$.(5$ points$)($LO$2)$ Two of my sons $($Boy $1$ and $$Boy $2)$ are negotiating over how to divide a pile of $20$ chocolates. Boy $1$ will engage Boy $2$ in up to three rounds of negotiations. The order of events is: FIRST ROUND: Boy $1$ makes Boy $2$ an initial offer. Boy $2$ accepts or rejects. If he accepts, the game ends and the two boys get their chocolates. If Boy $2$ rejects, I punish them for not working together by eating $5$ chocolates myself. The game then continues with $15$ chocolates to be divided. SECOND ROUND: Boy $2$ makes an offer. Boy $1$ accepts or rejects. If he accepts, the game ends and the two boys get their chocolates. If Boy $1$ rejects, I eat $5$ chocolates myself and the game continues with $10$ chocolates to be divided. THIRD ROUND: Boy $1$ makes a final offer. Boy $2$ accepts or rejects. If he accepts, the two boys get their chocolates. If Boy $2$ rejects, the game ends and I eat all the remaining chocolates. Note that I am expecting you to make an assumption of spiteful players, all else equal. Said another way, I assum