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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

*,*,*

A Variant