A Study on Two-Person Games in Fuzzy Environment

A Study on Two-Person Games in Fuzzy
Environment
Thesis submitted to the Bharathidasan University, Tiruchirappalli
in partial fulfillment of the requirements for the Degree of
DOCTOR OF PHILOSOPHY IN MATHEMATICS
Submitted By
T.PORCHELVI
(Ref. No. 15925/Ph.D/Mathematics/PT/January 2010)
Under the Supervision of
Dr. D. STEPHEN DINAGAR, M.Sc., M.Phil., Ph.D.,
Ph.D. Research Supervisor and Associate Professor,
PG and Research Department of Mathematics,
T.B.M.L. College, Porayar609 307
South India
P.G. AND RESEARCH DEPARTMENT OF MATHEMATICS
TRANQUEBAR BISHOP MANICKAM LUTHERAN COLLEGE
PORAYAR - 609307
TAMIL NADU, INDIA.
February 2013
Certificate
Dr. D. STEPHEN DINAGAR, M.Sc., M.Phil., Ph.D.,
Ph.D., Research Supervisor and Associate Professor,
P.G. and Research Department of Mathematics,
T.B.M.L. College, Porayar 609 307, South India
Date:
This is to certify that the thesis entitled
A Study On Two-Person Games in Fuzzy Environment
is a research work done by Ms. T. PORCHELVI, a part-time scholar
of Ph.D., Degree in the Department of Mathematics, T.B.M.L. College, Porayar, South India.
The subject on which the thesis has been prepared is her original work and has not been previously formed the basis for award to
any candidate of any Degree, Diploma, Associate ship, Fellowship
or other similar title. The thesis represents entirely an independent
work of the candidate under the general guidance given by me.
(Dr. D. STEPHEN DINAGAR)
Research Supervisor and Guide
Declaration
T.PORCHELVI
Ph.D., Scholar (Part time),
Department of Mathematics,
T.B.M.L. College,
Porayar 609 307, Tamil Nadu.
I do hereby declare that the thesis entitled A Study On TwoPerson Games In Fuzzy Environment, submitted by me, for
the degree of DOCTOR OF PHILOSOPHY (Ph.D.) in Mathematics of the Bharathidasan University, Tiruchirappalli is the record of
the research work done by me and it has not previously formed the
basis for the award of any Degree, Diploma, Associateship, Fellowship or other similar award of titles.
Place:
(T.PORCHELVI)
Date:
COUNTER SIGNED
Acknowledgements
Foremost, I would like to express my sincere gratitude to my supervisor, Dr.
D. STEPHEN DINAGAR, M.Sc., M.Phil.,
Ph.D., Associate Professor of Mathematics, T.B.M.L. College, Porayar, South India, for the continuous support of my Ph.D study
and research, for his patience, motivation, enthusiasm, and immense
knowledge. His guidance helped me in all the time of research and
writing of this thesis. I could not have imagined having a better
advisor and mentor for my Ph.D study.
I am thankful to the management of T.B.M.L. College, especially
to the college Secretary Rt. Rev. H.A. MARTIN, B.Sc., B.D.,
D.S.W., D.Th., 11th Bishop of Tranquebar for providing all the
necessary facilities to pursue this research programme.
I would like to record my heart felt thanks to Dr. G. JONAS
GUNASEKARAN, M.Sc., M.Phil., Ph.D., Principal, T,B.M.L.
College, Porayar, who permitted me to undergo my research in the
Department of Mathematics as a part time research scholar.
I
wish
to
acknowledge
my
ii
heartfelt
thanks
to
Prof. M.P. GNANAVEL, M.Sc., M.Phil., Head of the Department of Mathematics, T.B.M.L. College, Porayar, for his kind
encouragement.
I take up this opportunity to thank all the staff members of the
Department of Mathematics, T.B.M.L. College, Porayar, who have
supported me in various aspects towards the completion of this thesis.
It is my pleasant duty to thank my family members, who have
shown their patience, love and all possible support.
I thank Dr. C. Bhooma, M.A, M.Ed, M.Phil,Ph.D. for
helping me through proof reading the text and making necessary
corrections syntactically.
I thank the editors and unknown referees of various national
and international Journals who have published my research articles.
Finally, I would like to thank everybody who was important to the
successful realization of my thesis, as well as expressing my apology
that I could not mention personally one by one.
(T.PORCHELVI)
iii
List of Publications
1. D. Stephen Dinagar, P. Martin Deva Prasath and T.Porchelvi,
The Value of the games using fuzzy oddments method, Journal
of Current Sciences, ISSN 0972-6101, 15(1) 207-210 (2010).
2. D. Stephen Dinagar and T.Porchelvi, An Approximate Method
for Solving Fuzzy Games, International Journal of Advances in Fuzzy Mathematics, ISSN 0973-533X, 5(3) 295300 (2010).
3. D. Stephen Dinagar and T.Porchelvi,Fuzzy Dominance Principle, International Journal of Algorithms, Computing
and Mathematics, ISSN 0974-3367, 3(3) 17-20 (2010).
4. D. Stephen Dinagar and T.Porchelvi, A Study on Fuzzy Games
Using Fuzzy Number Matrices, International Journal of Mathematics Research, ISSN 0976-5840, 3(5) 475-482 (2010).
5. D. Stephen Dinagar and T.Porchelvi, The Value of the games
iv
using Fuzzy Nash Equilibrium, International Journal of Computational Science and Mathematics, ISSN 0974-3189, 3(4)
447-453 (2011).
6. D. Stephen Dinagar and T.Porchelvi, Solving Fuzzy Bi-Matrix
Games, Proceedings of the International Conference on
Mathematics and Computer Science, Loyola College, Chennai, January 2011.
7. D. Stephen Dinagar and T.Porchelvi, On Two Person Zero-Sum
Fuzzy Game With TOPSIS Ranking Procedure, International
Electronic Journal of Pure and Applied Mathematics,
ISSN 1314-0744, 4(1) 53-58 (2012).
8. D. Stephen Dinagar and T.Porchelvi,Constrained Fuzzy Games
Using Trapezoidal Fuzzy Numbers, Journal of Fuzzy Mathematics, International Fuzzy Mathematics Institute, Los
Angeles, ISSN 1066-8950, 21(1) 173-180 (2013).
9. D. Stephen Dinagar and T.Porchelvi, Solving Bi-Matrix Fuzzy
Games Using Fuzzy Determinant Procedure, Proceedings of
the IEEE
International Conference on Advances in
v
Engineering, Science and Management, ISBN: 978-81909042-2-3, (5) 88-90 (2012).
10. D. Stephen Dinagar and T.Porchelvi, Equilibrium Analysis of
Two Person Zero-Sum Fuzzy Game With OPNR Procedure,
Proceedings of the International Conference on Mathematics in Engineering and Business Management, Stella
Maris College, Chennai, March 2012.
vi
List of Notations
ã
I˜
Trapezoidal fuzzy number
0̃
Zero Trapezoidal fuzzy number
FG
Fuzzy Two person zero sum game
FBG
Fuzzy Bi-matrix games
CFG
Constrained fuzzy games
FV
Fuzzy value
FLPP
Fuzzy Linear programming problem
IVFM
Interval valued fuzzy matrix
TU Game
Transferable Utility Game
NTU Game
Non Transferable Utility Game
F-Mag
Magnitude of a fuzzy matrix
Ũ
Universal trapezoidal fuzzy number
Unit Trapezoidal fuzzy number
(x0 (ã); y0 (ã)) Centroid of ã
CM (ã)
Compoz and Munoz’s Ranking value of ã
M ag(ã)
Abbasbandy’s ranking value of ã
ITα (ã)
Liou and Wang’s Ranking value of ã
vii
Abstract
A Study On Two-Person Games In Fuzzy
Environment
By
T.Porchelvi
Research Scholar (PT) in Mathematics,
Department of Mathematics,
Tranquebar Bishop Manickam Lutheran College, Porayar.
Abstract
Game theory extends its applications in many areas like Management Sciences, Economics, Political Sciences, Psychology etc. with
known precise estimates. Most of real life situations cannot be precisely expressed, such situations have enabled the emergence of fuzzy
games. In the literature of Fuzzy games the players may have fuzzy
viii
Abstract
ix
goals or fuzzy payoffs or associated linguistic terms. Different types
of fuzzy estimates are available now. For example, triangular fuzzy
estimates, trapezoidal fuzzy estimates, bell-shaped fuzzy estimates
etc. They are being used to express a situation possibly well. Although, each fuzzy estimate has its own merits, trapezoidal fuzzy
estimates have been used in this research work.
The thesis entitled, ”A Study On Two-Person Games In
Fuzzy Environment” consists of seven chapters. The research
work states what is fuzzy two person game?their different types, the
procedures involved in solving them and how they can be applied
to the real life situations. Chapter 1 gives the chronological survey of game theory and summary of results. Chapter 2 consists of
the preliminary results and arithmetics defined on trapezoidal fuzzy
numbers. Chapter 3 deals with Two person zero sum fuzzy games
and proposes four types of solution procedures with relevant numerical examples. Chapter 4 discusses two person non zero sum fuzzy
games in details and proposing its solution procedures. Chapter 5
unfolds Constrained Fuzzy games with zero sum and non zero sum
fuzzy payoffs. In Chapter 6 the
TOPSIS
procedure has been exam-
ined with trapezoidal fuzzy payoffs and a new procedure for ordering
Abstract
x
alternatives has also been provided. Chapter 7 states a procedure
to reduce the size of a fuzzy payoff matrix and a real life implementation of the proposed techniques, it also includes the conclusion
section which comprises the overall summary of the thesis.
Table of Contents
Declaration
i
Acknowledgements
ii
List of Publications
iv
List of Notations
vii
Abstract
viii
Table of Contents
xi
1 Introduction
1
1.1
Decision Making in OR . . . . . . . . . . . . . . . . .
2
1.2
Uncertainty . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Game Theory . . . . . . . . . . . . . . . . . . . . . .
7
1.4
Vagueness and Fuzzy Logic . . . . . . . . . . . . . . .
12
xi
CONTENTS
xii
1.5
Motivation . . . . . . . . . . . . . . . . . . . . . . . .
20
1.6
Literature Review . . . . . . . . . . . . . . . . . . . .
21
1.7
Future Directions . . . . . . . . . . . . . . . . . . . .
28
1.8
Organization of the Thesis . . . . . . . . . . . . . . .
29
2 Preliminaries
2.1
34
Trapezoidal fuzzy number . . . . . . . . . . . . . . .
35
2.1.1
Definition . . . . . . . . . . . . . . . . . . . .
35
2.1.2
Interpretation . . . . . . . . . . . . . . . . . .
35
2.1.3
Operations on Trapezoidal fuzzy numbers . .
36
2.1.4
Universal trapezoidal fuzzy number . . . . . .
37
2.1.5
Comparison of Trapezoidal fuzzy numbers . .
38
2.2
Fuzzy Convex Combination . . . . . . . . . . . . . .
38
2.3
Important Notations . . . . . . . . . . . . . . . . . .
39
2.4
Operations on Fuzzy Matrices . . . . . . . . . . . . .
39
2.4.1
Fuzzy Matrix . . . . . . . . . . . . . . . . . .
39
2.4.2
Operations on Fuzzy Matrices . . . . . . . . .
39
2.4.3
F-Magnitude . . . . . . . . . . . . . . . . . .
40
Ranking Functions . . . . . . . . . . . . . . . . . . .
40
2.5.1
Novel Ranking function
. . . . . . . . . . . .
40
2.5.2
Compos and Munoz Ranking function . . . .
41
2.5
CONTENTS
2.6
2.7
xiii
2.5.3
Magnitude method proposed by Abbasbandy .
41
2.5.4
Liou and Wang’s Ranking function . . . . . .
41
Linguistic Variables . . . . . . . . . . . . . . . . . . .
42
2.6.1
Definition . . . . . . . . . . . . . . . . . . . .
42
2.6.2
Example . . . . . . . . . . . . . . . . . . . . .
43
Using Linguistic Variables . . . . . . . . . . . . . . .
44
3 Two Person Zero-Sum Fuzzy Games
47
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . .
48
3.2
A two person zero-sum fuzzy game . . . . . . . . . .
52
3.3
Fuzzy Nash Equilibrium . . . . . . . . . . . . . . . .
52
3.3.1
Definition . . . . . . . . . . . . . . . . . . . .
53
3.3.2
Numerical Example . . . . . . . . . . . . . . .
56
Fuzzy determinant Procedure . . . . . . . . . . . . .
59
3.4.1
Definition . . . . . . . . . . . . . . . . . . . .
59
3.4.2
Algorithm for Fuzzy Determinant Procedure .
59
3.4.3
Numerical Example . . . . . . . . . . . . . . .
60
Matrix Difference Procedure . . . . . . . . . . . . . .
63
3.5.1
Procedure . . . . . . . . . . . . . . . . . . . .
63
3.5.2
Numerical Example . . . . . . . . . . . . . . .
64
An Approximate Method . . . . . . . . . . . . . . . .
66
3.4
3.5
3.6
CONTENTS
xiv
3.6.1
Algorithm . . . . . . . . . . . . . . . . . . . .
66
3.6.2
Numerical Example . . . . . . . . . . . . . . .
67
4 Two Person Non Zero-Sum Fuzzy Games
72
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . .
73
4.2
A Fuzzy Bi - Matrix Game . . . . . . . . . . . . . . .
77
4.3
Fuzzy Nash Equilibrium . . . . . . . . . . . . . . . .
77
4.3.1
Definition . . . . . . . . . . . . . . . . . . . .
78
4.3.2
Numerical Example . . . . . . . . . . . . . . .
81
Fuzzy determinant Procedure . . . . . . . . . . . . .
92
4.4.1
Definition . . . . . . . . . . . . . . . . . . . .
92
4.4.2
Procedure . . . . . . . . . . . . . . . . . . . .
92
4.4.3
Numerical Example . . . . . . . . . . . . . . .
93
Matrix Difference Procedure . . . . . . . . . . . . . .
98
4.5.1
Procedure . . . . . . . . . . . . . . . . . . . .
98
4.5.2
Numerical Example . . . . . . . . . . . . . . .
99
4.4
4.5
4.6
An Approximate Method . . . . . . . . . . . . . . . . 102
4.6.1
Algorithm . . . . . . . . . . . . . . . . . . . . 102
4.6.2
Numerical Example . . . . . . . . . . . . . . . 103
5 Constrained Fuzzy Games
107
CONTENTS
xv
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . 108
5.2
A Constrained zero sum fuzzy game . . . . . . . . . . 111
5.2.1
Optimal strategies . . . . . . . . . . . . . . . 112
5.2.2
Remark . . . . . . . . . . . . . . . . . . . . . 112
5.3
Theorems . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4
A Constrained non zero sum Bi-matrix fuzzy game . 117
5.4.1
Optimal Strategies . . . . . . . . . . . . . . . 118
6 A new approach to solve fuzzy games
121
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . 122
6.2
TOPSIS procedure . . . . . . . . . . . . . . . . . . . 126
6.3
6.2.1
Algorithm . . . . . . . . . . . . . . . . . . . . 126
6.2.2
Numerical Examples . . . . . . . . . . . . . . 128
Preference Ordering of Alternatives using ranking functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.3.1
Algorithm with a general ranking function . . 133
6.3.2
The proposed algorithm with Novel Ranking
function . . . . . . . . . . . . . . . . . . . . . 135
6.3.3
The proposed algorithm with Compos and Munoz
Ranking function . . . . . . . . . . . . . . . . 136
CONTENTS
6.3.4
xvi
The proposed algorithm with Abbasbandy’s
Magnitude function . . . . . . . . . . . . . . . 137
6.3.5
Numerical Example . . . . . . . . . . . . . . . 140
6.3.6
Solution Procedure with Novel Ranking Function . . . . . . . . . . . . . . . . . . . . . . . 140
6.3.7
Solution Procedure with Compos and Munoz
Ranking function . . . . . . . . . . . . . . . . 145
6.3.8
Solution Procedure with Abbasbandy’s Ranking function . . . . . . . . . . . . . . . . . . . 147
6.3.9
Solution Procedure with Liou and Wang’s Ranking function . . . . . . . . . . . . . . . . . . . 149
7 Fuzzy Dominance Procedure
151
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . 152
7.2
Important Theorems . . . . . . . . . . . . . . . . . . 155
7.3
Numerical Example . . . . . . . . . . . . . . . . . . . 164
7.4
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 165
Bibliography
168
List of Figures
1.1
The Concept of Defuzzification . . . . . . . . . . . .
19
1.2
Membership Function of a fuzzy set . . . . . . . . . .
20
2.1
Universal trapezoidal fuzzy number . . . . . . . . . .
38
2.2
Example of a Linguistic Variable . . . . . . . . . . .
44
2.3
Usage of Linguistic variable . . . . . . . . . . . . . .
46
xvii
List of Tables
3.1
Simplex Table . . . . . . . . . . . . . . . . . . . . . .
58
3.2
58
3.3
Simplex Table(Continued) . . . . . . . . . . . . . . .
P P
i
j M̃ãij . . . . . . . . . . . . . . . . . . . . . . .
3.4
Row Iteration Table
. . . . . . . . . . . . . . . . . .
70
3.5
Column Iteration Table . . . . . . . . . . . . . . . . .
71
4.1
Simplex Table . . . . . . . . . . . . . . . . . . . . . .
85
4.2
Simplex Table(Continued) . . . . . . . . . . . . . . .
86
4.3
Simplex Table(Continued) . . . . . . . . . . . . . . .
87
4.4
Simples Table . . . . . . . . . . . . . . . . . . . . . .
88
4.5
Simplex Table(Continued) . . . . . . . . . . . . . . .
89
4.6
Simplex Table(Continued) . . . . . . . . . . . . . . .
90
4.7
Simplex Table(Continued) . . . . . . . . . . . . . . .
91
4.8
Fuzzy values and Fuzzy strategies for Player A . . . .
96
4.9
Fuzzy values and Fuzzy strategies for Player B . . . .
97
xviii
61
LIST OF TABLES
xix
4.10 Comparison of row elements . . . . . . . . . . . . . . 105
4.11 Row Iterations for Player I . . . . . . . . . . . . . . . 106
6.1
dij values Zero-Sum Game . . . . . . . . . . . . . . . 129
6.2
dij Values for Player A . . . . . . . . . . . . . . . . . 131
6.3
dij Values for Player B . . . . . . . . . . . . . . . . . 132
6.4
Linguistic terms and their values . . . . . . . . . . . 133
6.5
The interval approximations of alternatives . . . . . . 142
6.6
Mid point values . . . . . . . . . . . . . . . . . . . . 144
6.7
Ranking values in Compos and Munoz method . . . . 146
6.8
Ranking values in Abbasbandy method . . . . . . . . 148
6.9
Ranking values in Liou and Wang’s Method . . . . . 150
Chapter 1
Introduction
Abstract
In this introductory chapter the theoretical background of
the thesis is given. It also contains the detailed chronological survey of the related literature. It gives special attention to the motivations and further scope of the thesis.
1
Ch.1 Introduction
1.1
2
Decision Making in OR
Operations research, or operational research in British usage, is a discipline that deals with the application of advanced analytical methods which helps to make better decisions. It is often considered to
be a sub-field of mathematics. The terms management science and
decision science are sometimes used synonymously.
Decision making can be regarded as the mental processes (cognitive process) resulting in the selection of a course of action among
several alternative scenarios. Every decision making process produces a final choice. The output can be an action or an opinion of
choice.
One must keep in mind that most decisions are made un-
consciously that is ”we simply decide without thinking much about
the decision process.” In a controlled environment, such as a classroom, instructors encourage students to weigh pros and cons before
making a decision. However in the real world, most of our decisions
are made unconsciously in our mind because frankly, it would take
too much time to sit down and list the pros and cons of each decision
we must make on a daily basis.
Logical decision making is an important part of all science-based
professions, where specialists apply their knowledge in a given area
Ch.1 Introduction
3
to make informed decisions. For example, medical decision making often involves making a diagnosis and selecting an appropriate
treatment. Some research using naturalistic methods shows, however, that in situations with higher time pressure, higher stakes, or
increased ambiguities, experts use intuitive decision making rather
than structured approaches, following a recognition primed decision
approach to fit a set of indicators into the expert’s experience and immediately arrive at a satisfactory course of action without weighing
alternatives. Recent robust decision efforts have formally integrated
uncertainty into the decision making process. However, Decision
Analysis, recognized and included uncertainties with a structured
and rationally justifiable method of decision making since its conception in 1964.
Making a decision without planning is fairly common, but does
not often end well. Planning allows for decisions to be made comfortably and in a smart way. Planning makes decision making a
lot more simpler than it is. Decision will get four benefits out of
planning:
1. Planning give chance to the establishment of independent goals.
It is a conscious and directed series of choices.
Ch.1 Introduction
4
2. Planning provides a standard of measurement. It is a measurement of whether you are going towards or further away from
your goal.
3. Planning converts values to action. You think twice about the
plan and decide what will help to advance your plan best.
4. Planning allows limited resources to be committed in an orderly
way. It always governs the use of what is limited to you (e.g.
money, time, etc.)
Each step in the decision making process may include social, cognitive and cultural obstacles to negotiate dilemmas successfully. It
is suggested that becoming more aware of these obstacles allows one
to better anticipate and overcome them.
In [16], Dr. Pam Brown breaks decision making into seven steps:
1. Outline your goal and outcome
2. Gather Data
3. Develop alternatives (i.e., brainstorming)
4. List pros and cons of each alternative
5. Make the decision
Ch.1 Introduction
5
6. I immediately take action to implement it
7. Learn from and reflect on the decision
Decision making can be influenced by various external forces. The
results or outcomes of the decision making and the events that occur
during the course are quiet uncertain. This may even be called as
decision making under uncertainty.
1.2
Uncertainty
Uncertainty is a term used in subtly different ways in a number
of fields, including Physics, Philosophy, Statistics, Economics, Finance, insurance, Psychology, Sociology, Engineering and Information science. It applies to predictions of future events, to physical
measurements already made, or to the unknown.
Although the terms are used in various ways among the general
public, many specialists in decision theory, statistics and other quantitative fields have defined uncertainty, risk, and their measurement
in a different way. For example, in [37] Hubbard defined these terms
as:
1. Uncertainty: The lack of certainty, A state of having limited
Ch.1 Introduction
6
knowledge where it is impossible to exactly describe the existing
state, a future outcome, or more than one possible outcome.
2. Measurement of Uncertainty: A set of possible states or
outcomes where probabilities are assigned to each possible state
or outcome this also includes the application of a probability
density function to continuous variables
3. Risk: A state of uncertainty where some possible outcomes
have an undesired effect or significant loss.
4. Measurement of Risk: A set of measured uncertainties where
some possible outcomes are losses, and the magnitudes of those
losses this also includes loss functions over continuous variables.
Uncertainty may be purely a consequence of a lack of knowledge
of obtainable facts. That is, you may be uncertain about whether
a new rocket design will work, but this uncertainty can be removed
with further analysis and experimentation. At the sub-atomic level,
however, uncertainty may be a fundamental and unavoidable property of the universe. In quantum mechanics, the Uncertainty Principle puts limits on how much an observer can ever know about the
position and velocity of a particle. This may not just be ignorance
Ch.1 Introduction
7
of potentially obtainable facts but that there is no fact to be found.
There is some controversy in physics as to whether such uncertainty
is an irreducible property of nature or if there are hidden variables
that would describe the state of a particle even more exactly than
the uncertainty principle allows. Game theory enables manipulations that arises due to uncertainty.
1.3
Game Theory
Game theory is a mathematical method for analyzing calculated
circumstances, such as in games, where a persons success is based
upon the choices of others. More formally, in [58] Myerson defined
it as ”the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.” In [5], Aumann
suggested an alternative term as ”a more descriptive name for the
discipline” is interactive decision theory. Game theory is mainly
used in Economics, Political science, and Psychology, and other,
more prescribed sciences, like logic or Biology. The subject first addressed zero-sum games, such that one person’s gains exactly equal
net losses of the other participant(s). Today, however, game theory
applies to a wide range of class relations, and has developed into an
Ch.1 Introduction
8
umbrella term for the logical side of science, to include both human
and non-humans, like computers. Classical uses include a sense of
balance in numerous games, where each person has found or developed a tactic that cannot successfully better his results, given the
other approach.
Mathematical game theory had beginnings with some publications by Emile Borel in [13], which led to his book Applications aux
Jeux de Hasard. However, his results were limited, and the theory regarding the non-existence of blended-strategy equilibrium in
two-player games was incorrect. Modern game theory began with
the idea regarding the existence of mixed-strategy equilibria in twoperson zero-sum games and its proof by John von Neumann. Von
Neumann’s original proof used Brouwer’s fixed-point theorem[25],
on continuous mappings into compact convex sets, which became
a standard method in game theory and mathematical economics.
His paper was followed by his book Theory of Games and Economic
Behavior[61], with Oskar Morgenstern, which considered cooperative
games of several players. The second edition of this book provided
an axiomatic theory of expected utility, which allowed mathematical
Ch.1 Introduction
9
statisticians and economists to treat decision-making under uncertainty.
This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s,
although similar developments go back at least as far as the 1930s.
Game theory has been widely recognized as an important tool in
many fields.
Early discussions of examples of two-person games
occurred long before the rise of modern, mathematical game theory.
James Madison[46] made what we now recognize as a game-theoretic
analysis of the ways states can be expected to behave under different
systems of taxation.
In [13], EmileBorel proved a minimax theo-
rem for two-person zero-sum matrix games only when the pay-off
matrix was symmetric. Borel conjectured that non-existence of a
mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false.
Game theory experienced a flurry of activity in the 1950s, during
which time the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed.
In addition, the first applications of Game theory to philosophy and
political science occurred during this time. In [35], John Harsanyi
Ch.1 Introduction
10
developed the concepts of complete information and Bayesian games.
Nash, Selten and Harsanyi became Economics Nobel Laureates in
1994 for their contributions to economic game theory.
In 2005, game theorists Thomas Schelling and Robert Aumann
followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling
worked on dynamic models, early examples of evolutionary game
theory. In [6], Aumann contributed more to the equilibrium school,
introducing an equilibrium coarsening, correlated equilibrium, and
developing an extensive formal analysis of the assumption of common knowledge and of its consequences.
The payoffs of the game are generally taken to represent the utility of individual players. Often in modeling situations the payoffs
represent money, which presumably corresponds to an individual’s
utility. This assumption, however, can be faulty.
A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic
situation. One or more solution concepts are chosen, and the author
Camerer in [19] demonstrates which strategy sets in the presented
game are equilibria of the appropriate type. Naturally one might
wonder to what use should this information be put. Economists and
Ch.1 Introduction
11
business professors suggest two primary uses (noted above): descriptive and prescriptive.
The application of game theory to political science is focused
in the overlapping areas of fair division, political economy, public
choice, war bargaining, positive political theory, and social choice
theory. In each of these areas, researchers have developed gametheoretic models in which the players are often voters, states, special
interest groups, and politicians.
Game theory has also challenged philosophers to think in terms
of interactive epistemology: what it means to have common beliefs
or knowledge, and what are the consequences of this knowledge for
the social outcomes resulting from agents’ interactions. Philosophers who have worked in this area include Bicchieri, Skyrms and
Stalnaker [11].
In ethics, some authors have attempted to pursue the project, of
deriving morality from self-interest. Since games like the Prisoner’s
dilemma present an apparent conflict between morality and selfinterest, explaining why cooperation is required by self-interest is
an important component of this project. This general strategy is a
component of the general social contract view in political philosophy.
Ch.1 Introduction
12
Other authors have attempted to use evolutionary game theory
in order to explain the emergence of human attitudes about morality
and corresponding animal behaviors. These authors look at several
games including the Prisoner’s dilemma, Stag hunt, and the Nash
bargaining game as providing an explanation for the emergence of
attitudes about morality.
Some assumptions used in some parts of game theory have been
challenged in philosophy; psychological egoism states that rationality reduces to self-interest which is claim debated among philosophers. (see Psychological egoism Criticisms)
1.4
Vagueness and Fuzzy Logic
The term vagueness denotes a property of concepts (especially predicates). In everyday speech, vagueness is an inevitable, often even
desired effect of language usage. However, in most specialized texts
(e.g., legal documents), vagueness is distracting and should be avoided
whenever possible.
Vagueness is philosophically important. Suppose one wants to
come up with a definition of ”right” in the moral sense. One wants
a definition to cover actions that are clearly right and exclude actions
Ch.1 Introduction
13
that are clearly wrong, but what does one do with the borderline
cases? Surely, there are such cases. Some philosophers say that one
should try to come up with a definition that is itself unclear on just
those cases. Others say that one has an interest in making his or
her definitions more precise than ordinary language, or his or her
ordinary concepts, themselves allow; they recommend one advance
precising definition.
Vagueness is also a problem which arises in law, and in some
cases judges have to arbitrate regarding whether a borderline case
does, or does not, satisfy a given vague concept. Examples include
disability (how much loss of vision is required before one is legally
blind?), human life (at what point from conception to birth is one
a legal human being, protected for instance by laws against murder?), adulthood (most familiarly reflected in legal ages for driving,
drinking, voting, consensual sex, etc.), race (how to classify someone
of mixed racial heritage), etc. Even such apparently unambiguous
concepts such as gender can be subject to vagueness problems, not
just from transsexuals’ gender transitions but also from certain genetic conditions which can give an individual both male and female
biological traits (see intersexual).
Ch.1 Introduction
14
Many scientific concepts are of necessity vague, for instance species
in biology cannot be precisely defined, owing to unclear cases such
as ring species. Nonetheless, the concept of species can be clearly
applied in the vast majority of cases. As this example illustrates, to
say that a definition is ”vague” is not necessarily a criticism. Consider those animals in Alaska that are the result of breeding Huskies
and wolves: are they dogs? It is not clear: they are borderline cases
of dogs. This means one’s ordinary concept of doghood is not clear
enough to let us rule conclusively in this case.
The philosophical question of what the best theoretical treatment
of vagueness is - which is closely related to the problem of the paradox of the heap - has been the subject of much philosophical debate.
One theoretical approach is that of fuzzy logic, developed by
American mathematician Lotfi Zadeh in [97]. Fuzzy logic proposes a
gradual transition between ”perfect falsity”, for example, the statement ”Bill Clinton is bald”, to ”perfect truth”, for, say, ”Patrick
Stewart is bald”. In ordinary logics, there are only two truth-values:
”true” and ”false”. The fuzzy perspective differs by introducing an
infinite number of truth-values along a spectrum between perfect
truth and perfect falsity. Perfect truth may be represented by ”1”,
Ch.1 Introduction
15
and perfect falsity by ”0”. Borderline cases are thought of as having
a ”truth-value” anywhere between 0 and 1 (for example, 0.6).
Fuzzy concepts may generate uncertainty (they do not provide a
clear orientation for action or decision-making) and reducing fuzziness may generate more certainty. However, this is not necessarily
always so, insofar as a concept, although it is not fuzzy at all and
very exact, could equally well fail to capture the meaning of something adequately. A concept can be very precise, but not - or insufficiently - applicable or relevant in the situation to which it refers.
A fuzzy concept may indeed provide more security, because it provides a meaning for something when an exact concept is unavailable
- which is better than not being able to denote it at all. A concept
such as God, although not easily definable, for instance can provide
security to the believer.
The origin of fuzzy concepts is partly due to the fact that the
human brain does not operate like a computer. While computers
use strict binary logic gates, the brain does not; i.e., it is capable
of making all kinds of neural associations according to all kinds
of ordering principles (or fairly chaotically) in associative patterns
which are not logical but nevertheless meaningful. Something can
Ch.1 Introduction
16
be meaningful although we cannot name it, or we might only be able
to name it and nothing else. The human brain can also interpret
the same phenomenon in several different but interacting frames of
reference, at the same time, or in quick succession.
In part, fuzzy concepts arise also because learning or the growth
of understanding involves a transition from a vague awareness, which
cannot orient behaviour greatly, to clearer insight, which can orient
behavior.
Some logicians argue that fuzzy concepts are a necessary consequence of the reality that any kind of distinction we might like to
draw has limits of application. As a certain level of generality, it
works fine. But if we pursued its application in a very exact and
rigorous manner, or overextend its application, it appears that the
distinction simply does not apply in some areas or contexts, or that
we cannot fully specify how it should be drawn. An analogy might
be that zooming a telescope, camera, or microscope in and out reveals that a pattern which is sharply focused at a certain distance
disappears at another distance.
In psychophysics it has been discovered that the perceptual distinctions are drawn in the mind are often more sharply defined than
Ch.1 Introduction
17
they are in the real world. Thus, the brain actually tends to ”sharpen
up” one’s perceptions of differences in the external world. Between
black and white, they are able to detect only a limited number of
shades of gray, or colour gradations. If there are more gradations and
transitions in reality than our conceptual distinctions can capture,
then it could be argued that how those distinctions will actually apply must necessarily become vaguer at some point. If, for example,
one wants to count and quantify distinct objects using numbers, one
needs to be able to distinguish between those separate objects, but if
this is difficult or impossible, then, although this may not invalidate
a quantitative procedure as such, quantification is not really possible
in practice; at best, one may be able to assume or infer indirectly a
certain distribution of quantities.
The ”centroid” method is very popular, in which the ”center of
mass” of the result provides the crisp value. Another approach is the
”height” method, which takes the value of the biggest contributor.
The centroid method favors the rule with the output of greatest area,
while the height method obviously favors the rule with the greatest
output value. The diagram below demonstrates max-min inferencing
and centroid defuzzification for a system with input variables ”x”,
Ch.1 Introduction
18
”y”, and ”z” and an output variable ”n”. Note that ”mu” is standard
fuzzy-logic nomenclature for ”truth value” fig 1.1.
The membership function of a fuzzy set is a generalization of
the indicator function in classical sets. In fuzzy logic, it represents
the degree of truth as an extension of valuation. Degrees of truth
are often confused with probabilities, although they are conceptually distinct, because fuzzy truth represents membership in vaguely
defined sets, not likelihood of some event or condition.
For any set X, a membership function on X is any function from X
to the real unit interval [0, 1]. Membership functions on X represent
fuzzy subsets of X. The membership function which represents a
fuzzy set Ã is usually denoted by µA . For an element x of X, the
value µA (x) is called the membership degree of x in the fuzzy set Ã.
The membership degree µA (x) quantifies the grade of membership
of the element x to the fuzzy set Ã. The value 0 means that is not a
member of the fuzzy set; the value 1 means that is fully a member
of the fuzzy set. The values between 0 and 1 characterize fuzzy
members, which belong to the fuzzy set only partially.
Ch.1 Introduction
Figure 1.1: The Concept of Defuzzification
19
Ch.1 Introduction
20
Figure 1.2: Membership Function of a fuzzy set
1.5
Motivation
Game theory has already proved its tremendous potential for conflict
resolution problems in the fields of Decision Theory and Economics.
In the recent past, there have been attempts to extend the results
of crisp game theory to those conflict resolution problems which are
fuzzy in nature e.g. Nishizaki and Sakawa [62] and references cited
there in. These developments have lead to the emergence of a new
area in the literature called fuzzy games. Another area in the fuzzy
decision theory, which has been growing very fast is the area of fuzzy
mathematical programming and its applications to various branches
of sciences, Engineering and Management.
In the crisp scenario, there exists a beautiful relationship between
Ch.1 Introduction
21
two person zero sum matrix game theory and duality in linear programming. It is therefore natural to ask if something similar holds in
the fuzzy scenario as well. This discussion essentially constitutes the
core of Fuzzy Mathematical Programming and Fuzzy Matrix Games
[9].
Most commonly, fuzzy numbers have been ranked by its center
of gravity or by the real number with its maximal membership or
by using some different kinds of ranking functions when they been
discussed in the equilibrium strategy. In doing so, fuzzy games were
transformed into crisp games, so the fuzzy equilibrium strategy was
equal to crisp one. Thus, much uncertain information had been lost
and the fuzzy games were not the real fuzzy games yet. To overcome
these types of limitations, the arithmetics on fuzzy numbers have
been defined accordingly and being used. This emerges from the
trend of finding fuzzy equilibrium strategies as a fuzzy entity itself
and motivates lots of researchers.
1.6
Literature Review
Game theory was firstly proposed by Von Neumann and Morgenstern in the book named Theory of Games and Economic Behavior
Ch.1 Introduction
22
[61]. Then the game theory had become a subject, and many people
had been absorbed in the study of game theory. So far, many new
investigative directions have been created on game theory, one of
which is fuzzy game theory.
It is known that sets can be turned into fuzzy sets and relations
can be turned into fuzzy relations. Thus, it seems to create a fuzzy
game, we have many choices. Each choice stems from a different
story about how the game is to be played. In [85]Song and Kandel
examines a variation on the prisoners dilemma where the degree to
which the player wishes to help or harm his partner is fuzzy. In
[33], Garagic and Cruz uses a fuzzy fixed-point theorem to show
that certain fuzzy matrix games have a Nash equilibrium. Another
way of making a game fuzzy is explored by Arfi in [3] where outcomes of a variation of the prisoners dilemma are broken down into
finite, discrete values based on finite, discrete levels of cooperation
and defection. Another work by Nishizaki and Sakawa in [62] explores cooperative fuzzy games as a method of conflict resolution
and concentrates on numerical solutions to such game.
Another very similar construction called an equilibrium solution
with respect to the degree of attainment of the aggregated fuzzy
Ch.1 Introduction
23
goal is discussed in [62] by Nishizaki and Sakawa . Their work concentrates on computational methods for finding such solutions. We
turn to a more sophisticated conception of a fuzzy game, and define
a Nash equilibrium concept based upon it. A generalized version of
this is due to Butnariu[17]. Exploring this work will give another
example of the importance of fixed points in analyzing equilibria in
game theory. As Butnarius formulation is more general than that
of some other games found in more recent research, including that
of Nishizaki and Sakawa, examples and ideas discussed in that work
could be adapted to Butnarius concept.
In [43], F. Kacher and M. Larbani discussed the existence of
equilibrium solution for a noncooperative game with fuzzy goals
and parameters. In [99], Zhang and etal defined the safe point
as the solution of mixed fuzzy multi-objective many-person noncooperative game and also showed the existence of safe point. In [32],
Garagic studied n-person static fuzzy non-cooperative games. In
[63], Nishizalki and Sakawa studied the equilibrium solutions for the
multiobjective bi-matrix games with fuzzy payoff and fuzzy goals.
In [85], Song and Kandel proved the equilibrium strategy for fuzzy
Ch.1 Introduction
24
games by tools of multi-criteria decision making and fuzzy set theory. In [48], Kim and Lee proposed fuzzy equilibrium of fuzzy games
by introducing the fuzzy fixed point. In [17], Butnariu described
general playing rules in the fuzzy two-person games and formulated
non-cooperative fuzzy games. In [65], Ostrowski made a study on
two-person games with fuzzy strategy set. This kind of fuzzy games
is defined to be a set of rules that determine the possible exchange of
information between the participants in the games. In [77], Ragade
investigated fuzzy two-person games whose options were uncertain.
Almost all of the above researchers ranked fuzzy number by its
center of gravity or by the real number with its maximal membership when they discussed the equilibrium strategy. In doing so, fuzzy
games were transformed into crisp games, so the fuzzy equilibrium
strategy was equal to crisp one. By reducing fuzzy number into a
real number, one loses much fuzzy information that should be kept
during the operations between fuzzy numbers. The fuzzy quantities
or alternatives are ordered directly by Yuans[96] binary fuzzy ordering relation. Based on this binary fuzzy ordering, the existence of
fuzzy Nash equilibrium for fuzzy non-cooperative games has been
proved by Xiaohui Yu in[95]
Ch.1 Introduction
25
In the area of standard coalition TU games, consistency is a crucial property which has been applied comprehensively. If a solution
is not consistent, then a subgroup of agents might not respect the
original compromise but revise the payoff distribution within the
subgroup. The fundamental property of solutions has been investigated in various classes of problems by applying reduced games
always. The core is, perhaps, the most intuitive solution concept
in game theory. Relating to the core of standard coalition games,
there are two different types of imaginary reduced standard coalition
games in the literature, the max-reduced game proposed by Davis
and Maschler in [26] and the complement-reduced game proposed
by Moulin[57]. Based on the max-reduced games, Peleg [75] characterized the core on the domain of standard coalition games whose
core is non-empty. Subsequently, Serrano and Volij[81] characterized the core on the domain of all standard coalition games. Based
on the complement-reduced games, Tadenuma [86] characterized the
core on the domain of standard coalition games whose core is nonempty. Related results may be found in Peleg [74], Voorneveld and
Nouweland [90], and so on.
In this area of Coalition TU games, the notions of a fuzzy game
Ch.1 Introduction
26
and the core of a fuzzy game are introduced by Aubin [42, 4].Further,
Hwang in [39] extended the core and the max-reduced game to fuzzy
NTU games. Inspired by Serrano and Volij [81], Hwang offered
axiomatizations of the core of fuzzy NTU games. Different from the
works of Hwang, Liao and Yu-Hsier in [53] provides an extension of
the reduced game introduced by Voorneveld and van den Nouweland
in [90] to fuzzy NTU games.
In [87], Thomason has introduced the concept of fuzzy matrices.
After that a lot of works have been done on fuzzy matrices and its
variants by Ragab and Emam in [76], by Thomason in [87] and by M.
Pal et al in [68]].In his paper, Anita Pal[67] introduced interval valued fuzzy matrices (IVFMs) as the generalization of interval-valued
fuzzy sets. Some essential unary and binary operations of IVFM and
some special types of IVFMs i.e., symmetric, reflexive, transitive and
idempotent, constant, etc. are defined. The idea of convergence, periodicity, determinant and adjoint of IVFMs are also defined. Lot of
properties of IVFMs are also presented in that paper.
The concept of interval approaches to linear programming problem and decision processes were developed by Ishibuchi in [40], Shaochang
Ch.1 Introduction
27
in [82] and Kurano in [50]. Using these results Masami in [84] developed Interval matrix games and its extensions to fuzzy and stochastic games.
In [34], Chunyan Han and et al developed three kinds of minimax
equilibrium strategies based on a fuzzy max order for fuzzy games
with two players.
In the subject of Bi-matrix games, Milchtaich in [56] presented a
formula for computing the equilibrium payoffs when the support of
the equilibrium is known. Ostrowski in [65] presented formulas for
computing the completely mixed strategy and the equilibrium payoff in a symmetric bimatrix game. In [56], Igal Milchtaich focused
on saddle point matrices with two vector blocks and their applications in game theory. Necessary and sufficient conditions for the
existence and uniqueness of a completely mixed Nash equilibrium in
a Bimatrix game are presented.
Lots of research articles have been published in the area of bimatrix games, such as Maeda [84], Nishizaki and Sakawa [62] and
Vijay, Chandra and Bector [89] have done their contributions in a
considerable amount. Those includes, bi-matrix games with fuzzy
goals: Nishizaki and Sakawas model, bi-matrix games with fuzzy
Ch.1 Introduction
28
goals: another approach, bi-matrix games with fuzzy pay-offs: a
ranking functionapproach, bi-matrix games with fuzzy goals and
fuzzy pay-offs, and bi-matrix games with fuzzy pay-offs: a possibility measure approach.
There are certain matrix game theoretic problems in real life
where the strategies of the players are constrained to satisfy general linear inequalities rather than being in the strategy spaces only.
These decision problems give rise to constrained matrix games which
have initially been studied by Charnes in [20] and then later in some
what more generality by Kawaguchi and Maruyama in [45].
The work in fuzzy related areas wouldn’t be completed without
considering Dumitrescu, Lazzerini and Jain’ [29], Klir and Yuan’
[49], Lin and Lee’ [54] and Zimmermann’[100].
1.7
Future Directions
There are many opportunities for researchers to explore in this area
of important ”Fuzzy Games”.
1. Fuzzy games can be applied to a wide range of subjects like,
Economics, Psychology, Computational areas etc. Unlike the
crisp game which has some limitations that all practical life
Ch.1 Introduction
29
situations cannot be written as precise mathematical terms,
Fuzzy games will be a better option.
2. In this thesis, the fuzzy terms are taken as trapezoidal fuzzy
numbers. But it is possible of using all types of fuzzy notions
and their significant properties. To do so, it is necessary to
define fuzzy arithmetics on them relevantly.
3. As type - 2 fuzzy set are one of the developing area of research
and having its own special characteristics it can also be taken
as the fuzzy entities here. Other more useful fuzzy notions if
any that can also be used possibly.
4. With the knowledge of a new type of Fuzzy games, called constrained fuzzy games, some of complicated real life problems can
be converted into a fuzzy game with some set of constraints in
addition to the ordinary constraints.
1.8
Organization of the Thesis
The organization of the thesis is explained in this section. The thesis entitled ”A Study On Two-Person Games In Fuzzy Environment” consists of seven chapters. In chapter 1, the theoretical
Ch.1 Introduction
30
backgrounds of the thesis are given in detail. Also the origin and
the chronological developments of Game theory and Fuzzy logic have
been briefed. In addition to that, the usage of fuzzy concepts in some
areas of sciences are also given. Chapter 2 gives all the preliminary
concepts that are needed through out this work.
In Chapter 3 Fuzzy games in particular, two person zero-sum
fuzzy games are introduced. Examples are also given to explain how
a real life situation has been considered as a fuzzy game and some
new methods have been proposed to solve such a fuzzified game.
The contents of this chapter has been published in the International
Journals:
• International Journal of Computational Science and Mathematics, ISSN 0974-3189, 3(4) 447-453 (2011).
• International Journal of Mathematics Research, ISSN
0976-5840, 3(5) 475-482 (2010).
• Journal of Current Sciences,15(1) 207-210 (2010).
• International Journal of Advances in Fuzzy Mathematics, ISSN 0973-533X, 5(3) 295-300 (2010).
In Chapter 4 another type of fuzzy games called non-zero sum
Ch.1 Introduction
31
Bi-matrix fuzzy games are discussed and their solving procedures
are also proposed. The contents of this chapter has been published
in the International Journals:
• International Conference on Mathematics and Computer Science, Loyola College, Chennai, January 2011.
• IEEE International Conference on Advances in Engineering, Science and Management, ISBN: 978-81-9090422-3 2012 IEEE, 5.
Chapter 5 detailing about a different kind of Fuzzy games known
as constrained fuzzy games. In this kind, both zero - sum and non
zero sum types have been defined and explained in detail. Some solving procedures have also been given. The contents of this chapter has
been published in the Proceedings of the International Conferences:
• Journal of Fuzzy Mathematics,International Fuzzy Mathematics Institute, Los Angeles ISSN 1066-8950, 21(1)
173-180 (2013).
Ch.1 Introduction
32
Chapter 6 proposes two different kind of approaches namely, TOPSIS and a new ranking-function based algorithm to order alternatives. Relevant illustrations have been presented for better understanding. The contents of this chapter has been published in the
International Journal:
• International Electronic Journal of Pure and Applied
Mathematics, ISSN 1314-0744 4(1) 53-58 (2012).
and also been published in the Proceedings of the International Conference:
• Proceedings of the International Conference on Mathematics in Engineerng and Business Management, Stella
Maris College, Chennai, March 2012.
In the previous chapters, some of the solving procedures for zero
sum fuzzy games, non zero sum bi-matrix fuzzy games and Constrained fuzzy games have been proposed. In chapter 7 a common
method called Fuzzy Dominance which proposes the rules to reduce the size of a fuzzy payoff matrix without affecting the final
payoff is discussed. The contents of this chapter has been published
in the International Journal:
Ch.1 Introduction
33
• International Journal of Algorithms, Computing and
Mathematics, ISSN 0974-3367, 3(3) 17-20 (2010).
The Concluding section contains the overall summary of the thesis.
In the present scenario, it is evident that fuzzy logic and its
applied fields have acquired a wide range of attention because of
its extensive behaviors. Having been applicable to most of common
life problems, Game Theory itself is a subject of importance together
with fuzzy logic, will attract more researchers towards this literature
of Fuzzy Games in future.
Chapter 2
Preliminaries
Abstract
This chapter contains all the definition and concepts that
have been used in this work. Since the payoffs of the players are taken as trapezoidal fuzzy numbers, its meaning in
various aspects has also been discussed here.
34
Ch.2 Preliminaries
2.1
2.1.1
35
Trapezoidal fuzzy number
Definition
A trapezoidal fuzzy number ã ≈ (a1 , a2 , a3 , a4 )is defined by the
membership function,
µã (x) =

x−a1


a2 −a1 , if









if

1,
a 1 ≤ x ≤ a2 ;
a2 ≤ x ≤ a3 ;


x−a4


, if a3 ≤ x ≤ a4 ;

a
3 −a4








0,
otherwise.
2.1.2
Interpretation
A manufacturing company wants to find the quantity it has to produce so as to get an optimum profit. Based on the past experiences
it is observed that the demand for a particular product is around the
number r. It is also observed that in most of the times the possible
demand is between a2 and a3 and in some rare occasions the demand
lies between a1 and a2 or between a3 and a4 where
a1 +a2 +a3 +a4
4
= r.
This situation can better be represented by the trapezoidal fuzzy
number ã ≈ (a1 , a2 , a3 , a4 ). A sure demand of ’a’ units can better be
Ch.2 Preliminaries
36
represented by the trapezoidal fuzzy number (a,a,a,a).
For instance, I˜ ≈ (1, 1, 1, 1) denotes a sure 1 unit demand and
0̃ ≈ (0, 0, 0, 0) denotes a sure 0 unit demand i.e., no demand.
2.1.3
Operations on Trapezoidal fuzzy numbers
For ã ≈ (a1 , a2 , a3 , a4 ) and b̃ ≈ (b1 , b2 , b3 , b4 ) the following operations
are defined.
Addition:
ã ⊕ b̃ ≈ (a1 + b1 , a2 + b2 , a3 + b3 , a4 + b4 )
Subtraction:
ã b̃ ≈ (a1 − b4 , a2 − b3 , a3 − b2 , a4 − b1 )
Multiplication:
ã b̃ ≈

a1 (b1 +b2 +b3 +b4 ) a2 (b1 +b2 +b3 +b4 ) a3 (b1 +b2 +b3 +b4 )

(
,
,
,

4
4
4



a4 (b1 +b2 +b3 +b4 )

), if b̃ < 0̃;

4



a4 (b1 +b2 +b3 +b4 ) a3 (b1 +b2 +b3 +b4 ) a2 (b1 +b2 +b3 +b4 )


,
,
,
(

4
4
4



a1 (b1 +b2 +b3 +b4 )
), if b̃ 4 0̃;
4
Ch.2 Preliminaries
37
Division:
ã b̃ ≈



( (b1+b24a+b1 3+b4) , (b1+b24a+b2 3+b4) , (b1+b24a+b3 3+b4) ,




4a4

), if b̃ < 0̃;

(b
+b

1 2 +b3 +b4 )


4a3
4a2
4a4


,
,
,
(

(b
+b
+b
+b
)
(b
+b
+b
+b
)
(b
+b

1 2 3 4
1 2 3 4
1 2 +b3 +b4 )


4a1

), if b̃ 4 0̃;
(b1 +b2 +b3 +b4 )
Scalar Multiplication:
If k 6= 0 is a scalar, k ã is defined as,



(ka1 , ka2 , ka3 , ka4 ), if
kã ≈


(ka , ka , ka , ka ), if
4
3
2
1
2.1.4
k > 0;
k < 0;
Universal trapezoidal fuzzy number
The universal trapezoidal fuzzy number Ũ for a given set of trapezoidal fuzzy numbers has been defined as,
Ũ ≈ (min1 , min2 , max2 , max1 ),
where the minimums and maximums are taken over the universe
of discourse. If ã ≈ (a1 , a2 , a3 , a4 ) and b̃ ≈ (b1 , b2 , b3 , b4 ) be two
trapezoidal fuzzy numbers then Ũ ≈ (a1 , a2 , b3 , b4 ) which has been
shown in the following figure.
Ch.2 Preliminaries
38
Figure 2.1: Universal trapezoidal fuzzy number
2.1.5
Comparison of Trapezoidal fuzzy numbers
Let ã ≈ (a1 , a2 , a3 , a4 ) and b̃ ≈ (b1 , b2 , b3 , b4 ) be two trapezoidal fuzzy
numbers. Then ã b̃ iff a1 > b1 , a2 > b2 , a3 > b3 and a4 > b4 or
d(Ũ , ã) > d(Ũ , b̃), where
q
d(Ũ , ã) ≈ 2 (x0 (Ũ ) − x0 (ã))2 + (y0 (Ũ ) − y0 (ã))2 ,
q
d(Ũ , b̃) ≈ 2 (x0 (Ũ ) − x0 (b̃))2 + (y0 (Ũ ) − y0 (b̃))2
4 a3 −a1 a2
] and
x0(ã) = 31 [a1 + a2 + a3 + a4 − a4a+a
3 −a1 −a2
−a2
y0(ã) = 31 [1 + a4+aa33−a
].
1 −a2
2.2
Fuzzy Convex Combination
A fuzzy number vector X̃ ∈ F m (R) is called a fuzzy convex combination of the fuzzy vectors X̃i ∈ F m (R), (i = 1, . . . , n) if there exist
Ch.2 Preliminaries
39
fuzzy numbers λ̃i ∈ F (R) satisfying λ̃i < 0̃ and
P
that ni=1 λ̃i X̃i = X̃.
2.3
Pn
i=1 λ̃i
= I˜ such
Important Notations
• F(R)is the set of all trapezoidal fuzzy numbers, F (R) = {x̃ |
x̃ ≈ (x1 , x2 , x3 , x4 )}.
• F m (R) = {X̃ | X̃ ≈ (x̃1 , x̃2 , . . . , x̃m ), x̃i ∈ F (R), i = 1, 2, . . . , m}
the elements of F m (R) are called fuzzy column vectors.
2.4
Operations on Fuzzy Matrices
2.4.1
Fuzzy Matrix
A matrix having all its elements as fuzzy entities is called a fuzzy
matrix.
2.4.2
Operations on Fuzzy Matrices
Addition [ãij ] + [b̃ij ] ≈ [ãij + b̃ij ]
Subtraction [ãij ] − [b̃ij ] ≈ [ãij − b̃ij ]
Multiplication [ãij ]X[b̃ij ] ≈ [c̃ij ] where,
c̃ij ≈
n X
n
X
i=1 k=1
ãik b̃ki
(2.1)
Ch.2 Preliminaries
2.4.3
40
F-Magnitude
Fuzzy Magnitude(F-Mag) of a fuzzy matrix Ã ≈ [ãij ] of order n is
denoted and defined by,
F M ag(Ã) ≈ ã11 M̃ã11 ⊕ ã12 M̃ã12 ⊕ · · · ⊕ ã1n M̃ã1n
n
X
≈
ã1j M̃ã1j where,
(2.2)
j=1

M̃ãij
ã11
ã21
..
.
ã12
ã22
..
.
...
...
..
.
ã1,j−1
ã2,j−1
..
.
ã1,j+1
ã2,j+1
..
.
...
...
..
.





i+j
≈ (−1) F M ag 
 ãi−1,1 ãi−1,2 . . . ãi−1,j−1 ãi−1,j+1 . . .
 ãi+1,1 ãi+1,2 . . . ãi+1,j−1 ãi+1,j+1 . . .

 ..
..
..
..
..
..
 .
.
.
.
.
.
ãn1
ãn2 . . . ãn,j−1
ãn,j+1 . . .
ã1n
ã2n
..
.
ãi−1,n
ãi+1,n
..
.












ãnn
i.e., the matrix obtained by deleting its ith row and j th column.
2.5
2.5.1
Ranking Functions
Novel Ranking function
A pair of fuzzy number ãi and ãj can be ranked as ,
n
1 X
D(ãi , ãj ) ≈
(xik − xjk ),
n+1
k=0
where xik = 12 (lik + rik ), lik = inf {x ∈ < | µ̃ãi (x) ≥ αk } and rik =
sup{x ∈ < | µ̃Ãi (x) ≥ αk }, αk = nk , k = 0, 1, 2, . . . , n
Ch.2 Preliminaries
2.5.2
41
Compos and Munoz Ranking function
Ranking function proposed by Compos and Munoz for a trapezoidal
fuzzy number ã ≈ (a1 , a2 , a3 , a4 ) is given by
CM1λ (ã) = a2 + λ[(a3 − a2 ) +
2.5.3
(a4 −a3 )+(a2 −a1 )
]
2
−
(a2 −a1 )
.
2
Magnitude method proposed by Abbasbandy
For an arbitrary trapezoidal fuzzy number ũ ≈ (x0 , y0 , σ, β) the magnitude is defined by,
Z
1 1
M ag(ũ) ≈ ( (u(r) + ū(r) + x0 + y0 )f (τ ) dτ ),
2 0 ¯
where f (τ ) is a non-negative and increasing function on [0, 1] with
R1
f (0) = 0, f (1) = 1 and 0 f (τ ) dτ = 21 .
2.5.4
Liou and Wang’s Ranking function
Let Ã ≈ (a1 , a2 , a3 , a4 ) be a trapezoidal fuzzy number with membership function


µLÃ (x),



1,
µÃ (x) ≈

µR
(x),


Ã


0,
where µLÃ (x) ≈
x−a1
a2 −a1
and µR
(x) ≈
Ã
if a1 ≤ x ≤ a2 ;
if a2 ≤ x ≤ a3 ;
if a3 ≤ x ≤ a4 ;
otherwise.
x−a4
a3 −a4 .
Then the ranking value of Ã is defined by, ITα (Ã) ≈ αIR (Ã) +
Ch.2 Preliminaries
42
R1
(1 − α)IL (Ã), α ∈ [0, 1]where IL (Ã) ≈ 0 gAL (y) dy and IR (Ã) ≈
R1 R
L
R
L
0 gA (y) dy where gÃ (y), gÃ (y) are the inverse functions of µÃ (x)
and µR
(x) respectively.
Ã
2.6
Linguistic Variables
When fuzzy numbers represent linguistic concepts, such as very
small, small, medium, and so on, as interpreted in a particular contest, the resulting constructs are usually called linguistic variables.
Each Linguistic variable the states of which are expressed by linguistic terms interpreted as specific fuzzy numbers is defined in terms of
a base variable, the values of which are real numbers within a specific
range. A base variable is a variable in the classical sense, exemplified
by any other numerical variable, [e.g., temperature, pressure, speed,
voltage, humidity, etc.] as well as any other numerical variable, [e.g.
age, interest rate, performance, salary, etc.]
2.6.1
Definition
A linguistic variable is fully characterized by a quintuple (v,T,X,g,m)
in which
1. v is the name of the variable
Ch.2 Preliminaries
43
2. T is the set of linguistic terms of v that refer to a base variable
3. X is the universal set
4. g is a syntactic rule i(a grammar) for generating linguistic terms
5. m is a semantic rule that assigns to each linguistic term t ∈ T its
meaning, ,m(t) which is a fuzzy set on X(i.e., m : T → F (X)).
2.6.2
Example
An example of a linguistic variable is shown in fig. Its name is
performance. This variable expresses the performance(which is the
base variable in this example) of a foal-oriented entity (a person,
machine, organization, method, etc.) in a given context by five
basic linguistic terms - very small, small, medium, large, very large
- as well as other linguistic terms generated by a syntactic rule (not
explicitly shown in fig.) such as not very small, large or very large,
very very small, and so forth. Each of the basic linguistic term is
assigned one of give fuzzy numbers by a semantic rule, as shown in
the figure. The fuzzy numbers, whose membership functions have
the usual trapezoidal shapes, are defined on the interval [0,100], the
range of the base variable. Each of them expresses a fuzzy restriction
on this range.
Ch.2 Preliminaries
44
Figure 2.2: Example of a Linguistic Variable
2.7
Using Linguistic Variables
IF room IS cold THEN heat IS on; IF room IS hot THEN heat IS
off; Simple thermostats have been doing this for a hundred years
or more. Why would we need linguistic variables and fuzzy logic
to operate a simple switch? How do we evaluate a crisp temperature under such vague terms as hot and cold? What are hot and
cold anyway? The heating control problem sounds quite simple: we
measure a temperature for the room, and use two fuzzy logic rules to
control a furnace switch (heat). When the meanings of cold and hot
Ch.2 Preliminaries
45
are not precise opposites, the outcome becomes more complex and
useful. Linguistic variables associate a linguistic condition with a
crisp variable. A crisp variable is the kind of variable that is used in
most computer programs: an absolute value. A linguistic variable,
on the other hand, has a proportional nature: in all of the software
implementations of linguistic variables, they are represented by fractional values in the range of 0 to 1.
In the above example, room and heat are crisp variables, and hot,
cold, on and off are linguistic variables. The linguistic variables on
and off in the above example are represented in the crisp variable
heat as a 1 and a 0 respectively. The hot and cold linguistic variables
represent a range of values corresponding to the crisp variable room.
This relationship can be represented as shown in the graph fig.2.3.
Most linguistic variables can be represented in software with coordinates of 4 points. A crisp variable room is associated with a
linguistic variable hot, defined using four break points from the
graph. LINGUISTIC room TYPE unsigned int MIN 0 MAX 100
MEMBER HOT 60, 80, 100, 100 A lot of literature has been written on representation of linguistic variables, but implementations
Ch.2 Preliminaries
46
Figure 2.3: Usage of Linguistic variable
for most applications utilize four points as above. There are arguments for smooth curves to represent linguistic variables for accuracy, and against smooth curves because of computational intensity.
The worst-case error in 4-point presentation is in the corners. The
robust nature of fuzzy logic rules in applications compensates for
the simplistic representation of linguistic variables.
Chapter 3
Two Person Zero-Sum Fuzzy
Games
Abstract
This chapter unfolds a two person zero-sum fuzzy game
and its solution procedures. As fuzzy, the zero-sum means
that it is a neither loss nor profit situation. No ranking
methods have been used here to convert the fuzzy game
into its crisp counterpart. Instead, it has been solved in
fuzzy environment itself.
47
Ch.3 Two Person Zero-Sum Fuzzy Games
3.1
48
Introduction
It is because of the seminal works by Neumann-Morgenstern in [61]
and Nash in [59, 60], Game theory has played an important role in
the fields of decision making such as economics, management, and
operations research, etc. There is a vast literature on the theory and
applications of (crisp) matrix games, and some of which have been
very well documented in the excellent text books e.g. Jianhua [41],
Karlin [44], Parthasarathy and Raghavan [73], and Owen [66].
In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which a participant’s gain
(or loss) of utility is exactly balanced by the losses (or gains) of the
utility of other participant(s). If the total gains of the participants
are added up, and the total losses are subtracted, they will sum
to zero. Thus cutting a cake, where taking a larger piece reduces
the amount of cake available for others, is a zero-sum game if all
participants value each unit of cake equally .
Some more zerosum Games:
• Blotto games
• Dictator game
Ch.3 Two Person Zero-Sum Fuzzy Games
49
• Kuhn poker
• Matching pennies
Zerosum games and particularly their solutions are commonly
misunderstood by critics of game theory, usually with respect to
the independence and rationality of the players, as well as to the
interpretation of utility functions. Furthermore, Binmore in [12]
defined that the word ”game” does not imply the model is valid
only for recreational games.
In [14] Bowels and Samuel defined the zero-sum property (if one
gains, another loses) as any result of a zero-sum situation is Pareto
optimal (generally, any game where all strategies are Pareto optimal
is called a conflict game). A zero-sum game is also called a strictly
competitive game.
Zero-sum games are most often solved with the minimax theorem
which is closely related to linear programming duality, or with Nash
equilibrium.
Let us recall the discussion on (crisp) two-person zero-sum matrix
game theory and take note of one of the most celebrated and useful
result which asserts that every two person zero sum matrix game is
equivalent to two linear programming problems which are dual to
Ch.3 Two Person Zero-Sum Fuzzy Games
50
each other. Thus, solving such a game amounts to solving any one of
these two mutually dual linear programming problems and obtaining
the solution of the other by using linear programming duality theory.
When game theory applied to model some practical problems
which are encountered in real situations, the values of payoffs have
to be known exactly. However, it is difficult to know the exact values
of payoffs and could only be possible to know the values of payoffs
approximately. In such situations, it is useful to model the problems
as games with fuzzy payoffs or as games with fuzzy goals.
Although various attempts have been made in the literature to
study two person zero sum fuzzy matrix games (for example, Campos [83], Nishizaki and Sakawa [62] and Sakawa and Nishizaki [80])
but they do not take into consideration the fuzzy linear programming duality aspects. In this context it may be noted that the fuzzy
linear programming duality results are available in the literature
but unlike their crisp counter parts, they have not been used for the
study of fuzzy matrix game theory until very recently.
Now, similar to fuzzy linear programming problems, fuzziness in
matrix games can also appear in so many ways but two cases of fuzziness seem to be very natural. These being the one in which players
Ch.3 Two Person Zero-Sum Fuzzy Games
51
have fuzzy goals and the other in which the elements of the pay-off
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 pay-offs respectively. However, in this study the
matrix games with fuzzy pay-offs have only been taken for consideration and the term fuzzy matrix game will be used. In this case,
since the expected payoffs of the game should be fuzzy-valued, there
are no concepts of equilibrium strategies to be accepted widely. So,
it is an important task to define the concept of equilibrium strategies
and investigate their properties.
In this chapter, Two-Person Zero sum Fuzzy Games are discussed,
where the number of players are two with fuzzy payoffs and which
sums up to a zero fuzzy number. Further, this chapter is divided
into four main sections, namely, Fuzzy Nash Equilibrium, Fuzzy
Determinant Procedure, Matrix Difference Procedure and An Approximate method. All of these will suggest some different kinds of
concepts to find equilibrium strategies.
Ch.3 Two Person Zero-Sum Fuzzy Games
3.2
52
A two person zero-sum fuzzy game
A two person zero-sum fuzzy game can be defined as
F G ≈ (S̃1 , S̃2 , K̃, Ã) where,
S̃1 ≈ {X̃ ≈ (x̃1 , . . . , x̃m ) | x̃i < 0̃f or all i,
m
X
˜ and
x̃i ≈ I}
i=1
S̃2 ≈ {Ỹ ≈ (ỹ1 , . . . , ỹn ) | ỹj < 0̃f or all j,
n
X
˜
ỹj ≈ I}
j=1
are the strategy spaces for player I and II respectively. Then the
fuzzy payoff or the fuzzy gain for maximizing player1 is given by,
K̃(X̃, Ỹ ) ≈
n
m X
X
ãij x̃i ỹj ≈ X̃ T Ã Ỹ
i=1 j=1
where Ã is said to be the fuzzy gain2 (payoff) matrix for the maximizing player.
3.3
Fuzzy Nash Equilibrium
3
This section has been devoted to define a fuzzy nash equilibrium and
an FLPP based solution procedure to solve a Two person zero-sum
fuzzy game.
1
Throughout this work, player I will be considered as the maximizing player and player II
will be considered as the minimizing player.
2
The fuzzy payoff or the fuzzy loss of the minimizing player is K̃(X̃, Ỹ ) and the fuzzy loss
matrix is Ã.
3
The contents of this section form the substance of the paper published in the ”International Journal of Computational Science and Mathematics”, ISSN 0974-3189,Vol.3 ,
No.4(2011),447-453.
Ch.3 Two Person Zero-Sum Fuzzy Games
3.3.1
53
Definition
A pair of fuzzy strategies (X̃ ∗ , Ỹ ∗ ) is said to be a fuzzy nash equilibrium for the F G ≈ (S̃1 , S̃2 , K̃, Ã) if,
K̃(X̃ ∗ , Ỹ ∗ ) < K̃(X̃, Ỹ ∗ ), f or all f uzzy strategies X̃ f or player I and
K̃(X̃ ∗ , Ỹ ∗ ) 4 K̃(X̃ ∗ , Ỹ ), f or all f uzzy strategies Ỹ f or player II.
and K̃(X̃ ∗ , Ỹ ∗ ) ≈ Ṽ , the FV of player I.
Lemma 3.3.1. For any F G ≈ (S̃1 , S̃2 , K̃, Ã), its FV can be given
as follows:
1. Ṽ ≈ minỸ ∈S̃2 maxX̃∈S̃2 X̃ T Ã Ỹ
2. Ṽ ≈ minX̃∈S̃1 maxỸ ∈S̃1 X̃ T Ã Ỹ
Proof(1): By the definition,
Ṽ ≈ K̃(X̃ ∗ , Ỹ ∗ )
≈ X̃ ∗T Ã Ỹ ∗
< X̃ T Ã Ỹ ∗
f or allX̃ ∈ S̃1
< max X̃ T Ã Ỹ ∗
X̃∈S̃1
≈ min max X̃ T Ã Ỹ (as Y is the minimizing player)
Ỹ ∈S̃2 X̃∈S̃1
i.e., Ṽ < min max X̃ T Ã Ỹ
Ỹ ∈S̃2 X̃∈S̃1
(3.1)
Ch.3 Two Person Zero-Sum Fuzzy Games
54
Conversely, as X̃ ∗ ∈ S̃1 ,
X̃ ∗T Ã Ỹ ∗ 4 max X̃ T Ã Ỹ
∀ Ỹ ∈ S̃2
X̃∈S̃1
4 min max X̃ T Ã Ỹ
(3.2)
ỹ∈S̃2 X̃∈S̃1
From equations (3.1) and (3.2) it has been observed that,
Ṽ ≈ minỹ∈S̃2 maxX̃∈S̃1 X̃ T Ã Ỹ
Proof (2):
Ṽ ≈ K̃(X̃ ∗ , Ỹ ∗ )
≈ X̃ ∗T Ã Ỹ ∗
4 X̃ ∗T Ã Ỹ
f or allỸ ∈ S̃2
4 min X̃ ∗T Ã Ỹ
Ỹ ∈S̃2
4 max min X̃ T Ã Ỹ (as X is the maximizing player)
X̃∈S̃1 ỹ∈S̃2
i.e., Ṽ 4 max min X̃ T Ã Ỹ
(3.3)
X̃∈S̃1 ỹ∈S̃2
Conversely, as Ỹ ∗ ∈ S̃2 ,
X̃ ∗T Ã Ỹ ∗ < min X̃ T Ã Ỹ
∀X̃ ∈ S̃1
Ỹ ∈S̃2
< max min X̃ T Ã Ỹ
X̃∈S̃1 Ỹ ∈S̃2
From equations (3.3) and (3.4) it has been observed that,
Ṽ ≈ maxX̃∈S̃1 minỹ∈S̃2 X̃ T Ã Ỹ .
(3.4)
Ch.3 Two Person Zero-Sum Fuzzy Games
55
Theorem 3.3.2. Consider the F G ≈ (S̃1 , S̃2 , K̃, Ã), where
P
˜
S̃1 ≈ {X̃ ≈ (x̃1 , . . . , x̃m )T | x̃i < 0̃ for all i, m
i=1 x̃i ≈ I}
P
˜
S̃2 ≈ {Ỹ ≈ (ỹ1 , . . . , ỹn )T | ỹj < 0̃ for all j, nj=1 ỹj ≈ I}
and Ã ≈ [ãij ] . Then the fuzzy value Ṽ of the FG can be obtained
by solving an FLPP.
Proof:
Since, Ṽ ≈ minỸ ∈S̃2 maxX̃∈S̃2 X̃ T Ã Ỹ , Player II will select his
fuzzy strategy Ỹ ∈ S̃2 so as to minimizes his loss, i.e., to minimize
P Pn
maxX̃∈S̃1 m
i=1
j=1 x̃i ãij ỹj .
Since every strategy X̃ ∈ S̃1 is a fuzzy convex combination of m
pure fuzzy strategies for player I, we have
P Pn
Pn ˜
maxX̃∈S̃1 m
i=1
j=1 x̃i ãij ỹj ≈ max1≤i≤m [
j=1 I ãij ỹj ].
Then we have,
Ṽ ≈ minz̃
subject to
n
X
I˜ ãij ỹj 4 z̃,
j=1
n
X
i = 1, 2, . . . , m
ỹj ≈ I˜
j=1
ỹj < 0̃, j = 1, 2, . . . , n
(∵ ỹ ∈ S̃2 )
(3.5)
Ch.3 Two Person Zero-Sum Fuzzy Games
56
By taking ỹj0 ≈ ỹj z̃ we have,
Ṽ ≈ max ỹ10 ⊕ ỹ20 ⊕ . . . ỹn0
(3.6)
subject to,
n
X
˜
I˜ ãij ỹj0 4 I,
j=1
ỹj0 <
3.3.2
i = 1, 2, . . . , m
0̃, j = 1, 2, . . . , n
Numerical Example
Game Description:
A soft-drink company calculated the market share of two products
against its major competitor also having two products and found out
the impact of additional advertisement in any one of its products
against the other. The available data can be converted into the
payoff matrix,
Company B
B1
CompanyA
A1 (1, 2, 3, 4)
B2
(2, 5, 7, 9)
A2 (3, 4, 8, 10) (6, 9, 15, 17)
Ch.3 Two Person Zero-Sum Fuzzy Games
57
Solution Procedure
Step 1 Consider the payoff matrix,
(1, 2, 3, 4) (2, 5, 7, 9)
(3, 4, 8, 10) (6, 9, 15, 17)
Step 2 Then the relevant FLPP can be written as
M axz̃ ≈ ỹ1 ⊕ ỹ2
(1, 2, 3, 4)ỹ1 ⊕ (2, 5, 7, 9)ỹ2
4 (1, 1, 1, 1)
(3, 4, 8, 10)ỹ1 ⊕ (6, 9, 15, 17)ỹ2
4 (1, 1, 1, 1)
ỹ1 , ỹ2 < 0̃
Step 3 Solving these FLPP’s by simplex method given in tables 3.1
and 3.2 one can find,
I˜ Ṽ ≈ (0.16, 0.16, 0.16, 0.16)
Ṽ ≈ I˜ (0.16, 0.16, 0.16, 0.16), where I˜ ≈ (1, 1, 1, 1)
≈ ˜(1, 1, 1, 1) (0.16, 0.16, 0.16, 0.16)
≈ (6.25, 6.25, 6.25, 6.25)
S̃Ã ≈ ((1, 1, 1, 1), (0, 0, 0, 0))
S̃B̃ ≈ ((0, 0, 0, 0), (1, 1, 1, 1))
ỹB
s̃1
ỹ1
c̃B
0̃
I˜
s̃1
s̃2
(0,0,0,0)
(0,0,0,0)
(-1,-1,-1,-1)
(0,0,0,0)
(-1,-1,-1,-1)
(6,9,15,17)
(2,5,7,9)
ỹ2
0̃
(0.16,0.16,0.16,0.16)
(0,0,0,0)
(0,0,0,0)
(0.88, 0.88, 0.88, 0.88)
(0.96, 1.44, 2.4, 2.72)
(-5.52, -0.64, 3.24, 7.12)
ỹ2
s̃1
(1,1,1,1)
Table 3.2: Simplex Table(Continued)
(0.48,0.64,1.28,1.6)
(-3,-1,1,3)
ỹ1
Table 3.1: Simplex Table
(3,4,8,10)
(1,2,3,4)
ỹ1
(1,1,1,1)
(1,1,1,1)
x̃B
(0.16, 0.16, 0.16, 0.16)
(0.36, 0.52, 0.68, 0.84)
x̃B
ỹB
c̃B
0̃
(0.16, 0.16, 0.16, 0.16)
(0.16, 0.16, 0.16, 0.16)
(-0.64, -0.48, -0.32, -0.16)
I˜
0̃
s̃2
s̃1
(0,0,0,0)
(1,1,1,1)
(0,0,0,0)
s̃2
Ch.3 Two Person Zero-Sum Fuzzy Games
58
Ch.3 Two Person Zero-Sum Fuzzy Games
3.4
Fuzzy determinant Procedure
59
4
In this section, a procedure has been given to find the FV and Fuzzy
strategies of a fuzzy two person zero sum game by using F - Mag
values.
3.4.1
Definition
Let Ã be the given fuzzy matrix of order n. Then define Ṽ , X̃ and Ỹ
as follows:
X̃ ≈
F − M ag(Ã)
,
Ṽ ≈ Pn Pn
M̃
ã
ij
i=1
j=1
!
Pn
Pn
j=1 M̃ã1j
j=1 M̃ãnj
, . . . , Pn Pn
Pn Pn
i=1
j=1 M̃ãij
i=1
j=1 M̃ãij
and
Ỹ ≈
3.4.2
!
Pn
Pn
M̃ã
M̃ã
Pn i=1
Pn i1 , . . . , Pn i=1
Pn in
j=1 M̃ãij
j=1 M̃ãij
i=1
i=1
Algorithm for Fuzzy Determinant Procedure
Step 1 Let Ã be the fuzzy payoff matrix for player I of order n.
Step 2 Let Ã1 , Ã2 , Ã3 , . . . be the fuzzy square sub-matrices of Ã .
4
The contents of this section form the substance of the paper published in the ”International Journal of Mathematics Research”, ISSN 0976-5840,Vol.3, No.5 (2010),pp.475-482.
Ch.3 Two Person Zero-Sum Fuzzy Games
60
Step 3 For each Ãk find ṼÃk and, X̃Ãk and ỸÃk .
Step 4 Write X̃k from X̃Ãk by adding zeros corresponding to each
eliminated row and Ỹk from ỸÃk by adding zeros corresponding
to each eliminated column.
Step 5 Define SÃ and SB̃ as follows:
SÃ ≈ {X̃k ≈ (x̃k1 , x̃k2 , . . . , x̃km ) | x̃km < 0̃,
m
X
˜ (3.7)
x̃kj ≈ I}
k=1
SB̃ ≈ {Ỹk ≈ (ỹk1 , ỹk2 , . . . , ỹkn ) | ỹkn < 0̃,
n
X
˜
ỹkj ≈ I}
(3.8)
j=1
Step 6 Define ṼÃ as follows:
ṼÃ ≈ max{ṼÃk }
k
for the X̃k0 s and Ỹk0 in S̃1 and S̃2 respectively.
3.4.3
Numerical Example
Game Description:
Two companies Fumco and Tabacs are competing for their business.
The Strategies for each firm are don’t advertise or advertise. Also,
Tabacs is a newly opened one and so not known to all. Assume
that the following payoff matrix describes the increase in business
for Fumco and the decrease for Tabacs.
Ch.3 Two Person Zero-Sum Fuzzy Games
61
Tabacs
F umco
don0 t advertise
advertise
don0 t advertise
(0, 11, 12, 13)
(−6, −2, 1, 3)
advertise
(1, 5, 6, 8)
(−4, 4, 8, 20)
Solving procedure
Step 1 The payoff matrix is
(0, 11, 12, 13) (−6, −2, 1, 3)
Ã ≈
(1, 5, 6, 8)
(−4, 4, 8, 20)
Step 2 It is clear that there is no instant solution and the only
square matrix is the matrix itself.
Step 3 The corresponding M̃ãij are
M̃ãi1
P
M̃ãi2
j
M̃ãij
M̃ã1j
(-4,4,8,20)
(-8,-6,-5,-1) (-12,-2,3,19)
M̃ã2j
(-3,-1,2,6)
(0,11,12,13) (-3,10,14,19)
P
i M̃ãij
(-7,3,10,26)
(-8,5,7,12)
Table 3.3:
P P
i
j
(-15,8,17,38)
M̃ãij
Ch.3 Two Person Zero-Sum Fuzzy Games
Step 4 Then we can find F-Mag of Ã as follows:
F − M ag(Ã) ≈ 7(0, 11, 12, 13) 5(−6, −2, 1, 3)
≈ (0, 77, 84, 91) ⊕ (−15, −5, 10, 30)
≈ (−15, 72, 94, 121)
Step 5 Then we obtain the following:
Ṽ ≈ (−1.245, 5.976, 7.802, 10.043)
X̃ ≈ ((−1, −0.167, 0.25, 1.583)(−0.25, 0.833, 1.167, 1.583))
Ỹ ≈ ((−0.583, 0.25, 0.833, 2.167)(−0.667, 0.417, 0.583, 1))
62
Ch.3 Two Person Zero-Sum Fuzzy Games
3.5
63
Matrix Difference Procedure
5
This section proposes an algorithm to find FV and fuzzy strategies
of a fuzzy two person zero-sum game based on the fuzzy difference
values.
3.5.1
Procedure
Step 1: Let Ã be the fuzzy payoff matrix of player I .
Step 2: Write the matrices of Column and Row Differences as follows:
C̃ ≈ [C̃1 C̃2
C̃2 C̃3
. . . ] and R̃ ≈ [R̃1 R̃2
R̃2 R̃3
...]
where C̃j are the columns of Ã and R̃i are the rows of Ã.
Step 3: Now find
p̃i ≈ FM ag (Ãi ) and q̃j ≈ FM ag (B̃j ),
where Ãi is the matrix obtained by eliminating the it h row of the
matrix C̃ and B̃j is the matrix obtained by eliminating the j t h column of the matrix R̃
Step 4: Check whether
n
n X
n
n
X
X
X
p̃i ≈
Mãij ≈
q̃j .
i=1
5
i=1 j=1
i=1
The contents of this section form the substance of the paper published in the ”Journal of
Current Sciences”,Vol.15,No. 1(2010),pp.207-210.
Ch.3 Two Person Zero-Sum Fuzzy Games
64
If satisfied go to next step. Otherwise the method will fail.
Step 5: Take
Ṽ ≈
n
X
ãij p̃i , f or any j = 1, 2, . . . , n or
i=1
Ṽ ≈
n
X
ãij q̃j , f or any i = 1, 2, . . . , n
j=1
X̃ ≈
Ỹ ≈
p̃1
Pn
i=1 p̃i
q̃
Pn 1
p̃2
, Pn
j=1 q̃j
i=1 p̃i
p̃n
, . . . Pn
q̃2
, Pn
j=1 q̃j
i=1 p̃i
q̃n
, . . . Pn
and
!
i=1 q̃j
.
3.5.2
Numerical Example
Game Description:
Assume that two firms are competing for market share for a particular product. Each form is considering what promotional strategy
to employ for the coming period. Assume that the following payoff
matrix describes the increase in market share for Firm A and the
decrease in market share for Firm B.
Ch.3 Two Person Zero-Sum Fuzzy Games
65
Firm B
N o P romotion P romotion
F irmA
N o P romotion
(−3, 1, 8, 10)
(3, 5, 8, 12)
P romotion
(2, 3, 8, 11)
(−2, 1, 8, 13)
Solution Procedure
Step 1 Consider the payoff matrix
(−3, 1, 8, 10) (3, 5, 8, 12)
(2, 3, 8, 11) (−2, 1, 8, 13)
Step 2 Find C̃ and R̃, the matrices of column differences and row
differences respectively. Here
(−11, −5, 7, 13)
C̃ ≈
and
(−7, −3, 7, 15)
R̃ ≈ (−10, −3, 7, 14) (−8, −5, 7, 14)
Step 3 It is clear that
n
X
i=1
p̃i ≈
n
n X
X
i=1 j=1
Mãij ≈
n
X
q̃j .
i=1
Step 4
∴ X̃ ≈ ((−2.75, −1.25, 1.75, 3.25), (−1.75, −0.75, 1.75, 3.75))
Ỹ ≈ ((−2.5, −0.75, 1.75, 3.5), (−2, −1.25, 1.75, 3.5)) and
Ṽ ≈ (0.75, 2.5, 8, 10.75)
Ch.3 Two Person Zero-Sum Fuzzy Games
3.6
An Approximate Method
66
6
This section provides an algorithm to get an approximate solution
of a fuzzy game.
3.6.1
Algorithm
Step 1 Player I arbitrarily selects any row and places it under the
matrix.
Step 2 Then player II examines this row and chooses a column
corresponding to the smallest number in the row. This column
is then placed to the right of the matrix.
Step 3 Player I now examines this column and chooses a row corresponding to the largest number in this column. This row is
then added to the first row last chosen and the sum of the two
rows is placed beneath the row last chosen
Step 4 Player II now chooses a column corresponding to the smallest number in the new row and adds this column to the column
last chosen. In the case of a tie the player will select a row (or)
column different from his last choice.
6
The contents of this section form the substance of the paper published in
the ”International Journal of Advances in Fuzzy Mathematics”, ISSN 0973533X,Vol.5,No.3(2010),295-300.
Ch.3 Two Person Zero-Sum Fuzzy Games
67
Step 5 The smallest element in rows and largest element in columns
are encircled. This procedure may be continued for a required
number of times in the like manner.
Step 6 The approximate strategies are found by dividing the number of encircled elements in each row and column by the number
of iterations. Increase in the number of iterations will increase
the accuracy. Using this method, the approximate value of a
fuzzy game can be evaluated up to any desired degree of accuracy.
3.6.2
Numerical Example
Game Description:
In a small town there are only two stores and that handle sundry
goods A and B. The total number of customers is divided between
the two, because price and quality are equal. Both stores have reputation in community for the equally good services they render.Both
the stores plan to run a pre-Diwali sales during first week of October.
Sales are advertised through local press, radio and TV with the aid
of an advertising firm. Here the estimates are not precise and hence
taken as fuzzy. The payoff matrix is given below:
Ch.3 Two Person Zero-Sum Fuzzy Games
68
Firm
[0.05in]P ress
F irm A
B
P ress
(0, 2, 4, 5)
Radio
(−3, 1, 2, 9)
TV
(5, 9, 10, 12)
Radio
(−1, 0, 3, 6) (−5, −2, 4, 5)
(1, 3, 7, 9)
TV
(5, 6, 10, 13) (7, 9, 10, 15)
(1, 5, 9, 13)
Solution procedure
Step 1: Player I selects row 3 arbitrarily.
Step 2:Now Player II examines this row and has to select a column
corresponds to the minimum in this row. Using the comparison
technique defined in the Preliminaries these fuzzy numbers has been
compared.
Row 3 (5,6,10,13) (7,9,10,15) (1,5,9,13) (1,5,13,15)
x0
8.556
10.444
7
8.455
y0
0.444
0.370
0.417
0.455
d
0.102
1.992
1.455
Since, the minimum corresponds to the first column and so player
II selects first column.
Ch.3 Two Person Zero-Sum Fuzzy Games
69
Step 3:Now player I examines this column and selects the row corresponds to the maximum in this column. Here the maximum corresponds to the second row. Then this is added to the previous row
and taken to the next iteration.
Step 4:Player II again examines this row and selects the maximum
one and adds it to the previous column which then be used for the
next iteration. This procedure is being repeated for ten times. The
iteration tables (Rows and Columns) have been given below:
Step 5:From the iteration tables 3.4 and 3.5, we can find the strategies as follows:
p1 = 0.8, p2 = 0, p3 = 0.2 and q1 = 0, q2 = 1 and q3 = 0.
To find the approximate fuzzy value take the average of all p1 a1j ⊕
p2 a2j ⊕ p3 a3j , j = 1, 2, 3 and q1 ai1 ⊕ q2 ai2 ⊕ q3 ai3 , i = 1, 2, 3.
The approximate fuzzy value ≈ (0.533, 3.6, 5.767, 9.667)
(2,7,14,20)
(-3,5,18,25)
(-13,1,26,35)
(-18,-1,30,40)
(3,6,16,25)
(2,6,19,31)
(1,6,22,37)
(0,6,25,43)
(-1,6,28,49)
(-2,6,31,55) (-28,-5,38,50)
(-3,6,34,68) (-33,-7,42,55)
(-4,6,37,74) (-38,-9,46,60) (10,32,72,94) (-38,-9,74,94)
3
4
5
6
7
8
9
10
(6,20,44,58)
(5,17,37,49)
(4,14,30,40)
Table 3.4: Row Iteration Table
(9,29,65,85)
(8,26,58,76)
(-23,-3,34,45) (7,23,51,67)
(-8,3,22,30)
(3,11,23,31)
(2,8,16,22)
(-33,-7,68,85)
(-28,-5,58,76)
(-4,3,59,78)
(-18,-1,44,58)
(-13,1,37,49)
( -8,2,31,40)
(-3,3,25,31)
(2,4,20,22)
(1,5,13,15)
(4,6,13,19)
(1,5,9,13)
2
(7,9,10,15)
Ũ
(5,6,10,13)
III column
1
II column
I column
Iteration No.
Ch.3 Two Person Zero-Sum Fuzzy Games
70
(-1,3, 29,39)
(11,17, 29,39)
(10,28, 40,49)
(5,19, 30,37)
(10,30, 44,54)
8
(10,32, 48,59)
9
(-2,3, 39,52)
(16,23, 39,52)
(10,34, 52,64)
10
(-3,3, 49,65)
(21,29, 49,65)
(27,40, 68,91)
(-3,6, 68,91)
(26,35, 59,78)
(-4,3, 59,78)
(-4,6, 78,104)
(-5,6, 88,117)
(-6,6, 98,130)
(32,46, 78,104) (37,52, 88,117) (42,58, 98,130)
(-4, 3, 22,39) (-3, 6, 29,48) (-4, 6, 32,54) (-5, 6, 35,60) (-6, 6, 38,66)
7
6
Table 3.5: Column Iteration Table
(0,3, 19,26)
(6,11, 19,26)
(-1,0, 10,13)
(5,17, 26,32)
Ũ
(5,15, 22,27)
5
(5,6, 10,13)
(5,13, 18,22)
4
Row 3
(5,11, 14,17)
3
(-1, 0, 3,6) (0,3, 10,15) (-1, 3, 13,21) (-2, 3, 16,27) (-3, 3, 19,33)
(0,2, 4,5)
Row 1
2
Row 2
1
Iteration No.
Ch.3 Two Person Zero-Sum Fuzzy Games
71
Chapter 4
Two Person Non Zero-Sum Fuzzy
Games
Abstract
This chapter defines a two person non zero-sum fuzzy game
and its solution procedures. It should be noted that both
players can be considered as gain player as the sum is non
zero fuzzy. In other words, it means an increase in the
range of profit for one player implies an associated increase
in the other’s range also. The fuzzy nature of the payoffs
has been kept unchanged.
72
Ch.4 Two Person Non Zero-Sum Games
4.1
73
Introduction
In the earlier Chapter, Two-Person Zero-sum games were analysed
in which the gain of one player is the loss of the other player. But
there may be situations in which the interests of two players may
not be exactly opposite. Such situations give rise to two person nonzero sum games, also called bi-matrix games . For example, most of
the economic situations are not zero-sum, since valuable goods and
services can be created, destroyed, or badly allocated, and any of
these will create a net gain or loss. Assuming the counterparties are
acting rationally with symmetric information, any commercial exchange is a non-zero-sum activity, because each party must consider
the goods it is receiving as being at least fractionally more valuable
than the goods it is delivering. Economic exchanges must benefit
both parties enough above the zero-sum such that each party can
overcome its transaction costs.
Situations where participants can all gain or suffer together are
referred to as non-zero-sum. Thus, a country with an excess of bananas trading with another country for their excess of apples, where
both benefit from the transaction, is in a non-zero-sum situation.
Other non-zero-sum games are games in which the sum of gains and
Ch.4 Two Person Non Zero-Sum Games
74
losses by the players are sometimes more or less than what they
began with. In contrast to the zero-sum games, non-zero-sum describes a situation in which the interacting parties’ aggregate gains
and losses are either less than or more than zero.
The theory of zero-sum games is vastly different from that of nonzero-sum games because an optimal solution can always be found.
However, this hardly represents the conflicts faced in the everyday
world. Problems in the real world do not usually have straightforward results. The branch of Game Theory that better represents
the dynamics of the world one lives in is called the theory of nonzero-sum games. Non-zero-sum games differ from zero-sum games
in that there is no universally accepted solution. That is, there is no
single optimal strategy that is preferable to all others, nor is there a
predictable outcome. Non-zero-sum games are also non-strictly competitive, as opposed to the completely competitive zero-sum games,
because such games generally have both competitive and cooperative elements. Players engaged in a non-zero sum conflict have some
complementary interests and some interests that are completely opposed.
Some examples of Non-zero sum Games:
Ch.4 Two Person Non Zero-Sum Games
75
• Battle of sexes
• Centipede game
• Chicken
• Diner’s dilemma
• Prisoner’s dilemma
• Traveler’s dilemma
It has been theorized as the society becomes increasingly nonzero-sum as it becomes more complex, specialized, and interdependent. As former US President Bill Clinton states:
”The more complex societies get and the more complex the
networks of interdependence within and beyond community and national borders get, the more people are forced
in their own interests to find non-zero-sum solutions. That
is, winwin solutions instead of winlose solutions.... Because we find as our interdependence increases that, on
the whole, we do better when other people do better as
well so we have to find ways that we can all win, we have
to accommodate each other....”
Ch.4 Two Person Non Zero-Sum Games
76
Bill Clinton, Wired interview, December 2000.
In [61] John von Neumann and Oskar Morgenstern proved that
any zero-sum game involving n players is in fact a generalized form
of a zero-sum game for two players, and that any non-zero-sum game
for n players can be reduced to a zero-sum game for n + 1 players;
the (n + 1) player representing the global profit or loss.
Also, one can easily verify that a two person zero sum matrix
game is a special case of the bi-matrix game BG with B = -A.
Therefore for B = -A, the definition of an equilibrium solution reduces to a saddle point for the two person zero sum game G. This
can easily be verified by putting B = -A in definition, where A and
B are the payoff matrices for player I and II respectively.
Similar to a zero sum game, here also the case of fuzzy will arise
if the values of the payoff for players are not known exactly. So, it is
an important task to define the concept of equilibrium strategies and
investigate their properties which has been done in this chapter. This
chapter contains four main sections and will suggest some different
kinds of concepts to find equilibrium strategies.
Ch.4 Two Person Non Zero-Sum Games
4.2
77
A Fuzzy Bi - Matrix Game
A fuzzy two person non - zero sum game or a fuzzy bi-matrix game
can be defined as, F BG ≈ (Ã, B̃, S̃1 , S̃2 , K̃Ã , K̃B̃ ) where,
S̃1 ≈ {X̃ ≈ (x̃1 , . . . , x̃m ) | x̃i < 0̃f or all i,
m
X
˜
x̃i ≈ I}
i=1
is the strategy space for player I and
S̃2 ≈ {Ỹ ≈ (ỹ1 , . . . , ỹn ) | ỹj < 0̃f or all j,
n
X
˜
ỹj ≈ I}
j=1
is the strategy space for player II. Then the fuzzy payoff or the fuzzy
gains of the players will be given by,
K̃Ã (X̃, Ỹ ) ≈
n
m X
X
ãij x̃i ỹj ≈ X̃ T Ã Ỹ f orplayerIand
i=1 j=1
K̃B̃ (X̃, Ỹ ) ≈
n
m X
X
b̃ij x̃i ỹj ≈ X̃ T B̃ Ỹ f orplayerII
i=1 j=1
where, Ã ≈ [ãij ] and B̃ ≈ [b̃ij ], having equal dimensions are the
fuzzy gain matrices1 for player I and II respectively.
4.3
Fuzzy Nash Equilibrium
2
This section will define a fuzzy nash equilibrium for fuzzy bi-matrix
games and will suggest a FLPP based solution procedure.
1
In a fuzzy Bi-matrix game both players will be considered as maximizing players i.e., one
player will help the other to maximize his own profit or to minimize his own loss.
2
The contents of this section form the substance of the paper published in the Proceedings
of the International conference on Mathematics and Computer Science, January 2011
Ch.4 Two Person Non Zero-Sum Games
4.3.1
78
Definition
A pair of fuzzy strategies (X̃ ∗ , Ỹ ∗ ) is said to be a fuzzy nash equilibrium for the F BG ≈ (Ã, B̃, S̃1 , S̃2 , K̃Ã , K̃B̃ ) if,
K̃Ã (X̃ ∗ , Ỹ ∗ ) < K̃Ã (X̃, Ỹ ), ∀X̃ ∈ S̃1 andỸ ∈ S̃2
K̃B̃ (X̃ ∗ , Ỹ ∗ ) < K̃B̃ (X̃ ∗ , Ỹ ), ∀X̃ ∈ S̃1 andỸ ∈ S̃2
Also,K̃Ã (X̃ ∗ , Ỹ ∗ ) ≈ ṼÃ and K̃B̃ (X̃ ∗ , Ỹ ∗ ) ≈ ṼB̃ .
Lemma 4.3.1. For any F BG ≈ (Ã, B̃, S̃1 , S̃2 , K̃Ã , K̃B̃ ),
1. ṼÃ ≈ maxX̃∈S̃1 maxỸ ∈S̃2 X̃ T Ã Ỹ
2. ṼB̃ ≈ maxỸ ∈S̃2 maxX̃∈S̃1 X̃ T B̃ Ỹ
Proof(1): By the definition,
ṼÃ ≈ K̃Ã (X̃ ∗ , Ỹ ∗ )
≈ X̃ ∗T Ã Ỹ ∗
< X̃ T Ã Ỹ
∀ X̃ ∈ S̃1 and Ỹ ∈ S̃2
∴ ṼÃ < max max X̃ T Ã Ỹ
(4.1)
X̃∈S̃1 Ỹ ∈S̃2
Conversely, as X̃ ∗ ∈ S̃1 and Ỹ ∗ ∈ S̃2
X̃ ∗T Ã Ỹ ∗ max max X̃ T Ã Ỹ
X̃∈S̃1 Ỹ ∈S̃2
(4.2)
Ch.4 Two Person Non Zero-Sum Games
79
From equations (4.1) and (4.2) it has been observed that,
Ṽ ≈ maxX̃∈S̃1 maxỸ ∈S̃2 X̃ T Ã Ỹ
Proof (2):
ṼB̃ ≈ K̃B̃ (X̃ ∗ , Ỹ ∗ )
≈ X̃ ∗T B̃ Ỹ ∗
< X̃ T B̃ Ỹ
∀X̃ ∈ S̃1 andỸ ∈ S̃2
∴ ṼB̃ < max max X̃ T B̃ Ỹ
(4.3)
X̃∈S̃1 Ỹ ∈S̃2
Conversely, as X̃ ∗ ∈ S̃1 and Ỹ ∗ ∈ S̃2
X̃ ∗T B̃ Ỹ ∗ max max X̃ T B̃ Ỹ
(4.4)
X̃∈S̃1 Ỹ ∈S̃2
From equations (4.3) and (4.4) it has been observed that,
Ṽ ≈ maxỸ ∈S̃2 maxX̃∈S̃1 X̃ T B̃ Ỹ .
Theorem 4.3.2. Consider the F BG ≈ (Ã, B̃, S̃1 , S̃2 , K̃Ã , K̃B̃ ), where
P
˜
S̃1 ≈ {X̃ ≈ (x̃1 , . . . , x̃m )T | x̃i < 0̃ for all i, m
i=1 x̃i ≈ I}
P
˜
S̃2 ≈ {Ỹ ≈ (ỹ1 , . . . , ỹn )T | ỹj < 0̃ for all j, nj=1 ỹj ≈ I}
and Ã ≈ [ãij ] and B̃ ≈ [b̃ij ]. Then the fuzzy values ṼÃ and ṼB̃ of the
FBG can be obtained by solving the two FLPPs.
Proof:
Since, ṼÃ ≈ maxX̃∈S̃1 maxỸ ∈S̃2 X̃ T Ã Ỹ , Player I will select his
fuzzy strategy X̃ ∈ S̃1 so as to maximize his profit, i.e., to maximize
Ch.4 Two Person Non Zero-Sum Games
maxỸ ∈S̃2
Pm Pn
i=1
80
ãij ỹj .
j=1 x̃i
Since every strategy Ỹ ∈ S̃2 is a fuzzy convex combination of m pure
fuzzy strategies, we have
P Pn
Pm
maxỸ ∈S̃2 m
x̃
ã
ỹ
≈
max
[
ij
j
1≤j≤n
i=1
j=1 i
i=1 x̃i ãij ] and
hence,
ṼÃ ≈ max z̃
subject to
m
X
x̃i ãij 4 z̃, j = 1, 2, . . . , n
i=1
m
X
x̃i ≈ I˜
i=1
x̃i < 0̃, i = 1, 2, . . . , m
(4.5)
By taking x̃0i ≈ x̃i z̃ we have,
ṼÃ ≈ min x̃01 ⊕ x̃02 ⊕ . . . x̃0m
subject to,
m
X
˜ i = 1, 2, . . . , m
x̃0i ãij 4 I,
i=1
x̃0i <
0̃, i = 1, 2, . . . , m
(4.6)
Ch.4 Two Person Non Zero-Sum Games
i.e.,ṼÃ ≈ max x̃01 x̃02 . . . x̃0m
81
(4.7)
subject to,
m
X
˜ i = 1, 2, . . . , m
x̃0i ãij 4 I,
i=1
x̃0i <
0̃, i = 1, 2, . . . , m
Similarly,ṼB̃ ≈ max ỹ10 ỹ20 . . . ỹn0
(4.8)
subject to,
n
X
˜ i = 1, 2, . . . , m
I˜ b̃ij ỹj0 4 I,
j=1
ỹj0 <
4.3.2
0̃, j = 1, 2, . . . , n
Numerical Example
Game Description:
Two sisters Iris and Julia, students at nearby college, where all the
classes are graded on the curve. Since they are two best students in
their class, each of them will top the curve unless they enroll in the
same class. Iris and Julia each have to choose one more this term,
and one each of them can choose among mathematics, science and
literature. Their grade value matrices are given below:
Ch.4 Two Person Non Zero-Sum Games
82
Julia
M athematics
Iris M athematics
Science
Literature
(1, 4, 6, 9)
(−2, 3, 4, 7) (4, 5, 6, 13) and
Science
(3, 4, 8, 13)
(5, 7, 9, 15) (−2, 0, 2, 4)
Literature
(5, 9, 11, 15)
(2, 4, 7, 11)
(0, 1, 3, 4)
Julia
M athematics
Science
Literature
(3, 6, 10, 17)
(−6, −2, 3, 5)
(2, 4, 5, 9)
Science
(−3, 0, 2, 5)
(2, 4, 7, 11)
(−2, 1, 3, 6)
Literature
(−1, 4, 5, 8)
(−1, 2, 4, 7)
(0, 7, 11, 14)
Iris M athematics
Solution Procedure
Step 1 Take the payoff matrices for player I and II as Ã and B̃
respectively.

(1, 4, 6, 9)
Ã ≈  (3, 4, 8, 13)
(5, 9, 11, 15)

(3, 6, 10, 17)
B̃ ≈  (−3, 0, 2, 5)
(−1, 4, 5, 8)

(−2, 3, 4, 7) (4, 5, 6, 13)
(5, 7, 9, 15) (−2, 0, 2, 4)  and
(2, 4, 7, 11) (0, 1, 3, 4)

(−6, −2, 3, 5) (2, 4, 5, 9)
(2, 4, 7, 11) (−2, 1, 3, 6) 
(−1, 2, 4, 7) (0, 7, 11, 14)
Ch.4 Two Person Non Zero-Sum Games
83
Step 2 Write the corresponding FLPP’s as follows:
ṼÃ ≈ max ỹ1 ⊕ ỹ2 ⊕ ỹ3
(4.9)
subject to the constraints,
(1, 4, 6, 9)ỹ1
⊕ (−2, 3, 4, 7)ỹ2 ⊕ (4, 5, 6, 13)ỹ3 4 (1, 1, 1, 1)
(3, 4, 8, 13)ỹ1 ⊕ (5, 7, 9, 15)ỹ2 ⊕ (−2, 0, 2, 4)ỹ3 4 (1, 1, 1, 1)
(5, 9, 11, 15)ỹ1 ⊕ (2, 4, 7, 11)ỹ2 ⊕ (0, 1, 3, 4)ỹ3
4 (1, 1, 1, 1)
ỹ1 , ỹ2 , ỹ3 < 0̃
ṼB̃ ≈ max x̃1 ⊕ x̃2 ⊕ x̃3
(4.10)
subject to the constraints,
(3, 6, 10, 17)x1 ⊕ (−3, 0, 2, 5)x2 ⊕ (−1, 4, 5, 8)x3 4 (1, 1, 1, 1)
(−6, −2, 3, 5)x1 ⊕ (2, 4, 7, 11)x̃2 ⊕ (−2, 1, 3, 6)x̃3
(2, 4, 5, 9)x̃1
4 (1, 1, 1, 1)
⊕ (−2, 1, 3, 6)x̃2 ⊕ (0, 7, 11, 14)x̃3 4 (1, 1, 1, 1)
x̃1 , x̃2 , x̃3 < 0̃
Step 3 Solve these FLPP’s by Simplex Method given in tables 4.1
- 4.7.
Ch.4 Two Person Non Zero-Sum Games
84
ṼÃ and S̃Ã : From tables 4.1, 4.2 and 4.3 it can be found that
ṼÃ ≈ I˜ z̃
(4.11)
≈ (5, 5, 5, 5)
S̃Ã ≈ ((0.415, 0, 415, 0.415, 0.415), (0.33, 0.33, 0.33, 0.33),
(0, 0, 0, 0))
ṼB̃ and S̃B̃ . From tables 4.4, 4.5, 4.6 and 4.7 it can be found
that
ṼB̃ ≈ I˜ z̃
(4.12)
≈ (3.953, 3.953, 3.953, 3.953)
S̃B̃ ≈ ((−0.043, 0.174, 0.257, 0.826), (0.13, 0.455, 0.676, 1.004),
(−0.739, −0.071, 0.261, 1.071))
(2,4, 7,11)
(0,0,0,0) Y6 (1,1,1,1) (5,9, 11,15)
(0,1, 3,4)
(-2,0, 2,4)
(4,5, 6,13)
Y3
S2
S3
(0,0,0,0) (0,0,0,0) (1,1,1,1)
(0,0,0,0) (1,1,1,1) (0,0,0,0)
(1,1,1,1) (0,0,0,0) (0,0,0,0)
S1
Table 4.1: Simplex Table
(0,0,0,0) (-1,-1,-1,-1) (-1,-1,-1,-1) (-1,-1,-1,-1) (0,0,0,0) (0,0,0,0) (0,0,0,0)
(5,7, 9,15)
(-2,3, 4,7)
Y2
(3,4, 8,13)
Y1
(0,0,0,0) Y5 (1,1,1,1)
XB
(1,4, 6,9)
YB
(0,0,0,0) Y4 (1,1,1,1)
CB
Ch.4 Two Person Non Zero-Sum Games
85
YB
S1
Y2
S3
CB
0̃
I˜
0̃
0̃
(-0.222, -0.222, -0.222, -0.222)
(-0.777, -0.444, -0.333, 0.222)
I˜
0̃
0̃
0̃
(3.223, 4.556, 5.667, 13.222)
(-0.222, 0, 0.222, 0.444)
(-1.221, 0.223, 2.556, 3.778)
(-0.889, -0.889, -0.889, -0.889)
(0.111, 0.111, 0.111, 0.111)
(-1.221, -0.777, -0.444, -0.222)
(0.111, 0.111, 0.111, 0.111)
S2
S1
Y3
0̃
I˜
0̃
0̃
S3
(-9,-3,3,9)
(0.556, 0.778, 1, 1.667)
(-9,-1,1,9)
Y2
(-3.547, 3.561, 7.892, 13.446)
(0.333, 0.444, 0.889, 1.444)
(-4.446, 0.888, 3.666, 10.556)
Y1
Table 4.2: Simplex Table(Continued)
(-0.221, 0.223, 0.556, 0.778)
(0.111, 0.111, 0.111, 0.111)
(0.223, 0.556, 0.667, 1.222)
XB
Ch.4 Two Person Non Zero-Sum Games
86
( 0.105, 0.105, 0.105, 0.105)
(0.2, 0.2, 0.2, 0.2)
(-0.067, -0.033, 0, 0.033)
(-0.567, -0.383, -0.033, 0.183)
(0.083, 0.083, 0.083, 0.083)
(-0.666, -0.222, 0.222, 0.666)
(-4.999, -2.333, 2.333, 4.999)
0̃
S2
0̃
(-9,-3,3,9)
(0.066, 0.066, 0.066, 0.066)
(-1.032, -0.649, -0.433, -0.283)
(0.1, 0.111, 0.122, 0.133)
0̃
I˜
0̃
0̃
S3
(0.556, 0.778, 1, 1.667)
(-1.35, -0.15, 0.15, 1.35)
Y2
(-0.117, -0.067, -0.05, 0,033)
Table 4.3: Simplex Table(Continued)
(-6.003, 1.9, 7.747, 14.24)
(-0.599, -0.033, 0.534, 0.9)
(0.044, 0.3, 0.889, 1.588)
(0.15, 0.15, 0.15, 0.15)
S3
0̃
(0.067, 0.089, 0.111, 0.133)
(-0.667, 0.133, 0.55, 1.583)
(0.483, 0.683, 0.85, 1.983)
Y2
I˜
(0.033, 0.083, 0.1, 0.183)
Y1
S1
Y3
I˜
XB
Y3
YB
CB
Ch.4 Two Person Non Zero-Sum Games
87
YB
Y4
Y5
Y6
CB
(0,0,0,0)
(0,0,0,0)
(0,0,0,0)
(-1,-1,-1,-1)
(0,0,0,0)
(-1,-1,-1,-1)
(-2,1, 3,6)
(2,4, 7,11)
(-3,0, 2,5)
Y2
(-1,-1,-1,-1)
(0,7, 11,14)
(-1,2, 4,7)
(-1,4, 5,8)
Y3
Table 4.4: Simples Table
(2,4, 5,9)
(-6,-2, 3,5)
(3,6, 10,17)
Y1
(1,1,1,1)
(1,1,1,1)
(1,1,1,1)
XB
(0,0,0,0)
(0,0,0,0)
(0,0,0,0)
(1,1,1,1)
S1
(0,0,0,0)
(0,0,0,0)
(1,1,1,1)
(0,0,0,0)
S2
(0,0,0,0)
(1,1,1,1)
(0,0,0,0)
(0,0,0,0)
S3
Ch.4 Two Person Non Zero-Sum Games
88
YB
Y1
S2
S3
CB
I˜
0̃
0̃
S1
(0.111, 0.111, 0.111, 0.111)
(-0.556, -0.222, 0, 0.333)
(-0.889, -0.556, -0.444, 0.111)
(0.889, 0.889, 0.889, 0.889)
Y3
(0.222, 0.444, 0.556, 1)
(-4.78, -0.112 ,3, 7.668)
(-4.448, 4.22, 8.776, 14.556)
(-0.444,-0.444,-0.444,-0.444)
0̃
0̃
I˜
0̃
I˜
0̃
0̃
0̃
S3
(-0.889, -0.889, -0.889, -0.889)
0̃
S2
(-1.888, 1.445, 3.556, 7.111)
(1.445, 3.778, 7, 11.333)
(-0.667, 0.222, 1.111, 1.889)
Y2
(-9,-1,1,9)
(-8,-2,2,8)
(0.333, 0.667, 1.111, 1.889)
Y1
Table 4.5: Simplex Table(Continued)
(0.111, 0.445, 0.556, 1.111)
(0.445, 0.778, 1, 1.333)
(0.111, 0.111, 0.111, 0.111)
XB
Ch.4 Two Person Non Zero-Sum Games
89
(-0.095, -0.038, 0, 0.057)
(-0.925, -0.529, -0.376, 0.246)
(0.094, 0.094, 0.094, 0.094)
(-0.813, -0.019, 0.51, 1.304)
(-6.197, 3.345, 8.421, 15.02)
(-0.227, -0.227, -0.227, -0.227)
(0.093, 0.093, 0.093, 0.093)
(-1.209, -0.605, -0.246, 0.321)
(0.17, 0.17, 0.17, 0.17)
(-0.095, -0.057, -0.038, 0.113)
0̃
I˜
0̃
0̃
S3
0̃
0̃
S2
(-8.999, -2.111, 2.111, 8.999)
(0.246, 0.642, 1.19, 1.927)
(-1.223, -0.111, 0.111, 1.223)
Y2
(-9,-1,1,9)
Table 4.6: Simplex Table(Continued)
(-0.963, -0.092, 0.338, 1.396)
(-1.36, -0.34, 0.34, 1.36)
(0.098, 0.115, 0.117, 0.122)
S3
0̃
(0.076, 0.132, 0.17, 0.227)
(0.333, 0.667, 1.111, 1.889)
(0.085, 0.362, 0.501, 1.164)
Y2
I˜
(0.027, 0.061, 0.077, 0.212)
Y1
S1
Y1
I˜
XB
Y3
YB
CB
Ch.4 Two Person Non Zero-Sum Games
90
0̃
(0.253, 0.253, 0.253, 0.253)
(-0.157, -0.039, 0.039, 0.157)
(-0.179, -0.103, -0.073, 0.048)
(0.077, 0.077, 0.077, 0.077)
(-2.117, -0.529, 0.529, 2.117)
(-1.204, 0.65, 1.636, 2.918)
0̃
Y2
(0.13, 0.13, 0.13, 0.13)
(-0.24, -0.12, -0.05, 0.06)
(0.1, 0.17, 0.21, 0.28)
S3
(0.04, 0.04, 0.04, 0.04)
(0.19, 0.19, 0.19, 0.19)
(-0.253, -0.01, 0.004, 0.16)
(-0.23, -0.1, -0.07, -0.02)
0̃
(-1.746, -0.41, 0.41, 1.746)
(0.246, 0.642, 1.19, 1.927)
(-1.223, -0.111, 0.111, 1.223)
(-0.09, -0.03, 0.004, 0.21)
S2
Table 4.7: Simplex Table(Continued)
(-1.746, -0.194, 0.194, 1.746)
(-0.187, -0.018, 0.066, 0.271)
(-1.36, -0.34, 0.34, 1.36)
(0.105, 0.143, 0.155, 0.211)
Y3
I˜
(0.033, 0.115, 0.171, 0.254)
Y1
(0.333, 0.667, 1.111, 1.889)
(-1.079, -0.139, 0.139, 1.079)
Y2
I˜
(-0.011, 0.044, 0.065, 0.209)
S1
Y1
I˜
XB
Y3
YB
CB
Ch.4 Two Person Non Zero-Sum Games
91
Ch.4 Two Person Non Zero-Sum Games
4.4
Fuzzy determinant Procedure
92
3
This section proposes a F - Mag based solution algorithm to solve a
fuzzy bi-matrix game.
4.4.1
Definition
Let Ã be the given fuzzy matrix of order n. Then Ṽ , X̃ and Ỹ as
follows:
X̃ ≈
F − M ag(Ã)
,
Ṽ ≈ Pn Pn
M̃
ã
ij
i=1
j=1
!
Pn
Pn
j=1 M̃ãnj
j=1 M̃ã1j
, . . . , Pn Pn
Pn Pn
i=1
j=1 M̃ãij
i=1
j=1 M̃ãij
and
Ỹ ≈
4.4.2
!
Pn
Pn
M̃ã
M̃ã
Pn i=1
Pn i1 , . . . , Pn i=1
Pn in
i=1
j=1 M̃ãij
i=1
j=1 M̃ãij
Procedure
Step 1 Let Ã be the fuzzy payoff matrix of player I and B̃ be that
of player II and both having order n.
3
The contents of this section form the substance of the paper published in the ”Proceedings ofIEEE International Conference on Advances in Engineering, Science and
Management”, ISBN: 978-81-909042-2-3 2012 IEEE,Vol. 5
Ch.4 Two Person Non Zero-Sum Games
93
Step 2 Let Ã1 , Ã2 , Ã3 , . . . be the fuzzy square sub-matrices of Ã
and B̃1 , B̃2 , B̃3 , . . . be that of B̃.
Step 3 For each Ãk find ṼÃk and X̃Ãk and for each B̃l find ṼB̃l and ỸB̃l .
Step 4 Write X̃k from X̃Ãk by adding zeros corresponding to each
eliminated row. Similarly, write Ỹl from ỸB̃l by adding zeros
corresponding to each eliminated column.
Step 5 Define SÃ and SB̃ as follows:
SÃ ≈ {X̃k ≈ (x̃k1 , x̃k2 , . . . , x̃kn ) | x̃km < 0̃,
SB̃ ≈ {Ỹl ≈ (ỹl1 , ỹl2 , . . . , ỹln ) | ỹlm < 0̃,
n
X
˜ (4.13)
x̃km ≈ I}
m=1
n
X
˜
ỹlm ≈ I}
(4.14)
m=1
Step 6 Define ṼÃ and ṼB̃ as follows:
ṼÃ ≈ max {ṼÃk }andṼB̃ ≈ max {ṼÃk }
X̃k ∈SÃ
4.4.3
Ỹl ∈SB̃
Numerical Example
Game Description:
For this example, the problem of two departmental stores: Gacy’s
and Mimbel’s has been taken. Each of the two stores has to choose a
location for its one store in Gotham City. Each store will choose one
of three location strategies: Uptown, Center City and West Side.
Ch.4 Two Person Non Zero-Sum Games
94
The payoff matrices for both of them are given below where Mimbel
is the row player and Gacy is the column player.
Gacy’s
U ptown
U ptown
CenterCity
W estSide
(0, 7, 9, 12)
(−1, 0, 2, 3)
(1, 6, 8, 13) and
CenterCity (1, 10, 11, 14) (−4, −1, 0, 1) (−3, 1, 2, 4)
W estSide
(−2, 5, 7, 10)
(1, 6, 8, 13)
(−1, 6, 8, 11)
Gacy’s
U ptown
CenterCity
W estSide
U ptown
(0, 3, 6, 7)
(−1, 0, 2, 3) (−5, −4, −2, −1)
CenterCity
(1, 2, 4, 5)
(−1, 0, 2, 3)
(−1, 6, 8, 11)
W estSide
(−5, −4, −2, −1)
(0, 3, 6, 7)
(0, 1, 3, 4)
Solution Procedure
Step 1 Consider the payoff matrices for player I and II.


(0, 7, 9, 12)
(−1, 0, 2, 3) (1, 6, 8, 13)
Ã ≈  (1, 10, 11, 14) (−4, −1, 0, 1) (−3, 1, 2, 4)  and
(−2, 5, 7, 10) (1, 6, 8, 13) (−1, 6, 8, 11)


(0, 3, 6, 7)
(−1, 0, 2, 3) (−5, −4, −2, −1)
B̃ ≈ 
(1, 2, 4, 5)
(−1, 0, 2, 3)
(−1, 6, 8, 11) 
(−5, −4, −2, −1) (0, 3, 6, 7)
(0, 1, 3, 4)
Ch.4 Two Person Non Zero-Sum Games
95
Step 2 Consider Ã and all its square sub matrices and B̃ and all
its square sub matrices.
Step 3 Tabulate the corresponding ṼÃi X̃i andṼB̃j andỸj as given in
tables 4.8 and 4.9
Step 4 Then S̃Ã and S̃B̃ satisfying the conditions are taken as follows:
S̃A ≈ X2 , X3 , X5 , X6 , X7 , X8 , X9 , X10
S̃B ≈ Y1 , Y2 , Y3 , Y4 , Y5 , Y7 , Y9 , Y10
Step 5
ṼÃ ≈ max VAi ≈ (−0.375, 6.75, 8.125, 14.625) and
i
ṼB̃ ≈ max VBj ≈ (0.3, 2.4, 4.8, 5.7)
j
Step 6 Optimal fuzzy strategies are:
X̃ ≈ ((−0.4, 1, 1.3, 2.1)(−1.4, −0.4, 0.1, 1.6), 0̃)
Ỹ ≈ ((0, 0.8, 1.2, 1.6), (−0.5, −0.1, 0.4, 0.6), 0̃)
(-2.25,1.75,6.75,9.75)
(0.17,5.83,6.83,9.83)
(-1.875,4.875, 7.875,11.125)
(-4,-2.5,-1.5,0)
(-9,2.33,6.67,17.33)
(-1.57,4.287,8,13.87)
(-0.375,6.75, 8.125,14.625)
(-1.56,5.56,6.78,11)
(-65,2,24,67)
(-1.3,1,2.5,5)
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
((-0.1,0.17,0.22,0.36) (-0.2,0,0.1,0.2) (-0.1,0.04,0.09,0.191))
((-11,-1,3,13),0̃,(-13,-1,3,11))
(0̃,(-0.6,-0.1,0.2,0.7), (-0.2,0.4,0.6,0.9))
((-0.4,1, 1.3,2.1), (-1.4,-0.4, 0.1,1.6),0̃)
((-1.43,-0.3,0.3,2), 0̃, (-0.3,0.6,1.1,2))
(0̃, (-3.33,-0.7,0.7,4.7), (-1.3,0.3,1,2.667))
((-2, -0.8,-0.3,1)(-0.5, 1,2,3.5),0̃)
((-1.1,-0.1, 0.4,1.9), 0̃, (-0.4,0.6,1.1,1.63))
(0̃, (-0.8,-0.1,0.3,1.3), (0,0.8,1,1.5))
((0,2.5,3,4.5), (-3.3, -2.3, -1.3,0.8), 0̃)
X̃Ã
Table 4.8: Fuzzy values and Fuzzy strategies for Player A
ṼÃ
Matrix
Ch.4 Two Person Non Zero-Sum Games
96
(-9,-3,6,10)
(0.11,0.88,2.44,3.22)
(-0.3,1.2,3,3.7)
(-0.55,0.22,1.88,2.55)
(-1.43,2.89,4.56,6.43)
(-1.2,1.6,4,5.2)
(0.3,2.4,4.8,5.7)
(0.5,13,20,26.5)
(-1.25,-0.5,0.5,0.92)
(-0.2,1.1,2.9,3.6)
B1
B2
B3
B4
B5
B6
B7
B8
B9
B10
((-0.4,0,0.6,0.9) (-0.1,0.5,1.1,1.4) (-0.5,-0.2,0.3,0.5))
((0.1,0.3,0.6,0.8), 0̃, (0.1,0.4,0.8,1))
(0̃, (-5.5,-3.5,-1.5,2.5), (1,2,4,5))
((0,0.8,1.2,1.6), (-0.5,-0.1,0.4,0.6), 0̃)
((0.5,1.5,3.5,4.5), 0̃, (-4,-3,-0.5,1.5))
(0̃, (-0.7,0.4,1,1.6), (-0.4,0.1,0.9,1.1))
((0,0.9,1.3,1.8), (-0.4,-0.2,0.2,0.4), 0̃)
((-0.3,0.1,0.6,0.8), 0̃, (0.1,0.5,1,1.2))
(0̃, (-0.3,0.1,0.7,0.9), (0.2,0.4,0.9,1.1))
((-4,-2,2,4), (-5,-1,4,6), 0̃)
ỸB̃
Table 4.9: Fuzzy values and Fuzzy strategies for Player B
ṼB̃
Matrix
Ch.4 Two Person Non Zero-Sum Games
97
Ch.4 Two Person Non Zero-Sum Games
4.5
98
Matrix Difference Procedure
4.5.1
Procedure
Step 1 Let Ã be the fuzzy payoff matrix of player I and B̃ be that
of player II.
Step 2 Write the matrices of Column and Row Differences as follows:
C̃Ã ≈ [C̃1 C̃2
C̃2 C̃3
. . .]andR̃B̃ ≈ [R̃1 R̃2
R̃2 R̃3
...]
where C̃j are the columns of Ã and R̃i are the rows of B̃.
Step 3 Now find p̃i ≈ f det(Ãi ), where Ãi is the matrix obtained by
eliminating the it h row of the matrix C̃Ã and find q̃j ≈ f det(B̃j ),
where B̃j is the matrix obtained by eliminating the j t h column
of the matrix R̃B̃
Step 4 Check whether
n
n X
n
n
n X
n
X
X
X
X
p̃i ≈
Mãij and
q̃j ≈
Mb̃ij .
i=1
i=1 j=1
i=1
i=1 j=1
If satisfied go to next step. Otherwise the method will fail.
P
Step 5 Take ṼÃ ≈ ni=1 ãij p̃i , for any j = 1, 2, . . . , n and X̃Ã ≈
Pn
Pnp̃1 , Pnp̃2 , . . . Pnp̃n
and
Ṽ
≈
j=1 b̃ij q̃j , for any i =
B̃
i=1 p̃i
i=1 p̃i
i=1 p̃i
q̃2
q̃n
q̃1
P
P
P
, n q̃j , . . . n q̃j .
1, 2, . . . , n and ỸB̃ ≈
n
q̃j
j=1
j=1
i=1
Ch.4 Two Person Non Zero-Sum Games
4.5.2
99
Numerical Example
Game Description:
Professor Heffalump and Dr. Boingboing are the authors of rival
textbooks of game theory. Their books are of equal quality in every way except length. Both authors know that, given a choice
between two well-written books, professors will usually choose the
longer book. Each would like to get the larger audience, but writing
a longer book is a bigger effort, so neither author wants to write a
book longer than necessary to capture the bigger audience. Each
of the two authors can choose among the following three strategies:
write a book of 400 pages, 600 pages or 800 pages. The increasing
percentage of audiences in either side is given in the following payoff
matrices.
Dr. Boingboing
400pages
600pages
800pages
400pages (4, 6, 8, 10) (−3, 1, 2, 4) (4, 7, 8, 9)
P rof.Haf f
600pages (6, 9, 10, 11) (−8, 0, 1, 3) (−3, 1, 2, 4)
800pages (0, 4, 6, 10) (4, 7, 8, 9) (4, 5, 7, 8)
Dr. Boingboing
400pages
600pages
800pages
400pages (−4, 1, 3, 4) (−3, 2, 4, 5) (1, 2, 3, 6)
P rof.Haf f
600pages (1, 2, 3, 6) (−3, 1, 2, 4) (0, 1, 3, 4)
800pages (−1, 0, 2, 7) (0, 3, 4, 5) (−3, 1, 2, 4)
Ch.4 Two Person Non Zero-Sum Games
100
Solution Procedure


(4, 6, 8, 10) (−3, 1, 2, 4) (4, 7, 8, 9)
Step 1 Let Ã ≈  (6, 9, 10, 11) (−8, 0, 1, 3) (−3, 1, 2, 4)  and
(0, 4, 6, 10) (4, 7, 8, 9) (4, 5, 7, 8)


(−4, 1, 3, 4) (−3, 2, 4, 5) (1, 2, 3, 6)
B̃ ≈  (1, 2, 3, 6) (−3, 1, 2, 4) (0, 1, 3, 4)  be the fuzzy
(−1, 0, 2, 7) (0, 3, 4, 5) (−3, 1, 2, 4)
payoff matrices for player I and II respectively.
Step 2 Write the matrices of column and row differences as follows:


(0, 4, 7, 13)
(−12, −7, −5, 0)
C̃Ã ≈  (3, 8, 10, 19)
(−12, −2, 0, 6)  and
(−9, −4, −1, 6)
(−4, 0, 3, 5)
(−10, −2, 1, 3) (−7, 0, 3, 8) (−3, −1, 2, 6)
R̃B̃ ≈
(−6, 0, 3, 7) (−8, −1, −3, 4) (−4, −1, 2, 7)
Step 3 Then the values of p̃i s and q̃j are given as follows:
p̃1 ≈ (−15, 0, 8, 31),
p̃2 ≈ (−49, −1, 20, 54)
p̃3 ≈ (−8, 34, 52, 114)
q̃2 ≈ (−17, −5, 1, 9)
Step 4 Here
Pn
i=1 p̃i ≈
and
q̃1 ≈ (−11, 3, 4, 16),
q̃3 ≈ (−13, −5, 4, 26)
and
Pn Pn
i=1
and
j=1 Mãij ,
Pn
i=1 q̃j ≈
Pn Pn
i=1
j=1 Mb̃ij
Step 5 Then ṼÃ ≈ (1, 4.7, 6.6, 10.1) and ṼB̃ ≈ (−2, 1.667, 3.33, 5)
Ch.4 Two Person Non Zero-Sum Games
101
with fuzzy optimal strategies
X̃Ã ≈ ((−0.25, 0, 0.133, 0.517), (−0.817, −0.017, 0.33, 0.9),
(0.133, 0.567, 0.867, 1.9))
(4.15)
ỸB̃ ≈ ((−1.22, 0.33, 0.44, 1.778), (−1, 0.11, 0.556, 1.889),
(−1.44, −0.556, 0.44, 2.889))
(4.16)
Ch.4 Two Person Non Zero-Sum Games
4.6
An Approximate Method
102
4
This section provides an algorithm to get an approximate solution
of a fuzzy game.
4.6.1
Algorithm
Step 1 Player I(Player II) arbitrarily selects any row(column).
Step 2 Then player II(Player I) examines this row(column) and
chooses a column(row) corresponding to the largest number in
the row.
Step 3 Player I (Player II) now examines this column (row) and
chooses a row (column) corresponding to the largest number in
this column(row). This row(column) is then added to the one
previously chosen and the sum of the two rows(columns) will
be taken for next iteration.
Step 4 Player II(Player I) now chooses a column(row) corresponding to the largest number in the new row(column) and adds this
column(row) to the column(row) last chosen. In the case of a
4
The contents of this section form the substance of the paper published in the Proceedings of the International Conference on Mathematics and Computer Science, Loyola
College, Chennai, 2011.
Ch.4 Two Person Non Zero-Sum Games
103
tie the player will select a row (or) column different from his
previous choice.
Step 5 The largest element in rows and columns are encircled. This
procedure may be continued for a required number of times in
the like manner.
Step 6 The approximate strategies are found by dividing the number of encircled elements in each row and column by the number
of iterations. Increase the number of iterations the increased accuracy. Using this method, the approximate value of a fuzzy
game can be evaluated up to any desired degree of accuracy.
Step 7 For Player I take the row iteration table and calculate only
the row strategies. For column player follow the same steps
for his transposed payoff matrix and calculate only the column
strategies.
4.6.2
Numerical Example
Game Description:
There are two radio stations, WIRD and KOOL. Each station can
choose among three broadcast formats: rock-n-roll, country music
and all talk. These are their three strategies. The payoff for these
Ch.4 Two Person Non Zero-Sum Games
104
two radio stations are expressed as the percent of the potential audience they obtain. The following table shows the payoffs for WIRD
stations:
KOOL
rock − n − roll country music
W IRD
rock-n(0, 7, 9, 12)
rool
country (1, 10, 11, 14)
music
all talk (−2, 5, 7, 10)
all talk
(−1, 0, 2, 3)
(1, 6, 8, 13)
(−4, −1, 0, 1)
(−3, 1, 2, 4)
(1, 6, 8, 13)
(−1, 6, 8, 11)
Solving Procedure
Step 1:


(0, 7, 9, 12)
(−1, 0, 2, 3) (1, 6, 8, 13)
T ake Ã ≈  (1, 10, 11, 14) (−4, −1, 0, 1) (−3, 1, 2, 4) 
(−2, 5, 7, 10) (1, 6, 8, 13) (−1, 6, 8, 11)
Player I now selects row 2 arbitrarily.
Step 2:Now player II examines this row and selects a column corresponds to the maximum in this row4.10.
Here the maximum corresponds to the second column.
Step 3:Now player II examines this column and selects the row corresponds to the maximum in this column and add this row with the
previous one which then be used for the next iteration.
Ch.4 Two Person Non Zero-Sum Games
105
Table 4.10: Comparison of row elements
(1,10,11,14)
(-4,-1,0,1)
(-3,1,2,4)
(-4,-3,11,14)
x0
8.571
-1.111
0.875
4.521
y0
0.357
0.389
0.375
0.479
d
4.052
5.633
3.647
Step 4:Player II again examines this row and selects the column
corresponds to the maximum in this row which will be added to the
column previously chosen. This new column will be taken for the
next iteration. This procedure is repeated for ten times and the iteration table 4.11 is given. It is enough to take row iteration table
for player I and column iteration table for player II as both of them
have a different payoff matrices.
Step 5:Then the strategies for player I are: p1 = 0, p2 = 1, p3 = 0.
To find the fuzzy value take the average of all p1 a1j ⊕ p2 a2j ⊕ p3 a3j ,
j = 1, 2, 3.
Approximate fuzzy value for player I is (-1,1.667,2.167,3.167)
Similarly we can find the strategies and the approximate fuzzy value
for Player II using the payoff matrix for player II.
(2,20,22,28)
(3,30,33,42)
(4,40,44,56)
(5,50,55,,70)
(6,60,66,84)
(7,70,77,98)
(8,80,88,112)
(9,90,99,126)
2
3
4
5
6
7
8
9
(-36,-9,0,9)
(-32,-8,0,8)
(-28,-7,0,7)
(-24,-6,0,6)
(-20,-5,0,5)
(-16,-4,0,4)
(-12,-3,0,3)
(-8,-2,0,2)
(-4,-1,0,1)
Column II
(-27,9,18,36)
(-24,8,16,32)
(-21,7,14,28)
(-18,6,12,24)
(-15,5,10,20)
(-12,4,8,16)
(-9,3,6,12)
(-6,2,4,8)
(-3,1,2,4)
Column III
(-36,-28,99,126)
(-32,-24,88,112)
(-28,-21,77,98)
(-24,-18,66,84)
(-20,-15,55,70)
(-16,-12,44,56)
(-12,-9,33,42)
(-8,-6,22,28)
(-4,-3,11,14)
Ũ
Table 4.11: Row Iterations for Player I
(10,100,110,140) (-40,-10,0,10) (-30,10,20,40) (-40,-32,110,140)
(1,10,11,14)
1
10
Column I
Iteration No.
Ch.4 Two Person Non Zero-Sum Games
106
Chapter 5
Constrained Fuzzy Games
Abstract
This chapter will discuss a new and more useful topic in
fuzzy games called Constrained fuzzy games. This will
arise when in addition to the normal strategy requirements,
some other constraints like market uncertainties affects the
payoffs.The fuzzy nature of the payoffs has been kept unchanged.
107
Ch.5 Constrained Fuzzy Games
5.1
108
Introduction
In many current literature, the application of games are mostly limited to problems without constraints, i.e., users’ sets of strategies
are independent of one another. However, in most engineering scenarios, such restrictions lead to impractical solutions, as most of
the existing systems are constrained in an interactive fashion. For
example, capacity constraint is one of the notable constraints in a
resource allocation problem. Therefore, it is essential to establish a
theoretical basis to study games with constraints.
Attempts for such extension were made in [5] and [6], where the
payoff functions of the players are modified to include penalty terms
that reduces the payoff when the constraints are violated. This approach circumvents the problem of a constrained game by turning it
into an unconstrained game and thus solved in a known way. Since
the penalty functions are commonly chosen to be a reciprocal function of the constraints or a logarithmic function of multi-variables,
it becomes easily intractable when multiple constraints are involved
and utilities become nonlinear and strongly coupled. Though such
approach employs the current results on games, it is still challenging to perform analysis on the model. On the other hand, such
Ch.5 Constrained Fuzzy Games
109
approach is strongly problem dependent. A different problem may
need a different construction of modified payoff functions.
There are lots of situations where practitioners have to cope with
a bottleneck in a production system. Constraints or bottlenecks can
take the form of limited capacity, a customer requirement such as
quantity or due date, or the availability of a material.
A common practice in solving a constrained game involves the
process of embedding the constraints as a penalty into the utility
function. This approach offers a way to circumvent the constrained
problem by turning it into an unconstrained game proposed by Pan
and Pavel in [69] and Basar and Srikant in [8]. However, the problem becomes untractable when utilities are nonlinear and strongly
coupled. Furthermore, it is extremely challenging to find analytical
solutions as the complexity of the functions grows. On the other
hand, this approach is strongly problemdependent. Different problems may need a different construction of modified utility functions.
As a result, this approach doesnt provide a unifying theory to solve
a class of problems.
The recent work of Pavel[55] develops an extension of duality theory to solve a general class of constrained games. The idea centers
Ch.5 Constrained Fuzzy Games
110
around the hierarchical decomposition of the original constrained
problem into two unconstrained subproblems. This work gives a
general approach in dealing with constraints for convex cost functions. However, in practice, the theory needs to be further reduced
to a certain form for any particular application. As a result, there
exists a gap between engineering application and the theory. It will
be more useful if a theory can be found that is tailored for a special
class of functions, i.e., such as affine functions, commonly appearing
in wireless and optical networks.
Motivated by this, a class of Fuzzy games with linear fuzzy constraints and a linear programming approach to solve for its Nash
equilibrium have been designed. Fuzzy games without constraints
have been seen in many areas and there are lots of different approaches are also available to solve such games. However, it is a
challenge when it comes to fuzzy games with constraints. In this
regard, the researcher developed a theoretical framework to characterize a solution for this specific type of Fuzzy games.
Constrained Fuzzy games are a type of matrix games with fuzzy
payoffs and sets of players’ strategies which are constrained (Fuzzy).
So far as it is known that not much attempts have been made for
Ch.5 Constrained Fuzzy Games
111
constrained fuzzy games since there is no effective way to simultaneously incorporate the payoffs’ fuzziness and strategies’ constraints
into fuzzy matrix game methods. The aim of this chapter is to develop an effective methodology for solving constrained matrix games
with fuzzy payoffs and fuzzy constraints. Thus, a variety of engineering problems with practical concerns of constraints can be solved.
This chapter has two main sections, namely, Constrained zerosum fuzzy games and constrained non-zero sum fuzzy games. Both
of this sections will discuss a fuzzy linear programming solutions
for their respective types of Constrained fuzzy games. Numerical
examples have also been given to better understand the methods.
5.2
A Constrained zero sum fuzzy game
1
Let the fuzzy strategy spaces for player I and II be,
P
˜
S̃1 ≈ {X̃ ≈ (x̃1 , . . . , x̃m ) | x̃i < 0̃f or all i, m
i=1 x̃i ≈ I}
P
(x)
(x)
∪{ m
p=1 c̃pq x̃p rc̃q }
S̃2 ≈ {Ỹ ≈ (ỹ1 , . . . , ỹn ) | ỹj < 0̃f or all j,
Pn
j=1 ỹj
˜
≈ I}
P
(y)
(y)
∪{ ns=1 c̃rs ỹs rc̃r }
1
The contents of this section form the substance of the paper published in the ”International Journal of Fuzzy Mathematics”, ISSN 1066-8950, Vol.21 ,No.1 (20), 173-180.
Ch.5 Constrained Fuzzy Games
(x)
(x)
112
(y)
(y)
where C̃ (x) ≈ (c̃pq | c̃q ), C̃ (y) ≈ (c̃rs | c̃r )andr ∈ {≺, 4, ≈, , <},q is the number of constraints in x̃ and r is the number of
constraints in ỹ. Then a constrained zero sum fuzzy game can be
defined by CFG ≈ (Ã, S̃1 , S̃2 , K̃, C̃ (x) , C̃ (y) ) . Here Ã is the payoff
matrix and
K̃(X̃, Ỹ ) ≈
5.2.1
Pm Pn
i=1
j=1 ãij
x̃i ỹj ≈ X̃ T Ã Ỹ
Optimal strategies
A pair of fuzzy strategies (X̃ ∗ , Ỹ ∗ ) which satisfies the constraints
C̃ (x) andC̃ (y) is said to be a fuzzy Nash equilibrium for the CF G ≈
(Ã, S̃1 , S̃2 , K̃, C̃ (x) , C̃ (y) ) if,
K̃(X̃ ∗ , Ỹ ∗ ) < K̃(X̃, Ỹ ∗ ), ∀ f uzzy strategies X̃ f or player I and
K̃(X̃ ∗ , Ỹ ∗ ) 4 K̃(X̃ ∗ , Ỹ ), ∀ f uzzy strategies Ỹ f or player II
Here,K̃(X̃ ∗ , Ỹ ∗ ) ≈ Ṽ .
5.2.2
Remark
Since a constraint fuzzy game is a fuzzy game with some more set
of constraints every theorem which has been proved for fuzzy games
can also be proved for CFG’s.
Ch.5 Constrained Fuzzy Games
5.3
113
Theorems
Theorem 5.3.1. Let S̃1 and S̃2 be the strategy spaces for player I
and II respectively. Then,
min max K̃(X̃, Ỹ ) < max min K̃(X̃, Ỹ )
Ỹ ∈S̃2 X̃∈S̃1
X̃∈S̃1 Ỹ ∈S̃2
Theorem 5.3.2. The following statements are equivalent:
1. There exist fuzzy strategies x̃∗ ∈ S̃1 andỸ ∗ ∈ S̃2 and a fuzzy
number Ṽ such that,
X̃ ∗ Ã Ỹ T < Ṽ f orallỸ ∈ S̃2
X̃ Ã Ỹ ∗T 4 Ṽ f orallX̃ ∈ S̃1
2. minỸ ∈S̃2 maxX̃∈S̃1 K̃(X̃, Ỹ ) ≈ maxX̃∈S̃1 minỸ ∈S̃2 K̃(X̃, Ỹ )
3. (X̃ ∗ , Ỹ ∗ ) is a fuzzy solution of the CFG ≈ (Ã, S̃1 , S̃2 , K̃, C) and
Ṽ ≈ X̃ ∗ Ã Ỹ ∗T
Theorem 5.3.3. The fuzzy strategies and values of a constrained
zero sum fuzzy game can be obtained by solving a fuzzy LPP.
Proof
Since, Ṽ ≈ maxX̃∈S̃1 minỸ ∈S̃2 X̃ T Ã Ỹ , Player I will choose his
P Pn
fuzzy strategy X̃ ∈ S̃1 to maximize, minỸ ∈S̃2 m
i=1
j=1 x̃i ãij ỹj .
Ch.5 Constrained Fuzzy Games
114
Since every strategy Ỹ ∈ S̃2 is a fuzzy convex combination of n pure
fuzzy strategies for player II, we have
P Pn
Pm
minỸ ∈S̃2 m
i=1
j=1 x̃i ãij ỹj ≈ min1≤j≤n [
i=1 x̃i ãij ].
T hen Ṽ ≈ maxz̃
m
X
x̃i ãij < z̃, j = 1, 2, . . . , n
i=1
m
X
x̃i ≈ I˜
i=1
(∵ x̃ ∈ S̃1 )
x̃i < 0̃, i = 1, 2, . . . , m
(5.1)
P
(x)
(x)
In addition to that we also have the constraints m
p=1 c̃pq x̃p rc̃q .
P
(x)
(x)
Now suppose that r is <2 . i.e., m
p=1 c̃pq x̃p < c̃q .
By taking x̃0i ≈ x̃i z̃ we have,
Ṽ ≈ minx̃01 ⊕ x̃02 ⊕ . . . x̃0m
2
(5.2)
subject to,
m
X
˜
x̃0i ãij < I,
j = 1, 2, . . . , n
i=1
x̃0i <
i = 1, 2, . . . , m and
0̃,
If r is 4 then it can be written as
(x)
p=1 [c̃q
Pm
(x)
c̃pq ] x̃0p < 0̃∀q.
Ch.5 Constrained Fuzzy Games
m
X
p=1
m
X
p=1
m
X
115
(x)
c̃(x)
pq x̃p z̃ < c̃q z̃
c̃(x)
pq
x̃0p
<
c̃(x)
q
m
X
∀q
x̃0p
∀q
p=1
(x)
0
[c̃(x)
pq c̃q ] x̃p < 0̃
∀q
p=1
Hence the above FLPP will become,
Ṽ ≈ minx̃01 ⊕ x̃02 ⊕ . . . x̃0m
subject to,
m
X
˜
x̃0i ãij < I,
i=1
m
X
(x)
0
[c̃pq
c̃(x)
q ] x̃p < 0̃
p=1
x̃0i <
(5.3)
j = 1, 2, . . . , n
∀q
0̃, i = 1, 2, . . . , m
Then the dual of the FLPP in 5.3 is,
0
Ṽ ≈ maxỹ10 ⊕ ỹ20 ⊕ . . . ỹn0 ⊕ 0̃ ỹn+1
⊕ ...
subject to,
n
X
(x)
0
˜
ãij ỹj0 ⊕ [c̃pq
c̃(x)
q ] ỹn+1 ⊕ · · · 4 I,
j=1
ỹj0 <
(5.4)
i = 1, 2, . . . , n
0̃, j = 1, 2, . . . , n, n + 1, . . . q
Now we can easily include the constraints for y as follows and here
Ch.5 Constrained Fuzzy Games
116
suppose that ris 43
n
X
s=1
n
X
s=1
n
X
s=1
m
X
(y)
c̃rs
ỹs 4 c̃(y)
r
∀r
(y)
c̃rs
ỹs z̃ 4 c̃(y)
r z̃
∀r
(y)
c̃rs
ỹs0
4
c̃(y)
r
n
X
ỹs0
∀r
s=1
(y)
0
[c̃(y)
rs c̃r ] ỹs 4 0̃
∀r
(5.5)
p=1
Combining 5.4 and 5.5 we will get the FLPP corresponding to the
CFG.
0
Ṽ ≈ maxỹ10 ⊕ ỹ20 ⊕ . . . ỹn0 ⊕ 0̃ ỹn+1
⊕ ...
subject to,
n
X
(x)
0
˜
ãij ỹj0 ⊕ [c̃pq
c̃(x)
q ] ỹn+1 ⊕ · · · 4 I,
j=1
m
X
(y)
0
[c̃(y)
rs c̃r ] ỹs 4 0̃
p=1
ỹj0 <
3
(5.6)
i = 1, 2, . . . , n
∀r
(5.7)
0̃, j = 1, 2, . . . , n, n + 1, . . . q
For different r , it is easy to change the constraints so that all of them in same category
by multiplying .
Ch.5 Constrained Fuzzy Games
5.4
117
A Constrained non zero sum Bi-matrix fuzzy
game
Let the fuzzy strategy spaces for player I and II be,
S̃1 ≈ {X̃ ≈ (x̃1 , . . . , x̃m ) | x̃i < 0̃f or all i,
m
X
˜
x̃i ≈ I}
i=1
P
(x)
(x)
∪{ m
p=1 c̃pq x̃p rc̃q }
S̃2 ≈ {Ỹ ≈ (ỹ1 , . . . , ỹn ) | ỹj < 0̃f or all j,
n
X
˜
ỹj ≈ I}
j=1
P
(y)
(y)
∪{ ns=1 c̃rs ỹs rc̃r }
(x)
(x)
(y)
(y)
whereC̃ (x) ≈ (c̃pq | c̃q ), C̃ (y) ≈ (c̃rs | c̃r )andr ∈ {≺, 4, ≈, , <},q is the number of constraints in x̃ and r is the number of
constraints in ỹ. Then a constrained non zero sum bi-matrix fuzzy
game can be defined by CFBG ≈ (Ã, B̃, S̃A , S̃B , K̃A , K̃B , C̃ (x) , C̃ (y) ) .
Here Ã and B̃ are the payoff matrices for player A and B respectively.
Also,
K̃A (X̃, Ỹ ) ≈
Pm Pn
K̃B (X̃, Ỹ ) ≈
i=1
j=1 ãij
Pm Pn
i=1
j=1 b̃ij
x̃i ỹj ≈ X̃ T Ã Ỹ
x̃i ỹj ≈ X̃ T Ã Ỹ
and
Ch.5 Constrained Fuzzy Games
5.4.1
118
Optimal Strategies
A pair of fuzzy strategies (X̃ ∗ , Ỹ ∗ ) which satisfies the constraints
C̃ (x) , C̃ (y) is said to be a fuzzy nash equilibrium for the non zero sum
constrained fuzzy game CF BG ≈ (Ã, B̃, S̃1 , S̃2 , K̃Ã , K̃B̃ , C̃ (x) , C̃ (y) )
if,
K̃Ã (X̃ ∗ , Ỹ ∗ ) < K̃Ã (X̃, Ỹ ), ∀X̃ ∈ S̃1 andỸ ∈ S̃2
K̃B̃ (X̃ ∗ , Ỹ ∗ ) < K̃B̃ (X̃ ∗ , Ỹ ), ∀X̃ ∈ S̃1 andỸ ∈ S̃2
Also, K̃Ã (X̃ ∗ , Ỹ ∗ ) ≈ ṼÃ and K̃B̃ (X̃ ∗ , Ỹ ∗ ) ≈ ṼB̃ .
Theorem 5.4.1. For any CFBG,
1. ṼÃ ≈ maxX̃∈S̃1 maxỸ ∈S̃2 X̃ T Ã Ỹ
2. ṼB̃ ≈ maxỸ ∈S̃2 maxX̃∈S̃1 X̃ T B̃ Ỹ
Theorem 5.4.2. The fuzzy values and fuzzy strategies ṼÃ and ṼB̃
of a CFBG can be obtained by solving the two FLPPs.
Proof:
Since, ṼÃ ≈ maxX̃∈S̃1 maxỸ ∈S̃2 X̃ T Ã Ỹ , Player I will select his
fuzzy strategy X̃ ∈ S̃1 so as to maximize his profit, i.e., to maximize
P Pn
maxỸ ∈S̃2 m
i=1
j=1 x̃i ãij ỹj .
Since every strategy Ỹ ∈ S̃2 is a fuzzy convex combination of m pure
Ch.5 Constrained Fuzzy Games
119
fuzzy strategies, we have
P Pn
Pm
maxỸ ∈S̃2 m
x̃
ã
ỹ
≈
max
[
i
ij
j
1≤j≤n
i=1
j=1
i=1 x̃i ãij ] and
hence,
ṼÃ ≈ max z̃
subject to
m
X
x̃i ãij 4 z̃, j = 1, 2, . . . , n
i=1
m
X
x̃i ≈ I˜
i=1
x̃i < 0̃, i = 1, 2, . . . , m
(5.8)
ṼB̃ ≈ max z̃
subject to
n
X
ỹj b̃ij 4 z̃, i = 1, 2, . . . , m
j=1
n
X
ỹj ≈ I˜
j=1
x̃i < 0̃, j = 1, 2, . . . , n
(5.9)
In addition to that we also have the constraints
Pm (x)
Pn (y)
(x)
(y)
c̃
x̃
rc̃
and
pq
q
p
p=1
s=1 c̃rs ỹs rc̃r .
As in the zero sum case, this constraints can be written as
Pm (x)
Pn
(x)
(y)
(y)
0
0
p=1 [c̃q c̃pq ] x̃p 4 0̃ ∀ q and
s=1 [c̃r c̃rs ] ỹs 4 0̃ ∀ r.
Ch.5 Constrained Fuzzy Games
120
By taking x̃0i ≈ x̃i z̃ we have,
i.e.,ṼÃ ≈ max x̃01 x̃02 . . . x̃0m
(5.10)
subject to,
m
X
˜ i = 1, 2, . . . , m
x̃0i ãij 4 I,
i=1
x̃0i <
m
X
0̃, i = 1, 2, . . . , m
0
[c̃q(x) c̃(x)
pq ] x̃p 4 0̃
∀q
(5.11)
p=1
Similarly,ṼB̃ ≈ max ỹ10 ỹ20 . . . ỹn0
(5.12)
subject to,
n
X
˜ i = 1, 2, . . . , m
I˜ b̃ij ỹj0 4 I,
j=1
ỹj0 <
n
X
0̃, j = 1, 2, . . . , n
0
[c̃r(y) c̃(y)
rs ] ỹs 4 0̃
s=1
∀r
(5.13)
Chapter 6
A new approach to solve fuzzy
games
Abstract
This chapter proposes two ordering procedures for fuzzy
alternatives. The fuzzy payoffs will be converted into crisp
by making use of some known ranking functions and then
be used in the proposed algorithms to identify the best
range of profit nothing but the fuzzy alternative.
121
Ch.6 A new approach to solve fuzzy games
6.1
122
Introduction
In most of real world situations, usually decision makers are confronted with multiple criteria to be considered before any decision
can be made. This is the case of Multi Criteria Decision Making
(MCDM ); a case with the aim to find the overall preferences among
the available alternatives. The goal of the MCDM method is to aid
decision-makers in integrating objective measurements with value
judgments that are based not on individual opinions but on collective group ideas which is suggested by Belton and Stewart in [10].
Further, there are situations in which information is incomplete
or imprecise or views that are subjective or endowed with linguistic characteristics creating a fuzzy decision- making environment.
Therefore, a fuzzy MCDM problem with group decision accounts for
raising some evaluation points, which are evaluation criteria/subcriteria, feasible alternatives, decision-makers, and decision ranking
rules. Then, a set of alternatives is both feasible to the decisionmakers and known during the decision process.
The feasibility of an alternative is defined by a variety of constraints such as physical availability, monetary resources, information constraints, and so on. Later, the evaluation criteria of every
Ch.6 A new approach to solve fuzzy games
123
available alternative should be found out to evaluate the attractiveness of alternatives in terms of criteria values or performance value.
Finally, a choice from two or more alternatives requires a decision
rule or ranking rule in which the decision-makers can obtain the
information available to make a best choice.
There are a variety of multiple criteria techniques to aid selection
in conditions of multiple criteria. One of the most popular methods
in MCDM is the Technique for Order Preference by Similarity to the
Ideal Solution or TOPSIS . TOPSIS was initially presented by Hwang
and Yoon [38], Lai et al . [51], and Yoon and Hwang[94]. TOPSIS is
a multiple criteria method to identify solutions from a finite set of
alternatives based upon simultaneous minimization of distance from
an ideal point and maximization of distance from a nadir point. TOPSIS can incorporate relative weights of criterion importance. TOPSIS
is attractive in that limited subjective input is needed from decision
makers. The only subjective input needed is weights.
There are a number of specific procedures that can be used for developing weights and for distance measures. A number of distance
metrics can be applied. Traditional TOPSIS applied the Euclidean
norm (minimization of square root of the sum of squared distances)
Ch.6 A new approach to solve fuzzy games
124
to ideal and nadir solutions. A relative advantage of TOPSIS is the
ability to identify the best alternative quickly . TOPSIS has been
comparatively tested with a number of other multiattribute methods in [98] by zanakis et al . The other methods like Brans et al’s
method in [15] and Roy’s method in [78] primarily focused on generating weights with one method including a different way to combine weights and distance measures. TOPSIS was found to perform
almost as well as multiplicative additive weights and better than
analytic hierarchy process suggested by Saaty in [79] in matching a
base prediction model. When there were few criteria, TOPSIS had
proportionately more rank reversals. When there were many criteria, TOPSIS differed more from simple additive weight results, and
TOPSIS was also affected more with diverse sets of weights. TOPSIS
performed less accurately than AHP on both selecting the top ranked
alternative and in matching all ranks in this set of simulations.
TOPSIS has been applied to a number of applications by Hwang et
al in[38] and Yuan in [93], although it is not nearly as widely applied
as other multiattribute methods proposed by Zanakis et al in [98].
A variant of TOPSIS was used by Agrawal et al in [1] and [2] for
selection of grippers in flexible manufacturing. TOPSIS was applied
Ch.6 A new approach to solve fuzzy games
125
to financial investment in advanced manufacturing systems by Kim
et al in [47]. In other manufacturing applications, it has been used
in a case selecting manufacturing process by Chau and Parkan in
[21] and in an application selecting robotic processes by Parkan and
Wu in [70, 71, 72]. Neural network approaches to obtain weights
for TOPSIS have been proposed by Kim et al in [47]. TOPSIS has
also been used to compare company performances [27] and financial
ratio performance within a specific industry [31]. As opposed to the
original application of TOPSIS where the weight of the criteria and the
ratings of alternatives are known precisely, many real-life decision
problems are confronted with unquantifiable, incomplete and nonobtainable information that make precise judgment impossible as
discussed in [64] by Olcer and Odabasi . This is when fuzzy TOPSIS
comes into play where the criteria weights and alternative ratings
are given by linguistic variables, expressed by fuzzy numbers.
Many ranking methods based on the fuzzy concepts have been
proposed to solve the multiple criteria decision-making (MCDM)
problems, e.g. Ball and Korukolu [7], Bykzkan et al. [18], Chen [22],
Chou [23], Chou and Liang [24], Ding [28], Erturul and Karakaolu
[30], Lee and Chou [52], Liang [53], Tsaur et al. [36], Valls and
Ch.6 A new approach to solve fuzzy games
126
Vicenc [88], Wang et al. [92], Wang and Lee [91], etc. Currently,
Fuzzy TOPSIS presents a solution for decision makers when dealing
with real world data that are usually multi attributes and involves
a complex decision making process. This chapter contains two main
sections, namely,TOPSIS procedure and An ordering procedure using
Ranking functions. In the first section, TOPSIS procedure has been
used to find the best alternative when the payoffs are trapezoidal
fuzzy numbers. A numerical example has also been illustrated. In
the second section, a new ordering procedure has been proposed
using a general ranking function and the same is verified with some
four well-known ranking functions. The method has been used for
both zero sum and non zero sum fuzzy games.
6.2
6.2.1
TOPSIS procedure1
Algorithm
Step 1: For ã ≈ (a1 , a2 , a3 , a4 ) be a trapezoidal fuzzy number such
that its ranking function R(ã) =
a1 +a2 +a3 +a4
4
.
N
Step 2:Normalize each fuzzy numberÃi into ÃN
i such that Ãi ≈
( ak1 , ak2 , ak3 , ak4 ) where k denotes the maximum value of the universe of
1
The contents of this section form the substance of the paper published in the ”International Electronic Journal of Pure and Applied Mathematics”, Vol.4,No.1(2012), 53-58.
Ch.6 A new approach to solve fuzzy games
127
discourse. Let the normalized decision matrix be Ñ .
Step 3: Find the weight of each fuzzy Number as follows: Consider
each fuzzy number (a1 , a2 , a3 , a4 ) as the vertices of a trapezium given
by [(a1 , 0), (a2 , 0), (a3 , 0), (a4 , 0)] and find the area of each trapezium.
Divide the area of each trapezoidal fuzzy number by the maximum
area among all the areas to fix the weight of each fuzzy number [7].
Associate a linguistic variable vector to each weight vector to form,
the weighted matrix W̃ of Ñ .
Step 4:Calculate the weighted normalized decision matrix. The
weighted normalized value
[Ṽij ] ≈ Ṽ ≈ Ñ W̃ , i = 1, 2,
...,
m and j = 1, 2,
...,
n.
Step 5:Let the positive ideal and negative ideal solution be I˜+ ≈
(1, 1, 1, 1) and I˜− ≈ (0, 0, 0, 0)
˜−
Step 6:Calculate the separation measures d˜+
ij , dij using the distance
function defined as follows: If ã and b̃are any two trapezoidal fuzzy
numbers, the distance between them is given by
˜ b̃) = 1 M ax[| a1 − a2 | + | d1 − d2 |, | b1 − b2 | + | c1 − c2 |]
d(ã,
2
Step 7:Calculate the coefficient of closeness to the ideal solution.
The closeness coefficient to the alternative Ãij is defined as
ccij =
d˜−
ij
˜−
d˜+
ij + dij
Ch.6 A new approach to solve fuzzy games
128
Since d˜ij ≥ 0 clearly ccij ∈ [0, 1].
Step 8: According to closeness coefficient, the alternatives are
ranked in the descending order and the best alternative is the one
with the longest distance to the fuzzy positive ideal solution and
with the shortest distance to the fuzzy negative ideal solution.
6.2.2
Numerical Examples
TOPSIS for zero - sum game
Step 1: Consider the following fuzzy payoff matrix of a two person
zero-sum game.


(2, 4, 6, 8)
(0, 2, 4, 6)
(1, 5, 9, 13)
Ã ≈  (1, 5, 9, 13) (3, 7, 11, 15) (0.25, 0.75, 1.25, 1.75) 
(7, 9, 11, 13) (3, 5, 7, 9)
(0.5, 1.5, 2.5, 3.5)
Step 2:The corresponding normalized fuzzy decision matrix is given
by,


(0.13, 0.26, 0.4, 0.5) (0, 0.13, 0.26, 0.4) (0.06, 0.33, 0.6, 0.87)
Ñ ≈  (0.06, 0.3, 0.6, 0.87) (0.2, 0.47, 0.7, 1) (0.02, 0.05, 0.08, 0.12) 
(0.47, 0.6, 0.7, 0.87) (0.2, 0.3, 0.47, 0.6) (0.03, 0.1, 0.17, 0.2)
Step 3: The associated weighted fuzzy matrix using the linguistic
variables is given by


(0.15, 0.25, 0.35, 0.5) (0.15, 0.25, 0.35, 0.5) (0.2, 0.3, 0.5, 0.7)

W̃ ≈  (0.2, 0.3, 0.5, 0.7)
(0.75, 0.85, 0.9, 1)
(0, 0, 0, 0.1)
(0.15, 0.25, 0.35, 0.5)
(0.3, 0.5, 0.7, 0.9)
(0, 0, 0, 0.1)
Step 4:The weighted normalized matrix Ṽ is given by,
Ch.6 A new approach to solve fuzzy games
129

(0.06, 0.23, 0.42, 0.6) (0.186, 0.54, 0.94, 1.37) (0.08, 0.15, 0.2, 0.26)
Ṽ ≈  (0.12, 0.13, 0.51, 0.74) (0.21, 0.53, 0.86, 1.22) (0.04, 0.14, 0.28, 0.4) 
(0.25, 0.35, 0.47, 0.59) (0.35, 0.51, 0.73, 0.92) (0.21, 0.27, 0.32, 0.4)

Step 5:Let the positive ideal and negative ideal solution be I˜+ ≈
(1, 1, 1, 1) and I˜− ≈ (0, 0, 0, 0).
˜−
˜+
Step 6:Then the values of d˜+
ij and dij can be found using dij =
˜ ij , I˜+ ) and d˜− = d(ã
˜ ij , I˜− ). The values are as follows:
d(ã
ij
Table 6.1: dij values Zero-Sum Game
d˜ij
ã11
ã12
ã13
ã21
ã22
ã23
ã31
ã32
ã33
d˜+
ij
0.675
0.592
0.83
0.68
0.505
0.79
0.59
0.38
0.705
d˜−
ij
0.33
0.778
0.175
0.43
0.715
0.22
0.42
0.635
0.305
Step 7:The corresponding closeness coefficients are given below:
cc11 = 0.328 cc12 = 0.568 cc13 = 0.174
cc21 = 0.387 cc22 = 0.586 cc23 = 0.218
cc31 = 0.416 cc32 = 0.626 cc33 = 0.302
Step 8: Then the ranking order of the alternatives according to the
closeness coefficient is given by,
Ã32 ≺ Ã22 ≺ Ã12 ≺ Ã31 ≺ Ã21 ≺ Ã11 ≺ Ã33 ≺ Ã23 ≺ Ã13 .
Ch.6 A new approach to solve fuzzy games
130
Hence the best alternative for player I is Ã13 .Obviously, the best
alternative for player II is Ã32 .
TOPSIS for Non-zero sum games
Step 1: Consider the following fuzzy payoff matrices for Player I
and II.


(3, 5, 11, 13) (7, 10, 11, 16) (10, 11, 19, 20)
Ã ≈  (1, 10, 30, 39) (2, 4, 6, 12)
(0, 8, 14, 18) 
(−2, 1, 3, 6) (2, 4, 10, 12) (3, 6, 10, 17)


(1, 5, 6, 12) (0, 8, 11, 13) (7, 10, 15, 20)

B̃ ≈  (1, 2, 6, 7) (3, 6, 10, 17)
(1, 4, 5, 9)
(2, 4, 10, 12) (−1, 2, 4, 7) (10, 11, 19, 20s)
Step 2:The corresponding normalized fuzzy decision matrix is given
by,


(0.08, 0.13, 0.28, 0.33) (0.18, 0.26, 0.28, 0.41) (0.26, 0.28, 0.49, 0.51)
ÑA ≈  (0.03, 0.26, 0.77, 1)
(0.05, 0.1, 0.15, 0.31)
(0, 0.21, 0.36, 0.46) 
(−0.05, 0.03, 0.08, 0.15) (0.05, 0.1, 0.26, 0.31) (0.08, 0.15, 0.26, 0.44)


(0.05, 0.25, 0.3, 0.6)
(0, 0.4, 0.55, 0.65)
(0.35, 0.5, 0.75, 1)
ÑB ≈  (0.05, 0.1, 0.3, 0.35) (0.15, 0.3, 0.5, 0.85) (0.05, 0.2, 0.3, 0.45) 
(0.1, 0.2, 0.5, 0.6) (−0.05, 0.1, 0.2, 0.35) (0.5, 0.55, 0.95, 1)
Step 3: The associated weighted fuzzy matrix using the linguistic
variables is given by


(0.2, 0.3, 0.5, 0.7)
(0, 0.05, 0.1, 0.3)
(0.2, 0.3, 0.5, 0.7)
W̃A ≈  (0.75, 0.85, 0.9, 1) (0.15, 0.25, 0.35, 0.5) (0.3, 0.5, 0.7, 0.9) 
(0, 0.05, 0.1, 0.3)
(0.2, 0.3, 0.5, 0.7)
(0.2, 0.3, 0.5, 0.7)
Ch.6 A new approach to solve fuzzy games
131

(0.3, 0.5, 0.7, 0.9) (0.5, 0.75, 0.85, 0.9) (0.5, 0.75, 0.85, 0.9)
W̃B ≈  (0.2, 0.3, 0.5, 0.7) (0.5, 0.75, 0.85, 0.9) (0.2, 0.3, 0.5, 0.7) 
(0.5, 0.75, 0.85, 0.9) (0.2, 0.3, 0.5, 0.7) (0.5, 0.75, 0.85, 0.9)

Step 4:The weighted normalized matrix Ṽ is given by,
ṼA ≈ ÑA W̃


(0.22, 0.31, 0.42, 0.56) (0.17, 0.21, 0.33, 0.38) (0.25, 0.33, 0.5, 0.61)
≈  (0.05, 0.22, 0.5, 0.75) (0.02, 0.15, 0.29, 0.41) (0, 0.09, 0.15, 0.2) 
(0.03, 0.12, 0.29, 0.38)
(0.04, 0.1, 0.2, 0.3)
(0.04, 0.14, 0.3, 0.44)
ṼB ≈ ÑB W̃


(0.29, 0.7, 0.98, 1.39) (0.19, 0.7, 0.96, 1.36) (0.3, 0.73, 1.02, 1.48)
≈  (0.13, 0.34, 0.62, 0.91) (0.17, 0.39, 0.73, 1.09) (0.14, 0.35, 0.66, 0.96) 
(0.41, 0.58, 1.1, 1.26) (0.25, 0.46, 0.93, 1.14) (0.43, 0.61, 1.17, 1.35)
Step 5:Let the positive ideal and negative ideal solution be I˜+ ≈
(1, 1, 1, 1) and I˜− ≈ (0, 0, 0, 0).
˜−
˜+
Step 6:Then the values of d˜+
ij and dij can be found using dij =
˜ ij , I˜+ ) and d˜− = d(ã
˜ ij , I˜− ). The values are given below in the
d(ã
ij
following tables.
Table 6.2: dij Values for Player A
d˜ij
ã11
ã12
ã13
ã21
ã22
ã23
ã31
ã32
ã33
d˜+
ij 0.634 0.729 0.588 0.638 0.788 0.585 0.798 0.851 0.783
d˜−
ij 0.388 0.279 0.428 0.401 0.218 0.424 0.208 0.171 0.239
Ch.6 A new approach to solve fuzzy games
132
Table 6.3: dij Values for Player B
d˜ij
ã11
ã12
ã13
ã21
ã22
ã23
ã31
ã32
ã33
d˜+
ij 0.164 0.226 0.123 0.523 0.444 0.493 0.164 0.306 0.111
d˜−
ij 0.839 0.828 0.888
0.52
0.631
0.55
0.836 0.694 0.889
Step 7:The corresponding closeness coefficients are given below:


0.386 0.277 0.421
˜ A ≈  0.386 0.216 0.42 
CC
0.207 0.167 0.233


0.836 0.786 0.878
˜ B ≈  0.499 0.587 0.528 
CC
0.836 0.694 0.889
Step 8: Then the ranking order of the alternatives for Player I
according to the closeness coefficient is given by, Ã32 ≺ Ã31 ≺ Ã22 ≺
Ã33 ≺ Ã12 ≺ Ã11 ≺ Ã21 ≺ Ã23 ≺ Ã13 .Hence the best alternative
for player I is Ã13 . The ordering of alternatives for Player II is,
B̃21 ≺ B̃23 ≺ B̃23 ≺ B̃32 ≺ B̃12 ≺ B̃31 ≺ B̃11 ≺ B̃13 ≺ B̃33 . The best
alternative for player II is B̃33 .
Ch.6 A new approach to solve fuzzy games
133
Table 6.4: Linguistic terms and their values
6.3
6.3.1
Linguistic term
Value
Very Low
(0,0,0,0.1)
Low
(0,0.05,0.1,0.3)
Medium Low
(0.15,0.25,0.35,0.5)
Medium
(0.2,0.3,0.5,0.7)
Medium High
(0.3,0.5,0.7,0.9)
High
(0.5,0.75,0.85,0.9)
Very High
(0.75,0.85,0.9,1)
Preference Ordering of Alternatives using
ranking functions
Algorithm with a general ranking function
The general procedure to rank the alternatives in game theory is
developed and presented as:
Step 1: Define a trapezoidal fuzzy number.
Step 2: Let Ã ≈ (ãij ) be given fuzzy payoff matrix.
Step 3: Let R(ã, αk ) be any ranking function of trapezoidal fuzzy
Ch.6 A new approach to solve fuzzy games
134
numbers, where αk = nk , k = 0, 1, 2, . . . , n
Step 4: Find Rk (ãij ) for each ãij and for each k.
Pn
1
k
k
Step 5: Define D(ãij , ãmn ) = | n+1
k=0 [R (ãij ) − R (ãij )]|.
Step 6: Let Ũ be the universal trapezoidal fuzzy number.
Step 7: Calculate the coefficient of importance(COI) of each alternative with Ũ as follows:
COI = D(ãij , Ũ ) ⊕
where d(ã, b̃) =
p
area of ãij
⊕ d(Ũ , ãij ),
area of Ũ
(x0 (a) − x0 (b))2 + (y0 (a) − y0 (b))2 and
1
a4 a3 − a1 a2
x0 (ã) = [a1 + a2 + a3 + a4 −
],
3
a4 + a3 − a1 − a2
y0 (ã) = 31 [1 +
a3 −a2
a4 +a3 −a1 −a2 ].
Step 8: Write the matrix of COI’s and the alternatives will be
ranked in the descending order. In other words, the best alternative
is the one with minimum COI.
Ch.6 A new approach to solve fuzzy games
6.3.2
The proposed algorithm with Novel’s function
135
2
Step 1: Let ã ≈ (a1 , a2 , a3 , a4 ) be a trapezoidal fuzzy number with
membership function

x−a1


a2 −a1 ,


1,
µÃ (x) ≈ x−a
3


a4 −a3 ,



0,
if a1 ≤ x ≤ a2 ;
if a2 ≤ x ≤ a3 ;
if a3 ≤ x ≤ a4 ;
otherwise.
Step 2: Let Ã ≈ (ãij ) be given fuzzy payoff matrix. Then for
each alternative ãij find ãkij , k = 0, 1, . . . , n where ãkij ≈ {x ∈ < |
k
, rijk ] is the αkth level set of ãij where
µ̃Ãi (x) ≥ αk }. Clearly,ãkij ≈ [lij
k
= inf {x ∈ < | µ̃Ãi (x) ≥ αk } and rijk = sup{x ∈ < | µ̃Ãi (x) ≥
lij
αk }, αk = nk , k = 0, 1, 2, . . . , n
Step 3: Calculate xik = 12 (lik + rik ), for each ãij .
Pn
1
Step 4: Define D(ãij , ãmn ) ≈ | n+1
k=0 (xik − xjk )|.
Step 5: Let Ũ be the universal trapezoidal fuzzy number.
Step 6: Calculate the coefficient of importance(COI) of each alternative with Ũ as follows:
COI = D(ãij , Ũ ) ⊕
2
area of ãij
⊕ d(Ũ , ãij ),
area of Ũ
The contents of this section form the substance of the paper published in the Proceedings of the International Conference on Mathematics in Engineerng and Business
Management, Stella Maris College, Chennai,March 9-10,2012.
Ch.6 A new approach to solve fuzzy games
where d(ã, b̃) =
136
p
2
(x0 (a) − x0 (b))2 + (y0 (a) − y0 (b))2 and
a4 a3 − a1 a2
1
x0 (ã) = [a1 + a2 + a3 + a4 −
],
3
a4 + a3 − a1 − a2
y0 (ã) = 31 [1 +
a3 −a2
a4 +a3 −a1 −a2 ].
Step 7: Write the matrix of COI’s and the alternatives will be
ranked in the descending order. In other words, the best alternative
is the one with minimum COI.
6.3.3
The proposed algorithm with Compos and Munoz
Ranking function
Step 1: Let ã ≈ (a1 , a2 , a3 , a4 ) be a trapezoidal fuzzy number with
membership function

x−a1

,

a
2 −a1


1,
µã (x) ≈ x−a
1


a2 −a1 ,



0,
if a1 ≤ x ≤ a2 ;
if a2 ≤ x ≤ a3 ;
if a3 ≤ x ≤ a4 ;
otherwise.
Step 2: Let Ã ≈ (ãij ) be given fuzzy payoff matrix.
Step 3: Calculate
CMijk
=
a2ij
+
λk [(a3ij
−
a2ij )
(a2ij − a1ij )
(a4ij − a3ij ) + (a2ij − a1ij )
]−
+
2
2
for each ãij , where λk = nk , k = 0, 1, 2, . . . , n.
Pn
1
k
k
Step 4: Define D(ãij , ãmn ) ≈ | n+1
k=0 (CMij − CMmn )|.
Ch.6 A new approach to solve fuzzy games
137
Step 5: Let Ũ be the universal trapezoidal fuzzy number.
Step 6: Calculate the coefficient of importance(COI) of each alternative with Ũ as follows:
COI = D(ãij , Ũ ) ⊕
area of ãij
⊕ d(Ũ , ãij ),
area of Ũ
p
where d(ã, b̃) = 2 (x0 (a) − x0 (b))2 + (y0 (a) − y0 (b))2 and
1
a4 a3 − a1 a2
],
x0 (ã) = [a1 + a2 + a3 + a4 −
3
a4 + a3 − a1 − a2
y0 (ã) = 31 [1 +
a3 −a2
a4 +a3 −a1 −a2 ].
Step 7: Write the matrix of COI’s and the alternatives will be
ranked in the descending order. In other words, the best alternative
is the one with minimum COI.
6.3.4
The proposed algorithm with Abbasbandy’s Magnitude function
Step 1: The trapezoidal fuzzy number ũ ≈ (x0 , y0 , σ, β), with two
defuzzifiers x0 , y0 and left fuzziness σ > 0 and right fuzziness β > 0
is a fuzzy set where the membership function is as follows:

i


σ (x − x0 + σ), if x0 − σ ≤ x ≤ x0 ;


1,
if x ∈ [x0 , y0 ];
ũ(x) ≈ 1


β (y0 − x + β), if y0 ≤ x ≤ y0 + β;



0,
otherwise.
Ch.6 A new approach to solve fuzzy games
138
and its parametric form is, u(r) = (x0 − σ + σr) and ū(r) = (y0 +
¯
β + βr).
Step 2: Let Ã ≈ (ãij ) be given fuzzy payoff matrix.
Step 3: Calculate
Z 1
1
M agijk (ãij ) ≈ ( (aij (rk ) + āij (rk ) + x0 + y0 )f (τ ) dτ ),
2 0 ¯
for each ãij , where rk = nk , k = 0, 1, 2, . . . , n.
Pn
1
k
k
Step 4: Define D(ãij , ãmn ) ≈ | n+1
k=0 (M agij − M agmn )|.
Step 5: Let Ũ be the universal trapezoidal fuzzy number.
Step 6: Calculate the coefficient of importance(COI) of each alternative with Ũ as follows:
COI = D(ãij , Ũ ) ⊕
where d(ã, b̃) =
area of ãij
⊕ d(Ũ , ãij ),
area of Ũ
p
2
(x0 (a) − x0 (b))2 + (y0 (a) − y0 (b))2 and
1
a4 a3 − a1 a2
x0 (ã) = [a1 + a2 + a3 + a4 −
],
3
a4 + a3 − a1 − a2
y0 (ã) = 31 [1 +
a3 −a2
a4 +a3 −a1 −a2 ].
Step 7: Write the matrix of COI’s and the alternatives will be
ranked in the descending order. In other words, the best alternative
is the one with minimum COI.
Ch.6 A new approach to solve fuzzy games
139
The proposed algorithm with Liou and Wang’s Ranking function
Step 1: Let ã ≈ (a1 , a2 , a3 , a4 ) be a trapezoidal fuzzy number with
membership function


µLÃ (x), if a1 ≤ x ≤ a2 ;



1,
if a2 ≤ x ≤ a3 ;
µÃ (x) ≈

µR
(x), if a3 ≤ x ≤ a4 ;


Ã


0,
otherwise.
4
and µR
(x) ≈ ax−a
.
Ã
3 −a4
R1
R1
Step 2: Define IL (ã) ≈ 0 gãL (y) dy and IR (ã) ≈ 0 gãR (y) dy where
x−a1
a2 −a1
where µLÃ (x) ≈
gãL (y), gãR (y) are the inverse functions of µLã (x)andµR
ã (x) respectively.
Step 3: Calculate
ITk (ãij ) ≈ αk IR (ãij ) + (1 − αk )IL (ãij ),
for each ãij , where αk = nk , k = 0, 1, 2, . . . , n.
Pn
1
k
k
Step 4: Define D(ãij , ãmn ) ≈ | n+1
k=0 (IT (ãij ) − IT (ãmn ))|..
Step 5: Let Ũ be the universal trapezoidal fuzzy number.
Step 6: Calculate the coefficient of importance(COI) of each alternative with Ũ as follows:
COI = D(ãij , Ũ ) ⊕
where d(ã, b̃) =
area of ãij
⊕ d(Ũ , ãij ),
area of Ũ
p
2
(x0 (a) − x0 (b))2 + (y0 (a) − y0 (b))2 and
1
a4 a3 − a1 a2
x0 (ã) = [a1 + a2 + a3 + a4 −
],
3
a4 + a3 − a1 − a2
Ch.6 A new approach to solve fuzzy games
y0 (ã) = 31 [1 +
140
a3 −a2
a4 +a3 −a1 −a2 ].
Step 7: Write the matrix of COI’s and the alternatives will be
ranked in the descending order. In other words, the best alternative
is the one with minimum COI.
6.3.5
Numerical Example
Game description:
Two firms are competing for business whatever firm A gains firm B
loses. The table shows advertising strategies of both firms and the
utilities to firm A for various market shares in percentage (assuming
this a zero sum game).
Firm B
P ress
Radio
TV
P ress (1, 3, 3.5, 4.5) (6, 6.7, 6.8, 7) (5, 8, 8.1, 9)
F irmA
Radio (3, 3.1, 5.1, 6) (4, 5, 7, 9)
(1, 2, 3, 4)
TV
(2, 3, 8, 9) (0.1, 2, 3.1, 4) (8, 9, 12, 13)
6.3.6
Solution Procedure with Novel Ranking Function
Step 1: Consider the following fuzzy payoff matrix of the given two
person zero-sum game.


(1, 3, 3.5, 4.5) (6, 6.7, 6.8, 7) (5, 8, 8.1, 9)
Ã ≈  (3, 3.1, 5.1, 6) (4, 5, 7, 9)
(1, 2, 3, 4) 
(2, 3, 8, 9) (0.1, 2, 3.1, 4) (8, 9, 12, 13)
Ch.6 A new approach to solve fuzzy games
141
Step 2:Calculate ãkij , k = 0, 1, . . . , n.
Step 3: Calculate xkij for each ãij .
Step 4:Here Ũ ≈ (0.1, 1, 12, 13).
Step 5:The associated matrix of COIs is given below:


7.282 0.299 2.020
COI(Ã) ≈  4.647 0.815 8.219 
2.553 8.708 8.284
Step 6: The preference order in descending order is,
Ã32 ≺ Ã33 ≺ Ã23 ≺ Ã11 ≺ Ã21 ≺ Ã31 ≺ Ã13 ≺ Ã22 ≺ Ã12 .
Hence by this method, the best alternative for player I is Ã12 and so
the best alternative for player II is Ã32 .
4
3
2
1
0
k
4
0.1
13
8
3.02 4.2 1.2 2.2 0.48
6.2
3.04 4.4 1.4 2.4 0.86
8.4
Table 6.5: The interval approximations of alternatives
rij 4.1 6.92 8.64 5.64 8.2 3.6 8.6 3.73 12.6
lij 1.8 6.28
lij 1.6 6.21 5.9 3.03 4.3 1.3 2.3 0.67 8.3
rij 4.2 6.94 8.73 5.73 8.4 3.7 8.9 3.82 12.7
rij 4.3 6.96 8.82 5.52 8.6 3.8 8.8 3.91 12.8
5.6
8.2
9
2
ãk33
lij 1.4 6.14
4
1
ãk32
13
9
4
ãk22 ãk23 ãk31
rij 4.4 6.97 8.91 5.91 8.8 3.9 8.9 3.91
6
3
ãk21
8.1
5.3
9
5
ãk13
3.01 4.1 1.1 2.1 0.29
7
rij 4.5
lij 1.2 6.07
6
ãk12
1
lij
ãk11
Ch.6 A new approach to solve fuzzy games
142
10
9
8
7
6
5
k
4
rij
6.9
6.35
ãk12
6.8
ãk22 ãk23 ãk31
ãk32
8.5
ãk33
8.6
3.5 8.5 3.64 12.5
3.06 4.6 1.6 2.6 1.24
8
3.05 4.5 1.5 2.5 1.05
ãk21
8.55 5.55
6.5
ãk13
7.1
3.07 4.7 1.7 2.7 1.43
8.7
7.4
3.08 4.8 1.8 2.8 1.62
8.8
7.7
3.09 4.9 1.9 2.9 1.81
8.9
6.7
6.8
3
rij 3.5
lij
8.1
8
5.1
3.1
7
5
3
2
8
3
3.19
2
12
9
rij 3.6 6.82 8.19 5.19 7.2 3.1 8.1 3.28 12.1
lij 2.8 6.63
rij 3.7 6.84 8.28 5.28 7.4 3.2 8.2 3.37 12.2
lij 2.6 6.56
rij 3.8 6.86 8.37 5.37 7.6 3.3 8.3 3.46 12.3
lij 2.4 6.49
rij 3.9 6.88 8.46 5.46 7.8 3.4 8.4 3.55 12.4
lij 2.2 6.42
2
lij
ãk11
Ch.6 A new approach to solve fuzzy games
143
6.58
6.60
6.68
6.7
6.73
6.75
2.80
2.85
2.9
2.95
3
3.05
3.1
3.15
3.2
1
2
3
4
5
6
7
8
9
10 3.25
6.65
6.63
6.55
6.53
6.5
2.75
0
xk12
xk11
k
8.05
7.95
7.84
7.74
7.63
7.53
7.42
7.32
7.21
7.11
7
xk13
4.1
4.14
4.18
4.22
4.26
4.3
4.34
4.38
4.42
4.46
4.5
xk21
6
6.05
6.1
6.15
6.2
6.25
6.3
6.35
6.4
6.45
6.5
xk22
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
xk23
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
xk31
Table 6.6: Mid point values
2.55
2.5
2.45
2.4
2.35
2.3
2.25
2.2
2.15
2.1
2.05
xk32
10.50
10.50
10.50
10.50
10.50
10.50
10.50
10.50
10.50
10.50
10.50
xk33
Ch.6 A new approach to solve fuzzy games
144
Ch.6 A new approach to solve fuzzy games
6.3.7
145
Solution Procedure with Compos and Munoz Ranking function
Step 1:Consider the fuzzy payoff matrix of the given two person
zero-sum game.


(1, 3, 3.5, 4.5) (6, 6.7, 6.8, 7) (5, 8, 8.1, 9)
Ã ≈  (3, 3.1, 5.1, 6) (4, 5, 7, 9)
(1, 2, 3, 4) 
(2, 3, 8, 9) (0.1, 2, 3.1, 4) (8, 9, 12, 13)
Step 2:Calculate C̃Mijk , k = 0, 1, . . . , n. Take n = 10 for convenience. Step 3:Here Ũ ≈ (0.1, 1, 12, 13).
Step 4:The associated matrix of COIs’ is given below:


7.282 0.285 2.018
COI(Ã) ≈  4.647 0.815 8.219 
2.553 8.708 8.284
Step 5: The preference order in descending order is, Ã32 ≺ Ã33 ≺
Ã23 ≺ Ã11 ≺ Ã21 ≺ Ã31 ≺ Ã13 ≺ Ã22 ≺ Ã12 .
Hence by using compos and Munoz’s Ranking function, the best
alternative for player I is Ã12 . Obviously, the best alternative for
player II is Ã32 .
6.405
6.515
6.570
2.200
2.400
2.600
2.800
3.000
3.200
3.400
3.600
3.800
4.000
1
2
3
4
5
6
7
8
9
10
8.550
8.345
8.140
7.935
7.730
7.525
7.320
7.115
6.910
6.705
6.500
k
C̃M13
5.550
5.300
5.050
4.800
4.550
4.300
4.050
3.800
3.550
3.300
3.050
k
C̃M21
8.000
7.650
7.300
6.950
6.600
6.250
5.900
5.550
5.200
4.850
4.500
k
C̃M22
3.500
3.300
3.100
2.900
2.700
2.500
2.300
2.100
1.900
1.700
1.500
k
C̃M23
8.500
7.900
7.300
6.700
6.100
5.500
4.900
4.300
3.700
3.100
2.500
k
C̃M31
3.550
3.300
3.050
2.800
2.550
2.300
2.050
1.800
1.550
1.300
1.050
k
C̃M32
Table 6.7: Ranking values in Compos and Munoz method
6.900
6.845
6.790
6.735
6.680
6.625
6.460
6.350
2.000
0
k
C̃M12
k
C̃M11
k
12.500
12.100
11.700
11.300
10.900
10.500
10.100
9.700
9.300
8.900
8.500
k
C̃M33
Ch.6 A new approach to solve fuzzy games
146
Ch.6 A new approach to solve fuzzy games
6.3.8
147
Solution Procedure with Abbasbandy’s Ranking function
Step 1: Consider the fuzzy payoff matrix of the given two person
zero-sum game.


(1, 3, 3.5, 4.5) (6, 6.7, 6.8, 7) (5, 8, 8.1, 9)
Ã ≈  (3, 3.1, 5.1, 6) (4, 5, 7, 9)
(1, 2, 3, 4) 
(2, 3, 8, 9) (0.1, 2, 3.1, 4) (8, 9, 12, 13)
Step 2:Calculate M̃ agijk , k = 0, 1, . . . , n. Take n = 10 for convenience.
Step 3:Here Ũ ≈ (0.1, 1, 12, 13).
Step 4:The associated matrix of COIs’ is given below:


7.145 0.36 2.293
COI(Ã) ≈  4.835 0.928 8.207 
2.541 8.571 8.296
Step 5: The preference order in descending order is, Ã32 ≺ Ã33 ≺
Ã23 ≺ Ã11 ≺ Ã21 ≺ Ã31 ≺ Ã13 ≺ Ã22 ≺ Ã12 .
Hence by using Abbasbandy’s Ranking function, the best alternative
for player I is Ã12 . Obviously, the best alternative for player II is
Ã32 .
6.638
6.663
6.675
3.025
3.050
3.075
3.100
3.125
3.150
3.175
3.200
3.225
3.250
1
2
3
4
5
6
7
8
9
10
8.050
7.998
7.945
7.893
7.840
7.788
7.735
7.683
7.630
7.578
7.525
k
M̃ ag13
4.100
4.120
4.140
4.160
4.180
4.200
4.220
4.240
4.260
4.280
4.300
k
M̃ ag21
6.000
6.025
6.050
6.075
6.100
6.125
6.150
6.175
6.200
6.225
6.250
k
M̃ ag22
2.500
2.500
2.500
2.500
2.500
2.500
2.500
2.500
2.500
2.500
2.500
k
M̃ ag23
5.500
5.500
5.500
5.500
5.500
5.500
5.500
5.500
5.500
5.500
5.500
k
M̃ ag31
Table 6.8: Ranking values in Abbasbandy method
6.750
6.738
6.725
6.713
6.700
6.688
6.650
6.625
3.000
0
k
M̃ ag12
k
M̃ ag11
k
2.550
2.525
2.500
2.475
2.450
2.425
2.400
2.375
2.350
2.325
2.300
k
M̃ ag32
10.500
10.500
10.500
10.500
10.500
10.500
10.500
10.500
10.500
10.500
10.500
k
M̃ ag33
Ch.6 A new approach to solve fuzzy games
148
Ch.6 A new approach to solve fuzzy games
6.3.9
149
Solution Procedure with Liou and Wang’s Ranking
function
Step 1: Consider the fuzzy payoff matrix of the given two person
zero-sum game.


(1, 3, 3.5, 4.5) (6, 6.7, 6.8, 7) (5, 8, 8.1, 9)
Ã ≈  (3, 3.1, 5.1, 6) (4, 5, 7, 9)
(1, 2, 3, 4) 
(2, 3, 8, 9) (0.1, 2, 3.1, 4) (8, 9, 12, 13)
Step 2:Calculate ITk (ãkij ), k = 0, 1, . . . , n. Take n = 10 for convenience.
Step 3:Here Ũ ≈ (0.1, 1, 12, 13).
Step 4:The associated matrix of COIs’ is given below:


7.282 0.285 2.018
COI(Ã) ≈  4.647 0.815 8.219 
2.553 8.708 8.284
Step 5: The preference order in descending order is, Ã32 ≺ Ã33 ≺
Ã23 ≺ Ã11 ≺ Ã21 ≺ Ã31 ≺ Ã13 ≺ Ã22 ≺ Ã12 .
Hence by using Liou and Wang’s Ranking function, the best alternative for player I is Ã12 . Obviously, the best alternative for player
II is Ã32 .
6.350
6.405
6.515
6.570
2.000
2.200
2.400
2.600
2.800
3.000
3.200
3.400
3.600
3.800
4.000
0
1
2
3
4
5
6
7
8
9
10
8.550
8.345
8.140
7.935
7.730
7.525
7.320
7.115
6.910
6.705
6.500
5.550
5.300
5.050
4.800
4.550
4.300
4.050
3.800
3.550
3.300
3.050
I˜Tk (ãk13 ) ITk (ãk21 )
8.000
7.650
7.300
6.950
6.600
6.250
5.900
5.550
5.200
4.850
4.500
ITk (ãk22 )
3.500
3.300
3.100
2.900
2.700
2.500
2.300
2.100
1.900
1.700
1.500
8.500
7.900
7.300
6.700
6.100
5.500
4.900
4.300
3.700
3.100
2.500
ITk (ãk23 ) ITk (ãk31 )
3.550
3.300
3.050
2.800
2.550
2.300
2.050
1.800
1.550
1.300
1.050
ITk (ãk32 )
Table 6.9: Ranking values in Liou and Wang’s Method
6.900
6.845
6.790
6.735
6.680
6.625
6.460
ITk (ãk11 ) ITk (ãk12 )
k
12.500
12.100
11.700
11.300
10.900
10.500
10.100
9.700
9.300
8.900
8.500
ITk (ãk33 )
Ch.6 A new approach to solve fuzzy games
150
Chapter 7
Fuzzy Dominance Procedure1
Abstract
When considering the fuzzy entities a major problem to be
faced is its comparison technique. Based on a new comparison technique defined in this work an attempt has been
made to reduce the size of a fuzzy payoff matrix by eliminating the Fuzzy strategies which will not give a best payoff. This will be more useful when both the players have
more number of alternatives.
1
The contents of this Chapter form the substance of the paper published in the ”International Journal of Algorithms, Computing and Mathematics”, ISSN 09743367,Vol.3,No.3 (August 2010),pp.17-20.
151
Ch.7 Fuzzy Dominance Procedure
7.1
152
Introduction
In game theory, dominant strategy (commonly called simply dominance) occurs when one strategy is better than another strategy for
one player, no matter how that player’s opponents may play. Many
simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse
than another strategy for one player, depending on how the player’s
opponents may play.
Terminology When a player tries to choose the ”best” strategy
among a multitude of options, that player may compare two strategies A and B to see which one is better. The result of the comparison
is one of:
• B dominates A: choosing B always gives as good as or a better
outcome than choosing A. There are 2 possibilities:
– B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other
player(s) do.
– B weakly dominates A: There is at least one set of opponents’ action for which B is superior, and all other sets
Ch.7 Fuzzy Dominance Procedure
153
of opponents’ actions give B at least the same payoff as A.
• B and A are intransitive: B neither dominates, nor is dominated by, A. Choosing A is better in some cases, while choosing
B is better in other cases, depending on exactly how the opponent chooses to play.
• B is dominated by A: choosing B never gives a better outcome
than choosing A, no matter what the other player(s) do. There
are 2 possibilities:
– B is weakly dominated by A: There is at least one set of
opponents’ actions for which B gives a worse outcome than
A, while all other sets of opponents’ actions give A at least
the same payoff as B. (Strategy A weakly dominates B).
– B is strictly dominated by A: choosing B always gives a
worse outcome than choosing A, no matter what the other
player(s) do. (Strategy A strictly dominates B).
This notion can be generalized beyond the comparison of two
strategies.
• Strategy B is strictly dominant if strategy B strictly dominates
every other possible strategy.
Ch.7 Fuzzy Dominance Procedure
154
• Strategy B is weakly dominant if strategy B dominates all
other strategies, but some are only weakly dominated.
• Strategy B is strictly dominated if some other strategy exists
that strictly dominates B.
• Strategy B is weakly dominated if some other strategy exists
that weakly dominates B.
If a strictly dominant strategy exists for one player in a game, that
player will play that strategy in each of the game’s Nash equilibria.
If both players have a strictly dominant strategy, the game has only
one unique Nash equilibrium. However, that Nash equilibrium is
not necessarily Pareto optimal, meaning that there may be nonequilibrium outcomes of the game that would be better for both
players.
Strictly dominated strategies cannot be a part of a Nash equilibrium, and as such, it is irrational for any player to play them. On
the other hand, weakly dominated strategies may be part of Nash
equilibria.
A comparison technique is needed if we consider about fuzzy dominance. Once if we have an effective comparison technique then fuzzy
dominance rules can be derived from its crisp counter part. That has
Ch.7 Fuzzy Dominance Procedure
155
been done in this chapter. This chapter itself will conclude the study
after presenting a real life implementation of fuzzy game techniques.
7.2
Important Theorems
Theorem 7.2.1. Let K̃ be the payoff kernel of an mxn matrix game
whose value is Ṽ . Then a necessary and sufficient condition for
X̃ ∗ ∈ S̃1 to be optimal fuzzy strategies for player I is,
Ṽ 4 K̃(X̃ ∗ , Ỹ )f orallỸ ∈ S̃2 .
Similarly, a necessary and sufficient condition for Ỹ ∗ ∈ S̃2 to be
optimal fuzzy strategies for player II is,
K̃(X̃, Ỹ ∗ ) < Ṽ f orallX̃ ∈ S̃1 .
Proof:
If X̃ ∗ is an optimal fuzzy strategy for player I, then there exist Ỹ ∗
such that,
max K(X̃, Ỹ ∗ ) ≈ K(X̃ ∗ , Ỹ ∗ ) ≈ min K(X̃ ∗ , Ỹ )
X̃∈S̃1
∗
∗
∗
Ỹ ∈S̃2
∗
⇒K(X̃, Ỹ ) 4 K(X̃ , Ỹ ) 4 K(X̃ , Ỹ )f orallX̃ ∈ S̃1 , Ỹ ∈ S̃2
⇒K(X̃, Ỹ ∗ ) 4 Ṽ 4 K(X̃ ∗ , Ỹ )f orallX̃ ∈ S̃1 , Ỹ ∈ S̃2
∴ Ṽ 4 K(X̃ ∗ , Ỹ )f orallỸ ∈ S̃2
Ch.7 Fuzzy Dominance Procedure
156
Conversely, let Ṽ 4 K(X̃ ∗ , Ỹ ) for all Ỹ ∈ S̃2 .
Suppose that (X̃ 0 , Ỹ 0 ) be a solution to the fuzzy game. Then,
X̃ T Ã Ỹ 0 4 X̃ 0T Ã Ỹ 0 4 X̃ 0T Ã Ỹ
(7.1)
⇒Ṽ ≈ X̃ 0T Ã Ỹ 0 ≈ K̃(X̃ 0 , Ỹ 0 )
By assumption it is clear that,
Ṽ 4 K̃(X̃ ∗ , Ỹ 0 )
⇒K̃(X̃ 0 , Ỹ 0 ) 4 K̃(X̃ ∗ , Ỹ 0 )
(7.2)
Equation 7.1 gives
K̃(X̃, Ỹ 0 ) 4 K̃(X̃ 0 , Ỹ 0 )f orallX̃ ∈ S̃1
⇒K̃(X̃ ∗ , Ỹ 0 ) 4 K̃(X̃ 0 , Ỹ 0 )
(7.3)
Equation 7.2 and Equation 7.3 gives,
K̃(X̃ ∗ , Ỹ 0 ) ≈ K̃(X̃ 0 , Ỹ 0 ) ≈ Ṽ
⇒K̃(X̃, Ỹ 0 ) 4 K̃(X̃ 0 , Ỹ 0 ) ≈ K̃(X̃ ∗ , Ỹ 0 ) 4 K̃(X̃ ∗ , Ỹ )
⇒Ṽ 4 K̃(X̃ ∗ , Ỹ ) f or all Ỹ ∈ S̃2
∴X̃ ∗ is an optimal f uzzy strategy f or playerI.
Second part is similar.
(7.4)
Ch.7 Fuzzy Dominance Procedure
157
Theorem 7.2.2. Let Ẽ be the payoff kernel of an mxn fuzzy matrix
game whose value is Ṽ . Then a necessary and sufficient condition
for X ∗ ∈ S̃1 to be an optimal fuzzy strategy for player I is,
Ṽ 4 Ẽ(X̃ ∗ , β̃j ), j = 1, . . . , n.
Similarly, a necessary and sufficient condition for Y ∗ ∈ S̃2 to be an
optimal fuzzy strategy for player II is,
Ẽ(α̃i , Ỹ ∗ ) 4 Ṽ , j = 1, . . . , n.
Proof:
If X̃ ∗ is an optimal fuzzy strategy for player I, then by theorem(7.2.1),
Ṽ 4 K̃(X̃ ∗ , Ỹ )f orallỸ ∈ S̃2
⇒Ṽ 4 K̃(X̃ ∗ , β̃j ), j = 1, 2, . . . , n.
Conversely, let Ṽ 4 K̃(X̃ ∗ , β̃j ), j = 1, 2, . . . , n.
Every Ỹ ∈ S̃2 can be written as,
Ỹ ≈
n
X
j=1
ỹj β̃j and
n
X
j=1
ỹj ≈ I˜
(7.5)
Ch.7 Fuzzy Dominance Procedure
158
Hence we have,
K̃(X̃ ∗ , Ỹ ) ≈ X̃ ∗ Ã Ỹ
m X
n
X
≈
x̃∗i ãij ỹj
<
≈
≈
<
i=1 j=1
n
m
X
X
j=1
i=1
n
X
∗T
X̃
j=1
n
X
j=1
n
X
!
x̃∗i ãij I˜ ỹj
Ã β̃j ỹj
K̃(X̃ ∗ , β̃j ) ỹj
Ṽ ỹj
j=1
≈ Ṽ n
X
ỹj
j=1
≈ Ṽ I˜
≈ Ṽ
K̃(X̃ ∗ , Ỹ ) < Ṽ f orallỸ ∈ S̃2
Hence, by theorem 7.2.1,
(7.6)
X̃ ∗ is the optimal fuzzy strategy for
player I. Second part is similar.
Theorem 7.2.3. A necessary and sufficient condition forṼ to be
the fuzzy value of a fuzzy matrix game and X̃ ∗ ∈ S1 andỸ ∗ ∈ S2 to
be the optimal fuzzy strategies for player I and player II respectively
Ch.7 Fuzzy Dominance Procedure
159
is,
Ẽ(α̃i , Ỹ ∗ ) 4 Ṽ 4 Ẽ(X̃ ∗ , β̃j ), i = 1, . . . , m, j = 1, . . . , n.
Proof: If X̃ ∗ ∈ S̃1 and Ỹ ∗ ∈ S̃2 are the optimal fuzzy strategy for
player I and player II, then by theorem(7.2.2), we have,
Ẽ(α̃i , Ỹ ∗ ) 4 Ṽ 4 Ẽ(X̃ ∗ , β̃j ), i = 1, . . . , m, j = 1, . . . , n.
Conversely, suppose that the condition is satisfied. Then by theorem(7.2.2),
K̃(X̃, Ỹ ∗ ) ≈ X̃ Ã Ỹ ∗
n
m X
X
x̃i ãij ỹj∗
≈
≈
≈
≈
4
i=1 j=1
n
X
m
X
x̃i i=1
n
X
i=1
n
X
i=1
n
X
!
I˜ ãij ỹj
j=1
x̃i α̃i Ã ỹj
x̃i K̃(α̃i , Ỹ ∗ )
x̃i Ṽ
i=1
≈ I˜ Ṽ
4 Ṽ
K̃(X̃, Ỹ ∗ ) 4 Ṽ f orallX̃ ∈ S̃1
(7.7)
Ch.7 Fuzzy Dominance Procedure
160
Then, by Equations 7.6 and 7.7,
K̃(X̃ ∗ , Ỹ ∗ ) 4 Ṽ 4 K̃(X̃ ∗ , Ỹ ∗ )
⇒Ṽ ≈ K̃(X̃ ∗ , Ỹ ∗ )
⇒K̃(X̃, Ỹ ∗ ) 4 K̃(X̃ ∗ , Ỹ ∗ ) 4 K̃(X̃ ∗ , Ỹ )
(7.8)
∴ X̃ ∗ and Ỹ ∗ are the optimal fuzzy strategies for player I and player
II respectively.
Theorem 7.2.4. Let Ṽ be the value of a fuzzy matrix game. Then
if Ỹ ∗ ≈ (ỹ1∗ , . . . , ỹn∗ )T is an optimal fuzzy strategy for player II with
ỹj 0̃, every optimal fuzzy strategy X̃ ∗ for player I must have the
property,
∗
K̃(X̃ , β̃j ) ≈
m
X
x̃∗i ãij ≈ Ṽ
i=1
Similarly, if the optimal fuzzy strategy X̃ ∗ has x̃i 0̃ then every
optimal fuzzy strategy Ỹ ∗ must be such that,
∗
K̃(α̃i , Ỹ ) ≈
n
X
ãij ỹj∗ ≈ Ṽ
j=1
Proof:
If X̃ ∗ is the optimal fuzzy strategy for player I then by theorem
(7.2.2) we have,
Ch.7 Fuzzy Dominance Procedure
161
K̃(X̃ ∗ , β̃j ) < Ṽ f orallk = 1, . . . , n
Suppose that, K̃(X̃ ∗ , β̃j ) Ṽ K̃(X̃ ∗ , β̃j ) ỹj∗ Ṽ ỹj∗
K̃(X̃ ∗ , Ỹ ∗ ) ≈ X̃ ∗ Ã Ỹ ∗
m X
n
X
x̃∗i ãij ỹj∗
≈
<
i=1 k=1
n
m
X
X
i=1
k=1
≈
≈
n X
k=1
n
X
!
x̃∗i ãij I˜ ỹj∗
X̃
∗T
Ã β̃j ỹj∗
K̃(X̃ ∗ , β̃j ) ỹj∗
k=1
∗
≈ K̃(X̃ , β̃j ) ỹj∗
⊕
X
K̃(X̃ ∗ , β̃k ) ỹk∗
k6=j
Ṽ ỹj∗ ⊕
X
Ṽ ỹk∗
k6=j
≈ Ṽ n
X
ỹj∗
j=1
≈ Ṽ I˜
≈ Ṽ
i.e, K̃(X̃ ∗ , Ỹ ∗ ) Ṽ
which is a contradiction since Ṽ is the fuzzy value of the game.
Hence, K̃(X̃ ∗ , β̃j ) ≈ Ṽ for optimal X̃ ∗ .
Second part is similar.
Ch.7 Fuzzy Dominance Procedure
162
Definition 7.2.1. An n-tuole (ã1 , . . . , ãn ) ∈ F n (R) dominates (or
strictly dominates) the n-tuple (b̃1 , . . . , b̃n )inF n (R) if ãi < b̃i or ãi b̃i for all i=1,. . . ,n.
Theorem 7.2.5. If it h row of a fuzzy payoff matrix Ã is strictly
dominated by a fuzzy convex combination of the other rows in the
fuzzy matrix Ã, then the deletion of the it h row of Ã does not change
the set of optimal fuzzy strategies for player I. Further if the it h
column of Ã strictly dominates some fuzzy convex combination of
the other columns, then the deletion of the it h column of Ã does not
change the set of optimal fuzzy strategies for player II.
Proof: By the hypothesis, there exist (x̃1 , . . . , x̃m )T ∈ S̃1 with x̃i ≈
0̃ such that,
ãij ≺
X
x̃k ãkj ≈
k6=i
n
X
x̃k ãkj f orj = 1, . . . , n.
k=1
Let Ṽ be the value of the game and Ỹ ∗ be any optimal fuzzy strategy
for player II. Then, it follows that
n
X
j=1
ãij ỹj∗
≺
n X
n
X
x̃k ãkj ỹj∗ ≈ K̃(X̃, Ỹ ) 4 Ṽ
j=1 k=1
n
X
j=1
ãij ỹj∗ ≺ Ṽ
Ch.7 Fuzzy Dominance Procedure
∗
i.e., K̃(α̃i , Ỹ ) ≈
163
n
X
ãij ỹj∗ ≺ Ṽ
j=1
Then by theorem 7.2.4, x̃i ≈ 0̃ for every optimal fuzzy strategy X̃ ∗
for player I.
Second part is similar.
Results: The following are the results obtained from previous theorems.
• If the it h row of a fuzzy payoff matrix strictly dominates the
j t h row of the fuzzy matrix then delete the j t h row from the
matrix.
• If the it h column of a fuzzy payoff matrix strictly dominates
the j t h column of the fuzzy matrix then delete it h column.
• If the it h row of a fuzzy payoff matrix is strictly dominated by
a fuzzy convex combination of other rows, delete the it h row.
• If the j t h column of a fuzzy payoff matrix strictly dominates
some fuzzy convex combination of the other columns, delete
the j t h column.
where maximizing player is the row player and minimizing player
is the column player.
Ch.7 Fuzzy Dominance Procedure
7.3
164
Numerical Example
In the following example the row player is the maximizing player
and the column player is the minimizing one. However, if the row
player is the minimizing player and vice versa we will use the same
set of rules for the transposed pay-off matrix.
Player II


(2, 20, 38, 56) (7, 17, 27, 37)
(8, 9, 10, 11)
(6, 8, 10, 12)
P layerI  (5, 10, 15, 20) (6, 13, 20, 27) (13, 14, 15, 16) (9, 10, 11, 12) 
(1, 9, 17, 25) (12, 19, 26, 33) (20, 22, 24, 26) (17, 19, 21, 23)
Rule 1: Here second row strictly dominated by third row and so
delete second row.
Rule 2: Here second and third columns strictly dominates the fourth
column and so delete second and third columns. Further reduction
is not possible and the reduced fuzzy matrix is,
(2, 20, 38, 56) (6, 8, 10, 12)
(1, 9, 17, 25) (17, 19, 21, 23)
This fuzzy payoff matrix will then be solved by any one of the proposed methods.
Ch.7 Conclusion
7.4
165
Conclusion
• Fuzzy set theory is not certainly a philosopher’s stone which
solves all the problems that was confronted today. But it has
a considerable potential for practical as well as mathematical
application.
• In this work, the primary focus is on Trapezoidal fuzzy numbers
and their arithmetics, FLPP’s with trapezoidal fuzzy numbers
and finally, its real life application as they comprises the basic
needs of this thesis.
• The secondary focus is on the literature of Fuzzy game theory
and its chronological developments until recently. The main
focus of this work is on Fuzzy Game theory and proposes some
solving techniques on it.
• Three types of Fuzzy games have been discussed namely, zero
sum Fuzzy games, Non-zero sum Bi-matrix fuzzy games and
Constrained fuzzy games.
Ch.7 Conclusion
166
• First of all, a two person zero sum fuzzy game has been discussed and a detailed study has been put on its solving techniques. In four of the proposed techniques, one is an approximate method which will give only an approximate solution and
all the other methods will give somewhat required accuracy.
• The second type is a non zero sum bi-matrix fuzzy game, whose
solving techniques have also been discussed.
• The third type somewhat more useful, is a constrained fuzzy
game with zero sum and nonzero sum categories.
• A different kind of approach called TOPSIS, has been examined
with trapezoidal fuzzy alternatives, which is originally an area
based ranking technique.
• A new algorithm has been developed by making use of ranking functions so as to order the alternatives. Different Ranking
functions have been used to prove the efficiency of the given
procedure. This method shows good results which are discriminative and close to the intuition.
• With the help of a new comparison technique that has been
proposed between two trapezoidal fuzzy numbers, the concept
Ch.7 Conclusion
167
of domination is being generated for trapezoidal fuzzy payoffs.
• In the study the researcher introduced an original unified approach by which a number of new and yet unpublished results
have been acquired. This approach to fuzzy games, in this
work is mathematical oriented as the author is a mathematician. However, there exist different approaches and different
area of implementation putting more stress on other aspects of
the subject.
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