Dear all,

This email to comment some old discussions we had in this mailing list
around phantom-go.
I'm interested in this as I've spent time trying to extend MCTS towards
partially observable games,
and succeeded only for restricted sets of problems.

It was said that phantom-rengo (japanese rules) is a good candidate for
being undecidable, as there's a known
undecidability result (by Bob Hearn) stating that a game with
- finite state
- two teams (with no communication)
- unbounded horizon
can be undecidable. The undecidable question is
"is there a strategy for winning with probability 1 in situation X".
and an instance is a position X.

However, answering the question above is useless for playing well in a
partially observable game.
Most partially observable game never or almost never have a strategy for
winning whatever the opponent
can do: at best, they have a good probability of winning. But with the
question above, you can not
distinguish between a position in which you have 3% of probability of
winning in case of perfect play,
and a position in which you have 97% of probability of winning in case of
perfect play.

A more usefull question is
"is move A better than move B in situation X ?"
or
"is situation X leading to a win with probability at least z% ?"

Or these questions are undecidable, as a consequence of the paper by Madani
et al on randomized
one player games (randomized one player games can simulate non-randomized
two player games).
This consequence is not written in their paper, but one can prove it
mathematically.

Therefore, my claim is that phantom-Go (and not only phantom-rengo) is a
good candidate for undecidability.
Maybe Kriegskiel also, if we remove the rule which limits the length of
games.

All comments welcome :-)

Best regards,
Olivier
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