There are 213 students in ECOS3012. Each student must pick a real number x

between 0 and 100 (including 0 and 100). Let y be equal to 3

4 the average of everyone's

number. A student's payoff is equal to 1 if his number x is equal to y, and 0 if it isn't.

(a) Is there a Nash equilibrium in which all students pick the same number x = 50?

Explain why it is or isn't a Nash equilibrium. (4 points)

(b) Find all Nash equilibria of this game. Explain your answer. (8 points)

(c) Let's now change the rule. Suppose that a student's payoff is equal to 1 if his number

x is equal to y + 20, and 0 if it isn't. Find all Nash equilibria of this game. Explain your

answer. (8 points)