Lanchester's Laws in AoE2

[@marcogarcia9715]

After 2 years, finally we have a Spirit of the Law video about an actuel Law

[@UnDeaDCyBorg]

Well, one could argue it's actually more of a definition or theory to explain an observation. Maybe that was the meaning behind SotLs name all along?

[@chrisc3825]

That’s because he’s the Spirit of the Law and not the Letter of the Law 🙃

[@B0zZz3]

I AM THE LAW!

[@ShinFahima]

I thought his name was Law. Like the Bruce Lee lookin' guy from Tekken.

[@linkinboss2]

@@B0zZz3 LAAAAAAAAAAAAAAAW

[@TheGloriousLobsterEmperor]

Spirit, you just did more to explain Lanchester's laws than my maths teacher. Granted...they never covered Lanchester's laws since it wasn't in our curriculum. But you get the point.

[@Cythil]

What? They do not teach you how to play RTS games in school? What as this world come to!?

[@ethanboardman873]

Cythil That wasn’t the point of his comment

[@shaunwasher922]

Ethan Boardman its a joke

[@Cythil]

Correct. It was a joke. Though teaching math in school and relating it to a activity that student do do help them understanding concepts and make math seem relevant. Which we all know it is or else we would likely not be here. ^_^

[@TheGloriousLobsterEmperor]

@@Cythil I know, shocking.

[@xotl2780]

Interestingly enough, the legendary general of the Mongols, Subotai the Conqueror, learned all he had to know about world domination using research he gained from watching Spirit of the Law's videos on military tactics.

[@stylesrj]

He then proceeded to pimp out his rams to be 50% faster.

[@bananan7]

Coming after the Ant's video!

[@_antares031]

oh hi, Spirit of the Lanchester's Law :wave:

[@LightGreenCorona]

Your content still gets better and better and more and more mathematical. Prof. SOTL in 2038: "Applying quantum logarithm spirals in the build-up patterns of Byzantine Culture in Age of Empires 2 DE to prove the Hawking radiation in far-away black holes" You should start Online Courses on AoE2.

[@SY-jd7dm]

Spirit will be proving that Constantinople was conquered because of an inside job. And he will do it with F'N MATH!

[@oscars7948]

Yes Spirit. That is nice. I will just pretend I followed the math part.

[@satyakisil4289]

At least you learnt 14 knights survive in 20 v 15 fights.

[@nelsonmejiaslozada9362]

HAHAHAHAHAHAHAHAHA

[@physixger]

C'mon, it's not that hard. (Disclaimer: first two years of a physics degree you'll feel the way you do, then it changes.)

[@shelldie8523]

A math teacher rolls in his grave somewhere

[@arnau9638]

As a physicist, I am impressed. Damn good analysis with those differential equations

[@theninjainblack1237]

I am always happy to find other physics majors or physicists "in the wild" :)

[@davidgustavsson4000]

The modified formula was poorly motivated though.

[@user-kf7qg3hz2v]

Also as a physicist, this is 1st year bachelor level of math. Still nicely made though.

[@marinmaths3826]

As a physics student, I’d say Newton would be proud!

[@user-kf7qg3hz2v]

​@@marinmaths3826 I don't want to take anything away from sotl but seriously, this is by far the easiest kind of differential equation. 1st order separable DE are simply solved by integrating both sides, which he did in this vid.

[@fabriziomariano1332]

Me: I don't want graphic explanations. I demand the equations! 4:42 Also Me: Yeah baby, give me more. 5:09 Also me: *Oh shιτ*

[@MichaelMeyerRS]

Do any math at university and you realize that is simple obvious stuff. (The proffesor would go over it in the speed of Spirit here and you would be Like: WUT)

[@zwiebelas]

@@MichaelMeyerRS Most people get puzzled in advanced calculus due to small details step by step. Teaching at this level is a complex and challenging task. The division in 5:09 is a simple procedure, but it can be mindblowing if you are not familiar with the Leibniz's notation for f'(x) to begin with. You can give props to SotL for trying to explain it in simple terms (rate etc.) for an aoe2 video. High quality job as always.

[@arrano7087]

@@zwiebelas Here is how I solve it: docdro.id/J9HTMXq Tell me if anyone wants further explanation. I know many people would have done it differently. For me, the derivative of a function at any given point is equal to the slope of the tangent at that point. I have never used any other interpretation. Therefore, when people "arrange" differentials in equations like if they were variables I don't really understand what they are doing. Does it make any sense from a mathematical point of view?

[@hesseldekraai]

@@arrano7087 late answer here. I'm a physicist, not a mathematician, but you can't really do it. It is however common practice to see differentials as divisions within a lot of fields (including physics) and usually you will get the right answer. You can also get the right answer in a more mathematical rigorous way and see that it is the same but that takes more effort. You get into trouble however when the thing that you are differentiating has a dependency on multiple variables. In the video however when all of the starting conditions are given the only variable is time.

[@wilburdemitel8468]

@@MichaelMeyerRS you can barely write coherent english and you claim that this is simple obvious stuff. Lacking verbal IQ.

[@AS.47]

Don't even play aoe2 but I always find your videos interesting and entertaining :)

[@levelinguponthenarrowpath]

Lmao wth

[@Yamaazaka]

Wow crazy. This my childhood nostalgia

[@robertlewis6915]

same

[@meliocurie1809]

Same here

[@andremilanimartin3338]

Yeah, me too. I have played like 5 games of AoE in my entire life.

[@Markus-zb5zd]

All hail the lanchester! That's why militaries all around the world like to only fight fights with an overwhelming power.

[@DuckieMcduck]

Sun Tzu did say "Victorious warriors win first, and then go to war".

[@planescaped]

@@DuckieMcduck Zhuge Liangs "The wise win before they fight, while the ignorant fight to win" was always a more eloquent version of that.

[@johnapple6646]

A Lanchester always pays their debts. ... Wait

[@Chepicoro]

Not the Germans before WW2

[@marioanothlp]

@@Chepicoro Obviously, I mean, Lanchester started to think about his equations in 1916.....in the middle of WW1.... Everyone knew more is better, but noone actually thought it'd be THAT much better.....so noone had that until after WW1 and Europe had no bigger wars before WW2, because of the repairments that were necessary....so, yeah, noone had that idea before WW2.

[@MrToddino]

Love your content! Always makes my day 100% better, keep it up!

[@TorreFernand]

Don't you mean 168% better?

[@noidontwantthat7237]

@@TorreFernand 169%*

[@FlammieLL]

I want Spirit of the Law to be my future childrens Math teacher.

[@aelix56]

Wow ice cold.

[@someguy3987]

I want Spirit of the Law to be the father of my future children.

[@nobleactual7616]

I want spirit of the law to have my children

[@someguy3987]

I want Spirit of the Law to be my children

[@rainbowunicode8352]

Another awesome example of why I wish the concept of differential equations should be a mandatory topic in schools. You don't need to be able to solve them, just to understand how someone might create them in the first place. It's the real power hidden inside calculus.

[@ashina2146]

CONCENCRATE YOUR FORCES *Laugh in Mangonels*

[@UnknownSquid]

That's why a modern military mantra is instead "CHECK YOUR SPACING". Mangonels got pretty advanced.

[@khanch.6807]

@Krzysztof Milański Nukes, artillery, precision air strike and other AOE exist now in modern military. Concentrating can only be done if you're enemy doesn't know your location which in neigh impossible with satellites and recon drone.

[@feelthepony]

@Krzysztof Milański absolutely right. to try to make it sound clearn and simple. the name of the game is about who can reinforce faster. and it is not exactly about JUST gathering more resources than your enemy into certain area. defenders win wars by losing battles(and not losing badly) while your actual main force goes for a big counter attack. at its very core warfare didnt change that much,but still some things are approached in a different way.

[@northDSX]

iñigo montoya war. war never change

[@Nekrocow]

MANGO!

[@AmmarFarisA]

In the future, we will implement the Art of War book into AOE sequels. Our lord Bill Gates.

[@angelosreiper4051]

They did that and named it Rome: Total War.

[@ExcelonTheFourthAvalonHeirs]

I'm from the future and oh boy, they really did. They did put Art of War "mission" on AoE II DE. You're mad lad!!!

[@thestriker1595]

@@ExcelonTheFourthAvalonHeirs I was just about to reply that but you beat me to it! (:

[@wanderingskeleton52]

@@angelosreiper4051 What makes you say that?

[@ARojas-dg2ce]

Bill Gates is trash

[@BizarreWords]

Me in HS: I hate math! i hope I never use it again! Me rn: Wow, this is fascinating and useful! Great video, Spirit of the Law! love it!

[@Y.M...]

High school maths is often taught as a means without an end, divorced of purpose or elegance, or even suggestions of real life application. If you engage students with problems they actually care about solving, things end up being much more fun.

[@mralpaca2150]

@@Y.M... High school maths demands you to learn it and know how to solve problems yourself. After watching this video is anybody going to know how to differentiate and integrate and change constant to solve any problems themselves ? If SOTL says hey kids here are 5 similar problems you can try it yourself, is anybody going to do it ? Yeah I thought so. That would take a lot of work and people would give up just like they did in HS

[@mortache]

@@mralpaca2150 but even in university i got bored of maths for this same reason. Yeah those maths did come to use in 3rd/4th year, but its still a lot of meaningless jargon when you first learn it.

[@ganknujnal]

5:05 My mum gave me a thumb up when she saw this in on my monitor, thanks Spirits of The Law.

[@calebbridges4748]

I love this channel so much. I got into AoE2 again aftet YEARS because I took lots of maths and love the way you explain a game that I didn't understand in 1999.

[@ramandian]

I’ve played AoE2 for years and it makes me sad that I’ve only just found you recently. Amazing videos!

[@kln5705]

Spirit, you never fail to delight me (and others, I'm sure) with this sort of high quality content. Cheers, mate!

[@bbcmotd]

SotL: more maths and formulas Me as linguistics major: ah shit here we go again

[@GregTom2]

In ancient warfare, the square law also applies to most scenarios that are not a choke point or a fortress, because a more numerous army will flank or envelop the enemy, and because the morale of an army is proportional to its size. Basically, a moderate difference in numbers will mean that the losing army will rout first, and stop dealing damage, while still being cut down as they flee.

[@Skaitania]

The result of this analysis is not too different than what you found out in the 'Focus Fire' video, but with a new approach of explanation. Well done, I feel like my mind has come full circle.

[@jurgnobs1308]

well yea, essentially the whole point is to lower the enemies damage per second faster than he lowers yours. focus fire decides who wins in equal battles. the lancaster law explains why you have a lot more suriviving units than proportional in an unequal engagement. of course, if you are fast enough, you can combine the two principles and use decent focus fireing in a situation where you outnumber the enemy, and you will get an even more extreme result.

[@chrisc3825]

Had my Calc classes taught important examples like this I might’ve become an Engineer

[@simonroger565]

Hahaha probably

[@kotzpenner]

Wow, I was just nearly done with rewatching ALL your AoE2 videos, started with Civ overviews, then other tips, then gameplay vids, then vids I've watched already. So I really thank you for this fresh video so I can get my kick!

[@darrenr49]

I was sent this video by a friend. I am building my own game but havn't heard of this law (and i don't know any mathmatics - not enough to properly play with formulas at least!!) Now using this video I can work off it and learn all I need to learn. Thanks a lot for this fantastic learning tool. It's really given me a springboard to jump off so I can confidently take my understanding to the next level! Thanks again!

[@tvremote9394]

Not gonna lie, at first I read the title as "Lannister Law" and thought this was about always paying your debts

[@mikeyfisher4256]

A Lanchester always formulates his debts.

[@cameronsmith3047]

Game of loans

[@leonardozayasm.4983]

Fun fact Lanchester family and the War of the Roses inspired the creation of the Lanissters and the plot in GoT

[@charlierenard1221]

@@leonardozayasm.4983 Lancaster, but you're close.

[@leonardozayasm.4983]

@@charlierenard1221 Goddamn i got those two mixed up

[@qwerasdf-oy6uo]

This is super apparent in Total War, since it fills all the assumptions. The auto-resolve battle function also takes this into account as well.

[@princessbinas]

This is super interesting. I never really thought about this. Always knew that more units on your side and not being divided would help you win more often. I will try to amass more high quality units than my opponent for sure. Hopefully I can get better at my efficiency at executing the correct build order for booming despite using hotkeys. I am freaking slow at it. Maybe a video at how to become faster would be nice.

[@mattshepherd8762]

Yeah I hadn’t thought about it either. I think the thing that stood out to me the most was that I may be investing too much in upgrades and not enough in sheer numbers of units. I think this may be good evidence that more unupgraded units might often be better than fewer upgraded units in some situations

[@DoomWalker42]

I love the nostalgia from these videos even though I don't play AOE anymore. Still cool to actually understand the tactics and strategy behind it now. I was never a really competitive player.

[@samueljett7807]

It's simple: spirit makes a video, I get notification, I stop everything I'm doing, like, comment, and watch.

[@vegannegan9652]

I was working in a bar. When I got the notification I immidiately dropped my tray and started watching. Lost my job but it was worth it.

[@DatBrasss]

You should do a top 5 defensive civilizations list. I'd love that!

[@ingolifs]

I have also experimented with Lanchester laws in various games (AoE, starcraft 2 etc.) and this video mostly agrees with what I've found. The switch between square and linear doesn't come with melee vs ranged, it comes when there's no more 'surface area' for additional units to engage in combat with. In my tests starcraft 2 melee units fared better with the square law than Aoe2 units, because they are more mobile and have better pathing, which means they have less downtime after finishing off an enemy. The deviations you see are actually due to the discrete nature of the attacks and due to micro. Fascinatingly, if units lost strength based on their health, micro actions (specifically removing a wounded soldier then returning him a few seconds later to battle after he is no longer targeted) are a lot less effective. The units-always-do-full-damage thing has inadvertently resulted in the retention of much tactical depth in strategy games, and it's not immediately obvious that that would be the case! The difference between perfect micro and perfectly bad micro is huge. Perfect micro is essentially where all units are retained up to the point where they all have 1 hp before any of them die, and during the battle they continue to attack with maximum dps. Perfectly bad micro is where each unit dies before the next one begins to lose hp and attack effectiveness is lost at the maximal rate. Perfect micro vs perfectly bad micro with an identical army is really interesting, because the result depends on the health to damage ratio of the unit (or how many shots the unit can take before it dies). If units one or two-shot each other, micro doesn't matter much. If units are tanks with weak damage output, the damage mitigation that perfect micro achieves approaches 50%. If you want to make a very micro-heavy game, give the troops heaps of health and weak attacks. If you want to make it macro heavy, give them low health and strong attacks. Look at the difference in playstyles of Warcraft 3 and Aoe2. One's mainly about micromanaging your small band of units, the other's about maintaining a strong economy so you can continue to pump out masses of expendable units who are waypointed at the enemy's base. This is partly due to other game elements, but the health pool and therefore the effectiveness of micro plays a heavy part in the different styles of games. I find it fascinating that we are only just scratching the surface of the game theory of Strategy games. From a very simple and easily-understood system comes a surprisingly large amount of strategic depth!

[@knightsdiege]

Had my teacher used Aoe to teach math. Amazing work bro.

[@kaitokatsuki6862]

Ah, the intro is finally back. Glorious.

[@ChrisJones-xd1re]

"Divide and conquer" ("Divide and rule") is not simply waiting for the enemy's forces to be split. It is using diplomacy to gain favor with smaller groups inside a larger group, eg a Roman general bribing one or more Germanic tribes to not fight with the others, or even against them.

[@isorukuyamamoto9168]

Pretty nice stuff. I like how this also covers why a 4x stronger unit can actually only take up 2 enemy units at once. If we define strength as the time to kill an enemy vs how long it takes him to kill you, which i always think is useful. But in direct 2 v 1 melee combat the law is fully quadratic, except maybe for a tiny bit of targeting delay. So the power of this relation for melee, as you said, really depends on the situation. My estimate though is that in harsh cavalry vs archer or cavalry vs infantry fights it generally will tend really close to 2.0

[@DanteSenior04]

I have been waiting for a video about Lanchester's Laws for a very long time thanks a lot

[@frankfs8415]

I always loved that about Civilization games I've played. Your opponent could have a swarm of better tech units, but if you managed to catch one of those at low hp, even the base Warrior could take them down. *I'd rather let Civ do the math though*

[@BlackWhiteCater]

Thank you for this great video! For those of you interested in this kind of stuff: check out the military principles of force concentration and defeat in detail. They both have quite an impact in a lot of situations in RTS games. Super entertaining video!

[@Trollvolk]

Damn, your intro still gives me goosebumps. Together with an almost lovecraftian equation damn.. XD

[@huzaifaliaqat1491]

Did you have a math major or are you generally interested in math? I was wondering because your ability to apply so many formulas to obscure situations in the game is highly impressive

[@Hyproxious]

He didn't come up with these, he's just good at explaining them

[@huzaifaliaqat1491]

@@Hyproxious i know, my question was how does he apply them on aoe2 so casually

[@samueljett7807]

@Stephen Thomas I was first exposed to calculus in the 9th grade and I was hooked. Simple as that

[@soko-ban]

He is majoring in math I believe, don't know where he's at right now tho

[@runemaster96]

Where has this channel been hiding - quality work

[@gravijta936]

Lanchester would be proud of you Spirit.

[@alex_zetsu]

This is why it is very important for 1 HP units to be as effective as health ones in most games. I admit that in Advance Wars and AW2, it is actually a good thing damaged units are weaker, but this is because it leads to direct units having an incentive to attack and get that first strike. In general, you do not want damaged units to suffer damage output or it will lead to very unfun games.

[@MarSHornetsandWasps-v26]

I already learned to use this to get rid of my enemies in a different RTS game, and I didn't even know it was called Lanchester's laws... now I'm so proud of it, thanks a lot (I need something to be proud of in quarantine...)

[@markdowse3572]

SOTL, please don't shoot me..... :) Until just now, I had NEVER even heard of "Lanchester's Laws", so thanks for this great video. But as a D&D player and DM with decades' of experience I already knew the 'theory', because a basic rule of play was -- "DON'T SPLIT THE PARTY!" ;)

[@tylerfehr7883]

This video is so cool. I'd love to see more formal analysis of this game. I wanna go check out that guy's paper now too. I had no idea something that existed already.

[@Dinnye01]

"To quickly derive the equations..." And he starts actual derivations.

[@ProCreedify]

I rewatched this masterpiece today

[@kneight4927]

I'm pretty sure this falls under that portion of the video which explains how certain things like bonuses or strategy can change the battle, but I feel there's more weight on focused fire than unit number. This is a thing that I always use even if my numbers are lower and I result victorious. It does require paying attention to the whole thing, seeing which enemies are weaker and stuff, but still effective enough to do more with less. Great video! I don't play AoE, but I love how everything is explained here

[@sushantkulkarni9346]

I simply cannot imagine the amount of research and work you've put in into this video. Love your content. Kudos!

[@andycheung22ac]

Watching vids like these is like that Bale and Freeman moment in Batman. Great stuff

[@ducktape-3470]

New Video and The Awesome Title Track. Ahh... I just saw your channel a week ago and I am binging all the videos since then. My Fav Bonus - +1 Arrowguys Attacc vs Palisade Walls and Uprotected Homes

[@shmoequey2038]

This video is still one of the best AoE tip videos of all time

[@tominis4444]

Just when I needed more numbers in my life! Much Love, One with the spirit of Lanchester

[@meyes1098]

But this kind of breaks down when you micromanage. For example, in the 5 vs 10 archers, if we assume the 10 archers are split up such that 2 of them will attack 1 of the other 5, and the 5 archers all focus on one target, then they manage to kill 3 of them before they're wiped.

[@chiquita_dave]

Do you have a background in mathematics or something related? You did a very good job of explaining the differential equations to derive Lanchester's Laws EDIT: Also, a note about the error encountered on smaller size armies: you touched on it in the video but this can also possibly be attributed to the fact that Lanchester's Laws are based on continuous functions, whereas the damage dealt and the number of soldiers are discrete. As your army grows in size, the army size and damage dealt behave more continuously because the error introduced by each individual soldier is canceled out and it approximates the behavior of a continuous function, in the same way a normally distributed sample of 10,000,000 looks much more like a bell curve than a normally distributed sample of 10. I would be interested to see if an army of significant size (~1000) would yield results similar to the square law. Thanks for sharing this super interesting bit of math!!

[@RickyMuzakki]

Finally a LAW with formula and equation. Now that's some real LAW channel :)

[@xl1ry]

Its insane how people cant understand this law almost applies to every single game out there. THANK YOU FOR THIS AMAZING UPLOAD

[@YourFavoriteAerospaceProf]

As a professor of aerospace engineering who loves AoE, I really much enjoyed this video :)

[@bananenzijndebeste2]

Didn’t someone comment on one of the palladin videos that Spirit of the Law is not math-y? I guess that controversy has been solved with this video...

[@McdoTomato]

Spirit of the Law... talking about law... in a law titled video... this is what perfection looks like.

[@ShadowDragon1848]

A video about tech vs. numbers would be very interesting in my opinion. Btw. you should make military theory videos ^^

[@GummieI]

Might have to narrow it a bit more than that though, since there is a LOT of different techs and civ bonuses, so maybe numbers vs blacksmith techs specifically would be interesting

[@Jonathan-bu7iv]

Wouldn't the difference largely be due to the fact that 10 skirms can shoot at a skirm with 1hp left and thus, lose out most of their damage? Like, what this fails to predict is that sometimes you'll overkill, and sometimes you'll underkill. If a skirm survives with 1 hp, I still have to throw another full javelin at him.

[@michaelqiu9722]

The difference between Lanchester's model and experiment is most due to the fact the model uses continuous functions to model the armies’ size and their attack strengths, whereas in reality an army is made of discrete units, each of them attack in discrete strikes instead of dealing damage continuously every moment. In the video SOTL’s way of explaining this is by saying “the model assumes an unit at half up deals half he damage”, which isn’t exactly right, but gets the idea across. Overall, the take home message is that Lanchester's laws are limiting laws that are most true when you have a prolonged battle between two massive armies, where the individual discreteness of things is negligible. Just like water feels like a continuous fluid only when you have something like mols of them, instead of a few molecules

[@-Raylight]

Wow I realized never learned this at school, now I know what Lanchester's law is Thanks SOTL xD

[@thanhlevan2185]

Spirit! Aoe 2 DE has 4 new civilizations. I need a video review as soon as possible!

[@AyushPandey]

Oh man! Always had it in my mind that it would be so so nice to develop the math for all that happens in AoE2. Thanks! It's a great start.

[@Daxter250]

dunno why but those skirmishers fighting against each other at 5:00 could make a damn good screen saver!

[@apresmidi153]

This is one of the most interesting videos on RTS gaming I've watched! Bravo!

[@GamerZakh]

This was great, appreciate the explanation, thanks!

[@Kapuist1]

I'm curious as to how--sometimes--the opposite happens: outnumbered bands of troops successfully standing up to overwhelming forces.

[@Daedwartin2]

So I presume you have heard of the concept of Force Multiplers. Well, the primary reason they are called that is quite literally this law. A Force Multipler is something that in effect multiplies the effective number of troops you have in comparison to the enemy. So things from Artillery to Aircraft to just plain old better morale can all change the effective numbers. However, the issue is you dont have a clear idea of what your number is, and the enemy is even harder to know. With the right actions, one can in effect trick the enemy into doing a suboptimal action that allows you to win despite being outnumbered.

[@DoomDutch]

I've been aware of how and why this is happening for ages. Just didn't know it was called the Lanchester's Law :D

[@arthurthiele979]

This is an awesome video- more of these please!

[@hbarudi]

Some people don't know calculus, but I do. Interesting function about finding out about how units in rts games count when it comes to damage.

[@Athielmon]

Brillant. Didnt know the Lanchester laws, but by looking it, I can see that you can even optimize your number of units with their effectiveness in order to approach a closer exponential constant to equal enemy forces, like in the Knights vs Paladins example. You can do it by some extra calculus with Lagrange's optimization method.

[@kalpitpatel1314]

I feel like I'm taking a Calculus Exam with all of those differential equations :P Great Video!! It'd be awesome to include more such equations in future videos because they make sense to a student like me :D

[@TheVarrio]

Spirit of the Law video after almost two months. HYPE

[@Ibrahim_Orhan]

Nice video. Keep mathy bro. I love your contents. Can you make a video about rams which can tank a lot of damage in combat and how they are useful even they don't attack to a single unit. Second suggestion is a video about how monks are really powerful? Healing really worth it? In an unlimited resource condition how many percentage of your army should consist of monks?

I always see your videos, but this in particular I like a lot, because you explain math in a simple manner, and very useful way! thanks!

[@dalelerah]

There is no other video of yours that I have shared more than this one, simply because of all the math 😍

[@qubex]

I was doing my thesis on this! I always assumed that he’d have solved over the time domain, and I did that and then generalised it by making the rate of loss depend on the ratio between two armies’ strengths.

[@DisasterLord]

It's amazing to see how many thesis there are on starcraft and Doom, it's such an amazing rabbit hole.

[@blessedbethelord4835]

Literally calculus for the win...

[@HansLemurson]

Warfare is all about balancing the competing goals of Concentration of Force, and Protection of Assets. If you concentrate your forces into a single unbeatable army, your opponent can refuse to engage and destroy your lands with raiding. But if you try to defend all of your outposts, then you'll lose decisive battles and can be Defeated in Detail. No guaranteed winning strategy. You end up with a sort of "Rock Paper Scissors" dynamic with Small, Medium, and Large force concentrations. Medium beats small (outnumber), Large beats Medium (outnumber) , Small beats Large (large wasted their time, and you can afford to outnumber enemy elsewhere). Victory ends up following the principles of "Gerrymandering" where you make sure that you win in as many single places as possible, and then give no contest everywhere else to not waste resources on lost fights.

[@apprentice2101]

It's very impressive to see legit scientific research quality in your AoE videos. They are the best on KZbin by a great margin. Thank you so much for making these!

[@jqerty]

Godsj this was a great video. I love that you even cite papers and make your own mathematical models

[@IPromiseTomorrow]

Every video I see advanced math become useful the more I am convinced Spirit of the Law was a Mathematician/Strategist in another life.

[@pauloat]

Simpleton in my differential equation class: Why we learn this? when we are gone to use it?? Me, an intellectual:

[@cuprous7426]

r/iamverysmart

[@thelegendarypotatoman6188]

Keep up the great work ive been loving your videos. and I've been using what I've learned from you against my friends to piss them off.

[@ericjam6346]

"If slightly inferior in numbers, we can avoid the enemy". - Sun Tzu... That is so much more easier for me to remember! LOL!

[@joedemelfie5509]

Eric James though Sparta proved this wrong time and time again in ancient Greece world history. and he lost far more than he won anyway too. he still has some useful advice in war tactics sure.

[@lucianomantovani3567]

this is brilliant. more of this please

[@LamirLakantry]

I have never seen anything more nerdy in my life. I love it. XD

[@saminyead1233]

This is what I needed! Finally Sotl bringing calculus into Aoe!

[@weylin6]

Because slight numerical advantages are such a huge benefit that can cause things to snowball, I think it's good that there's units like the Scorpion, Onager, and Demolition Ship that can counteract those effects. Some games, like the browser-based OGame, have no such units, and so the strongest players are literally untouchable by any smaller force.

[@TheDarkDan12]

My day just got better

[@Recoil1808]

Also in many strategy games, concentrating your fire allows you to not only take out otherwise even foes, but even noticeably larger ones if they are spread out enough.. Especially, though not necessarily, when you get into hero units and/or healers.

[@Benjamin1986980]

I would like to point out an even better example. In Start Trek Armada, the Borg Fusion Cube is made by putting together 8 cubes from Next Generation. It doesn't get an attack bonus from this. In fact, I think it gets a decrease in stats versus the original 8 cubes. But this means that it maintains its full attack power throughout the entire battle, while the enemy is reduced. If a fusion cube battles a force of 8 cubes, it survives with a full half of its HP left.

[@michaelcross7665]

Yup, fully over my head this time when it comes to the raw math xD concept I get just fine though, very interesting good work and I love that you applied it to the paladin video