The paper is concerned with a two-person zero sum matrix game with fuzzy payo s, with saddle-point being de ned with a classic de nition in matrix game. In order to compare fuzzy numbers some di erent ordering operators can be used. The original game can be associated with the bimatrix game with crisp payo s which are the operators value on a fuzzy payo . We propose that every player use its ordering operator. The following statement can be proposed: when the ordering operators are linear, the same equilibrium strategy pro le can be used for the matrix game, with fuzzy payo s being the same for the bimatrix crisp game. We introduce and employ the algorithm of constructing a saddle-point in a two-person zero sum matrix game with fuzzy payo s. In the instances of matrix games we use such ordering operators and construct the saddle-point.
Game theory is well known to take a signi cant part in decision making, this theory being often used to model the real world. Applied in real situations the game theory is di cult to have strict values of payo s, because it is di cult for players to analyze some data of game. Thus, the players' information cannot be considered complete. Besides the players can have vague targets.
This uncertainty and lack of precision may be modeled as fuzzy games. Fuzzy sets are known to be initially used in non-cooperative game theory by Butnariu [3] to prove the belief of each player for strategies of other players. Since then fuzzy set theory has been applied in cooperative and non-cooperative games. The results of fuzzy games overview are given in [10]. Recently there were made Copyright c by the paper's authors. Copying permitted for private and academic purposes. In: S. Belim et al. (eds.): OPTA-SCL 2018, Omsk, Russia, published at http://ceur-ws.org various e orts in fuzzy bi-matrix game theory namely Maeda [11], Nayak [13], Dutta [6], Seikh [14], Verma and Kumar [16].
In [9], we represented the approach which generalizes some other ideas ([4], [5],[6] at al.). 2
In this part, some basic de nitions and results of fuzzy numbers and fuzzy arithmetic operations will be reminded. Here we will follow to [21].
A fuzzy set can be considered as a subset A~ of universal set X R by its membership function A~( ) with a real number A~(x) in the interval [0; 1] and assigns to each element x 2 R.
De nition 2.1. A fuzzy subset A~ de ned on R, is said to be a fuzzy number if
its membership function A~(x) comply with the following conditions:
(
The -cut of a fuzzy number A~ plays an important role in parametric ordering of fuzzy numbers. The -cut or -level set of a fuzzy number A~, denoted by A~ , is de ned as A~ = fx 2 R j A~(x) > g for all 2 (0; 1]. The support or 0-cut A~0 is de ned as the closure of the set A~0 = fx 2 R j A~(x) > 0g. Every -cut is a closed interval A~ = [gA~( ); GA~( )] R, where gA~( ) = inffx 2 R j A~(x) > g and GA~( ) = supfx 2 R j A~(x) > g for any 2 [0; 1].
The sets of fuzzy number is denoted as F. Then, two types of fuzzy numbers are used.
De nition 2.2. Let A~ be a fuzzy number. If the membership function of A~ is given by
8 x a+l ; f or x 2 [a l; a]; A~(x) = < a+rrl x ; f or x 2 [a; a + r];
: 0; otherwise; where, a, l and r are all real (crisp) numbers, and l, r are non-negative. Then A~ is called a triangular fuzzy number, denoted by A~ = (a; l; r).
The sets of triangular fuzzy number are denoted as F3.
De nition 2.3. Let A~ be a fuzzy number. If the membership function of A~ is given by
A~(x) = 8>>< x 1la;+l ; ff oorr xx 22 [[aa; b]l; a];
b+r x ; f or x 2 [b; b + r]; >>: 0r; otherwise; where, a, b, l and r are all real (crisp) numbers, and l, r are non-negative. Then A~ is called a trapezoidal fuzzy number, denoted by A~ = (a; b; l; r). [a; b] is the core of A~.
The sets of trapezoidal fuzzy number are denoted as F4.
If A~ = (a1; l1; r1) and B~ = (a2; l2; r2) are two triangular fuzzy numbers, arithmetic operations on A~ and B~ are de ned as follows:
Addition: A~ + B~ = C~ = (a1 + a2; l1 + l2; r1 + r2), C~ 2 F3.
Scalar multiplication: 8 k > 0, k 2 R,
kA~ = (ka1; kl1; kr1); kA~ 2 F3:
If A~ = (a1; b1; l1; r1) and B~ = (a2; b2; l2; r2) are two trapezoidal fuzzy numbers, arithmetic operations on A~ and B~ are de ned as follows:
Addition: A~ + B~ = C~ = (a1 + a2; b1 + b2; l1 + l2; r1 + r2), C~ 2 F4. Scalar multiplication: 8 k > 0, k 2 R,
kA~ = (ka1; kb1; kl1; kr1); kA~ 2 F4: Generaly, if A~ and B~ are two fuzzy numbers and A~ + B~ = C~, A~ = D~ and = const > 0, then
C~ = [gA~( ) + gB~ ( ); GA~( ) + GB~ ( )];
D~ = [ g ~( ); GA~( )]
A and for any 2 [0; 1].
To compare fuzzy numbers is crucial issue. There are plenty of various methods for comparing fuzzy numbers. For instance, fuzzy numbers can be ranked with the help of defuzzi cation methods.Defuzzi cation is the process of producing a real (crisp) value which correspond to a fuzzy number, the defuzzi cation approach being used for ranking fuzzy numbers. Fuzzy numbers are initially defuzzi ed and then the obtained crisp numbers are organised using the order relation of real numbers.
A function for ranking fuzzy subsets in unit interval was introduced by Yager in [17]. It was based on the integral of mean of the -cuts. Yager index is Y (A~) =
1 with Y is a subset of the unit interval and P is a probability distribution on Y , the de nition of fA~ being subjective for decision maker. The ordering operator was proposed by Ukhobotov in [15]
The defuzzi cation operator T is dependant on the order relation T . Then, we give the example.
Example 2.1. Let T ( ) = U ( ; ) and A~; B~; C~ 2 F3, A~ = (40; 8; 10), B~ = = (45; 20; 10), C~ = (42; 6; 4).
If X~ = (a; l; r) 2 F3, then
U (X~ ; ) = a + r Next, if = 0, then U (A~; 0) = 36, U (B~; 0) = 35, U (C~; 0) = 39. If = 12 , then U (A~; 12 ) = 40; 5, U (B~; 21 ) = 42; 5, U (C~; 12 ) = 41; 5. If = 1, then U (A~; 1) = 45, U (B~; 1) = 50, U (C~; 1) = 44.
Therefore, De nition 2.5. If 8 A~; B~ 2 F, 8 ;
= const ~ B ~ A ~ C
~ U( ;0) A
~ U( ; 12 ) C
~ U( ;1) A
U( ;0) C~; U( ; 12 ) B~;
U( ;1) B~: T ( A~ +
B~) =
T (A~) +
T (B~); then the defuzzi cation operator T ( ) is the linear defuzzi cation operator.
Clear, Yager index Y ( ) and operator U ( ; ) is linear. 3
In this part, some basic de nitions of non-cooperative game theory are presented.
Noncooperative N-Person Games Let us consider a non-cooperative N -players game in the class of pure strategies = hN; fXigi2N; ffi(x)gi2Ni; where N = 1; :::; N is the set of players' serial numbers; each player i chooses and applies its own pure strategy xi 2 Xi Rni , with no coalition with the others, which induces a strategy pro le being formed x = (x1; :::; xN ) 2 X = Y Xi
Rn (n = i2N
X ni);
i2N
for each i 2 N, a payo function fi(x) is de ned on the strategy pro le set X,
which gives the payo of player i. fi(x) is payo function of player i (i 2 N). In
addition, denote (xkzi) = (x1; :::; xi 1; zi; xi+1; :::xN ), f = (f1; :::; fN ).
De nition 3.1. A strategy pro le xe = (xe1; :::; xeN ) 2 X is called a Nash
equilibrium in the game (
Bimatrix Games A bimatrix game de ned by a pair (A; B) of real m n matrices being considered, matrices A and B are payo s to play I and II, respectively. M denotes the set of pure strategies of player I (matrix rows) and N stands for the set of pure strategies of player II (columns).
M = (1; : : : ; m);
N = (1; : : : ; n): The sets of mixed strategies of the two players being called X and Y , we want to write expected payo s as matrix products xAy and xBy, for mixed strategies x and y, so that x is a row vector and y is a column vector. That is, and De nition 3.2. A pair (xe; ye) 2 X game (A; B) if
X = f(x1; : : : ; xm) j xi > 0 (8i 2 M ); X xi = 1g
i2M Y = f(y1; : : : ; yn) j yj > 0 (8j 2 N ); X yj = 1g:
j2N xeAye > xAye 8x 2 X; xeBye > xeBy 8y 2 Y: The set of Nash equilibrium for a game (A; B) is proved to be non-empty in [12].
If matrix B = A, a bimatrix game is considered as a zero-sum matrix game. A solve of a zero-sum matrix game being a saddle-point.
Y is called a Nash equilibrium for the
(
Internally Instable Set of Nash Equilibrium
Now, consider internal instability of Xe. A subset X Rn is internally instable,
if there exist at least two strategy pro les x(j) 2 X (j = 1; 2) such that
hf (x(
Example 3.1. Consider a bimatrix game of the form (A; B), where
There are 3 Nash equilibrium strategy pro les x(i) = (x(1i); x(2i)) (i = 1; 2; 3):
Consequently, the set Xe is internally instable in the game (A; B), as for x(
1 2 1 2
f2(x(
1 2 1 2
f2(x(
We note that in the non-antagonistic setting of the game (
Associate the following auxiliary N -criterion problem with the game (
De nition 3.3. An alternative xP 2 Xe is Pareto optimal (e cient) in the
problem (
fi(x) > fi(xP ) (i 2 N); is infeasible, with at least one being a strict inequality.
Designate by XP the set of all fxP g. fxP g
XP .
According to De nition 3.3, the set XP satis es the inclusionXP is internally stable.
The following statement is obvious: if for all x 2 Xe we have
X fi(x) 6
X fi(xP ); i2N i2N
Xe and
(
Further, a non-cooperative N -person game is considered
e = hN; fXigi2N; ffei(x)gi2Ni:
(
Determining the concept of optimality, we have to compare payo s. We can used some defuzzi cation operator T (T ( ) = Y ( ), VP ( ), U ( ; ) etc.). In [9] we proposed the following de nition.
De nition 4.1. [9] A strategy pro le xe = (xe1; :::; xeN ) 2 X is called a T ( )-Nash
equilibrium in the game (
T fi(xe) (i 2 N): We note that the solutions, which de ned in [11], [5] and [6], are particular cases of De nition 4.1.
Next, we consider the associated crisp game for (
For example, we consider one bimatrix game with a triangular fuzzy payo s. Example 4.1. Consider a bimatrix game with fuzzy payo s e of the form (Ae; Be), where Ae and Be are the triangular fuzzy matrixes:
0 (30; 6; 12) (10; 8; 6) (15; 10; 5) 1
Ae = @ (20; 10; 5) (22; 6; 10) (30; 5; 10) A
(10; 8; 12) (30; 20; 4) (20; 8; 16)
and
The operator U ( ; ) is used. As a result, we obtain the associated crisp game
(
If = 21 , then the associated crisp game (
If = 1, then the game (
Another example: one zero-sum matrix game with a trapezoidal fuzzy payo s is considered .
Example 4.2. Let Ae be the trapezoidal fuzzy payo matrixes of the fuzzy zerosum matrix game e, which is given as follows:
Ae =
The mixed Y ( )-Nash equilibrium is xe = (xe1; xe2), where xe1 = ( 189 ; 1119 ), xe2 = ( 3187 ; 3281 ). 5
In this section, we consider a two-person zero sum game
ea = hf1; 2g; fXigi=1;2; fe1(x)i;
(
In addition, let Xi contains only a nite number of elements. In this case, ea is a two-person zero sum matrix game with fuzzy payo s. This game is determined by a fuzzy matrix Ae.
In the last section, we considered that the players prefer the same defuzzication operator. However, the players can have the di erent preferences. For example, it can be caused by the various attitude to the risk. In this case, the players can use the di erent defuzzi cation operators. The main idea of this paper is following:
Suppose that the player 1 has decided to use a defuzzi cation operator T1( ).
And the player 2 chose to use a defuzzi cation operator T2( ) (T1( ) 6= T2( )).
De nition 5.1. A strategy pro le x = (x1; x2) 2 X is called a T1( )T2(
)saddle-point in the game (
T1 fe1(x1; x2)
T2 fe1(x1; x2)
8 x1 2 X1; x2 2 X2:
Next, we consider the associated crisp game for (
Theorem 5.1. Let x is a Pareto-optimal Nash equilibrium in (
For example, we consider zero-sum matrix game with a trapezoidal fuzzy payo s from Example 4.2..
Example 5.1. Let Ae be the trapezoidal fuzzy payo matrixes of the fuzzy zerosum matrix game e, which is given as follows:
((250;9;;3200;;142); 8) ((110;;52;68;;84;)12) :
The operator Y ( ) is used for player 1, and the operator U ( ; 0) is used for player
2.
As a result, we obtain the associated crisp bimatrix game (
Acknowledgement. The work was supported by Act 211 Government of the Russian Federation, contract N 02.A03.21.0011 and by Grant of the Foundation for perspective scienti c researches of Chelyabinsk State University (2018).