The Mathematics of Voting and Apportionment: An Introduction (Compact Textbooks in Mathematics)

Description

This textbook contains a rigorous exposition of the mathematical foundations of two of the most important topics in politics and economics: voting and apportionment, at the level of upper undergraduate and beginning graduate students. It stands out among comparable books by providing, in one volume, an extensive and mathematically rigorous treatment of these two topics. The text’s three chapters cover social choice, yes-no voting, and apportionment, respectively, and can be covered in any order, allowing teachers ample flexibility. Each chapter begins with an elementary introduction and several examples to motivate the concepts and to gradually lead to more advanced material. Landmark theorems are presented with detailed and streamlined proofs; those requiring more complex proofs, such as Arrow’s theorems on dictatorship, Gibbard’s theorem on oligarchy, and Gärdenfors’ theorem on manipulation, are broken down into propositions and lemmas in order to make them easier to grasp. Simple and intuitive notations are emphasized over non-standard, overly complicated symbols. Additionally, each chapter ends with exercises that vary from computational to “prove or disprove” types. The Mathematics of Voting and Apportionment will be particularly well-suited for a course in the mathematics of voting and apportionment for upper-level undergraduate and beginning graduate students in economics, political science, or philosophy, or for an elective course for math majors. In addition, this book will be a suitable read for to any curious mathematician looking for an exposition to these unpublicized mathematical applications. No political science prerequisites are needed. Mathematical prerequisites (included in the book) are minimal: elementary concepts in combinatorics, graph theory, order relations, and the harmonic  and geometric means. What is needed most is the level of maturity that enables the student to think logically, derive results from axioms and hypotheses, and intuitively grasp logical notions such as “contrapositive” and “counterexample.” Preface Acknowledgments Contents 1 Social Choice 1.1 Introduction 1.1.1 Plurality vs Majority 1.1.2 The Plurality Procedure 1.1.3 The 2000 US Presidential Election and the Shortcomings of the Plurality Procedure 1.2 Elimination Procedures 1.2.1 Plurality with Run-Off 1.2.2 The Hare Procedure 1.2.3 The Coombs Procedure 1.2.4 Monotonicity 1.2.5 Dealing with Non-monotonicity 1.3 Condorcet Ideas and Related Procedures 1.3.1 The Condorcet Tournament 1.3.2 Condorcet Winner and Condorcet Loser 1.3.3 Condorcet Winner vs Majority Candidate 1.3.4 The Condorcet Criteria 1.3.5 Agenda Voting 1.3.6 Weak Pareto Efficiency 1.3.7 Killer Amendment 1.3.8 The Copeland Procedure 1.3.9 Monotonicity Revisited 1.3.10 McGarvey's Theorem 1.4 Scoring Procedures: Borda Count 1.4.1 Borda Count 1.4.2 Shortcomings of the Borda Count Procedure 1.4.3 More on Borda Count 1.4.4 Hare-Scoring Procedures 1.4.5 Borda Meets Condorcet 1.5 A Glimpse Into Social Welfare Theory 1.5.1 Rankings and Semi Rankings 1.5.2 Independence of Irrelevant Alternatives (IIA) 1.5.3 Transitivity 1.5.4 Black's Theorem 1.5.5 May's Theorem 1.5.6 Dictatorship and Arrow's Theorem 1.5.7 Oligarchy and Gibbard's Theorem 1.6 Social Choice Procedures: Indifference and Ties Allowed 1.6.1 Standard Social Choice Procedures 1.6.2 More Condorcet-Type Classifications of the Alternatives 1.6.3 The Social Choice Set of the Hare–Borda Procedure 1.7 Manipulability of Social Choice Procedures: Indifference and Ties Allowed 1.7.1 Comparing Sets of Alternatives 1.7.2 Anonymity and Neutrality 1.7.3 Gärdenfors' Theorem 1.7.4 Manipulability of Social Choice Functions Not Covered by Gärdenfors' Theorem Exercises 2 Yes-No Voting 2.1 Introduction 2.1.1 The Basics 2.1.2 The +/- Table 2.1.3 Parity of Banzhaf Scores 2.2 Quantification of Power in a Yes-No Voting System 2.2.1 The Banzhaf Index of Power 2.2.2 The Felsenthal–Machover Example 2.2.3 The Shapley–Shubik Index of Power 2.2.4 Banzhaf vs Shapley–Shubik Computations 2.3 Some Combinatorics 2.3.1 Permutations and Combinations 2.3.2 Cartesian Products 2.4 Banzhaf and Shapley–Shubik Indices in One View 2.4.1 Computing the Shapley–Shubik Indices from Winning Coalitions 2.4.2 Veto-Powered Voters, Dominant Voters and Dictators 2.5 Weightable Yes-No Voting Systems 2.5.1 Trades Among Coalitions 2.5.2 Trade-Robustness and the Taylor–Zwicker Theorem 2.5.3 The Magic Square Voting System Exercises 3 Apportionment 3.1 Introduction 3.2 Axioms of Apportionment 3.3 Quota Procedures 3.3.1 Hamilton's Procedure 3.3.2 Lowndes' Procedure 3.3.3 The Alabama Paradox 3.3.4 The Population Paradox 3.3.5 The Balinski-Young Theorem 3.4 Divisor Procedures 3.4.1 The General Framework of Divisor Procedures 3.4.2 Jefferson's Procedure 3.4.3 Adams' Procedure 3.4.4 Webster's Procedure 3.4.5 The Hill–Huntington Procedure 3.5 Equity Criteria of Divisor Procedures 3.5.1 Measures of Inequity 3.5.2 Postulates of a Measure of Inequity 3.5.3 Webster's Procedure and the δ-Inequity 3.5.4 The Hill–Huntington Procedure and the ρ-Inequity 3.5.5 The φ-Inequity Measure 3.5.6 The Harmonic Mean 3.5.7 Dean's Procedure 3.5.8 A Collective View of Divisor Procedures 3.5.9 The Threshold Divisors of Dean's Procedure 3.6 Apportionment Paradoxes 3.6.1 Monotonicity and the Divisor Procedures 3.6.2 Consistency 3.7 Applications of Priority Formulas 3.7.1 One-By-One Seat Apportionment 3.7.2 The Quota-Divisor Procedures Exercises Answers to Selected Exercises Chapter 1 Chapter 2 Chapter 3 References Index Answers to Selected Exercises References Index