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

[futurehermit]

YES! That is exactly what I needed -- now I get it, thanks

 

[futurehermit]

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.)

So...let's see if I get this right...

The formula you provide is for COST of the next policy. If we want to know about whether or not to found the next city (i.e., the question of diminishing returns), how about the following:

1) TURNS1 = COST1/CPT1

where 1 = staying put at the current number of cities and where TURNS1 = the number of turns to complete the next policy while staying put

2) TURNS2 = COST2/CPT2

where 2 = adding another city such that COST2 increases the cost as per the formula you provided and CPT2 increases the cpt by the amount immediately provided by the new city (i.e., including any rushed buildings, but excluding any future buildings); and where TURNS2 = the number of turns to complete the next policy while adding another city

3) If TURNS2 < TURNS1, found another city

Now, an alternative to 3) could be to accept temporary diminishing returns as an investment on the city eventually breaking even (i.e., paying for itself) and then giving a net return on that initial investment, but I'm happy to leave *that* point of discussion until I wrap my head around this initial hurdle

Thanks everyone!

 

[sanabas]

So...let's see if I get this right...

The formula you provide is for COST of the next policy. If we want to know about whether or not to found the next city (i.e., the question of diminishing returns), how about the following:

You're making it too complicated.

Take your empire's current total culture per turn.

Divide that by (x+9), where x is your number of cities.

If your new city will produce more cpt, found it. If not, don't.

So, if you have a 10 city empire, producing 50 cpt, then you get 50/11 = 4.55 If your new city is going to have just a monument + liberty opener, that's 3 cpt, so slows you down. If your new city is going to have an amphitheatre too, then it'll speed things up.

 

[Matthew.]

Here ya go, to bring some in-game numbers to the thread I clicked through a king CV game quick.

3 cities
Full Piety/Freedom/Liberty
Sistine, Alhambra, Hermitage, Golden Age
7 planted artists in capital
4 allied culture city-states
CPT = 855

Just finished 25th policy. 26 policy cost = 5865 with 3 cities. 5865/855=6.86. 26 policy cost = 6335 with 4 cities. 6335/6.86=923. 923-855= 68 culture needed in new city to maintain previous culture pace.

68 culture in a city is possible, because both my 2nd and 3rd city have 64 and 72 culture each. But here is the catch: Each one needs a world wonder, full culture buildings, representation from Liberty, and full artist slots worked.

When I had looked into this before I wasn't considering Liberty start, hence why I recall being unable to get enough culture in additional cities.

And so I will repeat again: BWS's summary is correct. You can settle as many cities as you want provided you 1. Take Liberty 2. Build a world wonder in each city 3. Get Sistine 4. Acquire all culture buildings and enough pop. to fill slots. There is an obvious soft cap to these requirements, as after a certain number of cities it becomes too difficult to fulfill each requirement. Although with enough additional culture bonuses through religion or a Civ's UA/UB, one may be able to bypass the world wonder in each city requirement--or at least do so without it harming CPT too much.

 

[futurehermit]

You're making it too complicated.

Take your empire's current total culture per turn.

Divide that by (x+9), where x is your number of cities.

If your new city will produce more cpt, found it. If not, don't.

So, if you have a 10 city empire, producing 50 cpt, then you get 50/11 = 4.55 If your new city is going to have just a monument + liberty opener, that's 3 cpt, so slows you down. If your new city is going to have an amphitheatre too, then it'll speed things up.

This is a great rule of thumb -- thanks!

 

[futurehermit]

You're making it too complicated.

Take your empire's current total culture per turn.

Divide that by (x+9), where x is your number of cities.

If your new city will produce more cpt, found it. If not, don't.

So, if you have a 10 city empire, producing 50 cpt, then you get 50/11 = 4.55 If your new city is going to have just a monument + liberty opener, that's 3 cpt, so slows you down. If your new city is going to have an amphitheatre too, then it'll speed things up.

Twilight Zone Moment:

Using this formula, with the French UA, if you never build a culture building or get cpt from any source other than the capital (1), liberty (1*x), and UA (2*x), it seems that there is NEVER (at least up to where I tested at 184 cities ) a point at which the new city's culture per turn (3) is not greater than the outcome of the formula!!!

Now that is no diminishing returns on a hardcore ICS

(situation changes drastically just adding in monuments btw...more to follow)

 

[futurehermit]

Ok, complicating things a bit further:

Building on this formula:

Take your empire's current total culture per turn.

Divide that by (x+9), where x is your number of cities.

If your new city will produce more cpt, found it. If not, don't.

Let's say I build 14 cities as France under Liberty, but don't build any monuments or other culture buildings, don't have a pantheon yet, and don't have any other sources of culture (e.g., CSs).

According to the formula, I'm all good to have founded that 14th city.

However, once we add the +2 from monuments to each city, the formula says -- uh oh, shouldn'tve built that 14th city...

Am I all good, based on the pre-monuments outcome, or do I have diminishing returns on the 14th city, based on the post-monuments outcome?

In other words, do culture buildings/sources added *after* impact the decision whether to found or not, relative to this formula?

Thanks!

 

[sanabas]

Twilight Zone Moment:

Using this formula, with the French UA, if you never build a culture building or get cpt from any source other than the capital (1), liberty (1*x), and UA (2*x), it seems that there is NEVER (at least up to where I tested at 184 cities ) a point at which the new city's culture per turn (3) is not greater than the outcome of the formula!!!

Now that is no diminishing returns on a hardcore ICS

Let's see, total cpt = 3x + 1

limit of 3x+1/x+9 as x increases is 3, so yeah, it'll always be slightly cheaper.

However, with any other civ, total cpt = x + 1

x + 1/x + 9 will always be less than 1, so same deal. Likewise 5x + 1/x + 9 for france with a monument.

Basically, if all your cities have equal, non-zero culture, a new city will have the same as the rest, and so will always make future policies slightly quicker. But since all cities don't have equal culture, and in particular because of the extra from wonders/CS/policies (2 happiness = 1 cpt, founding a new city drops the happiness, as well)/planted artists/puppet cities, continual expansion won't work. (actually, it would work if you had no limit on number of cities. 1 million 40 cpt cities + 4000 cpt extra, and every 40 cpt city you add will drop the number of turns for each policy.) *edit* sorry, that last bit is wrong. Let's say every normal city produces y cpt. Then, if the extra culture over and above that base average is less than 9y, then infinite expansion is good. If not, it's not. So with 40 cpt, 40 x 9 = 360. If your 'bonus' culture above that, more default cities won't help. And that applies even for just your 2nd city. Which is a good way to use your guesstimation rule of thumb, too.

 

[sanabas]

Let's say I build 14 cities as France under Liberty, but don't build any monuments or other culture buildings, don't have a pantheon yet, and don't have any other sources of culture (e.g., CSs).

According to the formula, I'm all good to have founded that 14th city.

However, once we add the +2 from monuments to each city, the formula says -- uh oh, shouldn'tve built that 14th city...

Your maths is bad. 13 cities = 66 cpt with liberty, monuments, france special, and 1 extra cpt in the capital. 66/22 = 3, so the 5 cpt from the new city says build it.

Am I all good, based on the pre-monuments outcome, or do I have diminishing returns on the 14th city, based on the post-monuments outcome?

In other words, do culture buildings/sources added *after* impact the decision whether to found or not, relative to this formula?

Just guesstimate your future numbers. Say your guess for later is 500 cpt from your 13 cities, thanks to more buildings, wonders, etc, etc. 500/22 = 22.7 cpt. Do you think your 14th city will produce 23 cpt at that point, just from the basic culture buildings + specialists? The more accurate your guesstimations, the more accurate your answer for whether to found the city or not.

 

[sanabas]

That edit I put above is interesting. If, for some reason, your 'bonus' culture will never reach 9 times the base city culture, then ICS away.

Until your bonus culture reaches 9 times the base city, then more cities will make policies cheaper for now, but more expensive once you cross that threshold. It's also possible to move backwards and forward across that threshold. Until amphitheatres, the threshold is 45 cpt for France (9 x 5). Add Amphitheatres, and it jumps to 90 cpt? (add 2 for building, 3 for specialist). Add cathedrals, and it's now 144 cpt. Build sistine chapel, it increases by 25%. And so on.

After you reach it, more cities are worse. So for a city to be worth founding, it either needs to be:

-early enough that the benefit to getting the next policies outweighs the longer time it takes to get later policies

or

-contribute enough to 'bonus' culture, by generating artists to plant/trigger golden ages, build wonders, etc

or

-be indirectly useful in increasing base/bonus culture, by increasing research speed, helping conquer more puppets, etc.

 

[Wargizmo]

For those interested in trying it out, here is the ICS france map that Hammer Rabbi posted a few weeks ago:

ICS_France_Emperor_0000 BC-4000.Civ5Save

It's a pretty nice map on Emperor difficulty, with enough room to settle pretty much as many cities as you want.

I would be interested to see what the fastest culture wins people can get are and how many cities they used.

 

[futurehermit]

Your maths is bad. 13 cities = 66 cpt with liberty, monuments, france special, and 1 extra cpt in the capital. 66/22 = 3, so the 5 cpt from the new city says build it.



Just guesstimate your future numbers. Say your guess for later is 500 cpt from your 13 cities, thanks to more buildings, wonders, etc, etc. 500/22 = 22.7 cpt. Do you think your 14th city will produce 23 cpt at that point, just from the basic culture buildings + specialists? The more accurate your guesstimations, the more accurate your answer for whether to found the city or not.

Well, I wasn't assuming a monument in the newly built city, so it was 3 vs. 3. A better question I guess would've been the next city up:

14 cities = 14+28+28+1 = 71/23 = 3.09. New city produces 3 culture w/out monument, so...don't build it? Build it assuming a monument will be built soon and will pay itself back? (probably)

Just trying to get a feel for things, since the math is so complex.

I'm starting to think that our final conclusion will simply be:

Expand to as many cities as your happiness and neighbours will allow while building monuments and acquiring as much other culture as possible. Then boom your empire while knocking out your neighbours.

Pretty simplistic, but, ultimately, effective, at least on Emperor where I'm playing.

 

[futurehermit]

That edit I put above is interesting. If, for some reason, your 'bonus' culture will never reach 9 times the base city culture, then ICS away.

Until your bonus culture reaches 9 times the base city, then more cities will make policies cheaper for now, but more expensive once you cross that threshold. It's also possible to move backwards and forward across that threshold. Until amphitheatres, the threshold is 45 cpt for France (9 x 5). Add Amphitheatres, and it jumps to 90 cpt? (add 2 for building, 3 for specialist). Add cathedrals, and it's now 144 cpt. Build sistine chapel, it increases by 25%. And so on.

After you reach it, more cities are worse. So for a city to be worth founding, it either needs to be:

-early enough that the benefit to getting the next policies outweighs the longer time it takes to get later policies

or

-contribute enough to 'bonus' culture, by generating artists to plant/trigger golden ages, build wonders, etc

or

-be indirectly useful in increasing base/bonus culture, by increasing research speed, helping conquer more puppets, etc.

This seems pretty critical to our final conclusion -- can you break down this point a bit further?

Until your bonus culture reaches 9 times the base city, then more cities will make policies cheaper for now, but more expensive once you cross that threshold.

 

[DaveMcW]

I'm not sure why you would want to build a city when you can just puppet one.

 

[beezany]

Very cool analysis, sanabas! Especially that 9× rule.

Here ya go, to bring some in-game numbers to the thread I clicked through a king CV game quick.

3 cities
Full Piety/Freedom/Liberty
Sistine, Alhambra, Hermitage, Golden Age
7 planted artists in capital
4 allied culture city-states
CPT = 855

Just finished 25th policy. 26 policy cost = 5865 with 3 cities. 5865/855=6.86. 26 policy cost = 6335 with 4 cities. 6335/6.86=923. 923-855= 68 culture needed in new city to maintain previous culture pace.

Thanks for the concrete example! I am having trouble matching these numbers, though. If I recall correctly, with those policies and wonders, the 26th-policy cost is 6,370 with three cities and 6,900 with four:

Base: 25 + (3×25)^2.01 = 5898.2
Three cities: ×1.2 = 7077.8, –10% discount = 6370.0
Four cities: ×1.3 = 7667.6, –10% discount = 6900.9

Using the exact numbers, you would need 855 × (6,900 ÷ 6,370) = 926.1 culture to maintain pace, or 71 more from the new city. Using sanabas's rule of thumb, you also need 855 ÷ (3 + 9) = 71 from the new city.

Note that the city screen does not show the Golden Age bonus, so I'm not sure whether the 64+72 culture for your second and third city includes it. I suspect not, but please correct me if I'm wrong. If I'm correct, they provided 77+86 culture during the Golden Age, leaving 692 "base" culture. By the "10% of base" rule, you need 69 culture from a new city for it to be worthwhile. I estimate the "bonus" culture for this civ (including landmarks, Alhambra, and Hermitage) at 606 cpt; by the "1/9 of bonus" rule, you need 67 culture from a new city.

So conservatively, you'd need a city with 71 cpt during a Golden Age, which is 59 cpt normally. That requires 37 culture without a wonder, which you could just barely manage with a strong cultural religion, but it's much more likely with a wonder built. Practically speaking, in a game like this, it's probably only worthwhile to add cities that you can use to build a wonder.

Some players have had good success with Rationalism culture games (instead of Piety), in which case there's much stronger incentive to expand infinitely.

 

[sanabas]

Using the exact numbers, you would need 855 × (6,900 ÷ 6,370) = 926.1 culture to maintain pace, or 71 more from the new city. Using sanabas's rule of thumb, you also need 855 ÷ (3 + 9) = 71 from the new city.

That bit isn't rule of thumb, it's exact numbers.

Well, I wasn't assuming a monument in the newly built city, so it was 3 vs. 3. A better question I guess would've been the next city up:

14 cities = 14+28+28+1 = 71/23 = 3.09. New city produces 3 culture w/out monument, so...don't build it? Build it assuming a monument will be built soon and will pay itself back? (probably)

I'd just assume your basic culture buildings would go up fairly quickly, or be purchased. But if you plan to leave it without a monument for some time thanks to no cash/minimal production, then yeah, work it out without a monument.

Just trying to get a feel for things, since the math is so complex.

I'm starting to think that our final conclusion will simply be:

Expand to as many cities as your happiness and neighbours will allow while building monuments and acquiring as much other culture as possible. Then boom your empire while knocking out your neighbours.

Pretty simplistic, but, ultimately, effective, at least on Emperor where I'm playing.

I think expanding will slow you down. I don't think it's that tough to reach 9x a city's base culture in bonuses from alhambra/hermitage/wonders/planted artists/CS/puppets. And once you do, the ideal number of cities is just 1. So the only reason to build more cities is to generate more artists/build more wonders/research faster. Go beyond about 4, and I don't think the extra cities will generate any more artists. Play on a sufficiently high level, and wonders are tough to get. Biggest reason for more cities is more science, to unlock Freedom faster, to build hermitage earlier, etc.

 

[beezany]

That bit isn't rule of thumb, it's exact numbers.

Nah, the exact ratio is (6,900 – 6,370) / 6,370. Which is a smidge different from 1/12 because of rounding. But I'll grant that your rule is exact enough.

I agree that for a conventional Piety-based culture game that you're probably best off in the long run with as few cities as you can manage. There are enough factors that it may not be amenable to numerical analysis. For a Rationalism-based culture game, you're probably best off with indefinite expansion. Unfortunately, I don't think there has been enough serious experimentation in the community to say which approach is superior.

 

[Matthew.]

I used to play CV's with Rationalism; the problem was finishing spaceship before Utopia.

I'd say regardless of victory, the best way to go about the game seems to be to start with 3-4 of your own. Then as you acquire more, either through settling or conquering, focus on getting 6-8 cities annexed with buildings and specialists while keeping the rest of the cities as puppets. Domination games will, of course, put higher priority on annexing enemy capitals for the wonders, but other victories I'd base it off city location and population.

 

[budweiser]

I'm not sure why you would want to build a city when you can just puppet one.

The answer is Great Artist generation. Each city you make should give you one during the game.

 

[budweiser]

I agree that for a conventional Piety-based culture game that you're probably best off in the long run with as few cities as you can manage. There are enough factors that it may not be amenable to numerical analysis. For a Rationalism-based culture game, you're probably best off with indefinite expansion. Unfortunately, I don't think there has been enough serious experimentation in the community to say which approach is superior.

I trust that game design actually makes them equal, depending on the land and the civ.

What I wonder is, can you win cultural before you have to advance into the industrial era? And could you do it in about 250 turns before the AI wins.