David R. MacIver's Blog: Small elections comparing Majority Judgement, Range Voting and Condorcet winners

Small elections comparing Majority Judgement, Range Voting and Condorcet winners

1 December 2016

I was interested in how range voting, majority judgement, and Condorcet winners interacted, so I thought I’d put together some small elections to demonstrate this. Perhaps unsurprisingly the answer is they’re fairly orthogonal.

The setup is that we’re considering 5 point scores, 0 through 4 (or if you prefer you can interpret these as “Terrible, bad, OK, good, great” for the majority judgement case).

Here is an election where the range winner is the Condorcet winner. We have three voters voting between two candidates, and they assign scores as follows:

3, 4
3, 2
0, 1

The first candidate has a range voting score of 6 and the second has a range voting score of 7, so the second wins the range vote. But the median score for the first is 3, while the median for the second is 2, so the first candidate wins the majority judgement vote. However, two out of three voters (the first and the third voter) prefer the second candidate to the first, so the second is also the Condorcet winner.

Conversely, in the following election the majority judgement winner is the Condorcet winner:

0, 1
0, 1
3, 0

The first candidate is the range voting winner because it totals 3 rather than 2, but it’s the majority judgement loser because it has a median score of 0 rather than 1. The Condorcet winner is also the second candidate, because the first two voters prefer it.

Sometimes range voting and majority judgement agree but they agree on someone who isn’t the Condorcet winner:

0, 1
1, 2
2, 0
2, 0
2, 3

The first candidate scores 7 vs the second’s 6, so it’s the range winner. It also has a median of 2 vs the second candidate’s 1, so it’s the majority judgement winner. But the first two and the last voters strictly prefer the second candidate, so the second candidate is the Condorcet winner.

Finally, here’s an example where all three disagree:

1, 2, 0
2, 0, 3
3, 4, 3
3, 4, 3
4, 0, 3

The first candidate is the range voting winner (13 vs 10 and 12), the second candidate is the Condorcet winner (3 out of 5 voters strictly prefer it) and the third candidate is the majority judgement winner (median 3 vs 2 for the second. It ties on the first median with the first candidate, but on removing one 3 score from each candidate the first candidate breaks downwards to 2 while the third stays at 3).

I don’t have strong opinions on what the right answers for any of these elections are, other than “it’s complicated”. In particular I don’t think any of the three options (Condorcet, Range and Majority Judgement) obviously produce the best outcome in all of these examples.

My rough intuitive judgement (formed without explicitly reminding myself of which systems won which, though I’m probably still biased by that knowledge) goes:

There’s not much in it but the second is slightly better (range vote, which is also Condorcet)
This election sucks, but the second candidate sucks less (majority judgement, which is also Condorcet)
First candidate seems a much better compromise (majority judgement and range voting)
Third candidate seems pretty universally popular other than the first voter who kinda hates everyone, so seems the best option (majority judgement only).

So on this blatantly biased sampling of four elections I like the range voting winner on three of them, the majority judgement winner on three of them, and the Condorcet winner on two of them. I don’t think that significantly updates my opinions about any of the three mechanisms, but it’s still interesting to see the comparison.