civilization-5
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?
Although bwarner's answer was helpful as always, no one verified beyond "I think so" that puppet cities are affected by each of the civilization wide effects, so I did some testing and:
Yes, puppet cities are affected normally by all civilization-wide effects.
I can't verify that there are no exceptions, but I verified at least one of each of the things I was concerned about...
Civilization Special Trait
I loaded up France on easy, and went about conquering. Japan fell, giving me Kyoto to play with. As soon as I conquered the city and took it as a puppet, it was producing 2 culture. This matches perfectly with France's ability, so CHECK!
More Test Setup
Having acquired my victim, I setup the test: social policy ready, golden-age ready, and wonder ready all on the same turn! (Save file available upon request, whenever I get around to it after that.)
Before
Social Policy
I enacted Liberty->Republic, which gives +1 production/city. That matches, so CHECK!
Golden Age
I already had culture working for me, so I started the golden age by enacting the social policy Piety->Reformation. Clearly shows an increase in both gold and production, CHECK!
World Wonder
If you take a look at all of my cities from the same turn, you can see Orleans is about to finish the Sistine Chapel, which gives +33% culture/city. I went to the next turn, and without any growth or buildings completing, the after shows increased culture on Kyoto. (Lyon also started with 7 culture and ended with 9 as well, so the rounding definitely matches.) CHECK!
Everything checks out. I suppose puppets really are just normal cities that you don't control. I would still love to hear if anyone finds any exceptions!
Is "Micro-Empire" a viable strategy for one or all of the victory conditions?
Micro-Empires don't work for all victories. Almost by definition, you probably won't win a domination victory with a micro-empire. To do so you would need to raze all cities you capture (to stay micro), which would just give your opponents space to expand, which makes your job harder. It would also be difficult to balance advancing your cities and cranking out military units without falling behind on both fronts. I think a minimum of 4 - 6 "core" cities is a must for a domination victory. It may be a great challenge to try it with two though.
Having said that, in addition to @WillfulWizards hints:
Best Answer
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:
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).
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.
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.