# A professor presents the following game to Elsa and her 49 classmates. Each of them simultaneously..

A professor presents the following game to Elsa and her
49 classmates. Each of them 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.

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A professor presents the following game to Elsa and her 49 classmates. Each of them simultaneously..
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(a) Show that choosing the number 80 is a dominated
strategy.

(b) 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?

(c) What would the set of best responses be for Elsa if
she knew that all of her classmates would submit the number 10?

(d) Find a symmetric Nash equilibrium to this game. That
is, what number is a best response to everyone else submitting that same
number?

(e) Which strategies are rationalizable in this game?

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