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CMU CS 15251 - Lecture

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The Mathematics Of 1950’s Dating: Who wins the battle of the sexes?Slide 2Slide 3Dating ScenarioSlide 5There is more than one notion of what constitutes a “good” pairing.Slide 7Rogue CouplesWhy be with them when we can be with each other?Stable PairingsSlide 11What use is fairness, if it is not stable?Some unsolicited social and political wisdomThe study of stability will be the subject of the entire lecture.Given a set of preference lists, how do we find a stable pairing?A better question…One question at a timeIdea: Allow the pairs to keep breaking up and reforming until they become stable.Slide 19An Instructive Variant: Bisexual DatingInsightThe Traditional Marriage AlgorithmSlide 23Traditional Marriage AlgorithmDoes the Traditional Marriage Algorithm always produce a stable pairing?Does TMA always terminate?Slide 27Improvement 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).Slide 29Corollary: Each girl will marry her absolute favorite of the boys who visit her during the TMALemma: No boy can be rejected by all the girlsTheorem: The TMA always terminates in at most n2 daysGreat! We know that TMA will terminate and produce a pairing. But is it stable?Theorem: Let T be the pairing produced by TMA. Then T is stable.Slide 35Opinion PollForget TMA for a momentThe Optimal GirlThe Pessimal GirlDating Heaven and HellSlide 41Slide 42The Naked Mathematical Truth!Theorem: TMA produces a male-optimal pairingSome boy b got rejected by his optimal girl g because she said “maybe” to a preferred b*. b* likes g at least as much as his optimal girl.What proof technique did we just use?Theorem: The TMA pairing T is female-pessimal.The largest, most successful dating service in the world uses a computer to run TMA!REFERENCESGreat Theoretical Ideas In Computer ScienceAnupam GuptaCS 15-251 Fall 2006Lecture 18 Oct 26, 2006 Carnegie Mellon UniversityThe Mathematics Of 1950’s Dating: Who wins the battle of the sexes?WARNING: This lecture contains mathematical content that may be shocking to some students.3,5,2,1,415,2,1,4,34,3,5,1,231,2,3,4,542,3,4,1,5513,2,5,1,421,2,5,3,434,3,2,1,541,3,4,2,551,2,4,5,32Dating ScenarioThere are n boys and n girlsEach girl has her own ranked preference list of all the boysEach boy has his own ranked preference list of the girlsThe lists have no tiesQuestion: How do we pair them off? What criteria come to mind?3,5,2,1,415,2,1,4,324,3,5,1,231,2,3,4,542,3,4,1,5513,2,5,1,421,2,5,3,434,3,2,1,541,3,4,2,551,2,4,5,3There is more than one notion of what constitutes a “good” pairing. Maximizing total satisfaction Hong Kong and to an extent the United StatesMaximizing the minimum satisfactionWestern EuropeMinimizing the maximum difference in mate ranksSwedenMaximizing number of people who get their first choiceBarbie and Ken LandWe will ignore the issue of what is “equitable”!Rogue CouplesSuppose we pair off all the boys and girls. Now suppose that some boy and some girl prefer each other to the people to whom they are paired.They will be called a rogue couple.Why be with them when we can be with each other?Stable PairingsA pairing of boys and girls is called stable if it contains no rogue couples.3,5,2,1,415,2,1,4,34,3,5,1,231,2,3,4,542,3,4,1,5513,2,5,1,421,2,5,3,434,3,2,1,541,3,4,2,551,2,4,5,32Stable PairingsA pairing of boys and girls is called stable if it contains no rogue couples.3,5,2,1,415,2,1,4,34,3,5,1,231,2,3,4,542,3,4,1,5513,2,5,1,421,2,5,3,434,3,2,1,541,3,4,2,551,2,4,5,32What use is fairness, if it is not stable?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.Some unsolicited social and political wisdomSustainability is a prerequisite of fair policy.The study of stability will be the subject of the entire lecture.We will:Analyze various mathematical properties of an algorithm that looks a lot like 1950’s datingDiscover the nakednaked mathematical truth mathematical truth about which sex has the romantic edgeLearn how the world’s largest, most successful dating service operatesGiven a set of preference lists, how do we find a stable pairing?Wait! We don’t even know that such a stable pairing always exists!A better question…Does every set of preference lists have a stable pairing?Is there a fast algorithm that, given any set of input lists, will output a stable pairing, if one exists for those lists?One question at a timeDoes every set of preference lists have a stable pairing?Idea: Allow the pairs to keep breaking up and reforming until they become stable.Can you argue that the couples will not continue breaking up and reforming forever?An Instructive Variant: Bisexual Dating132,3,41,2,4243,1,4*,*,*a.k.a. roommate selectionInsightAny proof that heterosexual couples do not break up and re-form forever must contain a step that fails in the bisexual caseIf you have a proof idea that works equally well in the heterosexual and bisexual versions, then your idea is not adequate to show the couples eventually stop.The Traditional Marriage AlgorithmThe Traditional Marriage AlgorithmWorshipping malesFemaleStringTraditional Marriage AlgorithmFor each day that some boy gets a “No” do:MorningMorningEach girl stands on her balconyEach boy proposes under the balcony of the best girl whom he has not yet crossed offAfternoon (for those girls with at least one suitor)Afternoon (for those girls with at least one suitor)To today’s best suitor: ““Maybe, come back tomorrowMaybe, come back tomorrow””To any others: “ “No, I will never marry youNo, I will never marry you””EveningEveningAny rejected boy crosses the girl off his listIf none of the boys gets a “No”Each girl marries the boy to whom she just said “maybe”Does the Traditional Marriage Algorithm always produce a stable pairing?Wait! There is a more primary question!Does TMA always terminate?It might encounter a situation where algorithm does not specify what to do next (a.k.a. “core dump error”)It might keep on going for an infinite number of daysTraditional Marriage AlgorithmFor each day that some boy gets a “No” do:MorningMorningEach girl stands on her balconyEach boy proposes under the balcony of the best girl whom he has not yet crossed offAfternoon (for those girls with at least one suitor)Afternoon (for those girls with at least one suitor)To


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