1. (10 points) There are three men m1, m2, m3 and three women w1, w2, w3 in a ma
ID: 1112538 • Letter: 1
Question
1. (10 points) There are three men m1, m2, m3 and three women w1, w2, w3 in a marriage market. Their preferences are as follows:P(m1):w1 w2 m1 w3, P(m2):w1 w3 w2 P(m3):w3 w1 w2
P(w1):m3 m1 w1 m2 P(w2):m3 m2 m1 P(w3):m2 m3 m1
(d) Suppose that agents submit their preferences to a matchmaker who implements the man-proposing deferred acceptance algorithm using the submitted preferences. Assuming that the rest of the agents are reporting their preferences truthfully to the matchmaker, what is the best woman that man m1 can get by submitting a list of (not-necessarily truthful) preferences? (e) Suppose that agents submit their preferences to a matchmaker who implements the man-proposing deferred acceptance algorithm with the submitted preferences. Assuming that the rest of the agents are reporting their preferences truthfully to the matchmaker, what is the best man that woman w1 can get by submitting a list of (not-necessarily truthful) preferences? 1. (10 points) There are three men m1, m2, m3 and three women w1, w2, w3 in a marriage market. Their preferences are as follows:
P(m1):w1 w2 m1 w3, P(m2):w1 w3 w2 P(m3):w3 w1 w2
P(w1):m3 m1 w1 m2 P(w2):m3 m2 m1 P(w3):m2 m3 m1
(d) Suppose that agents submit their preferences to a matchmaker who implements the man-proposing deferred acceptance algorithm using the submitted preferences. Assuming that the rest of the agents are reporting their preferences truthfully to the matchmaker, what is the best woman that man m1 can get by submitting a list of (not-necessarily truthful) preferences? (e) Suppose that agents submit their preferences to a matchmaker who implements the man-proposing deferred acceptance algorithm with the submitted preferences. Assuming that the rest of the agents are reporting their preferences truthfully to the matchmaker, what is the best man that woman w1 can get by submitting a list of (not-necessarily truthful) preferences? 1. (10 points) There are three men m1, m2, m3 and three women w1, w2, w3 in a marriage market. Their preferences are as follows:
P(m1):w1 w2 m1 w3, P(m2):w1 w3 w2 P(m3):w3 w1 w2
P(w1):m3 m1 w1 m2 P(w2):m3 m2 m1 P(w3):m2 m3 m1
(d) Suppose that agents submit their preferences to a matchmaker who implements the man-proposing deferred acceptance algorithm using the submitted preferences. Assuming that the rest of the agents are reporting their preferences truthfully to the matchmaker, what is the best woman that man m1 can get by submitting a list of (not-necessarily truthful) preferences? (e) Suppose that agents submit their preferences to a matchmaker who implements the man-proposing deferred acceptance algorithm with the submitted preferences. Assuming that the rest of the agents are reporting their preferences truthfully to the matchmaker, what is the best man that woman w1 can get by submitting a list of (not-necessarily truthful) preferences? (d) Suppose that agents submit their preferences to a matchmaker who implements the man-proposing deferred acceptance algorithm using the submitted preferences. Assuming that the rest of the agents are reporting their preferences truthfully to the matchmaker, what is the best woman that man m1 can get by submitting a list of (not-necessarily truthful) preferences? (e) Suppose that agents submit their preferences to a matchmaker who implements the man-proposing deferred acceptance algorithm with the submitted preferences. Assuming that the rest of the agents are reporting their preferences truthfully to the matchmaker, what is the best man that woman w1 can get by submitting a list of (not-necessarily truthful) preferences?
Explanation / Answer
d) In this type of problem, we need to find the best match for a man; this option is called stable if there is no other match by which man and woman would be individually better off than they are with their current partner.
Based on the preferences of man 1, his option is stay with woman 1; so, the optimal choice is (M1,W1). This is so because the man firstly prefers woman1, then woman 2, and lastly woman 3. The woman also prefers to stay with man 1 instead of the other men.
e) This solution implies the application of the Gale-Shapley algorithm.
For the case of the woman 1, she prefers to be with man 1 and in second place to be with man 2; so, she will get her best match with man 1 (W1,M1).
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