a longer example: stable matching unc chapel hillz. guo

A Longer Example: Stable Matching

UNC Chapel Hill Z. Guo

Matching Residents to Hospitals• Goal. Given a set of preferences among n

hospitals and n medical school students, design a self-reinforcing admissions process.

• Suppose four doctors q,r,s,t, and four hospitals A,B,C,D rank each other as follows:


q r s tA A B DB D A BC C C CD B D A

A B C Dt r t ss t r rr q s qq s q t

Matching Residents to Hospitals• Goal. Given a set of preferences among n hospitals

and n medical school students, design a self-reinforcing admissions process.

• Unstable pair: doctor x and hospital A are unstable if:– x prefers A to its assigned hospital, and– A prefers x to its admitted student.

• Stable assignment. Assignment with no unstable pairs.– Natural and desirable condition.– Individual self-interest will prevent any applicant/hospital

deal from being made.


Example• Suppose four doctors q,r,s,t, and four hospitals

A,B,C,D rank each other as follows:

• (A, s), (B, t), (C, q), (D, r) (stable)• (A, t), (B, q), (C, s), (D, r) (un-stable pair: (B, t))




A Simple Approach

• Function Simple-Proposal-But-Invalid– Start with some assignment between doctors and

hospitals– While unstable pair exists • “swap” to satisfy the pair

– end while




• (A, t), (B, q), (C, s), (D, r) (un-stable pair: (B, t))• -> (A, q), (B, t), (C, s), (D, r)




A Simple Approach

• Function Simple-Proposal-But-Invalid– Start with some assignment between doctors and

hospitals– While unstable pair exists • “swap” to satisfy the pair

– end while


This will NOT work since a loop can occur.Swaps might continually result in new “dissatisfied” pairs.

• The Boston Pool algorithm proceeds in rounds until every position has been filled.

• Each round has two stages:• 1. An arbitrary unassigned hospital A offers its position to the best

doctor x (according to the hospital’s preference list) who has not already rejected it.

• 2. Each doctor ultimately accepts the best offer that she receives, according to her preference list. Thus, if x is currently unassigned, she (tentatively) accepts the offer from A. If x already has an assignment but prefers A, she rejects her existing assignment and (tentatively) accepts the new offer from A. Otherwise, x rejects the new offer.


The Boston Pool Algorithm

Example• Suppose four doctors q,r,s,t, and four

hospitals A,B,C,D rank each other as follows:




Example• Suppose four doctors q,r,s,t, and four

hospitals A,B,C,D rank each other as follows:

• (A, s), (B, t), (C, q), (D, r)






• (A, s), (B, t), (C, q), (D, r)• The matching (A, r), (B, s), (C, q), (D, t) is also stable




Correctness - Termination• Observation 1. Hospitals propose to doctors in

decreasing order of preference.• Observation 2. Once a doctor accepts an offer he/she

never becomes unmatched, he/she only "trades up“.

• Claim. Algorithm terminates after at most n2 rounds.– Pf. Each hospital makes an offer to each doctor at

most once (only make offer to “new” doctor), so the algorithm requires at most n2 rounds.

– The average (expected) case is O(n lg n).


Correctness• Claim. The algorithm always computes a matching

(to all hospitals and doctors)– Pf. It’s obvious that throughout the process, no doctor

can accept more than one position, and no hospital can hire more than one doctor.

– Pf. (by contradiction) Suppose that Hospital A is not matched upon termination of algorithm. Then some doctor x is not matched upon termination.By Observation 2, x never received an offer.But, A made offers to everyone, since A ends up unmatched.


Correctness

• Claim. No unstable pair.– Pf. Suppose doctor x is assigned to hospital A in

the final matching, but prefers B. Because every doctor accepts the best offer she receives, x received no offer she liked more than A. In particular, B never made an offer to x. On the other hand, B made offers to every doctor they like more than y. Thus, B prefers y to x, and so there is no instability.


More properties• Def. x is a feasible doctor for A if there exists a stable

matching that assigns doctor x to hospital A. • Claim. Each hospital A is rejected only by doctors that are

infeasible for A. (Hospital-Optimal)• Pf. (by induction) Consider an arbitrary round of the algorithm, in which

doctor x rejects A for B (B is offering x, x prefers B to A). Every doctor that appears higher than x in B’s preference list has already rejected B and therefore is infeasible for B (why?).Now consider an arbitrary matching that assigns x to A: B prefers x to its partner => unstable. B prefers its partner to x => its partner is infeasible (why?), and again the matching is unstable.


More properties• Def. x is a feasible doctor for A if there exists a stable

matching that assigns doctor x to hospital A. • Claim. Each hospital A is rejected only by doctors that are

infeasible for A. (Hospital-Optimal)• Claim. Each doctor x prefers every other feasible match to its

final assignment A. (Doctor-Pessimal)• Pf. Consider an arbitrary stable matching where A is not matched with

x but with another doctor y.The previous Claim implies that A prefers x to y (why?). Because the matching is stable, x must therefore prefer her assigned hospital to A.


More properties• No matter which unassigned hospital makes an offer in

each round, the algorithm always computes the same matching ( why?)

• A doctor can potentially improve her assignment by lying about her preferences

• NRMP reversed its matching algorithm in 1998– So that potential residents offer to work for hospitals in

preference order, and each hospital accepts its best offer.

– The precise effect of this change on the patients is an open problem.


About its history• Until 1950’s– Competition among hospitals for the best doctors

led to earlier and earlier offers of internships, along with tighter deadlines for acceptance.

– In the 1940s, medical schools agreed not to release information until a common date during their students’ fourth year. In response, hospitals began demanding faster decisions.

– Interns were forced to gamble if their third-choice hospital called first.


In Academia

• For graduate school admission, we have the “April 15 Resolution”.

• However, the academic job market involves similar gambles, at least in computer science.– Some departments start making offers in February

with two-week decision deadlines; others don’t even start interviewing until late March;

– MIT notoriously waits until May, when all its interviews are over, before making any faculty offer.


About its history• In the early 1950’s– A central clearinghouse for internship

assignments, now called the National Resident Matching Program (NRMP), was established

– Each year, doctors submit a ranked list of all hospitals where they would accept an internship, and each hospital submits a ranked list of doctors they would accept as interns.

– The NRMP then computes a stable assignment of interns to hospitals.


It’s not the end of the story

• In reality, most hospitals offer multiple internships, each doctor ranks only a subset of the hospitals and vice versa, and

• There are typically more internships than interested doctors.

• And then it starts getting complicated, moreover, e.g.,– There are couples that willing to stay in the same

city/hospital whenever possible…UNC Chapel Hill Z. Guo

This course

• This class is ultimately about learning two skills that are crucial for all computer scientists:– Intuition: How to think about abstract computation.


This course

• This class is ultimately about learning two skills that are crucial for all computer scientists:– Intuition: How to think about abstract computation.– Language: How to talk about abstract computation.


This course

• This class is ultimately about learning two skills that are crucial for all computer scientists:– Intuition: How to think about abstract computation.– Language: How to talk about abstract computation.

• You can only develop good problem solving skills by solving problems. You can only develop good communication skills by communicating.


Example - Process• Suppose four doctors q,r,s,t, and four hospitals A,B,C,D

rank each other as follows:

• (A,t); (A,t)(B,r); (C,t)(B,r); (C,t)(B,r)(D,s);(A,s)(C,t)(B,r); (A,s)(C,t)(D,r); (A,s)(B,t)(D,r);+(C,r?);+(C,s?); (A,s)(B,t)(D,r)(C,q);




a longer example: stable matching unc chapel hillz. guo

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