Consider the following two$-$player game in which player $1$ can choose an action

called $$top$($T $)$ or an action called $$bottom$($B$),$ and player $2$ many choose

between $$left$($L$)$ and $$right$($R$).$ The payoffs are given by the following way:

u$1($T$,$ L$)=0$; u$1($T$,$ R$)=2$; u$1($B$,$ L$)=1$; u$1($B$,$ R$)=0$;

u$2($T$,$ L$)=4$; u$2($T$,$ R$)=0$; u$2($B$,$ L$)=0$; u$2($B$,$ R$)=1.$

$(1)$ Draw the game matrix of this game.

$1$ Remember that the only authorized material under the exam condition is your own hand$-$

written notebook.

$1$

$(2)$ Is there a pure strategy Nash equilibrium of this game? If so$,$ write down

all of pure strategy Nash equilibria in this game. If no$,$ write down the reason

why no pure strategy Nash equilibrium exists.

$(3)$ Consider the mixed strategy extension of this game.

$(3-$a$)$ Then, dfine the expected payoff functions of players A and B respec$-$

tively.

$(3-$b$)$ Derive the best response correspondences of players A and B respec$-$

tively.

$(3-$c$)$ Compute what mixed$-$strategy Nash equilibria of this game are