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)
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.
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).
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
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
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
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
Advanced Lecture 2: Proof of Arrow's Theorem
Variants of Arrow's Theorem
Week 4: Topics in Social Choice Theory
Domain Restrictions: Single-Peakedness
Sen’s Value Restriction
Manipulating Voting Methods
Advanced Lecture 3: Lifting Preferences
The Gibbard-Satterthwaite Theorem
Sen's Liberal Paradox
Week 5: Aggregating Judgements
Voting in Combinatorial Domains
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
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
The Stromquist Procedure
The Selfridge-Conway Procedure
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 readings will include a selection of articles and other material available online.