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Stable Matching

Stable Matching. A Special Case of. Stable Marriage. 3,5,2,1,4. 3,2,5,1,4. 1. 5,2,1,4,3. 1,2,5,3,4. 2. 4,3,2,1,5. 4,3,5,1,2. 3. 1,3,4,2,5. 1,2,3,4,5. 4. 2,3,4,1,5. 1,2,4,5,3. 5. Dating Scenario. 1. There are n men and n women.

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Stable Matching

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  1. Stable Matching A Special Case of Stable Marriage

  2. 3,5,2,1,4 3,2,5,1,4 1 5,2,1,4,3 1,2,5,3,4 2 4,3,2,1,5 4,3,5,1,2 3 1,3,4,2,5 1,2,3,4,5 4 2,3,4,1,5 1,2,4,5,3 5 Dating Scenario 1 There are n men and nwomen Each woman has her own ranked preference list of all the men 2 Each man has his own ranked preference list of the women 3 4 The lists have no ties 5 Question: How do we pair them off?

  3. Wait! Here is a problem! What Constitutes A “Good” Matching Maximizing total satisfaction … ?

  4. Blocking Pairs Suppose 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

  5. Why be with them when we can be with each other? blocking pair

  6. 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)

  7. Stable Matching A matching of men and women is called stable if it contains no blocking pairs

  8. Stable Matching A matching of men and women is called stable if it contains no blocking pairs 3,2,1 3,2,1 1 1 2,1,3 1,2,3 2 2 3,1,2 3,2,1 3 3

  9. Example

  10. National Resident Matching Program

  11. National Resident Matching Program

  12. 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 25,000 available residency positions. In 2010, the NRMP enrolled 4,176 programs in the Match, which altogether offered 25,520 positions. A total of 37,556applicants participated in the Match.

  13. A Variant 2,3,4 3,1,4 1 2 1,2,4 *,*,* 3 4

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