Game on Chessboard

There is a chessboard of size M \times N. $K$ fairy chess pieces called $(p,q)$-leapers (p < q) are placed on some squares of this board. The leaper's move is similar to a regular chess knight's move, with specific constraints. A $(p,q)$-leaper can move $p$ squares horizontally and $q$ squares vertically, or $q$ squares horizontally and $p$ squares vertically. In each case, the move of $q$ squares must be either strictly upward or strictly leftward (that is, in the direction of decreasing respective coordinate). Moves that would take a piece off the board are not allowed, but several leapers may occupy the same square.

Two players play the game, alternating moves. On each turn, a player must choose one leaper and make a permitted move. The player with no possible move loses the game. Assuming both play optimally, determine who will win.