[nimih]

With this variant, since there are a finite number of legal board positions in chess, and Human^n chess is simply chess with a finite number of moves prohibited at each board position (claiming is the same as resigning, in this formulation), there are a finite number of distinct Human^n chess, so the mapping Human^n-1 chess -> Human^n chess must eventually reach a cycle (potentially of length 1, i.e. a fixed point).

[ratorx]

I’m not sure about a straight-up cycle. I think it requires determinism, whereas the engine is not necessarily deterministic (eg. NN based ones like AlphaZero).

I think it’s possible that at least 1 position (and probably a lot more) will have more than 1 optimal solution. If the engine is not fully deterministic, then it’s possible instead of a normal cycle, there are a fixed set of strategies at each n that form a cycle, but no single sequence that repeats.

The other thing that would compound this is that no current chess engine solves the game fully. There would be even more positions that have multiple “optimal” solutions if the engine only looks ahead to bounded x.

[nimih]

That's a good point about non-determinism, although I wonder if there are known convergence/stability results in the ML literature that allow you to effectively ignore that detail (i.e. it seems plausible that you could get something like "human^n chess always stabilizes at some fixed amount of training time/computational power"). You can also just fix the randomization seed, but that's obviously a less satisfying result.

> The other thing that would compound this is that no current chess engine solves the game fully. There would be even more positions that have multiple “optimal” solutions if the engine only looks ahead to bounded x.

I'm not sure this is an obstacle; we're explicitly excluding the computer's preferred move, rather than the necessarily optimal move, after all. You could easily play human chess with the engine from (e.g.) Battle Chess, which honestly is sort of an interesting idea in its own right.