辅导 ECON 395 Economics Spring 2024 Problem Set - 9讲解 Prolog

Economics Spring 2024

ECON 395

Problem Set - 9

DUE 10pm Friday April 26th

PROBLEM (1) (Two Sided Matching) Consider 4 men (A, B, C, D) and 4 women (1, 2, 3, 4) in a dating market with the following preferences for each person on the other side of the market;

What matching would the men-proposing Gale Shapley Deferred Acceptance Algorithm (GSM) result in?

What matching would the women-proposing Gale Shapley Deferred Acceptance Algorithm (GSW) result in?

For each of these matchings, verify that indeed no unmatched couple (a man and a woman not currently matched under the matching) can “improve” over it by being together rather than being with the partner that the matching has assigned them (check for a few such possible pairs to convince yourselves).

PROBLEM (2) (Two sided matching) Consider 4 men (1, 2, 3, 4) and 4 women (A, B, C, D) in a dating market with the following preferences for each person on the other side of the market;

(a) Derive the men-proposing Gale Shapley Deferred Acceptance Algorithm (GSM) matching.

(b) Show that the matching (1, 2, 3, 4, 5, 6) (A, B, C, D, E, F) (1 matched to A, 2 matched to B, ..etc) is not stable, finding an unmatched pair that would rather be with each other than their assigned matches in (a).

(c) Suppose now man - 5 and woman - E have left the market. What would have been the men-proposing Gale Shapley Deferred Acceptance Algorithm (GSW) matching for the remaining 4 men- 4 women market? For the remaining 8 people, who is/are better off now compared to the matching in (a)? Who is/are worse off?

PROBLEM (3) (Two sided matching) Below are the preferences of men over women, and the “before” and “after” preferences of women over men. Note that man 3 has unambiguously improved in women’s rankings moving from “before” to “after” preferences, and other men’s relative rankings did not change: