The Condorcet Matrix

The Matrix

The result

After you have cast your vote in a condorcet poll, you will see the result in a form of a matrix like the following one:

Which of these capital cities would you rather live in?

	1	2	3	4
1 - Berlin	-	44	65	62
2 - Paris	39	-	55	51
3 - London	22	41	-	38
4 - Roma	38	48	41	-

Total Votes : 100

The winner(s) is(are):
Berlin

You don't have to know how to read the matrix: the name is automatically computed and displayed below it. However, if you wish to understand how to read such a matrix, you have to remember that condorcet elections amount to the same as a series of elections between only two candidates, i.e. pairwise elections. The matrix above is summing up the result of all such pairwise elections that take place within one condorcet election.

Let's see how to read the information presented in the matrix

The candidates

In our example the candidates are the capital cities themselves.

The numbers in the top row correspond to the candidates' names in the first column on the left. It was not convenient to print the names of the candidates on the top row so we are using this shortcut.

Which of these capital cities would you rather live in?

	1	2	3	4
1 - Berlin	-	44	65	62
2 - Paris	39	-	55	51
3 - London	22	41	-	38
4 - Roma	38	48	41	-

Total Votes : 100

The winner(s) is(are):
Berlin

Pairwise Election

Each city vs each of the other cities

The matrix is the result of a condorcet Election. As explained elsewhere in this site, such an election is considered very fair because it allows each candidate (capital city) to face each of the other candidates, one on one. Only this way can we find out the people's (the voters') true preferences.

Thus, in our example, we have the result of 6 one-on-one elections (pairwise election):

Berlin vs Paris
Berlin vs London
Berlin vs Roma
Paris vs London
Paris vs Roma
London vs Roma

Obviously, it doesn't make sense for one city to compete with itself , that's why the diagonal highlighted below doesn't have any value in it.

Which of these capital cities would you rather live in?

	1	2	3	4
1 - Berlin	-	44	65	62
2 - Paris	39	-	55	51
3 - London	22	41	-	38
4 - Roma	38	48	41	-

Total Votes : 100

The winner(s) is(are):
Berlin

We are now going to take one of those pairwise election as an example.

One side of the election: London vs. Paris

Suppose we wish to see how London compares to Paris.

Which of these capital cities would you rather live in?

	1	2	3	4
1 - Berlin	-	44	65	62
2 - Paris	39	-	55	51
3 - London	22	41	-	38
4 - Roma	38	48	41	-

Total Votes : 100

The winner(s) is(are):
Berlin

We can see that London (candidate 3) wins 41 votes against Paris (candidate 2). It means that among the 100 voters, 41 preferred to live in London rather than Paris. In their ballot, they ranked London higher than Paris.

The other side of the election: Paris vs. London

In the pairwise election (London vs. Paris), did London win or loose? Let's see the other side:

Which of these capital cities would you rather live in?

	1	2	3	4
1 - Berlin	-	44	65	62
2 - Paris	39	-	55	51
3 - London	22	41	-	38
4 - Roma	38	48	41	-

Total Votes : 100

The winner(s) is(are):
Berlin

Among the 100 voters, 55 voters ranked Paris (candidate 2) higher than London (candidate 3). Clearly, Paris won by 55 votes against 41. Apparently 4 voters ranked both cities the same, having no preferences between the two (55 + 41 + 4 = 100 voters).

Symetry

So, for each of the six pairwise elections, we can compare one side of the matrix to the other side of the matrix, so that we can determine the winners.

Notice the symetry of the matrix.

Which of these capital cities would you rather live in?

	1	2	3	4
1 - Berlin	-	44	65	62
2 - Paris	39	-	55	51
3 - London	22	41	-	38
4 - Roma	38	48	41	-

Total Votes : 100

The winner(s) is(are):
Berlin

A bunch of duels

This win from Paris over London by 55 votes against 41 is represented by a darker cell in the final matrix. We reprint here the original matrix: see how the winner of each single pairwise election is highlighted.

Which of these capital cities would you rather live in?

	1	2	3	4
1 - Berlin	-	44	65	62
2 - Paris	39	-	55	51
3 - London	22	41	-	38
4 - Roma	38	48	41	-

Total Votes : 100

The winner(s) is(are):
Berlin

Reading along the matrix, we can note the following results:

Berlin wins over Paris by 44 votes against 39 with 17 undecided voters (44 + 39 + 17 = 100 votes)
Berlin wins over London by 65 votes against 22 with 13 undecided voters (65 + 22 + 13 = 100 votes)
Berlin wins over Roma by 62 votes against 38 with no undecided voter (62 + 38 + 0 = 100 votes)
Paris wins over London by 55 votes against 41 with 4 undecided voters (55 + 41 + 4 = 100 votes)
Paris wins over Roma by 51 votes against 48 with 1 undecided voter (51 + 48 + 1 = 100 votes)
Roma wins over London by 41 votes against 38 with 21 undecided voters (41 + 38 + 21 = 100 votes)

And the winner is...

Now that we clearly know the winners of the 6 pairwise elections, how do we dertermine the overall winner? We shall see that further below. But first, let's have a quick look at how a single ballot would look like in the matrix.

My vote

From the ballot to the matrix

In the previous chapter, we have seen how to fill in a ballot, and in this one we get to understand how the result is presented in the form of a matrix. In the example below, we see how one vote (left) translates into a matrix (right).

Whom would you prefer to be your class delegate?

	1	2	3	4
1 - Susan	-	1	1	1
2 - Humphrey	0	-	0	0
3 - Jennifer	0	1	-	0
4 - Arthur	0	1	0	-

Total Votes : 100

Susan is ranked first, so she scores a win against each of the other three candidates. Arthur and Jennifer are ranked the same, so neither scores a win against the other but both win against Humphrey. The latter is the least prefered by the voter and is not given any point.

Finding the overall winner

The Ideal Democratic Winner

The Marquis de Condorcet devised the election that now bears his name to ensure that the winner of the election would be what is called the Ideal Democratic Winner (IDW). The IDW is the candidate who would beat all the other candidates in one/one elections.

Let's have a look again at the matrix we studied above:

Which of these capital cities would you rather live in?

	1	2	3	4
1 - Berlin	-	44	65	62
2 - Paris	39	-	55	51
3 - London	22	41	-	38
4 - Roma	38	48	41	-

Total Votes : 100

The winner(s) is(are):
Berlin

Remember that the winners of all the pairwise elections are highlighted this way. We can clearly see that Berlin wins all the pairwise elections against each of the other three candidate cities:

Berlin wins by 44 votes against 39 votes for Paris.
Berlin wins by 65 votes against 22 votes for London.
Berlin wins by 62 votes against 38 votes for Roma.

Berlin has won all of its pairwise elections. It is obviously the Ideal Democratic Winner that we were looking for. Berlin is therefore declared the overall winner of this election.

Cyclic ambiguities

What happens when there is no single candidate who can beat each of the others in pairwise election? Imagine a situation where A beats B, B beats C and C beats A... This is a cyclic ambiguity. Different methods can be used to resolve such uncertainties. The Beat Path method is one of them.

Beat Path

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