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 given as running comments in the main text.

Outline of the book The book is structured in two parts. The first part (Chapters 1–4) introduces what can be considered the established body of the theory: the motivating examples behind judgment aggregation and its place within the field of social choice theory (Chapter 1); the logic-based framework for judgment aggregation (Chapter 2); some results, with proofs, on the impossibility of finding ‘ideal’ aggregation procedures (Chapter 3); and various ways that have been explored in the literature to work around the limits imposed by those results (Chapter 4). The second part (Chapters 5–7) touches upon topics that are, to a greater or lesser extent, still under development in the research agenda of the field: the issue of manipulation and strategic behavior in judgment aggregation (Chapter 5); the design of non-resolute aggregation procedures (Chapter 6); and the modeling of deliberative processes of aggregation, and of processes of pre-vote deliberation (Chapter 7). All chapters present an overview of key concepts and results and conclude with a section pointing to further topics and readings and, sometimes, open issues.

Davide Grossi and Gabriella Pigozzi

December 2013

Acknowledgments

The present book has grown out of materials developed for a course entitled “Introduction to Judgment Aggregation” taught at ESSLLI’11 (Ljubljana, Slovenia) and considerably extends in both depth and coverage the lecture notes we wrote for that course, published in [GP12]. Parts of the book have also been presented at the Stanford Logic Group Seminar of Stanford University (Spring 2011), and at the Computer Science Seminar of the Royal Holloway University of London (Autumn 2013). We are much indebted to the students who attended the ESSLLI’11 course and the participants of the above events for their questions, comments and suggestions. They provided essential feedback for the writing of the book.

We are greatly indebted to Denis Bouyssou, Umberto Grandi, Eric Pacuit and the anonymous reviewers arranged by the publisher, who thoroughly read earlier drafts of the book and gave us detailed and extremely helpful feedback to produce this final version. The book largely benefited also from discussions with Franz Dietrich, Ulle Endriss and Umberto Grandi. Any remaining errors and omissions are of course the sole responsibility of the authors. Finally, we would like to thank Michael Wooldridge for encouraging us to embark in this project, and Morgan & Claypool—in particular our editor Mike Morgan—for assisting us along the way and making the writing of this book such a pleasant journey.

Davide Grossi and Gabriella Pigozzi

December 2013

CHAPTER 1

Logic Meets Social Choice Theory

Judgment aggregation is a recent theory that combines aggregation problems previously studied by social choice theory with logic. Social choice theory is a vast subject, including not only the study of preference aggregation and voting theory but also topics like social welfare and justice. Given the tight links between judgment aggregation and preference aggregation, in this first chapter we give a concise survey on some historical aspects of preference aggregation and then introduce and motivate the more recent field of judgment aggregation. The chapter builds on [Sen99, Sen86, Bla58] for the historical overview on social choice theory, and on [KS93, Kor92] for the informal introduction to judgment

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