Kidney Exchange

1
Kidney Exchange

• 4th Barcelona Economics LectureHospital Clinic,
  Barcelona
• 8 November 2004

2

• Roth, Alvin E., Tayfun Sönmez, and M. Utku Ünver,
  Kidney Exchange, Quarterly Journal of
  Economics, 119, 2, May, 2004, 457-488.
• ____ Pairwise Kidney Exchange, June 2004.
• _____ The Importance of Three Way Kidney
  Exchange, in preparation

3
ProposalNew England Center for Kidney Donor
Exchange

• Presented to the ROTC
• September 20, 2004
• (Approved!)

Frank Delmonico, MD (NEOB and MGH) Susan
Saidman, PhD (MGH Histocompatibility Lab) Al
Roth, PhD (Prof. of Economics Business Admin,
Harvard) Tayfun Somnez, PhD (Dept. of Economics,
Koc University Utku Unver, PhD (Dept. of
Economics, Koc University
4

• On Saturday I gave a companion talk as the Pareto
  Lecture, at a conference of economic theorists
• In that talk I emphasized some of the theoretical
  issues that arise in designing a Kidney Exchange.
• Today Ill speak more of practical issues, and
  how those shape what can be done (and what kind
  of theory is needed).

5
Economists As Engineers

• In recent years, game theorists have become
  usefully involved in the design of markets.
• See e.g. Roth and Peranson (1999), Roth
  (2002,medical labor markets) Wilson (2002,
  electricity markets), Abdulkadiroglu and Sönmez
  (2003, schools), Milgrom (2004, auctions),
  Niederle and Roth (2004, gastroenterologist labor
  market)
• A certain amount of humility is called for
  successful designs most often involve incremental
  changes to existing practices, both because
• It is easier to get incremental changes adopted,
  rather than radical departures from preceding
  practice, and
• There may be lots of hidden institutional
  adaptations and knowledge in existing
  institutions, procedures, and customs.

6
Kidney transplants

• There are over 60,000 patients on the waiting
  list for cadaver kidneys in the U.S.
• In 2003 there were over 8,500 transplants of
  cadaver kidneys performed in the U.S. (and over
  2,000 in Spain, which has one of the most
  effective cadaver organ donation systems in the
  world)
• In the same year, about 3,500 patients died while
  on the waiting list.
• In 2003 there were also over 6,000 transplants of
  kidneys from living donors in the US, a number
  that has been increasing steadily from year to
  year.
• (I dont know the local statistics, but I
  understand that the Hospital Clinic is one of the
  places at which live donor transplants are done
  here.)

7
Live-donor transplants are much less organized
than cadaver transplants

• The way such transplants are typically arranged
  is that a patient identifies a willing donor and,
  if the transplant is feasible, it is carried out.
  
• Otherwise, the patient remains on the queue for a
  cadaver kidney, while the donor returns home.
• Recently, however, in a small number of cases,
  additional possibilities have been utilized
• Paired exchanges exchanges between incompatible
  couples
• Indirect exchanges an exchange between an
  incompatible couple and the cadaver queue

8
Paired Exchange (still relatively rare)
9
Baltimore Center Carries Out Triple-Swap
TransplantsAugust 2, 2003, New York Times

• The triple-swap kidney transplant operation was
  announced in a news conference today at the Johns
  Hopkins Comprehensive Transplant Center, which
  said it believed that this was the first time
  three simultaneous kidney transplants have been
  performed
• Months in the making, the exchange was the only
  way all three recipients could have received a
  kidney, the lead surgeon, Dr. Robert A.
  Montgomery, said, because of tissue, blood or
  antibody incompatibilities among the donors and
  their originally designated recipients.
• Johns Hopkins has recently hired a paired kidney
  exchange coordinator to facilitate further
  exchanges

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How might more frequent and larger-scale kidney
exchanges eventually be organized?

• Building on existing practices in kidney
  transplantation, we consider how exchanges might
  be organized to produce efficient outcomes,
  providing consistent incentives (dominant
  strategy equilibria) to patients-donors-doctors.
• Why are incentives/equilibria important?
  (becoming ill is not something anyone chooses)
• But if patients, donors, and the doctors acting
  as their advocates are asked to make choices, we
  need to understand the incentives they have, in
  order to know the equilibria of the game and
  understand the resulting behavior.
• Experience with the cadaver queues make this
  clear

11
Incentives liver transplants

• Chicago hospitals accused of transplant fraud
• 2003-07-29 112007 -0400 (Reuters Health)
• CHICAGO (Reuters) Three Chicago hospitals were
  accused of fraud by prosecutors on Monday for
  manipulating diagnoses of transplant patients to
  get them new livers.
• Two of the institutions paid fines to settle the
  charges.
• By falsely diagnosing patients and placing them
  in intensive care to make them appear more sick
  than they were, these three highly regarded
  medical centers made patients eligible for liver
  transplants ahead of others who were waiting for
  organs in the transplant region, said Patrick
  Fitzgerald, the U.S. attorney for the Northern
  District of Illinois.
• These things look a bit different to economists
  than to prosecutors? it looks like these docs
  may simply be acting in the interests of their
  patients

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Incentives and efficiencyNeonatal heart
transplants

• Heart transplant candidates gain priority through
  time on the waiting list
• Some congenital defects can be diagnosed in the
  womb.
• A fetus placed on the waiting list has a better
  chance of getting a heart
• And when a heart becomes available, a C-section
  might be in the patients best interest.
• But fetuses (on Moms circulatory system) get
  healthier, not sicker, as time passes and they
  gain weight.
• So hearts transplanted into not-full-term babies
  may have less chance of surviving.
• Michaels, Marian G, Joel Frader, and John
  Armitage 1993, "Ethical Considerations in
  Listing Fetuses as Candidates for Neonatal Heart
  Transplantation," Journal of the American Medical
  Association, January 20, vol. 269, no. 3,
  pp401-403

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Kidney Matching

• Two genetic characteristics play key roles
• ABO blood-type There are four blood types A, B,
  AB and O.
• Type O kidneys can be transplanted into any
  patient
• Type A kidneys can be transplanted into type A or
  type AB patients
• Type B kidneys can be transplanted into type B or
  type AB patients and
• Type AB kidneys can only be transplanted into
  type AB patients.
• So type O patients are at a disadvantage in
  finding compatible kidneys.

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• 2. Tissue type or HLA type
• Combination of six proteins, two of type A, two
  of type B, and two of type DR.
• Prior to transplantation, the potential recipient
  is tested for the presence of antibodies against
  HLA in the donor kidney. The presence of
  antibodies, known as a positive crossmatch,
  significantly increases the likelihood of graft
  rejection by the recipient and makes the
  transplant infeasible.

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Goals of a structured method of direct kidney
exchange

1. Assemble a database of incompatible patient-donor
   pairs. (Right now, the incompatible donors are
   largely lost.)
2. Identify which exchanges are possible, and which
   sets of exchanges make best use of available
   donor kidneys
3. allow not only for paired-exchange but also other
   forms of exchange such as a three-way exchange.

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Some relevant economics papers

• Shapley, Lloyd and Herbert Scarf (1974), On
  Cores and Indivisibility, Journal of
  Mathematical Economics, 1, 23-37.
• Roth, Alvin E. and Andrew Postlewaite (1977),
  Weak Versus Strong Domination in a Market with
  Indivisible Goods, Journal of Mathematical
  Economics, 4, 131-137.
• Roth, Alvin E. (1982), Incentive Compatibility
  in a Market with Indivisible Goods, Economics
  Letters, 9, 127-132.
• Atila Abdulkadiroglu and Tayfun Sönmez 1999
  House allocation with existing tenants. Journal
  of Economic Theory 88, 233-260.

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DONOR KIDNEY EXCHANGE FOR INCOMPATIBLE RECIPIENTS

• by Francis L. Delmonico, MD 1, Paul E. Morrissey,
  MD 1, George S. Lipkowitz, MD 2, Jeffrey S.
  Stoff, MD 1, Jonathan Himmelfarb, MD 1, William
  Harmon, MD 1, Martha Pavlakis, MD 1, Helen Mah 1,
  Jane Goguen 1, Richard Luskin 1, Edgar Milford,
  MD 1 and Richard J. Rohrer, MD 1. 1, New England
  Organ Bank, Newton, MA and 2, LifeChoice Donor
  Services, Windsor, CT.
• Reports two live donor exchanges (4 recipients)
  and 8 list paired exchanges (16 recipients) from
  2001-02.

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House allocation

• Shapley Scarf 1974 housing market model n
  agents each endowed with an indivisible good, a
  house.
• Each agent has preferences over all the houses
  and there is no money, trade is feasible only in
  houses.
• Gales top trading cycles (TTC) algorithm Each
  agent points to her most preferred house (and
  each house points to its owner). There is at
  least one cycle in the resulting directed graph
  (a cycle may consist of an agent pointing to her
  own house.) In each such cycle, the corresponding
  trades are carried out and these agents are
  removed from the market together with their
  assignments.
• The process continues (with each agent pointing
  to her most preferred house that remains on the
  market) until no agents and houses remain.

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Theorem (Shapley and Scarf) the allocation x
produced by the top trading cycle algorithm is in
the core (no set of agents can all do better than
to participate)

• When preferences are strict, Gales TTC algorithm
  yields the unique allocation in the core (Roth
  and Postlewaite 1977).

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Theorem (Roth 82) if the top trading cycle
procedure is used, it is a dominant strategy for
every agent to state his true preferences.

• The idea of the proof is simple, but it takes
  some work to make precise.
• When the preferences of the players are given by
  the vector P, let Nt(P) be the set of players
  still in the market at stage t of the top
  trading cycle procedure.
• A chain in a set Nt is a list of agents/houses
  a1, a2, ak such that ais first choice in the
  set Nt is ai1. (A cycle is a chain such that
  aka1.)
• At any stage t, the graph of people pointing to
  their first choice consists of cycles and chains
  (with the head of every chain pointing to a
  cycle).

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Cycles and chains
Cycles and chains
i
22
The cycles leave the system (regardless of where
i points), but is choice set (the chains
pointing to i) remains, and can only grow
i
23

• Paired kidney exchanges similarly seek the gains
  from trade among patients with willing donors,
  but (with the recent Johns Hopkins 3-pair
  exchange being a notable exception) mostly among
  just two pairs.
• In the context of kidney exchange, if we consider
  exchange only among patients with donors, the
  properties of the housing market model
  essentially carry over unchanged (as long as
  donor preferences coincide with those of their
  intended recipient).
• However donors (unlike houses) have preferences.
  So all parts of a live-donor exchange are done
  simultaneously, to avoid incentive problems.

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How big are the welfare gains?

• Theory show us how to go from inefficient to
  efficient procedures, but it doesnt tell us how
  big the gains are likely to be.
• For that we turn to computational simulations,
  using data on the mismatch frequencies, patient
  demographics, etc.
• We first consider unrelated donor-patient pairs.
  (About 25 all living-donor transplants were in
  this category in 2001.)

25
Patient and Donor Characteristics

• Population Caucasian ESRD patient population
  between 18 and 79 years of age in the U.S. Renal
  Data System (USRDS).
• Blood-type and age distribution Distributions
  for new ESRD waitlist patients recorded between
  January 1995 and April 2003 in the USRDS
  database.
• Gender distribution Data recorded between 1992
  and 2001.
• HLA distribution The distribution reported in
  Zenios 1996 using the USRDS registration data
  for years between 1988 and 1991.
• We assume that all HLA proteins and blood type
  are independently distributed following Zenios
  1996.

26
Simulated patient preferences

• Preferences are determined using the graft
  survival analysis of Mandal et. al. 2003. We
  assume that the preferences of each patient
  depends on the donor age and the number of HLA
  mismatches. Using the graft survival analysis of
  Mandal et. al. 2003, MRS is determined as
• 5.14 years of younger donor age per each
  additional HLA mismatch for patients younger than
  60 years of age, and
• 5.10 years of younger donor age per each
  additional HLA mismatch for patients older than
  59 years of age

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How big are the benefits? N30
28
How big are the benefits? N100
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How about actual patient populations?

• While the simulated results look good, they are
  drawn from general patient distributions.
• Actual patient populations will consist of
  incompatible patient-donor pairs.
• Patients who are already known to be incompatible
  with one donor may be much harder to matche.g.
  they are more likely to be highly sensitized

30
MGH Dataset (constructed by Susan Saidman)

• MGH patients w/ incompatible (ABO or XM) donor(s)
• Data included
• ABO type of patient donor
• HLA type of patient donor
• Most recent class I and II PRAs
• Called abs or safe antigens
• Relationship of donor to recipient
• Reason donor was incompatible
• If donor not HLA typed, HLA types were assigned
  from list of UNOS deceased donors
• 44 patients and 68 donor/patient pairs
• 23 O 13 A 6 B 2 AB

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Example of two-pair exchange (B-O,O-B)
Rec ID ABO Cl I PRA Cl II PRA Called abs Called abs Donor ID Donor ID Relatn ABO Donor HLA type Reason incompat
R28 O 0 0 D28.2 D28.2 Sib B DR52 ABO
R45 B 0 41 DR53 DR53 D45 D45 Child O DR51, 53 Class II ab
Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45 Exchange D45 gives to R28 D28.2 gives to R45
Rec ID ABO Cl I PRA Cl II PRA Called abs Donor ID Donor ID Relatn Relatn ABO Donor HLA type
R28 O 0 0 D45 D45 - - O DR51, 53 -
R45 B 0 41 DR53 D28.2 D28.2 - - B DR52 -
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Example of three-pair exchange (A-B,B-B,B-A)
Rec ID ABO Cl I PRA Cl II PRA Called abs Called abs Donor ID Relatn ABO Donor HLA type Reason incompat
R19 B 0 50 DR12 DQ2,7 DR12 DQ2,7 D19 Child B DR2, 3 DQ1, DQ2 Pos B XM
R43 A 0 0 - - D43 Spouse B DR2, 8 DQ1, 4 ABO
R31 B 0 0 - - D31 Spouse A DR7, DQ2, 3 ABO
Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31 Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31 Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31 Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31 Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31 Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31 Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31 Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31 Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31 Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31 Exchange D43 gives to R19, D31 gives to R43, and D19 gives to R31
Rec ID ABO Cl I PRA Cl II PRA Called abs Donor ID Donor ID Relatn ABO Donor HLA type
R19 B 0 50 DR12 DQ2,7 D43 D43 B DR2, 8 DQ1, 4
R43 A 0 0 - D31 D31 A DR7, DQ2, 3
R31 B 0 0 - D19 D19 B DR2, 3 DQ1, DQ2
33
Note that

• The initial screening and computer match
  identifies potentially compatible donor and
  recipient pairs
• A crossmatch will always be required before pair
  can be confirmed to be compatible
• Extensive antibody screening of patients and
  careful identification of all antibody
  specificities by a sensitive and specific method
  can help prevent unexpected positive crossmatches

34
Summary of analysis of MGH dataset

• If only two way exchanges allowed
• 8 patient-donor pairs in the dataset can
  potentially exchange kidneys (2 ABO-O 3 ABO-A 3
  ABO-B)
• If three way exchanges allowed
• 11 patient-donor pairs in the dataset can
  potentially exchange kidneys (3 ABO-O 3 ABO-A 4
  ABO-B 1 ABO-AB)
• There is also a possible five way exchange
• Allows 12 patient-donor pairs to potentially
  exchange kidneys
• But logistics currently not practical

35
Properties of Cycles for n30
36
Properties of cycles for N100
37
Discussion of the Computational Results

• The computational results (for both the simulated
  data and the MGH data) suggest that adoption of
  the TTC mechanism will significantly improve the
  utilization rate of potential living-donor
  kidneys.
• But under the TTC mechanism, average/maximal
  sizes of exchanges grow as the population grows.
  For large populations of patient-donor pairs,
  some of the efficient exchanges may be
  impractically large.

38
Suppose exchanges involving more than two pairs
are impractical?

• Our New England surgical colleagues have 0-1
  (feasible/infeasible) preferences over kidneys.
• Initially, exchanges may be restricted to pairs.
  (see also Bogomolnaia and Moulin (2004)
• This involves a substantial welfare loss compared
  to the unconstrained case
• But it allows us to tap into some elegant graph
  theory for constrained efficient and incentive
  compatible mechanisms.

39
Pairwise matchings and matroids

• Let (V,E) be the graph whose vertices are
  incompatible patient-donor pairs, with mutually
  compatible pairs connected by edges.
• A matching M is a collection of edges such that
  no vertex is covered more than once.
• Let S S be the collection of subsets of V such
  that, for any S in S, there is a matching M that
  covers the vertices in S
• Then (V, S) is a matroid
• If S is in S, so is any subset of S.
• If S and S are in S, and SgtS, then there is
  a point in S that can be added to S to get a set
  in S.

40
Pairwise matching with 0-1 preferences

• All maximal matchings match the same number of
  couples.
• If patients have priorities, then a greedy
  priority algorithm produces the efficient
  (maximal) matching with highest priorities.
• Any priority matching mechanism makes it a
  dominant strategy for all couples to
• accept all feasible kidneys
• reveal all available donors
• So, there are efficient, incentive compatible
  mechanisms in the constrained case also.

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(No Transcript)
42
Gallai-Edmonds Decomposition
43
Summary

• There are several potential sources of increased
  efficiency from assembling a database of
  incompatible pairs (aggregating across time and
  space), including
• More couple exchanges
• longer cycles of exchange, instead of just pairs
• If longer cycles of exchange arent (initially)
  feasible, constrained efficient matches can still
  be achieved with good incentive properties

44
Why 3-way exchanges add so much

• Example Consider a population of 9 incompatible
  patient donor pairs consisting of
• O-A, O-B (difficult to match O patients)
• A-B, A-B, B-A (more A-B than B-A pairs)
• A-A, A-A, A-A (odd number of A-A pairs)
• B-O (scarce O donor)
• 3 two-way exchanges are possible 6 transplants
• (A-B,B-A) (A-A,A-A) (B-O,O-B)
• If three-way exchanges are also feasible 8
  transplants
• (A-B,B-A) (A-A,A-A,A-A) (B-O,O-A,A-B)

45
Four-way exchanges add less

• In connection with blood type (ABO)
  incompatibilities, 4-way exchanges add less, but
  make additional exchanges possible when there is
  a (rare) incompatible patient-donor pair of type
  AB-O.
• (AB-O,O-A,A-B,B-AB) is a four way exchange in
  which the presence of the AB-O helps three other
  couples
• Incompatibilities involving positive cross
  matches may sometimes generate larger exchanges,
  but it appears that these are relatively rare

46
Summary (for surgeons)What do the economists
bring to the table?

• To arrange exchanges efficiently in a population
  of patients with incompatible donors, there are
  distributional issues, not just issues of medical
  compatibility.
• For example, consider four incompatible
  patient-donor pairs P1, P2, P3, P4, and suppose
  pairwise exchanges are possible between P1 and
  P2 P2 and P3, and P1 and P4.
• Then the exchange P1-P2 results in two
  transplantations, but the exchanges P1-P4 and
  P2-P3 results in four.

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Summary (for economists)

• As game theorists start to take a more active
  role in practical market design, we have to deal
  with constraints, demands, and situations
  different than those that arise in the simplest
  theoretical models of mechanism design
• Here we address some of the issues that have come
  up as we try to help surgeons implement an
  organized exchange of live-donor kidneys
• Not only do these issues appear to allow
  satisfactory practical solutions, they suggest
  new directions in which to pursue the underlying
  theory.