Chapter 0

Introduction

Game Theory is a branch of Applied Mathematics that is used in Social

Sciences, notably in Economics, Political Science, International Relations,

Social Psychology, Philosophy and Management as well as in Biology,

Engineering and Computer Science. Theory of Games provides a se­

ries of mathematical models that may be useful in explaining interactive

decision-making concepts where two or more competitors are involved

under certain conditions of conflict and competition. Today "Game the­

ory is a sort of umbrella" or 'unified field theory' for the rational side

of Social Science, where 'social' is interpreted broadly, to include human

as well as non human players(computers, animals, plants). The Great

Mathematician John Von Neumann is known to be the founder of Game

Theory. Representation of games is a major element of Game Theory.

Games are situations in which at least a single agent is allowed to maxi­

mize his utility through anticipating the responses of one or more agents.

The agents are known as Players. Each game includes strategies, where

4

Introduction 5

strategy is a predetermined situation according to which the Player who

maximizes the profit, makes his move. The study by Antonie Cournot

in 1838 was the earliest case of formal game theoretic analysis. Further,

the history of Game Theory has attained two land marks - one is by Emile

Borel's (in 1921) "Theory of Parlour Games" and other is by the renowned

mathematician Jon Von Neumann's "Theory of Games and Economic Be­

haviour " .written jointly by Oskar Morgenstern which is a further devel­

opment of Borel studies. The field of Game Theory formally cam~ into

being with the book" Theory of Games and Economic Behaviour" by

Jon Von Neumann and Oskar Morgenstern in 1944 (Neumann and Mor­

genstern 1944). It has been widely recognized as an important tool in

many fields. It has incredible prospective for conflict resolution prob­

lems in the field of Decision Theory. It has come to play an increasingly

important role in Logic and Computer Science.

Game Theory got its present status by the seminal work of John

Nash in 1951 [Nash 1951]. He proved that every finite game always has

an equilibrium point at which all players choose actions which are best

for them, when their opponent's choices are given. The 1950's ·and 1960's

saw the Game Theory, which had earlier had an alliance only with eco­

nomics, taking a gaint leap by getting ramified into the sphere of war,

politics etc. Since 1970's game theory has been extensively applied to So­

ciology, Psychology and many other areas of Social Sciences. At present

Game Theory serves as a prime tool for modeling several decision mak­

ing processes in interactive environments. The theory deals with the pro­

cess of formally modeling a situation, taking into consideration the fact

0.1 Fuzzy Mathematics 6

that any game requires the decision-maker to identify the role and nature

of the players and their strategic options, preferences and relations. The

methodology pursed in constructing such a model provides the decision­

maker with a potential for a clearer and broader view of the situation ly­

ing ahead. This strictness in methodology applied in structuring and an­

alyzing the problem of strategic choices, serve as the strong hold of Game

Theory. A formal model of an interactive situation becomes the target of

studying in Game Theory. In many branches of Game Theory, it is taken

for granted that each player is a rational one who opts for an action it

produces the best outcome for him, given what he expect his opponents

to do. Hence the game theory analysis enables prediction of how best the

game will be played by rational players or gives advice to the player as to

how best the game can be played against rational opponents. When the

game theory is applied to model some practical problem which we en­

counter in real situations, we have to know the values of payoffs exactly.

However, it is difficult to know the values of payoffs approximately or

with some imprecise degree. In such cases, games with fuzzy payoff in

which payoffs are represented as fuzzy numbers are often employed.

0.1 Fuzzy Mathematics

Fuzzy mathematics form a branch of Mathematics related to Fuzzy Set

. Theory and Logic. It started in 1965 after the publication of Lotfi Asker

Zadeh's seminal work' Fuzzy Set'. A fuzzy set can be defined mathemat­

ically by assigning to each possible element in the universe of discourse,

0.1 Fuzzy Mathematics 7

a value representing its grade of membership in the fuzzy set. This grade

corresponds to the degree to which that element is similar or compatible

with the concept represented by the fuzzy set (Klir and Yuan 2001). Sets,

which are not fuzzy, are termed as classical sets or crisp sets or exact sets.

The roots of fuzzy sets can be traced back to the second half of the

19th century, more precisely, to the well-known controversy between

G. Cantor and L. Kronecker on the mathematical meaning of infinite sets.

Another source of fuzzy set lies in the imprecision in human decision

making, which was Zadeh's main motivation. Since the very inception

of the theory, several people all over the world have explored its various

facets and a large number of results have been generated. Fuzzy Set The­

ory has now become a major area of interest for modern scientists. The

capability of fuzzy sets to express gradual transitions from membership

to non-membership and vice versa has a wide utility.

The German Mathematician George Cantor (1845-1918) described

a crisp set as a well defined collection of objects. Given a set A and an

element a in the universe, there can be only two possibilities.

(i) either a is an element of A,

(ii) or, a is not an element of A.

The membership value 1 is assigned to a in the first case and 0 is assigned

to a in the second case. In other words the two element set {O, I} repre­

sents the membership of a with regard to A and the set A is characterized

by the characteristic function AA : X -7 {O, I} and it is defined as

0.1 Fuzzy Mathematics

AA(X) = 1:i.e., A = {(x, AA(X))jX E X}

ifxEA

8

In the actual life situations, the membership of elements are better rep­

resented by extending the value {O, 1} to the unit interval [0,1]. This is

the basic characteristic of fuzzy sets.

The theory of fuzzy sets deals with a subset A of the universal set

X where the transition between full membership and non membership

of a is gradual rather than abrupt. The fuzzy set has no well defined

boundaries where the universal set X covers a definite range of objects.

For example let A be the set of cups of "tasty coffee ". The class of "tasty

coffee" is not sharply defined. In fact, most of the objects encountered in

the real physical world are of this type, not sharply defined. They do not

have precisely defined criteria for membership. In such cases, it is not

necessary for an object to belong or not belong to a class, and there might

be intermediate grades of membership. This is the concept of Fuzzy Set,

which is a class with a continuum of grades of membership. Now we

give a precise definition of a fuzzy set.

Definition 0.1. Let X be a set, then afuzzy set A in X is afunction from X to

the unit interval [0,1]. That is afuzzy set A in X is defined by

A = {(x, MA(X))jx E X}; where MA(X) is the grade of membership of x in A.

0.2 Fuzzy game theory

Obviously afuzzy set is ageneralized subset ofcrisp set.

0.2 .Fuzzy Game Theory

9

Fuzziness in Matrix Games can appear in so many ways, but two cases of

fuzziness seem to be natural. These being the one in which players have

fuzzy goals and the other in which the elements of the payoff matrix are

given by fuzzy numbers. These two classes of fuzzy matrix games are

referred as matrix games with fuzzy goals and matrix games with fuzzy

payoff respectively.

Fuzziness in two person zero-sum game was studied by various au­

thors. (Campos 1989), (Campos and Gonzalez 1992), (Dubois and Prade

1980), (Butnariu and Klement 1993) and so on. Campos and Gonzalez

considered the payoff value in a game as fuzzy numbers and other as­

pects of the game were retained as crisp itself. In their paper "On the use

of ranking functions approach to solve fuzzy matrix games in a direct

way" (Campos and Gonzalez 1992), fuzzy numbers are defined as trape­

zoidal. Fuzzy numbers are defined as convex normalized functions on lR

with values in the unit interval [0,1] by Dubois and Prade. Another defi­

nition of fuzzy numbers like "triangular" are also used by Campos (Cam­

pos 1989). Kandel used a fuzzy number as convex normalized piece­

wise continuous fuzzy sets on the unit interval [0,1] (Kandel 1986). Cam­

pos dealt with matrix games with fuzzy payoffs and formulated a prob­

lem yielding a max-min solution by applying fuzzy mathematical pro­

gramming (Buckley 1989). Sakawa and Nishizaki examined fuzzy ma-

0.2 Fuzzy game theory 10

trix games incorporating fuzzy goals in single and multiple objective en­

vironments (Sakawa and Nishizaki 1994). Fuzziness in bi-matrix games

was studied by various authors (Bector and Chandra 2000), (Nishizaki

and Sakawa 1996), (Maeda 2000). They considered the payoff values as

fuzzy numbers in a bi-matrix game. In Fuzzy Set Theory (Leung 1984)

the goals are viewed as fuzzy sets and it is assumed that their member­

ship function are known.

We cannot deal with uncertainty, ambiguity or vagueness in ordi­

nary linear programming problems (LPP). However, there are many such

imprecise factors in our real world. For this reason various methods have

been proposed in order to deal with them. For example, dealing with

probabilistic uncertainty (randomness) in our problems, the stochastic

programming methods have been offered. And, we have been handling

ambiguity and vagueness in our problems by the fuzzy mathematical

programming methods.

Concerning fuzzy linear programming problems (FLPP), in recent

years, many researchers have proposed various formulations and new

concepts for solutions, for example, Buckley (Buckley 1988,1989), Inuiguchi

and Ichihashi (Inuiguchi and Ichihashi 1990), Lai and Hwang (Lai and

Hwang 1992), Leuent (Leuent 1984), Luhandjula )(Luhandjula 1986) and

Sakawa (Sakawa 1993). In FLP problems, formulated under fuzzy cir­

cumstances, the solutions do not contain fuzziness, (e.g.,Buckley and Sakawa).

Bector, Chandra and Vidyothama (Bector et al. 2004) aim to have a re­

look at the fuzzy matrix parameters. Recent efforts in this field are by

Bector, Maede (Maeda 2003) and Li (Li 1999). Bector has extended the use

0.2 Fuzzy game theory 11

of linear programming duality to solve matrix game with fuzzy goals to

matrix game with fuzzy payoffs. Vijay, Chandra and Bector (Bector et al

2004) had an attempt to study duality theory based on certain defuzzifi­

cation function F and its role in explaining an equivalence between ma­

trix games with fuzzy payoffs and certain primal-dual pair of fuzzy lin­

ear programming problems. Studies by Bector and Chandra (Bector and

Chandra 2002), Hamacher, Liberling and Zimmermann (Hamacher et al

1978) and Rodder and Zimmermann (Rodder and Zimmermann 1980)

have proved fuzzy linear programming duality results. Mangasarian and

Stone '(Mangasarian and Stone 1964) obtained a solution of a bi-matrix

game by solving a suitable quadratic programming problem. Maede,

Nishizaki and Sakawa also have studied fuzzy matrix games. Garagic

and Cruz. Jr (Garagic and Cruz 2003) have pointed out the fact that Nash

equilibrium strategy is said to effect the results of the multi criteria deci­

sion making and games.

In this thesis we deal with matrix games and bi-matrix games with fuzzy

payoffs and fuzzy goals.

We now give the summary of each chapter.

The thesis is organized in 8 chapters. The opening chapter, Chapter 0 is

the Introduction. It gives an outline of the entire thesis. The last chapter,

Chapter 7 is the conclusion which gives a brief summary of the work.

Chapter 1

Preliminary definitions of the terms like fuzzy set, fuzzy number, mono­

tonic decreasing function, fuzzy arithmetic, ranking function approach,

saddle point and equilibrium solutions of the game, required for the later

0.2 Fuzzy game theory

chapters are given in the chapter.

A fuzzy set A in the universal set X is defined as,

A = {(x, J.LA(X); X EX)}; whereJ.LA : X ---+ [0,1].

12

Apart from the presentation of definitions and results in the introductory

chapter, each chapter is provided with a brief section called' preliminar­

ies " which is the review of the literature for the particular chapter.

Chapter 2

In this chapter we deal with a two person zero-sum game with fuzzy

payoffs. Here we develop a descriptive theory to analyze games with

imprecise characteristics using fuzzy tools. We provide illustra~onsand

numerical examples to study the adequacy of the theory. For this we use

a special type of fuzzy number which is monotonic decreasing. Here we

consider maximin and minimax method to solve fuzzy matrix game.

Chapter 3

In chapter 3 we study a more general type of matrix games with trape­

zoidal fuzzy numbers as payoffs, and we use suitable defuzzification

function to convert them as primal-dual pairs in linear programming

problem. Thus, regards to the crisp matrix game, G = (sm, sn, A), where

sm = {x E Rr;, eT x = I}, sn = {y E R~, eT y = I} and A is an rn x n

real matrix, we take the concept of double fuzzy constraints, which are

expressed as fuzzy inequalities involving trapezoidal fuzzy numbers. Fi­

nally, a numerical example is provided to illustrate the method of solu­

tion.

Chapter 4

In this chapter we consider Fuzzy Bi-Matrix Games which mean bi-matrix

0.2 Fuzzy game theory 13

games with fuzzy payoff matrices. Here, we solve a fuzzy bi-matrix

game with fuzzy payoff by using fuzzy non linear programming method.

Chapter 5

Chapter 5 deals with matrix games with fuzzy goals and fuzzy payoffs.

When the goals and payoffs are both fuzzy, a ranking function approach

is developed to solve such fuzzy matrix games. In this chapter we choose

an appropriate ranking function for the given fuzzy scenario to arrive at

a new solution.

Chapter 6

In chapter 6, we discuss Bi-matrix games with fuzzy goals and fuzzy pay­

offs. Whenever the goals and payoffs are both fuzzy, a ranking function

approach is developed to solve such fuzzy bi-matrix games. Here also we

use the same method which is used in chapter 5, to arrive at a solution.

o