# Shang-Hua Teng

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Lecture 23: Stable Marriage ( Based on Lectures of Steven Rudich of CMU and Amit Sahai of Princeton). Shang-Hua Teng. Dating Scenario. There are n “men” and n “women” Each woman has her own ranked preference list of ALL the men Each man has his own ranked preference list of ALL the women - PowerPoint PPT Presentation

### Text of Shang-Hua Teng

• Lecture 23:Stable Marriage(Based on Lectures of Steven Rudich of CMU and Amit Sahai of Princeton)Shang-Hua Teng

• Dating ScenarioThere are n men and n womenEach woman has her own ranked preference list of ALL the menEach man has his own ranked preference list of ALL the womenThe lists have no ties

• The Matching (Marriage) ProblemQuestion: How do we pair them off? Which criteria come to mind?

• There is more than one notion of what constitutes a good pairing. Maximizing the number of people who get their first choiceMaximizing average satisfactionMaximizing the minimum satisfaction

• Couple of Mutually Cheating HeartsSuppose we pair off all the men and women. Now suppose that some man and some woman prefer each other to the people they married. They will be called a couple of MCHs.

• Why be with them when we can be with each other?

• Stable PairingsA pairing of men and women is called stable if it contains no couple of MCHs.

• Stability is Primary.Any list of criteria for a good pairing must include stability. (A pairing is doomed if it contains a shaky couple.)

Any reasonable list of criteria must contain the stability criterion.

• The study of stability will be the subject of the entire lecture.We will:Analyze an algorithm that looks a lot like dating in the early 1950sDiscover the naked mathematical truth about which sex has the romantic edge Learn how the worlds largest, most successful dating service operates

• How do we find a stable pairing?

• How do we find a stable pairing?Wait! There is a more primary question!

• How do we find a stable pairing?How do we know that a stable pairing always exists?

• A Traditional Marriage AlgorithmChoose the best among those who propose, get a commitmentPropose to the best who is still available

• A Todays Dating Algorithm

• Traditional Marriage AlgorithmFor each day each single man does the following:MorningEach bachelorette sits in her favorite coffee houseEach single men proposes to the best bachelorette to whom he has not yet proposed Afternoon (for those bachelorettes with at least one suitor)To todays best suitor: Maybe, come back tomorrow, and ask me againTo any others: Not me, but good luckEveningEach single man is ready to propose next bachelorette on his list The day WILL COME that no bachelor is rejectedEach bachelorette marries the last bachelor to whom she said maybe

• Does the Traditional Marriage Algorithm always produce a stable pairing?

• Does the Traditional Marriage Algorithm always produce a stable pairing?Wait! There is a more primary question!

• Does the TMA work at all?Does it eventually stop?

Is everyone married at the end?

• Theorem: The TMA always terminates in at most n2 daysEach day that at least one Bachelor gets a No.

There can be at most n2 Nos.

• Does the TMA work at all?Does it eventually stop?

Is everyone married at the end?

• Improvement Lemma: If a bachelorette has a committed suitor, then she will always have someone at least as good the next day (and on the final day).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 bachelorette will marry her absolute favorite of the bachelors who proposed to her during the TMA

• Lemma: No bachelor can be rejected by all the bachelorettesProof by contradiction.Suppose Bob is rejected by all the women. At that point:Each women must have a suitor other than Bob (By Improvement Lemma, once a woman has a suitor she will always have at least one) The n women have n suitors, Bob not among them. Thus, there must be at least n+1 men!Contradiction

• Great! We know that TMA will terminate and produce a pairing.

But is it stable?

• Theorem: The pairing produced by TMA is stable.

This means Bob likes Mia more than his partner, Alice.Thus, Bob proposed to Mia before he proposed to Alice.Mia must have rejected Bob for someone she preferred.By the Improvement lemma, she must like her parnter Luke more than Bob.

Proof by contradiction: Suppose Bob and Mia are a couple of MCHs.Contradiction!

• Opinion PollWho is better off in traditional dating, the men or the women?

• Forget TMA for a momentHow should we define what we mean when we say the optimal woman for Bob?

Flawed Attempt: The woman at the top of Bobs list

• The Optimal MatchA mans optimal match is the highest ranked woman for whom there is some stable pairing in which they are matchedShe is the best woman he can conceivably be matched in a stable world. Presumably, she might be better than the woman he gets matched to in the stable pairing output by TMA.

• The Pessimal MatchA mans pessimal match is the lowest ranked woman for whom there is some stable pairing in which the man is matched to.

She is the least ranked woman he can conceivably get to be matched to in a stable world.

• Dating DilemmasA pairing is man-optimal if every man gets his optimal match. This is the best of all possible stable worlds for every man simultaneously.

A pairing is man-pessimal if every man gets his pessimal match. This is the worst of all possible stable worlds for every man simultaneously.

• Dating DilemmasA pairing is woman-optimal if every woman gets her optimal match. This is the best of all possible stable worlds for every woman simultaneously.

A pairing is woman-pessimal if every woman gets her pessimal match. This is the worst of all possible stable worlds for every woman simultaneously.

• The Naked Mathematical Truth! The Traditional Marriage Algorithm always produces a man-optimal, and woman-pessimal pairing.

• Theorem: TMA produces a man-optimal pairingSuppose not: i.e. that some man gets rejected by his optimal match during TMA. In particular, lets say Bob is the first man to be rejected by his optimal match Mia: Lets say she said maybe to Luke, whom she prefers. Since Bob was the only man to be rejected by his optimal match so far, Luke must like Mia at least as much as his optimal match.

• We are assuming that Mia is Bobs optimal match Mia likes Luke more than Bob. Luke likes Mia at least as much as his optimal match.We now show that any pairing S in which Bob marries Mia cannot be stable (for a contradiction).Suppose S is stable:Luke likes Mia more than his partner in SLuke likes Mia at least as much as his best match, but he is not matched to Mia in SMia likes Luke more than her partner Bob in SContradiction!

• We are assuming that Mia is Bobs optimal match.Mia likes Luke more than Bob. Luke likes Mia at least as much as his optimal match.Weve shown that any pairing in which Bob marries Mia cannot be stable.Thus, Mia cannot be Bobs optimal match (since he can never marry her in a stable world).

So Bob never gets rejected by his optimal match in the TMA, and thus the TMA is man-optimal.

• Theorem:The TMA pairing is woman-pessimal.We know it is man-optimal. Suppose there is a stable pairing S where some woman Alice does worse than in the TMA.Let Luke be her partner in the TMA pairing. Let Bob be her partner in S.By assumption, Alice likes Luke better than her partner Bob in S Luke likes Alice better than his partner in SWe already know that Alice is his optimal match !Contradiction!

• Lessons???

• The largest dating service in the world uses a computer to run TMA !

• The Match:Doctors and Medical ResidenciesEach medical school graduate submits a ranked list of hospitals where he/she wants to do a residencyEach hospital submits a ranked list of newly minted doctors that it wantsA computer runs TMA (extended to handle polygamy)Until recently, it was hospital-optimal

• An Instructive Variant:Team Projects

• An Instructive Variant:Team Project132,3,41,2,4

• An Instructive Variant:Team Project132,3,41,2,4

• An Instructive Variant:Team Project132,3,41,2,4

• No Stable Teaming Possible132,3,41,2,4

• Stability

Amazingly, stable pairings do always exist in the bipartite relations.Insight: We must make sure our arguments do not apply to the class project case, since we know all arguments must fail in that case.

• REFERENCESD. Gale and L. S. Shapley, College admissions and the stability of marriage, American Mathematical Monthly 69 (1962), 9-15

Dan Gusfield and Robert W. Irving, The Stable Marriage Problem: Structures and Algorithms, MIT Press, 1989

• History: When Markets fail 1900Idea of hospitals having residents (then called interns)Over the next few decadesIntense competition among hospitals for an inadequate supply of residentsEach hospital makes offers independentlyProcess degenerates into a race. Hospitals steadily advancing date at which they finalize binding contracts

• History: When Markets fail 1944 Absurd Situation. Appointments being made 2 years ahead of time!All parties were unhappyMedical schools stop releasing any information about students before some reasonable dateDid this fix the situation?

• History: When Markets fail 1944 Absurd Situation. Appointments being made 2 years ahead of time!All parties were unhappyMedical schools stop releasing any information about students before some reasonable dateOffers were made at a more reasonable date, but new problems developed

• History: When Markets fail 1945-1949 Just As CompetitiveHospitals started

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