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The Mathematics Of Dating and The Mathematics Of Dating and Marriage: Marriage: Who wins the battle of the sexes? Who wins the battle of the sexes? (adapted from a lecture by CMU Prof. Steven Rudich) Prof. Amit Sahai Lecture Lecture 17 17

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The Mathematics Of Dating and Marriage: Who wins the battle of the sexes? (adapted from a lecture by CMU Prof. Steven Rudich). Lecture 17. Prof. Amit Sahai. Interesting Algorithms. So far: Seen algorithms for sorting, network protocols Today: An algorithm for dating! - PowerPoint PPT Presentation

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Page 1: Prof. Amit Sahai

The Mathematics Of Dating and The Mathematics Of Dating and Marriage:Marriage:

Who wins the battle of the sexes?Who wins the battle of the sexes?

(adapted from a lecture by CMU Prof. Steven Rudich)

Prof. Amit SahaiLecture 17Lecture 17

Page 2: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Interesting AlgorithmsInteresting Algorithms

• So far:

• Seen algorithms for sorting, network protocols

• Today:

• An algorithm for dating!

• Many interesting mathematical properties

• Warm-up for next few lectures on“What computers can’t do” and “Cryptography”

Page 3: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

WARNING: This lecture contains mathematical content that may be shocking to some students.

Page 4: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Dating 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 ties

Page 5: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

3,5,2,1,4

1

5,2,1,4,3

4,3,5,1,2

3

1,2,3,4,5

4

2,3,4,1,5

5

1

3,2,5,1,4

2

1,2,5,3,4

3

4,3,2,1,5

4

1,3,4,2,5

5

1,2,4,5,3

2

Page 6: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Dating 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 ties

Question: How do we pair them off? Which criteria come to mind?

Page 7: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

There is more than one notion of There is more than one notion of what constitutes a “good” what constitutes a “good”

pairing. pairing. • Maximizing the number of people who get their

first choice– “Barbie and Ken Land” model

• Maximizing average satisfaction– “Hong Kong / United States” model

• Maximizing the minimum satisfaction– “Western Europe” model

Page 8: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

We will ignore the issue of

what is “equitable”!

Page 9: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Rogue CouplesRogue Couples

Suppose 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 they married.the people they married. They will They will be called a be called a rogue couplerogue couple..

Page 10: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Why be with them Why be with them when we can be with when we can be with

each other?each other?

Page 11: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Stable PairingsStable Pairings

A 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.

Page 12: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Stability is primary.Stability is primary.

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 couple.)doomed if it contains a rogue couple.)

Any reasonable list of criteria must Any reasonable list of criteria must contain the stability criterion.contain the stability criterion.

Page 13: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

The 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 an algorithm that looks a lot

like dating in the early 1950’s•Discover the naked naked mathematical truthmathematical truth about which sex has the romantic edge

•Learn how the world’s largest, most successful dating service operates

Page 14: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

How do we find a stable pairing?How do we find a stable pairing?

Page 15: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

How do we find a stable pairing?How do we find a stable pairing?

Wait! There is a more primary

question!

Page 16: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

How do we find a stable pairing?How do we find a stable pairing?

How do we know that a stable

pairing always exists?

Page 17: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

An Instructive Variant:An Instructive Variant:BisexualBisexual Dating Dating

1

3

2,3,4

1,2,4

2

4

3,1,4

*,*,*

Page 18: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

An Instructive Variant:An Instructive Variant:BisexualBisexual Dating Dating

1

3

2,3,4

1,2,4

2

4

3,1,4

*,*,*

Page 19: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

An Instructive Variant:An Instructive Variant:BisexualBisexual Dating Dating

1

3

2,3,4

1,2,4

2

4

3,1,4

*,*,*

Page 20: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

An Instructive Variant:An Instructive Variant:BisexualBisexual Dating Dating

3

2,3,4

1,2,4

2

4

3,1,4

*,*,*

1

Page 21: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Unstable roommatesUnstable roommatesin perpetual motion.in perpetual motion.

3

2,3,4

1,2,4

2

4

3,1,4

*,*,*

1

Page 22: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

StabilityStability• Sadly, stability is not always possible for

bisexual couples!

• Amazingly, stable pairings do always exist in the heterosexual case.

• Insight: We must make sure our arguments do not apply to the bisexual case, since we know all arguments must fail in that case.

Page 23: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

The Traditional Marriage The Traditional Marriage AlgorithmAlgorithm

Page 24: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

The Traditional Marriage The Traditional Marriage AlgorithmAlgorithm

String

Worshipping males

Female

Page 25: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Traditional Marriage AlgorithmTraditional Marriage Algorithm

For each day that some boy gets a “No” do:For each day that some boy gets a “No” do:• MorningMorning

– 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)Afternoon (for those girls with at least one suitor)

– To today’s best suitor: “Maybe, come back tomorrow”“Maybe, come back tomorrow”– To any others: “No, I will never marry you” “No, I will never marry you”

• EveningEvening– Any rejected boy crosses the girl off his list

The day that no boy is rejected by any girl,The day that no boy is rejected by any girl,

Each girl marries the last boy to whom she said Each girl marries the last boy to whom she said “maybe”“maybe”

Page 26: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Does the Traditional Marriage Does the Traditional Marriage Algorithm always produce a Algorithm always produce a

stablestable pairing? pairing?

Page 27: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Does the Traditional Marriage Does the Traditional Marriage Algorithm always produce a Algorithm always produce a

stablestable pairing? pairing?

Wait! There is a more primary

question!

Page 28: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Does the TMA work at all?Does the TMA work at all?

• Does it eventually stop?

• Is everyone married at the end?

Page 29: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Traditional Marriage AlgorithmTraditional Marriage Algorithm

For each day that some boy gets a “No” do:For each day that some boy gets a “No” do:• MorningMorning

– 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)Afternoon (for those girls with at least one suitor)

– To today’s best suitor: “Maybe, come back tomorrow”“Maybe, come back tomorrow”– To any others: “No, I will never marry you” “No, I will never marry you”

• EveningEvening– Any rejected boy crosses the girl off his list

The day that no boy is rejected by any girl,The day that no boy is rejected by any girl,

Each girl marries the last boy to whom she said Each girl marries the last boy to whom she said “maybe”“maybe”

Page 30: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

TheoremTheorem: The TMA always : The TMA always terminates in at most terminates in at most nn22 days days

• Consider the “master list” containing all the boy’s preference lists of girls. There are n boys, and each list has n girls on it, so there are a total of n X n = n2 girls’ names in the master list.

• Each day that at least one boy gets a “No”, at least one girl gets crossed off the master list

• Therefore, the number of days is at most the original size of the master list

Page 31: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Does the TMA work at all?Does the TMA work at all?

• Does it eventually stop?

• Is everyone married at the end?

Page 32: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Traditional Marriage AlgorithmTraditional Marriage Algorithm

For each day that some boy gets a “No” do:For each day that some boy gets a “No” do:• MorningMorning

– 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)Afternoon (for those girls with at least one suitor)

– To today’s best suitor: “Maybe, come back tomorrow”“Maybe, come back tomorrow”– To any others: “No, I will never marry you” “No, I will never marry you”

• EveningEvening– Any rejected boy crosses the girl off his list

The day that no boy is rejected by any girl,The day that no boy is rejected by any girl, Each girl marries the last boy to whom she said Each girl marries the last boy to whom she said

“maybe”“maybe”

Page 33: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Improvement LemmaImprovement Lemma: If a girl has a : If a girl has a boy on a string, then she will boy on a string, then she will always have someone at least as always have someone at least as good on a string, (or for a good on a string, (or for a husband).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 . . . . . . . . . . . . .

Page 34: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

CorollaryCorollary: Each girl will marry : Each girl will marry her absolute favorite of the boys her absolute favorite of the boys

who visit her during the TMAwho visit her during the TMA

Page 35: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

LemmaLemma: No boy can be rejected : No boy can be rejected by all the girlsby all the girls

Proof by contradiction.Proof by contradiction.

Suppose Suppose BobBob is rejected by all the girls. is rejected by all the girls. At that point:At that point:• Each girl must have a suitor other than Bob

(By Improvement Lemma, once a girl has a suitor she will always have at least one)

• The n girls have n suitors, Bob not among them. Thus, there are at least n+1 boys!

Contradiction

Page 36: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

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

But is it stable?But is it stable?

Page 37: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

TheoremTheorem: The pairing produced: The pairing produced by TMA is stable. by TMA is stable.

• This means Bob likes Mia more than his wife, 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

husband LukeLuke more than Bob.

BobBobAliceAlice

MiaMia

LukeLuke

• Proof by contradiction: Suppose Bob and Mia are a rogue couple.

Contradiction!

Page 38: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Opinion PollOpinion Poll

Who is better off

in traditio

nal

dating, the boys

or the girls

?

Page 39: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Forget TMA for a momentForget TMA for a moment

How should we define what How should we define what we mean when we say “the we mean when we say “the optimal girl for optimal girl for BobBob”?”?

Flawed Attempt:Flawed Attempt: “The girl at the top of “The girl at the top of Bob’sBob’s list” list”

Page 40: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

The Optimal GirlThe Optimal Girl

A boy’s A boy’s optimal girloptimal girl is the highest is the highest ranked girl for whom there is ranked girl for whom there is somesome stable pairing in which the boy gets stable pairing in which the boy gets

her.her.

She is the She is the bestbest girl he can girl he can conceivably get in a stable world. conceivably get in a stable world. Presumably, she might be better than Presumably, she might be better than the girl he gets in the stable pairing the girl he gets in the stable pairing output by TMA.output by TMA.

Page 41: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

The Pessimal GirlThe Pessimal Girl

A boy’s A boy’s pessimal girlpessimal girl is the lowest is the lowest ranked girl for whom there is ranked girl for whom there is somesome stable pairing in which the boy gets stable pairing in which the boy gets

her.her.

She is the She is the worstworst girl he can girl he can conceivably get in a stable world. conceivably get in a stable world.

Page 42: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Dating Heaven and HellDating Heaven and Hell

A pairing is A pairing is male-optimalmale-optimal if if every every boy gets boy gets his his optimaloptimal mate. This is the best of all mate. This is the best of all possible stable worlds for every boy possible stable worlds for every boy simultaneously.simultaneously.

A pairing is A pairing is male-pessimalmale-pessimal if if every every boy boy gets his gets his pessimalpessimal mate. This is the worst mate. This is the worst of all possible stable worlds for every boy of all possible stable worlds for every boy simultaneously.simultaneously.

Page 43: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Dating Heaven and HellDating Heaven and Hell

A pairing is A pairing is female-optimalfemale-optimal if if everyevery girl girl gets her gets her optimaloptimal mate. This is the best of mate. This is the best of all possible stable worlds for every girl all possible stable worlds for every girl simultaneously.simultaneously.

A pairing is A pairing is female-pessimalfemale-pessimal if if everyevery girl girl gets her gets her pessimalpessimal mate. This is the worst mate. This is the worst of all possible stable worlds for every girl of all possible stable worlds for every girl simultaneously.simultaneously.

Page 44: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

The Naked The Naked Mathematical Mathematical

Truth!Truth!The Traditional Marriage The Traditional Marriage

Algorithm always Algorithm always produces a produces a male-optimalmale-optimal, , female-pessimalfemale-pessimal pairing. pairing.

Page 45: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

TheoremTheorem: TMA produces a : TMA produces a male-optimal pairingmale-optimal pairing

• Suppose not: i.e. that some boy gets rejected by his optimal girl during TMA.

• In particular, let’s say Bob is the first boy to be rejected by his optimal girl Mia: Let’s say she said “maybe” to Luke, whom she prefers.

• Since Bob was the only boy to be rejected by his optimal girl so far, Luke must like Mia at least as much as his optimal girl.

Page 46: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

We are assuming thatWe are assuming that Mia Mia isis Bob’s Bob’s optimal optimal girlgirl..

Mia Mia likes likes LukeLuke more than more than BobBob. . LukeLuke likes likes Mia Mia at least as much as his at least as much as his

optimal girl.optimal girl.

We’ll show that any pairing S in which We’ll show that any pairing S in which BobBob marries marries MiaMia cannot be stable (for a cannot be stable (for a contradiction).contradiction).•Suppose S is stable:Suppose S is stable:• Luke likes Mia more than his wife in S

– Luke likes Mia at least as much as his best possible girl, but he does not have Mia in S

• Mia likes Luke more than her husband Bob in S

LukeLuke MiaMiaContradiction!

Page 47: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

We’ve shown that any pairing in which We’ve shown that any pairing in which BobBob marries marries MiaMia cannot be stable. cannot be stable.• Thus, Mia cannot be Bob’s optimal girl

(since he can never marry her in a stable world).

• So Bob never gets rejected by his optimal girlin the TMA, and thus the TMA is male-optimal.

We are assuming thatWe are assuming that Mia Mia isis Bob’s Bob’s optimal optimal girlgirl..

Mia Mia likes likes LukeLuke more than more than BobBob. . LukeLuke likes likes Mia Mia at least as much as his at least as much as his

optimal girl.optimal girl.

Page 48: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

TheoremTheorem: : The TMA pairing is The TMA pairing is female-pessimal.female-pessimal.

We know it is male-optimal. We know it is male-optimal. Suppose there Suppose there is a stable pairing S where some girl is a stable pairing S where some girl AliceAlice does worse than in the TMA.does worse than in the TMA.Let Let Luke be her mate in the TMA pairing. be her mate in the TMA pairing. Let Let Bob be her mate in S. be her mate in S.• By assumption, Alice likes Luke better

than her mate Bob in S • Luke likes Alice better than his mate in S

– We already know that Alice is his optimal girl !

LukeLuke AliceAliceContradiction!

Page 49: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Advice to femalesAdvice to females

Learn to make the first move. Learn to make the first move.

Page 50: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

The largest, most The largest, most successful dating service successful dating service

in the world uses a in the world uses a computer to run TMA !computer to run TMA !

Page 51: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

““The Match”:The Match”:Doctors and Medical ResidenciesDoctors and Medical Residencies

• Each medical school graduate submits a ranked list of hospitals where he/she wants to do a residency

• Each hospital submits a ranked list of newly minted doctors that it wants

• A computer runs TMA (extended to handle polygamy)

• Until recently, it was hospital-optimal

Page 52: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

History: When Markets fail History: When Markets fail 19001900• Idea of hospitals having residents (then called

“interns”)

Over the next few decadesOver the next few decades• Intense competition among hospitals for an

inadequate supply of residents– Each hospital makes offers independently– Process degenerates into a race. Hospitals steadily

advancing date at which they finalize binding contracts

Page 53: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

History: When Markets fail History: When Markets fail

1944 Absurd Situation. Appointments 1944 Absurd Situation. Appointments being made 2 years ahead of time!being made 2 years ahead of time!• All parties were unhappy• Medical schools stop releasing any

information about students before some reasonable date

• Did this fix the situation?

Page 54: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

History: When Markets fail History: When Markets fail

1944 Absurd Situation. Appointments 1944 Absurd Situation. Appointments being made 2 years ahead of time!being made 2 years ahead of time!• All parties were unhappy• Medical schools stop releasing any

information about students before some reasonable date

• Offers were made at a more reasonable Offers were made at a more reasonable date, but new problems developeddate, but new problems developed

Page 55: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

History: When Markets fail History: When Markets fail

1945-1949 Just As Competitive1945-1949 Just As Competitive• Hospitals started putting time limits on

offers

• Time limit gets down to 12 hours

• Lots of unhappy people

• Many instabilities resulting from lack of cooperation

Page 56: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

History: When Markets fail History: When Markets fail

1950 Centralized System1950 Centralized System• Each hospital ranks residents

• Each resident ranks hospitals

• National Resident Matching Program produces a pairing

Whoops! The pairings were not always stable. By 1952 the algorithm was the TMA (hospital-optimal) and

therefore stable.

Page 57: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

History Repeats Itself!History Repeats Itself!NY TIMES, March 17, 1989NY TIMES, March 17, 1989

• “The once decorous process by which federal judges select their law clerks has degenerated into a free-for-all in which some of the judges scramble for the top law school students . . .”

• “The judges have steadily pushed up the hiring process . . .”

• “Offered some jobs as early as February of the second year of law school . . .”

• “On the basis of fewer grades and flimsier evidence . . .”

Page 58: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

NY TIMESNY TIMES• “Law of the jungle reigns . . “• The association of American Law Schools

agreed not to hire before September of the third year . . .

• Some extend offers from only a few hours, a practice known in the clerkship vernacular as a “short fuse” or a “hold up”.

• Judge Winter offered a Yale student a clerkship at 11:35 and gave her until noon to accept . . . At 11:55 . . he withdrew his offer

Page 59: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

Marry Well!Marry Well!

Page 60: Prof. Amit Sahai

Steven Rudich: www.discretemath.com www.rudich.net

REFERENCESREFERENCES

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

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