Understanding Empire Growth Model

[Cymsdale]

I've been playing around with numbers to try to better understand how total empire growth is affected by total population size. Saying "It takes 11+ years for a pop to grow" doesn't really seem to tell the whole picture.

This post is not meant to be an argument for or against how the growth model works, I'm simply trying to get a better understanding of how it currently works in game.

I've worked out some tables below (and if my math is wrong, please feel free to correct me) that display what empire growth looks like at different number of planets and different population sizes per planet. Obviously this is simplified as not all planets are the same, and this doesn't account for any type of growth modifiers from technology or traits. I was simply trying to get an understanding of what the "base" looks like.

The table is broken down into three different types of empire. One has planets built up to somewhat of a "min" level of 25. We are assuming all planets have the capacity to keep the +3 growth bonus, and the empire is built such that when the planet has that many pops it is done. The same is repeated for 35 and 45 pop planets. The final column basically tells you for each empire size, if you were to build a new colony, how fast would you be able to fill it with pops.

The results are not too surprising. Empires with more planets have faster growth, and empires that keep their planets "leaner" on average have faster growth on top of that. An 18-planet empire can grow a new colony about 2-2.5 times faster than a 3-planet empire. So bigger empires definitely have an advantage growing pops, but not a huge one. Once again, this isn't factoring in any of the tech/trait modifiers and is only looking at a baseline. You might assume for instance that the larger empire is later in the game and has higher modifiers over the 3 planet one.

I hope some of you find this useful.

Spoiler: Tables

25	<- Pops per Planet
# Planets	Total Pop	Req. Growth	Total M Growth	Total T Growth	New Pop / Year
3	75	137.5	18	216	1.570909091
6	150	175	36	432	2.468571429
9	225	212.5	54	648	3.049411765
12	300	250	72	864	3.456
15	375	287.5	90	1080	3.756521739
18	450	325	108	1296	3.987692308
21	525	362.5	126	1512	4.171034483
24	600	400	144	1728	4.32
30	750	475	180	2160	4.547368421
40	1000	600	240	2880	4.8
35	<- Pops per Planet
# Planets	Total Pop	Req. Growth	Total M Growth	Total T Growth	New Pop / Year
3	105	152.5	18	216	1.416393443
6	210	205	36	432	2.107317073
9	315	257.5	54	648	2.516504854
12	420	310	72	864	2.787096774
15	525	362.5	90	1080	2.979310345
18	630	415	108	1296	3.122891566
21	735	467.5	126	1512	3.234224599
24	840	520	144	1728	3.323076923
30	1050	625	180	2160	3.456
40	1400	800	240	2880	3.6
45	<- Pops per Planet
# Planets	Total Pop	Req. Growth	Total M Growth	Total T Growth	New Pop / Year
3	135	167.5	18	216	1.289552239
6	270	235	36	432	1.838297872
9	405	302.5	54	648	2.14214876
12	540	370	72	864	2.335135135
15	675	437.5	90	1080	2.468571429
18	810	505	108	1296	2.566336634
21	945	572.5	126	1512	2.641048035
24	1080	640	144	1728	2.7
30	1350	775	180	2160	2.787096774
40	1800	1000	240	2880	2.88

 

[sillyrobot]

The way I look at the last column is that's how many new resources I have to work with to augment the whole existing empire. "Gee my 40-planet empire expects 4 new pops this year. That means I can afford to construct 2 tier-1 buildings or upgrade 2 buildings over the next year! Hooray! OK, I'm done. Wake me next year."

 

[Cymsdale]

The way I look at the last column is that's how many new resources I have to work with to augment the whole existing empire. "Gee my 40-planet empire expects 4 new pops this year. That means I can afford to construct 2 tier-1 buildings or upgrade 2 buildings over the next year! Hooray! OK, I'm done. Wake me next year."

That is a good point, the ideal pace you need to construct new districts or buildings across your whole empire will remain relatively flat through the game, only raising steadily as your empire expands, and shrinking a bit if you stop to let things fill. Your required APY (actions per year) in the late game essentially won't be much different than early on.

 

[Yourss]

The way I look at the last column is that's how many new resources I have to work with to augment the whole existing empire. "Gee my 40-planet empire expects 4 new pops this year. That means I can afford to construct 2 tier-1 buildings or upgrade 2 buildings over the next year! Hooray! OK, I'm done. Wake me next year."

Optimistically, we can hope that this is PDX trying to scale back empire infrastructure and pop management to add additional mechanics to keep the player's attention?

Realistically? Something went very wrong.

 

[sillyrobot]

Optimistically, we can hope that this is PDX trying to scale back empire infrastructure and pop management to add additional mechanics to keep the player's attention?

Realistically? Something went very wrong.

It feels like the dev group either has a strong bias towards or is trying to push the player base into a more militaristic game. Limit the actions possible outside of war while increasing the value of conquest. You want to grow this century? Take stuff from your neighbor.

What's weird is I would expect such a push would include tools for breaking up defensive blocs since they are a good part of why the galaxy becomes static. Although Nemesis had the opportunity to introduce operations to undermine treaties and/or federations, they didn't pursue it. Perhaps, the multi-player game doesn't suffer from treaty stasis as much?

 

[Olterin]

Notably, this means there's an ideal mineral surplus number that you want to maintain to keep "building" growth going at the same speed as "pop" growth, and this number doesn't change much unlike in the previous version of the game.

Edit: thank you for the tables, it showcases quite nicely at which point it's more efficient to just farm colony ship spam and automatic offworld resettlement via habitats. The cheese is as follows:
1. Set up habitats that you intend to colonize and have pops auto-resettle off of
2. Send colony ships
3. Disable all jobs on habitats as soon as they exit the colonization phase
4. Pre-build next wave of colony ships
5. ???
6. Profit!

At 20 habitats dedicated to the cause (taking 40-80 years to set up, depending on your influence income) and a megashipyard you could easily have way more than +4 pops per year even in a 5000 pop empire

 

[sillyrobot]

Notably, this means there's an ideal mineral surplus number that you want to maintain to keep "building" growth going at the same speed as "pop" growth, and this number doesn't change much unlike in the previous version of the game.

And is very small. At 2 tier-1 buildings a year, you're looking at maybe 1,000 minerals a year or 80 a month.

 

[Olterin]

Just to add to this thread. I wrote down a quick and dirty mathematical model of how the number of pops behaves depending on the time passed. The short of it is that you need to grow your number of pop growth centers linearly with time in order to maintain overall linear pop growth (i.e. a static empire-wide per-month rate). This seems in line with the table presented further above.

This has the problem of course that while such a pace can be maintained for a long time (there are plenty of planets in the galaxy to spam habitats on) - each individual planet will take ever-longer to grow a pop (unless one can still somehow manipulate the actual migration mechanic to funnel growth to a select few growth centers), to a point where it doesn't really matter that you're keeping linear empire-wide growth, as the next batch of pops will take longer to appear than it will take for the game to end. This is a similar issue as what existed with multiple pops growing simultaneously under the tile system.

Therefore, a sort of empire-wide growth should mitigate the effects of the granularity of this current system.

Such is the bane of approximating discrete functions with smooth ones, thus failing to take into account the specifics of the system in question. Because without looking too closely, this is quite an elegant solution to the ballooning growth rate that a galaxy would experience with pure "logistic" growth. In fact, previously the growth curve of the overall population was roughly quadratic in nature, due to the potential habitat spam (until one ran out of places to build habitats, which let's face it, is virtually never).

 

[King Harkinian]

It feels like the dev group either has a strong bias towards or is trying to push the player base into a more militaristic game. Limit the actions possible outside of war while increasing the value of conquest. You want to grow this century? Take stuff from your neighbor.

What's weird is I would expect such a push would include tools for breaking up defensive blocs since they are a good part of why the galaxy becomes static. Although Nemesis had the opportunity to introduce operations to undermine treaties and/or federations, they didn't pursue it. Perhaps, the multi-player game doesn't suffer from treaty stasis as much?

You can absolutely run the Smear Campaign operation against both sides of an alliance to potentially break them up. People have reported success with it, though time and luck is needed.

 

[Cymsdale]

I translated the tables into graph form:

Pops Per Year

www.desmos.com

P slider changes the pops-per-planet value.

 

[fourteenfour]

While the chart presents the well known fact that who has more planets grows faster is does not address the real bogeyman in the room,

Even if my empire grows two, three, or four, pops per year they are not necessarily where I need them to grow and does not account for the disparity of population growth across your planets. Essentially your charts both hide and show how bad the current solution is.

 

[Cymsdale]

While the chart presents the well known fact that who has more planets grows faster is does not address the real bogeyman in the room,

Even if my empire grows two, three, or four, pops per year they are not necessarily where I need them to grow and does not account for the disparity of population growth across your planets. Essentially your charts both hide and show how bad the current solution is.

I'm not trying to hide anything, these are just numbers. I don't present this information to argue for or against how population works, it's just to provide a better understanding so people who read it can better talk about what they are debating for/against. Statements like "you reach 400 population and then your population just stops growing" is not helpful (and just wrong).

As to your specific point about where pops grow, the auto-resettlement in the game more or less takes care of that. Yes, there is a bit of inefficiency in that you have to wait for them to move, but they eventually (and fairly quickly) go where they need to be. It's almost like an organic representation of the inefficiency of empire sprawl (much better than what the *actual* sprawl mechanic does...)

So if you build your late game ringworld, it can still fill up, at about a rate of 4-5 pops per year [keep in mind the graph is showing base numbers, so it will actually be faster than that when you account for modifiers].

 

[Yourss]

I'm not trying to hide anything, these are just numbers. I don't present this information to argue for or against how population works, it's just to provide a better understanding so people who read it can better talk about what they are debating for/against. Statements like "you reach 400 population and then your population just stops growing" is not helpful (and just wrong).

As to your specific point about where pops grow, the auto-resettlement in the game more or less takes care of that. Yes, there is a bit of inefficiency in that you have to wait for them to move, but they eventually (and fairly quickly) go where they need to be. It's almost like an organic representation of the inefficiency of empire sprawl (much better than what the *actual* sprawl mechanic does...)

So if you build your late game ringworld, it can still fill up, at about a rate of 4-5 pops per year [keep in mind the graph is showing base numbers, so it will actually be faster than that when you account for modifiers].

Regardless of the correctness, pops=economy is kind of drilled into our brains at this point. Even without automatic resettlement, 4-5 pops per year just isn't interactive. Since players are bored, they tend to prebuild their planets, which amplifies the problem. As I have put in other threads, it's great that we don't have the mid-late game micro hell anymore, but you can't flip it all the way to having nothing to manage in an empire management game.

 

[Olterin]

Yoinking @Cymsdale 's link to the tool, here's the graph representation of what I had looked at (well, part of, anyway): (EDIT: had to adjust the planet acquisition rate, it was ridiculously low, I had originally plotted over number of months passed, forgot to adjust the 'x' for planet number growth rate)

Total Pops over Time

www.desmos.com

(r...monthly growth rate; G...pop growth target; a...Empire Pop slowing coefficient; N_0...starting number of pops; P...number of planets)
(for simplicity the assumption was made that each planet grows at the same rate, otherwise one has to do a sum over all the planetary growth rates instead of just multiplying by the number of planets)

As you can see, your number of pops grows (almost) perfectly linearly if you keep growing your number of planets at a certain linear rate (this is exactly the habitat spam scenario). I would like to reiterate that this is only an approximation because in actuality the total number of pops (and planets...) is not a smooth function, but more a ... step function, but it should still be a useful aid to picture the total population of a non-conquering empire.

EDIT:

Spoiler: Test Run

Doing an experimental Inward Perfection run seems to corroborate these numbers, largely. In 2381 I am the not-so-proud owner of 34 colonies, of which 30 are habitats, 1 is Wenkwort (Resort World), and 3 are the starting ring segments of Shattered Ring origin; the current pop count is 582. I have been building habitats as fast as I can spend influence (+5.8 monthly income) and alloys, as soon as I got them, with breaks for a Dyson Sphere and Science Nexus; Unity Ambitions came up rather late. There are some issues with this test, namely that I'm running out of alloys by now so can't keep up the habitat spam (sitting on an L-Gate with the Gray Tempest doesn't help). I suspect this has to do with trying to actually fill said habitats (the first ten, anyway) to a point where they can get as close to max growth as possible (5 housing buildings, 30 jobs, 4/4 housing/resource districts). Psionic Ascension, too.

Sidenote: This is distinctly worse than my other runs, also without warfare.

 

[Cymsdale]

Yoinking @Cymsdale 's link to the tool, here's the graph representation of what I had looked at (well, part of, anyway): (EDIT: had to adjust the planet acquisition rate, it was ridiculously low, I had originally plotted over number of months passed, forgot to adjust the 'x' for planet number growth rate)

Total Pops over Time

www.desmos.com

(r...monthly growth rate; G...pop growth target; a...Empire Pop slowing coefficient; N_0...starting number of pops; P...number of planets)
(for simplicity the assumption was made that each planet grows at the same rate, otherwise one has to do a sum over all the planetary growth rates instead of just multiplying by the number of planets)

As you can see, your number of pops grows (almost) perfectly linearly if you keep growing your number of planets at a certain linear rate (this is exactly the habitat spam scenario). I would like to reiterate that this is only an approximation because in actuality the total number of pops (and planets...) is not a smooth function, but more a ... step function, but it should still be a useful aid to picture the total population of a non-conquering empire.

This makes me wish Stellaris had graphs that would let me see things like my empire population over time. It would be interesting to see how much it lines up with the theoretical model.