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Coverage Vs Integrity Formula

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SimonScience:
Coverage Vs Integrity Formula
Quick notes:
My math may be wrong here, so feel free to correct me if i'm wrong.
I'm not that good at the game yet! I've only gotten to reaserch recently. So if anything I say dosent make sense for the late game, thats why.

In the game, how long an item is going to last (while in use) is decided by two variables:

Integrity is the amount of damage that part can take before it is destroyed
Coverage is the chance that item takes damage

Therefore, there is a tricky question that comes up on how to compare the "durability" of items that have inverse relations of integrity and coverage.

For example, take the Compact Whell vs the Wheel (the comparison that made me do this in the first place)

Wheel:
Integrity:40
Coverage:50

Compact Wheel:
Integrity:30
Coverage:30

The Wheel is more durable, AKA it can tank more damage, but the Compact Wheel has less coverage, AKA it is less likeley to take damage in the first place. So, how do we find out which will last longer?

Awnser: A lot of math.

So first off, because the odds of getting hit is proportional to the rest of the coverage on Cogmind, this measure assumes the rest of the coverage on Cogmind will stay the same. Of course, this isint really true. Other parts can break and lower your coverage drastically. But, if we assume that you are doing this calculation for an item you dont want to break, we can assume you have spares or a good suply of those other items (or at least items of similar coverage). If you attach a new part that has drastically more or less coverage, you can just recalculate.

I'm going to name this new value "Expected Durability", as it represents how much damage you can expect to take before that item breaks:


Where C is combination (nCr, see: https://en.wikipedia.org/wiki/Combination)
And L is a large number. The larger L is, the more accurate your reading of Expected Durability. L should allways be bigger then your durability. You can think of L as up to how much damage you are checking.

Here is an example to help:

Here, i've plotted the Expected Durability of Wheels(Black) and Compact Wheels(Red) for the rest of the coverage on Cogmind(c). Because our L is only 250 (due to the difficulty of plotting reapeated formulas), it is definetly only accurate up to c = 250, less if we want to be rigourous (I would guesstimate c = 200, but if you want to be strict, c = 100 works too) . The lower numbers of the plotting are still accurate enough though. I accidentally closed the window that had those plots, but I can at least say that somewhere between c = 70 and c = 80, compact wheels overtake wheels for expected durability. Therefore, if you have that much coverage, compact wheels are just better then wheels, period (as they have superior stats in mobility and support).

Kyzrati:
Welcome SimonScience! And interesting observations :D

On a related note, not sure if you've seen it yet but in game we have a "relative vulnerability" graph (press 'c' twice--it's paired with the coverage graph in the UI), which compares all of your parts by how likely they are to be destroyed next given their current remaining integrity, and it's generally pretty accurate (of course, there are too many variables to predict it perfectly, not to mention there's the RNG involved, but it's good for quick comparisons!). The formula there is essentially just... (coverage / integrity) :P

SimonScience:

--- Quote from: Kyzrati on December 12, 2018, 05:29:43 PM ---Welcome SimonScience! And interesting observations :D

On a related note, not sure if you've seen it yet but in game we have a "relative vulnerability" graph (press 'c' twice--it's paired with the coverage graph in the UI), which compares all of your parts by how likely they are to be destroyed next given their current remaining integrity, and it's generally pretty accurate (of course, there are too many variables to predict it perfectly, not to mention there's the RNG involved, but it's good for quick comparisons!). The formula there is essentially just... (coverage / integrity) :P

--- End quote ---
Thankyou! I did not notice that relative vulnerability graph (So thats what the red thignys mean! :P ). Though for a more in depth analysis Coverage/Integrity ratio wont give an accurate rating, as seen with Compact wheels vs Wheels. As a quick comparison tool though Coverage/Integrity seems to work fine  ;D .

I would suggest using the formula (if it is correct) when planning out a build or for trying to analyse stuff like Compact Wheel vs Wheel. I often use wheels in my runs and knowing which wheel to keep in a limited inventory helps (yes you can get them from neutral bots but attacking them withought a datajack is risky due to the distress signal. Datajacks are about as sturdy as dry spaghetti so swapping it out is the only realistic way of preserving it, which can be a problem because of matter). I currently have two builds I tinker with in my runs, which I may post to get some help with the game :P.

Valguris:
Hi SimonScience!

If I understand it correctly, you calculate the probability of an item surviving up to D damage by multiplying the probabilities that the item survives exactly d damage (let us call their corresponding events A_d). However, events A_d are dependent on eachother, so simple multiplication won't yield the desired probability. I also do not understand where is the division by (L - Integrity) coming from.

I believe that "which point of damage will destroy the part" follows https://en.wikipedia.org/wiki/Negative_binomial_distribution + the number of "failures". Then the expected value equals pr/(1-p) + r, where p=(1-PartCoverage/TotalCoverage) and r=Integrity, which results in (Integrity/PartCoverage)*TotalCoverage. So it turns out that Integrity to PartCoverage ratio is not the exact expected durability (using your definition), but it's closely related (note that TotalCoverage depends on PartCoverage).

Heh, I was expecting to get exactly Integrity/PartCoverage.

SimonScience:
The division is so that it calculates the Expected Value (https://en.wikipedia.org/wiki/Expected_value), which is given by finding the mean of each result. Tbh it took me awhile to understand it too: I had to do a lot of trial and error before I got an equasion that spat out numbers that made sense. Essentially, it calculates the chance of survival at a damage, then multiplies that by the chance of survival at the next damage and so on, which gives you the chance of survival of those damages combined. So the reapeated multiplication gives the chance that it survives up to that damage, and then it multiplies that value by the turn number and means it to find the expected value, which is therefore the expected damage at wich it survives!

 :P (still not sure though lol)

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