City Sizes

[Urtica dioica]

A few months ago I had an insight on how an efficient empire might be built, but my interest in Civ was waning (like my interests do, only to come back again). I wanted to play-test it before posting, but that never happened. That's the caveat.

I've read many times that one should build the biggest cities possible, or at least try to make size 20, because it saves gold on upkeep for city improvements. There's a class of improvements for which this is true, in particular the multipliers -- marketplace, bank, library, university, factory, manufacturing plant, and the various power plants. These structures work the same on a given number of squares regardless how many cities they're divided into, but with more cities, and more multipliers, upkeep rises, killing efficiency.

There's also a force in the opposite direction. The happiness improvements -- temple, cathedral, and colosseum -- become more efficient the more cities they're divided into because you get to build more of them. The same effect is even more potent with born contents and happiness wonders, which don't require upkeep.

Efficiency can be expressed by the ratio of squares worked to upkeep required. By assuming a constant amount of multiplier upkeep (two libraries covering ten squares each work just as well as one library covering twenty, with a small penalty for rounding error), you can make a chart of city sizes to find the optimum.

Here's an example, assuming 8 base upkeep (marketplace 1, library 1, bank 3, university 3), 2 born contents, Bach, and Michelangelo, meeting happiness requirements in the most efficient way possible:

Size Price/Squares Decimal Explanation
1 8/2 4.00
2 8/3 2.67
3 8/4 2.00
4 8/5 1.60 Freebies up to here from born contents and Bach
5 9/6 1.50 Add a Temple
6 9/7 1.29
7 11/8 1.38 Add Cathedral sell temple, or keep Temple add 2 luxury
8 11/9 1.22 Ignore the luxuries
9 11/10 1.10 Michelangelo keeping it going
10 11/11 1.00
11 14/12 1.17 Aqueduct and Temple and Cathedral
12 14/13 1.08
13 16/14 1.14 Luxuries beat Colosseum
14 18/15 1.20 Colosseum catches up
15 18/16 1.13 Colosseum finally wins
16 20/17 1.18 Luxuries here on out
17 22/18 1.22
18 24/19 1.26
19 26/20 1.30
20 28/21 1.33

In this scenario, selling the temple when it wasn't needed allowed size 10 to win, with only one upkeep/square. I've examined lots of these tables, changing base upkeep (say you want factories), removing Michelangelo, changing freebies (born-content plus Bach plus HG plus Cure), and selling the aqueduct when the city reaches target size (hope you don't get a famine or a siege!) In general, when you keep the aqueduct, size 10 wins a lot, and you have to add a lot of base upkeep to make colosseums worthwhile.

Now some more caveats:

This system applies when you have a certain area and you need to decide how many cities to stuff into it. I'm not suggesting you build cities without overlap and then use half the squares.

These ideas go out the window if you have a city limit, like if you're playing OCC or building a mega-empire that hits 127 cities. In those cases, it's most important to work every square you can.

On Chieftain, and maybe Warlord, you may be able to reach Robotics and launch before anyone reaches Religion. In that case, you could use Oracle to build size 9-10 cities with extreme efficiency.

Note that the efficiencies at sizes 8-16 and beyond in my chart are actually very similar. Colosseums and aqueducts are just barely not worthwhile, and easily could be worthwhile depending on the terrain. The main point is that you can grow faster, finish growing earlier, and be efficient with 0% luxuries.

In patch 3 and beyond, you lose born contents after a point. I don't think this makes a big difference though. Notice in the scenario I gave, if you lost both born contents, you could still reach size 10 just by keeping the temples. Even on Emperor, that would be fine for up to 36 cities.

Having a larger number of cities might be more expensive to defend. This is where play-testing would help.

 

[Osvaldo Manso]

Congratulations Urtica Dioica! You could write a thesis for a Ph.D. on Sid Meier's Civilization!

For me, it's hard to follow you on such a theoretical explanation. But I'll read it again later.

 

[Urtica dioica]

Yeah, that post was pretty abstract. I tried to make it as clear as I could, but an example would help.

Let's assume you've got 20 squares you want to work in an Emperor game. You want them to be efficient, so you want marketplaces, libraries, banks, and universities. You've got Bach and Michelangelo to help.

First, a really stupid scenario.

You could work those 20 squares with 10 cities of size 1. The great thing here is that all your cities will stay out of disorder just by virtue of the two free content citizens every city gets in an Emperor game. The problem is, to get the benefit of a marketplace, library, bank, and university on every square, you need one in every city. That's 8 gold of upkeep in 10 cities, for 80 total upkeep. It's true that 10 cities of size 1 wouldn't build these improvements, and that's the problem. But notice that the total luxury, science, and gold output is the same as it would be with one big city covering the same squares.

Next, what most players would do.

Those same 20 squares could be worked by one huge size 19 city. This time, the advantage is that I need only one marketplace, library, bank, and university, for 8 upkeep total. The problem is, keeping this city out of disorder is a nightmare. The most efficient way to do it is with a temple, a cathedral (remember Michelangelo), a colosseum, and 8 luxuries, in addition to the 2 free content citizens and 2 from Bach. That's 16 trade (gold plus luxuries) just to keep the city running. With 8 upkeep for the multipliers, 16 to keep the people from rioting, and 2 to keep the aqueduct flowing, this city eats 26 gold per turn, or 1.3 per square. It's an improvement over the size 1 cities, at least.

Finally, lets see if we can do better.

You could work 20 squares with two size 9 cities. You'll need two of each of the multipliers, so 16 upkeep already. To keep the people working, with born contents, Bach, and Michelangelo, all you need is a cathedral in each city, for six upkeep total. No temples, no colosseums, and no luxuries. That's 22 upkeep total (no aqueducts required), for 1.1 upkeep per square.

The problem with the single huge city is that born-contents, Bach, and happiness improvements can only be used once per city. When I split the city in two, I got twice the born-contents, I got to use Bach twice, and I got to build each happiness improvement twice. Compare that to luxuries, and you can save a lot of money.

 

[Osvaldo Manso]

Thank you! Your example made it easier to understand.

I would just like to say that sometimes those 20 squares you want to work when divided by two cities, instead of just one, will not allow a well balanced city.

Example: imagine that out of those 20 you have only 1 good square for production (shields), like hills+mine or swamp+oil. If you choose founding 2 cities, one of them will be very slow in building units or city improvements. My feeling is that most of the times the terrain is not suitable to be "divided" between 2 or more cities but could host one good city.

 

[Urtica dioica]

Your argument about production squares probably applies in certain cases. Of course, if two cities have one production square, they could just trade it back and forth. When you pack cities, you can mix up city geometries in some unusual ways.

Beyond that, there are ways to share resources among cities. For a long stretch of the game, from about Trade to Industrialization, trade arrows are as good at filling the shield box as shields. From Trade on, caravans can transfer shields, not just for wonders, but for anything else, by changing your building project. And most projects smaller than a caravan are units, which can be re-homed.

Hills haven't impressed me for a long time anyway. Not only do they take ages to mine (without the settler cheat), but a food square with a road often produces almost as well, and might allow me to make another worker, which would definitely make up the production gap. Also, with good trade multipliers, an ocean can produce as well as a mine, and there's usually no shortage of water.

Another argument is that building extra improvements for many cities will increase the total cost of shields across the empire. But usually, I'll build something at a particular stage of development anyway, like marketplace and library at size 4-5 or so, regardless how many cities I have. While I may have to make more improvements total, I'll be working more squares at the time they're built, making up the difference.

 

[trada]

Why does the long stretch end at Industrialization? Factories don't exactly leave caravans in the dust. Interesting statistics Urtica. I whipped up a graph: Fixed. Am I doing it right? If so its interesting that a size ten city is the most optimal (not taking into account the actual exponential returns on larger cities).

 

[jmas]

I'm not as well-versed in the intricacies of Civ I as others, but I do like math and I have enjoyed the discussion so far, so I'll definitely be lurking at least. Thank you for starting this thread Urtica dioica! The only thing I'm not clear about is what "re-homing" is--I've seen the term before but don't know what it means, and it's not in "Rome on 640K a Day" (in the index, at least).

 

[trada]

I'm not as well-versed in the intricacies of Civ I as others, but I do like math and I have enjoyed the discussion so far, so I'll definitely be lurking at least. Thank you for starting this thread Urtica dioica! The only thing I'm not clear about is what "re-homing" is--I've seen the term before but don't know what it means, and it's not in "Rome on 640K a Day" (in the index, at least).

Re-homing is where you change a unit's home in civ1. Every unit has a 'home' that pays the upkeep for it.

Yeah, statistics can be hella fun!

 

[jmas]

Okay, thank you. That's what I thought from the term itself, but I thought that only became possible in CivII. I'm not sure if the version of CivI I used to play didn't allow that, or if I just missed the fact that re-homing was possible! I'll see whether I can do it in my recently-acquired CivDOS sometime..

 

[lndm]

I guess I'd probably use it most when under democracy and a city is well into an expensive improvement like a wonder or a cathedral. Some other civ lands there and I only have a phalanx there so I build a musketeer somewhere else and send it there.

 

[Tristan_C]

Why does the long stretch end at Industrialization? Factories don't exactly leave caravans in the dust. Interesting statistics Urtica. I wipped up a graph: http://forums.civfanatics.com/newreply.php?do=postreply&t=318399 of them. Am I doing it right? If so its interesting that a size ten city is the most optimal (not taking into account the actual exponential returns on larger cities).

Urtica's statistics are geared towards the long midgame. If you incorporate factories into these schemes, the balance shifts dramatically towards large cities IMHO since factories are an end-game addition to the list of his "multipliers" ...multipliers that happen to be prohibitively expensive to build and maintain. At 200 shields apiece, it's a considerable outlay that you want to be making the most of; bedecking your massive roster of size-10 failproduction cities with 200, 4 facilities is an extravagant waste of resources. If the civ seems like it will be going deep into the modern era before conquering the world, or you simply wish to go deep into the modern era to boost your score, I would THINK that the raw statistics would favor size-19 cities. Powered factories can top a typical large city off at around 50 production per turn, meaning that you can distribute this production to other cities efficiently. I guess on the other hand, a powered MFG Plant can bring a smaller city's production to 50 if it has some hills to bolster it. On the tree, MFG plants are a deceptively long trek ahead of factories, though.

Unfortunately, you can't set a unique luxuries rate for a city. I wonder if this will EVER get implemented, properly, in a civ game. If you could do it here, we could build a sparing number of size-19's that feed fast and furious resources to our smaller cities.


EDIT: can't see your graph for some reason. I really want to.

 

[jmas]

Good points, Tristan_C. I'll defer to more knowledgeable others to comment on them--I'm focused on Civ4 BtS at the moment.

I can't see the graph either, trada. I get a blank page. I'm using Firefox if it matters.

 

[Urtica dioica]

Opinions and experiments are not required. It's easy to make your own chart and find out exactly what size is ideal. As Tristan_C points out, adding extra multipliers tends to increase the optimal city size, but maybe not as much as he suggests. There's no one optimal city size, since the chart you need depends on four variables, and the upkeep on your multipliers is only one. The other three are whether you sell your aqueduct when a city has reached its ultimate size, whether you have Michelangelo and don't expect it to go obsolete, and how many content citizens you get for free (born contents plus Bach plus Cure).

Here's a chart similar to the first one I posted, but with base upkeep at 12, accounting for a factory in each city. Of course, building factories without power plants is silly, but I'll assume you're using Hoover Dam:

1 12/2 6.00
2 12/3 4.00
3 12/4 3.00
4 12/5 2.40
5 13/6 2.17
6 13/7 1.86
7 15/8 1.88
8 15/9 1.67
9 15/10 1.50
10 15/11 1.36
11 18/12 1.50
12 18/13 1.38
13 20/14 1.43
14 22/15 1.47
15 22/16 1.38
16 24/17 1.41
17 26/18 1.44
18 28/19 1.47
19 30/20 1.50
20 32/21 1.52

Size 10 still wins. You can certainly tweak the variables to make a bigger optimal size, but you'll have trouble finding any scenario where luxuries make sense.

Or you could just not build factories. Industrialization comes so late in the game that, if you're efficient, it could be almost over. Of course, if you're laying out your empire for size 20 cities, and thus wasting half your space for most of the game, you may have a way to go.

When I referred to a long stretch from Trade to Industrialization, I was comparing the construction ability of shields with arrows. And I did it wrong. With three bad foreign trade routes per city (multiplying total trade by ~50%), a marketplace, and a bank, an arrow can turn into 3.375 coins, or 1.6875 shields. It would take a factory and a power plant to make a shield beat that, so the stretch in which arrows can outproduce shields is even longer than I thought.

Lastly, trada mentioned exponential returns from large cities. This leaves me confused, because the only exponential growth possible in the game is from outward expansion, which tends to involve small cities. I'd like to know what this refers to.

 

[trada]

Interesting read, Urtica. I fixed the graph link. Exponential growth in the sense if you add up the total returns on a large city (from birth to when the sample is taken).

 

[Tristan_C]

Opinions and experiments are not required. It's easy to make your own chart and find out exactly what size is ideal. As Tristan_C points out, adding extra multipliers tends to increase the optimal city size, but maybe not as much as he suggests. There's no one optimal city size, since the chart you need depends on four variables, and the upkeep on your multipliers is only one. The other three are whether you sell your aqueduct when a city has reached its ultimate size, whether you have Michelangelo and don't expect it to go obsolete, and how many content citizens you get for free (born contents plus Bach plus Cure).

...

When I referred to a long stretch from Trade to Industrialization, I was comparing the construction ability of shields with arrows. And I did it wrong. With three bad foreign trade routes per city (multiplying total trade by ~50%), a marketplace, and a bank, an arrow can turn into 3.375 coins, or 1.6875 shields. It would take a factory and a power plant to make a shield beat that, so the stretch in which arrows can outproduce shields is even longer than I thought.

I do realize that costs are but parallel to upkeep, and the stats were addressing upkeep.

I should have been more emphatic about my worries over the production cost of creating small cities. That was my beef with them when we first mentioned factories; the 200 shield-cost, and not so much the luxury and maintenance per turn, is where the size-10 strategy feels pain. For instance, three bad foreign trade routes, a marketplace, and a bank costs 350 shields to produce initially... so it damn well better be able to convert a trade arrow into more than one shield. When you are running a few cities (say, twelve) that are only working the ten best tiles on their way to size-19, 350*12 shields is a more manageable upfront cost than 350*24, particularly when we consider the quality of the tiles being worked and that ultimately, all the same tiles will be getting worked. It will take many turns for the small cities to produce, and then recover the 4200 extra shields, equivalent to half of a fully-loaded spaceship, we had to spend to make them operational. Although they will recover it, eventually, in the forms you've described: lower maintenance and lower luxury rates. Whether that full recovery point occurs by Railroad or by Future Tech 8 is an important consideration to weigh.

 

[lndm]

What if we decide to go past communism and we need temples again?

 

[hannurabi]

Great research Urtica dioica!
If I understood everything correct, the ultimate strategy would be somekind of "twin-city" placement: You build cities in groups of 2, two cities very close to each other, so they can share those 20 squares and both grow size 10. Later in the game you select other city and let it grow.The other unfortunate city will decrease in size and eventually disapperar to create room for 1 big city size of 20. This way don't loose tiles in early/mid game and get the full factory bonuses in late game.And you can defend these twin cities with less chariots cause they are very close. I need to test this strategy.

 

[Walt]

Great Thread, Urtica! Thanks for posting.
I often get many cities up to size 3-5, then go 100% luxuries to grow them one per turn. This leverages the benefits of larger cities over a longer time. i.e. the math changes (presumably) if you look at TOTAL cost vs. TOTAL production ($+shields) over the life of the city (cumulative efficiency), vs efficiency per turn. But I haven't done any analysis of it, so I really can't say.

 

[Walt]

I'd also like to see you do a similar analysis of military units. For example, it's obviously not worth paying to keep a barracks on line in a city building a battleship. How many turns to build a battleship? And barracks are, what, $2 a turn to maintain at that point? If a non-veteran battleship is built, it's fairly easy to turn it veteran by picking an easy first target.

 

[ignatius]

The analysis neglects some very important factors. The amortization of production costs being only the most obvious. Other issues are:

• Specialists: A typical inland city with just enough overlap to work all tiles can usually support more than one pop per tile, so you get specialists which are content by default. When you assume, say, a modest 5 specialists per town, the picture changes dramatically as they can either produce all the luxuries you need or generate enough coins to pay for most of the towns upkeep. With 10, you can do both, so all income from tiles is your's to keep.
• Coastal cities: To maximize the number of worked tiles, coastal cities need to work as many of those lucrative 3-trade sea-tiles as possible (typically 8 to 12). Grassland supports 2 and plains support only one sea-tile which means that coastal cities need to be big or you have to let reachable ocean tiles go unused.
• Garrisons: Obviously, the less cities you have, the easier they are to defend.
• Additional improvements like trade-routes, city walls, barracks or city-wonders you might want in some of your cities.

The argument that larger spaced cities let many of their tiles unused for much of the game is also bogus - at least if you're playing a representative govt (and if not, any discussion of optimality is misplaced anyway). While improvements can be bought and pop can be boosted by WLTPDs, what really limits grow in civ is how fast you can max-out your tiles, as working unimproved land is hardly worth it. And this has nothing to do with cities but with the number of settlers you employ to transform your land into those ugly but optimal terraformed, irrigated or mined rail-road grids.

ignatius