In the course content, we explained how we can solve two-player zero-sum games using linear programming. One of the games we described is called Rock-Paper-Scissors. In this problem, we are going to examine this game more closely. Suppose we have the following loss matrix for Player 1 (i.e., we are showing how much Player 1 loses rather than gains, so reverse the sign):

1. What is the expected loss for Player 1 when Player 1 plays a mixed strategy x = (x1, x2, x3) and Player 2 plays a mixed strategy y = (y1, y2, y3)?

2. Show that Player 1 can achieve a negative expected loss (i.e., an expected gain) if Player 2 plays any strategy other than y = (y1; y2; y3) = (1/3, 1/3, 1/3).

3. Show that x=(1/3, 1/3, 1/3) and y=(1/3, 1/3, 1/3) form a Nash equilibrium 4. Let x=(1/3, 1/3, 1/3) as in part 3. Is it possible for (x,y) to be a Nash equilibrium for some mixed strategy y'(1/3, 1/3,1/3)? Explain.

0 1 -1 0 1 0