Proof that you can win at least half the money.
First player can ascertain to obtain $a_1+a_3+a_5+a_7+\cdots$ or $a_2+a_4+a_6+\cdots$, whatever they prefer.
First player can ascertain to obtain $a_1+a_3+a_5+a_7+\cdots$ or $a_2+a_4+a_6+\cdots$, whatever they prefer.