Tracing the math in the U.S. presidential elections

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.

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.

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:

• • Majority Rules, a documentary about open primaries and ranked choice voting.

  • Instant runoff explained with Legos.

  • A nice video summarizing the advantages and disadvantages of instant runoff.

  • 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?

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 on Kenneth Arrow.

  • 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.

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.

  • Evaluating voting systems based on voter satisfation.

  • An organization that advocates for approval voting.

Class notes:

Multi-winner elections, covering

• • Plurality block voting

  • Discrete cumulative voting

Proportional representation, covering

• Multi-winner districts

• Single transferable vote

• D'Hondt method

• Sainte-Laguë method

Other materials:

• • What is the fairest electoral system?

  • More on discrete cumulative voting.

  • More on single transferable vote. And a video explaining it.

  • More on the D'Hondt method, including where it is used. And same for the Sainte-Laguë method.

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:

• • Various scenarios for the 2024 Electoral College outcomes.

  • 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.

  • You can read more about the 2016 elections here and here. 

  • 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. 

  • Does your vote count or not?

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.

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:

• • The Capitol can handle a larger House

  • 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).

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:

• • John Oliver show on gerrymandering

  • Slay the Dragon, a documentary about gerrymandering.

  • Mapmapker: The Gerrymandering Game, fun for the whole family.

  • An article about some of the most gerrymandered districts.

  • Is Massachusetts gerrymandered?

  • How gerrymandered is your district?

  • Are you any good at gerrymandering?

  • An article about the "Goofy kicking Donald Duck" district.

  • Some ways to ensure districting is fair and unbiased.

  • 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.

  • Compactness scores by state.

  • 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.

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.

• Prisoner's dilemma and environmental policy.

• More on the Nash equilibrium.

• A Beautiful Mind, movie about John Nash (and the famous scene when he comes up with the equilibrium theorem).

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.

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.

  • What is cryptography and why do we need it?

  • A brief history of number theory.

  • First crypto wars.

  • An article on Crypto War II.

  • About backdoors, and a backdoor in a random number generator.

  • History of the Consumer Privacy Bill of Rights.

  • Recent article about the continuing attempts at regulation of cryptography.

  • Links to many more materials (from a cryptography class I taught in spring '20).

• • Motivated by the Isoperimetric Inequality which has a new application in the Polsby-Popper score, we'll look at similar cool applications of "old math." This topic will only be covered if we have extra time left at the end of the term. This is just for fun, and there will be no homework on this topic.

• • 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...