The Pascal's triangle game

Multiple tokens are placed somewhere on Pascal's triangle.  Two players take turns.

A move is select a token and slide it a token a single step up and to the left, or alternatively up and to the right, (exactly the reverse of the Pascal triangle recursion).

A token is out of play when it reaches a "1" in the triangle (ie the left boundary, the right boundary, or the apex of the triangle).

In normal play, this is simple to work out.  Every position in the triangle is a *0, *1, or *2 and the pattern isn't hard to establish.

What about misere play?

Our existing code doesn't really handle this well because it's not an octal game.  We can probably coax out some genera easily enough though.