# 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.

Preview Lectures

## Instructors

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)
Preferences
How Likely is the Condorcet Paradox?
Condorcet Consistent Voting Methods
Approval Voting
Combining Approval and Preference

Choosing How to Choose
Should the Condorcet Winner be Elected?
Failures of Monotonicity
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

Week 5: Aggregating Judgements
Voting in Combinatorial Domains
The Condorcet Jury Theorem
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?

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 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.

## FAQ

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

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: