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How corrupt is your goverment?
Anywhere from none to as many as we need to give jobs to waste taxpayer money.
My point is, for example Great Britain managed a LOT of colonies around the globe just fine because of low population count therein.
Thus, empire sprawl penalty from small outposts in outlying systems is disproportionally large. Should be 0.5 or something.
It's actually just as you said.. they're small outposts and a bunch of low colonies.. that doesn't really support itself for an efficient society towards unity or science.
Also the new mechanic they already have is admin costs from pops.. so.. they are already doing what you said in a way.
Each system adds 2. Each colony adds 2.
So a system without a colony costs 2, while a system with a colony adds 4.
EDIT: Also 1 for each disyrict on those planets.
Speaking of wide vs tall,
this is kinda nonsense. Because more administrators are unlocked by tech, unity and civics, not the empire size.
Also, from what i know about Stellaris lategame, you either capture half the map or lose anyway because AI federations are rubbish vs endgame threats.
Empire sprawl and administrative capacity have exactly one purpose: to slow down your expansion and prevent one empire from locking up half the map and becoming a "runaway". Sprawl makes absolutely NO sense within the game universe. You have to break the fourth wall, and view the concept from OUTSIDE the game. As the PLAYER.
Anyone here ever heard of Master Of Orion? Hopefully everybody has. :) Anyway, that game had a much better rule for reducing the chance of a runaway: your ships had limited operational range. The game called it "fuel cell technology". At the start of the game, your ships had a range of (if I remember correctly) four light-years--and it was impossible for your ships to move further than four light-years away from one of your colonies. So you had to colonize stuff in order to get further out into space. And colony ships were expensive. There was technology to give your ships bigger fuel cells, but it took a LONG time to get to where you could travel anywhere on the map.
Old-school for the win, eh?
Also, MoOII had a LOT more colonizable planets, so the reference is not quite relevant to the discussion.
What i am trying to communicate is that administrative capacity should be affected less by claimed systems and more by meaningful power-wise objects.
If an empire claims a lot of systems it would cost alloys meaning lesser fleet, it would be cut down to size by more powerful rivals who would be infuriated by the border friction, so this is already covered.
On the topic of a "runaway" there are already end-game threats to deal with...
So the administrative capacity cost could be adjusted based on the number of habitable planets settings, leaving default as is but shifting on higher numbers towards less costs from systems and more from colonies.
Not in the early game. A lot of the planets CANNOT be colonized (at all) until the proper technology is researched. With some of the harsher planets, that takes a LONG time.
It would definitely be a better answer. Sprawl is annoying and immersion-breaking.