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

Learn about different voting methods and fair division algorithms, and explore the problems that arise when a group of people need to make a decision.

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Course at a Glance

About the Course

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? 

Course Syllabus

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)    
    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

Recommended Background

No background is required; all are welcome!  

Suggested Readings

Suggested readings will include a selection of articles and other material available online.

Course Format

The class will consist of lecture videos, which are between 8-15 minutes in length.  
Each video will contain 1-2 integrated quizzes. There will also be standalone quizzes that are not part of the video lectures  and a (not optional) final exam.  


Will I get a Statement of Accomplishment after completing this class? 

      Yes. Students who successfully complete the class will receive a Statement of
      Accomplishment signed by the instructor.  

What resources will I need for this class?

For this course, all you need is an Internet connection, copies of the texts
       (most of which can be obtained for free), and the time to read, write, discuss,
       and think about this fascinating material.  

What is the coolest thing I'll learn if I take this class?

      In addition to learning about the many different types of voting methods that
      can be used the next time you are running an election, you will also learn
      the best way to cut a birthday cake!  

Why do some lectures have an asterisk (*) next to them? 

      If a lecture is labeled with an asterisk (*), then this means that the lecture is 
      considered an "advanced" lecture.   These lectures will discuss somewhat
      more advanced topics and go into a bit more detail than what is found in
      the regular lectures (for instance,  I may give a proof of a theorem discussed
      in other lectures).   These lectures are part of the course and everyone is
      encouraged to view them; however, you will not be tested on this material. 

Do I need to watch the supplemental lectures?  

      The supplemental lectures are intended to be general introductions to some of the
      mathematical notions that  come up  in the course.   These lectures are not
      intended to be watched one after another.   They are there to supplement my
      regular lectures (for example, if you find that I am using some mathematical notion
      or some notation that you are unfamiliar with).    One of my goals in this course is
      to try to pitch the material (some of which can get quite technical) to a general
      audience (many of whom may not have a background in math).    There are a
      variety of resources on the internet that can be used to supplement these
      lectures.  For instance, it may be useful to consult the following
      Wikepedia pages: