Combat Algebra

[emptysands]

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

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

In the 0% fortification situation,

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:

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:

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.