Multiple Games

You are given a strictly increasing sequence of positive integers $A_1, A_2, \ldots, A_N$. It is also guaranteed that for each valid $i$, $A_i+A_1 \ge A_{i+1}$. Alice and Bob want to play $Q$ independent games using this sequence. Before they play these games, Alice should choose a positive integer $G$ between $1$ and $M$ (inclusive). The rules of each game are as follows: - There is a set $S$