Making Better Group Decisions: Voting, Judgement Aggregation and Fair Division

Much of our daily life is spent taking part in various types of what we might call “political” procedures. Examples range from voting in a national election to deliberating with others in small committees. Many interesting philosophical and mathematical issues arise when we carefully examine our group decision-making processes. 

There are two types of group decision making problems that we will discuss in this course. A voting problem: Suppose that a group of friends are deciding where to go for dinner. If everyone agrees on which restaurant is best, then it is obvious where to go. But, how should the friends decide where to go if they have different opinions about which restaurant is best? Can we always find a choice that is “fair” taking into account everyone’s opinions or must we choose one person from the group to act as a “dictator”? A fair division problem: Suppose that there is a cake and a group of hungry children. Naturally, you want to cut the cake and distribute the pieces to the children as fairly as possible. If the cake is homogeneous (e.g., a chocolate cake with vanilla icing evenly distributed), then it is easy to find a fair division: give each child a piece that is the same size. But, how do we find a “fair” division of the cake if it is heterogeneous (e.g., icing that is 1/3 chocolate, 1/3 vanilla and 1/3 strawberry) and the children each want different parts of the cake? 

Week 1:  Voting Methods     The Voting Problem     A Quick Introduction to Voting Methods (e.g., Plurality Rule, Borda Count,  
          Plurality with Runoff, The Hare System, Approval Voting)         Preferences     The Condorcet Paradox     How Likely is the Condorcet Paradox?     Condorcet Consistent Voting Methods     Approval Voting     Combining Approval and Preference     Voting by Grading
Week 2: Voting Paradoxes     Choosing How to Choose     Condorcet's Other Paradox     Should the Condorcet Winner be Elected?     Failures of Monotonicity     Multiple-Districts Paradox     Spoiler Candidates and Failures of Independence     Failures of Unanimity     Optimal Decisions or Finding Compromise?     Finding a Social Ranking vs. Finding a Winner
Week 3: Characterizing Voting Methods     Classifying Voting Methods     The Social Choice Model     Anonymity, Neutrality and Unanimity     Characterizing Majority Rule     Characterizing Voting Methods     Five Characterization Results     Distance-Based Characterizations of Voting Methods     Arrow's Theorem     Proof of Arrow's Theorem     Variants of Arrow's Theorem
Week 4: Topics in Social Choice Theory     Introductory Remarks     Domain Restrictions: Single-Peakedness     Sen’s Value Restriction     Strategic Voting     Manipulating Voting Methods     Lifting Preferences     The Gibbard-Satterthwaite Theorem     Sen's Liberal Paradox
Week 5: Aggregating Judgements     Voting in Combinatorial Domains     Anscombe's Paradox     Multiple Elections Paradox     The Condorcet Jury Theorem     Paradoxes of Judgement Aggregation     The Judgement Aggregation Model     Properties of Aggregation Methods     Impossibility Results in Judgement Aggregation     Proof of the Impossibility Theorem(s)
Week 6: Fair Division      Introduction to Fair Division     Fairness Criteria     Efficient and Envy-Free Divisions     Finding an Efficient and Envy Free Division     Help the Worst Off or Avoid Envy?     The Adjusted Winner Procedure     Manipulating the Adjusted Winner Outcome
Week 7:  Cake-Cutting Algorithms    The Cake Cutting Problem    Cut and Choose    Equitable and Envy-Free Proocedures    Proportional Procedures    The Stromquist Procedure    The Selfridge-Conway Procedure    Concluding Remarks