Chapter 3
Solution of Matrix Games with
Fuzzy Payoffs using Duality in
Linear Programming
3,,1 Introduction
One of the most useful results in the matrix game theory asserts that ev
ery two person zero-sum matrix game is equivalent to two linear pro
gramming problems which are dual to each other. The earliest study of
two person zero-sum matrix game with fuzzy payoffs is due to Cam
pos (Campos 1989) which remains the most basic reference on this topic.
Later Nishizaki and Sakawa ( ishizaki and Sakawa 2001) extended the
ideas of Campos (Campos 1989) to multi-objective matrix games. Cam
pos (Campos 1989) introduced a number of different types of linear pro
gramming (LP) models to solve zero-sum fuzzy normal form games. In
42
3.2 Preliminaries 43
his formulation, each player's strategy set is a crisp set, but players have
imprecise knowledge about the payoffs. He considered five different
ways of ranking fuzzy numbers, and for each case he formulated the
constraints using fuzzy triangular numbers. Two of these are based on
the work of Yager (Yager 1981) and involve the use of a ranking function
that maps the fuzzy numbers on to R A third approach involves the use
of a- cuts and is based on the work of Adamo (Adamo 1980). The last
two approaches rank fuzzy numbers using possibility theory. This stems
from the work of Dubois and Prade (Dubois and Prade 1983). Finally,
the five different parametric LP models obtained through this transfor
mation process are solved using conventional LP techniques to identify
their fuzzy solutions. This exercise is performed with different numeri
cal examples. Later Bector (Bector et a12004) presented a defuzzification
function method to solve matrix games in fuzzy environment using lin
ear programming method. In this chapter we study a more general type
of matrix games with trapezoidal fuzzy numbers as payoffs as in (Bector
and Chandra 2005), we use a suitable defuzzification function to convert
them as primal-dual pairs in linear programming problem. We study the
model which deals with the same problem but from totally different ap
proaches. We expound our approach to solve a fuzzy matrix game by
prime-dual method and a numerical example is also given.
3.2 Preliminaries
3.2 Preliminaries
44
Let lRn denote the n- dimensional Euclidean space and lR+ n be its non
negative orthant set. Let A E lRmxn be an m x n matrix and e be the
column matrix with all entries equal 1. A two person zero-sum matrix
game G is defined as G = (sm, sn, A), where sm = {x E lR+m : iI'x = I}
and sn = {y E lR+n : eT y = I} are the strategy space for the player I
and player II respectively. A is called the payoff matrix. The quantity
K(x, y) = xT Ay is called the expected payoff of player I by player II.
Now the concept of double fuzzy constraints means constraints which
are expressed as fuzzy inequalities involving fuzzy numbers. Let N(lR)
be the set of all fuzzy numbers. Let A, b, crespectively be m x n matrix,
m x 1 and n x 1 vectors having entries from N(lR) and the double fuzzy
constraints under consideration be given by Ax ~p band ATy ~ij cwith
adequacies pand qrespectively. Based on a resolution method proposed
in (Yager 1981) the constraint Ax ;Sp bis expressed as Ax (::s;) b+p(l - ),),
), E (0,1] where for i = 1,2, ... ,m the ith component of the fuzzy vec
tor p, namely Pi, measures the adequacy between the fuzzy numbers Aix
and bi which are the ith component of fuzzy vectors Ax and brespectively.
Similarly the constraint ATy ~ij cis expressed as ATy(2)c - q(l - '1]);
'I] E [0,1] where for j = 1,2, ... , n the lh component of the fuzzy vec
tor q, namely r]j measures the adequacy between the fuzzy numbers Aj T ?J
and cj which are the jth component of the fuzzy vectors ATy and crespec
tively. Here (:::;), and (2) are relations between fuzzy numbers which pre
serve the ranking when fuzzy numbers are multiplied by positive scalars.
3.2 Preliminaries 45
This could be with respect to any ranking function F : N(lR) ---7 lR taken
in Campos (Campos 1989) such that a(-::;)b implies F(a) -::; F(b). Since in
subsequent sections the function F is used to defuzzify the given fuzzy
linear programming problems. Therefore the double constraints of the
type Ax ;:5ii band AyT 2:,q Care to be understood as Aix(-::;)~ +Pi(l - ),),
for 0 -::; ). -::; 1 and i = 1,2, ... ,m and Aj T y(2.)S - qj(l - 'TJ); for 0 -::; 'TJ -::; 1
and j = 1,2, ... , n which are in turn means,
and
Now let aij, ~, Pi, sand ib are trapezoidal fuzzy numbers (TrFNs) and
F is the Yagers (Yager 1981) first index given by
du
Jxf-tD(x)dxF(D) = =d~-u-_-
Jf-tD(x)dxdl
where dl and du are the lower and upper limits of the support of the
fuzzy number D.
The constraints Ax;:5ii band AT 2:,q Crespectively means
3.2 Preliminaries 46
n - -l: [(aij )I+Qij;aij +(aij )u] X j ~ [(bi)I+Qi;bi +(bi)u] +(1 - A) (Pi) I) -+ Ei ;15, +(Pi)uj=l
and
~[( ) Qij+aij ()] [() £j+Cj ()] ( )()) iJ.jHj ()LJ aij 1+-2-+ aij u Yi ~ Cj 1+-2-+ Cj u + I-TJ qj I + 2 + qj ui=l
for A E [0, I], TJ E [0, I], i = 1,2, ..... , m and j = 1,2, ..... , n.
Here, aij = ((aij)l' Qij;aij
, (aij)u), b;j = ((bi)l, Qi;bi
, (bi)u), Pi((Pi)l, Ei;15i, (Pi)u)
and rb = ((qj)l' iJ.j;qj, (qj)u) are Trapezoidal Fuzzy numbers.
3.2.1 Duality in Linear Programming
In the crisp theory of primal-dual linear programming we have the pri
mal as
subject to
Ax ~ b;
x~O
and its dual pair is
3.2 Preliminaries
subject to
where x E lRn , Y E lRm , C E lRn , b E lRm , and A is an m x n real matrix.
47
Definition 3.1. Fuzzy Number
A fuzzy set A in lR is called afuzzy number if it satisfies the following con
dition
(i) A is normal
(ii) a A is a closed interval for every Cl' E (0,1]
(iii) The support ofA is bounded
Definition 3.2. (Trapezoidal fuzzy number) (TrFN)
A fuzzy number Ais called a trapezoidal fuzzy number if its membership
function is given by
°
1
3.3 Primal-dual fuzzy linear programming
The TrFN A is denoted by quadruplet A = (al' g, a, au).
Theorem 3.1.
48
Let (x, -\) and (y, ''7) are feasible fuzzy pair of primal-dual linear program
ming problem then
F(CTx) - F(bT y) :::; (1 - A)F(pry) + (1 - 'TJ)F(qrx).
3.3 Primal-Dual Fuzzy Linear Programming
In the crisp pair of primal-dual linear programming problems, the fuzzy
version of the usual primal and dual problem as
max cTx
subject to
- -Ax:::; b;
and
subject to
A?y> c·- ,
3.3 Primal-dual fuzzy linear programming 49
Here A is an m x n matrix of fuzzy numbers and band crespectively
are m x 1 and n x 1 vectors of fuzzy numbers. The symbols ;S and 2:
are fuzzy versions of the symbols :::; and ~ respectively and have the
interpretation" essentially less than or equal to " and " essentially greater
than or equal to" as explained in Zimmerman (Zimmerman 1978).
Theorem 3.2.
The triplet (x, fj, v) E sm x sn x lR is a solution of the game G ifand only ifx
is optimal to PIt fj is optimal to DI and v is the common value ofPI and its dual
DI . The double fuzzy constraint Ax :::; b and ATy ~ care to be in mind with
respect to a suitable defuzzification function F and adequacies pand q. Again,
the defuzzification function F once chosen is to be kept fixed. If F : N (lR) -+ lR
is the chosen defuzzification function fuzzy numbers for constraints in P2 and
D2 then the same defuzzification function F for the objective function in P2 and
D2 we get P3 and D3 as
max F(cT x)
subject to
F(Ax) :::; F(b) + (1 - A)F(p);
A < l'- ,
X,A ~ a
and
3.3 Primal-dual fuzzy linear programming
Da
mm F(6T y )
subject to
F(JFy) 2 F(c) - (1 - 'TJ)F(ij);
'TJ :S 1;
y,'TJ 2 a
50
Here p and ij measures the adequacies in the primal and dual constraints.
The pair Pa and D a are termed as fuzzy pair of primal-dual linear programming
problems.
In case A, cand bare crisp and A = 1 and 'TJ = I, then the pair Pa
and Da reduce to the usual crisp primal-dual pair and the theorem 3.2
becomes the usual weak duality theorem.
Definition 3.3.
Let v, wEN (IR). Then, (v, w) is called a reasonable solution of the fuzzy
matrix game FG if there exists x* E sm, y* E sn satisfying
(i) (x*fAy 2: v V Y E sn
(ii) xT Ay ;S w V x E sm
If (v, w) is a reasonable solution of FG then v (respectively w) is called a
reasonable value for player I (respectively player II)
3.3 Primal-dual fuzzy linear programming 51
Definition 3.4. (Bector and Chandra 2005)
Let T1 and T2 be the set ofall reasonable values vand 'W for player Iand player
II respectively where v, w E N(~). Let there exists v* E TIt w* E T2 such that
F(v*) 2 F(v) V v E T1 and F(w*) ::; F(w)Vw E T2 . Then (x*, y*, v*, w*) is
called the solution of the game FG, where v* (respectively w*) is the value of the
game FG for player I (respectively player II) and x* (respectively y*) is called an
optimal strategy for player I (respectively player II).
From the above definition for the game FG we construct the pair of
fuzzy linear programming problems for player I and player II as
max F(v)
subject to
and
mm F(-w)
subject to
Now, using the double fuzzy constraints and applying the relations
(S) and (2) preserve the ranking when fuzzy numbers are multiplied by
3.3 Primal-dual fuzzy linear programming 52
positive scalars, we have to consider only the extreme points of sets smand sn in the constraints of P4 and D4 • Then the problem P4 and D4 will
be converted into
Ps
max F(v)
subject to
TA-· > - -. . - 1 2x J rvP v, J - , , ... , n.
eT x = l',
and
Ds
subject to
Aiy ;:::;ij ow; i = 1,2, ... ,m.
Here Ai (respectively Aj ) denotes the i th row (respectively jth column)
of A(i = 1,2, ... ,m; j = 1,2, ... ,n) using the resolution procedure for the
double fuzzy constraints in Ps and Ds, we obtain
3.3 Primal-dual fuzzy linear programming
max F(v)
subject tom
LaijXiC~)V - (1- >.)p;j = 1,2, ... ,noi=1
>. < l'- ,
x,>' 2: a_ Qij + (iij V+v
where aij = ((aij)l, 2 ' (aij)u), v = (V)l, 2' (v)u),- p+pP= (P)l, 2' (p)u)
and
m'ln F(w)
subject ton
L aijYj(5:)iiJ + (1 - 'T})q; i = 1,2, ... ,m.j=1
Y,'T} 2: a
53
By applying the defu.zzification function F : N(JR) ---t JR for the con
straints P6 and D6 these problems can further be written as
3.3 Primal-dual fuzzy linear programming
max F(v)
subject tom
L F(aij)Xi ~ F(v) - (1 - A)F(p); j = 1,2, ... , n.i=l
A < l'- ,
X,A ~ a
and
mm F(w)
subject tom
L F(aij)Yj ~ F(w) + (l-17)F(q); i = 1,2, ... ,m.j=l
17 ~ 1;
Y, 17 ~ a
54
From the above we observe that for solving the fuzzy matrix game FG
we have to solve the crisp linear programming problems P7 and D7 for
player I and player II respectively. Also (x*, A*, v*) is an optimal solution
of P7 then for player I, x* is an optimal strategy, v* is the fuzzy value and
(1 - )..*)p is the measure of the adequacy level for the double fuzzy con
straints P7 . Similar interpretation can also be given to optimal solutions
3.3 Primal-dual fuzzy linear programming
(y*, TJ*, w*) of the problem D7 .
55
Theorem 3.3.
The pair P7 - D7 constitutes a fuzzy optimal-dual pair in the sense of the
theorem [3.11
Theorem 3.4. (Bector and Chandra 2005)
The fuzzy matrix game FG determined by FG = (sm, sn, A) is equivalent to
two crisp linear programming problems P7 and D7 which constitutes a primal
dual pair in the sense ofduality for linear programming with fuzzy parameters.
It is important that the crisp problems P7 and D7 which do not con
stitutes a primal-dual pair in the sense of duality in linear programming
but are dual in " fuzzy" sense. Therefore if (x*, A*, v*) is optimal to P7
and (y*, TJ*, w*) is optimal to D7 , then in general we should not expect
that F(v*) = F(w*).
If all the fuzzy numbers are to be taken as crisp numbers that is,
(iij = aij, bi = bi , Cj = Cj and in the optimal solutions of P7 and D7,
A* = TJ* = 1, then the fuzzy game FC reduces to the crisp two person
zero-sum game G. Thus if A, b, care crisp numbers and A* = TJ* = 1 FC
reduces to G, the pair P7 and D 7 reduces to the pair primal-dual and the
theorem 3.3 reduces to theorem 1.5
It is generally difficult to obtain exact membership functions for
fuzzy values v* and w* because of the large number of parameters. For
example if:ij is a TrFN (Vi, V, Q, vu ) then to determine:ij completely we need
all of the four variables. Therefore from the computational point of view
3.3 Primal-dual fuzzy linear programming 56
it becomes easier to take F(v) and F(w) as real variables V and W respec
tively and modify P7 and D7 as
Ps
max V
subject tom
L F(aij)xi 2: V - (1- A)F(p); j = 1,2, ... , n.i=l
eT x = l',
A < l'- ,
x, A 2 a
and
mm ltV
subject ton
L F(aij)Yj ~ W + (1 - 7])F(q); i = 1,2, ... , m.j=l
7] ~ 1;
Y,7] 2: a
Thus, although we know that value for player I (respectively player
II) is fuzzy with certain membership function we shall get only numeri
cal values v* (respectively w*) for player I (respectively player II) is fuzzy
3.3 Primal-dual fuzzy linear programming 57
with certain membership function we shall get only numerical values v*
(respectively w*) for player I (respectively player IT) and the actual fuzzy
value for player I and player II will be "close to" v* and w* respectively.
Thus, it can be concluded that we cannot get exact membership func
tion for the fuzzy values of player I and player II, even though these
are very much desirable. When F is Yagers first index (Yager 1981) the
numerical values v* (respectively w*) will represent the " centroid " or
"average" value for player I (respectively player II). The following illus
tration makes the procedure more clear. The matrix A has been taken
from (Bector and Chandra 2005) and the TrFN is defined accordingly to
avoid tedious steps of calculations.
Illustration
Consider the fuzzy game defined by the matrix of fuzzy numbers
A = [1~0 1~6]90 180
where,
180 (170,175,185,195)
156 (150,154,156,159)
90 (80,85,95,100)
Assuming that player I and player II have margins
PI P2 = (0.08,0.05,0.15,0.11)
elI q'2 = (0.14,0.05,0.25,0.17)
v (155,160,170,175)
if; (170,175,185,190)
3.3 Primal-dual fuzzy linear programming 58
According to theorem (3.4) to solve this game we have to solve the fol
lowing two crisp linear programming problems (H) and (D1) for player
I and player II respectively.
and
maxv+v
Vl+ 2 +VU
3
subject to
545xl + 270X2 ~ 495 - (1 - .\)(0.29)
464xl + 545x2 ~ 495 - (1 - .\)(0.29)
.\ < l'- ,
w+wWl+~+Wu
max 3
subject to
545Yl + 464Y2 ~ 540 + (1 - 77)(0.46)
270Yl + 545Y2 ~ 540 + (1- 77)(0.46)
Yl + Y2 = 1,
77 :S 1;
3.3 Primal-dual fuzzy linear programming 59
Now to get the full membership representations of the value for player
I (respectively player IT) one needs that in the optimal solution of (PI) re
spectively (D I ) all variables (VI,12.,V,Vu ) (respectively (WI,W,W,Wu ) come
out to be non zero that is they are basic variables. This seems to be most
unlikely as there are much less number of constraints and therefore many
of the variables are going to be nonbasic and hence take zero values only.VI + Q+V + Vu
This observation motivates us to take V = ; ,
w+wW = WI +~ + W
u and consider the following problem of P2 and D2 for3
the variables V and W
max V
subject to
545xI + 270X2 2 495 - (1 - '>')(0.29)
464xI + 545x2 2 495 - (1 - '>')(0.29)
.>. < l'- ,
and
3.3 Primal-dual fuzzy linear programming
maxvV
subject to
545Yl + 464Y2 ~ 540 + (1 - 7])(0.46)
270Yl + 545Y2 ~ 540 + (1 - 7])(0.46)
Yl + Y2 = 1,
7] ~ 1;
Solving the above linear programming problem, we obtain
(xi = 0.7725, x; = 0.2275, v = 160.91, A* = 0) and
(y{ = 0.2275, Yz = 0.7725, w = 160.65,7]* = 0).
60
Therefore we obtain optimal strategies for player I and player II as
(xi = 0.7725, x; = 0.2275) and
(y{ = 0.2275, Yz = 0.7725) respectively.
Also the fuzzy value of the game for player I is close to 160.91. In a simi
lar manner, the fuzzy value of the game for player II is also close to 160.65.
Note 3.1.
This chapter presented a study of matrix games with fuzzy payoffs. We first
established duality for linear programming problems with fUZZY parameters and
then the same was employed to develop a solution procedure for solving two
person zero-sum matrix game with fUZZY payoffs. Here we used the method of
analyzing the system of double fuzzy inequalities of the type Ax ;S b. If Q co-