4 100 athletes are competing in a race. Each athlete has the same ability and can complete the race in 1 minute by running or in 2 minutes by walking. Running, however, takes effort. Effort e = 1 if the athlete runs and G if the athlete walks so that the time in the race is t = 2 e. The payoff for each athlete is the following function of the athlete's effort and place in the race: V(e,ploce) = 2H0 e 2ploce (1) The place in the race is one if they come rst. 2 if the come second and so on until 100 is the place for the last person in the race. In the event of a tie all runners are assigned the lowest place in the tie divided by 2. For example, if 50 athletes walk and 50 run the place of those who walk is Ill-WE = 50 since the lowest plane in the tie is l, and the place of those who run is f = 25 since the lowest place in the tie for the runners is 50. Let n he the number of athletes who walk and so that 100 - n is the number of students who run. {a} When 2 3' n. E 98, what is the payoff for an athlete who runs and for an athlete who walks? Write these as functions of n. Hint: Everyone ties in this case. (12).] Show that there is no Nash Equilibrium when 3 5 n E 100. Hint: Show for 3 5 n 5 93 separately. and check casm n = 99 and n = 100. (c) Find all Pure Strategy Nash Equilibria of the game. Show your work. Hint: Checkcasesn=,n=1,n=2. (d.] Are the Nash Equilibria Pareto elcient? Explain