A calculation of the appropriate population for cities

Lily_Lancer

Lily_Lancer

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Population costs amenties while producing bonus, so I'd like to check their actual yield to see which number of population is the best.
Suppose in the mid-late game, a typical city consists of Campus(average +9 with 1 adjacent and 2 Science city-state Suzerain), library, university and a monument . Our calculation is based on this setting. Maybe a Commercial Hub is added if you're using Zimbabwe strategy. Suppose there are 2 districts for >=4 pop cities. The other district is a +10 Commercial Hub(2 Adjacent +1 City-State Suzerain).

Consider the typical policy is (+100% Science output) is used.

Typically we have 2 amenties for a city. One for luxury and one from policy. Suppose every population can work on a mine to produce +2 Food and +4 Production.

Code: 
Pop    Aments-Influence    Production_base    Production    Science_base    Science    Culture_Base    Culture    Gold_base    Gold    Overall_yield(Treat Sci/Cul=2Gold/Production)   
1    1.05    5    5.25    21.7    22.785    3.3    3.465    0    0    57.75  
2    1.05    9    9.45    22.4    23.52    3.6    3.78    0    0    64.05  
3    1.05    13    13.65    23.1    24.255    3.9    4.095    0    0    70.35  
4    1.05    17    17.85    23.8    24.99    5.2    5.46    10    10.5    89.25  
5    1    21    21    24.5    24.5    5.5    5.5    10    10    91  
6    1    25    25    25.2    25.2    5.8    5.8    10    10    97  
7    0.95    29    27.55    25.9    24.605    6.1    5.795    10    9.5    97.85  
8    0.95    33    31.35    26.6    25.27    6.4    6.08    10    9.5    103.55  
9    0.95    37    35.15    27.3    25.935    6.7    6.365    10    9.5    109.25  
10    0.95    41    38.95    28    26.6    7    6.65    10    9.5    114.95  
11    0.85    45    38.25    28.7    24.395    7.3    6.205    10    8.5    107.95  
12    0.85    49    41.65    29.4    24.99    7.6    6.46    10    8.5    113.05

So 10 pop for every city yields the most.

But when we consider, every population cost 1 housing, food and a mine to work. The cost can be transferred into about 2 worker charges+food cost.
Suppose 1 HPT=1FPT = 25 H, and we already have 6 housing for each city (5+1 for university)
Code: 
Pop    Aments-Influence    Production_base    Production    Science_base    Science    Culture_Base    Culture    Gold_base    Gold    Overall_yield    Worker charge needed    Food needed    Overall food needed    Transfer Value    Actual Output
1    1.05    5    5.25    21.7    22.785    3.3    3.465    0    0    57.75    1    15    0    1.2    56.55
2    1.05    9    9.45    22.4    23.52    3.6    3.78    0    0    64.05    2    24    15    3    61.05
3    1.05    13    13.65    23.1    24.255    3.9    4.095    0    0    70.35    3    33    39    5.16    65.19
4    1.05    17    17.85    23.8    24.99    5.2    5.46    10    10.5    89.25    4    44    72    7.68    81.57
5    1    21    21    24.5    24.5    5.5    5.5    10    10    91    5    55    116    10.64    80.36
6    1    25    25    25.2    25.2    5.8    5.8    10    10    97    7    66    171    15.24    81.76
7    0.95    29    27.55    25.9    24.605    6.1    5.795    10    9.5    97.85    9    77    237    20.28    77.57
8    0.95    33    31.35    26.6    25.27    6.4    6.08    10    9.5    103.55    11    89    314    25.76    77.79
9    0.95    37    35.15    27.3    25.935    6.7    6.365    10    9.5    109.25    13    101    403    31.72    77.53
10    0.95    41    38.95    28    26.6    7    6.65    10    9.5    114.95    15    114    504    38.16    76.79
11    0.85    45    38.25    28.7    24.395    7.3    6.205    10    8.5    107.95    17    126    618    45.12    62.83
12    0.85    49    41.65    29.4    24.99    7.6    6.46    10    8.5    113.05    19    139    744    52.56    60.49
So this rough calculation shows that 6 pop is the best amount for a city to get the most yield.​
 
It is quite hard to read the columns.

Anyway most of this seems to be of limited scope, it make alot of assumptions which may not be true. Zimbabwe in itself have the limitation that it use bonus resources which you otherwhise may be better of haveresting as bonus resource harvest increase in value with the same formula as districts increase in cost and there are other reasons to get rid of bonus resources such as being in the way for farms or districts so that is something to keep in mind with that wonder.

My guess is that optimal population would be somewhere there you can build all districts you need for that city, yes amenties are also important because of their multiplier effect but the big reason to grow cities in my opinion are to build more districts (which also increase the value of internal trade routes).

So my guess is you should base your population on the number of district you want the city to have. Here is a list on how many citizens (C) and amnities (A) you need for a certain amount of district slots. EA stands for Ecstatic Amnities, that is how many amnities you need for maximum happiness bonus
  1. C = 1 A = 0 EA = 3
  2. C = 4 A = 1 EA = 4
  3. C = 7 A = 3 EA = 6
  4. C = 10 A = 4 EA = 7
  5. C = 13 A = 6 EA = 9
  6. C = 16 A = 7 EA = 10
  7. C = 19 A = 9 EA = 12
One interesting thing we can see is that even citizen numbers are more amenities efficient than odd citizen numbers. We see that going from 4 to 7 citizens cost 2 amenities while going from 7 to 10 only cost 1 amenity which is something to keep in mind.

Then we can ask how many districts each city need. Campus, holy site and theater square all produce resources that tiles seldom yield so if you lack a unique improvement that can produce science, culture or faith these districts become an important source of these resources and even if you do have such improvement they are still useful for additional resources as well as for great people points.

Only the first trade route district in a city, harbour or commerical hub provide a trade route so that great reduce the value of the second one in that city, there are still reasons to get the other one but these reasons are diminished because both harbour and commerical hub produce common resources unlike the 3 districts mentioned before.

The area districts like the trade districts have also been nerfed from what they was in the release version as now only one area of effect building of each type can effect a city. Industrial zone suffer from being a production investment that produce production so in generall you probably want as few of these as possible but it also depend on how good yield you can expect from the industrial zone, can it pay back itself and how long do that take? You still want them for the factories and they can be useful for great engineer points given that districts and their buildings may be the most cost efficient way to produce great people. Entertainment complex is similar to the industrial zone with its area buildings. A entertainment complex and its local arena produce only 2 amenities and cost a district slot which we know cost or 2 amenities which mean growing the city to build a entertainment complex is really doubtful move. However you probably want entertainment complex for their global buildings.

Encampment is a bit like industrial zone if you have the CS. Great general points are very valuable because you can stack great general and get some super powered units and the boost to military units and defence are nice as well but unfortunaly they do not provide as many rare resources as campus, holy site and theater square.

So if we assume you build 2 out of encampment, campus, holy site and theater square. 1 out of harbour or commerical hub. And 1 out of industrial zone and entertainment complex. That is 4 districts in total which mean you need 10 citizens and 7 amenities to keep that city at peek productivity. 16 population may however be a better target because 10 may leave many tiles unworked but going from 10 to 16 population is quite expensive given you need to grow 6 more citizens, likely build a neighbourhood, get 3 more amenities and improve 6 more tiles (although you also use these 6 more citizens as specialists to cut some costs and get some rare resources.
 
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Lily_Lancer said:
So this rough calculation shows that 6 pop is the best amount for a city to get the most yield.
Click to expand...
Interesting, one extra pop and you can build another District though. Each major city is probably going to have an IZ and either a Commercial Zone or a Harbour so you really want to get to 7 to build your campus. theatre square, holy site etc.
 
Industrial zone is a quite doubtful district to build in every city. Its problem is that you invest production to produce production so you should caculate how long it take for the IZ to pay itself back. Yes there are great engineer points which may be actually more valuable than the production benefit of the IZ but is it worth one of your limited district slots to produce more of a common resource (production)?.

It was good everywhere then you could stack factories but that have been patched. Entertainment complex is even more doubtful in every city given to growth to the pop needed for EC it will cost one or 2 amenities and EC can at best give 2 local amenities. EC are however still useful for its global effect like IZ are.
 
Denkt said:
It is quite hard to read the columns.
Click to expand...

If a district values less than campus research grant, or builder, we'd better not build that district.

So in most cities, we only build campus, and maybe a commercial hub /harbor, then that's all.
 
Lily_Lancer said:
If a district values less than campus research grant, or builder, we'd better not build that district.
Click to expand...
How do you come to that conclusion?

Anyway how do you value a builder vs a district as you may already have improved all the tiles you want and research grant is also tricky to compare to another district as it produce science and great scientist points.

Yes it is very likely that campus research grant will produce more great people point than a new district will do in its lifetime if that is what you are talking about. However such argument can be used for any district. Build the district that give you the needed great people points and then just build its project and maybe purchase its buildings while doing so.

Holy districts and theatre square actually produce rare resources than science. Yes faith have limited scope but very nice for culture and religious victories. The square is pretty useful in any game given how much additional culture it can add. I mean a citizen produce 0.3 culture and 0.7 science so a great work of writing who produce 4 culture per turn is worth more culture than 13 citizens who don't work any culture. Even a theatre square specialist produce 2 culture which is worth nearly the same as 7 citizens. I probably say theatre square is about equal to commerce hub in importance due to the rarity and importance of culture given big effects of policies and governments have on everything including science and trade with right policies.

It is really the late game in which projects may really be the best production investment given the short time left on the game.
 
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Denkt said:
It is really the late game in which projects may really be the best production investment given the short time left on the game.
Click to expand...

I'm referring to a standard setting of Deity Science Victory, which is completed in ~T160~170 Standard Speed. (Since this one is the most stable victory, its strategy and finishing time do not vary much from map to map) Also, SV need yields from all sources.

For other victory conditions, yields are not important and any population is good enough.

After T120, maybe T110 if you have a good starting, you'll find yourself only producing builders or projects, except for some late-game wonder cities.
 
So just translate for anyone feeling a bit lost.

A 2-3pop city with a science district is great but getting it up to about 6 will only cost you one amenity and is much more efficient in creating the project required for victory as well as allowing a second useful district like a commercial zone. All for the price of one amenity. This is assuming you have enough hills to be of value.

This thought process is part of the best overall strategy of beating up everyone else as much as possible and getting a sub 200 victory... the optimal strategy as proven.
 
This sounds great on paper but the maps usually aren't that kind to me. Rarely will I have bunches of low pop cities that can reliably produce anything let alone districts in the late game. The hammers just aren't there.

This is especially true if I went early war and 2/3 of my cities are conquered ones that the AI has just randomly sprawled in the oddest of locations.
 
How would these calculations be affected for Civs which have a unique that grants an amenity, like Rome or Aztec? I'd assume they could comfortably go beyond 7 pop and get another district?
 
Victoria said:
So just translate for anyone feeling a bit lost.

A 2-3pop city with a science district is great but getting it up to about 6 will only cost you one amenity and is much more efficient in creating the project required for victory as well as allowing a second useful district like a commercial zone. All for the price of one amenity. This is assuming you have enough hills to be of value.

This thought process is part of the best overall strategy of beating up everyone else as much as possible and getting a sub 200 victory... the optimal strategy as proven.
Click to expand...

How does this account for all the captured cities where the AI placed a holy site as one of the districts?
 
mckinney156 said:
How does this account for all the captured cities where the AI placed a holy site as one of the districts?
Click to expand...

I missed this post, apologies.
So like last night on Deity I had 19 cities at T100 and maybe 12 of them had holy bits.
This is why I often talk about a city of 7 pop.... campus, commercial and holy site.

I was making a mistake for a while and building campuses first but getting 3 commercial down is much better.. and an encampment if possible although I missed that last night... 2 commercials with Markets... so you get the eureka s for 4 trade routes and 2 markets.
Then every city goes Campus crazy and only in cities that naturally have god prod and grow to fit them ... additional commercials. The holy sites are a cross we have to bear. Normally you can get a city to 4 pop so you can get a campus and 2 by chopping jungle often naturally happens. To be fair 12 holy sites can get you the odd great person but sometimes I just do not bother repairing due to low production in the city.

I did use autocracy as a government last night and really felt the difference.... thanks @Lily_Lancer
 
Yea I mean I'm too lazy to look at your math. More districts outweigh almost all other issues though - optimal population for a city is almost certainly 4, 7 or 10 depending on terrain.
 
If you only need the city to give you a campus and a trade route what third district is worth the cost of getting to 7 pop (both food and amenities) and the production needed to build it? A riverside city with only a granary in the way of housing improvements also "naturally" tends to end the game at 6 pop as this is when growth will slow.
 
stinkubus said:
If you only need the city to give you a campus and a trade route what third district is worth the cost of getting to 7 pop
Click to expand...
If you captured the city it will often have a holy city
If you want to go for the three factory eureka
You will need one ED for Coliseum
Sometimes you need a museum to get rid of artifacts... and one museum can bring in a fair amount of culture.
When a city gets to 7 you do not have to build a 3rd district, just place everyone on prod tiles and thrash out the projects
Often you want to chop jungle but know you are gonna grown more than you want. Normally is fine if you have a coliseum
 
Victoria said:
If you captured the city it will often have a holy city
If you want to go for the three factory eureka
You will need one ED for Coliseum
Sometimes you need a museum to get rid of artifacts... and one museum can bring in a fair amount of culture.
When a city gets to 7 you do not have to build a 3rd district, just place everyone on prod tiles and thrash out the projects
Often you want to chop jungle but know you are gonna grown more than you want. Normally is fine if you have a coliseum
Click to expand...

Of course you shall prioritize campus.
I usually ignore the eureka of Medival Fairs and Guilds. Except for some certain situations when I can capture some CHs or Harbors.
 
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