Yeah, but if you go the right three it will take you FOREVER to get the fourth policy without some other culture steroid. It's the biggest weakness of the tree.
I tend to go with the free worker followed by the two on the left, unless my resources are seafood. (Then I save the worker.)
The amphitheater sounds pretty early, much earlier than I would build it. Maybe try to specialize cities more, particularly with specialists? I usually skip science buildings in some and military buildings in others. I've recently noticed that libraries are really weak unless you use that specialist, so if you don't work him till your pop is bigger I wouldn't research writing till I'm in classical. With Carthage I like to rush the Great Cothon and often try for Colossus. The Great Cothon is all the "late" game Carthage ever gets, so I would try to go into those late eras competitive in population or at least city number.
I fit in the amphitheater if I can, only because I've been so weirdly culture-shy. In my current game, I've done some specialization -- basically a military city -- and it worked out well. Holding off on Writing in most cases may be a good idea, and I've already been doing that to a degree. I already hold off building them until I can put a specialist in them.
City number is my biggest issue, second only to that strange happiness plunge. I started a new game with Indonesia, and found myself alone on a small continent with only 4 iron. I built 5 cities, and prepared for war with China, who built two island cities behind me. That naval war was fast and easy, putting me at 7 cities. later came a longer one against China, going for 2 major cities on her continent -- again a naval battle, with the Kris playing defense against her counterattacks. This ended in my taking two major cities (7 iron), and making her my vassal.
At this point (close to 200 turns), I had 9 cities, crazy happiness, the lead in science, and Friendship with Assyria (strong) and India (weak). This was with Progress and Piety (heading toward Rationalism and Freedom). I also got a religion: Festivals/Transcendence (pre-Classical, Mastery, Inspiration, and Evangelism (mistake). I was surrounded by religions (everyone but India!), so I focused on what would help my cities.
Unfortunately, the Aztecs were about 20 turns from conquering their entire continent: 3 civs! So what to do about the ultimate runaway? Assyria suggested a war earlier, but I was still busy with China. By the time China capitulated, Assyria had denounced me for religious reasons, and war was brewing. I struck first, taking a strategic island near my coast, and then settled into a brutal 50-turn naval war, where my tech and promotions incrementally ground down his numbers. So how was Indonesia doing , halfway through Rationalism?
Bottoming out at -50 unhappiness.
Part of this has to do with the reduced number of civs, but I lost a net 80 or so points. I was sick of the game when I began to get a handle on the unhappiness, reducing it to -30 or so, as the war improved. Finally I made peace with Assyria on favorable terms, exchanged a bunch of luxuries, and found myself still #1 in science on turn 287.
Unfortunately, the Aztecs chose what I did with them -- Authority/Statecraft/Rationalism -- and are aiming for a diplo victory if domination doesn't come first. I'm not sure whether they can be stopped. It makes me think fondly of my usual control of the WC... and yet I know Piety is the main difference in wy I'm leading in Science.
I think the safe thing to say is that Progress isn't easy on Immortal. (To be continued.)