A follow on from the last post with some more small elections.

Here are some small examples of elections demonstrating some preferential voting systems (two popular and one a little obscure) are not Condorcet. I can easily add more if you want me to add your favourite preferential voting system to this, but there seem to be a sparsity of interesting non Condorcet preferential voting systems.

Each of these are minimal amongst examples I’ve been able to find. Where minimal is taken under the lexicographic order of number of candidates then number of votes. (i.e. if one election has fewer candidates it’s regarded as smaller. If they have the same number of candidates the one with fewer voters is smaller). I’ve no idea if they’re minimal in absolute terms, but I’ve made a reasonable attempt to find smaller ones and I think it likely that they are. In particular they’re all locally minimal in the sense that they will no longer demonstrate the desired behaviour if you drop a candidate or drop fewer than two votes from them, but it’s possible some smaller subset does demonstrate the desired behaviour.

### Borda count

- 2, 3, 1
- 2, 3, 1
- 2, 3, 1
- 3, 1, 2
- 3, 1, 2

The Borda winner is 3 but the Condorcet winner is 2.

### IRV and Contingent vote

- 1, 2, 3
- 1, 2, 3
- 2, 1, 3
- 3, 2, 1
- 3, 2, 1

The IRV winner is 1 but the Condorcet winner is 2. The contingent vote winner for this is also 1 (unsurprisingly because the two will always agree when there are only three candidates).

### Coombs

Coombs is basically the reverse of IRV: At each step you remove the candidate who the most voters have ranked last of the remaining candidates and rerun until you only have one candidate left.

- 1, 3, 2
- 2, 1, 3
- 2, 1, 3
- 2, 3, 1
- 2, 3, 1
- 3, 1, 2
- 3, 1, 2

The Coombs winner is 3 but the Condorcet winner is 2.

### Borda-majority judgment

This is Majority Judgement applied to the Borda scores. i.e. a preferential ranking is implicitly assigning a grade of n to its highest candidate, n – 1 to its second highest, etc. This isn’t my invention – it’s from the standard text on the subject – but I’m not aware of this being a system that anyone actually uses.

- 1, 2, 3
- 1, 2, 3
- 2, 1, 3
- 2, 1, 3
- 2, 3, 1
- 3, 1, 2
- 3, 1, 2

The Borda-majority judgment winner is 2 but the Condorcet winner is 1.