SAN JOSÉ STATE UNIVERSITY
ECONOMICS DEPARTMENT
Thayer Watkins

Lanchester's Theory of Warfare

Frederick Lanchester, a British mathematician, tried to apply mathematical analysis to warfare. Mathematics in the form of operational research or logistics has some very practical applications for the military. Lanchester was interested in more abstract analysis of warfare. For example, there is the Principle of Concentration that says that the best strategy is to concentrate the whole of a belligerent forces on a definite objective. Lanchester's analysis provides a justification of that principle.

Lanchester developed his analysis from a set of differentinal equations. Let n1 and n2 be the numerical strengths of two military forces. Then the rate of casualities for the two forces are:

dn1/dt = -c2n2 dn2/dt = -c1n1

where c1 and c2 are coefficients that reflect the effectiveness of forces 1 and 2, respectively.

Lanchester now asks the question as what condition determines the fighting strength of two forces. He argues that the two strengths are equal if both suffer the same proportional losses; i.e.,

(dn1/dt)/n1 = (dn2/dt)/n2

This condition, combined with the differential equations, then implies that

-c2n2/n1 = -c1n1/n2 or -c2n22 = -c1n12or c2n22 = c1n12

Thus the fighting strengths of the two forces are equal when the products of the squares of the numerical strengths times the coefficients of effectiveness are equal. In other words, the strength of a fighting force is equal to the product of the square of numerical strength times the effectiveness of an individual fighting unit, cini2.

This justifies the Principle of Concentration. There are economies of scale in military strength.

Lanchester illustrates the implications of this deduction by considering the case in which a machine gunner has the effectiveness of sixteen riflemen. He then asks how many machine gunners would be required to replace 1000 riflemen. By his calculation the number is

1000/(16)1/2 = 1000/4 = 250.

Lanchester also considers alternate conditions. Suppose firepower is directed at positions rather than individual soldiers. The casaulities would then be proportional to the density of the force as well as the rate of fire. Thus,

dn1/dt = -(c2n2)(n1/a1) dn2/dt = -(c1n1)(n2/a2)

where ai is the area over which force i is deployed.

An application of the previous analysis indicates that under these conditions the strength of a force is proportional to numerical strength rather than the square of the numerical strength.

In other conditions, such as the defense of a narrow pass, the strength of a force may have little to do with its numerical strength. This would be the situation in mountainous terrain.