A professor presents the following game to Lisa and her 19 classmates. Each of them simultaneously and privately writes down a whole number between 0 and 50 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 (1/2X+10) wins $50. If multiple students tie, they split the prize equally.

a. Show that choosing the number 9 is a dominated strategy. b. Show that choosing the number 37 is a dominated strategy. c. What would the set of best responses be for Lisa if she knew that all of her classmates would submit the number 30? That is, what is the range of numbers for which each number in the range is closer to the winning number than 30? d. What would the set of best responses be for Lisa if she knew that all of her classmates would submit the number 24? e. Find a symmetric Nash equilibrium to this game. That is, what number is a best response to everyone else submitting that same number? f. What are all of the dominated strategies? g. Suppose Lisa believes that none of her classmates will play the dominated strategies found in part f. Given these beliefs, what strategies are never a best response for Lisa? h. Which strategies do you think are rationalizable in this game? Explain your reasoning.