If they want to they will. If there is something in it for them they will.

Favoring any child to the extreme will definitely cause problems. Mainly narcs love to triagulate children. Now you are always vying for their attention and it makes them feel so very special.

Duke it out though sounds like she just didn't care. In those cases I would think of that almost as her probably not wanting do deal with either of you. If she did she would have given it to the favorite. Tossing something at too kids and letting them deal with from my perspective is being completely indifferent, not wanting to bother. If you pass something off to other to 'deal with' are you caring about it or are you not caring about it? I would say this was not caring which narc are always, only they seem to care when they are working for supply. When they don't need it, you are tossed to the wind until they do. Telling you to duke it out means she had her supply and therefore enough of both of you and wanted you both our of her hair.