Test it yourself! A new Score PR method from Sass

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 onefifth 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!

@sass said in Test it yourself! A new Score PR method from Sass:
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.
This sounds kind of unmathematical. Maybe you should be cubing instead of squaring, because that preserves negativity and does not require case analysis. But I don't understand the overall design, so have no idea whether cubing fits into it.

@jackwaugh But cubing gives a different result. It's perfectly reasonable to absolute value one of them before multiplying.
https://www.mathworks.com/matlabcentral/answers/508591keepthenegativevaluewhilesquaring

@jackwaugh For actual implementation, you would never bother squaring scores. You would always do SxS.
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 twentyfive, sixtyfour, twentyseven, eight, one, zero = 17 syllables
twentyfive, 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.

What about STARPOWER Voting?
Score Then Assign Representatives  Proportional Option With Exponential Ratings

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 30voter 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.

@sass said in Test it yourself! A new Score PR method from Sass:
Divide each candidate's Power Score by their Unity Score to get their Critical Score.
I haven't fully wrapped my head around this method yet, but division by the unity score seems extremely problematic. For instance, suppose Rita Writein gives herself a 5 and every single other voter gives her a zero. This gives her a Critical Score of 5^2/5 = 5 and causes her to immediately win a seat.

@marcusogren 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.