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 games have a finite and odd
number of Nash equilibria. In this note, we derive a sufficient condition under which the number
of Nash equilibria is odd. In doing so, we provide a new version and an elementary proof of the
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 assumption
JEL classification Numbers: A1, C7
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 distribution1 behind Oddness Theorem are also
developed.
The notations in this note 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 G  {S1 ,
, S n ; u1 ,
, un } is a n  person simultaneous game in strategic
form, where n is a positive integer and each payoff function ui is a real-function of the n
strategic variables s1  S1 ,
S1 ,
, Sn
u j ( s1 , s2 ,
are
all
finite
, sn  Sn . G  {S1 ,
sets.
G  {S1 ,
, S n ; u1 ,
, S n ; u1 ,
, sn )  c j , si  Si , i  1, 2,
, un } is said to be finite if
, un }
is
said
, n , and j  1, 2,
to
be
trivial
if
, n ; G is non-trivial
otherwise.
Obviously,
when
P(G )   u j ( s1 , s2 ,
G  {S1 ,
, S n ; u1 ,
, sn ) : si  Si , i  1, 2,
, un }
is
a
, n ; j  1, 2,
finite
and
simultaneous
game,
, n  is a finite set. We call
P (G ) the payoff set of Game G for simplicity. Denoting by k i the number of pure strategies
available for Player i , the number of the elements of the payoff set P (G ) is then
K ( P(G))  n k1
G  {S1 ,
, S n ; u1 ,
kn , or simply K , which will be called the pseudo-dimension of
, un } in this note.
A real-function f : R  R is said to be an affine transformation if f ( x)  a x  b , where
a , b are real numbers. Furthermore, f : R  R is said to be a positive affine transformation
if in addition a  0 . As is well known, Nash equilibrium is a concept concerning the strategy
profiles, not the payoff of a game. Thus, if we impose a positive affine transformation upon each
“a.e. continuous distribution” is an abbreviation for “almost everywhere continuous
distribution”. We will clarify the exact meaning of this term later.
1
2
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
structure if and only if it remains to be a Nash equilibrium after the transformation.
Hence, for any finite and simultaneous game G , we can impose an arbitrary positive affine
transformation upon all the elements of the payoff set P (G ) to make it satisfy the following
normalization condition: 0  u j ( s1 , s2 ,
 j  1, 2,
, sn )  1 ，
, n ; and  si  Si , i  1, 2,
m  m (G )  min u j ( s1 , s2 ,
, sn ) : si  Si , i  1, 2,
M  M (G )  max u j ( s1 , s2 ,
then the real-function
, n . For instance, if we let
, sn ) : si  Si , i  1, 2,
, n ; j  1, 2,
, n ; j  1, 2,
,n ,
, n ,
fG : R  R defined by fG ( x )  ( x  m  1) / ( M  m  2) is a
desired positive affine transformation, which can equilibrium-equivalently transform a non-trivial
finite and simultaneous game G into the one meeting the normalization 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 payoff set are between 0 and 1 in the following discussion.
Denote the set of so-normalized finite and simultaneous games of pseudo-dimension K by
G K . Let I K  {x  R K : 0  xl  1; l  1,
, K } be the K -dimensional, open unit cube of
K
the K -dimensional Euclidean Space R . Define the mapping
 :G K  I K
by
 (G )  {(u j ( s1 , , sn ) : s1  S1 , , sn  Sn ; j  1, , n }
for any G  {S1 ,
, Sn ; u1, , un }G K . With the mapping  : G K  I K , all the normalized
finite and simultaneous games of pseudo-dimension K are characterized as the points of the
K -dimensional open unit cube I K .
It should be noted that this characterization is not a one-to-one correspondence. For example,
a 2  3 2 game G with
P(G ) { 0 . 1 , 0 . 2 , 0 . 3 ;
0 . 4 , 0 . 5 , 0 . 6 ; 50; . 1 05 ., 405. ,2 05 ., 505. ,3, 0 . 6 5 }
and another 2  2  3 game G ' with
P(G ')  {0.1, 0.2; 0.3, 0.4; 0.5, 0.6; 0.15, 0.25; 0.35, 0.45; 0.55, 0.65}
are characterized by the same point
(0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65)
3
of the 12 -dimensional open cube I 12 .
However, on the other hand, it is obvious that for any point within the K -dimensional
open unit cube, there exists a finite and simultaneous game of pseudo-dimension K that can be
represented by that point. Hence, the non-one-to-one characterization will “fulfill” the
K -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 G  {S1 ,
all
the
, S n ; u1 ,
elements
P(G )  u j ( s1 , s2 ,
, un } is said to be non-degenerated if
of
, sn ) : si  Si , i  1, 2,
its
, n ; j  1, 2,
payoff
set
, n  are different from each
other; Otherwise, Game G 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 G  {S1 ,
, S n ; u1 ,
, un } has
a finite and odd number of Nash equilibria.
Proof: See the Appendix.
As
R K  {(t1 ,
is
well
known,
, t K ) : tk  R, k  1,
in
the
K
-dimensional
Euclidean
Space
, K } , the points with two identical coordinates tk1  tk2
K
constitute a ( K  1) -dimensional hyperplane, which Lebesgue measure in R is zero. There
are totally K !/( K  2)!2! such K  1 -dimensional hyperplanes with two identical coordinates
K
in R . Since countable unions (saying nothing of finite unions) of sets of zero measure remains
K
to be of zero measure, we know immediately that the subsets of R with two identical
K
coordinates are of zero measure in R . Similarly, the subsets of the K - dimensional open cube
I K with two identical coordinates are sets of zero measure of I K .
By definition, a finite and simultaneous game is degenerated if it has at least two identical
coordinates when characterized by a point of the cube. Thus, we get the following lemma.
Lemma 2: Among all the finite and simultaneous games of pseudo-dimension K
K
characterized by the open K - dimensional unit cube I , the subset consisting of degenerated
games is of Lebesgue measure zero.
As is well known, when expressing all the objects under discussion as a “measurable subset”
of a Euclidean Space, we call a proposition valid for “almost all” the objects if the set consisting
of the objects that violate the proposition is only a subset of zero measure of the underlying
4
measurable set. Thus, summarizing lemma 1 and lemma 2 leads to the following theorem.
Theorem 1: Almost all finite and simultaneous games 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 games have a finite and odd number of Nash equilibria, 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
4
1
UP
p
2
3
2
1
DOWN
1- p
1
1
2
It’s easy to check that there are totally two Nash equilibria for this game: ( p, q)  (1, 1) ,
or (UP, LEFT), and ( p, q)  (0, 0) , 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
4
1
UP
p
3
2
2
1
DOWN
1- p
1
1
2 
It’s easy to check that all the Nash equilibria for this game are ( p, q )  (1,1) and
( p, q )  ( 0, t ) , where t  [ 0, a /(2  a)] . The game has infinite Nash equilibria.
5
In spite of the conclusion of Oddness Theorem, we are frequently encountered with finite
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, without 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
distribution2. It should be noted that the specific 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
K
infinite. If we look upon I 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 each other is zero. Because of this, the
main result of this paper should be re-paraphrased as follows.
Theorem 2: Almost all finite and simultaneous games with their payoffs conforming to a
uniform distribution or an a.e. continuous distribution in the cube have a finite and odd number of
Nash equilibria
5. Concluding 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 terminologies
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 problem 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 continuous distribution if its probability
density function is continuous almost everywhere, for example, within the interval [0, 1] . That is,
the set of discontinuous points is of zero measure.
6
Appendix: Proof of Lemma 1
We just give a proof for a 2  2 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 t1 ，t 2 ，
，t8 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
t2
A
U
p
t4
t1
t3
t6
D
1- p
t8
t5
t7
The expected payoff of player A is
EU A  pqt1  p(1  q)t3  (1  p)qt5  (1  p)(1  q)t7
 p [q(t1  t5  t7  t3 )  (t7  t3 )]  q(t5  t7 )  t7
,
from which we can derive player A’s reaction function p  p (q ) given player B’s choice of q
as follows:

1,

When t7  t3  t1  t5  0 ， p (q)  [0,1], q  (t7  t3 ) /(t1  t5  t7  t3 ) ，
0,


When t7  t3  0  t1  t5 ， we have [q(t1  t5  t7  t3 )  (t7  t3 )]  0 ，which implies
p (q)  0 ，

0,

When 0  t7  t3  t1  t5 ， p (q)  [0,1], q  (t7  t3 ) /(t1  t5  t7  t3 ) ，
1,



1,

When 0  t7  t3  t1  t5 ， p (q)  [0,1], q  (t7  t3 ) /(t1  t5  t7  t3 ) ，
0,


When t7  t3  0  t1  t5 ， we have [q(t1  t5  t7  t3 )  (t7  t3 )]  0 ，which implies
7
p (q)  1 ，

0,

When t7  t3  t1  t5  0 ， p (q)  [0,1], q  (t7  t3 ) /(t1  t5  t7  t3 ) ，
1,


In the coordinate plane p  q , the shapes of the reaction function p  p (q ) can be
summarized by Figure 1 below:
q
q
q
q
p
p
FIG. 1.
p
p
Similarly, the reaction function of player B given player A’s choice of p can also be
summarized by Figure 2 below:
q
q
p
q
p
q
p
p
FIG. 2.
The intersections of the two players’ reaction functions give the Nash Equilibria for the
2  2 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:
8
q
q
p
q
p
p
p
p
p
q
p
q
p
q
p
q
q
q
p
q
q
p
q
p
q
p
q
q
p
q
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 2  2
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.
9
References
Debreu, G., Economies with a finite set of equilibria, Econometrica, 38 (1970), 387-392.
Debreu, G., The Rate of Convergence of the Core 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 Applied Economists, Princeton University Press, 1992.
Wilson, Robert, Computing equilibria of N-person games, SIAM J. Appl. Math., 21 (1971), 80-87.
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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 games
have a finite and odd number of Nash equilibria. In this note, we derive a sufficient condition
under which the number of Nash equilibria is odd. In doing so, we provide a new version and an
elementary proof of the theorem. Finally, some comments and treatment on the implicit
assumption behind the theorem are developed.
11