1 Introduction
The vector space structure of finite games is firstly proposed by can11 . Then it has been merged as an isomorphism onto a finite Euclidean space che16 . As a result, the decomposition of the vector space of finite games becomes a natural and interesting topic. Since the potential game is theoretically important and practically useful, a decomposition based on potential games and harmonic games has been investigated can11 ; che16 . Symmetric game is another kind of interesting games alo13
, which may provide useful properties for applications. Hence symmetrybased decomposition is another interesting topic. Decomposition may help to classify games and to reveal properties of each kind of finite games.
To provide a clear picture of the decompositions we first give a survey for the vector space structure of finite games.
Definition 1
A (normal form noncooperative) finite game consists of three ingredients:

Player, : which means there are players;

Profile, : where is the set of strategies (actions) of player ;

Payoff, : where is the payoff function of player .
Assume , i.e., is the th strategy of player . Instead of , we denote this strategy by , which is the
th column of identity matrix
. This expression is called the vector form of strategies. Since each payoff function is a pseudological function, there is a unique row vector , where , such that (when the vector form is adopted) the payoffs can be expressed as(1) 
where is the semitensor product of matrices which is defined in next section.
The set of finite games with , , , is denoted by . Now it is clear that a game is uniquely determined by
(2) 
which is called the structure vector of . Hence has a natural vector space structure as .
The potentialbased decomposition of finite games was firstly proposed by Candogan and Menache, using the knowledge of algebraic topology and the Helmholtz decomposition theory from graph theory can11 . The decomposition is shown in (3), where , , and are pure potential games, nonstrategic games, and pure harmonic games respectively. Unfortunately, the inner product used there is not the standard one in .
(3) 
The vector space structure of potential games has been clearly revealed in che14 by providing a basis of potential subspace. Using this result, che16 reobtained the decomposition (3) with standard inner product through a straightforward linear algebraic computation.
The concept of symmetric game was firstly proposed by Nash nas51 . It becomes an important topic since then alo13 ; bra09 ; cao16 . We also refer to chepr for a vector space approach to symmetric games. The symmetrybased decompositions have been discussed recently as for four strategy matrix games sza15 , as well as for general twoplayer games sza16 .
In this paper, the skewsymmetric game is proposed. First, we show that twoplayer games have an orthogonal decomposition as in (4). That is, the vector subspace of skewsymmetric games is the orthogonal complement of the subspace of symmetric games:
(4) 
where , and are symmetric and skewsymmetric subspaces of respectively.
Furthermore, certain properties of skewsymmetric games are also revealed. The bases of symmetric and skewsymmetric games are constructed. Due to their orthogonality, following conclusions about the decomposition of finite games are obtained:

if then
(5) 
if then
(6) where is the set of asymmetric games.
Finally, for statement ease, we give some notations:

: the set of real matrices.

: the set of Boolean matrices, (: the set of dimensional Boolean vectors.)

(): the set of columns (rows) of . (): the th column (row) of .

.

: the th column of the identity matrix .

.

.

: a matrix with zero entries.

A matrix is called a logical matrix if the columns of are of the form . That is, . Denote by the set of logical matrices.

If , by definition it can be expressed as . For the sake of compactness, it is briefly denoted as .

: th order symmetric group.

: the standard inner product in .

: th order Boolean orthogonal group.

(or ): general linear group.

: the set of finite games with , .

: , .

: the set of (ordinary) symmetric games. Denote by a symmetric game.

: the set of skewsymmetric games. Denote by a skewsymmetric game.

: the set of asymmetric games. Denote by an asymmetric game.
The rest of this paper is organized as follows: In section 2, a brief review of semitensor product of matrices is given. After introducing a symmetrybased classification of finite games, Section 3 presents mainly two results: (1) the orthogonal decomposition of two player games; (2) the linear representation of skewsymmetric games. Some properties of skewsymmetric games are discussed in Section 4. A basis of is also constructed. Section 5 is devoted to verifying the orthogonality of symmetric and skewsymmetric games. Section 6 provides a symmetrybased orthogonal decomposition of finite games. In Section 7, an illustrative example is given to demonstrate this decomposition. Section 8 is a brief conclusion.
2 Preliminaries
2.1 Semitensor Product of Matrices
In this section, we give a brief survey on semitensor product (STP) of matrices. It is the main tool for our approach. We refer to che11 ; che12 for details. The STP of matrices is defined as follows:
Definition 2
Let The STP of and is defined as
(7) 
where is the least common multiple of and and is the Kronecker product.
STP is a generalization of conventional matrix product, and all computational properties of the conventional matrix product remain available. It has been successfully used for studying logical (control) systems lht16 ; zgd17 . Throughout this paper, the default matrix product is STP. Hence, the product of two arbitrary matrices is well defined, and the symbol is mostly omitted.
First, we give some basic properties of STP, which will be used in the sequel.
Proposition 3

(Associative Law:)
(8) 
(Distributive Law:)
(9) (10)
Proposition 4
Let be a dimensional column vector, and a matrix. Then
(11) 
Definition 5
A swap matrix is defined as
(12) 
The basic function of a swap matrix is to swap two vectors.
Proposition 6
Let and be two column vectors. Then
(13) 
The swap matrix is an orthogonal matrix:
Proposition 7
is an orthogonal matrix. Precisely,
(14) 
Given a matrix , its row stacking form is
its column stacking form is
Proposition 8
Given a matrix . Then
(15) 
and
(16) 
Next, we consider the matrix expression of logical relations. Identifying
then a logical variable can be expressed in vector form as
which is called the vector form expression of
A mapping is called a pseudoBoolean function.
Proposition 9
Given a pseudoBoolean function , there exists a unique row vector , called the structure vector of , such that (in vector form)
(17) 
Remark 10
In previous proposition, if is replaced by , , then the function is called a pseudological function and the expression (17) remains available with an obvious modification that and .
Definition 11
Let and . Then the KhatriRao product of and , denoted by , is defined as follows:
Proposition 12
Assume
where Then
3 Symmetric and Skewsymmetric Games
3.1 Classification of Finite Games
This subsection considers the symmetrybased classification of finite games. First, we give a rigorous definition for symmetric and skewsymmetric games.
Definition 13
Let .

If for any , we have
(18) where then is called a symmetric game. Denote by the set of symmetric games in .

If for any , we have
(19) where then is called a skewsymmetric game. Denote by the set of skewsymmetric games in .
It is well known that is a vector space can11 ; che16 . It is easy to figure out that both and are subspaces of . Hence, they are also two subspaces of .
Then, we can define the following asymmetric subspace.
Definition 14
is called an asymmetric game if its structure vector
(20) 
The set of asymmetric games is denoted by which is also a subspace of .
Example 15
Consider A straightforward computation shows the following result:
It follows that , , and . Moreover, it is ready to verify the orthogonality:
We conclude that
which verifies (6).
111  112  121  122  211  212  221  222  

a  b  b  d  c  e  e  f  
a  b  c  e  b  d  e  f  
a  c  b  e  b  e  d  f 
111  112  121  122  211  212  221  222  
0  g  g  0  0  h  h  0  
0  g  0  h  g  0  h  0  
0  0  g  h  g  h  0  0 
111  112  121  122  211  212  221  222  
3.2 Two Player Games
In this subsection we consider . Let and be the payoff matrices of player and player respectively. According to Definition 13, it is easy to verify the following fact:
Lemma 16

is a symmetric game, if and only if,

is a skewsymmetric game, if and only if,
Note that for we have its structure vector as
where is the row stacking form of matrix .
Lemma 17

is a symmetric game, if and only if,
(21) 
is a skewsymmetric game, if and only if,
(22)
According to Lemma 17, the following result can be obtained via a straightforward computation.
Theorem 18
Let . Then can be orthogonally decomposed to
(23) 
where and .
Proof. Denote the structure vector of as . We construct a symmetric game by setting
and a skewsymmetric game by
where
Then, it is ready to verify that

;

.
The conclusion follows.
Example 19
Consider .

is symmetric, if and only if, its payoff functions are as in Table 4.

is skewsymmetric, if and only if, its payoff functions are as in Table 5.

Let with its payoff bimatrix as in Table 6.
Then it has an orthogonal decomposition into and with their payoff bimatrices as in Table 4 and Table 5 respectively with
a, a b, c c, b d, d Table 4: Payoff bimatrix of a symmetric game in a’, a’ b’, c’ c’, b’ d’, d’ Table 5: Payoff bimatrix of a skewsymmetric game in , , , , Table 6: Payoff bimatrix of a game in
3.3 SkewSymmetric Game and Its Linear Representation
First, we present a necessary condition for verifying skewsymmetric games.
Proposition 20
Consider . If , then
(24) 
Next, we consider another necessary condition: If , then what condition should verify? An argument similar to the one used in Proposition 20 shows the following result.
Proposition 21
Consider . If , then
(25) 
where
Proof. Assume satisfies . Let , . Then, we have
Since are arbitrary, we have
Setting , we have (25).
Note that the symmetric group is generalized by transpositions chepr . That is,
This fact motivates the following result.
Theorem 22
Proof. We need only to prove the sufficiency.

if (22) implies the sufficiency.

if we divide our proof into two steps.
First, we prove the condition for a single payoff function . For any and without loss of generality, we assume
where
From (25), it can be calculated that
(26) 
Applying (24) to (26), we have
(27) 
(27) implies that for any and we have
(28) 
Obviously, according to (19), (28) is the necessary and sufficient condition for a single payoff function to obey in a skewsymmetric game.
Next, we consider the condition for cross payoffs.
For any without loss of generality, we assume
where
Example 24
Consider From Proposition 21 we have
One can easily figure out that
where are real numbers. According to Proposition 20, we can calculate that
According to Definition 13, a straightforward verification shows that
As a byproduct, we have .
In the following, we consider the linear representation of in
Definition 25
ser77 Let be a group and a finite dimensional vector space. A linear representation of in is a group homomorphism .
Consider a profile of a . We define two expressions of as follows:

STP Form: The STP form of is expressed as

Stacking Form: The strategy stacking form of is expressed as
Denote
Comments
There are no comments yet.