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
.
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