Judgment Aggregation. Gabriella Pigozzi

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       ABSTRACT

      Judgment aggregation is a mathematical theory of collective decision-making. It concerns the methods whereby individual opinions about logically interconnected issues of interest can, or cannot, be aggregated into one collective stance. Aggregation problems have traditionally been of interest for disciplines like economics and the political sciences, as well as philosophy, where judgment aggregation itself originates from, but have recently captured the attention of disciplines like computer science, artificial intelligence and multi-agent systems. Judgment aggregation has emerged in the last decade as a unifying paradigm for the formalization and understanding of aggregation problems. Still, no comprehensive presentation of the theory is available to date. This Synthesis Lecture aims at filling this gap presenting the key motivations, results, abstractions and techniques underpinning it.

       KEYWORDS

      judgment aggregation, collective decision-making, logic, social choice theory, computational social choice, preference aggregation, voting paradoxes, aggregation rules, impossibility results, manipulability, ultrafilters, opinion pooling, deliberation

       DG—to my grandmother Alice, in memoriam

       GP—to my mother and to the memory of my father

       Contents

       Preface

       Acknowledgments

       1 Logic Meets Social Choice Theory

       1.1 A Concise History of Social Choice Theory

       1.1.1 The Early History

       1.1.2 Modern Social Choice Theory

       1.2 A New Type of Aggregation

       1.2.1 From the Doctrinal Paradox to the Discursive Dilemma

       1.2.2 Preference Aggregation and Judgment Aggregation

       1.3 Further Topics

       2 Basic Concepts

       2.1 Preliminaries

       2.1.1 Agendas in Propositional Logic

       2.1.2 Judgment Sets and Profiles

       2.1.3 Aggregation Functions

       2.1.4 Examples: Aggregation Rules

       2.2 Agenda Conditions

       2.2.1 How Interconnected is an Agenda?

       2.2.2 Comparing Agenda Conditions

       2.3 Aggregation Conditions

       2.3.1 How Should an Aggregation Function Behave?

       2.3.2 On the ‘Meaning’ of the Aggregation Conditions

       2.4 Further Topics

       2.4.1 Abstract Aggregation

       2.4.2 General Logics

       3 Impossibility

       3.1 What is the Majority Rule Like?

       3.1.1 Properties of Propositionwise Majority

       3.1.2 Characterizing Propositionwise Majority

       3.2 An Impossibility Theorem

       3.2.1 Winning Coalitions

       3.2.2 Winning Coalitions as Ultrafilters

       3.2.3 Dictators

       3.2.4 The Theorem

       3.3 (Ultra)filters, Dictators and Oligarchs

       3.3.1 Impossibility of Non-Oligarchic Aggregation

       3.3.2 Proof: from Ultrafilters to Filters

       3.3.3 Impossibility via (Ultra)filters

       3.4 Further Topics

       3.4.1 Other Impossibility Results

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