is given in Nishizaki and Sakawa [61] the results of crisp bi-matrix game do not seem to follow from the fuzzy bi-matrix game BFG.
2. As has been observed in Chapter 1, a crisp two person zero-sum matrix game is a special case of crisp bi-matrix game. More ex- plicitly, the quadratic programming problem given by Mangasar- ian and Stone [52] decomposes itself into a pair of two primal- dual linear programming problems for the case A = −B. Since assumption A = −B does not imply A
¯
a−a = B
b¯−b in general the above quadratic programming problem (QPP) given by Nishizaki and Sakawa may not always decompose in this manner. There is similar difficulty with the fuzzy bi-matrix (BGFG) as well. However if we choose a¯ = max
i max
j ai j, a = min
i min
j aij, b¯ = max
i max
j bi j and b= min
i min
j bi j then (QPP) decomposes itself into two linear programming problems which are dual to each other. In a similar manner if λ∗ = 1, αo = −βo then for the case B =−A, an optimal solution(x∗, y∗, λ∗=1)givesxTAy∗≤x∗TAy∗≤x∗TAyfor allx∈Sm and y∈ Sn, thereby implying that (x∗, y∗) is a saddle point of the (crisp) matrix game G=(Sm, Sn, A).
9.4 Bi-matrix games with fuzzy pay-offs: a ranking function approach
In Chapter 7, we have already seen a ranking function approach to matrix games with fuzzy pay-offs. Here we attempt to extend the same to bi-matrix games with fuzzy pay-offs . LetSm,Snbe as in Section 9.2 and,A˜ andB˜ be the pay-off matrices with entries as fuzzy numbers for Player I and Player II respectively. Then a bi-matrix game with fuzzy pay-offs, denoted byBGFP, is defined as
BGFP=(Sm, Sn, A,˜ B).˜
Now, we define the meaning of an “equilibrium solution” of the fuzzy bi-matrix gameBGFP. For this we first have the following definition.
Definition 9.4.1 (Reasonable solution of BGFP). Let v,˜ w˜ ∈ N(R). Then (v˜, w) is called a reasonable solution of the fuzzy bi-matrix˜ game BGFPif there exists x∗∈Sm, y∗∈Sn such that
186 9 Fuzzy Bi-Matrix Games
xTAy˜ ∗p˜ v,˜ ∀x∈Sm, x∗TBy˜ q˜w,˜ ∀y∈Sn, x∗TAy˜ ∗p˜ v˜,
x∗TBy˜ ∗q˜ w.˜
If (v˜, w) is a reasonable solution of˜ BGFP then v˜ (respectively w) is˜ called a reasonable value for Player I ( respectively Player II).
Definition 9.4.2 (Equilibrium solution of BGFP). Let T1 and T2 be the set of all reasonable values v˜ and w˜ for Player I and Player II respectively where v,˜ w˜ ∈ N(R). Let there exist v˜∗ ∈ T1, w˜∗ ∈ T2 such that
F( ˜v∗)≥F( ˜v), ∀v˜∈T1, and
F( ˜w∗)≥F( ˜w),∀w˜ ∈T2,
where F : N(R) →R is the chosen defuzzification function. Then the pair (x∗, y∗) is called an equilibrium solution of the game BGFP. Also
˜
v∗ (respectively w˜∗) is called the value of the game BGFP for Player I (respectively Player II).
By using the above definitions for the gameBGFP, we now construct the following fuzzy non-linear programming problem
(FNP1) max F( ˜v)+F( ˜w) subject to,
xTAy˜ p˜ v,˜ ∀x∈Sm, xTBy˜ q˜ w˜, ∀y∈Sn, xTAy˜ p˜ v,˜
xTBy˜ q˜w˜, x∈Sm, y∈Sn,
˜
v,w˜ ∈N(R).
Now recalling the explanation of the double fuzzy constraints as ex- plained in Chapter 7 and noting that the relations ≥ and ≤ preserve the ranking when fuzzy numbers are multiplied by positive scalars, it makes sense to consider only the extreme points of sets Sm and Sn in the constraints of(FNP1). Therefore the above problem(FNP1)will be converted into
(FNP2) max F( ˜v)+F( ˜w) subject to,
9.4 Bi-matrix games with fuzzy pay-offs: a ranking function approach 187
A˜iyp˜v,˜ (i=1,2, . . . ,m), xTB˜j q˜w˜, (j=1,2, . . . ,n), xTAy˜ p˜v,˜
xTBy˜ q˜w˜, x∈Sm, y∈Sn,
˜
v,w˜ ∈N(R).
Here A˜i denotes the ith row of A˜ and B˜j denotes the jth column of B, (i˜ =1,2, . . . ,m; j=1,2, . . . ,n).
By using the resolution procedure for the double fuzzy constraints in(FNP2)as discussed in Chapter 7, we obtain
(FNP3) max F( ˜v)+F( ˜w) subject to,
n j=1
˜
ai jyj ≤ v˜+(1−λ) ˜p, (i=1,2, . . . ,m), m
i=1
b˜i jxi ≤ w˜ +(1−η)˜q, (j=1,2, . . . ,n), xTAy˜ ≥ v˜−(1−λ) ˜p,
xTBy˜ ≥ w˜ −(1−η) ˜q, x∈Sm,
y∈Sn, λ, η∈[0,1],
˜
v,w˜ ∈N(R).
Now by utilizing the chosen defuzzification function for the con- straints in(FNP3), the problem can further be written as
(NLP1) max F( ˜v)+F( ˜w) subject to,
n j=1
F(˜aij)yj≤F( ˜v)+(1−λ)F( ˜p), (i=1,2, . . . ,m), m
i=1
F(˜bij)xi≤F( ˜w)+(1−η)F(˜q), (j=1,2, . . . ,n), F(xTAy)˜ ≥ F( ˜v)−(1−λ)F( ˜p),
188 9 Fuzzy Bi-Matrix Games
F(xTBy)˜ ≥ F( ˜w)−(1−η)F( ˜q), x∈Sm,
y∈Sn, λ, η∈[0,1],
˜
v,w˜ ∈N(R).
From the above discussion we observe that for solving the fuzzy bi- matrix game BGFPwe have to solve the crisp non-linear programming problem (NLP1). Also, if
x∗, λ∗, v˜∗, y∗, η∗, w˜∗ is an optimal solution of the crisp non-linear programming problem (NLP1), then (x∗, y∗) is an equilibrium solution of the game BGFP.
These results can now be summarized in the form of the following theorem.
Theorem 9.4.1 The fuzzy bi-matrix gameBGFPdescribed byBGFP= (Sm, Sn, A˜, B˜) is equivalent to the crisp non-linear programming prob- lem (NLP1) in which the objective as well as all constraint functions are linear except two constraint functions, which are quadratic.
Remark 9.4.1.If all the fuzzy numbers are to be taken as crisp num- bers i.e. a˜i j = ai j, b˜ij = bij, v˜ = v, w˜ = w and in the optimal solution of (NLP1),λ∗=η∗=1, then the fuzzy gameBGFPreduces to the crisp bi-matrix game BG. Thus if A˜, B˜, v˜ and w˜ are crisp and λ∗ =η∗ =1, thenBGFP reduces toBGand the crisp non-linear programming prob- lem (NLP1)reduces to the non-linear programming problem (NLP2)
(NLP2) max v+w
subject to, Ay≤ve, BTx≤we, xTAy≥v,
xTBy≥w, x∈Sm, y∈Sn, v,w∈R.
For the case B = −A, the bi-matrix game BG reduces to the matrix game G=(Sm, Sn, A) and the problem(NLP2)reduces to the system