I've never met an infinite decision tree in my life so far, and I doubt I ever will. It is a property of problems with an infinite solution space that they can't be solved optimally, and it doesn't reveal any decision theoretic inconsistencies that could come up in real life.
Consider this game with a tree structure: You pick an arbitrary natural number, and then, your opponent does as well. The player who chose the highest number wins. Clearly, you cannot win this game, as no matter which number you pick, the opponent can simply add one to that number. This also works with picking a positive rational number that's closest to 1 - your opponent here adds one to the denominator and the numerator, and wins.
The idea to use a busy beaver function is good, and if you can utilize the entire universe to encode the states of the busy beaver with the largest number of states possible (and a long enough tape), then that constitutes the optimal solution, but that only takes us further out into the realm of fiction.
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I admire the way this post introduces an ingenious problem and an ingenious answer.
Nonsense. The problem has posed has always been around, and the solution is just to avoid repeating the same state twice, because that results in a draw.