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:

`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:

`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?