Crit Pendants (Warning: Math)

Sagira

I did a mathematical analysis of the crit pendants and their value. Hopefully someone will find it useful!

Feel free to ask for clarifications, other specific calculations you'd like me to do, or point out places where I'm straight-up wrong in either my assumptions or my methodology.

https://drive.google.com/file/d/0B_g9wRjun0XuU01xdkdxVUtOVWs/view?usp=sharing

Cooper

That was pretty cool to read! Thanks for the in-depth look at this.

Penwize

I'm noticing this doesn't seem to account for the crit pendant's effect on the weights for each crit type.  The statement "The pendants give a flat increase to the crit rate, which we can call Δx" seems to indicate that you're completely ignoring this effect?  That's a mistake, I feel, because honestly that's probably the biggest effect the pendant has.  At the top end, you actually get slightly more world-shattering crits than annihilating crits, which is a pretty stark change from getting half as many.  I think that would significantly change the numbers you get, in both estimated damage multiplier and damage increase per credit.

Cooper

That's accounted for in the damage formula.

Penwize

I don't see how it does?  The document specifically states it applies a flat increase, but in fact the increase works very differently to that.  It's a flat increase to how many hits are crits, but it's a scaling increase to the strength of those hits.  That is, it'll affect WSC more than it'll affect just Crits.

Cooper

Well, the pendant does offer a flat crit bonus.

I'm not a math wiz but I assumed the index thing was the part taking care of that.

Penwize

I actually don't see an index thing either?  Are you mistaking the superscript in the expanded out section (which is an exponent) for an index subscript?

Cooper

Yeah, sorry, I assumed the exponent/thing meant something it didn't.

@Sagira Here's an example of how crits work in relation to crit pendants, with made up numbers from an old admin.

Again noting made up numbers for the example, and it's a straight copy/paste from an 8.5 year old email:

just over a 75% chance of getting any crit at level 91.
Then (100-75%) = 25% of those will be critical and the remaining 
something else.
Then (100-75%) = 25% of _those remaining_ will be crushing, and the 
remaining something else.
Then (100-75%) = 25% of _those remaining_ will be obliterating, etc.
Then (100-75%) = 25% of ... will be annihilatingly powerful,
and the rest will be WS.

Going quickly, I'd put that at
75.00% total for all crits
18.75% critical 
14.06% crushing
10.55% obliterating
07.91% annihilatingly
23.73% WS


Change the 75 to whatever the bonus is for the crit. In this case, 
that'd be +2% in every case.

So 77% get any crit
Then (100-77%) = 23% of those will be critical and the remaining 
something else.
Then (100-77%) = 23% of _those remaining_ will be crushing, and the 
remaining something else.
Then (100-77%) = 23% of _those remaining_ will be obliterating, etc.
Then (100-77%) = 23% of ... will be annihilatingly powerful,
and all the rest will be WS.

Again going quickly, I'd put that at
77.00% total for all crits
17.71% critical 
13.64% crushing
10.50% obliterating
8.08% annihilatingly
27.07% WS

Sagira

I am accounting for the pendant's effect on higher crit types. The idea is just that instead of the game rolling once for a crit/no crit, on a crit, it rolls again to see if that crit crits, itself. So that's why the index in the sum (Sorry, mixing math and programmer terminology, my bad) is there. For a hit, which is indexed as crit level 0, the chance is 1-x: because x are crits (or better). For a crit, index 1, the chance is x*(1-x), because of the x crit percent from before, only 1-x of those are crits - the rest are better than crits. And the damage is doubled, so that's where the 2 comes from.

An example, with a simple 25% crit rate:
Hit: (1-x) = 75%
Crit: x*(1-x) = .25*.75 = 18.75%
Crush: x^2*(1-x) = .25^2*.75 = 4.6875%
Oblit: x^3*(1-x) = .25^3*.75 = 1.171875%
AP: x^4*(1-x) = .25^4*.75 = 0.29296875%
WS: x^5 = .09765625%  [I actually just noticed that this term is wrong in the formula I put down; it doesn't get a (1-x) because there are no further rolls - it's correct in my MATLAB code, though, so the plots are correct. Consider the written formula fixed!]

Penwize

That doesn't match up.  If you follow that, then your method would say world-shattering crits can never  be more frequent than annihilating crits, which is not the case.

Sagira

Sure it does. Like I said, the formula I wrote down has an extra (1-x) on the WS term (So yes, it's wrong, and I'm currently unable to edit it for complicated reasons). So there does come a point where x^5 > x^4(1-x). Specifically, it comes when x>1-x or x>.5 - which actually isn't possible in-game. Dragons have something around 42.19% crit, plus 6% from the level 3 pendant, so they get very close, specifically 2.6% WS and 2.79% AP.

Paine

level 1 critical hit pendant at 200 credits is probably the first artefact you should purchase if you intend to do any bashing.

Sagira

Well, that depends, Paine. The crit pendants are lousy until the mid-70s. If you plan to have only 200 credits, then yes, the pendant is going to be your best bet, but if you're planning to get more than one artefact, a collar (or equivalent) is possibly a better bet for your first one. The level 1 crit pendant doesn't approach the collar for damage-per-credit until level 88, and it's NEVER as good as the collar for absolute damage.

Sena

Sagira said:

So there does come a point where x^5 > x^4(1-x). Specifically, it comes when x>1-x or x>.5 - which actually isn't possible in-game. Dragons have something around 42.19% crit, plus 6% from the level 3 pendant, so they get very close, specifically 2.6% WS and 2.79% AP.

It's possible with other critical boosts. If crit rate caps at level 100 (rather than 99, which my data seems to suggest), then just Lucky elixir is enough. At 99, Lucky elixir, Lucky trait, and human crit bonus should just barely bring you to 50% (maybe marginally less than 50%), then there are various less common bonuses (mostly from promotions) that can bring you over.

Sagira

Indeed. I was speaking with Penwize (Penny?), and he suspects that crit rate is not constant above 100, although the formula is obviously vastly different as the growth rate above 100 would be insane (as it's a cubic function). He's got a crit rate of something like 52% with a pendant and a lotus, so there's obviously something else at play, probably related to his level. If y'all want, I could make a plot of damage as a function of crit rate from 0 to 100% (Spoilers: It's 3200% at 100% crit rate), but that's not really as useful a metric as level, since you can't directly measure crit rate in-game.

Cooper

He's also human and probably has the crit trait too.

Sena

At level 126 as a human with Lucky and a level 3 pendant, Penwize was getting about 48% (24888/51835). As a human with no other bonuses, about 41.6% (12824/30809). About what you'd expect for level 99.

Sarathai

 Gonna tag @Penwize and @Seragorn to this thread, the topic of post-100 crits has come up a bit recently.

Szanthax

I tend to get an inordinate amount of crits. I don't have a stat tracker.. anyone got an easy crit tracker? 

103 lvl3 pendant lucky...

I hate bashing as non Dragon because I go sploot against the easy things

Sobriquet

104 with lvl 3 pendant here, happy to help with a stat tracker if anyone has one

Dochitha

Crit does not improve pass 100. I am at 110 it feels as lousy as 100. We need crit boost. 

I have L3 pendant, lucky trait, human, I sip lucky when I go on 12- hour hunt.

Eld

If a nearly 50% crit rate feels lousy, that sounds like a problem of unrealistic expectations, rather than with the rate itself...

Szanthax

Eld said:

If a nearly 50% crit rate feels lousy, that sounds like a problem of unrealistic expectations, rather than with the rate itself...

I just think mines higher but i dont have proof of numbers

Eld

Szanthax said:

Eld said:

If a nearly 50% crit rate feels lousy, that sounds like a problem of unrealistic expectations, rather than with the rate itself...

I just think mines higher but i dont have proof of numbers

I'd be very surprised if it were anything other than 46.5% in dragon (40.5% base at the level 99 cap, +6% for the pendant) or 47-47.5% in lesser (not sure if lucky is 1% or .5%, I think .5), with no other bonuses. 

Sobriquet

Eld said:

If a nearly 50% crit rate feels lousy, that sounds like a problem of unrealistic expectations, rather than with the rate itself...

Agreed. 104 with a lvl 3 pendant and my Crit rate is great.

Klendathu

I agree, the crit rate with a level 3 pendant at >100 level is fine (obscenely good)

Xaden

By my counter, at level 108, level 1 pendant, I'm circa 46.5%. And I only hunt in dragon and don't sip lucky or have a lotus (wtf is a lotus?)

Dochitha

Xaden said:

By my counter, at level 108, level 1 pendant, I'm circa 46.5%. And I only hunt in dragon and don't sip lucky or have a lotus (wtf is a lotus?)

It's Lily.

Qwyn

Eld said:

If a nearly 50% crit rate feels lousy, that sounds like a problem of unrealistic expectations, rather than with the rate itself...

It could just be that the person plays other IRE games. 65-75%+ are standard for high-end arti'd bashers in other IREs, so they might think their numbers are low here (even though they aren't, relatively speaking).

Eld

Xaden said:

By my counter, at level 108, level 1 pendant, I'm circa 46.5%. And I only hunt in dragon and don't sip lucky or have a lotus (wtf is a lotus?)

That's too high for a level 1 pendant and no other bonuses, even if the cap is 100 instead of 99. Sounds like either you've had an extremely lucky streak, your count includes some other bonuses that you've forgotten, or there's something wrong with your counter (such as counting all crits, but failing to include some attacks in the count of all hits - battlerage might be a likely culprit there).

Xaden

I thought the base rate was circa 44% at level 100?