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Maximum economic growth by OD

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7 years ago
How fast can your economy grow by reinvestment of overdrive if you have a limited number of mexes? I calculated it by solving a differential equation system. Here's the solution: The time t (in seconds) needed to multiply your economy by mu (= metal rate / base metal rate) is given by
gamma*t = mu-1+(delta-1)*ln(mu),
where
delta = 1/2*(base metal rate per mex)*(OD efficiency)² *(1+(energy efficiency)/(constructor efficiency)) = (base metal rate per mex)*179/2560
and
gamma = 1/2*(base metal rate per mex)*(OD efficiency)²*(energy efficiency) = (base metal rate per mex)*9/5120.
OD efficiency=0.25 comes from the overdrive formula 0.25*sqrt(energy per mex). Here you can see a graph of the time (in minutes) needed in dependency of the desired eco multiplication factor for base metal rate per mex=2. For this case, you need at least exactly 5 minutes to multiply your eco by 3 for example.

Due to idealized assumptions, this is an upper bound to economic growth. For example, limited grid and build range are not considered. That it is necessary to build generators and constructors to use the OD metal is considered, though. Also, continuity is assumed, which means that singus and caretakers can already produce partially if they are only finished partially, which is only a good assumption for many units. Otherwise it would make sense to build cheaper eco units first. But here, an energy efficiency of 225/4000 and constructor efficiency of 10/220 is used. You can also use the formula with other energy generators or constructors, though.

It can be seen that eco growth becomes linear at long time. However, Caretaker's own metal and energy rate is NOT considered here to only see how OD behaves. For even longer time, Caretaker spam allows exponential eco growth above this upper bound. To multiply your eco by mu with caretaker spam, you need time t (in seconds) with
K*t = ln(mu),
with caretaker efficiency K = 0.3/220. Now in practice, it would be interesting to know at which point caretaker farms become better than OD spam. I have calculated that, for base metal rate per mex=2, this is when you have multiplied your economy by a factor of 8.26, which means 3.75 singus per mex.
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7 years ago
Can I get an executive summary?
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quote:
Can I get an executive summary?

Once you have more than 3 singus per mex, spam caretaker. If you're out of space, spam Conch.
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7 years ago
Linear growth was our design goal when designing overdrive, so it's nice to see it should work as expected.
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7 years ago
That's good. For the old OD formula, eco growth seems to become linear, too, but with a slope that is bigger by a factor 4. Actually the above is already a summary because I spared you the calculation itself.

If base metal rate per mex is 4, then 20.5 sinugs per mex are better than caretakers. The dependency of the caretaker spam break even point in base metal rate per mex is nontrivial. For every mex constellation, it requires solving a nonlinear equation with Newton's method, which shows an untypically slow convergence. If anyone ever wants to make a widget for that, you can use the following algorithm:
F = gamma/K-delta+1
mu = 0
new_mu = big_enough(F)
while(|new_mu - mu|>0.01)
 mu = new_mu 
 new_mu = mu - (ln(mu)-(mu-1)/F)/(1/mu-mu/F)
end while
cost_effective_od_energy_per_mex = 16*(new_mu-1)²
[Spoiler]
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