A crime is observed by a group of n a 2 people. Everybody would like the police to be in- fo rmed about the crime but everybody prefers that someone else makes a call. They choose simultaneously whether to call the police or not. When nobody calls, everybody's payoff is {1. If anybody calls the police. those who call the police receive v c and those who do not receive it. where v :> c r: {1. {:11 Find all pure strategy Nash equilibria. [b] Using the following steps nd a symmetric mixed strategy Nash equilibrium. {I} Suppose that each player other than player i calls the police with probability p. What is the expected payoff player 1' gets if he calls the police and what is his expected payoff if he does not? {H} Find a symmetric mixed strategy Nash equilibrium in which each player calls with probability 311*. Is it unique? (III) In the Nash equilibrium you found in (ii), is the probability that a particular player reports the crime increasing or decreasing in n? Is the probability that the police is informed about the crime (i.e. at least one player calls) increasing or decreasing in n? What happens as n approaches infinity? Interpret the result. Hint: You can use the fact that Pr(no one calls) = Pr(i does not call) . Pr(no one else calls). Now, consider a model where player i's payoff when he reports is now v - c, instead of v-c and c, is only known to himself. Everybody believes that each c, is identically and independently distributed over [c, c]. Let F be the distribution function of c; and f is the associated density function with f > 0 for all celc, c]. (c) Give a realistic story(s) that would justify an uncertain c;. (d) Show that for any strategy profile of all others, a player's best response is given by the following cutoff strategy s;(c) with some c e [c, c] such that he calls the police if c c. (e) Find a symmetric pure strategy Bayesian Nash equilibrium in which every player plays s;(c*(n)). (ie. find a value of such c*(n)) What is the probability that a player does not call the police? What is the probability that nobody calls the police? (f) Compare your answer in part (e) above to part (b) in the previous model, when c and c are very close to one another. Comment