One day in Outremer: 'Lanchester' Battles Extended

'Year of Our Lord, 1185. You are Reynald of Châtillon, Lord of the Catle Kerak in Oultrejourdain. You hear of a large caravan with a 100-strong Saracen escort travelling northwards from Egypt, and now passing a short distance to the east. Though a truce in in existence, the treasure no doubt carried by the caravan is too tempting. You decide to attack it.' 

Such was the opening paragraph of an article I wrote thirty years ago for the local war games club magazine, Southern Sortie. I think I was the editor at the time, and the issue was one of the last. We won't go into the demise of Southern Sortie. I had first encountered Lanchester's Theory of Battles as a high school calculus problem.  It concerned two forces, one of 10,000 men, the other of 5000. For every one man any of the 10,000 might kill, a man of the 5000 would kill two. If the armies fought to a finish, would either of the armies still be in being, and how many would be the survivors? 

In a response to a comment in a red 'Mad Padre' posting, I mentioned using calculus for something to do with war gaming. The Padre expressed interest in seeing it. For a wonder the thing was still extant among the old SS issues I still have. The article doesn't actually show the calculus bits; just the results. At any rate, here it is.

'...Your followers are few, and the castle must retain an adequate garrison. On the other hand, nearly 90 years of crusading experience has shown that, in the sort of confused melee you anticipate, your knights and sergeants-at-arms are, man for man, twice as combat effective as the lighter armed Saracens. You wish to send the fewest that will ensure a better than even chance of victory. How many men do you send? Would 50 do the job? Perhaps we should send 60, just to make certain?

'This piece is unashamedly inspired by Graham Jones's articles on resolving melees in Wargames Illustrated No41 and 41 (Feb, Mar 1991). In the first of these, he discussed the theory, published in 1916 by one F.V. Lanchester, which postulated that the outcome of melees involving two disparate forces was determined by the squares of their respective numerical strength. Now, I had run across the idea before, once as a calculus problem when I was at school (where does a quarter century [now 55 years plus] go?), and again years later in a discussion of Nelson's planning for Trafalgar. It seems that Lanchester's interest was in deciding the outcome of air battles in World War One. Graham Jones mentioned limitations of the theory in other kinds of battle [we ware talking random free-for-all melees, here], and it was this that set me thinking.' 

Not the Crusades, but a free-for-all over the ramparts
of a British redoubt at Yorktown.

'Lanchester's Theory

'In symbolic terms, let us consider two forces: RED and BLUE. In a melee, each individual will get stuck in; anyone still standing will continue to fight until eliminated or until the enemy is no more.

'Let RED comprise R individuals, and BLUE comprise B individuals.

'Now, let us suppose that, for one or another reason, if in some time interval each BLUE man/unit expect to eliminate 1 RED, then in that same time interval, each RED man/unit would expect to eliminate a number of Blues that we shall call p. This p, which we might describe as RED's relative effectiveness, might relate to greater or lesser Proficiency, Protection, more or less Powerful weapons, or some other Pfactor. It will be a number greater than zero, but need not be a whole number (e.g. If a Red could eliminate 5 opponents in the time a Blue could eliminate 4, then we could say that BLUE's relative effectiveness is 1 and RED's is 5/4 = 1.25. Were the numbers reversed, RED's efffectiveness would be 0.8.

Now, RED will win if       pR² > B²  or, if you like R > B√ (p)/p

BLUE will win if                  pR² < B²  or, if you like B > B√(p) 

and both sides will be wiped out if     pR² = B² 

'Example:       
'Let us consider Reynald's situation. There are one hundred enemy, but since each of our men is worth [in a fight] two of any enemy, we don't need so many as 100, but somewhat fewer. How many is determined by substituting numbers as follows:

R must be at least equal to B√ (p)/p = 100√(2)/2 = 100 x 0.7071 = 70.71
0.71 of a man doesn't mean a lot, so we would send at least 71 men to deal with the infidel.

'Survival:

'We might be interested in knowing how many of our gallant lads we could expect to return after having wiped out the enemy. We will call this s(RED) and it is found by:

s(RED) = √(R²   -  B² /p)

Also not Crusades - but depictions of Japanese samurai
battles suggest a kind of randomness about them...

(If BLUE wins, then

s(BLUE) = √(B²   -  pR²)

'If we sent 71 men (REDs) then we would expect to see 

s(RED) = √(71²   -  100² /2) = √(5041 - 5000) = √(41) = 6 or 7 
i.e. six or seven survivors

'If we sent 70, thinking the fraction [0.71] unimportant, we would lose all our men, and the surviving enemy would equal 

s(BLUE) = √(100² - 2x70²) =√(10000 - 9800) = √(200) = 14.14'
Call it 14 survivors.

Assumptions: 
'Lanchester battles presuppose several things: 
1.Morale is not a consideration; battles are fought until one or both sides are annihilated. This unrealism is not, in my view, sufficient to invalidate the theory outright. Morale, if it is related to risk, might well be related in some way to the statistically expected outcome.
2. Chance variations are not addressed. When we arrive at a value for relative effectiveness, we are talking of statistical expectations, or mean. We will continue in this article to set this element aside.
3. These battles are general melees in which everyone is equally involved. All RED individuals have an equal random chance of being eliminated, and so do all BLUES, though the two need not be the same.'

Why should Lanchester's theory of battles not 
apply to this sort of battle?

Limitations.
'Concerning that last assumption, Graham Jones observed that the theory failed when confronted with a lack of randomness. In particular, he cited an instance in which, to use his numbers, 1200 formed infantry in 6 ranks fought 600 enemy in 3 ranks. Both sides have frontages of 200, and only these can kill or be killed. Hence the emphasis on melees.'


But need that be a limitation? Could Lanchester's ideas be extended to accommodate such a scenario? I rather thought they could...

To be Continued: Extending the Theory.  The Second half of my Southern Sortie article.