The level doesnt really matter, my moves arent affected by the difficulty. That was on Emperor, because for illustration purposes, I didnt want to have to worry about defense too much. Theres a scout on the map someplace.
Thanks for the numbers, but we need the costs of 2, 3, and 4 cities to make a better judgement.
I still think for the absolute lowest culture costs, its best to time representation with the founding of city 2. It will also give a GA for another small culture boost. I'll do more testing later.
Edit: Saying your moves aren't affected by difficulty is just weird. You can't build the Great Library on Deity. You can't afford to build the monument before the scout. You can't afford to not have that +50% city ranged attack strength when the early DoW comes at you. I dunno.
Also, I found my second and third cities at almost exactly the same time. The first settler comes out and I buy one at roughly that same time.
Here are the costs for all the various combos:
Two cities after Representation, then open and finish Tradition:
25,30,60,115,185,280,390,525,680
Three cities after Representation, then open and finish Tradition:
25,30,60,125,205,305,425,570,740
Three cities after the Tradition opener and Representation, then finish Tradition:
25,30,60,105,205,305,425,570,740
2 cities after Monarchy, finishing Tradition then Liberty to Representation:
25,30,60,120,195,290,410,550,710
3 cities after Monarchy, finishing Tradition then Liberty to Representation:
25,30,60,135,220,330,465,620,800
In all cases you end up with Representation and all of Tradition finished. Here are the total culture costs:
1 city: 2090 culture
2 cities after beelining Representation: 2290
3 cities after beelining Representation: 2485
3 cities after Tradition opener then Representation: 2465
2 cities after Monarchy, finishing tradition then Liberty to Representation: 2390
3 cities after Monarchy, finishing tradition then Liberty to Representation: 2685
Ignoring the OCC case for now, it's an extra 200 culture to finish tradition before opening liberty with 3 cities, and an extra 100 culture to finish tradition first with 2 cities. The case of going with the tradition opener then Representation, then finishing tradition is a tiny bit cheaper.
The thing is, and I think this is important... Getting Legalism earlier is a bunch of extra culture from the two free monuments. The free part is also important. Getting Monarchy earlier is more happiness to grow. Getting Landed Elite earlier is more growth. Getting free aqueducts earlier is more growth. More growth = more artist slots, more tech, more defense, more money and more production, plus it's easier to build wonders. Just the free monuments alone is 4 culture/turn until you have time to build them. Even if you built them right away, that's 40 culture. If you waited at all, it's more like 80-100 culture lost. I think the free Golden Age offsets this a bit, but that comes before you have significant culture generation. It's probably better to wait until later to use it.
I still maintain that even though it might take a few extra turns to complete these 9 policies by beelining through Tradition, those aqueducts are huge. They essentially amount to -40% on your growth times, which is the same as +66% food. (6/10 time to grow inverted is 10/6 growth)
The impact of getting those aqueducts 20-30 turns earlier... I think it outweighs the loss. However, that's probably the most debatable point. Monarchy is huge. Landed Elite is huge. Finishing Tradition before opening Representation is another matter though. Aqueducts 20-30 turns earlier means probably 2 or 3 extra pop in each city. Most importantly, you can plant artists sooner without impeding growth. (4 excess food/turn with aqueducts = 6 excess food/turn without aqueducts)
It will delay that 9th policy by a few turns at least. However, the extra tech from the extra growth means reaching the Industrial Era earlier, which means getting the Freedom bonuses earlier. This to me is the clincher. Extra production/tech/money/growth/defense = survival on Deity.
However, I am tempted to try it the other way now, just to see how it feels.