Putnam 2023 A6

Alice and Bob play a game in which they take turns choosing integers from $1$ to $n$. Before any integers are chosen, Bob selects a goal of "odd" or "even". On the first turn, Alice chooses one of the $n$ integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the $n$th turn, which is forced and ends the game. Bob wins if the parity of $\{k\colon \text{the number }{k}\text{ was chosen on the }{k}\text{th turn}\}$ matches his goal. For which values of $n$ does Bob have a winning strategy?