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![NRMP: Who proposes?
Failure to communicate (Gale-Sotomayor 1985)
“The question of course then arises as to whether these results can be
applied ‘in practice’. [Gale-Shapley ’62] had expressed some reservations
on this point,—and then came another surprise. Not only could the
method be applied, it had been more than ten years earlier!”
But who proposes?
Emily Riehl (Harvard University)
A solution to the stable marriage problem
6 March 2013
19 / 20](https://image.slidesharecdn.com/asolutiontothestablemarriageproblem-140126095115-phpapp01/75/A-solution-to-the-stable-marriage-problem-43-2048.jpg)
![NRMP: Who proposes?
Failure to communicate (Gale-Sotomayor 1985)
“The question of course then arises as to whether these results can be
applied ‘in practice’. [Gale-Shapley ’62] had expressed some reservations
on this point,—and then came another surprise. Not only could the
method be applied, it had been more than ten years earlier!”
But who proposes?
Prior to the mid-1990s the hospitals acted as the proposers.
After a review by Roth et al., the students propose.
Emily Riehl (Harvard University)
A solution to the stable marriage problem
6 March 2013
19 / 20](https://image.slidesharecdn.com/asolutiontothestablemarriageproblem-140126095115-phpapp01/75/A-solution-to-the-stable-marriage-problem-44-2048.jpg)
![NRMP: Who proposes?
Failure to communicate (Gale-Sotomayor 1985)
“The question of course then arises as to whether these results can be
applied ‘in practice’. [Gale-Shapley ’62] had expressed some reservations
on this point,—and then came another surprise. Not only could the
method be applied, it had been more than ten years earlier!”
But who proposes?
Emily Riehl (Harvard University)
A solution to the stable marriage problem
6 March 2013
19 / 20](https://crownmelresort.com/image.slidesharecdn.com/asolutiontothestablemarriageproblem-140126095115-phpapp01/75/A-solution-to-the-stable-marriage-problem-43-2048.jpg)
![NRMP: Who proposes?
Failure to communicate (Gale-Sotomayor 1985)
“The question of course then arises as to whether these results can be
applied ‘in practice’. [Gale-Shapley ’62] had expressed some reservations
on this point,—and then came another surprise. Not only could the
method be applied, it had been more than ten years earlier!”
But who proposes?
Prior to the mid-1990s the hospitals acted as the proposers.
After a review by Roth et al., the students propose.
Emily Riehl (Harvard University)
A solution to the stable marriage problem
6 March 2013
19 / 20](https://crownmelresort.com/image.slidesharecdn.com/asolutiontothestablemarriageproblem-140126095115-phpapp01/75/A-solution-to-the-stable-marriage-problem-44-2048.jpg)
Uploaded byTùng Thanh
A solution to the stable marriage problem
The document describes a solution to the stable marriage problem proposed by Gale and Shapley called the deferred acceptance algorithm. It begins with an overview of stable matchings and the stable marriage problem. It then describes the deferred acceptance algorithm, which uses a metaphor of individuals proposing and rejecting partners over multiple days or rounds until all are engaged in stable matches. The document provides an example run of the algorithm. It then proves various properties of the algorithm and stable matchings more generally.
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A solution to the stable marriage problem
- 1. A solution tothe stable marriage problem Emily Riehl Harvard University http://www.math.harvard.edu/~eriehl 6 March 2013 Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 1 / 20
- 2. Stable marriages A matchingis stable if no unmatched man and woman each prefers the other to his or her spouse. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 2 / 20
- 3. Stable marriages A matchingis stable if no unmatched man and woman each prefers the other to his or her spouse. The stable marriage problem Find a stable matching for any dating pool. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 2 / 20
- 4. The deferred-acceptance algorithm(Gale-Shapley) description via a metaphor Day 0: each individual ranks the opposite sex Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 3 / 20
- 5. The deferred-acceptance algorithm(Gale-Shapley) description via a metaphor Day 0: each individual ranks the opposite sex Day 1: (a.m.) each man proposes to his top choice (p.m.) each woman rejects all but her top suitor Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 3 / 20
- 6. The deferred-acceptance algorithm(Gale-Shapley) description via a metaphor Day 0: each individual ranks the opposite sex Day 1: (a.m.) each man proposes to his top choice (p.m.) each woman rejects all but her top suitor Day n + 1: (a.m.) each man rejected on day n proposes to his top remaining choice (p.m.) each woman rejects all but her top suitor Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 4 / 20
- 7. The deferred-acceptance algorithm(Gale-Shapley) description via a metaphor Day 0: each individual ranks the opposite sex Day 1: (a.m.) each man proposes to his top choice (p.m.) each woman rejects all but her top suitor Day n + 1: (a.m.) each man rejected on day n proposes to his top remaining choice (p.m.) each woman rejects all but her top suitor When each man is engaged, the algorithm terminates. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 4 / 20
- 8. An example Ken Leo Max Bev Ada Ada Cat Cat Bev Emily Riehl(Harvard University) Ada Bev Cat Ada Bev Cat A solution to the stable marriage problem Ken Leo Max Leo Max Leo Max Ken Ken 6 March 2013 5 / 20
- 9. An example Ken Leo Max Bev Ada Ada Cat Cat Bev Emily Riehl(Harvard University) Ada Bev Cat Ada Bev Cat A solution to the stable marriage problem Ken Leo Max Leo Max Leo Max Ken Ken 6 March 2013 5 / 20
- 10. An example Ken Leo Max Bev Ada Ada Cat Cat Bev Ada Bev Cat Ada Bev Cat Ken Leo Max Leo Max Leo Max Ken Ken Day 1: Leo& Max propose to Ada. Ken proposes to Bev. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 5 / 20
- 11. An example Ken Leo Max Bev Ada Ada ///// Cat Cat Bev Ada Bev Cat Ada Ken Leo Max ///// Bev Cat Leo Max Max Leo Ken Ken Day 1: Leo& Max propose to Ada. Ken proposes to Bev. Ada rejects Max. Ada & Leo and Bev & Ken are engaged. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 5 / 20
- 12. An example Ken Leo Max Bev Ada Ada ///// Cat Cat Bev Ada Bev Cat Ada Bev Cat Ken Leo Max Leo Max Leo Max Ken Ken Day 2: Maxproposes to Bev. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 6 / 20
- 13. An example Ken Bev //// Cat Ada Leo Max Ada Ada ///// Cat Bev Bev Cat Ada Bev Cat Ken Leo Max Leo Max Leo Max Ken ///// Ken Day 2: Maxproposes to Bev. Bev rejects Ken. Ada & Leo and Bev & Max are engaged. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 6 / 20
- 14. An example Ken Bev //// Cat Ada Leo Max Ada Ada ///// Cat Bev Bev Cat Ada Bev Ken Leo Leo Max Max Ken ///// Cat Max Leo Ken Day 2: Maxproposes to Bev. Bev rejects Ken. Ada & Leo and Bev & Max are engaged. Day 3: Ken proposes to Cat. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 6 / 20
- 15. An example Ken Bev //// Cat Ada Leo Max Ada Ada ///// Cat Bev Bev Cat Ada Bev Ken Leo Leo Max Max Ken ///// Cat Max Leo Ken Day 2: Maxproposes to Bev. Bev rejects Ken. Ada & Leo and Bev & Max are engaged. Day 3: Ken proposes to Cat. It’s a match: Ada & Leo, Bev & Max, Cat & Ken. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 6 / 20
- 16. A solution tothe stable marriage problem Theorem The deferred-acceptance algorithm arranges stable marriages. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 7 / 20
- 17. A solution tothe stable marriage problem Theorem The deferred-acceptance algorithm arranges stable marriages. Proof: Each of the women that a given man prefers to his wife rejected him in favor of a suitor she preferred. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 7 / 20
- 18. Heteronormativity Heteronormativity is anessential feature of the algorithm: the “stable roommates” problem might have no solution! Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 8 / 20
- 19. Heteronormativity Heteronormativity is anessential feature of the algorithm: the “stable roommates” problem might have no solution! Example Given “roommate” preferences: Ada Bet Cat Dot Bet Cat Ada - - Dot Dot Dot - then whomever is paired with Dot would rather swap to be with the suitor who ranks her first. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 8 / 20
- 20. Comparing stable matchings Saya man and a woman are possible for each other if some stable matching marries them. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 9 / 20
- 21. Comparing stable matchings Saya man and a woman are possible for each other if some stable matching marries them. Theorem (Gale-Shapley) In the solution found by the deferred-acceptance algorithm, every man gets his best possible match! Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 9 / 20
- 22. Comparing stable matchings Theorem(Gale-Shapley) Every man gets his best possible match. Proof: We show (by induction) that no man is rejected by a possible wife: O wb "b "b "b possible match "b "b " Emily Riehl (Harvard University) w’s preferences m A solution to the stable marriage problem 6 March 2013 10 / 20
- 23. Comparing stable matchings Theorem(Gale-Shapley) Every man gets his best possible match. Proof: We show (by induction) that no man is rejected by a possible wife: O o propose ˜ w ¤ /o /o /o /o 1 m "b "b "b reject "b "b " Emily Riehl (Harvard University) w’s preferences m A solution to the stable marriage problem 6 March 2013 10 / 20
- 24. Comparing stable matchings Theorem(Gale-Shapley) Every man gets his best possible match. Proof: We show (by induction) that no man is rejected by a possible wife: O wa ˜ m’s preferences ˜ wb Emily Riehl (Harvard University) !a !a possible match !a !a !a ! "b "b "b possible match "b "b " m ˜ O w’s preferences m A solution to the stable marriage problem 6 March 2013 10 / 20
- 25. Comparing stable matchings Theorem(Gale-Shapley) Every man gets his best possible match. Proof: We show (by induction) that no man is rejected by a possible wife: O m’s preferences ˜ O w¤ ˜ !a !a reject !a !a !a ! o /o /o /o /o 1 m w propose ˜ w’s preferences m Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 10 / 20
- 26. Partial order onstable matchings Define ≥M to mean preferred by every man and ≥W to mean preferred by every woman. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 11 / 20
- 27. Partial order onstable matchings Define ≥M to mean preferred by every man and ≥W to mean preferred by every woman. Theorem ≥M is equivalent to ≤W . Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 11 / 20
- 28. Better for menis worse for women Theorem ≥M is equivalent to ≤W . Proof: Suppose α ≥M ω but some woman w prefers α. O α w _ o /o /o /o / m _ _ _ ω _ Emily Riehl (Harvard University) w’s preferences m ˜ A solution to the stable marriage problem 6 March 2013 12 / 20
- 29. Better for menis worse for women Theorem ≥M is equivalent to ≤W . Proof: Suppose α ≥M ω but some woman w prefers α. O m’s preferences O w_ ˜ _ _ _ ω _ _ o /o α /o / m /o w_ _ _ _ ω _ w’s preferences m ˜ Stability of ω implies that ω matches m to some woman w he prefers, but ˜ then α M ω. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 12 / 20
- 30. The complete latticeof stable matchings Theorem (Conway) The set of stable matchings for any fixed dating pool is a complete lattice with partial order ≥M . Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 13 / 20
- 31. The complete latticeof stable matchings Theorem (Conway) The set of stable matchings for any fixed dating pool is a complete lattice with partial order ≥M . Proof: construction of α ∨ ω Suppose everyone joins right hands with their α-match and left hands with their ω-match (forming disjoint circles with men facing in and women facing out). Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 14 / 20
- 32. The complete latticeof stable matchings Theorem (Conway) The set of stable matchings for any fixed dating pool is a complete lattice with partial order ≥M . Proof: construction of α ∨ ω Suppose everyone joins right hands with their α-match and left hands with their ω-match (forming disjoint circles with men facing in and women facing out). If all drop hands and point at their preferred partner, in each circle, everyone will point in the same direction (so that the men and women have opposite preferences). Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 14 / 20
- 33. Sexism in themale-proposing algorithm Corollary In the stable matching found by the male-proposing algorithm, every woman gets her worst possible match! Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 15 / 20
- 34. Sexism in themale-proposing algorithm Corollary In the stable matching found by the male-proposing algorithm, every woman gets her worst possible match! Of course it’s a different matter if the women propose... Upshot: waiting to receive proposals is a bad strategy. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 15 / 20
- 35. Can the womenretaliate? Yes! Strategy: truncate preference lists Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 16 / 20
- 36. Can the womenretaliate? Yes! Strategy: truncate preference lists The women steal preference lists and compute their optimal matches. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 16 / 20
- 37. Can the womenretaliate? Yes! Strategy: truncate preference lists The women steal preference lists and compute their optimal matches. Each woman truncates her preference list below her best match. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 16 / 20
- 38. Can the womenretaliate? Yes! Strategy: truncate preference lists The women steal preference lists and compute their optimal matches. Each woman truncates her preference list below her best match. Male proposals will be rejected until the result is women-optimal. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 16 / 20
- 39. Men shouldn’t lie. Theorem(Dubins-Freedman) No man or consortium of men can improve their results in the male-proposing algorithm by submitting false preferences. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 17 / 20
- 40. Men shouldn’t lie. Theorem(Dubins-Freedman) No man or consortium of men can improve their results in the male-proposing algorithm by submitting false preferences. Real-world consequences At least on one side, the deferred-acceptance algorithm is strategy-proof. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 17 / 20
- 41. National Resident MatchingProgram: History Prehistory In the 1940s, hospitals competed for residents by making offers early in medical school and demanding immediate responses. General instability encouraged private negotiations between students and hospitals. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 18 / 20
- 42. National Resident MatchingProgram: History Prehistory In the 1940s, hospitals competed for residents by making offers early in medical school and demanding immediate responses. General instability encouraged private negotiations between students and hospitals. The medical match In 1952, after a few false starts, the National Intern Matching Program (NRMP) found a stable solution: ... the deferred-acceptance algorithm! Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 18 / 20
- 43. NRMP: Who proposes? Failureto communicate (Gale-Sotomayor 1985) “The question of course then arises as to whether these results can be applied ‘in practice’. [Gale-Shapley ’62] had expressed some reservations on this point,—and then came another surprise. Not only could the method be applied, it had been more than ten years earlier!” But who proposes? Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 19 / 20
- 44. NRMP: Who proposes? Failureto communicate (Gale-Sotomayor 1985) “The question of course then arises as to whether these results can be applied ‘in practice’. [Gale-Shapley ’62] had expressed some reservations on this point,—and then came another surprise. Not only could the method be applied, it had been more than ten years earlier!” But who proposes? Prior to the mid-1990s the hospitals acted as the proposers. After a review by Roth et al., the students propose. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 19 / 20
- 45. Real-world complications Emily Riehl(Harvard University) A solution to the stable marriage problem 6 March 2013 20 / 20
- 46. Real-world complications A problem Medicalstudents are often married (in real life) to each other. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 20 / 20
- 47. Real-world complications A problem Medicalstudents are often married (in real life) to each other. Solution: couples match Couples rank pairs of preferences. Only one problem: Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 20 / 20
- 48. Real-world complications A problem Medicalstudents are often married (in real life) to each other. Solution: couples match Couples rank pairs of preferences. Only one problem: Theorem (Ronn) The couples match is NP-complete. Emily Riehl (Harvard University) A solution to the stable marriage problem 6 March 2013 20 / 20