Rational lions and a piece of meat

A cage holds $n$ perfectly rational lions and one piece of meat. A lion that eats meat falls asleep and thereby becomes meat that another lion can eat. Every lion would like to eat, but above all wants not to be eaten. What happens, depending on $n$?

The meat is eaten if $n$ is odd and left alone if $n$ is even. One lion eats freely. With two, the eater would be eaten, so neither moves. With $n$ lions, eating leaves $n-1$ lions and fresh meat, which is safe exactly when $n-1$ is even; induction gives the parity rule.