Maybe Alchian and Demsetz would not be surprised:
A team of leading British and American scholars asked 108 sibling pairs in Colorado exactly what they fought about. Parental affection was ranked dead last. Just 9% of the kids said it was to blame for the arguments of competition.
The more common reason the kids were fighting was the same one that was the ruin of Regan and Goneril; sharing the castle's toys. Almost 80% of the older children, and 75% of the younger kids, all said sharing physical possessions — or claiming them as their own — caused the most fights.
Nothing else came close. Although 39% of the younger kids did complain that their fights were about…fights. They claimed, basically, that they started fights to stop their older siblings from hitting them.
I should add that I don't think the cited research settles the matter. Children might fight over toys as credible signals of parental affection, caring more about the signal than about the toy per se.