solution ofmatrix games with fuzzypayoffs usingdualityin...

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.


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


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


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)


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


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


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


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


(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


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


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)


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;


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


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-


incides with a, then the TrFN becomes TFN. So we can apply the same results

for triangular fuzzy numbers also. The next chapter deals with a new concept

solution in Bi-matrix games with fuzzy payoffs.

o

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