[1] Kidney Exchange By Alvin Roth, Tayfun Sonmez and M. Utku Unver Presented by Kevin Kurtz and Beisenbay Mukhanov February 5 th, 2015

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1 [1] Kidney Exchange By Alvin Roth, Tayfun Sonmez and M. Utku Unver Presented by Kevin Kurtz and Beisenbay Mukhanov February 5 th, 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)

2 [2] 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 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.

3 [3] 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

4 [4] 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 k i ith kidney; the kidney intended for the ith patient t i ith patient K Set of n available kidneys K i Feasible set of kidneys in K (K i K) which are compatible with patient i w Option of entering the waitlist for a cadaver kidney with a high priority P i The strict preferences of t i over K i { k i, w}, C i A cycle which constitutes a direct exchange between two or more patientdonor pairs W i 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, P i, there can never be an option ranked lower than k i. 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.

5 [5] w ki ki ti ki ti ti ti ki w 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 t i points either to his most preferred kidney or to the wait-list option, w. Each remaining kidney k i points to its paired patient, t i. 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.

6 [6] Example Suppose there is a list of patient-donor pairs with the following preferences. t 1 k 6, k 7, k 5 t 2 k 1, k 5, k 7, w t 3 k 1, k 9, w t 4 k 1, k 5 t 5 k 1, w t 6 k 3, k 4, k 8, k 7, k 1 t 7 k 9, w t 8 k 4, k 1, k 8, k 6 t 9 k 8, k 4, k 9 Round 1 k 2 k 5 t 5 k 4 t 4 t 8 t 2 k 8 k 6 t 1 k 1 t 9 w k 3 t 6 t 3 t 7 k 7 k 9 Fig. 2 Following Step 1, each patient points to the kidney they prefer most.

7 [7] Round 1 k 2 k 5 t 5 k 4 t 4 t 8 t 2 k 8 k 6 t 1 k 1 t 9 w k 3 t 6 t 3 t 7 Fig. 3 As a part of Step 2, a cycle C 1 = (k 1, t 1, k 6, t 6, k 3, t 3 ) is identified. k 7 k 9 Round 1 k 2 k 5 t 5 k 4 t 4 t 8 t 2 k 8 w t 9 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 k 7 t 7 k 9

8 [8] Round 2 k 2 k 4 t 4 t 8 t 2 k 5 t 5 k 8 t 9 w 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, W 1 = (k 7, t 7, k 9, t 9, k 8, t 8, k 4, t 4, k 5, t 5, w) and W 2 = (k 2, t 2, k 5, t 5, 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 W 1 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. k 7 Round 3 t 7 k 9 Fig. 6 As per Step 4, we repeat all previous steps once a W-chain is selected. As a result, t 2 switches to their next preferred kidney, k 7. This creates a single W- chain and is the end of the TTCC mechanism. Kidney k 2 will be given to someone from the cadaver queue, and patient t 5 will enter the cadaver queue with a high priority position k 2 t 2 t 4 t 8 k 5 k 4 t 5 k 8 w t 9 k 7 t 7 k 9

9 [9] 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 t 2 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, t 2 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

10 [10] 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 tailkidney for following round is the mechanism leading to Pareto efficient allocation at any given time. pairs. Example 1 Suppose there are the following preferences of five patient-donor 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 strategyproofness 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

11 [11] 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

12 [12] 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 Because the information about the willingness of donorpatient 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

13 [13] 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 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) = x-y/10 U(x,y) = x-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 t i are determined only over kidneys k j 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

14 [14] 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 Size of simulation Size of population (n) Exchange regimes Steps of the Simulations Monte-Carlo 100 trials a) 30 donor-patient pairs b) 100 donor-patient pairs c) 300 donor-patient pairs 1) no-exchange 2) paired exchange 3) TTC mechanism 4) Paired and indirect exchange 5) TTCC mechanism implemented with efficient and strategyproof 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 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.

15 [15] Table III Table IV Table V 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. 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. 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.

16 [16]

17 [17] 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, So nmez and U nver (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

18 [18] 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 donorpatient 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 [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 twoside 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, multihospital kidney exchange. Theoretical Economics, 9, Roth, Alvin E., Tayfon Sonmez and M. Utku Unver (2004), Kidney exchange. Quarterly Journal of Economics, 119, Roth, Alvin E., Tayfon Sonmez, and M. Utku Unver, Kidney Exchange. NBER Working Paper No , September, http//