Fixed Point Theorems

1
Fixed Point Theorems

• James Neel
• May 27, 2004

You Can't Comb the Hair on a Coconut Without
There Being a Whorl.
2
Presentation Overview

• Brouwer
• Kakutani
• Glicksburg-Fan-Debreu
• Nash
• Tarski
• Zhou
• Banach

3
Fixed Point Definition

• Given a mapping a point
  
• is said to be a fixed point of f if

In 2-D fixed points for f can be found by
evaluating where and
intersect.
1
How much information do we need to have to know
that a function has a fixed point/Nash
equilibrium?
f(x)
x
1
0
4
Visualizing Fixed Point Existence

• Consider continuous
• X compact, convex
• Fixed Point must exist

1
f(x)
x
1
0
5
Visualizing Fixed Point Existence

• Consider continuous
• X not compact, convex or X compact, not convex
• Fixed point need not exist

1
f(x)
x
1
0
6
Brouwers Fixed Point Theorem
Let f X? X be a continuous function from a
non-empty compact convex set X ? ?n, then there
is some x?X such that f(x) x. (Note
originally written as f B? B where B x ? ?n
x?1 the unit n-ball)
7
Visualizing Fixed Point Existence

• Consider f X? X as an upper semi-continuous
  correspondence
• X compact, convex

1
f(x)
x
1
0
8
Kakutanis Fixed Point Theorem

• Let f X? X be a upper semi-continuous convex
  valued correspondence from a non-empty compact
  convex set X ? ?n, then there is some x?X such
  that x ? f(x)

9
Quasi-concavity
A function f X ?X, X?? is quasi-concave if
for all ??0,1, x, y ? X
Upper Level Set
where a??
Equivalent definition of quasi-concave
f is quasi-concave if Pu is convex for all a
10
Glicksberg-Fan (Debreu) NE Existence

• Given
• is nonempty, compact, and
  convex
• ui is continuous in a, and quasi-concave in ai
  (implies BR A?A is upper-semi continuous)
• Then the game has a Nash Equilibrium

Note that these conditions are just an
application of Kakutanis. There also exists a
non Euclidean GF NE theorem.
11
Mixed Strategies (1/2)
Ai (Finite) Set of (Pure) Strategies (Actions)
Available to Player i
A mixed strategy of player i, ?i, is formed by
assigning a probability, pk, to each ai,k ?Ai.
Probability Assignment Rules 1. If Aim,
then 2. pk ?0,1
Support of ?i All ai,k for which pk gt 0
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Mixed Strategies (2/2)
?(?1, ?2,, ?n)
Mixed Strategy Profile
?(Ai) set of probability distributions for i
over Ai
Given a mixed strategy profile (?1, ?2, , ?n)
the probability that a particular action tuple a
(a1, a2, , an) will occur, p(a), is formed
from the product of the probabilities assigned to
(a1, a2, , an) by (?1, ?2, , ?n)
Ui(?) Expected utility to i for a mixed strategy
profile ?
ui the utility function of i in the strategic
form game occurring in each stage
13
A Little Math
Definition Concave Function f X?? is
concave if for any x, y ?X
for all ??0,1.
Note this is kinda (quasi- ha ha ha) similar to
the first definition of quasi-concave.
In fact any concave function is also
quasi-concave.
14
Key Properties of Expected Utility
Every players expected utility function is
multilinear. (O R)
This means, for any strategy profile ?, any mixed
strategies ?i, ?i of Player i, and any number
??0,1
Thus the following can be said 1. Ui is linear
(in its own probabilities) 2. Ui is thus
continuous (a linear function is continuous) 3.
Ui is concave in the mixed strategy set of i
(satisfies definition by equality) 4. Ui is
quasi-concave in the mixed strategy set of i
(concave is also quasi-concave.)
15
Mixed Extension to a Strategic Form Game
Consider a strategic form game
A mixed extension to G is given by
?(Ai) set of probability distributions over Ai
?i ? ?(Ai) Ui Expected utility to i for mixed
strategy profile ?
16
Nash
Every finite strategic form game has a
mixed-strategy equilibrium.
By introducing mixed strategies, X, the strategy
space (not the action space) becomes convex, and
ui is continuous in s, and quasi-concave in
si. Thus Nash is a special condition of
Glicksburg-Fan (Debreu)
17
Sublattice
Consider Si to be a subset (maybe convex) of
. Form S as
where
Definition Sublattice
A set is a sublattice if it is a partially
ordered (?) subset of and if the
operations ? and ? are closed on S. (i.e. if s,
s ? S then s ? s ? S and s ? s ? S)
Sublattice?
S (0,0), (0,0.5), (0.5,0), (1,0), (0,1)
no
yes
S (0,0), (0,1), (1,0), (1,1)
Sublattice property Every bounded sublattice
has a greatest and least element.
18
Visualizing Fixed Point Existence

• Consider f X? X as a nondecreasing function
• X compact, though not necessarily convex

1
f(x)
x
1
0
19
Lattices
(Copy-Pasted Pasted from my supermodular games
survey)
20
Complete Lattice

• If X is a lattice and every nonempty subset has a
  supremum and an infimum, then X is a complete
  lattice.
• Note that compact lattices are complete lattices.
  
• Note that compact sets are lattices so compact
  sets are complete lattices.

21
Tarskis Fixed Point Theorem
Copy-Pasted from Topkis
22
Visualizing Fixed Point Existence

• Consider f X? X as a nondecreasing
  correspondence
• X compact, though not necessarily convex

1
f(x)
x
1
0
23
Zhous Fixed Point Theorem
(Copy-Pasted from Topkis)
Note that Tarski is just a special case of Zhou.
24
Topkis NE Existence
(Copy-Pasted Pasted from my supermodular games
survey based on Topkis)
According to Topkis, this game then must have at
least one NE. Note that the supermodularity
requirement causes the BR function to be
a nondecreasing correspondence. Hence, this is
just an application of Zhou.
25
Contraction Mapping

• Let be a metric space,
  is a contraction if there is a
  such that

Consider a 2-D metric space with the usual
Euclidian metric and several iterations of a
contraction mapping.
26
Banach Fixed Point Theorem

• Let be a complete metric space, and
  be a contraction. Then f has a
  fixed point. Further, this fixed point is unique.
• So if BR A?A is a contraction, then not only is
  there a NE, but its a unique NE.

27
A Short Review Quiz
Given properties of a function f X? X and
properties of X identify if f has a fixed point.
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Fixed Point Guaranteed?
X
f
U.S.C.
Compact
No
X must also be convex.
Yes
Lattice
This lattice is a metric space
No
Continuous
Lattice
A lattice is not necessarily convex
Monotonically Decreasing
Lattice
No
f must be nondecreasing