3.
Begle, E. G., 1950, The Vietoris Mapping Theorem for Bicompact Spaces, Annals of Mathematics 51, No.3, 534-543.
4.
Berge, C., 1963, Topological Spaces (The MacMillan Company, N. Y.).
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