$3\mathrm{.}1[12$ points$]3-$Player Games.

Consider the $3-$player game shown below. The player going first $($at the top of the tree$)$ is the Left player, the player going second is the Middle player, and the player going last is the Right player, optimizing the left, middle and right components respectively of the utility vectors shown.

$($a$)$ Fill in the values at all nodes. Note that all players maximize their own respective utilities.

$($b$)$ Pruning for Zero$-$Sum $3-$Player Game. Based on the above game tree, now assume that we have the knowledge that the sum of the utilities of all $3$ players is always zero. Under this assumption is any pruning possible similar to $-$ pruning? If so mark the pruning on the tree above. If not, briefly explain why not below.