Part 15: (Game Mechanics) Fuckin' Bonus EXP, how does it work?

Let's Talk About Game Mechanics!
Episode the Fourth: Fuckin' Bonus EXP, how does it work?

WARNING: MATHEMATICS AHEAD

(and I'm not even certain I did it right myself)

Bonus EXP can be used to level up units between chapters, right from the Base menu. It takes more than 100 Bonus EXP to grant a level, however; the exact formula for how much you need to go the whole way from Lv X to Lv [X+1] is [50*X + 50] for beorc and [75*X + 50] for laguz (and remember that being promoted counts as being above Lv 20).

But that's not the interesting part. The interesting part is that levels gained from Bonus EXP are different from "natural" levels gained in combat.

Natural levels, we already know about. You have an X% chance to for each stat to grow by 1, where X is the stat's growth rate for the unit leveling up. For example, Edward's growth's are:

HP: 85%
Str: 60%
Mag: 5%
Skl: 65%
Spd: 60%
Lck: 50%
Def: 35%
Res: 20%

…so each time he gains a level, Edward has 85% chance of getting +1 to HP, 65% chance of +1 to Str, etc etc. Nothing but chance can stop you from getting +1 in all eight, although the chances of that are .03%, so you're probably not going to see it happen often.

Bonus EXP levels are different. When a unit gains a level using Bonus EXP, they will grow in three and only three stats, with the growth rates of each stat being used to determine the probabilities of a given stat being picked. For example, the total of all of Edward's growth rates is 380, so when he gains a bonus EXP level, he has an 85/380 chance of growing in HP, a 60/380 chance of growing in strength-

"HOLD IT!" I can already hear the Ace Attorney-playing math-nerds yelling in unison, "You're not accounting for the effect of one stat being chosen on the chances for the next!"

To them I say "HOLD IT yourself you impatient assholes, that's the whole point of why this is so important and we're just getting to that now, thanks."

In short, the whole system is positively revolutionized by the fact that you always get three growths. We could do this the long and boring and brain exploding way, eg:



X = collective of unit's growth rates for each stat
aX = unit's growth rate of stat a

'aX = shorthand for growth rates for stats other than a

∑ = sum of all
μ = mean of all

Mean probability of growth in stat a in a Bonus EXP Level =

(can you work it out yourself?)

 aX / ∑X

+

( aX / ( ∑X - μ('aX) ) ) * ( ∑('X) / ∑X )

+

( aX / (∑X - 2μ('aX)) * ( ∑('X) / ∑X ) * ( ( ∑('aX) - μ('aX) ) / ( ∑X - μ('aX) ) )

…I think. 

But by far a more succinct way of putting it would be to say that each stat's effective growth rate increases by virtue of the fact that the same stat can't be picked more than once.

Why is this such a big deal? Why should I bother with this system when I can take the risker but probably more rewarding system that lets me get more than three stat-ups per level?

Because of stat caps. It's been referenced a few times in the LP, but if you didn't already know, it's yet another Fire Emblem tradition that a unit's stats can only go so high, and how high is dependent on the unit and the stat. For example, once Micaiah's magic reaches 20, it stops going up, regardless of how lucky you are, though for Edward this happens at 10. Normally, capping one stat has no effect on the growth of the rest; you will always have X% chance of getting that +1. But with Bonus EXP, it makes a huge difference; capped stats are removed from the "draw". Thus, ∑X goes down, as does μX, if the stat that capped had a higher growth than that unit's mean growth (which, if it caps, it probably does), and therefore any interpretation of μ('aX) is also lower as a result. Since ∑X appears in every single denominator in that monstrous equation just now, that means that the mean probability of a stat going up in a bonus EXP level goes up every time ∑X goes down.

If you've been skipping over the maths, stop skipping here: This is most useful for "catching up" on stats in which units have only moderate growths; once one or two stats with high growth rates have been capped, the "draw" will come down to the few stats that aren't capped and that you want, and usually one stat that you don't need (such as magic for martial units), and the higher the growth rate of the capped stat, the greater the effective gain for the chances of other stats. The more stats you cap, the better the odds of being able to cap useful leftovers. We have seen this in action already; Edward capped both skill and speed, and then went on to gain +4 strength and +4 defense from 4 Bonus EXP levels.

Following this system to its logical conclusion; you can totally game level ups, and very easily. Cap one or two stats, binge on BEXP, and watch the rest of your stats go up too! The best part is that "lesser" stats tend to have lower caps, and you might be able to cap lesser stats by leveling up with BEXP!

Another side effect of this system is that you now have an increased incentive to use stat-up items before units reach their caps. By getting a unit to cap one of their "primary" stats sooner, you can massively increase the chances of being able to boost the rest with Bonus EXP.

Put bluntly: Bonus EXP might as well be considered a game-breaker.

Of course, you could use all your BEXP to level up a low-level character that just joined your team, safe in the knowledge that they will receive a set number of stat-ups, but who would do something boring like that when they could make their current dudes cap every stat ever (maybe)?

As a great "man" once said: "Only a yoghurt."