3.4.2 Infinite Agendas and Infinite Voters
3.4.3 Judgment Aggregation vs. Preference Aggregation
4.1 Relaxing Universal Domain
4.1.1 Unidimensional Alignment
4.1.2 Value-Restriction
4.2 Relaxing the Output Conditions
4.2.1 Abstention
4.2.2 Quota Rules
4.3 Relaxing Independence
4.3.1 The Premise-Based Approach
4.3.2 The Sequential Priority Approach
4.3.3 The Distance-Based Rules
4.4 Further Topics
4.4.1 More Domain Restrictions
4.4.2 Dropping Consistency
4.4.3 Other Distance-Based Rules
4.4.4 Judgment Aggregation and Abstract Argumentation
5.1 Types of Manipulation
5.1.1 Agenda Manipulation
5.1.2 Vote Manipulation
5.1.3 Manipulability: Definition and Characterization
5.1.4 Sincere and Insincere Manipulation
5.2 Non-Manipulable Aggregation: Impossibility
5.2.1 Auxiliary Results
5.2.2 The Impossibility Theorem
5.3 Further Topics: Manipulation Beyond Impossibility Results
5.3.1 The Possibility of Non-Manipulable Aggregation
5.3.2 Strategy-Proof Judgment Aggregation
5.3.3 Complexity as a Safeguard Against Manipulation
6.1 Introduction
6.2 Rules Based on the Majoritarian Judgment Set
6.3 Rules Based on the Weighted Majoritarian Judgment Set
6.4 Rules Based on the Removal or Change of Individual Judgments
6.5 Further Topics
7.1 Deliberation and Opinion Pooling
7.1.1 Probabilistic Judgments
7.1.2 A Stochastic Model of Deliberation
7.1.3 Opinion Pooling and Judgment Aggregation
7.2 Deliberation as Judgment Transformation
7.2.1 Deliberation and Voting
7.2.2 Judgment Transformation Functions
7.2.3 Examples of Transformation Functions
7.3 Limits of Judgment Transformation
7.3.1 Conditions on Transformation Functions
7.3.2 An Impossibility Result
7.4 Further Topics and Open Issues
Preface
This book concerns the aggregation of individual opinions into group opinions. When opinions exhibit logical structure (e.g., accepting that p is the case and accepting that p implies q compels me to also accept that q is the case) aggregation becomes difficult. Is it possible at all to find aggregation procedures that preserve compliance with logical principles, and that at the same time appeal to democratic criteria like, for instance, not being dictatorial? Are the methods we commonly use to aggregate our opinions (e.g., majority voting) appropriate, and under which conditions? And if, after all, ideal procedures turn out to be impossible, what are the reasons for such impossibility? Questions like these are the playground of judgment aggregation, and will be the topic of this book.
Before starting, the reader can find here some information about the main objectives we pursued by writing the book, the readership we aimed at, and an outline of the topics we are going to cover.
Objectives In writing this introductory book on judgment aggregation we had two main objectives in mind. First, we wanted to provide a compact and systematic exposition of the problems, definitions, results and proof techniques that drive the field. Survey papers appeared in philosophical and social sciences journals and volumes [LP09, Car11, Mon11, Lis12], but no comprehensive exposition of the field is available to date. Second, we wanted to make the theory of judgment aggregation accessible, in a ‘sympathetic’ format, to the disciplines of artificial intelligence and multi-agent systems, which in recent years have increasingly been concerned with the problems of aggregation and voting.
Readership and prerequisites The book is primarily meant as an introduction to the field of judgment aggregation for graduate students and researchers in computer science, artificial intelligence and multi-agent systems. At the same time, it has been our aim to make the book accessible also to mathematically minded graduate students and researchers in philosophy, the social and the political sciences. The material is presented in such a way to presuppose only familiarity with propositional logic and basic discrete mathematics. The book intends to put the reader at pace with the field, enabling the key conceptual, technical and bibliographical tools to understand (and possibly contribute to) its current developments. We have not included any exercises, but the reader will be asked at times to complete missing steps in proofs or try to prove statements
