Studies in Fuzziness and Soft Computing
Michael B. Gibilisco · Annie M. Gowen
Karen E. Albert · John N. Mordeson
Mark J. Wierman · Terry D. Clark
Fuzzy Social
Choice Theory
Studies in Fuzziness and Soft Computing
Volume 315
Series editors
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail: [email protected]
For further volumes:
http://www.springer.com/series/2941
About this Series
The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural
networks, evolutionary computation, probabilistic and evidential reasoning, multivalued logic, and related fields. The publications within “Studies in Fuzziness and
Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide
distribution. This permits a rapid and broad dissemination of research results.
Michael B. Gibilisco · Annie M. Gowen
Karen E. Albert · John N. Mordeson
Mark J. Wierman · Terry D. Clark
Fuzzy Social Choice Theory
ABC
Mark J. Wierman
Department of Computer Science
Creighton University
Omaha
Nebraska
USA
Michael B. Gibilisco
Rochester
New York
USA
Annie M. Gowen
Papillion
Nebraska
USA
Terry D. Clark
Department of Political Science
Creighton University
Omaha
Nebraska
USA
Karen E. Albert
Lincoln
Nebraska
USA
John N. Mordeson
Department of Mathematics
Creighton University
Omaha
Nebraska
USA
ISSN 1434-9922
ISBN 978-3-319-05175-8
DOI 10.1007/978-3-319-05176-5
ISSN 1860-0808 (electronic)
ISBN 978-3-319-05176-5 (eBook)
Springer Cham Heidelberg New York Dordrecht London
Library of Congress Control Number: 2014932428
c Springer International Publishing Switzerland 2014
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Springer is part of Springer Science+Business Media (www.springer.com)
Michael Gibilisco dedicates this book to his
parents whose moral, and, at times, financial
support, made the work possible. They have
always encouraged him and his research
throughout school and this project, and his
passion for learning began with them.
Preface
For almost a decade, three of the authors of this book (John N. Mordeson [Mathematics], Mark J. Wierman [Computer Science], and Terry D. Clark [Political Science]) have engaged in an extensive research agenda applying fuzzy set logic to
social choice theory. That collaboration has been rewarding on a number of dimensions. Among the most rewarding aspects has been the students who have joined
us in that collaboration. Michael Gibilisco, the primary author of this book, is one
of those students. Like Michael, many of our students have discovered the joys of
research and subsequently gone on to pursue the Ph.D. Even among those who have
not, the intellectual commitment and rigor that the effort has demanded has assisted
d virtually all of them in discovering their life’s vocation.
Of course, the discoveries that we have made along the way have been rewarding as well. While our research agenda has its genesis in the desire to apply formal
models to empirical problems, the theoretical work has necessarily consumed a substantial degree of our effort and attention. This book is in many ways a summary of
what we have discovered about theory. Nonetheless, at the conclusion of each of the
chapters that follow we make a conscious effort to discuss empirical applications.
The social choice issues that we address are those that one familiar with the research agenda would expect. We give consideration to the effects of applying fuzzy
logic to Arrow’s Impossibility Theorem, Black’s Median Voter Theorem, and the
Gibbard-Sattherthwaite Theorem. Along the way we consider varying definitions
of key concepts in social choice theory. As the chapters demonstrate, a fuzzy approach admits of a good deal more variation in these definitions than the customary
approach allows. It is therefore not surprising that many of the theorems no longer
hold under certain conditions. What is even more surprising, however, is how resilient the major social choice theorems are. While they no longer hold under certain
fuzzy definitions, they hold under most of them.
We admit that this is contrary to what we expected when we began our effort
almost a decade ago. At that time, it seemed to us that the problems that empiricists
were having with applying social choice theory to their work owed to the perverse
outcomes rooted in a mathematics that assumed too much precision in human thinking. The fuzzy approach intuitively seemed to offer a possible solution by modeling
VIII
Preface
less precision and clarity in human thinking on preferences and preference orders.
While this has turned out to be the case in a number of instances, thereby permitting
a marginal decrease in the estimation error on the part of fuzzy counterparts to familiar models in the comparative politics literature, the estimated outcome are still
not what we might like them to be. But we will hold that conversation for a subsequence volume on our empirical applications. In this volume, we focus on mostly
on our theoretical conclusions.
The volume’s primary author, Michael B. Gibilisco, is currently pursuing the
Ph.D. in political science at the University of Rochester. Michael wishes to acknowledge that his work benefitted from the faculty and students in the Fuzzy
Mathematics Research Colloquium throughout the years. In particular, he is grateful to Carly Goodman for her patience when reading drafts and listening to the
rough beginnings of ideas. Michael also extends his thanks to Creighton University’s
Graduate School, specifically, the International Relations department, for research
support. John N. Mordeson dedicates this book to his grandparents Katherine and
John Niece and Mary Ellen and Nels Mordeson. Mark J. Wierman dedicates this
book to Mary K. Dobransky. Annie Gowen thanks her co-authors, whose guidance
and patience made her work possible. She dedicates her contribution to her dearest
friend, Matthew Cockerill, for his unfailing encouragement. Karen Albert, who intends to pursue the Ph.D. in political science, would like to dedicate her work in this
book to her parents, James and Carol Albert. Terry D. Clark dedicates his work in
this book to his wife of thirty-seven years, whom he adores, Marnie.
Creighton University,
Omaha, NE,
December, 2013
John N. Mordeson
Mark J. Wierman
Terry D. Clark
Acknowledgements
This research grew out of the Fuzzy Spatial Modeling Colloquium. The colloquium
is indebted to Professor Bridget Keegan, Interim Dean of the College of Arts and
Sciences at Creighton University whose support has been invaluable in sustaining
our efforts. We are also indebted to Dr. George and Mrs. Sally Haddix for their
generous endowments to the Department of Mathematics at Creighton University.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IX
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV
1
Fuzzy Social Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Purpose and Plan of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Fuzzy Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Fuzzy Intersection and Union . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.5 Residuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
2
4
5
6
7
9
2
Classical Social Choice Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Arrows Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Gibbard-Sattherthwaite Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 The Median Voter Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 The Maximal Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
11
14
16
18
19
19
3
Rationality of Fuzzy Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 The Structure of Fuzzy Preference Relations . . . . . . . . . . . . . . . . . . .
3.2 Consistency of Fuzzy Preferences and the Fuzzy Maximal Set . . . .
3.3 Empirical Application I: Deriving an FWPR from a Fuzzy
Preference Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
22
34
45
50