Lanchesterâ€™s Laws in Modern Warfare
Presented by Jacob Choi Advised by Dr. Ali Enayat
Generally, an Artful Strategy must be supported with A thousand swift four-horse vehicles, A thousand armored four-horse vehicles, A hundred thousand armored troops, And provisions transported for a thousand miles. Sun Tzu, The Art of War, written between 480-221 B.C.
Our Age of Combat • Fourth Generation Warfare (4GW) • Asymmetric elements • Unconventional warfare • Unmanned vehicles • Satellites, real time information • Direct soldier-to-command communication
The Theory Behind Wargames •Combat models rely on differential equations (DE’s, or DiffEQ’s) •Differential Equations model how the world works and reflect rates of change •Unlike algebra, where we find solutions, DE’s are used to model known scenarios with various inputs/influences
Lanchester’s Laws Lanchester’s Linear Law (Unaimed Fire) • rR-bB=k, where rR is total number of blue troops slain by red forces, and bB is the converse. Unique Case: If k=0, then rR=bB •If k>0, then rR>bB, so red forces win •If k<0, then rR<bB, so blue forces win
Lanchester’s Laws Lanchester’s Square Law
•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 •What is a unit?
Conditions for Lanchester’s Square Law •Aimed Fire •Combat tactics, not strategies •Unit sizes may vary, but need to stay “small” —company sized •Combat units are comparable types •For example: infantryman vs. infantryman, or main battle tank vs. main battle tank
Lanchester’s Square Law dB dt = dB = −rR ⇒dB = rR dR dR −bB dR bB dt
bBdB = rRdR ⇒ bB = rR + k ∫ ∫ 2
bB2 - rR2 = k, where k is a constant
Application: Division of Forces “The other method of turning the enemy…entails the risk of attending a division of our own force, whilst the enemy… retains his forces united and therefore has the power of acting with superior numbers against one of our divisions.” General Carl von Clausewitz, Vom Krieg (On War)
Where R represents the total number r1 + r2 + ... + rn of Red units, composed of r , r , …., r 1 2 n 2
1 1 1 ∑ r + ... + r ≤ R 1 n 2
1 1 1 1 1 1 Ex : + = 2 = ≤ = 4 4 16 8 4 2
Application: Division of Forces •Suppose red is an inferior force, and blue units are three times as effective as red units b=3r •Red forces are two-fold the size of blue forces, R0 = 2B0 •rR2-bB2 = r(2B0)2 – 3rB02 = 4rB02 – 3rB02 = rB02 > 0 •Thus, red forces defeat blue forces, but in this situation, both groups are still unified
Application: Division of Forces •Now suppose that the blue units can divide the red forces into two approximately equal sized groups, and blue fights them in two sequential battles •Blue units are still three times more effective than red units In the first battle: • B0 blue versus R0 reds leads to B1 =
2 B0 3
•About 82% of the original blue units remain In the second battle, red forces have more troops to begin with, but blue still wins with 57% of its original units remaining
N (Battle Sequence)
β Effectiveness 2
87% 71% 50%
89% 77% 63% 45%
b β= r
How does the β-index apply to warfare? •β-index draws from multiple input variables according to a desired depth of accuracy •Generalized formula for Bn
n Bn = 1 − B0 β •The β-index is only a ratio, so we are only interested in keeping it high!
•US Army extends Basic Combat Training to ten weeks long, instead of eight weeks, as of October 2008, to emphasize “every soldier a warrior”
•Sea and air mobility allows the long-range capabilities of bringing heavy or large units to the battlefield. •An increase in force projection power corresponds to an increase in the β-index
Having more troops creates a target-rich environment for the enemyâ€Ś
…while small teams of covert forces are ideal for infiltration and have a much higher β-index thanks to rigorous training and many capabilities •Small vs. Large forces results in Lanchester’s Linear Law!
â€œThere is Local Intelligence; There is Inside Intelligence; There is Counter-intelligence; There is Deadly Intelligence; There is Secure Intelligenceâ€? Sun-Tzu
“You may fly over a land forever; you may bomb it, atomize it, pulverize it and wipe it clean of life—but if you desire to defend it, protect it, and keep it for civilization, you must do this on the ground, the way the Roman legions did, by putting your young men into the mud.” T. R Fehrenbach, This Kind of War, 1963