PC Wealth Table formula - help me!

[Laman Stahros]

Okay, I've almost got it figured out. The PC wealth table works off of an exponetial (or logrithymic) progression. But, I don't know enough about such things to be able to figure out the formula (and yes, there is an approximate formula). What I have is posted below.

Is there anyone out there who can tell me if I'm on to anything here? Please?

 

[Laman Stahros]

Okay, come on guys, I know that we have some math geniuses on this place. I've got the formula almost figured but I can't quite figure out how to do an exponential progression (or a log progression, for that matter). I know that it's one of the two, but I'm stuck.

 

[gizmo33]

The numbers are similar to the XP table. I don't have high hopes that you can find a formula for this based on the level. If you're trying to program this - you might want to think about an array.

 

[Cheiromancer]

Upper Krust recommends 100*level^3.

This gives you almost double the recommended treasure for 10th level characters, though. What you can do to compensate is multiply by a factor of max((level-1)/20, (20-level)/20) or thereabouts, which cuts the middle values in half, and trims a bit off the top end. Something like this:

        UK's    *factor As per DMG
1	100	95	
2	800	720	900
3	2700	2295	2700
4	6400	5120	5400
5	12500	9375	9000
6	21600	15120	13000
7	34300	22295	19000
8	51200	30720	27000
9	72900	40095	36000
10	100000	50000	49000
11	133100	66550	66000
12	172800	95040	88000
13	219700	131820	110000
14	274400	178360	150000
15	337500	236250	200000
16	409600	307200	260000
17	491300	393040	340000
18	583200	495720	440000
19	685900	617310	580000
20	800000	760000	760000

If folks can get something closer that's still reasonably simple, I'd love to see it.

 

[Thanee]

 2nd     900
 3rd   2,700 +  1,800 +   900
 4th   5,400 +  2,700 +   900
 5th   9,000 +  3,600 +   900
 6th  13,000 +  4,000 +   400
 7th  19,000 +  6,000 + 2,000
 8th  27,000 +  8,000 + 2,000
 9th  36,000 +  9,000 + 1,000
10th  49,000 + 13,000 + 4,000
11th  66,000 + 17,000 + 4,000
12th  88,000 + 22,000 + 5,000
13th 110,000 + 22,000 +     0
14th 150,000 + 40,000 +18,000
15th 200,000 + 50,000 +10,000
16th 260,000 + 60,000 +10,000
17th 340,000 + 80,000 +20,000
18th 440,000 +100,000 +20,000
19th 580,000 +140,000 +40,000
20th 760,000 +180,000 +40,000

This is in no way regular, the 12th and 13th level with the same difference, then the jump to almost twice the increase alone, makes it rather unlikely to find any easy formula for that.

Bye
Thanee

 

[der_kluge]

Given the non-conforming nature of the difference and variance, I don't think a simple formula will solve it. I think at best you'd need some kind of "if level < 3" kind of logic. I imagine it's probably not based on any kind of formula.

 

[orsal]

Okay, I've almost got it figured out. The PC wealth table works off of an exponetial (or logrithymic) progression.

It's clearly not exponential. It increases faster than a linear relationship, but not that much faster. Look at the ratios of consecutive terms. At the beginning of your chart, the ratios are: infinite, 3, 2, 1.67, 1.44, and so on, gradually getting smaller. At the end, the ratios are around 1.05. In an exponential progression, that ratio is constant.

It's even more clearly not logarithmic. It does increase faster than a linear progression (in other words, the difference between successive terms increases), whereas logarithmic growth slows down.

I'd think some sort of power function, y=b*x^n where b and n are constants. (Cheiromancer's suggestion is an example) They all grow. The higher n is, the faster they grow, so n>1 gives you something higher than linear (n=1 is linear). However, they are nowhere near exponential in their rate of growth. The ratio between successive terms in a power sequence eventually gets close to 1, which is what is happening here, although the difference between successive terms gets bigger and bigger.

My first guess would be to try a quadratic expression, because the first few terms exactly fit a quadratic pattern (the differences form a linear relationship). But for level 6 we get 13,000, instead of 13,500, and then start increasing rather faster. Extrapolating the formula 450*x*(x-1), which perfectly fits the first 6 terms, would give us only 171,000 at level 20 (less than 1/4 the official value) and 2,844,000 at level (less than 1/80 the correct value). But with regression on the logarithms (note: any time y is a power function of x, log is a linear function of log(x)), I get that the best fitting power function is y=14.318*x^3.74. (Here y=wealth, x=level.) The correlation is 0.994228, indicating a very good fit (an exact linear relationship would have correlation 1).

Maybe you'd prefer an integer exponent. In that case, the best formula of the form y=b*x^3 is with b=185.6658; the best formula of the form y=b*x^4 is with b=5.8109. Of course, you could round these numbers off as you see fit.

 

[wingsandsword]

Actually, the reason it's not easy to derive the formula for PC wealth by level is that the table itself is derived from other tables.

If you realize that the presumed average game is "X" encounters of "Y" Challenge Rating between levels, and what the average treasure for that Encounter level is multiplied by the number of presumed encounters, you get a number that gets pretty dang close (I might have made some math errors in the actual calculations, I figured this out like 6 months ago or so, or they might deduct a small percent to presume for eventual losses like cost of living, thefts and consumed items to date).

 

[Thanee]

...or just round to a more convenient number.

Bye
Thanee

 

[AntiStateQuixote]

Actually, the reason it's not easy to derive the formula for PC wealth by level is that the table itself is derived from other tables.

If you realize that the presumed average game is "X" encounters of "Y" Challenge Rating between levels, and what the average treasure for that Encounter level is multiplied by the number of presumed encounters, you get a number that gets pretty dang close (I might have made some math errors in the actual calculations, I figured this out like 6 months ago or so, or they might deduct a small percent to presume for eventual losses like cost of living, thefts and consumed items to date).

Furthermore, I believe that the wealth by character level assumes that the PCs spend some of their cash on consumable items that will not be reflected in the character's wealth as the items are now consumed.

The wealth by character level assumes the following:

1) Four PCs dividing treasure equally among each PC
2) Average encounter level equals party level at all times
3) Treasure generated from each encounter equals the the average treasure by encounter level
4) Some of the treasure is used in paying for consumable items and "maintenance" (food, lodging, etc.)
5) 13.33 encounters per level for advancement

So, upon achieving level 2 the party of 4 PCs has faced 13.33 encounters of 1st level generating an average of 300 gp per encounter. The total take from this treasure is 4,000 gp divided equally among the PCs is 1,000 gp each. Assume 100 gp spent on consumable items. Voila! 900 gp per character.

Now, to get to 3rd level the party will face 13.33 EL 2 encounters at a rate of 600 gp per encounter for a total take of 8,000 gp. Again, assuming divide equally and spend 200 gp each on items . . . 1,800 gp gained between level 2 and 3 for a total wealth of 2,700 gp.

Based on this quick look the average treasure by character level appears to be 13.33 (encounter count) multiplied by the average treasure per encounter by level divided by 4 (number of PCs) and then spend 10% on consumables. You must sum this value over each level to get the expected wealth at a given level.

So, if you can figure out a formula to derive the average treasure per encounter by level then you can use that value to figure average character wealth by level.

The wife is calling me to do chores. I'll figure the average treasure by encounter level later and get you your formula.