Stable Matching
Stable Matching
Matching Residents to Hospitals Goal.Given a set of preferences among hospitals and medical school students,design a self-reinforcing admissions process. Unstable pair:applicant x and hospital y are unstable if: x prefers y to its assigned hospital. y prefers x to one of its admitted students. Stable assignment.Assignment with no unstable pairs. Natural and desirable condition. Individual self-interest will prevent any applicant/hospital deal from being made
3 Matching Residents to Hospitals Goal. Given a set of preferences among hospitals and medical school students, design a self-reinforcing admissions process. Unstable pair: applicant x and hospital y are unstable if: x prefers y to its assigned hospital. y prefers x to one of its admitted students. Stable assignment. Assignment with no unstable pairs. Natural and desirable condition. Individual self-interest will prevent any applicant/hospital deal from being made
The Stable Matching Problem Goal.Given n men and n women,find a "suitable"matching. Participants rate members of opposite sex. Each man lists women in order of preference from best to worst. 。 Each woman lists men in order of preference from best to worst. favorite least favorite favorite least favorite 1st 2nd 3rd 1st 2nd 3rd Xavier Amy Bertha Clare Amy Yancey Xavier Zeus Yancey Bertha Amy Clare Bertha Xavier Yancey Zeus Zeus Amy Bertha Clare Clare Xavier Yancey Zeus Men's Preference Profile Women's Preference Profile 4
4 The Stable Matching Problem Goal. Given n men and n women, find a "suitable" matching. Participants rate members of opposite sex. Each man lists women in order of preference from best to worst. Each woman lists men in order of preference from best to worst. Zeus Amy Bertha Clare Yancey Bertha Amy Clare Xavier Amy Bertha Clare 1 st 2nd 3rd Men’s Preference Profile favorite least favorite Clare Xavier Yancey Zeus Bertha Xavier Yancey Zeus Amy Yancey Xavier Zeus 1 st 2nd 3rd Women’s Preference Profile favorite least favorite
The Stable Matching Problem Perfect matching:everyone is matched monogamously. Each man gets exactly one woman. Each woman gets exactly one man. Stability:no incentive for some pair of participants to undermine assignment by joint action. In matching M,an unmatched pair m-w is unstable if man m and woman w prefer each other to current partners. Unstable pair m-w could each improve by eloping. Stable matching:perfect matching with no unstable pairs. Stable matching problem.Given the preference lists of n men and n women,find a stable matching if one exists. 5
5 The Stable Matching Problem Perfect matching: everyone is matched monogamously. Each man gets exactly one woman. Each woman gets exactly one man. Stability: no incentive for some pair of participants to undermine assignment by joint action. In matching M, an unmatched pair m-w is unstable if man m and woman w prefer each other to current partners. Unstable pair m-w could each improve by eloping. Stable matching: perfect matching with no unstable pairs. Stable matching problem. Given the preference lists of n men and n women, find a stable matching if one exists
The Stable Matching Problem Q.Is assignment X-C,y-B,Z-A stable? favorite least favorite favorite least favorite 1st 2nd 3rd 1st 2nd 3rd Xavier Amy Bertha Clare Amy Yancey Xavier Zeus Yancey Bertha Amy Clare Bertha Xavier Yancey Zeus Zeus Amy Bertha Clare Clare Xavier Yancey Zeus Men's Preference Profile Women's Preference Profile 6
6 The Stable Matching Problem Q. Is assignment X-C, Y-B, Z-A stable? Zeus Amy Bertha Clare Yancey Bertha Amy Clare Xavier Amy Bertha Clare 1 st 2nd 3rd Men’s Preference Profile Clare Xavier Yancey Zeus Bertha Xavier Yancey Zeus Amy Yancey Xavier Zeus 1 st 2nd 3rd Women’s Preference Profile favorite least favorite favorite least favorite