Problem
A professor presents the following game to Elsa and her 49 classmates. Each of the students simultaneously and privately writes down a number between 0 and 100 on a piece of paper, and they all hand in their numbers. The professor then computes the mean of these numbers and defines X to be the mean of the students' numbers. The student who submits the number closest to one-half of X wins $50. If multiple students tie, they split the prize equally.
i. Show that choosing the number 80 is a dominated strategy.
ii. What would the set of best responses be for Elsa if she knew that all of her classmates would submit the number 40? That is, what is the range of numbers for which each number in the range is closer to the winning number than 40?
iii. What would the set of best responses be for Elsa if she knew that all of her classmates would submit the number 10?
iv. Find a symmetric Nash equilibrium for this game. That is, what number is a best response to everyone else submitting that same number?
v. Which strategies are rationalizable in this game?