Consider a two$-$player game where player A chooses $"$Up$,"$ or "Down" and player B chooses

"Left," "Center," or "Right". Their payoffs are as follows: When player A chooses $"$Up$"$ and

player B chooses "Left" player A gets $$5$ while player B gets $$5.$ When player A chooses

$"$Up$"$ and player B chooses "Center" they get $$6$ and $$1$ correspondingly, while when player

A chooses $"$Up$"$ and player B chooses "Right" player A loses $$2$ while player B gets $$4.$

Moreover, when player A chooses "Down" and player B chooses "Left" they get $$6$ and $$1,$

while when player A chooses "Down" and player B chooses "Center" they both get $$1.$ Finally,

when player A chooses "Down" and player B chooses "Right" player A loses $$1$ and player B

gets $$2.$ Assume that the players decide simultaneously $($or$,$ in general, when one makes his

decision doesn't know what the other player has chosen$).$

$($a$)$ Draw the normal form game.

$($b$)$ Is there any dominant strategy for any of the players? Justify your answer.

$($c$)$ Is there any Nash equilibrium in pure strategies? Justify your answer fully and discuss

your result.

A strategy is DOMINATED if there exists another strategy for the player that yields higher

payoff, regardless of which strategy the other player chooses. Dominated strategies are assig$-$

ned a probability of $0$ in any Nash Equilibrium in mixed strategies. Given this observation

answer the following parts of this problem:

$($d$)$ Find the best response functions and the mixed strategies Nash Equilibrium if each

player randomizes over his actions.

$($e$)$ Show graphically the best responses and the Nash Equilibria $($in pure and in mixed

strategies$).$