Pairwise Kidney Exchange ∗

In connection with an earlier paper on the exchange of live donor kidneys (Roth, Sönmez, and Ünver 2004) the authors entered into discussions with New England transplant surgeons and their colleagues in the transplant community, aimed at implementing a Kidney Exchange program. In the course of those discussions it became clear that a likely Þrst step will be to implement pairwise exchanges, between just two patient-donor pairs, as these are logistically simpler than exchanges involving more than two pairs. Furthermore, the experience of these surgeons suggests to them that patient and surgeon preferences over kidneys should be 0-1, i.e. that patients and surgeons should be indifferent among kidneys from healthy donors whose kidneys are compatible with the patient. This is because, in the United States, transplants of compatible live kidneys have about equal graft survival probabilities, regardless of the closeness of tissue types between patient and donor (unless there is a rare perfect match). In the present paper we show that, although the pairwise constraint eliminates some potential exchanges, there is a wide class of constrained-efficient mechanisms that are strategy-proof when patientdonor pairs and surgeons have 0-1 preferences. This class of mechanisms includes deterministic mechanisms that would accomodate the kinds of priority setting that organ banks currently use for the allocation of cadaver organs, as well as stochastic mechanisms that allow considerations of distributive justice to be addressed. ∗We would like to thank Anna Bogomolnaia, Frank Delmonico, Hervé Moulin, Muriel Niederle, Susan Saidman, and Jen Stack for helpful comments and conversations. Sönmez gratefully acknowledges the research support of KoçBank via the KoçBank scholar program and Turkish Academy of Sciences in the framework of the Young Scientist Award Program via grant TS/TÜBA-GEBú IP/2002-1-19, and Roth and Ünver acknowledge support from the NSF. Any errors are our own responsibility. E-mail: aroth@hbs.edu; address: Harvard University, Department of Economics, Cambridge, MA 02138 and Harvard Business School, Boston, MA 02163. E-mail: tsonmez@ku.edu.tr; address: Koç University, Department of Economics, College of Administrative Sciences and Economics, Rumeli Feneri Yolu, Sarõyer, ú Istanbul, 34450, Turkey. §E-mail: uunver@ku.edu.tr; address: Koç University, Department of Economics, College of Administrative Sciences and Economics, Rumeli Feneri Yolu, Sarõyer, ú Istanbul, 34450, Turkey.


Introduction
As of this writing, there are 58,480 kidney patients in the United States who are registered on a waiting list for a transplant of a cadaver kidney. The median waiting time is 1,144 days (for registrations in 1999), and in 2003, 4,233 patients died while on the waiting list, or were removed from it after having become too ill for a transplant. In 2003 there were 8,665 transplants of cadaver kidneys. 1 Because healthy people have two kidneys (and can remain healthy on only one), it is also possible for a kidney patient to receive a live-donor transplant. There were 6,464 live-donor transplants in 2003. However, a willing, healthy donor is not always able to donate to his intended patient, because of blood type or immunological incompatibilities between them. In this case, most often, the donor is sent home, and becomes once again invisible to the health care system. However, in a few cases, an exchange has been arranged between one incompatible patient-donor pair and another. In such an exchange, the donor from each pair gives a kidney to the patient from the other pair. Since 2001, there have been 6 such paired exchanges in the fourteen transplant centers that make up the New England region (Allen, 2004), and, in the United States, there have even been two exchanges among three incompatible patient-donor pairs. 2 These exchanges do not violate the 1984 National Organ Transplant Act (NOTA), which prohibits the sale or purchase of human organs. 3 While there is a national database of tissue types of kidney patients, used for allocating cadaver kidneys, there is no national (and few regional) databases of incompatible patient-donor pairs, despite earlier proposals to set up such databases (Rapaport, 1986, Ross et al 1997. 4 Nor is there a systematic method used for arranging exchanges between incompatible pairs. Roth, Sönmez, and Ünver (2004) showed how to identify efficient exchanges in a way that gave patients and their surgeons dominant strategy incentives to straightforwardly reveal their preferences (i.e. in a strategy-proof way). And, using tissue typing statistics from the Caucasian patient population, Roth, Sönmez and Ünver (2004) showed that the beneÞts of such an exchange could be very substantial, increasing live organ donations between unrelated donors from about 54% to as much as 91% if multiple-pair exchanges are feasible, and to as much as 75% even if only pairwise exchanges 1 United Network for Organ Sharing (UNOS) -The Organ Procurement and Transplantation Network (OPTN) national data, retrieved on 6/12/2004 from http://www.optn.org/data. 2 As of this writing, both of these have been arranged at the Johns Hopkins Comprehensive Transplant Center in Baltimore. Lucan et al (2003) also reports on three pair and four pair exchanges conducted in Romania. 3 See the legal opinion to this effect obtained by the UNOS at http://asts.org/ezefiles/UNOSSection_301_NOTA_.pdf. 4 Some hospitals have started to generate their own databases of incompatible patient-donor pairs, that could be used for exchanges. We are aware of such databases in Alabama, Baltimore (Johns Hopkins) and Ohio (Medical College of Ohio). are feasible. 5 In subsequent discussions aimed at organizing such exchange in the New England region of the transplant system, it became clear that a likely Þrst step will be to implement pairwise exchanges, between just two patient-donor pairs, as these are logistically simpler than exchanges involving more than two pairs. That is because, all transplantations in an exchange need to be carried out simultaneously, for incentive reasons, since otherwise a donor may withdraw her consent after her intended recipient receives a transplanted kidney (and so even a pairwise exchange involves four simultaneous surgical teams, operating rooms, etc.). Furthermore, the experience of American surgeons suggests to them that patient and surgeon preferences over kidneys should be 0-1, i.e. that patients and surgeons should be indifferent among kidneys from healthy donors that are blood type and immunologically compatible with the patient. 6 This is because, in the United States, transplants of compatible live kidneys have about equal graft survival probabilities, regardless of the closeness of tissue types between patient and donor (Gjertson and Cecka (2000) and Delmonico (2004)). 7 The present paper explores how to organize such exchanges. While the constraint that only pairwise exchanges be conducted means that the number of live donor transplants that can be arranged by exchange is smaller than if larger exchanges are feasible, it is still substantial. And, in the constrained problem, efficient and strategy-proof mechanisms will be shown to exist. This class of mechanisms includes deterministic mechanisms that would accommodate the kinds of priority setting that organ banks currently use for the allocation of cadaver organs, and which therefore may be especially appealing to transplant organizations. Also included are random matching mechanisms, such as the egalitarian mechanism, a stochastic mechanism that arises in connection with elementary notions of distributive justice. Interestingly, the constrained exchange problem is closely related to elegant results from graph theory, which will prove very useful, in ways recently pioneered by Bogomolnaia and Moulin (2004).

Related Literature
This paper intersects with the literatures on transplantation, graph theory, and mechanism design. The idea of paired kidney exchange between one incompatible patient-donor pair and another was Þrst proposed by Rapaport (1986) and then again by Ross et al (1997). UNOS initiated pilot testing of a paired kidney exchange program in 2000, and the same year the transplantation community issued a consensus statement indicating that the paired kidney exchange program is considered to be "ethically acceptable" (Abecassis et al 2000).
While the transplantation community approved the use of kidney exchanges to increase live kidney donation, it has provided little guidance about how to organize such exchanges. Roth, Sönmez and Ünver (2004) proposed an efficient and strategy-proof mechanism that uses both pairwise and larger exchanges. The present paper differs from Roth, Sönmez and Ünver (2004) in two major ways: 1. Here we only consider exchanges involving two patients and their donors.
2. In the current paper we adopt the assumption of many American transplant surgeons (including those we are working with in New England) that each patient is indifferent between all compatible kidneys (cf. Gjertson and Cecka (2000) and Delmonico (2004)).
These two assumptions considerably change the mathematical structure of the kidney exchange problem, and efficient exchange becomes an application of what is known in the graph theory literature as the cardinality matching problem (see for example Korte and Vygen (2002)). 8 For this purpose consider an undirected graph whose vertices each represent a particular patient and her incompatible donor(s), and whose edges connect those pairs of patients between whom an exchange is possible, i.e. pairs of patients such that each patient in the pair is compatible with a donor of the other patient. Finding an efficient matching then reduces to Þnding a maximum cardinality matching in this undirected graph (see Lemma 1), a problem well analyzed in this literature. More speciÞcally the Gallai (1963Gallai ( ,1964)-Edmonds (1965) Decomposition Lemma (henceforth GED Lemma) characterizes the set of maximum cardinality matchings. Technical aspects of our contribution heavily build on the GED Lemma.
We Þrst concentrate on deterministic outcomes, and show that there exists a wide class of efficient and strategy-proof mechanisms that accommodate the kinds of priority setting that organ banks currently use for the allocation of cadaver organs. We then allow stochastic outcomes as well, and show that there exists an efficient and strategy-proof mechanism, the egalitarian mechanism, which equalizes as much as possible the individual probabilities of receiving a transplant. If stochastic mechanisms can be accepted by the transplantation community, this mechanism can serve as a basis for discussion of how to address equity issues while achieving efficiency and strategy-proofness.
Our paper builds on the closely related recent paper by Bogomolnaia and Moulin (2004). They considered two-sided matching i.e. matching between two sets of agents that can be speciÞed exogenously (e.g. Þrms and workers), such that an agent on one side of the market can only be matched with an agent on the other side (cf. Roth and Sotomayor 1990), modeled as a bipartite graph, with 0-1 preferences. It was their paper that made us aware of some of the graph-theoretic results that we also use here. Our results on the egalitarian mechanism generalize their corresponding results to general, not necessarily bipartite graphs. Kidney exchange cannot be modelled as a two-sided market, since any patient with incompatible donors can potentially be matched with any other. 9 The extensions to the general case are of interest not only because of the importance of the application, but also because of the technical challenges that the generalization to arbitrary graphs presents.

Pairwise Kidney Exchange
Let N = {1, 2, ..., n} be a set of patients each of whom has one or more incompatible donors. Each patient is indifferent between all compatible donors and between all incompatible donors, except she strictly prefers her donor(s) to any other incompatible donor, and any compatible donor to her own donor(s). Since we are considering only pairwise exchanges in this paper, the above assumptions induce the following preference relation % i for patient i over the set of patients N: 1. For any patient j with a compatible donor for patient i we have j Â i i, 2. for any patient j without any compatible donor for patient i we have i Â i j, 3. for any patients j, h each of whom has a compatible donor for patient i we have j ∼ i h, 4. for any patients j, h neither of whom has a compatible donor for patient i we have j ∼ i h.
Here Â i denotes the strict preference relation and ∼ i denotes the indifference relation induced by % i . A (pairwise kidney exchange) problem is a pair (N, %) where %= (% i ) i∈N denotes the list of patient preferences. 9 Note that, since each donor comes to the exchange in the company of his incompatible patient, there isn't an option of modeling a two-sided market in which the sides are donors and patients. There is a very small population of undirected donors, who wish to donate but not to a speciÞc patient, and such a formulation might be applicable to them, but we do not consider such unattached donors here.
Throughout the paper with the exception of the proofs of the results on incentives, we Þx a problem (N, %).
We consider the case in which an exchange can involve only two pairs. Patients i, j ∈ N are mutually compatible if i Â j j and j Â i i. That is, two patients are mutually compatible if each one has a donor whose kidney is compatible for the other patient.
A matching µ : N → N is a function such that: Let M be the set of individually rational matchings for the problem (N, %). Throughout the paper we consider only individually rational matchings. That is, exchange is possible only between mutually compatible patients. A matching denotes an individually rational matching throughout the rest of the paper. For each matching µ ∈ M and patient i ∈ N, µ(i) = i means that the patient i remains unmatched. 10 For any matching µ ∈ M and pair of patients i, j ∈ N , µ(i) = j means that patient i receives a compatible kidney from a donor of patient j and patient j receives a compatible kidney from a donor of patient i. Since exchange is possible only among mutually compatible pairs, it is sufficient for our purposes to keep track of the symmetric |N| by |N| mutual compatibility matrix R = [r i,j ] i∈N,j∈N deÞned by for any pair of (not necessarily distinct) patients i, j ∈ N.
We will refer to the pair (N, R) as the reduced problem of (N, %). Occasionally it will be helpful to think of the reduced problem as a graph G = (N, R) whose vertices N are the patients (and their incompatible donors), and whose edges R are the connections between mutually compatible pairs of patients; i.e. there is an edge (i, j) ∈ R if and only if r i,j = 1. (It will be clear from the context whether R is the mutual compatibility matrix or the set of edges indicating mutual compatibility.) A matching then can be thought of as a subset of the set of edges such that each patient can appear in at most one of the edges. With this alternative representation 1. if (i, j) is an edge in the matching µ, patients i and j are matched by µ and, 2. if patient i does not appear in any edge in the matching µ, she remains unmatched.
A mechanism is a systematic procedure that selects a matching for each problem.

Efficient Exchange
A matching µ ∈ M is Pareto-efficient if there exists no other matching η ∈ M such that η(i) % i µ(i) for all i ∈ N and η(i) Â i µ(i) for some i ∈ N. In the present setting, µ is Pareto efficient if and only if the set M µ = {i ∈ N : µ(i) 6 = i} of patients matched by µ is maximal, i.e. if there does not exist any other matching η ∈ M such that M η ⊃ M µ . Let E be the set of Pareto-efficient matchings for the problem (N, %) .
A well known result from abstract algebra will help clarify the structure of the set of efficient matchings (see e.g. Lovasz and Plummer, 1986 on matchings, and Korte and Vygen, 2002 on matroids).
A matroid is a pair (X, I) such that X is a set and I is a collection of subsets of X (called the independent sets) such that

M1
if I is in I and J ⊂ I then J is in I; and

M2
if I and J are in I and |I| > |J| then there exists an i ∈ I\J such that I ∪ {i} is in I.
Proposition 1 Let I be the sets of simultaneously matchable patients, i.e. I = {I ⊆ N : ∃µ ∈ M such that I ⊆ M µ }. Then (N, I) is a matroid.
For any matching µ ∈ M, let |µ| = |M µ | = |{i ∈ N : µ(i) 6 = i}| denote the number of patients who are matched with another patient. The following well known property of matchings, which follows immediately from the second property of matroids, states that the same number of patients will receive a transplant at every Pareto-efficient matching.
If exchange is possible among more than two pairs, the conclusion of Lemma 1 no longer holds. Example 1 Let N = {1, 2, 3, 4} and suppose preferences are such that Consider the following two Pareto efficient trades: Two patients receive transplants if the Þrst trade is carried out whereas three patients receive transplants if the second trade is carried out. ¥

Priority Mechanisms
Because there is presently no organized exchange of live-donor kidneys, the experience of transplant centers is mostly with the priority allocation systems used to allocate cadaver organs. It is therefore natural to consider how priority mechanisms would function in the context of live kidney exchange. A priority ordering is a permutation of patients such that the k th patient in the permutation is the patient with the k th priority. Without loss of generality let the priority ordering of patients be the natural ordering (1, 2, ..., n), i.e. patient k is the k th priority patient for each k.
While we will concentrate on ordinal priorities here, priorities may depend on quantiÞable patient characteristics such as the patient's "percent reactive antibody" (PRA), which is correlated with how difficult it will be to Þnd a compatible kidney for that patient. (So it might be desirable, for example, for a high PRA patient to have a high priority for a compatible kidney in the relatively rare event that one becomes available.) In general, we will say that a non-negative function π : N → R + is a priority function if it is increasing in priority, i.e. if π(i) ≥ π(i + 1).
Consider a transplant center T whose decision makers wish to Þnd the set of exchanges that maximizes a preference Â T deÞned over matchings (more speciÞcally, over sets of matched patients). We will say that Â T is a priority preference if it is responsive to the priority ordering (Roth, 1985), i.e. if µ Â T ν whenever M µ ⊃ M ν , or when M µ and M ν differ in only one patient, i.e. M µ \M ν = {i}, M ν \M µ = {j}, for some i, j ∈ N, and i < j. That is, whenever M µ and M ν differ in only one patient, the matching with the higher priority patient is preferred, and adding additional matched patients to an existing matching always results in a preferred matching. (For a given priority ordering of patients, there remain many possible priority preferences Â T over sets of matched patients.) A priority mechanism produces a match as follows, for any problem (N, R) and priority ordering (1, 2, ..., n) among the patients: • Let E 0 = M (i.e. the set of all matchings).
• Let E 1 ⊆ E 0 be such that For a given problem (N, R) and priority ordering (1, 2, ..., n), we refer to each matching in E n as a priority matching, and a priority mechanism is a function which selects a priority matching for each problem. A priority matching matches as many patients as possible starting with the patient with the highest priority and following the priority ordering, never "sacriÞcing" a higher priority patient because of a lower priority patient.
By construction, a priority matching is maximal, and hence Pareto-efficient, i.e. E n ⊂ E. Proposition 1 implies, through the second property of matroids, that the "opportunity cost" of matching a higher priority patient will never be more than one lower priority patient who could otherwise have been matched. (For example, there might be two patients each of whom is mutually compatible only with the same third patient, and so matching the higher priority of the two patients will preclude matching the lower priority patient. But it cannot happen that, by matching a higher priority patient, two lower priority patients are excluded who otherwise could both have been matched.) And of course, by Lemma 1, the same total number of patients will be matched at each Pareto-efficient matching, so there is no trade-off between priority allocation and the number of transplants that can be arranged. In the matroid literature, a priority mechanism is called a greedy algorithm (since it "greedily" takes the highest priority remaining patient at each stage). The following proposition by Rado(1957) and Edmonds (1971) from the matroid literature will be helpful in understanding the resulting priority matchings.
Proposition 2 For a matroid (N, I) and any priority function π on N, a priority matching µ (obtained by a greedy algorithm with respect to π) identiÞes an element M µ of I that maximizes P i∈I π(i) over all I ∈ I.
Example 1 shows that if larger exchanges were permitted, the conclusions of Proposition 2 would not carry over (e.g. suppose that patient 4 has the highest priority, so the priority mechanism chooses the pairwise trade even if the 3-way trade has a higher sum of priorities).
For ordinal priorities, Proposition 2 allows us to quickly prove the following corollary, which helps explains the appeal that priority algorithms may have to transplant centers accustomed to prioritizing their patients.
Corollary 1 For any priority preference Â T a priority matching µ maximizes Â T on the set of all matchings, i.e. µ º T ν for all ν ∈ M.

Incentives in priority mechanisms
We turn next to consider the incentives facing patients (and the surgeons advocating for them) in a priority matching mechanism. Two apparently different issues arise that, upon examination, turn out to be closely related. The Þrst has to do with patients who have multiple incompatible donors willing to donate on their behalf. We show that a patient maximizes her chance of being included in an exchange by revealing all of her willing donors. The second issue involves revealing which compatible kidneys the patient is willing to accept. Again, we show that a patient maximizes her chance of being able to take part in an exchange by accepting her full set of compatible kidneys. That is, we show that with respect to both donors and kidneys, priority mechanisms do not give participants perverse incentives, but rather make it a dominant strategy to fully reveal which willing donors are available, and which kidneys are acceptable.
These two conclusions have the same cause. A patient enlarges the set of other patients with whom she is mutually compatible by coming to the exchange with more donors, and by being able to accept a kidney from more of those other patients' donors. And a patient's probability of being included in an exchange is monotonic in the set of other patients with whom she is compatible.
For a given set of patients and their available donors, the basic data for the problem (N, R) is determined by the tissue typing laboratory. So, once each patient has revealed a set of donors, the tissue typing lab establishes for each patient i a set of compatible kidneys K i = {j ∈ N : r ij = 1}. But a kidney exchange is a complicated event involving a patient and a donor on each side of the exchange, and so there will be no way to prevent a patient from declining a medically compatible kidney (e.g. for logistical reasons such as location of the other patient-donor pair, or, for that matter, for unspeciÞed reasons related e.g. to the preferences of the patient's donor). So the strategy set of each agent (i.e. each patient and donor, or each surgeon acting on behalf of a patient) is the set of all possible subsets of acceptable kidneys A i ⊆ K i that she might declare. (A kidney that is declared not acceptable can be thought of as being incompatible for reasons not initially revealed by the medical data.) A mechanism can only arrange exchanges between patient-donor pairs who are willing to accept each other's donor kidneys. A mechanism is strategy-proof in this dimension if a patient's probability of being included in an exchange with a compatible donor is maximized by declaring truthfully that A i = K i . 11 That is, a patient would have an incentive to declare a smaller acceptable set A i ⊂ K i only if this could cause the mechanism to include the patient in an exchange when truthful revelation of preferences (A i = K i ) would not. The Þrst part of Theorem 1 states that a patient can never beneÞt by declaring a compatible kidney to be unacceptable under a priority mechanism. 12 The second part states the similar result for revelation of available donors.
Theorem 1 A priority mechanism makes it a dominant strategy for a patient to reveal both a) her full set of acceptable kidneys; and b) her full set of available donors.
The proof of Theorem 1 a) is contained in the Appendix. The proof of part b) follows quickly from the proof of part a). In particular, a mechanism is donor-monotonic if a patient never suffers from the addition of one additional (incompatible) donor for her. The addition of one extra donor for a patient has the effect of enlarging her set of mutually compatible patients. But the proof of part a) of Theorem 1 shows that a patient never suffers from enlarging her set of mutually compatible patients in a priority mechanism, which therefore implies that priority mechanisms are donor monotonic in pairwise kidney exchange.

Corollary 2 A priority mechanism is donor-monotonic.
Looking ahead to implementation of actual kidney exchanges, it is worth emphasizing some limits on what these results establish. The motivations for donating a kidney to a loved one may be complex. When multiple potential donors prove to be incompatible, there may still be complex family preferences involved in going forward with an exchange, e.g. a family might prefer to have a sibling donate a kidney to its patient only after the possibility of a parent's donation had been exhausted, and such a family might prefer to initially look for an exchange by only revealing the availability of the parent as a donor. Theorem 1 b) does not rule out this possibility. Similarly, it might be that, even if a patient is willing to accept all compatible kidneys, her donor is for some reason unwilling to donate a kidney to some of the patients who are compatible with him (i.e. with his kidney), which would reduce the set of acceptable kidneys in a way not precluded by Theorem 1 a). 13 What Theorem 1 does establish is that a priority mechanism introduces no new reasons for available donors to be concealed, or compatible kidneys to be rejected.

Gallai-Edmonds Decomposition
The structure of Pareto-efficient pairwise matchings has been an active area of research in the combinatorial optimization literature. Understanding this structure will both enable us to gain added insight into priority mechanisms and allow us to apply a central concept in distributive justiceegalitarianism -to pairwise kidney exchange. The following partition of the set of patients is key to the structure of the set of Pareto-efficient matchings. Partition N as N U is the set of patients for each of whom there is at least one Pareto-efficient matching which leaves her unmatched. N O is the set of patients each of whom is not in N U (i.e., each of whom is matched with another patient at each Pareto-efficient matching) but is mutually compatible with at least one patient in N U . N P is the set of remaining patients (i.e., the set of patients who are matched with another patient at each Pareto-efficient matching and who are not mutually compatible with any patient in in N U ). Consider the reduced problem (N, R). For I ⊂ N, let R I = [r i,j ] i∈I,j∈I . We refer to the pair (I, R I ) as the reduced subproblem restricted to I.
We refer to a component (I, R I ) as an odd component if |I| is odd and as an even component if |I| is even. The following result due to Gallai (1963Gallai ( , 1964 and Edmonds (1965) is central to our paper: Lemma 2 (Gallai-Edmonds Decomposition Lemma) Let (I, R I ) be the reduced subproblem with I = N\N O and let µ be a Pareto-efficient matching for the original problem (N, R).

For any patient
2. For any even component (J, R J ) of (I, R I ) , J ⊆ N P and for any patient i ∈ J, µ(i) ∈ J\ {i}. Based on the Gallai-Edmonds Decomposition Lemma (GED Lemma), we refer to N U as the set of underdemanded patients, N O as the set of overdemanded patients and N P as the set of perfectly matched patients. 14 A Pareto-efficient matching matches each perfectly matched patient with another perfectly matched patient in the same even component; each overdemanded patient is matched with an underdemanded patient; and one patient in each odd component is either matched with an overdemanded patient or remains unmatched, whereas the remaining underdemanded patients in the same odd component are matched with one another. So each even component is self sufficient whereas the odd components compete for the overdemanded patients.
Let D = {D 1 , D 2 , ..., D p } be the partition of the set of underdemanded patients N U such that The following is an immediate implication of the GED Lemma: That is, the efficient matchings each leave unmatched |D| −¯N O¯p atients, each one in a distinct odd component. Note that Lemma 1 is an immediate corollary to the GED Lemma as well.

The Induced Two-Sided Matching Market
Loosely speaking there is competition at two levels: At the Þrst level, odd components (of underdemanded patients) compete for overdemanded patients. With the addition of an overdemanded patient to an odd component, all the patients in the augmented odd component can be matched to one another. The second level of competition is between the members of odd components that do not secure an overdemanded patient. If the odd component is of size one, its member will remain unmatched, but members of each larger odd component compete against each other not to be the one patient in the component who remains unmatched.
In this subsection we focus on the Þrst level of competition, between odd components for overdemanded patients. Since this competition does not involve perfectly matched patients (who are all matched to one another at every efficient outcome), we will not need to consider them for this discussion. Instead, we focus on an "induced" two-sided market, one side consisting of the overdemanded patients, and the other side consisting of the (odd) components of underdemanded patients. 15 For each odd component J ∈ D and overdemanded patient i ∈ N O , let we say there is a link between patient i and set J. Given the problem (N, %) , we refer to the triple ³ N O ,D,R´as the induced two-sided matching market.
A pre-matching is a functionμ : Let f M denote the set of pre-matchings.
LetẼ be the set of efficient pre-matchings. Note thatẼ is non-empty by the GED Lemma.

Priority Mechanisms Revisited
The GED Lemma allows us to see in detail how competition for compatible kidneys plays out in priority mechanisms. The outcome of a priority mechanism is Pareto-efficient and by the GED Lemma, each overdemanded as well as each perfectly matched patient is matched at each Paretoefficient matching. So the competition in a priority mechanism is among the underdemanded patients. Moreover, for any odd-component J and any patient j ∈ J, it is possible to match all patients in J\ {j} among themselves. So a priority mechanism determines which odd components will be fully matched (one member with an overdemanded patient and remaining patients with each other) and which odd components will have all but one of its patients matched (all with each other). This will depend on the relative priority ordering among the set of patients each of whom is the lowest priority patient at an odd component. That is, each overdemanded patient in C (J , I) is mutually compatible with at least one patient in S J∈J J. Which odd components will be fully matched and which ones will have all but its lowest priority member matched will be determined by the hierarchy among the odd components (where the priority of an odd component is determined by the priority of its lowest priority member) together with the following version of Hall's Theorem (Hall (1935)): Hall's Theorem: Let J ⊆ D. There exists a pre-matching which matches all odd components in J with a distinct overdemanded patient in N O if and only if Under the priority mechanism, odd components D 1 , D 2 , ..., D p are considered one at a time and the following iterative procedure will determine which odd components will have all its members matched and which ones will have all but its lowest priority member matched.
and in this case all members of D 1 will be matched.
then let J 1 = ∅ and in this case all members of D 1 except its lowest priority patient`1 will be matched.
In general, at Step k.
this case all members of D k will be matched.
all members of D k but its lowest priority patient`k will be matched.

Stochastic Exchange
So far our emphasis has been on deterministic exchange. One important tool to achieve equity in resource allocation problems with indivisibilities is using lotteries and allowing for stochastic outcomes. 16 A lottery λ = (λ µ ) µ∈M is a probability distribution over the set of matchings M. For each matching µ ∈ M, λ µ ∈ [0, 1] is the probability of matching µ in lottery λ, and P µ∈M λ µ = 1. Let L be the set of lotteries for the problem (N, %).
A stochastic mechanism is a systematic procedure that selects a lottery for each problem. Given a lottery λ ∈ L, the allocation matrix A (λ) = [a i,j (λ)] i∈N,j∈N summarizes the total probability that patient i will be matched with patient j for any pair of patients i, j ∈ N. Note that two distinct lotteries can induce the same allocation matrix. Let A be the set of all allocation matrices for the problem (N, %) . That is, Each lottery (and hence each allocation matrix) speciÞes the probability that each patient will receive a transplant. Given a lottery λ ∈ L, deÞne the utility of patient i to be the aggregate probability that she receives a transplant. Given λ ∈ L, the induced utility proÞle is a nonnegative real vector u (λ) = (u i (λ)) i∈N such that u i (λ) = P j∈N a i,j (λ) for any patient i ∈ N. Let U be the set of all feasible utility proÞles for the problem (N, %) . That is, U = {u (λ)} λ∈L .
A lottery is ex-post efficient if its support is a subset of the set of Pareto-efficient matchings. That is, lottery λ ∈ L is ex-post efficient if λ µ > 0 implies µ ∈ E.
An allocation matrix A ∈ A is ex-ante efficient if there exists no allocation matrix B ∈ A such that P j∈N b i,j ≥ P j∈N a i,j for all i ∈ N and An immediate implication of Lemma 1 (as well as the GED Lemma) is the "equivalence" of ex-ante and ex-post efficiency in the present context. This result is also stated by Bogomolnaia and Moulin (2004). Pre-matchings, pre-lotteries and pre-allocations will be very useful in our analysis.

The Egalitarian Mechanism
It is rare that a cardinal representation of preferences, i.e. a utility function, has a direct interpretation in a resource allocation problem. In the present context, however, a patient's utility corresponds to the probability that she receives a transplant, and thus equalizing utilities as much as possible becomes very plausible from an equity perspective. This approach is widely known as egalitarianism in distributive justice. 17 In this section we analyze the egalitarian mechanism. The GED Lemma will be key to the construction of egalitarian utilities.
Recall Step 1: In general, at 17 The two most related economic applications of egalitarianism to our setup are Bogomolnaia and Moulin (2004) and Dutta and Ray (1989). 18 For the sets of odd components J that we consider below, |J | > |C (J , I)|. 19 If there are multiple sets that minimizes f , their union minimizes f as well and we pick the largest such set as the arg min. See Lemma 4 in the Appendix for a proof of this result.
Step k: Let For each j ∈ N U , let k(j) be the step at which the odd component that contains patient j leaves the above procedure. That is, k(j) ∈ {1, 2, ..., q} is such that j ∈ J ∈ D k(j) .
We construct a vector u E = ¡ u E i ¢ i∈N as follows: • u E i = 1 for each i ∈ N \N U , and Theorem 2 The vector u E is a feasible utility proÞle.
Theorem 2 states that for each of the collections of under and overdemanded patients (D k , N O k ) in the above construction, the overdemanded patients can be probabilistically matched to all of the indicated odd components of underdemanded patients, in lotteries that divide the resulting probabilities equally among the underdemanded patients, and thus achieve the upper bound on the utility of the least fortunate patients.
For any utility proÞle u ∈ U, re-order individual utilities in an increasing order as ¡ u (t) ¢ t∈{1,2,...,n} A utility proÞle is Lorenz-dominant if and only if it Lorenz-dominates every other utility proÞle. If it exists, a Lorenz-dominant utility proÞle is efficient and as "evenly" distributes the probability of receiving a transplant among patients as possible constrained by the mutual compatibility constraints.
We refer to the utility proÞle u E as the egalitarian utility proÞle. We refer to any lottery that induces the egalitarian utility proÞle as an egalitarian lottery. Similarly we refer to any allocation matrix that induces the egalitarian utility proÞle as an egalitarian allocation matrix. We refer to a stochastic mechanism that selects an egalitarian lottery for each problem as an egalitarian mechanism.
Our next result states that, as for priority mechanisms, truthful revelation of private information is a dominant strategy under an egalitarian mechanism.
Theorem 4 An egalitarian mechanism makes it a dominant strategy for a patient to reveal both a) her full set of acceptable kidneys; and b) her full set of available donors.
As discussed when we proved the similar result for the priority mechanisms, the second part of the Theorem follows from the Þrst. Revelation of the full set of acceptable kidneys increases the set of patients with whom a given patient is mutually compatible, and, for the same reason, a patient never suffers because of an addition of one more incompatible donor. That is: Corollary 4 An egalitarian mechanism is donor-monotonic.

Concluding Remarks
One of the most challenging steps in implementing new market designs is addressing the constraints that arise in the Þeld. Because all surgeries in a kidney exchange must be done simultaneously, even the simplest exchange, between two patient-donor pairs, requires four simultaneous surgeries among the two donors and two recipients. It therefore seems overwhelmingly likely that, at least initially, some transplant centers are correct in anticipating that they will only be able to proceed with exchange among two pairs. Roth, Sönmez and Ünver (2004) showed that, even under this constraint, implementing kidney exchange could substantially increase the number of live organ kidney transplants. The present paper shows that when exchange is constrained in this way, and when the 0-1 nature of American surgeons' preferences regarding compatible/incompatible kidneys are taken into account, it is still possible to arrange exchange in an efficient and incentive compatible manner.
Another challenge in implementing new market designs arises in meeting the perceived needs and desires of the institutions, organizations, and individuals who must adopt and use the new design. Here too, the results are encouraging, and we show that the kinds of priority allocation that already govern the allocation of cadaver kidneys can be adapted to work effectively in organizing live donor kidney exchange. Since there are virtually no constraints on the kinds of priorities that can be used in an efficient and incentive compatible mechanism, we anticipate that priority mechanisms may appeal to different transplant centers that do not necessarily agree on how patients should be prioritized.
Indeed, there is lively discussion and disagreement about, and frequent revision of the priorities that different kinds of patients should have for cadaver organs, and we don't doubt that similar discussions about live organ exchange will take place as exchange becomes more common. The debate about cadaver organs frequently refers to considerations of distributive justice, and in this connection we have discussed stochastic mechanisms of exchange, and the egalitarian mechanism in particular. We do not anticipate or propose that this be taken as the last word on distributive justice in kidney exchange, but rather intend merely to show how the tools we assemble here can be used to address the kinds of distributional questions that arise. An interesting area of future research might be the extent to which egalitarian (and weighted-egalitarian) exchange mechanisms can be approximated by priority mechanisms with appropriately chosen priorities, taking into account different patients' likelihoods of Þnding compatible donors with whom to exchange, the size and frequency of exchanges, etc., in the dynamic environment in which new patients and donors arrive, and exchanges are carried out periodically.
More generally, as economists start to take a more active role in practical market design, we often Þnd we have to deal with constraints, demands, and situations different than those that arise in the simplest theoretical models of mechanism design. 20 In the present paper we address some of the issues that have arisen as we try to help surgeons implement an organized exchange of livedonor kidneys among incompatible donor-patient pairs. Not only do these issues appear to allow satisfactory practical solutions, they suggest new directions in which to pursue the underlying theory.

Appendix: Proofs
The proof of Proposition 1 is a standard exercise in combinatorial optimization theory, for example see Gommens (2004). Proposition 2 is due to Rado (1957) and Edmonds (1971). Lemma 1 and Lemma 3 are corollaries of the GED Lemma (Lemma 2) and see Lovász and Plummer (1986) for a proof of the GED Lemma.
Proof of Theorem 1: First, we introduce some notation. For each reduced problem R, consider the construction of a priority matching and sets of matchings E 0 (R), E 1 (R),...,E n (R) under the natural ordering; we deÞne sets of agents M 0 (R), M 1 (R), ..., M n (R) as Without loss of generality, we will prove the strategy-proofness of a priority mechanism which selects a priority matching under the natural ordering for each reduced problem. Let φ be a priority mechanism for the natural ordering and R = [r i,h ] i∈N,h∈N be a reduced problem. Construct sets of matchings E 0 (R), Any patient j ∈ M n (R) is matched at φ (R) ∈ E n (R), hence she cannot possibly beneÞt by underreporting the set of her compatible patients under φ. Let j ∈ N\M n (R) . Patient j is unmatched at φ (R). We will prove that patient j cannot receive a transplant by declaring a mutually compatible patient to be incompatible, and repeated application of this argument will conclude the proof.
We conclude the proof with a claim that implies M n (Q) = M n (R) and this together with j / ∈ M n (R) imply that j / ∈ M n (Q).
Claim: For each k ∈ {0, 1, ..., n}, That is because, in the LHS of the inequality not only the neighbors of G 3 = G 1 ∩ G 2 (i.e. members of C 3 ) are counted twice but also there may be additional common neighbors of G 1 and G 2 ; RHS of the inequality accounts for the double counting of members of C 3 but not the remaining common neighbors of G 1 and G 2 .
Since G 1 , G 2 each minimize the function f , and hence Adding the Þrst two lines and subtracting the third line ) and therefore f 1 n 4 ≥ n 4 − |G 4 | + C 4 or equivalently But since G 1 minimizes f , we shall have f 4 = f 1 and hence G 4 = G 1 ∪ G 2 minimizes f as well. ¥ We next present two lemmata that will be useful in our proofs for Theorem 2 and Theorem 3.
The following lemma is a part of Lemma 3.
ii. Consider the construction of the partition We prove Lemma 6 (ii) by iteration on steps k ∈ {1, 2, ..., q} of the construction.
• First consider k = 1. We have If D 1 = D, then q = 1 and we are done. Otherwise we proceed with k = 2.
• Next consider k = 2. Consider the reduced subproblem

Lemma 5 implies that the Gallai-Edmonds decomposition for the reduced subproblem
. Therefore, D\D 1 is the set of odd components of the reduced subproblem which is obtained by removing overdemanded patients from the reduced subproblem < 1 follows from the same sequence of arguments as in k = 1: If D 1 ∪ D 2 = D then q = 2 and we are done. Otherwise, we iteratively proceed in a similar way with k = 3, and so on. .
Note that f (D k , N O k ) < 1 by Lemma 6 (ii), and therefore f (D k , N O k ) ≥ α J . Also note that for each k ∈ {1, 2, ..., q} and J ∈ D k we have We will show that u E is a feasible utility proÞle in two major steps: In the Þrst step (Claim 1), we will show that it is possible to assign overdemanded patients N O to odd components D such that each odd component J ∈ D is assigned with an overdemanded patient with an aggregate probability of α J . In the second step (Claim 2) we will show that for each odd component J ∈ D k , it is possible to evenly divide the aggregate utility α J + (|J| − 1) among |J| members as f We will show this by deÞning an auxiliary task assignment problem and applying Hall's Theorem to the auxiliary task assignment problem. 21 Given N O k and D k , construct the task assignment problem (X, T , Γ) as follows: • For each overdemanded patient i ∈ N O k , introduce¯S J 0 ∈D k J 0¯i dentical agents. Let X i be the set of the identical agents associated with patient i, and X = S identical tasks. Let T J be the set of identical tasks associated with set J, and T = S J∈D k T J .
• Finally, introduce a matrix Γ = ¡ γ x,T ¢ x∈X,T ∈T such that γ x,T = 1 ifr i,J = 1 for x ∈ X i and T ∈ T J , and γ x,T = 0 otherwise. Intuitively agent x is Þt to perform task T if and only if patient i associated with agent x has a link with the odd component J that is associated with task T in the induced two-sided matching market Given N O k and D k we refer to (X, T , Γ) as the auxiliary task assignment problem. Note that |X| .
An auxiliary task assignment is a bijection ν : X −→ T . An auxiliary task assignment ν is feasible if and only if ν (x) = T implies that γ x,T = 1.
Here is the point of introducing the above auxiliary task assignment problem: Each agent or task in X∪T corresponds to a fraction Therefore if we show that there exists a feasible auxiliary task assignment ν, this would mean that it is possible to assign each agent in X to a distinct compatible task in T , and therefore the aggregate¯N O k¯.¯SJ 0 ∈D k J 0¯.
eight of the patients in N O k can be allocated among odd components of D k such that the share of the odd component J is We next prove that there exists a feasible auxiliary task assignment ν for the above task assignment problem. Given τ ⊆ T deÞne That is, C(τ , X) is the set of agents each of whom is Þt to perform at least one of the tasks in τ . By Hall's Theorem there exists a feasible auxiliary task assignment if and only if That is, no matter what subset of tasks is considered, the number of agents who are Þt for at least one of these tasks should be no less than the number of the tasks in this subset. We will prove this by contradiction. Suppose there exists a subset τ ⊆ T of tasks such that |τ | > |C(τ , X)|. Next construct the following set of tasks τ * ⊇ τ . For any task T ∈ τ , include all tasks which are identical to task T in set τ * . Note that since C(τ * , X) = C(τ , X), we have rearranging the terms, we have However N Ò´b y Lemma 6, and this together with Eq (6) imply that contradicting the deÞnition of D k and showing that for each τ ⊆ T we have |τ | ≤ |C(τ , X)|. Therefore there exists a feasible auxiliary task assignment ν by Hall's Theorem. We next construct matrixÃ k,k = [ã i,J ] i∈N O k ,J∈D k using the auxiliary task assignment ν. For each J ∈ D k and i ∈ N O k , deÞne By deÞnition |ν i,J | is the total number of tasks associated with odd component J each of which is assigned to an agent associated with the overdemanded patient i. For each J ∈ D k and i ∈ N O k , let Moreover for each overdemanded patient We conclude the proof of Claim 1 by constructing a pre-allocation matrixÃ ∈ A using the matrices For each pre-matchingμ ∈ f M in the support of pre-lotteryλ construct Q J∈D u (μ) |J| distinct matchings as follows: Pick one patient from each J ∈ D u (μ). Note that there are Q J∈D u (μ) |J| possible combinations. For each combination construct a Pareto-efficient matching µ such that: • each of the chosen patients is matched to herself, By the GED Lemma, there exists at least one such matching. Pick one and only one such matching for each of the Q J∈D u (μ) |J| possible combinations. Let M(μ) be the resulting set of matchings. Clearly |M(μ)| = Q J∈D u (μ) |J| . We are Þnally ready to construct a lottery λ E which induces the utility proÞle u E . The lottery λ E is constructed from the pre-lotteryλ by simply replacing each pre-matchingμ in the support of λ with the uniform lottery over M(μ). That is: Clearly, λ E is a lottery: Moreover by construction λ E is an ex-post efficient lottery. We conclude the proof of Claim 2 and Theorem 2 by showing that u(λ E ) = u E : Each patient in N\N U is matched with another patient in every efficient matching by the GED Lemma. Since λ E is ex-post efficient, for each patient i ∈ N\N U we have M be a pre-matching withλμ > 0.
1. If J ∈ D m (μ) then all patients in J are matched under every matching µ ∈ M (μ).

If
Therefore completing the proof of Claim 2 as well as the proof of Theorem 2. ¤¥ Proof of Theorem 3: Let J k be the set of patients in sets of D k , i.e. J k = S J∈D k J for any k ∈ {1, 2, ..., q}. Construct u E and Þx a feasible utility proÞle v ∈ U. We prove Theorem 3 through three claims.
and D k is the largest subset J ⊆ D / S k−1 =1 D`of satisfying this equality, we have Rearranging the terms in this inequality, we Þnd We conclude the proof of Claim 1 observing that by Lemma 6 (i). ¤ Claim 2: For any k ∈ {1, 2, ..., q}, we have

Proof of Claim 2:
i. For any k ∈ {1, 2, ..., q} , we have X i∈J k We prove part (i) by induction.
• For k = 1 : X i∈J 1 • For k > 1 : For the inductive step assume that for each k 0 < k we have concluding the proof Part (i).
ii. Fix k ∈ {1, 2, ..., q}. Consider the set of odd components S k =1 D`. By the GED Lemma, at most¯C( S k =1 D`, N O )¯of these odd components can have all its members matched under a Pareto-efficient matching. Therefore, the number of patients remaining unmatched in S k =1 Jù nder a Pareto-efficient matching is bounded below by¯S k =1 D`¯−¯C( is an upper bound for the number of patients in S k =1 J`matched under a Pareto-efficient matching and consequently the aggregate utility of patients in S k =1 J`under any ex-post efficient lottery is bounded above by This together with Part (i) of Claim 2 and the equivalence of ex-post and ex-ante efficiency by Lemma 3 imply for any feasible utility proÞle v completing the proof of Part (i). ¤ Claim 3: For any t ∈ {1, 2, ..., n} , we have Proof of Claim 3: We use indices k, k 0 , and`∈ {1, 2, ..., q} to denote indexes of sets in {D 1 , D 2 , ..., D q } , and indexes t, t 0 , t 00 , s ∈ {1, 2, ..., n} to denote the order statistics 22 of a feasible utility proÞle. Let n k = P k =1 |J`| for each k ∈ {1, 2, ..., q} and let n 0 = 0. Following equations will be useful in proving Claim 3: By the deÞnition of order statistics, we have That is because, since t 00 > t 0 , the average of (t + 1) th through (t 00 ) th lowest utilities cannot be less than the average of (t + 1) th through (t 0 ) th lowest utilities. By deÞnition of the order statistics, we also have For all k, k 0 ∈ {1, 2, ..., q} with k 0 < k, and for all i ∈ J k , j ∈ J k 0 we have u E i < u E j by Claim 1. Moreover for all k ∈ {1, 2, ..., q}, and for all i, j ∈ J k we have u E i = u E j by construction of u E . Hence, the t th order statistic of u E is given by 22 For a feasible utility proÞle v = ¡ v (t) ¢ t∈{1,2,...,q} , we refer to v (t) (i.e. the t th smallest utility under v) as the t th order statistic of v.
by |J k | = n k − n k−1 and reorganizing the terms by |J k | = n k − n k−1 and reorganizing the terms This concludes the proof of Claim 3 and Theorem 3. ¤¥ The next Lemma will be useful in proving Theorem 4.
Lemma 7 Let u, v ∈ U be such that u Lorenz-dominates v. Then for any α ∈ (0, 1), vector αu Proof of Lemma 7: Let u, v ∈ U be such that u Lorenz-dominates v, let α ∈ (0, 1) and w = αu + (1 − α) v. Since u ∈ U there is a lottery λ ∈ L that induces u, and since v ∈ U there is a lottery γ ∈ L that induces v. Let σ = αλ + (1 − α) γ. For each i ∈ N, we have This implies that σ induces w = (w i ) i∈N ∈ U.
Next, we show that w Lorenz-dominates v. By the deÞnition of order statistics, for any t ∈ {1, 2, ..., n} and any t member subset N 0 ⊆ N of patients we have ( 1 4 ) Moreover since u Lorenz-dominates v, P t s=1 v (s) ≤ P t s=1 u (s) for any t ∈ {1, 2, ..., n} and this inequality holds strictly for some t ∈ {1, 2, ..., n}. This together with the deÞnition of order statistics imply that for any t ∈ {1, 2, ..., n} and any t member subset N 00 ⊆ N of patients we have where the second inequality holds strictly for some t ∈ {1, 2, ..., n}. We have Therefore Eq 14 and Eq 15 imply that where the inequality holds strictly for some t ∈ {1, 2, ..., n}, completing the proof of Lemma 7. ¥ Proof of Theorem 4: First we introduce some additional notation that we will use in the proof of Theorem 4. Fix the set of agents N and hence each mutual compatibility matrix R deÞnes a distinct reduced problem. For any reduced problem R, let J and e (R) = max |µ| .
Recall that for any µ ∈ M(R), we have µ ∈ E(R) ⇐⇒ |µ| = e (R). For any reduced problem R and any two sets I, J ⊆ N, deÞne neighbors of J among I as For a singleton set J = {j}, we slightly abuse the notation and use C(j, I, R) instead of C({j}, I, R).

Proof of Claim 2:
(i) We prove Claim 2 (i) by contradiction. Suppose there exists a patient i ∈ N O (R) ∩ N U (Q).
Then there exists a Pareto-efficient matching µ ∈ E (Q) such that µ(i) = i. By Claim 1 (ii) E (Q) ⊆ E (R), and therefore µ ∈ E (R). This together with µ (i) = i imply that i ∈ N U (R) (ii) We prove Claim 2 (ii) in four steps: Step 1. We Þrst prove that N U (Q) ⊆ N U (R). Pick a patient i ∈ N U (Q). There exists a Paretoefficient matching µ ∈ E (Q) such that µ (i) = i. By Claim 1 (ii), E (Q) ⊆ E (R) and therefore µ ∈ E (R). This together with µ (i) = i imply that i ∈ N U (R).
Step 2. Let I ∈ D (R) \{J}. Clearly I ⊆ N U (R). We will show that I ∩ N O (Q) = ∅ by contradiction. Suppose there exists a patient i ∈ I ∩ N O (Q). Since i ∈ N O (Q), there exists a patient h ∈ N U (Q) ∩ C (i, N, Q). Since i 6 ∈ J, we have i 6 = j and since i 6 ∈ J and i 6 ∈ N O (R) we have i 6 = j 0 . Therefore C (i, N, Q) = C (i, N, R) and since I ∈ D (R), which in turn implies µ ∈ E (R). Since µ(h) = h and I ∈ D (R), we have µ (h 0 ) ∈ I\ {h, h 0 } for all h 0 ∈ I\ {h} by the GED Lemma. Construct the following Pareto-efficient matching ν ∈ E (R): Such a Pareto-efficient matching ν ∈ E (R) exists by the GED Lemma. Since µ ∈ E(Q), µ(j) 6 = j 0 and since j, j 0 ∈ N\I, ν (j) 6 = j 0 either. Therefore ν ∈ M (Q) and hence ν ∈ E (Q) by Claim 1(ii). This together with ν(i) = i imply i ∈ N U (Q) which in turn contradicts i ∈ N O (Q). Therefore I ∩ N O (Q) = ∅ as desired.
Step 3. Let I ∈ D (R) \{J}. We will show that Then there is a Pareto-efficient matching µ ∈ E(Q) such that µ (i) = i. By Claim 1 (ii) E (Q) ⊆ E (R), and therefore µ ∈ E (R). Since µ (i) = i and I ∈ D (R) , we have µ(h) ∈ I\ {i, h} for all h ∈ I\ {i} by the GED Lemma. Let h ∈ I. Construct the following Pareto-efficient matching ν ∈ E (R): Such a Pareto-efficient matching ν ∈ E (R) exists by the GED Lemma. Since µ ∈ E (Q), µ (j) 6 = j 0 and since j, j 0 ∈ N\I, ν (j) 6 = j 0 either. Therefore ν ∈ M (Q) and hence ν ∈ E (Q) by Claim 1 (ii). This together with ν(h) = h imply h ∈ N U (Q) and since h ∈ I was arbitrary, we have I ⊆ N U (Q) as desired.
Step 4. Let I ∈ D (R) \{J} such that I ⊆ N U (Q) . We will show that I ∈ D (Q). We need to show that there is no patient i ∈ C (I, N, Q) ∩ N U (Q). Suppose there is such a patient i. Since there is "utility transfer" from the higher utility patient j to the lower utility patient i, utility proÞle u (λ) Lorenz-dominates the egalitarian proÞle u E (Q) under Q contradicting Theorem 3. Therefore u E i (Q) ≥ u E j (Q) for all i ∈ J\ {j}. ¤ We are ready to complete the proof of Theorem 4. Suppose patient j strictly beneÞts from declaring patient j 0 incompatible. That is, suppose We will show that this will lead to a contradiction. Let J ∈ D k (R). Since φ E (Q) is an ex-post efficient lottery under Q, φ E (R) is an ex-post efficient lottery under R, and e (Q) = e (R) by Claim 1 (i), we have X i∈N u E i (Q) = e (Q) = e (R) = X i∈N u E i (R) .
Therefore, since u E j (Q) > u E j (R), there exists a patient h ∈ N such that u E h (Q) < u E h (R). This implies u E h (Q) < 1 which in turn implies h ∈ N U (Q). Since N U (Q) ⊆ N U (R) by Claim 2 (ii), h ∈ N U (R) as well. In a way of speaking, the utility of patient j increases under φ E (Q) with respect to φ E (R) at the expense of the utility of some other patients each of whom is underdemanded under R. That is, some utility is transferred from some underdemanded patients under R to patient j. We partition N U (R) as n S k =1 J`(R) , Consider patients in S q(R) =k+1 J`(R) . By the construction of u E (R), at any matching µ ∈ E (R) in the support of φ E (R), any patient in S q(R) =k+1 J`(R) either remains unmatched or she is matched with another underdemanded patient in S q(R) =k+1 J`(R) or is matched with an overdemanded patient in S q(R) =k+1 N Ò (R). Since patients in S k =1 J`(R) are handled before patients in S q(R) =k+1 J`(R) during the construction of u E (R), there is no patient in S k =1 J`(R) that is mutually compatible with any patient in S q(R) =k+1 N Ò (R). Since any patient in S k =1 J`(R) and any patient in S q(R) =k+1 J`(R) are in different odd components, there is no patient in S k =1 J`(R) that is mutually compatible with a patient in S q(R) =k+1 J`(R), either. Therefore for any i ∈ S k =1 J`(R) , we have Therefore, patients in S q(R) =k+1 N Ò (R) shall be committed for patients in S q(R) =k+1 J`(R) under the proÞle Q as well, and therefore the aggregate utility of patients in S q(R) =k+1 J`(R) cannot decrease under Q. Since aggregate utility remains constant at e (Q) = e (R), and since only patients in N U (Q) ⊆ N U (R) can have a utility reduction, Since j ∈ J k (R), we have u E h (R) ≤ u E j (R) by Claim 1 in the proof of Theorem 3. This together with u E j (Q) > u E j (R) imply that ( 1 6 ) Recall that patient j 0 is the patient that j declared as incompatible under Q although they were mutually compatible under R. Since j ∈ J ∈ D(R) and r j,j 0 = 1, we have either j 0 ∈ J\ {j} or j 0 ∈ N O (R). If j 0 ∈ J\ {j} then, u E j (Q) ≤ u E j 0 (Q) by Claim 3. If j 0 ∈ N O (R) then j 0 ∈ N O (Q) ∪ N P (Q) by Claim 2 (i) and we have u E j (Q) ≤ 1 = u E j 0 (Q). These together with Eq 16 imply that ( 1 7 ) Since ε ∈ (0, 1) is such that u j (γ ε ) > u j (λ ε ) > u h (γ ε ) and u j 0 (γ ε ) > u j 0 (λ ε ) > u h (γ ε ), neither j nor j 0 is one of the s lowest utility agents under γ ε . Therefore, since only patients j,j 0 are affected between lotteries γ ε and λ ε , (u (ϕ)) (`) for all t ≤ s.
( 1 8 ) We will show that Inequality 18 holds strictly for some t ≤ s which in turn will contradict ϕ is Lorenzdominant under Q completing the proof. Suppose not. Then Inequality 18 holds with equality which in turn implies (u (λ ε )) (t) = (u (ϕ)) (t) for all t ≤ s.
( 1 9 ) Observe that there is t ≤ s such that the t th smallest utility patient is different under these two lotteries. In particular this is the case for t = s, because (1) u h (λ ε ) = u h (γ ε ) > u h (ϕ) and (2) h is the s th smallest utility agent under one of the two lotteries λ ε ,ϕ although not in both since that would contradict Eq 19. Pick the smallest such t. That is under λ ε and ϕ, not only thet th smallest utility patient is the same patient for anyt < t, but she also has the same utility. On the other hand the t th lowest utility patient differs under the two lotteries although they have the same utility by Eq 19. Now consider the lottery 1 2 λ ε + 1 2 ϕ. Since λ ε and ϕ are both feasible under Q, the lottery 1 2 λ ε + 1 2 ϕ is feasible under Q as well. Under this lottery • the smallest utility patient is the same patient as in both λ ε ,ϕ and she has the same utility ¡ u ¡ 1 2 λ ε + 1 2 ϕ ¢¢ (1) = (u (λ ε )) (1) = (u (ϕ)) (1) ; . . . . . .
But the utility of the t th smallest utility patient under 1 2 λ ε + 1 2 ϕ is strictly larger than the utility of the t th smallest utility patient under ϕ. That is because, whoever she is, her utility is no less than (u (ϕ)) (t) under both λ ε , ϕ by Eq 19 and strictly larger in at least one, since the t th smallest utility patient differs under λ ε and ϕ. Hence ϕ does not Lorenz-dominate 1 2 λ ε + 1 2 ϕ, a feasible lottery under Q, a contradiction. Therefore Inequality 18 holds strictly for some t ≤ s. But then ϕ does not Lorenz-dominate λ ε , a feasible lottery under Q, leading to another contradiction and completing the proof. ¥