Players A and B simultaneously choose Blue or Green. If they both choose Blue, Player A loses 1 dollar, and B wins 1 dollar. If Player A chooses Blue and B chooses Green, Player A wins 3 dollars, and B loses 3 dollars. If Player A chooses Green and B chooses Blue, Player A wins 2 dollars, and B loses 2 dollars. If they both choose Green, Player A loses 4 dollars, and B wins 4 dollars. (1) List the pure strategies of both players.

(2) Make a payoff matrix for Player A.

(3) Is there an optimal pure strategy for each player? Which is it? Who wins? How much?

(4) Suppose the two players play this game 1000 times. What percentage of the time, randomly, should the winner choose Blue, in order to maximize the winning? What percentage of the time, randomly, should the loser choose Blue, in order to minimize the loss? Who is the expected winner, and how much is the expected winning amount? (HINT: This is a mixed strategy problem.) (For grading, I favor the graphical method over using computers, which still needs an explanation of the inputs, and an interpretation of the computer results. So, apply the graphical method with explanation for a better grade.)