Actually I saw this model when I was 12~13 in a science magazine, and I got so impressed that I thought of ideas of a game that uses this model...

I caught a whiff of (what I later discovered to be called the “Lancaster Model”, tracking references was tough in the early Internet era) on a documentary I saw in the late nineties and re-derived it all; later I speculated about the dynamics if effectiveness depended upon the *ratio* of number of soldiers and came up with a cube law - plus a very weird concept in phase space I ended up naming “homocyclicity” because I’d never seen a mention of it anywhere else.

I've first heard about this a decade ago in a thunderfoot video about the invasion of France. I've actually abused this in Ultimate Adrmiral: Dreadnoughts (video game) to win 130:24 engagements with like 129:5 losses. Ironically, I did it with quality (technology + big ships) over quantity. The thing is (as you rightly pointed out) that it's never purely square or linear. While a 12'' gun may have a range of 15km, it's accuracy/hit probability is going to fall off along 1/x or 1/x^2 outside point blank range. This means that with early (1890-1920) tech there is a significant gradient in accuracy across the space your own formations take up. At least in large battles. Also, naval warfare is quite a good example for the continuous model because while ships are large, partial damage will degrade speed, stability (accuracy/effective range/ exponent in this formula), rate of fire, etc... What that means is that you can form a dense formation, focus fire on exposed ships on the edge of their formation, match your focused (quadratic) firepower against that ship while most of their fleet is too far away for accurate fire support (near linear model). Once that ship is crippled, it's accuracy (exponent) is degraded to the point where you can safely sail past to single out the next ship. It's even better when you can get them to chase you because then you can just focus on the singular lead ship until flooding/ engine damage reduces it's speed to the point where it falls back and another ship takes it's place, basically spoonfeeding their fleet (waiting in line) into your guns. Basically you use research spending, design philosophy and maneuvering to outmatch them in exponent rather than numbers. It's also interesting to look at how warships evolved. Form ironclads and pre-dreadnoughts (armored omnidirectional gun bricks fighting in lines) to dreadnoughts (focus on central big, long range gun turrets for maximizing accurate fire on single targets across broad firing arcs instead of narrow broadsides), multi-layered fleets with smaller, short ranged screening ships outside the main formation, the shift to aircraft carriers during the 40s with a focus on long range force projection and front-loaded burst damage (see Pearl Harbor vs hour-long shore bombardment by battleship guns) and later the rise of cruise missiles and new models like salvo theory accounting for missile defense saturation/depletion and launch or loose. (if you ever want to do a collab with drachinifel)

I'm working on a video game that has battles that I think should follow the Lanchester's square model. Now I'm tempted to do some Monte Carlo simulations and see how well it holds! Thanks for introducing me to a cool model!

Very interesting, I've seen this phenomenon in startegy games lile Advanced Wars or Age of Empires and now I know why.

Great!

Great video professor !!!😊😊

Hi Dr. Bazett! I'll keep this in mind next time I'm on campaign with my troops.

This effect is actually pretty apparent in RTS games like StarCraft too. In that game, even looking battles will often be fought until it becomes clear that the numbers are avout to swing strongly in one direction (as the hyperbola approaches an axis in your model). In this case, as long as you cut your losses and disengage before the quadratic dropoff effect becomes too large, you mostly only feel the linear effects of the hyperbola and therefore it is not too costly. However, the difference in only a few seconds in deciding to disengage can sometimes have significant consequences as one side starts running away with the battle.

hmmm, feel like I've seen this somewhere... if only this video was out before the second homework due date :)

Sir please come up with a series on Numerical methods for ODE & PDE.

I dunno what room you're recording in, but the reverb is a lot. i see your lapel mic, but if it's omni-directional it'll still pick up a lot of the early reflections. if you're mixing with the camera mic, i recommend side-gating with

0:03 ain't the sound echo-ey??

Interesting. How would one verify the validity of the model? Also, one strong assumption, in the model, is independence. That is, a linear relationship is possible because of the assumption that each of the soldiers will act in a manner, in battle, that's consistent with, say, two equal militaries battling against each other, What if, though, you had a strong military versus a weak one. And each member of the military knows that its either strong or weak. Then, the soldiers may not act independently but may be highly correlated with each other. That is, the last solider in the weak military is not likely to fight against all of the remaining members of the strong military. Instead, there's likely a tipping point in which the remaining soldiers of a military know that they're going to lose and simply stop fighting. If this is true, then modeling militaries and how they fight should be more about modeling covariates and correlations and the likelihood that a solider is going to stop fighting.

Thank you trefor im first

sir i am actually working on math. And i have made some formula of polynomial equation, number theory, geometry etc.sir i want to talk with you.

I totally misread that title. You don't want to know what I thought it said mwahaha

Is this only one battle or is this the entire war?

sir i am actually working on math. And i have made some formula of polynomial equation, number theory, geometry etc.sir i want to talk with you.