the mathematics of 1950s dating: who wins the battle of the sexes? great theoretical ideas in...

The 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 Science

Steven Steven RudichRudich

CS 15-251 Spring CS 15-251 Spring 20032003

Lecture 17Lecture 17 Mar 14, 2003Mar 14, 2003 Carnegie Mellon Carnegie Mellon UniversityUniversity

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

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


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

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1,3,4,2,5

5

1,2,4,5,3

2


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? What criteria come to mind?


3,5,2,1,4

1

5,2,1,4,3

2

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


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 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


We will ignore the issue of

what is “equitable”!


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 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..


Why be with them when we can be with each

other?


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.

3,5,2,1,4

15,2,1,4,3

4,3,5,1,2

31,2,3,4,5

42,3,4,1,5

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


What 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’s social and political Steven’s social and political wisdom:wisdom:

SustainabilitySustainability is a is a prerequisite of prerequisite of fairfair

policy. policy.


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 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 operates


Given 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!



How could we change the question

we are asking?


Better Questions: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?


Think about this question:Think about this question:

Does every set of preference lists have a

stable pairing?


Idea: Allow the pairs to keep breaking up Idea: Allow the pairs to keep breaking up and reforming until they become stable. and reforming until they become stable.


Can you argue that the couples will not

continue breaking up and reforming forever?


An Instructive Variant:An Instructive Variant:BisexualBisexual Dating Dating

1

3

2,3,4

1,2,4

2

4

3,1,4

*,*,*


An Instructive Variant:An Instructive Variant:BisexualBisexual Dating Dating

3

2,3,4

1,2,4

2

4

3,1,4

*,*,*

1


Unstable roommatesUnstable roommatesin perpetual motion.in perpetual motion.

3

2,3,4

1,2,4

2

4

3,1,4

*,*,*

1


InsightInsight

Any proof that Any proof that heterosexual couples do heterosexual couples do not break up and reform not break up and reform forever must contain a forever must contain a step that fails in the step that fails in the bisexual casebisexual case


InsightInsight

If you have a proof idea that If you have a proof idea that works equally well in the works equally well in the hetero and bisexual versions, hetero and bisexual versions, then your idea is not then your idea is not adequate to show the adequate to show the couples eventually stop.couples eventually stop.


The Traditional Marriage The Traditional Marriage AlgorithmAlgorithm


The Traditional Marriage The Traditional Marriage AlgorithmAlgorithm

String

Worshipping males

Female


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

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


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

stablestable pairing? pairing?


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!


Does TMA always terminate?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


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

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


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 . . . . . . . . . . . . .

Informal Induction


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).

Formal Induction

PROOF: Let q be the day she first gets b on a string. If 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 the lemma is false, there must be a smallest k such that the girl has some bthat the girl has some b** inferior to b on day q+k. inferior to b on day q+k.

One day earlier, she has someone as good as b. One day earlier, she has someone as good as b. Hence, a better suitor than bHence, a better suitor than b** returns the next day. returns the next day. She will choose the better suitor contradicting the She will choose the better suitor contradicting the assumption that her prospects went below b on day assumption that her prospects went below b on day q+k.q+k.


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


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

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

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

Contradiction


TheoremTheorem: The TMA always : The TMA always terminates in at most nterminates in at most n22 days days

• A “master list” of all n of the boys lists starts with a total of n X n = n2 girls on it.

• 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 bounded by the original size of the master list In fact, since each list never drops below 1, the number of days is bounded by n(n-1) = n2.


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?


TheoremTheorem: Let T be the pairing : Let T be the pairing produced by TMA. T is stable.produced by TMA. T is stable.

ggbb

gg**



ggbb

gg**

I rejected you when you I rejected you when you came to my balcony, came to my balcony, now I got someone now I got someone

better.better.



• Let b and g be any couple in T.• Suppose b prefers g* to g. We will argue that

g* prefers her husband to b.• During TMA, b proposed to g* before he

proposed to g. Hence, at some point g* rejected b for someone she preferred. By the Improvement lemma, the person she married was also preferable to b.

• Thus, every boy will be rejected by any girl he prefers to his wife. T is stable.


Opinion PollOpinion Poll

Who is better off

in traditio

nal

dating, the boys

or the girls

?


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 boy b”?optimal girl for boy b”?

Flawed Attempt:Flawed Attempt: “The girl at the top of b’s “The girl at the top of b’s

list”list”


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.


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.


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.



A pairing is A pairing is male-optimalmale-optimal if if every every boy gets his boy gets his optimaloptimal mate. Thus, the mate. Thus, the pairing is simultaneously giving each pairing is simultaneously giving each boy his optimal.boy his optimal.

Is a male-optimal pairing always Is a male-optimal pairing always stable?stable?



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.


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.


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

• Suppose, for a contradiction, that some boy gets rejected by his optimal girl during TMA. Let t be the earliest time at which this happened.

• In particular, at time t, some boy b got rejected by his optimal girl g because she said “maybe” to a preferred b*. By the definition of t, b* had not yet been rejected by his optimal girl.

• Therefore, b* likes g at least as much as his optimal.


Some boy b got rejected by his optimal girl gSome boy b got rejected by his optimal girl g because she said “maybe” to a preferred bbecause she said “maybe” to a preferred b**. . b* likes g at least as much as his optimal girl.b* likes g at least as much as his optimal girl.

There must exist a stable paring There must exist a stable paring S in which b and g are married. S in which b and g are married. • b* wants g more than his wife in S

– g is as at least as good as his best and he does not have her in stable pairing S

• g wants b* more than her husband in S– b is her husband in S and she rejects him

for b* in TMA


There must exist a stable paring There must exist a stable paring S in which b and g are married. S in which b and g are married. • b* wants g more than his wife in S

– g is as at least as good as his best and he does not have her in stable pairing S

• g wants b* more than her husband in S– b is her husband in S and she rejects him

for b* in TMA

Some boy b got rejected by his optimal girl gSome boy b got rejected by his optimal girl g because she said “maybe” to a preferred bbecause she said “maybe” to a preferred b**. . b* likes g at least as much as his optimal girl.b* likes g at least as much as his optimal girl.

Contradiction of the stability of S.


What proof technique did we What proof technique did we just use?just use?


TheoremTheorem: : The TMA pairing, T, The TMA pairing, T, is female-pessimal.is 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 g is a stable pairing S where some girl g does worse than in T.does worse than in T.Let b be her mate in T. Let b be her mate in T. Let bLet b** be her mate in S. be her mate in S.• By assumption, g likes b better than her mate

in S • b likes g better than his mate in S

– We already know that g is his optimal girl

• Therefore, S is not stable. Contradict

ion


Advice to femalesAdvice to females

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


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 !


““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

• A computer runs TMA (extended to handle Mormon marriages)

• Until recently, it was hospital-optimal


History History 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


HistoryHistory

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?


HistoryHistory

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


HistoryHistory

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


HistoryHistory

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.


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 judge 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 . . .


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


Marry Well!Marry Well!


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

the mathematics of 1950s dating: who wins the battle of the sexes? great theoretical ideas in...

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