A professor presents a new game to Elsa and her 49 classmates. As before, each of the students simultaneously and privately writes down a number between 0 and 100 on a piece of paper, and the professor computes the mean of these numbers and calls it X. This time the student who submits the number closest to 2/3x(X+9) wins 50 dollars. Again, if multiple students tie, they spli the prize equally.
A) Find a symmetric Nash equilibrium to this game. That is, what number is a best response to everyone else submitting the same number?
B) Show that choosing the number 5 is a dominated strategy. (Hint:What would class average X have to be for the target number to be 5?)
C) Show that choosing the number 90 is a dominated strategy.
D) What are all of the dominated strategies?
E) Suppose Elsa believes that none of her classmates will play the dominated strategies found in part (d). Given these beliefs, what strategies are never a best response for Elsa?