Table for maintenance increase

[vicawoo]

Starting with n cities (columns), if you add a city d distance (row) away from your capital, how much more maintenance do you have to pay for the entire empire?
Remember that more cities increases maintenance in ALL cities.

Formula taken from the curious cat maintenance explained page 5. Settings Standard Deity.
Details:
Delta[n+1] = 3/5*n+0.3+8*Distance/29

Spoiler :

# city maintenance = N*(Pop+17)*WorldSize_m*Handicap_m/18+1/2 (1/2 is actually a modulo factor)
# city maintenance for all cities = N*((Total Pop+17*N)*WorldSize_m*Handicap_m/18+1/2)
Distance maintenance = Difficulty*Distance*(Pop+7)/29 for standard map
Delta[n+1] = (n+1+(P[total]+17*N)/18)*Worldsize_m*Handicap_m+1/2+8*Distance*Difficulty/29
Note that Total population (P_tot) + 17N ~ 18N early on when each city has size 1 to 5. If we liked we could get more accurate figures if we knew the empire population, but then we'd need a 3 dimensional table.
Delta[n+1] = 3/5*n+0.3+8*Distance/29

http://spreadsheets.google.com/pub?key=tne1Lse68ekMN-uQ-zheXgg&single=true&gid=0&output=html

Rule of thumb, every 3.5 extra distance you place your new city, empire wide maintenance will increase by 1.

Later I'll post how this affects empire growth optimization while switching to commerce tiles to maintain a constant research rate. That is, if a new city loses your empire 4 gold, you have to switch 2 mines to 2 cottage tiles and hence you lose 4 production. So when is it mathematically superior to grow your existing cities or to establish a new one?

 

[vicawoo]

Will edit this growth vs settling analysis later

Spoiler :

Let M(n,d[n+1]) be the increase in empirewide maintenance by adding an n+1st city at a distance of d[n+1] from the capital (In this notation d is the distance from the capital of the ith city).

In the long run, if we increase empire wide maintenance by 4, we would have to switch two tiles to 2 food 2 commerce tiles to maintain equilibrium.
However, we get 1 commerce from the new city itself; you may even get an additional 1 if you have a trade route.

So new city negatively impacts empire wide production by
( M(n,d[n+1]) - 1 )/tile_c tiles, which would have otherwise yielded tile_y (tile yield). So the cost of the new city in food/hammers is
M(n,d[n+1])*tile_y/tile_c
Note:

Spoiler :

If we give up a riverside mine for a riverside cottage, we are only netting one commerce, so in practice tile_c is increase in commerce from switching from our production oriented tile to our commerce tile. If by chance we have a domestic trade route to the new city, we can just pretend the new city costs the empire one less gold coin in maintenance

Profit = Revenue - Cost
Profit = tile_y[n+1] - ( (M(n,d[n+1] ) -1 )*tile_y/tile_c

We divide our profit by our initial investment, that is 100 production.

Now we compare this to growing our ith city from size p to p+1.

Again Profit = Revenue - Cost
Cost = d/29 + n/60
Note that in most cases this cost will usually be much less than 1.
Profit = yield'_fh + (yield'_c- d/29 + n/60)*tile_y/tile_c
In most of these cases we're debating working a mine or it's commerce equivalent.
And the investment is 2*(10+p) food, with an opportunity cost of
2*(10+p)*(f+h)/f food and hammers, where f is the food surplus and h is the hammer production of the ith city.

Ok, we have tons of variables. Let's simplify. Since the increase in maintenance from growing is much less than 1 and we're dealing with integer production values, we'll ignore that. We'll assume we are growing to work a mine, that is, a net +2 food/hammer production.

(tile_y[n+1] + ( 1+tile_c[n+1]- M(n,d[n+1] )*2/tile_c)/100 > 2/( 2*( 10+p )*( f+h )/f )

So what's the point of all this, since sometimes we may only care about maximizing hammers and other times we may care more about our tech rate? It provides a numerical value in the commerce adjusted value of new cities.

 

[Silu]

That post seems to predate BtS, which changed the formulas slightly. Main difference is making rounding errors 100 times smaller and omitting city rank from the formula (the "+ 1/2" from your post if I got that right). Also BtS obviously added Colony and Corporation maintenance but I guess those are omitted from this anyway.

 

[pi-r8]

Even if a new city temporarily loses you commerce, it will eventually pay for itself as long as it's in a decent location. And if you don't place a city quickly, another civ will probably grab that spot first. So anymore I just focus on getting as many cities as I possibly can, as quickly as I can.

 

[vicawoo]

That post seems to predate BtS, which changed the formulas slightly. Main difference is making rounding errors 100 times smaller and omitting city rank from the formula (the "+ 1/2" from your post if I got that right). Also BtS obviously added Colony and Corporation maintenance but I guess those are omitted from this anyway.

Yes the city rank is 1/2, can you show a link for the updated BTS formulas? I can update t if it's missing.
I ignored the rounding errors because it would be impossible to add anything. In actuality there's terms like [25*(pop+7)/100] or something similar.

 

[Silu]

I dove this a while back (not as simplified as in the old thread). Don't see much that has changed in the formulas except the deletion of city rank, though there's a cap on city number maintenance. Also the constants (difficulty & map size) might have changed from Vanilla, didn't check that. Probably they didn't, though.

 

[JammerUno]

It would be awesome if increase in city maintainance would appear in BUG though.

 

[vicawoo]

Ok updated the 1/2. The second post is hard to write in a way that a numbers oriented player could use. There's 4 major variables, 1) the empire maintenance increase, 2) the ratio of food to production of a city making workers/settlers, 3) the commerce yield of an workable commerce tile (1 if non-riverside cottage, 2 if lake or riverside cottage, 3 if financial lake or riverside cottage), 4) the yield of the new city.

pi-r8, if you want as many cities as you can try out the commerce free formula
If yield_newcity/100 > f/(f+h)/( 10+p )
then build a new settler, otherwise grow. I've tried doing that; you expand unbelievably fast but your economy tanks just as fast.