One model I have is that when things are exponentials (or S-curves), it's pretty hard to tell when you're about to leave the "early" game, because exponentials look the same when scaled. If every year has 2x as much activity as the previous year, then every year feels like the one that was the big transition.

For example, it's easy to think that AI has "gone mainstream" now. Which is true according to some order of magnitude. But even though a lot of politicians are talking about AI stuff more often, it's nowhere near the top of the list for most of them. It's more like just one more special interest to sometimes give lip service too, nowhere near issues like US polarization, China, healthcare and climate change.

Of course, AI isn't necessarily well-modelled by an S-curve. Depending on what you're measuring, it could be non-monotonic (with winters and summers). It could also be a hyperbola. And if we all dropped dead in the same minute from nanobots, then there wouldn't really be a mid- or end-game at all. But I currently hold a decent amount of humility around ideas like "we're in midgame now".

I think games sometimes go through something like a phase transition, where strategy heuristics that serve you well on one side of the border abruptly stop working.  I think this is typically because you have multiple priorities whose value changes depending on the circumstances, and the phase transitions are where the values of two priorities cross over; it used to be that X was more important than Y, but now Y is more important than X, and so heuristics along the lines of "favor X over Y" stop working.

I don't think that these phase transitions can be generalized to anything as useful as the concepts of solid/liquid/gas--or at least, I'm not aware of any powerful generalizations like that.  I don't have a set of heuristics that I deploy "in the mid-game" of most or all games.  Nor do I think that most games have exactly 3 phases (or exactly N phases, for any N).  I think of phrases like early/mid/late-game as meaning "the phase that this particular game is usually in at time X".

I do think you can make a general observation that some investments take a while to pay for themselves, and so are worth doing if-and-only-if you have enough time to reap those benefits, and that this leads to a common phase transition from "building up" to "scoring points" in many engine-building games.  But I think this particular observation applies to only one genre of game, and explains only a minority of the use of phrases like "early game" and "late game".

As an example of an unusually sharp phase transition:  In Backgammon, if you land on a single enemy piece, it sends that piece back to the start.  This is a big deal, so for most of the game, players spend a lot of effort trying to "hit" enemy pieces and defend their own pieces.  But players have a fixed number of pieces and they can only move forward, so there comes a point where all your pieces are past all of my pieces, and it's no longer possible for them to interact.  At that point, attack and defense become irrelevant, and the game is just about speed.

I once read about the development of Backgammon AI using early neural nets (I think this was in the 70s and 80s, so the nets were rather weak by today's standards).  They found the strategy changed so much at this point that it was easier to train two completely separate neural nets to play the two phases of the game, rather than training a single net to understand both.  (Actually 3 separate nets, with the third being for "bearing off", the final step of moving your pieces to the exact end point.  I think modern Backgammon AIs usually use a look-up table for bearing off, though.)

(Training multiple neural nets then caused some issues with bad moves right around the phase boundary, which they addressed by fuzzing the results of multiple nets when close to the transition.)

I don't think this story about Backgammon reveals anything about how to play Chess, or StarCraft, or Civilization.  Most games have phase transitions, but most games don't have the particular phase transition from conflict-dominant to conflict-irrelevant.

Another example:  I once told someone that, in a certain strategy game, 1 unit of production is much more valuable than 1 unit of food, science, or money, "at least in the early game."  The reason for that caveat was that you can use money to hurry production, and by default this is pretty inefficient, but it's possible to collect a bunch of stacking bonuses that make it so efficient that it becomes better to focus on money instead of regular production.  But it takes time to collect those bonuses, so I know you don't have them in the early game, so this heuristic will hold for at least a while (and might hold for approximately the whole game, depending on whether you collect those bonuses).

Again, I don't think this teaches us anything about "early game" in a way that generalizes across games.  Probably there are lots of games that have a transition from "X is the most important resource" to "Y is the most important resource", but those transitions happen at many different points for lots of different reasons, and it's hard to make a useful heuristic so general that it applies to most or all of them.

A third example:  The game of Nim has the interesting property that when you invert the win condition, the optimal strategy remains precisely identical until you reach a specific point.  You change only one move in the entire game:  Specifically, the move that leaves no piles larger than size 1 (which is the last meaningful decision either player makes).  You can think of this as a phase transition, as well (between "at least one large pile" and "only small piles").  And again, I'm not aware of any useful way of generalizing it to other games.

I think the most natural definitions are that early game is the part you have memorized, end game is where you can compute to the end (still doing pruning), and mid game is the rest.
So eg in Scrabble the end game is where there are no tiles or few enough tiles in the bag that you can think through all (relevant) combinations of bags.

I think perhaps the phases of a 4X game.

Explore: gain information that is relevant for what plan to execute 

Expand: Investment phase, you take actions that maximise your growth

Exploit: You slowly start depriotizing growth as the time remaining grows shorter.

Exterminate: You go for your win condition

what are some situations in real life, other than "AI takeoff", where the early/mid/late game metaphor seems useful?

I suspect it's easy to find games or situations that have nice-ish three phase maps, for example:

• Choosing a particular chess move: (I) assess the board, (II) generate some candidates moves, (III) find the best move of the candidates
• Getting new work as a contractor: (I) get a rough idea of what the potential client wants, (II) create a detailed specification and work plan, (III) finalize financial, ownership, termination, etc details in a contract
• Discovering a mathematical proof: (I) gather some foundational knowledge relevant to the problem/proposition, (II) search for ways to connect knowledge (and tricks) into a proof, (III) having found a good candidate strategy, try to develop a proof

In strategy board games, there's an explicit shift from "early game, it's worth spending the effort to build a longterm engine. At some point, you want to start spending your resources on victory points." And a lens I'm thinking through is "how long does it keep making sense to invest in infrastructure, and what else might one do?"

Maybe my examples miss the spirit of the early/mid/late game map.  Do you agree that in strategy games the three phases are roughly: (I) develop engine, (II) use engine to create relative advantage, (III) try to win.

Like, in Chess you start off with a state where many pieces can't move in the early game, in the middle game many pieces are in play moving around and trading, then in the end game it's only a few pieces, you know what the goal is, roughly how things will play out.

IMO one of the most important distinctions between middle and late game in chess is that the number of okay to good moves on any given turn is severely reduced in the late game compared to the middle game.