Car bonus renormalization

[Duplode]

Well, as Zak mentioned on the news, the bonus percentages are indeed quite strange. The balance between different cars in itself is fine, but overall the percentage values are growing at a rather alarming pace. Not only that makes for a bit strange-looking "compressed" scoreboard, but also having bonuses nearing 50% means that more often than not a pipsqueak will need to improve 0.10s in order to advance 0.05s on the scoreboard, which is quite undesirable and annoying. Since not rounding the corrected lap times to 0.05s precision (i.e, taking times like 0:45.7193...s) or adopting other esoteric manipulations (say, multiplying all times by 1.5 to widen the gaps beyond the 0.05s threshold) feel way too unnatural, I was left wondering whether reducing all percentages to more "sane" values would be a good idea, and it seems indeed to be the case.

The most natural procedure for bonus renormalization would be to reduce all percentages so that if two times achieved with different cars were equivalent before the changes they remain equivalent after them, save for rounding differences. Let's consider percentages for two arbitrary cars a and b, Pa and Pb, which we want to modify so that Pa becomes (presumably smaller) Pa' and Pb is corrected accordingly to Pb' so that equivalent times obtained with the old percentages remain equivalent with the new ones. A bit of math leads to an expression for Pb':

Pb' = 1 - (1-Pa')*(1-Pb)/(1-Pa)
(note percentages are expressed as decimals - i.e., 0.38 and not 38%)

I tested the above expression with the percentages of Z86. The Carrera percentage was used as the reference value Pa, which I arbitrarily reduced from 0.42 to Pa' = 0.32. The resulting percentages were:

Code Select Expand


Car Curr. % New % New %, rounded
Acura 0.38 0.273 0.27
Audi 0.33 0.214 0.21
ZR1 0.33 0.214 0.21
GTO 0.36 0.250 0.25
Jaguar 0.16 0.015 0.02
Count. 0.39 0.285 0.28
LM002 0.39 0.285 0.28
Lancia 0.28 0.156 0.16
P962 0.14 -0.008 -0.01
Carrera 0.42 0.320 0.32
Indy -0.14 -0.337 -0.34



The renormalization brought the larger percentage values to levels comparable to the ones at Z80. Of course, all the differences accumulated reflect on the lower values, culminating with the quasi-surreal -34% for Indy (unlike the Acura/Countach/LM002 soaring percentages we have now, this wouldn't be likely to get any lower, however). Moreover, the absolute differences between the values also were raised somewhat from the original set, since a difference of 0.02 in percentage when one is to correct times by a ~0.9 factor (~10% bonus) range is much less significant than 0.02 with ~0.5 factors (~50% bonus).

In order to see what would actually happen to the results, here's an example using times from the current skirmish between me and CTG on Z86:

Code Select Expand


Car Time Corrected by curr.% Corrected by new% Corrected by new, rounded%
Countach 01:14.95 45.72 53.59 53.96
Carrera 01:18.80 45.70 53.58 53.58



The first "Corrected by new%" column was calculated with percentages rounded to three digits, while the second one had the percentages truncated to two digits. With three-digit percentages the results are virtually equivalent. Truncating at two digits introduces some distortion (four tenths...), but this is a rather extreme case anyway (Countach percentage would be 28.48%, which is quite different from 28%), and since there was already a similar rounding before this truncation isn't a really serious issue.

There is, however, another non-obvious change induced by renormalization. As discussed before, absolute changes in percentages are more significant with large bonuses than with smaller ones. That means the catchup between cars from one round to another will be slowed down. Consider, for instance, a scenario where Carrera does 1-2-3 in Z86 and Indy gets unused. With current percentages, Z87 bonuses would be 32% (-10%) and -11% (+3%). Now, in Z86, a 1:15 Carrera lap (corrected time 43.50) would be matched by a 0:38.16 Indy lap. After the bonus updates in Z87, 1:15 with Carrera would mean 0:51.00 corrected time, which would be matched by 0:45.95 with Indy. Had we employed renormalized percentages on Z86, the percentages would be 32% and -34% for Z86, and 22% and -31% on Z87. Taking again a 1:15 lap with Carrera as reference, it would be matched by 0:38.06 on Indy at Z86, of course equivalent to the original situation except for rounding errors. On Z87, however, the equivalent Indy time would drop to 0:44.66 - a 1.3s change from the previous scenario.

Summarizing, renormalization would result in saner percentages with a bit extra tampering of balance and somewhat slowed down recovery of low percentage cars. Still, I think the gains outweigh any possible losses, and thus I actively suggest Zak renormalization of bonuses for Z87. So, your opinions?  

[zaqrack]

I planned to do this at the end of the year, but after reading your thorough analysis (and seeing the high percentages, which were caused by the revised and accelerated bonus changes) now I rather think we should normalize twice a year - the first of which was due end of june, just around the end of Z85 - so I'll do it now.
However, we should not define one car used for the normalization, as the original values could be bad too, but I'd rather normalize the car with the lowest positive coefficient to 0%.
This also makes the formula easier (let's not make excel sweat )

Pb'=1-((1-Pb)/(1-Pa)) as 1-Pa'=1-0=1

In this case this is the P962 going to 0% from 14%.

The results are:

Code Select Expand

OLD NEW ROUND
ANSX 38% 27,91% 28%
AUDI 33% 22,09% 22%
VETT 33% 22,09% 22%
FGTO 36% 25,58% 26%
JAGU 16% 2,33% 2%
COUN 39% 29,07% 29%
LM02 39% 29,07% 29%
PANC 28% 16,28% 16%
P962 14% 0,00% 0%
PCAR 42% 32,56% 33%
PMIN -14% -32,56% -33%

which values I'll apply after the current track (recalculated to related changes of course), as:
-I don't want to interfere with the race
-previously sent time are not corrected automatically, and we already have more than 30 replays sent - lots of recalculation needed.

In the future, this normalization will take place always in the mid-season and end of season - that means the 6th and 12th track of the year - (this year is special as we have only 11 tracks, it wont be the same again, we'll be back to monthly schedule)

[Duplode]

That looks perfectly fine  The only extra observation I could think of was on the end-of-year normalization. It seems the overall minimum% - maximum% gap is growing steadily along the season (it was 35% at Z80, now it's 56% and with renormalization it would reach 66%), so perhaps it would make for a more balanced season to reset the coefficients at the end of season (not necessarily to Z80 values, but to a set with a smaller minimum - maximum gap).

[zaqrack]

If you take the indy out, the gap is not that big. And the indy is down so much because of the two full indy podium tracks. With positive manual correction there'd be soon a track, where the indy could go pg, and dominate every other car. The solution could be maybe two separate bonus for the indy - one with PG and one without PG. The two options differ a lot, I guess a noPg indy bonus would not be below 0% right now.

[alanrotoi]

what is this? I don't understand  Is there a rule I don't know?

[Duplode]

what is this? I don't understand  Is there a rule I don't know?

No, there isn't - it's just the percentages are getting too high, making for a strange-looking scoreboard and situations when we need to improve a full tenth to improve 0.05s on the scoreboard. So what Zak will do is to lower all percentages for Z87, but in a way the balance remains essentially unchanged. Nothing to worry about 

If you take the indy out, the gap is not that big. And the indy is down so much because of the two full indy podium tracks. With positive manual correction there'd be soon a track, where the indy could go pg, and dominate every other car. The solution could be maybe two separate bonus for the indy - one with PG and one without PG. The two options differ a lot, I guess a noPg indy bonus would not be below 0% right now.

Well, considering IMSA instead of Indy the gap isn't growing that fast, indeed... still, I think we'll only be sure of that by the end of the season. As for two bonuses for Indy, the line between PG/non-PG looks very blurred. In Z84, for instance, even though the podium was 100% Indy only Ayrton actually performed a section of the track on full-PG - me and Mark only used it intermittently.

[CTG]

...or just go back to the old, conventional one car/track rule...

[alanrotoi]

aaaaaah Thank you Duploide!! I thought he did it every track, some cars got higher and others got lower.

[zaqrack]

...or just go back to the old, conventional one car/track rule...

no way. 
some improvements always have to be made to keep the race interesting. - sometimes they turn out to be wrong, but most of the time they bring a positive effect

[zaqrack]

bonus renormalization took place, the values used were the following:

Code Select Expand


OLD NEW ROUND
ANSX 34% 21,43% 21%
AUDI 35% 22,62% 23%
VETT 35% 22,62% 23%
FGTO 38% 26,19% 26%
JAGU 18% 2,38% 2%
COUN 39% 27,38% 27%
LM02 41% 29,76% 30%
LANC 30% 16,67% 17%
P962 16% 0,00% 0%
PCAR 39% 27,38% 27%
PMIN -12% -33,33% -33%


we will have the next renormalization after the end of this season. Next year we'll probably have two during the season.

[Akoss Poo a.k.a. Zorromeister]

bonus renormalization took place, the values used were the following:

Code Select Expand


OLD NEW ROUND
ANSX 34% 21,43% 21%
AUDI 35% 22,62% 23%
VETT 35% 22,62% 23%
FGTO 38% 26,19% 26%
JAGU 18% 2,38% 2%
COUN 39% 27,38% 27%
LM02 41% 29,76% 30%
LANC 30% 16,67% 17%
P962 16% 0,00% 0%
PCAR 39% 27,38% 27%
PMIN -12% -33,33% -33%


we will have the next renormalization after the end of this season. Next year we'll probably have two during the season.

Shouldn't we ignore this car percentage bonus next year, and use the standard 2001-07 system for racing? 

[zaqrack]

I don't think so (especially with USC restarting), but we can start a debate on that, you may change my mind.
It can be really annoying in the first days, but after all I think it makes the month more interesting.

[Akoss Poo a.k.a. Zorromeister]

I don't think so (especially with USC restarting), but we can start a debate on that, you may change my mind.
It can be really annoying in the first days, but after all I think it makes the month more interesting.

Well, I think we are not time millionaires, at least I am not. The second thing is that it is not funny to spend hours of driving to experience which would be the fastest car for the track taking the everchanging bonuses into consideration month by month. If it's going to be a debate or a poll about this, I'd vote for the old system. An other thing: with the old system, we can use all the cars in a year once, with this year's, there can be more frequent cars and cars which were never driven. And with the old system, we can write nice and interesting comparisons and analysises of the replays!

[zaqrack]

Well, I think we are not time millionaires, at least I am not. The second thing is that it is not funny to spend hours of driving to experience which would be the fastest car for the track taking the everchanging bonuses into consideration month by month. If it's going to be a debate or a poll about this, I'd vote for the old system.

You don't need hours of driving for that, you only need to watch the others in the first days. And from the tracks of this year we can conclude - the strong cars (or car) is usually well-known by the half of the track's lifespan. If you head for a win - just use the same car as the leader does. If you're not - you can have fun with your favourite car. That leaves a lot of freedom to pipsqueaks and makes the competition much more versatile.   

An other thing: with the old system, we can use all the cars in a year once, with this year's, there can be more frequent cars and cars which were never driven.

All the cars will be driven sooner or later, maybe not during a year but two years. The bonuses guarantee this. As the ones left out for a while are usually unpopular cars I guess everyone only wins in this situtation.

And with the old system, we can write nice and interesting comparisons and analysises of the replays!

That is certainly a drawback. But I have plans to overcome this: I'm considering on creating a splitscreen video of the best drivers on each track - you simply have to multiply the video's frame rate with the car bonus coefficient, and all the videos come into sync then. This is only theory by far, but it seems possible.