Kidney Exchange with Immunosuppressants

Transcription

1 Kidney Exchange with Immunosuppressants Youngsub Chun Eun Jeong Heo Sunghoon Hong February 29, 2016 Abstract We investigate the implications of using immunosuppressants as a part of a kidney exchange program. Immunosuppressants allow patients to receive transplants from any donor, thereby relaxing all immunological constraints. Given the limited availability of immunosuppressants, we identify which patients are to be provided immunosuppressants and then determine which patients are matched with which donors for the transplants. We ask whether there exist Pareto efficient solutions that satisfy additional requirements of monotonicity and maximal improvement. We propose modifications of the top-trading cycles (TTC) solutions to achieve these requirements. To quantify the improvement on current transplantation practice made possible by the use of immunosuppressants, we conduct counterfactual analyses using actual transplant data from South Korea. Our results suggest that it is possible to reduce the use of immunosuppressants by up to 55 percent from the current practice. JEL classification Numbers: C71; D02; D63; I10 Keywords: immunosuppressants; kidney exchange; top-trading cycles solutions; Pareto efficiency; monotonicity; maximal improvement File name: KEI tex Seoul National University. ychun@snu.ac.kr Vanderbilt University. heoeunjeong@gmail.com Korea Institute of Public Finance. sunghoonhong@kipf.re.kr

2 1 Introduction When a patient suffers from End Stage Renal Disease and needs to receive a kidney transplant, four options are available that depend on the immunological compatibility of the patient with her own donor. 1 First, if the patient is compatible with the donor, a direct transplant within this pair can be performed. Otherwise, she has to look for other ways to receive a transplant. The second option is to be registered on a waitlist to receive a transplant from a deceased donor. Another option, developed in the last decade, is to participate in a kidney exchange program in which donors are swapped between incompatible pairs. Unfortunately, the possibility of receiving transplants from deceased donors or through exchanges is quite limited relative to the increasing number of patients in the waitlist, as illustrated in Table 1 with data for the KONOS (Korean Network for Organ Sharing) program. Recent developments in immunosuppressive protocols have introduced a fourth option. Immunosuppressants (suppressants, for short) eliminate all immunological compatibility constraints both tissue-type and blood-type that patients might have against donors. If a patient uses a suppressant, then she becomes compatible with any donor, so is able to receive a transplant even from an incompatible donor (what we call an incompatible kidney transplant). 2 This protocol has been reported to be quite successful. Specifically, the long-term survival rate of incompatible transplants is equivalent to that of compatible transplants. 3 In recent years, the number of patients using suppressants for incompatible transplants has increased in many countries. In Korea, for example, the proportion of blood-type incompatible kidney transplants in total living-donor transplants has increased from 4.7 percent in 2009 to 21.2 percent in 2014, as shown in Table 2. This sharp increase is partly because the National Health Insurance Service (NHIS) of Korea has covered a large fraction of the total cost of suppressants since Patients pay a small share of 1 Immunological compatibility is mostly determined by biological characteristics of patients and donors, such as ABO blood types, Human Leukocyte Antigen (HLA) types, and the crossmatch. The ABO blood type is determined by the inherited antigenic substances on the surface of red blood cells. For example, if a patient s antigen is type A, then her blood type is type A and her antibody is type B; if a patient s antigen is type AB, then blood type is AB and she has no antibody. A patient with antibody of type X cannot receive a transplant from a donor with type X antigen. For example, if a person s blood type is A, then her antibody is type B, and thus, she cannot receive a transplant from any donor having a type B antigen, namely, blood types B and AB. Similarly, if a person s blood type is O, then her antibodies are types A and B; therefore, she cannot receive a transplant from any donor of blood types A, B, or AB. Tissue-type compatibility is determined by patient s and donor s HLAs, which are proteins on the surface of cells that are responsible for immunological responses. If a patient and a donor have the same HLA, then they are called an identical match, which is very rare because of the number of combinations of HLA proteins that are possible (for more information, see the Genetics Home Reference website, provided by the U.S. National Library of Medicine, at A patient s antibodies and a donor s HLAs also determine the crossmatch : if the crossmatch is positive, then then the patient s antibodies react to the donor s HLAs, thereby making a transplant unsuccessful. 2 A variety of immunosuppressive protocols have been reported in the medical literature. Most of these protocols include the use of rituximab (which is an immunosuppressant medicine that inactivates white blood cells) and plasmapheresis (which is a procedure used to remove harmful antibodies from the blood of patients). Rituximab is usually taken a month before transplant surgery and plasmapheresis is carried out every second day for two weeks before surgery. 3 According to the KONOS Annual Report in 2014, the five-year survival rate of ABO-incompatible living-donor kidney transplants is 95.2 percent and that of ABO-compatible living-donor kidney transplants is 96.4 percent. For other studies on the long-term outcomes of incompatible kidney transplants, see Takahashi et al. (2004), Tyden et al. (2007), Montgomery et al. (2012), and Kong et al. (2013). 1

3 Year Patients in waitlists Total transplants Transplants from deceased donors Transplants from living-donors ,769 1, ,857 1, ,426 1, ,245 1, , ,381 1, , ,612 1, ,000 Table 1. Kidney transplantations in Korea Year Transplants from living-donors Direct transplants of ABO-compatible pairs Direct transplants of ABO-incompatible pairs Transplants through exchanges (90.0%) 35 (4.7%) 40 (5.3%) (86.6%) 78 (9.8%) 29 (3.6%) (86.3%) 113 (11.8%) 18 (1.9%) , (81.1%) 193 (18.9%) 0 (0.0%) , (78.6%) 212 (21.0%) 4 (0.4%) , (78.3%) 212 (21.2%) 5 (0.5%) Table 2. Living-donor kidney transplants in Korea: percentage of cases to total shown in parentheses the total cost, as low as 20 percent depending on their medical conditions. In contrast, the proportion of transplants made using kidney exchanges has decreased from 5.3 percent to nearly 0 percent during the same period, as shown in Table 2. As can be seen, suppressants have largely replaced the kidney exchange program in South Korea. As the number of patients using suppressants increases, the expenditure of the NHIS to subsidize these incompatible transplants also increases. Because the NHIS has a limited budget each year, it cannot subsidize the use of suppressant for all incompatible pairs. Therefore, we should ask if there is room for improvement on the current practice of using suppressants. In this paper, we investigate how to optimally use suppressants in order to facilitate kidney transplants as a part of a kidney exchange program. As an illustration of how the suppressants can be used optimally, consider the following example. Assume that the compatibility is determined only by ABO blood type, rather than by both blood type and tissue type. A pair consists of a patient and a donor, which we represent type X-Y, where the patient s blood type is X and the donor s type is Y. Suppose that there are three pairs: A-B, B-AB, and O-AB. Due to the immunological constraints, a patient of type A can receive a transplant only from a donor of type A or O, a patient of type B can receive a transplant only from a donor of type B or O, and a patient of type O can receive a transplant only from a donor of type O. Because each patient is incompatible with her own donor, in the absence of suppressants, each of them can only receive a transplant from someone else. In a kidney exchange program, on the other 2

4 hand, compatible pairs are formed by swapping donors across pairs. Unfortunately, in this example, no such exchanges are possible, because the patient in pair A-B and the patient in pair O-AB are incompatible with all three donors. Now suppose that the health service can provide suppressants to at most two patients. One way to deal with this is to provide the patients in two incompatible pairs with suppressants and have them receive transplants directly from their own donors. In our example, the patients in A-B and O-AB pairs can be provided suppressants and receive direct transplants from their donors. The patient in the remaining pair, however, cannot receive a transplant. This procedure for allocating suppressants is currently used in South Korea. The number of direct transplants within ABO incompatible pairs for South Korea in the years 2009 to 2014 is shown in Table 2. However, there is a better way to use the limited supply of suppressants in our example, one which enables all patients to receive transplants. Note that the two pairs A-B and B-AB form a chain, (A-B) (B-AB), in which the donor in pair A-B is compatible with the patient in pair B-AB, while the remaining patient of type A and the remaining donor of type AB are not compatible. Such a chain can be viewed as a trading cycle with a missing link in the following sense: If the donor in pair B-AB and the patient in pair A-B were compatible, these pairs would have formed a cycle in a kidney exchange. Providing a suppressant to the patient of type B fills in this missing link and transforms the chain into a trading cycle, (A-B) (B-AB) (A-B). In this cycle, the patient of type B receives a transplant from the donor of type B and the patient of type A receives a transplant from the donor of type AB, with the latter transplant made possible due to the use of a suppressant. The other available unit of suppressant can be provided to the remaining patient, thereby enabling a direct transplant within the pair O-AB. A key feature of our proposal is that participation in a kidney exchange program is voluntary for patient-donor pairs who become compatible through the use of suppressants. Note that in the example, when the patient in pair A-B uses a suppressant, she can receive a transplant directly from her own donor, and so does not need to participate in the exchange program. Nevertheless, by participating in the program, as described above, the pair A-B can also benefit the pair B-AB. Provided that it does not matter from whom a patient receives a transplant when using a suppressant, the pair A-B would be willing to participate in the pool. Such altruistic compatible pairs have also been proposed and studied in the standard kidney exchange context (Sönmez and Ünver, 2014; Roth et al., 2005; Gentry et al., 2007). 4 Because the participation is voluntary, our proposal avoids a possible negative externality on a kidney exchange program that can arise with tissue-type suppressants. As Sönmez and Ünver (2013) have observed, the use of tissue-type suppressants may crowd out patient-donor pairs whose donors are in short supply in the exchange pool (usually, blood-type O donors). 5 In our proposal, howerver, all 4 We follow Sönmez and Ünver (2014) to use the altruistic to refer to compatible pairs who participate in a kidney exchange program even though a direct transplant between themselves is possible. 5 Simply put, the reason is as follows. Note that donors of blood type O are always on the short side of a kidney exchange program because any patient is blood-type compatible with them. Type O donors enter the exchange pool only when they are tissue-type incompatible with the other member of their pair. If type O donors do not participate in the 3

5 donors in incompatible pairs still remain in the exchange pool even after their patients use tissue-type suppressants. Therefore, the shortage of blood-type O donors does not get worse. In what follows, we formalize the idea of using suppressants to facilitate kidney transplants as part of kidney exchange program and investigate the implications of some fairness and efficiency requirements for such a program. As a benchmark, we begin with the standard kidney exchange model without suppressants. We define two Top-Trading Cycles (TTC) rules associated with a priority ordering over patients. 6 Both rules are defined by an algorithm in which patient-donor pairs form trading cycles in each step, as in the standard TTC algorithm. In our setting, however, there can be multiple overlapping cycles because preferences are very coarse: they are either compatible or incompatible. We propose two different ways of choosing trading cycles from among them and they result in two versions of TTC rule. We next extend the model by introducing suppressants. To accommodate the limited availability of suppressants, we assume that at most k patients can use suppressants. For each kidney exchange problem, we first determine who are to receive the suppressants. Because these recipients are then compatible with any other donors, we next update the compatibility profile. Based on this new profile, we choose which patients are matched with which donors. We regard Pareto efficiency as being a minimal requirement on matchings. A matching is Pareto efficient if there is no other matching that makes all patients weakly better off and at least one patient strictly better off. Note that this is an efficiency requirement on matchings for each given compatibility profile. In addition to requiring matchings to be Pareto efficient, we also require that suppressants be allocated fairly and efficiently. We consider two variants of our fairness condition. Both of them ensure that no patient gets worse off after suppressants are used to supplement the kidney transplantation program. Strong monotonicity requires all patients to be weakly better off after suppressants are introduced, no matter who receives them. In contrast, monotonicity requires that there be some way of allocating the suppressants that makes all patients weakly better off. We also consider two variants of our efficiency requirement for the allocation of suppressants. Both require that they be allocated so as to maximize transplants in the some sense. Consider two groups of potential recipients. Each group results in different sets of patients receiving transplants. Maximal improvement requires that if the first group results in more transplants than the other in terms of set inclusion, then the latter group should not be allocated the suppressants. Cardinally maximal improvement modifies this requirement by comparing the total number of transplants that they facilitate, rather than comparing them in terms of set inclusion. pool when tissue-type suppressants are available, the remaining patients suffer from the shortage of type O donors. Even if a tissue-type incompatible pair whose donor is type O uses a suppressant, in our proposal, this pair remains in the pool and participates in the kidney exchanges. 6 Shapley and Scarf (1974) propose the TTC rule for a general model with indivisible goods. Roth et al. (2004) develop a TTC rule for kidney exchange problems by considering chains formed with patients on the waitlist. As in Roth et al. (2004), we impose no constraint on the size of cycles in kidney exchanges. For more discussion on the size of exchanges, see Roth et al. (2007) and Saidman et al. (2006). 4

6 Based on these requirements, we present two impossibility results. The first shows that no rule jointly satisfies Pareto efficiency and strong monotonicity. The second shows that even if strong monotonicity is weakened to monotonicity, we cannot achieve it together with Pareto efficiency and cardinally maximal improvement. In view of these impossibilities, we then weaken cardinally maximal improvement to maximal improvement and ask if it is compatible with Pareto efficiency and monotonicity. We show that it is: We introduce two modifications of the TTC rules to show this. Each of these solutions operates in four steps. First, assuming that no one uses a suppressant, each of the TTC rules is applies in order to identify the set of patients who receive no transplant at the stage. Second, the initial priority ordering is modified so that the patients identified in the previous stage have lower priorities than any other patients. In the third stage, the available suppressants are allocated. This is done by selecting cycles and chains according to the modified priority ordering and assigning suppressants to the patients at the heads of these chains. Finally, the compatibility profile is updated based for the patients identified in the third stage and then the TTC rule associated with the modified priority ordering is applied to this profile. We next conduct counterfactual analyses to see how much suppressants can be reduced relative to the current practice. The use of suppressants for incompatible transplants is relatively new phenomenon and, as yet, there is little data available about how many transplants have been facilitated by the use of suppressants. However, good data is available in South Korea and so we use it for a quantitative analysis. The KONOS transplant data include all pairs who received living-donor kidney transplants, either compatible or incompatible, during the four year period ( ). This data set also includes the blood-types of all these pairs. Using this data, we determine the minimal quantity of suppressant that is needed to ensure that all patients, who actually received kidney transplants by using suppressants, still receive transplants. For our analysis, we assume that compatibility is only determined by the blood type. There are then seven types of incompatible pairs: A-B, B-A, A-AB, B-AB, O-A, O-B, and O-AB. During this period in South Korea, there were almost no living-donor transplants made through a kidney exchange. Therefore, a patient in any pair of one of these types in the data must have used a suppressant to receive a transplant. We propose an algorithm that minimizes the use of suppressants while ensuring that all of these patients receive transplants. This algorithm runs as follows. First, all trading cycles are identified and patients and donors are matched along these cycles. These pairs are then excluded from the exchange pool. Next, all 3-chains (a k-chain is a chain consisting of k pairs) are identified. In this setting, these are the longest chains. After assigning suppressants to the heads of these chains, patients and donors along these chains are matched and then excluded from the pool. This process is then repeated for 2-chains and then for 1-chains. We compare the outcomes obtained using this algorithm with the actual KONOS data. Using our algorithm, only 340 pairs need to use suppressants, whereas 757 patients actually used suppressants during the period studied. This is a reduction in usage of 55.1 percent. 5

7 As a robustness check, we redo our analysis for different specifications of the relevant patient-donor pairs, and obtain the similar results. The KONOS data only include pairs (either compatible or incompatible) whose patients have already received transplants. Some incompatible pairs do not perform transplants, so the KONOS data does not capture all of the individuals who would have benefited from the use of suppressants. To estimate the benefit of our proposal when all incompatible pairs are accounted for, we construct a hypothetical population of 1,001 pairs using the average distribution of ABO blood types in South Korea and apply our algorithm to this population. In this scenario, we show that it is possible to reduce the use of suppressants by 55.4 percent, while still obtaining the same benefit as above. The rest of this paper is organized as follows. Section 2 introduces the standard kidney exchange model without suppressants and defines two versions of a top-trading cycles rule. Section 3 extends the model to include suppressants and establish our two impossibility results. Section 4 considers two modifications of a top-trading cycles solution and determine which of the properties described above are satisfied. Section 5 presents our counterfactual analyses based on Korean experience. We provide some concluding remarks in Section 6. 2 Standard Kidney Exchange Model without Immunosuppressants Let N {1,..., n} be a set of patient-donor pairs. Pair i N consists of patient i and donor i. A patient is either compatible or incompatible with each other donor. If a patient is compatible with a donor, then she can receive a transplant from the donor. For each i N, let N i + {j N : patient i is compatible with donor j} and Ni N \N i +. Each patient i N has a weak preference R i over N as follows: Each patient prefers all elements in N i + to all elements in Ni. All elements in N i + are equally desirable and so are those in Ni. If a patient is incompatible with all donors, then she prefers to all other donors. Note that compatibility of a donor and a patient is determined by their immunological characteristics, independently of other pairs compatibility. Let P i and I i be the asymmetric and the symmetric part of R i, respectively. Let R be the set of all such weak preferences. A kidney exchange problem, or simply a problem, is defined as a list (N, R) with R = (R i ) i N. represent a problem as R. Let R N be the set of all problems. 7 Since we fix N, we We introduce a graph whose nodes are pairs in N. A directed arc from one node to another, say i to j, is denoted by i j. We allow j = i, in which case the arc is self-directed, i i. Let g be the set of all directed arcs over N. For each R R N and each i, j N, let i j if and only if donor i is compatible with patient j at R. Let g(r) g be the set of directed arcs defined as above at R. A problem R is uniquely represented by g(r). An ordered list of pairs (i 1,..., i k, i 1 ) forms a cycle in g(r) if i 1,..., i k are distinct pairs and i 1 i 2 i k i 1. As long as i 1,..., i k keep the order, it does not matter who comes first in 7 Bogomolnaia and Moulin (2004) study dichotomous preferences in a general matching context. They examine randomized matchings to achieve efficiency, fairness, and strategic requirements when only two-way exchanges are allowed. 6

8 the list (i 1,..., i k ). Note that a cycle can be a self-cycle, i i. Let C(R) be the set of all cycles in g(r). Two cycles c, c C(R) are jointly feasible if no pair is in c and c. Let C(R) 2 C(R) be the collection of all sets of jointly feasible cycles in C(R). For each C C(R), let N(C) be the set of pairs in a cycle in C. Let C be the number of the cycles in C. An ordered list of pairs (i 1,..., i k ) forms a chain if i 1,..., i k are distinct pairs and i 1 i 2 i k and i k i 1. Unlike a cycle we defined above, it does matter who comes first in the list (i 1,..., i k ). We call i 1 the head of this chain and i k the tail of the chain. Let L(R) be the set of all chains in g(r). Two chains l, l L(R) are jointly feasible if no pair is in l and l. Let L(R) 2 L(R) be the collection of all sets of jointly feasible chains in L(R). For each L L(R), let N(L) be the set of pairs in a chain in L. Let L be the number of the chains in L. A pair of c C(R) and l L(R) are jointly feasible if no pair is in c and l. Let H(R) C(R) L(R) be the set of all cycles and chains in g(r). Let H(R) 2 H(R) be the collection of all sets of jointly feasible cycles and chains in H(R). For each H H(R), let N(H) be the set of pairs in a cycle or a chain in H. A matching µ specifies which patient is matched to which donor. For each i, j N, µ(i) = j implies that patient i is matched to donor j. A matching µ is feasible if (i) for each i N, µ(i) N and {j N : µ(j) = i} = 1 and (ii) for each i N with µ(i) / N i +, µ(i) = i. A matching µ is individually rational at R if for each i N, µ(i) R i i. Let M(R) be the set of all feasible and individually rational matchings at R. For each µ M(R), let N(µ) be the set of patients receiving transplants, namely, N(µ) = {i N : µ(i) N i + }. We also introduce a priority ordering over pairs. 8 Denote a linear ordering over pairs by. For each pair i, j N, i j if and only if patient i has a higher priority than patient j. Example 1. Let N = {1, 2, 3}. Suppose that N 1 + = {2}, N 2 + = {1, 3}, and N 3 + = {2, 3}. The corresponding R and g(r) are given as follows: R 1 R 2 R 3 2 1, 3 2, 3 1, The set of cycles in g(r) is C(R) = {c (1, 2, 1), c (2, 3, 2), c (3, 3)}. Note that c and c are jointly feasible. Since pair 2 is in c and c, the two cycles c and c are not jointly feasible. The collection of jointly feasible cycles is C(R) = {{c}, {c }, {c }, {c, c }}. Let C {c, c } C(R). Then, N(C) = {1, 2, 3}. The set of chains in g(r) is L(R) = {l (1), l (2), l (1, 2, 3), l (3, 2, 1)}. Note that l and l are jointly feasible. Since pair 1 is in l and l, the two chains l and l are not jointly feasible. The collection of jointly feasible chains is L(R) = {{l}, {l }, {l }, {l }, {l, l }}. Let L {l, l } L(R). Then, N(L) = {1, 2}. Also, H(R) = {c, c, c, l, l, l, l } and H(R) = {{c}, {c }, {c }, {l},..., {l }, {l, l }, {c, l}, {c, l}, {c, l }, {c, l, l }}. 8 For more detailed discussion on these priorities, see Roth et al. (2005). 7

9 A matching rule is a mapping ϕ : R N R R N M(R) such that for each R R N, ϕ(r) M(R). A matching rule ϕ is essentially single-valued if for each R R N, each µ, µ ϕ(r), and each i N, µ(i) I i µ (i). If ϕ is essentially single-valued, then for each R R N and each µ, µ ϕ(r), N(µ) = N(µ ). 9 For each R R N, µ M(R) is Pareto efficient at R if there is no other µ M(R) such that for each i N, µ (i) R i µ(i) and for some i N, µ (i) P i µ(i). A matching rule ϕ is Pareto efficient if for each R R N and each µ ϕ(r), µ is Pareto efficient at R. We define Top-Trading Cycles (TTC) rules adapted to this model. 10 Without loss of generality, let 1 2 n. For simplicity, a priority ordering can be written as : 1, 2,..., n. Top-trading cycles rule associated with (simply, T T C ): For each R R N, let C 0 C(R). Step 1. Find a set of jointly feasible cycles C C 0 such that 1 N(C). If there is no such C, then C 1 C 0. Otherwise, let C 1 be the collection of all such sets of jointly feasible cycles. Step t 2. Find C C t 1 such that t N(C). If there is no such C, then C t C t 1. Otherwise, let C t be the collection of all such sets of jointly feasible cycles. Step n. Obtain C n. For each C C n, each patient in N(C) receives a transplant from another patient s donor along the directed arc of cycles in C. All other patients are assigned their own donors. Collect the resulting matchings for all C s in C n. Note that this rule can be viewed as a sequential priority rule since we first identify all matchings at which patient 1 receives a transplant, and then identify all matchings at which patient 2 does, and so on. If patients are restricted to make two-way exchanges, such a sequential priority rule is Pareto efficient and it also maximizes the number of transplants (Roth et al., 2005). If we allow more than two-way exchanges, however, Pareto efficient matchings do not necessarily maximize the number of transplants. Therefore, we can formulate another version of top-trading cycles rules. Top-trading cycles rule maximizing the number of transplants (simply, T T C ): For each R R N, let C 0 argmax C C(R) N(C). Step 1. Find a set of jointly feasible cycles C C 0 such that 1 N(C). If there is no such C, then C 1 C 0. Otherwise, let C 1 be the collection of all such sets of jointly feasible cycles. Step t 2. Find C C t 1 such that t N(C). If there is no such C, then C t C t 1. Otherwise, let C t be the collection of all such sets of jointly feasible cycles. Step n. Obtain C n. For each C C n, each patient in N(C) receives a kidney from another patient s donor along the directed arc of cycles in C. All other patients stay with their own donors. Collect the resulting matchings for all C s in C n. 9 Suppose, by contradiction, that there are R R N, i N, and a pair µ, µ ϕ(r) such that µ(i) N + i and µ (i) / N + i. Since all pairs in N + i are strictly preferred to those in N i, we have µ(i) Pi µ (i), contradicting that ϕ is essentially single-valued. 10 The TTC rule is proposed by Shapley and Scarf (1974) in a general model and applied to kidney exchange problems by Roth et al. (2004). 8

10 Remark. It is straightforward from the definitions that T T C and T T C are essentially single-valued. Proposition 1. For every priority ordering, T T C and T T C are Pareto efficient. Proof. Suppose, by contradiction, that T T C is not Pareto efficient. Then, there are R R N, µ T T C (R), and µ M(R) such that for each i N, µ (i) R i µ(i) and for some i N, µ (i) P i µ(i). Note that all patients who receive transplants at µ also receive transplants at µ. Furthermore, there is at least one patient who receives a transplant at µ, but not at µ. Among such patients, let patient i be the one with the highest priority under. According to the definition of T T C, this matching should be chosen so that patient i receives a transplant, a contradiction. Next, suppose, by contradiction, that T T C is not Pareto efficient. Then, there are R R N, µ T T C (R), and µ M(R) such that for each i N, µ (i) R i µ(i) and for some i N, µ (i) P i µ(i). Note that all patients who receive transplants at µ also receive transplants at µ and at least one patient receives a transplant at µ, but not at µ. Thus, the number of transplants at µ is greater than that at µ. Hence, µ / T T C (R), a contradiction. 3 Extended Kidney Exchange Model with Immunosuppressants We introduce suppressants to the standard model. If patient i receives a suppressant, she becomes compatible with all donors, including her own. We denote by Ri the preference of patient i after receiving a suppressant. For each S N, let RS (R i ) i S. For each R R N and each S N, let R(S) (RS, R S) denote the preference profile derived from R when patients in S receive suppressants. Since R(S) R N, all definitions in the previous section are carried over to this setting by replacing R with R(S). Example 2. (Example 1 continued) Consider R and g(r) given in Example 1: R 1 R 2 R 3 2 1, 3 2, 3 1, Now suppose that S = {2}. Then, R(S) = (R 1, R 2, R 3) and g(r(s)) are represented as follows: R 1 R 2 R 3 2 1, 2, 3 2, 3 1, The set of cycles in g(r 2, R 2) is C(R 2, R 2) = {c (1, 2, 1), c (2, 3, 2), c (3, 3), c (2, 2)}. Thus, the collection of jointly feasible cycles is C(R 2, R 2) = {{c}, {c }, {c }, {c }, {c, c }, {c, c }}. The set of chains in g(r 2, R 2) is L(R 2, R 2) = {(1), (1, 2, 3), (3, 2, 1)}. 9

11 To accommodate the limited availability of suppressants, we introduce a non-negative integer k to be an upper bound on the number of patients who can receive suppressants. Let Z + be the set of non-negative integers and for each k Z +, let 2 N (k) be the collection of subsets of N composed of at most k patients. When k = 0, the extended model coincides with the standard model. For each problem, we first decide at most k recipients of suppressants. We then update the preference profile and choose matchings between patients and donors. Let σ : R N 2 N (k) be a recipient choice rule. Let ϕ : R N R R N M(R) be a matching rule such that for each R RN, ϕ(r) M(R). We represent a solution Φ as a pair (σ, ϕ). With a slight abuse of notation, for each R R N, let Φ(R) ϕ(r(σ(r))). The following two requirements are defined for a matching rule, ϕ. Pareto efficiency: For each R R N, ϕ(r) is Pareto efficient at R. Note that Pareto efficiency does not say anything about how we assign suppressants. It simply requires Pareto efficiency of matchings at each given preference profile. A solution (σ, ϕ) is Pareto efficient if ϕ is Pareto efficient. Our next requirement says, no matter what a choice rule is, a matching rule should make all patients weakly better off after suppressants are introduced to the transplantation program. 11 Strong monotonicity: For each k Z +, each σ : R N 2 N (k), and each R R N, there are µ ϕ(r) and µ ϕ(r(σ(r)) such that for each i N, µ (i) R i µ(i). This requirement is convincing especially when we cannot choose a particular recipient set, for example, when patients individually decide whether they use suppressants or not. T T C and T T C violate this requirement. Unfortunately, Example 3. (T T C and T T C violate strong monotonicity.) Let N {1, 2, 3, 4}, : 1, 2, 3, 4, and R, R R N be given as follows: R 1 R 2 R 3 R , 3, 4 1, 2, 4 1, 2, 3 1, 3, 4 R 1 R2 R3 R , 3, 4 2, 3, 4 1, 2, 3 1, 3, 4 11 The two monotonicity requirements can be compared with welfare-dominance under preference replacement in allocation problems (for a complete survey about this requirement, see Thomson (1999)). It requires when a person changes her preferences, all the remaining people be affected in the same direction. Since the use of suppressants by a group of patients change their preferences, our requirements share the same sprit as welfare-domination under preference replacement, but there is no direct logical relation between them. In our requirements, first, at most k patients may change their preference at the same time. Second, when patient i s preference changes, it always changes to Ri. Third, when preferences change, all patients, including those who use suppressants, should be affected in the same direction. Lastly, all patients should be affected in a particular direction they become weakly better off. 10

12 Since there is only one cycle at g(r) and g( R), respectively, T T C (R) = {(µ(1) = 1, µ(2) = 3, µ(3) = 4, µ(4) = 2)} and T T C ( R) = {(µ(1) = 2, µ(2) = 1, µ(3) = 3, µ(4) = 4)}. Now suppose that k = 1 and consider a choice rule σ : R N 2 N (k) such that σ(r) = σ( R) = {2}. Patient 2 s preference changes to R2 as illustrated in the following graph: Let R (R 1, R2, R 3, R 4 ) and R ( R 1, R2, R 3, R 4 ). Then, T T C (R ) = {(µ(1) = 2, µ(2) = 1, µ(3) = 3, µ(4) = 4)}, T T C ( R ) = {(µ(1) = 1, µ(2) = 3, µ(3) = 4, µ(4) = 2)}. Under T T C, patients 3 and 4 are worse off, patient 1 is better off, and patient 2 remains indifferent. Under T T C, patient 1 is worse off, patients 3 and 4 are better off, and patient 2 remains indifferent. This observation generalizes to our first impossibility result. We cannot achieve strong monotonicity together with Pareto efficiency. Proposition 2. No matching rule ϕ jointly satisfies Pareto efficiency and strong monotonicity. Proof. We present an example of a problem with three patients. Let N {1, 2, 3}. Let ϕ be a Pareto efficient matching rule. Consider the following preferences: R 1 R 2 R 2 R , 3 2, 3 1, 2 1, 3 Suppose that k = 0. Let R (R 1, R 2, R 3 ) and R (R 1, R 2, R 3). By Pareto efficiency, ϕ(r) = {µ = (µ(1) = 2, µ(2) = 1, µ(3) = 3)} and ϕ(r ) = {µ = (µ (1) = 1, µ (2) = 3, µ (3) = 2)}. Now suppose that k = 1. Let σ : R N 2 N (k) be such that σ(r) = σ(r ) = {2}. The new preference profile resulting from σ is (R 1, R2, R 3) for both problems. To make everyone weakly better off at the new profile than at R, we should have µ ϕ(r 1, R2, R 3). Again, to make everyone weakly 11

13 better off at the new profile than at R, we should have µ ϕ(r 1, R2, R 3). Altogether, {µ, µ } ϕ(r 1, R2, R 3). However, patient 1 is worse off at µ than at µ and patient 3 is worse off at µ than at µ, a contradiction to strong monotonicity. Remark. Since a matching rule ϕ is not single-valued, there is another way to define strong monotonicity: for each k Z +, each σ : R N 2 N (k), each R R N, each µ ϕ(r), each µ ϕ(r(σ(r))), and each i N, µ (i) R i µ(i). It is easy to check that this implies strong monotonicity that we define above. Therefore, Proposition 2 still holds with this formulation. The implication of Proposition 2 is that, as long as we impose Pareto efficiency as a minimal requirement on matchings, there is no way to make all patients weakly better off for all possible recipient choice rules. In what follows, we instead ask if it is possible to make all patients weakly better off for some recipient choice rules. Before we proceed, let us take a closer look at Example 3 above. At R, suppose that patient 1, instead of patient 2, receives a suppressant. Independent of the cycle (2, 4, 3, 2), patient 1 forms a self-cycle. This results in all patients being weakly better off. Similarly, at R, suppose that patient 4, instead of patient 2, receives a suppressant. Independent of the cycle (1, 2, 1), patients 3 and 4 form a cycle (3, 4, 3). This again results in all patient being weakly better off. In this example, it is possible to make all patients weakly better off if we choose a particular recipient for each problem. We formulate this possibility to the following requirement. Consider a solution (σ, ϕ). Monotonicity: For each k Z +, each R R N, each µ ϕ(r), each µ ϕ(r(σ(r))), and each i N, µ (i) R i µ(i). We note that this requirement can be trivially satisfied by many solutions. Consider any essentially single-valued matching rule ϕ. Suppose that σ(r) = for all R R N. The solution (σ, ϕ) trivially satisfies monotonicity simply by wasting all available suppressants. To avoid such ineffective use, we formulate another efficiency requirement on the assignment of suppressants. The following requirement says that the recipients of suppressants should be chosen to maximize the set of patients receiving transplants in terms of set inclusion. Maximal Improvement (MI): For each k Z +, each R R N, and each T, T 2 N (k), if there are µ ϕ(r(t )) and µ ϕ(r(t )) such that N(µ ) N(µ), then T σ(r). Example 4. (Maximal improvement) Let N {1, 2, 3, 4}. (σ, ϕ). Suppose that R R N is given as follows: Consider any Pareto efficient solution R 1 R 2 R 3 R , 3, 4 1, 2, 4 1, 2, 3 1, 2, 4 12

14 S = 1 2 S = {1} 1 2 S = {2} Consider three cases of assigning suppressants when k = 1. Suppose first that no patient receives a suppressant. Since the solution is Pareto efficient, it selects {µ = (µ(1) = 1, µ(2) = 2, µ(3) = 4, µ(4) = 3)} and N(µ) = {3, 4}. Next, suppose that patient 1 receives a suppressant. By Pareto efficiency, the solution selects {µ} and N(µ) = {1, 3, 4}. Now, suppose that patient 2 receives a suppressant. By the same argument, the solution selects {µ = (µ (1) = 2, µ (2) = 1, µ (3) = 4, µ (4) = 3)} and N(µ ) = {1, 2, 3, 4}. Because all patients can receive transplants when patient 2 is provided a suppressant, maximal improvement requires that σ(r) and σ(r) {1}. As discussed in Section 2, we can also focus on maximizing the number of transplants. Then, the aforementioned requirement is reformulated in terms of the number of transplants. Cardinally Maximal Improvement (CMI): For each k Z +, each R R N, and each T, T 2 N (k), if there are µ ϕ(r(t )) and µ ϕ(r(t )) such that N(µ ) < N(µ), then T σ(r). It is straightforward from the definitions that CMI implies MI. Unfortunately, it turns out that we cannot achieve CMI together with other requirements we consider. Proposition 3. No solution jointly satisfies Pareto efficiency, monotonicity, and cardinally maximal improvement. Proof. The proof is by means of an example with six patients. Let N {1, 2, 3, 4, 5, 6} and (σ, ϕ) be a solution satisfying the three requirements. Consider the following preferences: R 1 R 2 R 3 R 4 R 5 R , N \ {5} N N \ {5} N \ {1, 2} N \ {4} N \ {3} Suppose first that k = 0. By Pareto efficiency, ϕ(r) = {µ = (µ(1) = 5, µ(2) = 2, µ(3) = 3, µ(4) = 1, µ(5) = 4, µ(6) = 6)}. Next, suppose that k = 1. To satisfy CMI, it is easy to check that we should 13

15 have σ(r) = {2} and all patients except for patient 1 receive transplants at the resulting preference profile. Since patient 1 is made worse off, monotonicity is violated. 4 Top-Trading Cycles Solutions with Immunosuppressants We propose two solutions satisfying Pareto efficiency, monotonicity, and maximal improvement. They are based on the top-trading cycles rules defined in Section 2, but we add a major modification to assign suppressants. The extended top-trading cycles solution (simply, et T C ): Stage 1. For each R R N, apply T T C. Denote by N the set of patients who receive no transplant at any matching selected by T T C. Without loss of generality, let N {j 1,..., j n } and j 1 j n. Stage 2. Modify into a new priority ordering as follows. All patients in N \ N have higher priories than those in N, while the relative priorities within N and N \ N remain the same. Denote this new priority ordering by (R). Stage 3. Find a set of jointly feasible cycles and chains. Let H 0 {H H(R) : N(H) N \ N and the number of chains in H does not exceed k}, 12 and for each t {1,..., n}, {H H t 1 : j t N(H)}, if {H H t 1 : j t N(H)}, H t H t 1, otherwise. Choose any H H n and assign suppressants to the heads of all chains in H. Denote the set of these heads by σ (R). Stage 4. Apply to R(σ (R)) the top-trading cycles rule associated with (R). For each R R N, let et T C (R) T T C (R)(R(σ (R))). In the first stage, we apply T T C to the initial preference profile and find the set of patients N who do not receive transplants at the stage. Since T T C is essentially single-valued, the sets of patients receiving transplants remain the same across all matchings selected by T T C. In the second stage, we modify the initial priority ordering to (R) so that all patients N have lower priorities than the rest of patients. In the third stage, we choose the sets of jointly feasible cycles and at most k chains including all patients in N \ N and some patients in N chosen according to their priorities. We choose any one of these sets and assign suppressants to the heads of the chains in the set. In the last stage, we update the preference profile into R(σ (R)) and apply to it T T C associated with the modified priority ordering. 12 Such H 0 is always non-empty. This is because at g(r), (i) patients outside N form cycles among themselves and (ii) patients in N do not form cycles, but only chains, among themselves (if there were any cycle among patients in N, such a cycle should have been chosen in Stage 1 of the algorithm, making these patients not belong to N). Therefore, we can choose the cycles from (i) and at most k chains from (ii). 14

16 Note that to satisfy monotonicity, all patients in N \ N should receive transplants after suppressants are introduced. For this, we sort them out and give them higher priorities than all patients in N when we apply T T C for the new preference profile. As long as they have higher priorities than N, the cycles or chains including these patients will be selected ahead of the rest, guaranteeing that they still receive transplants at the new profile. To satisfy maximal improvement, on the other hand, we identify at most k chains including a largest set of patients and then assign suppressants to the heads of these chains. By doing this, we can improve all patients welfare in a maximal way. It is straightforward from the definition that et T C is essentially single-valued. Example 5. (et T C ) Let N {1,..., 9} and : 1,..., 9. Let R R N and the corresponding g(r) be given as follows: R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 2 1, 3, 4 5, 7 7 1, N \ {2} N \ {1, 3, 4} N \ {5, 7} N \ {7} N N \ {1, 9} N \ {2} N \ {1} N \ {8} The set of cycles is C(R) = {c = (1, 2, 1), c = (2, 7, 4, 2), c = (2, 7, 3, 2)}. Since any pair of these cycles is not jointly feasible, T T C chooses the cycle c including the patient with the highest priority, which is patient 1. Therefore, T T C (R) = {(µ(1) = 2, µ(2) = 1, µ(3) = 3, µ(4) = 4,..., µ(9) = 9)} and N = {3, 4,..., 9}. (1) Suppose that at most one patient can receive a suppressant, i.e., k = 1. Note that patient 3 has the highest priority in N. Note also that the two chains l = (5, 3, 2, 1, 8, 9, 6) and l = (7, 3, 2, 1, 8, 9, 6) are in the sets of jointly feasible cycles and chains including patients 1, 2, and 3. In this collection, we identify the sets including the patient with the next highest priority in N, which is patient 4. Because there is no such set, we move on to patient 5. We end up with a single chain {l} and assign the suppressant to patient 5, who is the head of l, i.e., σ (R) = {5}. We also modify to (R), which happens to be again. By definition, et T C (R) = T T C (R)(R(σ (R))). The resulting matching is (µ(1) = 2, µ(2) = 3, µ(3) = 5, µ(4) = 4, µ(5) = 6, µ(6) = 9, µ(7) = 7, µ(8) = 1, µ(9) = 8). (2) Suppose that at most two patients can receive suppressants, i.e., k = 2. As above, we identify the set of cycles and at most two chains including patients 1, 2, 3 and 4. We obtain two sets H = {(5, 3, 2, 1, 8, 9, 6), (7, 4)} and H = {(7, 4, 2, 1, 8, 9, 6), (5, 3)}. Thus, σ (R) = {5, 7} for both sets. We also modify to (R) as above. The two resulting matchings of et T C (R) are µ and µ such that 15

17 µ = (µ(1) = 2, µ(2) = 3, µ(3) = 5, µ(4) = 7, µ(5) = 6, µ(6) = 9, µ(7) = 4, µ(8) = 1, µ(9) = 8), µ = (µ (1) = 2, µ (2) = 4, µ (3) = 5, µ (4) = 7, µ (5) = 3, µ (6) = 9, µ (7) = 6, µ (8) = 1, µ (9) = 8). Note that all patients find µ and µ indifferent, confirming that et T C is essentially single-valued. Remark. When multiple suppressants are available, it is also possible to assign them one by one sequentially, instead of assigning them all at once as in et T C. Example 5 shows that when k = 2, the resulting matchings may be different from et T C (R). When suppressants are assigned one by one sequentially, the only resulting matching is µ. All patients, however, still find both cases indifferent. The extended top-trading cycles solution maximizing the number of transplants (simply, et T C ): Stage 1. For each R R N, apply T T C. Denote by N the set of patients who receive no transplant at any matching selected by T T C. Without loss of generality, let N {j 1,..., j n } and j 1 j n. Stage 2. Modify into a new priority ordering as follows. All patients in N \ N have higher priories than those in N, while the relative priorities within N and N \ N remain the same, respectively. Denote this new priority ordering by (R). Stage 3. Find a set of jointly feasible cycles and chains. Let H 0 {H H(R) : N(H) N \ N and the number of chains in H does not exceed k}, and for each t {1,..., n}, {H H t 1 : j t N(H)}, if {H H t 1 : j t N(H)}, H t H t 1, otherwise. Choose any H H n and assign suppressants to the heads of all chains in H. Denote the set of these heads by σ (R). Stage 4. Apply to R( σ (R)) the top-trading cycles rule associated with (R). For each R R N, let et T C (R) T T C (R)(R( σ (R))). Basically we make the same modifications of T T C as we did for T T C except that we use T T C, instead of T T C, in Step 1. Note that, because T T C is essentially single-valued, the sets of patients receiving transplants remain the same across all matchings selected by T T C for each problem. It is straightforward from the definition that et T C is essentially single-valued. Example 6. (et T C ) Consider the problem in Example

18 Recall that C(R) = {c = (1, 2, 1), c = (2, 7, 4, 2), c = (2, 7, 3, 2)}. Since any pair of these cycles is not jointly feasible, T T C chooses a cycle of the largest size; if there are many, it chooses a cycle including patients with the highest priorities in order. Thus, c is chosen and the resulting matching is T T C (R) = {(µ(1) = 1, µ(2) = 3, µ(3) = 7, µ(7) = 2, µ(4) = 4,..., µ(9) = 9)} and N = {1, 4, 5, 6, 8, 9}. (1) Suppose that at most one patient can receive a suppressant, i.e., k = 1. Note that patient 1 has the highest priority in N. Among all sets of jointly feasible cycles and a single chain, we find the sets including patients 1, 2, 3, and 7. Then, we obtain H = {(1, 8, 9, 6), (2, 7, 3, 2)} and H = {(7, 3, 2, 1, 8, 9, 6)} at the last step of the et T C algorithm. If we choose H, then σ (R) = {7}. We modify to (R) : 2, 3, 7, 1, 4, 5, 6, 8, 9. By definition, et T C (R) = T T C (R)(R( σ (R))). The resulting matching is µ such that (µ(1) = 2, µ(2) = 3, µ(3) = 7, µ(4) = 4, µ(5) = 5, µ(6) = 9, µ(7) = 6, µ(8) = 1, µ(9) = 8). If we choose H, then σ (R) = {1}. The resulting matching is different from µ, but all patient find it indifferent to µ. (2) Now suppose that at most two patients can receive suppressants, i.e., k = 2. Similarly as above, we identify the collection of sets of jointly feasible cycles and at most two chains including patients 2, 3, and 7 as well as patients 1 and 4. Then, we obtain H = {(5, 3, 2, 1, 8, 9, 6), (7, 4)} and H = {(7, 4, 2, 1, 8, 9, 6), (5, 3)} at the last step of the et T C algorithm. If we choose H, then σ (R) = {5, 7}. We modify to (R) as above. The resulting matching is µ such that (µ(1) = 2, µ(2) = 4, µ(3) = 5, µ(4) = 7, µ(5) = 3, µ(6) = 9, µ(7) = 6, µ(8) = 1, µ(9) = 8). If we choose H, then σ (R) = {5, 7}. The resulting matching is different from µ, but all patients find it indifferent to µ. Our main result follows. Theorem 1. et T C and et T C satisfy Pareto efficiency, monotonicity, and maximal improvement. Proof. Pareto efficiency: For each R R N, note that R(σ (R)), R( σ (R)) R N. By definition, et T C (R) T T C (R)(R(σ (R))) and et T C (R) T T C (R)(R( σ (R))). By Proposition 1, T T C and T T C are Pareto efficient for every. Therefore, for each R R N, et T C (R) and et T C (R) are Pareto efficient at R(σ (R)) and R( σ (R)), respectively. Monotonicity: Let R R N. Recall that T T C and et T C are essentially single-valued. Note that at each matching, any patient who receives a transplant cannot be made strictly better off and any patient who receives no transplant cannot be made strictly worse off. Therefore, it is enough to show that all agents who used to receive transplants at T T C (R) still receive transplants at et T C (R). Denote the set of those patients by N \ N. Note that they are the patients included in jointly feasible trading cycles in the T T C algorithm. Since suppressants make patients compatible with more donors, the jointly feasible trading cycles including N \ N still remain jointly feasible in g(r(σ (R))). Now, consider the algorithm T T C (R) for R(σ (R)). Since these patients have higher priorities than the remaining patients at (R), these patients are always included in the sets of jointly feasible trading cycles chosen in each step. Therefore, we end up with et T C (R) in which all of these patients receive transplants, completing the proof. The same argument shows that et T C is monotonic. 17