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.
Ah! I had a total brain fart and was still scoring the ballots like STAR — "in an election between A and J, A has 4 points and J has 3". -.-
> a condorcet winner can be chosen
A condorcet winner doesn't exist, if there is a circular relationship:
Voter 1: A > B > C
Voter 2: B > C > A
Voter 3: C > A > B
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 +/-):
Voter 1:
5 for A (+)
3 for B (+)
Voter 2:
5 for A (+)
3 for B (+)
Voter 3:
3 for B (+)
2 for A (-)
Voter 4:
2 for A (-)
1 for B (-)
Voter 5:
2 for A (-)
1 for B (-)
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.
>At the end of the day, any form of score voting would be a MASSIVE improvement over the status quo
Yeah, I think we're both of one mind. I just find the math (or logic) behind these voting methods interesting. And it is somewhat important.
>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
I think the interesting part of my work in these threads is that the non-condorcet winner example for RCV is much easier to produce than the non-condorcet winner example for STAR. The RCV version only takes 3 candidates, while the STAR version takes ~25 candidates. While this could be a failure of our ability to find scenarios where the non-condorcet winner is elected, I believe it says more about how resistant these systems are to the spoiler effect, which, in essence, is where a non-condorcet winner is elected.
I really like the idea of overlaying STAR and approval voting to show that STAR's expressiveness makes a big difference and I think your example is pretty realistic overall. But I have some issue with it as well. I made an assumption about voters in my example that you didn't make, which is that voters will try to impart their own will as much as possible. This means always using the 5 star vote in the STAR method. I thought this was a sane assumption to make about voters and that only using the 4 slot (never bubbling in 5 star) would be irrational. I think I could make a much less complicated example if I dropped this assumption. Also, I kinda feel uneasy about the cut off, but I'm not sure how to improve it. I just imagine that if a voter had any slight preference of one candidate over another, they would indicate that in the approval method.
>Approval is leaving information about those preferences on the table (edit: as is condorcet!).
How would a condorcet election leave out expressiveness? I think expressiveness is subjective and hard to define. I might've simplified analysis of these schemes by assuming that a condorcet winner system had the ultimate expressiveness (assuming there is a condorcet winner).
>A condorcet winner doesn't exist, if there is a circular relationship:
I meant a condorcet winner will be elected if it exists.
I think the biggest problem with STAR voting is that these stars are kinda hard to reason with. What exactly does it mean if I give a candidate 4 or 5 stars? What affect will that have on the election? RCV might be more complicated, I'm not sure, but it's a lot easier to understand as it's easier to rank your favorite candidates. This really comes down to preference though and it's where I kinda checkout because it can lead to some really tedious arguments. They're all good. It would be great to see STAR in action in Eugene!
Assorted thoughts, with no particular relationship or broader point:
> the STAR version takes ~25 candidates
I was just poking around on the star voting site, and "More than 5 candidates" was one of the criteria for whether there should be a primary or a single election.
> I just imagine that if a voter had any slight preference of one candidate over another, they would indicate that in the approval method.
Well, depending on the number of candidates, this isn't always possible — If there are 4 candidates who you love, like, dislike, and hate respectively, you don't have a way to express the difference between love/like and dislike/hate.
> I made an assumption about voters in my example that you didn't make, which is that voters will try to impart their own will as much as possible. This means always using the 5 star vote in the STAR method.
I don't know if it considers that someone might always vote 5 stars for their favorite candidate, but thinking about that, I agree it is probably likely. It would not make much sense to cast an approval voting ballot and vote against, or for, all candidates. Might as well not vote, at that point :P
> How would a condorcet election leave out expressiveness?
N voters rank A=5, B=1, C=1.
N+1 voters rank A=4, B=5, C=1.
B is the condorcet winner, but I think it's clear that A is the best choice. This a failure mode of STAR voting as well, which elects B, while vanilla score voting (and approval voting) would elect A.
> What exactly does it mean if I give a candidate 4 or 5 stars?
I like to work up from FPTP (if I'm talking to someone who's not into this, I'd just call it "our current system" or "single choice"):
- In our current system, you vote for one person. Count the votes, and whoever has the most, wins.
- In Approval voting, you vote for as many people as you want — anyone you'd be okay with. Count the votes, and whoever has the most, wins.
- STAR voting works like Approval voting, but instead of a yes or no, you rank the candidates with 0-5 stars, like Yelp/Amazon reviews. Add up the stars, and whoever has the most, wins.
At this point, I make a choice about whether to mention the runoff, because it is hard to explain with words alone. If the person I'm talking to is interested, intellectual, or if I have access to a computer where I can show them this picture[1] on https://www.starvoting.us/star, then I'll go ahead and mention the runoff. Otherwise, it causes too much confusion.
This seems like the easier parallel to make, but also, I don't like the idea of relating presidential elections to something as inconsequential as product reviews...
>B is the condorcet winner, but I think it's clear that A is the best choice.
Ah, thanks, that makes sense. To rephrase it in my words: a condorcet election could elect someone that a minority HATES rather than a candidate that everyone is indifferent to. This makes a lot more sense why we would have these voting methods like RCV/STAR.
I'm definitely going to look into this when I find some free time. Thanks.
I've simply enjoyed attacking these voting methods via these small details, but I think it's really important as well. We are putting into place a system that will be put to extreme tests and will need a ton of integrity. Millions of people will be affected by it and will spend time coming up with strategies to maximize their votes (as they do currently by not voting for 3rd party candidates). Campaign strategists will try to use the system to their advantage (as they do currently by campaigning in battleground states). And malicious outsiders will try to use misinformation to attack the system as well. I want to make sure it's done right.
I think I was a little angry in my last comment where I said these conversations were tedious. I'm just really hesitant to choose a method like STAR, or even RCV as they're so different from our current system. I want a safe removal of the spoiler effect, which I think approval voting might do best, but I'm not sure, and it may be best for America if we make a big leap to a more complex system as this could help us avoid problems in the future, rather than having to transition again. Definitely going to look into this more though.
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:
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.