Ok guys this is from labbing it to see whether it's cumulative or not(just to be sure, I don't trust any of you guys ).
Settings: Initial gold set to filthy rich, intitial war rank set to 10. I do not think this alters gold income Map was crucible.
I measured using a separate timer and just reading the values off so there's uncertainty here. It shouldn't be anywhere near the level of confusing cumulative with non cumulative, especially since the time interval is 60 seconds. Obviously more data points would improve things, but I don't think they're necessary here.
Base:
Time - 60 seconds
Initial Gold - 25100
Final Gold - 25368
Rate: ~4.5 gold/sec
Expected Cumulative Rate: 4 gold/sec
Expected Non-cumulative Rate: 4 gold/sec
Currency 1:
Time - 60 seconds
Initial Gold - 23720
Final Gold - 24252
Rate: ~ 8.9 gold/sec
Expected Cumulative Rate: 8 gold/sec
Expected Non-cumulative Rate: 8 gold/sec
Currency 2:
Time - 60 seconds
Initial Gold - 21448
Final Gold - 22118
Rate: ~11 gold/sec
Expected Cumulative Rate: 14 gold/sec
Expected Non-cumulative Rate: 10 gold/sec
Currency 3:
Time - 60 seconds
Initial Gold - 18274
Final Gold - 19066
Rate: ~13 gold/sec
Expected Cumulative Rate: 22 gold/sec
Expected Non-cumulative Rate: 12 gold/sec
Most definitely not cumulative. That kind of sucks :/.
Knowing this, it takes 7.5/(# of players on your team) minutes for currency 1 to pay off, 25/(# of players on your team) minutes for currency 2 to pay off, and 35/(# of players on your team) minutes for currency 3 to pay off. Yuck. Currency 1 is always worth it and should quite probably be the first citadel upgrade bought. Currency 2 and 3 are no where near worth it in 1v1 or if you have ai allies. In 2v2, currency 2 will probably pay off if you get it early enough(games can last a good 40 minutes and it takes twelve and a half to pay off), but currency 3 is useless(17 minutes to pay off and you only get it late anyways). In 3v3, currency 2 will probably pay itself off towards the end game but currency 3 still seems like it'll take too long. 4v4 is getting to the point where 3 might be viable if you can get it fast enough. They're all no brainers in 5v5.