At the start of round $r$, every remaining deadline is at least $r$. Alice eats one piece, Bob eats one piece, and then all remaining pieces with deadline $r$ disappear.

Describe a position by its sorted remaining deadlines. We can now reason about complete rounds instead of individual turns.

An empty position loses for Alice. A position with exactly one piece wins for Alice: she eats it, leaving Bob without a move.

Our first observation is that every position with an even number of pieces loses for Alice. Bob can maintain that parity from one round to the next.