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  1. #1
    CategoryTheory's Avatar Senior Member
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    Additive vs. Multiplicative Damage (and a formula for that darn Measured talent)

    Months back, Sircowdog1 mentioned that he still didn't "fully understand the difference between multiplicative and additive when calculating out damage" (though he may now). It doesn't help that The Division 2 is not entirely clear about what is additive and what is multiplicative, it often shows you damage numbers only after they've been affected by various weapon and gear buffs, and the talents and the like change from update to update. Since I recently finally figured out what's up with the Measured talent and why it isn't as bad as it looks, I thought I'd post a PSA here explaining what's going on. (Corrections are very welcome; I'm not entirely certain I have this down when it comes to this game.)

    To briefly explain the concept:

    • First, remember that percentages are just another way of expressing a fraction, i.e., 23% is the same as 0.23, and 145% is the same as 1.45. But be careful about how things are stated: a total of +20% damage or 20% additional damage is really saying 120% damage, i.e., 1.2 times damage.
    • Additive boosts take additional damage bonuses, add them together, and then add that total to the base 1.00 damage to give you the final multiplier. So for example, a +30% boost added to a +40% boost gives you 0.30 + 0.40 = 0.70 boost, which is then added to the base 1.00 damage to give you a final multiplier of 1.70 times your base damage.
    • Multiplicative boosts multiply each other after adding in the base 1.00 damage to each one first, so a 30% boost multiplied by a 40% boost gives you (1.00 + 0.30) × (1.00 + 0.40) = 1.30 × 1.40 = 1.82 times your base damage.

    To apply this to the Measured talent, it's important to note that the the damage nerf in the first half of the magazine and the damage buff in the second half of the magazine are different. (The exact figures for the talent have changed over time; these are for today, TU 12.1.)

    • 1st half: +20% rate of fire (ROF) and -30% weapon damage. The latter is additive with all other weapon damage modifiers. So to get our rate of damage (damage over time, or DOT; you can also get damage per second (DPS) if you get all your units correct) we multiply the rate of fire and the weapon damage multiplier.
    • 2nd half is -20% ROF, +30% total weapon damage. The latter is a modifier separate from the plain weapon damage modifier. So here we multiply three things together, the ROF, the weapon damage multiplier, and the total weapon damage multiplier.

    So, let's work an example where we have a 30% weapon damage bonus, i.e., the weapon damage multiplier is 1.3 (times the weapon's base damage):

    • Base without talent: 1.30 DOT
    • 1st half of mag: 1.20 ROF * (1.30 - 0.30) = 1.20 DOT
    • 2nd half of mag: 0.80 ROF * (1.30 * 1.30) = 1.35 DOT
    • Average: (1.20+1.35)/2 = 1.27 DOT (if you fire full mags)

    Now 1.27 DOT is less than 1.30 DOT, so that's not looking too good! Even in the best case where you're never doing a tactical reload (before the mag is completely empty) you're doing less damage over time.

    But wait! You are, however, getting a little bit more damage per mag: ((1.30-0.30) + (1.30*1.30)) / 2 = 1.34. Because the first half subtracts 30% from your additional damage buff[, but the second half multiplies your additional damage buff by 130%, you come out ahead there, by 3%. (And you always will, even if just by a tiny bit, so long as you have any additional damage buff above 0.)

    So what's happening here is that you take a longer to fire a mag (because it does half the bullets at the slow rate, not half the time at the slow rate) but get a bit more damage out of it. This might at least kill someone that wouldn't otherwise be dead if your mag did slightly less damage. Still, it's taking the spot that could be used by some other talent that gives you a lot more than such miniscule improvements as this.

    (As for the exact boost/nerf you get when taking into account the extra damage in a mag, I'm going to leave as an exercise for the reader. As well as being roughly a wash in this example, we will see later that it's perhaps not the biggest factor in the use of this talent anyway.)

    Let's re-run this example with an 80% weapon damage bonus. This is pretty easily achievable in-game with +15% from a specialization, +10% from a piece of gear that gives a specific weapon boost, ~10% on the weapon itself, and the rest from four or five attributes on gear, gear mods, or from skills.

    • Base without talent: 1.80 DOT
    • 1st half of mag: 1.20 ROF * (1.80 - 0.30) = 1.80 DOT
    • 2nd half of mag: 0.80 ROF * (1.80 * 1.30) = 1.87 DOT
    • Average: (1.80+1.87)/2 = 1.835 DOT (if you fire full mags)

    Now things are starting to look a bit better; the first half of the mag is still doing the same amount of damage over time as without the talent, so we've got a bit of a boost overall. And we're also now seeing a bit more increase in the damage per mag: ((1.80-0.30) + (1.80*1.30))/2 = 1.92; dividing 1.92/1.80 shows a 6.6% increase.

    So what's going on here? Because the damage modifier is subtractive for the first half, but multiplicative for the second half:

    • For the first half your weapon damage is changed linearly: it's always 30% of your base damage.
    • For the second half your weapon damage changes on a rising curve that's a function of your starting weapon damage buff: it's always 30% of the sum of your base weapon damage and weapon damage buff.

    To make life easier for people poking at this, I've worked out the equations for the DOT change and increase in damage per mag, and run up a little table of results for some weapon damage boost figures from 0% (i.e., no mods, gear, etc.) to 125% (possibly ridiculous to achieve, but I'm not yet too familar with what goes on past level 30/WT 5). In the equations below, w is your weapon damage boost, expressed as a percentage (rembering that 3% = 0.03, so e.g.15% from a specialization plus 10% from a gear brand gives 25% = 0.25). If someone wants to get fancy and put these in a spreadsheet, be my guest. (I probably won't look at it unless it's in Google Sheets, however; I don't run "productivity" software on my Windows machine.)

    Code:
            Damage per mag:       ( (1+w-0.3) + ((1+w)*1.3) ) / 2
            Damage over time:     ( 1.2*(1+w-0.3) + 0.8*((1+w)*1.3) ) / 2
    
            Weapon Damage   Dmg over Time    Dmg per Mag
            boost    mult    mult  boost     mult  boost
               0%    1.00    0.94  - 6%      1.00    0%
              25%    1.25    1.21  - 4%      1.28    3%
              50%    1.50    1.50  + 0%      1.57    7%
              75%    1.75    1.77  + 2%      1.86   11%
             100%    2.00    2.06  +15%      2.15   15%
             125%    2.25    2.33  +43%      2.43   18%
    So it seems that the designers did not put this talent in as a troll after all (or at least not a complete troll, and I don't recall ever seeing a weapon drop with it in the levels where you've little chance of making it useful, below at least the late 20s), and some of those people who think that Measured is working really well for them may not be mistaken if they've put sufficient resources into weapon damage boosts, especially if they anyway tend to fire full mags rather than doing tactical reloads.
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  2. #2
    Licher.Rus's Avatar Senior Member
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    TLDR: Measured is trash.
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  3. #3
    Strained & Optimist are still better, doubly so by the "fire full mags" metric. I have yet to find any justification for using Measured.

    Also, RPM has no effect on Weapon Damage, just DPS [aka, how fast you deal that damage].
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  4. #4
    CategoryTheory's Avatar Senior Member
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    Originally Posted by LateNiteDelight Go to original post
    RPM has no effect on Weapon Damage, just DPS [aka, how fast you deal that damage].
    Correct, but the Measured talent doesn't change just RPM, it also changes the actual amount of damage done per round. And, vitally, it changes the damge amount differently between the top and bottom halves of the mag, in a way that you can gain more damage per round in the bottom half than you lose in the top half, and even do so without losing DPS in the top half. Understanding how to configure a build to do this is the key to understanding Measured.

    Note that I'm not saying you should change your build to do this; obviously you want to do so only if it's compatible with your playstyle anyway. The ideal situation would be one where you discover that your build is actually already one in which Measured might give you more damage and DPS boost than whatever talent you're currently using.


    Originally Posted by LateNiteDelight Go to original post
    Strained & Optimist are still better, doubly so by the "fire full mags" metric. I have yet to find any justification for using Measured.
    Strained, being about critical hit builds and (perhaps deliberately) taking armor loss, is entirely unrelated (and seems to have nothing to do with whether or not you fire full mags). There are plenty of builds and playstyles where Strained would be "better" than Measured not in doing any good but only in that it never reduces your DPS, and some where Strained actually would be straight-out worse than Measured.

    Optimist is a very good one to bring up, though, becuase it's quite comparable to Measured, also affecting damage based on magazine usage and not being dependent on unrelated parts of the build (such as critical hit chance).

    Optimist gives 3% extra damage for every 10% ammo missing from the magazine. That's a maximum of 27% extra damage per round, but the average over the whole magazine will be 27% / 2 * 10, or 13.5%. But at 90% weapon damage boost, Measured not only gives you 13.5% more damage per magazine, but also 4.8% more DPS. So we can say roughly that once you're at about 90% weapon damage boost, you should be using Measured instead of Optimist. The advantage from Measured over Optimist will only increase as you raise your damage boost beyond that due to the multiplicative nature of the talent's damage output vs. weapon damage boost.

    There are of course more subtlties when you consider at what points during the course of a magazine each reaches certain amounts of damage, what happens when you fire only partial mags, and so on.

    EDIT: The above is not quite correct: I didn't take into account that, as Optimist maintains the fire rate, it will of course also do 13.5% more DPS over the course of the mag (rising from start to end). So Measured at 90% still falls short of Optimist on DPS. As you can see from my tables in the original post, it's around 100% that Measured starts pulling ahead, with a bit more DPS and damage per mag; the gap just grows larger as the weapon damage modifier increases.
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  5. #5
    I think your calculations are wrong - but I'll double check and report back when I get a chance [I could easily be wrong].

    I thought:
    Total Weapon Damage = Additive
    Amplify = Multiplicative

    Additive would simply be WD [Cores/Brand/Spec] + Crit [WD*(CHC*CD] + HS [WD*(HS*HS%)].. then add the Talent [WD*Talent]

    Multiplicative takes that total number and then multiplies it. This is also how Damage To [Armor/Health/OutOfCover] work as well.

    Also, sDPS [the one that actually matters in gameplay], from my understanding, required using Mag Size to calculate Time to Empty. Pairing that with Damage per Mag, then Reload Speed is how you arrive at that figure [again, maybe I'm wrong].

    Lastly, Strained hasn't been predicated on armor missing in over a year [unless they switched it back post Faye? I haven't logged in since I finished that]

    I'll try to get around to looking at this on Monday.
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  6. #6
    Imagine_Brata's Avatar Senior Member
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    I have always liked Measured. The Apartment MPX build with obliterate and perfect clutch rocks
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  7. #7
    echolecter's Avatar Senior Member
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    Originally Posted by Imagine_Brata Go to original post
    I have always liked Measured. The Apartment MPX build with obliterate and perfect clutch rocks
    Same here, pre gear 2.0 I use ACR-E with measured on level 30 and had a blast with it. Armored NPCs where taken down with single magazine. After damage bonus kicks in you did serious damage with single rounds. Have yet to find another decent one, but now use it on FAL and works like a charm.
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  8. #8
    CategoryTheory's Avatar Senior Member
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    Originally Posted by LateNiteDelight Go to original post
    Lastly, Strained hasn't been predicated on armor missing in over a year
    Oops! I was looking at an old description on the web. It's now "+10% criticial hit damage for every 0.5 s you are firing." (I guess it's helpful here to put in the exact descriptions one is using when one mentions it.) At any rate, it's still a vastly different talent from ones like Optimist or Measured, requring a rather different build that brings criticial hit chance up to some reasonable level. I've never used Strained myself because it seems to me harder to keep triggered and used with the rifle builds that I usually use. (Maybe I'm wrong and I should try a critical hit rifle build one day; the game seems to point to that working because all rifles above green have a critical hit damage boost. But that doesn't mean Strained would work with it.)

    I think your calculations are wrong - but I'll double check and report back when I get a chance [I could easily be wrong].
    I welcome corrections!

    Additive would simply be WD [Cores/Brand/Spec] + Crit [WD*(CHC*CD] + HS [WD*(HS*HS%)].. then add the Talent [WD*Talent]
    At the end there where you said "add," did you not also mean to write not "WD * Talent" there but "WD + talent"? Regardless, it's different for the two halves of the magazine; "WD * Talent" is only the second half. (The first half might be written, in your terms, "WD [Cores/Brand/Spec/Talent] + ...."

    At any rate, the key point about Measured is that its damage penalty and the damage boost are calculated differently. The penalty subtracts from your weapon damage modifier, but the boost does not add to it, it multiplies it. E.g., if your weapon damage modifier before the talent was 100% (doubling your damage), it will be 70% in the first half (subtract 30% from the modifier, and multiply just the base and the modifier), but 160% in the second half (multiply each of the base, the modifier, and 1.3—two multiplications instead of just one). I'm not sure I'm explaining this in the clearest possible way (it may just be difficult to explain) so I welcome help clarifying that, too.

    I look forward to your input!
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  9. #9
    Imagine_Brata's Avatar Senior Member
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    Originally Posted by CategoryTheory Go to original post
    Oops! I was looking at an old description on the web. It's now "+10% criticial hit damage for every 0.5 s you are firing." (I guess it's helpful here to put in the exact descriptions one is using when one mentions it.) At any rate, it's still a vastly different talent from ones like Optimist or Measured, requring a rather different build that brings criticial hit chance up to some reasonable level. I've never used Strained myself because it seems to me harder to keep triggered and used with the rifle builds that I usually use. (Maybe I'm wrong and I should try a critical hit rifle build one day; the game seems to point to that working because all rifles above green have a critical hit damage boost. But that doesn't mean Strained would work with it.)
    Strained is not designed for Rifles, Its 1 of the best talent for AR/smg/Lmg up to 750-800 rpm and optimist it like "strained" for AR/smg when rpm start at 850 and up
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  10. #10
    Originally Posted by Imagine_Brata Go to original post
    Strained is not designed for Rifles, Its 1 of the best talent for AR/smg/Lmg up to 750-800 rpm and optimist it like "strained" for AR/smg when rpm start at 850 and up
    Yeah the rate of fire is the real indicator on how much use you will get from Strained besides what your Crit Chance is. On high rate of fire weapons, quite often you will run out of bullets before Strained can reach its maximum level.

    Cold relations, without using something like True Patriot's added ammo buff will often never reach its full 8th level damage buff because it runs out of bullets. The highest I have been able to get that thing to stack up to is 7.
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