Helpful Resources and Balance

[TheGeometer]

Helpful Resources and Balance

This section's main purpose is to explain why the system was designed the way it was, and to give players information that they're likely to find helpful.


Balance and Probability

The spoiler below contains probability tables that show how likely you are to roll any given formula, depending on your number of elemental dice. It also describes the balance decisions that were made, which are closely related to these probabilities.

(click to show/hide)

Formula Sets

To start, we divide the formulas into sets, where all the elements in each set map to each other. For instance, the formula AE is in the same set as the formula AF, since both have 2 different elements. This set has 6 members: AE, AF, AW, EF, EW, and FW. Note that this 6-fold symmetry only holds for the Elementalist and Chemoturge; since the Alchemagus and Elemental Blade always roll at least one of their signature element, formulas that contain that element are of course more likely. For the discussion that follows, we'll stick to the case of the Elementalist. We'll come back to the Alchemagus and Elemental Blade later.

I named each set by a representative formula, like AE. I always picked the member of the set that comes first in alphabetical order. That is, the one where A is the most numerous, followed by E, then F, then W. AEEEFW is in set AAAEFW, for example (3 of one element, then 1 each of the other 3). The latter is the name of the set since its most numerous element is A.


Probability Table (Elementalist and Chemoturge)

Since all formulas in the same set have the same distribution of probabilities, they're all equally easy or difficult to cast. That's why every formula in a set has the same level. However, some levels consist of several different sets, and it's not obvious why I grouped them together. For instance, both AAAA and AAAEEFW are the same level (7), even though one is size 4 and the other is size 7. The table below should make that decision a lot clearer:

Table: Probabilities (%)
Formula Level 0 — — 1 — — 2 — — 3 — 4 — — — 5 — — 6 — — — 7 — —	Set Name (Set Size) A (4) AE (6) — AA (4) AEF (4) — AAE (12) AEFW (1) — AAEF (12) — AAA (4) AAEE (6) AAEFW (4) — AAAE (12) AAEEF (12) — AAAEFW (4) AAEEFF (4) AAAEE (12) — AAAA (4) AAAEEFW (12) AAEEFFWW (1)	Number of Dice   3 4 5 6 7 8 9 57.8 68.4 76.3 82.2 86.7 90.0 92.5 28.1 43.0 55.7 66.0 74.1 80.4 85.2 — — — — — — — 15.6 26.2 36.7 46.6 55.5 63.3 70.0 9.4 23.4 38.1 51.3 62.3 71.1 78.1 — — — — — — — 4.7 13.3 23.9 35.1 45.7 55.2 63.5 0 9.4 23.4 38.1 51.3 62.3 71.1 — — — — — — — 0 4.7 13.7 24.9 36.6 47.6 57.3 — — — — — — — 1.6 5.1 10.4 16.9 24.4 32.1 39.9 0 2.3 7.8 15.9 25.5 35.6 45.4 0 0 5.9 16.1 28.2 40.2 51.2 — — — — — — — 0 1.6 5.4 11.3 18.6 26.8 35.3 0 0 2.9 9.5 18.8 29.4 40.0 — — — — — — — 0 0 0 2.9 9.0 17.2 26.4 0 0 0 2.2 7.7 16.0 26.0 0 0 1.0 3.8 8.7 15.4 23.3 — — — — — — — 0 0.4 1.6 3.8 7.1 11.4 16.6 0 0 0 0 2.6 8.1 16.0 0 0 0 0 0 3.8 11.5

For each set and each number of dice, the table lists how likely you are to roll that set with that number of dice. I considered any probability above 30% to be likely and colored the cell yellow. Somewhat-likely (orange) is 20 to 30%, rare (red) is 5 to 19.9%, and negligible (dark red) is anything below that. The rows are roughly ordered from most to least probable, and notably, the orange cells form an extremely regular progression. Each level N becomes somewhat-likely when there are exactly N+3 dice. This is what lets me use dice progression in place of formula level progression.

As a quick check, I wrote out 6 formulas like a 1st-level Elementalist might have, and rolled 4d4 several times. Orange formulas seemed to come up dependably often, the one red formula I took was a rare treat, and yellow formulas were there when I had nothing better to do. This is approximately the feel I'm going for. Hopefully, this table suggests that this feeling continues, but with higher-level formulas as the character levels up. Given the Elementalist's dice progression, 7th-level formulas should have roughly the power of 8th or 9th-level spells, and the rest of the formulas should scale in kind.

The Chemoturge has the same progression as the Elementalist, but with 1 fewer die. This means that a Chemoturge's first roll will tend to be pretty bad. However, the ability to preserve the roll and change 1 element at a time allows her to quickly get to much more powerful effects. I'm not as confident about this class's balance as I am about the others', so adjustments may be necessary in the future.

The last thing of note from this table is the size of each set, in parentheses next to the formula. I wanted each level to have a total of 8-14 formulas in it, and for the first few levels, I have barely enough. At later levels, due to the sheer number of combinations, I have more than enough. Since I wanted to make use of small formulas as well as large ones, and needed at least 8 formulas per level, the first few levels and the need to start at 4 dice were established immediately from the probability table. From there, I continued until the formulas became unwieldy, using natural gaps in the probability table to segment the formula levels. Adding a 10th die and an 8th formula level was possible, but using a Bard progression for the dice was simple and AAEEFFWW was a good point to end on, letting me have one final, unique formula in the vein of Wish.


Probability Table (Alchemagus and Elemental Blade)

The Alchemagus and Elemental Blade have a unique mechanic where they have a free element of their chosen type. Where Elementalists are incentivized to learn formulas containing all elements equally, so that they can act no matter what roll they get, Alchemagus and Elemental Blade have to balance that incentive with being much more proficient at formulas that all involve the same element.

In the case of the Elemental Blade, this results in strictly less reliability overall and less power for all formulas that don't involve the chosen element, but this is balanced by the fact that the class is a gish, giving out significant combat abilities as class features. The Alchemagus is a caster-type class like the Elementalist, and she compromises for the lack of reliability with a faster dice progression. While Alchemagi are strictly weaker than Elementalists at formulas that don't involve the signature element, they're correspondingly stronger with those that do, making them a more specialized counterpart to the Elementalist's generalist play style.

The probabilities for the Alchemagus and Elemental Blade are much more complicated. Since their signature element is defined ahead of time, it breaks the symmetry of the sets. So for an Earth Magus, the probability of producing EEW is higher than EFF, which is higher still than AFF, even though all of these formulae are in the same set (AAE). Rather than making a much larger table of new sets and probabilities, below is the "best-case" table, where the Alchemagus or Elemental Blade chooses to learn the member of the set that has the most of their element in it. This is what players are likely to do anyway. Due to the way the sets are labeled, this means that the table below is effectively the correct table for a Stormblade (Elemental Blade of Air) or Air Magus.

Table: Probabilities with Signature Element (%)
Formula Level 0 — — 1 — — 2 — — 3 — 4 — — — 5 — — 6 — — — 7 — —	Set Name (Set Size) A (4) AE (6) — AA (4) AEF (4) — AAE (12) AEFW (1) — AAEF (12) — AAA (4) AAEE (6) AAEFW (4) — AAAE (12) AAEEF (12) — AAAEFW (4) AAEEFF (4) AAAEE (12) — AAAA (4) AAAEEFW (12) AAEEFFWW (1)	Number of Dice   2 3 4 5 6 7 8 100 100 100 100 100 100 100 43.8 57.8 68.4 76.3 82.2 86.7 90.0 — — — — — — — 43.8 57.8 68.4 76.3 82.2 86.7 90.0 12.5 28.1 43.0 55.7 66.0 74.1 80.4 — — — — — — — 12.5 28.1 43.0 55.7 66.0 74.1 80.4 0 9.4 23.4 38.1 51.3 62.3 71.1 — — — — — — — 0 9.4 23.4 38.1 51.3 62.3 71.1 — — — — — — — 6.2 15.6 26.2 36.7 46.6 55.5 63.3 0 4.7 13.3 23.9 35.1 45.7 55.2 0 0 9.4 23.4 38.1 51.3 62.3 — — — — — — — 0 4.7 13.3 23.9 35.1 45.7 55.2 0 0 4.7 13.7 24.9 36.6 47.6 — — — — — — — 0 0 0 5.9 16.1 28.1 40.2 0 0 0 2.9 9.5 18.8 29.4 0 0 2.3 7.8 15.9 25.5 35.6 — — — — — — — 0 1.6 5.1 10.4 16.9 24.4 32.1 0 0 0 0 4.4 12.8 23.5 0 0 0 0 0 3.8 11.5

It looks messier than the Elementalist table, but not by too much. And note that while the Elemental Blade finishes at 7 dice with only slightly better command of her chosen element than an Elementalist has with all elements, the Alchemagus with 8 dice can cast those formulas substantially more easily than an Elementalist with 9.

These tables were generated by a combination of the multinomial theorem and loops in Python. There may still be off-by-one errors in there, so if you see anything that looks obviously wrong, let me know.


Formula Scaling with Level

The final issue to sort out with a system like this is how the formulas' power should scale with elemancer level. Spells do this in a relatively straightforward way; damaging effects in particular generally scale as 1d6/level, with caps at 5, 10, 15, or 20 to keep low-level spells from remaining as good as high-level ones forever. We can't do this with formulas, because then there would be no incentive to reach for an unlikely high-level effect when it would deal the same number of d6 as your low-level formulas. Imagine if you had a 1st-level formula that dealt 1d6/level (max 5d6) and a 3rd-level formula that dealt 1d6/level (max 15d6). Why would you try to cast that high-level formula at character level 4, for instance? Sure, it has a higher DC and maybe an interesting effect on a failed save, but it doesn't seem like much reward for the high risk of learning such an unlikely formula.

The natural solution is a shallower linear scale with an offset that increases with level. So at level 1, while a 1st-level formula will deal 2d6, a 3rd-level formula will definitely be worth using since it deals 6d6. This also helps make the round-to-round randomness (which is the main mechanic of this system) feel more pronounced.

The only question left is, why did I pick the specific scaling that I did (2 * [formula level] + 1 per 3 elemancer levels)? The table below should illustrate it (0th-level formulas have been omitted since none of them are designed to scale):

Table: Formula Scaling

Level 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Formula Level   1st 2nd 3rd 4th 5th 6th 7th 2 4 6 8 10 12 14 2 4 6 8 10 12 14 3 5 7 9 11 13 15 3 5 7 9 11 13 15 3 5 7 9 11 13 15 4 6 8 10 12 14 16 4 6 8 10 12 14 16 4 6 8 10 12 14 16 5 7 9 11 13 15 17 5 7 9 11 13 15 17 5 7 9 11 13 15 17 5 8 10 12 14 16 18 5 8 10 12 14 16 18 5 8 10 12 14 16 18 5 8 11 13 15 17 19 5 8 11 13 15 17 19 5 8 11 13 15 17 19 5 8 11 14 16 18 20 5 8 11 14 16 18 20 5 8 11 14 16 18 20

In green, I highlighted the character levels where another die is added, and the level of formula that becomes practical at that point (7th-level ideas never become practical, so they're a bit of an outlier). Note that the scale keeps very good pace with character level - it's always exactly 1 higher! I consider this an acceptable concession for putting up with all this randomness and having far fewer options than wizards.

In red, I highlighted the point where that formula level's progression stops. Though it happens later than it does for spellcasters at comparable levels, it stops at roughly the same point. For instance, 3rd-level formulas might deal a maximum of 11d6 damage, while 3rd and 4th-level spells typically max out at 10d6. Similarly, 5th-level formulas reach 14d6, while 5th and 6th-level spells reach 15d6. The scaling isn't 1-to-1, but it's close,  and the fact that it keeps level with the level progression and caps at 20 is very appealing to me.

And that's about it! That's how balance works in elemental magic! 

Formulas Known

This section unpacks the mechanics for learning formulas. The system allows for multiple possible progressions, which are outlined here for players looking for suggestions or who would rather follow a table than keep track of their formula levels independently.

(click to show/hide)

Basic Rules

As described in the introductory post, there are 2 rules for choosing formulas known:

• You gain 1 high-level formula known per level, or 2 low-level formulas (where low-level is up to and including 1/3 your elemancer level, rounded up). At 1st level, you gain 6 high-level formulas, which can each be swapped for 2 low-level formulas.
• Your number of formulas known of one level must be less than or equal to your formulas known of the level below it.

There are a few possible progressions allowed by these rules (we'll stick to the Elementalist progression for now). You can, for instance, maximize your total number of formulas by almost always choosing low-level formulas. This results in 6-7 formulas per level for most levels, as shown in the table below:

Table: Broad Progression
Level 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20	Formula Level   0th 1st 2nd 3rd 4th 5th 6th 7th 6 6 0 0 0 0 0 0 7 7 0 0 0 0 0 0 7 7 1 0 0 0 0 0 7 7 3 0 0 0 0 0 7 7 5 0 0 0 0 0 7 7 7 0 0 0 0 0 7 7 7 2 0 0 0 0 7 7 7 4 0 0 0 0 7 7 7 6 0 0 0 0 7 7 7 6 2 0 0 0 7 7 7 6 4 0 0 0 7 7 7 6 6 0 0 0 7 7 7 6 6 2 0 0 7 7 7 6 6 4 0 0 7 7 7 6 6 6 0 0 7 7 7 6 6 6 2 0 7 7 7 6 6 6 4 0 7 7 7 6 6 6 6 0 7 7 7 6 6 6 6 2 7 7 7 6 6 6 6 4

While an elemancer using this progression has many options for each formula level, she almost never knows any effects more powerful than a wizard of her level would. This type of build is essentially a wizard who, rather than prepare her spells, has all of them available to her semi-randomly. This progression is "broad" as opposed to "deep" since it has only a few levels, but each is very diverse.

A slightly deeper build has 5 formulas known of each level and always knows 1 higher-level formula:

Table: Medium Progression
Level 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20	Formula Level   0th 1st 2nd 3rd 4th 5th 6th 7th 5 5 1 0 0 0 0 0 6 6 1 0 0 0 0 0 6 6 2 0 0 0 0 0 6 6 2 1 0 0 0 0 6 6 4 1 0 0 0 0 6 6 6 1 0 0 0 0 6 6 6 1 1 0 0 0 6 6 6 3 1 0 0 0 6 6 6 5 1 0 0 0 6 6 6 5 1 1 0 0 6 6 6 5 3 1 0 0 6 6 6 5 5 1 0 0 6 6 6 5 5 1 1 0 6 6 6 5 5 3 1 0 6 6 6 5 5 5 1 0 6 6 6 5 5 5 1 1 6 6 6 5 5 5 3 1 6 6 6 5 5 5 5 1 6 6 6 5 5 5 5 3 6 6 6 5 5 5 5 5

Similarly, you could have 4 formulas known of each level and up to 2 high-level formulas at a time. This progression is very consistent from level to level and gives a good chance of producing a high-level formula, but it comes at the cost of not having as many formulas, especially at levels 19 and 20, which are largely wasted:

Table: Deep Progression
Level 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20	Formula Level   0th 1st 2nd 3rd 4th 5th 6th 7th 4 4 2 0 0 0 0 0 4 4 2 1 0 0 0 0 4 4 2 2 0 0 0 0 4 4 4 2 0 0 0 0 4 4 4 2 1 0 0 0 4 4 4 2 2 0 0 0 4 4 4 4 2 0 0 0 4 4 4 4 2 1 0 0 4 4 4 4 2 2 0 0 4 4 4 4 4 2 0 0 4 4 4 4 4 2 1 0 4 4 4 4 4 2 2 0 4 4 4 4 4 4 2 0 4 4 4 4 4 4 2 1 4 4 4 4 4 4 2 2 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 5 5 4 4 4 4 4 4 5 5 5 5 4 4 4 4

From my testing, the deep build maximizes the expected level of the highest-level formula you can cast in any given turn.

You can, of course, use an even deeper distribution of formulas, but at this point, the costs outweigh the benefits as high-level formulas become too unlikely to be worth using on average. This leads to gameplay that has high risk, but potentially high reward as well, since you'll occasionally be producing effects far above your normal power level. An ultra-deep distribution might look like this:

Table: Ultra-deep Progression
Level 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20	Formula Level   0th 1st 2nd 3rd 4th 5th 6th 7th 3 3 3 0 0 0 0 0 3 3 3 1 0 0 0 0 3 3 3 2 0 0 0 0 3 3 3 2 1 0 0 0 3 3 3 2 2 0 0 0 3 3 3 3 2 0 0 0 3 3 3 3 2 1 0 0 3 3 3 3 2 2 0 0 3 3 3 3 3 2 0 0 3 3 3 3 3 2 1 0 3 3 3 3 3 2 2 0 3 3 3 3 3 3 2 0 3 3 3 3 3 3 2 1 3 3 3 3 3 3 2 2 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 4 4 3 3 3 3 3 3 4 4 4 4 3 3 3 3 4 4 4 4 4 4 3 3 4 4 4 4 4 4 4 4

The difference between these progressions can be seen most clearly in when they gain their first 4th-level formula. The broad progression gains it at 10th, which is when 4th-level formulas are appropriate (as in, the system is balanced so that producing formulas of that level every turn at 10th wouldn't be too powerful). The broad progression doesn't know enough 4th-level formulas at this level to cast them reliably, but instead has a variety of 3rd-level formulas and is likely to get multiple of them to choose from every turn. The medium progression gets her first 4th-level formula at level 7, 3 levels ahead. Since she only has 6 dice, this formula isn't particularly likely to come up, but it is much more significant when it does.

The deep progression gets her first 4th-level formula at level 5, and the ultra-deep progression gets her first 4th-level formula at level 4. At ECL 4, a mere owlbear is an appropriate encounter. A typical 4th-level formula like Crossfire deals 9d6 damage, which is nearly certain to deal more than half the owlbear's health. This is far stronger than the 4d6 fire damage a Wizard of that level might be able to produce with a Scorching Ray, but it is of course unlikely to occur very often.

A Helpful Python Script

I wrote some simple code that rolls the dice, translates it into elements, and then tells the player which formulas they can cast (in order, from highest-level to lowest), from out of a list that the player provides. This should eliminate a lot of the hassle of figuring out what you can do each turn, especially as your formula list gets really long. Formula formulas can be input with their elements in any order. It might also be a good idea to input formulas affected by meta-element feats you know.

If you don't know Python at all, don't worry! Just copy the code below, go to this site, delete the 3 example lines of code in the box on the left, and paste. Then you can run the code by clicking "Run." You can easily customize your number of dice and your formula list by changing the variable N and swapping out the example formulas for your own list, respectively.

(click to show/hide)

Code: [Select]

"""
Randomly rolls N dice, converts the rolls into AEFW, and then searches a table
of formulas to see which can be cast.
"""

#---------- Player Data ----------#
N = 4  # how many dice you have
Signature_Element = 'X'  # if you have a signature element to add to the roll,
                         # replace 'X' with 'A', 'E', 'F', or 'W'. Otherwise,
                         # leave it unchanged.

# Input your formulas using the example format:
Level_0_Formulas = (
'A (Gust)',
'EW (Muddy Ground)',
'AF (Heat Haze)',
)

Level_1_Formulas = (
'WW (Icicles)',
'EE (Barrier)',
'AA (Supersonic Sprint)',
)

Level_2_Formulas = (
'AEFW (Creation, Lesser)',
'FWW (Dispelling Rain)',
)

Level_3_Formulas = (
'AEEW (Anchor)',
)

Level_4_Formulas = (
)

Level_5_Formulas = (
)

Level_6_Formulas = (
)

Level_7_Formulas = (
)
#---------------------------------#









#--------- Roll Elements ---------#

import random

Str = ''
for n in range(N):
    roll = random.randint(1,4)
    if roll == 1:
        Str = Str + 'A'
    elif roll == 2:
        Str = Str + 'E'
    elif roll == 3:
        Str = Str + 'F'
    else:
        Str = Str + 'W'

if Signature_Element.lower() == 'a':
    Str = Str + 'A'
elif Signature_Element.lower() == 'e':
    Str = Str + 'E'
elif Signature_Element.lower() == 'f':
    Str = Str + 'F'
elif Signature_Element.lower() == 'w':
    Str = Str + 'W'   

Str = "".join(sorted(Str))
print('')
print('Roll: ' + Str)
print('-----------------------')
print('Available Formulas:')
print('')
#---------------------------------#




#-------- Find Formulas ---------#

def find_formulas(Formula_List,level,Str):
    first_formula = 1
    for R in Formula_List:
        found = 1
        F = R.split()[0]
        dummyStr = Str
        for element in F:
            place = dummyStr.find(element)
            if place == -1:
                found = 0
                break
            elif place == len(dummyStr)-1:
                dummyStr = dummyStr[:place]
            else:
                dummyStr = dummyStr[:place] + dummyStr[place+1:]
           
        if found:
            if first_formula == 1:
                print('Level ' + str(level) + ': - ' + R)
                first_formula = 0
            else:
                print('         - ' + R)


if len(Level_7_Formulas):
    find_formulas(Level_7_Formulas,7,Str)
if len(Level_6_Formulas):
    find_formulas(Level_6_Formulas,6,Str)
if len(Level_5_Formulas):
    find_formulas(Level_5_Formulas,5,Str)
if len(Level_4_Formulas):
    find_formulas(Level_4_Formulas,4,Str)
if len(Level_3_Formulas):
    find_formulas(Level_3_Formulas,3,Str)
if len(Level_2_Formulas):
    find_formulas(Level_2_Formulas,2,Str)
if len(Level_1_Formulas):
    find_formulas(Level_1_Formulas,1,Str)
if len(Level_0_Formulas):
    find_formulas(Level_0_Formulas,0,Str)

#---------------------------------#

Here's what the output looks like:

Code: [Select]


Roll: AEFW
-----------------------
Available Formulas:

Level 2: - AEFW (Creation, Lesser)
Level 0: - A (Gust)
         - EW (Muddy Ground)
         - AF (Heat Haze)