A Note on Oddness Theorem

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Oct 10, 2013 (4 years and 29 days ago)

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A Note on Oddness Theorem


关于博弈论的奇数定理



by


Jie Li and Zeke Wang


Lingnan (University) College,
Zhongshan

University,

Guangzhou 510275, China



March

2004


Abstract

The oddness theorem dictates that almost all finite and simultaneous game
s

have a finit
e and odd
number of Nash
e
quilibria. In this
note
, we derive a sufficient condition under which the number
of Nash
e
quilibria is odd.
In doing so
, we provide
a new
version and an elementary proof of th
e

theorem. Finally, some comments and treatment on the
implicit assumption behind the theorem

are developed
.


中文摘要

博弈论的奇数定理论定,几乎所有有限同时博弈的
Nash
均衡的数目有限,并且
是奇数。本文在具体给出
Nash
均衡的数目是奇数的一个充分条件的基础上,给出奇数定理
的另一表述,并且提供一个初等的证明。最后,我们对于奇数定理的隐含假设提出讨论。




Key words:

finite and simultaneous game; reaction function; Oddness Theorem; set

of
zero

measure; a.e.

continuous distribution ass
umption

JEL classification Numbers:

A1, C7





2

1.

Introduction

Wilson’s (1971) Oddness Theorem dictates that almost all finite strategic games have a finite
and odd number of Nash equilibria. A game is called finite if the number of players is finite and
the
number of pure strategies available for each player is also finite.

In this note, we provide
a new version of
Oddness

Theorem and an elementary proof

as well,
which are
much
more concise and intuitive compared to Wilson’s original arguments with
so
-
called

almost
-
complementary paths. Furthermore, comments on the implicit assumption of
uniform distribution or at least a.e. continuous distribution
1

behind Oddness Theorem are also
developed.

The notations in this no
t
e are standard, which can be referred, for
example, to Gibbons (1992),
and Fudenberg and Tirole (1991).


2.

Cube Characterization of Finite and Simultaneous Games

Suppose

is a

simultaneous game in strategic
form, where

is a positive integer and
e
ach payoff function

is a real
-
function of the

strategic variables
.

is said to be
finite

if


are all finite sets
.

is said to be trivial if
,
,
, and
;

is non
-
trivial
otherwise
.

Obvio
usly, when

is a finite and simultaneous game,

is a finite set
. We

call


the payoff set of Game

for simplicity. Denoting by

the number of pure strategies
available for Player
, the number of the elements of the payoff set

is then
, or simply
, which
will be

called
the
pseudo
-
dimension of

in this note
.

A real
-
function

is said to be an affine transformation if
, where
,

are real nu
mbers. Furthermore,

is said to be a positive affine transformation
if
in addition
. As is well known, Nash equilibrium is
a concept
concerning the strategy
profile
s
, not
the
payoff of a game. Thus, if
we impose
a positive

affine transformation upon each



1

“a.e. continuous distribution” is an abbreviation for “almost everywhere continuous
distribution”. We will clarify the exact meaning of this term later.


3

element of the payoff set in a finite and simultaneous game, the equilibrium structure of the game
will not alter. That is, a strategy profile constitutes a Nash equilibrium under the original payoff
str
ucture if and only if it remains to be a Nash equilibrium after the transformation.

Hence,
for any
finite and simultaneous game

,

we can impose
an arbitrary

positive
affine
transformation upon all the elements of the payoff set

to make it satisfy the following
normalization
condition:

. For instance, if we let

,

,

then the real
-
function

defined by

is a
desired
positive affine transformation, which can
equilibrium
-
equivalently

transform

a

non
-
trivial
finite and simultaneous game


into the one
meeting
the normalizat
ion condition
.
To be clear
, a
transformation of a game is said to be
equilibrium
-
equivalent
if it does not
change
the equilibrium
structure of the game.

In this way, we can just focus on the finite and simultaneous games where
all the
elements of the payof
f set are between 0 and 1 in the following discussion.

Denote the set of so
-
normalized
finite and simultaneous games

of pseudo
-
dimension

by
. Let

be the
-
dimensional
,

open

unit

cube

of
the

-
dimensional Euclidean Space
.

Define the mapping


by


for any
. With the mappi
ng
,
all the
normalized
finite and simultaneous games

of pseudo
-
dimension

are characterized as
the points of the
-
dimensional open
unit
cube
.

It s
hould be noted that this characterization is not a one
-
to
-
one correspondence. For example,
a


game

with


,

and
ano
the
r


game

with



are
characterized

by the same point




4

of the 12
-
dimensional open cube
.

However,
on the other hand, it is obvious that for any point within the
-
dimensional

open unit cube, there exists a finite and simultaneous game of pseudo
-
dimension

that can be
represented by that point. Hence, the non
-
one
-
to
-
one characterization will
“fulfill”

the
-
dimensional

open unit cube and
will not change our main result that the non
-
degenerated,
finite and simultaneous games have a finite and odd number of Nash equilibria. In fact, it will
contribute to making the treatment more clear.


3.

Main Results

A

finite and simultaneous game


is said to be non
-
degenerated if
all the elements of
its

payoff set

are different from each
other; Otherwise, Game

is said to be degenerated.

Clearly, a non
-
degenerated game must be a non
-
trivial game.


Lemma 1:
A

non
-
degenerated, finite and simultaneous game


has
a finite and odd number of Nash equilibria.


Proof:

See the Appendix.


As is well known, in
the

-
dimensional Euclidean Space
, the points with two identical coordinates

constitute a
-
dimensional hyperplane
, which

Lebesgue measure in

is zero.

There
are totally

such
-
dimensional hyperplanes with two identical coordinates
in
. Since countable unions (saying nothing of finite unions) of sets of ze
ro measure remains
to be of zero measure, we know
immediately
that the subsets of

with two identical
coordinates are of zero measure in
. Similarly, the subsets of
the

-

dime
nsional open cube

with two identical coordinates are sets of zero measure of
.

By definition, a finite and simultaneous game is degenerated if it has at least two identical
coordinates when characteri
zed by a point of the cube. Thus, we get the following lemma.


Lemma 2:

Among
all
the
finite and simultaneous games

of pseudo
-
dimension

characterized by the open
-

dimensional unit cube
, the subset consisting of degenerated
games is
of
Lebesgue measure zero.


As is well known, when express
ing

all the objects under discussion as a “measurable subset”
of a Euclidean Space,
we call
a proposition valid for “almost all” the objec
ts if the set consisting
of the objects that violate the proposition is only a subset of zero measure of the underlying

5

measurable set. Thus, summarizing lemma 1 and lemma 2 leads to the following theorem.


Theorem 1:

Almost all finite and simultaneous gam
es have a finite number of Nash
equilibria, and this number is odd.


4.

The Implicit Assumption behind Oddness Theorem

Although Wilson’s Oddness Theorem and the above Oddness Theorem dictate that
almost all
finite and simultaneous game
s

have a finite and odd

number of Nash
e
quilibria
,
quite

often
we
encounter finite and simultaneous games with an even number of or infinite Nash equilibria. The
following is an easy example with an even number of Nash equilibria.

A simultaneous game has two players, where each
player has two pure strategies respectively.
The strategies for player 1 are UP and DOWN, and those for player 2 are LEFT and RIGHT.


TABLE 1.

2


LEFT

RIGHT


q

1
-

q


1


U P

p



4



3



2



2



DOWN


1
-

p



1



1



1



2


I t ’ s e a s y t o c h
e c k t h a t t h e r e a r e t o t a l l y t wo N a s h e q u i l i b r i a f o r t h i s g a me:
,
or
(UP, LEFT), and
, or
(DOWN, RIGHT).


Now let’s move on to see another example. Making a small change on the
payoff

structure
shown in
Table 1, we get a game represented by Table 2 below, where

is a small positive.


TABLE 2.

2


LEFT

RIGHT


q

1
-

q


1


U P

p

4

3

2

2


DOWN


1
-

p

1

1

1

2


I t ’ s e a s y t o c h e c k t h a t a l l t h e N
a s h e q u i l i b r i a f o r t h i s g a me a r e

and
, where
. The game has infinite Nash equilibria.


6

In spite of the conclusion of Oddness Theorem, we are frequently encountered with finit
e
and simultaneous games whose numbers of Nash equilibria are even or infinite. To explain this
puzzle, we must dig into the implicit assumption behind Oddness Theorem.

The validity of Oddness Theorem stands on an easily ignored assumption. That is, when
characterizing all finite and simultaneous games
in

a measurable set, the assumption that
the
payoffs of
these games conform to a uniform distribution or an a.e. continuous distribution within
the underlying measurable set must be implicitly embedded, with
out which Oddness Theorem
cannot hold. Specifically, the payoff of each player under each possible strategy profile is a
stochastic variable that conforms to a uniform distribution or at least an a.e. continuous
distribution
2
. It should be noted that the s
pecific probability distributions of such stochastic
variables can be different from each other. By lemma 1 we know that only when a finite and
simultaneous game is degenerated, the number of Nash equilibria may be an even number or
infinite. If we look up
on

as a
sample

space, then it can be easily shown that under the
assumption of uniform distribution or a.e. continuous distribution, the probability (or the Lebesgue
measure) that two independent stochastic variables equal to ea
ch other is zero. Because of this, the
main result of this paper should be re
-
paraphrased

as follow
s
.


Theorem 2:
Almost all finite and simultaneous games with their payoffs conforming to a
uniform distribution or an a.e. continuous distribution in the cu
be have a finite and odd number of
Nash equilibria


5.

Conclu
ding Remarks

Measurement is an extension of the concept of volume. In this sense, people can look upon a
set of zero measure as a set of volume zero. Concept of zero measure and related terminologi
es
such as “almost hold everywhere” are core concepts in real analysis, which is also of great
importance in
the application of
economics. For example, G. Debreu (1970, 1975, 1976)
incorporated these concepts into mathematical economics and solved the prob
lem of locally
uniqueness of
Walrasian
equilibrium in a competitive economy.

In this paper, we re
-
derive Oddness Theorem in a more concise and intuitive way. In doing so,
we provide a new characterization of finite and simultaneous games. Furthermore, we
also cast
some treatment on the implicit assumption of uniform

distribution or at least a.e. continuous
distribution behind Oddness Theorem, which in turn deepens our understanding about it.





2

We call a

stochastic variable conforming to an a.e conti
nuous distribution if its probability
density function is continuous almost everywhere, for example, within the interval
.

That is,
the set of discontinuous points is
of
zero measure
.





7

Appendix: Proof of Lemma 1

We just give a proof for a

game in which there are two players and each player has
two pure strategies. Two players are A and B. The pure strategies of A are U and D, and those of B
are L and R. For simplicity, we use




to denote all the 8 payoffs of the two players
under different pure strategy profiles. Suppose player A chooses U with probability
p

and D with
probability
1
-

p
, and player B chooses L with

probability
q

and R with probability
1
-

q
.




TABLE 3.

B



L

R


q

1
-

q


A




U
p



t
2


t
1



t
4


t
3





D

1
-

p



t
6


t
5



t
8


t
7




T h e e x p e c t e d p a y o f f o f p l a y e r A i s

,


f r o m w h i c h w e c a n
d e r i v e

p l a y e r A’ s r e a c t i o n f u n c t i o n

g i v e n

p l a y e r B ’ s c h o i c e
o f

a s f o l l o w s:

When




When



we have


which imp
l
ies




When




When




When



we have


which
implies


8



When




In the coordinate plane
, the shapes of t
he reaction function

can be
summarized by Figure 1 below:



q

q

q

q






p p p p


F
I
G.

1
.


Similarly, the reaction function of player B
given

player A’s choice

of

can also be
summarized by Figure 2 below:


q

q

q

q






p p

p p


F
IG.

2
.


T
he intersections of the two players’ reaction functions give the Nash Equilibria
for

the

simultaneous
game. Under the condition of non
-
degeneration, there are 16 cases of Nash
equilibria for
the game, which
can be

summarized by Figure 3 below:



9


q

q

q

q






p p p p



q

q

q

q







p p p p



q

q

q

q






p p p p



q

q

q

q






p p p p


FIG.

3
.


The number of Nash equilibria in all the 16 cases are:
3, 1
,
1, 1

1
,
1, 1
,
1

1, 1
,
3, 1

1, 1, 1
,
1
, respectively. Obviously, they are all odd number. Hence, Oddness Theorem holds for

simultaneous
games.

The key of the above proof lies in the linearity of the reaction function with respect to

the
strategy probabilities. Therefore, when discussing the number of Nash equilibria for a finite and
simultaneous game, increasing the number of the players or the number of the pure strategies
available for each player will not alter the basic result of

Oddness Theorem, but increases only the
technical difficulty.

We will present a full proof somewhere else.



10

Reference
s

Debreu, G.,

Economies with a finite set of equilibria,
Econometrica
,
38
(
1970
),
387
-
392
.

Debreu, G.,

The Rate of Convergence of the Cor
e of an Economy,
Journal of Mathematical
Economics
, 1975, 2, 1
-
7.

Debreu, G., Regular Differentiable Economies,
American Economic Review
, 66

(
1976), 280
-
287.

Fudenberg, Drew & Jean Tirole,
Game Theory
, MIT Press, 1991.

Gibbons, Robert,
Game Theory for Appl
ied Economists
, Princeton University Press, 1992.

Wilson, Robert, Computing equilibria of N
-
person games,
SIAM J. Appl. Math.
, 21 (1971), 80
-
87.






































11



题目:
关于博弈论的奇数定理



中文摘要

博弈论的奇数定理论定,几乎所有有限同时博弈的
Nash
均衡的数目有限,
并且是奇数。本文在具体给出
Nash
均衡的数目是奇数的一个充分条件的基础上,给出奇数
定理的另一表述,并且提供一个初等的证明。最后,我们对于奇数定理的隐含假设提出讨论。


Abstract
The oddness theorem dictates that almost all finite and simultaneous game
s

have a finite and odd number
of Nash
e
quilibria. In this
note
, we derive a sufficient condition
under which the number of Nash
e
quilibria is odd.
In doing so
, we provide
a new
version and an
elementary proof of th
e

theorem. Finally, some comments and treatment on the implicit
assumpti
on behind the theorem

are developed
.