# Discrete Fixed Point Theorems and Simplicial Decompositions

Electronics - Devices

Oct 10, 2013 (4 years and 7 months ago)

83 views

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)