LetSm,Sn,A andB be as introduced in the Section 9.2. Letv0,w0 be scalars representing the aspiration levels of Player I and Player II respectively. Then the bi-matrix game with fuzzy goals under consid- eration here is, denoted by BGFG, and is defined as
BGFG=(Sm, Sn, A, B, v0, , w0, ),
where ‘’ and ‘’ are the fuzzified versions of ‘≤’ and ‘≥’ respectively.
Therefore the game BGFG gets fixed only when the specific choices of membership functions are made to define fuzzy inequalities ‘’ and
‘’. Here we shall interpret ‘’ and ‘’ in the sense of Zimmerman [91]
although a more general interpretation in terms of modalities and fuzzy relations can also be taken.
Let t be a real variable and a ∈ R. Let pˆ > 0. We now recall the notation “t pˆ a” from Chapter 6 and note that this is to be read as
“t is essentially greater or equal to a with tolerance p”, and is to beˆ understood in terms of the following membership function
9.3 Bi-matrix games with fuzzy goals: another approach 181
àD(t)=
⎧⎪⎪⎪⎪
⎪⎪⎨⎪⎪⎪⎪
⎪⎪⎩
1 ,t≥a,
1− a−t
ˆ p
,(a−p)ˆ ≤t≤a, 0 ,t<(a−pˆ).
We also recall the below given lemma from Chapter 6.
Lemma 9.3.1.Let t1pˆa, t2pˆa, α ≥ 0, β ≥ 0 and α+β = 1. Then αt1+βt2pˆ a.
In view of the above discussion we include tolerancesp0,p0for Player I, and,q0andq0for Player II respectively in our definition of the fuzzy bi-matrix gameBGFGand therefore takeBGFGas
BGFG=(Sm, Sn, A, B, v0, p0, p0, w0, q0, q0, , ).
Now we define the meaning of an “equilibrium solution” of the fuzzy bi-matrix gameBGFG.
Definition 9.3.1 (Equilibrium solution ofBGFG).A point(x∗, y∗)∈ Sm ×Sn is called an equilibrium solution of the fuzzy bi-matrix game BGFG if
xTAy∗p0 v0, ∀x∈Sm, x∗TByq0 w0, ∀y∈Sn, x∗TAy∗p0 v0,
x∗TBy∗q0 w0.
Remark 9.3.2.For the crisp scenario above inequalities become xTAy∗≤v0,∀x∈Sm,
x∗TBy≤w0,∀y∈Sn, x∗TAy∗≥v0, and
x∗TBy∗≥w0.
Therefore for all x ∈Sm, y∈ Sn we have xTAy∗ ≤x∗TAy∗ and x∗TBy≤ x∗TBy∗ which is the same as the definition for an equilibrium solution for the (crisp) bi-matrix gameBG.
182 9 Fuzzy Bi-Matrix Games
In the Definition 9.3.1, the setsSm andSn are convex polytopes, there- fore in view of Lemma 9.3.1, for the specific choice of membership functions of type àD(t), it is sufficient to consider only the extreme points (i.e. pure strategies) ofSm and Sn. This observation leads to the following fuzzy non-linear programming problem (FNP)
(FNP) Find (x, y)such that
Aiyp0 v0, (i=1,2, . . . ,m), BjTxq0 w0,(j=1,2, . . . ,n), xTAyp0 v0,
xTByq0 w0, x∈Sm, y∈Sn,
where for i = 1,2, . . . ,m, Ai denotes the ith row of A and for j = 1,2, , . . . ,n, Bj denotes the jth column of B.
Now as per the requirement for the use of Lemma 9.3.1, we have to define the specific linear membership functions of type àD(t) for all the fuzzy constraints. Therefore membership function ài(Aiy), (i= 1,2, . . . ,m), which gives the degree to which ysatisfies fuzzy constraint Aiypo voandνj(BTjx), (j=1,2, . . . ,n), which gives the degree to which x satisfies the fuzzy constraintBTjxpo wo, are given as
ài(Aiy)=
⎧⎪⎪⎪⎪
⎪⎨⎪⎪⎪⎪
⎪⎩
1 ,Aiy≤v0, 1−Aiy−v0
p0 ,v0≤Aiy≤v0+p0, 0, ,Aiy≥v0+p0, and
νj(BjTx)=
⎧⎪⎪⎪⎪
⎪⎪⎨⎪⎪⎪⎪
⎪⎪⎩
1 ,BjTx≤w0, 1−BjTx−w0
q0 ,w0 ≤BjTx≤w0+q0, 0 ,BjTx≥w0+q0, respectively.
Similarly, linear membership functions for the fuzzy constraints xTAyp0 v0 and xTByq0 w0 are defined as follows
à0(xTAy)=
⎧⎪⎪⎪⎪
⎪⎨⎪⎪⎪⎪
⎪⎩
1 ,xTAy≥v0,
1−v0−xTAy
p0 ,v0≥xTAy≥v0−p0, 0 ,xTAy≥v0−p0,
9.3 Bi-matrix games with fuzzy goals: another approach 183
and
ν0(xTBy)=
⎧⎪⎪⎪⎪
⎪⎨⎪⎪⎪⎪
⎪⎩
1 ,xTBy≥w0,
1− w0−xTBy
q0 ,w0 ≥xTBy≥w0−q0, 0, ,xTBy≤w0−q0.
Now employing the above mentioned membership functions and fol- lowing Zimermann’s approach [90], we obtain the crisp equivalent of the fuzzy non-linear programming (FNP) as
(NLP) max λ
subject to,
λ≤1−Aiy−v0
p0 , (i=1,2, . . . ,m), λ≤1−BjTx−w0
q0 , (j=1,2, . . . ,n), λ≤1+xTAy−v0
p0 , λ≤1+xTBy−w0
q0 , x∈Sm,
y∈Sn, λ∈[0,1].
The above discussion leads to the following theorem:
Theorem 9.3.1 Let (x∗, y∗, λ∗) be an optimal solution to the problem (NLP). Then (x∗, y∗) is an equilibrium solution of the fuzzy bi-matrix game BGFG and λ∗ is the least degree up to which the respective aspi- ration levels (goals)v0 and w0 of Player I and Player II are met.
Remark 9.3.3.Let (x∗, y∗, λ∗) be a solution to the (NLP) with λ∗= 1. Then BGFG becomes the (crisp) bi-matrix game BG and (x∗, y∗) becomes its equilibrium solution. Therefore various results of (crisp) bi-matrix game theory follow as a special case of fuzzy bi-matrix game theory. Further, forλ∗=1, the non-linear programming problem (NLP) reduces to the system
184 9 Fuzzy Bi-Matrix Games
Aiy≤v0, (i=1,2, . . . ,m), BjTx≤w0,(j=1,2, . . . ,n), xTAy=v0,
xTBy=w0, x∈Sm, y∈Sn, which implies Theorem 1.6.1.
Special Cases:
It has already been explained in Remark 9.3.3 that various results of crisp bi-matrix game theory follow as a special case of fuzzy bi-matrix game theory. In the following certain other special cases are presented so as to bring out differences/similarities between the results of this section with that of Nishizaki and Sakawa as presented in Section 9.2.
1. Let us recall from Section 9.2 that if(x∗, y∗, p∗, q∗)is a solution of the quadratic programming problem (QPP) then it is also an equi- librium solution of the fuzzy bi-matrix gameBFG, where
(QPP): max xT
A
¯ a−a
y+xT
B b¯−b
y−p−q subject to,
A
¯ a−a
y≤pe, BT
b¯−b
x≤qe, eTx=1, eTy=1, x,y≥0.
Now from Mangasarian and Stone [52] and also from results of Section 9.2, it is known that if (x∗, y∗, p∗, q∗) is an optimal so- lution to the above quadratic programming problem (QPP) then x∗TAy∗=p(¯a−a) andx∗TBy∗=q(¯b−b). Therefore to obtain (crisp) bi-matrix game as a special case of the fuzzy bi-matrix game (BFG), the membership function values à1(x∗TAy∗) and à2(x∗TBy∗) should be equal to 1. Therefore we should havex∗TAy∗≥a¯andx∗TBy∗≥b;¯ which can be written as p(¯a −a) ≥ a¯ and q(¯b−b) ≥ b. Since no¯ relationship betweena¯andp(¯a−a), and also betweenb¯ and q(¯b−b),