Beauty contest based question:

'2". consider the following version of' the beauty contest game. There are n > 2 players. Each player 'i covertly submits a real number 33,; E [0, 100]. Let 5: be the mean of the numbers submitted. (a) Assunle player 's payoff is given by 200 (32,; 5:)2, i.e., he/ she would like to write numbers as close to the average as possible (note that their objective is not to target two thirds of the average). Assume that it is common knowledge in this game that every player is maximizing his / her own payoff. 1. Explain why everyone writing 50 is a Nash equilibrium (10 pts). (b) 2. Does player i have a strictly dominated or dominant strategy in this game? Explain your answer. (10 pts). 3. Find all pureestrategy Nash equilibria of this game (10 pts); explain why they are Nash equilibria (10 pts), and explain why there are no other purestrategy Nash equilibria (10 pts). Suppose among these n players, there are m players who will always submit the number 0, where 0