Class notes:

Introduction to MATH123 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:

Electoral College, covering

• • What is the Electoral College?

  • 2016 presidential elections

  • Popular vs. electoral 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.

  • 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.

  • Under very reasonable assumptions, 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?

  • Could the Civil War have been avoided if Borda count was used?

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)

  • Spoiler effect (2000 presidential race)

  • Runoff election

Ranked choice 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

  • The problem of extracting group preferences from individual preferences

  • The rise of social choice theory

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)

Desirable voting method properties, covering

• • 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 interpretation

  • Outline of proof

  • Gibbard-Satterthwaite Theorem

  • Now what?

Cardinal methods, covering

• • Ordinal vs cardinal methods

  • Approval voting

  • Range voting

  • Cumulative voting

Electing more than one candidate, covering

• • Discrete cumulative voting

  • Single transferable vote

Class vote on Super Tuesday, covering

• • Our votes on the Democratic candidates, tallied in eight different ways. (Spoiler: Warren won most of the time!)

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. And here are some more.

  • A video introducing various voting methods and the Condorcet paradox.

  • Crazy/fun election facts from around the world. And some more. And some more.

  • Some strange ways in which ties are broken in the U.S.

  • 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.

  • 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.

  • More on discrete cumulative voting.

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

  • Jusr in time for our class Super Tuesday vote, here is an article from the New York Times arguing for instant runoff in primaries.

Class notes:

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

  • 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 and videos:

Weighted voting

• • Lecture notes,

  • Video summary of what we did in class (we transitioned to online instruction after this class, so this video is intended to remind you what happened in that class)

  • Topics covered:

    • 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

• • Lecture notes

  • Lecture videos: Banzhaf power 1, Banzhaf power 2, Banzhaf power 3

  • Topics covered:

    • Banzhaf Power Index

    • Power in the European Economic Community

    • Power in the Electoral College

Shapley-Shubik Power Index

• • Lecture notes

  • Lecture videos: Shapley-Shubik power 1, Shapley-Shubik power 2

  • Topics covered:

    • Shapley-Shubik Power Index

    • Power of the U.S. President

Calculating the power indices of the UN Security Council

• • Lecture notes

  • Lecture video

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 and video:

Introduction to gerrymandering

• • Lecture notes

  • Lecture video

  • Covering

    • Why gerrymandering is sometimes good

    • Recent examples of bad gerrymandering

    • Racial v. political gerrymandering

    • Gerrymandering in the courts

Efficiency gap

• • Lecture notes

  • Lecture video

  • Covering

    • Packing and cracking

    • Stacking, hijacking, and kidnapping

    • Wasted votes and the efficiency gap

    • Problems with the efficiency gap

Geometry of gerrymandering

• • Lecture notes

  • Lecture video (access password: d2?@$m6@)

  • 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

  • An article about some of the most gerrymandered districts.

  • How gerrymandered is your district?

  • 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 the latest 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 or 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.

Class notes and videos:

• • Lecture notes

  • Lecture video

  • 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 am currently teaching).

Class notes and video:

• • Lecture notes

  • Lecture video

  • Covering

    • Basic ways of the misuse of statistics 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.

• • Notes including 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

      • Basics of social choice and game theory: fairness, conflict, bargaining, prisoner's dilemma, ...

      • Strategic voting

      • Revisiting some earlier topics using the language of social choice functions

      • Proofs of Arrow, Gibbard-Satterthwaite, and Balinski-Young Theorems

      • Nash Equilibrium

      • Median Voter Theorem

      • McKelvey-Schofield Chaos Theorem

      • 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...