A group of if} students are matched into pairs to play rock paper scissors. They must simultaneously make a shape with their hand; scissors beats paper, paper beats rock and rock heats scissors. The payoffs are I": [ii-n The game proceeds in multiple periods. At the beginning of every period. each student is endowed with a. strategy: R, P. or 5'. Each student sticks to this strategy throughout the period and meets every other student onee. Each student's payoff in this period is the average payoff of the 9 interactions. At the end of the period, the strategy being used by the students getting the highest average payoff is revealed to one third of the students using each of the other two strategies (when this number is a fraction, it is rounded to the nearest integer] and they switch their strategies to this one for the next period. Everyone else sticks to the same strategy for the next period. The state of the system, (r. p s). is the number playing each of the three strategies, R, P, or S for that period. For example, suppose we are in state [81 1. 1}. This means for this period. 8 students are in the group playing R, l is playing P and l is playing S. It is easy to check that the one student playing P will have the highest average payoff for this period f?) Then one third of the group playing R is 2%. and rounded to the nearest integer is 3. SO 3 students from this group will be given the information that P had the highest average payoff and switch to playing P in the next period. One third of the group playing 5' is 1 and rounded to the nearest integer is 0. So no student in the group playing 3 {which 3'. is just 1 student in this case} will receive the information. So with these adjustment rules, the state mov% to (5,4, 1) in the next period. For the transition we can use the notation (8.1.1) > {54,1}. (3.) Show that we have a mixed strategy Nash equilibrium where each strategy is played with probability %. Give a brief explanation to why this game does not have a pure strategy Nash equilibrium