I guess in the version with the infinitely many well-ordered fish in the bucket, the move is to decrease the number of fish in one bucket by any amount, and then add any number of fish to the buckets to the left? I think to terminate, we still need the restriction that the number of non-empty buckets at any stage of the game is finite.

I think the winning strategy for this game is to make sure that after your move, the first two buckets (0 and 1) have the same number of fish, same for buckets 2 and 3, 4 and 5, etc.

]]>I’m not sure this game will halt, though.

]]>Yesterday in our discussions at the Graduate Center we talked about allowing infinitely many fish in the buckets, but well-ordered, so that it is like transfinite Nim: http://jdh.hamkins.org/transfinite-nim/.

]]>1. At the beginning, there total number of fish in all the buckets is finite.

2. In each step, you pick a non-empty bucket, remove one fish, take a finite number of fish from the boat, and distribute them to the buckets to the left (ie buckets indexed by ordinals smaller than the one corresponding to the current bucket).

Since the total number of non-empty buckets is finite, the strategy “keep all buckets even” works, and the proof is similar (there’s always the unique non-empty bucket with the largest index).

Termination:

Start with all buckets “uncovered”.

Pick the non-empty uncovered bucket with the largest index. If such a bucket doesn’t exist: stop. As the game progresses, the number of fish in that bucket can only decrease. After some finite number of turns (maybe 0) the number of fish in that bucket will have changed for the last time ever. When this happens, cover the bucket and repeat the process described in this paragraph.

Each iteration of the previous paragraph takes a finite number of turns. The indexes of the buckets considered form a decreasing sequence of ordinals, so the number of iterations must be finite as well. After all these iterations, all non-covered buckets are empty. Also, each covered bucket was covered after the number of fish in it changed for the last time. So there are no moves available, so all buckets must be empty.

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