Comparative Analysis of Damage Mitigation...
By
Deacon_NA,
Posted on: 2007-04-13 17:44 in Player Guides
A few things:
First of all I like the guide, and it clearly demonstrates how defense and resistance work together to mitigate damage in idealized circumstances. I think the conclusions may not give the whole story, though. In most cases I find that any tank, if built properly, can handle sustained steady damage in most solo and teamed situations. More specifically, any tank can survive the sustained DPS that they will face solo, or teamed with support.
Problems arise when faced with debuffs, spike damage, and enemy buffs (such as Nems and their Vengeance). Certain mob types have the ability to almost completely negate one aspect of damage mitigation. For example, sappers make the resistance aspect of damage mitigation for invul (sans unstoppable) less useful than the defense aspecs. Similarly, DE quartz eminators almost completely negate the defense aspect.
Spike damage is also a problem for defense based damage mitigation. Imagine a tank with 1800 hp and 3 attacks of 1000 damage each incoming. The tank with 50% resistance will survive every time, while a tank with 25% defense will survive approximately 84% of the time (but will be unscathed 42% of the time). If this happens every fight the defense based tank is gonna debt cap pretty fast. Defense does more than mitigate damage., though so it balances out.
<QR>
The following spreadsheet takes account for level and mob type modifiers. What you've stated is basically only true for +0 Minions. Unfortunately it's more complicated than that.
Spreadsheet by MrQuizzles
You may also want to reference: Arcana's Guide to Defense v1.4 - Updated for I7
Interesting. Lots of numbers. Happily, your conclusions agree with my previous understanding and I found your examples and explaination easy enough that I could follow it.
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<QR>
The following spreadsheet takes account for level and mob type modifiers. What you've stated is basically only true for +0 Minions. Unfortunately it's more complicated than that.
Spreadsheet by MrQuizzles
You may also want to reference: Arcana's Guide to Defense v1.4 - Updated for I7
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Level modifiers definitely contribute to your chance of being hit, but I don't believe that invalidates anything I've concluded.
The formula for determining the effectiveness of defense or resistance against every possible permetation of level modifiers, damage type, accuracy bonuses, regeneration rates, recharge rates, status effects, etc would be staggering. Possibly too cumbersome to actually be useful (thank god the game handles all of that on it's own). However, I believe the end result would be similiar.
Defense is only effective damage mitigation if it can keep the attack from hitting. It counts toward every potential attack, but it's a binary result.
Resistance only works as damage mitigation on attacks that actually hit you. Therefore resistance is a more useful form of damage mitigation when facing an enemy that has a higher chance of hitting you, either through a larger to-hit chance or accuracy modifiers. The ratio supports that.
Most of the comparisons I see about defense or resistance are made against an even level minion with a 50% chance to hit. I'm merely pointing out that the ratio of defense to resistance being 2:1 that's held as a universal truth actually isn't a constant. Even before considering level mods or accuracy bonuses it fluctuates.
I'm confident that the conclussions I've come to would still hold up when compared using all of the accuracy and level modifiers. Still, the idea of integrating MrQuizzle's formulas into my compartive tables is very interesting. That would give a range of comparisons at the same time rather than only one at a time.
Formerly "Back Alley Brawler"
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Spike damage is also a problem for defense based damage mitigation. Imagine a tank with 1800 hp and 3 attacks of 1000 damage each incoming. The tank with 50% resistance will survive every time, while a tank with 25% defense will survive approximately 84% of the time (but will be unscathed 42% of the time). If this happens every fight the defense based tank is gonna debt cap pretty fast. Defense does more than mitigate damage., though so it balances out.
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True, but throw 4 of the same attacks at you and resistance's chance to survive is no longer guaranteed. Try 10 or 50. At certain a point, resistance is no longer relying purely on it's own damage mitigation, it's also relying on the base to-hit chance of the incoming attacks (ie, some are going to miss whether you have any defense or not)
In the other extreme case, say it's a two attacks that do 2000 points of damage each. The resistance tank is going to survive ~50% of the time (as long as they get lucky and at least one of the attacks misses). Likewise for Defense. It's similiar to how saying a coin fipped 100 times will come up heads 50% of the time. That has very little to do with concluding that if I flip a coin 3 times and it comes up heads the first two times, then it's more likely to be tails the 3rd. Or making the assumption that if it comes up heads twice and tails once, that means a 66% chance for heads and 33% chance for tails. Doesn't work that way.
I'm also not meaning to present this as the end all be all of comparisons, just a more accurate representation of the most common baseline comparison (versus an even con minion with a flat 50% chance to hit) in a much simpler formulaic way.
Formerly "Back Alley Brawler"
I think what you said is right. But on the other hand, what I really care is whether I live or not. While the damage that I avoided, which you called damage mitigation, is nice to know, what really matters is "survivability".
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I think what you said is right. But on the other hand, what I really care is whether I live or not. While the damage that I avoided, which you called damage mitigation, is nice to know, what really matters is "survivability".
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Of course, but the question is will adding Tough increase your "survivability" more than adding Weave? Which power is better to add slots to?
Survivability starts with damage mitigation. There's a ton of factors to consider in what makes you survive longer, but that's not the point of this. I'm simply trying to explain one way of comparing the relative merits of adding more defense versus more resistance to your particular build.
Formerly "Back Alley Brawler"
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I think what you said is right. But on the other hand, what I really care is whether I live or not. While the damage that I avoided, which you called damage mitigation, is nice to know, what really matters is "survivability".
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Actually, theres a really easy way to calculate the increase in suvivability for straight resistance/defense calculations:
HP * (100% + Total Dmg Mit%)
Or, a tank with 2000 HP and 90% dmg mitigation can actually absorb about 3800 dmg.
In regards to the OP, You might like this formula a little better T.T.:
(50acc%+Tacc%)*(100% - Tres%)=Total % dmg taken
The reason I say this formula specifically is it shows why the Defense and Resist values aren't just added together, they're really multiplied.
25% def is actually -25% Tacc, so with 50% defense we have (50% - 25%)*50% = 12.5% damage taken, or 87.5% dmg mitigation.
20% def and 20% resist = (50%-20%)*(100%-20%) = 24% dmg taken, or 76% dmg mitigation.
Tacc% is the full accuracy/defense formula, and Tres% is the total resistance, so the complete formula actually is:
((50% + ((def debuff - def) + (acc - acc debuff)))*acc bonus)*(dmg * (100% - Tres%)) = Total dmg
The other reason this is important is powers like damage debuffs: 25% dmg reduction from EF + 20% resist doesn't actually = 45% dmg mitigation.
Its actually (100%-25%)*20% = 40% mitigation.
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In regards to the OP, You might like this formula a little better T.T.:
(50acc%+Tacc%)*(100% - Tres%)=Total % dmg taken
The reason I say this formula specifically is it shows why the Defense and Resist values aren't just added together, they're really multiplied.
25% def is actually -25% Tacc, so with 50% defense we have (50% - 25%)*50% = 12.5% damage taken, or 87.5% dmg mitigation.
20% def and 20% resist = (50%-20%)*(100%-20%) = 24% dmg taken, or 76% dmg mitigation.
Tacc% is the full accuracy/defense formula, and Tres% is the total resistance, so the complete formula actually is:
((50% + ((def debuff - def) + (acc - acc debuff)))*acc bonus)*(dmg * (100% - Tres%)) = Total dmg
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Interesting, but shouldn't that be "((50% + ((def - def debuff) + (acc - acc debuff)))*acc bonus)*(dmg * (100% - Tres%)) = Total dmg"?
Other than that, I think we're arriving at the same point, just from opposite directions.
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The other reason this is important is powers like damage debuffs: 25% dmg reduction from EF + 20% resist doesn't actually = 45% dmg mitigation.
Its actually (100%-25%)*20% = 40% mitigation.
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Yes, but I haven't said anything about damage debuffs. It does parrellel my conclusion that defense reduces the total amount that resistance adds to damage mitigation though. Damage debuffs would do the same. Anything that reduces the amount of damage you take cuts into resistance doing it's job.
Formerly "Back Alley Brawler"
I think you are missing the logic as to why mitigation is better when added to higher base amounts. The amount of added mitigation is the same on a low base or a high base only if you are comparing it against a baseline of 0% mitigation. Using your example
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30% Defense = 60% Damage Mitigation
35% Defense = 70% Damage Mitigation(+10% Damage Mitigation)
and
05% Defense = 10% Damage Mitigation
10% Defense = 20% Damage Mitigation (+10% Damage Mitigation)
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You are correct that adding 10% damage mitigation to 60% or 10% results in the same amount of additional damage being avoided. However, the incremental value of that mitigation is much greater when applied to the higher base. Going from 10% damage mitigation to 20% means you are going from taking 90% of damage to 80%. While the absolute change is -10 percentage POINTS, those 10 points are an 11.1% reduction from the 90% you started at. When you go from 60% damage mitigation to 70%, those 10 points are a 25% reduction from the 60% you started at. This is why people feel small amounts of mitigation have greater value when added to a higher base.
The OP is technically correct, because damage mitigation should be the amount of damage being avoided, either resisted or dodged by defense. So, adding 3% defense has more bang for the buck for the 35%-defense SR scrapper than a 0%-defense blaster should be rephrased using a different language, rather than explaining in terms of damage mitigation.
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Actually, theres a really easy way to calculate the increase in suvivability for straight resistance/defense calculations:
HP * (100% + Total Dmg Mit%)
Or, a tank with 2000 HP and 90% dmg mitigation can actually absorb about 3800 dmg.
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Damage mitigation is the percentage of damage that you resisted or dodged, so if you want to find the amount of damage that the tank can absorb, you should multiply the damage mitigation percentage to the amount of incoming damage, not multiplying to his hit point.
I appreciate the efforts of the OP, this is a question that's been vexing me for a while. I'm very interested in these questions, not only as a math geek, but also because I'd like to see what "the perfect duo" is.
This is a nit-pick, but I suggest the OP balance the parentheses in the damage mitigation formula. There is currently an imbalance, as I found out by typing the formula into Excel.
For example, is it better to pair a /SR scrapper (DEF based) with a FF (DEF) or a Sonic (RES) defender? My instinct tells me that RES + DEF makes for better survivability than RES + RES or DEF + DEF.
When I get motivated, I'll dig up some old textbooks on calculating the probability of ruin. (See http://en.wikipedia.org/wiki/Gambler's_ruin )
Ultimately, that's the question we're trying to solve, what makes for higher survivability. Because that's the real question, I'm in the camp that finds it a flawed conclusion to say that small increments added to a high base is better than small increments added to a low base. Ok, someone's made up a stat that's effectively "rate of increase in damage reduction." Using the above example, going from 90% to 80% is a (10/90) 11.11% reduction and 40% to 30% reductions is (10/40) 25% reduction.
I don't see how that's a more valid stat than "rate of decrease in damage taken". To go from 10% to 20% reduction is a 100% increase in reduction ((20-10)/10). Going from 60% to 70% is just a 16.67% increase in reduction ((70-60)/60). In this case, it looks like adding a little to a little is better than adding a little to a lot.
Please note I'm not espousing the latter technique as more valid than the former. I'm just saying both seem pretty irrelevant to me, except as trivia, when you're trying to figure out how to increase survivability.
Global = Hedgefund (or some derivation thereof)
You probably have to calculate survivability to understand that. Since you're a math geek, you can do the math.
Suppose a toon has 2000 hp, each hit the mob makes is 1000, so 2 successful hits the toon will die. If you calculate the probability of the toon still surviving after ten attempts, you'll find that adding 5% defense to a 35% defense toon helps a lot more than adding 5% to a 0% defense toon.
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For example, is it better to pair a /SR scrapper (DEF based) with a FF (DEF) or a Sonic (RES) defender? My instinct tells me that RES + DEF makes for better survivability than RES + RES or DEF + DEF.
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You're mostly wrong based on Tic Toc's post. It really depends on the Scrapper's build. An SR Scrapper can hit the effective Defense cap if built for it, at least in I9.
(Toggle (13.785%) + Passive (5.625%) + Weave (3.75%) + Maneuvers (2.275%) + Combat Jumping (1.875%)) * Enhancements (1.57) + Steadfast Protection IO in Tough (3%) = 45.9% Defense
At that point, mobs have a 5% chance ToHit. More Defense is only useful to counter ToHit buffs or Defense debuffs. So in this case, the Sonic Defender would add to the Scrapper's survivability by reducing the little damage that did get through.
That said, your average SR scrapper without Weave, Maneuvers, and the IO, and with Combat Jumping one slotted, is at 32.7% Defense. An FF Defender will give them effectively capped Defense, as the mob ToHit drops from 17.3% ToHit to 5% ToHit, so they'll take only 29% of the damage that they used to at 32.7% Defense. A Sonic could provide 55% Damage Resistance, which means they'd be taking 45% of the damage. In this case the FF is significantly better.
In general, you're better off building on the protection you already have. But if you're lucky enough to hit a cap, it's time to start working on the other sort of protection.
Goodbye and thanks for all the fish.
I've moved on to Diablo 3, TopDoc-1304
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Damage mitigation is the percentage of damage that you resisted or dodged, so if you want to find the amount of damage that the tank can absorb, you should multiply the damage mitigation percentage to the amount of incoming damage, not multiplying to his hit point.
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Yep, almost exactly. Measuring survivability means we want to know the exact amount of damage it takes to deplete all the HP. Unfortunatly, the formula I gave was actually completely incorrect, sorry about that.
The formula is actually:
HP/(100%-mit%) = total dmg
So 1000 hp with 90% mitigation would be the equivalent of: 1000/(1 - 0.9) = 10,000 dmg
Heres how it works:
Basic resist formula: tdmg = dmg *(100%-mit%)
Basic kill formula: HP - tdmg = 0 hp
Combining the two: HP - dmg * (100%-mit%) = 0 hp
Thus we get:
HP = dmg * (100%-mit%)
HP/(100%-mit%) = dmg
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Interesting, but shouldn't that be "((50% + ((def - def debuff) + (acc - acc debuff)))*acc bonus)*(dmg * (100% - Tres%)) = Total dmg"?
Other than that, I think we're arriving at the same point, just from opposite directions.
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Actually there was a goofy reason why I wrote the original dmg mitigation formula as: 50% + (defdebuff - Def).
The formula is an accuracy formula and not a defense formula. Reason being, that particular notation accomodates for player and AV accuracy being larger then 50%. For example, the formula for a players atk power would look something like this:
((75% + (Defdebuff - def) + (tohit - tohitdebuff))*acc so's) * ((dmg * (100%-dmgdebuff%))*(100%-tres%)) = total damage %
So, with a mob that hits 50% of time as a base, the player automatically has a 50% damage mitigation. However, a player that hits 75% of the time, that only means 25% damage mitigation.
This is also why 1 Def = 2 res, the accuracy is reduced from 50% to 49% hits landing. That formulates to 49/50 = 98% less damage, as opposed to Res which reduces from 100% to 99% dmg. If base accuracy was 100%, 1 point of defense would = 1% dmg mitigation. For players its 1 def to approx 1.5 res, since the player hits 74/75 = 98.666%
Realistically the formula could just as well be written 75% - (def - defdebuff), except in terms of accuracy, Def functions exactly the same as Tohit Debuff. In any case, it's most likely the exact formula your using except retermed to show the accuracy and resistance differences.
As for the dmg debuff, I just figured you might want to look at that eventually too. The reason dmg resist is less effective for every point of dmg debuff and defense is actually due to the fact that dmg resist only works on damage taken, while debuff and defense prevent the damage completely. On the other hand, the advantage of resist is that it automatically reduces damage, whereas defense does not reduce damage and debuffs must be applied to be effective.
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I think you are missing the logic as to why mitigation is better when added to higher base amounts. The amount of added mitigation is the same on a low base or a high base only if you are comparing it against a baseline of 0% mitigation. Using your example
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30% Defense = 60% Damage Mitigation
35% Defense = 70% Damage Mitigation(+10% Damage Mitigation)
and
05% Defense = 10% Damage Mitigation
10% Defense = 20% Damage Mitigation (+10% Damage Mitigation)
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You are correct that adding 10% damage mitigation to 60% or 10% results in the same amount of additional damage being avoided. However, the incremental value of that mitigation is much greater when applied to the higher base. Going from 10% damage mitigation to 20% means you are going from taking 90% of damage to 80%. While the absolute change is -10 percentage POINTS, those 10 points are an 11.1% reduction from the 90% you started at. When you go from 60% damage mitigation to 70%, those 10 points are a 25% reduction from the 60% you started at. This is why people feel small amounts of mitigation have greater value when added to a higher base.
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No actually that's exactly my point. The benefits of mitigation are not linear. However, how stacking defense or resistance translates into migitaion is very linear. That make sense?
Other than that, the rest of your comparisons are misleading. 60% to 70% is a 10% increase. It's really that simple, there's no reason to render that down into another percentage of a percentage. That's just an attempt to manipulate numbers and make them seem more significant.
Formerly "Back Alley Brawler"
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Actually there was a goofy reason why I wrote the original dmg mitigation formula as: 50% + (defdebuff - Def).
The formula is an accuracy formula and not a defense formula. Reason being, that particular notation accomodates for player and AV accuracy being larger then 50%. For example, the formula for a players atk power would look something like this:
((75% + (Defdebuff - def) + (tohit - tohitdebuff))*acc so's) * ((dmg * (100%-dmgdebuff%))*(100%-tres%)) = total damage %
So, with a mob that hits 50% of time as a base, the player automatically has a 50% damage mitigation. However, a player that hits 75% of the time, that only means 25% damage mitigation.
This is also why 1 Def = 2 res, the accuracy is reduced from 50% to 49% hits landing. That formulates to 49/50 = 98% less damage, as opposed to Res which reduces from 100% to 99% dmg. If base accuracy was 100%, 1 point of defense would = 1% dmg mitigation. For players its 1 def to approx 1.5 res, since the player hits 74/75 = 98.666%
Realistically the formula could just as well be written 75% - (def - defdebuff), except in terms of accuracy, Def functions exactly the same as Tohit Debuff. In any case, it's most likely the exact formula your using except retermed to show the accuracy and resistance differences.
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I think incorporating accuracy, level mods, and classes into my spreed sheets is a great idea. When I get the chance to I'll do that. It'll give you an indication at a glance as to how effective the defense/resistance you're considering adding is against a variety of situations.
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As for the dmg debuff, I just figured you might want to look at that eventually too. The reason dmg resist is less effective for every point of dmg debuff and defense is actually due to the fact that dmg resist only works on damage taken, while debuff and defense prevent the damage completely. On the other hand, the advantage of resist is that it automatically reduces damage, whereas defense does not reduce damage and debuffs must be applied to be effective.
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Defense and to-hit debuffs affect all attacks (in pretty much the same way), damage debuffs only affect the attacks that hit, and resistances only affect actual damage taken. So it's a 3 tiered level of damage mitigation. Defense/To-Hit debuffs cut into damage debuffs, damage debuffs cut into resistance.
I'm not sure if it would be really helpful to include damage debuffs though. That's generally not something you can opt into. You're either playing a powerset that has it or your not.
Formerly "Back Alley Brawler"
[u]Comparative Analysis of Damage Mitigation and Ratio of Favorable Gains[u]
This is a theory that I've been working on for the past week or so. It was prompted by a post that stated adding a small amount of defense on to a large amount of defense was better than adding the same amount on to a small amount. Basically that adding +5% defense to 30% defense was better than adding +5% to 5%.
This didn't make sense to me, but the math being used to prove this theory was pretty hard to dispute. However, I've invested the time into figuring out exactly why that math still makes sense, despite many of the basic assumption being incorrect.
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[u]Defense/Resistance versus Damage Mitigation[u]
First, it's important to make a distinction between damage mitigation from defense and resistance and survivability being related to total damage mitigation. This theory does not address damage mitigation's affect on survivability. To do that we have to consider a lot of other factors such as regeneration rate, total number of hit points, recharge/activation time of incoming attacks, damage per attack, etc. There is a correlation between higher levels of damage mitigation and how long you can survive (which I believe can be proven to be non-linear). Once we separate damage mitigation gained from defense and resistance from damage mitigation's affect on survivability, it becomes much easier to focus purely on the best way to increase that damage mitigation percentage (and consequently, increase your survivability)
<ul type="square">[*]30% Defense = 60% Damage Mitigation[*]35% Defense = 70% Damage Mitigation(+10% Damage Mitigation)[/list]and
<ul type="square">[*]05% Defense = 10% Damage Mitigation[*]10% Defense = 20% Damage Mitigation (+10% Damage Mitigation)[/list]-------------------------------------------------------------------------------------------------------
Second, the common assumption is that defense is twice as effective at mitigating damage versus resistance. Unfortunately, it's not this simple and straightforward. The value of defenses addition to total damage mitigation fluctuates based on the to-hit chance of the mob attacking you. Also, since resistance only adds damage mitigation to attacks that actually hit you, the total contribution that resistance make to the total percentage of damage mitigation is affected by the amount of defense that you have. This means that the more defense you have, the less resistance adds to damage mitigation. This also means (in an indirect way) that adding equal amounts defense to different base amounts of resistance results in less gains to total damage mitigation as well. So there are two things that need to be considered when making comparative analysis between adding defense or adding resistance to a character; the current level of damage mitigation and the ratio of favorable gains.
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[u]The Damage Mitigation Formula[u]
(1-((ToHit%-DEF%)-((ToHit%-DEF%)*RES%)/ToHit%)*100 = Percentage of Damage Mitigation
Examples (vs. 50% To-Hit, 100 Attacks, 10 points of damage per attack):
<ul type="square">[*]00% Defense + 00% Resistance = 500.0 points of damage taken[*]05% Defense + 00% Resistance = 450.0 points of damage taken[*]00% Defense + 10% Resistance = 450.0 points of damage taken[*]10% Defense + 30% Resistance = 280.0 points of damage taken[*]25% Defense + 60% Resistance = 100.0 points of damage taken[*]35% Defense + 15% Resistance = 127.5 points of damage taken[/list]<ul type="square">[*]05% Defense + 00% Resistance = 10.0% Damage Mitigation[*]00% Defense + 10% Resistance = 10.0% Damage Mitigation[*]10% Defense + 30% Resistance = 44.0% Damage Mitigation[*]25% Defense + 60% Resistance = 80.0% Damage Mitigation[*]35% Defense + 15% Resistance = 74.5% Damage Mitigation[/list]A couple of things to note about this is that notice how the 2:1 ratio doesn't hold up when you start mixing mitigation types. If 5% of defense always added 10% of damage mitigation and 10% of resistance always added the same, we'd see different percentages.
<ul type="square">[*]10% Defense + 30% Resistance = 44.0% Damage Mitigation (Expected 50%)[*]25% Defense + 60% Resistance = 80.0% Damage Mitigation (Expected 110%)[*]35% Defense + 15% Resistance = 74.5% Damage Mitigation (Expected 100%)[/list]Obviously we can't have 100% (or more) total percentage of damage mitigation or we'd be gaining health from every attack. It's also important to note that because there is always a 5% chance of any attack landing, the maximum amount of damage mitigation that can be achieved through defense alone is 90.0%, the same amount as the highest resistance caps for players (Tanks/Brutes). This would also suggest that a non tank AT could achieve tank like levels of damage mitigation through defense, despite having lower damage resistance caps.
Also note that though the percentage gained through incremental increases decrease in effectiveness, but they always remain linear.
<ul type="square">[*]15% Defense + 00% Resistance = 350.0 points of damage taken[*]15% Defense + 10% Resistance = 315.0 points of damage taken[*]15% Defense + 20% Resistance = 250.0 points of damage taken[*]15% Defense + 30% Resistance = 245.0 points of damage taken[*]15% Defense + 40% Resistance = 210.0 points of damage taken[/list]<ul type="square">[*]15% Defense + 00% Resistance = 30.0% Damage Mitigation[*]15% Defense + 10% Resistance = 37.0% Damage Mitigation[*]15% Defense + 20% Resistance = 44.0% Damage Mitigation[*]15% Defense + 30% Resistance = 51.0% Damage Mitigation[*]15% Defense + 40% Resistance = 58.0% Damage Mitigation[/list]Notice how every 10% of Resistance adds 7% to Damage Mitigation. This is clearly regular and linear gains, regardless of the base amount.
<ul type="square">[*]05% Defense + 30% Resistance = 315.0 points of damage taken[*]10% Defense + 30% Resistance = 250.0 points of damage taken[*]15% Defense + 30% Resistance = 245.0 points of damage taken[*]20% Defense + 30% Resistance = 210.0 points of damage taken[*]25% Defense + 30% Resistance = 175.0 points of damage taken[/list]<ul type="square">[*]05% Defense + 30% Resistance = 37.0% Damage Mitigation[*]10% Defense + 30% Resistance = 44.0% Damage Mitigation[*]15% Defense + 30% Resistance = 51.0% Damage Mitigation[*]20% Defense + 30% Resistance = 58.0% Damage Mitigation[*]25% Defense + 30% Resistance = 65.0% Damage Mitigation[/list]In this example, the amount of Resistance remains constant and incremental increases of defense (+5% Def) amount to +7% Damage Mitigation, rather than the 10% wed expect. Its still linear gains. Again, this is separate from the logarithmic scale of total damage mitigation increasing survivability time. Defense and Resistance create Damage Mitigation; Damage Mitigation in turn increases the amount of time you can survive. Theyre related but separate.
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How did I arrive at this formula? First we calculate the damage from attacks considering just defense. (Defense affects all attacks)
((ToHit% - Def%) * NumberOfAttacks) * DamagePerAttack = DamageTaken
Then we calculate any damage resisted (Resistance only applies to damage from attacks that hit you):
DamageTaken * Res% = DamageResisted
-or-
(((ToHit% - Def%) * NumberOfAttacks) * DamagePerAttack) * Res%
Then we subtract the resisted damage from the total damage taken:
DamageTaken - DamageResisted = MitigatedDamage
-or-
(((ToHit% - Def%) * NumberOfAttacks) * DamagePerAttack) - ((((ToHit% - Def%) * NumberOfAttacks) * DamagePerAttack) * Res%)
Then we want to figure the damage you'd take without any defense or resistance (for comparison):
(ToHit% * NumberOfAttacks) * DamagePerAttack = PotentialDamage
Then to calculate the % of damage mitigation provided by defense and resistance:
1 - (MitigatedDamage / PotentialDamage)
-or-
(1 - (((ToHit% - Def%) * NumberOfAttacks) * DamagePerAttack) - ((((ToHit% - Def%) * NumberOfAttacks) * DamagePerAttack) * Res%) / (ToHit% * NumberOfAttacks) * DamagePerAttack)
However, neither the number of attacks nor the damage per attack affect the percentage of damage mitigation, so those can be assumed to be 1 and calculated out of theformula. This gives us our final formula:
(1 - (ToHit% - Def%) - (ToHit% - Def%) * Res%) / (ToHit%)
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[u]Ratio of Favorable Gains[u]
The second part of this theory deals with what I'm calling the Ratio of Favorable Gains. This is basically what tells you whether it's better to add defense or resistance to your build.
The damage mitigation value of Defense fluctuates based on the to-hit chance:
Defense is equivalent (1 * ToHit%) per 1% of Resistance.
So the final ratio is:
(1 *ToHit%)% of Defense : 1% of Resistance
This defines the ratio of favorable gains, the point at which adding an equivalent amount of defense or resistance results in the same bonus to damage mitigation. This ratio also defines what that equivalent amount is:
For example:<ul type="square">[*]50% To-Hit chance means that 1% of Defense is equivalent to 2% Resistance in terms of Damage Mitigation[*]75% To-Hit chance means that 1% of Defense is equivalent to 1.5% Resistance in terms of Damage Mitigation[*]100% To-Hit chance means that 1% of Defense is equivalent to 1% Resistance in terms of Damage Mitigation[/list]-------------------------------------------------------------------------------------------------------
Examples (vs. 50% To-Hit):
<ul type="square">[*]25% Defense + 50% Resistance = 75% Damage Mitigation[*]30% Defense + 50% Resistance = 80% Damage Mitigation (+5% Damage Mitigation)[*]25% Defense + 60% Resistance = 80% Damage Mitigation (+5% Damage Mitigation)
At this point, the ratio between Def and Res is 2:1. This means that adding 5% of Defense is equivalent to adding 10% Resistance[/list]
Compared to:
<ul type="square">[*]30% Defense + 50% Resistance = 80% Damage Mitigation[*]35% Defense + 50% Resistance = 85% Damage Mitigation (+5% Damage Mitigation)[*]30% Defense + 60% Resistance = 84% Damage Mitigation (+4% Damage Mitigation)
In this case, Defense beats the ratio (it's more than half of the amount of resistance), so adding an equivalent amount of Defense would be better than adding Resistance.[/list]
This holds true for Resistance as well:
<ul type="square">[*]25% Defense + 60% Resistance = 80% Damage Mitigation[*]25% Defense + 70% Resistance = 85% Damage Mitigation (+5% Damage Mitigation)[*]30% Defense + 60% Resistance = 84% Damage Mitigation (+4% Damage Mitigation)
In this Case, Resistance beats the ratio of favorable gains (it's more than twice the amount of defense), so adding an equivalent amount of Resistance would be better than adding Defense[/list]
The larger the gulf between the ratio of favorable gains, the larger the difference is between the bonus to damage mitigation
<ul type="square">[*]35% Defense + 5% Resistance = 71.5% Damage Mitigation[*]40% Defense + 5% Resistance = 81% Damage Mitigation (+9.5% Damage Mitigation)[*]35% Defense + 15% Resistance =74.5% Damage Mitigation (+3% Damage Mitigation)[/list]================================================== ===========
[u]Defense/Resistance versus Damage Mitigation[u]
So whats all this mean? We can draw a few conclusions from this:[*]The goal is to achieve the highest percentage of damage mitigation possible. Whether thats done by adding defense or resistance isnt relevant.[*]Defense being twice as effective as resistance in terms of damage mitigation isnt a constant. Its based on the to-hit chance of incoming attacks[*]Adding any amount of defense to any base amount adds the same amount to the total percentage of damage mitigation.[*]Likewise, adding any amount of resistance to any base amount adds the same amount to the total percentage of damage mitigation.[*]The amount of defense you have diminishes resistances contribution to the total percentage of damage mitigation (ie, if you get hit less, you resist less damage). This does not simply mean that defense is always better than resistance.[*]There is a way to determine whether adding more defense or more resistance would result in more damage mitigation. (see Ratio of Favorable Gains above)
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