Defense demystified: or taming the scary math

jolimor · April 2017

So I have been trying to build a tank using nothing but defense. It can't be done. But proving it caused me to learn how to calculate defensive values for many different situations, and I think I can show you how as well.
You need enough math to know that 1 = 1/1 =100%. Also that if you divide any number by 0 the universe will collapse and other people will laugh at you. Last one: if you divide any real number by infinity the answer will always be 0. let that one sink in for a minute, and I promise you that it will be much easier than I have made it sound

Step 0. make sure you are level 40 before you continue, then open your character sheet. divide your defense number by 4.25
Q: Why 4.25?
A: this is where Cryptic makes adjustments for your level so that lower level players get more defensive resistance per point of defense. The number is 4.25 for level40 and I am too lazy to figure the values for the other levels.
The number you are looking at right now should be the defensive resistance already on your character sheet. Congratulations now we have gained something we already had.
Q: Why is this important?
A: Because that shows that defensive resistance is not a percentage and never was. It is just a number.
Q: why are mathematicians everywhere horrified?
A: Because percentages act freaky weird when they approach 0 or 100, while regular numbers do not.
Q: Why would the Wise and Benevolent Gods of the CO universe (the Cryptic developers) do such a thing?
A: Because most people will say, ".5 increase in damage reduction?? What does that mean??" and also say, "50% damage reduction? I'm awesome!".
So we need to call these numbers something to remind us that, while Cryptic calls it defensive resistance%, it's not. We shall call it the defensive resistance (not really)%
Q: Is that why this is called step 0? Because we can skip it from now and just use the number on the character page?
A: Why yes, how very perceptive of you.

Step 1: ADD the defensive resistance (not really)% from your character sheet to ALL other sources of defensive resistance (not really)%, such as defiance auras, invulnerability, being in tank mode, targeting computer or blocking; even resistances which add to a specific damage type. Whatever adds you want to for the situation you are imagining.
Q: Surely you must mean multiply rather than add since we are dealing with percentages?
A: No. It is set up so that 1%(not really) of damage reduction from any 1 source is equal to 1%(not really) of damage reduction from any other source, but since it is not really a %(read step 0, silly), we can add it all together without worrying about what happens when we approach 100. so if you want to assume a defiance aura of 20(not really)% (which is achievable with 500 con) and 6 stacks, we can go ahead and add 120 to the number on your character sheet and all other sources of defensive resistance (not really)% without a problem.
Q: Surely the 250% from R1 Blocking applies to your defense and should be multiplied by -
A: No. it could have been done that way but it wasn't, if I have read the other forum threads correctly. all sources of damage resistance are added together.

Step 2: divide that number by 100
Q: Why?
A: we are converting it from a nasty pretend percentage into a nice clean decimal
now we have a number we can work with. it should be between 0(no defense at all) and maybe 6(insane tank with defiance aura and R3 blocking)

Step 3. add 1 to that number
Q: Why?
A: Cryptic wanted an equation to produce the right sort of curve. that equation is 1/DD +1, where DD is all those damage resistances added together and converted into a decimal

Step 4. on your calculator type in 1/(that number from step 3). this will give you a long hairy decimal between 0 and 1

Step 5. convert from a decimal into a really real percentage by multiplying by 100. that is the percent of damage that you take, so if you want the mitigation%,

Step 6th subtract that number from 100. that is your mitigation %
Q: could you put all that together without the stupid questions?

Step 1 add together all sources of defensive resistance %
Step2 divide by 100
Step3 add 1
Step 4 do 1/(step 3)
Step 5 multiply by 100
Step5 is the damage you take. for the % damage that you don't take, do 100 -(step5) as a %mitigation

this is the information I wish I'd had a while ago. if I am wrong please please correct me

pantagruel01 · April 2017

This information is slightly wrong. There are actually three types of damage resistance (normal, block, and avoidance), and each type is additive (as you note), but they multiply with one another. To make life more confusion, the avoidance shown on your character sheet is a true percentage (so 50% avoidance means a successful dodge halves damage), but avoidance percentages shown on powers are the normal cryptic math (so if you have 50% avoidance on your character sheet, and gain +50% avoidance, your character sheet will now show 60%); I don't recall the link between avoidance rating points and percentages.

I thought there was a sticky thread that discussed this, but I can't find it.

jolimor · April 2017

I was completely ignoring dodge/avoidance, assuming the damage had already passed through all the layers as described in the sticky above and was actually hitting the defensive resistance layer. as far as blocking goes I conducted an experiment - if blocking modifies the other factors rather than being added to them, then a person with 0 resistance while blocking should not receive any benefit. try it. instead we get a benefit that is proportionally even greater, since defense goes from 0 to 250 with a button push. Also I am pretty sure that it says to add rather than multiply on threads elsewhere in the forum. im sorry that the post was so big, I was gonna explain more about the formula then got too tired, figured most ppl wouldn't care . just got so excited when I realized that defensive resistance% was different from the other % in the game, had to share it with the world immediately. anyway, if the formula adds in dodge/avoidance and perhaps gets adjusted to properly deal with blocking It could be useful for the majority of players who can calculate dodge/avoid mitigation easily but cannot when it comes to the defensive resistance layer. ty for responding, btw, its my first confirmation that the math is at least partially correct. and yes there are a fair number of relevant postings on the subject hidden away. kaiserin confirmed the formula in 1 of them(if he hadn't i'd probably still think that defensive resistance was a %) . i'll get back to you on the blocking issue

flowcyto · April 2017

Welcome to Cryptic Math! Get comfortable with the idea of 'layers' and diminishing returns, and try not to go (too) crazy.

Jim's stickied thread did briefly tread the topic of defense and the diff layers when talking about what layers some of the diff block lingers were on (before I even knew of the sep layers)- maybe that's the one Panta was referring to since he helped clear that up:
https://www.arcgames.com/en/forums/championsonline#/discussion/250970/character-sheet-calculating-in-game-combat-values

The block and defense talk is about 2/3rds down on the 1st page.

Also, ur last reply is a bit confusing. The block layer does appear to multiply on the general layer; the 0% hypothetical you posted is actually not an applicable interpretation of how it works, since the formula is determining what final damage one will take. For example, taking this data from Jim's post:

(1/(1+0.33+0.4)*1500)*(1/(1+2.5))-2.3 = 245 (246)

the "(1/(1+2.5))" part is the block layer being multiplied (in this case using a +250% dmgRes block, or +2.5); the -2.3 at the end is the flat dmg shield from the non-stated Invuln used for the test (and can be ignored for the purpose of this topic), and 1500 is the size of incoming base damage being sampled (the blue laser test in the Powerhouse). Having 0% total dmgRes doesn't mean the formula before it becomes zero (it would actually be undefined as 1/0), since its measuring dmg taken. Rather, 0% dmgRes simply reduces it down to (1/(1 + 0)), or just 1 (ie. 100% dmg taken before blocking). Adding 200% block resistance (the standard block) to 0% base dmgRes would be (1/(1+0))*(1/(1+2.0)), or reduced to just 1/(1+2) = 1/3 = 33.3% dmg taken while blocking (ex. a 1500 base hit becomes ~500 dmg taken when blocking).

Hope that clears it up a bit.

pantagruel01 · April 2017

jolimor said:
I was completely ignoring dodge/avoidance, assuming the damage had already passed through all the layers as described in the sticky above and was actually hitting the defensive resistance layer. as far as blocking goes I conducted an experiment - if blocking modifies the other factors rather than being added to them, then a person with 0 resistance while blocking should not receive any benefit.

Blocking doesn't 'modify other factors'; it's just that you divide by (1 + total defense resistance/100%) * (1 + total block resistance/100%), rather than dividing by (1 + total resistance/100%).

Thus, for example, if you have 100% damage resistance from Defense, 120% from Defiance, rank 3 block with +360% damage resistance, and get hit for 1500 damage, you take 102 damage (1500 / (1 + 1 + 1.2) / (1 + 3.6)), not 220 (1500 / (1 + 1 + 1.2 + 3.6)).

spinnytop · April 2017

Panta knows the things u3u

jolimor · April 2017

my brain hurts, but I'm loving it

I think you(pantagruel) are saying that for the bottom half of the equation all sources of defensive resistance are added together, converted into a decimal then add 1. if there is no blocking, we are done(with that part). if there is, we take the block%, turn it into a decimal, add 1, then multiply the 2 numbers together. which predicts the same high damage reduction that I saw. my experiment was flawed.

in my defense let me point out that blocking generates defensive resistance expressed as a %, exactly the same labeling as the other, actual defensive resistances.

rw intrudes out of time ttfn

jolimor · April 2017

that first line should have read "all sources of defensive resistance except blocking

how's this?

add together all applicable defensive resistance bonuses except blocking, then divide by 100 and add 1
IF you are not blocking, calculate 1 divided by that number. multiply by 100. that is the % damage that you take
IF you are blocking, take the blocking %, divide by 100 then add one. multiply this number by the number from line 1 then calculate 1 divided by that number. multiply by 100. this is the % damage that you take
to get from the damage you take to the damage you don't take, calculate 100 minus the damage you take

now that I have the attention of ppl smarter than myself, my next question is: Is the 10% bonus from tank mode another exception? will rewrite if so

flowcyto · April 2017

Hm, I think in the last line you meant to put "%" in front of each mention of "damage". Also its prob easier to just use the formula and sub in the digits for the example you are using, since ur likely going to be using a calculator anyways. Could even be lazy and cut and paste the edited formula with your sample numbers to a general search engine like Google and get it automatically computed that way.

afaik, the Tank's dmgRes bonus is in the general defense layer, but I'm not sure if anyone has fully tested that yet.

jolimor · April 2017

I spent the largest part of my time with CO with a rather confused idea about how defense worked. the two of you have helped me considerably. I only wish I had asked for help during the early stage of my quest for knowledge. thanks

Defense demystified: or taming the scary math

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