Discussion in 'Civ5 - Strategy & Tips' started by alpaca, Oct 8, 2010.

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2. ### alpacaKing of Ungulates

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Yes, I wasn't aware that the culture costs changed with map sizes and such until somebody pointed it out to me, but you can find the analysis with variables a little further down and the rest of the culture part is done with the variable version.

3. ### Dizzy75Chieftain

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Interesting...

This is a pretty counterintuitive result. It means that, if you're getting zero culture from city states, then ANY increase in cities will reduce your time between policies, as long as cT is fixed. For example, even if all of your cities produce only 1 culture, adding a city will decrease your time, as long as the new city produces at least one culture and you're getting no culture from CSes.

[That's because cT is always positive, and pm is always greater than 0 and less than 1, so 1/pm > 1.]

For what it's worth, I differentiated the time function with respect to n, and found the exact same first-order condition, although I haven't checked my math closely yet. It's interesting that the condition doesn't depend on pB or n at all (except for the fact that pB must be positive).

I'd like to look at the derivative wrt n when cT is a function of n, but for now, it's time to play...

4. ### Dizzy75Chieftain

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For reference:

t = p/c = pB(1 + pm(n - 1)) / (n(cT) + cC)
= (pB + n(pB)(pm) - pB(pm)) / (n(cT) + cC)
= u/v

By the quotient rule,
dt/dn = (du/dn(v) - dv/dn(u)) / (v^2)
= (pB(pm)(n(cT + cC)) - cT(pB + n(pB)(pm) - pB(pm))) / (n(cT) + cC)^2
= (n(cT)(pm)(pB) + cC(pm)(pB) - cT(pB) - n(cT)(pm)(pB) + cT(pm)(pB)) / (n(cT) + cC)^2
= (cC(pm)(pB) - cT(pB) + cT(pm)(pB)) / (n(cT) + cC)^2
= pB(cC(pm) - cT + cT(pm)) / (n(cT) + cC)^2 = 0

The denominator is always positive and nonzero for us, so the first-order condition would be pB(cC(pm) - cT + cT(pm)) = 0; i.e., whether dt/dn is positive or negative depends solely on the numerator. We also assume that pB (policy cost) is positive, so that means cC(pm) - cT + cT(pm) = 0. Since we are looking for a decrease in time, we want cC(pm) - cT + cT(pm) < 0.

So:
cC(pm) - cT + cT(pm) < 0
cC(pm) - cT(1 - pm) < 0
cC(pm) < cT(1 - pm)
cC(pm)/cT < 1 - pm
cC/cT < 1/pm - 1

So dt/dn < 0 when cC/cT < 1/pm - 1, the same condition that alpaca found. I feel like the fact that we find the same condition using different methods means it's right, assuming the original functions are right.

5. ### KrikkitTwoImmortal

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Well the issue is, except for France/the Liberty Policy, each newly founded city produces 0 culture.

There is a lag time from founding of the city (and the policy price boost) to the city reaching cT.

Just like founding a city post colluseums will Always increase your happiness... the city makes an initial hit to happiness, followed by an increase once the buildings are built.

In Civ IV AND Civ V

6. ### Ulti_meciaChieftain

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Alpaca,

As others have said, many thanks for exposing the formula to the Social Policies.

I was very impressed with your audacity to do ICS. I was convinced it couldn't work, but after reading just a couple lines where you mentioned 'Meritocracy', I set off to try some things on my own and my conclusions so far seem to be the same as yours:

- Forbidden Palace is a beast. After getting the basic techs to work the lands, and then steel; I tend to beeline Banking to get it
- Rome's UA is amazing for ICS. I also very much like Ballista/Legions to carve out your piece of land (also after conquering a city it's nice not to have to go and get workers from home to build roads). I'm yet to try China but I can see them being very good as well. I refuse to play France out of principle
- Social Policy speed is a problem. I worked around this problem by REXing in waves. (Hoard culture, spend, expand)
(details in Spoiler tags, don't want to derail too much as this is more a reply to the ICS tread)
Spoiler :
First I'll play very OCC-like: get Stonehenge-National College.
Only after you've gotten Liberty-Citizenship-Meritocracy-Representation will you plop down additional cities to fill your capital's surroundings (usually 6-7ish cities). Something you should definately try to do is sell your luxury resources as much as you can so you can rushbuy all monuments, it'll give you a ton more momentum as the 'drag period' before your cities actually start to contribute is much lower.
Second wave occurs after getting Theocracy to greatly raise your happy cap so you can expand further (Classic-Middle age) and some war mongering to make room for this next wave. This wave will typically fill the continent as the next worthwile polices are Order and I don't like waiting with only 'two waves' of ICS until Steam Power to unlock it to grow further.
Typically I'll keep expanding untill everything is filled to the brim at this point and amazingly, with Rome's UA, all the infrastructure is usally built by this time, especially in the 'first wave' cities. This gives time to finally pump up culture in these as well, coinciding nicely with the railroad/factory production explosion

- Food costs are way too ramped over +pop8-10 (again, promoting ICS - yes I like it )
- I was doubtfull about not going Freedom-Civil Service - Rationalism-Secularism as +5 beaker scientists are prety amazing. But I found there was really no need to do this (and waste 4 SP's on getting it running) when you're getting techs every 3 turns already
So this opens up Piety for Theocracy. I was constantly at +50 Happy in my last game, even with the entire continent filled to the brim, and half of them didn't even have Collosea up yet. I'm very impressed by Theocracy.
- The main fear of not being able to get SP's reliably was not getting the crucial Order SP's (most importantly Planned Economy, which coupled with Forbidden Palace negates the unhappiness factor from ICS ). As your findings pointed out though, Represnetation-Monument will get you into Order very reliably. I added more culture infrastructure in the older cities as they had nothing else to do, and it brought my SP pace upto 14; which is pretty amazing considering I had 30ish cities and no CR or Sistine yet.

After throwing your formulas in excel I was relieved that my gut feeling in the culture matter was somewhat correct. To elaborate:

As you get deeper into the policies, it'll cost more and more culture/city on average. However, when you increase the city count (ae, expanding) it takes considerably less culture in order to 'keep the pace'. (Linear scaling, as opposed to Exponential)

Some basic calculations showed that to keep the pace, while expanding after getting Representation, new cities will need to contribute 2.5 cpt. Which is why I feel Representation is extremely good, it'll allow you to fill the 'drag time' much quicker. And, as you mentioned, hard build Monument after that. As Rome, I like to rush buy the monument and build the temple (Rome UA makes this very easy).
(Applying this to my personal SP preferences for ICS)
Spoiler :

The full tree is:
Liberty-Citizenship-Meritocracy-Representation
Piety-Org Religion-Theocracy
Order-Socialism-Planned Economy-Communism

First 4 policies with just the capital will take 108 turns (Palace-Monument) / 36 turns (+Stonehenge) to complete.
During this time what you do is prepare to REX 5-6 cities (or more, I prefer to play Tiny as my laptop overheats on bigger maps )
After you get Liberty-Citizenship-Meritocracy-Representation, plop down your cities (and see the next SP cost boom with it )
The next SP will cost you 690 culture. Now whatever I do with Alpace's formula's; the weight every new city will have to pull is 2.5 cpt. This is perfect as it means Representation+Monument per city. So, 2.5 cpt to get Piety in 20 turns (not unreasonable I'd say, you're not going for culture victory here, merely preventing SP stagnation)
Next SP's will cost significantly more (3.5-Org Religion, 4.6-Theocracy) so give or take 100 turns before you'll be able to get Order opened up.
That means the next 100 turn you can focus on clearing your continent, teching to Steam Power and maybe even getting culture buildings up in your older cities (who should have finished their Monument-Library-Colloseum-Market-Uni-Bank cycle by now)

- Once you get into Order, your ICS empire is more solid than a solid rock. (Order production bonus + Rome UA + Railroad bonus (+ factory). Also, my trade routes alone were paying for all my expenses last game

All that to say:
- Representation is worth it for ICS.
- Expanding won't kill your SP pace if you bother to build a monument. (add temples by SP nr 7-8)

Apologies for my long, and ICS focussed reply. This is more a 'thank you' to Alpaca for pointing me to ICS; as I'm assuming he started this crunching after he bumped into the same cultural wall I did when perfecting ICS

Ulti

7. ### Ulti_meciaChieftain

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Also, I'd like to add this here table, which shows you how much culture a new city would have to contribute to NOT slow you down.
The first column is the base cost of the policy, accounting for a small, standard Emperor map.

For example:
- How much cpt does a city need to produce when getting the 5th SP, assuming a pace of 15 turns (4th is usually Representation, after which I'll transition into REX mode): 5 cpt.

- How much cpt does a city need to produce when getting my 8th SP, assuming my SP pace is 20 turns (In my prefered SP path, the 8th would be 'Order'): 9 cpt.

Code:
```                       Desired Pace
5	10	15	20	25	30
Policy Nr    Base Cost
1	25	2	1	0.7	0.5	0.4	0.3
2	45	3	1.5	1	0.8	0.6	0.5
3	95	5	2.5	1.7	1.3	1	0.8
4	160	10	5	3.3	2.5	2	1.7
5	245	15	7.5	[B]5[/B]	3.8	3	2.5
6	350	21	10.5	7	5.3	4.2	3.5
7	465	28	14	9.3	7	5.6	4.7
8	600	36	18	12	[B]9[/B]	7.2	6
9	745	45	22.5	15	11.3	9	7.5
10	905	55	27.5	18.3	13.8	11	9.2
11	1080	65	32.5	21.7	16.3	13	10.8
12	1265	76	38	25.3	19	15.2	12.7
13	1460	88	44	29.3	22	17.6	14.7
14	1670	100	50	33.3	25	20	16.7
15	1890	114	57	38	28.5	22.8	19
16	2125	128	64	42.7	32	25.6	21.3
17	2370	142	71	47.3	35.5	28.4	23.7
18	2625	157	78.5	52.3	39.3	31.4	26.2
19	2890	173	86.5	57.7	43.3	34.6	28.8
20	3165	189	94.5	63	47.3	37.8	31.5
```
Quick reminder after all these numbers: the number you see is the amount of culture a new city would have to produce to at least keep you at the same SP speed.
Quick reference: a Museum city produces 16cpt assuming Representation and no specialists.
You can see Representation + Monument (3cpt) will allow you to get upto the 6th SP fairly quickly. Adding a temple will only be needed after a couple of SP's where the numbers really start to ramp up. This is not accounting for modifiers such as CR, Sistine, Oracle, etc etc
(Damnit, France comes out really good in this analysis, why! )

Ulti

Can't say this enough, but ty Alpaca for finding this very usefull formula I'll be keeping this table handy in my next game for sure.

8. ### PaeanblackChieftain

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Yeah, I'm pretty sure it's dead on. The caveat is that the simplifying assumptions I made in my model are not applicable for every possible strategy. Alpaca and I talked about this further up.

If you are playing REX/ICS, and your expansions are building monuments extremely slowly (or doing something else first), then your effective cT won't even be 2. Once cT gets low enough, then your rC will climb above the 7/3 threshold, just due to the Palace culture bonus.

9. ### alpacaKing of Ungulates

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I'm glad you like it but you're giving me more credit than I deserve: I copied most of the basic formulas from other people and merely checked that they were correct. As I stated, the culture analysis was a kind of a joint work with Paeanblack in a Songhai-related thread where I wanted to argue that the Mud Pyramid Mosque is awesome. It doesn't directly relate to ICS, I had the idea of trying an ICS strategy in the demo, although back then I planned to base it on the Egyptian Burial Tomb.

Your table does not directly pertain to new cities, by the way, but to any city in your empire. To not slow your policy speed over time you have to increase the amount of culture your empire generates, obviously. It shows some of the deficiencies of my approach, however, because building the city up to high standards takes quite a lot of time.

@dizzy75: We did not actually use different methods. You looked for a minimum in the continuous function by way of the derivative, I looked for a change in the amount of turns in the discrete function (which means I also looked for zeroes in the "derivative" analogon for discrete functions). Two sides of the same coin, they simply show that the continuous approach is applicable here. Also, the calculations are simple enough to check that I'm sure our math is correct.

10. ### alpacaKing of Ungulates

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Post 16 updated with info for f_c <= 2

11. ### ehrgeixChieftain

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Thanks a ton for the update. Reading that I think it seems reasonable to say that early granaries are somewhat viable for games where (you have 0-2 maritime states and) having cities with strong early production/income/early GS matters more than lategame city growth (but you do still want some city development through turns 50-150, say). I guess it's also an option to skip granaries and focus on working more food tiles if you want to put the production into immediate military. Would you agree with that assessment?

Again - thanks for your time.

12. ### alpacaKing of Ungulates

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To be honest, I haven't drawn any conclusions yet as to whether you should keep production cities growing or cap them. I tend to work a lot of plains farms in my production cities unless I absolutely need that production now. Farms are a viable alternative to granaries as long as you can convert gold into happiness at an exchange rate of 1 (so as long as you can build theatres in a reasonable amount of time). Building a granary might make sense to improve growth speed if you can afford to take that city off producing what it was for a few turns. If you can afford to switch the citizens to farm tiles, that would also give you a big growth boost, though, and you can switch back later.

I typically try to let the city grow as long as growth seems worth it in the amount of times the game is probably still going to take. An example:

Our city is at size 4, f_c = 1 and f_csq = 4 (two maritime CS). Switching to a tile that produces three additional food costs you two hammers per turn. Three food at size 4 will let the city grow in 14 turns so cost us 28 hammers. On the other hand, the new citizen can then be put onto the hill we worked earlier, leaving us at an additional 1 hammer, 1-1.5 science and 1 food per turn. If the game will continue for, say, 50 turns, I'd say that that is definitely a good return on investment if you don't have anything very pressing to build.

The point when it actually makes sense to stop your city growth is probably a lot higher but you can do a similar calculation yourself fairly easily for the actual scenario in your game using the food cost table in the OP. Obviously, if your city is tile-limited because we're looking at a secondary city in an ICS strategy, or you're happiness-limited, you have to look at things a bit differently.

13. ### ehrgeixChieftain

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I think that if you have civil service early the farms definitely look much more appealing. I tend to think colosseum and theatres are quite bad early in a game vs allying more city states and hunting lux resources, though obviously to maintain an empire's growth past t100~ or so you do need to start hitting colosseum in every city and founding more cities off the colosseum, but it takes an awful lot of turns for say, a 6 pop city founded on the back of two colosseum to start giving you a good RoI.

I definitely agree with your point on growing cities relative to the length of the game, though I think it is not always possible on deity. For me the real quandry is in the early game where you have to build enough military to maintain a good defense, but otherwise pretty much only want a monument and a library in each city. I think the options here are something like:

a) Rush HBR and spam horsemen (not always viable on deity, and not necessarily optimal if there's a superempire growing on the other side of the world).

b) Use the extra production to build granaries.

c) Found cities in different places and look at picking up more food centric tiles than production, skip granaries entirely, grow, and then try to transition into heavier production when you need more military later (not always feasible, but might be depending on start location, rivers, and whether you can take a detour to get civil service).

d) Use the extra production to build colosseum and get more cities active for max RoI before the game ends.

My intuition says something like B is > C unless you have civil service, A is > all others in some situations dependant on map size and strength of neighbour/terrain/whether you eat an early DoW, etc, and D is a very strong option on a larger map, but probably suboptimal on smaller maps because it takes so long for a city to generate the p/gpt required to exceed the initial investment on two colosseum (300p up front and 6gpt), and especially theatre!. I might be wrong though. Especially on B vs C. Or it might also vary depending on game eta. Thanks though. It's really interesting trying to figure this stuff out. I think it'd be very fun to play a ridiculous deity ICS on a large or huge map with mass early colosseum funded expansion.

edit: e) Play ramesses, pick up philosophy early and build burial tombs in all cities + found more cities? That also seems quite strong. Though not china strong.

14. ### alpacaKing of Ungulates

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I like to buy the Colosseums. Buying one is 680, the gold/hammer ratio is significantly better than for libraries or monuments.

I actually think that for higher difficulty settings and empires with more cities, a scenario where you are limited by happiness is much more likely than a scenario where you are limited by growth. Obviously, in a happiness-limited scenario, the additional citizen will have to also make up for the additional happiness building you need. For example 680/4 = 170 gold for a quarter of a colosseum. This does indeed reduce the RoI significantly but often you don't have anything very useful to build in a city because an additional military unit is unfeasible due to maintenance and 1upt requirements.

390 hammers is actually not that much for a new city of size 2. The city tile will generate 2 hammers and 1 gold and the two citizens will generate, say, 2 hammers and four gold or something like that. So while you need something like 100 turns to break even production-wise in that scenario, it becomes a lot better if you pick up things like the forbidden palace or meritocracy, and you generate 500 gold and 200 science in the process. With only meritocracy, you get an additional citizen for another 2 gold and 1 science per turn.

Why not try an ICS game? If you balance expansion with military-build up you might find yourself surprised.

15. ### Ulti_meciaChieftain

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I played an Egyptian game yesterday and I found the Burial tomb a bit underwhelming, mostly due to no specialist slots and lowered culture when compared to a temple. I also lost every wonder to the AI even with marble/UA, ah well -- Currently playing Emperor, winning about 80% so I'm considering moving up, but I'm still deterred by how daunting it was in CIV -- All in all I'm a bit dissapointed by Egypt (for ICS) and I really feel the production hit when lacking Rome's UA. (As a sidenote I don't really get the hulabaloo about horsemen, as Legions/swordsmen are far better for actually pushing forward imo; especially so when you get Blitz so you can move after attacking. Don't get me wring though, they're very good. They're hardly the backbone of my army though, that spot is reserved for medic swordsman so one can pull the horsies behond them to recuperate while you push)

And you're ofc correct, if you want to speed your SP gain up, you need to get more than the break even point obviously.
The way I calculated this table is I took the difference between the policy cost and policy cost with c=c+1; and see how much cpt the new city would need to generate to fill the gap at the same pace. I think this is what you mean when you say 'pertain to any city in the empire' which is correct because it is an indirect consequence of the presumption you're allready at a certain pace (implying the culture yield of the empire is at that level of avg culture per city).

I played a Songhai game and yes, the Mosques are pretty damn good.
Also, if you're impatient and decide to REX before Meritocracy/Representation, you're screwed SP-wise, found that out the hard way yesterday (I was feeling a little overconfident and it bit me in the ass, no Theocracy for aaaages)

16. ### BiborDoomsday Machine

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Please do clarify, Napoleon with his +2 culture per city accompanied with +1 culture per city from Representation will see no hit in policy gain speed from the moment the cities are founded?

17. ### Ulti_meciaChieftain

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Well, let's dig in the table and find out

Napoleon's Representation cities generate 3gpt. Representation implies you already have at least 3 policies down (Liberty-Citizenship-representation) so you're at policy nr 4.

Code:
```SP Pace:	5	10	15	20	25	30
Policy:	1	2	1	0.7	0.5	0.4	0.3
2	3	1.5	1	0.8	0.6	0.5
3	5	2.5	1.7	1.3	1	0.8
4	10	5	3.3	[B][U]2.5[/U][/B]	2	1.7
```
Let's assume you have only Paris and want to expand. Paris has a monument so we're generating 1 (Palace) + 2 (UA) + 2 (Monument) = 5 cpt.
Our next policy costs 160 culture, so that's a pace of 18. To keep up with that you'd need to get 2.5 cpt in the new city to keep the pace. It might in fact even speed you up, as 3 is pretty close to a pace of 16-17 (Pace 15 value is 3.3 )
So yea, as I grudgingly mentioned before, France comes out good in this

I'd say you can expand very aggresivelly with France, and not be SPless. Add a monument and you'll be at 5cpt which will support bearable progress even upto the 7th SP; which is Theocracy (In ICS at least it should be: last game I went from -1 to +11 Happy from Theocracy alone)

(I'm actually at work, so I can't readily verify wether or not Representation also adds 1 culture to the capital, so I assumed it didn't - as per the logic it shouldn't "gains +1cpt for every city connected to the capital". Unless a capital can be connected to itself... )

18. ### alpacaKing of Ungulates

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What I mean is that your table is a bit confusing because the assumption of a constant social policy speed implies that your existing cities can keep pace with the increased policy base cost, which they mostly won't. The high amount of culture a new city has to bring in isn't an effect of it being a new city but of pretty high culture demands that you're imposing on the city.

For France, if you don't have a significant amount of additional culture buildings in your old cities, and no cultural CS allies, then yes, a new city will speed up policy gain. I can't quantify how much "a significant amount" is at the moment.

19. ### Ulti_meciaChieftain

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Indeed, the assumption is made that you already have a SP pace, or SP pace goal (if I want to gain SP's every 20 turns, how much cpt do I need? )
Your analysis is spot on in regards that the number the table produces is indeed the culture any city will have to produce. Also, having a pace indirectly implies your old cities are at this level as well, otherwise you can't get there.

The average culture per city needed to keep the same pace goes up with every passing policy. The way to read the table is: if you want to go faster, new cities need to accrue more than the number the table shows you, at the current SP level. As you go down (as more SP's are picked, as time passes), you'll see numbers rising for keeping the same pace, this indicates the rising basic cost for policies as yuo get more and the need for your cities to get their cpt up as the game progresses.

(I recommend looking at my first overview table to follow this reasoning)
Spoiler :

Code:
```		Base	5	10	15	20	25	30
Policy:	1	25	2	1	0.7	0.5	0.4	0.3    Liberty
2	45	3	1.5	1	0.8	0.6	0.5    Citizenship
3	95	5	2.5	1.7	1.3	1	0.8    Meritocracy
4	160	10	5	3.3	2.5	2	1.7    Representation
5	245	15	7.5	5	3.8	3	2.5    Piety
6	350	21	10.5	7	5.3	4.2	3.5    Organised Religion
7	465	28	14	9.3	7	5.6	4.7    Theocracy
8	600	36	18	12	9	7.2	6       Order
9	745	45	22.5	15	11.3	9	7.5    Building upkeep one
10	905	55	27.5	18.3	13.8	11	9.2    Planned economy

```

As an example, let's assume a monumented Paris: 5cpt. With this culture, your pace is 5 turns per SP; for the first policy! If you were to expand at this point, the increase in cost to get the first SP will be 10 (for a total cost of 35). In order to keep up, any new city would need 2cpt. (French UA, how nice) However, this is only valid for the first SP.
So, let us assume you settle your second city, as the table tells you, you won't be slowed down and you get your first SP at turn 6. (5 turns is fast, so let's assume you are extremely lucky and get a hypothetical settler from ruins - even though I haven't had that happen to me yet)

The second SP will NOT be gotten in 5 turns. In order to get SP nr 2, the empire needs 60 culture. At this point, the empire has 7 cpt (5 from Paris, 2 from Bordeaux? I don't know the French city order ) so it'll be 9 turns. As you went down an SP level (1 => 2) your pace slowed down. However, if you want to keep THIS pace, you won't be slowed down if your new cities produce at least: 1.5 cpt (by settling the 3rd city, the SP cost rose from 60 to 75; meaning you'll need 1.5cpt to get this 15 culture in 10 turns)
This is what I mean by 'keeping the pace'. Every SP level is different, and rises pretty fast. So in this example expanding while keeping pace at the 2nd SP level would mean getting the new city 1.5cpt (rounding off to a pace of 10).
After getting the second at a pace of 10; your pace will now drop to (150 cost / 9cpt empire wide) = 17 for the third.

As you see, you have to look ahead to a certain goal before you expand. Let's assume you keep expanding and expanding (gogo ICS ) as France with monuments because you like culture: netting you 5cpt per city on average. Your SP pace will 15 by the time of the 5th SP, 20 by the 6th, ~28 by the 7th and >30 by the 8th (I'd say practically this is extremely good as nr 7 is theocracy, which is what you need for ICS, nr 8 is Order so you have time to play catch up while you tech. Thank you Bibor, for making me go in depth into ICS France. I say it'll extremely well as they won't really SP-stagnate if you monument right upto Theocracy.

To answer your original question Bibor: Short term you'll have very little impact in SP pace. However if you don't keep up with the increases per SP level, you'll eventually start lagging.

(Pretty hard to translate numbers in my head to something clear for other people to understand, thank god I'm not a teacher )

20. ### zzzzChieftain

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Anyone know what's the effect of a WLTK-day on the growth numbers?