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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 PairingsWhat use is fairness, if it is not stable?Steven’s social and political wisdom:The study of stability will be the subject of the entire lecture.Given a set of preference lists, how do we find a stable pairing?Slide 15Slide 16Better Questions:Think about this question:Idea: Allow the pairs to keep breaking up and reforming until they become stable.Slide 20An Instructive Variant: Bisexual DatingSlide 22Slide 23Slide 24Unstable roommates in perpetual motion.InsightSlide 27The Traditional Marriage AlgorithmSlide 29Traditional Marriage AlgorithmDoes the Traditional Marriage Algorithm always produce a stable pairing?Slide 32Does TMA always terminate?Slide 34Improvement 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 36Corollary: 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. T is stable.Slide 42Slide 43Opinion PollForget TMA for a momentThe Optimal GirlThe Pessimal GirlDating Heaven and HellSlide 49Slide 50The 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.Slide 54What proof technique did we just use?Slide 56Theorem: The TMA pairing, T, is female-pessimal.Advice to femalesThe largest, most successful dating service in the world uses a computer to run TMA !REFERENCESThe Mathematics Of 1950’s The Mathematics Of 1950’s Dating: Who wins the battle of the Dating: Who wins the battle of the sexes?sexes?Great Theoretical Ideas In Computer ScienceGreat Theoretical Ideas In Computer ScienceSteven Steven RudichRudichCS 15-251 Spring CS 15-251 Spring 20052005Lecture 11Lecture 11Feb 15, 2005Feb 15, 2005Carnegie Mellon Carnegie Mellon UniversityUniversitySteven Rudich: www.discretemath.com www.rudich.netWARNING: This lecture contains mathematical content that may be shocking to some students.Steven Rudich: www.discretemath.com www.rudich.net3,5,2,1,415,2,1,4,34,3,5,1,231,2,3,4,542,3,4,1,5513,2,5,1,421,2,5,3,434,3,2,1,541,3,4,2,551,2,4,5,32Steven Rudich: www.discretemath.com www.rudich.netDating ScenarioDating 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 tiesQuestion: How do we pair them off? What criteria come to mind?Steven Rudich: www.discretemath.com www.rudich.net3,5,2,1,415,2,1,4,324,3,5,1,231,2,3,4,542,3,4,1,5513,2,5,1,421,2,5,3,434,3,2,1,541,3,4,2,551,2,4,5,3Steven Rudich: www.discretemath.com www.rudich.netThere is more than one notion of There is more than one notion of what constitutes a “good” what constitutes a “good” pairing. pairing. •Maximizing total satisfaction– Hong Kong and to an extent the United States•Maximizing the minimum satisfaction–Western Europe•Minimizing the maximum difference in mate ranks–Sweden•Maximizing the number of people who get their first choice–Barbie and Ken LandSteven Rudich: www.discretemath.com www.rudich.netWe will ignore the issue of what is “equitable”!Steven Rudich: www.discretemath.com www.rudich.netRogue CouplesRogue CouplesSuppose we pair off all the boys and Suppose we pair off all the boys and girls. girls. Now suppose that some boy Now suppose that some boy and some girl prefer each other to and some girl prefer each other to the people to whom they are paired.the people to whom they are paired. They will be called a They will be called a rogue couplerogue couple..Steven Rudich: www.discretemath.com www.rudich.netWhy be with them when we can be with each other?Steven Rudich: www.discretemath.com www.rudich.netStable PairingsStable PairingsA pairing of boys and girls is called A pairing of boys and girls is called stablestable if it contains no rogue if it contains no rogue couples.couples.3,5,2,1,415,2,1,4,34,3,5,1,231,2,3,4,542,3,4,1,5513,2,5,1,421,2,5,3,434,3,2,1,541,3,4,2,551,2,4,5,32Steven Rudich: www.discretemath.com www.rudich.netWhat use is fairness, What use is fairness, if it is not stable?if it is not stable?Any list of criteria for a good pairing Any list of criteria for a good pairing must include must include stabilitystability. (A pairing is . (A pairing is doomed if it contains a rogue doomed if it contains a rogue couple.)couple.)Any reasonable list of criteria must Any reasonable list of criteria must contain the stability criterion.contain the stability criterion.Steven Rudich: www.discretemath.com www.rudich.netSteven’s social and political Steven’s social and political wisdom:wisdom:SustainabilitySustainability is a is a prerequisite of prerequisite of fairfair policy. policy.Steven Rudich: www.discretemath.com www.rudich.netThe study of stability will be the The study of stability will be the subject of the entire lecture.subject of the entire lecture.We will:We will:•Analyze various mathematical properties of an algorithm that looks a lot like 1950’s dating•Discover the naked mathematical naked mathematical truthtruth about which sex has the romantic edge•Learn how the world’s largest, most successful dating service operatesSteven Rudich: www.discretemath.com www.rudich.netGiven a set of preference lists, Given a set of preference lists, how do we find a stable pairing?how do we find a stable pairing?Steven Rudich: www.discretemath.com www.rudich.netGiven a set of preference lists, Given a set of preference lists, how do we find a stable pairing?how do we find a stable pairing?Wait! We don’t even know that such a pairing always exists!Steven Rudich: www.discretemath.com www.rudich.netGiven a set of preference lists, Given a set of preference lists, how do we find a stable pairing?how do we find a stable pairing?How could we change the question we are asking?Steven Rudich: www.discretemath.com www.rudich.netBetter Questions:Better Questions:Does every set of


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