International Journal of Analysis and Applications Volume 16, Number 3 (2018), 353-367 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-353 CHARACTERIZATION OF NASH EQUILIBRIUM STRATEGY FOR HEPTAGONAL FUZZY GAMES F. MADANDAR1, S. HAGHAYEGHI1 AND S. M. VAEZPOUR2,∗ 1Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran 2Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran ∗Corresponding author: vaez@aut.ac.ir Abstract. In this paper, the Nash equilibrium strategy of two-person zero-sum games with heptagonal fuzzy payoffs is considered and the existence of Nash equilibrium strategy is studied. Also, based on the fuzzy max order several models in heptagonal fuzzy environment is constructed and the existence of their equilibrium strategies is proposed. In the sequel, the existence of Pareto Nash equilibrium strategies and weak Pareto Nash equilibrium strategies is investigated for fuzzy matrix games. Finally, the relation between Pareto Nash equilibrium strategy and parametric bi-matrix games is established. 1. Introduction Modern game theory was developed by the mathematician John Von Neumann in the Mid-1940‘s and in 1944, he published the book of ”Theory of games and economic behavior” joint with Morgenstern [9]. The most important categories are as cooperative and non-cooperative games. In 1951, non-cooperative games was presented by John Nash [8]. In this article we focus on a class of non-cooperative games namely two-person zero-sum matrix games. Moreover, one of the most important concepts in game theory is the Nash equilibrium. Nash proves that if we approve mixed strategies, then every game with a finite number of players has at least one Nash equilibrium. Received 2017-12-13; accepted 2018-02-05; published 2018-05-02. 2010 Mathematics Subject Classification. 91A05. Key words and phrases. zero-sum matrix game; fuzzy payoffs; Nash equilibrium; heptagonal fuzzy number. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 353 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-353 Int. J. Anal. Appl. 16 (3) (2018) 354 In 1978, the idea of fuzziness was exhibited by Zadeh [15], that is a type of subjective uncertainty. After then fuzzy numbers found many applications in various fields with incomplete information such as engineering, social and economics. In many science such as economics, business competition, auction and etc., the payoff for games is not realistic indeed the payoffs are fuzzy. Many of mathematicians and reseachers study the fuzziness. Zimmermann [16] in 1985, Yazenin [13] in 1987 and Sakawa [12] in 1993 applied the fuzzy theory to optimization problems. In 1989, Campos [2] transformed the fuzzy games into fuzzy optimization problems. In 1999, Liu [5] founded theory in the uncertain environments. In 2000, Meada [7] constructed kind of concepts of minimax equilibrium strategies. In 2005, C. R. Bector and S. Chandra [1] provided fuzzy mathematical programming and fuzzy matrix games. In 2011, Cunlin and Zhang Qiang [4] investigated two-person zero-sum games in the symmetric triangular fuzzy environment. They obtained Nash equilibrium of two-person zero-sum games with fuzzy payoff. They also obtained Pareto Nash equilibrium strategy for fuzzy matrix game. In 2014, Bapi Dutta [3] extended their work in trapezoidal fuzzy environment and he introduced two special types of fuzzy games: constant and proportional fuzzy game. In [3, 4, 7] the uncertainty and imprecision in payoffs have been represented by either triangular or trapezoidal fuzzy numbers. The most frequently used fuzzy numbers in the different problems are triangular or trapezoidal fuzzy numbers. But, it is not possible to restrict the membership functions to take either triangular or trapezoidal form. Therefore this paper focus on fuzzy payoffs of decision makers by heptagonal fuzzy numbers. In 2014, the arithmetic operations of heptagonal fuzzy numbers are defined by K. Rathi and S. Balamohan [10]. The heptagonal fuzzy number gives additional possibility to represent imperfect knowledge what leads to model many problems. Heptagonal fuzzy numbers have different applications in optimization problems and decision making problems which need seven parameters. In this paper we define the k-heptagonal fuzzy numbers and generalize Cunlin and Qiang [4] and Bapi Dutta [3] Nash equilibrium solution concepts. The paper is organized as follows: In section 2, the basic definitions and notations of fuzzy numbers are given. In section 3, we introduce the notation of two-person zero-sum matrix games with heptagonal fuzzy payoffs, the different types of equilibrium strategies and investigate their existence conditions for the fuzzy games. In section 4, parametric bi-matrix games are introduced and then the relation between parametric bi-matrix games and Nash equilibrium strategies is studied. In section 5, we present some illustrative exampes. 2. Preliminaries In this section, we suggest some basic definitions and concepts of fuzzy numbers, which were proposed by Zadeh [14] in 1965. Also, we introduce some notations of fuzzy sets, such as α-cut for heptagonal fuzzy numbers. Int. J. Anal. Appl. 16 (3) (2018) 355 Definition 2.1. [4] A fuzzy number ã is a fuzzy set on the real line R if its membership function µã(x) satisfies the following conditions. (i) µã(x) is a mapping from R to the closed interval [0, 1]; (ii) There exist a unique real number c, called the center of ã , such that ; (a) µã(c) = 1 ; (b) µã(x) nondecreasing on (−∞,c]; (c) µã(x) nonincreasing on [c, +∞). The α-cut or α-level of fuzzy number have an important role in parametric ordering of fuzzy numbers. The α-cut set of a fuzzy number ã, denoted by [ã]α. Every α-cut is a closed interval [ã]α = [a L α,a U α ] ⊂ R, where aLα = inf{x ∈ R|µã(x) ≥ α} and aUα = sup{x ∈ R|µã(x) ≥ α} for any α ∈ [0, 1]. For more details see [4, 6]. Definition 2.2. A fuzzy number ã = (a,c,b,h,l,r,m) is called a k-heptagonal if its membership function is defined as µã(x) =   k(x−a+h h ) ,a−h ≤ x ≤ a, k ,a ≤ x ≤ c− l, k + ( x−c+l l ) ,c− l ≤ x ≤ c, k + ( c+r−x r ) ,c ≤ x ≤ c + r, k ,c + r < x < b + m, 0 ,otherwise. where c is the center of ã and h,l,r,m are non-negative. In the rest of the paper, for simplicity, the k-heptagonal fuzzy number is denoted by k-HFN. Let ã = (a1,c1,b1,h1, l1,r1,m1) and b̃ = (a2,c2,b2,h2, l2,r2,m2) are two k-HFN then • Addition: ã + b̃ = (a1 + a2,c1 + c2,b1 + b2,h1 + h2, l1 + l2,r1 + r2,m1 + m2) • Subtraction: ã− b̃ = (a1 −m2,c1 −r2,b1 − l2,h1 −h2, l1 − b2,r1 − c2,m1 −a2) • Scalar Multiplication : λ > 0,λ ∈ R; λã = (λa1,λc1,λb1,λh1,λl1,λr1,λm1) λ < 0,λ ∈ R; λã = (λm1,λr1,λl1,λh1,λb1,λc1,λa1). Int. J. Anal. Appl. 16 (3) (2018) 356 By definition of α-cut we have the following lemma. Lemma 2.1. Let ã = (a1,c1,b1,h1, l1,r1,m1) be a k-HFN. Then for α ∈ (0, 1], the α-cut of ã is, [ã]α =   [ a− (1 − α k )h,b + (1 − α k )m ] ; α ∈ (0,k] [c− ( 1−α 1−k ) l,c + ( 1−α 1−k ) r] ; α ∈ [k, 1]. Definition 2.3. [4] Let x = (ξ1,ξ2, ...,ξn) and y = (η1,η2, ...,ηn) be vectors in Rn. Then (i) x = y if and only if ξi ≥ ηi for all i = 1, 2, ...,n, (ii) x ≥ y if and only if x = y and x 6= y, (iii) x > y if and only if ξi > ηi for all i = 1, 2, ...,n. Definition 2.4. [4] Let ã and b̃ be two fuzzy numbers.Then, (i) ã v b̃ if and only if (aLα,a U α ) = (b L α,b U α ), for all α ∈ [0, 1], (ii) ã % b̃ if and only if (aLα,a U α ) ≥ (bLα,bUα ), for all α ∈ [0, 1], (iii) ã � b̃ if and only if (aLα,aUα ) > (bLα,bUα ), for all α ∈ [0, 1]. The following theorem characterize the orders for heptagonal fuzzy numbers. Theorem 2.1. Let ã = (a1,c1,b1,h1, l1,r1,m1), b̃ = (a2,c2,b2,h2, l2,r2,m2) be two k-HFN. Then (i) ã w b̃ if and only if max{h2 −h1, 0}≤ a2 −a1 , max{m1 −m2, 0}≤ b2 − b1 , max{l2 − l1, 0}≤ c2 − c1 and max{r1 −r2, 0}≤ c2 − c1, (ii) ã ≺ b̃ if and only if max{h2 −h1, 0} < a2 −a1 , max{m1 −m2, 0} < b2 − b1 , max{l2 − l1, 0} < c2 − c1 and max{r1 −r2, 0} < c2 − c1. Proof. By using Definition (2.4) ã w b̃ if and only if for all α ∈ [0, 1], (aLα,a U α ) 5 (b L α,b U α ) or equivalently a L α 6 b L α and a U α 6 b U α . But by Lemma (2.1) a L α 6 b L α if and only if a1 − (1 − αk )h1 6 a2 − (1 − α k )h2 for all α ∈ [0,k] and c1 − ( 1−α 1−k ) l1 6 c2 − ( 1−α 1−k ) l2 for all α ∈ [k, 1], which are equivalent to (1 − α k )(h2 −h1) 6 a2 −a1 for all α ∈ [0,k], and ( 1 −α 1 −k )(l2 − l1) 6 c2 − c1 for all α ∈ [k, 1]. which are equivalent to max{h2 − h1, 0} ≤ a2 − a1 and max{m1 − m2, 0} ≤ b2 − b1. Similarly, by using Lemma(2.1) it can be conclude aUα 6 b U α if and only if max{l2−l1, 0}≤ c2−c1 and max{r1−r2, 0}≤ c2−c1 and the proof of part (i) is complete. Part (ii) can be proved, similarly. � Int. J. Anal. Appl. 16 (3) (2018) 357 3. Two-person Zero-sum Matrix Fuzzy Games In this section, we shall consider two-person zero-sum games with fuzzy payoffs. Let P = {1, 2, ...,p} and Q = {1, 2, ...,q} be the sets of pure strategies of player I and player II , respectively. Let A = (aij)p×q be the payoff matrix whose entries aij denote the payoff that player I gains and player II loses. In the zero-sum games −aij is the amount paid by player I to player II i.e. the gain of one player is the loss of other player. The mixed strategies of players I and player II are probability distributions on the set of pure strategies, represented by X = {(ξ1,ξ2, ...,ξp) ∈ Rp|ξi ≥ 0, i = 1, 2, ...,p, p∑ i=1 ξi = 1}, Y = {(η1,η2, ...,ηq) ∈ Rq|ηj ≥ 0, i = 1, 2, ...,q, q∑ j=1 ηj = 1}. respectively. In this section, the payoffs of the pair (x,y) ∈ X × Y are modeled by k-heptagonal fuzzy number ã = (a,c,b,h,l,r,m). Let player I choose a mixed strategy x ∈ X and player II choose mixed strategy y ∈ Y . The k-heptagonal fuzzy number ãij = (aij,cij,bij,hij, lij,rij,mij) indicates the payoffs that player I receives and player II loses, the fuzzy payoff matrix of the game is given by à =   ã11 · · · ã1q ... . . . ... ãp1 · · · ãpq   . The fuzzy two-person zero-sum games is denoted by Γ̃ ≡ (X,Y,Ã). The fuzzy payoffs of the players I and II are xTÃy = p∑ i=1 q∑ j=1 ξiãijηj =(xTAy,xTCy,xTBy,xTHy,xTLy,xTRy,xTMy), which is a k-heptagonal fuzzy number, for more details see [11]. In the rest of this paper, we set à = (ãij), A = (aij), C = (cij), B = (bij), H = (hij), R = (rij), M = (mij), where Ã, A, B, H, L, R and M are p × q matrix. Also à is a fuzzy k-heptagonal payoff matrix. Definition 3.1. [3] A pair (x∗,y∗) ∈ X ×Y is called a Nash equilibrium strategy for a game Γ̃ if (i) xTÃy∗ w x∗TÃy∗, ∀x ∈ X, (ii) x∗TÃy∗ w x∗TÃy, ∀y ∈ Y. Int. J. Anal. Appl. 16 (3) (2018) 358 Theorem 3.1. Let Γ̃ = (X,Y,Ã) be a two-person zero-sum game with fuzzy payoffs, the pair (x∗,y∗) is the expected Nash Equilibrium strategy of Γ̃ if and only if : (i) xTAy∗ 6 x∗TAy∗ 6 x∗TAy, (ii) xTBy∗ 6 x∗TBy∗ 6 x∗TBy, (iii) xTCy∗ 6 x∗TCy∗ 6 x∗TCy, (iv) xT (A−H)y∗ 6 x∗T (A−H)y∗ 6 x∗T (A−H)y, (v) xT (B −L)y∗ 6 x∗T (B −L)y∗ 6 x∗T (B −L)y, (vi) xT (B + R)y∗ 6 x∗T (B + R)y∗ 6 x∗T (B + R)y, (vii) xT (C + M)y∗ 6 x∗T (C + M)y∗ 6 x∗T (C + M)y. Proof. Let Γ̃ be a two-person zero-sum game with the fuzzy k-heptagonsl payoff matrix à = (A,C,B,H,L,R,M). Let (x∗,y∗) ∈ X×Y be the Nash equilibrium strategy of the game Γ̃. Therefore by Definition (3.1) we have x∗TÃy∗ w x∗TÃy, ∀y ∈ Y. Since xTÃy∗ = (xTAy∗,xTCy∗,xTBy∗,xTHy∗,xTLy∗,xTRy∗,xTMy∗), and x∗TAy∗ = (x∗TAy∗,x∗TCy∗,x∗TBy∗,x∗THy∗,x∗TLy∗,x∗TRy∗,x∗TMy∗), So, by Theorem (2.1), xTÃy∗ w x∗TÃy∗ if and only if max{x∗THy∗ −xTHy∗, 0} 6 x∗TAy∗ −xTAy∗, max{xTMy∗ −x∗TMy∗, 0} 6 x∗TBy∗ −xTBy∗, max{x∗TLy∗ −xTLy∗, 0} 6 x∗TCy∗ −xTCy∗, max{xTRy∗ −x∗TRy∗, 0} 6 x∗TCy∗ −xTCy∗. Consequently xTÃy∗ w x∗TÃy∗ if and only if xT (A−H)y∗ 6x∗T (A−H)y∗, xTAy∗ 6x∗TAy∗, (3.1) xT (C −L)y∗ 6x∗T (C −L)y∗, xTCy∗ 6x∗TCy∗, (3.2) xT (C + R)y∗ 6x∗T (C + R)y∗, xTCy∗ 6x∗TCy∗. (3.3) Also, since x∗TÃy∗ = (x∗TAy∗,x∗TCy∗,x∗TBy∗,x∗THy∗,x∗TLy∗,x∗TRy∗,x∗TMy∗), Int. J. Anal. Appl. 16 (3) (2018) 359 and x∗TÃy = (x∗TAy,x∗TCy,x∗TBy,x∗THy,x∗TLy,x∗TRy,x∗TMy), similary by Theorem (2.1), x∗TÃy∗ w x∗TÃy if and only if x∗T (A−H)y∗ 6x∗T (A−H)y, x∗TAy∗ 6x∗TAy, (3.4) x∗T (B + M)y∗ 6x∗T (B + M)y, x∗TBy∗ 6x∗TBy, (3.5) x∗T (C −L)y∗ 6x∗T (C −L)y, x∗TCy∗ 6x∗TCy, (3.6) x∗T (C + R)y∗ 6x∗T (C + R)y, x∗TCy∗ 6x∗TCy. (3.7) Now, from (3.1) and (3.5) we get xTAy∗ 6 x∗TAy∗ 6 x∗TAy, xT (A−H)y∗ 6 x∗T (A−H)y∗ 6 x∗T (A−H)y, from (3.2) and (3.6) we obtain xTBy∗ 6x∗TBy∗ 6 x∗TBy, xT (B + M)y∗ 6x∗T (B + M)y∗ 6 x∗T (B + M)y, from (3.3) and (3.8) we have xTCy∗ 6x∗TCy∗ 6 x∗TCy, xT (C −L)y∗ 6x∗T (C −L)y∗ 6 x∗T (C −L)y. and from (3.4) and (3.8) we get xTCy∗ 6x∗TCy∗ 6 x∗TCy, xT (C + R)y∗ 6x∗T (C + R)y∗ 6 x∗T (C + R)y. Hence, we have the required inequalities (i)-(vii). � In the rest of the paper, we purpose the following notations: AL0 = A−H, C L 0 = C −L, C U 0 = C + R, B U 0 = B + M, where A,C,B,H,L,R,M are the p×q matrix. Using these notations Theorem(3.1) can be rewrite as follows. Int. J. Anal. Appl. 16 (3) (2018) 360 Corollary 3.1. Let Γ̃ be a two-person zero-sum game with fuzzy payoffs, the pair (x∗,y∗) is the Nash Equilibrium strategy of Γ̃ if and only if the followings hold xT (A,C,B,AL0 ,C L 0 ,C U 0 ,B U 0 )y ∗ 6 x∗T (A,C,B,AL0 ,C L 0 ,C U 0 ,B U 0 )y ∗ 6 x∗T (A,C,B,AL0 ,C L 0 ,C U 0 ,B U 0 )y. In the view of Theorem 3.1, we understand that to solve the fuzzy game Γ̃, it is enough to consider seven crisp two-person zero-sum games and attempt to determine a point (x∗,y∗) ∈ X × Y which is simultaneously a saddle point of them. Definition 3.2. A two-person zero-sum fuzzy game Γ̃ = (X,Y,Ã) is called to be a proportional fuzzy game if and only if there exists γn ∈ (0, 1]; n = 1, ..., 4 such that hij = γ1aij, lij = γ2cij, rij = γ3cij and mij = γ4bij for all i = 1, 2, ...,p and for all j = 1, 2, ...,q. Theorem 3.2. A pair of mixed strategies (x∗,y∗) ∈ X×Y is a Nash equilibrium strategy of the proportional fuzzy matrix game Γ̃ = (X,Y,Ã) if and only if (x∗,y∗) ∈ X ×Y is the Nash equilibrium of crisp two-person zero-sum games Γa = (X,Y,A), Γb = (X,Y,B) and Γc = (X,Y,C). Proof. Let Γ̃ = (X,Y,Ã) be a proportional fuzzy matrix game. Therefore by Definition (3.2) à = (A,C,B,γ1A,γ2C,γ3C,γ4B) is the payoff matrix of the game. By Theorem (3.1), (x ∗,y∗) ∈ X ×Y is a Nash equilibrium of Γ̃ if and only if (i) xTAy∗ 6 x∗TAy∗ 6 x∗TAy, (ii) xTBy∗ 6 x∗TBy∗ 6 x∗TBy, (iii) xTCy∗ 6 x∗TCy∗ 6 x∗TCy, because the other inequalities came to these one. Equivalently, (x∗,y∗) ∈ X × Y is a Nash equilibrium of crisp two-person zero-sum games Γa = (X,Y,A), Γb = (X,Y,B) and Γc = (X,Y,C). The proof is complete. � The following corollary is a direct result of Theorem(3.2). Corollary 3.2. Let à = (ãij) be a payoff matrix of proportional fuzzy game Γ̃. Suppose that bij = γ5aij,cij = γ6aij for all i,j with γ5,γ6 ≥ 1. Then a pair of mixed strategies (x∗,y∗) ∈ X ×Y is the Nash equilibrium strategy for Γ̃ if and only if (x∗,y∗) is a Nash equilibrium of crisp two-person zero-sum game Γa = (X,Y,A). Definition 3.3. Let Γ̃ be a two-person zero-sum fuzzy game. It is called constant fuzzy game if and only if there exist h,l,r,m > 0 such that hij = h,lij = l , rij = r and mij = m for all i = 1, 2, ...,p and j = 1, 2, ...,q. Lemma 3.1. Let Γ̃ = (X,Y,Ã) be a constant fuzzy game. A pair of mixed strategies (x∗,y∗) ∈ X×Y is the Nash equilibrium strategy for Γ̃ if and only if (x∗,y∗) is a Nash equilibrium of crisp two-person zero-sum games Γa, Γb and Γc. Int. J. Anal. Appl. 16 (3) (2018) 361 Proof. By Definition(3.3) H,L,R and M are constant matrices which all the entries are h,l,r and m , respectively. Hence xTHy = h,xTLy = l,XTRy = r and xTMy = m for all x ∈ X,y ∈ Y . By Theorem(3.1) the result can be obtained, directly. � Definition 3.4. [3] A pair of mixed strategies (x∗,y∗) ∈ X×Y is called a Pareto Nash equilibrium strategy of the game Γ̃ if (i) there does not exist any x ∈ X such that x∗TÃy∗ - xTÃy∗, (ii) there does not exist any y ∈ Y such that x∗TÃy - x∗TÃy∗. Theorem 3.3. Let Γ̃ ≡ (X,Y,Ã) be a fuzzy two-person zero-sum game. A pair (x∗,y∗) ∈ X ×Y is the Pareto Nash equilibrium strategy for the game Γ̃ if and only if (i) there exist no x ∈ X such that x∗TAy∗ ≤ xTAy∗, x∗TBy∗ ≤ xTBy∗, x∗TCy∗ ≤ xTCy∗ and (x∗TAL0 y ∗,x∗TCL0 y ∗,x∗TCU0 y ∗,x∗TBU0 y ∗) ≤ (xTAL0 y ∗,xTCL0 y ∗,xTCU0 y ∗,xTBU0 y ∗); (3.9) (ii) there exist no y ∈ Y such that x∗TAy ≤ xTAy∗, x∗TBy ≤ xTBy∗, x∗TCy ≤ xTCy∗ and (x∗TAL0 y,x ∗TCL0 y,x ∗TCU0 y,x ∗TBU0 y) ≤ (x∗TAL0 y ∗,x∗TCL0 y ∗,x∗TCU0 y ∗,x∗TBU0 y ∗).(3.10) Proof. By contradiction, let (x∗,y∗) ∈ X ×Y be the Pareto Nash equilibrium strategy of Γ̃. Assume that there exist x1 ∈ X such that following relationships are established (x∗TAL0 y ∗,x∗TCL0 y ∗,x∗TCU0 y ∗,x∗TBU0 y ∗) ≤ (xT1 A L 0 y ∗,xT1 C L 0 y ∗,xT1 C U 0 y ∗,xT1 B U 0 y ∗), and x∗TAy∗ ≤ xT1 Ay ∗, x∗TBy∗ ≤ xT1 By ∗, x∗TCy∗ ≤ xT1 Cy ∗. It implies that x∗TAL0 y ∗ ≤ xT1 A L 0 y ∗, x∗TCL0 y ∗ ≤ xT1 C L 0 y ∗, x∗TCU0 y ∗ ≤ xT1 C U 0 y ∗, x∗TBU0 y ∗ ≤ xT1 B U 0 y ∗. But, by Definition (2.3) the above inequalities do not occur simultaneously. Therefore, we get Int. J. Anal. Appl. 16 (3) (2018) 362 ( x∗T ( α k A+(1 − α k )(A−H))y∗,x∗T ( α k B + (1 − α k )(B + M))y∗ ) ≤( xT1 ( α k A + (1 − α k )(A−H))y∗,xT1 ( α k B + (1 − α k )(B + M))y∗ ) , for α ∈ [0,k] and obtain( x∗T ( α − k 1 − k )C+(1 − α − k 1 − k )(C − L))y∗,x∗T ( α − k 1 − k C + (1 − α − k 1 − k )(C + R))y∗ ) ≤ ( xT1 ( α − k 1 − k )C + (1 − α − k 1 − k )(C − L))y∗,xT1 ( α − k 1 − k C + (1 − α − k 1 − k )(C + R))y∗ ) , for α ∈ [k, 1]. By rearranging, it follows that ( x∗T (A− (1 − α k )H)y∗,x∗T (B + (1 − α k )M)y∗ ) ≤( xT1 (A− (1 − α k )H)y∗,xT1 (B + (1 − α k )M)y∗ ) , and ( x∗T (C − ( 1 −α 1 −k )L)y∗,x∗T (C + ( 1 −α 1 −k )R)y∗ ) ≤( xT1 (C − ( 1 −α 1 −k )L)y∗,xT1 (C + ( 1 −α 1 −k )R)y∗ ) . Using Definition (3.4) it implies that x∗TÃy∗ � xT1 Ãy∗. This is a contradiction. Conversely, we assume that the pair of mixed strategy (x∗,y∗) ∈ X×Y be satisfy (3.9) and (3.10). Suppose that there exists a strategy x1 ∈ X such that x∗TÃy∗ � xT1 Ãy∗. By Definition 2.4, we have for all α ∈ [0, 1] (x∗TALαy ∗,x∗TAUαy ∗) ≤ (xT1 A L αy ∗,xT1 A U αy ∗) which (A− (1 − α k )H) = ALα, (B + (1 − α k )M) = AUα for α ∈ [0,k], and (C − ( 1 −α 1 −k )L) = ALα, (C + ( 1 −α 1 −k )R) = AUα for α ∈ [k, 1]. Set α = 0, then x∗T (AL0 ,C L 0 ,C U 0 ,B U 0 )y ∗ ≤ xT1 (A L 0 ,C L 0 ,C U 0 ,B U 0 )y ∗, and x∗TAy∗ ≤ xTAy∗,x∗TBy∗ ≤ xTBy∗,x∗TCy∗ ≤ xTCy∗. This is contradict (i). Similarly, we can show that there does not exist any y ∈ Y such that x∗TÃy � x∗TÃy∗. Then proof of the Theorem is complete. � Int. J. Anal. Appl. 16 (3) (2018) 363 Definition 3.5. A pair of mixed strategies (x∗,y∗) ∈ X ×Y is a weak Pareto Nash equilibrium strategy of the game Γ̃ if (i) there does not exist any x ∈ X such that x∗TÃy∗ ≺ xTÃy∗, (ii) there does not exist any y ∈ Y such that x∗TÃy ≺ x∗TÃy∗. Following theorem is obtaine directly from Definition (3.5) and Theorem (3.3). Theorem 3.4. Let Γ̃ ≡ (X,Y,Ã) be a fuzzy two-person zero-sum game. A pair (x∗,y∗) ∈ X ×Y is the weak Pareto Nash equilibrium strategy for the game Γ̃ if and only if (i) there exist no x ∈ X such that (x∗TAL0 y ∗,x∗TCL0 y ∗,x∗TCU0 y ∗,x∗TBU0 y ∗) < (xTAL0 y ∗,xTCL0 y ∗,xTCU0 y ∗,xTBU0 y ∗) and x∗TAy∗ < xTAy∗, x∗TBy∗ < xTBy∗, x∗TCy∗ < xTCy∗; (ii) there exist no y ∈ Y such that (x∗TAL0 y,x ∗TCL0 y,x ∗TCU0 y,x ∗TBU0 y) < (x∗TAL0 y ∗,x∗TCL0 y ∗,x∗TCU0 y ∗,x∗TBU0 y ∗) and x∗TAy < xTAy∗, x∗TBy < xTBy∗, x∗TCy < xTCy∗. 4. Parametric Bi-Matrix Games In this section we characterize parametric bi-matrix games and investigate other types of Nash equilibrium strategies for parametric bi-matrix games. Let Sp = {η1,η2, ...,ηp} and Sq = {ξ1,ξ2, ...,ξq} be sets of pure strategies of player I and player II , respectively. We set U = (uij)p×q to be payoffs matrices of player I and V = (Vij)p×q to be payoffs matrices of player II , respectively. Suppose β,γ ∈ [0, 1] and let (1 −β)(aij + cij −hij − lij) + β(cij + rij + bij + mij) be the gain of player I and (1 −γ)(aij + cij −hij − lij) + γ(cij + rij + bij + mij) be the losses of player II when player I emploing pure strategy i and player II emploing pure strategy j. Then the game Γ = (X,Y,U,V ) is called a bi-matrix game. The notation of parametric as follow; Suppose β,γ ∈ [0, 1], then we set A(β) = (1 −β)(A + C −H −L) + β(C + R + B + M), (4.1) Int. J. Anal. Appl. 16 (3) (2018) 364 and −A(γ) = − [(1 −γ)(A + C −H −L) + γ(C + R + B + M)] . (4.2) Now, we consider the parametric bi-matrix game Γ(β,γ) = (X,Y,A(β),A(γ)). Definition 4.1. [8] Let Γ(β,γ) be a parametric bi-matrix game. A pair of mixed strategies (x∗,y∗) ∈ X×Y is a Nash equilibrium strategy of Γ if (i) xTA(β)y∗ ≤ x∗TA(β)y∗ for all x ∈ X, (ii) x∗TA(γ)y∗ ≤ x∗TA(γ)y for all y ∈ Y . Theorem 4.1. Let Γ(β,γ) be a prametric bi-matrix game and the pair of mixed strategy (x∗,y∗) ∈ X×Y be Nash equilibrium strategy of Γ. Then (x∗,y∗) ∈ X×Y is the Pareto Nash equilibrium strategy of the fuzzy two-person zero-sum game Γ̃. Proof. Let (x∗,y∗) ∈ X × Y be the Nash equilibrium strategy of the parametric bi-matrix game Γ(β,γ), which β,γ ∈ [0, 1]. By Definition (4.1) we obtain (1 −β)xT (A + C −H −L)y∗ + βxT (C + R + B + M)y∗ ≤ (1 −β)x∗T (A + C −H −L)y∗ + βx∗T (C + R + B + M)y∗, and (1 −γ)x∗T (A + C −H −L)y∗ + γx∗T (C + R + B + M)y∗ ≤ (1 −γ)x∗T (A + C −H −L)y + γx∗T (C + R + B + M)y. In order to show that (x∗,y∗) ∈ X×Y is Pareto Nash equilibrium strategy of Γ̃, we have to prove that there exist x1 ∈ X such that x∗TÃy∗ � xT1 Ãy∗ holds. From Definition (2.4), we get (x∗TAL0 y ∗,x∗TCL0 y ∗,x∗TCU0 y ∗,x∗TBU0 y ∗) ≤ (xT1 A L 0 y ∗,xT1 C L 0 y ∗,xT1 C U 0 y ∗,xT1 B U 0 y ∗). Moreover, by Definition (2.3) x∗TAL0 y ∗ = xT1 A L 0 y ∗, x∗TCL0 y ∗ = xT1 C L 0 y ∗, x∗TCU0 y ∗ = xT1 C U 0 y ∗, x∗TBU0 y ∗ = xT1 B U 0 y ∗, do not occur simultaneously. Then we have (1 −β)x∗T (A + C −H −L)y∗ + βx∗T (C + R + B + M)y∗ < (1 −β)xT1 (A + C −H −L)y ∗ + βxT1 (C + R + B + M)y ∗. Int. J. Anal. Appl. 16 (3) (2018) 365 This is a contradiction. The condition (ii) can de proved, similarly. � Theorem 4.2. Let the pair of mixed strategies (x∗,y∗) ∈ X ×Y be Nash equilibrium strategy of prametric bi-matrix game Γ(β,γ) with β,γ ∈ [0, 1]. Then (x∗,y∗) ∈ X×Y is the weak Pareto Nash equilibrium strategy of fuzzy two-person zero-sum game Γ̃. The following corollary is direct result of Theorem (4.1) and Theorem (4.2). Corollary 4.1. A fuzzy two-person zero-sum game Γ̃ satisfies the following properties: (i) There exsist at least one Pareto Nash equilibrium strategy of fuzzy game Γ̃, (ii) There exsist at least one weak Pareto Nash equilibrium strategy of fuzzy game Γ̃. 5. Illustrative Examples Example 5.1. Let Γ̃ be a fuzzy two-person zero-sum game and à be the fuzzy payoff matrix of Γ̃ given as follows: à =   (20, 40, 60, 2, 8, 12, 24) (70, 140, 210, 7, 28, 42, 84) (50, 100, 150, 5, 20, 30, 60) (10, 20, 30, 1, 4, 6, 12)   . Find the Nash equilibrium strategy for the game Γ̃. Obviously, Γ̃ is a proportional fuzzy game. Note that γ1 = 0.1, γ2 = 0.2, γ3 = 0.3 and γ4 = 0.4. Let x∗T = ( p, 1 −p ) and y∗T = ( q, 1 −q ) be the mixed strategy of player I and II , respectively. By theorem(3.2), the Nash equilibrium strategy of game Γ̃ can be obtined by solving a bi-matrix game whose payoff matrices are A =  20 70 50 10   ,C =   40 140 100 20   ,B =   60 210 150 30   . We have ( 1 0 )20 70 50 10     q 1 −q   ≤ (p 1 −p)  20 70 50 10     q 1 −q   , and ( p 1 −p )20 70 50 10     q 1 −q   ≤ (p 1 −p)  20 70 50 10    0 1   . It is easy to obtain that the Nash equilibrium strategy of the crisp matrix game Γa is (x ∗,y∗) =( ( 4 9 , 5 9 ), ( 2 3 , 1 3 ) ) and similarly the Nash equilibrium strategy of the crisp matrix games Γb and Γc can be obtained. So expected value of the gasme Γ̃ is ( 4 9 , 5 9 )Ã( 2 3 , 1 3 )T = ( 990 27 , 1980 27 , 2870 27 , 99 27 , 396 27 , 594 27 , 1188 27 ). Int. J. Anal. Appl. 16 (3) (2018) 366 Example 5.2. Let à be the payoff matrix of the fuzzy two-person zero-sum game Γ̃, given as follows: à =   (50, 100, 150, 10, 15, 10, 40) (80, 160, 240, 10, 15, 10, 40) (100, 200, 300, 10, 15, 10, 40) (20, 40, 60, 10, 15, 10, 40)   . Find the Nash equilibrium strategy for the game Γ̃. by definition(3.3) Γ̃ is a proportional fuzzy game and h = 10, l = 15, r = 10 and m = 40. Let x∗T = ( p, 1 −p ) and y∗T = ( q, 1 −q ) be the mixed strategy of player I and II , respectively. By Theorm(3.1), it is easy to show that the Nash equilibrium strategy of Γ̃ is (x∗,y∗) = ( ( 8 11 , 3 11 ), ( 6 11 , 5 11 ) ) and the expected value of Γ̃ is given by ( 8 11 , 3 11 )Ã( 6 11 , 5 11 )T = ( 7700 121 , 15400 121 , 2310 121 , 10, 15, 10, 40) Example 5.3. Consider the fuzzy two-person zero-sum game Γ̃ with heptagonal fuzzy payoff matrix à given by à =  (90, 100, 120, 10, 5, 10, 15) (70, 80, 100, 15, 5, 10, 20) (60, 90, 100, 15, 10, 5, 10) (170, 180, 210, 20, 5, 20, 10)   . Find the Nash, Pareto Nash and weak Pareto Nash equilibrium strategy of the game Γ̃. Let x∗T = ( p, 1 −p ) and y∗T = ( q, 1 −q ) be the mixed strategy of player I and II , respectively. Since there is no (x,y) ∈ X ×Y satisfying the conditions of Theorem(3.1), so there is no Nash equilibrium strategy for the game Γ̃. By Theorem(4.2) to finding the Pareto Nash equilibrium strategy, it is enough to find the Nash equilibrium strategy of parametric bi-matrix game Γ̃. So, we construct the bi-matrix game Γ(β,γ) from fuzzy matrix game Γ̃. Using relations (4.1) and (4.2) we obtain A(β) =  175 + 70β 130 + 80β 125 + 80β 325 + 95β   , A(γ) =  175 + 70γ 130 + 80γ 125 + 80γ 325 + 95γ   , where β,γ ∈ [0, 1]. It is easy to see that (x∗,y∗) is the Nash equilibrium strategy for the parametric bi-matrix game Γ(β,γ) if it satisfies the following: (1, 0)A(β)y∗ ≤ x∗TA(β)y∗, (0, 1)A(β)y∗ ≤ x∗TA(β)y∗, x∗TA(γ)y∗ ≤ x∗TA(γ)(0, 1)T , x∗TA(γ)y∗ ≤ x∗TA(γ)(1, 0)T , which are equivalent to   (245 + 5β)(1 −p)q − (195 + 15β)(1 −p) ≤ 0, (245 + 5β)pq − (195 + 15β)p ≥ 0. Int. J. Anal. Appl. 16 (3) (2018) 367   (245 + 5γ)(1 −q)p− (200 + 15γ)(1 −q) ≥ 0, (245 + 5γ)pq − (200 + 15γ)q ≤ 0. Thus for β,γ ∈ [0, 1] , Nash equilibrium strategy for the parametric game Γ(β,γ) are as follows (x∗1,x ∗ 2) = ( 200 + 15γ 245 + 5γ , 45 − 10γ 245 + 5γ ) , (y∗1,y ∗ 2 ) = ( 195 + 15β 245 + 5β , 50 − 10β 245 + 5β ) . 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