Lanchester's Law: Too Few American Soldiers?

Whatever one's opinion about the justification (or lack thereof) for the war, the failure of the Iraqi regime to collapse immediately has surprised many people.

To some it indicates that the American military strategy is based on excessively optimistic assessments. Retired general turned TV pundit Barry McCaffrey, for example, recently stated that Defense Secretary Donald Rumsfeld has sent too few troops and too little equipment to Iraq to dislodge Saddam Hussein.

Struck by such second thoughts among some in the military, I offer the following marginal contribution to the discussion.

It concerns Lanchester's Square Law, which was formulated during World War I and has been taught in the military ever since. Used to model conflicts from ancient times to the Battle of Trafalgar and Iwo Jima, it may be relevant to the Iraq situation as well.

Although usually couched in terms of differential equations (the context in which I first came across it), Lanchester's Law can be paraphrased as follows: "The strength of a military unit — planes, artillery, tanks, or just soldiers with rifles — is proportional not to the size of the unit, but to the square of its size."

What does this mean?

The Effect of Quantity

Before returning to its application to Iraq, let me illustrate Lanchester's Law with a simplified conflict between two armies, denoted army QN (for quantity) and army QL (for quality), each of which has 500 pieces of artillery.

Assume furthermore that the two sides' artilleries are more or less equivalent in effectiveness and are capable of destroying each other at a rate of, say, 6 percent per day. That is, after one day each side will have 94 percent of what they had the day before.

Neither side has an advantage, but let's alter the balance of power in a way suggested by a nice example from a new book, What the Numbers Say, by Derrick Niederman and David Boyum. What happens if we assume that army QN can increase its artillery to 1,500 pieces, three times as many as army QL has?

There are two consequences. One is that each of QL's artillery pieces will receive three times as much fire from QN's artillery as before because QN now has three times as many guns as QL. Because of this QL will lose artillery at a rate of 18 percent (3 x 6 percent) per day.

The other consequence is that each piece of QN's artillery will receive one third as much fire from QL's artillery as before because QL now has one third as many guns as QN. Because of this QN will lose artillery at a rate of 2 percent (1/3 x 6 percent) per day.

Lanchester's Law in this case: Tripling the number of pieces of army QN's artillery leads to a nine-fold advantage (18 percent versus 2 percent) in its relative effectiveness.

The Effect of Quality

Armies can increase not only the number of their artillery (or planes, tanks, or soldiers) but can also increase their quality, and so we alter the balance of power again.

Let's assume that army QL counters QN's numerical superiority with better technology. It does so by upgrading its 500 pieces of artillery to make them each 9 times as accurate as QN's 1,500 pieces of artillery.

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