Reckoning/Weapon Speed (Math)

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Reckoning/Weapon Speed (Math)

Postby Phasek » Thu Jun 14, 2007 8:05 am

I pulled some of this from another post I made to make it easier on me. I'm going to prove mathematically that weapons faster than 2.0 are actually WORSE for reckoning.

When a mob attacks, a paladin has roughly a 50% chance of taking damage varying with gear. When taking damage, there's a 10% chance to double the swings over the next 8 seconds. The trickiness with the math comes in because of procs that refresh reckoning while it's already active. I think the best way to approach this is to figure out what the chance that reckoning has NOT procced over the last 8 seconds. So:
First the mob has a 0.5 chance to hit.
Second the mob has a 0.1 chance on a hit to proc reckoning.
0.1 chance of reckoning * 0.5 hit chance = 0.05 so a mob has a 5% chance to proc reckoning per swing.

Therefor a mob has a 95% chance to not proc reckoning per swing. Let's now assume that y is your attack speed x4, but not exceeding 8, making it the amount of seconds that reckoning will stay active. If x is the mob's attack speed, the amount of swings the mob performs per y seconds is y/x.
therefor, the chance that reckoning is not up at any given time is 0.95^y/x at 50% miss/dodge/parry.

examples with a 2.0 attack speed weapon, making y=8:
mob attack speed 1.0: chance to not have reckoning up is 66.3% or 33.7% to have it up.
mob attack speed 2.0: chance to not have reckoning up is 81.4% or 18.6% to have it up.
mob attack speed 3.0: chance to not have reckoning up is 87.2% or 12.8% to have it up.
Once the chance to have reckoning up is determined, the math is simple. Since the swings are doubled, your damage will increase by 33.7% against a 1.0 attack speed mob.

This is where it gets scary, the highest % returns come when y=8, which only happens if weapon speed is 2.0 or higher. Thinking logically, a 1.8 speed weapon will yield the best result since you're guaranteed 4 extra swings. This is innaccurate. During reckoning, a 2.0 speed weapon will attack 8 times, and a 2.67 weapon will attack 6 times. However, if their dps is even at 100 dps: 2.0 speed 8 times = 1,600 damage. 2.67 speed 6 times = 1,600 damage.

According to my math, any weapon with a 2.0 attack speed or higher will receive the full benefit from reckoning over a large enough sample. Conversely, as the weapon speed drops below 2.0, the return from reckoning also drops. The one thing that could kill this math would be if seal of righteousness doesn't scale correctly with weapon speed, but I think it does. Unless there is a mistake somewhere in my logic, my vote is that we petition blizzard to increase continuum blade and crystalforged sword to a 2.0 attack speed.

EDIT: fixed some numbers.
Last edited by Phasek on Thu Jun 14, 2007 8:18 am, edited 1 time in total.
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Re: Reckoning/Weapon Speed (Math)

Postby Thels » Thu Jun 14, 2007 8:09 am

Phasek wrote:I pulled some of this from another post I made to make it easier on me. I'm going to prove mathematically that weapons faster than 2.0 are actually WORSE for reckoning.


That's pretty simple actually. With a 2.0 or slower weapon, your melee DPS is doubled for the full 8 seconds. With a faster weapon, your melee DPS is doubled for a shorter amount of time.
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Postby Phasek » Thu Jun 14, 2007 8:12 am

I'm on crack, I had 10 seconds and 5 swings on the brain. It's really 8 seconds and 4 swings, but the math is the same regardless.

I editted my original post with the correct numbers.
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Postby Nidal » Fri Jun 15, 2007 8:13 am

I've thought about this alot and ran my own numbers, which I didn't keep...but the end result is that Reckoning will double your damage per proc regardless of weapon speed. (ignoring for the moment procs during the first proc period).

So as far as damage/threat and Reckoning goes the weapon speed doesn't matter.

But, a faster weapon is better than a slower weapon for tanking or AoE grinding because of the procs of JoL or JoW.

A faster weapon will also have a smoother threat curve--ie. missing your target will have less negative impact on your threat generation.

It is easier with a faster weapon to spread your hits amongst multiple targets if you are in a position in which that is necessary.

For me (and I'm far from an expert) the faster the better! I'd prefer petitioning for Continuum to be 1.0.... :D I don't even consider a weapon now unless it is faster than 2.0.

Granted I'm still a lowbie but that's my 2 cents...
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Postby Brighid » Tue Jul 17, 2007 7:37 am

Well if you've got 50% avoidance, you must have around 20% parry, no? A well timed parry, or just a well timed reckoning in the swing cycle (like perhaps, just before a swing) would mean you're still getting the full charges worth with a slightly slower weapon speed.

Just putting that out there for you theory lovers. I'm partial to continuum/fast weapons myself, but I've seen the above argument on the WoW forums a bit
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Postby corc » Tue Jul 17, 2007 10:15 am

while you have a point, I'm going to stick to my fast weapons.

also, we miss/getparried/etc too, so having a faster weapon lets us have possibly another chance to use a charge of reckoning in those 8s. If you miss a slow weapon's swing you are taking out a huge chunk of your dps/threat possible from reckoning.
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Re: Reckoning/Weapon Speed (Math)

Postby jere » Tue Jul 17, 2007 11:29 am

Phasek wrote:Therefor a mob has a 95% chance to not proc reckoning per swing. Let's now assume that y is your attack speed x4, but not exceeding 8, making it the amount of seconds that reckoning will stay active. If x is the mob's attack speed, the amount of swings the mob performs per y seconds is y/x.
therefor, the chance that reckoning is not up at any given time is 0.95^y/x at 50% miss/dodge/parry.

examples with a 2.0 attack speed weapon, making y=8:
mob attack speed 1.0: chance to not have reckoning up is 66.3% or 33.7% to have it up.
mob attack speed 2.0: chance to not have reckoning up is 81.4% or 18.6% to have it up.
mob attack speed 3.0: chance to not have reckoning up is 87.2% or 12.8% to have it up.
Once the chance to have reckoning up is determined, the math is simple. Since the swings are doubled, your damage will increase by 33.7% against a 1.0 attack speed mob.

This is where it gets scary, the highest % returns come when y=8, which only happens if weapon speed is 2.0 or higher. Thinking logically, a 1.8 speed weapon will yield the best result since you're guaranteed 4 extra swings. This is innaccurate. During reckoning, a 2.0 speed weapon will attack 8 times, and a 2.67 weapon will attack 6 times. However, if their dps is even at 100 dps: 2.0 speed 8 times = 1,600 damage. 2.67 speed 6 times = 1,600 damage.

According to my math, any weapon with a 2.0 attack speed or higher will receive the full benefit from reckoning over a large enough sample. Conversely, as the weapon speed drops below 2.0, the return from reckoning also drops. The one thing that could kill this math would be if seal of righteousness doesn't scale correctly with weapon speed, but I think it does. Unless there is a mistake somewhere in my logic, my vote is that we petition blizzard to increase continuum blade and crystalforged sword to a 2.0 attack speed.

EDIT: fixed some numbers.


The problem with this is that you are looking at the damage increase, but forgetting about the window of time it occurs in. That 33.66% dmg increase occurs over an 8 second window so your dps increased by 33.66/8 = 4.21% increase in dps for a 2.0 speed weapon.

If you run the calculations for a 1.8 speed weapon with a mob attacking at 1.0 speed you get:

95^(1.8*4/10) = 69.12% chance to be down = 30.88% chance to be up.

That's 30.88% more dmg for a 1.8s weapon which, on the surface looks less than a 2.0s weapon. However, if you divide it by the window of time to get the dps: 30.88/(1.8*4) = 4.29% dps increase which is better than a 2.0 weapon.

Take a 1.6 speed weapon:
95^(1.6*4/1.0) = 72.02% chance to be down = 27.98% chance to be up.

Increase in dps = 27.98/(1.6*4) = 4.37% increase in dps.

So while yes, you get a bigger dmg output over your timeframe, because the time frame is longer, the overall dps is lower the higher your weapon speed. The quicker your weapon, the higher your dps.
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Postby Phasek » Wed Jul 18, 2007 11:09 am

I'm not entirely sure that I understand your math on increase in DPS.

I'm seeing your equation as y/x where y is the chance to have reckoning up and x is the duration of the reckoning window. That will give you a %damage/sec which seems fine. However, you're mixing equations. When reviewing the increase on a specific reckoning window, the chance that reckoning is up is not a factor. If you're looking at damage in a specific reckoning window, damage will be doubled for that duration, so exactly a 100% bonus on attacks with double swings.

For simplicity, we'll assume 1.0 dps from the weapon. What you're really getting is 200%*1.8damage*4swings/1.8seconds*4swings which ends up being a 200% dps bonus. For 2.0, 200%*2.0damage*4swings/2.0 seconds*4swings ends up being a 200% dps bonus.

The chance to have reckoning up is applied to the increase in dps once the increase in dps for the reckoning window is already determined. The way you've set it up favors a smaller reckoning window as a hyperbolic curve. The closer the weapon speed gets to 0, the smaller the reckoning window gets (which you've set up as the divisor). Setting the limit to 0 shows the dps bonus approaching infinity.

I hadn't thought when I first made this thread that people would write it off as "well, obviously." I had heard from many sources that a weapon slightly faster than 2.0 will give the best return, which inspired me to take this project up.
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Postby jere » Wed Jul 18, 2007 12:12 pm

Phasek wrote:I'm not entirely sure that I understand your math on increase in DPS.

I'm seeing your equation as y/x where y is the chance to have reckoning up and x is the duration of the reckoning window. That will give you a %damage/sec which seems fine. However, you're mixing equations. When reviewing the increase on a specific reckoning window, the chance that reckoning is up is not a factor. If you're looking at damage in a specific reckoning window, damage will be doubled for that duration, so exactly a 100% bonus on attacks with double swings.

For simplicity, we'll assume 1.0 dps from the weapon. What you're really getting is 200%*1.8damage*4swings/1.8seconds*4swings which ends up being a 200% dps bonus. For 2.0, 200%*2.0damage*4swings/2.0 seconds*4swings ends up being a 200% dps bonus.

The chance to have reckoning up is applied to the increase in dps once the increase in dps for the reckoning window is already determined. The way you've set it up favors a smaller reckoning window as a hyperbolic curve. The closer the weapon speed gets to 0, the smaller the reckoning window gets (which you've set up as the divisor). Setting the limit to 0 shows the dps bonus approaching infinity.

I hadn't thought when I first made this thread that people would write it off as "well, obviously." I had heard from many sources that a weapon slightly faster than 2.0 will give the best return, which inspired me to take this project up.


Actually what I was doing was taking your math and dividing by the window of time it occurred in.

From what I read, you calculate the chance that Reckoning is up in a given swing speed window and say it provides ____.___% dmg increase. I think that is misleading as the windows you provide with the higher dmg increase are larger windows of time than the ones with a lower dmg increase. Of course larger windows of time will provide "more dmg increase"...you have more time in order to achieve reckoning procs. What was doing was taking your statistical damage increase and giving it scope (i.e. the window of time in which it occurs). I will phrase it another way. Your math in the opening post essentially says:

A 2.0 speed weapon provides an average 33.7% dmg increase over an average 8.0 second window ( 4*2.0 )
A 1.8 speed weapon provides an average 30.9% dmg increase over an average 7.2 second window( 4*1.8 ).

However your conclusion only compares the dmg increases without accounting for the differences in time it took to accumulate that "dmg increase"...i.e. the 8.0s versus the 7.2s.

Sure if you talk in absolutes, Reckoning will give 200% dmg over its window, however, you were presenting math about the "average window of time" in order to account for reckoning procs that overlap. That average window of time may or may not have a full reckoning duration in it.

I understand that the dmg goes to infinity as that is the nature of any time based rate as you reduce the denomitor towards zero: miles per hour, meters per second, shoot even gallons per flush (assuming timed flushings). However, weapon speeds are capped at some number, so there are boundary conditions to the equation anyways.

My point was simply saying "a 2.0s weapon produces 33.7% more damage and a 1.8% speed weapon produces 30.9% more damage so a 2.0 is better" is misleading and not quite correct. That 33.7% dmg happens over a larger average window of time given the way you have laid out your math and equations.
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Postby corc » Wed Jul 18, 2007 1:46 pm

All of our +Dam weapons are measured in the same DPS. (and pretty much get the same overall dps from righteousness too)

So, if one has double damage for 4x2s and the other has double damage for 4x1.8s than the it's a no brainer. Dam/s * (reckoning time) = damage increase. 100 Dam/s * 8s = 800 Dam; where, 100 Dam/s * 7.2s = 720 Dam.

Say you have a 2.2s weapon, reckoning will last 6.6s because it won't get the last charge. 100 Dam/s * 6.6s = 660 Dam.

Say you have a 1.6s weapon (arena mace), reckoning will last 6.4s with room for one more swing. 100 Dam/s * 6.2s = 620 Dam with the added bonus of getting a second try on one swing!

In a perfect world, get a 2.0 weapon, get a 1.8 if you can't and get a 1.6 instead of a 2.2 or greater if you find out your world isn't perfect and you miss a swing with your slow weapon and only use 2 procs instead of 3...
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Postby jere » Wed Jul 18, 2007 1:57 pm

Ok, I have been playing with your math and post a bit and I think I understand where you are coming from. I think you needed to word it different as I read what you were saying very differently than what you posted.

What you really want to calculate is the probability that a particular swing consumes a reckoning charge for each weapon speed and enemy speed combination and then compare them over a 10 minute window (or any set duration) to see how many reckoning procs you get in that window for each combination of weapon speed and enemy speed.

The math you started with was a good start, but you need to consider one thing. In order for a particular swing to consume a reckoning charge, Reckoning had to be proc'd no more than 3 swings ago. When you chose 4 that made me think you were trying to find the probability that reckoning is up over a window of 4 swings since it would be up on the current swing if it proc'd 4 swings ago. I will modify your equation to accomodate this:

0.95^(weapons_speed*3/enemy_speed) = P[this swing can consume a reckoning charge]

Mind you a swing is a time slice relative to your weapon speed (2s for a 2.0 speed weapon, 1.8s for a 1.8 speed weapon, etc.)

Once you have this number, you have the probability that this current swing will consume a reckoning charge (well almost). One thing it doesn't take into consideration is the fact you yourself can have your attack missed, dodged, parried, etc..but this will be a static modifier and will favor faster weapons as you have more time to make up for a miss if your weapon is faster than 2.0s (a 1.6s weapon has 5 chances to proc reckoning for example so it would actually have a better chance at consuming a reckoning charge than I am showing than a slower weapon).

Over a 10 minute fight (600 seconds) a 2.0 speed weapon can hit 300 times and a 1.6 speed weapon can hit 375 times. Each of those times has a respective "chance to consume a reckoning charge". These are the tables I came up with in Excel:



Code: Select all
This table calculates the probability that a given swing            
will consume a Reckoning charge (that reckoning has            
proc'd in the last 3 swings.            

          1.0           1.5           2.0           2.5
1.4   0.193806779   0.133784213   0.102117368   0.082564255
1.5   0.206117509   0.142625000   0.108999163   0.088193908
1.6   0.218240252   0.151375556   0.115828213   0.093789015
1.7   0.230177878   0.160036803   0.122604923   0.099349790
1.8   0.241933214   0.168609651   0.129329692   0.104876441
1.9   0.253509045   0.177095002   0.136002919   0.110369180
2.0   0.264908109   0.185493750   0.142625000   0.115828213



Notice here the increasing chance that a slower weapon will consume a reckoning charge but also bear in mind that a slower weapon will have less hits and take up a longer time slice that a faster weapon so that evens out.

Now the total number of reckoning charges consumed in a 10 minute fight:
Code: Select all
This table calculates the number of reckoning charges            
consumed over a 10 minute fight.            
            

          1.0           1.5           2.0           2.5
1.4   83.06004797   57.33609110   43.76458615   35.38468086
1.5   82.44700355   57.05000000   43.59966529   35.27756319
1.6   81.84009444   56.76583366   43.43558004   35.17088078
1.7   81.23925110   56.48357745   43.27232562   35.06463164
1.8   80.64440483   56.20321684   43.10989728   34.95881378
1.9   80.05548775   55.92473745   42.94829031   34.85342524
2.0   79.47243281   55.64812500   42.78750000   34.74846403



Notice here that faster weapons actually get more reckoning procs. The equation used to get these numbers is:

duration/weapon_speed*P[a reckoning charge was consumed]

duration was 600 seconds for all of the above.

The above table shows your assertion that a slower weapon will indeed get a better DPS boost than a faster weapon from reckoning because even though faster weapons get slightly more procs they have less total hits and the % of hits that get double damage will be more for slower weapons.

Anyways, sorry again for the confusion, I totally misread your post at the beginning.
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Postby kurros » Wed Jul 18, 2007 1:59 pm

jere wrote:My point was simply saying "a 2.0s weapon produces 33.7% more damage and a 1.8% speed weapon produces 30.9% more damage so a 2.0 is better" is misleading and not quite correct. That 33.7% dmg happens over a larger average window of time given the way you have laid out your math and equations.


Your view is somewhat skewed though.

Over the course of a 5 minute fight, you might proc reckoning 10 times. It will be 10 times regardless of your weapon speed, because it procs upon being hit not upon hitting.

The 2.0 weapon will produce 33.7% more damage per proc while the 1.8 will produce 30.9% more per proc. The 1.8 will do it's damage "faster" in that it will do it before the proc would wear off otherwise, but when you are looking at overall damage that is NOT an advantage. Overall, over the whole fight, the 2.0 will benefit more from reckoning.
Anyway, lets just agree to disagree. Go away, or I'll just start reporting you to the mods for being a troll. In exchange, I'll stop pointing out your stupid in public.
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Postby jere » Wed Jul 18, 2007 2:03 pm

corc wrote:All of our +Dam weapons are measured in the same DPS. (and pretty much get the same overall dps from righteousness too)

So, if one has double damage for 4x2s and the other has double damage for 4x1.8s than the it's a no brainer. Dam/s * (reckoning time) = damage increase. 100 Dam/s * 8s = 800 Dam; where, 100 Dam/s * 7.2s = 720 Dam.

Say you have a 2.2s weapon, reckoning will last 6.6s because it won't get the last charge. 100 Dam/s * 6.6s = 660 Dam.

Say you have a 1.6s weapon (arena mace), reckoning will last 6.4s with room for one more swing. 100 Dam/s * 6.2s = 620 Dam with the added bonus of getting a second try on one swing!


This is a bad way to look at it though.

Over a 10 minute fight (600s) both weapons will do exactly 600*100 = 60000 dmg regardless of weapon speed (Assuming both are the same dps and have no misses, dodges, parries, etc.). Over 8 seconds a 1.8 speed weapon will still do 800 damage (on average) and over 7.2 seconds a 2.0 speed weapon will do 720 dmg (on average) if both are 100 dps weapons.
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Postby jere » Wed Jul 18, 2007 2:11 pm

kurros wrote:Your view is somewhat skewed though.

Over the course of a 5 minute fight, you might proc reckoning 10 times. It will be 10 times regardless of your weapon speed, because it procs upon being hit not upon hitting.

The 2.0 weapon will produce 33.7% more damage per proc while the 1.8 will produce 30.9% more per proc. The 1.8 will do it's damage "faster" in that it will do it before the proc would wear off otherwise, but when you are looking at overall damage that is NOT an advantage. Overall, over the whole fight, the 2.0 will benefit more from reckoning.


Actually your view is somewhat skewed. When reckoning procs it does 100% more damage every proc regardless. The 33.7% seems to indicate the chance that reckoning had proc'd in the last 4 swings period. This doesn't tell the dmg increase though because if the reckoning proc happend 4 swings ago only then the current swing has no reckoning charge to consume. He should have measured it from the last 3 swings instead (but even this has holes as there is no way to synchronize the times you swing with the times the enemy swings).

Also faster weapons will consume more charges over time than slower weapons, so both weapons won't necessarily proc 10 times. One might proc 11 and the other 10 for example.

Also, please notice that I said:
Jere wrote:Ok, I have been playing with your math and post a bit and I think I understand where you are coming from. I think you needed to word it different as I read what you were saying very differently than what you posted.


and

Jere wrote:Anyways, sorry again for the confusion, I totally misread your post at the beginning.


So I understand that I made a mistake and misread his post. That kind of stuff happens as I am human and am prone to mistakes. In either case I apologized for my hastiness to him.
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Postby Phasek » Wed Jul 18, 2007 2:28 pm

Ok, maybe I wasn't entirely clear, so I'm going to try this again.

The % I came up with is based purely on the mob's attack speed. The calculations show the chance that reckoning is NOT up at any given moment. I then found the chance that reckoning is up by subtracting the calculation from 100.

When calculating the chance that reckoning is not up, that's when the paladin's weapon speed is factored in. If the paladin uses only a 1.5 speed weapon, for example, reckoning will last 4 charges * 1.5 seconds = 6 seconds. Therefor, calculating the chance that reckoning is not up at any given moment with a 1.5 speed weapon calculates the chance that reckoning didn't proc in the last 6 seconds. The cap for the calculation is 8 seconds since 8 seconds is the maximum duration of a single reckoning proc.

Since a paladin doubles his damage when reckoning is up, his damage is increased (on average) exactly the percentage that reckoning is up during his fight. Blocks, parries, misses, dodges, etc can be thrown out the window, since the chance of any of these things is equal between any attack and the extra attack from reckoning as well. The extra attack will occur even if the first attack misses. Therefor, a weapon that reduces the time that reckoning is up per proc will only reduce the overall return from reckoning over an extended period such as in a boss fight.

The argument about a 2.2 weapon speed weapon is an interesting one, saying that 3 attacks will cover 6.6 seconds. However, the timing of reckoning procing is completely random, and the paladin's first swing could come 1.0 seconds after reckoning procced, therefor providing him all 4 attacks in a span on 7.6 seconds. Over a huge sample size, the chance that a paladin receives 3 or 4 extra attacks will average out to provide the exact same damage boost as a 2.0 speed weapon.

One final note, since I can forsee this argument coming. The logic in the last paragraph does not pertain to the amount of time that reckoning is up. A 1.5 speed weapon will have attacked exactly 4 times in a 6 second window regardless of when the first attack came. I hope this clears everything up.
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