NMC 2021 Problem 3

Let $n$ be a positive integer. Alice and Bob play the following game. First, Alice picks $n + 1$ subsets $A_1,...,A_{n+1}$ of $\{1,... ,2^n\}$ each of size $2^{n-1}$ . Second, Bob picks $n + 1$ arbitrary integers $a_1,...,a_{n+1}$ . Finally, Alice picks an integer $t$ . Bob wins if there exists an integer $1 \le i \le n + 1$ and $s \in A_i$ such that $s + a_i \equiv t$ (mod $2^n$ ). Otherwise, Alice wins. Find all values of $n$ where Alice has a winning strategy.