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):

A =

(a) [10 points] 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)?

(b) [10 points] 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 ) .

(c) [10 points] Show that x = ( 1/3, 1/3, 1/3 ) and y = ( 1/3, 1/3, 1/3 ) form a Nash equilibrium.

(d) [10 points] Let x = ( 1/3, 1/3, 1/3 ) as in part (c). Is it possible for (x, y) to be a Nash equilibrium for some mixed strategy y ( 1/3, 1/3, 1/3 )? Explain.

o 1 -1 |-1 0 || |11 0 o 1 -1 |-1 0 || |11 0