1 / 60

Prof. Amit Sahai

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. Interesting Algorithms. So far: Seen algorithms for sorting, network protocols Today: An algorithm for dating!

Download Presentation

Prof. Amit Sahai

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related