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 LecturesMuch 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?

The Voting Problem

A Quick Introduction to Voting Methods (e.g., Plurality Rule, Borda Count,

Plurality with Runoff, The Hare System, Approval Voting)

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

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

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

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

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)

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

The Cake Cutting Problem

Cut and Choose

Equitable and Envy-Free Proocedures

Proportional Procedures

The Stromquist Procedure

The Selfridge-Conway Procedure

Concluding Remarks

No background is required; all are welcome!

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

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? **

Accomplishment signed by the instructor.

(most of which can be obtained for free), and the time to read, write, discuss,

and think about this fascinating material.

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!

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.

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:

- https://en.wikipedia.org/wiki/Table_of_mathematical_symbols
- https://en.wikipedia.org/wiki/Naive_set_theory