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 and plot one against the other.



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.



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.



(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. 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.