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<ul><li><p>Studies in Fuzziness and Soft Computing</p><p>Peter C. Casey Michael B. GibiliscoCarly A. Goodman Kelly Nelson PookJohn N. Mordeson Mark J. WiermanTerry D. Clark</p><p>Fuzzy Social Choice ModelsExplaining the Government Formation Process</p></li><li><p>Studies in Fuzziness and Soft Computing</p><p>Volume 318</p><p>Series editor</p><p>Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Polande-mail: kacprzyk@ibspan.waw.pl</p><p>For further volumes:http://www.springer.com/series/2941</p><p>http://www.springer.com/series/2941</p></li><li><p>About this Series</p><p>The series Studies in Fuzziness and Soft Computing contains publications onvarious topics in the area of soft computing, which include fuzzy sets, rough sets,neural networks, evolutionary computation, probabilistic and evidential reasoning,multi-valued logic, and related fields. The publications within Studies in Fuzzi-ness and Soft Computing are primarily monographs and edited volumes. Theycover significant recent developments in the field, both of a foundational andapplicable character. An important feature of the series is its short publication timeand world-wide distribution. This permits a rapid and broad dissemination ofresearch results.</p></li><li><p>Peter C. Casey Michael B. GibiliscoCarly A. Goodman Kelly Nelson PookJohn N. Mordeson Mark J. WiermanTerry D. Clark</p><p>Fuzzy Social ChoiceModels</p><p>Explaining the GovernmentFormation Process</p><p>123</p></li><li><p>Peter C. CaseyDepartment of Political ScienceWashington University in St. LouisSt. Louis, MOUSA</p><p>Michael B. GibiliscoDepartment of Political ScienceUniversity of RochesterRochester, NYUSA</p><p>Carly A. GoodmanWest CorporationCreighton UniversityOmaha, NEUSA</p><p>Kelly Nelson PookTerry D. ClarkDepartment of Political ScienceCreighton UniversityOmaha, NEUSA</p><p>John N. MordesonDepartment of MathematicsCreighton UniversityOmaha, NEUSA</p><p>Mark J. WiermanComputer Science and InformaticsCreighton UniversityOmaha, NEUSA</p><p>ISSN 1434-9922 ISSN 1860-0808 (electronic)ISBN 978-3-319-08247-9 ISBN 978-3-319-08248-6 (eBook)DOI 10.1007/978-3-319-08248-6Springer Cham Heidelberg New York Dordrecht London</p><p>Library of Congress Control Number: 2014942067</p><p> Springer International Publishing Switzerland 2014This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed. Exempted from this legal reservation are briefexcerpts in connection with reviews or scholarly analysis or material supplied specifically for thepurpose of being entered and executed on a computer system, for exclusive use by the purchaser of thework. Duplication of this publication or parts thereof is permitted only under the provisions ofthe Copyright Law of the Publishers location, in its current version, and permission for use mustalways be obtained from Springer. Permissions for use may be obtained through RightsLink at theCopyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility forany errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained herein.</p><p>Printed on acid-free paper</p><p>Springer is part of Springer Science+Business Media (www.springer.com)</p></li><li><p>We dedicate this book to Rose Hill, who isalways there when you need her</p></li><li><p>Preface</p><p>John N. Mordeson, Mark J. Wierman, and Terry D. Clark began working 8 yearsago on the application of fuzzy Mathematics to public choice models. Their initialintent was to explain the formation of governments in parliamentary systems.However, they soon discovered that they would have to devote significant atten-tion to matters of theory. As a consequence, they ended up producing what canonly be described as a very large volume of work on public choice theory. Whilethey also produced some empirical work, the best of which has appeared on thepages of the journal Public Choice, the theoretical work has clearly overshadowedthe empirical work in sheer volume.</p><p>This book partially redresses that imbalance. It is a compilation of most, if notall, of the effort that went into the empirical question that motivated the initialproject. In it we present the results of several attempts to predict the outcome ofthe government formation process in parliamentary systems using fuzzy publicchoice models. However, even in this volume we present a substantial amount ofthe theoretical work related to the fuzzy models. While much of it has appearedpreviously in print, most readers would find understanding the approach that wetake with each model incomprehensible had we not restated many of our maintheoretical findings here.</p><p>As those who are familiar with our work are aware, we have engaged manybright young students over the last 8 years. Several of them have gone on to pursuePh.D. in Political Science or Mathematics. This is as true for our empirical work asit is for our theoretical work. Peter Casey and Michael Gibilisco took the lead inthe work presented here. They were assisted by a very large number of theirstudent colleagues, one of whom Carly Goodman was particularly instrumental inhelping to formulate the approaches and undertake the tests that led to the resultsthat we report in the book in front of you. Carly also took the lead in editing theresulting papers, which included checking the models and re-verifying the results.Kelly Pook finished the task, which Carly passed on to her before departing for alucrative position as a business analyst.</p><p>Peter C. Casey, is presently pursuing Ph.D. in Political Science at theUniversity of Washington in St. Louis. Peter dedicates this book to his mother,Virginia Casey, for her guidance and support. Michael Gibilisco is pursuing Ph.D.in Political Science at the University of Rochester. Michael dedicates this book tohis parents whose moral, and, at times, financial support, made the work possible.</p><p>vii</p></li><li><p>They have always encouraged him and his research throughout school and thisproject, and his passion for learning began with them. Carly Goodman is gratefulto her co-authors and mentors in the Fuzzy Research Colloquium, without whomthis book would not have been possible. She dedicates her contribution to herparents and to Eric Norrgard for their constant encouragement and support. KellyPook dedicates her work to Ryan, for the first year of many. She is at present thehead research assistant for the Social Network Analysis Working Group. As such,she supervises the work of 20 students. John Mordeson dedicates this book to hisloving wife Pat. Mark J. Wierman dedicates this book to Mary K. Dobransky.Terry D. Clark dedicates his work in this book to his wife of 37 years, Marnie,whom he adores, and to his granddaughter, Zoey, whose coming into this worldhas brought such joy.</p><p>Omaha, USA, March 2014 Terry D. ClarkJohn N. MordesonMark J. Wierman</p><p>viii Preface</p></li><li><p>Acknowledgments</p><p>This research grew out of the Fuzzy Spatial Modeling Colloquium. The collo-quium is indebted to Prof. Bridget Keegan, Interim Dean of the College of Artsand Sciences at Creighton University whose support has been invaluable in sus-taining our efforts.</p><p>We are also indebted to Dr. George Haddix and his late wife Sally Haddix fortheir generous endowments to the Department of Mathematics at CreightonUniversity.</p><p>Finally, we thank the journal, New Mathematics and Natural Computation, forits support of research using fuzzy Mathematics in Political Science and forallowing us to reuse some of our work for this book.</p><p>ix</p></li><li><p>Contents</p><p>1 A Fuzzy Public Choice Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Improving the Prediction of Public Choice Models:</p><p>A Fuzzy Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Fuzzy Maximal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11</p><p>2 Fuzzy Preferences: Extraction from Data and Their Usein Public Choice Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Extracting Preference Measures from Empirical Data . . . . . . . . 14</p><p>2.2.1 Extracting Preferences from the CMP Data . . . . . . . . . . 162.2.2 A Method for Extracting Fuzzy Preference Profiles</p><p>from CMP Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 The Conventional Public Choice Model . . . . . . . . . . . . . . . . . . 222.4 A Fuzzy Public Choice Model . . . . . . . . . . . . . . . . . . . . . . . . 272.5 Advantages of the Fuzzy Public Choice Model . . . . . . . . . . . . . 30References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31</p><p>3 Fuzzy Single-Dimensional Public Choice Models . . . . . . . . . . . . . . 333.1 Intransitivity in Collective Preference . . . . . . . . . . . . . . . . . . . 333.2 Fuzzy Aggregation Preference Rules . . . . . . . . . . . . . . . . . . . . 343.3 Fuzzy Voting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Single-Peaked Fuzzy Profiles . . . . . . . . . . . . . . . . . . . . . . . . . 45References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50</p><p>4 Predicting the Outcome of the Government Formation Process:Fuzzy Single-Dimensional Models . . . . . . . . . . . . . . . . . . . . . . . . . 514.1 An Overview of the Comparative Manifesto Project Data . . . . . 514.2 Traditional Models of Government Formation. . . . . . . . . . . . . . 56</p><p>xi</p><p>http://dx.doi.org/10.1007/978-3-319-08248-6_1http://dx.doi.org/10.1007/978-3-319-08248-6_1http://dx.doi.org/10.1007/978-3-319-08248-6_1#Sec1http://dx.doi.org/10.1007/978-3-319-08248-6_1#Sec1http://dx.doi.org/10.1007/978-3-319-08248-6_1#Sec1http://dx.doi.org/10.1007/978-3-319-08248-6_1#Sec2http://dx.doi.org/10.1007/978-3-319-08248-6_1#Sec2http://dx.doi.org/10.1007/978-3-319-08248-6_1#Sec3http://dx.doi.org/10.1007/978-3-319-08248-6_1#Sec3http://dx.doi.org/10.1007/978-3-319-08248-6_1#Sec4http://dx.doi.org/10.1007/978-3-319-08248-6_1#Sec4http://dx.doi.org/10.1007/978-3-319-08248-6_1#Bib1http://dx.doi.org/10.1007/978-3-319-08248-6_2http://dx.doi.org/10.1007/978-3-319-08248-6_2http://dx.doi.org/10.1007/978-3-319-08248-6_2http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec1http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec1http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec2http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec2http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec3http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec3http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec4http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec4http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec4http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec5http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec5http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec6http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec6http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec7http://dx.doi.org/10.1007/978-3-319-08248-6_2#Sec7http://dx.doi.org/10.1007/978-3-319-08248-6_2#Bib1http://dx.doi.org/10.1007/978-3-319-08248-6_3http://dx.doi.org/10.1007/978-3-319-08248-6_3http://dx.doi.org/10.1007/978-3-319-08248-6_3#Sec1http://dx.doi.org/10.1007/978-3-319-08248-6_3#Sec1http://dx.doi.org/10.1007/978-3-319-08248-6_3#Sec2http://dx.doi.org/10.1007/978-3-319-08248-6_3#Sec2http://dx.doi.org/10.1007/978-3-319-08248-6_3#Sec3http://dx.doi.org/10.1007/978-3-319-08248-6_3#Sec3http://dx.doi.org/10.1007/978-3-319-08248-6_3#Sec4http://dx.doi.org/10.1007/978-3-319-08248-6_3#Sec4http://dx.doi.org/10.1007/978-3-319-08248-6_3#Bib1http://dx.doi.org/10.1007/978-3-319-08248-6_4http://dx.doi.org/10.1007/978-3-319-08248-6_4http://dx.doi.org/10.1007/978-3-319-08248-6_4http://dx.doi.org/10.1007/978-3-319-08248-6_4#Sec1http://dx.doi.org/10.1007/978-3-319-08248-6_4#Sec1http://dx.doi.org/10.1007/978-3-319-08248-6_4#Sec2http://dx.doi.org/10.1007/978-3-319-08248-6_4#Sec2</p></li><li><p>4.3 A Fuzzy Model of Government Formation . . . . . . . . . . . . . . . . 584.3.1 A Fuzzy Maximal Set Approach</p><p>in One-Dimensional Models . . . . . . . . . . . . . . . . . . . . . 624.3.2 A Fuzzy Pareto Set Approach</p><p>in One-Dimensional Model . . . . . . . . . . . . . . . . . . . . . 644.4 Demonstrating the Two Fuzzy Modeling Approaches . . . . . . . . 684.5 Comparing the Fuzzy and Conventional Models</p><p>of Government Formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80</p><p>5 Issues in Fuzzy Multi-dimensional Public Choice Models. . . . . . . . 815.1 Stability in Multi-dimensional Models . . . . . . . . . . . . . . . . . . . 815.2 Majority Rule Maximal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3 The Existence of a Majority Rule Maximal Set</p><p>in Arbitrary n-Dimensional Public Choice Models. . . . . . . . . . . 895.4 Relation Spaces and Majority Rule . . . . . . . . . . . . . . . . . . . . . 1025.5 A Consideration of Different Definitions of Fuzzy</p><p>Covering Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.5.1 Covering Relations and Minimal Generating Sets . . . . . . 1065.5.2 Covering Relation C4 with n 3. . . . . . . . . . . . . . . . . . 1085.5.3 Covering Relation C2 . . . . . . . . . . . . . . . . . . . . . . . . . 1125.5.4 C2-Uncovered Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.5.5 Covering Relation C7 . . . . . . . . . . . . . . . . . . . . . . . . . 1205.5.6 A Symmetrical Covering Relation. . . . . . . . . . . . . . . . . 1235.5.7 The Implication of Missing n-Tuples. . . . . . . . . . . . . . . 125</p><p>References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127</p><p>6 Predicting the Outcome of the Government Formation Process:A Fuzzy Two-Dimensional Public Choice Model . . . . . . . . . . . . . . 1296.1 A Two-Dimensional Fuzzy Model . . . . . . . ....</p></li></ul>

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