100 Rational Tigers and 1 Sheep Backward Induction

Scenario

There are 100 perfectly rational, identical tigers and 1 defenseless sheep on an isolated island with abundant grass. A tiger can eat the sheep at any time, but upon eating the sheep, the tiger magically falls asleep and transforms into a sheep, making it vulnerable to being eaten by the remaining tigers. Tigers share common knowledge of rationality and strictly prioritize:

1. Survival: Avoid being eaten under all circumstances.
2. Hunger: Prefer eating mutton over grass.

Your Task

1. Using backward induction, analyze the game for $N = 1, 2, 3, 4$ tigers.
2. Establish the induction invariant characterizing the fate of the sheep as a function of the parity of $N$.
3. Conclude whether the first tiger will eat the sheep when $N = 100$.

Constraints

• Simultaneous opportunity; tigers are identical rational utility maximizers.

Examples With 1 tiger, it eats the sheep immediately. With 2 tigers, neither eats because eating leads to the 1-tiger state where the newly formed sheep is eaten.