Consider the following two$-$player game. The payoff of the row player $(1)$ is the first of

the two numbers, while the payoff of the column player $(2)$ is the second.

$($a$)[10$ marks$]$ Are any of the three actions of player $2$ strictly dominated? $[$Hint: you

have to show the strategy that strictly dominates it$]$ Are any of the three actions of

player $1$ strictly dominated?

$($b$)[10$ marks$]$ After eliminating the strictly dominated actions of the two players

identified in $($a$),$ do any of the remaining actions of player $1,$ or of player $2,$ become

strictly dominated?

$($c$)[5$ marks$]$ Explain $-$ very briefly $-$ why in a Nash equilibrium a player never plays a

strategy that is strictly dominated.

$($d$)[10$ marks$]$ After eliminating all the strictly dominated strategies of the two players

identified in $($a$)$ and $($b$),$ find all pure strategy Nash equilibria of the game.

$($e$)[15$ marks$]$ Find a mixed strategy Nash equilibrium of the game.