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Stable MatchingA Special Case of
Stable Marriage
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Question: How do we pair them off?
Dating ScenarioThere are n men and n women
Each woman has her own ranked preference list of all the men
Each man has his own ranked preference list of the women
The lists have no ties
3,2,5,1,4
1,2,5,3,4
4,3,2,1,5
1,3,4,2,5
1,2,4,5,3
1
2
3
4
5
1
2
3
4
5
3,5,2,1,4
5,2,1,4,3
4,3,5,1,2
1,2,3,4,5
2,3,4,1,5
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What Constitutes A “Good” Matching
Maximizing total satisfaction … ?
Wait! Here is a problem!
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Blocking PairsSuppose we match all the men and women
Now suppose that some man and some woman prefer each other to the people to whom they are matched
They are called a blocking pair
blocking pair
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Why be with them when we can be with each other?
blocking pair
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What use is satisfaction, if it is not stable?
Any list of criteria for a good matching must include stability. (A matching is doomed if it contains a blocking pair)
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A matching of men and women is called stable if it contains no blocking pairs
Stable Matching
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3,2,1
1
2,1,3
2
3,1,2
3
1
3,2,1
2
1,2,3
3
3,2,1
A matching of men and women is called stable if it contains no blocking pairs
Stable Matching
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Example
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National Resident Matching Program
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National Resident Matching Program
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National Resident Matching Program
…Each year approximately 16,000 U.S. medical students participate in the Main Residency Match. In addition, another 20,000 applicants complete for the approximately 25,00025,000 available residency positions. In 2010, the NRMP enrolled 4,176 programs in the Match, which altogether offered 25,520 positions. A total of 37,55637,556 applicants participated in the Match.
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1
3
2,3,4
1,2,4
2
4
3,1,4
*,*,*
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