The document describes the stable matching problem and the Gale-Shapley deferred acceptance algorithm. It provides definitions for stable matching, blocking pairs, and Pareto optimality. The deferred acceptance algorithm is presented as matching doctors to hospitals in rounds, with doctors proposing to their most preferred hospitals and hospitals either accepting the doctor or retaining their current match if they prefer them. The algorithm is proven to always produce a stable matching in O(n2) time. Examples are provided to illustrate how the algorithm works in practice.

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questions!
CS269I
Lecture 2: Stable Matching
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2. Please ask
questions!
Last time: Matching students to dorms
Sandra ‘50
Sally ‘73
Susan ‘86
Toyon Stern Branner
Key point:
Only Sandra knows* how
happy she would be in
each dorm. So we rely on
her to tell the algorithm.
* Does Sandra know that?
How do you measure happiness anyway? 2

3. Please ask
questions!
Last time: Matching students to dorms
Definition (mechanism):
Solicit inputs + algorithm + action
Definition (Strategyproof / Truthful):
A mechanism is strategyproof/truthful if misreporting preferences can never make a
participant better off.
Definition (Pareto-optimal):
An assignment 𝐴 is Pareto-optimal if for any* other assignment 𝐵,
there is a participant that (strictly) prefers 𝐴 over 𝐵.
Theorem:
Serial Dictatorship is a truthful mechanism that makes Pareto-optimal assignments.
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* any other assignment 𝐵 such that at least
one participant strictly prefers 𝐵 to 𝐴

4. Please ask
questions!
Today: the hospital residency problem
• After completing 4 years of undergrad and 4 years of med school,
med students are finally ready to start their internship (“residency”)
• Each applicant has a preference over different residency programs
• Each program has a preference over the applicants
How should you match applicants to residencies?
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Simplifying assumption today:
Each program has 1 slot

5. Please ask
questions!
Today: the hospital residency problem
• After completing 4 years of undergrad and 4 years of med school,
med students are finally ready to start their internship (“residency”)
• Each doctor has a preference over different hospitals
• Each hospital has a preference over the doctors
How should you match doctors with hospitals?
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Simplifying assumption today:
Each hospital has 1 slot

6. Please ask
questions!
Dorms vs Hospitals
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Sandra ‘50
Sally ‘73 Stern
Branner
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questions!
Dorms vs Hospitals
Main difference: dorms don’t have preference over students.
Another important difference: centralized vs optionally centralized
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Must be incentivized
to participate!

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questions!
Stable Matchings
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questions!
Unstable matching
Def (blocking pair):
Given Matching M, (Doctor i, Hospital j) are a blocking pair
if they prefer each other to their assignment in M
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n n
Stanford
wants
Doctor n
n really
wants
Stanford

10. Please ask
questions!
Stable Matching
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n n
Def (blocking pair):
Given Matching M, (Doctor i, Hospital j) are a blocking pair
if they prefer each other to their assignment in M
Def (stable matching):
M is a stable matching if there are no blocking pairs.

11. Please ask
questions!
Stable Matching
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Def (blocking pair):
Given Matching M, (Doctor i, Hospital j) are a blocking pair
if they prefer each other to their assignment in M
Def (stable matching):
M is a stable matching if there are no blocking pairs.
For every unmatched pair (i,j):
• Doctor i prefers Hospital M(i) over Hospital j, or;
• Hospital j prefers Doctor M(j) over Doctor i
equivalent

12. Please ask
questions!
Stable Matching
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Def (blocking pair):
Given Matching M, (Doctor i, Hospital j) are a blocking pair
if they prefer each other to their assignment in M
Def (stable matching):
M is a stable matching if there are no blocking pairs.
For every unmatched pair (i,j):
• Doctor i prefers Hospital M(i) over Hospital j, or;
• Hospital j prefers Doctor M(j) over Doctor i
equivalent
#mustknowthisdef!

13. Please ask
questions!
Stable Matching
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Def (blocking pair):
Given Matching M, (Doctor i, Hospital j) are a blocking pair
if they prefer each other to their assignment in M
Def (stable matching):
M is a stable matching if there are no blocking pairs.
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In a stable matching,
hospitals can’t find better
doctors outside the match
Key point: centralized vs optionally centralized

14. Please ask
questions!
Stable Matching
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Def (blocking pair):
Given Matching M, (Doctor i, Hospital j) are a blocking pair
if they prefer each other to their assignment in M
Def (stable matching):
M is a stable matching if there are no blocking pairs.
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In a stable matching,
hospitals can’t find better
doctors outside the match
Key point: centralized vs optionally centralized
Stable matchings are important (because of )
But can we find them efficiently?

15. Please ask
questions!
Deferred Acceptance Algorithm
[Gale Shapley ‘62]
An algorithm that finds stable matchings!

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questions!
Main idea: try to match each doctor to most favorite choice;
if you discover a blocking pair, just switch the matching!
Deferred Acceptance Algorithm
Almost-pseudo-code*:
While there is an unmatched doctor i:
Try to match Doctor i to next-favorite hospital in her list;
If this hospital doesn’t have a doctor yet:
Both Doctor i and hospital are happy with this new match ☺
Else-if this hospital prefers its current match i’ over i:
Doctor i remains unmatched
Else-if this hospital prefers i over i’:
Unmatch i’; Match (i, hospital)
* there is a hidden slide with more
detailed pseudocode and runtime analysis

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questions!
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freeDoctors ← Doctors
for all d in Doctors:
d.current ← 0
for all h in Hospitals:
h.D ← NIL
Deferred-Acceptance(Doctors,Hospitals):
// initialize
Deferred Acceptance Algorithm
while (exists d in freeDoctors)
h ← d.rank(d.current++)
if (h.free = true)
h.D ← d; h.free ← false
remove d from freeDoctors
else-if (h.rank(d) < h.rank(h.D))
add h.D to freeDoctors
h.D ← d
remove d from freeDoctors
return (h,h.D) for all h in Hospitals
// main loop
// h prefers d to
previous match
// h is d’s
next favorite
Running time:
Each iteration of
while loop = O(1)
Each iteration:
We +1 d.current
for some doctor
We always have:
d.current ≤ 𝑛
for every doctor
Therefore, total
run-time = 𝑂 𝑛2

18. Please ask
questions!
Example run-through
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19. Please ask
questions!
DA Example Run 1
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Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, B, C
B, C, A

20. Please ask
questions!
DA Example Run 1
20
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, B, C
B, C, A

21. Please ask
questions!
DA Example Run 1
21
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, B, C
B, C, A

22. Please ask
questions!
DA Example Run 1
22
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, B, C
B, C, A

23. Please ask
questions!
DA Example Run 1
23
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, B, C
B, C, A

24. Please ask
questions!
DA Example Run 1
24
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, B, C
B, C, A

25. Please ask
questions!
Another example
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26. Please ask
questions!
DA Example Run 2
26
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, C, B
B, C, A

27. Please ask
questions!
DA Example Run 2
27
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, C, B
B, C, A

28. Please ask
questions!
DA Example Run 2
28
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, C, B
B, C, A

29. Please ask
questions!
DA Example Run 2
29
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, C, B
B, C, A

30. Please ask
questions!
DA Example Run 2
30
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, C, B
B, C, A

31. Please ask
questions!
DA Example Run 2
31
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, C, B
B, C, A

32. Please ask
questions!
DA Example Run 2
32
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, C, B
B, C, A

33. Please ask
questions!
33
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, C, B
B, C, A
DA Example Run 2

34. Please ask
questions!
Main idea: try to match each doctor to most favorite choice;
if you discover a blocking pair, just switch the matching!
Deferred Acceptance Algorithm
Almost-pseudo-code:
While there is an unmatched doctor i:
Try to match i to next-favorite hospital;
If this hospital doesn’t have a doctor yet:
Match Doctor i and this hospital
Else-if this hospital prefers its current match i’ over i:
Doctor i remains unmatched
Else-if this hospital prefers i over i’:
Unmatch i’; Match (i, hospital)
Run-time analysis
• Each iteration: 𝑶(𝟏) time
• Every iteration, doctor “proposes” to
new hospital
• Never try to match same doctor and
hospital twice
• 𝑛2 possible (doctor,hospital) pairs
• Therefore, at most 𝒏𝟐 iterations
• Therefore, total 𝑶 𝒏𝟐

35. Please ask
questions!
Questions about DA before we analyze it?
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Deferred Acceptance Almost-pseudo-code:
While there is an unmatched doctor i:
Try to match i to next-favorite hospital;
If this hospital doesn’t have a doctor yet:
Match Doctor i and this hospital
Else-if this hospital prefers its current match i’ over i:
Doctor i remains unmatched
Else-if this hospital prefers i over i’:
Unmatch i’; Match (i, hospital)
Main idea: try to match each
doctor to most favorite choice;
if you discover a blocking pair,
just switch the matching!
Def (blocking pair):
(Doctor i, Hospital j) are a
blocking pair if they prefer each
other to their assignment in M
#mustknowthisdef!

36. Please ask
questions!
Deferred Acceptance works!
Theorem: Given n doctors and n hospitals,
DA algorithm outputs a complete stable matching.
Corollary: A stable matching exists.
(This is not obvious!)
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questions!
Proof of Theorem
Theorem: Given n doctors and n hospitals,
DA algorithm outputs a complete stable matching.
Proof: Follows from Claims 1+3 below…
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Claim 1: At every iteration, current match is stable
w.r.t. non-free doctors and hospitals.
Claim 2: Once a hospital is matched, it remains matched
(possibly to a different doctor) until end of algorithm.
Claim 3: At the end of algorithm, every doctor/hospital is matched.

38. Please ask
questions!
Proof of claims
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Claim 1:
At every iteration, current match is stable
w.r.t. non-free doctors and hospitals.
Proof by contradiction:
Suppose (d,h) blocking pair.
→ d is currently matched worse than h.
→ d already tried to match to h.
→ h either refused d or left d later. Why?
→ h must be matched to better than d –
contradiction!
Think-pair-share:
Prove these!
Claim 3: At the end of algorithm, every
doctor/hospital is matched.
Proof by contradiction: Suppose (d,h) free.
End of algorithm → d already tried h.
→ after that step, h wasn’t free
→ by Claim 2, contradiction!
Claim 2: Once a hospital is matched,
it remains matched (possibly to a different
doctor) until end of algorithm.
“Proof”: obvious from algorithm

39. Please ask
questions!
Deferred Acceptance works!
Theorem: Given n doctors and n hospitals,
DA algorithm outputs a complete stable matching.
Proof: Follows from Claims 1+3 below…
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Claim 1: At every iteration, current match is stable
w.r.t. non-free doctors and hospitals.
Claim 2: Once a hospital is matched, it remains matched
(possibly to a different doctor) until end of algorithm.
Claim 3: At the end of algorithm, every doctor/hospital is matched.

40. Please ask
questions!
Deferred Acceptance Algorithm recap
Blocking Pair: A doctor and hospital that prefer each other over their
respective matches.
Stable Matching: A matching without blocking pairs!
Deferred Acceptance Algorithm
“Tentatively match each free doctor to best interested hospital.
Allow the hospital to leave match when a better doctor arrives.”
Runs in time 𝑂 𝑛2 = linear in input size ☺
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41. Please ask
questions!
Deferred Acceptance as a mechanism
We have an efficient algorithm!
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42. Please ask
questions!
The optimal matching?
DA algorithm found a stable matching…
• Is it optimal?
• What does optimality mean?
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Theorem: Every stable matching is Pareto-optimal
Reminder (Pareto-optimal):
An assignment 𝐴 is Pareto-optimal if for any* other assignment 𝐵,
there is a participant (doctor or hospital) that prefers 𝐴 over 𝐵.

43. Please ask
questions!
The optimal matching?
Proof:
Suppose 𝐴 is stable, and let 𝐵 be any other matching.
Consider a doctor-hospital pair matched in 𝐵, and not matched in 𝐴:
By stability of 𝐴, either the doctor or the hospital must prefer their match in 𝐴.
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Theorem: Every stable matching is Pareto-optimal
Reminder (Pareto-optimal):
An assignment 𝐴 is Pareto-optimal if for any* other assignment 𝐵,
there is a participant (doctor or hospital) that prefers 𝐴 over 𝐵.

44. Please ask
questions!
The optimal stable matching?
Theorem: The matching returned by DA is doctor-optimal,
In other words, every doctor is matched
to favorite hospital possible in any stable matching.
Corollary: DA is doctor-strategyproof
I.e. doctors cannot gain from misreporting their preferences.
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45. Please ask
questions!
The optimal stable matching?
Theorem: The matching returned by DA is hospital-worst,
In other words, every hospital is matched
to least favorite doctor possible in any stable matching.
Caution: DA is NOT hospital-strategyproof
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46. Please ask
questions!
The optimal stable matching?
Caution: DA is NOT hospital-strategyproof
46
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, C, B
B, C, A
Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, B, C
B, C, A
Example 1 (true preferences):
Hospital Y matched to 2nd choice doctor
Example 2 (Hospital Y misreporting):
Hospital Y matched to top choice doctor

47. Please ask
questions!
Recap: DA as a mechanism
Pareto optimality:
Every stable matching is Pareto-optimal.
Optimality-among stable matchings
DA’s matching is doctor-optimal + hospital-worst
Deferred Acceptance and strategic behavior
1. DA is doctor-strategyproof
2. DA is not hospital-strategyproof
3. Stable matching = no (doctor,hospital) prefer to match outside
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Think-Pair-Share:
How would you change DA to
make it hospital-optimal (and
hospital-strategyproof*)?
Which one (doctor-optimal or
hospital-optimal) is better?
* Everything we discussed so far also applies to hospitals with >1 slot.
But with >1 slot, there are no stable hospital-strategyproof mechanisms.

48. Please ask
questions!
Deferred Acceptance in practice
48

49. Please ask
questions!
College admission
US system:
• Universities in a decentralized system
Q: How many students to admit?
A: Estimate based on historical data
• “Whole person review” -> application fees $$ -> limit # of apps/student
• Students strategize over where to apply (“safety choices”)
In many other countries:
• Admission mostly determined by standardized test scores
• DA-like mechanisms used in Brazil, China, Hungary,
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Aka apply to some “safety” (less competitive) schools
instead of your favorite-but-competitive choices

50. Please ask
questions!
Doctors-hospitals
How do doctors rank hospitals?
• 1950’s: DA-like National Resident Match Program (NRMP)
• At the time, very few female doctors (not to mention openly gay doctors)
• 1960’s: more couples - want to be near each other (allegedly ;))
• Aka doctors no longer have a ranked preference over hospitals
• 1980’s: Negative theory results on stable matching with couples
• Stable matching may not exist 
• NP-complete computational problem 
• 1990’s: Practical successful extension of DA for accommodating couples ☺ 50

51. Please ask
questions!
Doctors-hospitals
How do hospitals rank doctors?
• Interviews
Costly process: strategize over which doctors to interview (“safety choices”)
• Standardized tests?
(Probably not anymore - USMLE switched to pass/fail in Jan 2022…)
• Letters of rec, med school eval, personal statement and CV
Hospitals also strategically use “safety choices” when ranking doctors post-interview.
DA is not hospital-strategyproof, but…
Theorem: In DA, using safety choices is never safer for hospitals! 51

52. Please ask
questions!
Doctors-hospitals
How do hospitals rank doctors?
DA is not hospital-strategyproof, but it’s generally not easy to game the match:
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Alice
Bob
Charlie
X, Y, Z
Y, X, Z
Y, Z, X
X
Y
Z
B, A, C
A, C, B
B, C, A
Recall example:
Y switched B and C to match with A.
Needed detailed knowledge of
everyone’s preferences for this to work!

53. Please ask
questions!
Doctors-hospitals
How do hospitals rank doctors?
Theorem: In DA, using safety choices is never safer for hospitals!
Formally, manipulating true rank to move i-th doctor (“safety choice”)
to higher position cannot help matching with doctor of rank i-or-better.
Proof:
• Until Doctor i tries to match with hospital, manipulation has no effect.
• After Doctor i tries to match, they can only be replaced by a better doctor.
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54. Please ask
questions!
Doctors-hospitals
How do hospitals rank doctors?
Theorem: In DA, using safety choices is never safer for hospitals!
Hospitals still strategically use “safety choices” when ranking doctors post-interview.
Why?!
Possible explanations:
1. They didn’t pay attention during CS269I lecture
2. Reputation/ego
54

55. Please ask
questions!
Doctors-hospitals
How do hospitals rank doctors?
Theorem: In DA, using safety choices is never safer for hospitals!
Hospitals still strategically use “safety choices” when ranking doctors post-interview.
Why?!
Possible explanations:
1. They didn’t pay attention during CS269I lecture
2. Reputation/ego - “number needed to fill” statistic
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“Number needed to fill” –
the lowest rank in hospital’s list
of a doctor matched to hospital.

56. Please ask
questions!
Doctors vs Packets
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57. Please ask
questions!
Doctors vs Packets
• Suppose that instead of doctors and hospitals, you
want to match packets to servers on the internet.
57
n n
Lecture2.pptx
Lecture2.pdf

58. Please ask
questions!
Doctors vs Packets
• Suppose that instead of doctors and hospitals, you
want to match packets to servers on the internet.
• When you own all the servers, you don’t have to worry
about them matching outside your algorithm...
• But it turns out that Deferred Acceptance is just very
fast in practice ☺
58
n n
Lecture2.pdf

59. Please ask
questions!
Doctors vs Packets
• Suppose that instead of doctors and hospitals, you
want to match packets to servers on the internet.
• When you own all the servers, you don’t have to worry
about them matching outside your algorithm...
• But it turns out that Deferred Acceptance is just very
fast in practice ☺
59

60. Please ask
questions!
Stanford Marriage Pact
60

61. Please ask
questions!
Stanford Marriage Pact
• Matches between Stanford students who want to make a pact:
“If we don’t get married by time X, we’ll marry each other.”
• Historically, Gale-Shapley’s original paper talked about Stable Marriage
• Men = doctors; women = hospitals.
• Original Marriage Pact used variant of Deferred Acceptance
• It doesn’t any more…
61

62. Please ask
questions!
DA Applications recap
Challenges:
• Costly evaluations (US colleges, medical interviews)
• Strategic choice of candidates/applications
• Preference assumption doesn’t always hold
• Couples want close assignments
• Reputation/ego (“number needed to fill”)
Theorem: In DA, “safety choices” are never “safer”.
Opportunities:
• Distributed implementation, short lists – fast in practice! 62

63. Please ask
questions!
Recap
63

64. Please ask
questions!
Pre-recap announcements
Next lecture: Top trading cycles
Housing for graduate students (+ saving lives!)
To-do:
• Week 1 short quiz!
64

65. Please ask
questions!
Stable Matching lecture recap (1/2)
Last lecture: 1-sided matching (dorms don’t have preferences!)
vs
Today: 2-sided matching (hospitals, colleges, etc have preferences!)
Blocking Pair: A pair that prefer each other over their respective matches.
Stable Matching: A matching without blocking pairs.
Deferred Acceptance Algorithm
“Tentatively match each free doctor to best interested hospital.
Allow the hospital to leave match when a better doctor arrives.”
Runs in time 𝑂 𝑛2
. Very fast in practice!
65

66. Please ask
questions!
Stable Matching lecture recap (2/2)
Deferred Acceptance in theory:
• Pareto optimal among all matchings
• Doctor-optimal and hospital-worst among stable assignments
• Doctor-strategyproof, but hospital gaming is possible
Deferred Acceptance in practice:
• Costly evaluations (US colleges, medical interviews)
• Preferences not captured by our model (e.g. couples)
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