Question 4

(cl) A mechanism 99 is said to be (strongly) group strategy-proof for students if there is no coalition of students 5" C S and their (true or) misreported preferences (lms: such that every student in S' is matched to a weakly more preferred (according to his or her true preferences] outcome to the one under truthtelliug, and at least one student. in S' is made strictly better off under the misreporting. Show that DA is not strongly group strategy-proof using the above example. 4. (Two-sided matching) Consider the following one-to-one two-sided matching problem: Let S = {$1, $2, 83), C = {ci, 2), and preferences given by * 83 :C1, C2, *9: 81, 83, ca:83, 82, 81. (a) Calculate the result of the student-proposing DA. (b) We say that a matching is strongly Pareto efficient for students if there is no other individually rational matching in which every student weakly prefers to the former and at least one student strictly prefers. Is the result of the student- proposing DA strongly Pareto efficient for students? (c) We say that a matching is weakly Pareto efficient for students if there is no other individually rational matching that every student strictly prefers. Is the result of the student-proposing DA weakly Pareto efficient for students