Worth of new Cities in terms of Culture/Science

[Voremonger]

There was a discussion in the thread for the 8-16 release regarding how good a City needs to be in terms of Culture/Science to offset the increased cost for Policies/Technologies.
I understand the cost of Policies/Technologies total_cost to increase as follows:

total_cost = 1 + (n - 1) * increase_per_city,

where n is the number of non-puppeted Cities and increase_per_city is the map-dependent scalar that increases the cost of Policies/Technologies by a constant amount for each non-puppeted City (0.1 for Duel, Tiny, Small, 0.7 for Standard, 0.05 for Large, Huge).
total_cost=1 means Policies/Technologies have their "regular" cost if you have only your Capital, total_cost=2 means they're twice as expensive, and so on.
Another way to think about the increased cost for Policies/Technologies is to imagine that each new City reduces the effective Culture/Science output of all of your Cities.
We can calculate a factor beta that would have the same effect as follows:

beta = (1 + increase_per_city * (n - 1)) / (1 + increase_per_city * n) .

Note that beta is not linear in nature.
If we already have n Cities we get the following amount of Culture/Science per turn:

y_0 = n / n * y_0 = n * y_avg,

where y_0 is the total amount of Culture/Science that we are produce and y_avg is the total amount of Culture/Science that we produce per turn divided by the number of Cities.
If we were to gain a new City with Culture/Science output y_new our effective Culture would change as follows:

y_1 = (n * y_avg + y_new) * beta .

If we look for the point where a new City exactly breaks even in terms of Science/Culture we can set y_0=y_1 and rearrange it as:

y_new / y_avg = n * (1 / beta - 1) .

The left-hand side is equal to the yields from the new City relative to the average that we are getting without the new City.
The right hand side starts at a low value between 0 and 1 and converges to 1 for a very large amount of Cities.
I have visualized the results in the attached image.
The x axis represents the total number of Cities before a new City is founded/annexed.
The metric for the y axis is y_new / y_avg, meaning how the output from the new City compares to the total output of your empire divided by the number of non-puppeted Cities that you own.
The graphs show how good a new City would need to be in terms of Science/Culture to break even: to not slow down the acquisition of new Policies/Technologies.
We can see that if you have a very low amount of Cities gaining a new City is relatively worthwhile in terms of Culture/Science while the threshold for Cities that are worthwhile rises the more Cities you gain.
Please note that there are some sources of Culture/Science that do not scale with the number of Cities (Palace, guilds, killing Units with Authority) but that are for the purposes of this analysis still counted towards the average production of Culture/Science in your empire.

 

[CrazyG]

I understand the cost of Policies/Technologies total_cost to increase as follows:

total_cost = 1 + (n - 1) * increase_per_city,

I'm not sure if this is correct:
https://forums.civfanatics.com/thre...la-for-determining-social-policy-cost.657988/

 

[Voremonger]

I'm not sure if this is correct:
https://forums.civfanatics.com/thre...la-for-determining-social-policy-cost.657988/

It is as long as we are only talking about the number of Cities and the map size.
The factors for speed and difficulty are just constants and do not matter in this context.
The cost increases from the number of Policies/Tenets also do not matter because you're going to suffer these the same regardless of how many Cities you have.
Notice how the formula you posted is proportional to the formula I'm using.
(I'm neglecting the rounding because it has almost no effect.)

 

[Voremonger]

This graph isn't right.

Additional social policy and tech costs scale additively. It doesn't matter if a new city is city #2 or city #100, it needs the same amount of culture to break even.

I'm going to make a new thread for this topic, it's getting enough discussion.

You are correct in that the amount of Culture/Science that you need scales linearly with the number of non-puppeted Cities that you have: for every new City you need a constant amount of Culture/Science per turn to break even.
However, this does not mean that the amount of Culture/Science of the new City relative to the average output of your existing Cities stays constant.

Let us consider the following example:
If you had 1 City that produces 100 Culture per turn and the next Policy cost 1000 Culture you would need 10 turns to unlock it.
Obviously the average CPT of your Cities is 100 CPT.
If you were to gain a new City on a small map you would need 1100 Culture to unlock the same policy, meaning that the new City would need to produce 10 CPT to break even, in other words 10% of the average CPT of your Cities.
Notice how the average CPT of your Cities has decreased from 100 CPT to 55 CPT.
If you were to now gain a third City it would still need to produce 10 CPT to break even.
However, it would now need to produce ~18% of the average CPT of your Cities to break even.

Let us consider another example:
Let us first again assume that you have 1 City that produces 100 CPT and needs 10 Turns to unlock a Policy that costs 1000 Culture.
As before, a new City would need 10 CPT to break even.
If however, you instead had 11 Cities producing 100 CPT and you still needed 10 Turns to unlock a Policy that now costs 11000 Culture things would be different.
Obviously the average CPT of your Cities is still 100 CPT.
Notably getting a mere 10 CPT from a new City would no longer let you break even.
As the Policy cost would increase to 11550 Culture you would now need to produce 55 CPT (55% of your average CPT) to break even.

The math behind the increase in Policy/Technology cost is designed to take the average production per City into account but to also be more forgiving for the first few Cities that you found.
For example on a small map:
1 City producing 100 CPT is the same as 2 Cities producing 55 CPT is the same as 3 Cities producing 40 CPT is the same as 50 Cities producing ~12 CPT is the same as 100 Cities producing ~11 CPT is the same as 1000 Cities producing ~10 CPT.
The problem with founding/annexing new Cities is that they are typically underdeveloped and that they will therefore drag your average down; if however the number of Cities that you have is not too high you can still end up with a net benefit.
You have correctly pointed out that the amount of CPT that you need to generate in a City in order to break even does not depend on whether it's City #2 or City #100.
But the amount of CPT that you would need to actually break even is virtually unobtainable with a newly founded/annexed City (once you're no longer in the early game).
Those Cities start with less Population/fewer Buildings than you core Cities and therefore cannot immediately produce the necessary CPT.
They are typically able to produce that amount after a few turns but at that point the target value for CPT has already increased so they are still lagging behind.
I therefore think it makes more sense to think of new Cities in comparison to the rest of your Cities who got a head start; I think it takes a lot of time to sufficiently close the gap between a new City and your older Cities.

 

[Voremonger]

You would only be correct if the Culture/Science scaling was directly proportional to the number of Cities that you have.

I've read this like 5 times and I'm not sure what you mean.

I used this equation to get the social policy costs:
https://forums.civfanatics.com/thre...la-for-determining-social-policy-cost.657988/


A new city increases the cost proportionally. It's always about the same, just +5 or -5 due to rounding. Your 10th city increase the cost of your 10th social policy by 310. Same as your 2nd city.

A basic model would be just assume you want a social policy every X turns. Then divide the new cost by X, and see if that city is making enough. In my most recent game I had new policy roughly every 15 turns, so if I'm at 9 policies, that city needs about 20 culture to break even.

Tech is harder to analyze in detail, but a decent rule of thumb would just be that you want science equal to the culture. I don't think you need gold in the analysis (and your model will be flawed unless you decrease the weight of gold compared to science and culture, Stalker is right).

What I meant:
If the cost of Policies/Technologies were to increase additively by 100% for each City (i.e. if it were directly proportional to the number of Cities that you have), then to break even every new City would always need to produce at least as much Culture/Science as the average of your Cities to break even.
The graph I posted would be completely flat in that case.

 

[CrazyG]

However, this does not mean that the amount of Culture/Science of the new City relative to the average output of your existing Cities stays constant.

Isn't this relationship a bit circular?
If I have 10 cities, average culture per city is much lower than if I have 1 city. So I need 10% of a large number or 60% of a smaller number. The % needed rises with more cities because the base is getting smaller.

The actual culture I need doesn't change. I think this graph makes it seem like your 11th city needs more culture per turn than your 2nd, but it doesn't. They both have the same requirement to break even.

 

[Voremonger]

Isn't this relationship a bit circular?
If I have 10 cities, average culture per city is much lower than if I have 1 city. So I need 10% of a large number or 60% of a smaller number. The % needed rises with more cities because the base is getting smaller.

The first example was simply to illustrate the nonlinearity; it does not accurately reflect a real game because in a real game the CPT of Cities and their target values are constantly increasing.
The second example was to show how the need for a fixed number of turns per Policy changes the requirements for new Cities based on the number of Cities that you already have.

The actual culture I need doesn't change. I think this graph makes it seem like your 11th city needs more culture per turn than your 2nd, but it doesn't. They both have the same requirement to break even.

Your calculation gives you a CPT target value that you need to meet for X number of turns per Policy.
My calculation gives the CPT value that a new City needs to have to not increase the number of turns per Policy that you currently, actually have for your empire.
These are two very different things: your number tells you how to meet your target of X number of turns per policy while my number tells you how good a City needs to be to not make your current situation worse.
If you are doing better than X number of turns per Policy and a new City has the exact amount of CPT that your calculation prescribes you will actually do worse.
Conversely, If you are doing worse than X number of turns and you gain the same City you will do better afterwards.

You are correct that to meet some pre-decided target the 2nd and 11th City are the same.
But in a real game you are never going to exactly hit your target.
You might have Luxury Resources that give you a lot of CPT and the question in that situation would not be "How do I meet my original target?" but rather "How do I keep unlocking Policies faster than I originally planned?".
You might also for some reason have almost no CPT so you are unlocking Policies much slower than you planned initially.
Gaining a City that does not meet the X turns per Policy target value for CPT can under such circumstances still be an improvement and get you closer to your goal.

 

[Stalker0]

Another way to think about the increased cost for Policies/Technologies is to imagine that each new City reduces the effective Culture/Science output of all of your Cities.
We can calculate a factor beta that would have the same effect as follows:

beta = (1 + increase_per_city * (n - 1)) / (1 + increase_per_city * n) .
...
y_0 = n / n * y_0 = n * y_avg,
...
y_1 = (n * y_avg + y_new) * beta

So I am going to try and define things using an increase model (instead of the decay method you used), to see if I can validate Beta).

First, I am going to remove the decay component of y_1. I am going to assume that y_1 is the new per turn culture value after a city has been added.

y_1 = (n * y_avg + y_new)

Now, my core assumption. For each increase in n, we know that the total culture cost is increased by X. Therefore, in order for a new city to break even, the total CPT must equally increase to ensure the number of turns to get a policy does not drop. That would mean:

y_1 = Xy_0

Now apply some math manipulations:
y_1 = (n * y_avg + y_new)
y_1 = Xy_0
Xny_avg = (n * y_avg + y_new)
Xny_avg = ny_avg + y_new
Xny_avg - ny_avg = y_new
(X-1)ny_avg = y_new

(X-1) * n = y_new / y_avg

If both methods are equally valid, then

1 / beta = X

Which I found to be true, is we assume our base CPT is 1, than the percentage increase (X) is defined as.

X = (1+1*city increase*n) / (1+1 * city increase * (n-1)
which is in fact

X = 1/ beta.


So looks like the math checks out.

 

[Stalker0]

So now the question, what can this model do for us?

So I can easily obtain my current CPT by adding my base CPT (given to me in the toolbar) + the instant yield bonus (from the instant yield toolbar), and then divided by the number of cities.

From there I can calculate using the chart (or if you want to get precise you can calculate beta), how much culture my new city would need to use.


In comparison to CrazyG's model, it has the advantage of a bit more precision at any given time but at the cost of more calculation. Both models suffer from the same time fallacy...it assumes that a new city could get up and running with the expect culture yields instantly, which is highly flawed assumption. So again this model can give you a goal post on whether a new city could be profitable, but the actual profitability require a much more complicated, time driven model.

 

[Voremonger]

Just in case anyone actually want to use the model presented here: I've written the results as a table to a csv file for easier use and attached it to the OP.
(CivFanatics doesn't allow the upload of csv files for some reason so I packaged it as a zip file.)