Instructor’s Lecture Notes

for Math Investigations

Contents

Problem Solving

Growth Models

Financial Mathematics

Logic & Set Theory

Probability & Counting

Statistics

Voting Theory

Voting Methods

In basic contests (elections) where everyone simply votes for their preferred candidate, there are two common rules by which a winner is chosen.

Majority Rule
A candidate must receive >50% of the vote, a majority, to win. In cases where there are more than two candidates, this condition may not be met, in which case a run-off election of some design must be held to determine a winner.
Plurality Rule
A candidate must receive the most votes — a plurality, not necessarily a majority — to win. Withholding ties, one candidate will always receive a plurality.

There are however more sophisticated voting methods that require voter to rather than just pick their single preferred candidate, to rank all the candidates from more preferred to least preferred. Such a ballot that ranks candidates is called a preference schedule.

Single Run-Off
In case there are more than two candidates and no candidate receives a majority of most-preferred votes, all but the top two candidates are eliminated, and their votes redistributed to the top two. This is how elections in the US are often conducted, but since we don’t fill out preference schedules the run-off election requires that everyone votes again. The utility of having voters fill out a preference schedule is that in the case of a run-off election, the information of how votes would be redistributed can be read from the preference schedule.
Instant Run-Off
Only subtly different than a single run-off contest, instead of eliminating all but the top two candidates and redistributing their votes, losing candidates are eliminated, and their votes redistributed, one-by-one.
Borda Count Method
This one’s a bit more novel. Candidates are awarded points based on their positions in voters’ preference schedules, and the candidate with the most votes total is the winner. There is not canonical way to assign points, but a common method you’ll see is this: a candidate is awarded 1 point per ballot on which they’re ranked last, 2 points per ballot on which they’re ranked second-to-last, 3 points per ballot on which they’re ranked third-to-last, and so on.
Condorcet Method
Also called the method of pairwise comparison, or a round-robin tournament, every pair of candidates is considered independently in a head-to-head contest, and a winner is decided between those two. The overall winning candidate is the one who won the most head-to-head contests. If any candidate wins every head-to-head contest, they are called a Condorcet winner.

Let’s walk through each of these methods with an example. Suppose a first-grade class gets to choose a class pet. Their teacher presents them with four options — a fish, a gecko, a mouse, or a rock — and the 23 first-graders are asked to vote on which pet they’d prefer by filling out a preference schedule. With four candidates, there are 4! = 24 different ways that students could have ranked the possible pets. Lucky for us, only seven of those possibilities appeared. Their votes are recorded here:

First mouse mouse mouse gecko gecko fish rock
Second gecko fish rock fish mouse gecko fish
Third fish gecko gecko rock fish mouse gecko
Fourth rock rock fish mouse rock rock mouse
6 4 1 5 3 1 3

Fairness Criterion

A noble goal would be to establish objective, mathematical criterion that a contest should should satisfy if it is to be considered a “fair” contest. Here are four such criterion:

Condorcet Criterion
If a candidate is preferred to every other candidate when compared head-to-head — i.e. they are a Condorcet winner — then that candidate should win.
Majority Criterion
If a candidate has a majority of first-place (top-rank) votes, then that candidate should win.
Monotonicity Criterion
If any single voter were to change their ballot to rank a specific candidate higher, then that candidate’s position overall should not decrease.
Independence of Irrelevant Alternatives (IIA) Criterion
If a non-winning candidate is removed from the ballot, then the outcome of the contest should not change.

It would be nice if we could declare that a voting system is “fair” because it satisfies each of these criterion. Unfortunately, though these criterion appear innocuous enough, this cannot be done. The fact — a fact known formally as Arrow’s Impossibility Theorem (1951) — is that there is no ranked-voting method that satisfies all four of these criterion. In particular, each of the voting methods we’ve discussed violates at least one of these fairness criterion; there exists an example of a contest of each of these types that doesn’t satisfy some condition of being “fair”.

As examples let’s consider contests with three candidates: Aldous is a red-team candidate while Beatrix and Clive are blue-team, and Clive will “spoil” the fairness of each contest.