Scale Mechanics

[Yakk]

So I'm a bit of a mechanics junky. Thus I'm fiddling with mechanics in world building.

The hope is that this helps organize thoughts and makes societies make a bit of sense.

Scale is an integer that represents how "powerful" a given faction or society is overall. I'm going to use an exponential scale here; every 3 steps is 2x, and every 10 steps is x10.

So a scale 0 faction is 10x weaker/smaller than a scale 10 faction, and a scale 15 faction is 2x weaker/smaller than a scale 18 faction.

Next we can start with a baseline value. Scale 0 is "10 motivated soldier with a sword, shield and medium armor, enough food to keep them fed, and roads to walk on" to generate a baseline. (This is so scale 1 through 9 is not fractions-of-soldiers).

Scale 10 is then 100, scale 20 is 1000, scale 30 is 10k, scale 40 is 100k, scale 50 is 1 million soldiers, etc.

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D&D 5e specific:

We can map this to CR reasonably well. To go from monsters with CR to a scale, we can add up the CR in question and divide by 3, take the base-10 log and multiply by 10.

To go the other way, divide by 10, raise to the power 10, then multiply by 3.

(For this to work, note that monsters over CR 20 are actually 3 CR per point; so a CR 30 monster is "really" CR 50)

So a CR 30 (really 50) creature is a Scale 12 faction in and of itself from its raw combat power (and more if it isn't an idiot).

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Now, I'm using "able to project power with soldiers", but that is just one simple and easy axis to measure power on. A merchant guild is going to have less soldiers than its scale indicates, and more money and shipping capabilities than its scale indicates. But most power is fungible, in that the merchants could (if need arises) support a martial power (hire mercenaries, petition the emperor who they pay taxes to, or even set up their own puppet force) with their power (of course, doing so would weaken their other abilities, as it isn't cheap).

I hope to use this to set up factions quickly. I can create rivalries and ensure they are at least in the right ballpark, and break factions down into sub-factions in sensible ways.

A faction has a budget of 100 points to buy subfactions:
-1: 80
-2: 64
-3: 50
-4: 40
-5: 32
-6: 25
-7: 20
-8: 16
-9: 12
-10: 10
-11: 8
each of these subfactions can have subfactions; a subfaction is "part" of the larger faction, so its power adds to it.

Lets see if this works.

Albion is a major world power. I'll call that 40, because it is a round number.

Its major subfactions are:
The Crown, The South Seas Trading Company, the Admiralty and the Museum of Antiquities.

I can start off by making these equal, and assign another share to "other factions". Dividing 100 point budget into 5 gives 20 points each.

So each of these are a 33 point faction. And I'll make the "other" bag be "the Senate and People".

Albion: 40
Crown of Albion: 33
South Seas Trading Company: 33
Albion Admiralty: 33
Albion Museum of Antiquities: 33
Senate and People of Albion: 33

We can now drill down in one of these. Say, the Crown.

Under it are the Nobility (I'll treat this as a subfaction a sub faction), the Protectorate (a colony ruled by the Crown), the Crown Knights (order of Paladins), the Civil Service and the Palace (executive branch).

Again we have 5. So evenly divided they are all Power 26.
Albion: 40
*Crown: 33
**Nobility: 26
**Protectorate: 26
**Knights of the Crown: 26
**Administration: 26
**Palace: 26
*South Seas Trading: 33
*Admiralty: 33
*Museum: 33
*Senate and People: 33

We can do a sanity check. At 26 power, the Crown Knights have a power roughly equivalent to 4000 soldiers. That seems plausible as a strong military organization. Or a total CR of 1200 or so; 200 CR 3 "elites" each with a dozen CR 1/4 footsoldiers. Seems not insanely beefy.

The Knights of the Crown I can split into 3 roughly equal bits: the Royal Guard, the Blades and the Rangers, each (100/3 is 33, -5) Scale 21. The Royal Guard is the shiny bits that guard the palace and the capital city. The Rangers are the envoys and messengers of the Crown. The Blades are the secret police. Each have roughly 60 "elites" CR 3 plus support (or some combination like that) in terms of force available.

Next, let's play with the Museum. I want a scrappy smaller rival faction; Spiretop University. So I'll shave a point off the Museum (bring it down to 32) and Senate and People (32) to free up 8, which is enough to add a Spiretop University (29)

Albion: 40
*Crown: 33
**Nobility: 26
**Protectorate: 26
**Knights of the Crown: 26
*** Royal Guard: 21
*** Blades: 21
*** Rangers: 21
**Administration: 26
**Palace: 26
*South Seas Trading: 33
*Admiralty: 33
*Museum: 32
*Senate and People: 32
*Spiretop University: 29

3 points under the Museum, it is half the scale -- a smaller, scrappy rival.

The Museum is a power because in this D&D based world, ancient artifacts are magic, and researching the past is a way to find magic and learn magic. So the archeologists of the museum are now an institution that provides the most powerful spellcasters of Albion.

Spiretop University was a school that has grown to rival it. In the last century it has grown to rival the Museum as a center of magical research. It is less focused on the past, and more focused on applying what we know to the world.

Anyhow, I think this helps me understand the relative power scales of each sub organization, and the effects of what happens if I accidentally boost one or another too far. And as it is factal, I can sketch the main subfactions of a faction and leave the details for when I need it later. Like, at this point, I don't know what the main subfactions of the Museum or Spiretop are, but I can tell you roughly how beefy they have to be (they'll be in the 20s).

I should probably improve the currency system. A problem is "what if a I add another 34 size subfaction to Albion 'out of nowhere'"; doing the math to figure out what happens to Albion's total scale is a bit annoying right now. (The answer, by the way, is it becomes a Scale 41 faction; I can do the math, but the currency system only helps a bit; Albion is 100, and 41 would be 125, 42 would be 160, 43 200. Codifying how that works would feel better than winging it).

Thoughts? Is this just mathematical insanity?

[Yakk]

I have been thinking.

Factions can be more or less than the sum of their subfactions.

I can measure this via Discord and Unity.

Discord is a measure of how much lack of coordination/disagreement exists between subfactions. When a faction has discord, it is weaker than the sum of its parts.

Unity is a measure of how much the subfactions enhance each other. When a faction has unity, it is stronger than the sum of its parts.

Does Unity make sense? Should I just assume a certain level of Discord, then make Unity make up for it? Or should I just measure Discord, and assume it is non-zero?

As an example, I can assume that only 50% of subfaction's scale adds to the faction. Then Unity can be a subfaction up to 3 smaller than the faction, and represent how little of the subfaction's scale is lost.

Hmm.

[Yakk]

So, the next issue is can I use those Scale numbers as part of the game, not as world building?

Suppose you have a conflict between two different factions. Ideally I want a simple way to check who "wins".

I think it should be affine; so the difference is what matters. A difference of 3 means one is "twice as big" regardless of where you rae.

An easy affine check is "roll scale + some die" and see who is bigger.

Suppose they both roll 1d20. Then 1d20-1d20 is > 3 34% of the time, =3 4% of the time, and < 3 62% of the time.

Nice. So if we just take "Scale + 1d20" vs "Scale + 1d20" and whomever is bigger wins, the one 2x bigger wins about 2x as often.

The probability of winning when the other side is +3 is 34% (linear would be 33%), +10 is 11% (linear would be 9%), +5 is 26% (linear would be 25%).

(+3 is 2x bigger, +5 is 3x bigger, +10 is 10x bigger)

That is way better than I could have hoped.

[Yakk]

So the "zero point" of the above was CR 3 or so. I'm thinking mathematically it might be better to make it CR 1.

Ie, make Scale equal to log_10(Total CR)*10, which also matches how the Richter scale works. This means a 1 Scale faction has 5 CR 1/4 soldiers (or equivalent non-combat power).

Using this zero point moves Scale by 4 compared to the one in the OP; so Albion would be Scale 44. A 2.5x factor is just a translation.

An alternative is to give up on the logarithmic scale. Using "total CR it could bring to bear" as a measure of how big a faction is isn't a bad way to measure things.

In this system, Albion would be CR 25,000.

Using the linear-CR scale, we get:

Albion: 25k
*Crown: 5k
**Nobility: 1k
**Protectorate: 1k
**Knights of the Crown: 1k
*** Royal Guard: 0.4k
*** Blades: 0.3k
*** Rangers: 0.3k
**Administration: 1k
**Palace: 1k
*South Seas Trading: 5k
*Admiralty: 5k
*Museum: 4k
*Senate and People: 4k
*Spiretop University: 2k

this might be more intuitive to work with.

It does lose the scale-conflict resolution mechanics. "Take the log and x10" isn't as fun. But maybe it makes it easier to understand how big each faction is.

[Yakk]

Maybe I can efficiently map back to Scale from "total CR", and get the best of both worlds?

Count digits, subtract 1. Those are the 10s.

Then look at first two digits:
>= 10 is +0
>= 12 is +1
>= 16 is +2
>= 20 is +3
>= 25 is +4
>= 32 is +5
>= 40 is +6
>= 50 is +7
>= 64 is +8
>= 80 is +9

Using a random number generator I get a total CR of 3930 is 4 digits (so 30s); 39 >= 32, so Scale=35.

A faction with a total CR of 5 would have a Scale of 7. One with a total CR of 10 would have a Scale of 10.

I could simplify this, and just use digits plus the most significant digit. The inaccuracy might be worth the ease of use.

[Yakk]

An attempt at the "easier logarithm".

1) Every multiple of 10 adds +10 to your scale. So 0-9 is 0, 10-99 is +10, 100-999 is +20, etc.
2) Then take leading digit, and add that. So 67 is +10+6 = 16 scale.

The "proper" value is taking base 10 log and times 10. How far off is this?

    log_10*10   apporx       delta

10    10         11           +1
20    13         12           -1
30    15         13           -2
40    16         14           -2
50    17         15           -2
60    18         16           -2
70    18         17           -1
80    19         18           -1
90    20         19           -1
100   20         21           +1

not great but not horrible. And much faster.

[noob]

cr points above 20 are not truly worth 3 times as more, most cr 30 creatures would die to 2 level 20 characters which are worth one less organisation point.
There is no new different abilities(as in abilities that does stuff different from what was available before) above cr 17 (lower if you start taking account of genies and the like who get wish or other high ranking abilities earlier) and there is no more powerful abilities above cr 17, only more endurance and more damage on attacks (spells do not even scale above 17 and by the time you have cr 17+ creatures you have access to most stuff that mitigates attacks).
So yes: part of what you did is just mathematical insanity that does not fits 5e.
You should rather evaluate the organisation power of a creature not through its cr but on how useful its abilities are.
For example: a doppelganger that can enter people minds to eat their memories from within then replace their victims is going to be way stronger in terms of defeating organisations than a tarrasque: its mere existence will be sufficient to create distrust between everybody and once it gets active, the command structure breaks down. A balor is way better at destroying organisation than a tarrasque because the balor get good killing power and can disengage/engage at will with unlimited teleportation and thus fight only good engagements with low numbers and create constant fear (because anybody except the toughest in that organisation could just be stabbed to death in one hit by a teleporting balor at any moment).
Although a normal balor will quickly get an army because the ability to threaten people with teleporting next to them and killing them allows to recruit a lot so an organisation with a balor will likely not stay at a low organisation number for long.

[Yakk]

Compare a CR 17 to 19, and a CR 19 to a 21.

It is pretty clear that that size of the change is larger after 20. That is all. (both in actual monsters, and in the DMG CR guidelines, and in the XP values of the monsters)

And yes, this the "add up CR" is about "raw military power" scale. A faction with economic instead of military power wouldn't care as much about the total CR of its forces.

On the world building side, the idea is I want to be able to take a nation and generate factions, and have a good handle on how "big" a faction is at both the international and at the player character scale.

The mechanics -- the scale number -- is my attempt at mimicing ORE's Reign.

If you have a contest between two factions, rolling 1d20+scale for each to resolve it is simple, and it is sensitive to extra modifiers; an intervention by PCs could roll d4s or d6s or even more to swing the scale. One side can have advantage (for being better positioned, or specialized; a military faction attacking an economic faction with soldiers, for example). I can even roll the contest to start out, and then PC action can add dice to the side they support (so I can narrate the result moving in sensible ways), without making it a purely fiat thing.

This also means that I get a generator for how a contest goes if the PCs ignore it.