# Absorbative

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Absorbative is an Invention perk that gives a 20% (22% on a level 20 item) chance of reducing damage by 5% per rank. It can be created in armour gizmos.

Absorbative effectively provides a 1% damage reduction per rank. However, Absorbative does not reduce hard typeless damage (typeless damage that is unaffected by defensive abilities).

Rank Reduction per hit Average damage reduction
1 5% 1% (1.1%)
2 10% 2% (2.2%)
3 15% 3% (3.3%)
• All numbers in parentheses refer to level 20 gear.

## Calculations

• The average ratio of damage taken, ${\displaystyle r_{avg}}$, from Absorbative to that of without Absorbative is
${\displaystyle r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p\right)^{n}p\left[\left(\sum \limits _{i=1}^{n}d_{n,i}^{(0)}\right)+\lfloor {\left(1-.05\times R\right)\times d_{n}^{(1)}}\rfloor \right]}{\sum \limits _{n=0}^{\infty }\left(1-p\right)^{n}p\left[\left(\sum \limits _{i=1}^{n}d_{n,i}^{(0)}\right)+d_{n}^{(1)}\right]}}}$
• The average damage reduction is then, after some rearranging,
${\displaystyle 1-r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p\right)^{n}p\left[d_{n}^{(1)}-\lfloor {\left(1-.05\times R\right)\times d_{n}^{(1)}}\rfloor \right]}{\sum \limits _{n=0}^{\infty }\left(1-p\right)^{n}p\left[\left(\sum \limits _{i=1}^{n}d_{n,i}^{(0)}\right)+d_{n}^{(1)}\right]}}}$
Notes
• A calculator for this is at the top of this page.
• Assumption :
• The enemy is dealing damage with anything that is not hard typeless.
• ${\displaystyle R}$ is the rank of the Absorbative perk.
• ${\displaystyle p}$ is the proc chance of Absorbative to activate (.2 normally, .22 if the Absorbative perk is on level 20 gear).
• ${\displaystyle \textstyle \sum _{n=0}^{\infty }\left(1-p\right)^{n}p=1}$ when ${\displaystyle 0\leq p<1}$ or ${\displaystyle \left|1-p\right|<1}$. This represents summing over all possibilities where integer ${\displaystyle n\in [0,\infty )}$ represents the amount of hits prior to the proc of Absorbative. The probability of Absorbative proccing on hit ${\displaystyle {n+1}}$ is therefore ${\displaystyle \left(1-p\right)^{n}p}$.
• ${\displaystyle d_{n,i}^{(j)}}$ is the ${\displaystyle i^{th}}$ random value uniformly sampled between the enemy's minimum hit and maximum hit. The ${\displaystyle j}$ describes different sets of hits. The sets are regenerated for every new value of ${\displaystyle n}$.
• ${\displaystyle d_{n,i}^{(0)}}$ has ${\displaystyle n}$ elements with integer ${\displaystyle i\in [1,n]}$. This is the damage taken before the perk procs.
• Clarification : This does mean that for ${\displaystyle n=0}$ that there are no elements in this set.
• ${\displaystyle d_{n}^{(1)}}$ is the damage taken from a single hit when Absorbative procs.
• The set of values in ${\displaystyle d_{n}^{(j)}}$ for integer ${\displaystyle j\in [0,1]}$ is the same in both the numerator and denominator of the ratios for any given ${\displaystyle n}$.
Simplifications
• This can be simplified if the assumption is that every value ${\displaystyle d_{n,i}^{(0)}}$ ${\displaystyle \forall }$ integer ${\displaystyle i\in [1,n]}$, ${\displaystyle d_{n}^{(1)}}$ ${\displaystyle \forall }$ integer ${\displaystyle n\in [0,\infty )}$ is taken to be the same ${\displaystyle \left(d_{n,i}^{(0)}=d_{n}^{(1)}\rightarrow d\right)}$. In this scenario, there is no random element and the above ${\displaystyle r_{avg}}$ is then only dependent on the proc chance. This simplification is increasingly accurate in the limit of large ${\displaystyle d}$. A table is provided at the top of the page.

Using these simplifications, the damage reduction reduces to

${\displaystyle 1-r_{avg}={\frac {pR}{20}}}$

## Sources

MaterialRarityPerk ranks with X materials
12345
Fungal componentsRare01222–3
Strong componentsUncommon011–21–21–2
Plated partsCommon00111