A cooked piece with initial score $a$ is worth $a-t$ after $t$ pickups. A raw piece keeps its initial score.

Taking the largest current score can fail. Consider raw scores $6,6$ and a cooked score $5$.

If Alice takes a raw piece, Bob takes the other raw piece. Alice receives the cooked piece worth $3$, giving a difference of $3$.

Instead, Alice can take the cooked piece immediately. The raw pieces then cancel, giving a difference of $5$.

We need to account for future turns. We will replace changing scores with fixed weights, then simplify the resulting game without simulating every possible play.