How many cities should I build for a cultural victory?

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?

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.

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.

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.

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.

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?