Question re: new commerce cities

[Gr8scott]

Bah, too much math.

Relative to most things CIV, I thought the math was pretty easy. If a farm increases your food surplus by 25%, you can justify that if you are in the process of growing 4 or more cottages.

Here is a perhaps an easier metric:

At any point in time, try to adjust your farm/cottage mix until food surplus just equals remaining pop growth. (at least pre-biology)

Just download this game and tell me how much gold you get by 1 AD.

No need to add the randomness of a different player to the mix. Just replay your game, but if you find yourself in a situation where you want to do this:

farm everything and convert everything to cottages when you hit 15.

Use a cottage/farm mix to try and have food surplus = remaining growth. You should be better off.

GS

 

[spoooq]

Thanks for the maths, it makes a lot of sense



Screeenie!

Hehe, I will see if I get time to post a screenshot/savegame over the weekend, it's not very likely that I will, being dragged out for a drive in the countryside I had to spend a turn moving to get so many gems, FWIW.

 

[Gr8scott]

So, is the conclusion farm until you have a solid food surplus going then cottage the rest then cottage over the farms once you hit the size 15 health/happy caps?

Not quite. In the simplified setting I was working with, you are going for a mix of cottages and farms that makes:

(1) Food surplus = remaining growth

As you grow closer to your desired size, you are actually going back and converting previously built farms to cottages to try and ensure (1).

GS

 

[druidravi]

Its better to get farms/watermills early on till you reach population cap before cottaging for Granary>courthouse>forge>market>theatre>library etc and later start replacing watermills/farms with cottages /workshops and leave some farms to feed workshops. You should get some 8 to 10 production in the city even if it is a pure commerce city. Later with US you could cottage those farms/workshops.

 

[Gr8scott]

This all makes sense, but the assumption was that pop growth requirements are constant throughout. What happens when you have variable pop growth requirements?

I don't think there was any assumption regarding constant growth requirements.

In your example, the farm could take the time to growth down from 100 turns to 83 turns saving 17 turns.

The important thing to realize is that no matter how long it takes to grow each future population, you can achieve all future growth 17 turns faster because you are starting 17 turns earlier. To simplify things assume identical strategies for each approach after the first pop growth and I think this becomes clear.

The first farm lets you start your next cottage and all future cottages 17 turns earlier than growing without the farm. You need to know how many future cottages you have to decide if it was a good call or not, but I don't think constant/variable growth requirements matter. (again this is under the long-run assumption that all cottage turns equilibrate in the end)

Hopefully I undersood your question!

GS

 

[VirusMonster]

Not quite. In the simplified setting I was working with, you are going for a mix of cottages and farms that makes:

(1) Food surplus = remaining growth

As you grow closer to your desired size, you are actually going back and converting previously built farms to cottages to try and ensure (1).

GS

Yea, I agree that is the most simplified formula Make sure:

food surplus = population units to grow until you hit the health or happiness caps.

in other words,

food surplus = remained growth

As your city gets closer to the desired size, you will start switching farms to cottages. so for the 15 cottage or 15 farm scenario, at size size 6 with a surplus of 9 (4 initial + 5 farms), the remaining growth is 9.

Thus, you chould start working a cottage with the 7th citizen. 8 growth remains and your surplus is 9. Thus, you will turn 1 farm into cottage. You will work 2 cottages and 5 farms and the initial +4 rice.

Next, at size 8, 7 growth remains. Work the initial rice and 3 farms, for a total of 7 surplus and start working the 3rd & 4th cottages.

Next at size 9, 6 growth remains. Work the initial rice and 2 farms, for a total of 6 surplus and start working the 5th & 6th cottages.

Next at size 10, 5 growth remains. Work the initial rice and 1 farm, for a total of 5 surplus and start working the 7th & 8th cottages.

Next at size 11, 4 growth remains. Work only the rice for a total of 4 surplus and the 9th & 10th cottages.

For subsequent population growths 11-14, you will work the remaining cottages 11 through 14, switching farms to cottages if you haven't done earlier.

Here it goes

Remember that a city with only a 4 surplus rice should work farms until population 7. The 7th citizen starts to work the first cottage.

With a 5 surplus corn, work farms until population 6 for a total of 9 surplus. The 6th citizen starts working the first cottage.

With a 6 surplus pig, work farms until population 5 for a total of 10 surplus. The 5th citizen starts working the first cottage.

With a 6 surplus pig and a 4 surplus rice, for a total of 10 surplus through food resources, the 4th citizen starts working the first cottage.

Beautifully optimized!

PS. I would still work 2-3 farms more than in this scenario because of Slavery usage, to support 2 library specialists and 1 spy specialists, and the slight effect of growing getting more difficult with each new population unit. Artichoker's example was a huge exaggeration with 100 turns. In reality on marathon speed, going from size 10->11 with 5 surplus would take 63/5=12 turns, 1/8 of what Artichoker mentioned.

 

[Artichoker]

It all makes sense now...

Taking the extra step of turning a Farm into a Cottage so that the balance between growth and surplus is maintained.

You end up making 2 Cottages for each pop growth, once you hit the threshold.

If the food surplus is S, and the remaining turns is R, the original formula
was to make sure that

((S+1)/S - 1) * R > 1

But this is the same as:

(S+1-S)/S * R > 1

or

(1/S) * R > 1

R > S

when deciding whether to build a Farm.

However, the reverse also applies. Building a Cottage would apply when

S > R

And you could build extra cottages to make S = R. At small pop count, it's sometimes impossible to make S = R, because R is so much higher.

But once you reach the cutoff point, S = R the rest of the way, since you can reduce the unneeded surplus food by building 2 Cottages instead of 1.

 

[VoiceOfUnreason]

All of this is assuming that it doesn't matter when you get the commerce; ie that the that the expected value of marginal commerce is constant over the duration of the experiment.

How do people feel about that assumption? Is it true for the circumstances where we would apply these results, or simply convenient for calculation?

 

[Gr8scott]

If the food surplus is S, and the remaining turns is R

I like the math

I think you meant R is remaining pop growth not remaining turns.

One final point. Post biology, you would look for:

food surplus = 2*(remaining growth)

This accounts for the fact that farms are +2 food post biology instead of +1 food.

GS

 

[Gr8scott]

All of this is assuming that it doesn't matter when you get the commerce; ie that the that the expected value of marginal commerce is constant over the duration of the experiment.

How do people feel about that assumption? Is it true for the circumstances where we would apply these results, or simply convenient for calculation?

This is a good point to keep in mind. If we are looking to have a farm/cottage mix to make

Food surplus = remaining growth

We are adding cottages as we get closer to finishing up the growth. Considering the value of early commerce would cause you to err on the side of more cottages / less food surplus.

You are still confident about all the late stage cottaging, but this consideration could cause you to start the cottaging a little bit earlier.

VirusMonkey was also arguing the value of additional pop for slavery, etc, so he wanted to err on the side of more farms.

The true complexities of a particular game would clearly change the exact right answer, but at least the formula gives you a starting point for optimality.

GS

 

[VirusMonster]

This is a good point to keep in mind. If we are looking to have a farm/cottage mix to make

Food surplus = remaining growth

We are adding cottages as we get closer to finishing up the growth. Considering the value of early commerce would cause you to err on the side of more cottages / less food surplus.

You are still confident about all the late stage cottaging, but this consideration could cause you to start the cottaging a little bit earlier.

VirusMonkey was also arguing the value of additional pop for slavery, etc, so he wanted to err on the side of more farms.

The true complexities of a particular game would clearly change the exact right answer, but at least the formula gives you a starting point for optimality.

GS

Are you kidding with me? my name is VirusMonster, I eat monkeys alive On the other hand, I entirely agree with your last sentence.

 

[Gr8scott]

Are you kidding with me? my name is VirusMonster, I eat monkeys alive On the other hand, I entirely agree with your last sentence.

Eeek! my bad!

GS

 

[DaveMcW]

food surplus = population units to grow until you hit the health or happiness caps.

Thank you for stating it clearly.

 

[VirusMonster]

Thank you for stating it clearly.

You are welcome.

 

[Artichoker]

A general principle that I think can be derived from this concept is that regardless of whether you're planning to run a CE or an SE, it's wise to be working Farms for your first few pop growths.

The key turning point that determines the potential effectiveness of an SE is whether you can build the Pyramids successfully. If you can, then an SE becomes more attractive because of the option of having Representation as your government civic.

Otherwise, your Farms still provide the advantage of being able to work more Cottages earlier, if you happen to decide to run a CE.

 

[Artichoker]

All of this is assuming that it doesn't matter when you get the commerce; ie that the that the expected value of marginal commerce is constant over the duration of the experiment.

How do people feel about that assumption? Is it true for the circumstances where we would apply these results, or simply convenient for calculation?

"Marginal commerce" is indeed an important consideration in the early game.

You would need to have sufficient commerce in order to tech through at least Alphabet, at which point you gain the ability to trade techs, instantly boosting the actual value of your prior commerce.

The Palace gives a free +8 commerce, so there is some margin of error there for lacking early commerce.

The problem caused by low early commerce is why a ME (Military Economy) is so efficient in the early game. Because it allows you to make up for low early commerce by making it less of a drawback.

Even though your neighbors can tech faster in the early game, it takes only the presence of key resources and leader traits to make their tech advantage less relevant.

 

[VoiceOfUnreason]

"Marginal commerce" is indeed an important consideration in the early game.

Given the threshold of 15 happy, I think "early game" is already outside the bounds of the experiment....

 

[vicawoo]

There's a good chance the debate does not have the intuitive answers described, because of how towns grow. If they changed the parameters a little, there could be a large change in how the answer works.

With a granary, the growth rate is (10+n)/(food_source_bonus+n-1) with grassland farms, 2(n-1) with flood plains or biology, or with a ratio of k farms to 1-k cottages, (10+n)/(FSB+k*(n-1)).

Then we integrate from 1 to N, our target population, to obtain the total number of turns to grow with respect to k:

T_k=((N-1)/k+11*ln((F+k*(N-1))/F)/k)-F/k^2*ln((F+k*(N-1))/F).

Now the total amount of food to grow to size N is F_total=(N-1)/2*(10+1+10+N-1))=(N-1)*(N+20)/2

We can figure out the total number of worked cottage turns until max size C_k by the ratio of non food source farm food to cottage turns,
(F_total-FSB*T_k)/C_k = k/(1-k). This is exact, as long as T_k is exact. In the long run, if you work your size 15 city long enough to be all towns, all your cottages will have to go through the cottage/hamlet/village phase.
So

C_k=(F_total-FSB*T_k)*(1-k)/k
and total commerce up to T turns (assuming T > 40+T_k) is
(town or whatever bonus)*(C_k+N*(T-T_k)), that is, cottage turns until target size then all cottages at that target size.

Therefore the total commerce is bonus*((N+20)*(N-1)/2*(1-k)/k-FSB*T_k*(1-k)/k+N*T-N*T_k) or
bonus*(N*T + T_k*(-FSB*(1-k)/k-N) + (N+20)*(N-1)/2*(1-k)/k)
If we plot this, we find that cottage turns is maximized when you farm everything. This is assuming once you reach size 15 with a (5 food source) you cottage everything, and there is continuous growth. As for the total number of cottage turns till growth, it peaks at k=0.09, something like cottage everything except at size 11. I think I eliminated all the - sign mistakes. So farm to 15.

 

[VirusMonster]

wow, someone used integration to attack this problem I will try to understand your post later...

define your variables at start of your post plz... what are the axes of your graph? your initial formulas are not intuitive. what is n? what is N? what is k? how the they relate to each other? and start with the most basic scenario, ie all grasslands, attack the other problems later :-/ Why do I have to be the math grader here?

 

[ParadigmShifter]

I've got a degree in maths but can't be arsed doing it

I could also knock out some LISP or C++ code but I'm too lazy for that as well.