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Game theory is a mathematical theory studying competitive phenomena. Since John von Neumann proved the basic principles of game theory, modern game theory was formally established [1,2], which has been paid wide attention and applied to biology, economics, computer science, and many other fields. For example, biologists use game theory to predict certain outcomes of evolution. Economists regard the game theory as one of the standard analysis tools of economics.
The concept of symmetric games is first proposed by John von Neumann in [2]. The symmetry of a game means that all players have the same set of strategies and fair payoffs, that is, the payoffs depend only on the strategies employed, not on who is playing them. Because fair games are more realistic and acceptable, many common games are symmetric games such as the well-known games rock-paper-scissors and prisoner's dilemma. In recent years, many problems about symmetric games have been investigated in [3], [4], [5], and [6]. In addition, based on the definition of symmetric games, the concepts of skew-symmetric games, asymmetric games and the symmetric-based decomposition of finite games have been proposed in [4]. Although the bases of the symmetric game subspace and the skew-symmetric game subspace have been constructed in [4], the vector space structure of the asymmetric game subspace has not been revealed. Therefore, the motivation of this paper is to explore the vector space structure of the asymmetric game subspace and thoroughly solve the problem of symmetric-based decomposition of finite games. In our recent paper [6], a new method to construct a basis of the symmetric game subspace has been proposed, which gives us great inspiration for the study of skew-symmetric games, asymmetric games, and symmetric-based decomposition of finite games.
In the past decade, the semi-tensor product (STP) of matrices has been successfully applied to game theory by Cheng and his collaborators [7], which enables a game to be expressed in an algebraic form. In this paper, we still use the matrix method based on STP to investigate skew-symmetric games, asymmetric games and symmetric-based decomposition of finite games. First, by the semi-tensor product method based on adjacent transpositions, necessary and sufficient conditions for testing skew-symmetric games are obtained. Then, based on the necessary and sufficient conditions, a basis of the skew-symmetric game subspace is constructed explicitly. In addition, the discriminant equations for skew-symmetric games with the minimum number are derived concretely. According to the construction methods of the basis of the symmetric game subspace in [6] and the basis of the skew-symmetric game subspace in this paper, a basis of the asymmetric game subspace is constructed for the first time. Therefore, the problem of symmetric-based decomposition of finite games is completely solved.
The rest of this paper is organized as follows: Section 2 gives some preliminaries. Section 3 studies skew-symmetric games and skew-symmetric game subspace. Section 4 studies asymmetric games and solves the problem of symmetric-based decomposition of finite games. Section 5 is a brief conclusion.
In this section, some necessary preliminaries are given. Firstly, we list the following notations.
● D={0,1}: the set of values of logical variables;
● δik: the i-th column of Ik;
● Δk:={δik:i=1,2,⋯,k};
● δk[i1i2⋯in]:=[δi1kδi2k⋯δink];
● Mm×n: the set of m×n matrices;
● Lm×n:={L∈Mm×n|Col(L)⊆Δm};
● ⋉: the left semi-tensor product of matrices;
● 1n: the n-dimensional column vector of ones;
● 0m×n: the m×n matrix with zero entries;
● Sn: the n-th order symmetric group, i.e., a permutation group that is composed of all the permutations of n things;
● R: the set composed of all the real numbers.
Definition 2.1 ([7]). Let A∈Mm×n, B∈Mp×q. The left semi-tensor product of A and B is defined as
A⋉B=(A⊗Iαn)(B⊗Iαp), | (2.1) |
where ⊗ is the Kronecker product and α=lcm(n,p) is the least common multiple of n and p. When no confusion may arise it is usually called the semi-tensor product (STP).
If n and p in Definition 2.1 satisfy n=p, the STP is reduced to the traditional matrix product. So, the STP is a generalized operation of the traditional matrix product. Therefore, one can directly write A⋉B as AB.
Definition 2.2 ([7]). A swap matrix W[m,n]=(wIJij) is an mn×mn matrix, defined as follows:
Its rows and columns are labeled by double indices. The columns are arranged by the ordered multi-index Id(i1,i2;m,n), and the rows are arranged by the ordered multi-index Id(i2,i1;n,m). The element at the position with row index (I,J) and column index (i,j) is
wIJij={1, I=i and J=j,0, otherwise. |
When m=n, matrix W[m,n] is denoted by W[m].
Swap matrices have the following properties:
(Ik⊗W[k])(W[k]⊗Ik)(Ik⊗W[k])=(W[k]⊗Ik)(Ik⊗W[k])(W[k]⊗Ik). | (2.2) |
Definition 2.3. ([8]). A finite game is a triple G=(N,S,C), where
1) N={1,2,⋯,n} is the set of n players;
2) S=S1×S2×⋯×Sn is the set of strategy profiles, where Si={si1,si2,⋯,siki} is the set of strategies of player i;
3) C={c1,c2,⋯,cn} is the set of payoff functions, where ci:S→R is the payoff function of player i.
Denote the set composed of all the games above by G[n;k1,k2,⋯,kn]. When |Si|=k for each i=1,2,⋯,n, we denote it by G[n;k].
STP is a convenient tool for investigating games. Given a game G∈G[n;k], by using the STP method [9], each strategy xi can be written into a vector form xi∈Δk, and every payoff function ci can be expressed as
ci(x1,x2,⋯,xn)=Vci⋉nj=1xj,i=1,2,⋯,n, | (2.3) |
where ⋉nj=1xj∈Δkn is called the STP form of the strategy profile, and Vci is called the structure vector of ci.
Definition 2.4 ([10]). A game G∈G[n;k] is called a symmetric game if for any permutation σ∈Sn
ci(x1,x2,⋯,xn)=cσ(i)(xσ−1(1),xσ−1(2),⋯,xσ−1(n)) | (2.4) |
for any i=1,2,⋯,n.
Definition 3.1 ([4]). A game G∈G[n;k] is called a skew-symmetric game if for any permutation σ∈Sn
ci(x1,x2,⋯,xn)=sgn(σ)cσ(i)(xσ−1(1),xσ−1(2),⋯,xσ−1(n)) | (3.1) |
for any i=1,2,⋯,n.
The set composed of all the skew-symmetric games in G[n;k] is denoted as K[n;k].
Lemma 3.1 ([11]). The set of all the adjacent transpositions (r,r+1),1≤r≤n−1 is generator of the symmetric group Sn.
In the following, adjacent transpositions (r,r+1),1≤r≤n−1 are represented as μr.
Lemma 3.2. Consider G∈G[n;k]. For any σ1,σ2∈Sn, if σ1 and σ2 satisfy
ci(x1,x2,⋯,xn)=sgn(σ)cσ(i)(xσ−1(1),xσ−1(2),⋯,xσ−1(n)) | (3.2) |
for any i=1,2,⋯,n and any x1,x2,⋯,xn∈Δk, the compound permutation σ2∘σ1 also satisfies (3.2).
Proof. For any given xi∈Δk, i=1,2,…,n, let yi=xσ−11(i). Then
ci(x1,x2,⋯,xn)=sgn(σ1)cσ1(i)(xσ−11(1),xσ−11(2),⋯,xσ−11(n))=sgn(σ1)cσ1(i)(y1,y2,⋯,yn)=sgn(σ2)sgn(σ1)cσ2(σ1(i))(yσ−12(1),yσ−12(2),⋯,yσ−12(n))=sgn(σ2∘σ1)cσ2∘σ1(i)(xσ−11(σ−12(1)),xσ−11(σ−12(2)),⋯,xσ−11(σ−12(n))), |
which implies that σ2∘σ1 satisfies (3.2).
According to Definition 3.1, Lemma 3.1 and Lemma 3.2, the following lemma follows:
Lemma 3.3. Consider G∈G[n;k]. Game G is a skew-symmetric game if and only if
ci(x1,x2,⋯,xn)=−cμr(i)(xμr(1),xμr(2),⋯,xμr(n)) | (3.3) |
for any adjacent transposition μr, 1≤r≤n−1, i=1,2,⋯,n.
Proposition 3.1. Consider G∈G[n;k]. Game G is a skew-symmetric game if and only if
Vci=−Vcμr(i)Tμr,∀i=1,2,⋯,n,1≤r≤n−1, | (3.4) |
where Tμr=Ikr−1⊗W[k]⊗Ikn−r−1.
Proof. For any i=1,2,⋯,n and any 1≤r≤n−1, we have
cμr(i)(xμr(1),xμr(2),⋯,xμr(n))=Vcμr(i)xμr(1)xμr(2)⋯xμr(n)=Vcμr(i)(x1x2⋯xr−1)(xr+1xr)(xr+2⋯xn)=Vcμr(i)(x1x2⋯xr−1)(W[k]xrxr+1)(xr+2⋯xn)=Vcμr(i)Tμrx1x2⋯xn. | (3.5) |
From (2.3) and (3.5), it follows that (3.4) is equivalent to (3.3). Therefore, the proposition is proved.
Theorem 3.1. Consider G∈G[n;k]. Game G is a skew-symmetric game if and only if
[IknTμ1IknTμ2⋱⋱IknTμn−1Ikn+Tμ1Ikn+Tμ2⋮Ikn+Tμn−2](VG)T=0, | (3.6) |
where Tμr=Ikr−1⊗W[k]⊗Ikn−r−1, VG=[Vc1Vc2⋯Vcn], and the omitted elements in the coefficient matrix of (3.6) are all zeros.
Proof. Since (W[k])−1=W[k], we have (Tμr)−1=Tμr for any 1≤r≤n−1. Then, the equation Vci=−Vcμr(i)Tμr is equivalent to Vcμr(i)=−VciTμr. According to Proposition 3.1, G is a skew-symmetric game if and only if
[IknTμ1IknTμ2⋱⋱IknTμn−1B1B2⋱Bn−1Bn](VG)T=0, | (3.7) |
where
B1=[Ikn+Tμ2Ikn+Tμ3⋮Ikn+Tμn−1],B2=[Ikn+Tμ3Ikn+Tμ4⋮Ikn+Tμn−1], | (3.8) |
Bn−1=[Ikn+Tμ1Ikn+Tμ2⋮Ikn+Tμn−3],Bn=[Ikn+Tμ1Ikn+Tμ2⋮Ikn+Tμn−2], | (3.9) |
Br=[Ikn+Tμ1Ikn+Tμ2⋮Ikn+Tμr−2Ikn+Tμr+1Ikn+Tμr+2⋮Ikn+Tμn−1] (3≤r≤n−2). | (3.10) |
Let the coefficient matrix of equation (3.7) be
[A1A2B0(n−2)2kn×kn0(n−2)kn×(n−1)knBn] | (3.11) |
where
A1=[IknTμ1IknTμ2⋱⋱IknTμn−2Ikn], | (3.12) |
A2=[0(n−2)kn×knTμn−1], | (3.13) |
B=[B1B2⋱Bn−1]. | (3.14) |
Since A1 is an invertible matrix, we can perform the following row transformation on the coefficient matrix of (3.7)
[I(n−1)kn−BA−11I(n−2)(n−1)knI(n−2)kn][A1A2B0(n−2)2kn×kn0(n−2)kn×(n−1)knBn]=[A1A20(n−2)2kn×(n−1)kn−BA−11A20(n−2)kn×(n−1)knBn], | (3.15) |
where
−BA−11A2=[(−1)n−1B1Tμ1Tμ2⋯Tμn−1(−1)n−2B2Tμ2Tμ3⋯Tμn−1⋮−Bn−1Tμn−1]. | (3.16) |
Let
F1=In−2⊗(Tμn−1Tμn−2⋯Tμ1), |
Fr=In−3⊗(Tμn−1Tμn−2⋯Tμr),∀2≤r≤n−1. |
We perform the following row transformation on matrix −BA−11A2
[(−1)n−1F1(−1)n−2F2⋱−Fn](−BA−11A2)=[F1B1Tμ1Tμ2⋯Tμn−1F2B2Tμ2Tμ3⋯Tμn−1⋮Fn−1Bn−1Tμn−1]. | (3.17) |
Therefore, the equivalent form of (3.7) is as follows
[IknTμ1IknTμ2⋱⋱IknTμn−1F1B1Tμ1⋯Tμn−1F2B2Tμ2⋯Tμn−1⋮Fn−1Bn−1Tμn−1Bn](VG)T=0. | (3.18) |
From the property of W[k] shown in (2.2), it follows that
Tμn−1⋯Tμr+1TμrTμiTμrTμr+1⋯Tμn−1={Tμi∀1≤i≤r−2,Tμi−1∀r+1≤i≤n−1. | (3.19) |
Then, (3.18) is equivalent to (3.6). Thus, the proof is complete.
We see that the key of solving equation (3.6) is computing the solution space of the following linear equation:
[Ikn+Tμ1Ikn+Tμ2⋮Ikn+Tμn−2]x=0, | (3.20) |
where x is the kn-dimensional unknown vector. Considering
[Ikn+Tμ1Ikn+Tμ2⋮Ikn+Tμn−2]=[Ikn+W[k]⊗Ikn−2Ikn+Ik⊗W[k]⊗Ikn−3⋮Ikn+Ikn−3⊗W[k]⊗Ik]=[Ikn−1+W[k]⊗Ikn−3Ikn−1+Ik⊗W[k]⊗Ikn−4⋮Ikn−1+Ikn−3⊗W[k]]⊗Ik, | (3.21) |
we only need to solve the linear equations as follows:
[Ikn−1+W[k]⊗Ikn−3Ikn−1+Ik⊗W[k]⊗Ikn−4⋮Ikn−1+Ikn−3⊗W[k]]x=0, | (3.22) |
where x is the kn−1-dimensional unknown vector. Let x=(xl1l2⋯ln−1) be arranged by the ordered multi-index Id(i1,i2,…,in−1;k,k,…,k), that is,
x=(x11⋯11,x11⋯12,…,x11⋯1k,x11⋯21,x11⋯22,…,x11⋯2k,…,xkk⋯k1,xkk⋯k2,…,xkk⋯kk)T. | (3.23) |
Then, by the property of W[k], vector x is a solution of (3.22) if and only if, for any 1≤l1,l2,⋯,ln−1≤k, the following equations hold:
xl1l2l3⋯ln−1=−xl2l1l3⋯ln−1,xl1l2l3⋯ln−1=−xl1l3l2⋯ln−1,⋮xl1l2l3⋯ln−1=−xl1⋯ln−3ln−1ln−2, | (3.24) |
i.e.
xl1l2⋯ln−1=sgn(π)xπ(l1l2⋯ln−1),∀π∈Sn−1. | (3.25) |
Thus, for any 1≤r≤n−2, if lr=lr+1, then
xl1⋯lrlrlr+2⋯ln−1=−xl1⋯lrlrlr+2⋯ln−1, |
that is,
xl1⋯lrlrlr+2⋯ln−1=0. |
Therefore, all the free variables of the linear equations (3.22) are
xl1l2⋯ln−1,∀1≤l1<l2<⋯<ln−1≤k, | (3.26) |
whose number is Cn−1k. That is, the dimension of the solution space of linear equations (3.22) is Cn−1k.
For every given repeatable combination s1s2⋯sn−1,(1≤s1≤s2≤⋯≤sn−1≤k), denote by Ps1s2⋯sn−1 the set composed of all the repeatable permutation of s1s2⋯sn−1. For example, P122={122,212,221}. For every given unrepeatable combination l1l2⋯ln−1,(1≤l1<l2<⋯<ln−1≤k), denote by Rl1l2⋯ln−1 the set composed of all the unrepeatable permutation of l1l2⋯ln−1. For example, R123={123,132,213,231,312,321}. Let
Q=(⋃1≤s1≤s2≤⋯≤sn−1≤kPs1s2⋯sn−1)∖(⋃1≤l1<l2<⋯<ln−1≤kRl1l2⋯ln−1). |
Then, any permutation in Q is a repeated permutation.
Lemma 3.4. For every given unrepeatable combination l1l2⋯ln−1(1≤l1<l2<⋯<ln−1≤k), define a vector θl1l2⋯ln−1=x with the form (3.23) by
xt1t2⋯tn−1={sgn(t1t2⋯tn−1), t1t2⋯tn−1∈Rl1l2⋯ln−1,0, otherwise. |
Then the set
{θl1l2⋯ln−1| 1≤l1<l2<⋯<ln−1≤k} | (3.27) |
is a basis of the solution space ˉXn−1 of (3.22). For every l1l2⋯ln−1 (1≤l1<l2<⋯<ln−1≤k), we define |Rl1l2⋯ln−1|−1 number of vectors νr1r2⋯rn−1l1l2⋯ln−1=x with r1r2⋯rn−1∈Rl1l2⋯ln−1 and r1r2⋯rn−1≠l1l2⋯ln−1 by
xt1t2⋯tn−1={1,t1t2⋯tn−1=l1l2⋯ln−1,−sgn(t1t2⋯tn−1),t1t2⋯tn−1=r1r2⋯rn−1,0,otherwise. |
We define |Q| number of vectors λh1h2⋯hn−1=x (h1h2⋯hn−1∈Q) by
xt1t2⋯tn−1={1, t1t2⋯tn−1=h1h2⋯hn−1,0, otherwise. |
Then the set of νr1r2⋯rn−1l1l2⋯ln−1 (1≤l1<l2<⋯<ln−1≤k,r1r2⋯rn−1∈Rl1l2⋯ln−1,r1r2⋯rn−1≠l1l2⋯ln−1) and λh1h2⋯hn−1 (h1h2⋯hn−1∈Q) is a basis of the orthogonal complementary space ˉX⊥n−1. Denote by MW the matrix whose columns are composed of a basis of subspace W. Then the linear system (3.22) is equivalent to MTˉX⊥n−1x=0.
Proof. For any 1≤l1<l2<⋯<ln−1≤k, sgn(l1l2⋯ln−1)=1. From the equivalent equations (3.25) and the free variables shown by (3.26), it follows that the set of θl1l2⋯ln−1 (1≤l1<l2<⋯<ln−1≤k) is a basis of the solution space ˉXn−1. By the construction of νr1r2⋯rn−1l1l2⋯ln−1 and λh1h2⋯hn−1, it is straightforward to see that each νr1r2⋯rn−1l1l2⋯ln−1 and each λh1h2⋯hn−1 are orthogonal to ˉXn−1. The total number of νr1r2⋯rn−1l1l2⋯ln−1 is
∑1≤l1<l2<⋯<ln−1≤k(|Rl1l2⋯ln−1|−1)=∑1≤l1<l2<⋯<ln−1≤k|Rl1l2⋯ln−1|−Cn−1k. |
The total number of λh1h2⋯hn−1 is
∑1≤s1≤s2≤⋯≤sn−1≤k|Ps1s2⋯sn−1|−∑1≤l1<l2<⋯<ln−1≤k|Rl1l2⋯ln−1| |
=kn−1−∑1≤l1<l2<⋯<ln−1≤k|Rl1l2⋯ln−1|. |
Then the total number of νr1r2⋯rn−1l1l2⋯ln−1 and λh1h2⋯hn−1 is kn−1−Cn−1k, i.e. kn−1−dim(ˉXn−1). Therefore, we conclude that the set of νr1r2⋯rn−1l1l2⋯ln−1 (1≤l1<l2<⋯<ln−1≤k,r1r2⋯rn−1∈Rl1l2⋯ln−1,r1r2⋯rn−1≠l1l2⋯ln−1) and λh1h2⋯hn−1 (h1h2⋯hn−1∈Q) is a basis of ˉX⊥n−1. Then, the linear system (3.22) is equivalent to MTˉX⊥n−1x=0.
According to the above basis of the solution space of linear equations (3.22), we can construct a basis of skew-symmetric game subspace K[n;k].
Theorem 3.2. The dimension of the skew-symmetric game subspace K[n;k] is kCn−1k. A basis of K[n;k] is composed of the columns of matrix
[(−1)n−1W[kn−1,k](−1)n−2Ik⊗W[kn−2,k](−1)n−3Ik2⊗W[kn−3,k]⋮(−1)2Ikn−3⊗W[k2,k]−Ikn−2⊗W[k]Ikn](MˉXn−1⊗Ik), | (3.28) |
where MˉXn−1 is composed of the basis of the solution space of (3.22). Moreover, the linear equations with the minimum number to test skew-symmetric games in K[n;k] are
[IknTμ1IknTμ2⋱⋱IknTμn−1MTˉX⊥n−1⊗Ik](VG)T=0, | (3.29) |
where the omitted elements in the coefficient matrix of (3.29) are all zeros.
Proof. By Theorem 3.1 and Lemma 3.4, we can easily get the dimension of skew-symmetric game subspace K[n;k] is kCn−1k. Using MˉXn−1 whose columns are composed of a basis of the solution space of (3.22), we get a basis of the solution space of (3.6) as follows:
[(−1)n−1Tμ1⋯Tμn−1(MˉXn−1⊗Ik)(−1)n−2Tμ2⋯Tμn−1(MˉXn−1⊗Ik)(−1)n−3Tμ3⋯Tμn−1(MˉXn−1⊗Ik)⋮−Tμn−1(MˉXn−1⊗Ik)MˉXn−1⊗Ik]. | (3.30) |
By the property of swap matrices shown in (2.2), we have
TμsTμs+1⋯Tμn−1=Iks−1⊗W[kn−s,k] |
for each 1≤s≤n−1. Then, (3.30) is equivalent to (3.28). That is, the set of the columns of matrix (3.28) is a basis of K[n;k]. Since (3.29) is equivalent to (3.6) and the coefficient matrix of (3.29) has a full row rank, the equations in (3.29) have the minimum number for testing skew-symmetric games in K[n;k].
Remark 3.1. The coefficient matrix of (3.29) has nkn−kCn−1k number of rows and each row has at most two nonzero elements. Since Cn−1k≤kn−1, (n−1)kn≤nkn−kCn−1k≤nkn. Therefore, the computational complexity is just O(nkn) due to
limn→∞nkn(n−1)kn=limn→∞nn−1=1. |
Definition 4.1 ([4]). A game G∈G[n;k] is called an asymmetric game if its structure vector
VG∈[S[n;k]⊕K[n;k]]⊥. |
The set of asymmetric games is denoted by E[n;k].
Lemma 4.1 ([6]). The dimension of the symmetric game subspace S[n;k] is kCn−1k+n−2. A basis of S[n;k] is composed of the columns of matrix
[W[kn−1,k]Ik⊗W[kn−2,k]Ik2⊗W[kn−3,k]⋮Ikn−2⊗W[k]Ikn](MXn−1⊗Ik), | (4.1) |
where Xn−1 is the solution space of linear equations
[Ikn−1−W[k]⊗Ikn−3Ikn−1−Ik⊗W[k]⊗Ikn−4⋮Ikn−1−Ikn−3⊗W[k]]x=0, | (4.2) |
and MXn−1 is the matrix composed of a basis of Xn−1.
Let
A=[W[kn−1,k]Ik⊗W[kn−2,k]Ik2⊗W[kn−3,k]⋮Ikn−2⊗W[k]Ikn](MXn−1⊗Ik), | (4.3) |
B=[(−1)n−1W[kn−1,k](−1)n−2Ik⊗W[kn−2,k](−1)n−3Ik2⊗W[kn−3,k]⋮−Ikn−2⊗W[k]Ikn](MˉXn−1⊗Ik). | (4.4) |
It is easy to check that
[W[kn−1,k]Ik⊗W[kn−2,k]Ik2⊗W[kn−3,k]⋮Ikn−2⊗W[k]Ikn]T[(−1)n−1W[kn−1,k](−1)n−2Ik⊗W[kn−2,k](−1)n−3Ik2⊗W[kn−3,k]⋮−Ikn−2⊗W[k]Ikn]=n∑i=1(Iki−1⊗W[k,kn−i])((−1)n−iIki−1⊗W[kn−i,k])=n∑i=1(−1)n−iIkn | (4.5) |
Since the number of odd permutations of any combination is the same as the number of even permutations, according to the construction of a basis of Xn−1 in [6], we have
ATB=(MTXn−1⊗Ik)(n∑i=1(−1)n−iIkn)(MˉXn−1⊗Ik)=n∑i=1(−1)n−i(MTXn−1MˉXn−1⊗Ik)=0p×q, | (4.6) |
where p=kCn−1k+n−2,q=kCn−1k. That is, S[n;k] and K[n;k] are orthogonal. Therefore,
G[n;k]=S[n;k]⊕K[n;k]⊕E[n;k]. | (4.7) |
So far, we have constructed a basis of the symmetric game subspace and that of the skew-symmetric game subspace, respectively. Next, according to the two bases, we investigate the vector space structure of the asymmetric game subspace.
Consider the following linear equations
[ATBT]x=0, | (4.8) |
where A and B are shown in (4.3) and (4.4), composed of the bases of S[n;k] and K[n;k], respectively. Therefore, (4.8) is the discriminant equation with the minimum number for asymmetric games, and a basis of the solution space of (4.8) is also a basis of the asymmetric game subspace E[n;k].
Construct matrices MXn−1 and MˉXn−1 as follows:
MXn−1=[η1η2⋯ηβηβ+1⋯ηα], MˉXn−1=[θ1θ2⋯θβ], | (4.9) |
where α=Cn−1k+n−2, β=Cn−1k, and
∀1≤i≤β,∃1≤l1<l2<⋯<ln−1≤k,s.t.ηi=ηl1l2⋯ln−1, θi=θl1l2⋯ln−1, | (4.10) |
∀β+1≤i≤α,∃1≤l1≤l2≤⋯≤ln−1≤k,andl1l2⋯ln−1∈Q,s.t.ηi=ηl1l2⋯ln−1. | (4.11) |
Let
x=[(x1)T, (x2)T,…,(xn)T]T∈Rnkn, |
where xj∈Rkn. Then, (4.8) is equivalent to
{n∑j=1[(ηTi+(−1)n−j+1θTi)⊗Ik](Ikj−1⊗W[k,kn−j ])xj=0,n∑j=1[(ηTi+(−1)n−jθTi)⊗Ik](Ikj−1⊗W[k,kn−j ])xj=0,(1≤i≤β) | (4.12) |
and
n∑j=1(ηTi⊗Ik)(Ikj−1⊗W[k,kn−j])xj=0(β+1≤i≤α). | (4.13) |
According to the construction of ηi=ηl1l2⋯ln−1 and θi=θl1l2⋯ln−1 (1≤i≤β), we conclude that (4.12) is equivalent to
{∑t1t2⋯tn−1 ∈R l1l2⋯ln−1sgn(t1t2⋯tn−1 )=1 ∑1≤j≤njisoddxjt1t2⋯tj−1 lntj+1⋯tn−1+∑t1t2⋯tn−1 ∈R l1l2⋯ln−1sgn(t1t2⋯tn−1 )=−1 ∑1≤j≤njisevenxjt1t2⋯tj−1 lntj+1⋯tn−1=0,∑t1t2⋯tn−1 ∈R l1l2⋯ln−1sgn(t1t2⋯tn−1 )=−1 ∑1≤j≤njisoddxjt1t2⋯tj−1 lntj+1⋯tn−1+∑t1t2⋯tn−1 ∈R l1l2⋯ln−1sgn(t1t2⋯tn−1 )=1 ∑1≤j≤njisevenxjt1t2⋯tj−1 lntj+1⋯tn−1=0 | (4.14) |
for any 1≤l1<l2<⋯<ln−1≤k, 1≤ln≤k, and (4.13) is equivalent to
∑t1t2⋯tn−1 ∈P l1⋯ln−1 ∑1≤j≤nxjt1t2⋯tj−1 lntj+1⋯tn−1=0 | (4.15) |
for any 1≤l1≤l2≤⋯≤ln−1≤k, l1l2⋯ln−1∈Q and any 1≤ln≤k.
We first construct two sets of solution vectors of (4.14):
{μl1l2⋯ln;1t1t2⋯tn−1; j}, {μl1l2⋯ln;−1t1t2⋯tn−1; j}. |
If n is odd, let
μl1l2⋯ln;1t1t2⋯tn−1;j=[(x1)T, (x2)T,…,(xn)T]T∈Rnkn |
with each xp=(xpr1r2⋯rn),
xpr1r2⋯rn={1,p=n,r1r2⋯rn=l1l2⋯ln,−1,p=j,r1r2⋯rn=t1t2⋯tj−1lntj⋯tn−1,0,otherwise, | (4.16) |
where 1≤l1<l2<⋯<ln−1≤k, 1≤ln≤k, t1t2⋯tn−1∈Rl1⋯ln−1, 1≤j≤n satisfy one of following conditions:
(i) j=n, t1t2⋯tn−1≠l1l2⋯ln−1 and (−1)j+1=sgn(t1t2⋯tn−1),
(ii) j≠n, (−1)j+1=sgn(t1t2⋯tn−1).
Similarly, let
μl1l2⋯ln;−1t1t2⋯tn−1;j=[(x1)T, (x2)T,…,(xn)T]T∈Rnkn, |
with each xp=(xpr1r2⋯rn),
xpr1r2⋯rn={1,p=n,r1r2⋯rn=˜l1˜l2⋯˜ln−1ln,−1,p=j,r1r2⋯rn=t1t2⋯tj−1lntj⋯tn−1,0,otherwise, | (4.17) |
where t1t2⋯tn−1 and j satisfy one of the following conditions:
(i) j=n, t1t2⋯tn−1≠˜l1˜l2⋯˜ln−1=l2l1l3⋯ln−1 and (−1)j=sgn(t1⋯tn−1),
(ii) j≠n, (−1)j=sgn(t1t2⋯tn−1).
If n is even, let
μl1l2⋯ln;1t1t2⋯tn−1;j=[(x1)T, (x2)T,…,(xn)T]T∈Rnkn, |
with
xpr1r2⋯rn={1,p=n,r1r2⋯rn=l1l2⋯ln,−1,p=j,r1r2⋯rn=t1t2⋯tj−1lntj⋯tn−1,0,otherwise, | (4.18) |
where 1≤l1<l2<⋯<ln−1≤k, 1≤ln≤k, t1t2⋯tn−1∈Rl1l2⋯ln−1, 1≤j≤n satisfying one of the following conditions,
(i) j=n, t1t2⋯tn−1≠l1l2⋯ln−1 and (−1)j=sgn(t1t2⋯tn−1),
(ii) j≠n, (−1)j=sgn(t1t2⋯tn−1),
Similarly, let
μl1l2⋯ln;−1t1t2⋯tn−1;j=[(x1)T, (x2)T,…,(xn)T]T∈Rnkn, |
with
xpr1⋯rn={1,p=n,r1r2⋯rn=˜l1˜l2⋯˜ln−1ln,−1,p=j,r1r2⋯rn=t1t2⋯tj−1lntj⋯tn−1,0,otherwise, | (4.19) |
where t1t2⋯tn−1 and j satisfy one of following conditions:
(i) j=n, t1t2⋯tn−1≠˜l1˜l2⋯˜ln−1=l2l1l3⋯ln−1 and (−1)j+1=sgn(t1t2⋯tn−1),
(ii) j≠n, (−1)j+1=sgn(t1t2⋯tn−1).
Then we construct a set of solution vectors of (4.15):
{γl1l2⋯lnt1t2⋯tn−1;j}. |
Let
γl1l2⋯lnt1t2⋯tn−1;j=[(x1)T, (x2)T,…,(xn)T]T∈Rnkn |
with
xpr1⋯rn={1,p=n,r1r2⋯rn=l1l2⋯ln,−1,p=j,r1r2⋯rn=t1t2⋯tj−1lntj⋯tn−1,0,otherwise, | (4.20) |
where 1≤l1≤l2≤⋯≤ln−1≤k, l1l2⋯ln−1∈Q, 1≤ln≤k, t1t2⋯tn−1∈Pl1l2⋯ln−1, 1≤j≤n satisfy one of the following conditions:
(i) j=n, t1t2⋯tn−1≠l1l2⋯ln−1,
(ii) j≠n.
Theorem 4.1. The sets {μl1l2⋯ln;1t1t2⋯tn−1;j}, {μl1l2⋯ln;−1t1t2⋯tn−1;j} and {γl1l2⋯lnt1t2⋯tn−1;j} form a basis of the asymmetric game subspace E[n;k].
Proof. According to the construction method of μl1l2⋯ln;1t1t2⋯tn−1;j, μl1l2⋯ln;−1t1t2⋯tn−1;j and γl1l2⋯lnt1t2⋯tn−1;j, all the vectors in {μl1l2⋯ln;1t1t2⋯tn−1;j}, {μl1l2⋯ln;−1t1t2⋯tn−1;j} and {γl1l2⋯lnt1t2⋯tn−1;j} are linearly independent and satisfy both (4.14) and (4.15). Moreover, we have
|{μl1l2⋯ln;1t1t2⋯tn−1;j}|=|{μl1l2⋯ln;−1t1t2⋯tn−1;j}|=∑1≤l1<l2<⋯<ln−1≤kk(n∣Rl1l2⋯ln−1∣2−1)=∑1≤l1<l2<⋯<ln−1≤kk(n∣Rl1l2⋯ln−1∣2)−kCn−1k. |
|{γl1l2⋯lnt1t2⋯tn−1;j}|=∑1≤l1≤l2≤⋯≤ln−1≤kl1l2⋯ln−1∈Qk(n∣Pl1l2⋯ln−1∣−1)=∑1≤l1≤l2≤⋯≤ln−1≤kl1l2⋯ln−1∈Qk(n∣Pl1l2⋯ln−1∣)−(kCn−1k+n−2−kCn−1k). |
So,
∣{μl1l2⋯ln;1t1t2⋯tn−1; j}∣+∣{μl1l2⋯ln;1t1t2⋯tn−1; j}∣+∣{γl1l2⋯lnt1t2⋯tn−1; j}∣=2∑1≤l1<l2<⋯<ln−1≤kk(n∣Rl1l2⋯ln−1∣2)−2kCn−1k+∑1≤l1≤l2≤⋯≤ln−1≤kl1l2⋯ln−1 ∈Qk(n∣Pl1l2⋯ln−1∣)−(kCn−1k+n−2−kCn−1k)=∑1≤l1<l2<⋯<ln−1≤kk(n∣Rl1l2⋯ln−1∣)+∑1≤l1≤l2≤⋯≤ln−1≤kl1l2⋯ln−1 ∈Qk(n∣Pl1l2⋯ln−1∣)−kCn−1k+n−2−kCn−1k=nkn−kCn−1k+n−2−kCn−1k=nkn−dim(S[n;k])−dim(K[n;k]). |
Therefore, {μl1l2⋯ln;1t1t2⋯tn−1 ; j}, {μl1l2⋯ln;1t1t2⋯tn−1 ; j} and {γl1l2⋯lnt1t2⋯tn−1 ; j} form a basis of E[n;k].
Remark 4.1. We have given the bases of skew-symmetric game subspace K[n;k] and asymmetric game subspace E[n;k]. In our recently published paper [6], a basis of symmetric game subspace S[n;k] has also been given. Let the bases of S[n;k], K[n;k], E[n;k] be {α1,α2,⋯,αs}, {β1,β2,⋯,βt}, {γ1,γ2,⋯,γl} respectively, where s=kCn−1k+n−2, t=kCn−1k, l=nkn−kCn−1k+n−2−kCn−1k. For any G∈G[n;k], there are real numbers p1,⋯,ps,q1,⋯,qt,r1,⋯,rl such that
VG=p1α1+⋯+psαs+q1β1+⋯+ptβt+r1γ1+⋯+rlγl. |
Thus, [p1,⋯,ps,q1,⋯,qt,r1,⋯,rl]T is a solution of equation
[αT1⋯αTsβT1⋯βTtγT1⋯γTl]x=VTG. | (4.21) |
Since the coefficient matrix of (4.21) is a nonsingular matrix and each row has less than 3n! nonzero elements, the computational complexity of game decomposition is less than or equal to O(n!nkn).
Example 4.1. Consider G[3;2]. If 1≤l1<l2≤2, we have l1=1,l2=2. Then MˉX2=[0, 1, −1 0]T. If 1≤l1≤l2≤2, then l1l2=11, l1l2=12 or l1l2=22. Therefore,
MX2=[100010010001]. |
According to (3.28), a basis of K[3;2] is composed of the columns of matrix
[W[22,2]−I2⊗W[2,2]I23](MˉX2⊗I2) | (4.22) |
According to (4.1), a basis of S[3;2] is composed of the columns of matrix
[W[22,2]I2⊗W[2,2]I23](MX2⊗I2) | (4.23) |
According (4.18)-(4.20), the basis of E[3;2] and all non-zero elements in each vector are as follows:
μ121,112;1,x3121=1,x1112=−1; |
μ121,121;2,x3121=1,x2211=−1; |
μ122,112;1,x3122=1,x1212=−1; |
μ122,121;2,x3122=1,x2221=−1; |
μ121,−112;2,x3211=1,x2112=−1; |
μ121,−121;1,x3211=1,x1121=−1; |
μ122,−112;2,x3212=1,x2122=−1; |
μ122,−121;1,x3212=1,x1221=−1; |
γ11111;1,x3111=1,x1111=−1; |
γ11111;2,x3111=1,x2111=−1; |
γ11211;1,x3112=1,x1211=−1; |
γ11211;2,x3112=1,x2121=−1; |
γ22122;1,x3221=1,x1122=−1; |
γ22122;2,x3221=1,x2212=−1; |
γ22222;1,x3222=1,x1222=−1; |
γ22222;2,x3222=1,x2222=−1. |
It is easy to verify that the basis of E[3;2] are orthogonal to the columns of the matrices shown in (4.22) and (4.23).
This paper mainly investigates skew-symmetric game, asymmetric game and the problem of symmetric-based decomposition of finite games. By the semi-tensor product of matrices method with adjacent transpositions, necessary and sufficient conditions for testing skew-symmetric games are obtained. Based on the necessary and sufficient conditions of skew-symmetric games, a basis of skew-symmetric game subspace is constructed explicitly. In addition, the discriminant equations for skew-symmetric games with the minimum number are derived concretely, which reduce the computational complexity. Benefiting from the construction methods of the bases of symmetric game subspace and skew-symmetric game subspace given by us, a basis of asymmetric game subspace is constructed for the first time. Then, any game in G[n;k] can be linear represented by the bases of symmetric game subspace, skew-symmetric game subspace and asymmetric game subspace given by this paper and our previous work. Therefore, the problem of symmetric-based decomposition of finite games is completely solved. Some other kind of games can also be investigated in the frame of semi-tensor product of matrices [12,13,14,15]. We will try to generalize the obtained results in our future work.
The research work is supported by NNSF of China under Grant 62103194, Natural Science Foundation of Shandong Province of China under Grant ZR2020QA028.
The authors declare no conflict of interest.
[1] |
Aguilera-Caracuel J, Ortiz-de-Mandojana N (2013) Green innovation and financial performance: An institutional approach. Organ Environ 26: 365-385. doi: 10.1177/1086026613507931
![]() |
[2] | Al-Saidi M, Al-Shammari BA (2015) Ownership concentration, ownership composition and the performance of the Kuwaiti listed non-financial firms. Int J Commer Manage 25. |
[3] |
Al-Tuwaijri SA, Christensen TE, Hughes KE (2004) The relationship among Environmental Disclosure, Environmental Performance, and Economic Performance: A Simultaneous Equation Approach. Accounting Organ Society 29: 447-471. doi: 10.1016/S0361-3682(03)00032-1
![]() |
[4] |
Alexopoulos I, Kounetas K, Tzelepis D (2011) Environmental performance and technical efficiency, is there a link? Int J Prod Perform Manage 61: 6-23. doi: 10.1108/17410401211187480
![]() |
[5] | Arafat MY, Warokka A, Dewi SR (2012) Does Environmental Performance Really Matter? A Lesson from the Debate of Environmental Disclosure and Firm Performance. J Organ Manage Stud 2012. |
[6] |
Berrone P, Gomez-Mejia LR (2009) Environmental performance and executive compensation: An integrated agency-institutional perspective. Acad Manage J 52: 103-126. doi: 10.5465/amj.2009.36461950
![]() |
[7] | Bini L, Giunta F, Dainelli F (2010) Signalling Theory and Voluntary Disclosure to the Financial Market-Evidence from the Profitability Indicators Published in the Annual Report. Social Science Research Network 1930177. |
[8] | Brouwers R, Schoubben F, Hulle CV, et al. (2014) The link between corporate environmental performance and corporate value: a literature review. Rev Bus Econ Lit 58: 343-374. |
[9] |
Caixe DF, Krauter E (2013) The Influence of the Ownership and Control Structure on Corporate Market Value in Brazil. R Cont Financ 24: 142-153. doi: 10.1590/S1519-70772013000200005
![]() |
[10] | Chang K (2015) The Impacts of Environmental Performance and Propensity Disclosure on Financial Performance: Empirical Evidence from Unbalanced Panel Data of Heavy-pollution Industries in China. J Ind Eng Manange 8: 21-36. |
[11] |
Che-Ahmad A, Osazuwa NP (2015) Eco-efficiency and firm value of Malaysian firms. Int J Managerial Financ Accounting 7: 235-245. doi: 10.1504/IJMFA.2015.074902
![]() |
[12] |
Clarkson PM, Li Y, Richardson GD, et al. (2008) Revisiting the relation between environmental performance and environmental disclosure: An empirical analysis. Accounting Organ Society 33: 303-327. doi: 10.1016/j.aos.2007.05.003
![]() |
[13] |
Deegan C (2002) Introduction: The legitimising effect of social and environmental disclosures-a theoretical foundation. Accounting Auditing Accountability J 15: 282-311. doi: 10.1108/09513570210435852
![]() |
[14] |
Desoky AM, Mousa GA (2013) An empirical investigation of the influence of ownership concentration and identity on firm performance of Egyptian listed companies. J Accounting Emerging Econ 3: 164-188. doi: 10.1108/20421161311320698
![]() |
[15] |
Dye RA (1985) Disclosure of Nonproprietary Information. J Accounting Res 23: 123-145. doi: 10.2307/2490910
![]() |
[16] |
Farooque OA, Zijl TV, Dunstan K, et al. (2010) Co-deterministic relationship between ownership concentration and corporate performance. Accounting Res J 23: 172-189. doi: 10.1108/10309611011073250
![]() |
[17] |
Fontana S, D'Amico E, Coluccia D, et al. (2015) Does environmental performance affect companies' environmental disclosure? Meas Bus Excellence 19: 42-57. doi: 10.1108/MBE-04-2015-0019
![]() |
[18] |
Freeman RE, Evan WM (1990) Corporate governance: A stakeholder interpretation. J Behav Econ 19: 337-359. doi: 10.1016/0090-5720(90)90022-Y
![]() |
[19] | Handayati P, Rochayatun S (2015) The Effect Of Environmental Performance And Corporate Governance Mechanism On The Corporate Social Responsibility Disclosure. Int J Bus Econ Law 8. |
[20] |
Hart SL (1995) A natural resource-based view of the firm. Acad Manage Rev 20: 874-907. doi: 10.5465/amr.1995.9512280024
![]() |
[21] | Hart SL (1997) Beyond Greening: Strategies for a sustainable world. Harv Bus Rev 24: 66-76. |
[22] |
Iatridis GE (2013) Environmental disclosure quality: Evidence on environmental performance, corporate governance and value relevance. Emerging Markets Rev 14: 55-75. doi: 10.1016/j.ememar.2012.11.003
![]() |
[23] | Iqbal M, Sutrisno T, Assih P, et al. (2013) Effect of environmental accounting implementation and environmental performance and environmental information disclosure as mediation on company value. Int J Bus Manage Invent 2: 55-67. |
[24] |
Jaafar A, El-Shawa M (2009) Ownership concentration, board characteristics and performance: evidence from Jordan. Accounting Emerging Econ 9: 73-95. doi: 10.1108/S1479-3563(2009)0000009005
![]() |
[25] |
Jensen MC (2001) Value Maximization, Stakeholder theory, and the Corporate Objective Function. Eur Financ Manage 7: 297-317. doi: 10.1111/1468-036X.00158
![]() |
[26] |
Jo H, Harjoto MA (2012) The Causal Effect of Corporate Governance on Corporate Social Responsibility. J Bus Ethics 106: 53-72. doi: 10.1007/s10551-011-1052-1
![]() |
[27] |
Khlif H, Guidara A, Souissi M (2015) Corporate social and environmental disclosure and corporate performance. J Accounting Emerging Econ 5: 51-69. doi: 10.1108/JAEE-06-2012-0024
![]() |
[28] |
Krivogorsky V, Grudnitski G (2010) Country-specific institutional effects on ownership: concentration and performance of continental European firms. J Manage Gov 14: 167-193. doi: 10.1007/s10997-009-9097-6
![]() |
[29] |
Kuo L, Chen VY-J (2013) Is environmental disclosure an effective strategy on establishment of environmental legitimacy for organization? Manage Decis 51: 1462-1487. doi: 10.1108/MD-06-2012-0395
![]() |
[30] | Latan H, Ghozali I (2016) Partial Least Squares Concepts, Methods and Applications Using WarpPLS 50, Semarang: Diponegoro University Publisher Agency. |
[31] | Lin CJ (2011 ) An Examination Of Board And Firm Performance: Evidence From Taiwan. Int J Bus Financ Res 5. |
[32] |
Lindenberg EB, Ross SA (1981) Tobin's q Ratio and Industrial Organization. J Bus 54: 1-32. doi: 10.1086/296120
![]() |
[33] | Majumdar SK, Marcus AA (2001) Rules versus Discretion: The Productivity Consequences of Flexible Regulation. Acad Manage J 44: 170-179. |
[34] |
Marcus A, Geffen D (1998) The Dialectics of Competency Acquisition: Pollution Prevention in Electric Generation. Strategic Manage J 19: 1145-1168. doi: 10.1002/(SICI)1097-0266(1998120)19:12<1145::AID-SMJ6>3.0.CO;2-B
![]() |
[35] |
Mousa GA, Hassan NT (2015) Legitimacy Theory and Environmental Practices: Short Notes. Int J Bus Stat Anal 2: 41-53. doi: 10.12785/ijbsa/020104
![]() |
[36] |
Muhammad N, Scrimgeour F, Reddy K, et al. (2015) The Relationship between Environmental Performance and Financial Performance in Periods of Growth and Contraction: Evidence from Australian Publicly Listed Companies. J Clean Prod 102: 324-332. doi: 10.1016/j.jclepro.2015.04.039
![]() |
[37] |
Nguyen T, Locke S, Reddy K (2015) Ownership concentration and corporate performance from a dynamic perspective: Does national governance quality matter? Int Rev Financ Anal 41: 148-161. doi: 10.1016/j.irfa.2015.06.005
![]() |
[38] |
Plumleea M, Brownb D, Hayesa RM, et al. (2015) Voluntary environmental disclosure quality and firm value: Further evidence. J Accounting Public Policy 34: 336-361. doi: 10.1016/j.jaccpubpol.2015.04.004
![]() |
[39] | Porter ME, Van der Linde C (1995) Green and competitive: Ending the stalemate. Harv Bus Rev 73: 120-134. |
[40] | PROPER (2011) Report on the Results of the Company Performance Rating Assessment Program in Environmental Management. Report of the Ministry of Environment of the Republic of Indonesia. |
[41] | Rashida A, Zoysab AD, Lodh S, et al. (2010) Board Composition and Firm Performance: Evidence from Bangladesh. Australas Accounting Bus Financ J 4. |
[42] |
Ross SA (1977) The Determination of Finacial Structure:The Incentive Signalling Approach. Bell J Econ 8: 23-40. doi: 10.2307/3003485
![]() |
[43] |
Saka C, Oshika T (2014) Disclosure effects, carbon emissions and corporate value. Sustainability Accounting Manage Policy J 5: 22-45. doi: 10.1108/SAMPJ-09-2012-0030
![]() |
[44] | Sarumpaet S, Nelwan ML, Dewi DN (2017) The value relevance of environmental performance: evidence from Indonesia. Social Responsib J. |
[45] |
Sharma S, Pablo AL, Vredenburg H (1999) Corporate environmental responsiveness strategies: The importance of issue interpretation and organizational context. J Appl Behav Sci 35: 87-108. doi: 10.1177/0021886399351008
![]() |
[46] | Sholihin M, Ratmono D (2013) SEM-PLS Analysis with Warp PLS 3.0 for Nonlinear Relations in Social and Business Research, Yogyakarta: ANDI Publisher. |
[47] |
Stocken PC (2000) Credibility of Voluntary Disclosure. RAND J Econ 31: 359-374. doi: 10.2307/2601045
![]() |
[48] |
Utomo MN, Wahyudi S, Muharam H, et al. (2019) Linking Ownership Concentration to Firm Value: Mediation Role of Environmental Performance. J Environ Manage Tourism X: 182-194. doi: 10.14505//jemt.10.1(33).18
![]() |
[49] | Utomo MN, Wahyudi S, Muharam H, et al. (2018) Strategy to Improve Firm Performance Through Operational Efficiency Commitment to Environmental Friendliness: Evidence from Indonesia. Organ Markets Emerging Econ 9: 62-85. |
[50] |
Verrecchia RE (1983) Discretionary Disclosure. J Accounting Econ 5: 179-194. doi: 10.1016/0165-4101(83)90011-3
![]() |
[51] | Wahla K-U-R, Shah SZA, Hussain Z (2012) Impact of Ownership Structure on Firm Performance Evidence from Non-Financial Listed Companies at Karachi Stock Exchange. Int Res J Financ Econ. |
[52] | Warrad L, Almahamid SM, Slihat N, et al. (2013) The Relationship Between Ownership Concentration And Company Performance, A Case Of Jordanian Non-Financial Listed Companies. Interdiscip J Contemp Res Bus 4. |