Help with some complex math: Optimal # of cities for France opening

Hey everyone,

I'm wondering what the "break even" point is, in terms of # of cities, for a French liberty opening such that the next city would introduce diminishing returns in terms of the culture produced vs. the increase in social policy cost. I want to take into account representation, the French UA, etc.

Basically, I want to know when to stop building cities such that optimal culture/science victories are possible -- i.e., an alternative to full out ICSing.

Based on my calculations so far, I'm thinking it is around 7-8, but the math gets a little complicated and I'd like an exact number, preferably.

This is for standard speed/map size/Emperor.

I had a game last night opening with 6 cities and it was the strongest game I've ever had. Total power house, finishing the liberty tree at turn 66, puppeted two neighbours before turn 100, and spread enhanced Taoism far and wide across my continent within the first 150 or so turns. I want to build on the strategy, as I think I could squeeze in another city or two (or three).

Thanks!

The break even part is very simple. the socialpolycy cost increases by 10% per city with fully completed liberty. So as long as you can produce more then 10% of your highest culture city you will be ahead.

In practice wide can get cultural victory. especially if you get the sistine chapel +oracle.
Problem is wide usually wins by an other way then culture. If you go wide and focus on artist slots and cultural buildings then wide will give you a faster victory in culture then tall.

Yeah, contrary to popular wisdom, there is no break-even point or diminishing returns where culture gets harder. The main obstacle to wide culture is not how many cities you build, but rather how quickly you can get them to produce more than 10% of your base culture. You can settle as many cities as you want, as long as you can finish all of the culture buildings and fill all of their artist slots quickly.

Actually, you guys are both right and wrong, but you've helped me figure it out. You're right that it depends upon how fast you can get new culture buildings up (at least to a certain point); however, you're wrong that there isn't a break-even/diminishing returns point -- precisely because you do have to factor in that there will be a number of turns where the city is giving diminishing returns until the buildings are up and running. Of course, however, this presumes that you don't have the $$$ on hand to rush the building(s) the turn that you found the city. If you do, then by all means, add that city to the total number of cities you can found without diminishing returns -- and, with unlimited $$$, there would really be no break-even point. However, presuming that $$$ is a finite resource, especially during the opening of the game, there is a break-even/diminishing returns point -- and you've helped me figure out how to calculate it, so thanks!

Basically, I followed your simplified logic (I made it too complicated), I calculated that a new French city is going to produce 3 culture per turn (assuming no rushed buildings) and calculated that against a percentage of total empire culture per turn. Once the percentage dipped below 10%, I noted that as the point of diminishing returns.

I ran this calculation twice -- once for REXing without any monuments and once assuming that monuments are built asap during expansion. For the former, France can expand to 10 cities before diminishing returns and only 6 for the latter.

EDIT to add: For the latter, if you can rush monuments, you can again expand to 10.

EDIT 2 to add: Made some errors in my calculations and corrected my post as a result.

p.s., This is assuming Emperor/standard.

Cheers

BWS said:

Yeah, contrary to popular wisdom, there is no break-even point or diminishing returns where culture gets harder. The main obstacle to wide culture is not how many cities you build, but rather how quickly you can get them to produce more than 10% of your base culture. You can settle as many cities as you want, as long as you can finish all of the culture buildings and fill all of their artist slots quickly.

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That's his point though, I think. Diminishing returns may not be the exact term for it, but at some point, you simply won't be able to practically generate enough culture from the city to compensate for the increased cost in brings for every single future policy, which is a LOT when you add up the increased cost of all the later policies even with Representation.

A culture pantheon adds a whole other set of culture along with culture buildings like monestaries, cathedrals, etc.

Have you calculated the results you would get if you went to Monarchy via Tradition before or instead of Liberty?

I might take a break tonight from all these diety headaches and look for a wine start with France.

Edit: If you think about a game plan for France, you have access to culture from Tradition, right side Honor and the Liberty opener for starters. Two times nothing is still nothing. I think its better to add culture points to your cities before you start looking at percentages from Representation.

Light Cleric said:

That's his point though, I think. Diminishing returns may not be the exact term for it, but at some point, you simply won't be able to practically generate enough culture from the city to compensate for the increased cost in brings for every single future policy, which is a LOT when you add up the increased cost of all the later policies even with Representation.

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This largely depends on your ability to rush buy the buildings. You need tons of cash on hand per city 1000+.

Yeah, I'm looking at a France ***opening*** which means you're not going to have the cash on hand to do more than rush a small handful (at most) of monuments for your 7th-10th cities (approximately). Fortunately, that coincides with the point of diminishing returns while building monuments, which means that, if you can have the cash on hand to rush monuments 7-10, you can expand to 7-10 cities peacefully as France before hitting diminishing returns.

Of course, there are two main impediments to this REXing: Happiness and neighbours.

Similarly, this removes monasteries/cathedrals from the equation, since you won't have those in the first 70 turns of the game. Same with culture pantheon. However, for example, taking the pantheon that gives +1 per shrine dips the point of diminishing returns down to only 5 cities (assuming shrines in each city and assuming you don't have the $$$ to rush buy buildings).

So, the question really becomes -- how many cities can you build peacefully prior to turn 70???

budweiser said:

This largely depends on your ability to rush buy the buildings. You need tons of cash on hand per city 1000+.

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Well, yes, that's where the "practical" part comes in. I mean you have to ask "How much is this increasing my policy costs by?" for every future policy and take that amount into consideration. I think that's what people(or at least I) mean when they talk about diminishing returns even with the French bonus.

futurehermit said:

So, the question really becomes -- how many cities can you build peacefully prior to turn 70???

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You should be able to get the third policy well before turn 30 and just bang out settlers from the cap for the next 40 turns.

Edit: Or does this make sense or could the order be rearranged?

Start 3 build Monument +2>Tradition +3>Liberty +1>Right side Honor 123 +2 (max capital & max anti-barb variable) add cities>Lib>Lib (free city, free 3 or 5 culture & fast cities)>Legalism at D&P (free 3 +3) = 8 policies. Start Piety for bonus faith. Add Monarchy soon.

I'm not even sure it is possible to break even with additional cities. 1. Not only would you need all culture buildings ASAP, but filled specialist slots as well. That requires a heavy investment in gold/hammers and population. 2. Other sources indirectly become less effective, such as city-states. For example with three cities, a culture city state is contributing a certain percentage to overall culture, but with seven cities, the culture to policy ratio of city-states is significantly less.

I think at best you can greatly lessen the punishment of additional cities, maybe even to a negligible amount, but more cities will always result in longer policies. I'd rather play France normally and take the extra couple of policies.

futurehermit said:

Basically, I followed your simplified logic (I made it too complicated), I calculated that a new French city is going to produce 3 culture per turn (assuming no rushed buildings) and calculated that against a percentage of total empire culture per turn. Once the percentage dipped below 10%, I noted that as the point of diminishing returns.

Click to expand...

That is the correct way to calculate diminishing returns for an exponential increase in costs, but the increase is linear, not exponential. The correct way to calculate the value of additional cities with a linear increase is to figure how much culture you could produce with only your capital (including city-states, religion, and puppets). When your city produces more than 10% of that figure, it is a net asset to culture production. For the French example, every new city is immediately helpful unless your capital alone can produce more than 30 culture per turn.

Matthew. said:

I'm not even sure it is possible to break even with additional cities. 1. Not only would you need all culture buildings ASAP, but filled specialist slots as well.

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You don't need all of the culture buildings to exceed 10% of base culture production, as described above. Let's consider France in the early game. How much does it cost to adopt the 6th Liberty policy with varying numbers of cities, assuming you've already taken Representation?

Let's assume that the capital has a monument, amphitheater, and one specialist, but no wonders, puppets, or city state friends. Base culture is 2 (Ancien Regime) + 1 (Palace) + 1 (Liberty) + 2 (Monument) + 3 (Amphitheater) + 3 (Artist) = 12 culture per turn. The 6th policy with one city costs 255 culture, which takes 22 turns at that rate.

Three-city games are popular for cultural victories. With three cities, the 6th policy costs 305 culture. The second and third city add at least 3 culture each, for 18 total culture per turn. You will finish the policy in only 17 turns.

Most people would regard a ten-city empire as excessive for a cultural victory. With ten cities settled, the 6th policy costs 485 culture. But when you produce 39 culture per turn, it'll only take 13 turns! That's close to double the rate of using a single city.

The math changes once you reach a point where you could produce more than 30 culture per turn from your capital alone. Then again, monuments (and cathedrals) are very cheap, so you might want to figure the base culture for new cities at 5 per turn, 8 if you have a cathedral religion, 10 if you have cathedrals and the Sistine Chapel. It's quite far into the game before you could produce 100+ culture per turn from your capital alone, so rapid expansion makes a lot of sense as France.

More generally, there are three main concerns for empire size and culture:

• Will my capital ever reach a point where it produces so much culture that no city can beat the 10% penalty? If so, then eventually you reach the point Matthew. described, where it doesn't make sense to expand at all. However, because you can generate 36+ cpt from basic buildings alone, much more with religion and wonders, it's easy to pass the 10% mark even in the late game.
• Will it take so long to pass the 10% mark that my new cities will never pay for themselves? This is a more realistic concern in the late game. While even chump cities in an ICS game can eventually produce 36+ cpt, it might take a while, and the excess may be too little to pay for the set-up costs. If you have enough cash to fully develop new cities as you found them, this is not a problem.
• Can my new city pass the 10% at or very near to founding? This is the situation I describe in depth above, and it's quite possible for France. And if a city can beat the penalty at founding, it's very easy to maintain such that it continues to be a net asset.

So for a good general rule of culture and expansion, I would say that number of cities doesn't matter nearly as much as timing. At any point in the game where you can beat 10% of your capital culture right away, it pays to found as many cities as possible. At any point where you can buy yourself past the 10% penalty, it pays to found as many cities as you can afford the cash for culture buildings. After that, stop settling new cities and focus on development (or conquest) instead.

(Now I'm tempted to play a proof of concept game as France to demonstrate this.)

I looked into this a little while back when working on culture games, but it is difficult to find concrete numbers outside of the game and kind of lost interest. But I always got the impression that the increased policy cost was intended to keep relatively equal policy acquisition throughout the game. That is, a 4 city empire and a 16 city empire will roughly acquire policies at the same rate. The logic makes sense, because if such a system wasn't in place, only could fly through the policies by adding tons of cities.

The catch is culture games (CV's) rely on settled artists [logically] placed near the city with Hermitage, and to surpass or "break" the equalized acquisition of policies. In my above response, this is what I was thinking, and it was a mistake. For CV's (and fast CV times) I think it becomes more of a disadvantage to expand, because each settled city indirectly reduces the raw culture output of the super city (artists/hermitage). However when simply speaking of regular games and policy acquisition, the 10% penalty is likely a non-issue. I haven't checked the specific numbers, but I have no reason to doubt BWS's summary as being correct.

The other thing people are forgetting is the great artist generation. If you settle at the start of the game and work great artist slots as soon as you get them then each city will produce at least one great artist during the game. If that artist goes and settles in the capital that gives you a massive amount of culture, often 30+ per settlement when you take into account wonders, freedom finisher, hermitage and golden age bonuses.

There's also the science, gold and production benefits that having an extra city brings, which will enable you to unlock better culture buildings quicker, and then rush buy them quicker.

Remember, just because a city isn't producing that +10% right away, doesn't mean it isn't going to pay off down the track.

Very, very interesting, thanks!

Can you show me the math for where you get the 200 culture (1 city) and 245 culture (3 cities) figures from, so I can understand the math better? Alternatively, just point me in the direction of where I can get a current list of the policy costs?

Thanks!
Jason

Even for a CV game, more cities should be better: 10% is a very low bar. I'm not sure why the community here hasn't explored this more, but I have a few suspicions. I think a lot of people are operating under old or incorrect assumptions about the city penalty. Most culture guides I've seen are very old, from the era when the city penalty was much higher. I've also seen a lot of comments that "tall" is good for culture, but beyond having enough citizens to create Artists, population doesn't matter. Perhaps the biggest obstacle is that there simply isn't a lot of interest in culture, especially not among wide players who would rather use all those resources toward domination.

Also: Matthew. speculated that "each settled city indirectly reduces the raw culture output of the super city (artists/hermitage)." This is true to some extent, as additional cities can reduce the benefit of Mandate of Heaven, and to optimize Reformation you'll end up building fewer wonders in your capital. However, additional cities should also help with great artists as Wargizmo notes, and the benefit of Reformation is much higher than the cost.

futurehermit said:

Can you show me the math for where you get the 200 culture (1 city) and 245 culture (3 cities) figures from, so I can understand the math better? Alternatively, just point me in the direction of where I can get a current list of the policy costs?

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There's a lot of information in this thread about culture victores. Mondoflow created a Google spreadsheet with the standard culture costs (with no discounts). I use something similar to calculate things, with parameters for the discounts. Speaking of which, I accidentally included a 20% discount in my figures above. The correct culture costs with only the city discount are 255 (one city), 305 (three cities), and 485 (ten cities). With 12 base + 3 culture per additional city, the correct times to adopt are 22 turns (one city), 17 turns (three cities), and 13 turns (ten cities). I fixed my original post with these updated numbers.

Here's the thing I don't understand though... You say:

The correct way to calculate the value of additional cities with a linear increase is to figure how much culture you could produce with only your capital (including city-states, religion, and puppets). When your city produces more than 10% of that figure, it is a net asset to culture production.

Click to expand...

But, what about the 10% increase for each city previous to the one you are about to found???

In other words, isn't the 10% increase per city such that the policy cost penalty is:

x (base) * 1.10 (1st city beyond capital) * 1.10 (2nd city beyond capital) * 1.10 (ith city beyond capital)???

If so, how is it simply the case that your ith city just has to produce > 10% of your capital's culture???

futurehermit said:

In other words, isn't the 10% increase per city such that the policy cost penalty is:

x (base) * 1.10 (1st city beyond capital) * 1.10 (2nd city beyond capital) * 1.10 (ith city beyond capital)???

If so, how is it simply the case that your ith city just has to produce > 10% of your capital's culture???

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Because it's actually base * 1.1 for 2nd city, base * 1.2 with 3 cities, base * 1.3 for 4 cities, etc. And NOT base * (1.1)^n

The increase from 9 cities to 10 cities is from 1.8 * base to 1.9 * base, an increase of 5.56%.

So it's not that each city needs to produce 10% of the capital's culture, it's that your 10th city needs to produce 5.56% of your empire's culture. Your 20th city needs to produce 3.57% of your empire's culture. 10% of your capital + CS + policy culture is a much better rule of thumb than 10% of your empire's culture. But if you want to be accurate, the increase from x cities to x+1 cities needs the x+1st city to produce 100/(x+9) % of the total cpt for the rest of the empire, otherwise it's making future policies slower.

No, that is what I mean when I say that the cost increase is linear, not exponential. The correct formula is: (25 + 3k^2.01) × (1 + 0.10n) where k is number of policies already adopted and n is the number of additional cities. For the 6th policy and no additional cities, that's 25 + (3×5)^2.01 = 256.2, rounded to the nearest 5 = 255. Each additional city adds only 10% of that amount to the total cost. For ten cities, that's 256.2 ×1.9 = 486.7 culture, rounded to the nearest 5 = 485.

(Sniped by sanabas! Thanks for the alternate explanation.)