Will rational tigers eat the lone sheep

An island with plenty of grass holds $100$ tigers and $1$ sheep. Tigers can live on grass but prefer sheep. Only one tiger can eat a sheep at a time, and a tiger that eats a sheep turns into a sheep itself. Every tiger is perfectly rational and wants above all to survive. Does the sheep get eaten?

No. Work up from small cases: with $1$ tiger the sheep is eaten; with $2$, a tiger that eats becomes prey for the other, so none eats; with $3$, eating leaves a safe 2-tiger situation, so one eats. By induction the sheep is eaten exactly when the number of tigers is odd, and $100$ is even.