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In Civilization V, you attain a cultural victory by accumulating enough culture to purchase at least 36 social policies, and then building a wonder. The catch is that the more cities you build, the more culture you need to generate before you hit the next "plateau" of culture.

What is the ideal number of cities for a cultural victory? Does it change over time?

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This question has resulted in all kinds of fun math. (even if some of the/my math is wrong) Good work. –  WillfulWizard Oct 12 '10 at 17:31

5 Answers 5

up vote 58 down vote accepted

The question you're really asking is "When does the combined culture / turn of an additional city outstrip the increase in social policy cost incurred by founding that city?"

The short answer is...

We know that adding another city increases the culture costs by approximately 30% of the base cost (that of 1 city). Therefore:

If your maximum potential culture / turn won't increase by at least 30% due to the new city, you are hurting, not helping, the time till your next social policy.

(This may be slightly hard to calculate, and if you take too long to reach your "maximum potential culture / turn" you're actually wasting turns.)

The long answer is...

It depends

To begin with, we need to make some assumptions:

  1. When you found a new city, you can get its culture / turn maximized within a single turn by buying the necessary building improvements (monument, etc).

  2. Ignore city-states, leader specific abilities, +culture social policies, and wonders. These all help produce culture, and will shift the "ideal city count" down, but do so inconsistently. To produce an "ideal" city count, we limit ourselves by era and improvements alone.

  3. This list of social policy costs is accurate for the given parameters: medium map and normal speed.

alt text

And now, some math.

The 1st social policy costs 25 points with a single city. In the ancient era, your cities can generate 2 culture / turn due to the monument. (Remember, we're ignoring the palace for now)

This means that it will take 13 turns (Ceiling(25/2) to enact the policy, or 9 turns (ceiling (45/4) with two cities. We can continue this extrapolation -- 8 turns with 3 cities, 7 turns with 4 cities, 6 turns with 5 cities, and we finally reach diminishing returns at city 6 (also 6 turns).

For the second social policy, the ramifications of the # of cities gets magnified due to a larger starting value: One city takes 23 turns, two cities take 15 turns, three cities take 13 turns, four cities take 12, five cities take 10, and again, we run into diminishing returns cap out at at six cities (10 turns).

It is not until the 4th social policy that this trend is broken and diminishing returns end at the NINTH! city.

Remember -- this assumes that each city has a monument the minute it is founded.

Now let's say we've reached the classical age, and have temples in addition to monuments. Each city is now generating 5 culture.

The first policy takes 5 turns with a single city, 4 turns with two, and 3 with three.

What (hopefully) becomes clear is that we reached diminishing returns (4 cities as opposed to 6 cities) much faster when each individual city's contribution is higher. The more culture any one city is capable of producing, the more incentive there is to produce more cities. Even if you don't manage to build every +culture improvement immediately, you're still likely to come out ahead (as long as you're pre-diminishing returns).

So while the optimal number of cities changes due to any number of factors, you can probably safely not shoot yourself in the foot if you stay between three and six cities, with six being on the high end.

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+1 Wow... I got pwned. –  WillfulWizard Oct 12 '10 at 3:16
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I like how you start by rephrasing your own question in the second person... =) –  bwarner Oct 12 '10 at 11:46
    
You short statement "If your maximum potential culture / turn won't increase by at least 30% due to the new city, you are hurting, not helping, the time till your next social policy." is not entirely accurate. Counter example: Take 5 cities approaching policy 5, requirement is 540. If each city is producing 10 (50 total), that's 11 turns. +1 city would mean the target becomes 615 - to achieve 11 output must increase to a total of 56 per turn (615/11 = 55.9). Since 56/50 = 1.12 this means the requirement is a 12% increase in output. –  DMA57361 Oct 12 '10 at 12:40
    
@DMA57361 -- I think I'm having a little bit of trouble understanding you. You describe a situation where it takes 11 turns to get a policy. At that point, adding another city that also produces 10 culture / turn would still mean it takes 11 turns. (615/60 = 10.25 == 11 full turns). 60/50 = 1.2, which is less than 1.3, which implies that adding another city will not reduce the time till the next social policy. And it doesn't (it's exactly the same). Is there something about your argument that I'm not understanding? –  Raven Dreamer Oct 12 '10 at 14:10
    
@Raven the problem is the numbers are too small and get clipped by rounding errors. But the basic point is you do not need 60/turn to manage 11 turns as 615/11 = 55.9090 you only need 55 to maintain the same rate. Lets imagine you've 5 cities going for policy 30. You need 14225, at 50/turn that's 285 turns. +1 city requires 16165. 16165 / 285 = 56.719, so you need 57/turn to reach the next target (+14%). But, if you had the "full" 60/turn you'd only need 16165 / 60 = 270 turns for the next target - so adding a "full" city would actually speed up the next policy. –  DMA57361 Oct 12 '10 at 16:50

I wrote out this detailed answer, then realized I could not CITE how much each city increase the cost of social policies. I have not yet found it in either the Manual or the Civilpedia. The only reference I found quickly is that on duel maps the increase is 30%. Since we need to know for sure to properly answer this question, I've posted a new question on exactly that.

But, rather than not answer, I post this with the qualifier: this applies best to duel maps, until we know the numbers for larger maps.

There are two basic competing forces to your question: Socical Policies cost more for each city you control, but each city you control gains you more production/gold/science. So you're trying to balance. Where is the balance?

3 or 4 Cities

That's a pretty definite answer to what would usually get an "It depends!" answer. Why? Science! Math!

Because each city increases the cost of social policies by 30%, for your next city to be worth it, it needs to increase the cultural output of your civilization by more than 30%.

Now lets pretend every city in your empire is the same... they'll grow into having the same food, same gold, same science, same production, and, because of all that, the same culture as well. How much does each city improve your cultural output, eventually?

City | Culture | Change
2    |      2x |  +100%
3    |      3x |   +50%
4    |      4x |   +33%
5    |      5x |   +25%
6    |      6x |   +20%

Now its not looking to good for cities 5 & 6 there. All things cities being equal, they don't add at least 30% to your civilization's culture, not even considering culture that doesn't come from cities. (like from City-states)

But, of course, cities are not equal. If you've played any Civilization game for any amount of time, you know cities are not all equal, and you know which ones are better: the earlier cities. Exceptions do occur, but you try to build cities in the best spots first don't you? That makes city number 4 look not so good either, because if city 4 is off by just a little, its a loss.

Still, if city 4 is decent, even if its a net loss, it will still probably pull a lot of its own weight so as not to be too much of a loss, and it will be a win for production, gold, & science. So build it if you have a good spot, or don't if you're happy with 3 cities. And, of course, don't build city 4 if you are going for Bollywood.

BUT...

Conquer Puppet Cities

Puppet cities do not add to the cost of Social Policies, but can still add culture, when they get around to producing such buildings. Easy win! Especially good for such things as +Culture/city/turn (social policy & France). Note however, you can not achieve Bollywood with puppet cities.

City States are your Cities you don't own

I risk sounding like a broken record mentioning city-states for a cultural victory again, but I like to think about city-states in a way that relates directly to this: Each city-state is an extra in your empire, that isn't in your empire. All the benefits, none of the (normal) costs. Make friends.

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Interesting. We reach the same conclusion via slightly different trains of thought. Perhaps the biggest thing to take away from this is: if your cities themselves don't account for the majority of your culture gain (i.e., they're static values that can't be increased -- happiness overflow, cultural city states, etc) you don't want to found a new city. –  Raven Dreamer Oct 12 '10 at 3:02
    
When I first saw your answer (I started typing before you posteD), I was sad I had spent time on my answer with less sourced info, but now that I see my info is accurate (more or less, and thanks!), I'm glad we have the two answers, which I well balanced. I approached what is the most each city could do, and you approached what's the least each would have to do. Pleasantly complementary. –  WillfulWizard Oct 12 '10 at 5:40
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I think your math is a bit flawed in that you are assuming each city adds 30% of the current cost. Actually it adds 30% of the base cost, which isn't quite as bad. –  bwarner Oct 12 '10 at 15:08
    
@bwarner Sigh, yeah, I think you're right, thanks. I'll check raven's numbers and then run my scenario against against those. Sigh, another post I need to find time to rewrite. –  WillfulWizard Oct 12 '10 at 17:29

I've changed the approach of my answer, instead of trying to "solve" for a best number of cities in all situations (which isn't possible due to the number of factors) I'll provide a situational way to decide if adding a new city is a good idea, which will account for any wonders / policies / leader traits / specialists / city states / etc as well.

First Point

Counter intuitively the extra culture you need for an entire social policy actually increases by a smaller margin if you have lots of cities:

Cities   CostMulti    Change from Previous
1          100%         -
2          130%         +30%
3          160%         +23.08%
4          190%         +18.75%
5          220%         +15.79%
6          250%         +13.63%
7          280%         +12%
8          310%         +10.71%
9          340%         +9.68%
10         370%         +8.82%
et cetera

The Calculation

The first step is to work out the base policy cost = b, for this you need:
the current policy cost = p
and the number of cities = n.

we know that    p = ((0.3 * n) + 0.7) * b

this implies    b = p / ((0.3 * n) + 0.7)

Now work out what the new policy cost = q will be:

q =  roundUpToNearestMultipleOf5( ((0.3 * (n + 1)) + 0.7) * b)

We need to know the current turns to policy = t are left to go, the game will tell you this if necessary, but to calculate we need:
the current rate of culture = r
and the currently collected culture = c

t = ceiling( (p-c) / r );

Now work out the required rate of culture = R with the new policy cost to avoid any slow down:

R = ceiling( (q-c) / t );

And finally you can work out the required culture from your new city a:

a = R - r;

Worked Example

Here's an update using the real live data from my current game (I've had to start Civ to get them and am Alt-tabbed out the game as I type this).

Inputs

p = 1490
n = 5
r = 126
c = 785

Calculated Values (calculations omitted to save space!)

b = 677.2727 (Odd value, but I'm not on the middle difficultly and have the -33% policy cost policy)
q = 1695 (Checked this by getting a settler to found, and it's right) t = 6
R = 152
a = 26

So, if I added a new city I'd need it to generate 26 per turn to avoid slowing down my next social policy! That's unlikely to happen, so adding a new city right now would be a bad idea - I'm probably better to wait 6 turns for the next policy and build the city then when the difference between p and c is maximized.

Final Result

Here's the single combined formula. It uses just the four inputs already defined.

enter image description here

So, the question is then simple:
If I build/annex a new city, can I very quickly produce an extra a culture per turn to prevent slowing down my progress?

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@badp, ta for the effort with the Unicoded formula, but unfortunately it looks worse for me than it used to (on FF4, WinXP); the alignment is broken - see now vs before. And given that IE7 can't display all the characters I think we should stick with the lower fidelity version, to ensure that the maximum number of the site's visitors can see it correctly. Alternatively, I guess we need to get it draw up properly and insert an image instead... –  DMA57361 May 23 '11 at 8:16
    
I can't repro on Win7 + Chrome; I agree proper formula rendering would be better: ${\displaystyle a=\left\lceil \frac{\text{roundTo5}\left(\frac{p\left(0.3n+1\right)}{0.3n+0.7}\right)-c}{\left‌​\lceil \frac{p-c}{r}\right\rceil }\right\rceil }$ –  badp May 23 '11 at 11:50
    
@badp, ah, that's much better. Was going to give it a prod once I got home but, well, that does the job nicely :) –  DMA57361 May 23 '11 at 12:03

More about the Social Policies and general strategies

Maps affect the addition of city costs (large - 20%, Huge - 15%, all else 30%) and the game speed affects the base cost (Marathon is +300%, Epic 150%, quick is 67%). Really low difficulty levels also give a bonus. So in theory it is ok to expand more with larger maps.

There is an issue with Cristo though. it's only a 25% reduction in the cost of policies. I haven't checked Free Speech yet, but expect it's going to be calculated the same and end up only a 20% reduction. (calculation is Old cost / 1.33 (repeated) = new cost. So it ends up being Old * .75 = new)

Which is why going for culture victories is always a bad choice. Firaxis/2K always set the bar too high for them in each game, and never provide decent ways to speed it up.

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oh, and the +cost/city is added after the reduction of the Cristo Redentor, so it is worth it to consider expansion post attaining that wonder + Free Speech. –  MadDjinn Oct 18 '10 at 15:22

Not trying to compete with all the math here, but here are some points:

  • Build as little cities as possible yourself: Because getting them up to speed with culture buildings seems highly non-trivial to me.
  • Puppet as many cities as possible and don't annex them. Puppeted cites seem to like culture buildings, so you get extra culture from these with a decent rate.
  • An added bonus of Puppet Cities seems to be that they go for gold, so you'll have plenty to spend for ->
  • As others said, (cultural) City states are your friends!
  • The number of cities need to be sufficient to get your military up to strength or itherwise you will just be wiped off the map.
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