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The Definitive CritH vs CritD guide
cerealplayer
Member Posts: 214 Arc User
Disclaimer: this is a guide solely about choosing CritH and CritD. I will not discuss the larger picture relating to other gear mods ([DMG] etc.) nor will I discuss the larger STO meta-game (like things that trigger off of crits). My goal here is only to teach you the maths so that you can decide, for yourself, without a calculator, spreadsheet, some internet friend told you, etc., how to choose between, say, Spire consoles, or CritH and CritD weapons modifiers; assuming you know your current values of CritD and CritH.
On the other hand, these are handy maths that happen to apply to calculating crit damage in ANY game. They're also useful in RL, yo.
The Guide.
The purpose here is to compare only CritH to CritD. Hence, we will assume that our hit chance, and non-crit damage remain constant. In this scenario, our expected damage per landing shot is thus:
(Eq. 1): TED = BD * ( 1 + H * D),
where TED is our total expected damage, BD is non-crit damage H is CritH and D is CritD. The equation is dead simple, but we can ellaborate.
First, notice that H is a probability, hence we can only ever talk about the total expected damage. This is not going to be your actual damage per shot. Rather, if you were to shoot your gun from now until infinity, and you took the average damage per shot, then that would equal TED. In more useful terms, during the course of an ISE, your average damage per shot will, with high probability, be very close to TED.
Second, notice the 1. That is there because the base damage is always applied (when we land a shot), regardless of whether we crit or not. When/if we crit we get to do extra damage. The amount of extra damage is dertmined by our CritD, or D in the equation.
TED is what we want to maximise. How do we do that? Well, if you remember your high school calculus, you can see that if you take the partial derivative of TED with respect to H you get D, and vice versa. But if you remember your high school calculus you probably already know the right answer, so let's assume you don't.
Let's simplify the equation then. Since the BD is always applied, and we're assuming that other modifiers remain constant (weapon type, [DMG] modifiers, etc), we can consider BD to be constant. 1 is also, of course a constant. So we have only two variables H, and D, which together dermine the expected extra damage we get to inflict per shot (over our base, that is always applied on a hit). Let's remove the constants, and we're left with:
(Eq. 2): EED = H * D
Where EED is Expected Extra Damage (in units of base damage). This is what we wish to maximise with our choice of CritD and CritH. How do we proceed? suppose you currently have H amount of CritH, and D amount of CritD. You got a new, hypothetical, piece of gear, and can now add one extra CritD, or one extra CritH, which one do you go for?
Well, turns out we don't need high school calculus, we can use grade school arithmetic. Remember that multiplication is distributive. With that in mind, we have two choices. We can increase our D by 1 in which case our new EED is:
(Eq. 3): EED2 = H * (D +1)
= H * D + H
= EED + H
So, increasing our CritD by 1 increases our expected bonus damage by exactly H! Likewise, if we increase our CritH by one, we get a new EED of:
(Eq. 4): EED3 = (H + 1) * D
= H * D + D
= EED + D
So, increasing our CritH by 1 increases our expected bonus damage by exactly D!
Hence, at any point, if we can increase either CritH or CritD by one, we should always choose the one that is currently lower.
The Myth of the Magic Ratio
But, you say! You had heard of a magic ratio of 1:10. Where does that come from? Were you lied to? Or am I lying to you now? Well the math doesn't lie. This ratio comes from the fact that in STO you don't normally get to choose between ONE of CritH, and ONE of the other. There are, let's call them, exchange rates. For example, on weapons you can either get a CritH modifier for 2%, or a CritD modifier of 20%. So the exchange rate on weapons is 1:10. This is the same exchange rate found on universal consoles. In fact, the 1:10 rule came about at a time when the game only allowed you to choose, or exchange between, CritH and CritD at a ratio of 1:10. This is no longer the case, though.
So, what changes now? Well, imagine your company offered to pay you either British pounds, or Euro's. But they decide that if they pay you in Euros they'll give 10 times as many Euros as pounds. Which should you take? You might jump and say "Of course Euros". Well, that might be the case, but for a trully correct answer you need to look at the current exchange rate. Imagine, completely hypothetical I know, that the pound was worth 100 times as much as the Euro. Should you still go for the payment in Euros? Clearly not! If the pound is much more valuable than the Euro, you should choose the pound. But where is the cutoff point? Well, when the pound is worth exactly 10 times as much as the Euro, it doesn't matter how the company pays you. If the pound is any stronger, you should go for pounds, any weaker, and you should opt for Euros.
Same thing in STO. When you get the choice of 10 CritD or 1 CritH, you need to look at how valuable these two are for your current build. Remember the worth of a single point of CritH is exactly your current CritD, and the worth of a single CritD is exactly your current CritH. Hence, you should always go for CritD on a weapon, unless, your CritD is more than 10 times larger than your current CritH.
What about the new Spire Tac consoles? Well, there the exchange rate is different: it's 1:5 rather than 1:10. So, you should take the CritD only if your CritD is currently less than 5 times your CritH.
An important thing to keep in mind, though, is that each new piece of gear you add changes your current CritH :: CritD ratio. Where you start off, may not be where you end! For instance, you may decide, looking at your current numbers that locators are better than exploiters. But after you add one, or two, of these consoles, the ratio may change to favour the other!
If cryptic were to introduce a new item slot that gave you either X CritH, or Y CritD, the same maths would apply, with a new exchange rate of X:Y. The important thing here is that there is no magic ratio, and it is certainly not 1:10. There are merely item stat budgets, and conversion rates between stats. These conversion rates are what we have to pay attention to when picking an item, but remember that they apply only to that item or slot.
There, you no longer need a spreadsheet, or calculator. If you know your current CritH and CritD, then you IMMEDIATELY know which pieace of gear is better for you.
On the other hand, these are handy maths that happen to apply to calculating crit damage in ANY game. They're also useful in RL, yo.
The Guide.
The purpose here is to compare only CritH to CritD. Hence, we will assume that our hit chance, and non-crit damage remain constant. In this scenario, our expected damage per landing shot is thus:
(Eq. 1): TED = BD * ( 1 + H * D),
where TED is our total expected damage, BD is non-crit damage H is CritH and D is CritD. The equation is dead simple, but we can ellaborate.
First, notice that H is a probability, hence we can only ever talk about the total expected damage. This is not going to be your actual damage per shot. Rather, if you were to shoot your gun from now until infinity, and you took the average damage per shot, then that would equal TED. In more useful terms, during the course of an ISE, your average damage per shot will, with high probability, be very close to TED.
Second, notice the 1. That is there because the base damage is always applied (when we land a shot), regardless of whether we crit or not. When/if we crit we get to do extra damage. The amount of extra damage is dertmined by our CritD, or D in the equation.
TED is what we want to maximise. How do we do that? Well, if you remember your high school calculus, you can see that if you take the partial derivative of TED with respect to H you get D, and vice versa. But if you remember your high school calculus you probably already know the right answer, so let's assume you don't.
Let's simplify the equation then. Since the BD is always applied, and we're assuming that other modifiers remain constant (weapon type, [DMG] modifiers, etc), we can consider BD to be constant. 1 is also, of course a constant. So we have only two variables H, and D, which together dermine the expected extra damage we get to inflict per shot (over our base, that is always applied on a hit). Let's remove the constants, and we're left with:
(Eq. 2): EED = H * D
Where EED is Expected Extra Damage (in units of base damage). This is what we wish to maximise with our choice of CritD and CritH. How do we proceed? suppose you currently have H amount of CritH, and D amount of CritD. You got a new, hypothetical, piece of gear, and can now add one extra CritD, or one extra CritH, which one do you go for?
Well, turns out we don't need high school calculus, we can use grade school arithmetic. Remember that multiplication is distributive. With that in mind, we have two choices. We can increase our D by 1 in which case our new EED is:
(Eq. 3): EED2 = H * (D +1)
= H * D + H
= EED + H
So, increasing our CritD by 1 increases our expected bonus damage by exactly H! Likewise, if we increase our CritH by one, we get a new EED of:
(Eq. 4): EED3 = (H + 1) * D
= H * D + D
= EED + D
So, increasing our CritH by 1 increases our expected bonus damage by exactly D!
Hence, at any point, if we can increase either CritH or CritD by one, we should always choose the one that is currently lower.
The Myth of the Magic Ratio
But, you say! You had heard of a magic ratio of 1:10. Where does that come from? Were you lied to? Or am I lying to you now? Well the math doesn't lie. This ratio comes from the fact that in STO you don't normally get to choose between ONE of CritH, and ONE of the other. There are, let's call them, exchange rates. For example, on weapons you can either get a CritH modifier for 2%, or a CritD modifier of 20%. So the exchange rate on weapons is 1:10. This is the same exchange rate found on universal consoles. In fact, the 1:10 rule came about at a time when the game only allowed you to choose, or exchange between, CritH and CritD at a ratio of 1:10. This is no longer the case, though.
So, what changes now? Well, imagine your company offered to pay you either British pounds, or Euro's. But they decide that if they pay you in Euros they'll give 10 times as many Euros as pounds. Which should you take? You might jump and say "Of course Euros". Well, that might be the case, but for a trully correct answer you need to look at the current exchange rate. Imagine, completely hypothetical I know, that the pound was worth 100 times as much as the Euro. Should you still go for the payment in Euros? Clearly not! If the pound is much more valuable than the Euro, you should choose the pound. But where is the cutoff point? Well, when the pound is worth exactly 10 times as much as the Euro, it doesn't matter how the company pays you. If the pound is any stronger, you should go for pounds, any weaker, and you should opt for Euros.
Same thing in STO. When you get the choice of 10 CritD or 1 CritH, you need to look at how valuable these two are for your current build. Remember the worth of a single point of CritH is exactly your current CritD, and the worth of a single CritD is exactly your current CritH. Hence, you should always go for CritD on a weapon, unless, your CritD is more than 10 times larger than your current CritH.
What about the new Spire Tac consoles? Well, there the exchange rate is different: it's 1:5 rather than 1:10. So, you should take the CritD only if your CritD is currently less than 5 times your CritH.
An important thing to keep in mind, though, is that each new piece of gear you add changes your current CritH :: CritD ratio. Where you start off, may not be where you end! For instance, you may decide, looking at your current numbers that locators are better than exploiters. But after you add one, or two, of these consoles, the ratio may change to favour the other!
If cryptic were to introduce a new item slot that gave you either X CritH, or Y CritD, the same maths would apply, with a new exchange rate of X:Y. The important thing here is that there is no magic ratio, and it is certainly not 1:10. There are merely item stat budgets, and conversion rates between stats. These conversion rates are what we have to pay attention to when picking an item, but remember that they apply only to that item or slot.
There, you no longer need a spreadsheet, or calculator. If you know your current CritH and CritD, then you IMMEDIATELY know which pieace of gear is better for you.
Post edited by Unknown User on
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Comments
The previous analysis is meant to show how to make proper decisions for gear on the basis of your current CritD and CritH. But how do you find those?
DISCLAIMER: I do not purport to be an expert on STO game mechanics, unlike Maths. And, unlike Maths, STO mechanics are subject to change.
That said, I can tell you what I've gathered, for completeness sake.
Only look at your character sheet in space --but not sector space. That will give a starting point, but not the whole picture. Some things don't show up in your character sheet. What things? Well.
1) Weapon mods. The reason here is that these mods apply ONLY to the weapon that bears them. So, for each weapon, you're getting a different set of mods. Hence, it can't show up on your character sheet. You have to add them manually into your calculations.
2) Buffs, like AP:A, that are not up 100%. Some of these buffs you can trigger outside of combat, and you can check their value in your character sheet. However, the buffs normally don't have 100% uptime, so it would be incorrect for you to calculate your CritH/CritD values with buff on anyway. To add it properly, manually, you add the total buff, multiplied by it's uptime.(1) This gives you the average buff value.
3) Accuracy overflow. Accuracy that is not needed to hit your target gets overflowed into extra CritH and CritD, following a formula that has been debated on these forums ad nauseum; but has never, to the best of my knowledge, been made public in an official capacity by Cryptic (except, maybe, in a screen on a podcast, that can be briefly seen). Calculating overflow is, and will likely remain, outside the purview of this guide.
4) Weapon Specialization. The reason here is that each specialization applies only to either energy weapons or kinetic. And again, your character sheet only gives you the stats that apply to ALL attacks. The STO wiki has a good chart of what each point into these specs provide.
If you want to calculate your values manually, then these are things you have to consider properly.
Another approach, and one that I am fond of, is to simply download a parser that can calculate your Crit percentage, and Crit damage percentage. There's various out there. Play your favourite STF as you normally would, hitting all your self-buffs like your normally would, and parse yourself. Problem solved.
Notes:
1) How you calculate your buffs uptime is another matter entirely. If you care about your STF performance, then you calculate the percentage of time your buff can be maintained during the duration of an STF. For a PvP decloacking alpha vaper, who only ever decloacks when all buffs are off of cooldown, then you'd probably want to consider an artificial uptime of 100%.
For Faq
I have been going after Phasers Beam Array Mk XI [CritH] rather than [CritD] simply because I thought that was the better choice (w/o doing the math). Do all weapons have a base % for critical hits, or is the just limited to [CritH] weapons?
On the plus side I picked up a Chroniton Mine Launcher Mk XII [CritH]x2 [CritD] real cheap on the exchange.
Perhaps for my plasma weapon set I will focus on the [CritD] versions...
I've included some clarifications below just so people don't misunderstand the above statements.
1. The optimal crit chance is 100%. There is no optimal crit severity; you should aim for as high as you can get. Of course, much of time you have decide between more crit chance or more crit severity.
2. There is no equipment in the game that allows you to trade crit chance for an equal amount of crit severity. Because the trade-off isn't 1:1, the decision of whether to increase crit chance or crit severity is not simply a matter of increasing the lower value.
3. The optimal ratio of crit chance to crit severity is 1:1 if the sum of crit chance and crit severity must be a constant. But there is no such constraint in the game. Specific constraints are the number of tac consoles on a ship and the number of modifiers on a weapon.
If you use for example stuff with lower firing rate or abilities on a cooldown (overload), things look a little different: you need to kill your enemy in few shots, before they kill you.
Sometimes, CritH is your chance to survive :P
(an overstatement, of course)
D comes in 10% increments and H in 1%. Are you assuming those as a value of '1' for each delta? If you use the true delta the values are very different. And from my puttering around in Excel, H always seemed better under non-extreme conditions since D is dependent on H.
(Eq. 3): EED2 = H * (D +.1)
= HD + 0.1H
= EED + 0.1H
(Eq. 4): EED3 = (H + 0.01) * D
= HD + 0.01D
= EED + 0.01D
From his later statements, I think that statement was supposed to reflect the hypothetical scenario in which crit chance and crit severity came in the same increments. I do think the statement is misleading if one reads it isolated, out of context. That's why I tried to make some clarifications in my response above. The increments for crit chance and crit severity actually depend on what equipment you are talking about.
[CrtH]: 2% crit chance
[CrtD]: 20% crit severity
vulnerability locator: 1.6% crit chance
vulnerability exploiter: 8% crit severity
Universal consoles, traits, and rep passives all have different increments.
For all the other strange increment values on consoles and etc, I'm too lazy to bother with really but...
Just for fun, I upped the buff limit to 8 and removed the constraint of the values increasing by the standard increments (aka non-integer multipliers) and get... a equal number of buff increments. So I guess that's jiving with your numbers. Optimals are at CrtH/Ds values of X%/X0%
Now if 1 of the 4 buff slots are taken w/ something else Solver can't pick 2/2 and gives...
for integer increments H/D = 2/1 or 1/2 gives same value (added .2 damage to base of 100)
for non-integer increments H/D = 1.5/1.5 (added .3 damage to base of 100)
If I go to 7 available buff slots it goes:
for integer increments H/D = 4/3 or 3/4 gives same value (added 1.2 damage to base of 100)
for non-integer increments H/D = 3.5/3.5 (added 1.23 damage to base of 100)
Same kinda ratios but the difference in damage output is nearly the same.
So again, Solver goes for the balance and it's a coin flip on how you want the last odd 'point' of buff to go for max integer based damage.
Thank you for these, and your previous comments. I've tried to clarify my post, mostly by highlighting some of the text in red. I may add some further clarifications, but I'll play it by ear (I am loathe, for instance, to having to remind people that the optimal value for a probability is 1, or 100%).
I'm not sure I know what you're asking, but your question might be answered in FAQ I'll be posting when I have time later tonight. If it isn't then let me know.
Certainly we all want the highest crit chance and crit severity that we can get. The question being addressed in this thread, if I understand the original post correctly, is this:
When choosing between a CritH bonus and a CritD bonus, which will give the greater increase in total damage?
Obviously either one will increase your damage, but we are discussing what circumstances would make one clearly the better choice than the other. Will I get more damage from a CritDx2 CritH weapon, or from a CritHx2 CritD weapon, for example?
Hmm, not sure that's a dig or not...
In absence of other buffs; those 2 will do the same damage. 2x the bonus half the time or half the bonus twice the time.
It's not a dig. It was response to this statement by Frtoaster:
[QUOTE=frtoaster;15348221
I've included some clarifications below just so people don't misunderstand the above statements.
1. The optimal crit chance is 100%. There is no optimal crit severity; you should aim for as high as you can get. Of course, much of time you have decide between more crit chance or more crit severity.
[..]
[/QUOTE]
Of course, he's right. And no, I didn't mention that fact in my guide. I simply remarked that I would hope that I don't have to tell people that probabilities don't go over 100%.
You're clearly European (even without the later reference to GBP and Euros). In the US, the typical student will have had Alg1-Geo-Alg2 in high school, and that's it. Advanced students might have taken Trig or Calc, but didn't have to. Some students may have taken BS classes like "Business Math" to get their third math credit, and not even had Alg2. And yes, three math credits is all that's required. Why is the US trailing behind in STEM fields again?
(me personally, due to circumstances I wound up in four different schools over the course of my junior year, resulting in my almost failing Alg2. So I didn't take Calc. I still regret it.)
Anyway, great guide. It's always useful when somebody puts some actual numbers to all the FUD that spreads around. Numbers can be discussed and debated in a productive fashion.
I'm curious, are CritD and Severity the same thing. I just assumed it was a different way of describing the same property which is an increased amount of damage you ONLY get when you achieve a critical hit on a target?
/char
CritD is the name this game uses for severity, yes. I have chosen to keep to this convention in the hopes of reducing confusions. If I see that it instead creates more confusion, I'll redress my decision.
Oh, and thank you! I hope it's useful.
You'd be surprised, but I'm not! I could've just as easily used US vs Canadian dollar; for instance.
Thank you! I hope it's useful. And I hope it finally puts some of the really awful discussions going on to rest.
Thank you, that really confused me lol
I've calculated that (just going off my base stats always there, skill tree bonus, universal consoles and rep bonuses) is -
CritH 10.22
CritD 94.2
Given what you've said, which was brilliantly helpful, I will now use critH consoles and CritD based weapons to bring my H to 20 and D to between 160 and 170. Which should be a solid balance of getting good crits and good severity off those crits. I won't be a death machine in STF's but I think I'll be a lot more helpful to the team.
Thanks again, you've helped a lot of confused players.
Definitive Guide is an inappropriate title for something which glosses over Acc overflow. A better title would be "A narrow slice of damage mechanics which can be a good starting point for thinking about the topic."
I am on hiatus from the game, so maybe something changed in recent months, but the meta in Big Red Jedi's seminal spreadsheet is not in question.
Discussing CritH and CritD while willfully ignoring Acc and Def is like giving advice on how to salsa dance on top of a moving bus. You can't write the Definitive Guide without accounting for the bus.
This is a laudable goal. This is a philosophical stance which, irrespective of your talent with maths, takes your laudable goal and carefully, precisely, toe-by-toe, shoots it in the foot.
Should an army win a war because they have more soldiers, on average, in the region? Or should an army win a war because they have more soldiers at the actual battles?
If you are very fortunate the top salsa instructor will come by to explain why you can't call something Definitive if it coldly averages something that only exists in a mutable, sexy, hip shaking form. Perhaps Thissler will explain Maths + Soul = STO.
Thanks for the compliment. And don't worry. I have to deal with peer-review in my day job. Nothing you can say could possibly come off as harsher. So, let's take a look:
You're not wrong. The problem is, there's been some debate about how exactly acc overflow turns into critH/critD. So, yes, in your abscence the meta has been put into question. I simply don't have the time to test it myself, and honestly I don't know who to trust on this. Once there's a definitive answer, I'll consider updating my guide. There are two possible goals in any guide: completeness, and correctness. I tried to be complete, but when I couldn't personally verify the correctness of a claim I decided not to include it my guide. That way, I can vouch for the correctness of it all.
Not a bad point to make at all either. Clearly, my guide, as it currently stands, is biased towards the average PvE user. If you're a PvP alpha-vaper, or you're into the 1:30 minute ISE runs, then yes, my assertion would be somewhat incorrect. Thank you for pointing out my hidden assumption here. I have corrected the OP. I hate making it more verbose, but correctness is a higher virtue than succinctness.
BTW, Redricky? If it is you, despite your protestations to the contrary, I loved your A2B guide!
But on my spike dmg norgh, my numbers for quantums and DBB are: crth=10% and crtd=180% (holds and buffs excluded) even if math would tell me i'd be better with 15% crth and 155% crtd. From 10% or 15% to 100% is a long way as i flip the probability coin only once
I'd hate to see you when you're not trying to be polite.
Or
Does apologizing in advance mean you can be as big a prick as you can be? I'll have to keep that in mind for future postings!
And...
Damn it man, I admitted to leaving out Acc overflow and other stuff too! Where's my not at all prickish criticism? I feel left out!
Please, by all means, do go on.
Empirical evidence.