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 aggregation.
Chapter outline: We start by giving a brief overview of the history of social choice theory, from the contributions of Borda and Condorcet during the French Revolution (Section 1.1.1) until the general impossibility theorem by Arrow in 1951 that started modern social choice theory (Section 1.1.2). We then present the doctrinal paradox that originated the whole field of judgment aggregation (Section 1.2), and look at how judgment aggregation relates to the older theory of preference aggregation (Section 1.2.2). In the concluding section we point to literature at the interface of social choice theory, computer science and artificial intelligence, completing the sketch of the broad scientific context of the present book.
1.1 A CONCISE HISTORY OF SOCIAL CHOICE THEORY
Social choice theory studies how individual preferences and interests can be combined into a collective decision. An example of such a type of aggregation problems is a group electing one of many candidates on the basis of the preferences that the individuals in the group express over the candidates.
Collective decision-making is a constant feature of human societies. In his 1998 Nobel lecture, Amartya Sen [Sen99] recalled that already in the fourth century B.C., Aristotle in Greece and Kautilya in India explored how different individuals could take social decisions. However, the systematic and formal study of voting and committee decisions started only during the French Revolution, thanks to French mathematicians like Borda, Condorcet and Laplace.
Besides the problem of selecting the winning candidate in an election, social choice theory has its origins in the normative analysis of welfare economics [Sen86], a branch of modern economics that evaluates economic policies in terms of their effects on the social welfare of the members of the society. Welfare economics took inspiration from Jeremy Bentham’s utilitarianism rather than from Borda and Condorcet. According to Bentham’s utilitarianism fundamental axiom, “it is the greatest happiness of the greatest number that is the measure of right and wrong” [Ben76, Preface (2nd para.)]. Utilitarianisms assumed that individual preferences could be expressed by cardinal utilities and that they could then be compared across different individuals (interpersonally comparable preferences). Yet, in the 1930s both cardinality and interpersonal comparability of personal utility were questioned by Lionel Robbins [Rob38]. If utilities reflect individual mental states—Robbins argued—since it is not possible to measure mental states, then utilities cannot be compared across several individuals either. It was such “informational restriction” [Sen86] of social welfare to a n-tuple of ordinal (and interpersonally non-comparable) individual utilities that induced economists to look at methods developed in the theory of elections. As we shall see in Section 1.1.2, the informational restriction that followed Robbins’s criticisms made the problem of welfare economics look similar to the exercise of deriving a social preference ordering from individual orderings of social states, a problem addressed by Borda and Condorcet during the French Revolution.
1.1.1 THE EARLY HISTORY
On the history of the theory of elections McLean wrote:
The theory of voting is known to have been discovered three times and lost twice. The work of Condorcet, Borda, and Laplace was entirely ignored from about 1820 until 1952, with the sole exception of E. J. Nanson’s paper ‘Methods of Election’,
