Calculating Parry Chance in Cataclysm
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Re: Calculating Parry Chance in Cataclysm
Ok, more coffee imbibed. Just doublechecking your math (<edit> Switched back to leftmultiplication notification to be consistent with the wikipedia article, hopefully I didn't make any typos in the process):
We're solving the equation
qT = qI
where (using matlab notation here, so [a b; c d] represents a 2x2 matrix with rows [a b] and [c d].)
q=[q1; q2] is the state vector describing the system (probability q1 of being in S1, q2 of being in S2)
T=[T_11 T_12; T_21 T_22] is the transition matrix with elements T_ij representing the probability of transitioning from Si to Sj
I=[1 0; 0 1] is the identity matrix
In this form, your version gives us the transition vector:
Solving the equation given by q(TI)=0 gives me one expression:
q2=P*q1
The other expression we need is the constraint q1+q2=1 (conservation of probability). Rearranging that to q2=1q1, we have:
q1=1q1=P*q1
1=(P+1)*q1
q1=1/(P+1)
and then substituting that back into the first equation gives
q2=P*q1=P/(P+1)
Consistent with your findings.
My scenario only changes things slightly. State 1 is a normal attack, but State 2 is where you take reduced damage from a hit or block. In essence, State 1 is when you don't have the parrywall buff, state 2 is when you do. Since the only things that remove the buff are a block or hit,T_22 (the chance of staying in state 2) is M+D+P, which I'll call A for short. Similarly, T_21 is just B+H = 1MDP = 1A. So the matrix looks like this:
Solving this equation in the same fashion gives the following steadystate probabilities:
Sanity check:
Since 1MD can't go to zero, the first expression is strictly bounded ( 1A = 1MDP is always less than 1MD)
Since P=1MDBH, the second expression is strictly bounded (1MDBH always less than 1MD).
In the steadystate, q2 should describe the probability that you're under the effects of the parrywall debuff for any given incoming attack. So if we ignore blocking and other mitigation effects for the moment, the average damage reduction on a regular hit will be (10.5*q2). It's a little more difficult to calculate for a block, since we'd have to make assumptions on how the DR's interact (multiplicatively vs. additively vs. exclusively).
We're solving the equation
qT = qI
where (using matlab notation here, so [a b; c d] represents a 2x2 matrix with rows [a b] and [c d].)
q=[q1; q2] is the state vector describing the system (probability q1 of being in S1, q2 of being in S2)
T=[T_11 T_12; T_21 T_22] is the transition matrix with elements T_ij representing the probability of transitioning from Si to Sj
I=[1 0; 0 1] is the identity matrix
In this form, your version gives us the transition vector:
 Code: Select all
 1P P 
T =  1 0 
Solving the equation given by q(TI)=0 gives me one expression:
q2=P*q1
The other expression we need is the constraint q1+q2=1 (conservation of probability). Rearranging that to q2=1q1, we have:
q1=1q1=P*q1
1=(P+1)*q1
q1=1/(P+1)
and then substituting that back into the first equation gives
q2=P*q1=P/(P+1)
Consistent with your findings.
My scenario only changes things slightly. State 1 is a normal attack, but State 2 is where you take reduced damage from a hit or block. In essence, State 1 is when you don't have the parrywall buff, state 2 is when you do. Since the only things that remove the buff are a block or hit,T_22 (the chance of staying in state 2) is M+D+P, which I'll call A for short. Similarly, T_21 is just B+H = 1MDP = 1A. So the matrix looks like this:
 Code: Select all
 1P P 
T =  1A A 
Solving this equation in the same fashion gives the following steadystate probabilities:
 Code: Select all
q1=(1A)/(P+(1A)) = (1A)/(1MD)
q2=P/(P+(1A)) = P/(1MD)
Sanity check:
Since 1MD can't go to zero, the first expression is strictly bounded ( 1A = 1MDP is always less than 1MD)
Since P=1MDBH, the second expression is strictly bounded (1MDBH always less than 1MD).
In the steadystate, q2 should describe the probability that you're under the effects of the parrywall debuff for any given incoming attack. So if we ignore blocking and other mitigation effects for the moment, the average damage reduction on a regular hit will be (10.5*q2). It's a little more difficult to calculate for a block, since we'd have to make assumptions on how the DR's interact (multiplicatively vs. additively vs. exclusively).
"Theck, Bringer of Numbers and Pounding Headaches," courtesy of GrehnSkipjack.
MATLAB 5.x, Simcraft 6.x, Call to Arms 6.0, Talent Spec & Glyph Guide 5.x, Blog: Sacred Duty
MATLAB 5.x, Simcraft 6.x, Call to Arms 6.0, Talent Spec & Glyph Guide 5.x, Blog: Sacred Duty

theckhd  Moderator
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Re: Calculating Parry Chance in Cataclysm
I commend your math but why are you bothering before we've even seen how they plan to implement parry in cataclysm? Your dedication is great but you're basically saying "hi guys this is how I guess it will work, and heres the math proving my guess."

Soralin  Posts: 107
 Joined: Wed Jan 14, 2009 8:43 pm
Re: Calculating Parry Chance in Cataclysm
Soralin wrote:I commend your math but why are you bothering before we've even seen how they plan to implement parry in cataclysm? Your dedication is great but you're basically saying "hi guys this is how I guess it will work, and heres the math proving my guess."
I can't answer for Slept, but I wasn't going to bother working through the math until we had beta details. However, when someone posts an interesting math problem that intrigues me, I get easily distracted.
In addition, this method should be applicable no matter what mechanic they settle on for Parry, so having the foundations written out somewhere ahead of time doesn't hurt. Once we have details, we just tweak the T matrix to match the appropriate transition mechanisms and we get our answer.
"Theck, Bringer of Numbers and Pounding Headaches," courtesy of GrehnSkipjack.
MATLAB 5.x, Simcraft 6.x, Call to Arms 6.0, Talent Spec & Glyph Guide 5.x, Blog: Sacred Duty
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theckhd  Moderator
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Re: Calculating Parry Chance in Cataclysm
Soralin wrote:I commend your math but why are you bothering before we've even seen how they plan to implement parry in cataclysm? Your dedication is great but you're basically saying "hi guys this is how I guess it will work, and heres the math proving my guess."
Answering the easy question first while I digest Theckhd's posts. Why not? This stemmed from me wondering how parry would actually work in Cata. Once I had an answer, there didn't seem to be any point in just sitting on the the information.
As Theckhd mentioned, the technique should be applicable for nearly any version of parry Blizzard introduces. Perhaps it will give the Rawr guys a leg up on making their 4.0 models.

Slept  Posts: 72
 Joined: Fri Jul 11, 2008 7:46 am
Re: Calculating Parry Chance in Cataclysm
Slept wrote:Answering the easy question first while I digest Theckhd's posts.
Don't worry too much about it. They basically boil down to:
Your math is right
But I think that the model isn't likely to be the way it's implemented
Here's a model I think is more likely, which is a minor variant of yours (different probability paths and a slightly different meaning to each state, basically)
Here's the version of the math that applies to my model
"Theck, Bringer of Numbers and Pounding Headaches," courtesy of GrehnSkipjack.
MATLAB 5.x, Simcraft 6.x, Call to Arms 6.0, Talent Spec & Glyph Guide 5.x, Blog: Sacred Duty
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theckhd  Moderator
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Re: Calculating Parry Chance in Cataclysm
That's certainly a valid interpretation, Theckhd. The main benefit of a parrywallbuff system is that block can be handled much more reasonably. I touched on it earlier, but, with the version I proposed, block does become a bit of a mess. Critical blocks will be a minimum of 60% DR (from what we've been told), which is stronger than parry's 50% DR.
This gives two options:
1) Critical blocks (and all blocks for geared paladins that block more than 50% damage) stay on the state 2 combat table and get higher priority than parry. This will work in a sensible and intuitive fashion, but does require dynamic reordering of the combat table. I have no idea if Blizz can dynamically reorder combat tables to account for this.
2) Parry overrides block all the time. A quick and dirty way of dealing with block and parry clashing. Not something I would like to see happen.
The parrywallbuff system does have some holes too:
1) I'm pretty sure Blizz sold the parry mechanic as being 50% DR on two consecutive attacks. Of course they are free to change that, but it was a big factor in why I chose the system I did.
2) The difference between parrying and parrywalling (heh) is only the consumption of the parrywall buff. Looking at your examples on the previous page:
The third example shows two parries with a miss in between them. The first parry would move the tank to state 2 and grant them the parrywall buff. In your example, it's still possible to parry with the parrywall buff. A cleaner solution would be to have parrywall completely replace parry in state 2.
This gives two options:
1) Critical blocks (and all blocks for geared paladins that block more than 50% damage) stay on the state 2 combat table and get higher priority than parry. This will work in a sensible and intuitive fashion, but does require dynamic reordering of the combat table. I have no idea if Blizz can dynamically reorder combat tables to account for this.
2) Parry overrides block all the time. A quick and dirty way of dealing with block and parry clashing. Not something I would like to see happen.
The parrywallbuff system does have some holes too:
1) I'm pretty sure Blizz sold the parry mechanic as being 50% DR on two consecutive attacks. Of course they are free to change that, but it was a big factor in why I chose the system I did.
2) The difference between parrying and parrywalling (heh) is only the consumption of the parrywall buff. Looking at your examples on the previous page:
theckhd wrote:MDHPW (miss, dodge, hit, parry, regular attack that consumes parrywall debuff)
MDHPMDW (this time, you avoid 2 attacks after the parry, so the parrywall carries over)
MDHPMPDWM?.
The third example shows two parries with a miss in between them. The first parry would move the tank to state 2 and grant them the parrywall buff. In your example, it's still possible to parry with the parrywall buff. A cleaner solution would be to have parrywall completely replace parry in state 2.

Slept  Posts: 72
 Joined: Fri Jul 11, 2008 7:46 am
Re: Calculating Parry Chance in Cataclysm
Slept wrote:The third example shows two parries with a miss in between them. The first parry would move the tank to state 2 and grant them the parrywall buff. In your example, it's still possible to parry with the parrywall buff. A cleaner solution would be to have parrywall completely replace parry in state 2.
Or have miss/dodge consume the parrywall buff (which is liable to be invisible, even if implemented in that fashion).
 Meloree
 Maintankadonor
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Re: Calculating Parry Chance in Cataclysm
Slept wrote:A cleaner solution would be to have parrywall completely replace parry in state 2.
That's another possibility, yes.
Meloree wrote:Or have miss/dodge consume the parrywall buff (which is liable to be invisible, even if implemented in that fashion).
I intentionally didn't have it do that, because it devalues parry somewhat (I think we've had this discussion before, in fact). By letting the buff persist until a hit comes through, you don't waste a parry effect on an attack that would have been avoided anyhow.
Of course, unless you let the parry buff stack, you'll run into situations where you "waste" a parry on another parry, like the example I gave.
"Theck, Bringer of Numbers and Pounding Headaches," courtesy of GrehnSkipjack.
MATLAB 5.x, Simcraft 6.x, Call to Arms 6.0, Talent Spec & Glyph Guide 5.x, Blog: Sacred Duty
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theckhd  Moderator
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Re: Calculating Parry Chance in Cataclysm
well we will just have to see how it works out in beta i guess.
I wonder how the attack table will look for the second hit.
Good point about parry being chosen over 50%+ improved blocks, that sounds very worrying.
I wonder how the attack table will look for the second hit.
Good point about parry being chosen over 50%+ improved blocks, that sounds very worrying.

Awyndel  Posts: 672
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 Location: The Netherlands
Re: Calculating Parry Chance in Cataclysm
I think the important thing to take from this is that in terms of total overall damage taken, which may in fact be relevant again, parry is not as good as dodge, because parry will be "clipped" by a followon parry/dodge/miss.
If I was designing the hit table, I would use the same hit table for parry wall as I would for a normal hit, but I would cap the incoming damage at 50% if parry wall was active. So a hit, parry, or block would reduce damage by 50%, a critical block by 60%, and a dodge or a miss wouldn't take any damage. In any event, the "parry wall" would be used.
Intuitively, as parry% approaches 100%, the damage reduction from parry approaches 50% because every hit is reduced by 50%.
So it is fairly safe to say that parry will have a very steep diminishing returns, because it will quickly go from 100% damage reduced at no avoidance to its 50% eventual value. Every bit of parry you stack increases the chance of a followon parry that would "clip" the first one. Every other bit of avoidance as well as block crit % would decrease the overall value of parry by increasing the chance of "clipping" your parry.
In fact it seems to me that at desirable levels of avoidance, we can devalue parry to around 60%65% as effective as Dodge for reducing overall damage taken.
Depending on the healing model, it may be more or less desirable than that. In the current healing model, the damage reduction is better than avoidance. In Cataclysm, there is a lot of talk about not topping off tanks, which may mean they are attempting to design to reactive healing rather than proactive healing, and they are trying to make mana meaningful again. If that is the case, then reducing the healing requirement will make total damage taken relevant, and avoidance will be king.
There are a lot of assumptions in there and not a lot of math, but the purpose of the post was to draw some conclusions from the suppositions thus far, and to present my own take on how the hit table will perform. I will be quite surprised if the final model is much different.
If I was designing the hit table, I would use the same hit table for parry wall as I would for a normal hit, but I would cap the incoming damage at 50% if parry wall was active. So a hit, parry, or block would reduce damage by 50%, a critical block by 60%, and a dodge or a miss wouldn't take any damage. In any event, the "parry wall" would be used.
Intuitively, as parry% approaches 100%, the damage reduction from parry approaches 50% because every hit is reduced by 50%.
So it is fairly safe to say that parry will have a very steep diminishing returns, because it will quickly go from 100% damage reduced at no avoidance to its 50% eventual value. Every bit of parry you stack increases the chance of a followon parry that would "clip" the first one. Every other bit of avoidance as well as block crit % would decrease the overall value of parry by increasing the chance of "clipping" your parry.
In fact it seems to me that at desirable levels of avoidance, we can devalue parry to around 60%65% as effective as Dodge for reducing overall damage taken.
Depending on the healing model, it may be more or less desirable than that. In the current healing model, the damage reduction is better than avoidance. In Cataclysm, there is a lot of talk about not topping off tanks, which may mean they are attempting to design to reactive healing rather than proactive healing, and they are trying to make mana meaningful again. If that is the case, then reducing the healing requirement will make total damage taken relevant, and avoidance will be king.
There are a lot of assumptions in there and not a lot of math, but the purpose of the post was to draw some conclusions from the suppositions thus far, and to present my own take on how the hit table will perform. I will be quite surprised if the final model is much different.
 Koatanga
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Re: Calculating Parry Chance in Cataclysm
The mistake you're making, Koatanga, is in applying the current avoidance system to the 4.0 system, having no guarantees that it will be the same. The following vital questions must be answered before anyone can assign a value to parry:
1) How steep will DR on Dodge be for 4.0?
2) What are reasonable levels of Avoidance going to look like for raiding in Cata?
3) Will Parry have lower DR than Dodge? How close will the gain be compared to Block?
4) How will the 'Parry Wall' effect function?
My gut feeling is that dmg reduction is going to become a larger priority for Cataclysm, otherwise they wouldn't be turning Parry into an alternative to Block and making Block a fixed amount of dmg reduction. I would expect to see in beta a very low return on Dodge and generally low levels of avoidance in comparison to even miss+dodge today. I would also expect that either the parry wall is the only option on the attack table or it puts up a buff that isn't consumed until a full attack/reg block is taken. This is more in line with Blizzard's stated philosophy going into Cata of 'don't top off the tank.' It seems to me that we're meant to mitigate as many hits as possible while avoiding a few, rather than trying to avoid most of the hits while maybe mitigating the rest for a small fraction. This will make for more predictable damage which means healers will know exactly when to cast a heal.
1) How steep will DR on Dodge be for 4.0?
2) What are reasonable levels of Avoidance going to look like for raiding in Cata?
3) Will Parry have lower DR than Dodge? How close will the gain be compared to Block?
4) How will the 'Parry Wall' effect function?
My gut feeling is that dmg reduction is going to become a larger priority for Cataclysm, otherwise they wouldn't be turning Parry into an alternative to Block and making Block a fixed amount of dmg reduction. I would expect to see in beta a very low return on Dodge and generally low levels of avoidance in comparison to even miss+dodge today. I would also expect that either the parry wall is the only option on the attack table or it puts up a buff that isn't consumed until a full attack/reg block is taken. This is more in line with Blizzard's stated philosophy going into Cata of 'don't top off the tank.' It seems to me that we're meant to mitigate as many hits as possible while avoiding a few, rather than trying to avoid most of the hits while maybe mitigating the rest for a small fraction. This will make for more predictable damage which means healers will know exactly when to cast a heal.
 Raivnor
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Re: Calculating Parry Chance in Cataclysm
Raivnor wrote:The mistake you're making, Koatanga, is in applying the current avoidance system to the 4.0 system, having no guarantees that it will be the same. The following vital questions must be answered before anyone can assign a value to parry:
1) How steep will DR on Dodge be for 4.0?
2) What are reasonable levels of Avoidance going to look like for raiding in Cata?
3) Will Parry have lower DR than Dodge? How close will the gain be compared to Block?
4) How will the 'Parry Wall' effect function?
My gut feeling is that dmg reduction is going to become a larger priority for Cataclysm, otherwise they wouldn't be turning Parry into an alternative to Block and making Block a fixed amount of dmg reduction. I would expect to see in beta a very low return on Dodge and generally low levels of avoidance in comparison to even miss+dodge today. I would also expect that either the parry wall is the only option on the attack table or it puts up a buff that isn't consumed until a full attack/reg block is taken. This is more in line with Blizzard's stated philosophy going into Cata of 'don't top off the tank.' It seems to me that we're meant to mitigate as many hits as possible while avoiding a few, rather than trying to avoid most of the hits while maybe mitigating the rest for a small fraction. This will make for more predictable damage which means healers will know exactly when to cast a heal.
What you are saying addresses the "don't top off the tanks" part, but not the "let's make mana count" part. If mana really matters, then avoidance will matter quite a lot as well since an avoided hit doesn't require any mana use.
I believe the changes to parry are to do with making it easier to balance tanks because it gives DKs a damage reduction mechanic like the other plate classes have and make them less spiky.
I do agree that we will be mitigating more frequently and avoiding less. The change in parry assures that  I'll be dropping some 20% avoidance from that alone. I suspect that instead of dodge suffering from severe DR (bears would be screwed), it will just be a bit scarce on plate gear.
But the parry change I see as a way to equalise DKs. They will have more parry than we do, I imagine, to make up for their lack of block. Our base block is 30%, but with Mastery it will be more. Warriors will get more critical block with mastery, so the total damage reduction should be similar to our block and similar to extra classbenefit DKs will get from parry. That should give all three plate classes similar total damage mitigation with slightly different flavours.
If that's the way it is heading, we actually come out on top, being the least spiky of the plate tanks.
 Koatanga
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Re: Calculating Parry Chance in Cataclysm
It seems to me that this conversation has neglected to take into account a certain blue post:
That sounds to me like "if you parry a hit, that hit is reduced by 50% and the next hit can be dodged/parried/blocked/hit with an additional 50% reduction".
Also, Blizz moves to a direction where they want to simplify the game mechanics a bit (like removing stats from gear). Having multiple attack tables depending on situations is somewhat complicated that would interest only the experienced players and confuse new players.
Ghostcrawler wrote:The relative value of dodge vs. parry is something we'll have to play with. My gut reaction is that parry needs to be cheaper since avoiding 100% of one hit is more valuable than avoiding 50% of two hits, but I'm not sure how much cheaper. Avoiding spikiness (which dodge will contribute to) also has value, and if the second hit after a parry is dodged instead (such that you lose the parry "charge") that plays into the cost as well.
That sounds to me like "if you parry a hit, that hit is reduced by 50% and the next hit can be dodged/parried/blocked/hit with an additional 50% reduction".
Also, Blizz moves to a direction where they want to simplify the game mechanics a bit (like removing stats from gear). Having multiple attack tables depending on situations is somewhat complicated that would interest only the experienced players and confuse new players.

Marblehead  Posts: 202
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Re: Calculating Parry Chance in Cataclysm
Marblehead wrote:It seems to me that this conversation has neglected to take into account a certain blue post:Ghostcrawler wrote:The relative value of dodge vs. parry is something we'll have to play with. My gut reaction is that parry needs to be cheaper since avoiding 100% of one hit is more valuable than avoiding 50% of two hits, but I'm not sure how much cheaper. Avoiding spikiness (which dodge will contribute to) also has value, and if the second hit after a parry is dodged instead (such that you lose the parry "charge") that plays into the cost as well.
That sounds to me like "if you parry a hit, that hit is reduced by 50% and the next hit can be dodged/parried/blocked/hit with an additional 50% reduction".
Also, Blizz moves to a direction where they want to simplify the game mechanics a bit (like removing stats from gear). Having multiple attack tables depending on situations is somewhat complicated that would interest only the experienced players and confuse new players.
Well, it's not really a separate attack table. The attack table is generated dynamically  if you gain 100 defense rating from a proc, your avoidance chances go up immediately and stay that way until the buff drops off. Everything in this thread would work the same way in that context, in that after a parry you'd have a "buff" that pushed block and hit off the table (i.e. +100% parry). That buff could be visible or invisible depending on how they coded it, though visible makes the most sense to me.
The GC quote does indicate that their design model at that point had the "parrywall" buff become active after a parry, but had it being consumed on the next attack table roll regardless of outcome. However, it's a glimpse of the mechanic very early in the design process, and it wouldn't surprise me if they found that it was more convenient or consistent to make the buff carry over than to reduce the itemization cost of parry to compensate for missed opportunities.
"Theck, Bringer of Numbers and Pounding Headaches," courtesy of GrehnSkipjack.
MATLAB 5.x, Simcraft 6.x, Call to Arms 6.0, Talent Spec & Glyph Guide 5.x, Blog: Sacred Duty
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theckhd  Moderator
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Re: Calculating Parry Chance in Cataclysm
It does seem to indicate, however, that Parry will become cheaper than Dodge per point, or at the very least similar. This makes sense when you consider this Block 2.0. It is a little out of the purview of this forum, but as an aside DKs already have a block like mechanic in Blade Barrier and it wouldn't surprise me if they tuned that to be a little more in line with block, rather than have DKs with enough parry to compensate for their lack of block.
 Raivnor
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 Joined: Sat Jun 21, 2008 12:53 pm
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