You should accept whatever is offered, obviously, but humans have this instinct to make sure a split is "fair," even when accepting significantly less would actually benefit themselves. It's irrational, but it probably evolved from the ancient tribal days when resources were scarce and overt inequality would cause an intramural fissure.
True. It would depend on how much sibling rivalry existed. If the rivalry was heated enough, then the fact Anne was given the power to make the call would require that she give Bob more than 50% or he would likely reject it (If he was in a bad mood for not being given the power to divide it). I think the majority of siblings would reject less than half because of the competition and the fact that the scenario begins with an unequal power balance. If it was a large sum of money, then you could probably give Bob less than half and he would accept it, but he would probably be snarky and mean to Anne for the rest of their lives. One common problem for people is the price they are willing to pay personally to express their anger or feelings of personal injustice.
The first man to raise a fist is the man who's run out of ideas. H.G. WELLS
The first principle is that you must not fool yourself, and you are the easiest person to fool. FEYNMAN If this is monkey pee, you're on your own.SCULLY