Pairwise Kidney Exchange over the Blood Group Barrier

This article investigates how such technology should be implemented when designing pairwise kidney exchange programs. The possibility to receive a kidney transplant from a blood group incompatible donor motivates an extension of the preference domain, allowing patients to distinguish between compatible donors and half-compatible donors ( i.e. blood group incompatible donors that only become compatible after undergoing an immunosuppressive treatment). It is demonstrated that the number of transplants can be substantially increased by providing an incentive for patients with half-compatible donors to participate in kidney exchange programs. The results also suggest that the technology is beneﬁcial for patient groups that are traditionally disadvantaged in kidney exchange programs ( e.g. blood group O patients). The positive effect of allowing transplants over the blood group barrier is larger than the corresponding effects of including altruistic patient–donor pairs or of allowing three-way exchanges in addition to pairwise exchanges.


INTRODUCTION
Since the pioneering work by Roth et al. (2004Roth et al. ( , 2005b and the establishment of the first centralized multi-hospital kidney exchange program in New England in 2004, kidney exchange research has become an integral part of the market design literature. The research is motivated by the rapidly increasing shortage of kidneys 1 together with the key observation that even if 1094 REVIEW OF ECONOMIC STUDIES 1.9% and 8.5% of the patients can manipulate Model (b). Hence, Model (b) performs better than Model (a) in this respect as well. Here, it should also be noted that manipulation attempts are risky for patients because if they are unsuccessful, the patients will not receive any transplants at all. In this sense, an attempt to manipulate the outcome of the mechanism may ultimately come at the cost of the patient's own life.

Related literature
After the establishment of the kidney exchange program in New England 2004, new design features such as non-simultaneous extended altruistic donor chains  and larger cyclic exchanges (Saidman et al., 2006;Roth et al., 2007) were suggested in the literature and added to existing exchange programs. One of the most important insights from the early literature is that a clever method for increasing the number of transplants in existing kidney exchange programs is to increase the number of participating patient-donor pairs. This can be achieved in a number of different ways. For example, Roth et al. (2005a) advocate the inclusion of patients with compatible donors, as this would generate "the largest patient welfare gains in comparison to a number of other design modifications" (Sönmez and Ünver, 2014, p. 108). Considering that the participation of compatible pairs is purely altruistic, one may wish to minimize the number of compatible pairs involved in exchanges. This is the idea in Sönmez and Ünver (2014), where a pairwise kidney exchange problem with both compatible and incompatible patient-donor pairs is investigated. They introduce Pareto efficient matchings that maximize the number of transplants while minimizing participation of compatible pairs.
A crucial assumption in Sönmez and Ünver (2014) is that patients are indifferent between compatible donors. This assumption can be supported by medical practice in the United States, as the general tendency among U.S. doctors is to assume that two compatible living donor kidneys essentially have the same survival rates (Gjertson and Cecka, 2000;Delmonico, 2004;Sönmez and Ünver, 2014). A recent paper by Nicoló and Rodríguez-Álvarez (2017) also focuses on the inclusion of compatible pairs in kidney exchange programs. Based on a number of medical studies (e.g. Gentry et al., 2007;Øien et al., 2007), they argue that the age and general health status of a donor impacts graft survival. Given this observation, patients in their model have strict preferences over compatible donors based on kidney age. This provides an incentive for compatible pairs to participate in exchange programs as the patient may be assigned a different, strictly preferred donor. As already explained above and in similarity with Sönmez and Ünver (2014) and Nicoló and Rodríguez-Álvarez (2017), this article also considers exchanges involving "compatible pairs". However, their participation is not motivated by altruism or the possibility to receive a preferred kidney in terms of age. Instead, the main argument rests, as in Chun et al. (2015), on recent developments in immunosuppressive protocols and, more specifically, on the possibility to transplant kidneys over the blood group barrier (see Section 2 for a description of the immunological conditions and the medical requirements for transplantation across the blood group barrier). This also motivates the extension of the compatibility concept to also include half-compatibility.
The inclusion of half-compatible patient-donor pairs in a kidney exchange program plays a similar role to the inclusion of compatible pairs, as the inclusion increases the size of the patientdonor pool. However, these patients have an incentive beyond altruism to participate since they may be assigned a compatible donor and thereby avoid transplantation over the blood group barrier. Due to the distinction between compatible and half-compatible donors, some standard results in the literature will not continue to hold. For example, Roth et al. (2005b) consider a pairwise kidney exchange problem with no transplantation over the blood group barrier and introduce a class of Pareto efficient matchings called priority matchings (see Appendix A.2).
However, priority matchings are no longer Pareto efficient in settings that distinguish between compatible and half-compatible donors. For this reason, a specific subset of priority matchings is introduced in this article. They are called half-compatibility priority matchings and are guaranteed to be Pareto efficient. The article also provides a computationally efficient method for identifying such matchings.
The two papers that are most closely related to this article are Chun et al. (2015) and Sönmez et al. (2018). The former of these papers considers a kidney exchange program where transplants can be carried out over immunological barriers (both blood group and tissue type). In their model, it is assumed that patients are indifferent between crossing the immunological barrier and not crossing it and that cyclic exchanges of arbitrary length are possible. To reflect that there is a limited availability of immunosuppressants in South Korea, Chun et al. (2015) assume that at most k patients are allowed to use immunosuppressants. For each kidney exchange problem, they first determine which patients are to receive immunosuppressants. Based on this selection and the compatibility structure, a matching is chosen. Their counterfactual analysis shows that the current use of immunosuppressants in South Korea can be reduced by 55%. Sönmez et al. (2018) analyse a recent change in the United States where kidneys are transplanted over the blood group barrier using advanced blood subtyping. This new method allows a fraction of blood group A kidneys to be safely transplanted into a fraction of blood group B and O patients. Given their assumptions, Sönmez et al. (2018) demonstrate that the current implementation of this technology has some unintended consequences in the sense that it reduces the number of transplants from living donors, both in the overall population and for certain biologically disadvantaged groups. Their main results show that these unintended problems can be solved by making two small adjustments to the current practice. They suggest the establishment of an anti-A titre level history for blood group O patients and a delay in the subtyping tests until incompatible pairs are transferred to the kidney exchange pool.

Outline of the article
The remaining part of the article is outlined as follows: Section 2 provides a description of the immunological conditions and the medical requirements that enable transplantation across the blood group barrier. The formal kidney exchange framework is introduced in Section 3. Section 4 provides some properties of half-compatibility priority matchings and presents a computational method for finding them. Section 5 analyses the welfare implications of pairwise kidney exchange over the blood group barrier, both theoretically and by means of a simulation study. Section 6 concludes the article. Appendix A provides an equivalence between the set of priority matchings as defined in this article and the set of priority matchings as defined by Roth et al. (2005b), and some technical results relating to the matroid structure of pairwise kidney exchange problems. Appendix B contains the proofs of the theoretical results.

MEDICAL DETAILS OF BLOOD GROUP INCOMPATIBLE TRANSPLANTATION
This section provides a brief description of the ABO blood group classification system and a medical technology that enables transplantation across the blood group barrier. Throughout this section, the reader should keep in mind that transplantation across the blood group barrier is a medical reality not only in "regular" transplantation, but also within kidney exchange frameworks. As described in Section 1, European countries like Austria, the Czech Republic, Spain, the United Kingdom, Sweden, and Switzerland already use this technology within their exchange programs.
The central principle in the ABO blood group system is that antigens on red blood cells differ between individuals. Since there are two possible antigens (A and B), there are four possible red blood cell types (or blood groups); O, A, B, and AB, where O is standard notation for the absence of antigen A and B. A patient who only has antigen A (antigen B) cannot produce anti-A antibodies (anti-B antibodies) and will therefore only have anti-B antibodies (anti-A antibodies) in her blood plasma. For a patient to be blood group compatible with a donor, the patient must not have anti-A or anti-B antibodies in the plasma that correspond to the A or B antigens in the donor's red blood cells. Consequently, a patient with red blood cell type A (type B) is only blood group compatible with donors that have red blood cell types A and O (types B and O). Patients with red blood cell type O have neither antigen A nor antigen B while carrying both antibodies, and patients with red blood cell type AB have both antigens while carrying neither antibody. Hence, red blood cell type O patients are only blood group compatible with donors that have red blood cell type O, whereas red blood cell type AB patients are blood group compatible with all donors independently of their red blood cell types.
The incompatibilities between some blood groups clearly impose restrictions on organ transplantation as the patient's immune system rejects kidneys from incompatible blood groups. However, immunosuppressive protocols for removing anti-A and/or anti-B antibodies, also known as desensitization, have been known since the 1970s (Alexander et al., 1987). By removing antibodies, these protocols make transplants over the blood group barrier feasible. In 2001, the blood group antigen-specific filter GlycoSorb was introduced (Rydberg et al., 2005). This filter absorbs specific antibodies with the purpose of reducing the patient's antibody level below a certain threshold in order to enable transplantation over the blood group barrier. The antibody level (antibody titre) is determined by a blood serum sample and is diluted in serial ratios (1:1, 1:2, 1:4, 1:8, 1:16, 1:32, ...). Using an appropriate detection method, each dilution is tested for the presence of detectable levels of the antibody of interest. If the level of anti-A and/or anti-B antibodies in a patient's blood is below a threshold value after the filtering process and over a given period of time, a transplant over the blood group barrier is feasible. In Sweden, for example, the threshold is set to 1:32 and the time period is typically set between three and six months. 6,7 GlycoSorb is currently used in all Swedish transplant centres and at least 60 European centres spread across 17 countries. Between 2001 and 2012, more than 200 living donor kidney transplants over the blood group barrier were carried out in Sweden using this filter (Thydén et al., 2012).
There are no medical reasons related to graft and/or patient survival for not using GlycoSorb to conduct transplants over the blood group barrier. In fact, the GlycoSorb filtering process is completely non-toxic (as opposed to non-specific plasma exchange). Moreover, the five-year graft survival rate and patient survival rate for living donor kidney transplants across the blood group barrier are identical to the corresponding five-year survival rates for "normal" blood group compatible living donor kidney transplants (Thydén et al., 2007). Even though these arguments speak in favour of using this medical technology, there are also good reasons for not transplanting kidneys across the blood group barrier whenever alternative transplantation opportunities exist, e.g., if patients have alternative compatible donors available or if it is possible 6. These numbers were communicated to one of the authors of this article (Andersson) at a meeting in Stockholm (March, 2016) with immunologists and transplant surgeons from the four Swedish transplant centres (Karolinska institutet, Akademiska sjukhuset, Sahlgrenska sjukhuset, and Skånes universitetssjukhus). In the United States, for example, a threshold of 1:8 must be maintained for six months before the transplant .
7. The authors of this article are unaware of any studies that report the share of patients that can feasibly receive kidneys over the blood group barrier. Peter S. Björk at the "Immunotherapy Unit" at "Skånes universitetssjukhus" stated, in a telephone conversation with one of the authors (Andersson) in May 2016, that approximately 90% of patients can receive kidneys over the blood group barrier whenever the donor is tissue type compatible. Furthermore, Thydén et al. (2004) report that all patients in their sample with a titre value of at most 1:128 who were treated with GlycoSorb successfully received transplants over the blood group barrier and evidence in Dallaval et al. (2011) suggests that 86.9% of all blood donors with blood group O had antibody titre values strictly below 1:128.
Downloaded from https://academic.oup.com/restud/article-abstract/87/3/1091/5486585 by guest on 09 June 2020 to obtain kidneys from compatible donors by means of exchange. By taking advantage of such alternative transplantation opportunities, additional medical treatments before and after the transplant can be avoided, time to transplantation can be shortened and costs related to the purchase of immunosuppressants can be reduced.
Finally, we note that it can be argued that there are no cost-benefit reasons for not using GlycoSorb. 8 In Sweden, for example, the alternative to a transplant is to keep the patient on dialysis at an annual cost of SEK 650,000. The costs of the surgical procedure and the immunosuppressive protocol are SEK 2,000,000 and SEK 100,000, respectively. Hence, it only takes around three years to reach parity in expenses. In addition, sick leave costs are reduced as the patients no longer need to be on dialysis and patients often experience an increased quality of life after transplantation (Pinson et al., 2000). Similar evidence can be found in, e.g., the United States. In a recent debate article in the Washington Post, Cartwright and Roth (2018) concluded that a kidney transplant "pays for itself in less than two years".

THE MODEL
This section introduces the basic ingredients of the kidney exchange model together with a number of important concepts and definitions.

Agents, preferences and priorities
Let N ={1,...,n} be a finite set of patients participating in a kidney exchange program. Each patient i ∈ N has a living donor d i . Patient i is compatible with donor d j if patient i can receive a kidney from donor d j without crossing the blood group barrier. Patient i is half-compatible with donor d j if patient i can receive a kidney from donor d j only by crossing the blood group barrier. Patient i is incompatible with donor d j if patient i cannot receive a kidney from donor d j under any circumstances. No patient in N is compatible with her own donor since patients with compatible donors are assumed to receive kidneys from their own donors outside the kidney exchange program (except in Section 5.2). The patients in N are partitioned into two disjoint sets: N H and N I . A patient i belongs to N H if and only if she is half-compatible with her own donor d i . N I thus contains all patients who are incompatible with their own donors. The compatibility structure C describes the compatibility between patient i and donor d j for any patients i,j ∈ N.
For any patient i ∈ N, let i denote the patient's preferences over the set of donors. Let ≻ i and ∼ i denote the corresponding strict preference and indifference relations, respectively. Each patient in N strictly prefers any compatible donor to all half-compatible and incompatible donors, and any half-compatible donor to all incompatible donors. Each patient i ∈ N is indifferent between two donors (not including d i ) whenever both are compatible or both are half-compatible with i. Patients in N H also strictly prefer their own donors to all other half-compatible donors. Formally, for any i,j ∈ N and any k,l ∈ N \{i}: 8. The figures in this paragraph are based on Swedish data and they can be found in Thydén et al. (2012) and Wennberg (2010). SEK 1 = USD 0.11 (February, 2019).

REVIEW OF ECONOMIC STUDIES
The preferences of all patients in N are gathered in the list := ( i ) i∈N . Many existing kidney exchange programs give priority to patients that are highly HLA-sensitized (i.e, patients that are highly sensitized to Human Leukocyte Antigents), since it is particularly difficult to find compatible donors for such patients (see, e.g., . As in Roth et al. (2005b), this is modelled by a priority function π : N → R ++ assigning each patient i ∈ N a unique priority π (i). 9 Patient i has higher priority than patient j whenever π (i) >π(j). It is assumed that the priority π (i) of each patient i ∈ N is given by a fraction of the type π (i) = p(i) q for some p(i) ∈{1,...,p} and some p,q ∈ Z ++ , where p and q are fixed and equal for all patients. The interpretation of this assumption is that all patients are assigned a priority that takes a value on a predetermined scale (based on, e.g., Panel Reactive Antibody scores or some other measure of HLA-sensitization). 10 A kidney exchange problem is defined as a triple (N,C,π) and will, with a few exceptions, be held fixed throughout most of the article.

Matchings and properties of matchings
A pairwise kidney exchange between pairs (i,d i ) and (j,d j )isfeasible if and only if d j ≻ i d i and d i ≻ j d j . That is, whenever both patients strictly benefit from the exchange. For a given problem (N,C,π), a matching M consists of (i) a set of mutually exclusive feasible pairwise exchanges and (ii) a set of patients in N H that do not participate in any kidney exchanges. Informally, patients may either receive a transplant (i) through a kidney exchange or (ii) from their own half-compatible donors. A matching specifies which transplants to carry out. The set of all matchings for a given problem (N,C,π) is denoted by M. For any matching M, patients that receive a transplant are said to be matched and patients that receive kidneys from their own half-compatible donors are said to be self-matched. A patient that does not receive a transplant is said to be unmatched.If patient i receives a kidney from donor d j at a matching M, patient i is said to be matched to both

Priority matchings and half-compatibility priority matchings
There is a planner (or a market designer) with complete, transitive, and responsive preferences B over matchings in M. Let ≻ B and ∼ B denote strict preference and indifference, respectively. A matching M is strictly preferred to a matching M ′ if all patients matched at M ′ are also matched at M and some patients not matched at M ′ are matched at M. Moreover, M is strictly preferred to M ′ if the set of patients matched at M can be obtained from the set of patients matched at M ′ 9. R + and R ++ denote the set of non-negative real numbers and the set of positive real numbers, respectively. The same convention applies to the set of integers, Z.
10. This assumption on π (i) is made without loss of generality to get a "non-messy" upper bound on the constant ε defined in Section 4.2. All results presented in the article hold for any π (i) ∈ R ++ as long as the priorities are unique.
Half-compatibility priority preferences are closely related to the priority preferences introduced by Roth et al. (2005b). In fact, a preference relation π is a priority preference relation if it satisfies all of the conditions above, given that (3.2) and the last line in (3.1) have been replaced by a requirement that the planner always be indifferent between M and M ′ whenever N * (M) = N * (M ′ ). 11 The only difference between priority preferences and half-compatibility priority preferences is that whenever the same patients are matched at two different matchings, a planner with priority preferences is indifferent between the two matchings whereas a planner with half-compatibility priority preferences prefers the matching that minimizes the number of transplants over the blood group barrier. Note that in models where transplantation over the blood group is either disallowed or not considered an option by the transplant community. Hence, the two classes of preferences coincide in such settings. Consider some priority preferences π and some half-compatibility priority preferences B . A matching M is called a priority matching if M π M ′ for every matching M ′ ∈ M.Foragiven problem (N,C,π), all priority matchings are gathered in the set M * ⊆ M. A matching M is called a half-compatibility priority matching if M B M ′ for every matching M ′ ∈ M.Foragiven problem (N,C,π), all half-compatibility priority matchings are gathered in the set M B .

PROPERTIES OF HALF-COMPATIBILITY PRIORITY MATCHINGS
This section is divided into two parts. The first part discusses the properties of half-compatibility priority preferences and half-compatibility priority matchings. The second part provides a computational method based on graph theoretical techniques that can be used to find halfcompatibility priority matchings.

Properties
A first observation is that the definition of half-compatibility priority preferences does not induce a unique preference relation. For a given problem, there may be multiple half-compatibility priority preference relations. A natural question is then whether the set of half-compatibility priority matchings M B depends on the choice of preference relation B . Fortunately, the following result reveals that M B remains the same for any choice of half-compatibility priority preference relation B . Proposition 1. For a given problem (N,C,π), all half-compatibility priority preference relations induce the same set of half-compatibility priority matchings.
Consider some matching mechanism that, for every problem (N,C,π), makes use of some half-compatibility priority preference relation to locate a half-compatibility priority matching M. Proposition 1 then guarantees that M is a half-compatibility priority matching for all halfcompatibility priority preferences (such a mechanism is described in Section 4.2). Proposition 1 is closely related to the observation in Roth et al. (2005b, Corollary 1) that any priority matching (defined differently) is weakly preferred to every other matching by any priority preference relation. It is established in Appendix A.2 that the definition of priority matchings in this article is equivalent to the definition in Roth et al. (2005b) and that Corollary 1 in Roth et al. (2005b) can be extended to a biconditional statement both in settings with and without transplantation over the blood group barrier. Given the following result, it is not surprising that priority matchings and half-compatibility priority matchings share many properties.
Proposition 2. For a given problem (N,C,π), every half-compatibility priority matching is a priority matching.
Half-compatibility priority matchings can therefore be thought of as the subset of priority matchings that minimize the number of transplants over the blood group barrier. If patients do not distinguish between compatible and half-compatible matchings, then every maximal matching is Pareto efficient. Since priority matchings are maximal by construction, they are always Pareto efficient in such settings (Roth et al., 2005b). 12 However, priority matchings are no longer necessarily Pareto efficient when transplantation over the blood group barrier is possible. The next result shows that, contrary to priority matchings, half-compatibility priority matchings are guaranteed to be Pareto efficient. Furthermore, half-compatibility priority matchings (and priority matchings) maximize the number of transplants.
Proposition 3. For a given problem (N,C,π), every half-compatibility priority matching is a Pareto efficient maximum matching.
Proposition 3 is silent about how the technology enabling transplantation over the blood group barrier is implemented. The result merely states that if a matching is a half-compatibility priority matching in a given problem, then it is Pareto efficient and a maximum matching in that particular problem. As will be discussed in Section 5, immunosuppressants can be introduced in a kidney exchange program by finding a half-compatibility priority matching for the patients in N I and self-matching all the patients in N H . Then the outcome may not be Pareto efficient or maximize the number of transplants when considering all patients in N (see Proposition 7). However, Proposition 3 still implies that the matching is a Pareto efficient maximum matching in the reduced problem containing only the patients in N I .
Another implication of Proposition 3 is that all half-compatibility priority matchings result in the same number of transplants, i.e., |N * (M)|=|N * (M ′ )| for all M,M ′ ∈ M B , since all maximum matchings necessarily match the same number of patients. In fact, Proposition 4 shows that all priority matchings (including all half-compatibility priority matchings) match exactly the same patients. This result no longer holds if priorities are not required to be unique as in, e.g., Okumura (2014). To see this, imagine a situation with three patient-donor pairs; pairs 1, 2, and 3. There is a feasible kidney exchange between pair 1 and pair 2, a feasible exchange between pair 2 and pair 3, but no feasible exchange between pair 1 and pair 3. If pair 1 and pair 3 have the same priority, then each of the two feasible exchanges constitutes a priority matching. Only one of them can be selected by the planner and depending on this selection, different patients will be matched.
Many of the results in this section relate to the structure of pairwise kidney exchange problems in particular. One important aspect of pairwise kidney exchange problems is that the set of all patients N and a family I containing all sets of patients that can be matched simultaneously constitute a matroid (N,I). Such a structure ensures that every maximal matching is a maximum matching and that the same number of patients receive a transplant at every Pareto efficient matching. Thanks to this structure, the opportunity cost of matching a particular patient (e.g. a high-priority patient) will never be more than one patient (with lower priority) who could otherwise have been matched. Roth et al. (2005b) showed that the pairwise kidney exchange problem has a matroid structure when the compatibility structure is binary (no transplantation over the blood group barrier). Proposition 11 in Appendix A.1 demonstrates that this result continues to hold in settings that distinguish between compatibility and half-compatibility. This is not immediately obvious since self-matches alter the structure of the sets of simultaneously matchable patients.
A final remark is that the findings in this section provide justification for half-compatibility priority preferences. As argued in Section 2, there are good reasons for minimizing the use of immunosuppressants. Furthermore, as described in , maximizing the number of transplants is an objective in all existing European kidney exchange programs and all these programs (except in Austria and the Czech Republic) also prioritize patients in accordance with their HLA-sensitization levels. A planner with half-compatibility priority preferences selects a matching from the set of half-compatibility priority matchings. Consequently, the use of immunosuppressants is minimized, the number of transplants is maximized and patients receive priority based on, e.g., the degree of HLA-sensitization. In addition, the planner is guaranteed that any choice of half-compatibility priority preferences will result in the same set of half-compatibility priority matchings and that the same set of patients will receive transplants. The last point implies that a planner need not worry about the specific choice of halfcompatibility priority preference relation affecting various groups in a diverse patient population differently.

Identification of half-compatibility priority matchings
Given the desirable properties of half-compatibility priority matchings discussed in the previous section, the main purpose of this section is to investigate how these matchings can be computed. In contrast to the iterative method for identifying priority matchings introduced by Roth et al. (2005b), the method considered in this section takes a graph theoretical approach. More specifically, it is demonstrated that half-compatibility priority matchings can be identified in polynomial time by solving a maximum weight matching problem. This maximization technique is, in similarity with many algorithms like, e.g., the deferred acceptance algorithm (Gale and Shapley, 1962) and the top trading cycles mechanism (Shapley and Scarf, 1974), frequently adopted in the market design literature to solve various matching problems. For example, solution methods based on maximum weight matching problems have previously been applied to problems related to school choice (Kesten and Ünver, 2015), delegate pairings at meetings (Vaggi et al., 2014), kindergarten placements (Biró and Gudmundsson, 2017), and REVIEW OF ECONOMIC STUDIES kidney exchange (Biró et al., 2009). To describe this computational method, some graph theoretical notation needs to be introduced.
For any compatibility structure C, there exists a corresponding compatibility graph g = (N,E) comprising a set N of vertices and a set E of edges. It will sometimes be convenient to let N(g) and E(g) denote the vertex set and the edge set, respectively, of the compatibility graph g. Every vertex in a compatibility graph corresponds to a patient in N. There is an edge between two patients i,j ∈ N if and only if a pairwise exchange between the pairs (i,d i ) and (j,d j ) is feasible, and there is a loop at vertex i ∈ N if and only if patient i is half-compatible with her own donor d i . Let ij denote an edge between patients i and j and let ii denote a loop at patient i. Formally, the edges in a compatibility graph g = (N,E) have the following construction: For any compatibility graph g = (N,E), a matching M ⊆ E can be defined as a set of edges in the graph that are not incident to each other. That is, for any edge ij ∈ M, it must be the case that ik / ∈ M and jk / ∈ M for all k ∈ N \{i,j}. There is an edge ij ∈ M for some i,j ∈ N, i = j, whenever the pairs (i,d i ) and (j,d j ) are involved in a pairwise kidney exchange. Moreover, ii ∈ M for some i ∈ N H whenever i receives a kidney from her own donor. The non-incidence requirement on the edges ensures that each patient receives at most one kidney and each donor donates at most one kidney. This definition of a matching is thus equivalent to the definition given in Section 3.2.
A weighted graph (g,w) consists of a graph g and a set of edge weights w := (w ij ) ij∈E(g) where w ij is a weight assigned to edge ij ∈ E(g). Let (g,w) be a weighted graph and let S(M,w) := ij∈M w ij be the sum of all edge weights at matching M ∈ M. A matching M is a maximum weight matching in (g,w)i fS(M,w) ≥ S(M ′ ,w) for all M ′ ∈ M. Okumura (2014) demonstrated that priority matchings can be found by solving an appropriately defined maximum weight matching problem.
Lemma 1. (Okumura, 2014, Theorem 2). Consider a problem (N,C,π) with corresponding compatibility graph g. Suppose that transplantation over the blood group barrier is not possible. 13 If w ij = π (i)+π (j) for all ij ∈ E(g), then M is a priority matching if and only if M is a maximum weight matching in (g,w).
The maximum weight matching problem described in Lemma 1 is not directly applicable in the setting considered in this article since the result is based on the assumptions that patients cannot receive kidneys from their own donors (i.e. no loops in the compatibility graph) and that there is no distinction between compatible and half-compatible donors. However, the following theorem shows that even if these assumptions are relaxed, priority matchings can be identified by solving an almost identical maximum weight problem (the only difference being the presence of loops in the graph).
Proposition 5. Consider a problem (N,C,π) with corresponding compatibility graph g.Ifw ii = π (i) for all ii ∈ E(g) and w ij = π (i)+π (j) for all ij ∈ E(g) whenever i = j, then M is a priority matching if and only if M is a maximum weight matching in (g,w).
Recall that the set of half-compatibility priority matchings is the subset of priority matchings that minimize the number of transplants over the blood group barrier. Hence, solving the maximum weight matching problem defined in Proposition 5 will not necessarily identify a halfcompatibility priority matching. To address this issue, a modified maximum weight matching problem is presented, the solution to which is guaranteed to be a half-compatibility priority matching. Consider a problem (N,C,π) with corresponding compatibility graph g. Let 0 <ε< 1 2nq and let the weights w ε := (w ε ij ) ij∈E(g) for each i,j ∈ N be defined by: Proposition 6. Consider a problem (N,C,π) with corresponding compatibility graph g. Then a matching is a half-compatibility priority matching if and only if it is a maximum weight matching in (g,w ε ).
A solution to the maximum weight matching problem in Proposition 6 can be found in polynomial time by adopting the techniques in Edmonds (1965).

WELFARE IMPLICATIONS OF KIDNEY EXCHANGE OVER THE BLOOD GROUP BARRIER
This section analyses the consequences of introducing transplantation over the blood group barrier in kidney exchange programs. Even though the existence of some welfare effects can be proven theoretically, a simulation study is necessary to estimate their magnitudes. The section ends with an extended discussion on the possibility to manipulate the matching mechanisms induced by Model (a) and Model (b). As a benchmark in the analysis, we will use a model with pairwise exchanges, in which transplants over the blood group barrier are either not allowed or not an option considered by the medical community (as in, e.g. Roth et al., 2005b). It is then assumed that the technology enabling transplants across the blood group barrier gets implemented within kidney exchange programs, e.g., due to new legislation allowing such transplants, awareness, or changes in the attitude towards immunosuppressants in the transplant community. This will have the consequence that patient and planner preferences will distinguish between compatible and half-compatible donors and matchings. Two distinct "models" are introduced to represent two different ways in which the planner can implement this technology within kidney exchange frameworks. Both models can be thought of as extensions to the Benchmark Model. Let (N I ,C I ,π) denote a reduced problem containing only the patients in N I , where C I denotes the compatibility structure between patients in N I and donors of patients in N I . 14 • Benchmark Model. Transplants over the blood group barrier are either disallowed or not considered an option by the medical community. A priority matching is found for the problem (N,C,π). • Model (a). The technology enabling transplants over the blood group barrier is adopted by the medical community. Patients in N H (i.e. patients with half-compatible donors) are self-matched and do not participate in the kidney exchange program. A half-compatibility priority matching is found for the reduced problem (N I ,C I ,π). • Model (b). The technology enabling transplants over the blood group barrier is adopted by the medical community. A half-compatibility priority matching is found for the problem (N,C,π).
Even though there are variations in rules, regulations, techniques, etc., that differentiate kidney exchange programs from each other, many existing programs can, in general terms, be categorized within our framework. The Benchmark Model is the standard model in the theoretical kidney exchange literature and it has been analysed by, e.g., Roth et al. (2004Roth et al. ( , 2005aRoth et al. ( ,b, 2007, Okumura (2014), Saidman et al. (2006), Sönmez and Ünver (2014), and Sönmez et al. (2018). Note also that the initial program in the United States (Roth et al., 2005b) and the current practice in, e.g., France, India, and Italy (Jha et al., 2015;Biró et al., 2019), only considered/consider pairwise exchanges and transplants over the blood group barrier within the existing exchange programs were/are not allowed or not considered an option by the medical community. The same holds in, e.g., Belgium, the Netherlands, Poland, and Portugal, even if larger cyclical exchanges are allowed in these countries (Biró et al., , 2019. Transplantation across the blood group barrier is, however, allowed in Model (a). This model corresponds to the current program in, e.g., Sweden, where only pairwise exchanges are considered and patients with half-compatible donors receive kidneys from their own donors outside the exchange system. Although larger cyclic exchanges are allowed in, e.g., Austria, the Czech Republic, Spain, the United Kingdom, and Switzerland, it can be argued that the exchange programs in these countries correspond to Model (a). Patients in these countries are routinely referred to desensitization treatments outside their respective exchange programs, even though transplants over the blood group barrier are allowed within their corresponding exchange frameworks (Biró et al., , 2019. Model (b) also allows for transplantation across the blood group barrier. To the best of our knowledge, no country in the world has adopted an exchange program that corresponds to Model (b), i.e., a program that includes all incompatible and all half-compatible patient-donor pairs in a common exchange pool. The main difference between Model (b) and Model (a) is that the exchange program in the latter model does not include patients with half-compatible donors. These patients are, in Model (a), always self-matched with their own donors and will therefore never be part of an exchange.
When transplantation over the blood group barrier is not a possibility, all priority matchings are half-compatibility priority matchings. This means that, by Proposition 3, the Benchmark model would always select Pareto efficient matchings. However, if blood group incompatible transplants are feasible, priority matchings are no longer necessarily half-compatibility priority matchings. In such settings, there are good reasons for a planner concerned with patient welfare to only select half-compatibility priority matchings, as in Model (a) and Model (b). The following example shows that priority matchings may be Pareto dominated by the matching half-compatibility matchings selected in Models (a) and (b).
Example 1. Suppose that N ={1,2,3,4}, that each patient is incompatible with her own donor and that transplantation over the blood group barrier is permitted. Furthermore, suppose that Patient 1 is compatible with donor d 2 and that no other patient is compatible with any other donor. A possible pairwise exchange between pairs (i,d i ) and (j,d j ) is denoted by ij and it is assumed that only the pairwise exchanges 12, 23, 34, and 14 are feasible. 15 In this case, matchings M ={12,34} and M ′ ={14,23} are both priority matchings since all patients receive transplants. However, only M is a half-compatibility priority matching since more patients are matched to compatible donors at M than at M ′ . Since Patient 1 strictly prefers donor d 2 to donor d 4 and all other patients are indifferent between M and M ′ , it follows that M Pareto dominates M ′ . This demonstrates the importance of distinguishing between priority matchings and half-compatibility priority matchings and, as in Model (a) and Model (b), selecting matchings of the latter type.

Theoretical results
A noteworthy difference between the Benchmark Model and Model (b) on the one hand and Model (a) on the other is that (half-compatibility) priority matchings are identified for the entire set of patients in the former two models, whereas Model (a) only selects a half-compatibility priority matching for the patients in N I and self-matches the patients in N H . Proposition 3 implies that the half-compatibility priority matching which patients in N I are matched in accordance within Model (a) is a Pareto efficient maximum matching for the reduced problem (N I ,C I ,π). The matchings selected in the Benchmark Model and Model (b), on the other hand, are Pareto efficient maximum matchings for the problem (N,C,π) containing all patients in N (note that the matching selected in the Benchmark Model is only Pareto efficient under the constraint that transplantation over the blood group barrier is not possible).
By excluding the patients in N H when identifying a half-compatibility priority matching in Model (a), the aggregate outcome, defined by the transplants outside the kidney exchange program and the transplants generated by exchanges within the exchange program, need not maximize the number of transplants or be Pareto efficient. Intuitively, this failure hinges on the use of immunosuppressants that enable patients with half-compatible donors to receive kidneys from their own donors outside the kidney exchange program. A planner implementing Model (a) not only denies patients with half-compatible donors the possibility to find a compatible donor within the exchange framework, but also shrinks the size of the patient-donor pool when self-matching all patients with half-compatible donors. This reduces the likelihood that patients with incompatible donors participate in pairwise exchanges since the set of patients they could be matched to is smaller. 16 The following result is proven with the help of an example that will also be useful later in this section.
Proposition 7. The matching selected in the Benchmark Model may Pareto dominate the matching selected in Model (a). In addition, the total number of transplants may be higher in the Benchmark Model than in Model (a).
Proof. Suppose that N ={1,2} where N I ={1} and N H ={2}. Assume further that Patient 1 is compatible with donor d 2 and that Patient 2 is compatible with donor d 1 . In Model (a), Patient 2 is self-matched as 2 ∈ N H . Since Patient 1 is incompatible with donor d 1 , Patient 1 remains unmatched. In the Benchmark Model, no patients are self-matched. Patient 2 is therefore available for a mutually beneficial kidney exchange with Patient 1. Thus, both patients receive transplants in the Benchmark Model, whereas only Patient 2 receives a transplant in Model (a). Furthermore, 15. In the Benchmark model, no pairwise exchanges would be feasible. 16. It is well-known that larger patient-donor pools result in more transplants than smaller pools. See, e.g., Roth et al. (2006). Patient 2 is strictly better off in the Benchmark Model since Patient 2 is compatible with donor d 1 and only half-compatible with donor d 2 .
The above example shows that the introduction of transplantation over the blood group barrier could, in theory, reduce the number of transplants and make all patients worse off if implemented as in Model (a). In the simulation study in Section 5.2, the number of transplants was indeed lower in Model (a) than in the Benchmark Model in a small number cases for the smallest pool size. However, this problem vanished completely as the size of the patient-donor pool grew. Negative outcomes of this kind could never occur in Model (b) since it includes patients with half-compatible donors in the kidney exchange program and selects a Pareto efficient matching for all patients N in the problem (N,C,π). Given this observation, it is natural to ask whether Model (b) will generally result in a weakly larger number of transplants than the Benchmark Model and Model (a). According to the next result, it will.
The results above indicate that the manner in which medical technology enabling kidney transplants over the blood group barrier is used can have significant welfare implications. Even though one would suspect that this technology would increase the total number of kidney transplants, Proposition 7 reveals that this is not necessarily the case since a planner implementing Model (a) first maximizes the number of self-matches and only includes the remaining patientdonor pairs in the kidney exchange program. A planner implementing Model (b), on the other hand, regards self-matches as the last option for patients with half-compatible donors since these patients are first included in the exchange program in the hope of finding compatible donors for them. This inclusion means that a planner using Model (b) first aims to maximize the number of pairwise exchanges and, consequently, ensures that the number of patient-donor pairs participating in the kidney exchange program is maximized. As seen in Proposition 8, this strategy guarantees the total number of transplants to be (weakly) greater in Model (b) than it is in both the Benchmark Model and Model (a).
Recall that Model (a) corresponds to current practice in, e.g., the Swedish kidney exchange program. Given the findings above, a natural question is then whether to transition from Model (a) to Model (b). The answer not only depends on the number of additional transplants that the transition would result in, it also depends on how it would affect patients with incompatible donors. To make this point clear, recall that patients with half-compatible donors can always receive kidneys over the blood group barrier from their own donors outside the exchange program. From a welfare perspective, it is then important to ensure that a patient with a half-compatible donor is not involved in an exchange at the expense of a patient with an incompatible donor as patients of the latter type cannot receive kidneys outside the exchange program. The next proposition ensures that such situations never occur. More precisely, Proposition 9 shows that all patients that would have received transplants in Model (a) will still receive transplants if there is a transition from Model (a) to Model (b).
From a welfare perspective, it is reassuring that a transition from Model (a) to Model (b) is guaranteed to weakly increase the number of transplants (Proposition 8) and that patients receiving transplants before the transition are guaranteed to still receive transplants after the transition (Proposition 9). This does, however, not say anything about what type of donors the patients will be matched to. It is clear that patients with half-compatible donors are made weakly better off by the transition since the worst possible outcome for them is to be paired with their own halfcompatible donors, i.e., the same outcome as in Model (a). The story for patients with incompatible donors is a bit different. and there is no general theoretical prediction. For some problems, there is no half-compatibility priority matching such that all patients with incompatible donors weakly gain by the transition from Model (a) to Model (b), and for some problems there is. A situation where all patients with incompatible donors are made better off by the transition is illustrated in the Proof of Proposition 7 above, since the outcomes in the Benchmark Model and Model (b) coincide. A situation where some patient is made worse off by the transition is illustrated in the following example.
Example 2. Suppose that N ={1,2,3,4}, N I ={1,2,3}, N H ={4} and π (1) >π (3). A feasible pairwise exchange between pairs (i,d i ) and (j,d j ) is denoted by ij and it is assumed that only the pairwise exchanges 12, 14, and 23 are feasible. Suppose further that Patient 1 is half-compatible with donor d 4 and that all patients in the other three feasible pairwise exchanges are compatible with the donors they participate in the exchanges with. In Model (a), Patient 4 is matched to her own half-compatible donor and the pairwise exchange 12 is conducted since π (1) >π(3). In Model (b), the pairwise exchanges 14 and 23 are carried out. Even though Patient 1 receives a transplant in both models, Patient 1 is made worse off by a transition from Model (a) to Model (b) since the patient is compatible with donor d 2 and only half-compatible with donor d 4 .

Simulation results
This section aims to investigate the magnitudes of the theoretical findings presented in the previous section. It also attempts to shed light on some issues that are discussed extensively in the kidney exchange literature but have so far not been addressed in this article. For instance, this section will investigate how patients that are often proportionally disadvantaged in kidney exchange programs (specifically, blood group O patients) are affected by the introduction of transplantation over the blood group barrier.
In addition to the three models introduced earlier in this section, two additional models will be investigated. The first of these models is the Altruistic Model (Roth et al., 2005a;Sönmez and Ünver, 2014) in which compatible patient-donor pairs participate in the kidney exchange program. It is called the Altruistic Model because patients with compatible donors do not benefit from participation in exchange programs, as they can already receive kidneys from their own donors (without crossing the blood group barrier). The compatible pairs ("altruistic pairs") participate to help other incompatible or half-compatible pairs. The second model, called the Cycle Model (Roth et al., 2007), allows for three-way exchanges in addition to pairwise exchanges, i.e., cyclic exchanges involving three patient-donor pairs. 18 The reason for including these models in the simulation study is that the design features of both models are known to (weakly) increase the number of transplants (see, e.g. Roth et al., 2005a;Gentry et al., 2007;Sönmez and Ünver, 2014). The outcomes in these two models are estimated under the assumption that transplants over the blood group barrier are either not allowed or not considered an option by the transplant community, and compared to the outcomes in Models (a) and (b). This makes it possible to compare the effect of introducing medical technology enabling transplantation over the blood group barrier to the impact of other design features that are known to work well.
In order to analyse the Altruistic Model, a third type of patients must be added to the model, namely the patients with compatible donors. These patients are gathered in the set N C and all patients in N C ∪N are included in the simulations, where N C ∩N =∅. The patients in N C play no role in the Benchmark Model, Model (a), Model (b), or the Cycle Model since they are simply self-matched and unavailable for pairwise exchanges in all of these models. The patients in N C do, however, play a significant role in the Altruistic Model. Let C C be the compatibility structure between patients in N C ∪N and donors of patients in N C ∪N. Furthermore, let π C : N C ∪N → R ++ be a priority function assigning each patient in N C ∪N a unique priority.
• Altruistic Model. Transplants over the blood group barrier are either disallowed or not considered an option by the medical community. A priority matching is found for the problem (N C ∪N,C C ,π C ). 19 Unmatched patients in N C are then self-matched with their own compatible donors. • Cycle Model. Transplants over the blood group barrier are either disallowed or not considered an option by the medical community. Three-way exchanges are permitted in addition to pairwise exchanges. A maximum matching 20 is found for the problem (N,C,π).
The reported results for the Cycle Model are taken from Roth et al. (2007), but these results are directly comparable to the other simulation results provided in this section (see footnotes 23 and 25 for details). For the other four models, a population of patient-donor pairs is generated using medical data (e.g. blood group distributions, PRA distributions (i.e., Panel Reactive Antibody distributions), crossmatch probabilities, etc.) identical to the data described in Roth et al. (2007) and Saidman et al. (2006). Two pieces of information required for the simulations are missing in these articles; the share of patients that are female and have a spouse donor, and the share of patients for whom transplantation over the blood group barrier is feasible. The first of these numbers is based on Swedish medical data (Fehrman-Ekholm et al., 2011) and is set to 10%. For the second number, it is assumed that transplantation over the blood group barrier is feasible for 75% of patients. 21 The priority π (i) for each patient i is defined as in Keizer et al. (2005): The equation above captures the transplantation possibilities for patient i both outside and within the kidney exchange program. The higher the priority, the more difficult it is to find a suitable donor for the patient. Since both factors on the right hand side of equation (5.3) belong to the interval [0,1], patient priorities are guaranteed to take values between 0 and 1. Note also that the above 19. Since patients in N C are assumed to be "altruistic" in the Altruistic Model, their preferences must be amended slightly. More precisely, patients in N C are assumed to be indifferent between all compatible donors, including their own. Furthermore, it is sufficient that a patient i ∈ N C weakly gain in a pairwise exchange for the exchange to be feasible. Patients in N must, however, still strictly gain for an exchange to be feasible.
20. The reason the Cycle Model selects a maximum matching rather than a priority matching is that the matroid structure, discussed in Section 4.1 and Appendix A.1, is lost in the Cycle Model, giving rise to a trade-off between prioritization of HLA-sensitized patients and maximization of the number of transplants. See, e.g., Kratz (2019)o r Sönmez and Ünver (2014). 21. Recall from footnote 7 that this number is estimated to be around 90%. In the simulation study, however, a more conservative number is used to ensure that the results for Models (a) and (b) Biró et al. (2018). The remainder of this section analyses a Monte Carlo simulation based on 1,000 populations randomly drawn from the medical distributions discussed above for population sizes of 25, 50, 100, 200, and 500 patient-donor pairs. 22 To put these population sizes in perspective, the kidney exchange program in the United Kingdom is the largest in Europe with 250 participating patientdonor pairs per matching run. The second largest European program is found in Spain with 110 participating pairs per run. In the simulations, around 50% of the patients already have compatible donors (see column "Self-match" in Table 1) and will, consequently, not be part of the patient-donor pool. This means that generated populations of 500 and 200 patient-donor pairs correspond to kidney exchange programs with 250 and 100 participating pairs, respectively, like the U.K. program and the Spanish program. Many European countries, including Austria, Belgium, France, Italy, Poland, Portugal, and Sweden, have much smaller programs captured by the population sizes 100, 50, and 25. Table 1 displays the percentages of different types of transplants for each model and population size. In the table, "Exchange" only includes pairwise exchanges in all models except the Cycle Model, which includes three-way exchanges as well. Moreover, "ABOi" indicates that a patient receives a transplant over the blood group barrier. Note that such transplants, by assumption, are infeasible in the Benchmark Model, the Altruistic Model, and the Cycle Model. Furthermore, in all models except the Altruistic Model, patients with compatible donors receive kidneys from their own donors outside the kidney exchange program.
Recall Proposition 8, which states that Model (b) always generates a weakly larger number of transplants than both the Benchmark Model and Model (a). The exact magnitude of this difference can be seen in Table 1. For a population size of 50, an average of 34.2% of the patients in the Benchmark Model will not receive transplants. 23 The corresponding numbers for Models (a) and (b) are 11.1 % and 8.0 %, respectively. For a population size of 50, the simulation results also suggest that a transition from the Benchmark Model to Model (b) would on average result in 13.1 additional transplants (i.e. 34.2−8.0 = 26.2 % of 50 patients). To achieve this, 11.5 patients (i.e. 14.6+8.4 = 23.0 % of 50 patients) must on average receive transplants over the blood group barrier. An implied rule of thumb is that for every additional transplant achieved by a transition from the Benchmark Model to Model (b), one transplant must be carried out over the blood group barrier. This rule of thumb holds for all population sizes. Note also that the gain, measured in 22. The simulation makes use of Joris van Rantwijk's script for finding maximum weight matchings in graphs, ported to MATLAB by Daniel R. Saunders.
23. Roth et al. (2007) consider both the Benchmark Model and the Cycle Model. The patient-donor pool in their simulation study only contains patients with incompatible donors, whereas the patients in N C are included in this section. Since roughly 50% of the pairs included in this simulation study have a compatible donor (see the "Self-match" column for the Benchmark Model in Table 1), the case when n = 50 in this article roughly corresponds to the case when n = 25 in Roth et al. (2007). Table 2 in Roth et al. (2007) shows that an average of 8.86 patients are involved in exchanges when n = 25 and only pairwise exchanges are allowed. The corresponding number in this article is 8.6 patients (17.2% when n = 50). Furthermore, Roth et al. (2007) find that an average of 21.8 patients are involved in pairwise exchanges when n = 50. The corresponding number in this article is 21.5 (21.5% when n = 100). In this sense, the results in this article confirm the findings in Roth et al. (2007). As a consequence, the results in Roth et al. (2007) for the Cycle Model can safely be used as an approximation of the corresponding results in this article. total number of transplants, from introducing transplants over the blood group barrier increases with pool size. 24 It is also notable that not only do more patients receive transplants in Model (b) than in Model (a), the share of transplants over the blood group barrier is also lower. For a population size of 50, an average of 27.1 % (i.e. 20.9+6.2 %) of patients receive transplants over the blood group barrier in Model (a). The corresponding number in Model (b) is only 23.0 % (i.e. 14.6+8.4 %). This conclusion holds for all population sizes. The results in Table 1 also suggest that the impact on the number of transplants of introducing transplantation over the blood group barrier is larger than the impact of including altruistic donors or allowing three-way exchanges. 25 This conclusion holds for both Models (a) and (b).
24. This follows from the fact that both Models (a) and (b) select maximal matchings. More specifically, suppose that a set of patient-donor pairs are added to an existing exchange pool but that the number of transplants decreases. Then a contradiction is obtained immediately because the matching for the initial and smaller pool is feasible also for the larger pool and both Models (a) and (b) select maximal matchings. Consequently, the total number of transplants must increase with pool size (in fact, this conclusion holds for any matching mechanism that selects maximal matchings, including, e.g. the priority mechanism). 25. Note that the results in Table 1 for the Cycle Model are most likely marginally underestimated. This follows from the fact that the simulations in Roth et al. (2007) are based on population sizes of 25, 50 and 100, while the corresponding population sizes in this article are 25.7, 51.4, and 102.6 (see also footnote 23). Since Models (a) and (b) clearly outperform the Cycle Model in Table 1, the marginal difference in population sizes will not affect the general conclusions that can be drawn from the results.  The results for Models (a) and (b) in Table 1 are based on the assumption that all patients would accept receiving a kidney from any half-compatible donor. 26 Even though this is a reasonable assumption since most patients would prefer receiving a half-compatible kidney to not receiving any transplant at all 27 , a sensitivity analysis is provided in Tables 2 and 3. These tables provide the recalculated results for Model (a) and Model (b) under the assumption that 0, 25, 50, 75, or 100 % of all patients are willing to receive kidneys from all half-compatible donors. Note first that the results for 0% and 100% in Table 2, by definition, represent the Benchmark Model and Model (a) as presented in Table 1, respectively. Similarly, the results for 0% and 100% in Table 3 represent the Benchmark Model and Model (b) as presented in Table 1, respectively. As can be seen in Tables 2 and 3, for every additional 25% of patients that are willing to receive kidneys from half-compatible donors, between 3% and 7% more of the patients in the pool will receive transplants. This conclusion holds for both Model (a) and Model (b) independently of pool size. Furthermore, it suffices that at least 25% of the patients are willing to receive kidneys from half-compatible donors for Model (a) and Model (b) to perform better than the Cycle Model in terms of the number of transplants. However, more than 75% of the patients must be willing to 26. This assumption will be discussed further in Section 5.3 in relation to manipulability. Note also that even if all patients are assumed to be willing to receive kidneys from all half-compatible donors, not all patients are able to receive kidneys from all half-compatible donors as explained earlier (see footnote 7).
27. This assumption has also been informally confirmed by the Swedish transplant doctors and immunologists we have spoken to based on the observation that patients follow the recommendations made by their medical doctors in almost all cases and the fact that Sweden has had a well-functioning program for kidney transplantation across the blood group barrier for more than 10 years (see Fehrman-Ekholm et al., 2011). Unfortunately, we have been unable to confirm this informal statement with official statistics. receive kidneys from any half-compatible donors for Model (a) and Model (b) to perform better than the Altruistic Model. The two latter conclusions also hold independently of pool size. Proposition 7 showed that the Benchmark Model may generate a larger number of transplants than Model (a). The simulation results in Table 1 suggest that this is not the average case. Table 4 provides more details for the different population sizes. For population sizes 50, 100, 200, and 500, the Benchmark Model never generates more transplants than Model (a). For a population size of 25, the Benchmark Model only generates more transplants than Model (a) in 0.3% of the cases. In fact, in nearly 100% of all cases, Model (a) generates a strictly larger number of transplants than the Benchmark Model. Hence, the theoretical finding in Proposition 7 that fewer patients may receive transplants in Model (a) than in the Benchmark model appears to mostly be a theoretical possibility and not something a planner needs to worry about.
Proposition 9 showed that if there is a transition from Model (a) to Model (b), then all patients receiving transplants in the former model will still receive transplants in the latter. However, Example 2 revealed that some patients that would have been matched to compatible donors in Model (a) may only be matched to half-compatible donors in Model (b). For this reason, it is  (b) 6.4% 7.6% 8.8% 9.5% 9.7% Total 100.0% 100.0% 100.0% 100.0% 100.0% important to investigate how many patients are made better off and how many patients are made worse off by such a transition. Table 5 shows that, on average, between 1.2% and 2.4% of the patients (depending on the population size) who were matched to compatible donors in Model (a) were matched to half-compatible donors in Model (b). By comparison, an average of between 6.4 and 9.7 % of the patients (again, depending on the population size) who were matched to half-compatible donors in Model (a) were matched to compatible donors in Model (b). In this sense, a transition from Model (a) to Model (b) would improve the average "kidney quality" for patients receiving transplants in Model (a). Another indication of this result can be found in Table 1 where, for a population size of 50, an average of 27.1% of patients receive kidneys over the blood group barrier in Model (a), whereas the corresponding number for Model (b) is 23.0%.
So far, there has been no discussion of the patients who remain unmatched after a matching has been selected. It is, for example, well-known that patients with blood group O are often proportionally disadvantaged in kidney exchange programs (Roth et al., 2007;Sönmez and Ünver, 2013;Sönmez et al., 2018). The underlying reason for this is that there are typically more blood group O patients than there are blood group O donors in kidney exchange pools when transplants over the blood group barrier are infeasible. A blood group O patient is less likely to be involved in a kidney exchange than a patient with a different blood group since she can only receive kidneys from blood group O donors (see Section 2 for a description of the ABO blood group classification system). Hence, not only are blood group O patients expected to be over-represented in the kidney exchange pool, the proportion of blood group O patients is also expected to increase after the exchanges have been carried out. Table 6 sheds some light on the impact that the introduction of transplantation over the blood group barrier would have on this biological unbalance. In the table, the "Ex Ante" distribution is the blood group distribution in the kidney exchange pool, whereas the "Ex Post" distribution is the blood group distribution of the patients that remain in the pool after the matched patients have been removed.
As in Roth et al. (2007) and Saidman et al. (2006), the simulations are based on a blood group distribution where 48, 34, 14, and 4% of the patients have blood group O, A, B, and AB, respectively. The biological unbalance described above is confirmed for the Benchmark Model where between 58.5% and 59.5% (depending on population size) of patients included in the kidney exchange pool have blood group O. These numbers are even higher in the ex post distribution and range between 69.0% and 86.3. In fact, the proportion of all blood groups except blood group O is lower in the ex post distribution than in the ex ante distribution in the Benchmark Model. Hence, not only are blood group O patients over-represented in the ex ante distribution, they are even more over-represented in the ex post distribution. The ex ante blood group distribution in the Altruistic Model is expected to be close to the assumed underlying blood group distribution since all patients are included in the ex ante distribution, independently of whether they can receive kidneys from their own donors. However, blood group O patients are still clearly disadvantaged in the ex post distribution in the Altruistic Model as well. Again, this finding hinges on the fact that it is more difficult for blood group O patients to find donors within the kidney exchange pool compared to patients with other blood groups. When the medical technology that enables transplantation over the blood group barrier is introduced, blood group O patients become less disadvantaged than they were in the Benchmark Model and the Altruistic model, although they are still disadvantaged. The ex post proportion of blood group O patients in Models (a) and (b) for a population size of 50 is 59.1% and 65.4%, respectively. This can be compared to the corresponding numbers for the Benchmark Model and the Altruistic Model, which are 72.5% and 70.5%, respectively. This means that transplantation over the blood group barrier both increases the number of transplants and makes blood group O patients less disadvantaged. This conclusion holds for almost all population sizes. The only exception is for n = 200 and n = 500, in which the proportion of blood group O patients in the ex post distribution is 76.1% and 85.7%, respectively, in Model (b) and 72.4% and 74.4%, respectively, in the Altruistic Model. However, note that Table 6 also reports the average (absolute) number of patients for each blood group (in parenthesis). It can be seen that the higher percentages in Model (b) are a direct consequence of the fact that only an average of 3.38 and 1.12 patients remain unmatched in Model (b) for n = 200 and n = 500, respectively. That is, it hinges on the fact that the proportion of unmatched patients with a specific blood group in the ex post distributions is defined relative to the proportion of unmatched patients with other blood groups. Furthermore, a quick look at Table 6 may give the impression that blood group O patients are less disadvantaged in Model (a) than in Model (b) since the proportion of blood group O patients in the ex post distribution in Model (a) is lower than the corresponding proportion in Model (b) for all population sizes. However, Model (b) generates significantly more transplants than Model (a) and the patients that remain unmatched tend to be those that are the most difficult to find suitable donors for, e.g., blood group O patients. By consulting the absolute values (in parenthesis) in Table 6, it is clear that the skewed ex post distributions for Model (b) is a direct consequence of the fact that almost all patients receive transplants. The average number of blood group O patients that remain unmatched is lower in Model (b) than in all other models for all population sizes. To make this point clearer, Table 7 shows the average improvement in the number of blood group O patients receiving transplants compared to the Benchmark model. For example, with a pool size of 100, transitioning from the Benchmark model to Model (b) would on average help 20 additional blood group O patients receive transplants, which is 42.5% of all blood group O patients. Note that Model (b) helps a larger number of blood group O patients receive transplants than the other two Models for every pool size.
If a planner in a "small" program (e.g. the programs in Austria, Belgium, France, Poland, Portugal, and Sweden) is concerned about the outcome for blood group O patients in Models (a) and (b), the edge weights in the maximum weight matching problem (previously defined in Section 4.2) can be adjusted slightly, by adding a "sufficiently small" constant for blood group O patients to increase their likelihood of receiving transplants.

On manipulability
An important problem in all market design applications is whether or not agents can manipulate the outcome of the matching mechanism by misrepresenting their preferences over donors. When only pairwise exchanges are allowed and patient preferences are dichotomous, as in, e.g., the Benchmark Model, it is well-known that it always is in the best interest of the patients to truthfully report their preferences (Roth et al., 2005b). However, when expanding the preference domain from the dichotomous to the trichotomous domain, as in, e.g., Model (a) and Model (b), the positive findings relating to non-manipulability no longer hold. Specifically, if the planner insists on always selecting maximum matchings that minimize the use of immunosuppressants (such as half-compatibility priority matchings), it may be possible for patients to benefit by misrepresenting their preferences as illustrated in the following example.

REVIEW OF ECONOMIC STUDIES
Example 3. Suppose that N ={1,2,3,4} and that each patient is incompatible with her own donor. A possible pairwise exchange between the pairs (i,d i ) and (j,d j ) is denoted by ij and it is assumed that only the pairwise exchanges 12, 14, and 23 are feasible. Assume that Patient 1 is compatible with donor d 2 but only half-compatible with donor d 4 , i.e., that Patient 1 strictly prefers a pairwise exchange with the pair (2,d 2 ) over a pairwise exchange with the pair (4,d 4 ). Next, suppose that Patient 2 and Patient 3 are half-compatible with donor d 3 and donor d 2 , respectively. In this case, the unique maximum matching that minimizes the use of immunosuppressants is described by the pairwise exchanges 14 and 23. However, Patient 1 can manipulate the outcome by declaring herself incompatible with donor d 4 . In this case, the unique maximum matching that minimizes the use of immunosuppressants is described by the pairwise exchange 12.
The findings in Example 3 should come as no surprise as it is well-known that non-manipulability is incompatible with individual rationality and Pareto efficiency (or maximality) on preference domains more general than the dichotomous (Sönmez, 1999). 28 Note also that no priority function is needed to obtain the negative conclusion in Example 3., i.e., it is maximality in combination with minimal use of immunosuppressants that drives the result. Moreover, while Example 3 demonstrates that Model (b) can be manipulated, it is easy to construct an example showing that Model (a) can be manipulated as well. In fact, as will be demonstrated below, it is "easier" for patients to manipulate Model (a) than Model (b). Before evaluating which patients that can manipulate Models (a) and (b), we state a few general remarks regarding the above type of manipulation in kidney exchange.
It is by no means obvious that patients are allowed to or even should be allowed to declare themselves incompatible with specific donors. Such decisions are more likely to be made by their immunologists and medical doctors based on observable and verifiable medical data. This conclusion was recently stated by two of the researches that have pioneered kidney exchange research: "…manipulations of this sort [preference manipulation] do not play a significant role, since compatibility information is usually obtained from observable and verifiable medical data." (Sönmez and Ünver, 2014, p. 114) If preferences are solely based on medical information, the manipulation strategy adopted by patient 1 in Example 3 will no longer work. 29 However, even though the use of immunosuppressants can increase the number of transplants, it may also introduce new opportunities for manipulation. Whether or not a patient finds kidney transplants over the blood group barrier acceptable is the patient's private information (recall the discussion relating to Tables 2 and 3). Then, if patients are allowed to object to the use of immunosuppressive treatments, they may use this option to manipulate the matching mechanism. That is, even if a patient accepts half-compatible donors, she may claim that she does not as part of a strategy to be matched to a more preferred donor, exactly like Patient 1 did in Example 3. A similar discussion can, for example, be found in liver exchange frameworks (Ergin et al., 2018) where it is private information for living donors whether or not they are willing to donate the right lobe of their livers. This kind of strategy may then be adopted in an attempt to avoid donating the right lobe of the liver as 28. This conclusion has previously been reached in a kidney exchange framework by Nicoló and Rodríguez-Álvarez (2012). For similar results in other matching frameworks, see, e.g., Alcalde and Barberà (1994), Roth (1982), or Schummer (1999.
29. Unless patients are assisted by their immunologists and/or medical doctors in manipulating the matching mechanism. the mortality rate is 4-5 times higher for right lobe donors than for left lobe donors. As already argued in Section 2, no such medical risks are associated with kidney transplantation over the blood group barrier, but whether or not patients consider these types of transplants to be acceptable is nevertheless private information. Here, it should also be noted that this type of manipulation attempt is very risky. If it is unsuccessful, the patient will not receive a transplant at all. In this sense, an attempt to manipulate the outcome of the matching mechanism may ultimately come at the cost of the patient's own life.
Given the insight that patients may successfully manipulate matching mechanisms by declaring half-compatible donors unacceptable, it is next investigated under what circumstances patients can gain by such manipulation strategies. Since Roth et al. (2005b) and Sönmez and Ünver (2014) have already proved that the Benchmark Model and the Altruistic Model, respectively, cannot be manipulated, the analysis is restricted to Model (a) and Model (b). To make the analysis tractable, it will be based on two assumptions, namely that (i) all medical information is observable and can be verified, and that (ii) no patient i ∈ N can affect the priority function π by declaring certain donors (such as half-compatible donors) unacceptable. The first assumption can be justified by the arguments above. The second assumption is also standard in the literature, where priorities are assumed to be exogenously given (see e.g. Roth et al. (2005b) and Okumura (2014)). 30 The second assumption implies that patients cannot affect the edge weights in the maximum weight matching problem described in Section 4.2. Patients can, however, remove some edges from the graph by declaring half-compatible donors unacceptable. If some patient i declares half-compatible donors unacceptable, all edges between patient i and patients whose donors are half-compatible with i would be removed. Proposition 10 states that only patients matched to half-compatible donors have the potential to manipulate Model (a) and Model (b) in this way. Note that Proposition 10 provides a necessary, but not sufficient, condition for this type of manipulation. Any patients that could potentially manipulate the matching mechanism in this way are either (a) incompatible with their own donors and receive half-compatible donors through exchange or (b) matched to their own half-compatible donors. This means that a failed manipulation attempt would always result in the patient not receiving any transplant.
A simulation study is conducted to evaluate how "difficult" it is for patients matched to halfcompatible donors to gain by strategic misrepresentation of preferences, i.e., to gain by declaring half-compatible donors unacceptable. The simulation study is based on each of the 1,000×5 = 5,000 populations considered in Section 5.2. The following method is adopted for each population. First, all patients matched to half-compatible donors are identified. The proportion of these patients is stated in the column "Potential manipulation" in Table 8. For example, for Model (a) and population size 50, this group represents, on average, 27.1% of the patients (this can also be 30. If a patient can improve her priority by declaring certain donors unacceptable, it is easy to find problems in which the priority mechanism in Roth et al. (2005b) can be manipulated (see Appendix A.2 for a formal description of the priority mechanism). For instance, suppose that some unmatched patient i is compatible with some donor d j , but the patient matched to d j has higher priority than i when i reports truthfully. Clearly, if i could raise her priority above j's priority by declaring some donors (other than d j ) unacceptable, the priority mechanism would match i to donor d j , which she prefers to being unmatched. seen in Table 1 where 20.9+6.2 = 27.1 % of the 50 patients in the pool are matched to halfcompatible donors on average). Second, each of these patients will then, one by one, declare all half-compatible donors unacceptable. For each such unilateral declaration, the maximum weight matching problem is solved for the modified graph and the matching for the patient that declared all half-compatible donors unacceptable is compared to the original matching. 31 If the patient is matched to a compatible donor when declaring all half-compatible donors unacceptable, the manipulation is said to be successful. Note also that successful manipulation always comes at the cost of fewer transplants in total. The mean success rate among the patients that could potentially manipulate the matching mechanism is stated in the column "Success among potential" in Table 8. As can be seen in the table, patients in Model (a) are on average more successful than patients in Model (b) and the success rate increases monotonically with pool size. The success rates for the different pool sizes are between 24.8% and 56.0% in Model (a), and between 8.3% and 39.9% in Model (b). That it is more difficult to manipulate in Model (b) follows from the fact that patients with half-compatible donors are always given the opportunity to be matched to a fully compatible donor as they are always, by construction, included in the kidney exchange pool even when not declaring halfcompatible donors unacceptable. Patients with half-compatible donors in Model (a), however, are never given this opportunity as they are always, by construction, matched to their own halfcompatible donors. That it is easier to manipulate in larger pools follows from the fact that there are more transplantation opportunities (i.e. more edges in the graph) in larger pools and patients are therefore more likely to receive transplants even if they remove edges from the graph by declaring half-compatible donors unacceptable. The column "Success in pool" in Table 8 reports the mean percentage of all patients in the pool that can successfully manipulate the matching mechanism. Given the conclusions above, this number is also expected to be lower in Model (b) compared to Model (a) but increasing in pool size. This is confirmed in Table 8 where it is shown that the success rate among all patients in the pool is between 6.2% and 15.9% in Model (a), but only between 1.9% and 8.5% in Model (b).

CONCLUSIONS
This article has investigated pairwise kidney exchange programs using a medical technology for transplanting kidneys over the blood group barrier. In particular, the focus has been on the set of half-compatibility priority matchings and how the technology is best utilized. If a planner is only interested in maximizing the number of transplants, minimizing the number of transplants over the blood group barrier and, in addition, designing a program that is less biased against the biologically disadvantaged blood group O patients, the theoretical results and the findings in the simulation study suggest the following policy recommendations.
First, if the technology enabling transplantation over the blood group barrier not is allowed within the existing kidney exchange framework, like in, e.g., Belgium, France, India, Italy, the Netherlands, Poland, and Portugal, and there is a change in the legal framework or in the attitude towards immunosuppressants in the transplant community allowing for transplantation over the blood group barrier within the exchange program, then any existing kidney exchange program should be amended to make use of it. In the language of this article, a transition to Model (b) is recommended. This would generate a significantly larger number of transplants and, in addition, help the biologically disadvantaged blood group O patients.
Second, if the technology enabling transplantation over the blood group barrier is allowed but mainly used to obtain self-matches over the blood group barrier outside a kidney exchange program as in, e.g., Austria, the Czech Republic, Spain, Sweden, Switzerland, and the United Kingdom, a transition to a system where patients with half-compatible donors are first added to the exchange pool in search of a compatible donor is recommended. In the language of this article, a transition from Model (a) to Model (b) is recommended. This would generate more transplants in total and reduce the proportion of patients receiving transplants over the blood group barrier.
The conclusions above are only valid if the social planner can accept that some patients may be able to manipulate the outcome of the matching mechanism. More precisely, the introduction of the technology enabling transplants over the blood group barrier naturally extends the preference domain from the dichotomous to the trichotomous domain and, therefore, also opens up for manipulation possibilities. However, fewer patients can manipulate Model (b) than Model (a), so the former model performs better than the latter in this respect as well. Since less than 10 % of all patients can successfully manipulate Model (b) even in very large exchange pools (and they attempt to do so with their own lives at stake), the main take away message is that if the social planner can accept that a small fraction of all patients may be able manipulate the matching mechanism, then a transition to Model (b) is recommended independently of what the current exchange framework is.
It should be noted that Model (a) is an exact description of the exchange program currently used in Sweden. This program was initiated and designed by one of the authors of this article (Andersson) and will be expanded to also include Norway and Denmark in the spring of 2019 at the latest. 32 Patients with half-compatible donors are not routinely asked to join the Swedish exchange program, although there is a discussion within the transplant community about designing a protocol for investigating their interest in participating. The findings in this article strongly support any such initiative.
The theoretical results and the simulation results presented in this article are valid for kidney exchange programs where only pairwise exchanges are allowed. Considering the findings in this article, it is important that future research investigates how transplantation over the blood group barrier can be integrated into more sophisticated kidney exchange programs allowing for, e.g., non-simultaneous extended altruistic donor chains and larger cyclic exchanges. Such features will with certainty lead to even more transplants. In general, new medical technology and more potent immunosuppressants will most likely continue to affect kidney exchange programs in the future, making more research in this direction important. Apart from the results presented in this article, future research may also build on, e.g., Chun et al. (2015), Nicoló and Rodríguez-Álvarez (2017), and Sönmez et al. (2018).

A. PRIORITY MATCHINGS AND MATROIDS
This Appendix is divided in two parts. The first shows that pairwise kidney exchange problems have a matroid structure, even in settings with immunosuppressants. The second demonstrates that the set of priority matchings defined in this article is equivalent to the set of priority matchings defined by Roth et al. (2005b). The graph theoretical definition of matchings will be adopted throughout both Appendices A and B. That is, for any problem (N,C,π) with corresponding compatibility graph g, a matching M ⊆ E(g) is defined as a set of edges in g that are not incident to each other (see Section 4.2).

A.1. Matroids
Many of the results in Section 4.1 relate to the structure of pairwise kidney exchange problems in particular. One important aspect of pairwise kidney exchange problems is that the set of all patients N and a family I containing all sets of patients that can be matched simultaneously constitute a matroid (N,I). This was shown by Roth et al. (2005b) for settings with a binary compatibility structure (no transplantation over the blood group barrier). Proposition 11 below states that the matroid result in Roth et al. (2005b) continues to hold in settings that distinguish between compatibility and half-compatibility. This is not immediately obvious since self-matches alter the structure of the sets of simultaneously matchable patients.

Definition 1.
A pair (X,I) where X is a finite set (called the ground set) and I is a family of subsets of X (called the independent sets)isamatroid if it has the following two properties.
• If I ∈ I and J ⊂ I, then J ∈ I (the hereditary property).
• If I,J ∈ I, and |J| < |I|, then there exists some i ∈ I \J such that J ∪{i}∈I (the augmentation property).
The matroid structure ensures that every maximal matching is a maximum matching. Before stating the matroid result formally, note that if cyclic exchanges involving three or more patient-donor pairs are feasible, the matroid result no longer holds, giving rise to a trade-off between prioritizing patients and maximizing the number of transplants, see, e.g., Kratz (2019)orSönmez and Ünver (2014) for detailed discussions. Next, note that it must be the case that jk ∈ M for some k ∈ N * (M)\{j}. Otherwise, jj ∈ M and (M \{jj})∪{ij}∈M, which again is a contradiction.
Thus, both in case (a) and case (b), there exists exactly one "corresponding" patient in N

A.2. Priority matchings
This section finds an equivalence between the set of priority matchings defined in this article and the set of priority matchings defined by Roth et al. (2005b). To achieve this, Proposition 11 is used to derive a number of new lemmas. These lemmas will not only prove to be important in showing the equivalence mentioned above, they will also be useful in proving some of the results in Appendix B.
Let Ŵ : N →{1,...,n} be a permutation of N such that Ŵ(i) = j if i is the patient with the jth highest priority. That is, Ŵ −1 (1) is the top priority patient and Ŵ −1 (n) is the patient with lowest priority. Roth et al. (2005b) define priority matchings in terms of the following priority mechanism: The set E n is the set of priority matchings in Roth et al. (2005b). Note that E n is defined without reference to any preferences. To avoid confusion between the two definitions of priority matchings before they have been shown to be equivalent, E n will be used whenever discussing priority matchings as defined by Roth et al. (2005b). A first observation is that all priority matchings are maximal matchings by construction. One implication of Proposition 11 is then that all priority matchings are also maximum matchings.
Lemma 2. For a given problem (N,C,π), each priority matching is a maximum matching. Priority preference relations were defined informally in Section 3. Formally, a preference relation π is called a priority preference relation if it is complete, transitive, and satisfies the following conditions: The proof of the result that any priority matching is preferred to any other matching by any priority preference relation (Lemma 3) is included alongside the proof of the converse statement (Lemma 5) for completeness. Lemmas 3 and 5 imply that the set of priority matchings as defined in this article is identical to the set of priority matchings as defined by Roth et al. (2005b), i.e., that M * = E n .
Lemma 3. (Roth et al., 2005b). For any priority preference relation π and any M ∈ E n , M π M ′ for all M ′ ∈ M.
Proof. Consider some priority preference relation π and some M ∈ E n . To reach a contradiction, suppose that there exists some M ′ ∈ M such that M ′ ≻ π M. Note that matchings in E n are maximal by construction. Furthermore, since (N,I) is a matroid by Proposition 11, every maximal matching is a maximum matching by the augmentation property. 33. It should be noted that this requirement on priority preferences is only imposed implicitly in Roth et al. (2005b), but is nevertheless necessary for Lemma 3 to hold. such that N * (M ′ ) ⊂ N * (M 1 ) by the augmentation property. Then M 1 ≻ π M by transitivity, since M 1 ≻ π M ′ ≻ π M. Let M 1 denote some maximum matching such that M 1 ≻ π M. Note that N * (M)△N * (M 1 ) contains the same number of patients from N * (M) and N * (M 1 ) and that its cardinality is at least 2, since N * (M) = N * (M 1 )byM 1 ≻ π M.
Let j be the patient with the highest priority in N * (M)△N * (M 1 ). Such patient must exist since each patient have a unique priority and |N * (M)△N * (M 1 )| > 2. By definition of the set N * (M)△N * (M 1 ), it must be the case that j ∈ N * (M 1 ) or j ∈ N * (M). To reach the desired contradiction, it will be demonstrated that (a) j / ∈ N * (M 1 ) and (b) j / ∈ N * (M). Proof. Consider some priority preference relation π and some M 1 ∈ M such that M 1 π M ′ for all M ′ ∈ M. First note that E n =∅ as M =∅. Since π is a priority preference relation, Lemma 3 implies that M π M ′ for all M ∈ E n and all M ′ ∈ M. This implies that M 1 ∼ π M ′ for all M ′ ∈ E n . This, in turn, requires that |N * (M 1 )|=|N * (M ′ )| for all M ′ ∈ E n . To see why, first note that since priority matchings are maximal by construction and since (N,I) is a matroid by Proposition 11, every priority matching is a maximum matching by the augmentation property. Downloaded from https://academic.oup.com/restud/article-abstract/87/3/1091/5486585 by guest on 09 June 2020 M 1 or M, but not both. Again, note that the symmetric difference always contains the same number of patients from N * (M 1 ) and N * (M) and that its cardinality is at least 2, since N * (M 1 ) = N * (M) by assumption.
Let j be the patient in N * (M 1 )△N * (M) with highest priority. Without loss of generality, suppose that j ∈ N * (M 1 ). Let A = N * (M 1 )∩N * (M) be the (possibly empty) set of patients matched at both M 1 and M. Since A∪{j}⊂N * (M 1 ) and N * (M 1 ) ∈ I, A∪{j}∈I by the hereditary property. As |N * (M 1 )△N * (M)| > 2, it follows that |A∪{j}| < |N * (M)|. Hence, there exists some j ′ ∈ N * (M)\(A∪{j}) such that A∪{j,j ′ }∈I by the augmentation property. Patients can continue to be added like this until the union between A and the added patients has the same cardinality as N * (M). That is, there exists some A ′ ⊂ N * (M)\(A∪{j}) such that A∪A ′ ∪{j}∈I and |A∪A ′ ∪{j}| = |N * (M)|. Since A∪A ′ ∪{j}∈I, there exists some M 2 ∈ M such that A∪A ′ ∪{j}⊆N * (M 2 ). Furthermore, since |A∪A ′ ∪{j}| = |N * (M)| and M is a maximum matching by M ∈ E n , it follows that A∪A ′ ∪{j}=N * (M 2 ). Note that N

B. PROOFS OF THE THEORETICAL RESULTS
This Appendix contains the proofs of all theoretical results except Propositions 11 and 7. The Proof of Proposition 11 is found in Appendix A.1 and the Proof of Proposition 7 is found in Section 5.1. Many proofs make use of results from other lemmas and propositions. For this reason, the proofs are not necessarily presented in the same order as their corresponding results in the main text. To make it easier for the reader to find the proofs, this Appendix is divided into four parts that are named after the specific sections where the results are presented in the main text. As in Appendix A, the graph theoretical definition of matchings will be adopted in this Appendix as well (see Section 4.2).

B.1. Proofs of the results in Section 4.1
Proposition 2. For a given problem (N,C,π), every half-compatibility priority matching is a priority matching.
Proof. Consider some priority matching M ∈ M * . Then M π M ′ for all M ′ ∈ M and all priority preference relations π by Lemma 3. Note that if M ≻ π M ′ for some M ′ ∈ M and all priority preferences π , then M ≻ B M ′ for all halfcompatibility priority preferences B . To see this, first note that if M is preferred to M ′ by all priority preference relations, then this preference does not depend on the choice of priority preference relation. Thus, the preference is induced by the properties of priority preferences, i.e., the restrictions imposed on priority preference relations. Note that the same restrictions are imposed on both priority preference relations and half-compatibility priority preference relations when considering matchings M,M ′ ∈ M for which N * (M) = N * (M ′ ). Since M ∼ π M ′ , it follows that N * (M) = N * (M ′ ). The restrictions imposed on half-compatibility priority preference relations will therefore induce the same preferences over M and M ′ . Then, because M ≻ π M ′ for all M ∈ M * , all M ′ ∈ M\M * and all priority preference relations π , it follows that M ≻ B M ′ for all M ∈ M * , all M ′ ∈ M\M * and all half-compatibility priority preference relations B . That is, half-compatibility priority preference relations prefer all priority matchings to all non-priority matchings. Consequently, M B ⊆ M * , i.e., every half-compatibility priority matching is a priority matching.
Proposition 3. For a given problem (N,C,π), every half-compatibility priority matching is a Pareto efficient maximum matching. Proof. The proof follows directly from Lemma 4 and the fact that the set of priority matchings defined in this article is equivalent to the set of priority matchings defined by Roth et al. (2005b).

Proof. Consider any
Proposition 1. For a given problem (N,C,π), all half-compatibility priority preference relations induce the same set of half-compatibility priority matchings.

B.2. Proof of Proposition 5
The Proof of Proposition 5 is divided into two main parts. In the first part (Lemmas 7-10), a specific problem, denoted by (N,Ĉ,π M ), plays an important role. More specifically, a number of equivalences are derived between an arbitrary problem (N,C,π) and a specific problem (N,Ĉ,π M ), which has a simple corresponding compatibility graphĝ. For any problem (N,C,π), the corresponding compatibility graph g is a simple graph whenever it contains no loops, i.e., whenever N H =∅. While Lemma 1 is only applicable in problems with corresponding compatibility graphs that are simple graphs, this is not a sufficient requirement. It also requires that no transplantation over the blood group barrier be possible, i.e., that the compatibility structure is binary in the sense that any patient i and donor d j are either incompatible or compatible. The main idea in the second part of the proof (Lemma 11 to the end of Appendix B.2) is therefore to use the findings from the first part of the proof to demonstrate that an arbitrary problem (N,C,π) can be translated into an equivalent specific problem (N,Č,π) with a corresponding simple graph, whereČ is a compatibility structure at which no patients are half-compatible with any donors. Once this has been established, the proof of Proposition 5 follows from Lemma 1 and the findings in this Appendix. Note that this section (Appendix B.2) involves multiple compatibility structures (e.g. C,Ĉ,Č), each of which may induce a different set of matchings M. For this reason, the notation M(C) will be used throughout the section to denote the set of all matchings for a given compatibility structure C. The sets M * (C) and M B (C) are defined analogously.
Before introducing the problem (N,Ĉ,π M ), it is first proved that all patients in N H are matched at any maximal matching and, by extension, at any maximum weight matching or priority matching. Lemma 6. Consider a problem (N,C,π) with corresponding compatibility graph g, where N H =∅. Then each patient in N H is matched at each maximal matching.
Proof. By construction, ii ∈ E(g) for all i ∈ N H . To obtain a contradiction, suppose that there exists some patient i ∈ N H who is unmatched at some maximal matching M. Then M ′ := M ∪{ii}∈M(C) since i ∈ N H . This contradicts the assumption that M is a maximal matching. Hence, each patient in N H is matched at each maximal matching.
Consider a problem (N,C,π) with corresponding compatibility graph g and let M ∈ M(C). As mentioned above, the construction of the problem (N,Ĉ,π M ) is key in the first part of the proof of Proposition 5. For any compatibility structure C ′ , let N H (C ′ ) denote the set of patients that are half-compatible with their own donors at C ′ and define N I (C ′ ) analogously, i.e., N I (C ′ ) = N \N H (C ′ ). The priority function π M is defined by: where M is the set containing each patient in N H (C) not matched to any patient in N I (C) at matching M. Moreover, the compatibility structureĈ is defined by the following two conditions: • For any i,j ∈ N H (C), i and j are incompatible atĈ. This includes the cases when i = j.
• Consider two patients i,j ∈ N such that i / ∈ N H (C)orj / ∈ N H (C). Then i and j are compatible (half-compatible) at C if and only if they are compatible (half-compatible) at C.
Note that the first condition removes all the loops at patients in N H . Consequently, the resulting compatibility grapĥ g is a simple graph. Furthermore, for each matching in M(Ĉ), a patient in N H (C) is either unmatched or matched to a patient in N I (C). Let L(g) := {ij ∈ E(g) | i,j ∈ N H (C)} be the set of edges between patients in N H (C), including all loops. Then E(ĝ) and L(g) constitute a partitioning of E(g), i.e., E(g) = E(ĝ)∪L(g) and E(ĝ)∩L(g) =∅. A final observation is that if N H (C) =∅, then C =Ĉ and π = π M . In that case, the proofs of Lemmas 7-10, below, follow immediately. Hence, in the proofs of these four lemmas, it is assumed that N H (C) =∅. Let w(π) be defined by w ii (π) = π(i) for all i ∈ N and w ij (π ) = π(i)+π(j) for all i,j ∈ N such that i = j.
Lemma 7. Consider two problems (N,C,π) and (N,C,π M ) with corresponding compatibility graph g and let M ∈ M(C). Then M is a maximum weight matching in (g,w(π)) if and only if M is a maximum weight matching in (g,w(π M )).
Proof. Consider some patient i ∈ N H (C) such that M is a maximum weight matching in (g,w(π )). By definition, M is a maximum weight matching in (g,w(π )) whenever S(M,w(π ))−S(M ′ ,w(π )) ≥ 0 for all M ′ ∈ M(C). Since each maximum weight matching is a maximal matching, patient i is matched at M by Lemma 6. Consider an arbitrary matching M ′ where, without loss of generality, patient i is matched. Then the term π(i) is found in both the sum S(M,w(π)) and the sum S(M ′ ,w(π)). Thus, π(i) cancels out in the difference S(M,w(π ))−S(M ′ ,w(π )). Consequently, S(M,w(π))−S(M ′ ,w(π)) ≥ 0 for all M ′ ∈ M(C) and all values of π(i). That is, M is a maximum weight matching in (g,w(π )) for all values of π(i). Since this argument can be repeated for all i ∈ N H and since π (j) = π M (j) for all j ∈ N I ,it follows that M is a maximum weight matching in (g,w(π M )) as well. The same argument can be used in reverse to show that if M is a maximum weight matching in (g,w(π M )), then it is a maximum weight matching in (g,w(π)). Lemma 8. Consider two problems (N,C,π) and (N,C,π M ) with corresponding compatibility graph g and let M ∈ M(C). Then M is a priority matching at (N,C,π) if and only if it is a priority matching at (N,C,π M ).
Proof. Since almost identical arguments can be used in both directions of the proof, it is only shown that if M is a priority matching at (N,C,π), then it is a priority matching at (N,C,π M ). To reach a contradiction, suppose that M is a priority matching at (N,C,π) but not at (N,C,π M ). As M is a priority matching at (N,C,π), it is a maximum matching by Proposition 3. Furthermore, since the priority function does not impact whether a matching is a maximum matching, M is a maximum matching at (N,C,π M ) as well. Then there exists some pair of patients i,j ∈ N and some maximum matching M ′ such that N * (M)\N * (M ′ ) ={i}, N * (M ′ )\N * (M) ={j} and π M (j) >π M (i). First, suppose that i ∈ N H . This means that i can feasibly be self-matched. Hence, M ′ ∪{ii}∈M(C). N * (M) ⊂ N * (M ′ ∪{ii}) contradicts the observation that M is a maximum matching. Hence, i ∈ N I . Since π M (j) >π M (i), condition (B.2) in the definition of π M implies that j ∈ N I as well. As π M (i) = π (i) and π M (j) = π(j), this contradicts the assumption that M is a priority matching at (N,C,π). Consequently, if M is a priority matching at (N,C,π), then it is a priority matching at (N,C,π M ). Now recall that L(g) := {ij ∈ E(g) | i,j ∈ N H (C)} is defined to be the set of edges between patients in N H (C), including all loops.
Lemma 9. Consider two problems (N,C,π M ) and (N,Ĉ,π M ) with corresponding compatibility graphs g andĝ, respectively. Let M ∈ M(C) be a maximal matching. Then M is a maximum weight matching in (g,w(π M )) if and only if M ′ := M \L(g) is a maximum weight matching in (ĝ,w(π M )).
Proof. It will first be shown that M is a maximum weight matching in (g,w(π M )) only if M ′ := M \L(g) is a maximum weight matching in (ĝ,w(π M )). Suppose that M is a maximum weight matching in (g,w(π M )). Since M ⊆ E(g) and E(g)\L(g) = E(ĝ), by construction, it follows that M ′ ∈ M(Ĉ). As L(g) only contains edges between patients in N H (C), it is clear that N * (M)∩N I (C) = N * (M ′ )∩N I (C). Now, to reach a contradiction, suppose that M ′ is not a maximum weight matching in (ĝ,w(π M )). By definition of π M , it must be the case that π M (j) > i∈N H (C) π M (i) for any j ∈ N I (C). If M ′ is not a maximum weight matching at (N,Ĉ,π M ), then there must exist some maximum matchinĝ M ∈ M(Ĉ) such that N * (M)∩N I (C) = N * (M)∩N I (C) and some i ∈ (N * (M ′ )∩N H (C))\N * (M) such that π(i) < j∈(N * (M)∩N H (C))\N * (M ′ ) π (j). Next, note that i ∈ N H (C)\ M for all i ∈ (N * (M ′ )∩N H (C))\N * (M), since all patients in M are unmatched at M ′ by construction. Moreover, every patient in N H (C) matched to a patient in N I (C)atM is still matched to the same patient at M ′ and every patient in N H (C) that is matched at M ′ orM is matched to some patient in N I (C) sinceĝ contains no loops or edges between patients in N H (C). Thus, j ∈ M for all j ∈ (N * (M)∩N H (C))\N * (M ′ ). This contradicts condition (B.3). Hence, M is a maximum weight matching in (g,w(π M )) only if M ′ is a maximum weight matching in (ĝ,w(π M )).
It is next proved that M is a maximum weight matching in (g,w(π M )) if M ′ is a maximum weight matching in (ĝ,w(π M )). Suppose that M ′ is a maximum weight matching in (ĝ,w(π M )). Then condition (B.2) implies that there exists noM ∈ M(Ĉ) such that i∈N * (M)∩N I (C) π(i) > i∈N * (M ′ )∩N I (C) π(i). Recall that N * (M)∩N I (C) = N * (M ′ )∩N I (C). These two findings together with the observation that L(g) only contains loops and edges between patients in N H (C) imply that there exists noM ∈ M(C) such that i∈N * (M)∩N I (C) π (i) > i∈N * (M)∩N I (C) π(i). To reach a contradiction, suppose that M is not a maximum weight matching in (g,w(π M )). Then there exists someM ∈ M(C) such that S(M,w(π M )) > S(M,w(π M )). Considering that i∈N * (M)∩N I (C) π(i) ≤ i∈N * (M)∩N I (C) π(i), it must be the case that i∈N * (M)∩N H (C) π(i) > i∈N * (M)∩N H (C) π (i). However, this is a contradiction as all patients in N H (C) are matched at M by Lemma 6. Hence, M is a maximum weight matching in (g,w(π M )) if M ′ is a maximum weight matching in (ĝ,w(π M )).
Some definitions related to paths are helpful in some of the coming proofs.
Definition 2. An ordered list of (not necessarily unique) patients (i 1 ,...,i t )isapath of length t in a graph g if: A path is a maximal path if it is not contained in a longer path. Proof. It will first be shown that M is a priority matching at (N,C,π M ) only if M ′ is a priority matching at (N,Ĉ,π M ). Suppose that M is a priority matching at (N,C,π M ). Since M ⊆ E(g) and E(g)\L(g) = E(ĝ), by construction, it follows that M ′ ∈ M(Ĉ). To reach a contradiction, suppose that M ′ is not a priority matching at (N,Ĉ,π M ). Then there exists somê M ∈ M(Ĉ) such thatM ≻ π M M ′ . Furthermore, since M ′ is a maximal matching, there must exist some matchingM ∈ M(Ĉ) such that N * (M)\N * (M ′ ) ={i} and N * (M ′ )\N * (M) ={j} for some i,j ∈ N, where π M (i) >π M (j). By the definitions of N I (C) and N H (C) and by the existence of patient i, it must be the case that i ∈ N I (C)ori ∈ N H (C). Two different cases must be considered to reach the desired contradiction: (ii) Suppose that i ∈ N H (C). Then condition (B.2) implies that j ∈ N H (C) as well. Furthermore, i ∈ M and j ∈ N H (C)\ M by construction. This is a contradiction, since π M (i) <π M (j) by condition (B.3). Hence, i / ∈ N H (C).
In conclusion, i / ∈ N I (C) and i / ∈ N H (C), which contradict the existence of patient i. Thus, M is a priority matching at (N,C,π M ) only if M ′ is a priority matching at (N,Ĉ,π M ).
Finally, it will be shown that M is a priority matching at (N,C,π M ) if M ′ is a priority matching at (N,Ĉ,π M ). To reach a contradiction, suppose that M ′ is a priority matching at (N,Ĉ,π M ) and that M is not a priority matching at (N,C,π M ). Let i := i 1 and note that there must exist a maximal path (i 1 ,...,i t )i n( N,M ∪M ′ ), where i 1 = i t . Hence, t ≥ 2. To establish the contradiction, it will next be demonstrated that it cannot be the case that t ≥ 2.
34. In the case when ii ∈ E(g) for some i ∈ N, the list (i 1 ,i 2 ) is a path of length 2 where i 1 = i 2 , i.e., every path has a length of at least 2.
Downloaded from https://academic.oup.com/restud/article-abstract/87/3/1091/5486585 by guest on 09 June 2020 (i) Suppose that t = 2. Since i t is matched at M while i is not, this implies that i t i t ∈ M and consequently that i t ∈ M .
ThenM := M ′ ∪{i 1 i t }∈M(Ĉ). Since N * (M ′ ) ⊂ N * (M), this contradicts the assumption that M ′ is a priority matching at (N,Ĉ,π M ). (ii) Suppose that t = 3. Then either i t ∈ N I (C)\N * (M) and i t = j,ori t ∈ N H (C). Note thatM := (M ′ \{i 2 i t })∪{i 1 i 2 }∈ M(Ĉ). Furthermore, by π(i) >π(j) and condition (B.2), π (i 1 ) >π(i t ) both in the case when i t = j and the case when i t ∈ N H (C). This implies thatM ≻ π M M ′ , which contradicts the assumption that M ′ is a priority matching at (N,Ĉ,π M ). (iii) Suppose that t ≥ 4. Recall that there are no edges between patients in N H (C) or loops in M ′ orM. Two different cases arise: Hence, M is a priority matching at (N,C,π M )ifM ′ is a priority matching at (N,Ĉ,π M ).
The next result (Lemma 11) finds an equivalence between priority matchings and maximum weight matchings for any given problem (N,C,π). The problem (N,Ĉ,π M ) has a corresponding simple compatibility graphĝ, since all the loops have been removed. However, while no patients are half-compatible with their own donors atĈ, they may still be halfcompatible with other donors. Therefore, the key idea in the proof of Proposition 5 will be to demonstrate that for any problem (N,C,π), there exists an equivalent problem (N,Č,π) with a corresponding simple graphǧ, in which no patients are half-compatible with any donors. That is,Č is binary, in the sense that any pair of patients and donors is either compatible or incompatible atČ. Gather all compatibility structures at which no patients are half-compatible with any donors in the set C 02 . Note that the compatibility structures considered by Okumura (2014) belong to C 02 . Lemma 1 is only applicable in problems with compatibility structures belonging to C 02 . To understand Lemma 11, note that the problem (N,C,π) need not correspond to the conditions in Lemma 1 by Okumura (2014). Any problem (N,C,π)i n which N H =∅ has a corresponding simple compatibility graph. However, N H =∅ does not guarantee the requirement in Lemma 1 that no patients and donors are half-compatible to be satisfied. Lemma 11 is therefore not implied by Lemma 1, since there are compatibility structures not belonging to C 02 with corresponding compatibility graphs that are simple graphs.
Lemma 11. For any problem (N,C,π) with corresponding simple compatibility graph g, a matching M is a priority matching at (N,C,π) if and only if M is a maximum weight matching in (g,w(π)).
Proof. It will first be demonstrated that for any problem (N,C,π) with corresponding simple compatibility graph g, there exists a compatibility structureČ such that no patients and donors are half-compatible, a matching M is a priority matching at (N,C,π) if and only if it is a priority matching at (N,Č,π) and M * (C) = M * (Č). The corresponding compatibility graph ofČ is denoted byǧ. Let C be a compatibility structure for which each patient is incompatible with her own donor and compatible with every other donor. Then C ∈ C 02 and the corresponding compatibility graph, g, is a complete graph. Consider some arbitrary set of edges E ′ ⊆ E(g). LetĈ ′ be a compatibility structure for which each patient i ∈ N is compatible with some donor d j if and only if id j ∈ E(g)\E ′ and letĝ ′ be its corresponding compatibility graph. ThenĈ ′ ∈ C 02 , E(ĝ ′ ) ⊂ E(g) and E(g)\E(ĝ ′ ) = E ′ . Since C ∈ C 02 , g is a complete graph and E ′ is an arbitrary set of edges, there exists some C ′ ∈ C 02 with corresponding compatibility graph g ′ = (N,E) for any E ⊆ E(g). Hence, for any compatibility structure C / ∈ C 02 with corresponding simple compatibility graph g, there exists someČ ∈ C 02 with corresponding simple compatibility grapȟ g such thatǧ = g.Ifǧ = g, then M * (C) = M * (Č) since the set of priority matchings for a given compatibility structure only depends the priority function, which is fixed, and on information contained in its corresponding compatibility graph. Hence, for any problem (N,C,π) with corresponding simple compatibility graph g, there exists someČ ∈ C 02 such that a matching M is a priority matching at (N,C,π) if and only if it is a priority matching at (N,Č,π).
To conclude the proof, note that sinceČ ∈ C 02 , Lemma 1 implies that M is a priority matching at (N,Č,π) if and only if it is a maximum weight matching in (ǧ,w). Sinceǧ = g, M is a maximum weight matching in (ǧ,w) if and only if it is a maximum weight matching in (g,w). Hence, for any problem (N,C,π) with corresponding simple compatibility graph g, M is a priority matching if and only if M is a maximum weight matching in (g,w).
Proposition 5. Consider a problem (N,C,π) with corresponding compatibility graph g.Ifw ii = π(i) for all ii ∈ E(g) and w ij = π (i)+π (j) for all ij ∈ E(g) whenever i = j, then M is a priority matching if and only if M is a maximum weight matching in (g,w).
Proof. Let (N,C,π) be a problem with corresponding, not necessarily simple, compatibility graph g and suppose that M is a maximum weight matching in (g,w). Since M is a maximal matching in g, Lemmas 7 and 9 imply that there exists a compatibility matrixĈ with corresponding simple compatibility graphĝ such that M is a maximum weight matching in (g,w) if and only if M ′ := M \L(g) is a maximum weight matching in (ĝ,w(π M )). Sinceĝ is a simple compatibility graph, it follows from Lemma 11 that M ′ := M \L(g) is a priority matching at (N,Ĉ,π M ) if and only if it is a maximum weight matching in (ĝ,w(π M )). This implies that M is a maximum weight matching in (g,w) if and only if M ′ := M \L(g)isa priority matching at (N,Ĉ,π M ). This conclusion together with Lemmas 8 and 10 implies that for any problem (N,C,π), M is a maximum weight matching in (g,w) if and only if M is a priority matching.

B.3. Proof of Proposition 6
Proposition 6. Consider a problem (N,C,π) with corresponding compatibility graph g. Then a matching is a halfcompatibility priority matching if and only if it is a maximum weight matching in (g,w ε ).
Proof. (⇒) It is first proved that any maximum weight matching in (g,w ε ) is a half-compatibility priority matching, i.e., if a matching M is a maximum weight matching in (g,w ε ), then M ∈ M B . Let M be a maximum weight matching in (g,w ε ). The first step in this part of the proof is to show that M is also a maximum weight matching in (g,w), where w is defined as in Proposition 5. To obtain a contradiction, suppose that M is not a maximum weight matching in (g,w). This means that there is some other matching M ′ ∈ M such that S(M ′ ,w) > S(M,w). By the construction of w, it is clear that: Inequalities (B.6) and (B.7) then imply that S(M ′ ,w ε ) > S(M,w ε ). But this contradicts the assumption that M is a maximum weight matching in (g,w ε ), i.e., that S(M,w ε ) ≥ S(M ′ ,w ε ) for all M ′ ∈ M. Hence, M is a maximum weight matching in (g,w) and, consequently, a priority matching by Proposition 5.
To complete this part of the proof, it will next be shown that M ∈ M B . To obtain a contradiction, suppose that M / ∈ M B . Consider a matching M ′ that belongs to M B .A sM B ⊆ M * , M ′ is a maximum weight matching in (g,w) by Proposition 5. However, as demonstrated above, M is also a maximum weight matching in (g,w). Hence, S(M,w) = S(M ′ ,w). Furthermore, note that: (⇐) It will now be proved that any M ∈ M B is a maximum weight matching in (g,w ε ). To obtain a contradiction, consider a matching M ∈ M B and assume that M is not a maximum weight matching in (g,w ε ). Then there exists some other matching M ′ ∈ M such that: S Proposition 9. Consider a problem (N,C,π) and suppose that µ ′ and µ ′′ contain all patients that receive transplants in Model (a) and Model (b), respectively. Then µ ′ ⊆ µ ′′ .
Proof. Consider a problem (N,C,π) with corresponding compatibility graph g and letg be the compatibility graph corresponding to the reduced problem (N I ,C I ,π). That is, letg = (N I ,E(g)) where ij ∈ E(g) for all i,j ∈ N I such that ij ∈ E(g). Suppose that M is a maximum weight matching in (g,w ε ) andM is a maximum weight matching in (g,w ε ). By Proposition 6, M corresponds to a matching selected in Model (b) andM corresponds to a matching selected in Model (a). Moreover, µ ′′ = N * (M) and µ ′ = N * (M)∪N H . Note that while Model (a) and Model (b) only select a single matching each from a set of half-compatibility priority matchings in their corresponding problems, the selection is inconsequential since any half-compatibility priority matching is a maximum weight matching in the corresponding weighted compatibility graphs.
To prove the result, it will be demonstrated that any patient contained in a maximal path (see Definition 2)i nt h e graph (N,M ∪M) belongs to N * (M). 35 The result mentioned above is first proved for maximal paths of length t = 2, i.e., paths of the type (i 1 ,i 2 )in(N,M ∪M).

35.
Recall that a matching is defined as a set of non-incident edges in a compatibility graph g (org). M ∪M ⊆ E(g) is therefore a subset of the edges in the compatibility graph g, constituting the edge set in the graph (N, M ∪M). Maximal paths in (N,M ∪M) are of interest since any patient corresponding to an interior element (non-end point) in such a maximal path will be matched at both M andM. Thus, attention can be restricted to the end points of such maximal paths.
Proposition 10. Consider a problem (N,C,π) and a matching M selected in Model (b) (Model (a)). Suppose that a patient i ∈ N is either unmatched or matched to a compatible donor at M. Then, in Model (b) (Model (a)), patient i cannot benefit by declaring half-compatible donors unacceptable.
Proof. Consider a problem (N,C,π) with corresponding weighted compatibility graph (g,w ε ). First note that no patient can affect the weights by declaring half-compatible donors unacceptable by Assumption (ii). This means that a patient i may only influence the outcome of the maximum weight matching problem by removing edges incident to i in g. Let M be a maximum weight matching in (g,w ε ) and consider a patient i ∈ N. Furthermore, letĈ be the resulting compatibility structure when i removes some edges incident to i and let (ĝ,ŵ ε ) be the corresponding weighted compatibility graph. Note that, givenŵ ε , all edges inĝ are assigned the same weights as in (g,w ε ).
First, if i is unmatched at M, then there are no edges incident to i in M. Thus, M is still a feasible matching atĈ. Sinceĝ is a subgraph of g (i.e. it contains the same vertices and a subset of the edges) and bothŵ ε and w ε assign the same weights to the edges inĝ,ifM is a maximum weight matching in in (g,w ε ), it must also be a maximum weight matching in (ĝ,ŵ ε ). Thus, by Propositions 4 and 6, exactly the same patients are matched at both (N,C,π) and (N,Ĉ,π) in Model (b) and i remains unmatched. Since (N,C,π) is an arbitrary problem, the same arguments apply to the reduced problem considered in Model (a) as well.
Finally, suppose that i is matched to a compatible donor d j at M. Then there exists no donor in the exchange pool that i strictly prefers to d j . Consequently, i can only be made weakly worse off by declaring half-compatible donors unacceptable.