Income on a Curve


Rules Discussion


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The discussion on the thread Savings from Crafting reached a point where I wanted to perform a mathematical analysis of Table 4–2, Income Earned, before I commented further. The discussion died down, so I had plenty of time for the analysis. This post is merely to present the results.

I had performed this analysis before (Unskilled labor unable to live at Subsistence Cost of Living, comment #13) on the playtest version of the Income Earned table, but the real key to the table is a description Mark Seifter gave in Trickle Down Alchemy Economics, comment #16:

Mark Seifter wrote:
Mathmuse wrote:
I have been thinking about the crafting rules as written in the Playtest Rulebook and its updates and trying to analyze its patterns.
I don't think it's even mathematically possible to figure out the formula we used for item value / crafting, especially after the rounding (as you posited later), but even without the rounding, it's a piecewise-defined function on a logarithmic y axis, so one function won't cut it.

I used to break secret codes for a living, so that is a cakewalk for me. Explaining the math to other people, that's the challenge.

That "logarithmic y axis" is familiar to players who have played a character over several levels. I use the x axis and call it exponential instead, unless I use the old-fashioned phrase geometric progression. A moderate encounter (80 xp by Table 10-1, Encounter Budget, page 489) against a 1st level party would be 2 orc warriors, creature 1. To get the same threat against a 2nd-level party requires 3 orc warriors. Against a 3rd-level party requires 4 orc warriors, against a 4th-level party requires 6 orc warriors, against a 5th-level party requires 8 orc warriors, against a 6th-level party requires 12 orc warriors, and so on, except that the system breaks down for large differences between levels. That sequence 2, 3, 4, 6, 8, 12, 16, etc. is an exponential sequence with growth rate 1.4142, the square root of 2. The exact sequence is supposed to be 2, 2.83, 4, 5.66, 8, 11.31, 16, but 3 orc warriors are easier to play than 2.83. The error between the rounded values and the exact values is less than 7%.

Character Wealth also follows an exponential curve. Its growth rate is 1.4678, the sixth root of 10. Using a root of 10 means that the numbers will be rounder and 1.4678 is close enough to 1.4142 for practical purposes. Actually, Character Wealth starts that growth rate at 6th level. Before 6th level it grows even faster, pretty much doubling every level. I suspect the higher growth rate at the lower levels is because plenty of low-level foes have mundane gear, so controlling the acquisition of mundane-level wealth by restricting enemy gear is more difficult.

Comparing Character Wealth to 1.4678 growth:
Lump Sum column of Table 10-10, Character Wealth, page 511, compared to formula (100 gp)×(1.4678)^(level-2).
1st level: 15 gp (22% of 68.1 gp)
2nd level: 30 gp (30% of 100 gp)
3rd level: 75 gp (51% of 147 gp)
4th level: 140 gp (65% of 215 gp)
5th level: 270 gp (85% of 316 gp)
6th level: 450 gp (97% of 464 gp)
7th level: 720 gp (106% of 681 gp)
8th level: 1100 gp (110% of 1000 gp)
9th level: 1600 gp (109% of 1470 gp)
10th level: 2300 gp (107% of 2150 gp)
11th level: 3200 gp (101% of 3160 gp)
12th level: 4500 gp (97% of 4640 gp)
13th level: 6400 gp (94% of 6810 gp)
14th level: 9300 gp (93% of 10000 gp)
15th level: 13500 gp (92% of 14700 gp)
16th level: 20000 gp (93% of 21500 gp)
17th level: 30000 gp (95% of 31600 gp)
18th level: 45000 gp (97% of 46400 gp)
19th level: 69000 gp (101% of 68100 gp)
20th level: 112000 gp (112% of 100000 gp)

The wealth varies from the 1.4678 exponential curve by at most 10% between 6th level and 19th level.

Prices are harder to mathematically model, since we have a range of prices for items at a single level. However, we can find a reasonably smooth price curve by taking the maximum price of permanent items at each level.

Comparing Maximum Prices to 1.4678 growth:
Maximum Prices from Table 11-1, Treasure by Level, pages 536-542, compared to formula (100 gp)×(1.4678)^(level-4).
1st level: 18 gp (57% of 31.6 gp)
2nd level: 40 gp (86% of 46.4 gp)
3rd level: 60 gp (88% of 68.1 gp)
4th level: 100 gp (100% of 100 gp)
5th level: 160 gp (109% of 147 gp)
6th level: 250 gp (116% of 215 gp)
7th level: 360 gp (114% of 316 gp)
8th level: 500 gp (108% of 464 gp)
9th level: 700 gp (103% of 681 gp)
10th level: 1000 gp (100% of 1000 gp)
11th level: 1400 gp (95% of 1470 gp)
12th level: 2000 gp (93% of 2150 gp)
13th level: 3000 gp (95% of 3160 gp)
14th level: 4500 gp (97% of 4640 gp)
15th level: 7000 gp (103% of 6810 gp)
16th level: 11200 gp (112% of 10000 gp)
17th level: 15000 gp (102% of 14700 gp)
18th level: 24000 gp (112% of 21500 gp)
19th level: 40000 gp (127% of 31600 gp)
20th level: 90000 gp (194% of 46400 gp)

The maximum price varies from the 1.4678 exponential curve by at most 14% between 2nd level and 18th level.

Table 4-2, Income Earned, on page 236 has five columns of incomes. The four columns that represent successfully earning income are Trained, Expert, Master, and Legendary. For the first 4 rows, from 0th level to 3rd level, the four columns have the same incomes. At 4th level, the Trained column's income begins to fall short of the other three incomes, gradually falling further and further behind until it stabilizes near half the value of the Expert column beginning at 15th level. Similarly, the Expert income is the same as the Master and Legendary incomes until it falls behind at 10th level. And the Master income is the same as the Legendary income until it falls behind at 16th level.

Clearly the four columns are supposed to represent that in the long run, greater proficiency rank gives more income. A Master cook in a restaurant can earn more than a Trained cook. The rows tell that character level also affects income. While level and proficiency bonus are linked, I believe that the main reason to link income with level is to enable crafting. Since crafting involves working until the crafter achieves the full price of the item, income (crafting rate) ought to keep up with the rising prices or crafting time would become impossibly lengthy. As the table below shows, combining the maximum prices with legendary earned income keeps maximum crafting time from growing out of bounds. Er, sort of.

Longest Craft Time:
Craft time for Maximum Prices from Table 11-1, Treasure by Level, pages 536-542, assuming crafter is same level as the item, and ignoring the 4-day preparation time.
1st level: (1/2)(18 gp)/(0.2 gp/day) = 45 days
2nd level: (1/2)(40 gp)/(0.3 gp/day) = 67 days
3rd level: (1/2)(60 gp)/(0.5 gp/day) = 60 days
4th level: (1/2)(100 gp)/(0.8 gp/day) = 63 days
5th level: (1/2)(160 gp)/(1 gp/day) = 80 days
6th level: (1/2)(250 gp)/(2 gp/day) = 63 days
7th level: (1/2)(360 gp)/(2.5 gp/day) = 72 days
8th level: (1/2)(500 gp)/(3 gp/day) = 84 days
9th level: (1/2)(700 gp)/(4 gp/day) = 88 days
10th level: (1/2)(1000 gp)/(6 gp/day) = 84 days
11th level: (1/2)(1400 gp)/(8 gp/day) = 88 days
12th level: (1/2)(2000 gp)/(10 gp/day) = 100 days
13th level: (1/2)(3000 gp)/(15 gp/day) = 100 days
14th level: (1/2)(4500 gp)/(20 gp/day) = 113 days
15th level: (1/2)(7000 gp)/(28 gp/day) = 125 days
16th level: (1/2)(11200 gp)/(40 gp/day) = 140 days
17th level: (1/2)(15000 gp)/(55 gp/day) = 137 days
18th level: (1/2)(24000 gp)/(90 gp/day) = 134 days
19th level: (1/2)(40000 gp)/(130 gp/day) = 154 days
20th level: (1/2)(90000 gp)/(200 gp/day) = 225 days

Ignoring the off-the-curve low prices at 1st, 2nd. and 3rd levels and high prices at 19th and 20th level, the longer crafting time varies from 63 days at low levels to 140 days at high levels. The rising earned income does not keep up with the rising prices.

The growth rate of prices is close to 1.4678 from 4th level to 18th level, but the growth rate of crafting varies so much that I cannot model it as a single growth rate with a few exception at each end. Changing growth rates in the columns is inevitable when two columns that matched each other diverge. Yet I had hoped that the Legendary column would have a fixed growth rate. It doesn't. It appears to have at least three different growth rates.

Comparing Legendary Column of Income Earned Table to a Piecewise Curve:
Legendary column of Table 4-2, Income Earned, page 236 compared to piecemeal combination of three curves.
1st level: 2 sp (100% of transition point 2sp)
2nd level: 3 sp (95% of 1.585 curve 3.17 sp)
3rd level: 5 sp (100% of 1.585 curve 5.02 sp)
4th level: 8 sp (100% of 1.585 curve 7.96 sp)
5th level: 10 sp (79% of 1.585 curve 12.6 sp)
6th level: 20 sp (100% of transition point 20 sp)
7th level: 25 sp (96% of 1.308 curve 26.2 sp)
8th level: 30 sp (88% of 1.308 curve 34.2 sp)
9th level: 40 sp (89% of 1.308 curve 44.8 sp)
10th level: 60 sp (102% of 1.308 curve 58.5 sp)
11th level: 80 sp (105% of 1.308 curve 76.5 sp)
12th level: 100 sp (100% of transition point 100 sp)
13th level: 150 sp (106% of 1.414 curve 141 sp)
14th level: 200 sp (100% of 1.414 curve 200 sp)
15th level: 280 sp (99% of 1.414 curve 283 sp)
16th level: 400 sp (100% of 1.414 curve 400 sp)
17th level: 550 sp (97% of 1.414 curve 566 sp)
18th level: 900 sp (112% of 1.414 curve 800 sp)
19th level: 1300 sp (115% of 1.414 curve 1131 sp)
20th level: 2000 sp (125% of 1.414 curve 1600 sp)
20th crit: 3000 sp (133% of 1.414 curve 2263 sp)

The three curves start with a steady exponential growth of 1.585 from 2sp at 20 sp at 6th level, which means that the curve automatically fits the endpoints perfectly. Likewise, I used a steady exponential growth of 1.308 from 20 sp at 6th level to 100 sp at 12th level. Past that, the curve appeared to be the familar 1.414 growth rate used for experience points per level. That broke down at 18th level, but given the amount of rounding there, I can't determine the intended growth rate there.

The curves also show that earning 10 sp at 5th level is well below the intended curve. It should be 12 or 13 sp. That would not matter if the money were a price, since prices vary, but this 10 sp earning is also the reward for critical success at 4th level. An extra 2 sp is a disappointing reward when 3rd level gets an extra 3 sp.

Another oddity in the Income Earned table that if the columns are intended to pay off for higher proficiency rank, then why do they have a time delay. A character can become an Expert at 3rd level (2nd level for rogues) but the Expert column does not give more earnings until 4th level. A character can become a Master at 7th level, but the Master column does not give more earnings until 10th level. A character can become Legendary at 15th level, but the Legendary column does not give more earnings until 16th level.

I also found that the critical success rules for Crafting have an unintended consequence due to the 1.3 or higher rate of increases in Income Earned by level, but that is a matter for another post: Mathmuse's Houserules.


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I'm sorry I just had to have a big fat laugh at the "impossible to crack" vs "break secret codes" bit :D It's always good to see your work!

Sovereign Court

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I appreciate your skill in breaking down the numbers, but the most interesting thing to me is what meaning hides in those numbers.

Mathmuse wrote:
Another oddity in the Income Earned table that if the columns are intended to pay off for higher proficiency rank, then why do they have a time delay. A character can become an Expert at 3rd level (2nd level for rogues) but the Expert column does not give more earnings until 4th level. A character can become a Master at 7th level, but the Master column does not give more earnings until 10th level. A character can become Legendary at 15th level, but the Legendary column does not give more earnings until 16th level.

I think I can guess this one. You can be an Expert/Master/Legendary at a certain level, but in only one skill. Later, in another and then another. So every character has to make choices about which skills to prioritize. Skilling up enables taking some skill feats and makes some other skill feats do more. Some skills are really useful in other ways (rogue using Stealth for initiative).

By delaying the income bump for higher skill proficiency until the point where you could have gotten your second skill increase to that grade of proficiency, removes an incentive to raise Craft as your first skill for the moneyz.

It also puts more distance in between earning money from adventuring, and money from downtime activities. Which neatly illustrates why people go do something as risky as adventuring: to make a quick buck.

Paizo Employee Design Manager

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Ediwir wrote:
I'm sorry I just had to have a big fat laugh at the "impossible to crack" vs "break secret codes" bit :D It's always good to see your work!

The magic item value formula we used on the exponential y axis is significantly different from the one Mathmuse found, so it's still uncracked from my perspective. Getting a curve that works "close enough" is not hard if you are awesome at math like Mathmuse; with enough math you can always find something that runs through a set of points, especially if you can go piecewise. But I am still confident that figuring out the actual formula we used is virtually impossible (beyond good old-fashioned human spycraft of asking).


Ascalaphus wrote:

I appreciate your skill in breaking down the numbers, but the most interesting thing to me is what meaning hides in those numbers.

Mathmuse wrote:
Another oddity in the Income Earned table that if the columns are intended to pay off for higher proficiency rank, then why do they have a time delay. A character can become an Expert at 3rd level (2nd level for rogues) but the Expert column does not give more earnings until 4th level. A character can become a Master at 7th level, but the Master column does not give more earnings until 10th level. A character can become Legendary at 15th level, but the Legendary column does not give more earnings until 16th level.

I think I can guess this one. You can be an Expert/Master/Legendary at a certain level, but in only one skill. Later, in another and then another. So every character has to make choices about which skills to prioritize. Skilling up enables taking some skill feats and makes some other skill feats do more. Some skills are really useful in other ways (rogue using Stealth for initiative).

By delaying the income bump for higher skill proficiency until the point where you could have gotten your second skill increase to that grade of proficiency, removes an incentive to raise Craft as your first skill for the moneyz.

It also puts more distance in between earning money from adventuring, and money from downtime activities. Which neatly illustrates why people go do something as risky as adventuring: to make a quick buck.

My own opinion is that PF2 has too much emphasis on the proficiency ranks. Thus, I dropped the ranks from my homebrew Table 4-2 in my houserules. I don't mind a reduced emphasis through delay.

But to have the delay in higher income being just the next even level: 4th level for Expert and 16th level for Legendary, when skill increases occur at odd levels for most classes does not fit that idea.

For Master proficiency, the delay is longer, but another rule forces it earlier for Crafting.

PF2 Core Rulebook, Crafting & Treasure chapter, page 535 wrote:
Unless stated otherwise, creating items of 9th level and higher requires you to have the master proficiency rank in Crafting, and items of 16th level and higher require legendary Crafting.

So master proficiency is required for Craft at 9th level, but not rewarded with higher progress until 10th level.

Nevertheless, Ascalaphus's hypothesis is better than my own idea. I guessed that developers had spliced the different curves for different proficiency together at 3rd, 7th, and 15th levels, but since splicing matches numbers, they had no difference at those levels. The mathematical technique of splicing was not the technique they needed. And my guess does not explain why the Expert and Master lines don't separate until 10th level.


Mark Seifter wrote:
Ediwir wrote:
I'm sorry I just had to have a big fat laugh at the "impossible to crack" vs "break secret codes" bit :D It's always good to see your work!
The magic item value formula we used on the exponential y axis is significantly different from the one Mathmuse found, so it's still uncracked from my perspective. Getting a curve that works "close enough" is not hard if you are awesome at math like Mathmuse; with enough math you can always find something that runs through a set of points, especially if you can go piecewise. But I am still confident that figuring out the actual formula we used is virtually impossible (beyond good old-fashioned human spycraft of asking).

In that case, Let me ask.

Would you please explain the curves in the Income Earned table?

I am especially interested in the ratio 1.4678, the 6th root of 10. It is a cleverly chosen number, close enough to 1.4142 but naturally giving round numbers.

EDIT: I realized such an explanation could make a good Paizo Blog preview for whichever book introduces PF2 Downtime business rules to match the ones in PF1's Ultimate Campaign. I am willing to wait a year or two or three until the most marketable time for an overview.

Paizo Employee Design Manager

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Mathmuse wrote:
Mark Seifter wrote:
Ediwir wrote:
I'm sorry I just had to have a big fat laugh at the "impossible to crack" vs "break secret codes" bit :D It's always good to see your work!
The magic item value formula we used on the exponential y axis is significantly different from the one Mathmuse found, so it's still uncracked from my perspective. Getting a curve that works "close enough" is not hard if you are awesome at math like Mathmuse; with enough math you can always find something that runs through a set of points, especially if you can go piecewise. But I am still confident that figuring out the actual formula we used is virtually impossible (beyond good old-fashioned human spycraft of asking).

In that case, Let me ask.

Would you please explain the curves in the Income Earned table?

I am especially interested in the ratio 1.4678, the 6th root of 10. It is a cleverly chosen number, close enough to 1.4142 but naturally giving round numbers.

Sure! Income Earned is a derived table from the magic item price formula table. That table was created by going onto the exponential y-axis, as you said, but only one of the pieces is specifically a "straight line" with a given slope (it was actually a piece of an absolute value function, which since we only see part of it, is why I was confident the original function was more-or-less unguessable as it's almost indistinguishable from a line) whereas the other piece in the function is actually the bottom of an ellipse in the exponential y-axis that meets the absolute value function at a point.


Mark Seifter wrote:
Sure! Income Earned is a derived table from the magic item price formula table. That table was created by going onto the exponential y-axis, as you said, but only one of the pieces is specifically a "straight line" with a given slope (it was actually a piece of an absolute value function, which since we only see part of it, is why I was confident the original function was more-or-less unguessable as it's almost indistinguishable from a line) whereas the other piece in the function is actually the bottom of an ellipse in the exponential y-axis that meets the absolute value function at a point.

I had noticed the long straight line 3 gp, 4 gp, 5 gp, 6 gp, 7 gp, 8 gp in the Trained column from 9th level to 14th level. I had assumed that it was an artifact of trying for minimal growth in that column as it diverged from the Expert column, which was linear itself, 3 gp, 4 gp, 5 gp, 6 gp from 8th level to 11th level. The short linear sequence 3, 4, 5 is hard to avoid at a low growth rate without resorting to non-integers such as 3 gp, 5 sp. I saw that 1 gp, 5 sp and 2 gp, 5 sp showed up a few times, but those are very difficult to avoid due to Benford's Law.

I would have never guessed that a piece of an ellipse was used. An ellipse does not fit the exponential growth paradigm. An ellipse would be good for matching the slope of two curves smoothly in a splice. However, as I discussed with Ascalaphus, a smooth splice would delay the divergence of two columns. Which did happen. Hmm.

Grand Lodge

lol... I like math, but you guys are making my head hurt!

Good on both of you!

NiftyB

Paizo Employee Design Manager

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Mathmuse wrote:
Mark Seifter wrote:
Sure! Income Earned is a derived table from the magic item price formula table. That table was created by going onto the exponential y-axis, as you said, but only one of the pieces is specifically a "straight line" with a given slope (it was actually a piece of an absolute value function, which since we only see part of it, is why I was confident the original function was more-or-less unguessable as it's almost indistinguishable from a line) whereas the other piece in the function is actually the bottom of an ellipse in the exponential y-axis that meets the absolute value function at a point.

I had noticed the long straight line 3 gp, 4 gp, 5 gp, 6 gp, 7 gp, 8 gp in the Trained column from 9th level to 14th level. I had assumed that it was an artifact of trying for minimal growth in that column as it diverged from the Expert column, which was linear itself, 3 gp, 4 gp, 5 gp, 6 gp from 8th level to 11th level. The short linear sequence 3, 4, 5 is hard to avoid at a low growth rate without resorting to non-integers such as 3 gp, 5 sp. I saw that 1 gp, 5 sp and 2 gp, 5 sp showed up a few times, but those are very difficult to avoid due to Benford's Law.

I would have never guessed that a piece of an ellipse was used. An ellipse does not fit the exponential growth paradigm. An ellipse would be good for matching the slope of two curves smoothly in a splice. However, as I discussed with Ascalaphus, a smooth splice would delay the divergence of two columns. Which did happen. Hmm.

There were...a few reasons why I knew the actual two-piece curve was going to be unguessable. The use of an ellipse and absolute value function on the exponential y-axis (as opposed to say a horizontal line where it just always increases by the same multiplier each level) allows trends where moving into magic and high quality items can spike more quickly, then drop off to a slightly slower pace for a while, allowing for more experimentation and customization when selling an old or unwanted item gives more of a percentage of a new item, and then move up again more quickly at the highest levels for the biggest baddest items. Some of the values in the Earn Income Table are also smoothed off the curve due to rounding, which sometimes either obscured the progression or created its own apparent progression, which was another reason I was confident the original function was too hard to find.

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