| Edit to my previous comment: there are 10 candidates, not 12 I'm new to these terms, but when I look up "condorcet winner" in on wikipedia, it's defined as: "the candidate that wins a majority of the vote in every head-to-head election" So, I'm assuming "every head-to-head election" would be 10 factorial elections in our case (one for each pair). Let's look at a couple of these elections and then extrapolate on the rest of them: J vs A: Voters 2 and 3 didn't put A on the list, while they did indicate that they liked J, so they would vote for J. Only Voter 1 would vote for A. D vs A: Again, two voters didn't put D on the list (1 & 3), but did indicate that they liked J, while only one voter (voter 2) likes D over A. You can see that, for each candidate besides J, only one voters like them, while all voters at least like J a little. This means that J will win "every head-to-head election" and thus be the condorcet winner. The only problem with this proof currently is that STAR doesn't explicitly elect a winner, so therefore it is not "choosing the wrong winner" as it isn't choosing any candidate. It is eliminating the condorcet winner though, which is concerning, and I believe a more complicated example would show STAR choosing a non-condorcet winner. I like STAR more than RCV after reading up on STAR more. I believe that approval voting might be better though, as it completely elimintates the spoiler effect I believe. Though it may have other problems. Thinking about this more, I believe a condorcet winner can be chosen with RCV (and STAR with some assumptions) just by changing the tallying system. I was able to dicern from my example who the condorcet winner was, so why not do the same in an election? Just simulate all head-to-head elections to find the condorcet winner. I must be missing something. Here's a scenario where STAR elects a candidate that is not the condorcet winner: Voter 1:
5 for B !
4 for C
3 for D
2 for E
1 for A !
Voter 2:
5 for G !
4 for H
3 for I
2 for J
1 for A !
Voter 3:
5 for L
4 for M
3 for N
2 for A !
1 for B !
Voter 4:
5 for Q
4 for B !
3 for S
2 for T
1 for A !
Voter 5:
5 for V
4 for W
3 for G !
2 for Y
1 for A !
I've marked important votes with a '!'. All others are unique votes for candidates with no support.In this scenario, B and G get the top two scores with 9 and 8. B wins the election with 3 votes (voters 1, 3, and 4) over G's two votes (voters 2 and 5). A is the condorcet winner though as they win every head-to-head contest (with close elections (3-to-2), against B and G). A was eliminated though, as they only scored 6 points. This was complicated and too much fun to write, so it probably isn't practical in a real world scenario. |
> a condorcet winner can be chosen
A condorcet winner doesn't exist, if there is a circular relationship:
Yes, there are cases when STAR does not elect the condorcet winner. Of course, you could always add condorcet on top, and use STAR as the fallback. But you can do that for any system — basically all of which fail to elect the condorcet winner in some edge case — so it's not a particularly interesting insight. It's more about tradeoffs and what scenarios are most likely.Note that Approval voting is also a form of score voting — with a range of 0-1. The drawback here is limited expression — I can't say that I love or hate a candidate, and so it can elect a candidate with broad, weak approval and some strong opposition, over a candidate with broad, weak disapproval and some strong support. For example, let's draw the "I approve" line between 2 and 3 stars (approval indicated with +/-):
A is preferred by 4 out of 5 voters and 16 vs 11 stars, but B wins with 3 approvals, to A's 2.This is not a terrible failure mode — electing a mediocre, likely moderate, candidate over a decent one — considering it is the worst-case scenario for approval voting. Compare to the worst case scenario for FPTP, where sufficiently many candidates representing the most popular viewpoints split the vote, giving the win to a universally disliked, radical candidate. Of course, in actual elections we (the electorate) avoid this scenario by favorite betrayal, which has its own obvious shortcomings. To fix the Approval voting failure mode, scenario, we just have to be more stringent with our votes (ie, raise our standards).
Note how we were able to use the STAR ballot to derive what the outcome would have been in Approval. If we only had access to the Approval ballot, we couldn't tell that it was electing an inferior candidate. If the goal of the election is to choose the candidate who best reflects the preferences of the electorate, Approval is leaving information about those preferences on the table (edit: as is condorcet!). Aside: I remember reading (maybe on https://rangevoting.net ?) that increasing the range past about 7 sees quickly diminishing returns on this front.
STAR is more expressive, at the price of (mostly) complexity. edit: forgot to add that, while score voting is about the best you can do in terms of preferences among candidates, support of a candidate doesn't capture the full picture, either, since it bundles all the issues you may care about into a single decision (but putting that on the ballot moves us towards a different system of government).
At the end of the day, any form of score voting would be a MASSIVE improvement over the status quo. I would wholeheartedly support any of Approval, STAR, or vanilla score voting (on the same 5-point scale / STAR minus the runoff). I think we should put our efforts behind whichever one is most politically viable. STAR actually has a chance of being implemented (in Eugene, Oregon — see https://www.starvoting.us/campaigns); not sure where Approval stands.