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        Arimaa >> General Discussion >> the perfect game revisited
        
(Message started by: seanick on Aug 7^th, 2006, 11:51am)

Title: the perfect game revisited
Post by seanick on Aug 7^th, 2006, 11:51am

I know there have been discussions of perfect games. but what if two humans played, at postal speeds, where either player could roll the game backto a certain point and make a different move. there would be a repetition rule, but that would be it- otherwise you could roll back to any move, say move 5, because on move 10 your horse was trapped due to a mistake on move 5. so theoretically you could block any offense by the other player, by simply rolling the game back to before that point to make a move to block it.

the question is, would that game ever end, if the only rollbacks were to fix mistakes? If the game did end, it would mean either the players got bored at about move 4096 or so, or that there is such a thing as a perfect game, given a set of particular strategies.

could there be any value in such research?

oh and btw: this is partly in consideration of the plan window. if any of you arimaa gods/authors want a subject for the wikibook that would help some, (or at least me), immensely, a tutorial of how to use the plan window to ones advantage would be very useful... :)

NICK

Title: Re: the perfect game revisited
Post by Fritzlein on Aug 7^th, 2006, 1:05pm

My own conviction is that, with perfect play on both sides, Gold has a forced win based on the initiative of the first move. If I am right, it raises the further question of whether the repetition rule is necessary to enforce that win, and I suspect it is. More precisely, I believe that although Gold has a forced win from the opening position, Arimaa is so finely balanced at the start that the defensive option of being able to undo a move might be enough to turn that forced win into an infinite-move draw.

In practice, it seems to me that most positions could not be drawn by the side at disadvantage, even if they were allowed to repeat positions forever. However, the opening position is usually the most drawish position of the game. It's quite rare for a game to get more drawish as it progresses. Therefore, if it is true that the repeat rule is necessary to tip the game away from being a theoretical draw, then I suspect removing HD from each army in the opening setup would remove enough drawishness to restore the forced win for Gold, even in the absence of banning repetition.

Some have suggested that the advantage of setting up pieces second (responsively) could compensate Silver for having the second move. I understand that is a theoretical possibility, but I don't believe it is true based on my experience.

Fortunately, even if Arimaa is a forced win for Gold, the Gold advantage might be equivalent to only, say, 10 rating points of skill. In other words, even if you only know a tiny bit more about Arimaa strategy than the other guy, that makes you a favorite to win even with Silver.

The more I play Arimaa, the more I am convinced it is as deep as chess or deeper. As more people join the Arimaa playing pool, and as we learn more about the game, the ratings will spread further and further apart. Eventually more levels of skill (i.e. a wider gap in rating points) will separate the Arimaa World Champion from an Arimaa beginner than currently separate the chess World Champion from a chess beginner. That's my prediction, anyway. ;)