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Springer Texts in Business and Economics Satish Kumar Jain Social Choice Theory An Introductory Text Springer Texts in Business and Economics Springer Texts in Business and Economics (STBE) delivers high-quality instruc- tionalcontentforundergraduatesandgraduatesinallareasofBusiness/Management Science and Economics. The series is comprised of self-contained books with a broadandcomprehensivecoveragethataresuitableforclassaswellasforindividual self-study. All texts are authored by established experts in their fields and offer asolidmethodologicalbackground,oftenaccompaniedbyproblemsandexercises. More information about this series at https://link.springer.com/bookseries/10099 Satish Kumar Jain Social Choice Theory An Introductory Text SatishKumarJain Gurgaon,Haryana,India ISSN2192-4333 ISSN2192-4341 (electronic) SpringerTextsinBusinessandEconomics ISBN978-981-16-9660-2 ISBN978-981-16-9661-9 (eBook) https://doi.org/10.1007/978-981-16-9661-9 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Dedicated TotheMemoryof Dr.MahendraSagarPrachandia ShriShyamDuttPaliwal ProfessorJamesW.Friedman Preface Social choice theory is concerned with the problems of aggregating individual preferences into social preferences. This book presents the basics of the subject- matter of the social choice theory. The main motivation behind this introductory textbook is to make the social choice theoretic framework and some of the main findingsofthetheoryaccessibletoalargeraudience.Thisisattemptedbymaking the text essentially self-contained. No previous knowledge of mathematical logic or relational algebra is assumed. Whatever technical prerequisites are needed, are developed in the text itself. Although the text is at an introductory level, there has been no compromise on rigor. Proofs are provided for all propositions in a detailed and rigorous manner. To aid understanding, concepts and theorems have beenillustratedwithalargenumberofexamples.Alargenumberofexercisesare given in the text with full solutions provided at the end of the book, so that the reader can check her/his understanding of the material. Chapters 2–4 contain the required prerequisites of elementary logic and relational algebra, and Chaps. 5– 9 the basics of the social choice theory. Chapter 5 introduces the social choice framework, and contains the statement and the proof of the Arrow Impossibility Theorem.Chapter6isdevotedtoadiscussionofsomeimportantvalue-judgments and rules. Conditions of Pareto-criterion, monotonicity, neutrality and anonymity, among others, are discussed in detail. Chapter 7 examines the implications of replacing transitivity requirement of social preferences by the weaker conditions of quasi-transitivity and acyclicity. Chapter 8 is concerned with the derivation of conditionsunderwhichthemethodofmajoritydecisiongivesrisetotransitiveand quasi-transitivesocialpreferences.Chapter9isdevotedtoexaminingthestrategic aspectsoftwo widelyusedrules, namelythemethod ofmajoritydecision andthe plurality rule. I wish to thank Professor Subrata Guha and Dr. Rajendra Kundu for their comments and suggestions on an earlier draft of this book. I thank Shri Kaushal Kishore for his meticulous proof-reading of the entire text, which resulted in the vii viii Preface elimination of quite a few errors. This book is dedicated to the memory of three inspiringteachers;Dr.MahendraSagarPrachandia,ShriShyamDuttPaliwal,and Professor James W. Friedman. Agra, India Satish Kumar Jain September 2021 Contents 1 Introduction ........................................................ 1 References .......................................................... 7 2 ElementaryLogic ................................................... 9 2.1 Sentential Logic ................................................ 12 2.1.1 Logical Connectives .................................... 12 2.1.2 Well-Formed Sentential Formulas ....................... 15 2.1.3 Tautologies, Contradictions, Contingent Formulas, and Logical Equivalences ............................... 19 2.1.4 Three Standard Proof Methods .......................... 22 2.2 Predicate Logic ................................................ 24 2.2.1 Universally Valid Formulas .............................. 26 References .......................................................... 33 3 Sets,Relations,Functions ........................................... 35 3.1 Sets ............................................................ 35 3.2 Relations ....................................................... 40 3.3 Functions ...................................................... 41 4 BinaryRelations .................................................... 47 4.1 Binary Relations and Some Important Properties ................ 48 4.2 Some Elementary Propositions About Binary Relations ......... 51 5 SocialChoiceTheoreticFrameworkandArrowImpossibility Theorem ............................................................ 61 5.1 Social Choice Theoretic Framework ............................ 62 5.2 Arrow Impossibility Theorem .................................. 65 References .......................................................... 75 6 SomeImportantValue-Judgments,RulesandTheorems ........... 77 6.1 Pareto-Optimality and Paretian Value-Judgment ................. 78 6.2 Three Important Conditions on Social Decision Rules ........... 84 6.2.1 Monotonicity Conditions ................................ 84 6.2.2 Anonymity (A) ......................................... 88 6.2.3 Neutrality (N) ........................................... 89 ix x Contents 6.3 Characterization of the Method of Majority Decision ........... 90 6.4 Pareto-Rule .................................................... 91 References .......................................................... 94 7 ImplicationsofWeakeningofSomeofArrowConditions ........... 95 7.1 Implications of Weakening of Transitivity to Quasi-Transitivity ........................................... 96 7.2 Implications of Weakening of Transitivity to Acyclicity ......... 98 7.3 Acyclicity Under m−1-Majority Rules .......................... 99 m References .......................................................... 102 8 TheMethodofMajorityDecision:ConditionsforTransitivity andQuasi-Transitivity .............................................. 103 8.1 The Reduced Form of a Profile ................................. 107 8.2 Necessary and Sufficient Condition for Transitivity ............. 109 8.3 Necessary and Sufficient Condition for Quasi-Transitivity ....... 111 8.4 Domain Restriction Conditions for Transitivity and Quasi-Transitivity .......................................... 112 8.5 Latin Squares .................................................. 114 8.6 Restrictions on Preferences ..................................... 115 8.7 Domain Restriction Conditions for Transitivity and Quasi-Transitivity under the Method of Majority Decision ....................................................... 117 8.8 Single-Peaked Preferences ...................................... 125 References .......................................................... 129 9 StrategicAspects ................................................... 131 9.1 Definitions, Assumptions, and Notation ......................... 134 9.2 Manipulability of Rules ........................................ 136 9.2.1 The Two-Alternative Case ............................... 136 9.2.2 Manipulability of the MMD ............................. 137 9.2.3 The Plurality Rule ...................................... 138 9.2.4 Manipulability of the Plurality Rule ..................... 138 9.3 Existence of Nash Equilibria ................................... 139 9.4 Existence of Strong Equilibria Under the Method of Majority Decision ........................................... 140 9.5 Existence of Strong Equilibria Under the Plurality Rule ......... 142 10 SummaryandConcludingRemarks ................................ 151 SolutionsandAnswerstoExercises ..................................... 155 References ............................................................... 197 SubjectIndex ............................................................ 199