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) Solve for the pure strategy Nash Equilibrium using normal form.

b) Suppose player A chooses up with probability p and down with probability (1-p) and player B chooses left with probability q and right with probability (1-q). Determine player A's best response to any choice of q by player B, and player B's best response to any choice of p by player A. What happened to Center?

c) Find the mixed strategy Nash equilibrium. Show these best Response functions and all the possible equilibria on a graph.

d) Now suppose that player B moves first, and Player A moves second. Show the game information in extensive form. What is the Nash equilibrium in this case?