# Multiplicative PCS scoring?

#### silveruskate

On the Ice
I don't think that's a desirable thing to have, it just pushes people to homogenize their skating. Instead of trying to be brilliantly artistic or brilliant at jumps, it encourages people to just be "okay" at everything. 80-80 doesn't deserve to beat 60-100 or 100-60.

BTW, how are you arriving at those numbers of '75' and '80'? It should be 60 and 64 (which is the same on a percentage basis, so I'm curious where you got 75 and 80 from).

No, my numbers are correct.

I see it differently....why not incentivise skaters to be good at everything? That is the point of it all. If this is your grievance then this system is perfect since even though the discrepancy in my numbers is so great, the difference in points is not even that large (and no skater gets perfect 80 on PCS). Sasha if I take her, compared to a skater with 80-80 will need 7 more TES points on the old system to win in the new system (107-60 = 80.25), and if she increased her PCS by 4 points (100-64 = 80) then she would match 80-80.

The problem with the system for those that want a big impact is more that it does almost nothing at all. For a PCS difference of 5 for example, the added advantage for the person with more PCS compared to the old system is maybe 1 point or even less.

E.g.

80-75 = 75
85-70 = 74.375
86-70 = 75.25

#### Skatesocs

Final Flight
No, my numbers are correct.

I see it differently....why not incentivise skaters to be good at everything? That is the point of it all. If this is your grievance then this system is perfect since even though the discrepancy in my numbers is so great, the difference in points is not even that large (and no skater gets perfect 80 on PCS). Sasha if I take her, compared to a skater with 80-80 will need 7 more TES points on the old system to win in the new system (107-60 = 80.25), and if she increased her PCS by 4 points (100-64 = 80) then she would match 80-80.

The problem with the system for those that want a big impact is more that it does almost nothing at all. For a PCS difference of 5 for example, the added advantage for the person with more PCS compared to the old system is maybe 1 point or even less.

E.g.

80-75 = 75
85-70 = 74.375
86-70 = 75.25
I just have a question about why you were saying it values parity more? I'll state outright I haven't tested too many pairings, but in the original you take someone who did perfectly on PCS and pitted them against someone who did much better on tech but 25% behind on PCS? I guess I don't see how we can say someone who does perfectly on PCS but is only "good" on tech is "good at everything"? This current set of examples is what seems closer to being "good at everything" - and as you notice, there's nothing between them compared to the 100-60 example you chose.

I agree with the diminishing returns part of course - that makes sense to me. So it's possible the first part is just me quibbling over semantics, or that I'm focusing too much on one example

#### gkelly

Record Breaker
My recommendation for any proposed change to judging, or in this case scoring, would be to test it an entire JGP competition phase.

Preferably a ladies' SP for the judging proposals, to see how well it works with a very large group of skaters with a wide range of abilities in a random draw.

For scoring changes, using a free skate or a somewhat smaller (but still larger than a typical senior GP) event would probably suffice.

#### Mathman

Record Breaker
I just have a question about why you were saying it values parity more? I'll state outright I haven't tested too many pairings, but in the original you take someone who did perfectly on PCS and pitted them against someone who did much better on tech but 25% behind on PCS? I guess I don't see how we can say someone who does perfectly on PCS but is only "good" on tech is "good at everything"? This current set of examples is what seems closer to being "good at everything" - and as you notice, there's nothing between them compared to the 100-60 example you chose.

It will work out this way for every example. For all pairs of numbers that add up to a given total x, the harmonic mean is maximized when the two numbers are both x/2. This is essentially because (x+d) + (x-d) = 2x for all d, but (x+d)*(x-d) = x[SUP]2[/SUP] - d[SUP]2[/SUP], which is less than x[SUP]2[/SUP] unless d is zero.

The differences are probably too small to worry about, but the principle is this. If you are Alexandra Trusova, are you better off to do 5 quads and basically blow off the PCSs, or should you do 3 quads and work hard on your "in-betweens." The proposal would slightly push Sasha toward the second choice.

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#### Mathman

Record Breaker
My recommendation for any proposed change to judging, or in this case scoring, would be to test it an entire JGP competition phase.

Preferably a ladies' SP for the judging proposals, to see how well it works with a very large group of skaters with a wide range of abilities in a random draw.

I will try it. I expect that the proposed system will not make much difference in the outcome. It will somewhat work to the disadvantage of good skaters who, on that day, made a lot of mistakes (with the result that the TES is much lower than expected for that skater, but the PCSs do not suffer much).

#### Skatesocs

Final Flight
It will work out this way for every example. For all pairs of numbers that add up to a given total x, the harmonic mean is maximized when the two numbers are both x/2. This is essentially because (x+d) + (x-d) = 2x for all d, but (x+d)*(x-d) = x[SUP]2[/SUP] - d[SUP]2[/SUP], which is less than x[SUP]2[/SUP] unless d is zero.

The differences are probably too small to worry about, but the principle is this. If you are Alexandra Trusova, are you better off to do 5 quads and basically blow off the PCSs, or should you do 3 quads and work hard on your "in-betweens." The proposal would slightly push Sasha toward the second choice.

I missed the part about "adds to the same" even though I understood that yesterday. Time to leave the thread.

##### Skating is Art, if you let it be
Record Breaker
I see it differently....why not incentivise skaters to be good at everything? That is the point of it all.

I don't find that to be "the point of it all". The scoring system is open ended and everything is worth a specific value; skaters should just go out there and do whatever they can and get the points they deserve for it. Multiplying numbers together is counter intuitive, it changes the final scores for no real reason. The system is ideally supposed to be balanced with every technical element being worth a very specific amount of points for a reason and the PCS being worth a sufficient amount in a comparison.

A performance that is 8/10 on both technical and presentation is actually doing less to push the sport forward. A performance that's 10/10 technical 6/10 presentation, or a 6/10 technical 10/10 presentation, is going to be more of a landmark because of what it excels at and how different it is. Why should a competitor be punished for being masterful at something? People should strive to constantly improve in every area they can, of course, but they shouldn't be forced to be merely "good" at a variety of things, instead of exceptional in one thing and passable in another.

#### Mathman

Record Breaker
OK, I started in doing the ladies SP from the Riga Cup (JGP Latvia), 2019-20 season. Already the first two skaters pretty much isolated the effect of the proposed change.

The actual results were (TES + PCS = Total)

Usacheva 38.83 30.31 69.04
Khromykh 39.36 29.57 68.93

Usacheva is ahead by 0.11 points, or about 0.16 % of the highest total.

Multiplicative method (TESxPCS/40)

Usacheva 29.33
Khromykh 29.10

Usacheva increases her lead to 0.23 points. This is now 0.78 % of the highest total.

(The two skaters had virtually the same actual total scores, but Usacheva's scores are ever so slightly more "balanced" than Khromykh's, hence the tiny increase in Usacheva's lead.)

#### Mathman

Record Breaker
Here are the results for the ladies' Short Program at the Jr Grand Prix Latvia event last year. A few of the numbers are slightly different from the final protocols because of fall deductions, which are charged to the total but not to either the TES or the PCS. Ignoring fall deductions, in both cases, the number of skaters that placed differently with the propsed multiplication rule compared to the current adding rule is ... zero.

Rank ... Name.....TSS (adding) ... TSS (multiplying)

1. Usacheva 69.04 29.33
2. Khromykh 68,93 29.10
3. Lee 66.93 27.49
4. Matsuke 66.41 26.95
5. Izzo 59.39 21.63
6. Kurakova 58.65 21.39

7. Sumiyashi 57.74 20.61
8. Arkhipova 57.30 20.25 (placements 7 and 8 switch for both methods if fall penalties are included)
9. Levushkina 55.23 18.75
10. Inthisone 54.63 18.32
11. Reuter 53.96 18.14
12. Petrokina 53.47 17.59

13. Stolt 52.93 16.82
14. Schermann 51.40 16.19
15. Lace 47.56 14.05
16. Hong 46.23 13.19
17. Piegad 45.92 13.16 (Piegar almost caught up to Hong, but not quite, by the multiplying method because her TES and PCS were about equal to each other)
18. Aglinskyte 45.56 12.98

19. Somova 45.10 12.64
20. Nordquist 42.82 11.44
21. Mikutina 42.36 11.12
22.Wooden 41.83 10.94
23. Toktarova 41.58 10.79
24. Van Gorp 40.92 10.46

25. Berkovich 39.21 9.60
26. Isaacs 36.45 8.25
27. Eudine 35.73 7.98
28. Chan 34.55 7.46
30. Gomez 33.13 6.85

31. Konya 33.05 6.82
32. Somerville 32.08 6.42
33. Luneva 31.96 6.33
34. Petersen 28.36 5.00
35. Gari 25.87 3.70
36. Golding 24.72 3.67

Conclusion. Mathematically, the rankings using the two methods are identical in this sample.

Other factors. The adding method is better because with the multiplying method a skater might say indignantly, "I've been out here bustin' my butt for 2 minutes and 40 seconds and all I get is a lousy 3.7 points????!!!!"

#### Skatesocs

Final Flight
Other factors. The adding method is better because with the multiplying method a skater might say indignantly, "I've been out here bustin' my butt for 2 minutes and 40 seconds and all I get is a lousy 3.7 points????!!!!"

Yes that does seem like a valid reason to not switch

Not to mention there might be complaints about how close "3.67" and "3.70" are. Or "13.19" vs "13.16".

#### silveruskate

On the Ice
I don't find that to be "the point of it all". The scoring system is open ended and everything is worth a specific value; skaters should just go out there and do whatever they can and get the points they deserve for it. Multiplying numbers together is counter intuitive, it changes the final scores for no real reason. The system is ideally supposed to be balanced with every technical element being worth a very specific amount of points for a reason and the PCS being worth a sufficient amount in a comparison.

A performance that is 8/10 on both technical and presentation is actually doing less to push the sport forward. A performance that's 10/10 technical 6/10 presentation, or a 6/10 technical 10/10 presentation, is going to be more of a landmark because of what it excels at and how different it is. Why should a competitor be punished for being masterful at something? People should strive to constantly improve in every area they can, of course, but they shouldn't be forced to be merely "good" at a variety of things, instead of exceptional in one thing and passable in another.

Rather than seeing it as being punished, you should see it as there being an extra reward for being better at both things where we should remember that the PCS cap already puts it at a disadvantage. Now I actually prefer the technical side much more, but from my experience the way a lot of figure skating fans and commentators talk about it, it is like they don't want skaters to be too unbalanced. Everything really depends on what most fans want, I couldn't care less and like TES being weighted more.

This added bonus is very very small in most cases so what you said would never happen. It still pays highly to have massive TES. Frankly it is the other way around, since according to your logic the PCS cap means that skaters are forced to focus on TES and not PCS because of the scoring system. The old system is always about 10/10 technical and 6/10 performance....it is never 10/10 performance and 6/10 technical. The problem with this method is actually that if you want big change you're not going to get it, and it would really only break tie-break situations of the old scores. Really all it is doing is making it 9.5/10 technical and 6.5/10 performance in the eyes of the skaters.

#### silveruskate

On the Ice
Other factors. The adding method is better because with the multiplying method a skater might say indignantly, "I've been out here bustin' my butt for 2 minutes and 40 seconds and all I get is a lousy 3.7 points????!!!!"
You illustrated it perfectly! Yes I was going to say it's a bit of a faff for no difference except for negligible cases. And in the process you make it more confusing for fans too.

##### Skating is Art, if you let it be
Record Breaker
Frankly it is the other way around, since according to your logic the PCS cap means that skaters are forced to focus on TES and not PCS because of the scoring system.

There is a PCS cap but realistically nobody should ever hit it. Assuming PCS were actually be scored correctly, then there should plenty of points for people to chase there, in comparison to the competition.

If competitors did deserve to start hitting close to max on PCS - meaning we would be seeing some of the most stunningly beautiful and musical and emotionally evocative performances of all time from multiple people (which would be the best thing for the sport) - then yes they would need to increase TES if they want to find an extra advantage. That's just inherent to sport though. If Michelle Kwan had been able to develop a high quality 3Flip+3Toe without sacrificing anything else, then she would deserve to beat the version of herself that was just doing 3Toe+3Toe. If it were so simple to add technical content without losing quality on that element or in the overall program, though, then everyone would do it.

#### Mathman

Record Breaker
If Michelle Kwan had been able to develop a high quality 3Flip+3Toe without sacrificing anything else, then she would deserve to beat the version of herself that was just doing 3Toe+3Toe. If it were so simple to add technical content without losing quality on that element or in the overall program, though, then everyone would do it.

Well, that is kind of the point of the multiplication idea.

Let's say that Kwan can do a 3T+3T (8.40 points) and work it into a program that is 90% of the maximum in terms of PCSs.

Then she masters a 3F+3T (9.50 points), but finds that the extra tech difficulty causes the quality of the program as a whole to go down to 80% of perfect.

So under the multiplying idea she gets 90% of 8.40 = 7.20 in the first case and 80% of 9.50 = 7.60 in the second case. It is still worth it to do the flip combo, but since she had to "sacrifice something else" (PCS went down from 90% to 80y%), she does not gain as much as the difference in raw base values alone would dictate.

Of course if she could do the flip combo without the upgrade having any negative effect on PCS, then she would get the full 90% of the difference in base values. Flip combo = 8.55 versus Toe combo = 7.20.

In either system, adding or multiplying, if you gain in tech but lose some of it back in PCS, then you lose some of your expected gain. If you can upgrade the tech and still keep up the PCSs, than you gain the full meansure (in either system).

What the multiplying method does do. though, is make the rich get richer and the poor get poorer. Kwan at 90% gets more for doing a 3F+3T than a lesser skater (say, PCS 50% of maximum) gets for doing the same element. (This would seem to be a fatal flaw of the multiplication method.)

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##### Skating is Art, if you let it be
Record Breaker
Hmm, if someone's PCS dropped so much in that instance, the 3Flip wouldn't be worth it at all. It's not just that combo suffering in the math, but the whole program receiving the deduction as a result of lower PCS.

What is the point in having any kind of multiplying thing though? If someone's PCS is dropping, that's already reflected in the points, with subtraction. I don't see any need to add an extra random element to the process of it all.

#### silveruskate

On the Ice
Hmm, if someone's PCS dropped so much in that instance, the 3Flip wouldn't be worth it at all. It's not just that combo suffering in the math, but the whole program receiving the deduction as a result of lower PCS.

What is the point in having any kind of multiplying thing though? If someone's PCS is dropping, that's already reflected in the points, with subtraction. I don't see any need to add an extra random element to the process of it all.

If you disagree with the whole premise of this thread fair enough, but the original poster was trying to address what he perceived as an negative imbalance in PCS weighting....I really don't know what else to say to you!

#### Mathman

Record Breaker
Hmm, if someone's PCS dropped so much in that instance, the 3Flip wouldn't be worth it at all. It's not just that combo suffering in the math, but the whole program receiving the deduction as a result of lower PCS.

Good point. If going for the extra tech degraded the entire program, PCS-wise, better to leave it out. The same is probably true in the current adding method, too, though.

What is the point in having any kind of multiplying thing though?

I would put it this way. The point of the multiplying thing is to treat the situation on the global scale the same way that GOEs are used in the details.

For each element you get a base value. Then you multiply that base value by a factor ranging from 0.50 (-5 GOE) to 1.50 (+5 GOE) depending on whether you messed up or went beyond thre call of duty. You could apply the same principle globally. You do a bunch of elements and get a TES score. Then this is multiplied by a factor that reflects, overall, whether you "messed up" or "went beyond the call of duty" in terms of making a coherent program out of that list of elements.

If someone's PCS is dropping, that's already reflected in the points, with subtraction. I don't see any need to add an extra random element to the process of it all.

Adding is sort of a random element, just the same as multiplying is. We are just more used to it (from 6.0 judging, for instance). If we think of TES as being a more or less quantitative measure, while PCS is more qualitative, then it is questionable what you get when you add a quantity to a quality. I have 5 apples and a juicy steak. You have 7 apples and a steak that is 20% less juicy. What do we add together to determine an overall winner?

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#### Skatesocs

Final Flight
Adding is sort of a random element, just the same as multiplying is. We are just more used to it (from 6.0 judging, for instance). If we think of TES as being a more or less quantitative measure, while PCS is more qualitative, then it is questionable what you get when you add a quantity to a quality. I have 5 apples and a juicy steak. You have 7 apples and a steak that is 20% less juicy. What do we add together to determine an overall winner?

Oh, is this why you asked all the way back on page 1 if OP was going to fold GOEs into PCS (because GOE is also qualitative)?

#### Mathman

Record Breaker
If you disagree with the whole premise of this thread fair enough, but the original poster was trying to address what he perceived as an negative imbalance in PCS weighting....!

That is the attractive feature of the proposed method. The weighting factors (.8 and 1.6 for ladies) are washed out entirely. In the formula TESxPCS/80, the 80 plays no role (it is just a constant multiplier that every skater gets no matter what her TES and PCS are). If you changed it to 100, or left is out entirely, the results would still be the same (except that everyone's point total might turn out to be in the tens of thousands intead of hundreds. )

#### Mathman

Record Breaker
Oh, is this why you asked all the way back on page 1 if OP was going to fold GOEs into PCS (because GOE is also qualitative)?

Yes. If you have GOES and also a PCS multiplier it seems like that's double dipping on the quality side.

On the other hand, it would force skaters to pay especial attention both to the quality of each element and to the the overall quality of the program, rather than just piling on the tech willi-nilli.

I think that this is the motive behind the ISU's continuing struggle to define what the Short Program is supposed to do. Rules like "Ladies cannot include a quad, but instead are judged by the quality of their triple jumps" speaks to this question, but is heavy-handed. Maybe a subtle change in the scoring rules would accomplish the goal without the need for arbitrary rules that are hard to explain or justify.

It might even mitigate the rancor of the debate about age limits.

Actually, though, it is trickier that that. GOEs are about quality but also about objective technical features like height of jumps. PCS include SS and TR which lean more toward the quantitative side, but even the qualities that go into Performance, Composition and Interpretation have quantifiable aspects. As unfashionable as it might be to say so, I am actually pretty well satisfied with the current scoring method and I think that the ISU has done quite a good job of juggling the myriad aspects of the sport.

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