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

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Course Date: 25 August 2014 to 13 October 2014 (7 weeks)

Price: free

Course Summary

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.

Estimated Workload: 1-3 hours/week

Course Instructors

Eric Pacuit

Eric Pacuit is an Assistant Professor in the Department of Philosophy at the University of Maryland, College Park.   He has a masters degree in Mathematics from Case Western Reserve University and a PhD in computer science from the Graduate Center of the City University of New York. His primary research interests are in logic (especially modal logic), foundations of game theory, and social choice theory; he has secondary interests in (formal) epistemology and decision theory. Prior to coming to Maryland, he was a resident fellow at the Tilburg Institute for Logic and Philosophy of Science, a postdoctoral scholar at the Institute for Logic, Language and Information at the University of Amsterdam, and taught in the Philosophy and Computer Science departments at Stanford University. His work has been supported by a grant from the National Science Foundation and a VIDI grant from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO: the Dutch Science Foundation).

Course Description

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? 


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!  

What is the advanced track? 

      Each week there will be 1-2 lectures designated as "advanced track," and
      at least one quiz will be designated as "advanced track," These lectures
      will discuss somewhat more advanced topics and go into a bit more detail
      than what is found in the regular lectures (e.g., I may give a proof of a
      theorem discussed in other lectures). Of course, everyone is welcome to
      view these lectures and to attempt the more advanced quizzes. 


Week 1: Introduction to 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
    Advanced Lecture 1: 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
    Advanced Lecture 2: 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
    Advanced Lecture 3: 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
    Advanced Lecture 4: Proof of the Impossibility Theorem(s)

Week 6: Fair Division: Indivisible Goods
    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
    Advanced Lecture 5: Proof that Adjusted Winner is Envy Free, Efficient and Equitable

Week 7: Fair Division: 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


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.  

Suggested Reading

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

Course Workload

1-3 hours/week

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