Matching and Market Design Chapter 2: Design of Matching Markets Fuhito Kojima1

February 12, 2009

1

Yale University. http://sites.google.com/site/fuhitokojimaeconomics/. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

NRMP redesign

Today we will see how basic matching theory can be used for economic design.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

NRMP redesign

Today we will see how basic matching theory can be used for economic design. But also we will see why the basic theory is not enough, and

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

NRMP redesign

Today we will see how basic matching theory can be used for economic design. But also we will see why the basic theory is not enough, and Other approaches are useful, and

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

NRMP redesign

Today we will see how basic matching theory can be used for economic design. But also we will see why the basic theory is not enough, and Other approaches are useful, and What kind of new theories are called for to tackle complicated issues for economic design.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

NRMP redesign

Today we will see how basic matching theory can be used for economic design. But also we will see why the basic theory is not enough, and Other approaches are useful, and What kind of new theories are called for to tackle complicated issues for economic design. We will discuss redesign of NRMP algorithm in 1990s as a case study.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

A Brief History of NRMP

Began as a decentralized market around 1900.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

A Brief History of NRMP

Began as a decentralized market around 1900. By mid 20th century, the market suffered from unraveling and congestion, causing mismatches.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

A Brief History of NRMP

Began as a decentralized market around 1900. By mid 20th century, the market suffered from unraveling and congestion, causing mismatches. NRMP introduced the centralized matching mechanism (hospital proposing DA) in 1950s.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

A Brief History of NRMP

Began as a decentralized market around 1900. By mid 20th century, the market suffered from unraveling and congestion, causing mismatches. NRMP introduced the centralized matching mechanism (hospital proposing DA) in 1950s. Decline in participation rates in 1970s (especially among couples), and change in the design. High participation rate since the change.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

A Brief History of NRMP

Began as a decentralized market around 1900. By mid 20th century, the market suffered from unraveling and congestion, causing mismatches. NRMP introduced the centralized matching mechanism (hospital proposing DA) in 1950s. Decline in participation rates in 1970s (especially among couples), and change in the design. High participation rate since the change. Crisis in confidence in 1990s.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Crisis in Confidence in 1990s

Groups such as American Medical Students Association, Public Citizen Health Research Group, Medical Student Section of the American Medical Association advocated reconsideration of the algorithm.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Crisis in Confidence in 1990s

Groups such as American Medical Students Association, Public Citizen Health Research Group, Medical Student Section of the American Medical Association advocated reconsideration of the algorithm. The Board of Directors of NRMP commissioned the design of a new algorithm.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Crisis in Confidence in 1990s

Groups such as American Medical Students Association, Public Citizen Health Research Group, Medical Student Section of the American Medical Association advocated reconsideration of the algorithm. The Board of Directors of NRMP commissioned the design of a new algorithm. New algorithm by Roth and Peranson, based on student-proposing DA but accommodating couples and other complications, was introduced from 1998.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Crisis in Confidence in 1990s

Groups such as American Medical Students Association, Public Citizen Health Research Group, Medical Student Section of the American Medical Association advocated reconsideration of the algorithm. The Board of Directors of NRMP commissioned the design of a new algorithm. New algorithm by Roth and Peranson, based on student-proposing DA but accommodating couples and other complications, was introduced from 1998. The algorithm is in use now.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

A sketch of the Roth-Peranson algorithm The new (and current) NRMP algorithm, called the Roth-Peranson algorithm, is based on student-proposing DA, but try to accommodate couples. The algorithm allows couples to express preferences on pairs of hospital programs. First run DA without couples, and then add couples one at a time. If someone is displaced, then such an agent is allowed to apply later in the algorithm. The basic idea is based on Roth and Vande Vate (1989). Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Aside: Some open question Roth and Vande Vate (1989) showed that, starting from any matching, there is a sequence of blocking pairs that leads to a stable matching in one-to-one matching without couples. Kojima and Unver (2008) showed a similar result in many-to-many matching, when one side has substitutable preferences and the other side “responsive” preferences. One conjecture is that the same result holds when every agent has substitutable preferences (it is known that stable matching exists when every agent has substitutable preferences, and it is essentially the weakest such condition.)

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

What were the issues?

1

The NRMP algorithm favors hospital programs at the expense of doctors?

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

What were the issues?

1

The NRMP algorithm favors hospital programs at the expense of doctors? → Yes, since NRMP is the hospital-proposing DA.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

What were the issues?

1

The NRMP algorithm favors hospital programs at the expense of doctors? → Yes, since NRMP is the hospital-proposing DA.

2

NRMP is a “manipulable” system: students should report false preferences to get the best outcome?

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

What were the issues?

1

The NRMP algorithm favors hospital programs at the expense of doctors? → Yes, since NRMP is the hospital-proposing DA.

2

NRMP is a “manipulable” system: students should report false preferences to get the best outcome? → Yes, since both doctors and hospitals may have incentives to manipulate.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

NRMP “match variations”

As suggested before, NRMP has special features, called “match variations,” which is not present in the simple theory. Examples are:

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

NRMP “match variations”

As suggested before, NRMP has special features, called “match variations,” which is not present in the simple theory. Examples are: 1

Couples,

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

NRMP “match variations”

As suggested before, NRMP has special features, called “match variations,” which is not present in the simple theory. Examples are: 1

Couples,

2

Hospital programs that want to fill even number of positions,

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

NRMP “match variations”

As suggested before, NRMP has special features, called “match variations,” which is not present in the simple theory. Examples are: 1

Couples,

2

Hospital programs that want to fill even number of positions,

3

“Reversion”: Hospital programs with positions that revert to other programs if they remain vacant.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

NRMP “match variations”

As suggested before, NRMP has special features, called “match variations,” which is not present in the simple theory. Examples are: 1

Couples,

2

Hospital programs that want to fill even number of positions,

3

“Reversion”: Hospital programs with positions that revert to other programs if they remain vacant.

Some are easy to accommodate, and others pose fundamental problems.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Further problems with match variations

There are problems that happen because there are match variations. 1

Some people are unmatched because of the choice of the algorithm?

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Further problems with match variations

There are problems that happen because there are match variations. 1

Some people are unmatched because of the choice of the algorithm? → No such concern if no match variations are present, but possible otherwise.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Further problems with match variations

There are problems that happen because there are match variations. 1

Some people are unmatched because of the choice of the algorithm? → No such concern if no match variations are present, but possible otherwise.

2

Does NRMP find a stable matching in the first place?

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Further problems with match variations

There are problems that happen because there are match variations. 1

Some people are unmatched because of the choice of the algorithm? → No such concern if no match variations are present, but possible otherwise.

2

Does NRMP find a stable matching in the first place? → A stable matching exists if no match variations are present, but a stable matching may not exist otherwise.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Empirical study 1

Traditional theory points to potential problems even without match variations. 1 2

hospitals are be favored the system are be manipulable

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Empirical study 1

Traditional theory points to potential problems even without match variations. 1 2

2

hospitals are be favored the system are be manipulable

With complex reality (e.g., couples), positive theoretical results do not apply 1

2

different numbers of students may be matched in different stable matchings a stable matching may not exist

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Empirical study 1

Traditional theory points to potential problems even without match variations. 1 2

2

With complex reality (e.g., couples), positive theoretical results do not apply 1

2 3

hospitals are be favored the system are be manipulable

different numbers of students may be matched in different stable matchings a stable matching may not exist

Empirical and numerical studies are useful. So, look at the market!

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Empirical study 1

Traditional theory points to potential problems even without match variations. 1 2

2

With complex reality (e.g., couples), positive theoretical results do not apply 1

2 3

hospitals are be favored the system are be manipulable

different numbers of students may be matched in different stable matchings a stable matching may not exist

Empirical and numerical studies are useful. So, look at the market! → Roth and Peranson obtained data on NRMP such as submitted preferences, and did a number of simulations.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Descriptive statistics of NRMP

APPLICANTS Applicants with ROLs Applicants who are Coupled PROGRAMS Active Programs with ROL Programs with Even Match Total Quota Before Match

Fuhito Kojima

1987

1993

1994

1995

1996

20071 694

20916 854

22353 892

22937 998

24749 1008

3170 4 19973

3622 2 22737

3662 6 22801

3745 7 22806

3758 8 22578

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Difference between hospital proposing and college proposing DAs

APPLICANTS Number of Applicants Affected Applicant Proposing Preferred Program Proposing Preferred New Matched New Unmatched PROGRAMS Number of Programs Affected Applicant Proposing Preferred Program Proposing Preferred Prog. with New Position(s) Filled Prog. with New Unfilled Positions Fuhito Kojima

1987

1993

1994

1995

1996

20 12 8 0 1

16 16 0 0 0

20 11 9 0 0

14 14 0 0 0

21 12 9 1 0

20 8 12 0 1

15 0 15 0 0

23 12 11 2 2

15 1 14 1 0

19 10 9 1 0

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Magnitude of possible manipulations by students Upper limit of the number of applicants who could benefit by truncating their lists at one above their original match point (for students, truncation is known to be “exhaustive”2 )

Program-Proposing Algorithm Applicant-Proposing Algorithm

1987 12 0

1993 22 0

1994 13 2

1995 16 2

1996 11 9

As expected, more applicants can benefit from list truncation under the program-proposing algorithm than under the applicant-proposing algorithm. But both numbers are very small. 2

Roth and Vande Vate (1991). Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Magnitude of possible manipulations by hospitals

Upper limit of the number of hospital programs that could benefit by truncating their lists at one above their original match point (for hospitals, truncation is not exhaustive3 )

Program-Proposing Algorithm Applicant-Proposing Algorithm

1987 15 27

1993 12 28

1994 15 27

1995 23 36

1996 14 18

3

Kojima and Pathak (2009) show that the class of “dropping strategies” is exhaustive. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Magnitude of possible manipulations by capacities Hospitals can also misreport capacities (Sonmez 1997).4 Estimate of the Upper Bound of the Number of Programs That Could Improve Their Remaining Matches By Reducing Quotas

Program Proposing Algorithm Applicant Proposing Algorithm

1987 28 8

1993 16 24

1994 32 16

1995 8 16

1996 44 32

In fact, hospitals can manipulate both ranking and capacities, and it may not need to use truncation (but this was not done by Roth and Peranson). 4 Tayfun Sonmez (1997), “Manipulation via Capacities in Two-Sided Matching Markets,” Journal of Economic Theory, shows that no mechanism is immune to capacity manipulation. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Why is looking at data not sufficient? All results above are suggestive that prediction of simple theories are approximately correct, and some of the potential problems suggested by theories may not be important. But looking at data alone is only suggestive, and not conclusive. We will look at two additional approaches: 1 2

simulation on randomly generated data, and theoretical analysis.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Magnitude of conflict of interest/manipulations Simulation on randomly generated data. Simple model: n hospital programs, n doctors, (no couples). Preferences are drawn independently and uniformly. Each doctor applies to k hospitals. C (n) = number of doctors matched differently at hospital-proposing and doctor-proposing DAs.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Magnitude of possible manipulations

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Theory Based on Kojima and Pathak (see also Immorlica and Mahdian 2005). Finite sets S of students and C of colleges. Each student can be matched to at most one college, and college c can be matched with at most qc students (many-to-one matching). For now, assume there is no match variations (no couple, etc).

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Student-proposing DA Input: Each student submits her preference list. Each college submits its quota and preference list. Step 1: (a) Each student “applies” to her first choice college. (b) Each college tentatively holds the most preferred acceptable applicants up to its quota and rejects the rest. Step t: (a) Each student rejected in Step (t − 1) applies to her next highest choice. (b) Each college considers both new applicants and students held at Step (t-1), tentatively holds the most preferred acceptable students from the combined set of students up to its quota, and rejects the rest. DA terminates when every unmatched student has applied to all her acceptable colleges. Termination in finite time.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Example: DA is not Strategy-Proof Look at an example with manipulation possibilities. Two students {i, j} and two colleges {H(Harvard), Y (Yale)}, with one seat each. i : H, Y , j : Y , H, H : j, i, Y : i, j.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Example: DA is not Strategy-Proof Look at an example with manipulation possibilities. Two students {i, j} and two colleges {H(Harvard), Y (Yale)}, with one seat each. i : H, Y , j : Y , H, H : j, i, Y : i, j.

If everyone is truthful:

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Example: DA is not Strategy-Proof Look at an example with manipulation possibilities. Two students {i, j} and two colleges {H(Harvard), Y (Yale)}, with one seat each. i : H, Y , j : Y , H, H : j, i, Y : i, j.

If everyone is truthful: H is matched to i, Y is matched to j.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Example: DA is not Strategy-Proof Look at an example with manipulation possibilities. Two students {i, j} and two colleges {H(Harvard), Y (Yale)}, with one seat each. i : H, Y , j : Y , H, H : j, i, Y : i, j.

If everyone is truthful: H is matched to i, Y is matched to j. If H declares i unacceptable (0H : j):

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Example: DA is not Strategy-Proof Look at an example with manipulation possibilities. Two students {i, j} and two colleges {H(Harvard), Y (Yale)}, with one seat each. i : H, Y , j : Y , H, H : j, i, Y : i, j.

If everyone is truthful: H is matched to i, Y is matched to j. If H declares i unacceptable (0H : j): H is matched to j, Y is matched to i. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Example: DA is not Strategy-Proof Look at an example with manipulation possibilities. Two students {i, j} and two colleges {H(Harvard), Y (Yale)}, with one seat each. i : H, Y , j : Y , H, H : j, i, Y : i, j.

If everyone is truthful: H is matched to i, Y is matched to j. If H declares i unacceptable (0H : j): H is matched to j, Y is matched to i. College H successfully manipulates DA. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Large Matching Markets There are constants q¯, q˜, k (independent of n). G n is a game of incomplete information such that

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Large Matching Markets There are constants q¯, q˜, k (independent of n). G n is a game of incomplete information such that there are n colleges, with quota at most q¯.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Large Matching Markets There are constants q¯, q˜, k (independent of n). G n is a game of incomplete information such that there are n colleges, with quota at most q¯. there are at most q˜n students.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Large Matching Markets There are constants q¯, q˜, k (independent of n). G n is a game of incomplete information such that there are n colleges, with quota at most q¯. there are at most q˜n students. Preferences of colleges are common knowledge (the result holds under incomplete information as well). Utility uc (S 0 ) for college c of being matched with a set of students S 0 is additive: ( P = s∈S 0 uc ({s}) if |S 0 | ≤ qc , uc (S 0 ) < 0 otherwise. uc ({s}) is always positive (every student is acceptable). The value sup{uc ({s})|n ∈ N, s,c are in G n } is finite.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Large Matching Markets There are constants q¯, q˜, k (independent of n). G n is a game of incomplete information such that there are n colleges, with quota at most q¯. there are at most q˜n students. Preferences of colleges are common knowledge (the result holds under incomplete information as well). Utility uc (S 0 ) for college c of being matched with a set of students S 0 is additive: ( P = s∈S 0 uc ({s}) if |S 0 | ≤ qc , uc (S 0 ) < 0 otherwise. uc ({s}) is always positive (every student is acceptable). The value sup{uc ({s})|n ∈ N, s,c are in G n } is finite. Preferences of students are private information. A student’s preference list is drawn from a uniform distribution over preference lists of length k, independently across students (more general cases are analyzed in the paper.)

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Large Matching Markets There are constants q¯, q˜, k (independent of n). G n is a game of incomplete information such that there are n colleges, with quota at most q¯. there are at most q˜n students. Preferences of colleges are common knowledge (the result holds under incomplete information as well). Utility uc (S 0 ) for college c of being matched with a set of students S 0 is additive: ( P = s∈S 0 uc ({s}) if |S 0 | ≤ qc , uc (S 0 ) < 0 otherwise. uc ({s}) is always positive (every student is acceptable). The value sup{uc ({s})|n ∈ N, s,c are in G n } is finite. Preferences of students are private information. A student’s preference list is drawn from a uniform distribution over preference lists of length k, independently across students (more general cases are analyzed in the paper.) Timing of the game: Students and colleges submit their preference lists and quotas simultaneously. DA is applied under the reported preferences. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Main Result Given ε > 0, a strategy profile is an ε-Nash equilibrium if no player gains more than ε by unilateral deviation.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Main Result Given ε > 0, a strategy profile is an ε-Nash equilibrium if no player gains more than ε by unilateral deviation. Theorem For any ε > 0, there exists n such that truth-telling by every agent is an ε-Nash equilibrium for any game with more than n colleges.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Main Result Given ε > 0, a strategy profile is an ε-Nash equilibrium if no player gains more than ε by unilateral deviation. Theorem For any ε > 0, there exists n such that truth-telling by every agent is an ε-Nash equilibrium for any game with more than n colleges. The theorem may explain success of DA in applications.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Main Result Given ε > 0, a strategy profile is an ε-Nash equilibrium if no player gains more than ε by unilateral deviation. Theorem For any ε > 0, there exists n such that truth-telling by every agent is an ε-Nash equilibrium for any game with more than n colleges. The theorem may explain success of DA in applications. There is also a result regarding the “counting analysis” by Roth and Peranson.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Main Result Given ε > 0, a strategy profile is an ε-Nash equilibrium if no player gains more than ε by unilateral deviation. Theorem For any ε > 0, there exists n such that truth-telling by every agent is an ε-Nash equilibrium for any game with more than n colleges. The theorem may explain success of DA in applications. There is also a result regarding the “counting analysis” by Roth and Peranson. Theorem The expected proportion of colleges that can manipulate DA when others are truthful goes to zero as the number of colleges goes to infinity.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Main Result Given ε > 0, a strategy profile is an ε-Nash equilibrium if no player gains more than ε by unilateral deviation. Theorem For any ε > 0, there exists n such that truth-telling by every agent is an ε-Nash equilibrium for any game with more than n colleges. The theorem may explain success of DA in applications. There is also a result regarding the “counting analysis” by Roth and Peranson. Theorem The expected proportion of colleges that can manipulate DA when others are truthful goes to zero as the number of colleges goes to infinity. The expected proportion of colleges that are matched to the same set of students in all stable matchings goes to one as the number of colleges goes to infinity. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Intuition DA is strategy-proof for students, so truthtelling is an optimal strategy for students. Strategic rejection by a college causes a chain of application and rejections. Some of the rejected students may apply to the manipulating college, and the college may be made better off if these new applicants are desirable. In a large market, there is a high probability that there will be many colleges with vacant positions. So the students who are strategically rejected (or those who are rejected by them and so on) are likely to apply to those vacant positions and be accepted. So the manipulating college is unlikely to be made better off. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Sketch of Proof (Step 1): Dropping Strategy (0c , qc0 ) is a dropping strategy of (c , qc ) if (1) qc0 = qc , and (2) 0c drops some acceptable students from c , but does not change orders between remaining students. Lemma If c cannot manipulate student-proposing DA successfully by a dropping strategy, then c cannot manipulate it successfully by any strategy. This lemma simplifies analysis by narrowing down the class of strategies to consider.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Sketch of Proof (Step 2): Rejection Chains Given c and dropping strategy 0c , consider rejection chains algorithm, an algorithm similar to student-proposing DA: (1) First, run DA under true preferences. (2) Then let c reject students matched to c who are unacceptable under 0c . Each rejected student applies to next choice, just as in DA. The rest proceeds as in DA. The rejection chain returns to c if some student applies to c at Step (2). Lemma If no rejection chains return to c, then no dropping strategies are successful manipulations for c.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Sketch of Proof (Step 3): Vanishing Market Power

Lemma (Vanishing market power) For any ε > 0, if the number of colleges n in the market is sufficiently large, Pr(at least one rejection chain returns to c) < ε for any college c in the market. Intuition: In a large market, with high probability there are many colleges with vacant positions. So the rejected students (or those who are rejected by them and so on) usually apply to those vacant positions and are accepted, ending a rejection chain. Lemmas 1-3 show the theorem. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Couples

In the theory above, we have focused on simple matching markets and saw manipulations become unimportant in such markets, and conflict of interest between hospitals and students become small. But in NRMP and in other markets, couples (and other match variations) make it possible for nonexistence of stable matchings, and failure of the rural hospital theorem. Also the above conclusions, such as non-manipulability of DA in large markets, are not directly applicable. Even worse, DA may not be strategy-proof even for students.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Couples Here is a quote from Roth (2008):5 [An] empirical observation made in the resident match data, and in the other matches ... is that, even when couples are present, it is a very rare occurrence for the set of stable matchings to be empty. ... An open question is why this is so. I offer the following loose conjecture: Conjecture In the limit, as n goes to infinity in a regular sequence of random markets in which the proportion of couples is bounded [or, goes to zero], the probability that the set of stable matchings is empty goes to zero. 5 Roth, Alvin E. ”Deferred Acceptance Algorithms: History, Theory, Practice, and Open Questions,” International Journal of Game Theory, Special Issue in Honor of David Gale on his 85th birthday, 36, March, 2008, 537-569. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Matching with couples: Theory Kojima, Pathak and Roth (in progress) consider a model similar to Kojima and Pathak but assume there are a small number of couples. Theorem The probability that there exists a stable matching converges to one, as the size of the market (number of colleges) goes to infinity with the number of couples being fixed. Theorem For any ε > 0, there exists n such that truth-telling by every agent is an ε-Nash equilibrium under the Roth-Peranson algorithm for any game with more than n colleges. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Intuition for Existence

Roth-Peranson algorithm will find a stable matching if couples are not displaced by another couple or single doctors. In a large market, there is a high probability that there will be many colleges with vacant positions. So couples and singles are unlikely to apply and displace a couple in a hospital. So the algorithm is likely to terminate, producing a stable matching.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Some other match variations and Substitutable Preferences

Some variations are rather innocuous but we need more sophisticated models. Reversion makes hospitals’ preferences violate assumptions in the simple model, but it is still in the class of so-called “substitutable preferences” (Kelso and Crawford 1982, RS chapter 6). If hospital preferences are substitutable, then DA will find a stable matching (Kelso and Crawford 1982). Moreover, substitutability is the “weakest” condition that guarantees existence of a stable matching (Sonmez and Unver 2009).

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

A preference relation of a college c is substitutable if a student is chosen from a group of students S 0 , then she is still chosen from a subset S 00 of students in S 0 . Some match variations, such as couples and even positions, violate this condition, while others, such as reversion, still satisfy substitutability.6

6

To be more precise, couples and reversion is outside the simple matching model. The “matching with contracts” model of Hatfield and Milgrom (2005) subsumes these variations, and one natural generalization of substitutability guarantees existence of stable matchings. See Hatfield and Kojima (2007, 2008) too. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Manipulation of DA Stable matchings with couples

Large Matching Market Design: One-sided matching One-sided matching. Examples: (1) some student placement mechanisms (NYC supplementary round, Boston), (2) house allocation in colleges. The random serial dictatorship (RSD) is used in many markets. RSD is strategy-proof, but there is efficiency loss in RSD (ex ante). The probabilistic serial (PS) mechanism has better efficiency properties than RSD, but is not strategy-proof. Theorem (i) If the number of copies of each object is sufficiently large, then truthtelling is a dominant strategy for agents under PS. (ii) PS and RSD are asymptotically equivalent. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Summary DA is not strategy-proof, and there are also some other potential problems. Why is the mechanism adopted in applications? We took NRMP as case study and studied how big such problems are. Numerical studies based on real data and simulation suggest that they are not large problems. Inspired by observations in such markets, some new theories are developed to evaluate performance of stable mechanisms. Open questions: other match variations, relatively small markets, etc. Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M

NRMP design Theory Summary

Reading for next class

We will learn the one-sided matching problem next week, with such applications as college dorm allocation, kidney exchange and (later) school choice. Much of the class will be based on Sonmez and Unver, “Matching, Allocation, and Exchange of Discrete Resources,” available from the website, http://www2.bc.edu/ unver/ This is a survey article that covers a lot of material from the classical theory to recent applications.

Fuhito Kojima

Matching and Market Design Chapter 2: Design of Matching M