Problem 1 A crime is observed by a group of n :3 2 people. Everybody would like The police to be in- formed 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 El. Ifanybody calls the police. those who call the police receive v c and those who do not receive 1!. where y :3 c > t}. {a} Find all pure strategy Nash equilibria. (b) Using d'te 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 t' gets if he calls the police and what is his expected payoffif he does not? {II} Find a symmetric mixed strategy Nash equilibrium in which each player calls with probability p'. ls 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? [s the probability that the police is informed about the crime (Le. at least one player calls} increasing or decreasing in it? What happens as n approaches innity? lnterpret the result. Hint: You can use the fact that Prtno one calls] = Pr does not call} -Pr{no one else calls}. Now. consider a model where player is payoff when he reports is now u '3:- instead of 0 c' and q is only known to himself. Everybody believes that each 4:; is identically and independently distributed over [9.3]. Let F be the distribution function of c; and f is the associated density function with f :>l] for all cI E[,F]. {c} Give a realistic storyts] that would justify an uncertain q _ {III} In the Nash equilibrium you found in {ii}. is the probability that a particular player reports the crime increasing or decreasing in n? [s the probability that the police is informed about the crime (Le. at least one player calls} increasing or decreasing in it? What happens as n approaches innity? Interpret the result. Hint: You can use the fact that Prno one calls] = Pr does not call} -Pr{no one else calls). Now. consider a model where player E's payoff when he reports is now u 6,. instead of a c and q is only known to himself. Everybody believes that each d:J is identically and independently distributed over [9?]. Let F be the distribution function of c; and f is the associated density function with f 3|] for all cI E[.F]. (c) Give a realich storyl{s] that would justify an uncertain q . {d} Show that for any strategy prole of all others. a player's best response is given by the following cutoffstrategy site} with some c E [9?] such that he calls the police ifci