What is Lanchester's laws?

Lanchester's laws are a pair of simple mathematical models, first worked out in the early twentieth century, that describe how two opposing forces wear each other down. They are used in defense economics and operations research to reason about the effect of numbers versus quality. Their main lesson is that concentration of force can matter much more than a simple head count suggests.

How it works

There are two forms. In the linear law, used when forces engage one-to-one, each side loses strength in proportion to the other's effectiveness, and strength is simply numbers times quality. In the square law, used when forces can all bring their strength to bear on the other, losses depend on the opponent's current size. For forces A and B with effectiveness a and b, dA/dt = -b x B and dB/dt = -a x A.

The square law has a conserved quantity: a x A squared minus b x B squared stays constant. This means fighting strength grows with the square of numbers, so doubling a force's size makes it four times as strong. It also creates a feedback loop, because the side that loses units loses its ability to cause losses in return, and the decline accelerates.

Example

Say force A has 1,000 units and force B has 800, and each unit of either side is equally effective. Under the square law the quantity A squared minus B squared is 1,000,000 - 640,000 = 360,000, which stays constant. When B reaches zero, A squared equals 360,000, so A has about 600 units left. A force that is 25% larger ends with 60% of its strength intact, while a 1,000 against 1,000 contest ends in mutual exhaustion. These are illustrative numbers.

How to model it in Machinations

Use a Pool for each force and a Drain on each, with a state connection making the loss rate of one Pool depend on the size of the other, scaled by effectiveness set in Registers. Add an End condition when one Pool reaches zero. Make the starting sizes and effectiveness Levers, then Play and chart both Pools to see the surviving share.