I think it has to be one or the other. The path would look something like this:
Math > Do AI have Alpha? > Yes > Currency, try to trade for Alpha and backfill
> No > TW > Pottery > Currency
Alpha > back fill with trades > Currency > wait for Math trade
It's difficult to make a decision when missing so many relevant pieces of information. If we knew or could accurately predict what the AI might tech and when then this would be pretty simple. If the AI are going to be very slow to Alpha i think that needs to be the choice. If they get it at a reasonable time and we can pick it up in a trade then Math is clearly the way to go.
The best i can do is to offer a rough guess based on my experience with Immortal AI and the information that is available in our game.
-1000 BC is typically a reasonable
average date to trade for Alpha in an Imm Pangaea game with 6 potential trading partners. On standard continents with 2-3 potential partners this can take longer and self-teching Alpha is quite a bit stronger and more common.
-This isn't a standard continents game. There are 9 AI in this game and we could very well have a 4th somewhere on our continent. Perhaps unlikely at this point, but still possible.
-How are these AI compared to normal AI in terms of the start they were given by the map maker? Do they have abnormal advantages that will speed them up? We were given a worker. What was given to the AI? We were given one relevant piece of information here:
The overall difficulty level is immortal, but some AIs have slightly altered starts and so do you: We added a worker to your starting units to allow a quick start.
What does that mean? I don't know. It doesn't really seem like Willem's start was altered aside from possibly a great deal of food. Perhaps the ladies got strong starts? I don't know. I don't think that's enough to conclude that these AI will be faster than normal though.
-There are two typically competent techers here that will also tech Alpha. Shaka probably is of no value for this. Overall i would rate our chances slightly lower than on a typical Pangaea map so you can probably push that 1000 BC date back a little bit to like 750 BC. I think that's turn 85? Of course somebody might go for it early.. you just never know. Anyway we are 54 turns in so
maybe another 30 turns before we can reasonably expect to trade for Alpha.
-How does that compare to Math being available for trade? I think in this situation we can probably expect Math to be available at a similar date. Willem + another AI need to have it first so while it is a cheaper tech that the AI tend to prioritize more than Alpha, there are more restrictions on it.
-So is it worse to wait 30T on Alpha or 30T on Math? What will we be doing in those 30T? How will the lack of one or the other impact us during that time?
Potential timetable with Math first
-Current base BPT is 40 x 1.2 for pre-req bonus on math = 48 beakers. Math costs 405 which works out to about 8.5 turns. Our BPT should increase over the next 10T though so i would expect to be able to tech Math in 8T of 100% research... maybe less. I wouldn't expect to get it down to 7T, but the last turn might be doable at less than full research. It currently costs 88g to finance 8T of research, but we will settle one more city in 5-6T which will increase our expenses a good bit. We will need over 100g i think. We have 18g right now and can make 28 per turn so we
probably need 3T to collect gold and Math will finish 11T from now.
-After Math in 11T we will have to decide on TW and Pottery or waiting for Alpha. Our only clue will be watching Willem's research unless somebody actually has Alpha at that point. Our BPT should be a lot higher in 11T with our island city settled + deer city + maybe gems + library in Palace too. I think as much as 70 BPT is reasonable which is an increase of 75% from where we are now. We could get TW in 2T with big OF and then 1T Pottery. We would probably need 1T to save money.
We can expect TW 14T from now and Pottery 15T from now if we self-tech.
-Next up would be Currency which i think we could tech in as little as 7T by then. We would probably need a few turns to collect gold first so Currency in maybe 10T after Pottery would be reasonable. That would be
Currency roughly 25T from now. This should make Currency available to us right around the time we can reasonably expect to trade for Alpha. We shouldn't mind giving an AI like Willem Currency since that's more foreign trade routes for us.
Potential timetable with Alpha first
-More expensive than Math so about 10T of research vs 8T. It could potentially also require 4T of banking gold rather than 3T.
Alpha in 13-14T is reasonable.
-We could likely trade for TW on the turn we get Alpha and Pottery the next so
TW in 13-14T and Pottery in 14-15T. Also probable that we could get IW here.
-Currency should require about the same time to research as above, but here we would get started on it maybe 1T earlier so
Currency in maybe 24T turns vs 25T above.
Opinions based on that?
We can expect to get Pottery around the same time in both scenarios, and i think it's basically coming right about when we need it. The lower beaker cost of Math nearly gives us Math + TW for the same cost of Alpha. We have to research Pottery but it's really cheap here with 2 prerequisite techs. Math first also gives us the flexibility to tech TW and/or Pottery sooner if we wish.. at the cost of delaying Math a few turns.
IMO many advantages of Alpha faded away as i did this timeline. It basically gives us earlier IW and like 1T of research saved. Maybe seeing what techs the AI have is useful, but probably not. I think getting an extra 100+ chop hammers over the next ~30 turns is more important.
Now if we get unlucky and go beyond 500 BC and the AI still don't have Alpha for trade then going for Math will probably be a bad decision, but i think we should be willing to risk that.