What if the problem were entirely symmetric, and we both had exactly the same preferences. One of us has to stay, one has to go. Is it blackmail if I go first, and thus force you to stay?

Well, If I have a chance to react to that before your decision is final, I could say that if you do that then I'll go anyway (Thus leaving mother to burn) Then it seems to resolve to Chicken. (We both want the other person to back down first, so that we can go all the way to Olympos, but if neither backs down, mother burns in desert heat.)

And if I don't have a chance to react to that before your decision is final, then it's sort of like "What do you do if you are playing Chicken against someone who has already smashed his steering column with a rock?"

And I think that makes sense, since in both cases (blackmail, chicken) the goal is make the person back down because of some sort of threat. Does that help?