Econ 4415: Problem Set #4Qingmin LiuColumbia UniversityApril 8, 2021AbstractDue on 10PM at April 14,...

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Econ 4415: Problem Set #4Qingmin LiuColumbia UniversityApril 8, 2021AbstractDue on 10PM at April 14, 2015. See the syllabus for problem setpolicies.1 Defeed Acceptance Algorithm (DA)Consider the following one-to-one two-sided matching problem: There are threemen and two women. Let the set of men be M = fm1;m2;m3g and the set ofwomen be W = fw1; w2g. Their preferences are given by�m1 : w1; w2;�m2 : w1;�m3 : w2; w1;�w1 : m3;m2;m1;�w2 : m1;m3:Here, �x for x 2M[N; is the preference orderof individual x; when we write�m1 : w1; w2; we mean m1 strictly prefers w1 to w2; when we write �m2 : w1; wemean w2 is not acceptable to man m2: A matching � species who is matchedwith whom. For example, when we write � = f(m1; w1) ; (m2; w2) ; (m3;?)g ;we mean m3 is unmatched while man i is matched with woman i for i = 1; 2:We say that a matching � is individually rational if no individual strictlyprefers to stay single rather than take the partner assigned to him/her by thegiven matching �.1. Find all the individually rational matchings for this problem. [15pts]2. Find all the stable matchings for this problem. [15pts]3. Describe each round of the man-proposing DA as we have done in class(in each round, two kinds of events take place: men propose to women,and women reject o¤ers). Calculate the nal result of the man-proposingDA. [15pts]14. We say that a matching is weakly Pareto e¢ cient for men if thereis no other individually rational matching that every man strictly prefers.Explain in words the di¤erence between the denitions of weakly Par-ento e¢ cient matching for men and theman-optimal stable matching.[15pts]5. Is the result of the man-proposing DA you found above weakly Paretoe¢ cient for men? If your answer is yes, provide a proof. If you answer isno, explain why. [20pts]6. We say that a matching is strongly Pareto e¢ cient for men if there isno other individually rational matching which every man weakly prefersto the former and at least one man strictly prefers. Is the result of theman-proposing DA you found above strongly Pareto e¢ cient for men? Ifyour answer is yes, provide a proof. If you answer is no, explain why.[20pts]2

Rajeswari · Accepted Answer

80352 assignment
Given that 
m1 : w1; w2;
 m2 : w1; 
m3 : w2; w1; 
w1 : m3; m2; m1; 
w2 : m1; m3:
1) Hence individually rational matchings would be
I. (m1,w2) (m2,w1), (m3, Or
II. (m1,w1) (m2,  or
III. (m1,w2), (m2,,(m3, w1)
There cannot be any other individual rational matchings.
2) All stable matchings would be the matching where each individual does not want to switch on to other.
Of the rational matchings given above, (m1,w1) (m2, phi) (m3,w2) is stable as long as m2 is content with no wife, from men point of view.  But women w1 prefers m3 to m1 and w2 prefers m1 to m3.  So they want to switch over
(m3,w1) (m2, (m1, w2) would be ideal for women’s side.  Hence as such no stable matchings in this case
3) Here man proposes to woman.  M1 proposes to w1,

Econ 4415: Problem Set #4 Qingmin Liu Columbia University April 8, 2021 Abstract Due on 10PM at April 14, 2015. See the syllabus for problem set policies. 1 Deferred Acceptance Algorithm (DA) Consider...

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