Calculating Parry Chance in Cataclysm

[Slept]

TL;DR

Parry will be changed to a 50% damage reduction on the next two incoming attacks in Cataclysm.

The formula to calculate your total parry chance (Ps) is:

Code: Select all

Ps = P*(2-MD)/(P+1)


Where:
P is the parry chance on your character's paper doll.
MD is the combined miss and dodge chance on your character's paper doll.



[Edit]This analysis is based off what I consider to be the most sensible implementation of Blizzard's two-attack parry mechanic. If you disagree with the assumptions below, then disregard the results of the analysis.[/Edit]

Proof

Lets assume that the combat table for a regular attack looks like so (in order of decreasing importance):

Code: Select all

Miss
Dodge
Parry
Block
Hit

For an attack following a parry, we assume hit and block will be pushed off. Since dodge and miss provide greater damage reduction than parry, they remain on the combat table giving:

Code: Select all

Miss
Dodge
Parry

Therefore, the next attack following a parry can actually end up being a dodge or miss, reducing the effective value of parry because the "guaranteed" second parry is lost.

Since miss and dodge both function in the same manner, we can combine them to simplify the analysis. We can also combine block and hit as they follow the same rules for their appearance on the combat table. This gives a simplified combat table for a regular incoming attack:

Code: Select all

Miss & Dodge
Parry
Block & Hit

For an attack following a parry the combat table will simplify to:

Code: Select all

Miss & Dodge
Parry

Since our combat table changes depending on whether we parry or not, finding the actually probability of parrying requires some work. Note: because miss and dodge do not move from the top spot in the combat table, their behaviour is always consistent. 10% dodge on a character's paper doll will always yield 10% avoidance (ignoring the effects of diminishing returns).

To calculate our parry chance, we have to use a Markov Chain. Markov Chains can be represented with a state diagram, which I am going to attempt to do in ascii.

Code: Select all

           
               _____MD_____
        ____  |            |
     MD|   _|_V__       ___|__
       |->|      |  P  |      |
          |  S1  |---->|  S2  |
     BH|->|______|     |______|
       |____| ^            |
              |____________|
                    P'


This diagram shows how we move from a neutral state (S1) to a guaranteed parry state (S2). We start in S1. If the first incoming attack is either a miss, dodge, block, or hit, then we stay in S1. If the incoming attack is parried, we move to S2 for the next incoming attack. In S2, the incoming attack can either miss, dodge, or parry. Regardless of what happens, we move back to S1.

Since we are only concerned with parry, and block and hit only ever affect us in S1 and can never move us to S2, we can simplify the state diagram to:

Code: Select all

           
               _____MD_____
        ____  |            |
   MDBH|   _|_V__       ___|__
       |->|      |  P  |      |
          |  S1  |---->|  S2  |
          |______|     |______|
              ^            |
              |____________|
                    P'


P* indicates the probability of of a parry from S2. This can be defined as:

Code: Select all

P' = 1 - MD

In order to determine our parry probability, we need to determine a transition matrix (T) for our Markov Chain. Since we're only interested in parry, we're going to define the transition matrix in terms of our parry chance. P will be our chance to parry, (1-P) will be our chance to NOT parry.

Code: Select all

         S1   S2 
T = S1 | 1-P  P |
    S2 | 1    0 |


In order to calculate our parry chance, we need to calculate the eiganvector (q) of the transition matrix.

Code: Select all


T = | 1-P  P |
    | 1    0 |

qT = qI
q(T - I) = 0

q|| 1-P  P | - | 1  0 || = 0
|| 1    0 |   | 0  1 ||

q| -P   P | = 0
|  1  -1 |


Solving the simultaneous equations, where q1 is the steady state probability of being in S1 and q2 is the steady state probability of being in S2. Since q1 and q2 are probabilities: q1 + q2 = 1.

Code: Select all

[q1 q2]| -P  P  | = 0
       |  1  -1 |

q1 = 1/(P+1)
q2 = P/(P+1)


Our steady state parry chance (Ps) is given by the probability of being in S1 multplied by the probability of parrying while in S1 plus the probability of being in S2 multiplied by the probability of parrying while in S2. In equation form:

Code: Select all

Ps = q1*P + q2*P'
Ps = P/(P+1) + (1-MD)*P/(P+1)
Ps = P*(2-MD)/(P+1)


[Meloree]

I could have just gotten lost in the math, but is there any reason that from state S2 you assume that you couldn't parry again - staying in S2? Perhaps that parry would lead to the hit being 0.25% sized? They've stated that parry reduces the current hit and the next hit by 50%, but not how that interacts with the combat table on the next hit. That said, I'm not a math guy, and it may not materially effect the results.

Just intuitively speaking, 1% parry would seem to be to be necessarily less total damage reduction than 1% dodge, which is all well and good. However, in bursty scenarios I would personally value the parry significantly more highly. I would much rather take 2 0.5 strength hits than 1 fullsize hit and one dodge/miss. I think the relative value of the two stats will depend somewhat on the type of fight we're facing, and how well Blizzard actually implements "large healthpools relative to heal size and hit size". Total Damage Reduction could be the new name of the game, "damage smoothing" could be the new metric, or we could just end up forced to stick with good old stamina stacking for extremely bursty bosses.

[Slept]

I could have just gotten lost in the math, but is there any reason that from state S2 you assume that you couldn't parry again - staying in S2? Perhaps that parry would lead to the hit being 0.25% sized? They've stated that parry reduces the current hit and the next hit by 50%, but not how that interacts with the combat table on the next hit. That said, I'm not a math guy, and it may not materially effect the results.

That would set off a chain of almost guaranteed parries, which would be very OP. Basically, tanks would avoid dodge like the plague because dodge and miss would be the only things that could stop their parry chain.

[Meloree]

I could have just gotten lost in the math, but is there any reason that from state S2 you assume that you couldn't parry again - staying in S2? Perhaps that parry would lead to the hit being 0.25% sized? They've stated that parry reduces the current hit and the next hit by 50%, but not how that interacts with the combat table on the next hit. That said, I'm not a math guy, and it may not materially effect the results.

That would set off a chain of almost guaranteed parries, which would be very OP. Basically, tanks would avoid dodge like the plague because dodge and miss would be the only things that could stop their parry chain.

Err, what? I'm pretty certain "not parrying" would stop the chain, too. We're still talking about a maximum of ~20% chance to parry, there's no reason to assume guaranteed chains.

Allow me to illustrate:
State 1: Normal incoming damage, normal combat table. If a parry happens, move to State 2, otherwise remain in State 1.
State 2: 50% sized hit, normal combat roll applies. If a parry happens, this hit is actually 25%, remain in State 2. If a block happens, it's a 35% sized hit, move to State 1. If a dodge/miss happens, it's a 50% sized hit, move to State 1. If a "hit" happens, it's 50% sized, move to State 1.

In other words, a possible implementation is that a parry reduces the current hit by 50%, and puts up a 1-hit Shield Wall applying to the next hit. Functioning very much like a 1-charge Redoubt proc.

[Slept]

In S2 the chance to parry is more like 80%. If a parry in S2 doesn't remove you from S2, then you will have a 80% chance to parry until you finally dodge or miss.

[Raivnor]

I also wonder if you have any blue post evidence that dodge and miss are still on the combat table for the second strike? It could be that in S2 the only option on the table is a guaranteed 50% hit.

[Slept]

An interesting conundrum arises with block. Critical block for warriors will easily exceed the 50% DR of parry. Paladins could also conceivably get their block DR over 50% through mastery and talents. In both cases, block becomes more powerful than parry, but parry is pushing block off the combat tables in S2.

A potential solution would be to modify the combat table for warriors so that critical block is not pushed off in S2. Also, paladins could retain block on the combat table only if it exceeds the effectiveness of parry. I don't know if Blizz has the ability to dynamically alter the order of combat tables based on talent and gear changes. I guess we'll see.

[Slept]

I also wonder if you have any blue post evidence that dodge and miss are still on the combat table for the second strike? It could be that in S2 the only option on the table is a guaranteed 50% hit.

It's an assumption. The same method could be used to calculate the effect on total dodge and miss chances if parry does push them off the combat table in S2.

[Meloree]

In S2 the chance to parry is more like 80%. If a parry in S2 doesn't remove you from S2, then you will have a 80% chance to parry until you finally dodge or miss.

I understand that this is the way you have chosen to model it, and how you interpret the statements Blizzard has made as to how it works, but what evidence do you have that it works this way? I have seen nothing from Blizzard confirming the mechanics, and beta isn't open, so it's not testable yet. I have provided an alternate hypothesis for a potential implementation. Are we just talking past each other? Is there some failure of terminology?

[Edit since more posts have come in]:
I'm unsure why you are presenting your interpretation as fact, unless you have access to some information that the rest of us don't have. I understand how your model works, but you're presenting it as reality. There's no reason to assume that various different DRs can't stack in S2.

[Slept]

From the first line of the proof section:

Lets assume that the combat table for a regular attack looks like so (in order of decreasing importance):

I haven't misrepresented anything. Obviously this is all subject change since Blizz can change any mechanic they like between now and Cataclysm releasing. Do you really need me to state that explicitly?

[Meloree]

From the first line of the proof section:

Lets assume that the combat table for a regular attack looks like so (in order of decreasing importance):

I haven't misrepresented anything. Obviously this is all subject change since Blizz can change any mechanic they like between now and Cataclysm releasing. Do you really need me to state that explicitly?

What I've been asking is why you choose to model it that way? What has lead you to believe that S2 should be treated as a "second guaranteed parry" rather than a 50% strength hit on the normal combat table? Is it straight up assumption/interpretation? Is the model flexible enough to accommodate other interpretations?

The way that various different avoidance and mitigation stats will interact in Cataclysm is a fascinating subject, but I'm curious about some of your fundamental assumptions - specifically about how "state 2" works. In other words, I'm asking about the part that isn't labelled as an assumption.

[Slept]

All the combat table stuff is based on assumptions of how I think it will work. I'll edit that in more clearly so we can move onto more productive areas of discussion.

To address your proposed S2:

Allow me to illustrate:
State 1: Normal incoming damage, normal combat table. If a parry happens, move to State 2, otherwise remain in State 1.
State 2: 50% sized hit, normal combat roll applies. If a parry happens, this hit is actually 25%, remain in State 2. If a block happens, it's a 35% sized hit, move to State 1. If a dodge/miss happens, it's a 50% sized hit, move to State 1. If a "hit" happens, it's 50% sized, move to State 1.

In other words, a possible implementation is that a parry reduces the current hit by 50%, and puts up a 1-hit Shield Wall applying to the next hit. Functioning very much like a 1-charge Redoubt proc.

This doesn't make much sense to me. Why is a miss or dodge the same as a hit (50% DR). That's unintuitive. A miss or dodge should always give 100% DR.

It is possible that block isn't pushed off the combat table in S2, but then tanks will be parrying AND blocking the same attack. It's just a bit weird.

[Meloree]

This doesn't make much sense to me. Why is a miss or dodge the same as a hit (50% DR). That's unintuitive. A miss or dodge should always give 100% DR.

It is possible that block isn't pushed off the combat table in S2, but then tanks will be parrying AND blocking the same attack. It's just a bit weird.

Yes, right. I didn't reread clearly, I intended that to suggest that miss/dodge would be... a miss or a dodge. Either way, moving the thing back to P1.

Either way, the way I read the whole thing would be that you "parry" one attack, but in doing so your opponent is, perhaps, off balance, and so the next hit is weaker. It shouldn't prevent the whole combat table from applying normally, with the exception of the incoming hit being smaller than normal. So I wouldn't read state 2 as implying "guaranteed parry", just "guaranteed reduced hit size".

EDIT: That's personal interpretation, neither of us can make any progress on what are fundamentally just assumptions. But the question that arises is how would stacking DRs - S2+Block or S2+Parry (remain in S2) adjust the results. Does it make parry stronger, or weaker? By a little, or a lot?

[Slept]

In that case, the only difference between your implementation and mine is whether block is included in S2. If block doesn't override parry, then this doesn't actually change the total parry chance and the result is still valid for your proposed combat table.

[theckhd]

The math looks correct for the model, but I agree with Mel that the model itself is open to a bit of interpretation. Filling the combat table of S2 with Parry and having it behave the way you did is one possibility. But I think a model like Mel's is probably more realistic - a single state, with parry giving a shield-wall buff that gets consumed on the next unavoided hit.

<edit> I should probably add, I don't think the DRs will stack. In other words, two consecutive parries won't lead to a 25% damage unavoided hit. That's just my gut feeling though. So in the following examples, I'm assuming that you can't stack the DRs of consecutive parries or blocks.

<edit2> Upon thinking about it more, I think my guess works out the same way your model does, except that you'd have to make a Miss/Dodge/Hit/Block in S2 cycle back to S1, while a parry in S2 keeps the system in S2. I'd have to have some more coffee and map it out to be sure though.

<edit3> Actually, it's not exactly equivalent. See my next post on the subject.

In other words, your combat table is always the same:
Miss
Dodge
Parry
Block
Hit

On a successful parry, you take half damage and get the "ParryWall" buff, which reduces the damage from the next unavoided melee attack by 50%. So if you represented a combat series with the bolded letters, you would get something like this:

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?

In this last example, you get two consecutive parries without an unavoided attack between them. The parrywall buff carries over anyway. The ? represents an unavoided attack whose outcome depends on whether the parrywall buff can stack or not. I think it's highly unlikely to stack for balancing reasons, so that would normally be a H or B.

As for reasoning - I think this is more likely just because it's the most similar to the current attack table system and it's probably the most intuitive way to do it. But none of us will know for sure how they implement it until beta.