How are you going to make a cottage without a worker??? The point is that you get the worker out several turns early, and the same for the settler, meaning you get more commerce or whatever you set the worker to be building faster.
That said, it's generally a poor strategy to build cottages first - you need the more pop more than you need the commerce (again, the relative return of 1 commerce < relative return of one food/hammer in most cases early on, because of the 8c the palace gives you). I don't want to go to the extent of proving that, except to say that in my empirical testing - and I've played both ways - I do much, much better if I don't start cottage-first.
The 11 vs 10 number is simple:
Size 2 city, working the average 2f/1h/1c tile unimproved (actually hard to find that specific tile, but it's a good average), yields: 8c(palace) + 1c (city tile) + 2c (worked tiles) for 11c. Slave one of them away and you get 8+1+1 or 10. Relative cost, 9% of your commerce.
I think you're taking this argument out way too late in the game. Neither I nor anyone else will argue that there is a considerable tradeoff food/hammer vs commerce later in the game (even 100 turns into the game), and it is not necessarily better one versus the other. I think generally 100 turns into the game the tradeoff is to the direction of (more food/hammers), but not 100%; and once you get to the Medieval or even Ren period, I won't slave except in slave cities (tons of food not much else, not even good commerce - ie the 3 sugars 2 banana type city; you can use the 2f/Xc grassland all you want, you will always have tons of extra food with no other purpose) and in tiny new cities (like the OP's example) where, again, the f/h v c tradeoff is not much of one since the c being provided are minimal.
However, in the first sixty to one hundred turns, I think it would be very hard to argue that a slaved economy is inferior to a non-slaved one - even by the measure of commerce. Cities and improvements out faster is the whole of the point - if you can get more cities out, you can get more commerce, plain and simple; at that early stage in the game, f/h -> c directly, because 3 cities will always give more commerce than 2 (due to the happy cap). I'm not trading off any commerce - I'm at most trading commerce-now for commerce-in-a-few-turns. The new-city bonus (+2f1h1c) alone gives a significant offset to that, not counting the fact that the new city is essentially an extension of C to the happy cap (where C is the happy cap), meaning more commerce is possible at higher growth.
True worker first is of course an option, but you lose the flexibility of exploring with your first warrior - which is generally a very bad thing when it comes to AI diplomacy, and knowing where to settle that first settler. You also still are losing the 22f->30h exchange, although the slightly earlier worker might offset that (though the extra few turns of commerce you're losing hurts your commerce argument
) depending on the tiles you are working/improving.
That said, it's generally a poor strategy to build cottages first - you need the more pop more than you need the commerce (again, the relative return of 1 commerce < relative return of one food/hammer in most cases early on, because of the 8c the palace gives you). I don't want to go to the extent of proving that, except to say that in my empirical testing - and I've played both ways - I do much, much better if I don't start cottage-first.
The 11 vs 10 number is simple:
Size 2 city, working the average 2f/1h/1c tile unimproved (actually hard to find that specific tile, but it's a good average), yields: 8c(palace) + 1c (city tile) + 2c (worked tiles) for 11c. Slave one of them away and you get 8+1+1 or 10. Relative cost, 9% of your commerce.
I think you're taking this argument out way too late in the game. Neither I nor anyone else will argue that there is a considerable tradeoff food/hammer vs commerce later in the game (even 100 turns into the game), and it is not necessarily better one versus the other. I think generally 100 turns into the game the tradeoff is to the direction of (more food/hammers), but not 100%; and once you get to the Medieval or even Ren period, I won't slave except in slave cities (tons of food not much else, not even good commerce - ie the 3 sugars 2 banana type city; you can use the 2f/Xc grassland all you want, you will always have tons of extra food with no other purpose) and in tiny new cities (like the OP's example) where, again, the f/h v c tradeoff is not much of one since the c being provided are minimal.
However, in the first sixty to one hundred turns, I think it would be very hard to argue that a slaved economy is inferior to a non-slaved one - even by the measure of commerce. Cities and improvements out faster is the whole of the point - if you can get more cities out, you can get more commerce, plain and simple; at that early stage in the game, f/h -> c directly, because 3 cities will always give more commerce than 2 (due to the happy cap). I'm not trading off any commerce - I'm at most trading commerce-now for commerce-in-a-few-turns. The new-city bonus (+2f1h1c) alone gives a significant offset to that, not counting the fact that the new city is essentially an extension of C to the happy cap (where C is the happy cap), meaning more commerce is possible at higher growth.
True worker first is of course an option, but you lose the flexibility of exploring with your first warrior - which is generally a very bad thing when it comes to AI diplomacy, and knowing where to settle that first settler. You also still are losing the 22f->30h exchange, although the slightly earlier worker might offset that (though the extra few turns of commerce you're losing hurts your commerce argument

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