For PC classes, if the highest-level character indicated is 2nd level or higher, assume the community has twice that number of characters of half that level. If those characters are higher than 1st level, assume that for each such character, the community has two of half that level. Continue until the number of 1st-level characters is generated.
no gp limit applies when buying or selling goods in the City of Doors, and NPCs of any class and level combination can be found there.
So, trivially, there are infinite inhabitants of Sigil of each level.
We know that with each level you decrease, you double the number of characters you have. Another way of saying that is that with each level you increase, you halve the total number of characters of a given level. That is to say, you have half as many characters of each level as the previous sentence. If you were to start with, say, half your population, then one quarter of it would be 1 level higher, and one eighth would be 1 level higher than that and... hang on, this looks awfully familiar. This is a well-known series which can be represented by a function:
Proportion of the population by level: 2^(-x), for all x > 1, where x = the level in question.
1/2 of Sigil's infinite population is 1st level by this rule.
Intuitively, I feel like the rapidly increasing variety of possible builds at a given level is going to interact in some way with the decreasing proportion of the population that is allowed to be that level. I
want to say that at some sufficiently high level X, the number of possible builds exceeds the number of possible NPCs, so that you're likelier to encounter higher level NPCs than lower (breaking the previous rule, because Sigil has special rules for NPC generation), but on the other hand I
know that there are infinite NPCs of each level and I'm not sure how to get that infinity to go away (or if it's possible). It might just be that you have mandatory variety at higher levels (an ever smaller, but still infinite, proportion of the population at that level has levels that match those of another character of that level). I was working on something with an assumption of no PrCs, to simplify things since prereqs are all but impossible to deal with en masse, but got nowhere and really need to get back to studying.
EDIT: Oooh, when I get done with this lecture, I'm gonna see if I can work out the distribution of Sigil's infinite wealth.
EDIT: Okay, so, Sigil obviously has infinite wealth. Since we're talking about NPCs, I'm using their wealth table. Spoilered for large amounts of (possibly wrong and definitely badly presented) math.
|
| 1st: | 900 |
| 2nd: | 2000 |
| 3rd: | 2500 |
| 4th: | 3300 |
| 5th: | 4300 |
| 6th: | 5600 |
| 7th: | 7200 |
| 8th: | 9400 |
| 9th: | 12000 |
| 10th: | 16000 |
| 11th: | 21000 |
| 12th: | 27000 |
| 13th: | 35000 |
| 14th: | 45000 |
| 15th: | 59000 |
| 16th: | 77000 |
| 17th: | 100000 |
| 18th: | 130000 |
| 19th: | 170000 |
| 20th: | 220000 |
| 21st: | 240000 |
| 22nd: | 265000 |
| 23rd: | 290000 |
| 24th: | 320000 |
| 25th: | 350000 |
| 26th: | 390000 |
| 27th: | 430000 |
| 28th: | 470000 |
| 29th: | 500000 |
| 30th: | 570000 |
The ELH provides an algorithm for determining the wealth of an NPC of higher than 30th level, which is the earliest time it isn't just a table lookup. You add 10% of the wealth from the previous step.
Converted to a formula, you have 570000*1.1^(x-30), where x is your current level. This is a fairly good fit at level 21 and above, but a poor one below.
Below level 21, we'll be using 140000*1.3^(x-18.25), something I arrived at by trial and error because I suck. It gives good approximations, except at level 1, which is a huge outlier no matter what you do because you gain less than half of the gain from 1 to 2 when you go to 2 to 3.
So, anyway, the city's total wealth is obviously infinite. We can define it as the sum of the wealth contributed by each level of character that exists; each individual character contributes more than a character at the previous level, but there are half as many characters so contributing. In fact, we can already tell that characters at each level control, as a whole, less wealth than characters at each previous level, because no matter what level you're at, you contribute less than 100% more money than a given character of the previous level (fun fact: level 2 characters are the only exception, if you use the real table). But just how much less?
Through the miracle of algebra and things cancelling, all that really matters is the base of your exponent. 1.3 more at or before level 20, 1.1 after. This makes sense - you're contributing 30% more and 10% more, respectively, and everything else is just scaling things to the appropriate point on the graph. Multiplying your individual contribution relative to the previous level by the number of you doing it, you get the total part of the wealth that characters of your level contribute relative to all the characters of the previous level.
(1.3^x)(2^-x) = .65^x
(1.1^x)(2^-x) = .55^x
If you determine the sums of the portion of the wealth controlled by characters at each level, you'll find that while the series do converge, they don't converge at 1. You need to divide by the value they approach in order to determine the percentage of total wealth, I think. Below level 21, it appears this is 13/7, and at 21 or above it is 11/9. I'll be honest, you may need to combine these in order to ensure that your total winds up at 100%, but I can't be sure because, again, I suck.
So, for instance, 21st level characters control 9(.55^21)/11, or about .0003% of Sigil's total wealth, while 1st level characters control about 35% of it. Therefore, we can safely conclude that the Lady of Pain, charismatic leader that she may be, ultimately has no financial control over the city, because no matter how high her CR gets, she'll never have treasure comparable to the combined resources of the city's 1st level characters. Nevertheless, she's still an effective dictator obeyed by everyone in the city (whose wealth is controlled largely by the proletariat), who happens to dress in red and frequently purges the city of "undesirable" elements and wields power beyond opposition.
Sigil is a communist utopia.
EDIT: More on the Lady of Pain. She occupies an interesting position, because she is explicitly the most powerful creature in a city with arbitrarily powerful creatures. She is a mathematical impossibility - the level infinity character, and her statistics are appropriately undefined, though we can safely assume her to have every level (including racial hit dice, and so on, through various shenanigans that I'm sure are possible via creative application of level draining and such). Her effective omnipotence also suggests that she has more at her disposal than simply having infinite levels, that is to say, the ability to define new abilities for herself. Not even epic spellcasting has this degree of versatility. Thus, I propose that we have enough information to deduce the Lady of Pain's true nature, and her origin.
What other creature do we know possesses infinite levels? What other creature can grant itself new abilities, even ones that don't exist in the game? What other creature is, necessarily, the most powerful character?
That's right. The Lady of Pain began her life as a Kobold native to Toril.
EDIT: Now, it's known that the Lady of Pain was always in Sigil, and Sigil was always there. How, then, can she have an origin as a humble kobold, which must necessarily have been after she was known to exist? Teleport through Time. Essentially, the whole Sarrukh thing is only a formality, if one which she undoubtedly ensures through her various agents and superpowers, even if she never personally leaves Sigil. The Lady of Pain was already there*. She would also ensure that no other kobold ever carried out the ascension - one might hypothesize infinitely many Ladies Teleporting through Time to arrive at the beginning of the multiverse from infinite possible futures, of which only the known Lady of Pain survived and ensured that the others would never be. This is baseless speculation, however.
*Corollary: The Lady of Pain is Lord English.
