Discrete Fixed Point Theorems and Simplicial Decompositions
Hidefumi Kawasaki
Faculty of Mathematics, Kyushu University,
Fukuoka 819

0395
,
Japan
There are three types of discrete fixed point theorems:
t
ype (
M)
deals with
monotone
mapping
s
,
t
ype
(C)
deal
s with
(local)
contraction mapping
s,
and t
ype
(B)
is
based on
Brouwer's fixed point theorem.
In this talk we mainly consider t
ype (B)
and
show that
simplic
i
al decompositions of the convex hull of the domain of the mapping are
important for analyzing type
(B).
Further we
m
ake a comparative review of
types
(B)
and (M) by
applying them to bimatrix games.
Iimura [1] proposed an idea to derive a
discrete fixed point theorem from Brouwer
’
s theorem. Let
X
be a finite subset of
Z^n
,
and
f
a mapping from
X
into it
self. We denote by co
X
the convex hull of
X
.
Iimura
’
s idea
is as follows.
(1)
Give a simplic
i
al decomposition of
co
X
.
(2)
Extend
f
to a piecewise
linear mapping
g
.
(3)
Apply Brouwer's theorem to obtain a fixed point
y
*
of
g
on
co
X
.
(4)
Impose an assumption
for a vertex of the simplex including
y
*
be a fixed point of
f
.
When a s
implicial
decomposition of co
X
is given, two points
x
and
x
’
are
said to be
cell

connected
if they belong to a same simplex. Then we denote the relation by
x~x
’
.
f
is said to be
direc
tion preserving
if
(f_i(x)

x_i)(f_i(x')

x'_i)
is nonnegative for any
i=1,
…
,n
and
x~x
’
.
Iimura
[1]
gave a theorem
that any direction preserving mapping has
a fixed point.
His proof was corrected
in [2]
. The main purpose of this talk is to
characterize the
d
irection
preserving property in bimatrix games and n

person games
in
terms of best response mappings
.
[1]
T.
Iimura, A discrete fixed point theorem and its applications
(2003)
[2]
T.
Iimura, K.
Murota
,
A.Tamura, Discrete fixed point theorem reconsider
ed (2005)
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