Class notes and other materials
Tracing the math in the U.S. presidential elections
Most of the material we will cover arises by tracing what happens to a vote that is cast in the presidential elections. The main body of the flowchart on the left, shaded in blue, gives the sequence of steps that lead from the moment a ballot is cast to the official declaration of the winner. Behind each step is interesting matematics that often leads to even more interesting mathematics, as depicted in the green boxes. We will follow this flowchart and examine what lies behind each of the topics listed in it.
Introduction
Class notes:
Introduction to MATH 123 Mathematics and Politics, covering
Why this class?
What this class is about
Quantitative literacy in politics
Goals of the class
Topics we will try to cover
Other materials:
There are no additional materials for this class meeting.
Majority and plurality voting
Class notes:
Majority vote, covering
Simple majority and supermajority
Near-decisiveness
Quota method
Parity, monarchy, dictatorship methods
Anonymity, neutrality, monotonicity properties
May's Theorem
Plurality vote, covering
Plurality method
Vote splitting
2016 Republican primaries
1998 Minnesota gubernatorial race
2018 and 2020 Massachusetts Democratic state primaries
Spoiler effect
2000 presidential race
Runoff election
2021 Georgia senate runoff elections
Other materials:
Good summary of problems with plurality voting, some of which we talked about in class.
More on Duverger’s Law.
More on the center-squeeze effect.
Crazy example of vote splitting in a recent election in Massachusetts.
Crazy example of voting results in the UK.
More examples of vote splitting/spoiler effect.
A video introducing various voting methods and the Condorcet paradox.
Crazy/fun election facts from around the world. And some more.
Some strange ways in which ties are broken in the U.S.
Ranked voting
Class notes:
Ranked voting methods, covering
Instant runoff
Borda count
Condorcet method
An example using all the methods, covering
An example with a single profile producing six different outcomes depending on the method (done as a group activity)
The problem of extracting group preferences from individual preferences
The rise of social choice theory
Other materials:
Advocates of various voting methods: Instant runoff, range voting, approval voting, cumulative voting.
An interactive guide to the voting methods we’ve seen (and then some). And here is a more condensed version.
Another interactive visual guide to voting.
Summary of ranked voting methods and where they are used.
An article from the New York Times arguing for instant runoff in primaries.
Another article from the New York Times about how deciding who to rank first is not always straightforward.
Could the Civil War have been avoided if Borda count was used?
What went wrong with the Massachusetts referendum on ranked choice in 2020?
Social choice theory and Arrow Impossibility
Class notes:
Some strange examples, covering
Various counterintuitive and paradoxical voting outcomes such as
Condorcet paradox
Paradox of positive association
Failure of the majority criterion
Failure of independence of irrelevant alternatives (featuring the 1995 Figure Skating World Championship)
Social choice theory, covering
Definition of social choice and social welfare functions
Anonymity, neutrality, monotonicity, majority, and non-dictatorship criteria (all of which we already saw)
Condorcet Criterion
Pareto Criterion
Independence of Irrelevant Alternatives
Methods we have seen that do and do not satisfy these criteria
Arrow Impossibility Theorem, covering
Arrow Impossibility and its interpretation
Outline of proof
Gibbard-Satterthwaite Theorem
Now what?
Other materials:
Video explaining the Arrow Impossibility Theorem and outlining its proof.
More formal treatment of the Arrow Impossibility Theorem.
Proof of the Impossibility Theorem.
Arrow's original paper on the Impossibility Theorem.
More on the Gibbart-Satterthwaite Theorem, and here is a proof.
Cardinal voting
Class notes:
Cardinal methods, covering
Ordinal vs cardinal methods
Approval voting
Range voting
Cumulative voting
Other materials:
A good summary or ordinal and cardinal voting systems.
An organization that advocates for approval voting.
Electing more than one candidate
Class notes:
Electing more than one candidate, covering
Discrete cumulative voting
Single transferable vote
Other materials:
More on discrete cumulative voting.
More on single transferable vote. And a video explaining it.
Electoral College
Class notes:
Electoral College, covering
What is the Electoral College?
2016 and 2020 presidential elections
Popular vs. electoral vote
Representative democracy and "one person, one vote"
Big vs. small states and the “+2 effect”
Alternatives to the Electoral College
Other materials:
More details about the Electoral College.
A video explaining the Electoral College.
An excellent cartogram giving the Electoral College results for all the presidential elections.
Article on the history of the Electoral College and some ways to fix it.
Here is a site that encodes the difference between the popular votes and the electoral college votes for presidential elections since 1972.
What if the states allocated their electoral votes in different ways? Here is the answer.
All you need is 23% of the popular vote to win the Electoral College! Here is how.
Crazy example of the popular vote gone wrong in the UK elections.
More on the National Popular Vote Interstate Compact.
Quantification of Power
Class notes:
Weighted voting, covering:
Weighted voting
Electoral College revisited
UN Security Council as a weighted voting system
U.S. legislative system
Taylor-Zwicker Theorem
1991 U.S. Senate and the power of Jim Jeffords
Examples illustrating the importance of vote distribution
Banzhaf Power Index, covering
Banzhaf Power Index
Power in the European Economic Community
Power in the Electoral College
Shapley-Shubik Power Index, covering:
Shapley-Shubik Power Index
Power of the U.S. President
Calculating the power of the President of the United States (notes coming soon)
Calculating the power of the members of the UN Security Council
Other materials:
More on John Banzhaf, with links to materials on the Banzhaf power index.
Calculator that computes the Banzhaf power index of any weighted voting system.
More on the Shapley-Shubik power index. And here is another document that explains it and works out some example.
Apportionment
Class notes:
Why 435?, covering
The Constitutional Convention
1929 Reapportionment Act
Cube root law
Wyoming rule
Hamilton apportionment method, covering
apportionment problem
Hamilton apportionment method
Alabama paradox
New states paradox
Population paradox
Other apportionment methods, covering
Jefferson method
Adams method
Webster method
Dean method
Huntington-Hill method
Apportionment criteria and the Balinski-Young Theorem, covering
Quota rule
Neutrality
Balinski-Young Theorem
Other materials:
History and legislation of apportionment in the U.S. by the Census Bureau.
An NPR article about the history of the House getting stuck at 435 representatives.
The case for enlarging the House (includes an explanation of a typo in the Constitution).
Another case for making the House bigger: part one and part two.
More on the cube root law, and why it might be flawed.
How the 2000 election would have turned out with different House sizes. And here is the 2016 election.
Trial-and-error way to find the best House size.
Apportionment around the world.
Apportionment in the European Parliament.
Series of lectures on apportionment, voting, and gerrymandering by Michel Balinski (of the Balinski-Young Theorem).
Gerrymandering
Class notes:
Introduction to gerrymandering, covering
Why gerrymandering is sometimes good
Recent examples of bad gerrymandering
Racial v. political gerrymandering
Gerrymandering in the courts
Efficiency gap, covering
Packing and cracking
Wasted votes and the efficiency gap
Problems with the efficiency gap
Geometry of gerrymandering, covering
Isoperimetric Inequality
Polsby-Popper compactness score
Schwartzberg, Reock, convex hull, length-width, X-symmetry scores
Problem with compactness scores and possible solutions
Other materials:
Slay the Dragon, an excellent documentary about gerrymandering.
An article about some of the most gerrymandered districts.
An article about the "Goofy kicking Donald Duck" district.
Metric Geometry and Gerrymandering Group, a collection of cool people trying to use geometry and computing to counter gerrymandering in a rigorous way. And a talk on the mathematics of gerrymandering by its founder, Prof. Moon Duchin from Tufts.
An article explaining a recent Supreme Court decision against challenges to gerrymandering.
The original article introducing the efficiency gap.
More on the efficiency gap, and on some of its flaws. And a more technical paper on its flaws. And another paper about its flaws.
Atlas of redistricting, an interactive site where you can play with gerrymandering.
More on the Isoperimetric Inequality.
An article summarizing various geometric criteria for checking gerrymandering.
An article laying out issues with compactness scores.
A font created with congressional districts; use it to write to your representative!
Natural gerrymandering by a 100 million-old coastline.
Game theory
Class notes:
Prisoner's dilemma, covering
Classical prisoner's dilemma
Tragedy of the commons
Congress vs. the Fed
Arms race
Nash equilibrium, covering
Basics of game theory
Nash equilibrium
Game of chicken
1962 Cuban missile crisis
Hotelling's game
Median Voter Theorem
Fair division problem from the Talmud
Other materials:
Good introductory video on game theory.
Good summary of the main features of game theory.
An article referenced in class on the connection between the prisoner's dilemma and the strategy for COVID-19 vaccinations.
More on the Nash equilibrium.
A Beautiful Mind, movie about John Nash (and the famous scene when he comes up with the equilibrium theorem).
Statistics
Class notes:
Statistics in politics, covering
Basic ways in which statistics is used for political purposes
Examples of faulty visualisations, misuses of large numbers, cherry picking, etc.
Other materials:
Summary of some common misuses of statistics.
Examples of misleading graphs.
Some hilarious correlations (we showed some of these in class).
An oganization trying to improve statistical literacy.
An article about the growing mistrust in statistics.
A video about the misuses of statistics. And another one.
Cryptography
Class notes:
Cryptography and privacy, covering
Brief history of cryptography
Public key cryptography
Diffie-Hellman and RSA
Issues of cryptography and privacy
Regulation of cryptography vs. civil liberties
First Crypto Wars
Snowden revelations and cryptography
Cryptography as dual-use technology
Other materials:
Fun clip from the show Silicon Valley.
An article on Crypto War II.
About backdoors, and a backdoor in a random number generator.
Recent article about the continuing attempts at regulation of cryptography.
Links to many more materials (from a cryptography class I taught in spring '20).
Some final thoughts
Some final thoughts, covering
a reminder of why we must insist on political quantitative literacy
a list of topics we would cover if we had an infinite amount of time, such as
More social choice theory and game theory
Proofs of Arrow, Gibbard-Satterthwaite, and Balinski-Young Theorems
Strategic voting
Other measures of gerrymandering like mean-median score and partisan bias
More geometry of gerrymandering, closer look at compactness scores
More math behind some basic cryptosystems, more on the politics of privacy
More statistics in politics: methods, misuses, polling, ...
Bias of mathematical models
Graph theory of social networks and voter manipulation
Quantitative literacy and political bias
Political quantitive literacy in K-12 education
High-powered math in voting: representation theory, category theory, combinatorial topology, ...
And so much more...