Social Choice Theory: An Introductory Text

211 Pages·2022·1.963 MB·English

by  Satish Kumar Jain

Springer Texts in Business and Economics
Satish Kumar Jain
Social Choice
Theory
An Introductory Text
Springer Texts in Business and Economics
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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
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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