Question $3.$ Imagine a sequential moves version of rock$-$paper$-$scissors where player $2$ gets to pick what they will do after player $1$ picks. Please model the game in its extensive form $($as a game tree$).$ Assume both player $1$ and player $2$ only care about the result of the game and have the following preferences over the result of the game: win $>-$ tie $>-$ loss. ${?}^2$

$($a$)$ Answer the following questions:

i$.[2$ points$]$ How many nodes are there?

ii$.[2$ points$]$ How many branches are there?

iii. $[2$ points$]$ How many terminal nodes are there?

$($b$)[6$ points$]$ Prune the tree as much as possible. How many branches were you able to eliminate? $($A complete answer should include your drawing$($s$)$ of the game tree$)$

$($c$)[8$ points$]$ Use the same setup, but now imagine player $1\text{'}$s preferences change because they want to be seen as a "tough guy". Given that what they want to play remains the same, they still have the following preferences over the result of the game: win $>-$ tie $>-$ loss. However, they now would prefer to lose playing rock than win playing paper or scissors.

Please create a new game tree so the payoffs reflect these new preferences.

Prune the tree as much as possible. How many branches were you able to eliminate? $($Include your drawing$($s$))$