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

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