The Condorcet Numbers

Recently I was watching a video about voting, and I was struck by a voting system called the Condorcet method. The Condorcet method is where instead of picking one candidate you like the most, you have candidates "square off" against each other akin to boxing "matches". Take a simple ballot with candidates A, B, & C. The ballot would look like this: do you prefer A or B, A or C, B or C. If voters prefer A over B, A over C, & B over C, then candidate A wins the election. For larger elections you would simply have voters rank the candidates, then have them pair & compete in matches. This method is not perfect however, since if voters prefer A over B, B over C, & C over A, then no winner is selected.

I then began to think about the number of matches you would need for a certain number of candidates "n". The following integer sequence belongs to what I call the Condorcet Numbers. Although it's basically the triangular numbers offset.

The Condorcet numbers are the number of matches needed for a Condorcet election. For example, if there is only 1 candidate, there are no pairings so the number is 0. But for 2 candidates, a voter will need to at least vote once on either, so the second Condorcet number is 1. & in the example ballot I used, you need three matches so the third Condorcet number is 3.

You'll notice that each number n increases by the value of the previous n. Just like the triangular numbers in pascal's triangle. This is the formula for generating the triangular numbers n*(n 1)/2. However, n is offset ahead by 1. If n = 4, then if we use the formula it won't work: 4*(4 1)/2 = 10, which should be the answer to n = 5. So if we modify the formula to only use the previous number (n-1), then we get (n-1)*n/2. I prefer to write it as ((n-1)*n)/2.

1 = 0

2 1/2 = 1

3 1/2,1/3,2/3 =3

4 1/2,1/3,1/4,2/3,2/4,3/4 =6

5 1/2,1/3,1/4,1/5,2/3,2/4,2/5,3/4,3/5,4/5 =10

6 1/2,1/3,1/4,1/5,1/6,2/3,2/4,2/5,2/6,3/4,3/5,3/6,4/5,4/6,5/6 =15

0,1,3,6,10,15 = ((n-1)*n)/2

These are the first 25 Condorcet numbers: 0,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231,253,276,300