When to stop building cities » Forum Post by startno

Six cities is wrong. The optimal number of cities is a lot higher than that in the situations you'll typically be facing. I've used 15 as a rule-of-thumb number in my lengthy thread on this issue.

The optimal amount of cities depend on a huge number of factors, many more than your formula accounts for - one of the more important being how much infrastructure they have. The more developed your current cities are, the less attractive adding a new level 1 one without any buildings becomes. There are also considerations that aren't about the economy directly that might influence whether to expand or not such as denying enemy expansion.

Successfully updated karma reason!

So if you already own 15 cities and capture a 16th, then you should raze it?

Successfully updated karma reason!

as a simple addition to your formula, you should probably reduce/correct the amount of global unrest by 20% since effectively every city can get a belltower and townhall which provide -10% unrest each (and both are easy enough to unlock and build to make them generally available)

Successfully updated karma reason!

Quoting Apheirox,

reply 1

Six cities is wrong. The optimal number of cities is a lot higher than that in the situations you'll typically be facing.

I've used 15 as a rule-of-thumb number in my lengthy thread on this issue.

Well, according to my gut the optimal number of cities is reached when I've colored the whole map. Which is neither 6 or 15. I trust neither my gut, nor rules-of-thumb with these issues, I like formulas. Your thread was one thing that got me started to try to create one.

I understand I will never be able to calculate a specific number like 6 or 15, but I would like to identify a range where I need to doubt the benefit of new cities and control my greed.

Also I'd like to invite everyone to add to this formula if you can improve on it.

Quoting Apheirox,

reply 1

The optimal amount of cities depend on a huge number of factors, many more than your formula accounts for - one of the more important being how much infrastructure they have. The more developed your current cities are, the less attractive adding a new level 1 one without any buildings becomes.

I am aware of that. As I said, a new city is less productive than a new one. Or do you mean something else?

Quoting Apheirox,

reply 1

There are also considerations that aren't about the economy directly that might influence whether to expand or not such as denying enemy expansion.

Yes I didn't include anything that I could not hard to put a number on. Although it could also be argued that outposts may be better way to deny enemy expansion than cities. Maybe. Also I am wondering if the AI realizes it can choke itself to death by over-expanding.

Razing conquered cities or not is even more complicated as there can be world wonders and it can serve as a base of operation in a hostile environment. So I'd like to focus this discussion on actually building cities.

Successfully updated karma reason!

Quoting Azunai_,

reply 3

as a simple addition to your formula, you should probably reduce/correct the amount of global unrest by 20% since effectively every city can get a belltower and townhall which provide -10% unrest each (and both are easy enough to unlock and build to make them generally available)

I wondered about that one, too. I decided not to because Taxes rise unrest in about equal amount. But fiddling about with a global reduction does have a great effect on the result of this formula. If I add a reduction like this:

$\frac{3}{100}x -0.2 = \frac{x+1}{x}-1$

x resolves to 10! Makes me wonder if I underestimated the value of the trait 'noble'...

Am I missing something?

Successfully updated karma reason!

Primal_Savage

Join Date 07/2013

+184

N/A

Successfully updated karma reason!

It looks ugly and confusing this way, but sure:

(3/100)x -0.2 = ((x+1)/x)-1

The -0.2 is the global reduction as recommended by Azunai_.

Successfully updated karma reason!

parrottmath

Join Date 03/2008

+206

Simply put he has the formula in latex.

3 x / 100 = ((x + 1) / x) - 1

$\frac{3}{100} x = \frac{x + 1}{x} - 1$

This is the original formula he uses.

(3x / 100) - .2 = ((x + 1) / x) - 1

$\frac{3}{100}x - .2 = \frac{x+1}{x} - 1$

is the second formula that resolves to 10.

I for one do not quite understand exactly the right side, (x + 1) / x - 1 -- latex: $\frac{x + 1}{x} - 1$.

Why is this the relative benefit of a new city. I think a more justified formula for that side would be needed in my opinion since why is

1 / x --- latex: $\frac{1}{x}$

The added benefit of a new city is a 25% bonus to your current situation? The first city being a 100% bonus, and then a second city is a 50% bonus next, yielding from the beginning of the game a 150% bonus? I'm rather confused by this simple calculation you pose.

I could spend some time analyzing the benefit of having 1 city versus having 2 cities versus having 3 cities... First thing that comes to mind is the size of the map before you justify any formula. A small or tiny map optimally you do not want very many cities. Simply put the game may be over before you develop the cities beyond the first or second building and your first two cities have built a large enough and powerful enough army to win the game. In a huge map, 6 cities, may not be as optimal since travel time may become a factor to replace broken troops a long way off. (or simply having a source of healing nearby for troops, beyond the magic).

Successfully updated karma reason!

Am I the only one who can see my formulas as a pretty graphic? Or does it look alright to most of you? If not I'll create some screen shots, or something.

Quoting parrottmath,

reply 8

I for one do not quite understand exactly the right side, (x + 1) / x - 1 -- latex: $\frac{x + 1}{x} - 1$.

Why is this the relative benefit of a new city. I think a more justified formula for that side would be needed in my opinion since why is

1 / x --- latex: $\frac{1}{x}$

The added benefit of a new city is a 25% bonus to your current situation? The first city being a 100% bonus, and then a second city is a 50% bonus next, yielding from the beginning of the game a 150% bonus? I'm rather confused by this simple calculation you pose.

Well I am describing how many more cities you have - relatively speaking.

Examples:

One City, build a second city -> (1+1)/1-1 = 1 -> 100% more cities.
Two Cities, build a third city -> (2+1)/2-1 = 0.5 -> 50% more cities.
Three Cities, build a fourth city -> (3+1)/3-1 = 1/3 -> 33% more cities.
15 Cities, build a 16. -> (15+1)/15-1 = 6,67% more cities.

I am pretty sure the right part of the formula is correct, I am not that sure about the left part.

Quoting parrottmath,

reply 8

First thing that comes to mind is the size of the map before you justify any formula. A small or tiny map optimally you do not want very many cities. Simply put the game may be over before you develop the cities beyond the first or second building and your first two cities have built a large enough and powerful enough army to win the game. In a huge map, 6 cities, may not be as optimal since travel time may become a factor to replace broken troops a long way off. (or simply having a source of healing nearby for troops, beyond the magic).

I totally agree that there are other reasons to stop building cities. Many. For example having an Ogre walking through your territory. What I am trying to find is the amount of cities where I should really say to myself: Stop it now, You are hurting yourself by building this city, build an outpost instead.

Successfully updated karma reason!

Primal_Savage

Join Date 07/2013

+184

N/A

Successfully updated karma reason!

Sorry, I think I goofed up completely. As in, I am comparing two things that should not be compared. I'll try to come up with a better idea.

Edit:

Thanks Primal_Savage, I will refrain from including images from now on. Thanks parrottmath for getting me to double check my math.

Ok completely new and maybe just as wrong.

The amount of Research and Production of a Kingdom/Empire is = Amount of Cities * (100 - (Amount of Cities * 3))

Let's replace "Amount of cities with x for brevity and shorten the formula -> -3x²+100x

This is a parabola. Lets find it's highest point by deriving it and equalizing it with zero -> -6x+100 = 0 -> x = 16.67

Ridiculously close to 15. Kudos to Apheirox, I feel stupid now.

But let's see what happens if we add an unrest modifier like 10% of noble. (This will create incorrect results for x < 4)

-3x²+110x -> -6x+110=0 -> x = 18.33

Successfully updated karma reason!

parrottmath

Join Date 03/2008

+206

This formula looks better, although I'm not sure if it is the best one. It does achieve a maximum, which your first equation didn't.

But the conclusion may be exactly it that 15 is the sweet spot of building cities and anything more is a detriment and less is not advantageous. Of course I play a lot with rebels so my city max is skewed (although it turns out to be 15 from your formula).

Although, I question on how you are solving these equations in general...

Let us not forget as part of your equation to include a tax rate since you cannot funciton an empire without the tax revenue.

With no taxes you have a minimum of 10 unrest, and thus you really have that 90 instead of the 100. But a usual tax rate of normal yields an unrest of 21. Thus, you really want to make this calculation at 79 instead thus, somewhere around 13 cities is your target number.

With noble you get closer to 15 by your formula. Good job with the formula.

Successfully updated karma reason!

That first formula tried to describe benefits and drawbacks of another settlement as two lines with an intersection. But I think I described the drawbacks incorrectly.

What do you dislike about how I am solving the equations? I am always open to suggestions.

Quoting parrottmath,

reply 12

Let us not forget as part of your equation to include a tax rate since you cannot funciton an empire without the tax revenue.

With no taxes you have a minimum of 10 unrest, and thus you really have that 90 instead of the 100. But a usual tax rate of normal yields an unrest of 21. Thus, you really want to make this calculation at 79 instead thus, somewhere around 13 cities is your target number.

I am really undecided on what number to take there. Should I take possible improvements into account, like a certain amount of gallows, or town halls in every city?

Regardless, I would not say that 15 or any other number is a sweet spot. There are too many variables there that I can't put a number on. For example if you have fewer but faster producing settlements you can create turn advantage over your enemy. How do you put that into an equation?

Successfully updated karma reason!

parrottmath

Join Date 03/2008

+206

The purpose of your exercise here is to look strictly at the unrest penalty and determine what is the sweet spot for the unrest of a civilization and the penalty therein. Firstly the only issue with your current equation is that you did forget to take in the automatic tax unrest into your equation. Thus, instead of 100 it should have started at 90 (given absolutely no tax at all)

Secondly, if you want to define a sweet spot determine the minimum number of material a city must have to be a city. That is 3 materials. So, a new city with a production rate of 1 is useless for all intensive purposes. Each material generates 8 production. Thus, for 3 materials we are talking 24 production roughly.

Thus, a percentage penalty from this on the materials gives you a better judge on the next city usefulness. Note at 15 cities, we have a 75% unrest penalty just from the cities themselves. From the tax base of no taxes we get 10% more unrest. Leaving one with 85% unrest for the 15 city empire. 85% of 24 yields your new city to have 3 production. Not particularly a useful city at all since a market will take roughly 21 turns. (when at the beginning it would take roughly 4 turns at 3 materials).

Thus, we should consider at least this element otherwise you wouldn't be able to build unrest reducing buildings in order to make the city useful.

Successfully updated karma reason!

Quoting parrottmath,

reply 14

The purpose of your exercise here is to look strictly at the unrest penalty and determine what is the sweet spot for the unrest of a civilization and the penalty therein. Firstly the only issue with your current equation is that you did forget to take in the automatic tax unrest into your equation. Thus, instead of 100 it should have started at 90 (given absolutely no tax at all)

I am not sure if this really improves the results of that equation. I could subtract 21 for normal tax rate, add 20 for town hall and end up with 99. Or I could subtract 10 for low tax rate, add 20 for town hall and end up at 110. Or I could take 90 as you suggested. Or any other number. What number should I pick and why?

Quoting parrottmath,

reply 14

Secondly, if you want to define a sweet spot determine the minimum number of material a city must have to be a city. That is 3 materials. So, a new city with a production rate of 1 is useless for all intensive purposes. Each material generates 8 production. Thus, for 3 materials we are talking 24 production roughly.

Thus, a percentage penalty from this on the materials gives you a better judge on the next city usefulness. Note at 15 cities, we have a 75% unrest penalty just from the cities themselves. From the tax base of no taxes we get 10% more unrest. Leaving one with 85% unrest for the 15 city empire. 85% of 24 yields your new city to have 3 production. Not particularly a useful city at all since a market will take roughly 21 turns. (when at the beginning it would take roughly 4 turns at 3 materials).

Wait, why 75%? I assumed 15*3%=45%. Add 10% for low tax rate and get 55%.

Successfully updated karma reason!

If you take the level 4 town/level 5 fortress upgrades that lower unrest across your empire I believe you can theoretically keep an infinite number of cities productive, though I haven't crunched the numbers. The problem is getting them to 4 or 5 in a relevant timeframe, though you can mitigate that problem by creating only cities with at least 1 essence and prioritizing Sovereign's Call over everything. This also means Kingdoms can ICS better than empires in most games through more consistent access to Life magic. Bless City/Oppression further tilt the equation towards infinity if you can restrain yourself to only 2 essence locations.

Successfully updated karma reason!

parrottmath

Join Date 03/2008

+206

I had 25 cities on the mind when I suggested 75%.

Recall that your new cities do not get these improvements immediately and would have to be rushed in order to get their full effect of unrest reduction. Not a very cost effective transfer of funds. But yes 55% of your 24 production yields 12 production and gives you a build time for the merchant of 6 seasons. Or roughly 11 seasons for the bell tower (the first level of unrest reduction). All these things change with the production of your next city.

A material 3 city would do just fine with 15 cities, but a material 4 or 5 city can probably be built beyond the usual 15 city limit, one can maintain a decent production rate at those levels.

Successfully updated karma reason!

I would never go in trying to aim for a certain number of cities.

For me, once I have the basics of one city of each type covered things become a lot more about controlling key areas to allow myself easy movement and to deny the same to the AIs. I will put a Fortress or three down in key areas and snake to completely prevent passage through by the AIs once I have one of each city, but I am not doing that for its effect on my own economy, just to keep AIs contained.

Sometimes there are seemingly free expansion areas that I just won't bother with because I don't want to try to defend those places. Often it will be better for me to conquer and raze an AI city than to build another one of my own. Conquering an AI capital and making them lose a huge portion of their empire production and razing it so they have no hope of getting it back is a huge blow, better than making 5 more towns for yourself.

Successfully updated karma reason!

Interesting topic.

There are probably too many factors to put it in a nice formula, but I applaud the effort!

Basically, the new city must be 'worth more' than the burden of unrest on all your existing cities.The problem is, like Apheirox already mentioned, that older cities are more developed and a reduction in their research and production will hurt your kingdom much more than a new city will add when founding it.

(Global) unrest reduction buildings can compensate for the hit in research/production, but building those can take time. In conclusion: it largely depends on the pace of your game how many cities you can afford.

Successfully updated karma reason!

Quoting Fallenchar,

reply 19

The problem is, like Apheirox already mentioned, that older cities are more developed and a reduction in their research and production will hurt your kingdom much more than a new city will add when founding it.

Yes. So far I am not sure on what assumptions to make in that regard. We could assume that every city is 10% (made up number) less valuable than than the city that was build before it.

Example: First city has a value of 1, second 0.9, third 0.81 , fourth 0.729...

Or just take a flat amount off: First city has a value of 1, second 0.95, third 0.9.

So far I don't feel experienced enough to come up with my own, but if you have any suggestions you can make a good argument for I would try to add them into the formula.

For now it can be said that at some number < 16 building new cities can harm your economy more than you will ever get out of it. If this formula gets further improvements I expect this number to become smaller and more precise.

Quoting Fallenchar,

reply 19

(Global) unrest reduction buildings can compensate for the hit in research/production, but building those can take time. In conclusion: it largely depends on the pace of your game how many cities you can afford.

Yes and you could build something different instead which could help you more.

+1

Successfully updated karma reason!

I've had a go at assessing the value of adding cities later in the game.

Taking startno's formula from post 11, of total production = a.x.(100-3x), where x is the number of equal sized cities, and each city has a production of a.

The balance point at which the total production for the x cities is equal to that of (x+1) cities is

x.(100-3x) = (x+1)(100-(3x+1))

which resolves to the same answer as startno reached solving it as a differential, e.g. 16 cities is best.

However, as various people have commented, later cities will be smaller and less productive than earlier cities for most or all of their lives, so the penalty on those earlier cities is more damaging.

If we make a guess that an extra city starting mid-game is worth only half of a early game city then the formula becomes:

x.(100-3x) = (x+1/2)(100-(3x+1))

which resolves to a peak at 11 cities.

Later still, adding a city in the late game might be worth say 1/4 of an early city so:

x.(100-3x) = (x+1/4)(100-(3x+1))

which resolves to just 6.

So if you have 6 big cities late game, then adding a 7th new city will reduce your total output.

By contrast, captured cities that are size 3 or 4 already and come with a set of completed buildings are much nearer to being early game cities.

Of course there are loads of other strategic factors such as the resources captured by a new city or its location at a choke point etc, not to mention access to bless spells, level 4 fortresses etc

Successfully updated karma reason!

Borg999

Join Date 02/2006

+59

You also build the temple of forgetting to get a faction wide reduction in unrest of 15%

Successfully updated karma reason!

As I move into the end game I don't bother settling new cities unless they have Essence or helpful resources close by. More Mana is always nice. Up to that point (i.e. with 50+ turns to go) I will tend to settle any except the most marginal cities. This is as much to claim the territory as anything else, but also on the principle that I have time to build everything I need to make it productive.

High level cities captured from the AI are almost always worth keeping because they will probably be producing substantial amounts of gold, if nothing else. Sacrifice is good too if you have that available. However I have Razed marginal AI cities which it has settled on land which is nearly useless, where I can't ever see the city becoming useful and it's giving a 3% penalty to my other cities.

Successfully updated karma reason!

The math analysis from JustaMo sounds right in that I don't think it is a really great idea to build cities after about 6.

Maybe if you are playing on really large maps it might make sense, but still after that I would rather just conquer an AI city than build my own city. I wouldn't make any new city after that unless it had serious strategic value, like adding a town if my cities can't grow or adding a fortress to block a certain pass. I wouldn't even really expect much from these cities either, they will probably be doing nothing but making buildings the whole game.

If you are not trying to outfit 3+ heroes, gold will probably be fine for the most part and extra research helps very little in the short term, so conquered cities and towns often have very little value beyond denying their benefits to the opponents. Going from 1 high level conclave to 2 will add maybe 50% to your research and just push you into more costly techs so the value is really much less, but the enemy going from 1 high level conclave to 0 will pretty much bring their research to a grinding halt. They might as well be doing no more research at that point. Destroying it just prevents them from getting from 0 back to 1, or prevents anyone else from going 1 to 2. Your lead is achieved/maintained just by preserving the 1 you already have in your back lines while other people lose theirs.

If you want additional research, just build research buildings in your towns and fortresses, imho. If you already did that to the greatest possible extent you are probably good on research anyway. If you want more money there should be ways available to get it more quickly than that. Defeating a high level monster and selling whatever it gave can give you in single digit turns the gold it would take a hundred turns to get from a new town.

Successfully updated karma reason!

You guys are crazy!

Keep going... Im enjoying reading this!

Successfully updated karma reason!