MATHS: The relationship between XP and BR

[Hovercraft]

As a statistician by trade, I enjoy messing around with numbers. So given some idle time earlier today, but away from my gaming rig so I couldn't actually play Planetside 2, I decided to have a go at reverse-engineering the mathematical relationship between XP and Battle Rank. It turned out to be rather more complicated than I was expecting, but I've tried to explain things as clearly and non-technically as possible. Read on, if you dare!

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Right, the first thing to do is simply pull the data for the cumulative XP needed to reach each BR from here (http://wiki.planetside-universe.com/ps/Battle_Rank) and plot one against the other.

https://dl.dropbox.com/u/8165699/cumxp.jpeg

The amount of XP needed to get to the next BR increases with BR, as I'm sure everyone already knew. It's tempting to call it "exponential growth", but a couple of quick calculations proves it's not. Technically, exponential growth would be where the ratio of XP at each BR with the previous one is constant. Here we have e.g. BR3/BR2 = 1500/375 = 4, and BR4/BR3 = 3750/1500 = 2.5. Not constant. The curve looks too wiggly to be proper exponential growth anyway, and the gap between BR99 and BR100 is too big.

So if the difference in XP between successive BRs doesn't grow according to a constant ratio, how does it grow? Let's try plotting BR against the amount of XP needed to get to that BR from the previous one.

https://dl.dropbox.com/u/8165699/diffxp.jpeg

Aha, piecewise linear! The pattern here, at least for most of it, is that every 5 BRs, from those that end with 1 or 6 to those that end with 0 or 5 (e.g. 41->42->43->44->45), the amount of XP needed to get to the next BR increases linearly. Then in between these "blocks" of five (going from a BR ending 0 or 5 to the next one, ending 1 or 6), the amount of XP needed jumps up. You can also see that the linear increase within the blocks of five and the jumps in between both step up as BR increases.

The most noticeable thing about this plot though is the HUGE leap from BR99 to BR100. I was pretty surprised by that. Getting to 99 from 98 takes about 550,000 XP, but getting to 100 from 99 takes over 1.2 million!

We've already pretty much established the pattern with this plot, but I'm not done yet. We need to go deeper. This is the plot of BR against *deep breath* the difference between [the amount of XP needed to get to that BR from the previous BR] and [the amount of XP needed to get to the previous BR from the one before that]. The differences between the differences.

https://dl.dropbox.com/u/8165699/diffdiffxp.jpeg

(Lines between points added for readability; BR100 omitted because including it squishes all the other points to the bottom, making them hard to read.)

If your head is hurting at this point, consider that the y-values in this third plot are just the vertical distances between successive points in the second plot, just as the y-values in the second plot are the vertical distances between successive points in the first plot.

This final plot really shows the hidden subtleties of what, at first glance, looked like a simple exponential growth relationship. Firstly, we can see something that wasn't clear in the second plot: That between BR1 and 4, the amount of extra XP needed to get to the next rank, on top of that which was required to get to the current rank from the one before, itself increases linearly (don't worry, I'm not going to bother plotting those differences!) The amount of extra XP then remains constant across the blocks of five ranks, right up until BR65. Then it doubles for two blocks, and doubles again for the final five blocks (well, four and four fifths, without BR100). Meanwhile the between-block jumps increase in size more regularly, and, except for the last one, always by the same amount: 4500.

So there you go. I had fun doing this, I hope a few people find it at least slightly interesting. Comments/questions welcome.

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As a side note, I'm aware that the devs changed the XP/BR relationship for launch. Gaining BR was a lot slower during beta. Check out the version of that PSU wiki page from back then (http://wiki.planetside-universe.com/ps/index.php?title=Battle_Rank&oldid=14790). BR100 used to require over 1.4 BILLION XP! I might go back and do this same analysis for those old values at some point, it won't take long.

[JawsOfLife]

I'm still trying to wrap my tired brain around some of these calculations, but this is some cool stuff! I'm a big fan of numbers myself, and this is at the very least a useful academic look at level up statistics. Thanks for taking the time to do this!

[Wahooo]

...
. I had fun doing this
...

:huh:

http://timothyjones.typepad.com/photos/stuff/nerds.jpg

[Mathematics]

Beautiful! Nice work! :cheers:

[Captain1nsaneo]

THEMOREYOUKNOW!

Good stuff, interesting to see the break down. Any ideas on why they stopped with BR100 when BR doesn't seem to affect anything of serious consequence? Seems that a simple exponential curve (with maybe a slow growth at the start) with no BR cap could have easily filled in as well.

[KaskaMatej]

Any ideas on why they stopped with BR100 when BR doesn't seem to affect anything of serious consequence?

They ran out of names for ranks ^^

[Hovercraft]

Good stuff, interesting to see the break down. Any ideas on why they stopped with BR100 when BR doesn't seem to affect anything of serious consequence? Seems that a simple exponential curve (with maybe a slow growth at the start) with no BR cap could have easily filled in as well.

They ran out of names for ranks ^^

Sounds about right! Also I suppose it's simple MMO psychology. Players want a big but finite goal. So when they finally hit BR100, which is a nice round number, they can think "that's it, I made it." If they kept going forever, the ranks would be even more pointless than they currently are.

That might also the reason for the blocks of five ranks with bigger jumps in between blocks. Once you hit a nice round number ending in 0 or 5, the XP to the next BR includes an XP jump, so you stay on the nice round number for a fair bit longer than any previous rank.

Someone on the official forum raised the way more practical question of what BR you need to be to have earned enough certs to unlock everything. I hadn't thought of that, but I'll look into it next. I think there are already tables of all cert costs, so shouldn't be too hard.

[Qwan]

This is all interesting, with the numbers and battle ranks and all, but I have one question. Does battle rank mean anything in this game. I mean if you think about it, the only battle ranks that have a purpose are the once that give you load out slots, other than that you get cute little names that dont give you any xp boost, or more efficiency with any weapons. I mean this is not PS1 were BR and CR ment somthing. I like reading the numbers, but since I opened up the last load out slot I havent even paid attention to my BR since then. Im more concerned with certs resources. I think the only thing battle rank is for is just to show that you have alot of time to do alot of gaming. :D and those are some pretty big numbers for some guys.

[Wahooo]

Nope, as you said other than the load out slot it means nothing.
As Hovercraft points out it is a MMO psychology thing. Carrot on a string?
One of my outfit mates stayed up a few days after release and did a lot of these calculations on his own and figured at his current rate of XP gain it was going to take a couple years (iirc) to reach BR100. It gets pretty steep. This was coming from a world class rocket pod whore as well.
Funny thing about some gamers... they want it and will do whatever they can to reach that goal and reach it first NO MATTER WHAT IT TAKES. In order to hook them properly the goal has to be attainable but still very hard. Big mistake many games make is to give in to the pressure of making the goal worth either too much or making it literally impossible or simply too easy.

[Hovercraft]

Someone on the official forum raised the way more practical question of what BR you need to be to have earned enough certs to unlock everything. I hadn't thought of that, but I'll look into it next. I think there are already tables of all cert costs, so shouldn't be too hard.

Well I thought there were tables of ALL certs, but I guess I was mistaken. Best I can find is the official Wiki listing all class and vehicle certs (http://wiki.planetside-universe.com/ps/Certifications). There are way more not listed for weapon upgrades, and even more than that if you include the cert costs of everything that can be bought with Smedbucks. but it turns out that doesn't really matter.

I make the sum of all the class/vehicle cert costs on that Wiki page, omitting the dupes for the empire-specific vehicles, to be 134,897. Multiplied by 250 XP per cert point gives 33,724,250 XP. BR100 is 18,868,950 XP. So get to BR100, and you're still only a bit over halfway towards getting a subset of all the certs in the game. Higby wasn't lying when he said it'd take years to get everything!

[Captain1nsaneo]

This is all interesting, with the numbers and battle ranks and all, but I have one question. Does battle rank mean anything in this game. I mean if you think about it, the only battle ranks that have a purpose are the once that give you load out slots, other than that you get cute little names that dont give you any xp boost, or more efficiency with any weapons. I mean this is not PS1 were BR and CR ment somthing. I like reading the numbers, but since I opened up the last load out slot I havent even paid attention to my BR since then. Im more concerned with certs resources. I think the only thing battle rank is for is just to show that you have alot of time to do alot of gaming. :D and those are some pretty big numbers for some guys.

Gives access to implants of higher quality and load out slots. I also imagine they'll find some other use for them as well.

By the time you unlock everything currently available there will be new things available and then when you get those there will be other things and so on.

[TheBladeRoden]

Thanks to math, I now know that considering my XP rate over the past 30 days, I'll be BR 100 in approximately 6392 days, or June 24, 2030!

At least by then I'll hopefully have computer than can run it