Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 4415-251Great Theoretical Ideas in Computer ScienceforSomeCMU StudentCrush Scripthttp://1982087.com/crush.htmlThe Mathematics Of 1950’s Dating: Who wins The Battle of The Sexes?Lecture 10 (February 12, 2009)WARNING: This lecture contains mathematical content that may be shocking to some studentsQuestion: How do we pair them off? Dating 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 ties3,5,2,1,415,2,1,4,324,3,5,1,231,2,3,4,542,3,4,1,5513,2,5,1,421,2,5,3,434,3,2,1,541,3,4,2,551,2,4,5,3More Than One Notion of What Constitutes A “Good” PairingMaximizing total satisfaction Hong Kong and to an extent the USAMaximizing the minimum satisfactionWestern EuropeMinimizing maximum difference in mate ranksSwedenMaximizing people who get their first choiceBarbie and Ken LandWe will ignore the issue of what is “best”!Rogue CouplesSuppose we pair off all the boys and girlsNow suppose that some boy and some girl prefer each other to the people to whom they are paired They will be called a rogue coupleWhy be with them when we can be with each other?What 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.)A pairing of boys and girls is called stable if it contains no rogue couplesStable PairingsA pairing of boys and girls is called stable if it contains no rogue couplesStable Pairings3,2,112,1,323,1,2313,2,121,2,333,2,1The study of stability will be the subject of the entire lectureWe will:Analyze various mathematical properties of an algorithm that looks a lot like 1950’s datingDiscover the naked 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 pairing always exists!Does every set of preference lists have a stable pairing? Better Question:Idea: Allow the pairs to keep breaking up and reforming until they become stableCan you argue that the couples will not continue breaking up and reforming forever?132,3,41,2,4243,1,4*,*,*An Instructive Variant:Bisexual DatingInsightAny 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 hetero and bisexual versions, then your idea is not adequate to show the couples eventually stopThe Traditional Marriage AlgorithmWorshipping MalesFemaleStringThe Traditional Marriage AlgorithmFor each day that some boy gets a “No” do:Morning• Each girl stands on her balcony• Each boy proposes to the best girl whom he has not yet crossed offAfternoon (for girls with at least one suitor)• To today’s best: “Maybe, return tomorrow”• To any others: “No, I will never marry you”Evening• Any rejected boy crosses the girl off his listIf no boys get a “No”, each girl marries boy to whom she just said “maybe”3,5,2,1,415,2,1,4,34,3,5,1,231,2,3,4,542,3,4,1,5513,2,5,1,421,2,5,3,434,3,2,1,541,3,4,2,551,2,4,5,32Wait! There is a more primary question!Does Traditional Marriage Algorithm always produce a stable pairing?Does TMA Always Terminate?It might encounter a situation where algorithm does not specify what to do next (e.g. “core dump error”)It might keep on going for an infinite number of daysImprovement 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 betterShe would only let go of that guy for someone even betterShe would only let go of that guy for someone even betterAND SO ON…Corollary: Each girl will marry her absolute favorite of the boys who visit her during the TMAContradictionLemma: No boy can be rejected by all the girlsProof (by contradiction):Suppose boy b is rejected by all the girlsAt that point:Each girl must have a suitor other than b(By Improvement Lemma, once a girl has a suitor she will always have at least one) The n girls have n suitors, and b is not among them. Thus, there are at least n+1 boysTheorem: The TMA always terminates in at most n2 daysA “master list” of all n of the boys lists starts with a total of n x n = n2 girls on itEach day at least one boy gets a “No”, so at least one girl gets crossed off the master listTherefore, the number of days is bounded by the original size of the master listGreat! We know that TMA will terminate and produce a pairingBut is it stable?gbg*I rejected you when you I rejected you when you came to my balcony. came to my balcony. Now I’ve got someone Now I’ve got someone betterbetterTheorem: The pairing T produced by TMA is stableWho is better off in traditional dating, the boys or the girls?Opinion PollForget TMA For a Moment…How should we define what we mean when we say “the optimal girl for boy b”?Flawed Attempt: “The girl at the top of b’s list”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 TMAThe Optimal GirlA boy’s optimal girl is the highest ranked girl for whom there is some stable pairing in which the boy gets herA boy’s pessimal girl is the lowest ranked girl for whom there is some stable pairing in which the boy gets herThe Pessimal GirlShe is the worst girl he can conceivably get in a stable worldDating Heaven and HellA pairing is male-optimal if every boy gets his optimal mate. This is the best of all possible stable worlds for every boy simultaneouslyA pairing is male-pessimal if every boy gets his pessimal mate. This is the worst of all possible stable worlds for every boy simultaneouslyDating Heaven and HellA pairing is female-optimal if every girl gets her optimal mate. This is the best of all possible stable worlds for every girl simultaneouslyA pairing is female-pessimal if every girl gets her pessimal mate. This is the worst of all possible stable
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