**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 n_{1} and n_{2} be the numerical strengths of two military forces.
Then the rate of casualities for the two forces are:

#### dn_{1}/dt = -c_{2}n_{2}

dn_{2}/dt = -c_{1}n_{1}

where c_{1} and c_{2} 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.,

#### (dn_{1}/dt)/n_{1} = (dn_{2}/dt)/n_{2}

This condition, combined with the differential equations, then implies that
#### -c_{2}n_{2}/n_{1} = -c_{1}n_{1}/n_{2}

or

-c_{2}n_{2}^{2} = -c_{1}n_{1}^{2}

or

c_{2}n_{2}^{2} = c_{1}n_{1}^{2}

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, c_{i}n_{i}^{2}.

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,
#### dn_{1}/dt = -(c_{2}n_{2})(n_{1}/a_{1})

dn_{2}/dt = -(c_{1}n_{1})(n_{2}/a_{2})

where a_{i} 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.

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