EN$-$US

$$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 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

$)$

$.$