Posts made by Sass

I believe you mean "spoiled" as in "voided", not "spoiler".

With your assumptions, some of the "middling" problems can't happen because every voter fully filled out their ballot correctly. Basically the red and yellow boxes become the same.

Regardless, this exercise is fairly trivial once you wrap you head around the basic approach, which I'm happy to walkthrough

First off, all ranked ballots have an imaginary "last rank" that isn't shown on the ballot. For example, if I can only rank 3 candidates explicitly, then in practice I can explicitly rank all candidates if there are up to 4 of them because unranked candidates are considered ranked 4th. So, given your assumptions, if we want to create these kinds of problems, we need at least 2 more candidates than the number of ranks on the ballot. 3 ranks and 5 candidates is perfect.

Next up, we just make a ballot set that produces RCV results where no candidate is within a couple votes of beating another candidate. Here's one with little thought:

40: A>B>C
30: D>E>B
20: C>A>D
5: E>B>A

Now we run through the tally.

B, E, and C are all eliminated in the first round. All 25 votes transfer to A. A beats D 65 to 30. Looking back, I guess we need to make sure it comes down to a number of finalists no more than the difference between the number of candidates and the number of available ranks (so 5-3=2), which happened in this case.

Now, we add a voter who ranked all 3 of those eliminated candidates.

40: A>B>C
30: D>E>B
20: C>A>D
5: E>B>A
1: B>E>C

We also need a voter whose ballot stops transferring partway through their ranks but ultimately goes to a losing finalist. This means a losing finalist needs to be in the middle of their marked rankings. The only losing finalist is D and there are only 3 explicit ranks available, so the ballot needs to rank first a candidate who gets eliminated, then rank second D, and then rank third ideally a candidate who also lost, but the third rank could really be any candidate because RCV is nonmonotonic.

40: A>B>C
30: D>E>B
20: C>A>D
5: E>B>A
1: B>E>C
1: B>D>C

Boom. Those last two ballots show the problems you're looking for.

Of course, another voter intent issue in RCV is nonmontonicity, which this ballot set doesn't show. Again, it'd be fairly easy to come up with a ballot set that shows all of the problems (my TikTok series does that), but nonmontonicity is probably best presented in isolation because of how screwy it is. The following is the simplest ballot set demonstrating nonmontonicity in RCV:

8: A>B>C
7: C>B>A
6: B>A>C

If 2 C voters strategically rank A first, then A would lose.

And that's it.

I just manually checked mine (the first one) and it's correct. I suspect the second one counts equal preferences as half points for each candidate in the pairwise comparison. I haven't seen that before, but there is a metric that a voting method could use that would be screwed up by doing it: total number of pairwise preferences over all candidates. That would be sum of the entire row of a candidate. Candidates who are ranked equally to other candidates on more ballots would benefit. However, that's an unreliable metric generally and is really only useful for tiebreaking. So basically my conclusion is that both approaches are correct and should give you the same results, at least for Ranked Robin (not including the 3rd degree tiebeaker).

As I've thought about it more, if there's a Condorcet Winner, then cloning is irrelevant under Ranked Robin, making it an unreliable strategy.

Also, basically all Condorcet methods fail Participation. It comes with the territory.

Moreover, focusing on pass/fail criteria is the issue that caused voting enthusiasts not to achieve real-world progress for 200 years. The question is not "Does this method pass this criterion 100% of the time?"; the question is "How well does this voting method perform on this metric in practice?". Considering that cloning is only helpful under Ranked Robin when there's no Condorcet Winner and that scaled elections without Condorcet Winners are incredibly rare and difficult to predict, I see it as a nonissue.

And just to set the record straight, I think Approval and Score are great methods. I absolutely support them and would be very happy to see their use in public elections.

@Toby-Pereira I was on mobile, so the link didn't copy properly. Here's the section discussing frequency of ties:
https://electowiki.org/wiki/Ranked_Robin#Frequency_of_ties

I need to clean up the electowiki page, but the Equal Vote site on Ranked Robin is a much better reference:
https://www.equal.vote/ranked_robin

A few things, all.

@masiarek, if I rank candidate A first, candidate B second, and candidate C third, then I know the order I prefer each, but not by how much compared to each other. If I had a 5-star ballot, I might give A 5 stars, B 4 stars, and C 0 stars; or I might give A 5 stars, B 1 star, and C 0 stars. There’s no way for you to determine how I really feel about B based only on my rankings. However, if I start with a 5-star ballot and give A 5 stars, B 3 stars, and C 0 stars, then you know for sure that I would rank A first, B second, and C third. You can always extract a full set of rankings from a set of scores, but usually you cannot extract a full set of scores from a set of rankings. Scores contain strictly more information than ranks; therefore, a score ballot allows voters to express more information than a rank ballot, i.e. score ballots are more expressive.

In my opinion, a rank ballot that allows equal ranks allows for a sufficient degree of expression to consistently determine the candidate closest to the center of public opinion, and the simulations support my claim as good Condorcet methods perform on par with good score methods.

@Toby-Pereira, you are overestimating the frequency of Condorcet cycles in real-world elections. As more people have studied the question, the estimates have gone down and down.

Additionally, Ranked Robin fails clone independence in the opposite direction of Choose One Voting. This means that to gain an advantage, a party would have to support entire campaigns of multiple candidates who are seen as identical by the electorate. This is so difficult in practice that it legitimately can be dismissed as a real concern. Candidates like to differentiate themselves from each other, electorates do not behave predictably, and campaigns are egregiously expensive. These difficulties are further amplified under a method like Ranked Robin that incentivizes candidates to appeal evenly to the entire electorate (see @Marcus-Ogren’s new paper on Candidate Incentive Distribution).

Also, if there’s not a Condorcet winner, then there are multiple scenarios more likely than a top-3 cycle that Ranked Robin resolves simply. Check out the electowiki.

https://electowiki.org/wiki/Ranked_Robin

Furthermore, your claim that Ranked Pairs is simple is…absurd. I canvass for STAR Voting, a far simpler method, every day, and it truly is at the limit of what we can expect lay voters in America to digest.

Simplicity actually is the most important factor for a Condorcet method because…it’s a Condorcet method. By that metric alone, it excels at both accuracy and honesty, and is also sufficiently expressive.

@andy-dienes Considering that the list of signatories is almost exclusively academics who have spent zero time in the advocacy space beyond signing a letter to a body that will ignore it, I'm not inclined to believe that they understand how reform works in the US. I truly do write off that entire letter because the people who've signed it (except for maybe Lee Drutman) have done almost no actual work to improve elections in the US. We do know better than most of those signatories because we collect signatures and create educational materials and build coalitions and read constitutions and election codes and support volunteers and so much more.

And your -ism misses my most important point: don't blame voters. It is the math of the method, not a vice of the voters or a crime of the candidates. I understand perfectly that "drawing better districts" doesn't solve gerrymandering, but I recognize that our most pressing problem is polarization, which is a direct result of the center-squeeze effect.

We advocate for STAR, Approval, Ranked Robin, etc. because we know that there's a way to eliminate center-squeeze and that single-winner reform is the fastest way there. Americans don't know anything about voting science and the laws and constitutions in place are major obstacles to ProRep. Focusing on single-winner is a strategic decision that applies to the US right now. Hopefully in 20 years, it won't be, but for now it is.

@andy-dienes What we need are voting methods that are stable with more than two competitive candidates, and RCV is not one of them. We haven't seen dissipation in Fargo or St. Louis, which have both had two Approval elections. Granted, STL halved their number of wards in this last round, so I wouldn't call the evidence definitive, but PR is not the only way to get true candidate variety.

It's so devastating. I just want to fly out to Fargo and hug everyone there.

the forum should be run democratically, with participants able to vote on everything substantial. We can even include in that a process for people to vote...

Votes can happen on the forum itself, and should be very transparent in that everyone can see how everyone else votes, from the moment they cast their vote. We would have crystal clear rules for who is allowed to vote, which should be everyone who participates with any regularity at all. For any votes that have tangible consequences, we would always respect the outcome.

But it is important to me that people who are involved in any decision making be regular participants.

I actually would like to call a vote of the people on this forum.

...we can simply ask nicely.

There are processes already in place for voting in our bylaws and procedures. Fortunately, Sara's on top of scheduling a meeting where some votes can happen soon. Importantly, we should follow the protocols already in place. They're there for a reason.

The main thing that I proposed is that the forum will not just discuss voting methods, but also serve as a hub for developing resources that advance the general cause (of replacing plurality/FPTP with something better). So that means building voting “widgets”, building tools for testing methods, tools for holding internet votes, tools for visualizing how they work, for simulating elections, and so on. Also, it means hosting some static content, which can be web pages (that may link or embed external things, like youtube videos or CodePen apps), libraries of code (typically javascript since this is the web and all, but it can be any language), images, ChatGPT conversations (which are just web pages, but all of a certain type and format) and so on.

I'll repeat what I stated before, Rob. If you'd like to help build out features and tech improvements, then do so. No vote is needed for that until we're ready to implement. No control needs to change. Additionally, Jack informed Sara that he unilaterally added you to the tech committee for the forum that was formed by the council after a unanimous vote on Motion 4, which Sara highlighted here. Tech upgrades are something you can just start working on.

I'll note that the Equal Vote Software Development committee has been working on some of the features you've described in a modular way that should be easy to fold into the forum soon, including a tool that allows people to vote with many different voting methods. This work is already being done by a coalition of volunteers and I'm sure they would like your help. The best way to do so is to sign up to volunteer at equal.vote/join.

I would hope that all discussions about the forum take place on the forum itself, rather than in external meetings that everyone must attend at the same time, I think this is both more inclusive, better documented, and simply takes advantage of the fact that we actually have a discussion board. Votes can happen on the forum itself, and should be very transparent in that everyone can see how everyone else votes, from the moment they cast their vote.

If the participants here make decisions that are just as good as those that could be made in meetings, but without having meetings, that in itself will amount to a significant innovation in the practice and theory of social choice.

If we can have methods to arrive at decisions without everyone having to meet at the same time, and without a "council" of specially-anointed people having to meet and decide, I think that will be great and it is the way we should go if we can figure out how to make it work so that people will feel that the process is legitimate and fair.

I'm generally in favor of streamlining the democratic process. If anyone has specific ideas for changes to our bylaws or procedures, they should draft a formal proposal to take to the upcoming meeting to be voted on. I'll state that I would never vote in favor of any consequential change if that change is just some ideas in an informal discussion in a forum thread. A specific, written proposal is the minimum bar for me to consider seriously adopting a change to any consequential process, and I suspect many others feel that way, too. If we want to change the process, that's fine, but we have to go through our current process make that change happen. Attempting to circumvent that process is one of the least democratic things we could do in this forum.

In terms of what you've been proposing, it seems so far to be incredibly nebulous and full of holes. Who gets to vote? How do you define "actively participating in the forum"? How does it address accessibility for those spread thin across different, relevant platforms? How do we ensure every voter is informed? What kind of timeline do we use? Which method? What if someone has to take a leave of absence? Which things do we bother voting on? How many people need to vote? What's quorum? How do we categorize different changes in relation to the last few questions? The list goes on. Overall, it feels poorly considered in my opinion.

If I were handling the "social" side of things, this is how I'd typically handle things. One, I'd be sure that if anyone new posted to the forum, that they'd get a response welcoming them and engaging them in conversation, assuming they seemed sincere and interested in voting theory. This doesn't mean it will always be me, but if no one else did, I would. I think that is one of the most important things a person running a forum can do.... keep people engaged and coming back.

If there was a thread that seemed to get adversarial, I'd jump in and attempt to steer it back to a positive discussion, or maybe advise each party to wrap things up or take it to private messages, because it isn't positive.

I'd have a general "be nice, be respectful" policy. If someone seemed to violate it, I'd usually DM the person first and see if they were willing to edit their content. If it was egegious enough I'd remove it immediately, but always engage them via DM so they don't feel like I am running them off.

This also applies to divisive political content that isn't directly related to voting theory. I think it is fair to say that I differ with Jack on how this should be handled, but I think I am in agreement with several others here that allowing that sort of content is toxic and counter to our mission. ("mission" being getting better voting systems in use in political elections). Again, I'd typically DM the person who seemed to go out of bounds. We're not "cancelling" anyone, but we are saying you need to stay on topic or at least steer clear of the sort of divisive content that could drive people away, or make people suspect that our motives are partisan.

Finally, the third category of problematic content that I'd tend to moderate is that which slams too hard on ranked choice / IRV. There's nothing wrong with saying that there are way better systems (I don't think any current participant disagrees with that), but I would usually draw the line at statements such as "IRV is worse than plurality". If you really want to make that argument, I think you should make it elsewhere. IRV, Score, STAR, and Approval are ALL significantly better than plurality, and I think that is essentially the one thing that the forum can have a "guiding philosophy" on. If you want to call that a "bias", ok.

EDIT: three people disagreed with limiting discussion on IRV being worse than plurality, so I'll back off that one.

Rob, we already have an existing code of conduct. A quick scan through it makes me wary of your ability to enforce it. Your first post on this topic was pretty inflammatory and arguably akin to a personal or professional attack as defined in Section e. It could also be consider disrespectful according to Section a depending how one defines "disrespect".

Diversity and inclusion are specifically highlighted throughout the code multiple times, mostly in the first half. Some of the few people who are not white, male, or neurotypical but have spent time on this forum — regardless of whether they've posted — have told me that the forum does not feel welcoming to them, in part because of the nature of your posts. It might not feel like it to you, but some of your posts feel unnecessarily aggressive to some folks, particularly those who regularly experience that kind of behavior from others in their real life. I don't feel equipped to recount their experiences here, but I think you should start by asking what you can do differently.

I believe we should facilitate more active moderation, but I'd opt for moderators who would do a better job of making this forum feel welcoming and inclusive.

Beyond all that, from what I can tell, @Jack-Waugh seems to have some desire to pass on some responsibilities or keys. This was the case before your first post. That's part of what the upcoming meeting is for and a few of us are scouting for volunteers to help with that, including through the Software Development Committee. In my opinion, a bit more consolidation would be helpful to improve simplicity and consistency. It would also better enable you to help with feature improvements through streamlining the overall operations.

Overall, I think we need to engage constructively about this issue and commit to going through the processes that are already in place.

Full disclosure, I tweeted that from the STAR Voting twitter account in reply to a direct question about the different methods. Twitter has a tight character limit and I find value in keeping the core of a response to one tweet, so I had to be brief.

Keith said something very similar to that quote in an interview I did with him:
Youtube Video – [00:56..]

Otherwise, I think Keith spelled out the core reasonings behind Equal Vote's stance.

@bternarytau Thank you for bringing attention to this. I work with @SaraWolk everyday and I'm certain it was not the intention to dis you. I'll bring this to her attention and I'm confident she'll reach out and work with you to find a solution.

@rob,

As stated in your original post, "It’s not like there is much activity." So why, then, do you keep highlighting that "[Equal Vote hasn't] even visited the forum in ages." What do you count as Equal Vote?

Until 8 months ago, @SaraWolk was the only employee, and now I'm the second. But are we the only people who count as "Equal Vote"? I'd say no. @Andy-Dienes was recently the chair of our PR Research Committee. @Keith-Edmonds is a board member. @Marcus-Ogren is leading Equal Vote research. @Jameson-Quinn is doing the same. @masiarek helps with outreach and the Software Development Committee. I could go on, but I consider all of these fine folks to be a part of Equal Vote and when they post here, that counts as Equal Vote participation.

We promote the Forum in our slack regularly. It's linked on the Equal Vote site. And have you checked out similar forums recently? There hasn't been much text-based activity anywhere lately. r/EndFPTP is mostly news articles about RCV. The three voting theory channels in the CES discord are so dead that when I promote my Open Democracy Discussions in them every week, the most recent post is often my promotion from the previous week. The Forward Party discord server had some activity when it started up, but it's died down, too. The most in-depth text-based discussions about voting theory recently have been nonsense Twitter fights with people who think improving single winner elections in the US is completely pointless. There's just a general apathy in the theory space right now. I suspect it's temporary and will ramp back up when high-profile US election campaigns pick up steam, but for now, the issue isn't a lack of participation from Equal Vote.

Obviously, we all want the forum to be better. So why not offer that? There's no need to wrap it up in some weird package about you "running" it. As Sara stated, there are processes for all of this, agreed on by a council of active volunteers.

Though I wasn't around for the founding, I'm privy to the history. I'm included in the email threads. I wouldn't have moved 2,300 miles across the country if I didn't care enough to learn it all. Moreover, I care enough that I've cultivated a space where the in-depth voting theory discussions ARE happening: my Open Democracy Discussions. They're not text-based, but they've attracted some of the folks here like @stardrop, @last19digitsofpi, @Jack-Waugh, @robla, and even yourself once or twice. I think part of why people go there is because it's a space that is explicitly non-toxic. Sara has been talking about this for years and she's right. It's why she's been elected as the Executive Director of Equal Vote multiple times. Cooler heads have prevailed and realized that Sara is what this movement needs, and activists have responded positively to that.

I'll point out that I don't think Sara should be the dictator of the Forum or whatever. I work next to her every day. I promise you that she doesn't want to be. Sara's trying to cultivate a positive culture and I implore you take that to heart.

Fueling rage against Equal Vote only makes this forum worse. If you want to make this forum better, then start by making it better.

@marcus-ogren Good point. The idea behind the Critical Score is to boost candidates with isolated support. It's possible that it goes too far, as you've highlighted. I just found that to sort of achieve the effect I was looking for, I would need a really big exponent for the Power Score.

I wonder how it all washes out in more realistic scenarios, but it's valid to consider someone putting themselves on the ballot and literally not telling anyone just to then score themselves 5 stars. Perhaps there's some kind of balancing that can be built in, but I suspect, on the general advice of James Quinn, that that would likely cause new problems.

Also, for reweighing, I think I used P instead of U because I originally intended to perform the calculation using Power Scores instead of Unity Scores. I just realized that I think I switched in part because I was doing the math wrong in the first place. If we just use the S^2 and the original Power Scores instead of S and the Original Unity Scores, it should still work, and maybe even better. Otherwise, differentiating the Power Scores as their own set of total scores for each candidate is effectively useless.

Looking back at my (limited) testing, I actually tried this and it seemed to work just fine, so the new reweighing formula for each ballot would be 1-(Qx(S1)^2/P1)-(Qx(S2)^2/P2)-... where Q is the number of ballots in a Hare Quota, S1 is the original score given to the first winner on that ballot, P1 is the original Power Score of the first winner, S2 is the original score given to the second winner on that ballot, P2 is the original Power Score of the second winner, and so on.

After a candidate is elected in a given round, we give some influence to them from that winner's supporters before the next round because those voters now have some (more) representation. How much influence do we give? The more stars you originally gave a winning candidate, the more of your influence we give to them, and it's exactly proportional to how much of that candidate's total support you contributed to and the number of voters needed to support that candidate so they can win a seat in the first place. For example, in an election with 150 voters and 5 winners, each winner needs the equivalent of full support from 30 voters to be given a seat because 150 divided by 5 is 30. This quota of 30 voters stays the same throughout the entire tally.

Now, if you gave the first winner 4 stars, which translates to 16 stars, and that winner's original Power Score was 800 stars, then you contributed 2% to their success. Go you! So our 30-voter quota times your 2% means that we give 60% of your original influence to that first winner before the next round, which is not too bad of a loss for a candidate you gave 4 out of 5 stars to! You'd think it would be more like 80%, but this first winner was so popular that they gained more than a quota's worth of support across the electorate. We only give each winner what they need so you don't get punished for supporting a popular candidate.

I feel the point of clarification isn't even needed when I phrase influence in terms of giving yours to winners you supported.

What about STAR-POWER Voting?
Score Then Assign Representatives - Proportional Option With Exponential Ratings

@jack-waugh For actual implementation, you would never bother squaring scores. You would always do Sx|S|.

Mathematically, cubing would be more elegant, and I considered it -- the results would just be a little more biased toward "polarizing" candidates. If a particular jurisdiction preferred that, that would be fine.

The problem is that the masses don't have their cubes memorized and the general explanation would require more syllables:

one hundred twenty-five, sixty-four, twenty-seven, eight, one, zero = 17 syllables

twenty-five, sixteen, nine, four, one, zero = 9 syllables

It's much easier for a person to "feel" the difference between 25 and 4 than 125 and 16.

@andy-dienes Yes, I'll add clarification to my statement.

I don't think IRV inherently elects more polarizing candidates than Choose-one Voting because of its mechanics. What I've seen is that in the US right now, because of the way IRV is sold, after it's implemented somewhere, candidate (exit) strategy and voter strategy decreases (at least initially). That would cause either method to elect more polarizing candidates, and I'm pointing in particular to (rare) IRV elections like Burlington, VT 2009 and this ongoing Alaska Special General Election with 3 distinct front-runners. As voters and candidates better figure out IRV, I would expect it to "reduce" mostly back to Choose-one Voting with similar candidate and voter strategy. The transparency and familiarity of Choose-one Voting helps it to "stabilize".

@andy-dienes I agree that it's mostly speculation at this point, though I have seen other papers and reports suggesting it's not that I need to find again.

I think the point about voters feeling like they have a fair choice needs to be qualified: it's important that we use systems that won't cause that feeling to backfire down the road. If voters like it at first, great. But if we're lying to them to make it that way, then when they inevitably discover the truth, we may end up in a worse place than where we started. It's important that we set ourselves and society up for success the first time, otherwise morale for voting method reform could be destroyed for a generation or more.

@rob I want competitive elections. That's why I'm qualifying your statement -- because it matters. I would agree that Condorcet methods encourage people to run who have wide appeal even in competitive races because the candidates actually need to beat each other. IRV is not a Condorcet method and only appears to tend to elect Condorcet winners because it is so often used in elections that are not competitive. Part of the reason that IRV isn't used in competitive elections is because it doesn't create them, which we need voting method reform to achieve in the US right now. Under IRV, candidates only need to get a majority (really a plurality) of support and then the minimum number of votes they need will naturally flow to them. What you're describing sounds more to me like competition has decreased in favor of one ruling faction, which sounds like the trend across most of California. It doesn't appear to me based on your recounting that IRV has done anything to prevent monopoly rule in San Fransisco.

This new voting method is so fresh that I haven't even named it yet! And I haven't even run it through more than one election, so I'm excited to hear from you all the ways in which it is terrible!

Okay, so the the question is how do we use score voting, which is inherently biased to elect candidates with broad appeal, to elect candidates with more isolated support? My answer: amplify voters' support for candidates at the top end of their ballot. How? The simplest way is to square their scores!

• Voters score candidates from 0 up to 5 stars.
• Add up these scores for each candidate to get their Unity Score.
• Add up the squares of these scores for each candidate to get their Power Score.
• Divide each candidate's Power Score by their Unity Score to get their Critical Score.
• Elect the first winner based on who has the greatest Critical Score.
• Calculate the new weight of each ballot using the formula 1-(QxS1/P1) where Q is the number of ballots in a Hare Quota, S1 is the original score given to the first winner on that ballot, and P1 is the original Unity Score of the first winner.
• Recalculate the Critical Score for each remaining candidate using the new ballot weights.
• Elect the next winner based on who has the greatest recalculated Critical Score.
• Calculate the new weight of each ballot using the formula 1-(QxS1/P1)-(QxS2/P2) where Q is the number of ballots in a Hare Quota, S1 is the original score given to the first winner on that ballot, P1 is the original Unity Score of the first winner, S2 is the original score given to the second winner on that ballot, and P2 is the original Unity Score of the second winner.
• Recalculate the Critical Score for each remaining candidate using the new ballot weights.
• Repeat, using S3 and P3 for the third winner, and so on.

Important note.
Ballot weights are able to go negative after a few rounds. This is intentional. It means that a voter is overrepresented, so now their ballot starts counting against their interests. On the surface, this appears like a huge spot for tactical voting, but in my (admittedly limited) testing, it's hardly an issue and serves more as an incentive for voters to distinguish among a large group of favorites than plan for a possible tiny bit of influence they might have over candidates they probably don't care much about by the rounds it matters. Instead of squaring scores during the calculation of the Power Scores, be sure to multiply scores by the absolute value of the scores to retain their negative nature.

Optionally, the (recalculated) Power Scores can be used to elect candidates instead if a jurisdiction wants winners who are more unifying. The (recalculated) Unity Scores can also be used instead if the jurisdiction wants candidates who are even more unifying.

Really, any power >1 can be used. I picked 2 because it is the simplest and most familiar for voters.

If you give a candidate 0 stars, it's still like you're giving them 0 stars. If you give them 1 star, it's still like you're giving them 1 star. However, if you give them 2 stars, it's like you're giving them 4 stars. If you give them 3 stars, it's like you're giving them 9 stars. 4 stars is like 16 stars and 5 stars is like 25 stars. So you can see how we're amplifying the support you're giving a candidate. That's the whole point of proportional representation. There will be multiple winners and this helps to ensure at least one strongly represents you.

For ballot reweighing, we need to take away one quota's worth of ballot weight from the entire electorate. Because the sum of all S for all voters in any given round is just P, then (sum of all S)/P=P/P=1, so Qx(sum of all S/P)=Q. Subtract Q from the entire electorate and now you have 1 less quota's worth of ballot weight from the entire electorate.

After a candidate is elected in a given round, we take some influence away from that winner's supporters before the next round because those voters now have some (more) representation. How much influence do we take away? The more stars you originally gave a winning candidate, the more influence we take away from you, and it's exactly proportional to how much of that candidate's total support you contributed to and the number of voters needed to support that candidate so they can win a seat in the first place. For example, if you gave the first winner 2 stars and their original Unity Score was 400 stars, then you contributed one half of one percent to their success. Go you! Now, if there are going to be 5 winners in the election, then each winner needs support from one-fifth of the electorate. Let's say this election had 150 voters. That means each winner needs the equivalent of full support from 30 voters, so 30 voters times your one half of one percent means that we take away 15% of your influence from the next round. Not too bad of a loss for a candidate you bothered to give 2 out of 5 stars to! A point of clarification: let's say you lose an additional 40% of influence based on the original score you gave the second winner. That stacks simply with the previous round, so now you've lost 15% plus 40%, so 55%, of your original influence by the third round.

Let me know what's wrong with my new method!

@andy-dienes Do you believe those gains are sustainable?

"In the 2014 survey, the gaps between resident perceptions of three indicators of campaign negativity in RCV and non-RCV cities were narrower than they were in the 2013 version..."

I believe that when voters and candidates are honest and excited, such as the first few cycles after IRV is implemented and FairVote convinces everyone they can finally be positive and honest, then IRV does perform better than Choose-one Voting and exhibits some of the positive effects that are often sold. However, as time goes on and the voters and candidates experience that their shiny new method keeps electing the same politicians as their old one, things will revert. That seems to show up in analyses of voter turnout in IRV jurisdictions after enough time.

https://www.rangevoting.org/IRVturnoutSF.html