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“Kidney Exchange”

By Alvin Roth, Tayfun Sonmez and M. Utku Unver

Presented by Kevin Kurtz and Beisenbay Mukhanov

February 5th

, 2015

I. Introduction

Background:

Kidney transplantation is the treatment of choice for most kidney diseases, but

there are many more people in need of kidneys than there are kidneys available. Table 1

shows the extent of this demand, but also a staggering undersupply of kidneys. The result

is long waitlists of patients waiting, often for years, for a kidney to become available.

Due to this wait, thousands of patients die every year without receiving a transplant.

Kidneys for transplantation can come from two sources: donors who are willing to

give a kidney – usually to a loved one or relative; and cadavers. A problem with the

former is that not everyone who is healthy enough to donate a kidney and wishes to do so

can donate a kidney to his or her intended recipient. A successful transplant requires the

donor and recipient to be compatible in blood and tissue types. A similar problem exists

in the latter option as well, for the characteristics of cadavers cannot be guaranteed in

advance. This gives rise to the possibilities of a kidney exchange: where incompatible

patient-donor pairs can swap kidneys.

Source: (extracted from Roth 2004a)

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II.A Kidney Transplantation

While there is a distinct shortage of live donors for all the number of transplants

needed, the sale of organs is strictly prohibited by the National Organ Transplant Act

(NOTA) of 1984. This has prevented a market oriented solution to the chronic shortage

of kidneys available for transplant in the United States. Live donor kidneys are preferable

to cadaver kidneys due to a higher survival rate from surgery. Thus, there is a distinct

need for an exchange mechanism of kidneys from live donors.

Types of Exchanges:

Direct Exchange – Also called paired exchange, this involves two patient-donor

pairs in which a transplant from the donor to the intended patient is infeasible, but

successful transplants are possible using the kidney from the other patient-donor

pair.

o Example: Suppose there are two patients, A and B. Each of them have

donors, X and Y, respectively, who are willing to give them a kidney.

Furthermore, suppose patient A is compatible with kidney Y, but not

kidney X, and patient B is compatible with kidney X, but not Y. It is

possible for patients A and B to exchange their donor kidneys with each

other rather than be put on a waitlist, which is a Pareto Efficient outcome.

Indirect Exchange - Also called list exchange, this involves an exchange

between one incompatible patient-donor pair and the cadaver queue. In return for

donating a kidney to the cadaver queue, the patient in the pair receives a high

priority listing on the cadaver queue. This is welfare improving for both the

general public and for the patient in the patient-donor pair.

o Note: this could have a negative impact on O-bloodtype patients, as they

have the fewest kidneys available to them on the cadaver queue. Since

anyone can receive a transplant from an O-bloodtype donor, it is less

likely that O-bloodtype patients without a donor will be able to receive a

kidney in the cadaver queue.

o Example: Suppose there is one patient, A, who has someone willing to

donate their kidney, X. If patient A is not compatible with kidney X, (s)he

has the option of giving kidney X to the pool of cadaver kidneys in

exchange for a high placement on the cadaver queue.

Cadaver Queue:

Those who do not have a donor – or whose donor’s kidney is not compatible with

them – are referred to a waiting list for cadaver kidneys. This waiting list is highly

structured, with scores being assigned to each candidate based on several factors. Each of

these factors (blood type, tissue [or HLA], age, size) can significantly affect whether or

not a transplant is likely to succeed. Each of these preferences is given a numeric score.

This allows for strict preferences within any kidney exchange mechanism.

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II.B Mechanism Design

The Housing Market Analogy:

Roth compares the market for kidney exchange to that of a model of the housing

market created by Shapley and Scarf [1974]. In this model, each agent is endowed with

an indivisible good (their house), and has strict preferences across all houses, but there is

no money in the market. They use David Gale’s Top Trading Cycle (TTC) mechanism

to produce an allocation of houses. The TTC mechanism works in situations where the

quantity of goods is fixed and known, and where individual choice is ordered by strict

preferences. It works as follows:

Each agent points to the house that they want most.

Agents that point to their own house are removed from the market.

There is at least one cycle as a result of this. The trades in this cycle are carried

out, and both agents and houses are removed from the market.

Continue from the beginning of the process until no agents remain.

The result of the TTC process is a unique, Pareto efficient outcome. As Roth found in

later studies [1984], it does not pay for agents to lie in this mechanism, as they will be

rewarded with a house that is not their most preferred choice.

The College Dorm Analogy:

Another analogy that Roth makes is to a later study by Abdulkadiroĝlu and

Sönmez [1999] of the housing allocations on college campuses. The difference between

this and the housing market model mentioned above is that there is the introduction of

unallocated goods (unoccupied rooms) and agents who are not endowed with goods (new

students). So, the authors made some changes to the TTC model, which is as follows:

Each student reports their strict preferences over all rooms.

Assign the first student (based on priority) their first choice, and so on, until a

student requests a unit that is already owned.

Modify the ordering by moving the tenant of the requested unit to the front of the

line ahead of the person who requested that unit. Then continue with the

procedure once more.

If at any point a cycle forms, assign all students in that cycle the units they desire.

The key innovation here is that people who own a unit already are guaranteed to keep

that unit if they enter the market. By being placed in the front of the line before their unit

is allocated, they have a risk-free opportunity to upgrade. Thus, it is a Pareto efficient

system where every endowed agent enters the market.

There are also strong parallels between this system and the kidney exchange system.

The housing queue is a parallel to the cadaver queue. If a donor gives a kidney to the

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cadaver queue, his intended recipient jumps to the top of the queue, just as in the college

room scenario. The only difference, albeit a key one, is that the number of rooms in the

college scenario is fixed, whereas the number of kidneys is not. It is also not known how

long one will wait in the cadaver queue until a compatible kidney becomes available.

III.A Top Trading Cycles and Chains Mechanism (TTCC)

Model Variables:

n number of patient-donor pairs

ki ith kidney; the kidney intended for the ith patient

ti ith patient

K Set of n available kidneys

Ki Feasible set of kidneys in K (Ki ⊂ K) which are compatible with patient i

w Option of entering the waitlist for a cadaver kidney with a high priority

Pi The strict preferences of ti over Ki ∪{ ki, w},

Ci A cycle which constitutes a direct exchange between two or more patient-

donor pairs

Wi W-chain, assigned in instances of multiple W-chains occurring in a single

period

Model Terminology:

Head – Pair who donate a kidney to the cadaver queue

Tail – Pair who receive a high priority for a kidney from the cadaver queue

Cycle – A series of direct exchanges between patient-donor pairs which form a

closed loop. It is represented by the left diagram in Fig. 1.

W-chain – A series of direct exchanges between patient-donor pairs that do not

form a closed loop, but instead contain indirect exchanges at the ends (referred to

as Head and Tail). At the head of the W-chain, the donor gives their kidney to the

cadaver queue in exchange for another patient-donor pair’s kidney. At the tail of

the patient-donor queue, the patient receives high priority on the cadaver queue in

exchange for giving their donor’s kidney to another patient from the live donor

pool. It is shown by the right diagram in Fig. 1.

There are a few important things to note:

In a patient’s preference set, Pi, there can never be an option ranked lower

than ki. This ensures that the process is Pareto Efficient, as no one will end up

worse off than if they had not entered

Preferences are determined based on probability of survival

Lemma 1 - Consider a graph in which both the patient and the kidney of each pair are

distinct nodes as is the wait-list option w. Suppose that each patient points either toward

a kidney or w, and each kidney points to its paired recipient. Then either there exists a

cycle, or each pair is the tail of some w-chain.

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Fig. 1 – Visual representation of two conditions of lemma 1

III.B The Exchange Mechanism

Step 1 – All kidneys are available, and all agents are active. Each patient ti points

either to his most preferred kidney or to the wait-list option, w. Each remaining

kidney ki points to its paired patient, ti.

Step 2 – At this point, by Lemma 1, there is either a cycle or a w-chain, or both.

o If there is not a cycle, proceed to step 3.

o If there is a cycle, carry out the corresponding exchanges and remove all

patients and kidneys involved from the mechanism.

Once this has been done, have all remaining patients point to their

top choice among the remaining kidneys. Locate any new cycles,

remove them, and repeat until there are no more cycles.

Step 3 – At this point, all remaining pairs are the tails of w-chains.

o Select one of the chains according to whichever chain selection rule you

choose. This assignment is final for those in the selected w-chain.

Depending upon which chain selection rule you use, the w-chain is

either removed from the mechanism, or patients and kidneys

remain in the mechanism, but are passive instead of active (cannot

accept new assignments)

Step 4 – After the w-chain is selected, new cycles may form, so you must repeat

steps 2 and 3 until there are no patients or kidneys are left.

By the end of this procedure, every patient with a living donor has either been

assigned a living kidney for transplant or a high priority position on the waiting list.

w

ti

ki

ti

ki

ki

w

ti

ki

ti

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Example:

Suppose there is a list of patient-donor pairs with the following preferences.

t1: k6, k7, k5

t2: k1, k5, k7, w

t3: k1, k9,w

t4: k1, k5

t5: k1,w

t6: k3, k4, k8, k7, k1

t7: k9, w

t8: k4, k1, k8, k6

t9: k8, k4, k9

Round 1

Fig. 2 – Following Step 1, each patient points to the kidney they prefer most.

t2

t6 t3 t7

t4 t8

t9

t5

k6

:

k2

:

k3

:

k5

:

k1

:

k4

k8

:

k9

:

k7

:

w

t1

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Round 1

Fig. 3 – As a part of Step 2, a cycle C1= (k1, t

1, k6, t6, k3, t3) is identified.

Round 1

Fig. 4 – As part of Step 2, the cycle is removed from the pool of patient-donor pairs. This allocation is now

fixed, and those kidneys in it are no longer available for consideration

t2

t6 t3 t7

t4 t8

t9

t5

k6

:

k2

:

k3

:

k5

:

k1

:

k4

k8

:

k9

:

k7

:

w

t1

t2

t7

t4 t8

t9

t5 k2

:

k5

:

k4

k8

:

k9

:

k7

:

w

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Round 2

Fig. 5 – As per Step 2, those remaining pairs choose their most preferred kidney. Since there are no longer

any cycles, we proceed to Step 3. There are two W-chains remaining, W1 = (k7, t7, k9, t9, k8, t8, k4, t4, k5, t5, w)

and W2 = (k2, t2, k5, t5, w). Which you choose depends upon what chain selection rule you are using. For this

example, we will use the longest chain rule. Thus, we take W1 and switch its occupants from an active role

to a passive role. Thus, they are still in the pool of donor pairs, but cannot change their current assignments.

Round 3

t2

t7

t4 t8

t9

t5

k2

:

k5

:

k4

k8

:

k9

:

k7

:

w

t2

t7

t4 t8

t9

t5

k2

:

k5

:

k4

k8

:

k9

:

k7

:

w

Fig. 6 – As per Step 4, we

repeat all previous steps

once a W-chain is selected.

As a result, t2 switches to

their next preferred kidney,

k7. This creates a single W-

chain and is the end of the

TTCC mechanism. Kidney

k2 will be given to someone

from the cadaver queue,

and patient t5 will enter the

cadaver queue with a high

priority position

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Alternative W-Chain Rules:

Minimal w-chains:

o In this rule, the smallest w-chains to be formed are removed from the

mechanism first. The result from this will be an increased number of live

donor kidneys being allocated to the cadaver queue.

Maximum w-chains:

o In this rule, the largest w-chains to be formed are removed from the

mechanism first. If there are multiple w-chains of the same length which

are not unique, then a tiebreaker is used (compatibility, age, etc) to

determine which chain is removed.

o There is also the option of keeping the longest w-chain in the mechanism,

but having it remain in a passive role, rather than an active role. This

means that the selected patients have their choices locked, and cannot

choose a new kidney. This is done in hopes that after the mechanism goes

through one more iteration, the w-chain can be lengthened.

The benefit of keeping the w-chain in the mechanism for the next

round is that it can be a Pareto-improving option. In the example

above, patient t2 would have to move to the cadaver queue if the w-

chain was removed at the end of round 2. By keeping the w-chain

there, but in a passive role, t2’s welfare improved, but not at the

expense of any other patient or those in the cadaver queue (the

effect on which we will consider net neutral).

o The benefit of this chain rule is increased welfare among those in the live

donor pool as opposed to using the minimum w-chains rule.

Prioritized pairs:

o Choose the patient-donor pair that has the highest priority and remove it

from the mechanism. This has the benefit of making sure that the patients

with the most need are able to receive the kidney that best suits them.

o As with the maximum w-chains rule above, this can be altered so that

instead of removing the w-chain from the mechanism immediately, it can

remain in a passive role in hopes of lengthening it.

Prioritize O-type donor pairs:

o This is a special case of the prioritized pairs rule above. Patients whose

donor has O-type blood are given priority in this situation. The w-chain

starts with the highest priority pair, and if the donor in that pair has O-type

blood it is immediately removed.

o The effect of this is a significant increase of type O kidneys to the cadaver

queue, while there may be significant efficiency losses within the

mechanism itself.

IV. Efficiency and Incentives

The paper considers the Pareto efficiency of the kidney allocation obtained by the

TTCC mechanism. If there is no other patient-kidney matching which is not worse than

the initial matching for all patient-donor pairs and strictly better for at least one pair, then

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this initial matching is Pareto efficient. In the article a kidney exchange mechanism

regards as efficient if it always gives a Pareto efficient at any given time.

The authors claim the following two theorems about the TTCC mechanism’s

efficiency and when the TTCC is strategy- proof.

The Theorem of TTCC mechanism’s efficiency

Claim: The TTCC mechanism is efficient if it applies a chain selection rule in

which w-chain chosen at every intermediate round keeps in the procedure and its tail

stays available for the following round.

On the other hand the TTCC mechanism in which w-chain’s tail is not available

for the next round is not necessary Pareto-efficient matching.

Proof: Suppose that the TTCC mechanism is implemented with a chain rule such

that w-chain chosen at every intermediate round keeps in the procedure and its tail stays

available for the following round. Then, after Round 1 every patient that takes his final

assignment has his top-preferred kidney. In Round 2 every finalized patient takes his

top-preferred kidney among leftover grafts including the tail kidney from chosen in

Round 1 w-chain. Hence these patients can be made better off only making worse off the

patients finalized in Round 1. Continuing this procedure, there is no any patient that can

get better choice without making worse off another patient that was finalized in previous

rounds. Therefore, the TTCC mechanism implemented with a chain rule that keeps tail-

kidney for following round is the mechanism leading to Pareto efficient allocation at any

given time.

Example 1: Suppose there are the following preferences of five patient-donor

pairs.

t1: k5 k1

t2: k5 k3 k2

t3: k4 k5 w

t4: k5 w

t5: w

Suppose one uses the TTCC mechanism with the chain selection rule that chooses

the longest w-chain and extracts it.

Then, in Round 1 there is no any cycle and the longest w-chain is (k3, t3, k4, t4,

k5, t5, w). After removing it there will be two cycles (k1, t1) and (k2, t2). The final

outcome of the matching will be (t1-k1, t2-k2, t3-k4, t4-k5, t5-w). However, the Pareto

efficient matching will be (t1-k1, t2-k3, t3-k4, t4-k5, t5-w).

The Theorem of TTCC’s Strategy-Proofness

The second theorem lists the chain selection rules that guaranty the strategy-

proofness of the TTCC. In this aspect the paper limits the strategy space to the space of

declared preferences. At the same time the kidney transplant process might consist other

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strategic issues that are not considered. For example, patient might register in multiple

regional transplant centers and as result be on multiple queue lists.

According to Roth (1982), truly stated preferences in the housing model are the

necessary strategy to prevent the profit gain by an agent that misrepresents his

preferences. Because the static kidney exchange and the housing model are similar, the

authors refer Roth (1982) results for strategy-proofness of the TTC mechanism for the

kidney exchange without indirect exchanges.

The TTCC’ strategy-proofness depends from an implemented chain selection rule.

Claim: The TTCC mechanism applying any of the mentioned alternative w-

chain rules except the maximum w-chain rule is strategy-proof.

The proof of this theorem is skipped and can be found in the paper in question.

At the same time the most attractive chain selection rules are the prioritized pairs

and prioritize O-type donor pairs rules with the condition that w-chain-tail kidney keeps

for the next round.

Both of them lead to an efficient and strategy proof TTCC mechanism. Applying

of rule f, in addition, increases the number of deficit type kidney O available for patients

waiting cadaveric kidneys.

On the other side, the TTCC with a chain selection rule choosing the longest w-

chain is not strategy proof because a patient can benefit by influencing the w-chain

lengths via preference falsification.

Consider the following illustration of the preference misrepresentation.

Example 2.

Suppose there are seven donor-patient pairs (k1,t1), ….(k7,t7) with the following

truthful preferences.

t1: k2 k5 w

t2: k5 k4 k1 w

t3: k2 k6 k3

t4: k6 k1 k6

t5: k1 k3 k5

t6: k5 w

t7: k3 k7

In this example patient t3 stays with his donor for the next time, while patient t4

gets kidney k6. However, if patient t3 misrepresents his preferences as k6, k2,k3, he

receives kidney k6 instead of patient t4.

V. Simulations

The paper provides the simulations’ results in order to compare the welfare gains

that might be obtained by applying the TTCC exchange mechanism and other

mechanisms, and by this way support the TTCC concept.

V.A. Data

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The data for simulations was limited because of lack of wide detailed information

about patients and donors. The information that the authors could find is represented in

Table II (was taken without any changes form the paper). In addition, the authors used

the Zenios’s (1996) HLA characteristics’ distribution based on the UNOS registration

data for period of 1987-1991. Because the information about the willingness of donor-

patient pairs to exchange the donor’s kidney for priority on the cadaveric waiting list, the

authors tested the reliability of results by simulating a wide diapason of preferences. In

the current article they demonstrate the results of the simulations with two assumptions

that there is no the pair who is willing to exchange the donor’s kidney for priority on the

cadaveric graft queue and that 40% of the pairs are willing to make that exchange.

* The information about similar computations in constructions of other economic empirical designs might

be find in Roth [2002]

V.B. Assumptions

Assumption 1

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“All HLA proteins and blood type are independently distributed following

Zenios”. To simplify the simulations, the authors consider a scenario with unrelated pairs

of donors and patients (spouses, friends and so on). In 2001 the rate of this group was

about 25.3% among all living-donor grafts.

Assumption 2

The authors assume that all patients and donors are adults (of age 18-79). Hence

they calculate the conditional age distributions of the patients and the unrelated

nonspousal donors given that the donors and patients are adults. For this purposes Table

II was used.

Assumption 3

HLA characteristic and blood type distribution of patients and donors are the

same, “the characteristics of a nonspousal unrelated donor are independently distributed

with the patient, and the characteristics of a spouse are independently distributed with the

patient except his or her age”. The ages of spouses are the same.

Assumption 4 – Preference Determination

The authors make assumption that preferences of donors and patients over

available kidneys depend from the probability that the implanted graft will be not

rejected. They use the results of survival analysis published in Mandal et al. (2003) and

based on data obtained in 1995-1998 from the United States Renal Data System

(USRDS). The authors suggest two types of preference construction.

“Rational”

for patients

from 18 to 59

for patients

from 60 to 80

U(x,y) = -0.514x-y/10 U(x,y) = -0.510x-y/10

a monotone decreasing function, where

x – the number of HLA mismatches, x 0, 6 y – the donor age, y 18, 80

“Coutious” The patient ti prefers the kidney of donor kj if and only if - kidney of its own donor ki is incompatible with him, or - kidney of its own donor is compatible, but has more HLA

mismatches than kidney kj has. In both methods the preference of a patient ti are determined only over kidneys kj

that are ABO compatible with the patient.

HLA mismatches determine through pre-transplant crossmatching test. The test

may be positive or negative. The positive crossmatch means that patient’s antibodies will

attack donor’s HLA that increase the graft failure risk.

The Marginal Rates of Substitution of one additional HLA mismatch by

decreasing in the donor age were determined by using Mandal’s et al. (2003) estimations,

and are

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5.14 for patients from 18 to 59, and

5.10 for patients from 60 to 80

However, in Mandal et al. (2003) there are also such factors as patient race and

age, patient health history (especially the history of diabetes and the period of the

treatment with dialysis) that influence the failure risk.

Assumption 5

The authors used statistics from different papers that were based on data from

different periods. Therefore they assume that characteristics’ distributions of new patients

are independent from time period and the same for the same population, i.e. American

Caucasian ESRD patients represented in this paper.

V.C. Simulated Mechanism

Method of

simulation

Monte-Carlo

Size of simulation 100 trials

Size of population

(n) a) 30 donor-patient pairs

b) 100 donor-patient pairs

c) 300 donor-patient pairs

Exchange regimes 1) no-exchange

2) paired exchange

3) TTC mechanism

4) Paired and indirect exchange

5) TTCC mechanism implemented with efficient and strategy-

proof w-chain selection rule when patient-donor pairs prioritize

in a single list , and w-chain is chosen with the highest priority

pair and keeps for next Round

Steps of the

Simulations

1) Random simulation of a sample of n-size population using the

characteristics of donor-patient pairs

2) Determination of four preference sets for each patient

- two sets using “rational” utility function and assuming 0% and

40% of donor-patient accepting cadaveric wait-list option.

- two sets using “cautious” approach and also assuming 0% and 40%

of donor-patient accepting cadaveric wait-list option

3) Simulation of the five mechanisms using all four preference set

for every population size.

V.D. Results of the Simulations

The simulations’ results are represented in Tables III, IV and V taken from the

paper. The rows of the tables consist of different population sizes and exchange regimes

under the different preference constructions.

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Table III Column 4 is the percentage of living grafts that were received by patients

under every exchange regime.

Column 4 = Column 5 + Column 6

Column 7 is the number of HLA mismatches for an average graft.

The numbers in parentheses are standard errors of the evaluations.

Table IV The last five columns consist information about the percentage of the

population size that didn’t receive transplant and wasn’t willing to trade

their donor’s kidney for the priority in the waiting list.

Table V The columns of the table represent the number, the average length and

the maximum length of cycles and w-chains. The last columns called

“Longest” show the length of the longest cycle/w-chain among all

simulated 100 trials.

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The authors claim that the applying of the TTCC mechanism leads to significant

gains in the number of kidney exchanges and the quality of compatibility. There is some

interpretation of the obtained results in the paper.

1. The TTCC mechanism gives higher rate of adaptation of the kidneys from

unrelated living donors and decreases the number of HLA mismatches. This

positive effect improves as the donor-patient size rises.

2. The average and maximal lengths of cycles and w-chains increase with

increasing of the sample size.

3. The patients with the O type blood but without living donor advantage from

TTCC mechanism compared with indirect/paired-kidney exchange

mechanism. The TTCC decreases the number of the O type patients having

incompatible living donors and willing to change their kidneys for priority on

the cadaveric waitlist because there is no an available compatible kidney from

the other donor-patient pairs. This result might be explained by the fact that in

the TTCC mechanism A, B or AB patients that have donors with O type blood

but with some HLA mismatches can be matched with other pairs’ donors with

the same blood type and less HLA mismatches, and therefore it makes

available more O type donors for other O type patients including the O type

patients from the waitlist.

More detailed discussion consists in Roth, Sonmez and Unver (2003).

VI. Developments Since Publication

At the time of Roth’s paper, there was only the infrastructure in place for pairwise

exchanges to take place within the same hospital or treatment center, and this was the

extent of kidney exchange for a few years. But in subsequent years more papers came out

supporting the concept and by 2010 several infant kidney exchange programs had

cropped up, encompassing about 50 hospitals each. However, participation was not

uniform, as 20% of enrolled hospitals accounted for 50% of submitted kidney pairings.

In Ashlagi and Roth [2014], the authors studied some problems that have arisen

from the partial implementation of the kidney exchange system. As the program has

expanded, and hospitals began administering it rather than doctors, free riding began to

emerge. Hospitals would still do paired exchanges that arose within their system (“Easy

matches”) rather than submit them to the regional exchange network. Only “hard”

matches would be submitted to the exchange network. Hospitals would often join

multiple networks in an effort to free ride off of both of them. This led to efficiency

losses in the form of shortened w-chains and suboptimal kidney allocation, as well as

competing systems for kidney exchange.

More recently, a Johns Hopkins team has been able to nullify the effects of having

both positive crossmatch and different blood types on the probability of a successful

kidney transfer. The technique, called plasmapheresis, could potentially render a kidney

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exchange system obsolete, as one of the largest barriers to transplant compatibility would

be removed. Currently this practice is still in early trials, so there will still be a need for

an exchange system for the near future, at least.

VII. Extensions and critique

Applications:

Due to the many requirements for a TTCC mechanism (no currency, indivisible

goods, and strict preferences) there are few other situations where the TTCC mechanism

could be useful. Universities competing for students may be able to use this system to

optimize admission of candidates. If universities were to share lists of applications and

their strict preferences for each student, they could collude to selectively admit students

that they wish to accept – knowing that other universities would not accept them. This

would replace the current system where they accept many students and expect to have

some students turn them down. However, there are significant legal ramifications to this

and it is possible that lying about preferences may be beneficial for colleges in these

scenarios. Also, there would have to be a system created to determine the tiebreakers for

universities. This would work better in other countries which a central authority allocates

students to schools, rather than the individual choice that presides in the United States.

The TTCC mechanism could also be applied to the distribution of public goods, in

special scenarios. After natural disasters, certain public goods could be scarce, and the

different affected regions could have different needs for aid. For example while after a

hurricane all residents on an island may need temporary shelter, some may have no

access to fresh water. An aid drop of water to the shelter-less community could be

diverted to the community in need of water, in exchange for a promise to give the first

available building materials to them. So a barter system could be put in place where the

government receives the needs of each community and allocates a scarce supply of aid to

each. There are also similar problems with this application, as misrepresenting

preferences could be beneficial to communities.

Focus on patient preferences.

The suggested exchange algorithm ranges compatible with a patient kidneys

relying only on their survival rate. This approach may miscount preferences of other

players participating in the exchange process. Some of these omissions might encourage

blocking the complex exchange. For example, hospitals might accept only the exchange

schemes that don’t decrease the number of transplant surges inside of their patients.

Therefore, as mentioned in Ashlagi and Roth [2014] they will try to provide for TTCC

exchange only donor-patients hard-matching with their other pairs. This will lead to

significant decrease of TTCC mechanism’s efficiency. As another example, a donor-

patient pair might have preferences not only over compatible kidneys but also over

patients who can get their kidney. And therefore there might be situation when the pair

will want to change their preferences over compatible kidneys in order to make the

[19]

allocation where their more preferable patient will get their kidney. That means that a

donor-patient pair might have preferences over kidneys depending from preferences over

possible patients.

The length of the cycle/w-chain.

The application of the TTCC mechanism implies long cycles and w-chains.

Moreover, the length of the cycles and w-chains increase as the population grows that is

the necessary condition for the increase of the TTCC efficiency. For example, when the

population size is 30 pairs, the longest cycle of “cautious” TTCC mechanism involves 10

pairs. As population grows to 300 pairs, this index increases to 22 pairs. This fact

requires two times more operation rooms and surgeon teams because all operations

should be made at the same time to avoid the risk of donor’s rejection. The authors

mention that kind of problem that should be overcome. However, in order to address the

current economical capabilities this paper might be extend by considering the ways to

decrease potential cost from TTCC’s implementation.

VII. Conclusion

The authors suggest adopting the centralized TTCC mechanism that based on the

idea of getting higher potential gains from multi-side trades rather than from simple two-

side trades. Consequently this mechanism will increase the number and quality of kidney

exchanges in comparison with existing pair and indirect exchange practices. The new

mechanism is designed by the extension of Gale’s top trading cycle (TTC) mechanism

involving donor-patient pairs participating in indirect exchanges.

References

Ashlagi Itai, Roth, Alvin E. (2014), “Free riding and participation in large scale, multi-

hospital kidney exchange.” Theoretical Economics, 9, 817-863.

Roth, Alvin E., Tayfon Sonmez and M. Utku Unver (2004),“Kidney exchange.”

Quarterly Journal of Economics, 119, 457-488.

Roth, Alvin E., Tayfon Sonmez, and M. Utku Unver, “Kidney Exchange.” NBER

Working Paper No. 10002, September, 2003.

http://www.hopkinsmedicine.org/transplant/programs/kidney/incompatible/