by **theobserver** » Wed Oct 03, 2012 10:51 am

It's been a long time since I posted...

Anyway, I just want to point out a few things.

The first is something called the Lanchester's Laws. Basically, some English dude decided to invent a way to predict the casualties and victors of battles, and he came up with this.

In medieval/ancient warfare, the offensive capability of a military force is its number of troops times the offensive ability of each soldier. This kind of warfare follows Lanchester's Linear Law. Let A be the number of troops and a be the offensive ability of each soldier. The total offensive capability will be:

Total= Aa

In modern warfare, where aimed fire can attack multiple targets at a distance and attacks can come from multiple directions, the rate of attrition now only depends on the number of weapons firing. In this case, Lanchester's Linear Law no longer applies, and Lanchestor's Square Law is used. Lanchestor, the English dude who invented these equations, determined that the power of such a force is proportional not to the number of units it has, but to the square of the number of units. Let A be the number of troops and a be the offensive ability of each soldier. Total offensive capability will be:

Total= (A^2)a

Derived from these equations, the attrition rate of medieval battles will be:

Aa=-B and Bb= -A

where A is the number of troops on the first side, a is the offensive ability of each troop, B is the number of troops on the second side and b is the offensive capability of each troop. In this case, the survivors of the winning side would simply be the difference between the Total offensive capability of each side divided by the offensive ability of each troop on the winning side. For example, let A= 2000, a=2, B=3000, b=1, hence, the survivors for the winning side (The A side) is:

Total A - Total B= 2000*2 - 3000*1= 1000

Total A= Aa

1000= A(2)

A= 500 (The survivors for the winning, A, side)

And thus, the casualties are: 2000-500= 1500 casualties.

Also derived from these equations, the attrition rate for modern battles will be:

dA/dt= -Bb

and

dB/dt= -Aa

where dA/dt represents the change in the number of soldiers on side A in a particular instant of time, and the same applies for dB/dt.

In order to calculate the probable final survivors (the survivors at the end of the battle), you simply have to use the original Lanchester's Square Law equation and subtract the total offensive capability of both sides. Let A=2000, a=2, B=500, a=8. Because the formula for total capability is Total= (A^2)a, the survivors are:

Total A - Total B= (2000^2)2 - (500^2)8= 4000000*2 - 250000*8= 6000000

Total A= (A^2)a

6000000= (A^2)2

A^2=3000000

A= 1732 (The survivors for the winning, A, side)

And thus, the casualties are: 2000-1732= 268 casualties.

In relation to this discussion, these equations can be easily applied to several scenarios. First off, the Drow of Drowtales, no matter how impressive their tech, ultimately aren't capable of the sheer range, accuracy, and destructiveness of anything WW1 or onward. A Machinegun fires at several hundreds of rounds per minute and can hit targets one and a half kilometers away. A crossbow can barely reach 100 meters, and a Golem's cannon (one of the Val Ilhardo's) can't seem to fire past a single river, so they're range is probably limited to 500 meters tops. This just has no comparison to WW1 direct fire artillery, which can slam at targets accurately at over two kilometers away, not to mention indirect artillery. Therefore, the Drow's forces will still obey Lanchester's Linear Law in terms of their total offensive capability. Now, let's say A is the number of Drow, a is each Drow's offensive ability, B is the number of humans and b is each human's offensive ability.

The total offensive capability of the Drow's force will always be: Total= Aa

While the human's offensive capability will depend on the tech level. If it is medieval or ancient, it's Total= Bb, if it's modern, it's Total= (B^2)b.

Now, to illustrate the difference between each scenario, let A= 2000, a= 10, B= 2000, b=1. In other words, let's say one Drow can kill 10 humans before going down.

In medieval times, the drow win by a large margin. Total A= 20000 and Total B= 2000, the Drow have 18000 more units of offensive capability than the humans.

In WW1 type of tech, the humans win by an insane margin so damn large it's not even funny. Total A= 20000 and Total B= 4000000. In words, Total A is equals to two hundred thousand while Total B is equals to four million. The humans have 3800000 (Three million eight hundred thousand) more units of offensive capability than the Drow. So even if a Drow has ten times more experience and skill than a single human, the humans will murder nearly all of them before they can even use that skill.

Now what if the tech used by the humans is not medieval nor modern? Somewhere in the middle, like let's say 18th century technology (Napoleonic Era)? There are some technologies (grapeshot, for example) that allow attacking multiple targets at once, but the human footsoldiers will still be limited to attacking one person at a time (Assuming the musket ball even hits), and the range of the weapons are still limited. In this case, instead of using total number of soldiers, we have to use the ratio between artillery weapons and normal infantry, then use the Lanchester's square law. Let's say that there is one artillery battery (Plus Minus 4 Field Guns) per 500 soldiers. Assuming B is still equals to 2000, the ratio between artillery and soldiers is 1:4. Assuming we divide the army into two units each with a single artillery battery, each of those four units will have their offensive ability squared, thus the total offensive capability of the human army would be the sum of those four units offensive ability. Hence, assuming we divide both the Drow and Human army into four units, each with 500 soldiers:

Total A= (4 units)10= 40 units= 20000

Total B= (4 units^2)1= 16 units= 80000

The humans in this case still outgun the Drow by 60000 units of offensive capability, but you can see the battle being far more balanced. The ratio between artillery batteries and soldiers in this case is very important. If the ratio is 1:2, the human army would have lost by 16000 units. If it were 1:3, it would be something else.

However, these equations only apply to battles of attrition with no maneuver being conducted whatsoever. This is obviously unrealistic, but it gives a handy rule of thumb for anyone trying the balance things out between the humans and the Drow. Also, be aware that anything that causes deviation from the Lanchester Laws are called Force Multipliers. There are a lot of Force Multipliers, including morale, deception, coordination, etc. In real life, the Lanchester Laws are practically useless, because modern armies do not spread out and conduct battles of attrition, rather they focus on specific points for breakthrough, and coordinate their attacks. There are so many ways to criticize the Lanchestor Laws (see Col. Trevor N. Dupuy's works for further criticism of the Lanchestor Laws. Dupuy favored empirical analyses of historical battles, not numerical analyses).