1. Discrete Math for CS CMPSC 360 LECTURE 9 Last time: Strong induction
Well ordering principle
Induction and recursion
Today:
Application: Stable Matching
CMPSC
360
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

2. 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, and
y prefers x to one of its admitted students
Stable assignment: no unstable pairs.
Individual self-interest will prevent any applicant/hospital deal from being made.
2012 Nobel prize in economics for work on matching algorithms (“mechanisms”)
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

3. 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
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

4. Stable Matching Problem
Unstable pair: man m and woman w are unstable if
m prefers w to his assigned match, and
w prefers m to her assigned match
Unstable pairs have an incentive to elope
Stable matching: no unstable pairs.
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
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

5. Stable Matching Problem
Input: preference lists of n men and n women
Goal: find a stable matching if one exists
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
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

6. I-clicker question (frequency: BC)
Q. Is assignment X-C, Y-B, Z-A stable?
Stable.
Unstable.
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
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

7. Stable Matching Problem
Q. Is assignment X-C, Y-B, Z-A stable?
A. No. Bertha and Xavier will hook up.
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
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

8. I-clicker question (frequency: BC)
Q. Is assignment X-A, Y-B, Z-C stable?
Stable.
Unstable.
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
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

9. Stable Matching Problem
Q. Is assignment X-A, Y-B, Z-C stable?
A. Yes. X and Y got their first choice; Z is the last choice for every woman. No man can participate in an unstable pair.
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
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

10. Existence of Stable Matching
Q. Do stable matchings always exist?
A. Not obvious a priori.
Suffixes to check all 3 possible matchings: {A-B, C-D}, {A-C, B-D}, {A-D, B-C}
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

11. Stable Roommate Problem
2n people; each person ranks others from 1 to 2n-1.
Assign roommate pairs so that no unstable pairs.
Observation. Stable matchings do not always exist for stable roommate problem.
1st
2nd
3rd
Adam B C D
A-B, C-D  B-C unstable A-C, B-D  A-B unstable A-D, B-C  A-C unstable
Bob C A D
Chris A B D
Doofus A B C
Suffixes to check all 3 possible matchings: {A-B, C-D}, {A-C, B-D}, {A-D, B-C}
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

12. An Algorithm for Stable Matching
Propose-and-reject algorithm. [Gale-Shapley 1962]
Input: tables of preferences for men and women
while (there are rejected suitors) {
Start a new day.
Morning: Every man proposes to 1st women remaining on his list.
Afternoon: Every woman says ``Maybe’’ to the man she likes best among the proposals (she now has him on a string) and rejects the rest.
Evening: Each rejected suitor crosses off the woman that rejected him from his list. }
Each woman marries the man she has on the string.
Output: The set of married couples.
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

13. Example: Execution of the Algorithm
Men’s preferences
Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

14. Day 1: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Bertha.
Wyatt proposes to Diane.
Xavier proposes to Bertha.
Yancey proposes to Amy.
Zeus proposes to Bertha.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

15. Day 1: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Yancey.
Bertha says ``Maybe’’ to Xavier,
rejects Victor and Zeus.
Diane says ``Maybe’’ to Wyatt.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

16. Day 1: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor and Zeus cross off Bertha.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

17. Day 2: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Diane.
Xavier proposes to Bertha.
Yancey proposes to Amy.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

18. Day 2: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor,
rejects Yancey.
Bertha says ``Maybe’’ to Xavier.
Diane says ``Maybe’’ to Zeus,
rejects Wyatt.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

19. Day 2: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Wyatt crosses off Diane.
Yancey crosses off Amy.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

20. Day 3: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Bertha.
Xavier proposes to Bertha.
Yancey proposes to Diane.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

21. Day 3: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor..
Bertha says ``Maybe’’ to Xavier,
rejects Wyatt.
Diane says ``Maybe’’ to Zeus,
rejects Yancey.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

22. Day 3: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Wyatt crosses off Bertha.
Yancey crosses off Diane.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

23. Day 4: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Amy.
Xavier proposes to Bertha.
Yancey proposes to Clare.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

24. Day 4: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor.,
rejects Wyatt.
Bertha says ``Maybe’’ to Xavier.
Clare says ``Maybe’’ to Yancey.
Diane says ``Maybe’’ to Zeus.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

25. Day 4: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Wyatt crosses off Amy.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

26. Day 5: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Clare.
Xavier proposes to Bertha.
Yancey proposes to Clare.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

27. Day 5: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor.
Bertha says ``Maybe’’ to Xavier.
Clare says ``Maybe’’ to Wyatt,
rejects Yancey.
Diane says ``Maybe’’ to Zeus.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

28. Day 5: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Yancey crosses off Clare.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

29. Day 6: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Clare.
Xavier proposes to Bertha.
Yancey proposes to Bertha.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

30. Day 6: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor.
Bertha says ``Maybe’’ to Xavier,
rejects Yancey.
Clare says ``Maybe’’ to Wyatt.
Diane says ``Maybe’’ to Zeus.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

31. Day 6: Evening Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Yancey crosses off Bertha.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

32. Day 7: Morning Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Victor proposes to Amy.
Wyatt proposes to Clare.
Xavier proposes to Bertha.
Yancey proposes to Erika.
Zeus proposes to Diane.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

33. Day 7: Afternoon Men’s preferences Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Amy says ``Maybe’’ to Victor.
Bertha says ``Maybe’’ to Xavier.
Clare says ``Maybe’’ to Wyatt.
Diane says ``Maybe’’ to Zeus.
Erika says ``Maybe’’ to Yancey.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne

34. Day 7: Evening (Happy End)
Men’s preferences
Women’s preferences
Wyatt
Victor
1st B D
2nd A
3rd
Zeus
Yancey
Xavier C E
4th
5th
Bertha
Amy Z X W Y V
Erika
Diane
Clare
Output of our algorithm.
The following couples get married:
Victor and Amy,
Wyatt and Clare,
Xavier and Bertha,
Yancey and Erika,
Zeus and Diane.
stable matching is unique for worst-case instance
1/17/2019
S. Raskhodnikova; based on slides by E. Demaine, C. Leiserson, A. Smith, K. Wayne