Lanchester’s Laws

While playing an online multi-player wargame, someone attacked 10 units with 19 units of the same type.  How many attackers survived?  Contrary to what some might assume, it was not nine units.  They were equally matched and 16 units survived of the attacking force.

This is a very simplified version of reality, but essentially there are, initially, nearly twice as many attackers as defenders shooting, so a defender will be killed in half the time of an attacker.  Then the ratio of attackers to defenders is even greater: 19 to 9.  Now the likelihood of a defender being killed before an attacker is even greater than before.  Eventually, only a few defenders are being outnumbered 4:1 or 5:1 and so they are eliminated very quickly, with few if any losses to the attackers.

This is described in Lanchester’s Laws which say that when you have people shooting at one another at range, and each can fire on any other opponent (as opposed to one-on-one melee combat), the effectiveness of the forces are in proportion to the squares of their numbers meaning the attrition over time is far greater for the lesser force.  That is, the smaller force, will lose members faster and faster and the greater force lose them slower and slower.

One can imagine how ten people shooting at two people may manage to shoot both before any of the ten are themselves shot, contrary to what is usually represented in Wild West and action movies.

As examples, assuming a 1% chance of killing with any given shot, a battle between 100 attackers and 50 defenders will – statistically – end in victory after 55 rounds with 86 attackers surviving.  Some more analysis:

Attackers Defenders %likelihood Rounds Survivors
100 99 1% 265 13
100 90 1% 148 43
100 50 1% 55 86
100 14 1% 15 98
100 10 1% 11 99
100 99 50% 5 6
100 90 50% 3 28
100 50 50% 2 65
100 10 50% 1 95
100 99 90% 2 2
100 90 90% 2 19
100 50 90% 1 55
100 10 90% 1 91

The lower the likelihood of killing with one shot, the greater the Lanchester Law effect: the larger number of attackers will whittle away the defenders before the defenders can respond in a significant way.  At the other extreme, 100% likelihood of killing in one shot, it resembles melee combat and the number of survivors is simply the number of attackers minus the number of defenders.

What is the point of this?

  1. When playing tabletop wargaming, always attack with overwhelming numbers to maximise enemy losses while minimising your own.
  2. The Generals of the Great War were indeed incompetent buffoons, believing attrition would win them the war, by sending in wave after wave of small numbers of their own troops in short lines on the front to be massacred.

The second point is particularly so since the infantry were walking into defensive machine gun fire, where the likelihood of killing per unit of time was far greater than for the attackers.  Attacking in that way maximised the losses for the attacker.  And this would have been apparent for any player of wargames or mathematically minded person at the front.  The defence for the the donkeys running the war was that they knew no better – then they were idiots.

The Prussians had been playing wargames as a military training tool for over a century by the time of the Great War.

Why do I as a pacifist play wargames?  Partly recreation, partly research.  It helps one to understand the true horror of mechanised, organised, warfare.

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