Consensus Under Fuzziness PDF
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CONSENSUS 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
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