Stacking Multiplicatively

[Whitey]

Apparently I'm slow and can't figure out the formula of how to stack this way. Can someone show me the formula using the following values.

Nerys is getting a bonus of 2011% from Turiel's Voice of Authority perk.

This perk stacks 50% for every CON stat above 10. Nerys' CON is 15. So the stack should be 5.

Turiel's gear bonus is 154.2%.

And then Turiel's upgrades that give a 100% bonus to Voice of Authority each. (Levels 850 and 1100)

[Mobile Doll Cordelia]

Additive means you add 50% with each stack. Multiplicative means you multiply by 50% (or 1.5) with each stack.

[Whitey]

Originally posted by Milkman Dan:

Additive means you add 50% with each stack. Multiplicative means you multiply by 50% (or 1.5) with each stack.

Yes, I understand the concept, but I don't understand it in practice. That's why I asked for the formula given the above values.

[muljostpho]

Looking at my own numbers here... I've got Turiel's staff boosting Voice of Authority by 277.2%. Nerys (con 15) receives a 5969% boost from it when it's active. Drizzt (con 14) receives a 3678% boost from it. Celeste (con 13) receives a 2150% boost from it. Warden (con 12) receives a 1132% boost from it.
Who can I throw in to check who would only have con 11? That would probably be a good place to start picking apart what it's doing... (checks wiki) None of them. Alright, let's start with Warden's 1132% then...

Oh! I missed something! I have an upgrade on Turiel purchased at level 850 that boosts Voice of Authority by 100%. I think that's the only one boosting it?

Anyway, Warden: con 12, aka 2 stacks, and the gear will boost something by a multiplier of 3.772. Looking to see how it calculated 1132%, aka a 12.32 multiplier, from that...
Well... The square root of 12.32 is about 3.50999, so can the 3.772 multiplier from gear, in combination with the later upgrade's multiplier of 2, turn the two 50%s into two multipliers of 3.50999, aka 250.999%? Maybe if we take the 50% and call it 0.50 that might work out. (With my numbers it seems like it's actually starting from 0.465269, aka 46.5269%, instead of 0.50, aka 50%.) Doesn't seem like they'd do it that way, though.

Looking now at Celeste's 2150%, aka a 22.5 multiplier. The cube root of 22.5 is about 2.82311. I must be missing something because I don't think that we're going to be able to do that same math to turn the 50%s into three multipliers of 2.82311. If this was the right way to look at it I should have gotten 3.50999 again. So there's still something missing.

Hmmm.... 5969, 3678, 2150, and 1132... 5969 is 1.62 times larger than 3678, and 3678 is 1.71 times larger than 2150, and 2150 is 1.9 times larger than 1132. So something in the way it's calculating this is having a "diminishing returns" effect. Each next stack is having a less severe impact on how much the bonus increases than the one before it.

Fiddling around in a spreadsheet to graph those numbers and make a trendline... f(x) = 2.66 x ^ -0.31, where x = the number of stacks. That formula describes how much bigger a bonus is at x stacks than the bonus that was at x-1 stacks. I don't know if that helps find a generic formula for the bonus at any number of stacks, though.

Oh, I can graph the bonus and making a trendline for that, at least... f(x) = 391 x 1.74 ^ x, where x = the number of stacks. That formula gives the bonus at any number of stacks, but I don't know how someone would use their own bonus from gear (and whatever else) to derive the values in that formula for themselves.

The more I look at it, the more that I don't think that they did anything as straightforward with this boost as you'd normally expect to see on other boosts in both this game and in Codename's previous game Crusaders of the Lost Idols. There's more going on here than just adding or multiplying things.

[Whitey]

Originally posted by muljostpho:

Looking at my own numbers here... I've got Turiel's staff boosting Voice of Authority by 277.2%. Nerys (con 15) receives a 5969% boost from it when it's active. Drizzt (con 14) receives a 3678% boost from it. Celeste (con 13) receives a 2150% boost from it. Warden (con 12) receives a 1132% boost from it.
Who can I throw in to check who would only have con 11? That would probably be a good place to start picking apart what it's doing... (checks wiki) None of them. Alright, let's start with Warden's 1132% then...

Oh! I missed something! I have an upgrade on Turiel purchased at level 850 that boosts Voice of Authority by 100%. I think that's the only one boosting it?

Anyway, Warden: con 12, aka 2 stacks, and the gear will boost something by a multiplier of 3.772. Looking to see how it calculated 1132%, aka a 12.32 multiplier, from that...
Well... The square root of 12.32 is about 3.50999, so can the 3.772 multiplier from gear, in combination with the later upgrade's multiplier of 2, turn the two 50%s into two multipliers of 3.50999, aka 250.999%? Maybe if we take the 50% and call it 0.50 that might work out. (With my numbers it seems like it's actually starting from 0.465269, aka 46.5269%, instead of 0.50, aka 50%.) Doesn't seem like they'd do it that way, though.

Looking now at Celeste's 2150%, aka a 22.5 multiplier. The cube root of 22.5 is about 2.82311. I must be missing something because I don't think that we're going to be able to do that same math to turn the 50%s into three multipliers of 2.82311. If this was the right way to look at it I should have gotten 3.50999 again. So there's still something missing.

Hmmm.... 5969, 3678, 2150, and 1132... 5969 is 1.62 times larger than 3678, and 3678 is 1.71 times larger than 2150, and 2150 is 1.9 times larger than 1132. So something in the way it's calculating this is having a "diminishing returns" effect. Each next stack is having a less severe impact on how much the bonus increases than the one before it.

Fiddling around in a spreadsheet to graph those numbers and make a trendline... f(x) = 2.66 x ^ -0.31, where x = the number of stacks. That formula describes how much bigger a bonus is at x stacks than the bonus that was at x-1 stacks. I don't know if that helps find a generic formula for the bonus at any number of stacks, though.

Oh, I can graph the bonus and making a trendline for that, at least... f(x) = 391 x 1.74 ^ x, where x = the number of stacks. That formula gives the bonus at any number of stacks, but I don't know how someone would use their own bonus from gear (and whatever else) to derive the values in that formula for themselves.

The more I look at it, the more that I don't think that they did anything as straightforward with this boost as you'd normally expect to see on other boosts in both this game and in Codename's previous game Crusaders of the Lost Idols. There's more going on here than just adding or multiplying things.

Wow. Thank you. So you're telling me I picked the wrong skill as an example of how to stack multiplicatively.

Okay, let's say it is straightforward. Given all the bonuses and at 5 stacks, what would the formula be? Ignoring the final outcome.

[nemo_annonymous]

Okay i've spent far too long looking into this and re-written my response twice >_<

For Shandie's "Agile Allies" ability is definitely:

Buff = ((2^n)-1) * Buff Base * (1+GearBonus) * Buff1 * Buff2 ... * Buff4

The underlined part is the key because your normal "multiplicative" growth would be 1,2,4,8, 16 and instead your get 0, 1, 3, 7, 15

So, with 1 ally i get:

Buff = (2^1 -1) * 140 * (1+1.518) = 140 * 2.518 = 352.48%

I have no idea why the base buff is 140% - the text suggests it should be 200%, but it's consistent:

for 2 allies:

Buff = (2^2)-1 * 140 * (2.518) = 3 * 140 * 2.518 ~= 1,058%

for 3: (2^3)-1 * 140 * 2.518 = 7 * 140 * 2.518 = 2,468
etc.


The various buff-to-buffs do exactly as expected:
Shandie's Lvl 140 "increase Agile Allies by 200%" add a (1+200%) = x3 multiplier to the end of that equation.

TL;DR: For (most?) multiplicative abilities, only the "stack" portion of the bonus stacks. Each additional stack slightly-better-than doubles the buff.

[Psylisa]

Voice of Authority follows the formula of 1.5^n, where n = CON - 10.

If you take the Hearty Constitution specialization, it becomes 1.75^n rather than 1.5^n (it increases the base value by 50%).

Gear comes after the stack multiplication, not before. Same with the skill increases, but each is a separate multiplier. Remember to convert from a % to a multiplier.

If you're only getting 2000% with gear on Turiel, something is wrong. Even gearless with two upgrades to Voice of Authority, you should be well over 3000%.

So the final formula is 1.5^n * 2 * 2 * [1+Gear]

https://imgur.com/ecvMG7w

In this case, the formula is 1.5^5*2*2*1, or 30.375 multiplier. If you had 154% gear, that would increase it to 77 multiplier (or 7800%).

[nemo_annonymous]

Oh sorry, so at 5 stacks, i predict:

Buff = (2^5)-1 * 140 * 2.518 * 1+200% buff)
= 31 * 1057.56%
= 32,784.36%

and checking the game annnnd 3.28e4, just as planned.


For Turiel...

Embodiment of the Law:
0 slots = no buff
1 slot = 252.4% = (2^1 - 1) * (1+152.4% from gear) = 1 * 252.4%
2 slots = 757.4% = (2^2 -1) * (1+152.4%) = 3 * 252.4%
3 slots = 1767% = (2^3 -1) ... = 7 * ...
4 slots = 3786% = (2^4 -1)... = 15 * ...

Each buff to Embodiment of the law will do as advertised (e..g "+100%" will double the numbers above)



You're right though, Voice of Authority doesn't seem to behave properly. I'll try some more later.

[Whitey]

Originally posted by Psylisa:

Voice of Authority follows the formula of 1.5^n, where n = CON - 10.

If you take the Hearty Constitution specialization, it becomes 1.75^n rather than 1.5^n (it increases the base value by 50%).

Gear comes after the stack multiplication, not before. Same with the skill increases, but each is a separate multiplier. Remember to convert from a % to a multiplier.

If you're only getting 2000% with gear on Turiel, something is wrong. Even gearless with two upgrades to Voice of Authority, you should be well over 3000%.

So the final formula is 1.5^n * 2 * 2 * [1+Gear]

https://imgur.com/ecvMG7w

In this case, the formula is 1.5^5*2*2*1, or 30.375 multiplier. If you had 154% gear, that would increase it to 77 multiplier (or 7800%).

Okay, well I keep missing something. New run. Using Drizzt as DPS. New values.

CON stack is 4
Turiel base buff is 75%
Gear is 154.2%
850 upgrade 100%

So I'm doing 1.75^4 * 2 * 2.542 = 47.68 (4768% or 4868% I don't know how these numbers work)

But Drizzt is getting a bonus of 5112%. What am I doing wrong?

[nemo_annonymous]

Okay:
10 - Jim - No buff
12 - Quillek - 283.35%
13 - Celeste - 538.37% (190% of 12)
14 - Paultin - 920.89% (171 % of 13)
15 - Shandie - 1495% (162% of 14)
16 - Nayeli - 2355% (157% of 15)
17 - Jamilah - 3646% (155% of 16)
18 - Binwin - 5583% (153% of 17)

Pretty sure that doesn't match your 1.5^n * 2 * 2 * [1+Gear]..., that would result in a 150% increase per CON point, no?

For me, currently gear is +88.9% and there is no buff to VoA beyond the initial 50%

[adeyke]

I decided to record some data. My gear was only a green +89.6%, and I was running Vajra Safahr patron, so I could only use CON 14+ champions.

Base:
CON 14: +924.3% = x10.243
CON 15: +1500% = x16
CON 16: +2364% = x24.64
CON 17: +3660% = x37.6
CON 18: +5604% = x57.04

Hearty Constitution:
CON 14: +1906% = x20.06
CON 15: +3507% = x36.07
CON 16: +6308% = x64.08
CON 17: +1.12e4% = x113
CON 18: +1.98e4% = x199

Hearty Constitution + level 850:
CON 14: +3813% = x39.13
CON 15: +7014% = x71.14
CON 16: +1.26e4% = x127
CON 17: +2.24e4% = x225
CON 18: +3.96e4% = x397

I accidentally skipped over merely adding the level 1110 upgrade and went straight for adding the level 1350 upgrade as well.

Hearty Constitution + level 850 + level 1110 + level 1350:
CON 14: +1.53e4% = x154
CON 15: +2.81e4% = x282
CON 16: +5.05e4% = x506
CON 17: +8.97e4% = x898
CON 18: +1.58e5% = x1581

So, the first obvious conclusion is that the level 850, 1110, and 1350 bonuses indeed do just double the whole percentage boost each time. However, I still can't make sense of the other stuff.

[Whitey]

Originally posted by Whitey:

Originally posted by Psylisa:

Voice of Authority follows the formula of 1.5^n, where n = CON - 10.

If you take the Hearty Constitution specialization, it becomes 1.75^n rather than 1.5^n (it increases the base value by 50%).

Gear comes after the stack multiplication, not before. Same with the skill increases, but each is a separate multiplier. Remember to convert from a % to a multiplier.

If you're only getting 2000% with gear on Turiel, something is wrong. Even gearless with two upgrades to Voice of Authority, you should be well over 3000%.

So the final formula is 1.5^n * 2 * 2 * [1+Gear]

https://imgur.com/ecvMG7w

In this case, the formula is 1.5^5*2*2*1, or 30.375 multiplier. If you had 154% gear, that would increase it to 77 multiplier (or 7800%).

Okay, well I keep missing something. New run. Using Drizzt as DPS. New values.

CON stack is 4
Turiel base buff is 75%
Gear is 154.2%
850 upgrade 100%

So I'm doing 1.75^4 * 2 * 2.542 = 47.68 (4768% or 4868% I don't know how these numbers work)

But Drizzt is getting a bonus of 5112%. What am I doing wrong?

I forgot I did ask for the formula if it was straightforward.

So then, Psylisa, were you answering my straightforward situation or my original problem?

[Whitey]

Interesting discussion here anyway. I didn't realize this skill was going to garner so much response.

[Psylisa]

Originally posted by Whitey:

So then, Psylisa, were you answering my straightforward situation or my original problem?

Not sure which. I just know what it's supposed to be.

If you're getting different results, I'd suggest filing a bug report. But it should be as I stated it. CNE has known to make mistakes in math... :eyeroll:

[Whitey]

Originally posted by Psylisa:

Originally posted by Whitey:

So then, Psylisa, were you answering my straightforward situation or my original problem?

Not sure which. I just know what it's supposed to be.

If you're getting different results, I'd suggest filing a bug report. But it should be as I stated it. CNE has known to make mistakes in math... :eyeroll:

Yeah, because apparently everyone else is getting a different value than what it should be.

[SevenBlueSeven]

Props to everyone and their mad math skills!

In short, this discussion has taught me that 'Hearty Constitution' is the best specialization choice for a team of random mixed champions.

Also, I am furthermore grateful t(so far)hat my only epic on Turiel is Staff of the God of Dragons.