Combat Algebra

**emptysands** · 06-29-2010, 11:37 PM

I've been doing some analysis of the combat mechanics. Here is some basic combat maths:

Code:

CR = Critical Range
CM = Critical Multipler
B  = (Average) Base Damage
F  = Fortification 
S  = Seeker bonus

In the 0% fortification situation,

Code:

Average Damage (AvD)
  = B*(19 - CR)/20 + B*(CR/20*CM)
  = B/20 * (19 - CR + CR*CM)
  = B/20 * (19 + CR*(CM - 1))
  = B * (95% + 5% * CR*(CM - 1))
  = B * (95% + 5% * Z)

Where Z is the critical multiplicative factor.

So, if CM and CR = 1, then Z = 0 and the Average Damage is B * 95% - as you would expect. Or for a MinII Khopesh CR = 4 and CM = 3, so Z = 8 and AvD = B(GSKhopesh) * 135%. This applies through to the base damage modifiers. For example:

AvD with Strength 30 = 135% * (B + 10) = B * 135% + 10 * 135%

Thus for a minII Khopesh each two points of strength give another 1.35 points per swing of average damage against 0% fortification.

In comparison, minII Dwarfen Axe has CR = 2 and CM = 3, so Z = 4. Thus, AvD = B(GSDAxe) * 115%.

B(GSKhopesh) = 10.5 => AvD = 14.175
B(GSDAxe) = 14 => AvD = 16.1

The big difference is for each point of fixed damage boost the khopesh adds 1.35 and the daxe adds 1.15. So for fixed damage bonus Y,

B(GSKhopesh) = 14.175 + 1.35 * Y
B(GSDAxe) = 16.1 + 1.15 * Y

So B(GSKhopesh) = B(GSDAxe), when Y = 9.625, or when strength is 30. Or strength is 20 and you are using Power Attack. Or finally, when strength is 8 you are using Power Attack and Divide Might II.

If you include fortification, the equation becomes:

Code:

Average Fortification Damage 
  = B * (19 - CR)/20 + B*(CR/20)*(CM*(1 - F) + F)
  = B/20 * (19 - CR + F*CR + CR*CM - CR*CM*F)
  = B/20 * (19 + CR*(CM*F - CM - F +1))
  = B/20 * (19 + CR*(CM - 1)*(1 - F))
  = B * [95% + 5% * CR*(CM - 1)*(1 - F)]
  = B * [95% + 5% * (1 - F) * Z]

Fortification then for any given fixed F acts as a multiplicative reduction factor against Z. So say for 50% fortification, Y becomes 26.25.

B(GSKhopesh) = 12.075 + 1.15 * Y
B(GSDAxe) = 14.7 + 1.05 * Y

ie. Strength 64, or strength 44 and Power Attack or strength 36, power attack and divide might II.

The above numbers should allow one to create baselines for analysis. So for example, given a dwarf with 34 Strength and Rams Might - so a +15 fixed base damage boost - and +2 axe damage, when does it become better to use a khopesh? At 50% fort, the +2 axe damage addes 2.1 to the base damage and the base boost of 15 addes 15.75 for the axe and 17.25 for the khopesh.

B(GSKhopesh) = 1.15 * (10.5 + 15 + Y)
B(GSDAxe) = 1.05 * (14 + 15 + 2 + Y)

So 12.075+17.25+1.15Y = 14.7+2.1+15.75+1.05Y. Solve for Y, gives Y = 32.25, 21 of this comes from the axe damage boost. What this means is the same character would need an additional 32 points of damage boost before criticals in when using a Khopesh to match the DAxe against a 50% fort target.

The same calculation for 0% Fort is 1.35*(10.5+15+Y)=1.15*(14+15+2+Y) => 1.35Y-1.15Y=16.1+17.25+2.3-14.175-20.25. Gives Y=6.125.

We can add Seeker (Bloodstone) damage a modification to the equation:

Code:

Average Fortification Damage with Seeker
  = B * [95% + 5% * (1 - F) * Z] +  S * [CM * CR] * 5% * (1 - F)

The same 50% Fort scenario above with a bloodstone then:

B(GSKhopesh) = 1.15 * (10.5 + 15 + Y) + 6 * 30%
B(GSDAxe) = 1.05 * (14 + 15 + 2 + Y) + 6 * 15%

So 12.075+17.25+1.15Y+1.8 = 14.7+2.1+15.75+1.05Y+.9, or 1.15Y- 1.05Y = 14.7+2.1+15.75 +.9 - 12.075-17.25-1.8. Gives Y = 23.25.

Note this is just a basic model and ignores THAC0, critical burst damage (similar to seeker damage) and any DR.

**Kinerd** · 07-05-2010, 06:57 PM

Nice post.

**Backley** · 07-07-2010, 07:18 PM

Nice post, seems to match my excel sheet for comparing weapons' average damage per attack.

The next step would be to multiply by the number of attacks per second to get to Damage Per Second (so I can tell if my Bard should use Bow A or Crossbow B when plinking-away at Harry in the Shroud).

Anyone have any current research of Attacks Per Minute for different weapon types?

I've seen 67 shots per minute quoted in Longbow threads.

Crossbows are supposed to be less. Not sure if throwing weapons match one or the other.

Ranged should be fairly easy to test: time how long it takes you to use up a stack of arrows/bolts/darts. Any tips for timing melee swings per minute? Maybe have to set up some video capture?

**GhoulsTouch** · 07-07-2010, 07:27 PM

*Gag* *Puke*

Didn't anybody ever tell you that math cannot solve that which cannot be counted on? Chaos is the rule not the exception...and please, for the love of God, stop with the Math!!!

Everything is 50 50. You might have a better or worse chance but it's still just luck. You either will or won't. You can run numbers all day long and it will still come down to chance.

If I complimented someone's weapon and they gave me the breakdown that you just did, I would either fall asleep or run far far away.

**Backley** · 07-07-2010, 07:33 PM

Also, where would be the best place on the DDOWiki to place this?

**Backley** · 07-07-2010, 07:39 PM

Originally Posted by GhoulsTouch

Didn't anybody ever tell you that math cannot solve that which cannot be counted on? Chaos is the rule not the exception...and please, for the love of God, stop with the Math!!!

Everything is 50 50. You might have a better or worse chance but it's still just luck. You either will or won't. You can run numbers all day long and it will still come down to chance.

Quite the opposite. If you don't run the numbers, you have no idea if the "interesting and probably good" weapon you just pulled is better than your current weapon.

Chance you can do nothing about, but you can make sure you are equipped the best you can be.

**Backley** · 07-07-2010, 08:32 PM

Originally Posted by emptysands

Note this is just a basic model and ignores THAC0, critical burst damage (similar to seeker damage) and any DR.

<EDIT: you were right, fail-on-1 already accounted for>

It does ignore crit fail on a 1.

Code:

Average Damage (AvD), miss on a 1
  = B*(19 - CR)/20 + B*(CR/20*CM)*19/20 + B*(CR/20)*1/20
  = B/20 * (19 - CR + CR*CM*19/20 + CR*1/20)
  = B/20 * (19 + CR*(CM*19/20 + 1/20 - 1))
  = B/20 * (19 + CR*(CM*0.95 - 0.95))
  = B/20 * (19 + 0.95*CR*(CM - 1))
  = B * (95% + 4.75% * CR*(CM - 1))
  = B * (95% + 4.75% * Z)

My own Excel workbook also counts Fort, bonus damage (Holy, etc), bonus crit damage (Holy Burst, etc) and your Attack+Buffs vs. Monster AC.

I can post the full formula if you want.

**emptysands** · 07-07-2010, 08:32 PM

Originally Posted by Backley

Nice post, seems to match my excel sheet for comparing weapons' average damage per attack.

The next step would be to multiply by the number of attacks per second to get to Damage Per Second (so I can tell if my Bard should use Bow A or Crossbow B when plinking-away at Harry in the Shroud).

Anyone have any current research of Attacks Per Minute for different weapon types?

I've seen 67 shots per minute quoted in Longbow threads.

Crossbows are supposed to be less. Not sure if throwing weapons match one or the other.

Ranged should be fairly easy to test: time how long it takes you to use up a stack of arrows/bolts/darts. Any tips for timing melee swings per minute? Maybe have to set up some video capture?

For the same build it shouldn't matter when examining similar weapons, ie. ignoring feat and/or enhancement effects like IC:Slashing vs IC:Ranged. The action speed boost for a Khopesh is likely to have the same effect on dps as on a Longsword.

The difference is analysis of TWF vs THF for the same or similar builds. Say for a TWF vs THF dwarf paladin when using Dwarfen Axes/Great Axes to see if THF+S&B mode is worth considering to a DOS build.

Conversely, you might also consider different damage boosts and the effect on a build dps output. Will taking Dwarf Axe damage make a difference vs WF: Power Attack boost?

I found the follow posts about attack speeds:
http://forums.ddo.com/showpost.php?p...6&postcount=68 - this has a link to spreadsheet with numbers which are reused in the dps calc spreadsheet. Not sure how up to date.
http://forums.ddo.com/showpost.php?p...25&postcount=8 - this maybe out of date as well. But the attack chain mechanism might be the same, and then it's just figuring numbers with the U5 mechanics.
http://forums.ddo.com/showthread.php...46#post1871646 - lots of detail for different weapon types, but only up to BAB 16

Two threads which form the basis for a lot the weapon attack speed analysis.
http://forums.ddo.com/showthread.php?t=250610
http://forums.ddo.com/showthread.php?t=201535

**emptysands** · 07-07-2010, 09:10 PM

Originally Posted by Backley

It also ignores missing on a 1 (and crit fail on a 1). I think it would be much better to include that, since you can then say "this is your average damage per hit against a monster you hit on a 2 or better."

Code:

Average Damage (AvD), miss on a 1
  = B*(18 - CR)/20 + B*(CR/20*CM)*19/20 + B*(CR/20)*1/20
  = B/20 * (18 - CR + CR*CM*19/20 + CR*1/20)
  = B/20 * (18 + CR*(CM*19/20 + 1/20 - 1))
  = B/20 * (18 + CR*(CM*0.95 - 0.95))
  = B/20 * (18 + 0.95*CR*(CM - 1))
  = B * (90% + 4.75% * CR*(CM - 1))
  = B * (90% + 4.75% * Z)

My own Excel workbook also counts Fort, bonus damage (Holy, etc), bonus crit damage (Holy Burst, etc) and your Attack+Buffs vs. Monster AC.

I can post the full formula if you want.

I don't think I made that error, but I'm glad you looked closely.

The non-critical hits are included correctly. Read another way:

Code:

Average Damage (AvD)  
  = [B*1] * (19 - CR)/20 + [B*CM] * (CR/20)

I assumes CR > 0, ie 20 is always a critical. So say CM = 1 then [B*1] * (19 - CR)/20 + [B*CM] * (CR/20) = [B*1] * (19 - CR ) /20 + [B*1] * ( CR/20) = [B*1] (19 - CR + CR ) /20 = [B*1] * 19/20 = B * 95%. As you would expect. So 1d6 weapon with CM = 1 will do average 3.5 * 95% or 3.325 damage per swing, ignoring to hit but account for always fail on "1".

Or the equivalent from a different view:

Code:

 Average Damage (AvD)  
  = damage per non-critical swing that hits + critical damage per critical swing that hits
  = B*(95% - CR*5%) + B*CM*(CR*5%)

Code:

 B*(18 - CR)/20 + B*(CR/20*CM)*19/20 + B*(CR/20)*1/20

In your equation if CR = 0, just in this case, we get B * (18 - 0) / 20 + B *(0/20)*19/20 + B*(0/20)*1/20. Or B*18/20. This is wrong as you lose the damage from 1 hit.

Bonus damage, power attack, strength, etc can be basically be added to the base damage. Burst on critical damage, say B_c, can be added by adding.

Code:

AvDB_c = AvD + B_c * 5% * (1 - F) * Z

Seeker damage can be handled basically in a similar way. I haven't dealt with sneak attack damage, but that isn't affected by the critical multiplier - so it's just a linear boost at the to hit rate.

I didn't really look at the To Hit aspect. Of the top of my head it would reduce the 95% - CR*5% factor. The edge case is dealing with the situation where the miss range was greater than 95%-CR*5%. However, I think it's a fair assumption that any build that misses on a potential critical needs to be rebuilt, even if this is in epics.

There is the case where you roll a critical but fail to confirm. This is not an auto fail on 1[*] and seeker items are likely to make this even less likely. Even so the above is an attempt at a simple look.

I'd be interested to see your spreadsheet. I've got my own, but google docs has some trouble importing it at the moment so I haven't posted it. Plus I've been trying to work out a good way to deal with the attack speeds so I can do quick analysis. Work in progress. Second the dps spreadsheet that's on the codemasters forum site is pretty good. Their numbers don't quite match mine, and I assume I've made some errors. So I'm still working thru the details. Whatever the case, I'm sure the theory/formulas above are correct.
[*] http://ddowiki.com/page/Confirmation_roll

**emptysands** · 07-07-2010, 10:57 PM

Here is the dps calculator I mentioned above: http://community.codemasters.com/for...alculator.html

You can extract some swing info from this sheet as well.

**PopeJual** · 07-07-2010, 11:27 PM

Originally Posted by GhoulsTouch

Everything is 50 50. You might have a better or worse chance but it's still just luck.

Is there any chance you'd like to join my cash-only weekly poker game?

**h4x0r1f1c** · 07-22-2010, 04:20 AM

Originally Posted by emptysands

I've been doing some analysis of the combat mechanics. Here is some basic combat maths:

Code:

CR = Critical Range
CM = Critical Multipler
B  = (Average) Base Damage
F  = Fortification 
S  = Seeker bonus

In the 0% fortification situation,

Code:

Average Damage (AvD)
  = B*(19 - CR)/20 + B*(CR/20*CM)
  = B/20 * (19 - CR + CR*CM)
  = B/20 * (19 + CR*(CM - 1))
  = B * (95% + 5% * CR*(CM - 1))
  = B * (95% + 5% * Z)

Where Z is the critical multiplicative factor.

So, if CM and CR = 1, then Z = 0 and the Average Damage is B * 95% - as you would expect. Or for a MinII Khopesh CR = 4 and CM = 3, so Z = 8 and AvD = B(GSKhopesh) * 135%. This applies through to the base damage modifiers. For example:

AvD with Strength 30 = 135% * (B + 10) = B * 135% + 10 * 135%

Thus for a minII Khopesh each two points of strength give another 1.35 points per swing of average damage against 0% fortification.

In comparison, minII Dwarfen Axe has CR = 2 and CM = 3, so Z = 4. Thus, AvD = B(GSDAxe) * 115%.

B(GSKhopesh) = 10.5 => AvD = 14.175
B(GSDAxe) = 14 => AvD = 16.1

The big difference is for each point of fixed damage boost the khopesh adds 1.35 and the daxe adds 1.15. So for fixed damage bonus Y,

B(GSKhopesh) = 14.175 + 1.35 * Y
B(GSDAxe) = 16.1 + 1.15 * Y

So B(GSKhopesh) = B(GSDAxe), when Y = 9.625, or when strength is 30. Or strength is 20 and you are using Power Attack. Or finally, when strength is 8 you are using Power Attack and Divide Might II.

If you include fortification, the equation becomes:

Code:

Average Fortification Damage 
  = B * (19 - CR)/20 + B*(CR/20)*(CM*(1 - F) + F)
  = B/20 * (19 - CR + F*CR + CR*CM - CR*CM*F)
  = B/20 * (19 + CR*(CM*F - CM - F +1))
  = B/20 * (19 + CR*(CM - 1)*(1 - F))
  = B * [95% + 5% * CR*(CM - 1)*(1 - F)]
  = B * [95% + 5% * (1 - F) * Z]

Fortification then for any given fixed F acts as a multiplicative reduction factor against Z. So say for 50% fortification, Y becomes 26.25.

B(GSKhopesh) = 12.075 + 1.15 * Y
B(GSDAxe) = 14.7 + 1.05 * Y

ie. Strength 64, or strength 44 and Power Attack or strength 36, power attack and divide might II.

The above numbers should allow one to create baselines for analysis. So for example, given a dwarf with 34 Strength and Rams Might - so a +15 fixed base damage boost - and +2 axe damage, when does it become better to use a khopesh? At 50% fort, the +2 axe damage addes 2.1 to the base damage and the base boost of 15 addes 15.75 for the axe and 17.25 for the khopesh.

B(GSKhopesh) = 1.15 * (10.5 + 15 + Y)
B(GSDAxe) = 1.05 * (14 + 15 + 2 + Y)

So 12.075+17.25+1.15Y = 14.7+2.1+15.75+1.05Y. Solve for Y, gives Y = 32.25, 21 of this comes from the axe damage boost. What this means is the same character would need an additional 32 points of damage boost before criticals in when using a Khopesh to match the DAxe against a 50% fort target.

The same calculation for 0% Fort is 1.35*(10.5+15+Y)=1.15*(14+15+2+Y) => 1.35Y-1.15Y=16.1+17.25+2.3-14.175-20.25. Gives Y=6.125.

We can add Seeker (Bloodstone) damage a modification to the equation:

Code:

Average Fortification Damage with Seeker
  = B * [95% + 5% * (1 - F) * Z] + S * Z * 5% * (1 - F)

The same 50% Fort scenario above with a bloodstone then:

B(GSKhopesh) = 1.15 * (10.5 + 15 + Y) + 6 * 20%
B(GSDAxe) = 1.05 * (14 + 15 + 2 + Y) + 6 * 10%

So 12.075+17.25+1.15Y+1.2 = 14.7+2.1+15.75+1.05Y+.6, or 1.15Y- 1.05Y = 14.7+2.1+15.75 +.6 - 12.075-17.25-1.2. Gives Y = 26.25.

Note this is just a basic model and ignores THAC0, critical burst damage (similar to seeker damage) and any DR.

Please do all that **** you just said for GSKopesh and GSBastardSword.

Thanks.

**emptysands** · 07-22-2010, 05:15 AM

Originally Posted by h4x0r1f1c

Please do all that **** you just said for GSKopesh and GSBastardSword.

Thanks.

Z(GSBastard) = 4* [2 - 1] = 4. So AvD(GSBastard) = (14 + mod) * 1.15

So for the basic case, the same as the DAxe.

12.075+1.15Y = 14.7+1.05Y = > Y = 26.25. Khopesh is better after 26.25 damage modify for 50% Fort.

The main difference is Dwarf Axe damage bonus, which gives a DAxe a base of 14 + 2 = 16, and the affect of the critical range/multiplier on Seeker damage.

Seeker damage = [ S * CM ] * CR * 5% * (1 - F)

So for the GSBastard sword this is S * 4 * 2 * .5 * (1-F) = S * 40% * (1-F) and for the DAxe this is S * 3 * 2 * .5 * (1-F) = S * 30% * (1-F). So the BSword does more seeker and other burst effect damage.

The same scenario as in OP is then:
B(GSKhopesh) = 1.15 * (10.5 + 15 + Y) + 6 * 30%
B(GSBastard) = 1.05 * (14 + 15 + Y) + 6 * 20%

So 12.075+17.25+1.15Y+1.8 = 14.7+15.75+1.05Y+1.2, or 1.15Y- 1.05Y = 14.7+15.75 +1.2 - 12.075-17.25-1.8. Gives Y = 5.25.

So after fixed damage boost of 20.25 the Khopesh is better than the Bastard Sword for 50% Fort. ie. Power Attack and 40 Strength.

**h4x0r1f1c** · 07-22-2010, 04:40 PM

Originally Posted by emptysands

Z(GSBastard) = 4* [2 - 1] = 4. So AvD(GSBastard) = (14 + mod) * 1.15

So for the basic case, the same as the DAxe.

12.075+1.15Y = 14.7+1.05Y = > Y = 26.25. Khopesh is better after 26.25 damage modify for 50% Fort.

The main difference is Dwarf Axe damage bonus, which gives a DAxe a base of 14 + 2 = 16, and the affect of the critical range/multiplier on Seeker damage.

Seeker damage = [ S * CM ] * CR * 5% * (1 - F)

So for the GSBastard sword this is S * 4 * 2 * .5 * (1-F) = S * 40% * (1-F) and for the DAxe this is S * 3 * 2 * .5 * (1-F) = S * 30% * (1-F). So the BSword does more seeker and other burst effect damage.

The same scenario as in OP is then:
B(GSKhopesh) = 1.15 * (10.5 + 15 + Y) + 6 * 30%
B(GSBastard) = 1.05 * (14 + 15 + Y) + 6 * 20%

So 12.075+17.25+1.15Y+1.8 = 14.7+15.75+1.05Y+1.2, or 1.15Y- 1.05Y = 14.7+15.75 +1.2 - 12.075-17.25-1.8. Gives Y = 5.25.

So after fixed damage boost of 20.25 the Khopesh is better than the Bastard Sword for 50% Fort. ie. Power Attack and 40 Strength.

Thanks for doing that. So far everyone tells me GS Khopesh is better than GS Bastard Sword and guess it's true.

You're the 2nd person to mention it swapping into Khopesh's favor after around a 21 damage mod.

**Gol** · 07-22-2010, 04:52 PM

No offense, but this model is quite simplistic and doesn't take many (any?) of the possible weapon effects into account. DPS calcs involve dozens of variables you're completely ignoring, never mind all the class/race differences that heavily influence the viability of one weapon over another.

**emptysands** · 07-22-2010, 07:36 PM

Originally Posted by Gol

No offense, but this model is quite simplistic and doesn't take many (any?) of the possible weapon effects into account. DPS calcs involve dozens of variables you're completely ignoring, never mind all the class/race differences that heavily influence the viability of one weapon over another.

Depends on the purpose. The main focus when I did the initial analysis was which weapon was better for a single build style, not what the overall dps or if one build with one weapon is better than another weapon with another build. The relation between per swing and final dps is likely to very similar for a given build style.

As such the variables are on purpose simplifications - base damage can include all base damage and not just weapon base damage.

That said, we can build further. Too add critical burst effects to the equation, average B say, we have:

Code:

Average Fortification Damage with Seeker/Burst
  = B * [95% + 5% * (1 - F) * Z] +  [S * CM + B ] * CR * 5% * (1 - F)

Like any analysis these are just the equations, the data or character+gear that are used as input are the main interest. DDO like many RPG games is a very gear focused game, a utility build with the best gear will likely out perform a "dps-only" build without much gear.

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