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FAQ by C.LE

Version: 1.32 | Updated: 04/13/2021
FAQ of the Month Winner: November 2018 | Highest Rated Guide

Glossary and How Deadfire Does Math

Inversions

(Much of this research credit goes to MaxQuest of the Obsidian forums.)

If I could pinpoint the single-most confusing aspect of Deadfire, it's how negative modifiers are handled. TL;DR: deadfire is an additive system for positive modifiers but gets weird with negative modifiers.

Let's say you have a buff and a debuff. The buff gives you +20% damage. The debuff gives you -20% damage. What's the net effect on your damage?

Most games do one of two things: an additive combination or a multiplicative combination. Under an additive system, we just combine the two modifiers by addition: +20 + -20 = 0, so no net effect on damage. Under a multiplicative system, we convert the modifiers into multipliers, and multiply them together: so +20% becomes 1.2x and -20% becomes .8x, and then we do 1.2 * .8 = .96, so a -4% net effect on damage.

Deadfire does something very different, that we'll call "inversion". It's basically at its heart an additive system, but it treats negative modifiers specially.

Sidebar: Why inversions?

A weakness with additive systems is that negative modifiers are extremely easy to trivialize. This was a problem with PoE1. Grazes in PoE1 were hits made with a -50% modifier to damage. This was appropriately weak early on, but as you got stronger and got better weapons, you would get more and more damage buffs. Eventually, this might altogether erase the impact of a graze, making it relatively less painful than before. Indeed, in PoE1, much of a rogue's power came from their sneak attack essentially making grazes "as good" as a normal hit from anyone else. Another weakness with additive systems is that if you're not careful with designing it, it becomes easy to pile on enough negative modifiers that you end up with 0% net effect on stuff. A plus, however, is that once you know a system is additive it's pretty easy to reason about. You just take every modifier you have and combine them through addition. Really easy for a player to evaluate. Plus, because everything is added together, it's real hard to end up with surprisingly degenerate situations where you can combine buffs and blow out game balance.

By contrast, multiplicative systems mean that negative modifiers are always impactful. If Graze was instead .5x damage, then no matter how much base damage you would do, a Graze would always do half damage. Similarly, unless you have a negative modifier that is explicitly -100% damage, it is impossible to combine too many negative modifiers and end up with 0% net effect. However, multiplicative systems suffer from its own weakness: because all buffs multiply with each other, if you're not extremely careful you can have runaway buff-stacking and end up with huge numbers that blows away game balance. It also is in some cases unintuitive to the player; in our earlier example, a player might intuitively think that a +20% buff and a -20% buff will cancel out, but such is not the case in a multiplicative system.

You can try to design a combined additive/multiplicative system, but you have to be very careful; probably such a system would mostly have additive effects and only a handful of very special multiplicative effects. Diablo 3, for example, uses both additive and multiplicative modifiers but favors multiplicative multiplier; the end result is that people chase multiplicative modifiers really hard since there are so many that they blow out most additive effects you can find. So it's hard to get right.

I believe Deadfire's designers were trying real hard to come up with a system that combined the general intuitive and ease-of-balancing benefits of an additive system but still let negative modifiers be significant without the risk of 0% effects as in a multiplicative system. I think in terms of intuitivity they failed horribly, but if you learn and understand the system, I think there's a lot to be said in favor of it. It's just rather a significant "if."

Basically, in Deadfire's system, when it comes to negative modifiers, you "invert" them by doing: 1-1/(1+x), where x is your negative modifier. You end up with a negative number which you then additively combine with any other modifiers. Importantly, you do not pre-combine negative modifiers; you have to individually invert each of them before combining them.

After you combine all your modifiers, if the number you have remaining is positive, you add 1. If the number you have is negative, you "reinvert" it by doing: 1/(1-x) where x is the negative subtotal. The resulting number is a number less than 1. In both cases, the final number is a multiplier.

Find that confusing? Here are my attempts at simplified takeaways:

  • If all you have are positive modifiers, this system is indistinguishable from an additive system. Just add up all your buffs and you're good to go.
  • Negative modifiers are very close to a multiplicative penalty. A graze, for example, goes from being a -50% penalty to a -1 additive modifier. Which means you need a total of +100% positive buffs to cancel out the effect of a graze. Very similar situation to if graze was a .5x multiplier. But it does mean negative modifiers are much more punishing than they were in PoE1. (see below sidebar)
  • Multiple negative modifiers combine in a way that is similar to, but not as strong as in a multiplicative system. Like in a multiplicative system, the inversions mean that it is impossible to combine too many negative effects to reach 0% net effect. But unlike a multiplicative system e.g. two -50% penalties will combine in the inversion system to a net .33x multiplier, which is not as severe as in a strict multiplicative system (a .25x multiplier).

Note that this inversion happens everywhere negative modifiers are used with respect to damage, duration, area of effect, healing effectiveness, resolve-related duration influence, and even in places I suspect Obsidian's developers did not anticipate (i.e. paladin/priest dispositions and their effects). Note that action speed/recovery time is almost a whole other animal and will be discussed in the next section.

Inversions are so common that from now on, when I talk about some math, I'll use a notation like:

invert(x, y, z, ...)

where it's implied that each x, y, z are inverted if necessary, and then all of the values are combined together and the final result possibly re-inverted.

Sidebar: What does the inversion "mean"?

Mathematically, the inversion means that when you see "-50% damage" on a debuff it is not doing -50% damage per se. Rather it is saying "-50% damage rate" or in other words "Take twice as much time to accomplish the same thing."

This might seem like a semantics quibble but is the way to make sense of the inversion needed to combine multiple negative modifiers, because it is effectively how you would combine different "rates" of doing something: simply multiplying coefficients together fails to account for the fact that differences in magnitudes should be weighted differently because when it comes to "rates" different values mean that the "rate" is active for different periods of time. (Put in other terms, if you were in car traveling over one mile at 60 miles per hour, and then another mile at 30 miles per hour, and you wanted an average speed for the two miles, you can't just average those two numbers naively because it would fail to account for the fact that you spent more time at the slower speed than the faster speed.) This is essentially the difference between a geometric mean and a harmonic mean (a harmonic mean also does all sorts of inversions, and harmonic mean is notably used to calculate fleet-wide Miles Per Gallon for testing whether automakers are meeting efficiency standards).

Anyway, while I said that under the inversion system a negative modifier is "very close" to a multiplicative penalty, it's definitely not the same. Importantly, if e.g. a graze was a purely multiplicative .5x coefficient, positive modifiers would linearly help you climb out of that hole. With inversions, the relationship is not linear; it's polynomial (it curves up). This means the first few buffs do very little to help you climb out of the hole, but the impact of those buffs gradually accelerates until you have +100% and have completely erased the effect of the -50% graze.