Nadeo tax and other formulas

[Queen Jadis]

Hi everyone!

You have probably noticed that taxes do not only exist in the real world but also in Trackmania
Whenever you use your coppers to download from manialinks, receive wins from online bets or send ingame messages, there will always be a certain (usually not very high) amount of coppers being deducted and sent to Nadeo. This Nadeo tax is not meant to annoy you but it actually helps stabilise the financial system in Trackmania by avoiding inflation.

The transaction tax varies depending on the amount of coppers that are transmitted. For higher values, such as >10000 coppers, it is close to five per cent.

Although the tax rises with the number of transmitted coppers, the relative amount of the tax decreases and asymptotically approximates 5%.

Let y be the Nadeo tax (in %):

y=5+(250/x)

(x being the number of coppers transmitted)

In conclusion, it is more efficient to transmit high amounts of coppers in few messages

If you are also interested in skillpoints calculation, you can check out these posts:

viewtopic.php?f=42&t=19540&start=495

http://forum.maniaplanet.com/viewtopic. ... ts+formula

Jadis

[eXtreme34]

I want to add something, as the guy said on the maniaplanet forum
You said :

That means for rank n:

(x-100)/n=y

(y being the SP of n)

I think it's not (x-100)/n=y but rather (x+100)/n-100=y I tried on many tracks and it always worked, even for non top 10 ranks.
That means n=(x+100)/(y+100) and x=(y+100)*n-100. But for the number of players it's the right formula.
I have to add that this formula can show the number of SP you can take if you do -0.01 (but that's another story )

EDIT : plus I didn't know what was the tax formula. One always learns something

[Queen Jadis]

I think it's not (x-100)/n=y but rather (x+100)/n-100=y I tried on many tracks and it always worked, even for non top 10 ranks.
That means n=(x+100)/(y+100) and x=(y+100)*n-100.

Thank you for correcting this. This formula can be applied on any track provided that the records of rank 1 or n are unequaled. In practice, it may be not 100% precise though because TM only displays integers

The same applies for my Nadeo tax formula. It is precise in itself but in TM you only have the approximated value. I also have to add that, in theory, the nadeo tax would be pracically the same as 5% if the number were only high enough. However, the lowest percentage possible ingame is 5.0005% (due to the monthly limitation of copper transactions).

Cheers

[eXtreme34]

Equality shows the skillpoints you can have with -0.01 (just divide the WR amount +100 by the rank shown in-game and then substract 100 )
And the +100 explains why you CAN have 0 SP when the map is unfinished (because you're last of course ^^) so, if n is the number of players who have played (not necessarily finished) the map in official, (you have x the number of SP for the world record), the formula would be the following one without the -100 at end :
x=100n-100
(x+100)/n=(100n-100+100)/n=100n/n=100 potential SP for the last place.
Because an unfinished map must bring nothing, so -100 at the end

Did you follow ? Or am I unclear (I hope I ain't )

[Scorm]

The formula proposed by Jadis is an okay approximation of the In-Game Messaging Tax with the downside of being more and more false the lower your "coppers transmitted" are.
He does end up with the correct conclusion that the higher the coppers sent value is, the more the tax lowers down to 5%.

Indeed I have tested some coppers messaging up to about 250k at which point I cannot go higher. Here is the data sheet that I will use to explain the exact formula.

First of all we notice that for a regular message (no coppers sent) the cost is 2 coppers. But we do know Nadeo always take at least 5% of any copper transaction (not the message cost itself) and that what they take are always integers. In the data sheet we can see that low values of coppers sent gives us a 3 coppers tax each times. This means we pay the 2 coppers for the message and a Rounded Up 5% of the coppers we want to send : 1 copper since the values are low (it is not possible otherwise, for example with 5 coppers sent : 5% of 5 is 0.25 so Nadeo either takes 0 cp or 1 cp from you but 0 would mean they take 0% therefore making one of the conditions above not verified ; it then adds the 2cp price of the message itself).

Let x be an integer (number of cp sent) :

Number of cp payed = roundup(x*1.05)+2

The tax is nothing more than the cp payed - cp sent :
Tax = roundup(x*1.05)+2-x
= ⌈x*1.05⌉+2-x
= ⌊x*1.05 +1⌋+2-x
(This is the column D of the sheet which always match the experimental taxes in column C)

Now this is where it starts to get interesting : the conditions of Nadeo gives us a function with an Integer Part : ⌊⌋
This means that the tax = f(coppers sent) will have a stairs like shape. These "stairs" will also appear in the Tax function in % because it depends on the previous one.

By using an easy cross-multiplication :
tax(coppers)/tax(%) = sent(coppers)/sent(%) <- this is 100 because it is all the cp we want to send.

so tax(%) = (100*tax_cp)/sent_cp

But we do have tax_cp we determined just before ; and sent_cp is nothing more than x.

tax(%) = (100*( ⌊x*1.05 +1⌋+2-x ))/x
We end up with an inverse and stairs type of function in green here : http://i.imgur.com/NpLzAn3.jpg
The function in red is the one Jadis proposed which approximates our result better and better as the cp_sent gets higher the stairs get smaller and the scatter also.
However in the low transactions the average error keeps getting bigger as we approach 1.
I hope this showed that games uses some rounding quite a lot, also known as floor() and ceiling() programming functions in order to give us accessible integers numbers in their games

[eXtreme34]

Really well explained
I think there's nothing more to add Didn't believe mathematics were so present in TrackMania (I was a bit sure about that when I discovered the formula for SkillPoints, which I can improve of course ).
But yeah one has to consider integers before all, even with SkillPoints there's no number with commas.
If we add this, the SkillPoints formula is this one. First of all, with N the number of players which TRIED the track in official :
Let f be an integer of the WR's SkillPoints on a given track :
f=(N-1)*100
Now appear the integers. First of all to determine the SkillPoints x for a given rank r (still an integer) :
x=rnd((f+100)/r)-100=rnd((f+100)/r-100) (rnd function is really important, it's not the integer part function this time )
This gives for rank r :
r=rnd((f+100)/(x+100))
And for the first (here the formula is more complicated because of -00 thing, colors to prevent difficulties with parenthesis ) :
f=rnd(((x+100)*r-100)/100)*100
NB : the reason of /100, then rnd then *100 is that the number of SkillPoints will always finish with -00

[Scorm]

Thanks for the kind words.
You might be able to change rnd with ceiling() function I believe but that is simply to have the programming code aspect of it, nothing major.
I am not convinced that decompiling TM's code will give us access to the real functions because it would seem that the game sends server queries each time it wants information on coppers, skillpoints and ladder points : but we can figure them out anyways

My time spent in solo mode is pretty low now compared to the rest but I might take an interest in the SP formulas ^^

[Queen Jadis]

As there seems to be some interest in finding Trackmania formulas, I decided to re-name the thread so you can post your observations and mathematical proofs here.

I think there are plenty of unknown formulas in TM that we have not even thought about. For example: How do you calculate points in Stunt mode? or: How does acceleration work?

Maybe someone will take the challenge

See you online,
Jadis

[Mizu]

Thats a great topic, i would love to know about the stunt one tho