If by age, you mean “time in rank” you would be correct.
If there are only privates left, I imagine those guys would work it out, “Lord of the Flies” style. There have been plenty of instances in small unit combat where the privates had to fend for themselves. They either worked it out…or they died.
Privates also have a date when they made their rank, there is no Lord of the Flies chaos over rank.