2. The Mathematics Of Dating and Marriage: Who wins the battle of the sexes? The Mathematics Of Dating and Marriage: Who wins the battle of the sexes? (adapted from a lecture by CMU Prof. Steven Rudich) Prof. Amit Sahai Lecture 17

3. Steven Rudich: www.discretemath.com www.rudich.net Interesting Algorithms So far: Seen algorithms for sorting, network protocols Today: An algorithm for dating! Many interesting mathematical properties Warm-up for next few lectures on “What computers can’t do” and “Cryptography”

4. Steven Rudich: www.discretemath.com www.rudich.net WARNING: This lecture contains mathematical content that may be shocking to some students.

5. Steven Rudich: www.discretemath.com www.rudich.net Dating Scenario There are n boys and n girls Each girl has her own ranked preference list of all the boys Each boy has his own ranked preference list of the girls The lists have no ties

6. Steven Rudich: www.discretemath.com www.rudich.net 3,5,2,1,4 1 5,2,1,4,3 4,3,5,1,2 3 1,2,3,4,5 4 2,3,4,1,5 5 1 3,2,5,1,4 2 1,2,5,3,4 3 4,3,2,1,5 4 1,3,4,2,5 5 1,2,4,5,3 2

7. Steven Rudich: www.discretemath.com www.rudich.net Dating Scenario There are n boys and n girls Each girl has her own ranked preference list of all the boys Each boy has his own ranked preference list of the girls The lists have no ties Question: How do we pair them off? Which criteria come to mind?

8. Steven Rudich: www.discretemath.com www.rudich.net There is more than one notion of what constitutes a “good” pairing. Maximizing the number of people who get their first choice –“Barbie and Ken Land” model Maximizing average satisfaction – “Hong Kong / United States” model Maximizing the minimum satisfaction –“Western Europe” model

9. Steven Rudich: www.discretemath.com www.rudich.net We will ignore the issue of what is “equitable”!

10. Steven Rudich: www.discretemath.com www.rudich.net Rogue Couples Suppose we pair off all the boys and girls. Now suppose that some boy and some girl prefer each other to the people they married. They will be called a rogue couple.

11. Steven Rudich: www.discretemath.com www.rudich.net Why be with them when we can be with each other?

12. Steven Rudich: www.discretemath.com www.rudich.net Stable Pairings A pairing of boys and girls is called stable if it contains no rogue couples.

13. Steven Rudich: www.discretemath.com www.rudich.net Stability is primary. Any list of criteria for a good pairing must include stability. (A pairing is doomed if it contains a rogue couple.) Any reasonable list of criteria must contain the stability criterion.

14. Steven Rudich: www.discretemath.com www.rudich.net The study of stability will be the subject of the entire lecture. We will: Analyze an algorithm that looks a lot like dating in the early 1950’s naked mathematical truthDiscover the naked mathematical truth about which sex has the romantic edge Learn how the world’s largest, most successful dating service operates

15. Steven Rudich: www.discretemath.com www.rudich.net How do we find a stable pairing?

16. Steven Rudich: www.discretemath.com www.rudich.net How do we find a stable pairing? Wait! There is a more primary question!

17. Steven Rudich: www.discretemath.com www.rudich.net How do we find a stable pairing? How do we know that a stable pairing always exists?

18. Steven Rudich: www.discretemath.com www.rudich.net An Instructive Variant: Bisexual Dating 1 3 2,3,4 1,2,4 2 4 3,1,4 *,*,*

19. Steven Rudich: www.discretemath.com www.rudich.net An Instructive Variant: Bisexual Dating 1 3 2,3,4 1,2,4 2 4 3,1,4 *,*,*

20. Steven Rudich: www.discretemath.com www.rudich.net An Instructive Variant: Bisexual Dating 1 3 2,3,4 1,2,4 2 4 3,1,4 *,*,*

21. Steven Rudich: www.discretemath.com www.rudich.net An Instructive Variant: Bisexual Dating 3 2,3,4 1,2,4 2 4 3,1,4 *,*,* 1

22. Steven Rudich: www.discretemath.com www.rudich.net Unstable roommates in perpetual motion. 3 2,3,4 1,2,4 2 4 3,1,4 *,*,* 1

23. Steven Rudich: www.discretemath.com www.rudich.net Stability Sadly, stability is not always possible for bisexual couples! Amazingly, stable pairings do always exist in the heterosexual case. Insight: We must make sure our arguments do not apply to the bisexual case, since we know all arguments must fail in that case.

24. Steven Rudich: www.discretemath.com www.rudich.net The Traditional Marriage Algorithm

25. Steven Rudich: www.discretemath.com www.rudich.net The Traditional Marriage Algorithm String Worshipping males Female

26. Steven Rudich: www.discretemath.com www.rudich.net Traditional Marriage Algorithm For each day that some boy gets a “No” do: MorningMorning –Each girl stands on her balcony –Each boy proposes under the balcony of the best girl whom he has not yet crossed off Afternoon (for those girls with at least one suitor)Afternoon (for those girls with at least one suitor) “Maybe, come back tomorrow” –To today’s best suitor: “Maybe, come back tomorrow” “No, I will never marry you” –To any others: “No, I will never marry you” EveningEvening –Any rejected boy crosses the girl off his list The day that no boy is rejected by any girl, Each girl marries the last boy to whom she said “maybe” Each girl marries the last boy to whom she said “maybe”

27. Steven Rudich: www.discretemath.com www.rudich.net Does the Traditional Marriage Algorithm always produce a stable pairing?

28. Steven Rudich: www.discretemath.com www.rudich.net Does the Traditional Marriage Algorithm always produce a stable pairing? Wait! There is a more primary question!

29. Steven Rudich: www.discretemath.com www.rudich.net Does the TMA work at all? Does it eventually stop? Is everyone married at the end?

30. Steven Rudich: www.discretemath.com www.rudich.net Traditional Marriage Algorithm For each day that some boy gets a “No” do: MorningMorning –Each girl stands on her balcony –Each boy proposes under the balcony of the best girl whom he has not yet crossed off Afternoon (for those girls with at least one suitor)Afternoon (for those girls with at least one suitor) “Maybe, come back tomorrow” –To today’s best suitor: “Maybe, come back tomorrow” “No, I will never marry you” –To any others: “No, I will never marry you” EveningEvening –Any rejected boy crosses the girl off his list The day that no boy is rejected by any girl, The day that no boy is rejected by any girl, Each girl marries the last boy to whom she said “maybe” Each girl marries the last boy to whom she said “maybe”

31. Steven Rudich: www.discretemath.com www.rudich.net Theorem: The TMA always terminates in at most n 2 days Consider the “master list” containing all the boy’s preference lists of girls. There are n boys, and each list has n girls on it, so there are a total of n X n = n 2 girls’ names in the master list. Each day that at least one boy gets a “No”, at least one girl gets crossed off the master list Therefore, the number of days is at most the original size of the master list

32. Steven Rudich: www.discretemath.com www.rudich.net Does the TMA work at all? Does it eventually stop? Is everyone married at the end?

33. Steven Rudich: www.discretemath.com www.rudich.net Traditional Marriage Algorithm For each day that some boy gets a “No” do: MorningMorning –Each girl stands on her balcony –Each boy proposes under the balcony of the best girl whom he has not yet crossed off Afternoon (for those girls with at least one suitor)Afternoon (for those girls with at least one suitor) “Maybe, come back tomorrow” –To today’s best suitor: “Maybe, come back tomorrow” “No, I will never marry you” –To any others: “No, I will never marry you” EveningEvening –Any rejected boy crosses the girl off his list The day that no boy is rejected by any girl, Each girl marries the last boy to whom she said “maybe” The day that no boy is rejected by any girl, Each girl marries the last boy to whom she said “maybe”

34. Steven Rudich: www.discretemath.com www.rudich.net Improvement Lemma: If a girl has a boy on a string, then she will always have someone at least as good on a string, (or for a husband). She would only let go of him in order to “maybe” someone better She would only let go of that guy for someone even better AND SO ON.............

35. Steven Rudich: www.discretemath.com www.rudich.net Corollary: Each girl will marry her absolute favorite of the boys who visit her during the TMA

36. Steven Rudich: www.discretemath.com www.rudich.net Lemma: No boy can be rejected by all the girls Proof by contradiction. Suppose Bob is rejected by all the girls. At that point: Each girl must have a suitor other than Bob (By Improvement Lemma, once a girl has a suitor she will always have at least one) The n girls have n suitors, Bob not among them. Thus, there are at least n+1 boys! Contradicti on

37. Steven Rudich: www.discretemath.com www.rudich.net Great! We know that TMA will terminate and produce a pairing. But is it stable?

38. Steven Rudich: www.discretemath.com www.rudich.net Theorem: The pairing produced by TMA is stable. This means Bob likes Mia more than his wife, Alice. Thus, Bob proposed to Mia before he proposed to Alice. Mia must have rejected Bob for someone she preferred. LukeBy the Improvement lemma, she must like her husband Luke more than Bob. Bob Alice Mia Luke Proof by contradiction: Suppose Bob and Mia are a rogue couple. Contradiction!

39. Steven Rudich: www.discretemath.com www.rudich.net Opinion Poll Who is better off in traditional dating, the boys or the girls?

40. Steven Rudich: www.discretemath.com www.rudich.net Forget TMA for a moment How should we define what we mean when we say “the optimal girl for Bob”? Flawed Attempt: “The girl at the top of Bob’s list”

41. Steven Rudich: www.discretemath.com www.rudich.net The Optimal Girl A boy’s optimal girl is the highest ranked girl for whom there is some stable pairing in which the boy gets her. She is the best girl he can conceivably get in a stable world. Presumably, she might be better than the girl he gets in the stable pairing output by TMA.

42. Steven Rudich: www.discretemath.com www.rudich.net The Pessimal Girl A boy’s pessimal girl is the lowest ranked girl for whom there is some stable pairing in which the boy gets her. She is the worst girl he can conceivably get in a stable world.

43. Steven Rudich: www.discretemath.com www.rudich.net Dating Heaven and Hell A pairing is male-optimal if every boy gets his optimal mate. This is the best of all possible stable worlds for every boy simultaneously. A pairing is male-pessimal if every boy gets his pessimal mate. This is the worst of all possible stable worlds for every boy simultaneously.

44. Steven Rudich: www.discretemath.com www.rudich.net Dating Heaven and Hell A pairing is female-optimal if every girl gets her optimal mate. This is the best of all possible stable worlds for every girl simultaneously. A pairing is female-pessimal if every girl gets her pessimal mate. This is the worst of all possible stable worlds for every girl simultaneously.

45. Steven Rudich: www.discretemath.com www.rudich.net The Naked Mathematical Truth! The Traditional Marriage Algorithm always produces a male-optimal, female-pessimal pairing.

46. Steven Rudich: www.discretemath.com www.rudich.net Theorem: TMA produces a male-optimal pairing Suppose not: i.e. that some boy gets rejected by his optimal girl during TMA. In particular, let’s say Bob is the first boy to be rejected by his optimal girl Mia: Let’s say she said “maybe” to Luke, whom she prefers. Since Bob was the only boy to be rejected by his optimal girl so far, Luke must like Mia at least as much as his optimal girl.

47. Steven Rudich: www.discretemath.com www.rudich.net We are assuming that Mia is Bob’s optimal girl. Mia likes Luke more than Bob. Luke likes Mia at least as much as his optimal girl. We’ll show that any pairing S in which Bob marries Mia cannot be stable (for a contradiction). Suppose S is stable:Suppose S is stable: Luke likes Mia more than his wife in S –Luke likes Mia at least as much as his best possible girl, but he does not have Mia in S Mia likes Luke more than her husband Bob in S Luke Mia Contradiction!

48. Steven Rudich: www.discretemath.com www.rudich.net We’ve shown that any pairing in which Bob marries Mia cannot be stable. Thus, Mia cannot be Bob’s optimal girl (since he can never marry her in a stable world). So Bob never gets rejected by his optimal girl in the TMA, and thus the TMA is male-optimal. We are assuming that Mia is Bob’s optimal girl. Mia likes Luke more than Bob. Luke likes Mia at least as much as his optimal girl.

49. Steven Rudich: www.discretemath.com www.rudich.net Theorem: The TMA pairing is female-pessimal. We know it is male-optimal. Suppose there is a stable pairing S where some girl Alice does worse than in the TMA. Let be her mate in the TMA pairing. Let be her mate in S. Let Luke be her mate in the TMA pairing. Let Bob be her mate in S. By assumption, Alice likes Luke better than her mate Bob in S Luke likes Alice better than his mate in S –We already know that Alice is his optimal girl ! Luke Alice Contradiction!

50. Steven Rudich: www.discretemath.com www.rudich.net Advice to females Learn to make the first move.

51. Steven Rudich: www.discretemath.com www.rudich.net The largest, most successful dating service in the world uses a computer to run TMA !

52. Steven Rudich: www.discretemath.com www.rudich.net “The Match”: Doctors and Medical Residencies Each medical school graduate submits a ranked list of hospitals where he/she wants to do a residency Each hospital submits a ranked list of newly minted doctors that it wants A computer runs TMA (extended to handle polygamy) Until recently, it was hospital-optimal

53. Steven Rudich: www.discretemath.com www.rudich.net History: When Markets fail 1900 Idea of hospitals having residents (then called “interns”) Over the next few decades Intense competition among hospitals for an inadequate supply of residents –Each hospital makes offers independently –Process degenerates into a race. Hospitals steadily advancing date at which they finalize binding contracts

54. Steven Rudich: www.discretemath.com www.rudich.net History: When Markets fail 1944 Absurd Situation. Appointments being made 2 years ahead of time! All parties were unhappy Medical schools stop releasing any information about students before some reasonable date Did this fix the situation?

55. Steven Rudich: www.discretemath.com www.rudich.net History: When Markets fail 1944 Absurd Situation. Appointments being made 2 years ahead of time! All parties were unhappy Medical schools stop releasing any information about students before some reasonable date Offers were made at a more reasonable date, but new problems developedOffers were made at a more reasonable date, but new problems developed

56. Steven Rudich: www.discretemath.com www.rudich.net History: When Markets fail 1945-1949 Just As Competitive Hospitals started putting time limits on offers Time limit gets down to 12 hours Lots of unhappy people Many instabilities resulting from lack of cooperation

57. Steven Rudich: www.discretemath.com www.rudich.net History: When Markets fail 1950 Centralized System Each hospital ranks residents Each resident ranks hospitals National Resident Matching Program produces a pairing Whoops! The pairings were not always stable. By 1952 the algorithm was the TMA (hospital-optimal) and therefore stable.

58. Steven Rudich: www.discretemath.com www.rudich.net History Repeats Itself! NY TIMES, March 17, 1989 “The once decorous process by which federal judges select their law clerks has degenerated into a free-for-all in which some of the judges scramble for the top law school students...” “The judges have steadily pushed up the hiring process...” “Offered some jobs as early as February of the second year of law school...” “On the basis of fewer grades and flimsier evidence...”

59. Steven Rudich: www.discretemath.com www.rudich.net NY TIMES “Law of the jungle reigns.. “ The association of American Law Schools agreed not to hire before September of the third year... Some extend offers from only a few hours, a practice known in the clerkship vernacular as a “short fuse” or a “hold up”. Judge Winter offered a Yale student a clerkship at 11:35 and gave her until noon to accept... At 11:55.. he withdrew his offer

60. Steven Rudich: www.discretemath.com www.rudich.net Marry Well!

61. Steven Rudich: www.discretemath.com www.rudich.net REFERENCES D. Gale and L. S. Shapley, College admissions and the stability of marriage, American Mathematical Monthly 69 (1962), 9-15 Dan Gusfield and Robert W. Irving, The Stable Marriage Problem: Structures and Algorithms, MIT Press, 1989