Attacker Protection Build Banks/Armories

We assume the following buildings:

10% homes
10% farms
15% guilds
5% towers

and would like to determine the most efficient way to split the remaining 60% of buildings between banks and armories to maximize the number of troops you end up with OOP.

Banks increase income, so you can measure benefit by total income / natural income (the income you would have without banks), which measures the proportion of extra troops you can train. Armories decrease unit costs so you can measure benefit by the inverse of the amount by which they reduce to also measure the proportion of extra troops you can train. For 400 acres, and 240 to be split between the 2, let x denote your number of banks and 240-x denote your number of armories.

Income I(x)=(1+1.25x(1-x/400)/400)(20250+25x)
=-1.953125e-4x^3-0.080078125x^2+88.28125x+20250

Benefit1=I(x)/20250

Training Cost Reduction R(240-x)=1-1.5(240-x)(1-(240-x)/400)/400
=9.375e-6x^2-7.5e-4x+0.64

Benefit2=1/R(240-x)

Total Benefit B(x)=Benefit1*Benefit2=I(x)/(20250*R(240-x))

Then all you have to to is maximize total benefit or take dB/dx=0 and check that d^2B/dx^2<0. I've solved this equation to find x=116.67539. This means you can maximize the number of troops you can train if you get 29% banks and 31% armories. This is close enough to 30% banks and 30% armories for easier calculations.

Assumptions: 400 acres, 240 disposable acres, no unfilled jobs (This last assumption won't hold all the way, but it won't effect calculations by much because the majority of effects of both buildings vary together with the number of workers. The only thing that does not is the flat rate the banks give you, which is a function of the building efficiency, which is a different function of the number of workers. However, the changes are so small that we can simplify the model to what I have.

for 90% of people... 2:1 banks : arms ratio is a good start =)

lots of math mumbojumbo but +1 cookie for you doing some math on it.


Normally I just go slightly more banks than arms methode... never done any math on it.
Normally you dont have all your jobs filled either, which means homes added pop factors in too.

reguardless of it all... you cant go wrong with a good high mix of % in homes/banks/arms.

30% banks.
30% armories.

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Originally Posted by Utopia4life

lots of math mumbojumbo but +1 cookie for you doing some math on it.


Normally I just go slightly more banks than arms methode... never done any math on it.
Normally you dont have all your jobs filled either, which means homes added pop factors in too.

reguardless of it all... you cant go wrong with a good high mix of % in homes/banks/arms.

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seeing as all acres have 25 jobs/acre, and you wont go below 100% BE during protection, i dont see why you would want homes in there...

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Originally Posted by VT2

30% banks.
30% armories.

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depends ALOT on the race you are going and how high an elite count OOP you want...
for human/DE/gnome i would lean much more towards banks than arms
an orc with -50% draft, unless you want decent elite #'s oop, stick with just banks
elf and dwarf i could see more of an even split for... but even so, would still go more towards banks

As I said, the even split is designed to maximize the number of troops you get trained OOP. It actually doesn't matter whether or not what you're training, but I like a full elite build from the start. The thing that could make a difference is that you may get all of your troops trained, if you're a gnome with elites or something else with all ospecs, and still have leftover cash, in which case banks would beat armories by a long shot.

Intuition actually did lean me towards banks over armories and in the long run during the course of the age, banks pay better as long as you're not growing too heavily. I was actually expecting more like 35%-25% banks to armories rather than 29%-31%, so the solution surprised me as well, but it should be accurate and give us the best combination under the assumptions. As well, keep in mind that I'm allocating 60% of my buildings between the 2; if you use any other amount, the solution is different and may lean one day or the other.

For a good chunk of protection, your BE will be 100%; it will start to dwindle towards the end as your draft completes. Even so, whatever proportion of homes you will run as your war strat, it's probably good to leave the 10% homes you started out with and not raze them or build more during protection.

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Originally Posted by Eigenvector

As I said, the even split is designed to maximize the number of troops you get trained OOP. It actually doesn't matter whether or not what you're training, but I like going for the stronger unit for offense/defense, which would be elite/dspec for everyone except DE, in which case we can fight about what to train. The only thing that could make a difference is that you may get all of your troops trained as gnomes because they're so cheap and still have leftover cash, in which case banks would beat armories by a long shot.

Intuition actually did lean me towards banks over armories and in the long run during the course of the age, banks pay better as long as you're not growing too heavily. I was actually expecting more like 35%-25% banks to armories rather than 29%-31%, so the solution surprised me as well, but it should be accurate and give us the best combination under the assumptions. As well, keep in mind that I'm allocating 60% of my buildings between the 2; if you use any other amount, the solution is different and may lean one day or the other.

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as i said... there is only 2 races this works for... elf and dwarf... and even then, maybe not dwarf =)

Well if you have an argument for your the case, I'd like to hear it. My case is the result of calculations.

I'm assuming everyone knows this, which everyone should, but in case there's any misconception, % rate buildings have diminishing returns; the higher the proportion of the building you have, the lower the rate an extra unit of it will give you, or for effect rate y on percentage x, dy/dx->0 as x->50. That accounts for the tendency for the split to become more even, even if one is clearly better than the other at initial levels.

You are overcalculating and underthinking. OOP is not a magical time point for which you need to optimize, consider what kind of build you want to grow into and how you want to play post protection.

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Originally Posted by Eigenvector

Banks increase income, so you can measure benefit by total income / natural income (the income you would have without banks), which measures the proportion of extra troops you can train. Armories decrease unit costs so you can measure benefit by the inverse of the amount by which they reduce to also measure the proportion of extra troops you can train. For 400 acres, and 240 to be split between the 2, let x denote your number of banks and 240-x denote your number of armories.

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I didn't even read the whole thing seems to much of a load of crap. One can measure the benefits of banks by taking the difference of natural income from total income. Taking the difference will give you a more meaningful answer then "/". One can measure the benefit of armories by taking the different detween the normal cost of the unit and the new cost of the unit. For example, defspec. costs 300gc the new defpec cost is only 266gc. Take 266 from 300 and you get 34gc saved on defspec training.

This is just another case of someone trying to sound smart rather than be smart.

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Originally Posted by Eigenvector

Well if you have an argument for your the case, I'd like to hear it. My case is the result of calculations.

I'm assuming everyone knows this, which everyone should, but in case there's any misconception, % rate buildings have diminishing returns; the higher the proportion of the building you have, the lower the rate an extra unit of it will give you, or for effect rate y on percentage x, dy/dx->0 as x->50. That accounts for the tendency for the split to become more even, even if one is clearly better than the other at initial levels.

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why wouldnt a DE or human build more banks than someone else?

+30% income and ToG are 2 damn good reasons in my mind....

also, orc saves money on draft costs, so therefore has less need for arms
gnome's army is similarly dirt cheap, so higher bank % are better for them

also, at lower drafts banks, with minimal army, banks far far far outweigh arms, arms rely on 3 things to be profitable, reduction in wages (N/A) draft costs (see the point here) and troop training (doesnt happen until last 18 hours)

i can potentially see the point in building a decent % of arms so they come in just before training time, but no sooner...

EDIT:
your post also assumes draft + training, that only happens for 18 of the 50 hours after buildings come in... for 30 hours, you are only drafting... not training... for that you want a higher ratio of banks over arms

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Originally Posted by vines

I didn't even read the whole thing seems to much of a load of crap. One can measure the benefits of banks by taking the difference of natural income from total income. Taking the difference will give you a more meaningful answer then "/". One can measure the benefit of armories by taking the different detween the normal cost of the unit and the new cost of the unit. For example, defspec. costs 300gc the new defpec cost is only 266gc. Take 266 from 300 and you get 34gc saved on defspec training.

This is just another case of someone trying to sound smart rather than be smart.

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I like this argument, counter this one Eigenvector, i dare you.

eigenvector nice analysis, though i think your equations for I (x) has an error

I(x)=(1+1.25x(1-x/400)/400)(20250+25x) i believe should be I(x)=(1+x(1.25-x/400)/400)(20250+25x)

at least thats the what i reduced my bank income eqn down to... i hope you did this with a spreadsheet. or matlab. if you solved for dy/dx by hand my hats off to you, and you need another hobby.

PS, i find it easier and more useful to do these calculations in a per acre analysis. and then put the output in hours to train.

Too much maths, too many assumptions, too few real numbers and too little practice.

I think you forget that armories have draft costs reduction too and you draft like mad in protection.

Also, you forget stables and other buildings that are handy OOP.