Evaluating Voting Systems with Probability Models: Essays by and in Honor of William Gehrlein and Dominique Lepelley

Foreword
Acknowledgements
Contents
Contributors
Introduction
	1 Part I: The Condorcet Efficiency of Voting Rules and Related Paradoxes
	2 Part II: Other Voting Paradoxes
	3 Part III: Binary Voting in Federations
	4 Part IV: Resistance to Manipulations
	5 Part V: Game Theory
	6 Part VI: Techniques for Probability Computations
	References
The Condorcet Efficiency of Voting Rules and Related Paradoxes
Analyzing the Probability of Election Outcomes with Abstentions
	1 Introduction
	2 Probability of ACW and OCW Coincidence
		2.1 ACW and OCW Coincidence Results with IC
		2.2 ACW and OCW Coincidence Results with IAC
	3 Actual Condorcet Efficiency of Single-Stage Voting Rules
		3.1 Actual Single-Stage Rule Efficiency with IC
		3.2 Actual Single-Stage Rule Efficiency with IAC
	4 Borda Paradox Probabilities
		4.1 Borda Paradox Probabilities for Single-Stage Rules with IC
		4.2 Borda Paradox Probabilities for Single-Stage Rules with IAC
	5 Two-Stage Voting Rules
		5.1 Condorcet Efficiency of Two-Stage Voting Rules with IC
		5.2 Condorcet Efficiency of Two-Stage Voting Rules with IAC
		5.3 Borda Paradox Probability for Two-Stage Voting Rules with IC
		5.4 Borda Paradox Probabilities for Two-Stage Rules with IAC
	6 Results for Other Voting Rules When Indifference Between Candidates is Permitted
	7 Discussion and Explanation of These Disconcerting Results
		7.1 A Further Evaluation
		7.2 An Explanation of the Results: Hindsight is 20–20
	8 Conclusion
	References
Condorcet Efficiency of General Weighted Scoring Rules Under IAC: Indifference and Abstention
	1 Introduction
	2 The Scope
	3 Condorcet Efficiency of Weighted Scoring Rules When Indifference Is Observable
		3.1 With No Unconcerned Voters
		3.2 With Possibly Unconcerned Voters
	4 Condorcet Efficiency with Abstention Allowed
		4.1 Global Abstention
		4.2 Self-confident Abstention
		4.3 Pessimistic Abstention
	5 Conclusion
	References
The Effect of Closeness on the Election of a Pairwise Majority Rule Winner
	1 Introduction
		1.1 Related Literature
		1.2 Our Contribution
	2 Preliminaries
	3 Results and Discussions
	4 Conclusion
	5 Appendix
	References
Analyzing the Practical Relevance  of the Condorcet Loser Paradox  and the Agenda Contraction Paradox
	1 Introduction
	2 Related Work
	3 Models and Definitions
		3.1 Stochastic Preference Models
		3.2 Voting Rules
		3.3 Voting Paradoxes
	4 Quantifying Voting Paradoxes
		4.1 Exact Analysis via Ehrhart Theory
		4.2 Finding a Quasi-polynomial for MaxiMin
		4.3 Experimental Analysis
		4.4 Empirical Analysis
	5 Condorcet Loser Paradox
		5.1 An Upper Bound
		5.2 Results Under IAC
		5.3 Results Under Other Preference Models
	6 Agenda Contraction Paradox
		6.1 Varying Voting Rules
		6.2 Varying Preference Models
	7 Conclusion
	References
Other Voting Paradoxes
On the Probability of the Ostrogorski Paradox
	1 Introduction
	2 The Case of Three Issues
	3 The Case of Partial Agreement on Three Issues
	4 The Impact of the Party Parity Assumption
	5 The Case of Two Issues
	6 The General Case of M Issues
	7 The General Case of M Issues—Partial Agreement
	8 The Case of Four Issues with IC
	9 Conclusion
	References
Violations of Reversal Symmetry Under Simple and Runoff Scoring Rules
	1 Introduction
	2 Preliminaries
	3 Reversal Symmetry in Universal Domain
		3.1 Violations of Reversal Symmetry According to the Number of Voters
		3.2 Infinite Electorate and Three Candidates
	4 Restricted Domain: Bipolar Preferences
		4.1 Three Candidates
		4.2 Large Electorates and Three Candidates
		4.3 Four Candidates
	5 Conclusion
	References
Binary Voting in Federations
Majority Efficient Representation  of the Citizens in a Federal Union
	1 Introduction
	2 Normative Criteria for Two-Tier Voting Rules
		2.1 Equalizing Power and Influence: The Penrose-Banzhaf Model
		2.2 Alternative Models
		2.3 Utility Based Arguments
	3 Methodology
		3.1 The Model
		3.2 On Probability Assumptions
		3.3 Simulation Techniques
	4 The General Study of the δ-Rules Via Simulations
		4.1 Staircase Curves
		4.2 Toward a General Result Under IAC
		4.3 How Far from Optimality Is the Square Root Rule?
	5 Conclusion
	References
``One Man, One Vote'' Part 1: Electoral Justice in the U.S. Electoral College: Banzhaf and Shapley/Shubik Versus May
	1 Introduction
	2 Notations and Definitions
		2.1 Two-Tier Weighted Majority Mechanisms
		2.2 Probability Models
	3 Electoral Justice in the Electoral College
		3.1 The 2010 US Electoral College and Population Data
		3.2 Electoral Justice with Respect to Banzhaf
		3.3 Electoral Justice with Respect to Shapley-Shubik
		3.4 Electoral Justice with Respect to May
	4 The Twelve Amendment
	5 Concluding Remarks
	References
Resistance to Manipulations
Further Results on the Manipulability of Social Choice Rules—A Comparison of Standard and Favardin–Lepelley Types of Individual Manipulation
	1 Introduction
	2 Social Choice Rules
	3 Extended Preferences and Manipulation
	4 Results
	5 Conclusion
	References
Exploring the No-Show Paradox for Condorcet Extensions
	1 Introduction
	2 Related Work
	3 Preliminaries
		3.1 Condorcet Extensions
		3.2 Strategic Abstention
		3.3 Stochastic Preference Models
	4 Quantifying the No-Show Paradox
		4.1 Exact Analysis via Ehrhart Theory
		4.2 Experimental Analysis
	5 Results and Discussion
		5.1 Analytical Results Under IAC
		5.2 Experimental Results Under IAC
		5.3 Comparing Different Preference Models
		5.4 Empirical Analysis
	6 Conclusion
	References
Susceptibility to Manipulation by Sincere Truncation: The Case of Scoring Rules and Scoring Runoff Systems
	1 Introduction
	2 Notation and Definitions
		2.1 Preferences
		2.2 Voting Rules
		2.3 Illustrating the Truncation Paradox
	3 The Vulnerability of Scoring Runoff Rules  to the Truncation Paradox in Three-Candidate Elections
		3.1 The Case of One-Shot Scoring Rules
		3.2 The Case of Scoring Runoff Rules
	4 The Impact of Single-Peaked Preferences
	5 Concluding Remarks
	References
Game Theory
Dummy Players and the Quota in Weighted Voting Games:  Some Further Results
	1 Introduction
	2 Notation, Definitions and Assumptions
	3 Preliminary Results
	4 Numerical Results
		4.1 Finite Case
		4.2 Infinite Case
	5 Conclusion and Final Remarks
	References
Who Wins and Loses Under Approval Voting? An Analysis of Large Elections
	1 Introduction
		1.1 Related Literature
	2 Elections
	3 The Electoral Game
	4 Approval Voting and the Condorcet Winner
	5 On the Indeterminacy of Approval Voting
		5.1 Three Candidates
		5.2 Viable* Candidates and Many Candidates
	6 Concluding Comments
	References
Techniques for Probability Computations
Combinatorics of Election Scores
	1 Introduction
	2 Framework
		2.1 ANEC Enumeration Problem
		2.2 Three Alternatives Case
	3 Voting Situations Induced by Voting Rules
		3.1 The Plurality Rule
		3.2 The Borda Rule
		3.3 The Discernibility Potential of the Scoring Rules
		3.4 The Kemeny Rule
	4 Conclusion
	5 Appendix
	References
From Gehrlein-Fishburn's Method on Frequency Representation to a Direct Proof of Ehrhart's extended Conjecture
	1 Introduction
	2 A Direct Proof of the Ehrhart's Conjecture
		2.1 How Does the Gehrlein–Fishburn's Method Work?
		2.2 General Framework
	3 Algorithms
		3.1 The Subdomain-Search Procedure
		3.2 Summing a Polynomial over a Feasible Hierarchical System
		3.3 The Sum of a Polynomial over a Polytope
	4 Further Illustrations and Applications
		4.1 Further Illustrations
		4.2 Some Applications
	5 The Continuous Case
	6 Discussion and Conclusion
	7 Appendix
	References
IAC Probability Calculations in Voting Theory: Progress Report
	1 Introduction
	2 Probabilities Calculations Under the IAC Condition
	3 The Algebraic Approach
	4 The Geometric Approach for Limiting Probabilities
	5 Huang–Chua Method and EUPIA Procedure
	6 Ehrhart Theory-Based Methods
	7 Concluding Remarks
	References