$(14$ points$)$ Consider the infinitely repeated game in which the stage game is shown here. Each players payoff is the present value of her payoff stream, where the discount factor is $.$ Also assume that the two players share the same discount factor $$ of future payoffs.

$\backslash$table$[[$A$,$Player $2],[,$A$,$B$,$C$],[$B$,2.9,3.11,9.2],[\backslash$table$[[$Player $1],[$C$]],11\mathrm{.}3,5.5,8.4],[2.9,4.8,6.6]]$

$($a$)(1$ point$)$ Find every pure$-$strategy Nash equilibrium of the stage game.

$($b$)(1$ point$)$ If Player $1$ chooses $A,$ what is the best response of Player $2?$

$($c$)(1$ point$)$ If Player $1$ chooses B$,$ what is the best response of Player $2?(5$ points$)$ Consider the following punishment mechanism:

Each player will choose $A$ in stage $1$ and in any later stages as long as A was chosen by both players in all past stages.

If a player chooses any strategy other than $A$ in one stage, the opponent will chooses B in all later stages.

For what range of $$ does this punishment mechanism support $(A,A)$ to be played in every stage?

$($Hint: You can start by using your answers in parts $($b$)$ and $($c$)$ to figure out these two things: $(1)$ If Player $2$ defects in a stage by choosing something other than A $,$ what specific strategy would she choose? $(2)$ For this defecting player, what would be her resulting payoff stream in later stages?$)$

$($d$)(1$ point$)$ If Player $1$ chooses C$,$ what is the best response of Player $2?$

$($e$)(5$ points$)$ Consider an alternative punishment mechanism as follows:

Each player will choose C in stage $1$ and in any later stages as long as C was chosen by both players in all past stages.

If a player chooses any strategy other than C in one stage, the opponent will chooses B in all later stages.

For what range of $$ does this punishment mechanism support $(C,C)$ to be played in every stage? $($Hint: You can start by using your answers in parts $($b$)$ and $($e$)$ to figure out these two things: $(1)$ If Player $2$ defects in a stage by choosing something other than $C,$ what specific strategy would she choose? $(2)$ For this defecting player, what would be her resulting payoff stream in later stages?$)$