Oberlus wrote: ↑Sat Aug 13, 2022 12:34 pm
I think a good solution for this is to make influence upkeep grow slower. 0.5*N^1.33 instead of 0.4*N^1.5. But Geoff expressed his preference for simple exponents (it was already a concession to use 1.5).
If you adjust exponent on the cost side but not the earning side, you lose certain... proportionality? Not clear if that was intention. Such formula gives hard cap of 1319 planets, and soft cap of 555 planets, with 139 set to production. If you put cubic root instead of square root on the earning side as well, like 30^(1/3), then it gives 101 soft cap at 27 productive planets, 245 hard cap.
This all could be designed with math if you know exactly what you want, and math ending up complicated for the player I don't think is a good argument. I can attest that it is confusing at first anyway, but all a player should learn is that upkeep
per planet grows with amount of planets you have, so eventually you hit a cap where all planets would need to be set on influence. And that policies etc. help putting that cap away. No need for specifics I think.
Maybe I can offer again a perspective.
U(x) being upkeep for x planets, and I being average influence per planet, gives you y = U(x)/I of planets that need to be set to influence.
You can put current U(x) and I = sqrt(30) on a chart:
- InfluencePlanetsVsTotal.png (21.4 KiB) Viewed 757 times
I put f(x)=x there as well - you can see those intercept at about 190 i.e. the hard cap I've calculated earlier
Ultimately perhaps what is the most interesting thing is difference of those two functions, which is the number of "productive" planets
p(x) = x - y
p(x) = x - U(x)/I
for U(x) = 0.4 * sqrt(x) * (x-1) - 3 and I = sqrt(30) you have
p = x - (0.4 * sqrt(x) * (x-1) - 3) / sqrt(30)
and it charts as such:
- UsablePlanets.png (9.28 KiB) Viewed 757 times
Where this hits 0 (for positive x), it's the hard cap - you get zero productive planets, all get set to influence (at 190 planets). Where this has maximum, is the soft cap - it's the maximum number of productive planets, 28 productive planets at 83 total planets
You can put those calculations into Wolfram Alpha and see:
https://www.wolframalpha.com/input?i=0+ ... rt%2830%29
and for maximum, too, which gives what I came to earlier, maximum of 28 productive planets at 83 total planets:
https://www.wolframalpha.com/input?i=ma ... 2830%29%29
You can tweak your function to fit predetermined points more or less. p(x) = 0 for hard cap, so you can "reverse engineer" some factors if you plug in the expected hard cap to p(x) and equate to zero
U(x) = a * x^b * (x-1) - 3
If I wanted to keep b as square root, but have hard cap at 400, then
400 - (a* 400^(1/2) * (400-1) - 3)/sqrt(30) = 0 => a ≈ 0.274923587721888
But then soft cap comes at
max(x - (0.274923587721888 * sqrt(x) * (x-1) - 3) / sqrt(30) )
I.e. 60 productive planets at 177 total planets
If I wanted to keep b as square root, but have soft cap at 100 planets, then need to get derivative to find maximum:
p = x - (a * sqrt(x) * (x-1) - 3) / sqrt(30) => p'(x) = 1 + (a*(1 - 3*x))/(2*sqrt(30)*sqrt(x))
p'(100) = 0 => 1 - (299*a)/(20*sqrt(30)) = 0
a = 20*sqrt(30)/299 ≈ 0.36637
And then hard cap comes at 226 planets, and for the soft cap of 100 planets you get 34 productive ones (really looks like with current formula what you get is 34% productive planets at soft cap), Though thanks to discreet nature of things, rather than going the differential route, you may go the earlier route I shown i.e. U(x+1)-U(x) <= I
Ultimately I think a different shape of p(x) would be preferable. Maybe something skewed right, so initially it takes longer to get to soft cap, but then you hit hard cap at roughly same time?
Maybe instead of exponential growth, you'd rather see logistic growth? Planet population should preferably grow logistically, which is more or less exponential at first then slows down / becomes more or less logarithmic. Don't know the actual formula used in FO for pop growth, but presumably along those lines, and seems like Oberlus wishes to have something like that for number of usable planets, where hard cap is the entire galaxy but you want to get there slowly after hitting some form of soft and hard-ish cap (I can envision cost that eventually means every planet on influence, and every new planet on influence, but never going negative so able to colonize entire galaxy)