Answering the earlier question:

In the original example, we have three voters, each of whom ranks four candidates:

```
X: B C A D
Y: D A C B
Z: A C B D
```

(That is, voter X ranks B first and D last.)

The wikipedia page adds everything up with 0 meaning â€śloses toâ€ť and 1 meaning â€świns overâ€ť. However, we want to allow for the possibility of â€śindifferentâ€ť, so Iâ€™m going to use -1 and 1, with 0 for â€śindifferentâ€ť.

The sum matrix now looks like:

```
A B C D
A 0 1 1 1
B -1 0 -1 1
C -1 1 0 1
D -1 -1 -1 0
```

(To work through the math in the top right corner: voters X and Z rated A higher than D; voter Y rated A lower than D. So this entry is 1 + 1 + -1 = 1.)

This sum matrix gives us an unambiguous outcome. A beats every other candidate. If you remove A from the matrix, C beats B and D. If you also remove C, then B beats D. So the final ranking is `A C B D`

.

You asked what happens if voter X skipped playing D. The sum matrix would then look like

```
A B C D
A 0 1 1 0
B -1 0 -1 0
C -1 1 0 0
D 0 0 0 0
```

Now we have a problem. A beats B and C, but is indifferent versus D. In fact D is indifferent versus each other candidate. (Because voter Y ranked D above them all, but Z ranked D below them all.) Thereâ€™s no way to place D without disappointing one voter, so the whole ranking is stuck.

This sort of problem is likely when you have very few voters. If you have 50 voters, even if theyâ€™re very polarized about D, the D-top and D-bottom factions are not likely to be *perfectly balanced*. A slight majority will break the deadlock.