Welcome to Section 8-2: Voting Systems. As mentioned in our class introduction, Chapter 8 is widely considered one of the most challenging sections of our textbook; however, it is also one of the most applicable to the real world. In this section, we move beyond simple vote-counting to answer a complex question: How do we actually choose a winner?

It is highly recommended that you review the attached notes thoroughly and follow along with the video lecture. Attempting to master this section completely on your own can be difficult, so please reach out if you have questions!

Key Concepts in Voting Theory

A Voting System is a set of rules used to determine a winner. While we are used to the idea that "the person with the most votes wins," this section introduces several mathematical methods that can lead to completely different outcomes based on the same set of ballots.

  • Plurality Voting: The candidate with the most votes wins. Note that this does not necessarily mean a majority (which requires $> 50\%$ of the votes). A candidate can win a plurality with far less than a majority if the vote is split among many candidates.
  • The Spoiler Effect: This occurs when a candidate with no realistic chance of winning influences the outcome of the election, often by drawing votes away from a major candidate.

Preferential (Ranked) Voting Systems

When voters rank candidates (1st choice, 2nd choice, etc.), we can use more sophisticated methods to determine the will of the people:

  1. Top-Two Runoff: If no candidate achieves a majority in the first round, a second election is held between only the top two vote-getters.
  2. Elimination Runoff (Instant Runoff): If no candidate receives a majority, the candidate with the fewest first-place votes is eliminated. Their votes are redistributed to the voters' next choices. This process repeats iteratively until a candidate holds a majority.
  3. Borda Count: This method assigns a point value to every rank. As defined in your notes:
    $$ \text{Last place} = 0 \text{ points} $$ $$ \text{Next higher} = 1 \text{ point} $$ ...and so on. The winner is the candidate with the highest total Borda count.

The Perfect System?

We also look at the Condorcet Winner, which is a candidate who would defeat every other candidate in a head-to-head (one-on-one) matchup. Interestingly, a candidate can be a Condorcet winner but lose a Plurality election!

Finally, we discuss Arrow's Impossibility Theorem, which mathematically proves that when there are three or more candidates, it is impossible to design a voting system that satisfies every criterion of fairness perfectly. This is a crucial concept to grasp: every voting method has its own inherent flaws or biases.

Please download the PDF notes below and prepare for the quiz. Good luck!