Prof. Amit Sahai

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

2. 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” Steven Rudich: www.discretemath.com www.rudich.net

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

4. 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 Steven Rudich: www.discretemath.com www.rudich.net

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

6. 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? Steven Rudich: www.discretemath.com www.rudich.net

7. 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 Steven Rudich: www.discretemath.com www.rudich.net

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

9. 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. Steven Rudich: www.discretemath.com www.rudich.net

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

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

12. 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. Steven Rudich: www.discretemath.com www.rudich.net

13. 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 • Discover the naked mathematical truth about which sex has the romantic edge • Learn how the world’s largest, most successful dating service operates Steven Rudich: www.discretemath.com www.rudich.net

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

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

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

17. 3,1,4 2 *,*,* 4 An Instructive Variant:Bisexual Dating 2,3,4 1 1,2,4 3 Steven Rudich: www.discretemath.com www.rudich.net

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

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

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

21. 3,1,4 2 *,*,* 4 Unstable roommatesin perpetual motion. 2,3,4 1 1,2,4 3 Steven Rudich: www.discretemath.com www.rudich.net

22. 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. Steven Rudich: www.discretemath.com www.rudich.net

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

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

25. Traditional Marriage Algorithm • For each day that some boy gets a “No” do: • Morning • 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) • To today’s best suitor: “Maybe, come back tomorrow” • To any others: “No, I will never marry you” • Evening • 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” Steven Rudich: www.discretemath.com www.rudich.net

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

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

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

29. Traditional Marriage Algorithm • For each day that some boy gets a “No” do: • Morning • 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) • To today’s best suitor: “Maybe, come back tomorrow” • To any others: “No, I will never marry you” • Evening • 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” Steven Rudich: www.discretemath.com www.rudich.net

30. Theorem: The TMA always terminates in at most n2 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 nXn = n2 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 Steven Rudich: www.discretemath.com www.rudich.net

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

32. Traditional Marriage Algorithm • For each day that some boy gets a “No” do: • Morning • 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) • To today’s best suitor: “Maybe, come back tomorrow” • To any others: “No, I will never marry you” • Evening • 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” Steven Rudich: www.discretemath.com www.rudich.net

33. 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 • She would only let go of that guy for someone even better • AND SO ON . . . . . . . . . . . . . Steven Rudich: www.discretemath.com www.rudich.net

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

35. 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! Contradiction Steven Rudich: www.discretemath.com www.rudich.net

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

37. Luke Bob Alice Mia Theorem: The pairing produced by TMA is stable. • Proof by contradiction:Suppose Bob and Mia are a rogue couple. • 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. • By the Improvement lemma, she must like her husband Luke more than Bob. Contradiction! Steven Rudich: www.discretemath.com www.rudich.net

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

39. 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” Steven Rudich: www.discretemath.com www.rudich.net

40. 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. Steven Rudich: www.discretemath.com www.rudich.net

41. 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. Steven Rudich: www.discretemath.com www.rudich.net

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

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

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

45. 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 sayBobis thefirst boyto be rejected by his optimal girl Mia: Let’s say she said “maybe” toLuke, whom she prefers. • Since Bobwas the only boy to be rejected by his optimal girl so far, Lukemust likeMiaat least as much as his optimal girl. Steven Rudich: www.discretemath.com www.rudich.net

46. Mia Luke 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: • LukelikesMiamore than his wife in S • LukelikesMia at least as much as his best possible girl, but he does not have Mia in S • MialikesLukemore than her husbandBobin S Contradiction! Steven Rudich: www.discretemath.com www.rudich.net

47. 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’ve shown that any pairing in which Bob marries Mia cannot be stable. • Thus, Miacannot be Bob’s optimal girl(since he can never marry her in a stable world). • So Bob never gets rejected by his optimal girlin the TMA, and thus the TMA is male-optimal. Steven Rudich: www.discretemath.com www.rudich.net

48. Alice Luke 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 Luke be her mate in the TMA pairing. Let Bob be her mate in S. • By assumption, Alice likes Lukebetter than her mate Bobin S • Lukelikes Alice better than his mate in S • We already know that Alice is his optimal girl ! Contradiction! Steven Rudich: www.discretemath.com www.rudich.net

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

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