### Efficient assignment and the Israeli medical match

```Efficient assignment and
the Israeli medical match
25th Jerusalem School in
Economic Theory
Assaf Romm
Harvard University
Outline
• Israeli medical match: introduction
•
•
•
•
•
Random Serial Dictatorship
Probabilistic Serial
Competitive Equilibrium approach
Israeli medical match: mechanism
Birkhoff-von Neumann decomposition
• Israeli medical match: couples
Medical studies in Israel
• A long track: 6 years of school, 6 months of exams, one year of
internship, 3-6 years of residency.
• Highly desirable: requires the best psychometric exam scores,
some students study abroad (but still need to do internships),
most people’s moms are jewish.
• What do you do during internship: clerkship in various
departments, taking blood, paperwork, night shifts, minimum
wage.
Why do we have internships?
• Justifies the fact that the state of Israel pays for tuition.
• Prevents outside immigration of recently trained doctors.
• Helps the hospitals (bypasses the rural hospital theorem).
• Similar practices are common throughout the world.
Some numbers
• In Israel there are 23 hospitals participating in the match.
• In 2013, there were 448 interns that completed their studies
• The number of positions is determined by the number of
interns.
How should we assign doctors?
Why do interns care about their assignment?
• Let the doctors decide where they want to do their
internship?
– No. We won’t get many interns in the more distant hospitals.
• Two-sided matching according to preferences reported by
interns and hospitals?
– No. We will not get uniform quality across hospitals.
• Random permutation?
– No. Really wasteful to ignore doctors’ preferences.
Principles in assigning doctors
•
•
•
•
Uniform quality across hospitals
Capacities are determined according to size and geography
Fairness among doctors (flexible concept!)
Subject to the above, make interns as happy as possible.
The assignment problem
(not to be confused with “the assignment problem”)
• We have a set of objects  = 1 , … ,  to be assigned to a
set of agents  = 1 , … ,  .
• Each agent has preferences over which object she would like
to be assigned.
• How do we allocate the objects to the agents?
• (Important: transfers are not allowed)
• More motivation: medical internships, seats in schools, course
allocation, thesis advisers, NBA draft, and many more.
• Fairness
• Efficiency
(compare with two-sided matching)
Questions we should also ask ourselves when designing:
• How important is strategy-proofness? (and how difficult is
“cheating”?)
• Do items themselves have preferences? (here we assume they
do not have any preferences)
Random Serial Dictatorship
Agents are randomly sorted, and then each one in her turn picks
the object she likes best (among those that are still available).
• super-easy to understand and to implement (why is this
important?)
• fair
• strategy-proof
• ex-post efficient
• easily extended to more general environments.
Easy to implement?
• Given everybody’s preferences, it’s about two lines of code.
• However…
The French « choix des postes d’internes »
• Choosing medical internship in France
• Each candidate has to get a position, whose main
characteristics are specialization and location.
• (Non-random) serial dictatorship: interns are sorted according
to their result in the national exam.
• Up until 2011: everybody comes to Paris
to state their choice. 700 interns entering
the hall every day and pick among remaing
positions.
The French « choix des postes d’internes »
• Starting from 2011: the process got computerized!
• Every student in her turn enters the system and picks among
the remaining positions.
The French « choix des postes d’internes »
• Starting from 2011: the process got computerized!
• Every student in her turn enters the system and picks among
the remaining positions.
The French « choix des postes d’internes »
• Question: why won’t they take preferences from students and
run the serial dictatorship automatically?
• Answer: too many positions to rank!
Harvard housing
• Many incoming graduate students, many apartments.
• The selected method: random serial dictatorship with a twist.
• The twist: each student is randomly assigned to a time block.
Within time blocks, students compete for apartments.
• Question: why?
• Answer: too many apartments to rank,
and we don’t want it to take forever.
The Israeli medical match
• Only 23 possible hospitals, no specialization at that point.
• Up until 2011: Everybody gets in the same room, and, to make
things as random as possible...
we draw names out of a hat.
The Israeli medical match
• And yes, with about 400 interns, that’s one giant hat…
G-RRRRY-FIN-DDDDO…
Oh, sorry, I meant, Soroka
• Starting from 2011: computerized random serial dictatorship.
Back to theory…
One small problem with RSD: it is not ex-ante efficient.
Example:  = 1 , 2 , 3 , 4 and  = 1 , 2 , 3 , 4 , and:
1 , 2 : 1 ≻ 2 ≻ 3 ≻ 4
3 , 4 : 2 ≻ 1 ≻ 4 ≻ 3
Resulting distribution:
1
2
3
4
1
5/12
5/12
1/12
1/12
2
1/12
1/12
5/12
5/12
3
5/12
5/12
1/12
1/12
Suggested improvement:
4
1/12
1/12
5/12
5/12
1
2
3
4
1
2
3
4
1/2
0
1/2
0
1/2
0
1/2
0
0
1/2
0
1/2
0
1/2
0
1/2
Notions of efficiency
In laymen terms: after the RSD lottery nobody wants to switch,
but prior to the lottery people are willing to trade.
A mechanism  is
• Ex-post efficient at ≻ if  ≻ can be represented as a
probability distribution over deterministic efficient
assignments.
• Ex-ante efficient at  if   is Pareto optimal.
• Ordinally efficient at ≻ if the random assignment  ≻ is not
stochastically dominated by another random assignment.
• Rank efficient, rank distribution is not dominated.
Probabilistic Serial
• Based on Bogomolnaia and Moulin, JET, 2001.
Simultaneous eating algorithm with uniform eating speeds.
Probabilistic Serial – Step 1
Each object is a pizza. Portions of the pizza represent probability
to be assigned that object.
Object 1
Object 2
Object 3
Object 4
Probabilistic Serial – Step 2
Each agent eats pizzas at the same rate (one bite/sec) and each
second he takes a bite from the pizza he likes best.
Agent 1
Agent 2
Object 1
Agent 3
Object 2
Agent 4
Object 3
Object 4
Probabilistic Serial – Step 3
When all pizzas were consumed, the content of each agent’s
stomach represents the probabiliies with which he is assigned
each item.
Agent 1
0.51 + 0.53
Agent 2
0.51 + 0.53 0.52 + 0.54
Agent 3
0.52 + 0.54
Agent 4
Probabilistic Serial – Step 4
We conduct a lottery according to the probabilities.
In the example:
1
2
3
4
1
1/2
1/2
0
0
2
0
0
1/2
1/2
3
1/2
1/2
0
0
4
0
0
1/2
1/2
We perform a lottery between two deterministic assignments:
1
2
3
4
1
1
0
0
0
2
0
0
1
0
3
0
1
0
0
4
0
0
0
1
and
1
2
3
4
1
0
1
0
0
2
0
0
0
1
3
1
0
0
0
4
0
0
1
0
Probabilistic Serial – Step 4
But what if we got this matrix?
1
2
3
4
1
0.23
0.29
0.46
0.02
2
0.28
0.44
0.12
0.16
3
4
0.39 0.1
0.05 0.22
0.28 0.14
0.28 0.54
Note that we cannot just randomize for each agent
independently, we have to randomize between valid
deterministic assignments.
Theorem (Birkhoff-von Neumann): This is always possible.
Properties of Probabilistic Serial
• Not very easy to understand, not very easy to implement.
• “Fair” (envy-free, but we’re not going to dicuss that)
• Ordinally efficient.
Proof: Suppose agents  and  ′ want to trade the probabilities of
getting objects  and  ′ . It means that when  ate ,  ′ was busy
eating something better for him, and he only ate  ′ later. But
then, why did  eat  when we  ′ was available?
A similar argument is true for more general trades.
PS is not strategy-proof
1 : 1 ≻ 2 ≻ 3
2 : 1 ≻ 3 ≻ 2
3 : 2 ≻ 1 ≻ 3
Resulting PS assignment:
1
2
3
1
2
1/2 1/4
1/2
0
0
3/4
3
1/4
1/2
1/4
If, however, 3 reports 1 ≻ 2 ≻ 3 he gets:
Manipulated PS assignment:
1
2
3
1
2
3
1/3 1/2 1/6
1/3
0
2/3
1/3 1/2 1/6
PS is weakly strategy-proof
This means, no one can manipulate and get an assignment that
first-order stochastically dominates the PS assignment.
(and we are not going to prove that)
Large markets results
• Kojima and Manea (2008): PS is approximately strategy-proof
in large enough markets.
• Che and Kojima (2010): PS and RSD allocations converge as
the market grows large.
• Liu and Pycia (2013): if a mechanism is asymptotically
efficient, symmetric and asymptotically strategy-proof it
converges to RSD.
Large markets results
Here are the ranking distribution for RSD and PS in the Israeli
medical match from 2011 (378 doctors, 20 hospitals):
Large markets results
Here are the ranking distribution for RSD and PS in the Israeli
medical match from 2011 (378 doctors, 20 hospitals):
Competitive equilibrium approach
• Based on Hylland and Zeckhauser, JPE, 1979.
Algorithm:
1. agents submit their vNM utility functions.
2. Each agent is endowed with equal probabilities to the others
for getting any object.
3. Find a price vector that clears the market, and get the
equilibrium allocation.
4. A lottery is conducted according to the resulting assignment.
Competitive equilibrium approach
• Not very easy to understand
• Not very easy to implement
• Fair
• Pareto efficient with respect to the reported utilities
Proof: by the first welfare theorem
( Ordinally efficient  Ex-post efficient)
CE approach – small problem
Requires submitting vNM utility functions!
CE approach is not strategy-proof
Roughly: if an agent can change her report and affect the
equilibrium prices, she can manipulate the prices to her
CE approach in large markets
If no agent can substantially change the prices, the mechanism is
“almost strategy-proof”.
By Liu and Pycia (2013), this implies that for any pre-specified
utility structure (and only submission of ordinal preferences), the
CE approach also converges to RSD in large markets.
Israeli medical match – mechanism
(with Arnon Afek, Slava Bronfman and Avinatan Hassidim)
How did we redesign to do this match?
• Requirement: do no harm (compared to RSD).
– Looking to push efficiency even at the cost of strategy-proofness
– A good measure for efficiency is rank distribution
– Other mechanisms that were designed this way: Teach for America
and HBS Field 2 / global immersion (Featherstone and Roth).
Described in Featherstone (2014).
Israeli medical match – mechanism
Sketch of the mechanism:
1. Compute RSD probability shares
2. Solve a linear program to maximize “social welfare” with the
constraints that nobody gets less “utility” compared with
their RSD probability vector, and that hospitals’ capacities are
respected.
3. Run a lottery using BvN decomposition.
Properties of the mechanism
• Not easy to understand or to implement
• Fair
• Not strategy-proof: for example, if you like an unpopular
hospital (you rank it in the fifth place), you should rank a more
popular hospital in front of it so you would get better
probability to get to the unpopular hospital you prefer. (Think
• Moreover, it is not strategy-proof even in large markets!
• Why do we think it is ok?
– If people begin to cheat, we will know
– It is not easy to understand how to cheat (classic cryptography style)
Efficiency gains
On the other hand, not being strategy-proof means that we
avoid the large market results of Liu and Pycia (2013):
Open questions
There is an efficiency / strategy-proofness trade-off.
• Are there situations in which the trade-off can be shown to be
less pronounced? (different domains of preferences)
• Can we design a mechanism that will require a lot of
computing power to find profitable manipulations?
BvN decomposition
• Except for RSD, the three mechanisms presented require
decomposition of matrices containing probability shares into
lotteries over deterministic assignments.
• The matrices have two very special properties:
– All rows sum to 1 (that is, every doctor is guaranteed an internship)
– Every hospital’s column sums to that hospital’s capacity.
• This reminds us of bi-stochastic matrices, whose every row
and every column sum to 1, and all elements are in 0,1 .
BvN decomposition
Theorem: Any bi-stochastic matrix can be represented as a
convex combination of permutation matrices.
Proof: Suppose  is a  ×  bi-stochastic matrix that has
elements equal to zero. Think about a bipartite graph  with
edge ,  iff  > 0. Note that for every set of rows  we have:

=

=
∈ =1
≤
∈ ∈
=
∈  =1
Use Hall’s theorem to find a perfect matching, represented by
the permutation matrix . Let ′ =  − , where  =
min Aij  > 0 , and normalize ′ so it will be a bi-stochastic
matrix with  + 1 zero elements. Repeat until  = .
BvN decomposition - complexity
We can decompose a matrix quite quickly:
We do at most  2 steps, and find a perfect matching in each
step (can be done by Ford-Fulkerson algorithm in   2 ).
Run time is   4 .
Extensions of the theorem
• Easily extended to our case (hospitals with capacities), just by
splitting each hospital to independent positions.
• Works for more doctors than positions, or more positions
than doctors (adding dummy positions or doctors).
• Budish et al. (2013): multi-unit environments (course
allocation) and real-world constraints (group-specific quotas).
Also works for certain substitutable preferences.
• There are also some more mathematical generalizations (e.g.
Ellis et al., 2014)
Israeli medical match – couples
• There are medical couples in Israel as well!
• Even in tiny Israel, couples do not want to be assigned
hospitals that are too far apart. This is a form of
complementarity, and it’s going to be a problem again.
• You may expect more couples in Israel compared to the US,
but turns out there aren’t that many. In 2014 there were
about 15 couples (out of ~500 doctors). Possibly because
being a medical couple in Israel is quite a disasterous life
decision. Of those, some are actually not romantically-related
(but would love to be roommates).
Israeli medical match – couples
• In order to accommodate for couples, it has been decided
that they can participate by submitting one rank order list for
both of them.
• When doing RSD (as in the past), it is easy to give the couple
one lottery number, and give them their most preferred
hospital that still has two vacant internship positions (if there
are none, we split the couple).
• How can we give those couples a similar benefit in our
mechanism? (or in any of the ordinally-efficient mechanisms)
Couples and BvN decomposition
(with Noga Alon, Slava Bronfman and Avinatan Hassidim)
Example: Assume that couples cannot be split. Both hospitals
have two positions. There are two doctors in the market, 1 and
2 , and a couple, 1 . By naively trading probability shares we
could possibly get the following matrix:
1
2
1
ℎ1
ℎ2
3/4 1/4
1/4 3/4
1/2 1/2
This matrix cannot be decomposed to determinstic allocations.
Couples and BvN decomposition
Theorem: Deciding whether a given matrix with couples can be
decomposed to a convex combination of valid deterministic
assignments is NP-complete.
• Note : neither the fact that we only deal with couples (and not
groups) nor that we restrict their preferences to a certain
structure (diagonal elements) help.
Couples and BvN decomposition
Theorem: assuming that:
• there are more singles than couples,
• all doctors must be assigned,
• and couples must be assigned together,
we can find a decomposition that approximates a given matrix
1
up to
, where  = min ℎ .

ℎ
There are examples showing that this is tight, in the sense that
we cannot do better than
doctors.
1

or with more couples than
What’s next?
• The 2014 medical match went well (in the sense that we got
no complaints).
• We are in the process of running surveys to elicit participants’
satisfaction, and to let them compare their probability vectors
(RSD vs. our method).
• MoH people were happy with the procedure. As we speak,
Slava and Avinatan sit with them and give them the code so
they can use it on interns that come from non-Israeli schools.
THANK YOU!
Some interesting results from the
Israeli psychology match
•
•
•
•
Replacing crazy rounds mechanism
40+ programs in 12 institutions
970 students, 537 matched
Program preferences:
– Reserved minority slots, gender balance
– Scholarships
– Professor-specific quotas
• Still working on analyzing the data (ranking period ended a
week ago). But…
– Almost 20% failed sanity test (!)
– Scholarships seem to have only a minor effect (contrary to
departments’ beliefs).
```