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

[beezany]

Aha! Thanks, sanabas! That's a cool way to figure the Bonus/9 rule.

I was trying to figure out the equivalents for the full 15% penalty, but it's ugly. I realized that the general form is (1/penalty)–1. For the 10% penalty, that's a nice, round figure: N+9 and Bonus/9. For 15%, it becomes N+(17/3) or Bonus×3/17, which is ludicrous.

Then again, the exact number isn't too important, because the 15% mark is impossible to reach in a standard game. For the example above: 15% of core is 106 cpt, and the Bonus rule is even worse, 127 cpt. (The next-city rule breaks down when all of the existing cities are substandard.) It looks like optimal culture victories require Representation if you build more than one city.

Moving along: I played a second game with a Tradition quad start and four puppets. (I accidentally boxed in Catherine, so war was inevitable.) I won in 287 turns, about 10% faster than the wide game and a personal best for CVs.

Spoiler :

Breakdown:

• 519 in the capital
• 140 from puppets, excess happiness, and city-states.
• 659 core culture (capital + puppets + other sources)
• 264 in three settled cities (97, 89, 78 cpt)
• 923 total culture
• 1,107 Golden Age culture

This time I had Cathedrals + Monasteries, which produce the same 60 cpt base city.

• 923 total culture
• 240 base culture (four cities × 60 cpt)
• 683 bonus culture
• Bonus rule = 76 cpt
• Next-city rule = 71 cpt
• 10% of core = 66 cpt

Also, despite the much smaller empire, my final science was only about 12% lower. HOF score was much lower though: Despite the quicker finish, the total score was less than half of my wide effort.

I guess the moral of the story is: Wide and tall CVs are both viable. If you're going for fastest victory, wide is competitive, but tall is optimal. If you're going for highest score, tall is not even in the running. And either way, you'll need Representation.

 

[sanabas]

It looks like optimal culture victories require Representation if you build more than one city.

If you had just two cities, you could call the base city 97 cpt, the total 756. 756 - 97*2 = 562. 562 x 3/17 = 99 So 2nd city is more or less at the break-even point at the end, and likely ahead of it earlier in the game.

I guess the moral of the story is: Wide and tall CVs are both viable. If you're going for fastest victory, wide is competitive, but tall is optimal. If you're going for highest score, tall is not even in the running. And either way, you'll need Representation.

If you want highest score, speed stops being relevant. It's almost always max turns -1 as the finish date you want.

And just a random thought: I have a feeling that settler has a smaller than normal penalty for each extra city. Vaguely remember the tooltip saying 7.5 instead of 15. Which with representation would drop to 5, and you could likely go as wide as you want, provided you can afford to buy the culture buildings. Unless my memory is faulty, which is always possible.

I really should play a couple of games for this, haven't started any yet.

 

[futurehermit]

As I'm most interested in domination games with France (to utilize the two UUs), what I'm taking from this discussion is that, if I want to continue generating enough culture to complete 3-4 policy trees (or their equivalent in less than complete trees), I should keep my core (non-puppet) empire to about <= 8 cities -- unless I go into piety and build WWs in the cities, in which case I could expand up into the 10-12 city range.

This matches my original intuition and experience that 6-8 cities is an optimal opening with France.

One note from my recent games: I generally want to start into Commerce right after completing Liberty. If you want to do that (or Patronage), then it's important to balance culture against science. I'm finding that if I'm teching too broadly and building cities too rapidly that I end up having to "waste" a policy on Tradition, Honor, or Piety. Of course, if you plan to go further up one of these trees, that's no problem. But, it becomes a problem if you don't plan on it.

In contrast, I find that if I roughly beeline Theology or Guilds while keeping my core cities to about 4-6 that I can move smoothly into Commerce after Liberty. In my current game I've done this while keeping my core empire to I think 5 cities. I started with 4 cities, puppeted first Greece and India with warriors/archers/chariots/catapults then Sweden and Spain with upgrades to swords and CBs. I've finished Commerce and started into Order (from dynamite). I've got settlers dispersed across the rest of my continent and once I finish Order, I will simultaneously settle between 10-20 cities. I have quite a bit of excess happiness (have produced 2-3 golden ages), but may also need to buy up some happiness buildings. However, with the per-city bonuses, I should have enormous empire wide production in short order to close out the game once my second UU comes online.

USA and Celts are on one small continent, at war with each other, and Korea (my main rival) is alone on another continent. I'm currently assaulting USA with musketeers and gatling guns and cannons and will steamroll into Celts as well. Korea will get mauled with my second UU. Looks to be a sub-turn 300 domination victory. I'll let you know how it goes. Emperor/Standard/Continents.

 

[beezany]

Sorry if we hijacked your thread. If your goal is not the fastest possible cultural victory, then number of cities really doesn't matter much. The city penalty only becomes a factor when you pursue culture very aggressively and build a cultural super-city. Even then, wide vs tall is only the difference between winning at turn 250 instead of 300, sometimes not even that much.

Let's consider your actual goal now! What does it take to finish most of three or four policy trees in the course of a domination game? Let's pin down the goal: Domination games are pretty quick, so let's consider what you can do in 240 turns. I am guessing that you want to finish at least one Ancient tree and one Industrial tree (e.g., Liberty and Order), plus a handful of Medieval and Renaissance policies. Let's assume that you want 16–20 total policies in 240 turns. That requires a pace of 12–15 turns per policy, which is aggressive but not impossible.

First, let's consider the early game: Start with 3 cpt, produce 25 culture to open Liberty in 9 turns. Now we have 4 cpt and soon a monument for 6 cpt; make 30 culture for Republic in 5–7 turns. With 6 cpt, you get 60 culture for Collective Rule in 10 turns. France starts with three policies and a free settler in about 25 turns, well ahead of the curve even for a CV game.

With a second city you'll have 9 cpt to get to 120 culture for Citizenship. Depending on how long it takes to found your second city, that will take about 12–16 turns. By my experience and calculations, you are likely to have 3–4 cities and Citizenship by turn 40. Working on Representation, you will have roughly four cities and a monument or two: 12–17 cpt to get 250 culture in 15–20 turns. Now with the cheaper city penalty, Meritocracy costs 330 culture, and you'll have enough culture to adopt it in 15–20 turns, possibly faster if you have started building wonders or befriending city-states. Altogether, you should finish Liberty by turn 75, faster if you get lucky with ruins or religion.

Reality check: Last time I tried this, I founded Paris on turn 1, Orleans on 25, Lyon on 39, Troyes on 47, and Tours on 67. I adopted Liberty on turn 3 (culture ruin), Republic on 11, Collective Rule on 22, Citizenship on 36, Representation on 53, and Meritocracy on 67 (just before founding Tours).

Now for late game! First, let's figure bonus culture. After entering the Industrial Era, each cultured city-state ally grants 26 cpt. Figure that you can ally with three of them (liberating them from conquerors if necessary) for 78 cpt. You'll also have culture from puppets, although they put a very low priority on culture buildings. In my game above, about a quarter of the puppets never even built a monument, only a quarter built amphitheaters, and only one built past that. Puppets also have a –25% culture penalty, although by late game you should have (or capture) the Sistine Chapel, which offsets that. Overall, expect just 3 cpt per puppet, but in a domination game that could still be 50–100 cpt. Let's estimate 150 cpt total from city-states and puppets.

Finally, let's calculate what you need from your own cities in the end game. At the easy end, what do you need for 16 policies at a 15-turn pace? With four cities and Representation, the 16th policy costs 2,765 culture, so you'll need 185+ cpt to keep pace. That's barely more than the bonus culture, only 9 cpt per city. You can get that with just an amphitheater. With twelve cities, the 16th policy costs 4,465 culture, and you'll need 300+ cpt to keep pace, or about 12 cpt per city. That's still trivial. Let's say you want to build all 40 cities on the France city name list. With forty cities, the 16th policy costs 10,425 culture, 695 per turn, still just 14 cpt per city. If you only need 16 policies, build as many cities as you want.

On the harder side, how about 20 policies in 240 turns, a 12-turn pace. With four cities and Representation, the 20th policy costs 4,430 culture, 370 per turn, and about 55 cpt in each city. A full set of culture buildings, with four artists, Liberty, and Sistine Chapel produces 47 cpt. A staffed cathedral will get you over 55 cpt. More cities makes it easier: With twelve cities, the 20th policy costs 7,155 culture, 600 per turn, and less than 40 cpt per city. That's low enough that you can skip broadcast towers, Sistine Chapel, or religion entirely. How about lots of cities? With forty cities, the 20th policy costs 16,695 culture, 1,391 per turn, and only about 30 cpt per city. That's low enough that you only need one of the three major culture boosters.

Generally, unless you are going for the fastest possible culture victory, more cities will help with culture. There are no diminishing returns, just the opposite! With forty cities and cathedrals, you could finish four policy trees in 240 turns, just by building and staffing all the culture buildings. To do that with four cities, you'd need about 650 cpt, which you just can't get without significant bonus culture, enough that you're basically playing for a culture victory.

@sanabas: While doing the math for my last post, I realized that the 10% core rule is identical to the next-city rule, for the first extra city. 1/(N+9) = 10% when N=1. The next-city rule is interesting because the target number keeps rising as you exceed expectations, but it falls if you start adding weaker cities. When the next-city number is lower than the 10% core number, it means that you have already screwed up.

 

[futurehermit]

Sorry if we hijacked your thread. If your goal is not the fastest possible cultural victory, then number of cities really doesn't matter much. The city penalty only becomes a factor when you pursue culture very aggressively and build a cultural super-city. Even then, wide vs tall is only the difference between winning at turn 250 instead of 300, sometimes not even that much.

Let's consider your actual goal now! What does it take to finish most of three or four policy trees in the course of a domination game? Let's pin down the goal: Domination games are pretty quick, so let's consider what you can do in 240 turns. I am guessing that you want to finish at least one Ancient tree and one Industrial tree (e.g., Liberty and Order), plus a handful of Medieval and Renaissance policies. Let's assume that you want 16–20 total policies in 240 turns. That requires a pace of 12–15 turns per policy, which is aggressive but not impossible.

First, let's consider the early game: Start with 3 cpt, produce 25 culture to open Liberty in 9 turns. Now we have 4 cpt and soon a monument for 6 cpt; make 30 culture for Republic in 5–7 turns. With 6 cpt, you get 60 culture for Collective Rule in 10 turns. France starts with three policies and a free settler in about 25 turns, well ahead of the curve even for a CV game.

With a second city you'll have 9 cpt to get to 120 culture for Citizenship. Depending on how long it takes to found your second city, that will take about 12–16 turns. By my experience and calculations, you are likely to have 3–4 cities and Citizenship by turn 40. Working on Representation, you will have roughly four cities and a monument or two: 12–17 cpt to get 250 culture in 15–20 turns. Now with the cheaper city penalty, Meritocracy costs 330 culture, and you'll have enough culture to adopt it in 15–20 turns, possibly faster if you have started building wonders or befriending city-states. Altogether, you should finish Liberty by turn 75, faster if you get lucky with ruins or religion.

Reality check: Last time I tried this, I founded Paris on turn 1, Orleans on 25, Lyon on 39, Troyes on 47, and Tours on 67. I adopted Liberty on turn 3 (culture ruin), Republic on 11, Collective Rule on 22, Citizenship on 36, Representation on 53, and Meritocracy on 67 (just before founding Tours).

Now for late game! First, let's figure bonus culture. After entering the Industrial Era, each cultured city-state ally grants 26 cpt. Figure that you can ally with three of them (liberating them from conquerors if necessary) for 78 cpt. You'll also have culture from puppets, although they put a very low priority on culture buildings. In my game above, about a quarter of the puppets never even built a monument, only a quarter built amphitheaters, and only one built past that. Puppets also have a –25% culture penalty, although by late game you should have (or capture) the Sistine Chapel, which offsets that. Overall, expect just 3 cpt per puppet, but in a domination game that could still be 50–100 cpt. Let's estimate 150 cpt total from city-states and puppets.

Finally, let's calculate what you need from your own cities in the end game. At the easy end, what do you need for 16 policies at a 15-turn pace? With four cities and Representation, the 16th policy costs 2,765 culture, so you'll need 185+ cpt to keep pace. That's barely more than the bonus culture, only 9 cpt per city. You can get that with just an amphitheater. With twelve cities, the 16th policy costs 4,465 culture, and you'll need 300+ cpt to keep pace, or about 12 cpt per city. That's still trivial. Let's say you want to build all 40 cities on the France city name list. With forty cities, the 16th policy costs 10,425 culture, 695 per turn, still just 14 cpt per city. If you only need 16 policies, build as many cities as you want.

On the harder side, how about 20 policies in 240 turns, a 12-turn pace. With four cities and Representation, the 20th policy costs 4,430 culture, 370 per turn, and about 55 cpt in each city. A full set of culture buildings, with four artists, Liberty, and Sistine Chapel produces 47 cpt. A staffed cathedral will get you over 55 cpt. More cities makes it easier: With twelve cities, the 20th policy costs 7,155 culture, 600 per turn, and less than 40 cpt per city. That's low enough that you can skip broadcast towers, Sistine Chapel, or religion entirely. How about lots of cities? With forty cities, the 20th policy costs 16,695 culture, 1,391 per turn, and only about 30 cpt per city. That's low enough that you only need one of the three major culture boosters.

Generally, unless you are going for the fastest possible culture victory, more cities will help with culture. There are no diminishing returns, just the opposite! With forty cities and cathedrals, you could finish four policy trees in 240 turns, just by building and staffing all the culture buildings. To do that with four cities, you'd need about 650 cpt, which you just can't get without significant bonus culture, enough that you're basically playing for a culture victory.

@sanabas: While doing the math for my last post, I realized that the 10% core rule is identical to the next-city rule, for the first extra city. 1/(N+9) = 10% when N=1. The next-city rule is interesting because the target number keeps rising as you exceed expectations, but it falls if you start adding weaker cities. When the next-city number is lower than the 10% core number, it means that you have already screwed up.

This is incredible -- THANK YOU

I've learned a ton in this thread -- thanks to everyone who has contributed thus far

I just completed the domination victory I talked about at around turn 250, so this post I just quoted is hugely helpful.

I went Liberty -- Commerce -- Order, but, in the end, I don't think that Order was all that useful for a domination game that is already steamrolling. I think that Autocracy would have been a lot more helpful.

I'm going to try Liberty -- Honor -- Autocray next, adding Tradition/Oligarchy and Commerce/Naval ____ (can't remember the name, but the one that gives bonus movement to naval units).

That's 3 complete trees (6x3) + 2 from Tradition + 2 from Commerce for a total of 22 policies in ~250 turns. From the math you provided, if I keep building cities and emphasizing culture, this should be very doable. My only concern with not taking Order and the right side of Commerce will be maintaining sufficient happiness, but we'll see how it goes

Thanks again!

 

[sanabas]

It is extremely difficult for me not to take Freedom for the happiness, once I'm into the industrial era. I think that order is 1 happiness per city? But Freedom is 1 happiness per 2 specialists? For me, even if I'm doing a domination game with a lot of puppets, I'm very likely to average more than 2 specialists per city. Which makes for more happiness from freedom, and it's more flexible too, if you take a new puppet and accidentally hit -11 overall, easy enough to add a couple more specialists somewhere, even if it means a city is briefly starving, and stay at -9 or better.

 

[beezany]

Yeah, I had to do the specialist happiness trick in my wide culture game. After I captured Rome, Caesar gave up and handed me his whole sprawling empire. The unhappiness would've been crushing if not for the Democracy policy.

I hope my analysis helps you, futurehermit. I'm definitely interested in feedback. If you do run into trouble, I most expect it in the middle part of the game. Because policy costs are quadratic, and culture production is a growth function (exponential with plateau), there's normally a stretch in the mid-game where costs grow faster than you keep up, followed by a stretch where your culture production explodes past the costs, followed by the plateau where costs pass you up again. France's UA helps a lot to mitigate the earliest part, but you will definitely hit spots where you struggle to maintain the 11-turn pace you need to adopt 22 policies.

Religion can help you a lot with culture and happiness, often at the same time. I recommend Tithe or Ceremonial Burial for your founder policy, then Cathedrals and Pagodas for your follower policies, with Monasteries and Asceticism as alternates if either of those are taken. I hope this helps you! And if not, please let me know what went wrong!

 

[futurehermit]

Thanks! Started a test game. Expanded to 9 cities before running out of room to expand peacefully. Sweden to my East, Austria to my S and Songhai to my SE, most distant from me. Defensive war against a dogpile by Sweden and Austria early. They both paid me for peace some time later. Finished Liberty on turn 68. Finished Honor sometime around turn 130-140 give or take. I used a GE from liberty on the Oracle, which helped. Everyone hates me for being so large, but I just hit Gunpowder with barracks, the next military building, and workshops in most cities and am pumping musketeers out of 9 cities. Sweden is now for real pressuring my NE most city, but, even if it falls, I should be able to take it back and them overwhelm him with muskets. If one or both of Austria/Songhai dogpiles, it could make things dicey. But, if I'm able to get out enough Musketeers and Cannons before that happens, I should steamroll my continent in fine fashion. I'm currently working toward fertilizer to open up autocracy. I'm hoping to grab autocracy before conquering too far and wide, for the culture accumulation.

Main problem this game is not so much policy pace, which is fine (currently at about one every 20-22 turns and am at the 13th policy, with one free), but, rather, that I am bleeding gold. Commerce and Oligarchy should both help take the edge off, but I need to start conquering to really end my money woes.

Religion went well. MotG, CB, Cathedrals, Asceticism, Religious Texts all up and running around turn 130-150 and my religion has spread like wildfire through my empire.

I am really enjoying this boom game with France and feel like it plays to their strengths very well. However, hostile neighbours and bleeding money are my main challenges right now.

 

[Lord Dante]

Whenever I play France (which is pretty often on Emporer difficulty) I tend to go Tradition & Honor, playing as the sleeper, small, peaceful, and Tall Empire Stategy until Musketeers come around.

My strategy is to settle 4 cities in optimum locations (not to far spread), get massive populations through beelining Tradition, develope infrastructure and nab the important beginning wonders (Statue of Zeus, Temple of Artemis, Great Library if I can swing it). Meanwhile, I keep my science high and the AI's off my backs by simply keeping my head down. However, after gunpowder is unlocked I completely shift gears, slamming into full Musketeer production mode. Because they have such high power and dont have any special promotions, I can slightly neglect ranged units and just continuously crank out these guys, slugging them at the nearest civilizations without worrying about losing any "boons" in later upgrades.

If done right, by the age of riflery when my Musketeers become weak, I should have doubled to quadrupled the size of my empire, leaving most if not all of my continental foes under my crippeled, and have the continent (or whatever major landmass) under my rule.

From there, I turn to the Commerce branch to keep my people happy and keep the money rolling in (especially because of all the puppeting I do as France). And finally, I go to Order, which gives all those cities massive boons. Autorcracy isnt as important, because by this point money can win wars, and I can set my sights on a Diplomatic Victory. By the end game I have both a tall and wide empire, with massive internal cities and respectable outer cities and a massive gold supply. Religion has always been my weak point, but I like Tithe, Fertility Rights, and the one that gives up to +15% production.

I know you already figured out your strategy, but I thought maybe this view could add something to your playstyle, as this method too, really capitalizes on France's unique and powerful attributes.

 

[futurehermit]

I've thought a lot about Tradition and going tall. I agree it can work. I just think you end up leaving a lot of culture on the table from the UA if you stop at 4 cities and sit tight through much of the early-to-mid-game.

I've had the most success going liberty and settling to 4 cities while teching CBs and taking out a neighbour. From there, settling the land near my and the conquered AI's capitals. This usually brings me to around 10 cities. Meanwhile, after getting Drama, head to musketeers/cannons and wipe my continent.

 

[T-hawk]

In this example, the empire produces 704 "core" culture per turn from the capital and other non-penalized sources. To avoid a slowdown, the eleven settled cities must produce 70.4 cpt each or 774 cpt overall.

Not too surprisingly, every city with a world wonder beats the guideline, thanks to Reformation. What I do find surprising is that even the wonder cities can only pay for themselves because of Reformation, Representation, Cathedrals, and weak core culture (300+ turn pace). Remove any one of those factors, and even the best cities have trouble covering the city penalty.

What about Great Artists from the additional cities? Any city that can produce a Great Artist contributes a landmark for 27 additional late game extra culture (12 + 128%). This culture should count as BOTH that city's contribution (for purposes of determining if that city was worthwhile) AND core culture (for purposes of determining if the NEXT city is worthwhile.)

That goes a long way towards clearing the payback bar for the first few additional cities, right? Each city's natural production of ~55 plus adding 27 to the capital via great artist does clear the 10% added cost. So I feel that even going for the fastest possible culture victory can be speeded by almost surely a second city and likely even up to a fourth. But each additional Great Artist raises the bar of core culture and makes it harder for later cities to compete.

(Of course, this doesn't help for cities wide enough that they'll never reach a Great Artist with any time left in the game.)

 

[joncnunn]

What about Great Artists from the additional cities?

Great Artists tend to always be placed in tiles worked by the capital.
More direct bonus there from Hermitage than any other city.
It's more common in G&K if you actually run out of tiles for great people to use in the capital that any future artist for a Golden Age, especially if your civ is already in one.

 

[beezany]

That's right, and T-hawk seems to know this, although the +128% capital bonus doesn't look right. It should be 50% Hermitage + 33% Broadcast Tower + 33% Reformation + 25% Sistine Chapel = 141%, so each landmark contributes about 29 culture to the capital. Unfortunately, I haven't analyzed how the number of Great Artists changes with the number of cities. I don't think the math would be too horrible, I just haven't been motivated to figure it out yet.

 

[T-hawk]

Yes I knew that, settle GAs from other cities at the capital. The point is that some of what we call core culture actually does come from other cities, so the comparison becomes a bit more complicated.

I had +128% as 50% Hermitage + 20% Alhambra + 33% Reformation + 25% Sistine Chapel. Forgot about the Broadcast Tower, so that means +161% total and 31 culture per landmark.

 

[beezany]

And I forgot about Alhambra! I wondered what could account for a 13% difference in our numbers. That would do it!

 

[futurehermit]

Yeah, that does make things more complicated

Right now I'm happy with the 9-10 city opening rule of thumb that was eventually arrived at. It seems to work really well.

But right now I'm working on another France strat with Honor Can't hurt to have a few tools in the ol' warchest

 

[Deau]

Sorry I stopped searching for an exact math approach to your original question early on page4. Here's a pure math approach, sadly I couldn't find my old posts about it from when I was hardcore at civ5 years ago.

It is extremely simple to calculate


policy cost rule is base(Y)*(1+0.10*N)
where base(Y) is a function of Y that was presented by BWS on the first page of replies but that is irrelevent to calculating the "break point" to another city.


Let's define CPT_N and CPT_(N+1) respectively as the the culture per turn you currently generate and the culture per turn you WOULD generate if you built an extra city

Then, the equality breakpoint is such that

base(Y)*(1+0.10*N) /base(Y)*(1+0.10*(N-1)) = CPT_(N+1)/CPT_N


Or in it's simpler version
(1+0.10*N) / (1+0.10*(N-1)) = CPT_(N+1)/CPT_N



you know N and CPT_N, it's easy to calculate CPT_(N+1) or just delta_CPT = CPT_(N+1) - CPT_N which is what a new city would minimally need to generate to keep the same policy rate.


I won't go much into the math details of my following comments, I will *try* to revisit this post over the next few days but I'm sure there are other contributers within this thread already to explain in further details if necessary.

What does the above formula imply?
Under non CV conditions, unless you play at very low difficulty and have a CPT output leveraged by loads of Wonders, any new city that you will bring to your "base" culture building setup will be beneficial to your policy rate.

In particular, from a France/liberty OPENER perspective where you don't even bother with monument, there is no breakpoint. ICS will always net you faster policy rate.

The concept of "base" and "bonus" CPT brought up by Sanabas and BWS are actually very important. The bigger the bonus is in proportion to the base, the more likely an extra city will be detrimental to the policy rate. This being said, the bonuses come from hermitage, all Wonders, settled GAs and Culture CSs for the most part (few from policies). This means that the higher difficulty you play, the less bonus culture your empire will generate since you will get fewer wonders, likely won't get or settle any GA unless you play CV and you will also likely drop culture CSs for other CSs as the game progresses.


The real issue in non-CV games is that more often than not, you do not wish to spend the gold nor the hammers on bringing a new city to your base culture building setup as there are already so many other buildings to catch up on that appear more relevant to the victory goals you've chosen...

*edit* Just so that people understand not to take the formula as an absolute. The formulae calculates the new city CPT breakpoint under your current culture paradigm. If you plan on building hermitage and other single-city buffing wonders/buying CSs etc. (anything that increases only the BONUS CPT), you would need to guestimate ahead CPT_N under those conditions to truly evalute the longer-term impact of adding a new city.

Cheers!
Hope this helps.

 

[beezany]

Or in it's simpler version
(1+0.10*N) / (1+0.10*(N-1)) = CPT_(N+1)/CPT_N

The solution to this equation is the "N+9" rule I have mentioned previously; sanabas worked it out in another thread. Adding a new city will provide an immediate benefit if it can quickly provide more than your empire's current culture divided by N+9, where N is the current number of cities.

In particular, from a France/liberty OPENER perspective where you don't even bother with monument, there is no breakpoint. ICS will always net you faster policy rate.

Yes, that's correct in the ancient era, but it doesn't last long. Once you start building amphitheaters, and especially once you start staffing them with artists, then you start to see breakpoints where more cities hurt your culture. The turning point depends on the exact details of your game, but for France it should be around 9-12 cities in the classical & medieval eras. After that point, it's very difficult to build new cities with enough culture, because they don't have the population for specialists.

 

[Deau]

Eh good call, solution is indeed cpt_N/(N+9). I didn't have the time yesterday to quickly solve it and Sanabas had done a random obvious mistake on post#23 so I just figured it was one of those formulas lying around on the forums that people improperly use or adapt over time.


With that said, amphitheaters and staffing them should not alter the value of an additionnal city if you assume you will build the amphitheater AND staff it as well in your additionnal city. Neither is outside the realm of what your BASE city could be. Obviously since you can't instantly grow it enough to staff a rush purchased amphitheater, it's not as straight forward but pretend you do some warmongering and takeover cities of the appropriate size, annexing it could still be optimal. It's really only the BONUS culture that is creating a breakpoint.

 

[futurehermit]

Eh good call, solution is indeed cpt_N/(N+9). I didn't have the time yesterday to quickly solve it and Sanabas had done a random obvious mistake on post#23 so I just figured it was one of those formulas lying around on the forums that people improperly use or adapt over time.


With that said, amphitheaters and staffing them should not alter the value of an additionnal city if you assume you will build the amphitheater AND staff it as well in your additionnal city. Neither is outside the realm of what your BASE city could be. Obviously since you can't instantly grow it enough to staff a rush purchased amphitheater, it's not as straight forward but pretend you do some warmongering and takeover cities of the appropriate size, annexing it could still be optimal. It's really only the BONUS culture that is creating a breakpoint.

So what would your conclusion be in terms of when, or even if, to stop building/annexing cities with France?