Suppose we have an $n \times n$ board and we have $n^2 - 1$ square tiles. These tiles consist of a 8 vertices, two on each edge, and every vertex is connected to precisely one other vertex. These tiles connect together to make paths.

At the start of the game there are 8 pieces belonging to 8 players which begin at random points on the edge of the board and players take turns placing tiles in any orientation they desire. The pieces then move along the newly created paths to the end and players must place tiles so that their piece moves. Pieces are removed if they collide with one another or move off the edge of the board. ( http://en.wikipedia.org/wiki/Tsuro )

What is the probability that, with a randomly generated tileset, all pieces can survive 'til the end. I.e. that they're all clustered around the empty square? All players can see the entire tileset so they can play "perfectly". The original game is for a specific tile set and a $6 \times 6$ grid but I'm more interested in the general game.