Damage Block Order/Priority and EHP

[[GR]Schizofriend]

I'm currently trying to write a guide about Damage Reduction and EHP for a website I am on staff at and I need a little help regarding how block effects it. I have a lot figured out about it already but would like to know if there is a mathamatical equation to see how blocking affects your EHP.

What I have right now about Blocking is the Damage Reduction Calculation Order goes:

1. Evasion
2. Static Damage Buffer Sources (ie, Arcane Hide and Barrier Idol)
3. Damage Block/"Deflection" (ie, the static damage reduction from Iron Shield, Iron Buckler and Helm of the Black Legion)
4. Armor Type (Combat Type)
5. Armor Value (Damage reduction from armor)
6. Other Stuff (passive hero abilities like Armordillo, Electric Shield, Dissipation, Focus Buffer)

Because Damage Block is calculated before Armor Value, the chance to block becomes less effective as Armor Value increases. The resulting affect to the damage done can be calculated as:

Effective Block = DmgBlock*(1-.06*Armor/(1+.06*Armor))

In this way a Hero with Helm of the Black Legion and 10 armor would only block 25 damage 70% of the time.

Maybe there is no way of converting that into EHP, I certainly can't think of one, but since there are several people here who seem to know much more than I do when it comes to these things, I thought I would ask.

[[S2]ElementUser]

You can always ask Malle, he wrote the EHP guide on the DotA-Allstars forum.

The general equation for EHP is:

EHP = Max HP + (0.06*Armor*Max HP)

[Neber]

There isn't a constant EHP in regards to Damage Block, as EHP would go up or down depending on the damage range of the attacker/s. Using the Formulas from the above posts you could calculate it using averages.

EffectiveBlock = DmgBlock*(1-.06*Armor/(1+.06*Armor))

The NoAttacks formula works out how many attacks is needed to kill a player. The AvgDamageTaken is exactly as it says, the average damage taken from all sources per hit. You would round up the NoAttacks as well.

NoAttacks = (MaxHP + (0.06*Armor*MaxHP)) / (AvgDamageTaken - EffectiveBlock)

The last formula Totals the MaxHP, Armoured HP and Amount Blocked(On Average) to get the EHP.

EHP = MaxHP + (0.06*Armor*MaxHP) + EffectiveBlock * NoAttacks

[[GR]Schizofriend]

Cool. Maybe I'll be able to use that to make some sort of graph or chart to show how block affects your EHP less as you get more armor, and enemies get higher damage.

[edxs]

Effective Armor = (Damage Block/0.06) / (Incoming Damage - Damage Block)

HotBL = 28 average block
Single Shield = 12 average block

[Neber]

Effective Armor = (Damage Block/0.06) / (Incoming Damage - Damage Block)

HotBL = 28 average block
Single Shield = 12 average block

I don't get your formula?

[Malle1]

U - unreduced damage
At - armor type damage factor
B - actual blocked damage (as in 0 if there is no block, or Bd if there is)
Av - armor value damage factor
D - damage actually dealt
Bc - chance to block
Bd - damage blocked if Bc triggers
Db - damage dealt when a block triggers
Dn - damage dealt when no block triggers
Da - average damage dealt
Bf - block damage factor

Testing indicates that the game in general does the following calculation (assuming B ≤ U*At):

D = (U*At - B)*Av = U*At*Av - B*Av

Using a single source of blocking this can be put as two equations (or, in general, as n+1 equations for n different possible amounts of blocking)

With Bc probability:

Db = U*At*Av - Bd*Av

With 1-Bc probability:

Dn = U*At*Av

Averaging this we have:

Da = Bc*(U*At*Av - Bd*Av) + (1 - Bc)*U*At*Av = U*At*Av - Bc*Bd*Av

We manipulate this:

Da = U*At*Av - Bc*Bd*Av = U*At*Av - Bc*Bd*At*Av/At = At*Av*(U - Bc*Bd/At)

We then compare the average damage dealt with block to the average damage dealt without block to see the damage taken factor from the block:

Bf = Da / Dn = (At*Av*(U - Bc*Bd/At)) / (U*At*Av) = (U - Bc*Bd/At) / U = 1 - (Bc*Bd)/(U*At)

We do the same calculations but assuming B > U*At:

P: Bc : Db = 0
P: 1-Bc : Dn = U*At*Av
Da = Bc*(0) + (1 - Bc)*U*At*Av = (1 - Bc)*U*At*Av
Bf = Da / Dn = (1 - Bc)*U*At*Av / U*At*Av = (1 - Bc)

The true general formula for EHP would be

EHP = HP / DamageFactor

So, with Blocking, Armor and Evasion we have:

EHP = HP / ( (1 - (0.06*a)/(1+0.06*a)) * (1-Ev) * (1 - (Bc*Bd)/(U*At)) ) = HP * (1 + 0.06*a) / ( (1 - Ev) * (1 - (Bc*Bd)/(U*At)) )

where the armor a ≥ 0, the evasion probability Ev is such that 0 ≤ Ev < 1 and the block value B, unreduced damage U and armor type damage factor At are such that B ≤ U*At. (I think this should be the most common scenario in hero versus hero fights)


EDIT:
Since the armor factor is on the form (1 - (0.06*a)/(1+0.06*a) and the block factor is on the form (1 - (Bc*Bd)/(U*At)) or (1-Bc) we can relate it to armor:

1 - (0.06*a) / (1 + 0.06*a) = 1 - (Bc*Bd)/(U*At)
⇔
(0.06*a) / (1 + 0.06*a) = (Bc*Bd) / (U*At)
⇔ {solve for a}
(0.06*a)(U*At) = (Bc*Bd)(1 + 0.06*a)
⇔
0.06*a*U*At = Bc*Bd + 0.06*a*Bc*Bd
⇔
0.06*a*U*At - 0.06*a*Bc*Bd = Bc*Bd
⇔
a * 0.06 * (U*At - Bc*Bd) = Bc*Bd
⇔
a = Bc*Bd / (0.06 * (U*At - Bc*Bd))


1 - (0.06*a) / (1 + 0.06*a) = 1 - Bc
⇔
(0.06*a) / (1 + 0.06*a) = Bc
⇔ {solve for a}
0.06*a = Bc*(1 + 0.06*a)
⇔
0.06*a = Bc + 0.06*Bc*a
⇔
0.06*a - 0.06*Bc*a = Bc
⇔
a*0.06*(1 - Bc) = Bc
⇔
a = Bc / (0.06*(1 - Bc))

This produces the following graphs for current existing values:

[[GR]Schizofriend]

Wow. More than I was even hoping for.

If only it wasn't 2am here. I'll have to really look at all of that tomorrow. Nice Graphs too, that's pretty much what I wanted to make myself, just couldn't figure out how to relate the block to armor.

[Neber]

Ah too many variables to remember, i had to go up, down constantly to work out the formulas. I was mainly curious about the Blocking values, as your formula is basically the same as mine, just added in (1-Evasion) to fix it up to account for that. However i was wondering with the Blocked Damage, since mine has the same problem. Does your formula account for the times when Average Damage Taken <= Average Block Value, however the player will take damage when the block doesn't occur, but using averages, the formula would be calculating as if no damage is done.

Also With averages, if the Hi-range of the Incoming damage is greater than block value, and the average incoming damage <= Block value, then again the same problem occurs where averages removes damage that should have happened, but doesn't.

Hopefully you can answer those questions, if your formula takes that into account because i gave it a little thought, but my brain is on sleep mode now.

[Malle1]

I was a bit unclear, perhaps. Those is calculations for a single incoming damage value, not a damage range. Depending on which of the two EHP functions I listed (see below for easier to read version), it either applies when the damage is not less than what can be blocked or when it is.

Also, the graphs I posted are the equivalent of the armor increase at 0 armor. It should be possible to make it a function of armor as well, but I am not going to do that right now.

Here is an easier to read form of the EHP function:

Relevant variables are:
a - armor
Ev - evasion probability
Bc - chance to block
Bd - damage blocked when it triggers
U - incoming unreduced damage
At - armor type damage factor

The lower one is when the damage block blocks all damage when it triggers.

[Neber]

Okay that's much much easier to read, can just glance at it now and it makes sense.

You wouldn't be able to add the possibility for damage range would you, i'd be interested in the maths for it.

[Malle1]

I think I could, but I don't think it would be pretty and I can't do it right now.

Essentially you would have to divide the damage range so that you have two separate parts: one which falls in the range B < U*At and one range with B ≥ U*At, sum over all U and divide by the spread of the damage range.

If the whole damage range fulfills B ≥ U*At, then the EHP would be as given by the lower part in the image above.

If the whole range falls into B < U*At and U is large and/or the damage spread is low, then we can approximate U to the average of the incoming damage, but it will be just that: an approximation.

EDIT: Just a quick update so I don't forget it when I have time for this: damage uses real values, not integers, so you'd need to integrate.

[Malle1]

image files because I don't feel like typing this out in text right now >_> Warning for large sized images.




The following graphs are all with hero versus hero in mind.

Easier to read with logarithmic y-axis:





And linear y-axis

[[GR]Schizofriend]

Wow, I really appreciate the incredibly detailed responses Malle. You basically did all the math I was trying to figure out for me. I had a stack of papers with scribbled fail formulas on them until you cleared this up for me. Got me past what I was stuck on and then way past that.

[pugi]

Here are two more graphs showing the differences between vs hero and vs creep (melee and neutrals). No label on the y-axis because i can't get it vertically. Also my program doesn't do the gridlines as good.

[[Wolf]Trysaeder]

Moved to Guides and Lists for future reference.

[Meowmeister]

what does all this mean in plain english summary? I dont understand the graphs.

Basically are you trying to say that as you gain more armor the hotbl blocks less damage? because that doesnt seem to be true.

[Comma]

"Blocking Damage becomes less valuable as your opponent's physical damage increases"

[Meowmeister]

Ok, but what is the deal comparing hotbl to armor and stuff?

And I wouldnt say its less valuable, i mean it will shave off 40 damage off the top when it blocks, of course if your hit does like 300 physical it does 300-40 but thats to be expected its still going to do its job by blocking 40.

[archon_]

The point of comparing hotbl to armor and stuff, is to compare the damage reduction values between block and armor, letting you get the EHP (effective hp).
With this we can see what is more effective to purchase, armor or block value.
Graphs show that, at a certain point, block becomes less valuable then a given amount of armor.

Also, out of the graphs, I'd say that HotBL is a viable purchase against heavy physical damage teams. It is more effective then 10 armor (Platemail) before ~130 damage from heroes, in addition it gives 300 hp and 8hp/sec regen. Proves that HotBL is not a "noob" purchase at all, if you need the HP and Regen aswell.


After reading this, I will consider simply purchasing a Iron Buckler on Armadon.