Cartesian Chase

[Taken from Thinking Mathematically, p. 162. Modified by Sam Rebelsky]

Cartesian chase is a two-player game which is played on a rectangular grid with a a fixed number of rows and columns. Play begins in the bottom left hand square where the first player puts a mark. On each turn, a player may put a mark into a square that is

• directly above, or
• directly to the right of, or
• diagonally above and to the right of the last mark made by the previous player. Play continues in this fashion, and the winner is the player who puts a her mark in the upper right hand corner first. Find a way to win which even the computer could understand and use. You do not have to write this in algorithmic form, but it would help if you did.

Can you come up with a strategy for three-player game of cartesian chase?