Theorem 8.2.1 (Furukawa [19]). Let a˜=(a, α)T and b˜ =(b, β)T be two symmetric triangular fuzzy numbers. Then
(i) a˜b˜ ⇐⇒ |α−β|(a−b) (ii) a˜b˜ ⇐⇒ |α−β|<(a−b)
8.3 A bi-matrix game approach: Maeda’s model
Consider a two person zero-sum matrix game with fuzzy pay-offs de- scribed by FG = (Sm, Sn, A˜), where the symbols have their usual meanings as explained in Chapter 7. We now have the following three types of concepts of equilibrium strategies for the game FG (Maeda [50]).
Definition 8.3.1 (Minmax equilibrium strategy). An element ( ¯x, y)¯ ∈ Sm ×Sn is called a minmax equilibrium strategy of the game FG=(Sm, Sn, A˜) if
(i) xTA˜y¯ x¯TA˜y,¯ for all x∈Sm, and (ii) x¯TA˜y¯ x¯TAy,˜ for all y∈Sn.
In this case the scalar v˜ = x¯TA˜y¯ is said to be the (fuzzy) value of the game FG and the triplet ( ¯x, y,¯ v)˜ is said to be a solution of FG under the fuzzy max order ‘’.
Definition 8.3.2 (Non-dominated minmax equilibrium strat- egy). An element ( ¯x, y)¯ ∈ Sm ×Sn is said to be a non-dominated minmax equilibrium strategy of the gameFG=(Sm, Sn, A˜) if
(i) there does not exist any x∈Sm such that x¯TA˜y¯ xTA˜y, and¯ (ii) there does not exist any y∈Sn such that x¯TAy˜ x¯TA˜y.¯
Definition 8.3.3 (Weak non-dominated minmax equilibrium strategy.) An element ( ¯x, y)¯ ∈ Sm ×Sn is said to be a weak non- dominated minmax equilibrium strategy of the gameFG=(Sm, Sn, A˜) if
(i) there does not exist any x∈Sm such that x¯TA˜y¯ ≺xTA˜y, and¯ (ii) there does not exist any y∈Sn such that x¯TAy˜ ≺x¯TA˜y.¯
Remark 8.3.1.Minmax equilibrium strategy⇒non-dominated min- max equilibrium strategy⇒weak non-dominated minmax equilibrium strategy.
160 8 More on matrix games with fuzzy pay-offs
Remark 8.3.2.If A˜ is crisp, i.e. the game FG is the usual two per- son zero sum matrix game G = (Sm, Sn, A), then all the above three definitions coincide and become the definition of the usual saddle point.
We now assume that the elementsa˜ijof the pay-off matrixA˜ =(˜ai j) are symmetric triangular fuzzy numbers given by a˜i j = (ai j, hij)T and A=(ai j)andH=(hi j)are two(m×n)crisp matrices resulting from the fuzzy matrixA.˜
If we now agree to denote by[ ˜A]α =([˜ai j]α)where[˜aij]αis theα-level set of the fuzzy number a˜i j then for the symmetric TFN case
[ ˜A]α =
A−(1−α)H, A+(1−α)H .
Here it may be noted that [ ˜A]α is not a matrix in the conventional sense but rather it is an arrangement of (m×n) intervals of the type aij−(1−α)hij, aij+(1−α)hij , (i=1,2. . . ,m, j=1,2. . . ,n).
Further, let fuzzy matrix games with symmetric TFN’s pay-offs be denoted by SFG=(Sm, Sn, A˜).
The following theorem now gives a characterization of the minmax equilibrium strategy of the game FG = (Sm, Sn, A)˜ for the case of symmetric TFN’s, i.e. for the game SFG.
Theorem 8.3.1 A point ( ¯x, y)¯ ∈ Sm ×Sn is a minmax equilibrium strategy of the game SFG=(Sm, Sn, A)˜ , A˜ =(˜aij) with a˜i j =(ai j, hi j)T, if and only if
(i) xT(A+H) ¯y x¯T(A+H) ¯y x¯T(A+H)y, and (ii) xT(A−H) ¯y x¯T(A−H) ¯y x¯T(A−H)y, hold for all x∈Sm, y∈Sn.
Proof. Let ( ¯x, y)¯ ∈ Sm ×Sn be a minmax equilibrium strategy of the game SFG. Then from Theorem 8.2.1 we have
|x¯THy¯−xTHy¯ | x¯TAy¯−xTAy¯, and
|x¯THy−x¯THy¯ | x¯TAy−x¯TAy,¯ for all x∈Sm,y∈Sn.
The result now follows directly by employing the definition of the mod- ulus function and appropriate rearrangement of terms.
Remark 8.3.3.In view of the above theorem if we wish to solve the game SFG, then we have to consider a pair of crisp two person zero- sum games G1 = (Sm, Sn, A+H) and G2 = (Sm, Sn, A−H), and
8.3 A bi-matrix game approach: Maeda’s model 161
attempt to determine a point( ¯x, y)¯ ∈Sm×Snwhich is simultaneously a saddle point ofG1 andG2. Since this is not going to happen in general, the next best thing will be to look for solution in accordance with the non-dominated minmax equilibrium strategy (Definition 8.3.2) or weak non-dominated minmax equilibrium strategy (Definition 8.3.3).
The development given below asserts that the game SFGcertainly has a solution in these situations.
Theorem 8.3.2 Let xTAy be defined as the ordered pair xT(A+H)y, xT(A−H)y ∀x∈Sm, y∈Sn.
Then an element ( ¯x, y¯)∈Sm×Sn is a non-dominated minmax equilib- rium strategy of the game SFG if and only if
(i) there does not exist any x∈Sm such thatx¯TAy¯ ≤xTAy, and¯ (ii) there does not exist any y∈Sn such thatx¯TAy≤x¯TAy.¯
Proof. We shall first prove the direct part. For this let us assume that ( ¯x, y)¯ ∈Sm×Snis a non-dominated minmax equilibrium strategy of the gameSFG. If possible, let there existxˆ ∈Smsuch thatx¯TAy¯ ≤ ( ˆx)TAy.¯ This, by definition, implies
x¯T(A−H) ¯y, x¯T(A+H) ¯y T ≤ ˆ
xT(A−H) ¯y, xˆT(A+H) ¯y T. Now by appropriately rearranging terms in the above inequality we get
x¯TAy¯−xˆTAy¯, x¯TAy¯−xˆTAy¯ T≤
¯
xTHy¯−xˆTHy¯, xˆTHy¯−x¯THy¯ T. As both components on the L.H.S are same but on the R.H.S, they are negative of the other, we have
¯
xTAy¯−( ˆx)TAy¯ < 0, i.e
¯
xTAy¯ < xˆTAy¯.
Also in view of the specific understanding of the symbol ‘≤’, the above inequality with some obvious manipulations, gives that for allα∈[0,1], x¯T(A−(1−α)H) ¯y, x¯T(A+(1−α)H) ¯y T ≤
ˆ
xT(A−(1−α)H) ¯y, xˆT(A+
(1−α)H) ¯y T,
162 8 More on matrix games with fuzzy pay-offs
which by Definition 8.2.2 implies that x¯TAy˜ xˆTA˜y. But this is a¯ contradiction to the fact that ( ¯x, y)¯ is a non-dominated minmax equi- librium strategy. Similarly we can show that there does not exist any y∈Sn such thatx¯TAy ≤ x¯TAy.¯
Conversely, let ( ¯x, y)¯ ∈Sm×Sn be such that conditions (i) and (ii) of above theorem hold. We have to show that ( ¯x, y)¯ is a non-dominated minmax equilibrium strategy of the gameSFG.
For this, if possible, let there exist xˆ ∈ Sm such that x¯TA˜y¯ xˆTA˜y.¯ This, by definition, gives
x¯T(A−H) ¯y, x¯T(A+H) ¯y T ≤ ˆ
xT(A−H) ¯y, xˆT(A+H) ¯y T, which implies that
¯
xTAy¯ ≤ ( ˆx)TAy,¯
but this is a contradiction to the condition (i) of the theorem. Similarly we can show that if there exists yˆ ∈Sn such that x¯TA˜yˆ ≤( ¯x)TA˜y¯ then condition (ii) of the theorem is contradicted.
Theorem 8.3.3 An element( ¯x, y)¯ ∈Sm×Snis a weak non-dominated minmax equilibrium strategy of the game SFG if and only if
(i) there does not exist any x∈Sm such that x¯TAy¯ <xTAy, and¯ (ii) there does not exist any y∈Sn such that x¯TAy<x¯TAy.¯
The proof of Theorem 8.3.3 is similar to that of Theorem 8.3.2.
In view of Theorems 8.3.1-8.3.3, it is natural to define the bi-matrix game BG(λ, à) =
Sm, Sn, A(λ), −A(à) for λ, à∈[0,1], whereA(λ)= A+(1−2λ)H, and A(à) = A+(1−2à)H. Here it may be noted that A(0) = A+H, A(1) = A−H, and for λ = à, BG(λ, à) becomes the matrix game
Sm, Sn, A(λ) .
We now recall the following from Chapter 1.
Definition 8.3.4 (Nash equilibrium strategy). Let λ, à ∈ [0,1]. A point ( ¯x, y)¯ ∈ Sm ×Sn is called a Nash equilibrium strategy of the game BG(λ, à) if
(i) xTA(λ) ¯y x¯TA(λ) ¯y, for all x∈Sm, and (ii) x¯TA(à) ¯y x¯TA(à)y, for all y∈Sn.
Theorem 8.3.4 An element( ¯x, y¯)∈Sm×Snis a non-dominated min- max equilibrium strategy of the game SFG if and only if there exist
8.3 A bi-matrix game approach: Maeda’s model 163
positive real numbers λ, à ∈ (0,1) such that ( ¯x, y)¯ is a Nash equilib- rium strategy of the (crisp) bi-matrix game BG(λ, à).
Proof. Let ( ¯x, y¯) ∈ Sm×Sn be a non-dominated minmax equilibrium strategy of the gameSFG. Therefore, by Theorem 8.3.2, there does not exist anyx∈Sm such that x¯TAy¯ ≤ xTAy¯ i.e. there does not exist any x∈Sm such that
x¯T(A−H) ¯y, x¯T(A+H) ¯y T ≤
xT(A−H) ¯y, xT(A+H) ¯y T. But this implies (Steuer [71]) that there exist positive scalars λ1, λ2 withλ1+λ2=1 such that for allx∈Sm,
¯ xT
λ1(A+H)+λ2(A−H) y¯ xT
λ1(A+H)+λ2(A−H) y¯. Now by taking λ2=λ and λ1=(1−λ), the above inequality becomes
¯ xT
A+(1−2λ)H y¯ xT
A+(1−2λ)H y,¯ for allx∈Sm. Similarly the second condition of Theorem 8.3.2 gives (Steuer [71]) that there exists0< à <1 such that
¯ xT
A+(1−2à)H y¯ x¯T
A+(1−2à)H y, for all y∈Sn. The above two inequalities therefore imply that ( ¯x, y¯) is a Nash equi- librium strategy of the bi-matrix game BG(λ, à).
Conversely let λ, à∈ (0,1) and ( ¯x, y)¯ be a Nash equilibrium strategy of the game BG(λ, à). Then by Definition 8.3.4
¯ xT
A+(1−2λ)H y¯ xT
(A+(1−2λ)H y¯, for all x∈Sm, and
¯ xT
A+(1−2à)H y x¯T
A+(1−2à)H y,¯ for all y∈Sn. But
A+(1−2λ)H=λ(A−H)+(1−λ)(A+H) and
A+(1−2à)H=à(A−H)+(1−à)(A+H).
Therefore the above inequalities imply that forλ, à∈(0,1),
¯ xT
λ(A−H)+(1−λ)(A+H) y¯ xT
λ(A−H)+(1−λ)(A+H) y,¯ x∈Sm,
164 8 More on matrix games with fuzzy pay-offs
and
¯ xT
à(A−H)+(1−à)(A+H) y x¯T
à(A−H)+(1−à)(A+H) y¯, y∈Sn, which again means (Steuer [71]) that there is no x∈Sm such that
x¯T(A−H) ¯y, x¯T(A+H) ¯y T ≤
xT(A−H) ¯y, xT(A+H) ¯y T, and also there is no y∈Snsuch that
x¯T(A−H)y, x¯T(A+H)y T ≤
¯
xT(A−H) ¯y, x¯T(A+H) ¯y T. The above inequalities now imply that ( ¯x, y)¯ is a non-dominated min- max equilibrium strategy of the game SFG.
Theorem 8.3.5 An element( ¯x, y¯)∈Sm×Snis a weak non-dominated minmax equilibrium strategy, of the fuzzy matrix gameSFGif and only if there exist real numbers λ, à ∈ [0,1] such that ( ¯x, y)¯ is a Nash equilibrium strategy of the bi-matrix game BG(λ, à).
The proof of the above theorem is similar to that of Theorem 8.3.4.
Theorem 8.3.6 For the fuzzy matrix gameSFG the following is true (i) there exists at least one non-dominated minmax equilibrium strat-
egy,
(ii) there exists at least one weak non-dominated minmax equilibrium strategy.
The proof of Theorem 8.3.6 follows because the existence of a Nash equilibrium strategy for the (crisp) bi-matrix BG(λ, à) is always guar- anteed as stated in Theorem 1.5.1.
In view of Theorems 8.3.4 and 8.3.5, for finding a non-dominated (weak non-dominated) minmax equilibrium strategy of the fuzzy ma- trix gameSFGwe have to find Nash equilibrium strategies of the (crisp) bi-matrix game BG(λ, à).
Example 8.3.4. (Maeda [50]). Let us consider the fuzzy matrix game SFG where the pay off matrixA˜ is given by
A˜ =
(180,5)T (156,6)T
(90,10)T (180,5)T
.
Now as explained in Theorem 8.3.4, to solve the fuzzy matrix game SFG, we have to consider the (crisp) bi- matrix game BG(λ, à) =