Lanchester's laws
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Lanchester's laws are mathematical formulae for calculating the relative strengths of a predator/prey pair. This article is concerned with military forces.
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). 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).
In modern combat, however, with artillery pieces firing at each other from a distance, the guns can attack multiple targets and can receive fire from multiple directions. Lanchester determined that the power of such an army 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. It relies on the fact that when either side has more units it means that they have also more surface-area that opponent can hit (thus diluting the amount of fire hitting each unit, reducing their rate of attrition) and also more firepower (thus increasing the enemy's rate of attrition).
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.
Lanchester's Laws apply to all pairs of predator/prey in the natural world. It explains why the relative predominance varies from year to year of the ladybird and green fly, foxes and rabbits and many others.
(The majority of this article is copied, with permission, from an article by Ernest Adams appearing in the Gamasutra video game developers' webzine. See External Links below.)
[edit] External links
- "Kicking Butt By the Numbers: Lanchester's Laws", a Designer's Notebook column by Ernest Adams in the Gamasutra webzine
- 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
- Lanchester combat models, to appear in 'Mathematics Today'
- lanchester.com
