Handbook on Approval Voting

Preface

Acknowledgements

Contents

Contributors

1 Introduction to the Handbook on Approval Voting

References

Part I History of Approval Voting

2 Acclamation Voting in Sparta: An Early Use of Approval Voting

References

3 Going from Theory to Practice: The Mixed Success of Approval Voting

3.1 Background

3.2 Early History and Rhetoric

3.3 The Adoption Decisions in the Societies5

3.4 Does Approval Voting Make a Difference?

3.5 Does Approval Voting Elect the Lowest Common Denominator?

3.6 Is Voting Ideological?

3.7 Summary and Conclusions

References

Part II Axiomatic Theory

4 Collective Choice for Simple Preferences

4.1 Introduction

4.1.1 Preliminaries

4.1.2 Classical Impossibility Results

4.1.2.1 Arrow's Theorem

4.1.2.2 Gibbard–Satterthwaite Theorem

4.1.2.3 Sen's Paretian Liberal Paradox

4.2 Possibility Results on Some Dichotomous Domains

4.2.1 Two Alternatives

4.2.2 Two Fixed Indifference Classes

4.2.3 Two Indifference Classes

4.2.4 Two Fixed Classes Separated by Strict Preferences

4.3 Axiomatic Foundations for Majority Decisionand Approval Voting

4.3.1 Characterizations of Majority Voting Systems

4.3.1.1 Basic Axioms in the Fixed Population Model

4.3.1.2 Characterization Results: Voting Systems

4.3.1.3 Extension in the Variable Population Framework

4.3.2 Characterizations of Majority Voting in the Variable Population Framework

4.3.2.1 Basic Axioms of Voting Rules

4.3.2.2 Scoring Rules

4.3.2.3 Characterizations of Majority Voting

4.4 Strategic Voting and Condorcet Principle

4.4.1 Strategic Voting Under Anonymous Voting Systems

4.5 Unconstrained Multi-issue Problems and Voting by Committees

4.5.1 Strategy-Proofness and Separable Preferences

4.5.2 Strategy-Proofness Versus Efficiency and Domain Restrictions

4.6 Simple Opinion Aggregation and Decision by Powers and Consent

4.6.1 Social Decision by Powers and Consent for Dichotomous or Trichotomous Opinion Aggregation

4.6.2 Group Identification

4.6.2.1 Liberalism and Axiomatic Characterizations

4.6.2.2 Consent-Based Choice Functions

4.6.3 Simple Preferences and the Paradox of Paretian Liberal

References

5 Axiomatizations of Approval Voting

5.1 Introduction

5.2 Notation and Definitions

5.3 Variable Electorate

5.4 Fixed Electorate

5.5 Variable Electorate with a Single Ballot Response Profile

References

Part III Committees

6 Approval Balloting for Multi-winner Elections

6.1 Introduction

6.2 The Setting

6.3 Scoring Procedures

6.3.1 (Simple) Approval

6.3.2 Net Approval

6.3.3 Satisfaction

6.3.4 Net Satisfaction

6.3.5 Representativeness

6.3.6 Proportional Approval

6.3.7 Sequential Proportional Approval

6.4 Threshold Procedures

6.5 Centralization Procedures

6.6 Conclusions

References

7 Does Choosing Committees from Approval Balloting Fulfill the Electorate's Will?

7.1 Introduction

7.2 Voting from Approval Ballots

7.2.1 Candidate-Wise Voting Rules

7.2.2 Preferences Over Committees

7.3 Representativeness of Voting Rules Under Separable Preferences

7.3.1 Pareto Efficiency

7.3.2 Condorcet Properties

7.3.2.1 Voting Paradoxes

7.3.2.2 Avoiding Paradoxes

7.3.3 Compromising Through Majority Voting

7.4 Non-separable Preferences

References

Part IV Strategic Voting

8 The Basic Approval Voting Game

8.1 Introduction

8.2 The Normal Form Game

8.3 Admissibility and Sincerity

8.3.1 Admissible Strategies

8.3.2 Sincerity

8.4 Approval Voting Outcomes

8.4.1 Admissible and Sincere Outcomes

8.4.2 Equilibrium Outcomes

8.5 Conclusion

References

9 Approval Voting in Large Electorates

9.1 Introduction

9.2 The Myerson–Weber Framework

9.2.1 Voting Equilibrium in the Myerson–Weber Framework

9.2.2 Comparison of AV and PV in the Myerson–Weber Framework

9.3 The Score Uncertainty Model

9.3.1 Voting Equilibrium in the Score Uncertainty Model

9.3.2 Approval Voting on the Score Uncertainty Model

9.4 Large Poisson Games

9.4.1 Voting Equilibrium on Large Poisson Games

9.4.2 The Decision Process

9.4.3 Computing Magnitudes

9.4.4 Approval Voting and Plurality Voting on Large Poisson Games

9.4.5 AV Does Not Satisfy Condorcet Consistency on Large Poisson Games

9.5 Conclusion

References

10 Computational Aspects of Approval Voting

10.1 Introduction

10.2 Discussion of Models for Control of Domain-Sensitive Rules

10.3 The Control Complexity of Approval Voting and Sincere-Strategy Preference-Based Approval Voting

10.3.1 Introduction, Background, and Discussion

10.3.2 Notions: Elections and Complexity

10.3.3 44 Control Results

10.3.3.1 Adding and Deleting Candidates

10.3.3.2 The Partition Cases

10.3.3.3 Adding and Deleting Voters

10.3.4 Conclusions

10.4 The Complexity of Bribery for Approval Voting

10.5 Local Search Heuristics for Minimax Approval Voting

10.5.1 Minisum and Minimax Approval Voting

10.5.2 NP-Hardness and Approximability of Fixed-SizeMinimax Approval Voting

10.5.3 Local Search Heuristics

References

Part V Probabilistic Exercises

11 On the Condorcet Efficiency of Approval Voting and Extended Scoring Rules for Three Alternatives

11.1 Introduction

11.2 The Model

11.2.1 Preferences

11.2.2 Voting Rules

11.2.3 Extending the Impartial Culture Hypothesis

11.2.4 The Likelihood of a Condorcet Winner Under EIC

11.3 The Probability Calculations: Robustness of the Ranking of a Pair

11.3.1 The Optimal Extended Scoring Rule on Pairs

11.3.2 Comparing Borda Count and Approval Voting

11.3.3 Optimal Extensions for Plurality and Antiplurality

11.4 The Condorcet Efficiency

11.4.1 Deriving the Condorcet Efficiency for Extended Scoring Rules

11.5 Conclusion

References

12 Behavioral Heterogeneity Under Approvaland Plurality Voting

12.1 Introduction

12.2 Strategic Behavior under Approval and Plurality Voting

12.2.1 A Signal Extraction Model for the Pivot Probabilities

12.3 Simulation and Mixed Behavior Setups

12.4 Simulation Results

12.4.1 Non-systematic Behavioral Heterogeneity

12.4.2 Systematic Behavioral Heterogeneity

12.4.2.1 Engaging Setups: Plurality Voting

12.4.2.2 Engaging Setups: Approval Voting

12.4.2.3 Abstaining Setups

12.4.3 A Comparison

12.4.3.1 The Consequences of Intensity Information in the Signals

12.5 Conclusions

References

13 In Silico Voting Experiments

13.1 Introduction

13.2 Cultures

13.2.1 Rousseauist Cultures

13.2.2 Impartial Culture

13.2.3 Distributive Cultures

13.2.3.1 Consensual Redistributive Culture

13.2.3.2 Inegalitarian Distributive Cultures

13.2.4 Spatial Cultures

13.2.4.1 Uni-dimensional Spatial Culture

13.2.4.2 Multi-dimensional Cultures

13.3 The Decision Process: Voting Rule and Behavior

13.3.1 Sincere Voting

13.3.2 Responsive Voting

13.3.2.1 Plurality Voting

13.3.2.2 Approval Voting

13.3.2.3 Borda Voting

13.3.2.4 Example

13.3.2.5 Discussion

13.4 Results

13.4.1 Results for Rousseauist Cultures

13.4.2 Results for Impartial Cultures

13.4.3 Results for Consensual Distributive Cultures

13.4.4 Results for Inegalitarian Distributive Cultures

13.4.5 Results for Spatial Cultures

13.4.5.1 Results for Uni-dimensional Culture

13.4.5.2 Results for Multi-dimensional Culture

13.5 Conclusion

References

Part VI Experiments

14 Laboratory Experiments on Approval Voting

14.1 Introduction

14.2 Experiments on a Divided Society

14.2.1 The Preference Profile

14.2.2 First Protocol, on the Internet

14.2.3 Results of the Internet Protocol

14.2.4 Second Protocol, in the Laboratory

14.2.5 Results of the Laboratory Protocol

14.2.6 Conclusion on Divided Society Experiments

14.3 Experiments on a Split Majority

14.3.1 The Preference Profile

14.3.2 The Protocol

14.3.3 Results

14.3.4 Conclusion on Split Majority Experiments

14.4 Experiments on a Political Domain

14.4.1 The Preference Profile

14.4.2 The Protocol

14.4.3 Results: Electoral Outcomes

14.4.4 Results: Individual Behavior

14.4.4.1 Number of Approbations and Sincerity

14.4.4.2 Strategic Behavior

14.4.5 Conclusion on Political Domain Experiments

14.5 Conclusion

14.5.1 Consensual Outcomes

14.5.2 Voting Behavior

References

15 Framed Field Experiments on Approval Voting: Lessons from the 2002 and 2007 French Presidential Elections

15.1 Introduction

15.2 Experimental Design

15.2.1 The Experimental Voting Rules and Ballots

15.2.2 The Pilot Experiments

15.2.3 Information for Participants

15.2.4 The Experiments

15.2.5 Questionnaires

15.2.6 Participating in a Public Debate

15.3 General Lessons from AV Experiments

15.3.1 Participation

15.3.2 Positive Public Response to AV

15.4 Specific Lessons Regarding AV

15.4.1 Comparing the Outcomes of AV with the Official First-Round Vote

15.4.2 Examining the Conversion of Approval into Single-Name Balloting

15.4.3 Analyzing the French Political Context

15.5 Concluding Remarks

References

16 Approval Voting in Germany: Description of a Field Experiment

16.1 Introduction

16.2 The Official Election

16.3 The Experiment

16.3.1 District Election

16.3.2 State Election

16.3.3 The Messel Parliament

16.3.4 Further Remarks

16.3.4.1 Coalitions

16.3.4.2 Number of Votes

16.3.4.3 Visibility of the Small Parties

16.4 Afterword

16.4.1 Repetition of the Election in Hessen

16.4.2 A Second Experiment in Germany

16.4.3 Final Words

References

Part VII Electoral Competition

17 Classical Electoral Competition Under Approval Voting

17.1 Introduction

17.2 The Model

17.2.1 Candidates, Voters, and Preferences

17.2.2 Individual Voting Behavior

17.2.3 Electorate Voting

17.2.4 Equilibrium

17.3 Results

17.3.1 Condorcet-Consistency

17.3.2 Median Convergence

17.3.3 Comparison with Plurality Voting

17.4 Extensions

References

18 Policy Moderation and Endogenous Candidacy in Approval Voting Elections

18.1 Introduction

18.2 Model

18.3 Sincere Voting

18.3.1 One-Position Serious Equilibria

18.3.2 Two-Position Serious Equilibria

18.3.3 Multiposition Serious Equilibria and Spoiler Equilibria

18.4 Strategic Voting

18.4.1 One-Position Serious Equilibria

18.4.2 Two-Position Serious Equilibria

18.4.3 Multiposition Serious Equilibria

18.4.4 Spoiler Equilibria

18.5 Relative Sincerity

18.5.1 One-Position Serious Equilibria

18.5.2 Two-Position Serious Equilibria

18.5.3 Multi-Position Serious Equilibria

18.5.4 Spoiler Equilibria

18.6 Conclusion

References

Part VIII Meaning for Individual and Society

19 Describing Society Through Approval Data

19.1 Introduction

19.2 Approval Scores

19.3 Endogenous Electoral Proximity

19.4 The Canonical Representation

19.5 A Connection with Spatial Voting

19.6 What are the Results of an AV Election?

References

20 Approval as an Intrinsic Part of Preference

20.1 The Model

20.2 A Benchmark: The Majoritarian Approval Axiom

20.3 Four ``New'' Social Choice Rules

20.4 Monotonicity and Independence

20.5 Concluding Remarks

References