1. 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 $2. 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 gets $7 while player B gets $3. Moreover, when player A chooses "Down" and player B chooses "Left" they get $6 and $2, 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 $1. 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 strategic 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. When an action is never chosen by a player it is because this action is DOMINATED by another action (or by a combination of other actions). Dominated strategies are assigned 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).