In an effort not to derail the "Undead & Homes?" thread any further, I have decided to make a separate thread to discuss the merits of homes vs. TGs.
The calculation I posted in the other thread was indeed not entirely correct (I was lazy.) This is an attempt to answer the question a little more in-depth.
1. Minimizing draft rate for a given OPA + DPA combination:
The total amount of units you have available to train is given by (draft rate) * (max pop). Max pop does not depend on the ratio of forts to TGs, so we can treat it as a constant as we are only messing with this ratio.
I have assumed in the below calculation that you want to dedicate 30% of your land total to TGs and forts, and are trying to find the optimal combination - whether to build those structures at all is a different matter.
x = % of TGs
a = total acres; o_val = off value of your elite; d_val = def value of your spec; ME = base Mil. Eff.
Initially:
(draft rate)*(max pop) = opa*a/(o_val*ME*(1+1.5*x*(1-x)))+dpa*a/(d_val*ME*(1+1.5*(0.3-x)*(1-(0.3-x))))+thieves+wizards+soldiers+(stuff in training)
Simplifying, we get:
draft rate = (a*[opa/(o_val*(1+1.5*x*(1-x)))+dpa/(d_val*(1+1.5*(0.3-x)*(1-(0.3-x))))]+K)/(max pop)
To find the extremum, first differentiate for x:
DR' = (a/max pop)*[1.333*opa*(x-0.5)/(o_val*(x^2-x-0.6667)^2)+1.3333*dpa*(x+0.2)/(d_val*(x^2+0.4x-0.8767)^2)]
The next step is set DR' equal to zero and solve for x. This is incredibly complicated (I think unsolvable) with all the unknowns, so I solved the equation for two sets of concrete values: orc (highest off elite) and halfer (lowest off elite, excluding fairy). Both times, I used 100 OPA, 45 DPA.
Result:
For orc, the optimal [TG,forts] combination given the assumptions is [17.3%,12.7%]
For halfer, the optimal [TG,forts] combination is [21.6%,8.4%]
So looking from a pure "efficiency" perspective, it is indeed best to run a good mix of both forts and TGs. The optimal ratio of course varies by race and by the specific OPA/DPA values you are aiming for.
2. However, the more realistic scenario for an attacker is to maximize OPA at a given DPA and draft rate.
I kept messing up on the differentials, so I decided to take a more brute force approach. Note that DPA refers to static DPA, and draft rate means "elites & dspecs" in this context; no thieves, wizzies etc.
y = pop. per acre that can be trained as elites = (pop per acre)*(draft rate) - dpa/(base_ME*d_val*(1+0.15*(0.3-x)*(1-0.3+x)))
OPA = y*o_val*(1+0.15*x*(1-x)) = [(pop per acre)*(draft rate) - dpa/(base_ME*d_val*(1+0.15*(0.3-x)*(1-0.3+x)))] * o_val * (1+0.15*x*(1-x))
I simply iterated through this equation in steps of 0.01 for x. The results for x seemed to depend most heavily on ppa and base_ME, and fluctuated within a few points of [18%,12%]. For example:
(1) The "ideal" ratio for an orc with 26 PPA, 200% wages, draft rate of 65% and target DPA of 45 would be [18% TG, 12% Forts]
(2) The same ratio applies to an orc with 28 PPA and 100% wages.
(3) A halfling with the same numbers as (1) would have ratio of about [16%,14%]
(4) A human with 30 PPA, 200% wages, DR of 70% and target DPA of 40 maxes out at [25%,5%]
To be fair, the difference between, say, [13%,16%] and [16%,13%] is pretty small. There are other factors that should influence your decision between TGs and Forts - attacks made:suffered ratio, playing style of KD, cost of retraining,potentially being unbreakable, yadda yadda.
In conclusion, forts seem more viable than I thought they did, although TGs still have a clear edge.
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