Table Of ContentCONSENSUS
UNDER
FUZZINESS
INTERNATIONAL SERIES IN
INTELLIGENT TECHNOLOGIES
Prof. Dr. Dr. h.c. Hans-Jiirgen Zimmermann, Editor
European Laboratory for Intelligent
Techniques Engineering
Aachen, Germany
Other books in the series:
Applied Research in Fuzzy Technology
by Anca L. Ralescu
Analysis and Evaluation of Fuzzy Systems
by Akira Ishikawa and Terry L. Wilson
Fuzzy Logic and Intelligent Systems
edited by Hua Li and Madan Gupta
Fuzzy Set Theory and Advanced Mathematical
Applications
edited by Da Ruan
Fuzzy Databases: Principles and Applications
by Frederick E. Petry with Patrick Bose
Distributed Fuzzy Control of Multivariable Systems
by Alexander Gegov
Fuzzy Modelling: Paradigms and Practices
by Witold Pedrycz
Fuzzy Logic Foundations and Industrial Applications
by Da Ruan
Fuzzy Sets in Engineering Design and Configuration
by Hans-Juergen Sebastian and Erik K. Antonsson
CONSENSUS
UNDER
FUZZINESS
edited by
Jannsz Kacprzyk
Polish Academy ofS ciences
Warsaw, Poland
•
Hannn Nnrmi
University ofTurku
Turku, Finland
•
Mario Fedrizzi
University ofTrento
Trento, Italy
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
ISBN 978-1-4613-7908-9 ISBN 978-1-4615-6333-4 (eBook)
DOI 10.1007/978-1-4615-6333-4
Library of Congress Cataloging-in-Publieation Data
A C.I.P. Catalogue record for this book is available
ftom tbe Library of Congress.
Copyright © 1997 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers, New York in 1997
Softcover reprint of the hardcover Ist edition 1997
AlI rights reserved. No part ofthis publication may be reproduced, stored in a retrieval
system or transmitted in 80y form or by 80y me8Os, mech8Oical, photocopying,
recording, or otherwise, without the prior written permission ofthe publisher, Springer
Science+Business Media, LLC
Printed on acid-free paper.
Table of Contents
Preface
1. INTRODUCTORY SECTIONS
Consensus, negotiation and mediation 3
K. Lehrer
Fuzziness and the normative theory of social choice 17
P.K. Pattanaik
Types and measures of uncertainty 29
J. Klir and D. Harmanec
2. TOOLS AND TECHNIQUES FOR MEASURING
AND MONITORING CONSENSUS REACHING
"Soft" degrees of consensus under fuzzy preferences and 55
fuzzy majorities
J. Kacprzyk, M. Fedrizzi and H. Nurmi
An approach to the consensus reaching support in fuzzy environment 83
S. Zadrozny
The dichotomous approach to soft consensus measurement 111
S. Greco
Consensus based on fuzzy coincidence for group
decision making in linguistic setting 121
F. Herrera, E. Herrera-Viedma and J.L. Verdegay
Modeling preference relations and consensus in a linguistic 147
environment: an approach based on OWA operators
G. Bordogna, M. Fedrizzi and G. Pasi
vi
3. NEW PARADIGMS AND ARCHITECTURES FOR MODEUNG
CONSENSUS REACHING
Protocol for negotiations among multiple intelligent agents 165
R.R. Yager
The development of fuzzy consensus via neural modelling 175
W. Pedrycz
4. AUXILIARY FORMAL TOOLS AND TECHNIQUES
FOR MODEUNG CONSENSUS REACHING
Consensus for decomposable measures 191
J. Fodor, D. Dubois, H. Prade and M. Roubens
Construction of fuzzy utility functions in group decision making 211
F. Seo
Problem solving with multiple interdependent criteria 231
C. Carlsson and R. Fuller
Lexicographical solutions in n-person cooperative 247
games with multiple scenarios
M. Sakawa and I. Nishizaki
5. APPLICATIONS AND CASE STUDIES
Identification of ideological dimensions under fuzziness: 267
the case of Poland
J. Holubiec, A. Malkiewicz, M. Mazurkiewicz,
J. Mercik and D. Wagner
Determining weights of research topics on the basis of expert 285
judgments. The case of Systems Research Institute
D. Wagner
INDEX 301
PREFACE
We live, unfortunately, in turbulent and difficult times plagued by various political,
economic, and social problems, as well as by natural disasters worldwide. Systems
become more and more complicated, and this concerns all levels, exemplified first by
global political alliances, groups of countries, regions, etc., and secondly, by
multinational (global) corporations and companies of all sizes. These same concerns
affect all social groups. This all makes decision processes very complicated.
In virtually all decision processes in these complicated systems, there are various actors
(decision makers) who represent individual subjects (persons, countries, companies,
etc.) and their respective interest groups. To reach a meaningful (good) decision,
opinions of all such actors must be taken into account or a given decision may be
rejected and not implemented. Ideally, a decision would be made after a consensus
between the parties involved had been attained. So, consensus is a very desirable
situation.
In most real-world cases there is considerable uncertainty concerning all aspects of the
decision making process. Moreover, opinions, goals, constraints, etc. are usually
imprecisely known. This makes the decision making process difficult as one cannot
employ conventional "hard" tools.
Consensus is traditionally defmed as a strict and unanimous agreement wherein the
parties involved are collectively in agreement on all issues in question. However, since
various actors have different (or often conflicting) opinions and/or value systems, it
must be acknowledged that adherence to this traditional, strict meaning of consensus
is unrealistic. The human perception of consensus is much "softer," and people are
willing to accept that a "consensus" has been reached when "most" or ''the more
predominant" actors arrive at a "sufficient" agreement.
This book gathers relevant contributions from leading experts in the field which are
concerned with various issues related to the modeling and monitoring of consensus
reaching processes under fuzzy preferences and fuzzy majorities. Basically, a "soft"
meaning of consensus is advocated as realistic and humanly consistent.
viii
The first part contains some introductory contributions which discuss, general issues
related to consensus, negotiation, social choice, and related topics. Moreover, an
analysis of various measures of fuzziness, which may be of use for a formal treatment
of fuzziness related aspects of the process, is provided.
In the second part, the authors discuss various measures of "soft" consensus taking into
account fuzzy preferences and fuzzy majorities, and also some aspects of monitoring
the consensus reaching process within a group decision support system. The case of
traditional fuzzy preference relations is considered first, and then linguistic fuzzy
preference relations are assumed.
In the third part, some new paradigms for the modeling of consensus reaching are
presented and advocated, including groups of intelligent agents and neural networks.
In the fourth part, some tools which are useful for the analysis and modeling of
consensus reaching are discussed, including issues related to the construction of a
group utility function, solution concepts in multiperson cooperative games, etc.
In the fifth part, two interesting case studies are reported, one concerning the analysis
of ideological dimensions of parliament parties, and one discussing the research
planning and fund allocation process in a research institute employing experts'
testimonies.
It is hoped that the wide array of paradigms, tools and techniques presented in the
contributions will help develop better analytic tools for consensus reaching processes,
and will lead to more "human-consistent," realistic, and hence easier to implement
procedures and group decision support systems (GDSSs) for consensus reaching.
The editors wish to thank all of the contributors for their outstanding papers and
effective and efficient collaboration in this exciting editorial project. Mr. Alex Greene
from Kluwer Academic Publishers deserves our deepest appreciation for constant
encouragement and support, and a rare ability to create a synergistic collaboration
between the editors and the publisher.
J. Kacprzyk
H.Nurmi
M. Fedrizzi
INTRODUCTIONS
CONSENSUS. NEGOTIATION AND MEDIATION·
Keith Lehrer
University of Arizona
Karl-Franzens University, Graz
[email protected]
Carl Wagner and I articulated a mathematical model ofa ggregating vectors to
reach consensus which has been the subject ofs ubsequent controversy in the literature.
We formulated a model of convergence toward consensus applied to an allocation
matrix ofv ectors. In our collaboration, we laid great emphasis on the merits ofa ssigning
weights and weighted averaging that converged toward the consensual allocation. In this
paper, I will consider the application of the model to negotiation. I investigate the
rationality of blocking convergence toward consensus, most deCisively, by assigning a
weight of zero to all other members of the group. The basic rationale for blocking
convergence in this way is to prevent one from being co-opted in the process of
negotiation. Nevertheless, blocking convergence results in the decomposition ofs ociety
and failure to base policy on consensus. To prevent such decomposition, I consider
adopting a mediator who is a default referee in the aggregation process. The default
referee connects the group by receiving a standard positive weight from a/l involved and
giving positive weight to all others to yield convergence and consensus. The assignments
oft he default referee to others may be egalitarian differential and yet equallly effective
01'
in producing convergence.
Keywords. Aggregation, allocation, averaging, communication, connectedness,
consensus, convergence, decomposition, defection, fixed point, matrix, mediation,
negotiation, respect, vectors, and weights.
Carl Wagner and I articulated a mathematical model of aggregating vectors to
reach consensus which has been the subject of subsequent controversy in the literature.
I
We fonnulated a model of convergence toward consensus applied to an allocation matrix
of vectors. These vectors may include fuzzy representations of features or values, though
they are not limited to such representations. The basic idea of the model was that if
members of a group assign weights to all members of the group, where the weights are
nonnegative numbers which sum to one, then iterated weighted averaging will converge
toward a consensual allocation vector as the iterations go to infmity under plausible
conditions. A sufficient condition for the convergence is connectedness among members
combined with constancy in the assignment of weights at some stage of averaging, though
this condition is not necessary.
J. Kacprzyk et al. (eds.), Consensus Under Fuzziness
© Springer Science+Business Media New York 1997
