Thanks, Master Jz and asdfgh, for posting values.
I'm missing the values from someone with 0% Draft Sizes. (Master Jz?)
I upgraded draft sizes, and noticed I don't gain from drafting what I expected. There were some changes in the formula through the updates (for example, in the past drafting was affected by all armies gained since the previous draft - army camps, mercenaries, caches/arenas and also drafted armies; now armies drafted during the previous draft don't seem to influence current draft).
However, even once the drafting formula stabilised, the increase is not that high.
In the examples above, Afro Eurasia Gargantuan provided 1168.2B armies from camps/mercenaries/caches. If unupgraded drafting accounts for 17% as Master Jz said, it comes out to 198.6B. Upgraded drafting increased it to 269.3B, or +35.6%, about a quarter (26.2%) of the expected 136%.
In Orbis Veteribus Notus, 17% unupgraded drafting would amount to 19.75B. Upgrading drafting to +166% effectively added 43.8% (26.4%).
This suggests me that the Draft Sizes upgrade affects only the 26%-27% portion of drafting, leaving the remaining 73%-74% unaffected.
So, by raising Draft Sizes from 0 to 100% (upgrades plus artifacts), the drafted armies increase by 26-27%.
By maxing it out (300% upgrades + 64% legendary artifact), the drafts increase by 95-98%, close to doubling in size.
We need more data points in order to pin down these values. Unfortunately "turning off" Draft Sizes is costly, we have to rely on different players with different playing styles, causing uncertainities in our evaluation.

Players who think about investing AP in Draft Sizes should consider that the effective increase is little more than a quarter of the expected increase.

Here's my stats for Europe Huge.

Army counts in Billions.

Army camps 586.4233 7.9%
Mercs 1778.8 23.9%
Drafts 607.8218 8.2%
Caches 927.6203 12.4%
Joint strike 1256.7 16.9%
Hospitals 2299.2 30.8%
TOTAL 7456.5654

Or if you prefer to count w/o JS and hospitals like many seem to:
(but this is weird counting, b/c then your total army counts will be significantly less than the total actually present on the level)

Army camps 586.4233 15.0%
Mercs 1778.8 45.6%
Drafts 607.8218 15.6%
Caches 927.6203 23.8%
TOTAL 3900.6654

And based on the info from Fizzer here, this means that with my +250% draft Adv, these values would be approx 243B down from 607.8218B.

Edited 5/15/2021 20:13:29

sorry, those numbers still don't explain how to calculate the two different components of one draft.
Here are my numbers from Scandinavia:
armies per sec: 172K
armies earned : 72B
armies from camps: 38B
armies drafted so far: 12B
drafting modifier: +186%

armies per one draft: ~45-65M

so what now?

To determine the fixed portion:
For a while, don't buy mercenaries and don't claim Army Caches, and draft as often as possible. At some point the armies per individual draft should converge towards a (relatively low) value, meaning the varying portion should be almost zero. Put this value in contrast to your draft boosts and your army camp production and you have the fixed portion.
To determine the percentage of drafts that the varying portion converges towards:
After you have the formula for the fixed portion (and you have drafted regularly), look at the ratio of "total armies earned" and "drafted armies". This should apparently be around 20%.
To determine the varying portion:
Calculate the expected fixed portion based on army camp production and boosts, check the ratio of total armies and drafted armies, then draft and note down this value together with the observed ratio. Then try to interpolate several data points to find the formula for the varying portion. Ideally without any boosts first because the boost affects the varying portion.

The posted total stats won't help much, I'd assume. I will log out soon, so I can't try this today.

Well presumably the formula is something like:

- d = total draft size
- dm = drafting modifier
- fixed draft portion
- fdp% = fixed portion %
- sliding draft portion
- sdp% = sliding portion %
- aps = armies per second
- tagol = total armies gained on level
- tagol% = ratio, b/c we know "The total armies gained from the sliding portion will approach but not exceed a percentage of the total armies gained in that level."
- z = some alpha factor that we don't know about but is related to "how many armies you've drafted in the past", so this is some part of the formula that increases the sdp when you have acquired lots of armies from caches, drafts, mercs, and decreases as you draw down on them -- essentially this is the "pool" that everyone talks about

d = dm * (fdp + sdp)

fdp = fdp% * aps

sdp = min (sdp% * aps * z, tagol * tagol%)

There are too many variables to deal with, and the sdp is the tricky part. WZI rounding will likely not permit you to get a solid figure, but we could likely reasonably estimate fdp% by continuously drafting until you get to a stable #, which likely (but not definitely) means you've exhausted all or most of the sdp, and the resultant d value is largely coming from the fdp or the tagol*tagol% MIN clause of the sdp formula.

At that point you know d, dm, aps, assume sdp=0, and thus can estimate fdp & thus fdp%.

Maybe someone with better math skills could use the resultant estimated fdp% to figure out sdp%, z, tagol%. tagol is always known if you have the stats, or at least known well enough. I'm not sure armies saved from active artifacts like Field Hospital, Triple Strike, Quad Strike, Damage Territory are accounted for in the stats shown or if they factor into the tagol summation.

I drafted several times in a row on Europe Huge until the draft size stopped going down. I had no draft modifiers. My income was 219,244.30 and my smallest draft size was 38.4 Million. Taking the smallest draft size and dividing by income gives me the number 175. Part of that is the sliding portion. It's possible I could have seen a slightly lower number if I kept drafting. The formula for the maximum you can obtain from fixed drafts (by drafting exactly every 6 minutes) is:

Maximum Fixed Drafts (for constant drafting) = Total Armies From Camps x (Fixed Drafts Multiplier / 360 seconds) x (1 + Modifiers)

The multiplier for fixed drafts is less than 175.

In my previous post, I used the following to calculate my draft percentage:
Total armies drafted / Total armies earned (from army camps, mercenary camps, caches, and drafts).

Edit: Here are some additional numbers for the smallest drafts sizes I've seen (as a multiple of army income).

0% Boost: ~170.1
8% Boost: ~177.4 (Dividing by 1.08 gives 164.3)
40% Boost: ~212.4 (Dividing by 1.4 gives 151.7)
48% Boost: ~220.5 (Dividing by 1.48 gives 149.0)

Linear Regression Equation: y = 106.3462*x + 169.5769
(x is the boost to drafts, and y is the value you multiply your income by to get the draft amount)

If we round and reformat the equation, we get:
y = 106 * (1 + x) + 63

If the numbers are accurate and the relationship is linear, the contribution from the fixed portion is 106 times your income and the contribution from the sliding portion is 63 times your income. If you take 63 / 360 , you get 17.5% for the sliding portion. The sliding portion may be boosted by the sliding portion from the previous draft, and the fixed portion may be boosted due to the sliding portion obtained from the fixed draft. If so, this would put the sliding portion at around 15% of total armies earned and the fixed portion at around 90 times current income.

Edited 5/21/2021 14:33:28

I believe that the fixed portion is 90 * Armies Per Second and the Sliding portion maxes out at 15% of your total armies earned. I also believe that drafting increases the sliding draft pool.

Solving these equations in the opposite direction of my previous post:

Fixed Portion = 90 * (1 + sliding draft portion that results from past fixed drafts) * (1 + Modifiers)
Minimum Sliding Portion = 360 seconds * Armies Per Second * 15% * (1 + sliding draft portion due to past sliding drafts)
Sliding Draft From Past Drafts = 15% + 15%^2 + 15%^3 + ... + 15% ^ n = 15/85 = ~17.65%

Maximum Draft = 3690 * Armies Per Second * (1 + Modifiers)
Minimum Draft = Fixed Portion + Minimum Sliding Portion
Minimum Draft = 105.88 * Armies Per Second * (1 + Modifiers) + 63.53 * Armies Per Second

Moving around terms gives:
Minimum Draft = Armies Per Second * (105.88 * Modifiers + 169.41)
Minimum Draft (as a product of Armies Per Second) = (105.88 * Modifiers + 169.41)

I could probably figure out the draft sizes for non-minimum and non-maximum drafts from this information, but these are the values that really matter.

This is great work. Thank you to everyone contributing.

I also believe that drafting increases the sliding draft pool.

It should. As Fizz stated that the sliding portion is some % of total armies earned, and Drafting grants armies which increase the total # of armies earned, it follows logically that this increases the sliding portion.

That said, where did the 15% value for max sliding draft portion come from? Don't see the value calculation in your post.

15% seems too high. I see regular MAX draft values after rapidly expanding it after claiming caches or lots of mercs, etc, but I've never seen a draft in the range of 15% of the total armies earned.

Some of the final maps have total armies of 4-7T, so 15% of that would be 600-900B, but I've never seen anything that high. After buying even as high as 350B mercs, the drafts following are in the range of (going from memory) 2-6B, which assuming I was 50-75% finished the map (# of armies) at that point would make the value <1%.

15% is the max total of all sliding drafts. For my minimum draft size equation, the 15% are the new armies from the army camps over the 6 minutes between drafts. I'm able to do this because the drafts have leveled out to all be roughly equal.

I don't follow. Maybe an example would help ... what equation would I use to estimate the next few draft sizes under conditions of:

- 2 hrs since last draft
- last draft was at a minimum (pool was depleted)
- earned 4T armies so far on current level
- just bought 350B mercs & received 25B armies from a cache

I drafted over and over every 6 minutes for an hour with different draft modifiers on different levels. I did a linear regression on the minimum draft sizes (as a multiple of army income) to get an equation of the form y = ax + b. Part of b comes from the fixed portion so I removed that piece by subtracting a from b. I divide by 360 because it's the time between drafts.

I then solved the equation:
(b-a) / 360 seconds = sliding / (1 - sliding)

(169.5769 - 106.3462) / (360 Seconds) = sliding / (1- sliding)

It comes out to 14.94%

I only solved the maximum draft equation and minimum draft equation for fixed armies per second.

Edited 5/21/2021 18:06:16

Some of the final maps have total armies of 4-7T, so 15% of that would be 600-900B, but I've never seen anything that high. After buying even as high as 350B mercs, the drafts following are in the range of (going from memory) 2-6B, which assuming I was 50-75% finished the map (# of armies) at that point would make the value <1%.

To clarify, as far as I understand the situation so far, the 15% DOES NOT apply to one individual draft. You will never have a single draft that is the size of 15% (or whatever percentage) of your total armies earned. What the percentage is saying is, that the game tries to get the number of overall drafted armies as close as possible to 15% of the total armies earned. If you have the Level 1 Statistics, look at the numbers when hovering the current army count. The mentioned percentage is the ratio between "Total armies from drafting" and "Total armies earned". Whether 15% is in fact the correct number here, we haven't finally determined.

Regarding the 15%: Wasn't the idea that regardless of the individual draft, the fact that my drafts vary means that I haven't depleted the sliding portion? So, I use an uncommon Draft Boost artifact but I don't draft every whatever minutes they appear. Hence, my drafts are not the same but varying. Therefore, I haven't depleted my sliding portion, hence, I should still be below whatever percentage.

If I haven't made a logical error here, I doubt the 15%.
In my current level I earned 7.0942B armies. 15% of that would be 1.06413B. But I already drafted 1.2836B armies.

As I said, I'm quite sure that I should (always) be below the percentage that the sliding portion targets. But I have drafted more than 15% of the total armies. Hence, the percentage would have to be higher than that, at least 18.09% based on my current numbers.

The Fixed Draft is not affected by the Cap.

So, of that 1.28B total drafts, 1.06B is Sliding and 0.22B is Fixed.

The more often you Draft, the bigger the Fixed amount will be. Increased Draft Size increases the Fixed amount, but the Sliding Cap stays in place.

Then we have a totally different understanding of what the percentage exactly controls. Because, yeah, sure, you could divide the actual draft into two pods. But that doesn't change the fact that as long as the total draft is lower than the total overall, there should be a sliding portion, if you exceed the percentage, the sliding portion should be zero. At least this is how I interpreted Fizzer's statements.

So, of that 1.28B total drafts, 1.06B is Sliding and 0.22B is Fixed.

After all, both numbers combined make my "total drafts". And only my "total drafts" are compared to my "total overall". Not some magical portion of my drafts.

Can someone put this like Im a 2nd grader.

Is draft boost artifact worth it or not? I swap it in everytime i collect but if its not making any significant difference Ill consume it and move on.

tl;dr

If you Draft constantly, YES.

If you Draft infrequently, mildly useful.

If you draft every 6 minutes with no modifiers, your fixed drafts will be roughly equal to 1/4 of the amount gained from army camps. With an 8% Draft Boost, that means your fixed drafts will go from 25% what your camps give to 27% what your camps give.

Divide your draft modifier by 4 to see the percentage effect it will have in comparison to armies from camps, if you draft constantly.

The overall effect will be slightly higher because you'll get a few more armies from the sliding portion, but this is a good rule of thumb.

Edited 5/21/2021 20:06:37

@Mr Perfect
I also give up understand the proposed math ... BUT ... both the artifact & AP Adv are confirmed to increase the resultant amount, so Yes, it's worth it, and imho, it doesn't depend on how much you draft, but is especially beneficial if you buy lots of mercs as this gives a massive boost to it and gives you large drafts for a number of drafts.

Here some folks are saying that Drafts can be as large as 25% of your total armies earned from camps ... don't listen to that. LOL. Seriously, I upgraded AP Adv Increased Draft Size to 250% a long while back, and my draft sizes are ALWAYS more than my armies earned from camps -- and I have a SuperCamp, so that's even compared to an inflated army camp income. I also have +150% Army Camp Production AP Adv as well. I also use the Epic Draft Boots (+32%).

Of all army earning counts displayed at the end of levels, those provided by army camps are always my lowest figure. Order is typically: (lowest to highest) army camps, drafts, caches, hospitals, mercs. Sometimes the order of drafts/caches and hospitals/mercs change orders, but of the order AC, D, C, H, M, the D/C are always in the middle, and the H/M are always on the high end, AC always on the low end. I also have +45% additional mercs AP Adv. Lately I added +50% merc discount so that gives me both cheaper and more plentiful mercs, and this is turn will drive up the Drafts again.