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Unless you are connecting a bunch of crap cities or spamming roads across the map, it should be easy to always maintain a net-positive on maintenance.
Also no reason to wait until Renaissance. You are probably losing gold in that situation by not having the gold bonus from city connections. While building roads it can be decieving to see the GPT tank, but once the connection finishes you should be getting more gold than you previously were.
You're probably losing money because you dont have roads and miss out on trade, rather than saving money on road upkeep. Unless offcours you build your cities 10 tiles apart
Not to say that roads are essential if you wanna quikcly move units around in case there is an invasion.
In Civ IV, you needed a road to connect resources to your capital. then again ,roads didn cost anything so you spammed them on every tile :p
Exact formula for each connection.
For a comprehensive thread on it, check here:
http://forums.civfanatics.com/showthread.php?t=438745
For what I mentioned above, just the number of roads needed to connect to the route--so nearest city that already has a connection.
Capital size has influence, as can be seen by the formula.
Generally though, like I said, you should always be getting a net-positive amount so you really don't need to go into this much detail. Basically if you build them right away there may be a small window where you are paying more maintenance than gold, but after like turn 70, the size of the capital alone will cover all connections
Another side-note: Maintenance rules are if it is in your borders, you pay for it. If it is outside of your borders, whoever built them pays for it. So if you capture a city you pay for the road tiles inside your new borders, but anything outside of that you don't unless you were the original builder.
Like for example, in a current mp game the huns (also human)have settled a city far away from their start position (cause it had great luxuries, excellent location). But the road to it will cost him quite a bit
3 hexes apart might seem a bit too few, cause that will cause massive overlap (and looks quite ugly to me). Its not necessary to work all the tiles. And your border expansion will even before halfway go beyond 3 tiles.
Dont also forget that there is a 1unit/tile rule. You will need a place to stall your military units, especially lategame you'll need adequate space. Even if you dont use them they will act as a deterrent towards the AI
@pandoradog22: roads didnt cost upkeep in Civ IV, so its not that obvious :p
Also, in that mp-game, the huns would be better off building another coastal city close to the "great luxuries"-city, and make a road to the coast-city, with the harbor connecting to the capitol.
It seems your capitol needs either a harbor, or a road-connection to a harbor, for overseas "road" to work, though?