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Matching Markets. Jonathan Levin Economics 136 Winter 2010. National Residency Match. Doctors in U.S. and other countries work as hospital “residents” after graduating from medical school. In the US, about 15,000 US med students and many foreign-trained doctors seek residencies each year.

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Matching markets

Matching Markets

Jonathan Levin

Economics 136

Winter 2010


National residency match
National Residency Match

  • Doctors in U.S. and other countries work as hospital “residents” after graduating from medical school.

    • In the US, about 15,000 US med students and many foreign-trained doctors seek residencies each year.

    • About 4000 hospitals try to fill 20,000+ positions.

  • Market operates as a central clearinghouse

    • Students apply and interview at hospitals in the fall.

    • In February, students and hospitals state their preferences

      • Each student submits rank-order list of hospitals

      • Each hospital submits rank-order list of students

    • Computer algorithm generates an assignment.

  • Why run a market this way? Does it make sense?


History of nrmp
History of NRMP

  • It wasn’t always this way.

    • Historically, medical students found residencies through a completely decentralized process.

    • But there were problems: students and hospitals made contracts earlier and earlier, eventually in second year of med school!

  • Hospitals decided to change the system by adopting a centralized clearinghouse.

    • National Residency Matching Program (NRMP) adopted, after various adjustments in 1952.

    • System has persisted, though with some modification in late 1990s to handle couples and some recent debate about salaries.

  • In the early 1980s, it was realized that the NRMP was using an algorithm proposed by David Gale and Lloyd Shapley in 1962. Properties they discovered may help explain NRMP success.


School choice
School Choice

  • Most US cities have historically assigned children to neighborhood schools.

  • Recently, many cities have adopted school choice programs, that try to account for the preferences of children and their parents.

  • School authorities hope choice will lead to more efficient placements without sacrificing fairness or creating confusion.

  • Problem seems similar to residency assignment

    • The most commonly used mechanism, however, is quite different, and arguably has less desirable properties.

    • Maybe cities should redesign their programs? In fact, NYC recently adopted Gale-Shapley algorithm.

    • As it turns out, school choice is not exactly the same as the residency problem, and maybe improvements are still possible…


Kidney exchanges
Kidney Exchanges

  • More than 75,000 people in the United States are waiting to receive a kidney transplant.

  • There is a shortage of donors

    • Deceased donors (maybe 10,000 a year)

    • Living donors (maybe 7000 a year).

    • In 2005, 4200 patients died on the wait list.

  • Problem is not just straight supply and demand

    • Donor kidney needs to be compatible with the patient.

    • So sometimes patient has a living donor, but can’t use the kidney because of incompatibility.

    • Maybe two patients could trade donor kidneys, or several patients could engage in a kidney exchange.

  • It turns out that matching theory can also help us understand this problem, and make optimal use of a limited pool of donors.


Matching applications galore
Matching applications galore!

  • What are the common features of these problems?

    • Two sides of the market to be matched.

    • Participants on at least one side, and sometimes on both sides care about to whom they are matched.

    • For whatever reason, money cannot be used to determine the assignment. (Why not?)

  • Other examples

  • College admissions

  • Judicial clerkships

  • Military postings

  • NCAA football bowls

  • Housing assignment

  • Fraternity/sorority rush

  • MBA course allocation

  • Dating websites



Marriage model
Marriage Model

  • Participants

    • Set of men M, with typical man mM

    • Set of women W, with typical woman w W.

    • One-to-one matching: each man can be matched to one woman, and vice-versa.

  • Preferences

    • Each man has strict preferences over women, and vice versa.

    • A woman w is acceptable to m if m prefers w to being unmatched.


Matching
Matching

  • A matching is a set of pairs (m,w) such that each individual has one partner.

    • If the match includes (m,m) then m is unmatched.

  • A matching is stableif

    • Every individual is matched with an acceptable partner.

    • There is no man-woman pair, each of whom would prefer to match with each other rather than their assigned partner.

  • If such a pair exists, they are a blocking pairand the match is unstable.


Examples
Examples

  • Two men m,m’ and two women w,w’

  • Example 1

    • m prefers w to w’ and m’ prefers w’ to w

    • w prefers m to m’ and w’ prefers m’ to m

    • Unique stable match: (m,w) and (m’,w’)

  • Example 2

    • m prefers w to w’ and m’ prefers w’ to w

    • w prefers m’ to m and w’ prefers m to m’

    • Two stable matches {(m,w),(m’,w’)} and {(m,w’),(m’,w)}

    • First match is better for the men, second for the women.

  • Is there always a stable match?


Deferred acceptance algorithm
Deferred Acceptance Algorithm

  • Men and women rank all potential partners

  • Algorithm

    • Each man proposes to highest woman on his list

    • Women make a “tentative match” based on their preferred offer, and reject other offers, or all if none are acceptable.

    • Each rejected man removes woman from his list, and makes a new offer.

    • Continue until no more rejections or offers, at which point implement tentative matches.

  • This is the “man-proposing” version of the algorithm; there is also a “woman proposing” version.



Stable matchings exist
Stable matchings exist

Theorem.The outcome of the DA algorithm is a stable one-to-one matching (so a stable match exists).

Proof.

  • Algorithm must end in a finite number of rounds.

  • Suppose m, w are matched, but m prefers w’.

    • At some point, m proposed to w’ and was rejected.

    • At that point, w’ preferred her tentative match to m.

    • As algorithm goes forward, w’ can only do better.

    • So w’ prefers her final match to m.

  • Therefore, there are NO BLOCKING PAIRS.


Aside the roommate problem
Aside: the roommate problem

  • Suppose a group of students are to be matched to roommates, two in each room.

  • Example with four students

    • A prefers B>C>D

    • B prefers C>A>D

    • C prefers A>B>D

    • No stable match exists: whoever is paired with D wants to change and can find a willing partner.

  • So stability in matching markets is not a given, even if each match involves just two people.


Why stability
Why stability?

  • Stability seems to explain at least in part why some mechanisms have stayed in use.

    • If a market results in stable outcomes, there is no incentive for re-contracting.

    • Roth (1984) argues that stability of NRMP (which uses Gale-Shapley) helps explain why it has “stuck” as an institution.

    • When we look at related markets, many though not all unstable matching mechanisms have failed.

  • What would be an alternative?


Decentralized market
Decentralized market

  • What if there is no clearinghouse?

    • Men make offers to women

    • Women consider their offers, perhaps some accept and some reject.

    • Men make further offers, etc..

  • What kind of problems can arise?

    • Maybe w holds m’s offer for a long time, and then rejects it, but only after market has cleared.

    • Maybe m makes exploding offer to w and she has to decide before knowing her other options.

    • In general, no guarantee the market will be orderly…


Priority matching
Priority matching

  • Under priority matching, men and women submit preferences,

    • Each man-woman pair is given a priority based on their mutual rankings.

    • The algorithm matches all priority 1 couples and takes them out of the market.

    • New priorities are assigned and process iterates.

  • Example:

    • Assign priority based on product of the two rankings, so that priority order is 1-1, 2-1, 1-2, 1-3, 3-1, 4-1, 2-2, 1-4, 5-1, etc…

    • Algorithm implements all “top-top” matches, then conditional top-tops, etc. When none remain, look for 2-1 matches, etc.

  • Will this lead to a stable matching?


Failure of priority matching
Failure of priority matching

  • Roth (1991, AER) studied residency matches in Britain, which are local and have used different types of algorithms --- a “natural experiment”.

  • Newcastle introduced priority matching in 1967.

    • By 1981, 80% of the preferences submitted contained only a single first choice.

    • The participants had pre-contracted in advance!

  • This is the type of “market unraveling” that plagued the US residency market prior to the NRMP.

  • We’ll have more to say about unraveling later.



Roth kagel experiment
Roth-Kagel Experiment

  • Idea: use laboratory setting to “hold fixed” the environment and see if participants prefer stable mechanisms.

  • Structure of experiment:

    • Create a matching market that is inefficient and suffers from unraveling.

    • Offer participants the option of waiting to use a centralized clearinghouse.

    • See if participants choose to wait for the clearinghouse that leads to stable outcomes.

  • Reported in Kagel and Roth (QJE, 2000).


Kagel roth experiment
Kagel-Roth Experiment

  • The environment

    • Twelve subjects (6 “firms” and 6 “workers”)

    • Half of each are “high productivity”

    • Subject payoffs depend on their match.

    • Stability involves “high-high”, “low-low” matches.

  • The experimental market

    • Three periods, with small cost of early contracting.

    • Each period, a firm can make one offer.

    • If a worker accepts, match is made and is final.

    • All matches announced after each period.

  • After ten runs of the market, introduce a clearinghouse

    • Participants not matched before last period are matched using either DA (variant 1) or PM (variant 2).





Optimal stable matchings
Optimal stable matchings

  • A stable matching is man-optimal if every man prefers his partner to any partner he could possibly have in a stable matching.

    Theorem.The man-proposing DA algorithm results in a man-optimal stable matching.

  • This matching is also woman-pessimal (each woman gets worst outcome in any stable matching).

  • Note: the same result holds for woman-proposing GS with everything flipped.


Proof
Proof

  • Say that w is possiblefor m if (m,w) in some stable matching.

  • Proof: show by induction that no man is ever rejected by a woman who is possible for him.

    • Suppose this is the case through round n.

    • Suppose at round n+1, woman w rejects m in favor of m’.

    • Can there be a stable match that includes (w,m)?

      • If so, m’ must be matched with some w’ who he prefers to w and who is possible for him (or else w,m’ block).

      • But then m’ could not be making an offer to w in round n+1: m’ would have first extended an offer to w’ and would not have been turned away.

  • So in no round is a man rejected by a possible woman.


Rural hospital theorem
Rural Hospital Theorem

  • Some years ago, there were a set of hospitals, mostly in rural areas, that had trouble filling their positions and were not happy.

  • Question: would changing around the algorithm help these hospitals?

    Theorem. The set of men and women who are unmatched is the same in all stable matchings.


Proof of rh theorem
Proof of RH Theorem

  • Consider the man-optimal stable matching and some other stable matching.

  • Any man who is matched in the other matching, must be matched in the man-optimal matching, so at least as many men are matched in the man-optimal.

  • Any woman matched in the man-optimal matching must be matched in all other stable matchings.

  • In any stable matching, the number of matched men just equals the number of matched women.

  • So the same set of men and women are matched in the two matchings, although different pairings.


Strategic behavior
Strategic Behavior

  • The Gale-Shapley algorithm (and other mechanisms such as priority matching) asks participants to report their preferences.

  • What is a good strategy?

  • Should participants report truthfully?


Definitions
Definitions

  • A matching mechanismis a mechanism that maps reported preferences into an assignment.

  • A mechanism is strategy-proofif for each participant it is a dominant strategy to report true preferences (i.e. optimal regardless of the reports of others).


Da is not strategy proof
DA is not strategy-proof

  • Example (two men, two women)

    • m prefers w to w’

    • m’ prefers w’ to w

    • w prefers m’ to m

    • w’ prefers m to m’

  • Under man-proposing DA algorithm

    • If everyone reports truthfullly: (m,w),(m’,w’)

    • If w reports that m is unacceptable, the outcome is instead (m,w’),(m’,w) --- better for w!


Strategic behavior1
Strategic behavior

  • The example on the previous slide can be used to establish the following result (try it on your own, or see RS).

    Theorem. There is no matching mechanism that is strategy-proof and always generates stable outcomes given reported preferences.

  • Both version of DA lead to stable matches, so they are not strategy-proof!


Truncation strategies
Truncation strategies

  • In the “man-proposing” DA, a woman can game the system by truncatingher rank-order list, and stopping with the man who is the best achievable for her in any stable match.

    Theorem.Under the man-proposing DA, if all other participants are truthful, a woman can achieve her best “possible” man using the above strategy.

  • Question: how likely is it that one would have the information to pull off this kind of manipulation?


Proof1
Proof

  • The DA must yield a stable matching.

  • If participants report as in Thm, one stable matching is the woman-optimal matching under the original true preferences.

  • This gives the manipulator her best possible man.

  • Under the reported preferences, any other matching would have to give the woman either someone better, or leave her unmatched.

  • By the RH theorem, she can’t be unmatched in some other stable matching. She also can’t get someone better because whatever would block under the true preferences will block under the reports.

  • Therefore, she must get her best possible man!


How many stable matchings
How many stable matchings?

  • Evidently, the incentives and scope for manipulation depend on whether preferences are such that there are many stable matchings.

  • If there is a unique stable match given true preferences, there is no incentive to manipulate if others are reporting truthfully.

  • When might we have a unique stable match?

    • Ex: if all women rank men the same, or vice-versa.

    • In “large” markets? We’ll come back to this later.


Da is strategy proof for men
DA is strategy-proof for men

Theorem(Dubins and Freedman; Roth). The men proposing deferred acceptance algorithm is strategy-proof for the men.

Proof.

  • Fix the reports if all the women and all but one man.

  • Show that whatever report the man m starts with, he can make a series of (weak) improvements leading to a truthful report.


Proof2
Proof

Suppose man m is considering a strategy that leads to a match x where he gets w. Each of the following changes improves his outcome

  • Reporting that w is his only acceptable woman.

    • x is still unblocked.

    • By RH, m must get matched, and so must get w.

  • Reporting honestly, but truncating at w.

    • m being unmatched is still blocked (because it was blocked if m reported just w), so m must do at least as well as w.

  • Reporting honestly with no truncation.

    • This won’t affect DA relative to above strategy.



Many to one matching1
Many-to-One Matching

  • In the NRMP, the hospitals actually want to hire several doctors. How to extend the theory for account for this?

  • Simplest extension

    • Doctors have strict preference over hospitals

    • Hospitals have a quota of spaces and a strict ranking of doctors.

    • Stability defined similarly: no hospital can find a doctor post-match and make a mutually agreeable contract

    • Note: bilateral and “group” stability are the same here.


Extended da algorithm
Extended DA Algorithm

  • Doctors and hospitals submit rankings

  • Algorithm

    • Each hospital proposes to its preferred doctors.

    • Doctors make a “tentative match” based on their preferred offer, and reject other offers, or all if none are acceptable.

    • Each hospital receiving rejections removes these doctors from its list and makes new offers from lower down.

    • Continue until no more rejections or offers, at which point implement tentative matches.

  • There is also a doctor-proposing version.


Properties of many to one da
Properties of Many-to-One DA

  • Think of a hospital with q positions as q hospitals each with one position.

  • Many results carry over

    • At least one stable matching exists.

    • Hospital proposing DA results in hospital-optimal stable matching (same for doctor-proposing).

    • Rural hospital theorem: all hospitals fill the same number of positions across stable matchings and the same doctors are assigned a position.

  • But some do not

    • No stable mechanism is strategy-proof for the hospitals, even the hospital-proposing DA algorithm…


Hospital incentives in da
Hospital Incentives in DA

  • Example

Student 1: H3, H1, H2

Student 2: H2, H1, H3

Student 3: H1, H3, H2

Student 4: H1, H2, H3

Hospital 1: s1, s2, s3, s4(quota=2)

Hospital 2: s1, s2, s3, s4 (quota=1)

Hospital 3: s3, s1, s2, s4 (quota=1)

  • Unique stable matching: (H1,s3,s4), (H2,s2), (H3,s1).

  • If H1 submits preferences s1,s4, the unique stable matching becomes (H1,s1,s4), (H2,s2), (H3,s3).


More general preferences
More General Preferences

  • What if hospitals (or, say, schools) care about the composition of their class?

    • Maybe a hospital wants to balance research-oriented and clinically-oriented residents.

    • A public school may want to balance local students and high academic achievers.

  • Preferences are more complicated than a quota and a rank-order list.

    • In general, a preference ranking for a hospital is a an ordered list of sets of residents, e.g. {r1,r2}, {r1}, {r2}, .

    • Turns out to be subtle to extend the theory to this case.


Substitutable preferences
Substitutable preferences

  • Let ch(A) denote the set of students that hospital h would select given a choice of any set of students in A, and Rh(A)=A - ch(A).

  • Hospital h has substitutespreferences if A A’ implies that Rh(A)  Rh(A’).

    • That is: if the set of students available to h expands, so does the set of students that h rejects, i.e. h does not add students who it previously rejected.


Substitutes and no regret
Substitutes and “No Regret”

Theorem. Suppose hospitals have substitutes preferences. Then a stable match exists and can be found with the DA algorithm.

Proof.

  • Consider the student-proposing algorithm.

  • If a hospital rejects a student at round n, then if an any subsequent round the that same student made a new offer to the hospital, the hospital would still reject them

  • This holds after algorithm ends, so result is stable.

    Key idea: hospital never “regrets” making a rejection, which clearly is also the case in the one-to-one case.

  • Note that regret can occur if substitutes fails – e.g. if a hospital wants students 1 and 2 together, but neither individually.


Further issues couples
Further issues: Couples

  • In the residency match, there are a fair number of married couples (maybe 500).

    • Typically couples want to be in the same city or at the same hospital.

    • The DA algorithm doesn’t account for this; it might put a husband in Boston and wife in Chicago.

  • Problem: there may be no stable match!


Couples an example
Couples: an example

  • Couple c1,c2 and single student s

  • Two hospitals, each hiring one student

    • Hospital 1: c1, s

    • Hospital 2: s, c2

    • Single student: H1, H2

    • Couple prefers positions at H1, H2 or nothing.

  • There is no stable match! (Check)



Back to the nrmp
Back to the NRMP

  • Starting in the 1970s and accelerating into the 1990s, many couples started to go around the NRMP to find positions.

  • NRMP decided to investigate and ultimately re-design the match

  • Roth-Peranson (AER, 1998) describe this.

  • Let’s look at a few of the interesting findings.



Questions
Questions

  • Taking stated preferences as the truth…

    • Is there always a stable match?

    • Are there many possible stable matches?

    • Can students/hospitals gain from mistating?

    • Any difference between student proposing and hospital proposing DA? Hospital-proposing was in use.



What do we learn
What do we learn?

  • Few participants could gain by manipulating their ROLs

    • Between 0 and 22 doctors out of 20,000 depending on year and form of DA.

    • Between 12 and 36 hospitals out of 4,000 depending on year and form of DA.

  • Despite the possibility that having couples would lead to no stable match, one can be found each year.

  • Simulations with randomly drawn preferences confirm these findings

    • Each doctor draws k hospitals randomly to rank

    • Each hospital randomly ranks all students.


Large markets incentives
Large markets: Incentives

  • Consider a sequence of markets with n men and n women, where for each n, each participant’s preferences are drawn from a uniform distribution over the possible rankings.

    Theorem. As n, the expected proportion of females who can manipulate the man-proposing DA goes to zero.

  • This result is due to Immorlica and Mahdian (2005) and, for many-to-one, Kojima and Pathak (2009).


Proof sketch
Proof (sketch)

  • Can focus on manipulation by truncation

  • Truncation may help because…

    • w rejects offer of M-optimal match partner m

    • m makes an offer to some w’, who rejects m’

    • m’ makes an offer to some w’’, who rejects w’’

    • ….

    • mkmakes an offer to w, who w prefers to m.

  • What is the probability that for a given woman w, rejecting her M-optimal match partner creates such as cycle? In a large market, very very small.


Large markets couples
Large markets: Couples

  • Consider a sequence of n markets similar to above with a fixed number of couples, independent of n.

    Theorem.As n, the probability that a stable matching exists converges to one.

  • This result is due to Kojima, Pathak and Roth (in progress) – I think the intuition is similar.

    • Stability fails if a couple gets displaced by a single doctor d, leading the other couple member to leave her hospital, either directly or through a chain of offers to displace d, which causes the original hospitals to re-hire the couple.

    • In a large market, the probability of this kind of cycle becomes very very small.


Resident salaries
Resident Salaries

  • In 2002, former residents filed a class action suit, arguing that the NRMP violated antitrust laws and depressed resident’s wages.

  • Some background data (AAMC)

    • Average resident salary: $46,245 (37,383 in 2002)

    • Range: 25th percentile $44,055, 75th percentile $47,760.

    • Work hours are long, regularly 80+ hours a week.

  • Residents claimed the low and compressed salaries are due to the use of a centralized match.

  • But maybe it’s some other feature of the market, for instance scarcity of positions or the fact that hospitals pay all residents the same amount?


Resident salaries1
Resident Salaries

  • Bulow and Levin (AER, 2006) analyze salary competition.

  • Consider two alternative markets

    • Firms and workers with competitive eqm prices

    • Firms post salaries, then match with workers.

  • Bulow-Levin showed the latter situation leads to

    • Small efficiency losses

    • Larger wage decreases and salary compression

    • Increased profits for the firms.

  • The problem is not the use of the match per se, but the nontargeted nature of salary offers

  • Little incentive to offer a premium salary to attract Lebron if you might miss out and get matched with Wally Szczerbiak.


Outcome of antitrust case
Outcome of antitrust case

  • Antitrust case was dismissed after Ted Kennedy sponsored a bill in Congress exempting the NRMP from antitrust laws.

  • One side effect was that in 2003, the ACGME passed regulations limiting resident work hours to 80 hours a week.

    • Some states already had such a limit, in particular New York following the Libby Zion case.

    • Shetty and Bhattacharya (Annals of Internal Medicine, 2007) found that the 2003 reduction led to small decreases in mortality at teaching hospitals.



House allocation problems
House allocation problems

  • In some matching markets, only one side of the market has preferences – think of students picking housing on campus.

  • House allocation problem

    • N individuals and N houses

    • Each individual has strict preference over houses.

    • Goal: assign each individual to a house.


Random serial dictatorship
(Random) Serial Dictatorship

  • Each student gets a priority (perhaps randomly assigned).

  • Students pick houses in order of their priority.

    Theorem.Serial dictatorship is efficient (i.e. no mutually agreeable trades afterwards) and strategy-proof.


Proof3
Proof

  • Strategy-proof

    • Individual with first pick gets her preferred house, so clearly no incentive to lie.

    • Individual with second pick gets her preferred house among remaining houses, so again no reason to lie.

    • and so on…


Proof4
Proof

  • Efficiency

    • Individual with priority one doesn’t want to trade.

    • Given that she is out, individual with priority two doesn’t want to trade.

    • And so on….


Top trading cycles
Top Trading Cycles

  • Now imagine that individuals start with a house, but the original allocation might not be efficient.

  • Gale’s TTC algorithm

    • Each person points to most preferred house

    • Each house points to its owner

    • This creates a directed graph, with at least one cycle.

    • Remove all cycles, assigning people to the house they are pointing at.

    • Repeat using preference lists where the assigned houses have been deleted.



The core
The Core

  • In cooperative games, the core is a solution concept describing “stable” outcomes (like Nash eqm in noncooperative games).

  • Consider a candidate assignment in the house problem:

    • A coalition of agents blocks if, from their initial endowments, there is an assignment among themselves that they all prefer to the candidate assignment.

  • The core consists of all feasible unblocked assignments.

  • What’s the difference between core & stability?

    • Stability is ex post (no more trade), core is ex ante (diff. trade)

    • But closely related: no difference in the marriage or roommate problems, and in more generally, if there are no externalities, unstable allocations are blocked by coalition of the whole.


Properties of ttc
Properties of TTC

Theorem.The outcome of the TTC algorithm is the unique core assignment in the housing market.


Proof5
Proof

  • Core

    • Blocking coalition cannot involve only those matched at round one (all agents get first choice).

    • Blocking coalition cannot involve only those matched in first two rounds (can’t improve round one guys, and to improve round two guys, need to displace round one guy).

    • And so on by induction.

  • Uniqueness:

    • The assignments at round one are necessary or else these agents would block, and inductively for rounds 2, 3,….


Incentives in the ttc
Incentives in the TTC

Theorem.The TTC algorithm is strategy-proof.

Proof. For any agent j assigned at round n

  • No change in his report can given him a house that was assigned in earlier rounds.

  • No house assigned in a later round will make him better off.


Combining the problems
Combining the problems

  • What if some individuals start with houses but some do not?

  • This is a common problem in allocating student housing

    • Michigan, Duke, Northwestern, Penn, CMU all use a variation of random serial dictatorship.

    • Let’s see how it works.


Random serial dictatorship with incumbency
Random Serial Dictatorship with Incumbency

  • Each agent with a house decides whether to keep their house or enter a lottery.

    • Agents who keep their house are done.

    • Houses that are abandoned are available later on.

  • Lottery used to order newcomers and agents who gave up their house (can be completely random or favor particular agents).

  • Serial dictatorship applied using order from lottery and selection from available houses.


Rsd with incumbency
RSD with incumbency

  • There is a problem…

  • Existing tenants are not guaranteed to get at least as good a house as their current house!

    • This may cause existing tenants to avoid the lottery and the market may not exploit all the possible gains from trade.

    • Is there a way to protect existing tenants, while getting pareto efficient outcomes and maybe strategy-proofness?


Yrmh igyt
YRMH-IGYT

  • The “you request my house – I get your turn” mechanism.

    • All agents are ordered according to some priority (maybe random).

    • Agent with top priority chooses a house, then second agent, and so on, until someone requests the house of an existing tenant.

    • If the existing tenant has already chosen a house, continue. If not, insert that tenant above the requestor and re-start the procedure with the existing tenant.

    • If a cycle forms, it is formed exclusively by existing tenants – clear the cycle and proceed with the priority order.



Properties of yrmh igyt
Properties of YRMH-IGYT

Theorem. The YRMH-IGYT mechanism is pareto-efficient, strategy-proof and makes no existing tenant worse off.

  • Proof is similar to results we’ve shown (try it!)

  • Relationship with SD and TTC

    • If there are no existing tenants: SD = YRMH-IGYT

    • If everyone has a house: TTC=YRMH-IGYT

    • In fact, YRMH-IGYT, is just TTC except that every unoccupied house points to the agent with the (current) highest priority).


Summary on house allocation
Summary on House Allocation

  • We looked at several variations

    • House allocation problem

    • Housing market problem

    • House allocation with existing tenants

  • And mechanisms for each problem

    • Serial dictatorship

    • Top Trading Cycles

    • IRMH-IGYT (TTC with a twist)

  • We showed that these mechanisms have desirable properties: efficiency, strategyproof, core, etc.

  • Next, we’re going to tackle some applications….



Kidney exchange1
Kidney Exchange

  • Transplants are standard treatment for patients with failed kidneys.

    • There is a shortage of kidneys.

    • Currently over 70,000 patients are waiting.

  • Some statistics from 2006

    • 10,659 transplants from diseased donors

    • 6428 transplants from living donors

    • 3875 patients died on the waiting list.


Resolving the shortage
Resolving the shortage

  • Buying and selling kidneys is illegal.

  • Section 301 of National Organ Transplant Act

    • “it shall be unlawful for any person to knowingly acquire, receive or otherwise transfer any human organ for valuable consideration for use in human transplantation.”

  • There are probably ways to increase the supply of cadaveric kidneys (e.g. make donation the default).

  • We’re going to focus on ways to increase the supply of living donor kidneys.


Donor kidneys
Donor Kidneys

  • Deceased donors: a centralized mechanism has long been in use – prioritizing patients higher on the waitlist.

  • Living donors: mostly friends and relatives of a patient – numbers have been increasing


Compatibility
Compatibility

  • Donor kidney must be compatible with patient

  • Blood type match

    • O type patients can receive O kidneys

    • A type patients can receive O or A kidneys

    • B type patients can receive O or B kidneys

    • AB type patients can receive any blood type

  • Also tissue type match (HLA compatibility).

  • Potential inefficiency: if a patient has a donor but can’t use the donor’s kidney, the donor goes home.


Paired exchange
Paired Exchange

  • Paired exchange: match two donor-patient pairs, where

    • Donor 1 is incompatible with Patient 1, but compatible with Patient 2

    • Donor 2 is incompatible with Patient 2, but compatible with Patient 1.

  • List exchange: match one incompatible donor-patient pair and the waiting list

    • Donor of incompatible pair donates to patient at the top of the waiting list.

    • Patient of incompatible pair goes to the top of the wait list.


Do we know this problem
Do we know this problem?

  • Problem seems very similar to house allocation with existing tenants.

  • Roth, Sonmez and Unver (2004, QJE)

    • The problems are (essentially) equivalent

    • TTC can be used to efficiently assign kidneys.

  • In 2004, RSU and doctors in Boston established first clearinghouse for New England.


Exchange in practice
Exchange in practice

  • In practice, turns out the problem can be simplified a bit…

    • At first, doctors wanted to limit to pairwise trades, and rule out list exchange.

    • US doctors think of compatibility as 0-1, which makes preferences different than the strict ranking in the housing model.

    • Compatible donors may not participate.


Simplified case
Simplified Case

  • For a given set of donor-patient pairs, define a matching to be any set of compatible donations that includes only own-donor transplants and pairwise exchanges.

  • A matching is pareto eficient if no other matching makes every patient weakly better off and at least one strictly better off.

  • A mechanism is strategy-proof if no patient-donor pair wants to misrepesent compatibility.

    Lemma: All pareto optimal matchings match the same number of pairs (RSU, 2005).


Priority mechanism
Priority Mechanism

  • Order donor-patient pairs (perhaps random or due to medical priority)

  • If there is a matching in which top priority pair is matched, match that pair, else skip.

  • Match the second pair if there is a matching that also matches the first pair, else skip.

  • Continue…

    Theorem.The priority mechanism is Pareto efficient and strategy-proof (RSU, 2005).


Multi way exchanges
Multi-Way exchanges

  • It is possible, though tricky to do multi-way exchanges (recently a 10-way swap!)

  • RSU (2007) show that

    • Three way exchanges can add a lot of transplants

    • The gains to more-way exchanges are small.

  • Let’s see how this works..


Three way exchange
Three-Way exchange

  • A pair is x-y if the patient and donor have blood type x-y.

  • Consider a population consisting of

    • O-B, O-A, A-B, A-B, B-A (blood type incompatible)

    • A-A, A-A, A-A, B-O (HLA incompatible)

  • Assue there is no HLA problem across pairs

    • Two-way (A-B,B-A), (A-A,A-A), (O-B,B-O)

    • Three-way: (A-B,B-A), (A-A,A-A,A-A), (B-O,O-A,A-B).


Gains from three way
Gains from Three-Way

  • An odd number of A-A pairs can be trasplanted.

  • O-type donors can facilitate three transplants rather than two.

  • In practice, O-type donors are short relative to demand, so useful to leverage them.


Four way exchanges
Four-way exchanges

  • Consider the following population

    • O-A, A-B, B-AB (blood-type mismatch)

    • AB-O (HLA mismatch)

  • Again, assume no cross-pair HLA mismatch

    • Up to three way: (O-A, A-B, AB-O)

    • Four-way: (AB-O, O-A, A-B, A-AB)

  • However, this is not a typical scenario because

    • very few AB people (<4% in US)

    • AB-O problem comes from HLA mismatch, and these can be resolved with large numbers.


Donor chains
Donor Chains

  • In July 2007, Alliance for paired donations started an “Altruistic Donor Chain”

  • Altruistic donor in Michigan donated kidney to woman in Phoenix.

  • Husband of Phoenix woman gave kidney to woman in Toledo.

  • Her mom gave kidney to patient A in Columbus, whose daugher simultaneously gave kidney to patient B in columbus.

  • Now patient B’s sister is looking to donate….



Background
Background

  • In US and many other countries, children historically have gone to neighborhood schools.

  • Recently, many US cities have adopted school choice programs, designed to give families additional flexibility, and maybe create some competition between schools.

  • What are important considerations in design?

    • Aim for efficient placements

    • “Fair” procedure and outcomes

    • Mechanisms that are easy to understand and use.


Background cont
Background, cont.

  • Abdulkadiroglu and Sonmez (2003, AER) showed that placement mechanisms used in many cities, such as Boston, are flawed, and proposed alternative mechanisms.

  • This has led many cities, including Boston and New York, to adopt new mechanisms.

  • This is an active area of research

    • Designing improved mechanisms

    • Studying the performance of mechanisms in use.

    • Also, partially random nature of allocations has facilated studies of school effectiveness in training students.


School choice model
School Choice Model

  • Set of students S and schools C

    • Each student can go to one school

    • Each school can admit qcstudents

    • Each student has strict preferences over schools and over being unmatched.

    • Each school has a strict “priority order” over students.

  • This is the many-to-one model we looked at earlier.

  • Stability means: (a) no blocking individual, and (b) no blocking pair – no student that can find a school that will displace someone to accept that student.


Stability as fairness
Stability as Fairness

  • No blocking individual means no student can be forced to attend a school they don’t want to attend, and no school can be forced to take a student they view as unqualified.

  • No blocking pair means no justified envy. That is, there is no student s who gets a school they prefer less than c, only to see a student with lower priority end up at c.


Boston mechanism
Boston Mechanism

  • Each student submits a preference ranking.

    • Consider only the top choices of the students.

    • For each school, assign seats to students that ranked it first according to priority order.

    • Stop if all seats assigned or run out of students ranking it first.

    • Consider remaining students, and their second choices. Repeat the above process.

    • Continue with third choices and so on…


Problems with boston
Problems with Boston

  • Boston mechanism is not strategy-proof.

    • If you don’t put your priority school high on your rank list, you may lose it!

    • This was well-understood by Boston parents, and frequently showed up on parent message boards.

  • Boston mechanism is also unfair…

    • Doesn’t lead to stable outcomes.

    • Disadvantages families that don’t know how to game the system.


Boston in practice
Boston in Practice

  • Students in K,6,9 submit preferences

  • Students have priorities as follows

    • Students already at a school.

    • Students in the walk zone and siblings at school

    • Students with siblings at school

    • Students in the walk zone

    • Everyone else

  • Abdulkadiroglu et al. found that 19% listed two over-demanded schools as top two choices and about a quarter ended up unassigned – ugh.


Columbus oh schools
Columbus (OH) Schools

  • Each student applies to up to three schools.

    • For some schools, seats are guaranteed based on assignment area.

    • Once those are filled, a lottery is held at each school and offers are made.

    • A student with an offer has three days to decide. If she says yes, she’s removed from the system.

  • Process repeats.

  • Ugh again…


Nyc schools
NYC Schools

  • Each student (90,000 plus applying to high schools) can submit up to five applications.

    • Each school receives applications and makes offers, plus it makes a waiting list.

    • Students accept and reject offers.

    • Schools make offers from wait lists (three rounds)

  • Roughly 30,000 students would be unassigned at the end; they would be administratively assigned.

  • Ugh, again… there must be a better way.


Student proposing da
Student Proposing DA

  • What about the student proposing DA?

    • We know this leads to a stable match.

    • And the stable match that is best for all the students, and they are the ones whose welfare we care about.

    • Plus it’s strategy-proof for the students, and we may not be worried about schools if priorities are clearly stated.

  • So our earlier results indicate that student-proposing DA has attractive properties…

    • We’ll see shortly that Boston and NYC have adopted it.


But is it efficient
But is it efficient?

  • Stable matches can be inefficient

  • Consider students s1,s2,s3, and schools A, B

  • s1: B>A

  • s2: A

  • s3: A>B

  • A: s1>s2>s3

  • B: s3>s1

  • Schools have one slot

  • Student-proposing DA => (s1,A), s2, (s3,B)

  • But every student prefers: (s1,B), s2, (s3,A)


Ttc for school choice
TTC for School Choice?

  • Schools are different from houses because

    • A school has multiple positions not just one

    • A student can have priority at multiple schools.

  • Still, we can adapt TTC to this setting

    • Abdulkadiroglu and Sonmez (2003) explain how.


Ttc for school choice1
TTC for School Choice

  • TTC Algorithm

    • Each student points to it top-ranked school

    • Each school points to its top priority student

    • Cycles are identified and removed (note: a cycle always exists – why?)

    • Any school that is full, or student that is matched is removed from all the lists.

    • Point again and repeat the process…

  • TTC allows students to trade priorities.



Why ttc for schools
Why TTC for Schools?

Theorem.The outcome of TTC is efficient and TTC is strategy-proof.

  • Proof is just like the housing problem

    • Strategy-proofness is the same.

    • Efficiency is just like proof of core outcomes.


Reform of boston program
Reform of Boston Program

  • Abdulkadiroglu et al. (2008) describe the reform of the Boston school match.

    • They proposed either student-proposing DA or TTC; the school system chose DA in 2006.

    • They didn’t like the idea of “trading priorities”.


Reform of nyc program
Reform of NYC Program

  • Abdulkadiroglu et al. (2009) describe reform of the NYC school choice program.

    • Recall NYC has a really big system: 90,000 students enter high schools each year!

    • Unlike in Boston, schools do not have fixed priorities, but can be strategic (and frequently were strategic under the old system).

    • NYC decided to adopt student-proposing DA.


More on nyc
More on NYC

  • NYC now uses student-proposing DA, except

    • Students can rank only 12 schools

    • A few schools (Stuyvesant, Bronx Science) get filled first using examinations.

    • Some top students automatically get first choice.

    • Unmatched students go to a second round, which uses random serial dictatorship.

  • In first year of new program

    • More than 70,000 got one of their choice schools.

    • Another 7,600 got a choice school in RSD stage

    • Just 3,000 were left, down from 30,000.


Summary on school choice
Summary on School Choice

  • School choice is a new application of matching theory.

    • Stability is a fairness criterion: no justified envy.

    • Student-proposing DA makes sense if stability is important.

    • TTC generates efficiency gains if stability is not necessary.

  • NYC, Boston and other cities have switched to new mechanisms, mainly DA.


Weak preferences
Weak preferences

  • A new twist in school choice is that school priorities generally aren’t strict.

    • Example: in boston there were just four priority categories: (1) at the school, (2) walk zone + sibling, (3) sibling, (4) walk zone.

    • This means that in doing the DA algorithm, or TTC, one frequently has to break ties.

    • This leads to new issues, with trade-offs between efficiency, stability and strategy-proofness.


Student da with ties
Student DA with Ties

  • Start by resolving all ties randomly.

  • Run the student proposing DA.

  • But how to break the ties?

    • One lottery to decide order on all students, and all schools use this order.

    • Each school uses a separate lottery.

  • Either way, student-proposing DA is strategy-proof, but may not lead to student-optimal stable match.


Example
Example

  • Students s1,s2,s3; schools A,B,C, one seat.

  • s1: B > A > C

  • s2: C > B > A

  • s3: B > C > A

  • A: s1 > {s2,s3}

  • B: s2 > {s1,s3}

  • C: s3 > {s1,s2}

  • A: s1 > s2 > s3

  • B: s2 > s1 > s3

  • C: s3 > s1 > s2

  • Assume ties are broken s1 > s2 > s3 for all.

  • DA then finds (s1,A), (s2,B), (s3,C)

  • Students prefer (s1,A), (s2,c3), (s3,B) – and it’s stable for the original priorities!


What sort of tie breaking
What sort of tie-breaking?

  • In NYC, policy-makers thought having a single lottery was less fair – only gives students “one shot” at a high ranking.

  • Simulations using NYC data, reported in Abdulkadiroglu et al. suggested that single tie-breaking was (a bit) better for students.

  • NYC opted for single tie-breaking.

  • Can we do better on efficiency, while maintaining stability?


Stable improvement cycles
Stable Improvement Cycles

  • Given a stable matching, look for cycle of students s0, s1,…,sN=s0 such that for each k

    • sk prefers s(k+1)’s school to her own

    • sk’s priority for s(k+1)’s school is as high as anyone else who would like to switch to that school from their match assignment.


Stable improvement cycles1
Stable Improvement Cycles

Theorem(Erdil and Ergin, 2008 AER)

  • Suppose we start with a stable match and implement some SIC. The new match is stable and Pareto dominates the old one.

  • Suppose we start with a stable match that is not student-optimal. Then a SIC exists.

  • Implication: we can run DA with some tie-breaking, and then apply SICs repeatedly to reach a student-optimal match.


Unresolved issues
Unresolved Issues

  • SIC procedure isn’t strategy-proof

    • A student can rank a popular school high up, and then use it to “trade up” in SIC.

    • In fact, for any tie-breaking rule, there is no strategy-proof mechanism that always results in better matching than the student DA.

  • Open question: how best to trade off efficiency, stability and strategy-proofness?


Lotteries as evaluation tools
Lotteries as evaluation tools

  • Randomization in school choice algorithms provides an opportunity to study the effectiveness of different schools.

  • Usual problem: students who go to school A may be very different than those at B. If A’s students test better, was school A more effective or did it just have better students?

  • Solution: suppose a set of students are randomly assigned to schools A and B. By comparing the performance of those ending up at A and B, we can see which school is more effective.

    • Lotteries in school choice yield this randomization!


Boston evaluation
Boston evaluation

  • Boston since 1995 has introduced two alternatives to standard public schools

    • charter schools – public schools, but not regulated by the usual school boards or subject to collective bargaining with teacher’s unions.

    • pilot schools – public schools with flexibility to determine their own budgets, etc. but subject to collective bargaining agreements.

  • Charter schools appear to have the best standardized test scores.


Boston evaluation cont
Boston evaluation, cont.

  • Abdukadiroglu et al. (2009) try two approaches to comparing school performance.

    • “Statistical controls” – gather data on the students and control for observed student differences.

    • “Lottery control” – compares students who got into charter/pilot schools due to good lottery draw with thos who were rejected due to bad lottery draw.


Boston findings
Boston findings

  • Large positive effect for charter schools

    • For middle schools, raise student achievement by 0.09-0.17 std. deviations in English, 0.18-0.54 std devs in Math relative to traditional schools.

    • For high schools, raise student achievement by 0.16-0.19 std deviations in English, 0.16-0.19 std deviations in Math relative to traditional schools.

  • Results for pilot schools are ambiguous.


New york evaluation
New York evaluation

  • Hoxby and Murarka (2007) used lotteries to study the effectiveness of New York City charter schools.

  • New York has charter schools in four of five boroughs.

    • Students are generally disadvantaged (eg KIPP, Amistad schools).

    • This makes it difficult to compare performance of average charter school and traditional student.

    • Hoxby approach: compare “lotteried in” and “lotteried out” students.


New york findings
New York findings

  • Charter schools are quite effective

    • For middle schools, raise math scores by 0.09 std deviations per year, and reading scores by 0.04 std deviations per year.

  • Important question: what is it about charter schools that is making them more effective?

    • Neither Hoxby study nor Boston study addresses this question, although pieces of evidence.


More evidence on choice
More evidence on choice

  • Do families make informed choices?

  • Hastings and Weinstein (QJE, 2008) report on an experiment run in Charlotte, SC.

    • Base case: families submit preferences and information about schools is available but not explicitly emphasized.

    • Experiment case: detailed information about school test scores provided in an accessible way.


Charlotte findings
Charlotte findings

  • Absent explicit test score information, low-income families very likely to choose their local school.

  • Explicit information increases likelihood of students choosing a “high score” school over a “local” school by 5-7 percentage points, and increased average test score of schools chosen by 0.05 to 0.10 std devs.

  • Also report substantial improvement in student test scores from attending a “high score” school.

  • Lesson: part of school choice design (and market design more generally) is helping participants understand how things work!



Market unraveling1
Market Unraveling

  • Recall that one reason for the adoption of the NRMP was that resident hiring had unraveled – become earlier and earlier in med school.

  • Many markets with fixed appointment dates appear to suffer from this problem.

  • Examples

  • Medical fellowships

  • Judicial clerkships

  • College admissions

  • College football bowls

  • High school prom

  • NBA/NCAA basketball recruiting

  • Baseball free agency

  • Political campaigns/primaries

  • Airline advance purchases

  • Event tickets


Regulating market timing
Regulating market timing

Markets often adopt regulations that govern the timing of offers – with more and less success.

  • Council of graduate schools (graduate admissions)

  • National Association for College Admission Counseling (undergraduate admissions)

  • Nationa Residency Matching Program

  • Specialty Matching Services: medical fellowships

  • National Association for Law Placement (law firm hiring)

  • Judicial Conference of the US (judicial clerkships)

  • NCAA/BCS (for college bowls)

  • National Panhellic Conference (sorority matching)


Attempts to control the market
Attempts to control the market

  • Prior to 1992: Rose, Orange, Sugar and Cotton Bowls claimed at least one conference champion; Fiesta had at-large.

  • Bowl Coalition 1992-94: Rose kept Big-10/Pac-10 matchup, others kept conference affiliations but agreed to set up 1v2 game if possible.

  • Bowl Alliance 1995-97: Rose kept Big-10/Pac-10, Orange, Sugar, Fiesta agreed to pool conference teams to create 1-2 matchup.

  • BCS 1998-: Rose, Orange, Sugar, Fiesta and Big 6 conferences agree to pool and ensure 1-2 BCS match.


Why do markets unravel
Why do markets unravel?

  • Absence of stability if market runs late (recall Kagel-Roth experiment), perhaps due to congestion problems.

  • Uncertainty aversion – want to “lock in” a match.

  • Strategic pre-emption – if worried that market will start to unravel, want to pair up first.

  • Another way to put it:

    • Fine to wait if others are waiting and late market will be “thick and efficient”.

    • But if late market is either thin, or inefficient, why stick around?


Gastroenterology fellowships
Gastroenterology Fellowships

  • Gastroenterology – fellows train for 2-3 years after completing their internal medicine residencies.

  • Prior to 1986: decentralized market

    • Residents apply to programs, receive offers, etc.

    • Market frequently unraveled, exploding offers, etc.

  • 1986-1996: adopt centralized match a la NRMP.

  • Mid-1990s: the match collapses (unusual for a stable matchng algorithm), and has only recently been reconstituted.

  • What happened? (Niederle and Roth, many papers).


Collapse of gi match
Collapse of GI Match

  • Key event may have been 1996 study published in JAMA stating that there are “too many” GI docs, and there needs to be a 25-50% reduction in fellowships, which programs subsequently endorsed.



What happened
What happened?

  • In GI both sides appear to have felt they were on the short side of the market!

  • In fact, demand for positions appears to have fallen, but maybe applicants were not aware of it?


After the gi match died
After the GI Match Died

  • Two things seemed to have happened

    • The market unraveled

    • The market became very rushed

    • Matches were made “locally”

  • Niederle and Roth (2005, JPE) explain.

    • Based on surveys of programs in early 2000s.




Offer acceptance rush
Offer/Acceptance “rush”

  • 46% of programs made offers before they finished interviewing (6% had all slots filled)

  • 87% had applicants cancel interviews, and almost 40% had 5 or more cancellations.

  • 56% of programs give deadlines of a week, and 93% less than two weeks.

  • 31% took chance of acceptance into account in making offers.

  • 45% speed up offers for applicants with an exploding offer in hand.

  • 21% got all their acceptances in under one hour.



Law clerk market
Law Clerk Market

  • This market has also unraveled.

  • Exploding offers are a big problem, and very hard to organize judges and make them pay attention to voluntary regulations.

  • Avery et al. (2001)

    • Market for clerkships to start in Fall 2003 cleared in September 2001.

    • Market clears early => less information.

    • Market is thin, fast and chaotic => inefficient?


Attempts to reform
Attempts to reform

  • Judges have tried on many occasions to enforce hiring dates – so far, it hasn’t worked.

  • Here’s one example.

    • In March 2002, the Judicial Conference agreed to have a one-year moratorium on hiring, with hiring for Fall 2004 to commence in Fall 2003.

    • The moratorium was okay, but implementing the start date was a bit tricky…




Exploding offers continue
Exploding offers continue

  • “I received the offer via voicemail while I was in flight to my second interview. The judge actually left three messages

    First, to make and offer.

    Second, to tell me that I should respond soon

    Third, to rescind the offer.

    It was a 35 minute flight.

  • “I had 10 minutes to accept”

  • “I asked for an hour to consider the offer. The judge agreed; however thirty minutes later [the judge] called back and informed me that [the judge] wanted to rescind my offer.”



Exploding offer experiment
Exploding offer experiment

  • Niederle & Roth (2009)

    • 5 firms, 6 students, 9 periods

    • Each period, firm can make offer to one student and students decide on offers.

    • Each firm and applicant has a quality, payoff xy

    • Quality of students revealed over time (equal to sum of signals observed in periods 1,4,7).

    • Firms see quality of all students, but students just see their own, and then their ranking in period 7.


Experimental regimes
Experimental Regimes

  • Exploding Offers:Firms can make exploding offers and acceptances are binding.

  • Renege:Firms can make exploding offers, but applicants can renege on their acceptance, for a small fee.

  • Open Offers: Firms can only make open offers.





Modeling unraveling
Modeling Unraveling

  • What kind of economic model can help us understand unraveling?

  • Consider two simple models

    • Early contracting as “insurance”

    • Early contracting to avoid “congestion”


Labor market model
Labor market model

  • A market with W workers and F firms.

  • A firm-worker pair can produce one unit of output, valued at $100.

  • Labor market is competitive

    • If W>F, there is “excess supply” of labor. All firms hire a worker, and competitive wage is w=$0.

    • If W<F, there is “undersupply” of labor. All workers are hired, and competitive wage is w=$0.


Adding uncertainty
Adding uncertainty

  • Suppose that market has two periods

    • Initially, W0<F workers. A worker in the market may contract with a firm if it’s mutually desirable.

    • Then, a random number of new workers show up, and there is another opportunity for contracting.

    • Let W be final number of workers, Pr[W<F]=.


Early contracting
Early contracting?

  • A worker who waits will:

    • be hired at w=1 with probability 

    • receive w=0 (or be unemployed) with prob. 1-.

  • A firm that waits will:

    • pay w=1 (or vacancy) with probability .

    • hire a worker at w=0 with probability 1-.

  • Should a firm and worker contract early?

    • If contract now for w: get u(w) and 1-w

    • If they wait: get u(1)+ (1-)u(0), 0+(1-)1 = 1-


Early contracting cont
Early contracting, cont.

  • There are gains from early contracting if the firm can offer a wage w< such that

    u(w) > u(1)+(1-)u(0)

  • Suppose worker is risk-neutral: u(x)=x.

    • Early contracting requires >w=u(w)> . NO!

  • Suppose worker is risk-averse: u’’(x)<0.

    • Then u()> u(1)+(1-)u(0).

    • Worker would accept slightly less than w= to contract early, and as a result the initial workers will sign up early!


Unraveling and congestion
Unraveling and congestion

  • Unraveling can also arise because participants anticipate that if they wait until the last minute the market will clear inefficiently.

  • Example

    • Two identical firms, can hire one worker each.

    • Two identical workers

    • Any firm-worker pair gives both parties $1.

  • Efficient allocation: both workers get hired.


Unraveling and congestion1
Unraveling and congestion

  • Suppose market game is:

    • Each firm can make an offer to one worker.

    • Workers then accept or reject.

  • With one round of offers, if firms do not coordinate, they are equally likely to make an offer to either worker.

    • with probability 1/2, make offers to different workers, so two matches are made.

    • with probability 1/2, make offers to the same worker, so only one match is made.

  • With two rounds of offers, this problem is avoided!


Summary
Summary

  • Market unraveling is often an issue when there is a fixed appointment date.

  • Markets can unravel because of fears that final market will be thin, or otherwise inefficient.

  • Market regulations can sometimes solve these problems – NRMP, graduate school admissions.

  • Some markets are harder to fix…



College admissions
College Admissions

Admission to selective U.S. colleges is extraordinarily competitive and becoming more so….


Early admissions
Early Admissions

Ubiquitous feature of the market: early admissions programs that allow students to target a single school with an early application.

Early Action: admission but no commitment to attend.

Early Decision: other schools will not accept the student.

Prevalence of early admission

30 of 38 “most selective” US universities

24 of 25 “most selective” liberal arts colleges.

Very controversial

Early admit rates can be double regular admit rates.

Harvard & Princeton recently eliminated their programs.


History of early admissions
History of Early Admissions

Prior to WW II, colleges admitted pretty much all their applicants (applicant pool was highly self-selected…)

Admission becomes more competitive in 1950s.

Various forms of early admissions arise to manage yield.

“Modern” early admissions adopted in mid-1970s.

Prevalence of early admissions soars in 1980s-90s.

By 1995, Harvard admits more than half its class early.

More than 100 colleges adopt early admissions in 1990s.

Students become increasingly aware of admissions advantage for early applicants, leads to more early applications.

Modest backlash in last decade --- but few changes.


Evidence on early admissions
Evidence on Early Admissions

Data from College Admissions Project

Survey of 3294 students at top high schools (1998)

Data on student characteristics, where they applied, were admitted and enrolled.

Selected sample of highly qualified students.

Focus on most popular colleges receiving many apps.

Use data to establish some basic patterns in early application, admission, enrollment behavior.

Findings match what we know from admissions offices at particular schools.



Admissions advantage
Admissions Advantage

(F1) At top schools, early applicants have better observable characteristics than regular applicants. The reverse is true at lower ranked schools.

(F2) Colleges favor early applicants in their admissions decisions.


Differential enrollment
Differential Enrollment

(F3) Students who are admitted early are more likely to enroll than students admitted regular.


Student and school incentives
Student and School Incentives

(F4) Students have an incentive to strategize

May be optimal to apply early even if undecided.

May not be optimal to apply to one’s top choice.

(F5) Different schools may have an incentive to use different types of programs.

Currently, early action disproportionately used by the most selective school (top 5-10).

Early decision used by virtually all the others.


Why early admission
Why Early Admission?

Almost certainly many reasons…

Focus on one particular idea

Schools prefer students who are enthusiasts.

Early application lets students signal their enthusiasm.

Let’s see if this story is consistent with the patterns described above.


Model
Model

Three schools

A, B are selective and can each enroll K<1/2 of students.

C is not selective and admits all students.

Students have “types” (v,y) --- ability & preference.

All students prefer A and B to non-selective C.

u(y,A)/u(y,B) increasing in y, and y=0 indifferent.

Assume Pr(y<0) < 1/2, and v,y are independent.

School preferences

School A: A(v,y) increasing in (v,y)

School B: B(v,y) increasing in (v,-y)


Model cont
Model, cont.

Application game

Students apply.

Schools make admissions decisions.

Students decide where to enroll.

If early admissions, students can designate one “early” application.


Model cont1
Model, cont.

Look for “threshold” equilibria, in which

Schools admit students above an ability threshold (possibly different for early and late applicants)

Students use a threshold rule to decide where to apply early, if early application is offered.

Lots of “babbling” equilibria.

Focus on equilibria that aren’t sensitive to having a small fraction of the most enthusiastic students apply early to their preferred school.


Regular admissions
Regular Admissions

Proposition. With regular admissions only, there is a unique equilibrium. In equilibrium, the students apply to all schools and the schools use admission thresholds A, B with A>B.


Regular admission equilibrium
Regular Admission Equilibrium

v

Accepted

at both,

attend B.

Accepted at both, attend A.

A

Accepted only at B.

B

Not accepted at

a selective school.

y

0

Y


Early action students
Early Action: students

  • Suppose schools use cut-offs AE < ARand BE < BR, and that BE < AE.

    • Students who prefer B should apply there early. Why?

    • Students who prefer A should apply there early if

  • The higher is y the more reason to apply early to A

  • For a student who is just indifferent, apply early to B

  • Optimal to apply early to A if y>Y, for some Y>0.


  • Early action colleges
    Early Action: colleges

    Suppose students apply early to A iff y>Y>0

    • School A: optimal to set AE<ARbecause

      • Early admits have y>Y,

      • Late have y<Y (though maybe 0<y<Y).

    • School B: optimal to set BE<BRbecause

      • Early admits have y<Y (and maybe y<0)

      • Late have y>Y.


    Early action
    Early Action

    Proposition. If both colleges offer Early Action, there is at least one threshold equilibrium. In equilibrium,

    (i) students apply early to A if and only if y>Y>0;

    (ii)colleges strictly favor early applicants, BE<BR and

    (iii) early admission yields are higher then regular yields.


    Early action equilibrium
    Early Action Equilibrium

    v

    Accepted early at B, attend A.

    Accepted early at A, and attend.

    Accepted early at B and attend.

    AR

    A

    AE

    Accepted late at B.

    BR

    B

    BE

    Not accepted at

    a selective school.

    y

    0

    Y


    Early action welfare
    Early Action: Welfare

    • Proposition 3. Both schools benefit from early action relative to regular admission.

      • Proof. By revealed preference.

    • Caveats

      • Students don’t all benefit, although they benefit on average if u(y,A) increasing, u(y,B) decreasing in y.

      • Schools may not benefit from early action if some students do not apply early.


    Welfare cost of ea
    Welfare cost of EA

    v

    Attend B

    Attend A

    AR=BR

    Attend B if apply early

    Attend A if apply early

    A=B

    AE=BE

    Not accepted at

    a selective school.

    y

    0


    Sorting in application behavior
    Sorting in Application Behavior

    Suppose beyond knowing y, students have signal w that is informative about v (affiliated) so information is (w,y).

    Example: test scores or SATs, observable information.

    There is a robust equilibrium with the same structure except that the student threshold for applying to A is Y(w), and Y(w) is decreasing in w.

    So, early applicants at A will have higher scores than regular applicants; and the reverse will hold at B.


    Early decision
    Early Decision

    Proposition. A threshold equilibrium exists. In equilibrium,

    (i) students apply early to A if and only if y > YED > 0;

    (ii) schools favor early applicants, BED<BRD and B<AED<ARD.

    Actually, eqm outcome is the same if A uses EA or eliminates early admissions.

    Proposition. School B prefers ED to regular admissions, but school A may not. The comparison between ED and EA is ambiguous.


    Early decision equilibrium
    Early Decision Equilibrium

    v

    Accepted at A, and attend.

    Accepted early at B and attend.

    ARD

    A

    AED

    Accepted late at B.

    BRD

    B

    BED

    Not accepted at

    a selective school.

    y

    0

    YED

    Y


    Summary1
    Summary

    Signaling model of early admissions explains many of the empirical patterns, though likely not whole story.

    Early admissions can be viewed as form of market unraveling, a la Roth and Xing (1994)

    In their examples, unraveling reduces welfare because it limits information aggregated in market outcomes.

    Here, early admissions allows credible information transmission, though not always welfare enhancing.

    Other countries have very different mechanisms for matching students to schools -- why this one in US?



    Random assignment1
    Random assignment

    The mechanisms we have considered so far involved a limited amount of randomness

    A few exceptions

    random serial dictatorship

    random tie-breaking in school choice

    There is somewhat more to be said, however,….


    Random serial dictatorship1
    Random Serial Dictatorship

    House allocation problem.

    Recall random serial dictatorship

    Order is chosen randomly.

    Agents choose objects in order

    RSD has very nice properties

    Simple and “fair”

    Strategy-proof

    Ex post efficient (no gains from trade at the end).


    Is there a problem with rsd
    Is there a problem with RSD?

    People 1,2, 3, 4, and houses A, B

    1 and 2 like A > B > nothing

    3 and 4 like B > A > nothing.

    Under random priority

    All orderings equally likely (there are 24).

    1 gets A if picks first, or second behind 3 or 4. (10 cases)

    1 gets B if he picks behind 2 but ahead of 3,4. (2 cases)

    Otherwise 1 gets nothing (12 cases).

    We can do similar calculations for 2, 3, 4.


    Rsd is inefficient
    RSD is inefficient

    “Random assignment” under RP

    • There is a random assignment everyone prefers!


    Probabilistic serial
    Probabilistic serial

    Probabilistic serial, or “cake eating,” mechanism

    Each good is divided into “probability shares”

    Each agent “eats” his preferred good with speed one

    If a good is finished, agents eating switch to their next preferred good among what those that are left.

    When goods are done, each agent has eaten a certain amount of probability of one or more goods. This is his or her random assignment - have to flip coins to resolve.


    Revisiting the example
    Revisiting the example

    People 1, 2, 3, 4 and houses A, B

    1 and 2 like A > B > nothing

    3 and 4 like B > A > nothing.

    Consider the PS mechanism

    1 and 2 start off eating A; 3 and 4 are eating B.

    At time 1/2, both houses are “eaten”

    Agents 1,2 each eat 1/2 of A

    Agents 3,4 each eat 1/2 of B.


    Outcome of ps
    Outcome of PS

    “Random assignment” under PS

    • This is the assignment that we said dominated RSD!


    Properties of ps
    Properties of PS

    Theorem.For any reported preferences the PS mechanism leads to an “ordinally efficient” and “envy-free” assignment.

    Ordinally efficient: there no other random assignment that is better for everyone irrespective of their risk preferences.

    Envy free: everyone likes his or her random assignment as well as the random assignment of anyone else.


    Proof loosely
    Proof (loosely)

    At each moment in time, everyone is eating probability shares of their favorite available good, so

    No one accumulates a share of a good when something better remains (efficiency);

    Everyone has a chance to eat at least as well as anyone else (envy-free).


    Is ps better than rsd
    Is PS better than RSD?

    Probabilistic serial is not strategy-proof.

    May want to start on a popular good, then switch to your top choice later if not in high demand!

    So there is a trade-off

    PS is ordinally efficient, but not strategy-proof

    RSD is strategy-proof, but not ordinally efficient.

    No mechanism has both properties!

    Interestingly, PS is a new mechanism and not used much (yet), whereas RSD is common.