Week 7: Condorcet Method, Condorcet Winner/Loser Criterion, and Other Simple Elections

Welcome to the Week 7 blog post of the course! This week, I read chapters three and four of "The Mathematics of Elections and Voting," which focused on the Condorcet method, Condorcet voting systems, sequential pairwise voting, and polls.

CONDORCET METHOD

As early as 1299, a variation of the Condorcet method was proposed by a man named Ramon Llull, but in 1785, French mathematician and political theorist Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet formally developed the technique and published his work. The Condorcet method consists of conducting runoff elections between every possible pair of candidates. If any one candidate wins all of his or her runoffs, we can ensure that candidate will be the winner. Take the following preference profile for a hypothetical election.

For this election, we will compare candidates A and B, B and C, and A and C. Candidate B would beat Candidate A 10-9, Candidate B would beat Candidate C 12-7, and Candidate A would beat Candidate C 16-3. Therefore, Candidate B is the Condorcet winner.

There are times, however, when the rankings don't match up, such as in the case of the election below.

Candidate A beats B 8-4, Candidate B beats C 9-3, but Candidate C beats A 7-5. To address this paradox, Condorcet proposed the Condorcet extended method, in which one ranks the pairs in order from the greatest gap of winning votes to the smallest gap. B beats C with the greatest gap, then A vs. B, then C vs. A. From the first two rankings (B vs. C and A vs. B), we get ABC. We next look at C vs. A, but this contradicts the already-established order of ABC, so the result of C vs. A is ignored. The final list is ABC, meaning that Candidate A is the Condorcet winner.

CONDORCET WINNER/LOSER CRITERION

This part of the text was pretty confusing to me. "The Mathematics of Elections and Voting" states that an "electoral system is said to satisfy the Condorcet winner criterion if, whenever there is a Condorcet winner, then the electoral system in question will always choose the Condorcet winner." The Hare system, Borda count, and plurality voting all do not satisfy the criterion. However, the book also states that a system proposed by Scottish economist Duncan Black in 1958, which combines Condorcet's method and Borda count (if there is a Condorcet winner, the candidate wins, but if not, the Borda count is calculated and the Borda winner is elected), satisfies the winner criterion. I'm not quite sure how, if Black's method had to resort to the Borda count winner, it would satisfy the criterion because it was previously mentioned that Borda count did not satisfy it. This issue is something I have to think more about this weekend.

Wallis' book also stated that a "voting system satisfies the Condorcet loser criterion if it can never happen that a Condorcet loser [a candidate who would lose in a runoff against each other candidate] wins the election."

SEQUENTIAL PAIRWISE VOTING

In this system, candidates are paired in separate runoff elections, which follow an agenda (an ordered list of candidates). For example, if the agenda is D, C, B, A, then the election proceeds as follows:

1. Candidate D against Candidate C.

2. The winner of Candidate D vs. Candidate C against Candidate B.

3. The winner of the previous runoff against Candidate D.

The agenda is extremely important, as different orders can produce independent outcomes.

Here's another fascinating scenario brought up in the book in which a voter would need to place an unexpected candidate higher up on his/her preference list in order for his/her favorite candidate to win.

POLLS

Polls can change who people vote for, as someone whose first choice is not doing too well in the polls might just vote for their second choice when it comes time for the actual election because they believe their vote wouldn't make a difference in the outcome.

I also finished a first draft of my paper that covered Borda count, Condorcet, and sequential pairwise voting. This weekend, I'll be looking at Arrow's Impossibility Theorem and making revisions to my second paper.