# 1 Stable Matching A Special Case of Stable Marriage.

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1 Stable Matching A Special Case of Stable Marriage

2 Question: How do we pair them off? Dating Scenario There 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

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

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

8 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

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 25,000 20,000 applicants complete for the approximately 25,000 available residency positions. 37,556 In 2010, the NRMP enrolled 4,176 programs in the Match, which altogether offered 25,520 positions. A total of 37,556 applicants participated in the Match.

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

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