God and the Devil decide to play a game. God will start by picking an infinite sequence of moves of the form "left", "right", "up", and "down". The Devil responds by creating a finite maze with an exit and by placing God somewhere inside. God then follows His pre-selected sequence to traverse the maze. Unmakable moves are ignored; for example, if the next move is "left" and there is a wall to the left of the current square, God goes on to the next move in the sequence without moving.

If God escapes the maze in finite time, He wins. Otherwise, the Devil wins.

Assuming both agents act optimally, who will win?

(assume that the maze is formed by deleting some edges from a rectangular grid, and that it has no isolated regions; i.e., it is always possible to get to the exit from any point inside the maze)

The reason randomness is to be rejected as a strategy is that the outcome cannot be determined with absolute certainty.  For example, if God's random sequence turned out to be a sequence of all "downs", the Devil would win.  It is incorrect to say that it is impossible for God to pick such a sequence.  The best one can say is that the probability that God will pick such a sequence is zero.  These two ("impossible" vs. "with probability 0") are NOT the same thing.

Posted by David Shin on 2005-02-09 00:33:24