Great Theoretical Ideas In Computer Science Steven Rudich Lecture 21 CS 15 251 Mar 30 2003 Spring 2004 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 Steven Rudich www discretemath com www rudich net 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 Steven Rudich www discretemath com www rudich net 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 Steven Rudich www discretemath com www rudich net 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 Steven Rudich www discretemath com www rudich net 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 the number of people who get their first choice Barbie and Ken Land Steven Rudich www discretemath com www rudich net We will ignore the issue of what is equitable Steven Rudich www discretemath com www rudich net 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 Steven Rudich www discretemath com www rudich net Why be with them when we can be with each other Steven Rudich www discretemath com www rudich net Stable Pairings A pairing of boys and girls is called stable if it contains no rogue couples 3 5 2 1 3 2 5 1 4 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 Steven Rudich www discretemath com www rudich net 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 Steven Rudich www discretemath com www rudich net Steven s social and political wisdom Sustainability is a prerequisite of fair policy Steven Rudich www discretemath com www rudich net 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 Steven Rudich www discretemath com www rudich net Given a set of preference lists how do we find a stable pairing Steven Rudich www discretemath com www rudich net Given a set of preference lists 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 net Given a set of preference lists how do we find a stable pairing How could we change the question we are asking Steven Rudich www discretemath com www rudich net Better Questions 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 Steven Rudich www discretemath com www rudich net Think about this question Does every set of preference lists have a stable pairing Steven Rudich www discretemath com www rudich net Idea Allow the pairs to keep breaking up and reforming until they become stable Steven Rudich www discretemath com www rudich net Can you argue that the couples will not continue breaking up and reforming forever Steven Rudich www discretemath com www rudich net 2 3 4 An Instructive Variant 3 1 4 Bisexual Dating 1 2 1 2 4 3 4 Steven Rudich www discretemath com www rudich net 2 3 4 An Instructive Variant 3 1 4 Bisexual Dating 1 2 1 2 4 3 4 Steven Rudich www discretemath com www rudich net 2 3 4 An Instructive Variant 3 1 4 Bisexual Dating 1 2 1 2 4 3 4 Steven Rudich www discretemath com www rudich net 2 3 4 An Instructive Variant 3 1 4 Bisexual Dating 1 2 1 2 4 3 4 Steven Rudich www discretemath com www rudich net 2 3 4 Unstable roommates in perpetual motion 1 3 1 4 2 1 2 4 3 4 Steven Rudich www discretemath com www rudich net Insight Any proof that heterosexual couples do not break up and reform forever must contain a step that fails in the bisexual case Steven Rudich www discretemath com www rudich net Insight If 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 stop Steven Rudich www discretemath com www rudich net The Traditional Marriage Algorithm Steven Rudich www discretemath com www rudich net The Traditional Marriage Algorithm Female Worshipping males Steven Rudich www discretemath com www rudich net 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 Each girl marries the boy to whom she just said maybe Steven Rudich www discretemath com www rudich net Does the Traditional Marriage Algorithm always produce a stable pairing Steven Rudich www discretemath com www rudich net Does the Traditional Marriage Algorithm always produce a stable pairing Wait There is a more primary question Steven Rudich www discretemath com www rudich net Does TMA always terminate It might encounter a situation where algorithm does not specify what to do next core dump error It might keep on going for an infinite number of days Steven Rudich www discretemath com www rudich net Traditional Marriage …
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