The group's earlier discussion about double moves dealt mostly with impartial games. What about partizan games, like (Blue-Red) Hackenbush? I conjecture that a Hackenbush position P that under single-move play has value v still has value v under double-move play but in a weakened sense. For instance, if P is a beanstalk of value 1/2 and Q is a beanstalk of value -1 (under single-move play), then I conjecture that under double-move play mP+nQ is m,n-eventually a win for Left if m/n converges to any value > 2 and is m,n-eventually a win for RIght if m/n converges to any value < 2. Cincotti's theory of multi-player Hackenbush might be pertinent here (since double-move 2-player Hackenbush has similarities to 4-player Hackenbush), but I've never understood Cincotti's theory, and to the extent that I've been able to glean what he's up to, it seems to me that he skirts issues like this. Jim Propp