A version of the quite well-known game of having $2n$ numbers $x_1,x_2,\ldots, x_{2n}$ in a row, each player being allowed to pick one of the ends. First player compares $a_1+a_3+\cdots + a_{2n-1}$ to $a_2+a_4+\cdots + a_{2n}$, and selects the larger; then picks numbers with indexes of the parity of those in that larger sum. In our case, we may assume wlog that $a_1\geq a_2$. Now, if $a_1+a_3+a_5 \geq a_2+a_4+a_6$, Cpt. Jack starts with $A_1$, then picks the odd indexed chests, and wins; if instead $a_1+a_3+a_5 < a_2+a_4+a_6$, Cpt. Jack starts with $A_6$, then when one of $A_1/A_2$ is picked by the other person, he picks the other. He cannot be denied to pick $A_4$, and ends up with either $a_2+a_4+a_6 > a_1+a_3+a_5$ or $a_1+a_4+a_6 \geq a_2+a_4+a_6 > a_1+a_3+a_5 \geq a_2+a_3+a_5$, and wins.