In this paper we consider a bimatrix game with fuzzy payoffs. To compare a fuzzy numbers, some different ordering operators can be used. We define a Nash equilibrium in fuzzy game using the ordering operator. The game with crisp payoffs is associated with the original game. Here, a crisp payoffs are the operator's value on a fuzzy payoff. We propose the following statement: if the ordering operator is linear, then the game with payoffs have same Nash equilibrium strategy profile as the crisp game. We present an algorithm for constructing a Nash equilibriumin in a bimatrix game with fuzzy payoffs and we are using this fact. We use such ordering operators, and construct the Nash equilibrium in examples of bimatrix games.
Game theory takes an important role in decision making and actively is used for modeling the real world. Applying game theory in real situations, it is difficult to have strict value of payoffs, because players are not able to analyze some data of game and as a result, their information isn‘t complete. This lack of precision and certainty may be modeled by different ways such as fuzzy games. Initially, fuzzy sets were used by [Butnariu, 1978] in non-cooperative game theory. He used fuzzy sets to represent the belief of each player for strategies of other players. Since then, fuzzy set theory has been used in many non-cooperative and cooperative games. Overview of the results of fuzzy games are in [Larbani, 2009]. In recent past, various attempt have been made in fuzzy bi-matrix game theory namely [Maeda, 2003], [Nayak, 2009], [Dutta, 2014], [Seikh et al., 2015]
In this paper, we present an approach that generalizes some other ideas
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In this section, we recall some basic concepts and result of fuzzy numbers and fuzzy arithmetic operations. Here, we follow [Zimmermann, 1996].
A fuzzy set is defined as a subset A˜ of universal set X ⊆ R by its membership function A~(·) with assigns to each element x ∈ R, a real number A~(x) in the interval [0; 1].
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) satis es the following conditions: (1) A~(x) : R → [0; 1] is upper semi-continuous; (2) A~(x) = 0 outside some interval [a; d]; (3) There exist real numbers b, c such that a 6 b 6 c 6 d and (a) A~(x) is monotonic increasing on [a; b]; (b) A~(x) is monotonic decreasing on [c; d]; (c) A~(x) = 1, ∀x ∈ [b; c].
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 defined as A˜ = {x ∈ R | A~(x) > } for all ∈ (0; 1]. The support or 0-cut A˜0 is defined as the closure of the set A˜0 = {x ∈ R | A~(x) > 0}. Every -cut is a closed interval A˜ = [gA~( ); GA~( )] ⊂ R, where gA~( ) = inf{x ∈ R | A~(x) > } and GA~( ) = sup{x ∈ R | A~(x) > } for any ∈ [0; 1].
We denote the sets of fuzzy number as F. Next, we use two types of fuzzy numbers.
De nition 2.2. Let A˜ be a fuzzy number. If the membership function of A˜ is given by
x a+l ; f or x ∈ [a − l; a]; A~(x) = a+rrl x ; f or x ∈ [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).
We denote the sets of triangular fuzzy number as F3.
De nition 2.3. Let A˜ be a fuzzy number. If the membership function of A˜ is given by
A~(x) = x 1la;+l ; ff oorr xx ∈∈ [[aa;−b]l; a]; b+r0r; x ; oftohrerxw∈is[eb;; b + r]; 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˜.
We denote the sets of trapezoidal fuzzy number as F4.
Let A˜ = (a1; l1; r1) and B˜ = (a2; l2; r2) be two triangular fuzzy numbers. Then arithmetic operations on A˜ and B˜ are defined as follows:
Addition: A˜ + B˜ = C˜ = (a1 + a2; l1 + l2; r1 + r2), C˜ ∈ F3.
Scalar multiplication: ∀ k > 0, k ∈ R,
kA˜ = (ka1; kl1; kr1); kA˜ ∈ F3:
Let A˜ = (a1; b1; l1; r1) and B˜ = (a2; b2; l2; r2) be two trapezoidal fuzzy numbers. Then arithmetic operations on A˜ and B˜ are defined as follows:
Addition: A˜ + B˜ = C˜ = (a1 + a2; b1 + b2; l1 + l2; r1 + r2), C˜ ∈ F4.
Scalar multiplication: ∀ k > 0, k ∈ R, In general, let A˜ and B˜ be two fuzzy numbers. If A˜ + B˜ = C˜, A˜ = D˜ and = const > 0, then kA˜ = (ka1; kb1; kl1; kr1); kA˜ ∈ F4: ˜ C = [gA~( ) + g ~ ( ); GA~( ) + GB~ ( )];
B and ˜ D = [ g ~( ); GA~( )]
A for any ∈ [0; 1].
Comparing of fuzzy numbers is a very important question. Various methods for comparing fuzzy numbers have been proposed. For example, fuzzy numbers can be ranked using the defuzzification methods. A defuzzification is the process of producing a real (crisp) value corresponding to a fuzzy number. In order to rank fuzzy numbers are using the defuzzification approach, the fuzzy numbers are first defuzzified and then, the obtained crisp numbers are ordered using the order relation of real numbers.
Yager in [Yager, 1981] introduced a function for ranking fuzzy subsets in unit interval, which is based on the integral of mean of the -cuts. Yager index is
Jain in [Jain, 1977], Baldwin and Guild in [Baldwin, 1979] were also suggested methods for ordering fuzzy subsets in the unit interval.
Ibanez and Munoz in [Ibanez, 1989] have developed a subjective approach for ranking fuzzy numbers. In [Ibanez, 1989], Ibanez and Munoz defined the following number as the average index for fuzzy number A˜ where Y is a subset of the unit interval and P is a probability distribution on Y . The definition of fA~ could be subjective for decision maker.
Ukhobotov in [Ukhobotov, 2016] proposed the ordering operator
U (A˜; ) = ((1 − )gA~( ) +
GA~( )) d ; where crisp parameter ∈ [0; 1]. Different correspond to different behavior of the decision maker.
Some other defuzzification operators were given in [Basiura et al., 2015].
De nition 2.4. Let A˜ and B˜ are a fuzzy numbers, T : F → R is the operator of defuzzi cation (T (·) = Y (·), VP (·), U (·; ) etc.).
We say that B˜ is preferable to A˜ by the defuzzi cation operator T (A˜ ≼T B˜) if and only if
T (A˜) 6 T (B˜):
The order relation ≼T depends on the defuzzification operator T . Next we give the example. Example 2.1. Let A˜; B˜; C˜ ∈ F3, A˜ = (50; 10; 20), B˜ = (55; 30; 12), C˜ = (52; 30; 30), T (·) = U (·; ). If X˜ = (a; l; r) ∈ F3, then
U (X˜ ; ) = a + r − (1 − )l : 2 Next, if = 0, then U (A˜; 0) = 45, U (B˜; 0) = 40, U (C˜; 0) = 37. If = 12 , then U (A˜; 12 ) = 52; 5, U (B˜; 21 ) = 51; 5, U (C˜; 12 ) = 52. If = 1, then U (A˜; 1) = 60, U (B˜; 1) = 61, U (C˜; 1) = 67.
Therefore,
Y (A˜) =
(gA~( ) + GA~( )) d : VP (A˜) = ∫ Y f ~( )dP ( );
A 1 ∫ 0
1
0 De nition 2.5. If ∀ A˜; B˜ ∈ F, ∀ ; then the defuzzi cation operator T (·) is the linear defuzzi cation operator.
Clear, Yager index Y (·) and operator U (·; ) is linear.
˜ ˜ ˜ C ≼U( ;0) B ≼U( ;0) A; ˜ ˜ ˜ B ≼U( ; 12 ) C ≼U( ; 12 ) A; A ≼U( ;1) B ≼U( ;1) C˜: ˜ ˜ = const T ( A˜ +
B˜) =
T (A˜) +
T (B˜);
In this section, we present some basic definitions of non-cooperative game theory. where N = 1; :::; N is the set of players’ serial numbers; each player i chooses and applies his own pure strategy xi ∈ Xi ⊆ Rni , forming no coalition with the others, which induces a strategy profile x = (x1; :::; xN ) ∈ X = ∏ Xi ⊂ Rn (n = i2N ∑ ni); i2N for each i ∈ N, a payoff function fi(x) is defined on the strategy profile set X, which gives the payoff of player i. fi(x) is payoff function of player i (i ∈ N). In addition, denote (x∥zi) = (x1; :::; xi 1; zi; xi+1; :::xN ), f = (f1; :::; fN ).
De nition 3.1. A strategy pro le xe = (xe1; :::; xeN ) ∈ X is called a Nash equilibrium in the game (1) if max fi(xe∥xi) = fi(xe) (i ∈ N): xi2Xi The set of all {xe} in the game (1) will be designated by Xe. 3.2
Bimatrix Games We consider a bimatrix game defined by a pair (A; B) of real m × n matrices. Matrices A and B are payoffs to play I and II, respectively. The set of pure strategies of player I (matrix rows) is denoted by M and the set of pure strategies of player II (columns) is denoted by N . and De nition 3.2. A pair (xe; ye) ∈ X × Y is called a Nash equilibrium for the game (A; B) if X = {(x1; : : : ; xm) | xi > 0 (∀i ∈ M ); ∑ xi = 1}
i2M Y = {(y1; : : : ; yn) | yj > 0 (∀j ∈ N ); ∑ yj = 1}:
j2N xeAye > xAye xeBye > xeBy ∀x ∈ X; ∀y ∈ Y: (1) (2) (3) M = (1; : : : ; m);
N = (1; : : : ; n): The sets of mixed strategies of the two players are called X and Y . For mixed strategies x and y, we want to write expected payoffs as matrix products xAy and xBy, so that x should be a row vector and y should be a column vector. That is, From [Nash, 1950] implies that the set of Nash equilibrium for a game (A; B) is non-empty.
We recall, a bimatrix game is a zero-sum bimatrix game if matrix B = −A. A solve of a zero-sum bimatrix game is sadle-point. 4
Further, we consider a non-cooperative N -person game
Γe = ⟨N; {Xi}i2N; {fei(x)}i2N⟩; which differs from (1) only payoffs functions. In (3), a payoff function of player i is fei(x) : X → F. In addition, Xi contains only a finite number of elements. Γe is a finite game with fuzzy payoffs. If N = {1; 2}, then (3) is a bimatrix game with fuzzy payoffs.
To determine the concept of optimality, we must compare payoffs. We use some defuzzification operator T (T (·) = Y (·), VP (·), U (·; ) etc.). We propose the following definition.
De nition 4.1. A strategy pro le xe = (xe1; :::; xeN ) ∈ X is called a T (·)-Nash equilibrium in the game (3) if fi(xe∥xi) ≼T fi(xe) (i ∈ N): We note that the solutions, which defined in [Maeda, 2003], [Cunlin, 2011] and [Dutta, 2014], are particular cases of Definition 4.1.
Next, we consider the associated crisp game for (3) Γec = ⟨N; {Xi}i2N; {T (fei(x))}i2N⟩: (4) Theorem 4.1. Let xe is a Nash equilibrium in (4) and T (·) is a linear defuzzi cation operator, then xe is T (·)-Nash equilibrium in a game (3).
For example, we consider one bimatrix game with a triangular fuzzy payoffs.
Example 4.1. Let Ae and Be are the triangular fuzzy payoff matrixes of the fuzzy bimatrix game Γe, given as follows:
Ae = The mixed U (·; 0)-Nash equilibrium is xe = (xe1; xe2), where xe1 = ( 52 ; 35 ), xe2 = ( 130 ; 170 ).
If = 12 , then the associated crisp game (4) given as follows The mixed U (·; 12 )-Nash equilibrium is xe = (xe1; xe2), where xe1 = ( 173 ; 163 ), xe2 = ( 159 ; 1194 ).
If = 1, then the associated crisp game (4) given as follows
A = The mixed U (·; 1)-Nash equilibrium is xe = (xe1; xe2), where xe1 = ( 5 ; 38 ), xe2 = ( 2 ; 79 ). 8 9
To give another example: we consider one zero-sum bimatrix game with a trapezoidal fuzzy payoffs. Example 4.2. Let Ae is the triangular fuzzy payoff matrixes of the fuzzy zero-sum bimatrix game Γe, given as follows: (
(20; 30; 12; 8) (1; 5; 8; 4) (5; 9; 20; 4) (10; 30; 8; 12) ) : We use the operator Y (·). The associated crisp game (4) given as follows
A = The work was supported by Act 211 Government of the Russian Federation, contract 02.A03.21.0011 and Grant of the Foundation for perspective scientific researches of Chelyabinsk State University (2017)