Hundred rational tigers and a single sheep

An island has $100$ tigers and $1$ sheep. Tigers are perfectly rational: staying alive matters most, and eating a sheep beats eating grass. At any moment at most one tiger may eat one sheep, and a tiger that eats a sheep turns into a sheep itself. How many sheep remain on the island in the end?

$1$. Work up from small cases: one tiger eats freely; with two, whichever eats becomes a sheep and is eaten, so none eats. With three, eating leaves a safe two-tiger state, so the sheep is eaten. The pattern alternates, and with an even number such as $100$ no tiger eats, leaving the one sheep.