3. (40 points) This problem considers a special case of the citizen-candidate model we dis- cussed in lecture 17. Citizens vote sincerely, with ideal points distributed continuously on some interval of the real number line and unique median bliss point x,, = 1/2. Assume that the cost of entry is 6 = 1/4 and that the exogenous payoff from winning is v = 0. That is, holding office comes with no intrinsic benefits. Let the status quo policy be z = 0. Derive the set of one-candidate equilibria of this game following these two steps: (i) Find the set of citizens whose bliss points are such that, if any such citizen enters alone, all other citizens prefer not to enter. Show that all other citizens prefer to stay out of the race. (ii) Check that any citizen in this set indeed prefers to enter, given that no other citizen enters in equilibrium