A Fictitious Play Algorithm for Matrix Games with Fuzzy Payoffs

and Applied Analysis 3 2.1. Fuzzy Numbers In this section, we give certain essential concepts of fuzzy numbers and their basic properties. For further information see 2, 3 . A fuzzy set à on a set X is a function à : X → 0, 1 . Generally, the symbol μà is used for the function à and it is said that the fuzzy set à is characterized by itsmembership function μà : X → 0, 1 which associates with each x ∈ X, a real number μà x ∈ 0, 1 . The value of μà x is interpreted as the degree to which x belongs to Ã. Let à be a fuzzy set on X. The support of à is given as S ( à ) { x ∈ X : μà x > 0 } , 2.1 and the height h à of à is defined as h ( à ) sup x∈X μà x . 2.2 If h à 1, then the fuzzy set à is called a normal fuzzy set. Let à be a fuzzy set on X and α ∈ 0, 1 . The α-cut α-level set of the fuzzy set à is given by [ à ]α ⎧⎨ ⎩ { x ∈ X : μà x ≥ α } , if α ∈ 0, 1 clS ( à ) , if α 0, 2.3 where cl denotes the closure of sets. The notion of convexity is extended to fuzzy sets on R as follows. A fuzzy set à on R n is called a convex fuzzy set if its α-cuts Ãα are convex sets for all α ∈ 0, 1 . Let à be a fuzzy set in R, then à is called a fuzzy number if i à is normal, ii à is convex, iii μà is upper semicontinuous, and iv the support of à is bounded. From now on, we will use lowercase letters to denote fuzzy numbers such as ã and we will denote the set of all fuzzy numbers by the symbol F. Generally, some special type of fuzzy numbers, such as trapezoidal and triangular fuzzy numbers, are used for real life applications. We consider here L-fuzzy numbers. The function L : R → R satisfying the following conditions is called a shape function: i L is even function, that is, L x L −x for all x ∈ R, ii L x 1 ⇔ x 0, iii L · is nonincreasing on 0, ∞ , iv if x0 inf{x > 0 | L x 0}, then 0 < x0 < ∞ and x0 is called the zero point of L. 4 Abstract and Applied Analysis Let a be any number and let δ be any positive number. Let L be any shape function. Then a fuzzy number ã is called an L-fuzzy number if its membership function is given by μã x L ( x − a δ ) ∨ 0, x ∈ R. 2.4 Here, x∨y max{x, y}. Real numbers a and δ are called the center and the deviation parameter of ã, respectively. In particular, if L x 1 − |x|we get μã x ⎧⎨ ⎩1 − 1 δ |x − a|, x ∈ a − δ, a δ 0, otherwise 2.5 and ã is called a symmetric triangular fuzzy number. It is clear that for any shape function L, an arbitrary L-fuzzy number ã can be characterized by the its center a and the deviation parameter δ. Therefore, we denote the L-fuzzy number ã by ã ≡ a, δ L. In particular, if ã is a symmetric triangular fuzzy number, we write ã ≡ a, δ T . We also denote the set of all L-fuzzy numbers by FL. Let ã ≡ a, δ L be an L-fuzzy number then by 2.4 we see that the graph of μã x approaching line x a as δ tends to zero from the right. Therefore, we can write that μã x { 1, x a 0, x / a. 2.6 The function in 2.6 is just a characteristic function of the real number a. Hence, we get R ⊂ FL. From now on, we will call a fuzzy number ã as an L-fuzzy number if its membership function is given by 2.4 or 2.6 . Let ã, b̃ ∈ F and k be any real number. Then the sum of fuzzy numbers ã and b̃ and the scalar product of k and ã are defined as μã b̃ z sup z x y min { μã x , μb̃ ( y )} , μkã z max { 0, sup z kx μã x } , 2.7 respectively. In particular, if ã ≡ a, δ1 L and b̃ ≡ b, δ2 L are L-fuzzy numbers and k is any real number, then one can verify that ã b̃ ≡ a b, δ1 δ2 L, kã ≡ ka, |k|δ1 L. 2.8 Let ã be any L-fuzzy number. By the definition of the α-cut, ã α is a closed interval for all α ∈ 0, 1 . Therefore, for all α ∈ 0, 1 we can denote the α-cut of ã by aα, aα , where aα and a R α are end points of the closed interval ã . Abstract and Applied Analysis 5and Applied Analysis 5 For any symmetric triangular fuzzy numbers ã, b̃ Ramı́k and Řı́mánek see 4 introduced binary relations as follows: ã qb̃ ⇐⇒ aα b α, aα b α ∀α ∈ 0, 1 ( fuzzy max order ) , ã b̃ ⇐⇒ ã qb̃, ã / b̃ ∀α ∈ 0, 1 ( strict fuzzy max order ) , ã b̃ ⇐⇒ aα > b α, aα > b α ∀α ∈ 0, 1 ( strong fuzzy max order ) . 2.9 Following theorem is a useful tool to check fuzzy max order and strong fuzzy max order relations between symmetric triangular fuzzy numbers. Theorem 2.1 see 5 . Let ã ≡ a, δ1 and b̃ ≡ b, δ2 be any symmetric triangular fuzzy numbers. Then the statements ã qb̃ ⇐⇒ |δ1 − δ2| a − b, ã b̃ ⇐⇒ |δ1 − δ2| < a − b 2.10 hold. It is not difficult to check that the fuzzy max order is a partial order. Then we may have many minimal and maximal points with respect to fuzzy max order. Therefore, use of the fuzzy max order is not so efficient in computer algorithms. Furukawa introduced a total order relation which is a modification of the fuzzy max order with a parameter see 5, 6 . Let 0 ≤ λ ≤ 1 be arbitrary but a fixed real number. For any L-fuzzy numbers ã ≡ a, δ1 L and b̃ ≡ b, δ2 L we define an order relation with parameter λ by ã≤λb̃ ⇐⇒ ⎧⎪⎪⎪⎪⎨ ⎪⎪⎪⎩ i x0|δ1 − δ2| ≤ b − a, or ii λx0|δ1 − δ2| ≤ b − a < x0|δ1 − δ2|, or ii |a − b| < λx0|δ1 − δ2|, δ2 > δ1, 2.11 where x0 is the zero point of L. The simple expression of 2.11 is as follows: ã≤λb̃ ⇐⇒ ⎪⎨ ⎪⎩ i λx0δ1 a < λx0δ2 b, or ii λx0δ1 a λx0δ2 b, δ2 ≤ δ1. 2.12 It is clear that for any L-fuzzy numbers ã ≡ a, δ1 L and b̃ ≡ b, δ2 L, ã≤0 b̃ if and only if a ≤ b. Therefore, the relation ≤0 is the order among the centers of L-fuzzy numbers. On the other hand, ã≤1 b̃ if and only if b̃ ã or they are incomparable and δ2 > δ1. For 0 < λ < 1, the relation ≤λ determines the order with respect to their values of center and their size of 6 Abstract and Applied Analysis ambiguity. The smaller λ is, the larger the possibility of ordering by the value of center is, and the larger λ is, the larger the possibility of ordering by the size of ambiguity is. Theorem 2.2 see 5 . For every shape function L and for each λ ∈ 0, 1 , the relation ≤λ is a total order relation on FL. Let λ ∈ 0, 1 be fixed arbitrarily and let Ṽ ṽ1, ṽ2, . . . , ṽn be any L-fuzzy vector, that is, all components of Ṽ are L-fuzzy numbers and expressed by a common shape function L. Then maximum and minimum of Ṽ in the sense of the total order ≤λ are denoted as Max λ Ṽ , Min λ Ṽ , 2.13 respectively. Example 2.3. Let Ṽ −2, 0.1 L, 0, 0.1 L, −3, 0.3 L, −1, 0.4 L be L-fuzzy vector. Then Max λ ( Ṽ ) 0, 0.1 L, Min λ ( Ṽ ) −3, 0.1 L, 2.14 for all λ ∈ 0, 1 . Let ã and b̃ be any L-fuzzy numbers, then the Hausdorff distance between ã and b̃ is defined as d ( ã, b̃ ) sup α∈ 0,1 max {∣∣∣aLα − b α∣∣∣, ∣∣∣aRα − b α ∣∣∣}, 2.15 that is, d ã, b̃ is the maximal distance between α-cuts of ã and b̃. In particular, if ã ≡ a, δ1 and b̃ ≡ b, δ2 are any symmetric triangular fuzzy numbers, then d ã, b̃ |a − b|. 2.2. Two-Person Zero-Sum Game with Fuzzy Payoffs and Its Equilibrium Strategy In this section, we consider zero-sum games with fuzzy payoffs with two players, and we assume that player I tries to maximize the profit and player II tries to minimize the costs. The two-person zero-sum gamewith fuzzy payoffs is defined bym×nmatrix G̃whose entries are fuzzy numbers. Let G̃ be a fuzzy matrix game G̃ II1 II2 · · · IIn I1 I2 .. Im ⎛ ⎜⎜⎜⎝ g̃11 g̃12 · · · g̃1n g̃21 g̃22 · · · g̃2n .. .. . . . .. g̃m1 g̃m2 · · · g̃mn ⎞ ⎟⎟⎟⎠ 2.16 Abstract and Applied Analysis 7 and x ∈ Xm, y ∈ Yn, that is, x and y are strategies for players I and II. Then the expected payoff for player I is defined byand Applied Analysis 7 and x ∈ Xm, y ∈ Yn, that is, x and y are strategies for players I and II. Then the expected payoff for player I is defined by