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 = ? ? 0 1 ?1 ?1 0 1 1 ?1 0 ? ? .

(a) 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) 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) Show that x = ( 1 3 , 1 3 , 1 3 ) and y = ( 1 3 , 1 3 , 1 3 ) form a Nash equilibrium.

(d) 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

In the course content, we explained how we can solve two-player zero-sunm 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- 01 1 -1 0 What is the expected loss for Player 1 when Player 1 plays a mixed strategy x- Show that Player 1 can achieve a negative expected loss (i.e., an expected gain) if Show that x -( and y -(, form a Nash equilibrium. (zi,Z2, ?3) and Player 2 plays a mixed strategy y-(Y1, 3/2,ys)? Player 2 plays any strategy other than y (,23 Let x -(,as in part (c). Is it possible for (x, y) to be a Nash equilibrium for 3' 3' 3 some mixed strategy y' ??31)? Explain