Who would win in a fight: five twenty-first century US Marines or fifty doughboys from the First World War? Would you rather face a hundred duck-sized horses or one horse-sized duck? Everybody, even folks with no knowledge of military matters, understands the idea of quantity versus quality. But for most people, it remains a vague abstraction. Sometimes quantity is more important, sometimes quality is, and no one can really lay out the relationship between the two with much precision.
Frederick Lanchester, a British mathematician during the First World War, thought differently. He developed a series of equations designed to establish the connection between quantity and quality in modern warfare.
In premodern warfare, according to Lanchester, the only way to directly target an enemy combatant is by being right next to them. Only a few people can fight at any one time, so, an advantage in numbers provides no more benefit than an advantage in quality. If 300 Spartans are fighting a horde of Athenians, and the Spartans are twice as good as the Athenians (which in this case means twice as good at dealing damage), then the Spartans will be expected to kill 600 Athenians before heroically dying. To put it in numerical terms, the fighting power of a premodern force equals quantity x quality.
However, modern combat changes everything. With accurate firearms, many soldiers can shoot at a single target. This means that everyone involved in a combat can theoretically fight at once, making superior numbers far more powerful. Think of it this way: if you’re fighting against a bunch of attackers in a narrow hallway and everyone has swords, you can fight people one at a time and so you might stand a chance. If everyone has guns, your attackers can all shoot at you at once, and you’re in big trouble. Lanchester’s equation puts the combat power of a modern force at quantity squared x quality.
In this example, seven soldiers are fighting ten soldiers of the same quality. Instead of a close match, Lanchester’s model gives the team of seven a combat strength of 49, and the team of ten a combat strength of 100, more than double. Lanchester predicts that the red team will wipe out the blue team with the loss of only three soldiers. Napoleon may not have said that “quantity has a quality all its own,” but Lanchester would certainly agree all the same.
Lanchester’s model of modern warfare puts the combat effect of increasing quality as linear, but the combat effect of increasing quality as quadratic. In short, if the enemy has twice as many soldiers/planes/tanks as you, you need soldiers/planes/tanks that are four times better in order to reach parity. Lanchester applied his equations to model aerial warfare in the First World War. While German planes in the late war tended to be qualitatively superior, allied planes decisively beat them through numbers.
Now, you might think that Lanchester’s laws are so simple and abstract as to be basically useless, and you wouldn’t be wrong. In real warfare, there are always additional factors at play like terrain, morale, and tactics. However, what Lanchester’s laws are really good for is analyzing fictional combat.
Let’s leave the skies above the western front and turn to aerial warfare in a galaxy far far away. In one issue of the Darth Vader comics, Darth Vader unexpectedly runs into a squadron of X-Wing fighters. What follows is perhaps Vader’s most impressive display of his abilities in a pilot, as he mows through the poor rebels effortlessly. Although he doesn’t technically wipe out the squadron, the story makes it clear that if not for the intervention of a certain young Skywalker, he would have. With this information, we can use Lanchester’s law to determine exactly how much better Vader is than the average rebel pilot.
First, we need to assess each side. Vader is piloting a Tie Advanced, a ship roughly equivalent to a X-Wing in combat ability. Neither side has any tactical advantages. So far, we have an even field. As for numbers, I count at least 26 X-Wings against Vader. Because of the quadratic effect of quantity, we first square each side’s numbers:
- Rebel Quantity-Derived Combat Power: 676
- Vader Quantity-Derived Combat Power: 1
Now we multiply combat power by quality: assume that the rebels have a skill of one. We know that Vader wipes the floor with the rebel ships, so we know that Vader’s total combat power needs to exceed the rebels. Therefore, we conclude that Vader is, at a minimum, 677x better than the average rebel pilot. This leads to other interesting implications. How many lowly Tie Fighters would it take to defeat Vader? Assuming that a Tie Fighter is half as dangerous as an X-Wing (surely a fair comparison given how they get routinely massacred), then Vader in a Tie Advanced is therefore 1353x more dangerous than the average Tie Fighter. Therefore, it would require the square root of 1353, or about 36-37 tie fighters to even have a chance against Vader.
I’ll let the readers make their own calculations about how much better John Wick is than the gunmen he faces, or how many Agent Smiths it would take to defeat Neo. There’s only one limit: remember, Lanchester’s square law applies only when any combatant can target any other combatant. If the hapless minions are charging your protagonist one by one and getting stabbed in turn, you’re looking at Lanchester’s premodern model, not the one we used for Vader.
That’s it for now, apologies for my infrequent blogging, but while Peace Corps has many advantages, frequent WiFi is not one of them.