Diminishing science penalty of more cities

Just a small point, but having more cities i see players often calculate like 5% more science compared to your current output. But say you have 10 cities, then you are paying 150% research cost on your techs. Getting another city will put you at 155% which is only 155/150=31/30=1,0333... which is only 3.3 % science penalty. So say you are at 2000 beakers with 10 cities, the breakeven for a new city is not 100 beakers, but only 67 beakers.

Just something to keep in mind i guess when expanding a lot.

consentient said:

Thank you very much for starting this thread. I have been trying to work it out for a while. Sadly, my maths is terrible.

Can you expand a bit more on what each figure means, and how you arrive at these 'break even' figures?

What I mean is, can you use words as well as numbers, for those of us, like me, who are a bit 'maths dumb'?

Thank you.

Click to expand...

If you're not into the math side, here is the one game affecting fact about science and culture penalties:

Every city you found will in effect make the same raw amount of science penalty. This means that you should never be afraid to have too many cities because of the science penalty. The only thing about too many cities is that you should worry about being able to develop them. Assuming that each city you have makes about the same amount of science, it is ALWAYS better to have one more of those cities if your happiness can keep it.

The math proof:

With n cities, each making x science, your total teching will be:

nx/(1+0,05*(n))

Norming x at 1 means that this becomes:

n/(1+(0,05)n)

Lagrange rule gives that this function of n is increasing with a limit at 20.

Here is graph of science with up to 20 cities (not to be confused with 20 above.)

Thank you very much for starting this thread. I have been trying to work it out for a while. Sadly, my maths is terrible.

Can you expand a bit more on what each figure means, and how you arrive at these 'break even' figures?

What I mean is, can you use words as well as numbers, for those of us, like me, who are a bit 'maths dumb'?

Thank you.

The 5% per city penalty applies to the base beaker cost of The tech. So, for example, a tech that has a base cost of 300 beakers will cost 450 beakers if you have 10 cities (300 * (1 + (.05 * 10))). If you found 1 more city, that tech's cost will rise another 5% (15 beakers) to 465 beakers. However, that additional 15 beakers is only 3.333% of the previous tech cost (15/450).

Browd - does the cap count? I thought it was 5% for every city after the cap.

Thank you Browd! I understand now. 5% isn't much if you have lots of nice juicy cities with specialists

budweiser said:

Browd - does the cap count? I thought it was 5% for every city after the cap.

Click to expand...

Yes the capital counts

I'm guessing the same mechanism works for culture costs?

Yes, the arithmetic is similar for the per-city culture costs increase.

consentient said:

Thank you very much for starting this thread. I have been trying to work it out for a while. Sadly, my maths is terrible.

Can you expand a bit more on what each figure means, and how you arrive at these 'break even' figures?

What I mean is, can you use words as well as numbers, for those of us, like me, who are a bit 'maths dumb'?

Thank you.

Click to expand...

If you're not into the math side, here is the one game affecting fact about science and culture penalties:

Every city you found will in effect make the same raw amount of science penalty. This means that you should never be afraid to have too many cities because of the science penalty. The only thing about too many cities is that you should worry about being able to develop them. Assuming that each city you have makes about the same amount of science, it is ALWAYS better to have one more of those cities if your happiness can keep it.

The math proof:

With n cities, each making x science, your total teching will be:

nx/(1+0,05*(n))

Norming x at 1 means that this becomes:

n/(1+(0,05)n)

Lagrange rule gives that this function of n is increasing with a limit at 20.

Here is graph of science with up to 20 cities (not to be confused with 20 above.)

So I think the horizontal axis is number of cities, and vertical axis is total of empire science. But that implies that, after 20 cities, additional cities do not add science. Is this correct?

That conclusion somewhat contradicts the opening observation that the science penalty is relatively less severe with each city that is founded. OTOH, 20 cities with 5% science penalty per city, the 100% penalty cap at that point makes sense. It is a good game mechanic, as otherwise runaway AIs would be a real science monster. Also, 20 cities is pretty a pretty generous amount from a player’s perspective.

It is a little unrealistic that your 20th city has the same science output as your first. But would the diminishing returns ever start pulling the curve down? I think it could. Suppose every city produces only half as much science as the one before. What is the break even then?

beetle said:

So I think the horizontal axis is number of cities, and vertical axis is total of empire science. But that implies that, after 20 cities, additional cities do not add science. Is this correct?

That conclusion somewhat contradicts the opening observation that the science penalty is relatively less severe with each city that is founded. OTOH, 20 cities with 5% science penalty per city, the 100% penalty cap at that point makes sense. It is a good game mechanic, as otherwise runaway AIs would be a real science monster. Also, 20 cities is pretty a pretty generous amount from a player’s perspective.

It is a little unrealistic that your 20th city has the same science output as your first. But would the diminishing returns ever start pulling the curve down? I think it could. Suppose every city produces only half as much science as the one before. What is the break even then?

Click to expand...

No, every additional city adds science, but there is a limit at infinity which is 20 teching, like shown on the graph it would flat out at 20

Thanks, I misread the implied asymptote. Good to know I got the axis labels correct though!

One thing I like about this graph is that it show how your 4 city empire has a third of the science of the monster AI 20 city empire, despite being only a fifth of the size. That is 50%+ more than you might expect on the face, even making the unrealistic assumption about the AI working specialists the same and prioritize science buildings and SP as the player would. This really helps illustrate how the player can keep up.

I'm not a maths man. When a cities science output is more than %5 of the base cost of the currently researching tech then I'm happy. I find that once a city builds a university it starts producing more than %5 of the base cost of the currently researching tech.

When deciding to keep a puppet i do an even cruder check and that is "does the puppet produce more than %5 of my total science output?" If the answer is yes then it's a keeper.

I don't quite understand what you mean by the limit of the function at 20?

the function n/(1+0.05n) has a limit of 20 at n->infinity, but that doesn't mean the optimal point is 20 cities assuming equal beakers. The adjusted beakers is always better for each new city added. It only means that the best you can get is 20 times faster than a person with 1 city with the same amount of beakers per city.

Say I have 100 cities each producing 1 beaker, then the adjusted total beakers is 100/(1+5), which is much larger than with 20 cities, which is 20/(1+1).

Of course, the real game is more complicated with modifiers and secularism, but generally you can calculate the science from a city of 10 pop to be 10+10+4 (labs)+3 (schools)+5*4 (specialists)=47 raw * (100+50(labs)+50(free thought)) = 94.
I'm pretty sure all the players here can grow make a city like that no problem, then it's definitely worth it unless you have more than 20*94=1880 bpt already.

Sessy said:

I don't quite understand what you mean by the limit of the function at 20?

the function n/(1+0.05n) has a limit of 20 at n->infinity, but that doesn't mean the optimal point is 20 cities assuming equal beakers. The adjusted beakers is always better for each new city added. It only means that the best you can get is 20 times faster than a person with 1 city with the same amount of beakers per city.

Say I have 100 cities each producing 1 beaker, then the adjusted total beakers is 100/(1+5), which is much larger than with 20 cities, which is 20/(1+1).

Of course, the real game is more complicated with modifiers and secularism, but generally you can calculate the science from a city of 10 pop to be 10+10+4 (labs)+3 (schools)+5*4 (specialists)=47 raw * (100+50(labs)+50(free thought)) = 94.
I'm pretty sure all the players here can grow make a city like that no problem, then it's definitely worth it unless you have more than 20*94=1880 bpt already.

Click to expand...

I never said that 20 cities makes max science, just a coincidence that my graph has 20 max and that the function limit is 20, as i said, more cities is better always, thats why i made the point of the function being strictly increasing. Sorry for this confusion of 20. Should have chosen diff number for graph

klaskeren said:

I never said that 20 cities makes max science, just a coincidence that my graph has 20 max and that the function limit is 20, as i said, more cities is better always, thats why i made the point of the function being strictly increasing. Sorry for this confusion of 20. Should have chosen diff number for graph

Click to expand...

I see, maybe it would be easier to plot out the function for a greater range, although it's not hard to see the derivative is always positive for n>0. we should try to look at the relationship between science, culture and happiness to get a better picture. Just looking at the science penalties we would think that having more cities is always worth it.