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 “loses to” and 1 meaning “wins over”. However, we want to allow for the possibility of “indifferent”, so I’m going to use -1 and 1, with 0 for “indifferent”.

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.) There’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 they’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.