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An Introduction to Voting Theory

Session B: July 20 - July 30, 2021
1:30 p.m. - 4 p.m.
Philosophy and Society
Thomas Brazelton

An Introduction to Voting Theory will survey the mathematical underpinnings of the theory of voting and electoral systems, with a specific focus on developing the tools to assess and compare existing voting systems. Students will attend lectures, gain experience in proof-based mathematics, and have the option to work with real datasets to analyze voting systems.

Goals for students

  • To develop a mathematically rigorous definition of “fairness” as it pertains to election systems, and to develop an informed opinion about what constitutes a fair, democratic election
  • To gain a basic hands-on introduction to proof-based mathematics
  • To engage students in reflection about existing voting systems, and to be able to critically assess claims about voting
  • To become familiar with a broad history of voting systems, their criticisms, and their capacity to change

Possible assignments

Problem sets: Students will complete daily problem sets during the first week in order to “jump into the deep end” of proof-based mathematics. For the first few days, a lot of effort will be placed on peer and instructor feedback for logical arguments and developing a feeling for what constitutes a rigorous mathematical argument.

Research projects: Students may conduct individual or group research projects in which they investigate a voting system that currently exists in the world, and mathematically assess its attributes and its viability. Such projects may be presented to the class, or otherwise reviewed by fellow classmates, and then could be formulated into blog posts by students or groups.

Data analysis: Students with prior computer experience may have the opportunity to run analysis of existing voting data and will synthesize their results using the mathematical tools developed in this course.

Day 1: Mathematical introductions

  • Fundamentals of logical reasoning, introduction to mathematical proofs
  • Sets and functions

Day 2: Mathematical introductions (part 2)

  • Relations and orders
  • Reflexivity, symmetry, transitivity
  • Monotonicity
  • The Condorcet paradox

Day 3: What is a voting system?

  • Criterion for voting, voting strategies
  • Social welfare functions, Pareto efficiency
  • What do we expect out of a voting system?
  • Small examples: May’s Theorem
  • Plurality voting systems, ranked voting, approval voting, score voting

Day 4: The dictators win

  • Arrow’s Impossibility Theorem and its proof

Day 5: Picking up the pieces and rebuilding a democracy

  • Relaxing IIA (independence of irrelevant alternatives)
  • The Borda count
  • Multi-round voting, runoff elections

Day 6: Manipulating elections, voter fraud

  • Tactical voting, bullet voting, cloning, teaming
  • Which voting systems are susceptible to tactics?
  • Voter fraud, spoiled ballots, double voting, and the birthday problem

Day 7: Building an electoral system

  • Proportional representation
  • Parallel voting in bicameral legislatures
  • Digression: a brief discussion on gerrymandering

Days 8-9: Student presentations

  • Students present on existing voting theories, or develop their own. Presentations will touch on which criteria each voting system satisfies, which criteria it fails, and why these criteria are important. Presentations should also discuss the vulnerabilities of their voting system to tactical voting and fraud. Students are asked to juxtapose the voting system they present with those discussed in the course, and compare and contrast their various attributes. These may be turned into blog posts for a class blog.