Lanchester's laws

Lanchester's laws are mathematical formulae for calculating the relative strengths of a predator/prey pair. This article is concerned with military forces.

The Lanchester equations are differential equations describing the time dependence of attacker and defender strengths A and D as a function of time, with the function depending only on A and D. [ [http://www.rand.org/pubs/monograph_reports/MR638/app.html#fn11 Lanchester Equations and Scoring Systems ] ]

In 1916, during the height of World War I, Frederick Lanchester devised a series of differential equations to demonstrate the power relationships between opposing forces. Among these are what is known as "Lanchester's Linear Law" (for ancient combat) and "Lanchester's Square Law" (for modern combat with long-range weapons such as firearms).

Lanchester's Linear Law

In ancient combat, between phalanxes of men with spears, say, one man could only ever fight exactly one other man at a time. If each man kills, and is killed by, exactly one other, then the number of men remaining at the end of the battle is simply the difference between the larger army and the smaller, as you might expect (assuming identical weapons).

The linear law also applies to unaimed fire into an enemy-occupied area. The rate of attrition depends on the density of the available targets in the target area as well as the number of weapons firing. If two forces, occupying the same land area and using the same weapons, fire randomly into the same target area, they will both suffer the same rate and number of casualties, until the smaller force is eventually eliminated: the greater probability of any one shot hitting the larger force is balanced by the greater number of shots directed at the smaller force.

Lanchester's Square Law

In modern combat, however, with firearms engaging each other directly with aimed fire from a distance, they can attack multiple targets and can receive fire from multiple directions. The rate of attrition now depends only on the number of weapons firing. Lanchester determined that the power of such a force is proportional not to the number of units it has, but to the square of the number of units. This is known as Lanchester's Square Law.

More precisely, the law specifies the casualties a firing force will inflict over a period of time, relative to those inflicted by the opposing force. In its basic form, the law is only useful to predict outcomes and casualties by attrition. It does not apply to whole armies, where tactical deployment means not all troops will be engaged all the time. It only works where each man (or ship, unit or whatever) can kill only one equivalent enemy at a time (so it does not apply to machine guns, artillery or, an extreme case, nuclear weapons). The law requires an assumption that casualties build up over time: it does not work in situations in which opposing troops kill each other instantly, either by firing simultaneously or by one side getting off the first shot and inflicting multiple casualties.

Note that Lanchester's Square Law does not apply to technological force, only numerical force; so it takes an N-squared-fold increase in quality to make up for an N-fold increase in quantity. "Parts of this article are copied, with permission, from an article by [http://www.designersnotebook.com Ernest Adams] appearing in the [http://www.gamasutra.com Gamasutra ] video game developers' webzine. See External links below."

See also

* Attrition warfare
* Maneuver warfare

ources

* "Numbers, Predictions and War", Col T N Dupuy, Macdonald and Jane’s, 1979

Footnotes

External links

* [http://www.gamasutra.com/features/20040806/adams_01.shtml "Kicking Butt By the Numbers: Lanchester's Laws"] , a Designer's Notebook column by Ernest Adams in the [http://www.gamasutra.com Gamasutra] webzine
* [http://www.rand.org/publications/MR/MR638/app.html Lanchester Equations and Scoring Systems] , appendix to "Aggregation, Disaggregation, and the 3:1 Rule in Ground Combat" by Paul K. Davis, Rand Corporation publication MR-638-AF/A/OSD
* [http://arxiv.org/abs/math.HO/0606300 Lanchester combat models] , to appear in 'Mathematics Today'
* [http://www.lanchester.com lanchester.com]
* [http://www.tactica.org.uk/10.html "Lanchester Laws: In Sales Campaigns"] You can find a White Paper here describing the application of Lanchester Theory to High Value Sales Models in the Business World.