This course will introduce the key results (including proofs) and the main research themes in the study of judgement aggregation and the wisdom of the crowds. The course will focus on both logical and probabilistic models of judgement aggregation. The primary objective is to introduce the main mathematical methods and conceptual ideas found in this literature. In addition, I will state and prove the central results found in this literature. To facilitate understanding the main proof techniques and results, I will spend much of the time working through the theorems at the blackboard. The topics and theorems that I will discuss in the course include:
- The judgement aggregation model, Discursive Dilemma, and the Doctrinal Paradox
- Probabilistic opinion pooling
- The Condorcet Jury Theorem and its variants
- Opinion pooling (Lehrer-Wagner model, de Groot’s Theorem)
- Possibility and impossibility of aggregating judgements (Dietrich and List impossibility Theorem)
- Merging probabilistic opinions (Blackwell-Dubins Theorem)
- Aumann’s agreeing to disagree theorem and its generalizations
- Diversity trumps ability theorem (Hong-Page Theorem)
Of course, there is a large literature concerning each of the theorems and topics listed above. My goal will not to be to survey this entire literature. Instead, I present enough background material to state and prove each of the theorems mentioned above. This will provide students with a solid foundation to engage with this fascinating research area.