Great Theoretical Ideas In Computer Science Anupam Gupta Lecture 18 CS 15 251 Oct 26 2006 Fall 2006 Carnegie Mellon University The 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 2 5 1 4 3 5 2 1 4 1 1 5 2 1 4 3 1 2 5 3 4 2 2 4 3 5 1 2 4 3 2 1 5 3 3 1 2 3 4 5 1 3 4 2 5 4 4 2 3 4 1 5 1 2 4 5 3 5 5 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 What criteria come to mind 3 2 5 1 4 3 5 2 1 4 1 1 5 2 1 4 3 1 2 5 3 4 2 2 4 3 5 1 2 4 3 2 1 5 3 3 1 2 3 4 5 1 3 4 2 5 4 4 2 3 4 1 5 1 2 4 5 3 5 5 There is more than one notion of what constitutes a good 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 number of people who get their first choice Barbie and Ken Land We will ignore the issue of what is equitable 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 to whom they are paired They will be called a rogue couple Why be with them when we can be with each other Stable Pairings A pairing of boys and girls is called stable if it contains no rogue couples 3 5 2 1 4 3 2 5 1 4 1 1 5 2 1 4 3 1 2 5 3 4 2 2 4 3 5 1 2 4 3 2 1 5 3 3 1 2 3 4 5 1 3 4 2 5 4 4 2 3 4 1 5 1 2 4 5 3 5 5 Stable Pairings A pairing of boys and girls is called stable if it contains no rogue couples 3 5 2 1 4 3 2 5 1 4 1 1 5 2 1 4 3 1 2 5 3 4 2 2 4 3 5 1 2 4 3 2 1 5 3 3 1 2 3 4 5 1 3 4 2 5 4 4 2 3 4 1 5 1 2 4 5 3 5 5 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 Any reasonable list of criteria must contain the stability criterion Some unsolicited social and political wisdom Sustainability 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 dating Discover the naked mathematical truth about which sex has the romantic edge Learn how the world s largest most successful dating service operates Given 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 time Does 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 Dating 2 3 4 a k a roommate selection 1 3 1 4 2 1 2 4 3 4 Insight Any proof that heterosexual couples do not break up and re form forever must contain a step that fails in the bisexual case If 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 Algorithm The Traditional Marriage Algorithm Female Worshipping males String 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 If 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 days 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 If none of the boys gets a No Each girl marries the boy to whom she just said maybe 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 Informal Induction 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 Proof Let q be the day she first gets b on a string If the lemma is false there must be a smallest k such that the girl has some b inferior to b on day q k But then one day earlier she has someone as good as b Hence a better suitor than b returns the next day She will choose the better suitor contradicting the assumption that her prospects went below b on day q k Formal Induction Corollary Each girl will marry her absolute favorite …
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