Lt. Commander
Join Date: Feb 2013
Posts: 183
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.

Last edited by cerealplayer; 02-26-2014 at 09:57 AM.
Lt. Commander
Join Date: Feb 2013
Posts: 183
# 2
02-25-2014, 10:05 AM
Addenda: Finding your current CritH and CritD.

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%.

Last edited by cerealplayer; 02-27-2014 at 08:09 AM.
Lt. Commander
Join Date: Feb 2013
Posts: 183
# 3
02-25-2014, 10:09 AM
****reserved********

For Faq
Captain
Join Date: Jan 2014
Posts: 1,744
# 4
02-25-2014, 11:22 AM
Huh... Interesting...

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...
Captain
Join Date: Jun 2012
Posts: 1,813
# 5
02-25-2014, 12:02 PM
I've made comments in several previous threads with the same mathematical analysis. Those who understand mathematics will agree. Many who heard about 1:10 ratio from somewhere, but don't actually understand enough math to follow the discussion, will stubbornly refuse to agree.

Quote:
Originally Posted by cerealplayer View Post
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 optimal value is achieved when both are equal.
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.
Lt. Commander
Join Date: Sep 2013
Posts: 165
# 6
02-25-2014, 02:38 PM
Well, this applies for a large number of shots.

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
(an overstatement, of course)
Career Officer
Join Date: Jun 2012
Posts: 2,339
# 7
02-25-2014, 06:45 PM
Quote:
Originally Posted by cerealplayer View Post
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 optimal value is achieved when both are equal.
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
Sometimes I think I play STO just to have stuff to rant about on the forums!

Captain
Join Date: Jun 2012
Posts: 1,813
# 8
02-25-2014, 07:04 PM
Quote:
Originally Posted by dracounguis View Post
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.
Career Officer
Join Date: Jun 2012
Posts: 2,339
# 9
02-25-2014, 07:47 PM
Now when I use Excel Solver to find the best combo of H and D using the elite weapon number of buffs (4) it also finds that 2 of each is the optimal. This was for just the base increments of 1% and 10%.

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.
Sometimes I think I play STO just to have stuff to rant about on the forums!


Last edited by dracounguis; 02-25-2014 at 08:03 PM.
Lt. Commander
Join Date: Feb 2013
Posts: 183
# 10
02-25-2014, 07:58 PM
Quote:
Originally Posted by frtoaster View Post
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.
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%).
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