Li Shenron said:
That's a fair question. The short answer is: because otherwise the penalty doesn't grow at the same rate as the cost of a level. Sean K. Reynolds provides
an example of how fractional XP penalties fail in practice, but not an explanation.
The long answer involves math. If you don't have the appetite for it, that's fine, but in that case you should not try to change any of the crunchy parts of the game, because you won't be able to predict the consequences.
Let's say, for example, that you have the following XP chart:
Level 2: 1 000 XP
Level 3: 2 000 XP
Level 4: 4 000 XP
Level 5: 8 000 XP
...
Now, suppose you give Aaron the aasimar half XP (but calculate XP as if he were the same level as his teammates). When everyone else has 2,000 XP and reaches level 3, Aaron will have 1,000 XP and reach level 2. When everyone else reaches level 4, Aaron will reach level 3, and so on. In other words, a 50% XP penalty with this XP chart is exactly the same as a LA +1. In addition, this system has more granularity than the existing XP chart. If you decided that aasimar are underpowered with a 50% penalty, you could instead give them a 1/3 penalty (but still calculate XP awards as if Aaron had his unmodified XP). This would mean that Aaron always levels up when the rest of the party is halfway to its next level.
You
can't do this if the XP chart is non-geometric. It doesn't work. If you try giving Aaron a fractional XP penalty in 3E, regardless of what fraction you choose, you get the results that Sean K. Reynolds noted above, namely: the character frequently has no penalty at low levels, but falls more and more levels behind the higher you go.
So far, I've given you only a few examples. Here's a more rigorous treatment. By definition, each element in a geometric progression is equal to its predecessor times some constant
r. If the XP cost of each level increases,
r > 1. So, if the function
x(
n), denoting the XP needed to reach level
n, follows a geometric progression,
x(
n) =
kr^
n, where
k is some constant.
Now, suppose that we award the character only some proportion
p of the experience he would otherwise earn. Since we're discussing XP penalties, we assume that 0 <
p < 1. We'll denote the amount of XP he needs to reach a level as
x'(
n). Since he receives only
p XP for every XP he would otherwise earn,
px'(
n) =
x(
n) for all
n. Therefore,
x'(
n) =
x(
n)/
p = (
k/
p)
r^
n.
Now,
p is a constant between 0 and 1. This means that 1/
p is, like
r, a constant greater than 1, which in turn implies that 1/
p =
r^
j for some
j. Taking the log of both sides, we get: log (1/
p) =
j log
r. Solving for
j, we obtain:
j = - log
p/log
r.
Now, we determined two paragraphs up that
x'(
n) =
x(
n)/
p = (
k/
p)
r^
n. We can now take this a step further:
x'(
n) = (
k/
p)
r^
n =
k(
r^
j)
r^
n =
kr^(
n+
j) =
x(
n+
j) =
x(
n - log
p/log
r). We therefore see that a character with an XP penalty is always a fixed number of levels ahead or behind a character without one; to be precise, - log
p/log
r levels behind.
Revisiting our examples above, we initially used values of
k = 500,
p = 1/2 and
r = 2. This gives us a value of
j = -1, which tells us that Aaron will indeed always be one level behind. Next, we looked at
p = 2/3 and
r = 2. In this case,
x'(
n) = 1.5
x(
n) = 1.5
kr^
n = 1.5×500×2^
n = 500×2^(
n+0.58). We confirm that, indeed, -log (2/3)/log 2 = 0.58.
As Sean K. Reynolds has already shown a counterexample to the contention that the same will be true of the 3E XP table, there is no need for me to proceed further.
Because the AD&D XP charts used a close-to-geometric progression at low levels, fractional XP penalties happened, by total coincidence, to work most of the time. It is naïve to assume that the same approach would work in 3E.
