PROBLEM (1) (Volunteer's Dilemma) A task, if completed, benefits each citizen in a society $64. However, doing the task costs $1 to a citizen. The utility to a citizen is then $64 (if the task got done, otherwise no benefit), minus the cost of doing the task (if the citizen did the task, otherwise no cost). Each citizen is a risk neutral expected utility maximizer in the face of risk. a) If you were the only citizen, would you rather do the task or not? Why? b) If you (A) and another citizen (B) are choosing to do the task or not independently of each other. Potentially they can simultaneously do the task (costs each of them $1, in this case) or only one of them does the task, or neither of them does it. At least one of them doing the task would result in $64 benefit to each citizen. Calculate the mixed strategy NE (Nash Equilibrium) of this game. In this situation, what is the probability that you choose to do the task? What is the probability that the task gets done? What is your expected utility overall in this game? ) It is calculated that (you don't need to prove this nor memorize it!) if there are N citizens in the society, in the symmetric mixed NE of the game where each citizen independently chooses to do the task or not, 1 each citizen ends up choosing to do the task with probability p = 1 ()h'_l. Show by calculation that the probability that the task gets done in the mixed NE decreases as N increases; from N=2 to N=3 to N=4. Calculate the expected utility of a citizen when the mixed NE is played, in a society of i) N=2 citizens, or ii) N=3 citizens, or iii) N=4 citizens