Fair Allocation of Vaccines, Ventilators and Antiviral Treatments: Leaving No Ethical Value Behind in Health Care Rationing
Parag A. Pathak, Tayfun Sönmez, M. Utku Ünver, M. Bumin Yenmez
FFair Allocation of Vaccines, Ventilators and Antiviral Treatments:Leaving No Ethical Value Behind in Health Care Rationing ∗ Parag A. Pathak Tayfun S¨onmezM. Utku ¨Unver M. Bumin Yenmez † July 2020
Abstract
COVID-19 has revealed several limitations of existing mechanisms for rationing scarcemedical resources under emergency scenarios. Many argue that they abandon various ethicalvalues such as equity by discriminating against disadvantaged communities. Illustrating thatthese limitations are aggravated by a restrictive choice of mechanism, we formulate pandemicrationing of medical resources as a new application of market design and propose a reservesystem as a resolution. We develop a general theory of reserve design, introduce new conceptssuch as cutoff equilibria and smart reserves, extend previously-known ones such as sequentialreserve matching, and relate these concepts to current debates.
JEL codes: D45, D47, I14Keywords: ethical rationing, reserve system, COVID-19, vaccine, ventilator ∗ This version supercedes: “Leaving No Ethical Value Behind: Triage Protocol Design for Pandemic Rationing,”NBER Working Paper 26951, April 2020. We are grateful for input from several experts in bioethics, emergencyhealthcare, and webinar participants at Johns Hopkins University, Stony Brook International Conference on GameTheory, Australasian Microeconomic Theory Seminars, the HELP! (HEaLth and Pandemics) Econ Working GroupSeminar Series, Penntheon Webinar Series, Stockholm School of Economics Corona Economic Research NetworkWebinar Series, and UK Virtual Seminars in Economic Theory. Nikhil Agarwal, Alex Rees-Jones, Robert Truog,Iv´an Werning, and Doug White provided helpful comments. Feedback from Eric Budish, David Delacr´etaz, FuhitoKojima, Govind Persad, and Alex Teytelboym was particularly valuable. † Pathak: Department of Economics, Massachusetts Institute of Technology, and NBER,email: [email protected] , S¨onmez: Department of Economics, Boston College, email: [email protected] , ¨Unver:Department of Economics, Boston College, email: [email protected] , Yenmez: Department of Economics, BostonCollege, email: [email protected] . a r X i v : . [ ec on . T H ] A ug If you fail to plan, you are planning to fail. ” — Benjamin Franklin
Agencies responsible for public health, medical and emergency preparedness at the international,national, regional, and hospital level all have proposed guidelines for allocating scarce medicalresources in crisis situations. These situations range from wartime triage medicine to publichealth emergencies, such as influenza pandemics and COVID-19. In 2005, the World HealthOrganization warned of a severe influenza pandemic and advised countries to develop pandemicguidelines and triage protocols (WHO, 2005). While the nature of the resources in short supplydepends on the crisis situation, for influenza and COVID-19, they may include ventilators, anti-viral medications, and potential vaccines.How to implement a rationing system during a crisis situation presents a complicated ques-tion rife with ethical concerns. Rationing guidelines typically start by articulating several dif-ferent ethical principles. These principles include equity, which is fair distribution of benefitsand burdens; utilitarianism, which involves maximizing welfare; reciprocity, which is respectingcontributions others have made in the past; instrumental valuation, which is respecting con-tributions others could make in the future; solidarity, which is fellowship with other membersof society; non-discrimination, which is requiring that certain individual characteristics such asgender, race, and age play no role allocation. Rationing guidelines also emphasize proceduralvalues, such as accountability, reasonableness, and transparency (see, e.g., WHO (2007); Prehnand Vawter (2008); Emanuel et al. (2020); Truog, Mitchell and Daley (2020)).After articulating these values, regional or national guidelines propose a way to operationalizethem with an allocation mechanism. The most common system used in these guidelines is a priority system which places patients into a single priority order, and allocates all units basedon this single priority. For example, 2018 US Centers for Disease Control Vaccine Allocationguideline place patients into one of four tiers based on their role in providing homeland andnational security, providing health care and community support services, maintaining criticalinfrastructure, and being a member of the general population (CDC, 2018). In some casespriority orders are obtained through an objective scoring method, resulting in a priority pointsystem . This is especially common for rationing of ICU beds and ventilators. For example,Piscitello et al. (2020) reports that 19 states have adopted priority point systems based on theSequential Organ Failure Assessments (SOFA) score, a scoring method that measures severityof organ dysfunction in six organ systems where higher aggregate scores are associated withelevated mortality risk.The COVID-19 pandemic has spurred renewed interest in these guidelines and has revealedseveral important limitations of the existing allocation mechanisms. Whether it is about ra-tioning of ventilators, antiviral drugs or vaccines, a common theme in many of these debates isthat existing guidelines and allocation mechanisms have given up on various important ethicalvalues. For example, advocates for disadvantaged groups voice opposition to priority point sys-tems which use survival probabilities based on various proxies such as the SOFA score. They2rgue that these criteria are discriminatory for they fail to acknowledge pre-existing discrimi-nation in access to health care (Schmidt, 2020). From March-June 13, 2020, the age-adjustedCOVID-19 hospitalization rate of non-hispanic blacks is 178 per 100,000, which is more thanfour times that of non-hispanic whites (CDC, 2020). Moreover, disparate access to testing fordisadvantaged groups have also increased COVID-19 prevalence in these communities, furtherelevating their disadvantage. Similarly, disabilities’ advocates voice opposition to rationing plansbased solely on survival probabilities. Some of them reject a detailed triage protocol in favor ofrandom selection (Ne’eman, 2020). In their view, such criteria are inherently discriminatory forcertain types of disabled patients.In this paper, we argue that several shortcomings of the existing rationing systems are adirect consequence of restricting allocation mechanisms to priority systems. This restrictionimposes a single priority order of individuals for allocation of all units. This limiting featureof a priority system compromises its ability to represent a variety of ethical considerations. Aremedy must therefore break this characteristic. We propose an alternative way to accommodatemultiple ethical principles through a reserve system . In a reserve system, units are divided intomultiple categories each representing an ethical value or a balance of multiple ethical values.Rather than relying on a single priority order of individuals for allocation of all units, a distinctpriority order is used to prioritize individuals for units of each category. This heterogeneityallows for accommodating the desired ethical values without the need to aggregate them into asingle metric or into a strict lexicographic hierarchy.The flexibility of a reserve system over a priority system can help reach compromises forseveral ongoing polarizing debates. For example, many argue that disadvantaged groups orunder-represented minorities should be given priority access for a COVID-19 vaccine. This is asignificant departure from the recommended priority tiers in the 2018 CDC Vaccine allocationguideline. Others question whether prioritizing these groups would erode public trust in vacci-nation. For more on this debate, see Ducharme (2020), McCaughey (2020), and Twohey (2020).While there is no middle ground for compromise under a priority system, a compromise can bereached through a reserve system by giving disadvantaged communities preferential treatmentfor a fraction of vaccines. In Section 5, we present other examples of how a reserve system canhelp reach compromises in other debates.Reserve systems have an important precedent in medicine for deceased donor kidney allo-cation. Until 2014, the U.S. Organ Procurement and Transplantation Network (OPTN) used apriority system to allocate deceased donor kidneys (OPTN, 2014). After establishing medicalcompatibility, patients were ordered by priority type, and then priority points, with ties brokenby waiting time. In 2014, the OPTN system changed to include a reserve (Israni et al., 2014).In the reserve, 20% of the highest quality kidneys are reserved for adults with the highest 20%expected post-transplant survival score (EPTS). A priority points system is used for the remain-ing kidneys. Parallel to our motive for introducing reserve systems (i.e. better integration ofvarious ethical principles), OPTN added this reserve to better reflect the ethical principle ofutilitarianism, which was not as prominently captured in the prior system. The New Kidney Allocation System (KAS) Frequently Asked Questions, Organ Procurement and Transplan-tation Network. Available at https://optn.transplant.hrsa.gov/media/1235/kas_faqs.pdf (last accessed on
3n addition to formulating medical rationing as a new application of market design, our paperalso contributes to the theoretical matching literature by developing a general theory of reservesystems. Beyond our primary application, our results have direct implications for a wide rangeof applications such as affirmative action in allocation of public positions or public school seatsand assignment of immigration visas. In the remaining part of this section, we summarize ourcontributions to the theory of reserve design.In our formulation of medical rationing, there is a number of identical medical units to beallocated to a set of patients. These units can be ICU beds, ventilators, anti-virals, vaccines, orother scarce vital items. An allocation is a matching of patients either to a reserve category ifthe patient receives a unit or the empty set if she does not. Hence, by formulating the medicalrationing problem as an application of reserve design, we do away with the restriction of mappingall ethical considerations on to a single dimension. Under our formulation, units are placed intoone of a set of reserve categories, each with a distinct priority order of patients. A reserve systemis a generalization of a priority system because units in different reserve categories can prioritizepatients in different ways.Our formal approach is axiomatic and lays the foundations of reserve systems by specifyingthree basic principles any system has to satisfy. First, patients should only receive units forwhich they are qualified. Second, no unit from any reserve category should stay idle as long asthere is an eligible patient for that category. Finally, for each category, units should be allocatedbased on the priority order of individuals in the category. We see these principles as minimalrequirements, which can be justified on both normative and positive grounds.In many real-life applications of reserve systems it is customary to announce the outcome ofa reserve system though a specification of the cutoff individual at each category , i.e., the lowestpriority individual who has gained admission through each category. The vector of cutoffindividuals identify the budget set for each individual in the sense that they can see throughwhich (if any) category they can receive a unit. Motivated by this observation, we formalize thenotion of a cutoff equilibrium , a notion akin to a competitive equilibrium. A cutoff equilibriumis a vector of cutoff individuals together with a matching, where each patient is matched with acategory in her budget set, and any category that has not filled its quota has a cutoff of ∅ (thecounterpart of a price of zero in our model). Our first result not only rationalizes the prevalenceof the use of cutoffs in real-life applications but also the plausibility of our three axioms: Amatching satisfies these three properties if and only if it is supported at a cutoff equilibrium.Although our first main result provides a full characterization of matchings that satisfy ourthree properties, it leaves open the question of how to find matchings supported at cutoff equi-libria. Just like computing all competitive equilibria often presents challenges except in specialcases, so does the computation of all cutoff equilibria. Our second main result is that matchingsthat are supported at cutoff equilibria can be computed by constructing a hypothetical two-sidedmatching market where each patient has strict preferences between categories. This construc-tion is hypothetical because in the original problem the patient only cares about obtaining aresource and is therefore indifferent between all units. We show that a matching satisfies our See, for example, Appendix Figure A1. i is considered first foran unreserved category and then for an essential personnel category, whereas another patient j with similar characteristics is considered for these categories in the reverse order. We thereforefocus on an intuitive subclass of matchings, known as sequential reserve matchings, in whichreserve categories are processed in sequence in a particular order. The set of sequential reservematchings is a refinement of the set of cutoff matchings. Within this class, we show that theearlier a category is processed, the higher is the maximum cutoff. Intuitively this means thatthe earlier a category is processed the more competitive it becomes.We then turn to a special environment where there is a baseline priority order. This specialenvironment, which is the focus of much of the earlier literature on reserve systems, is widespreadin real-life applications where the baseline order may depend on scores on standardized exams,results of random lotteries, or time of application. For medical rationing the baseline order maydepend on an objective measure of expected health outcome such as the SOFA score. There isan unreserved category in which all patients are beneficiaries and the priority order is the same asthe baseline. Any other category is a preferential treatment category, with a set of beneficiaries,and all beneficiaries of a category are prioritized over patients who are not, but otherwise theirrelative priority order is the same as the baseline. If each patient is a beneficiary of at most onepreferential treatment category, and there are not more than five categories, every beneficiaryof the preferential treatment category who is matched when a category c is processed earlier isalso matched when it is processed later. This result substantially generalizes earlier results onreserve processing to several categories, while also showing the limit of obtaining sharp formalresults for more than five categories.Finally, we turn to one possible shortcoming of certain sequential reserve matchings: someof the sequential reserve matchings can be Pareto dominated by others. This shortcomingis a direct consequence of the mechanical allocation of patients into reserve categories undersequential reserve matching when they clear the cutoffs for multiple categories. We thereforeintroduce an additional principle requiring a “smart” allocation of reserves in these situations,maximizing the accommodation of intended beneficiaries of reserves. This property together withnon-wastefulness imply Pareto efficiency. We conclude our theoretical analysis by introducingan algorithm that produces a range of “smart” reserve matchings with the desired properties.The rest of paper organized as follows. Section 2 presents some additional background onwidespread use of priority systems for emergency rationing of medical resources and identifies The European Society of Intensive Care Medicine devised the SOFA score at a consensus meeting in October1994 in Paris, France (Vincent et al., 1996). Each of six organ systems – lungs, liver, brain, kidneys, bloodclotting, and blood pressure – is independently assigned a score of 1 to 4. The SOFA score sums these six scores,and sicker patients are assigned higher scores. While not initially designed as a prognostic score, subsequentresearch supports its use for that end (Jones, Trzeciak and Kline, 2009).
The most common allocation mechanism for medical rationing is a priority system where unitsare allocated to patients based on a single priority order. This priority order captures the ethicalvalues guiding the allocation of the scarce medical resource. An example is the 2005 US NationalVaccine Advisory Committee plan. This guideline places patients into the following tiers: Tier1A: Health Care Workers, Tier 1B: Highest-Risk Groups, Tier 1C: Household contacts andpregnancy, Tier 1D: Pandemic responders, Tier 2A: Other high-risk groups, Tier 2B: Criticalinfrastructure groups, Tier 3: Other key government health care decision-makers, and Tier 4:Healthy patients between 2 to 64 years without any high-risk conditions. Within each group,all patients have equal priority. These guidelines were drafted following the 2004-05 influenzavaccine shortage in the United States (Temte, 2006).In some of the applications, most notably for allocation of ventilators and ICU beds, theunderlying priority order is obtained through a monotonic scoring function. Such a refinementof a priority system is called a priority point system . Under this system, each ethical valueis represented with a monotonic function. Values are then integrated with an additive formula,which produces an aggregate point score for each patient. The claims of patients over medicalresources are determined based on their point scores, with a lower score typically indicating ahigher claim. Often a priority score is coarsened into tiers, and all patients in the same tier havethe same claim. Tie-breaking within a tier is typically based on clinical criteria or lotteries.A single-principle point system is a priority point system based on only one ethical value.The 2015 New York State Ventilator Guideline is a prominent example. In the system, as afirst step certain patients are deemed ineligible. The remaining patients are ordered based onestimated mortality risk, which is re-evaluated every 48 hours (Zucker et al., 2015). Mortalityrisk is measured by the Sequential Organ Failure Assessment (SOFA) score which places patientsinto priority tiers. In cases of excess demand among members of a given priority tier, New Yorkand other proposals recommend random allocation – a lottery – among equal-priority patients(Zucker et al., 2015; Emanuel et al., 2020).Several bioethicists and clinicians criticize single-principle priority point systems solely basedon SOFA for ignoring multiple ethical values. These critics emphasize the need to integrate avariety of ethical values and advocate for a multi-principle approach, see, e.g., White et al.(2009) and Daugherty-Biddison et al. (2017). White et al. (2009) describe a multi-principlepriority point system where several ethical values are placed on a numerical scale and summed6p across ethical values to arrive at a single number. Variants of the system shown in Table 1have become the leading multi-principle priority point system for ventilators. Table 1 shows how the ethical values of saving the most lives, saving the most years of life,and the life-cycle principle are integrated through an additive formula. As an example, considera hypothetical patient with a SOFA score of seven. She obtains two points based on the ethicalvalue of saving the most lives. If the patient has no chronic diseases or comorbidities and isbetween 61-74 years old, she obtains four more points based on the other two ethical valuesyielding a total of six. A patient with a lower total point score has a higher priority for theresource than a patient with a higher total point score. More than half of US states use eithera single- or multiple-principle priority point system (Whyte, 2020).
While practical, priority systems have a number of important limitations. A priority system mayfail to integrate different ethical values because of incommensurability. For example, the ethicalvalues of saving the most lives and the life-cycle principle in Table 1 are incommensurable values,making it hard to interpret the role of these values under the priority point system in Table 1. Inaddition, various ethical values have implications on group composition, and a priority systemlacks the flexibility to accommodate these considerations. In many cases, these challenges haveled to the exclusion of some of ethical values all together. For these reasons, priority systems are As of April 24, 2020, California, Colorado, Massachusetts, New Jersey, Oklahoma, and Pennsylvania haveadopted variants of this system. de Pu Kamp, Devine and Griffin (2020) reports that several hundred hospitalsaround the country have adopted this system. Most protocols specifies a tie-breaker between patients with identical total points, although the South Carolinaprotocol fails to provide one. In ethics, two values are incommensurable when they do not share a common standard of measurement. Many of these shortcomings are shared by priority systems in general.
A priority point system requires that ethical values be mapped to a single linear order. However,there are some ethical principles where the claims of patients cannot be represented with amonotonic function. One example is group-based policies, such as those related to regions orgender. For example, the European Union has proposed balanced participation of women andmen in political and public decision making by requiring that at least 40% of public offices areheld by women and at least 40% are held by men (Dittmar, 2018; Rankin, 2020). For medicalrationing, it is possible that a future pandemic is so devastating that it threatens a significantportion of the human race. In such a hypothetical crisis, a principle based on survival of thespecies may suggest a similar constraint. A guideline may recommend to allocate at least 40%of vital resources to female patients and at least 40% to male patients. Clearly, considerationsbased on group composition cannot be represented with a function that relies only on individualattributes. Similarly, a priority point system cannot accommodate a guideline that wishes toallocate resources to disabled citizens in proportion to their representation in society.When constructing priority points and incorporating multiple ethical values, a priority pointsystem norms or scales different and potentially incommensurable ethical principles into onedimension. These challenges are like the usual ones associated with aggregating social alterna-tives into a single ordering based on multiple inputs – a situation which involves “comparingapples to oranges.” The debate on how rationing guidelines should compare claims of childrenversus adults illustrates this issue. Massachusetts guidelines state that indicators that feed intoscores for adults are not reliable for children (Bateman et al., 2020). They explain that “scoringsystems that are meaningful for adult critical care patients do not apply to pediatric patients ornewborns.” As a result, the Massachusetts guidelines use a different scoring system for children.However, their point system then uses a single priority point system to evaluate all patientstogether. This decision ends up comparing the point scores of children with those of adults.Third, the fact that all resources are ordered using a single uniform priority order canresult in the exclusion of certain ethical values. An example of this phenomenon appears inthe debate about prioritizing essential personnel. Many groups argue that essential personnel,and especially frontline healthcare workers, should receive priority allocation of scarce resourcesunder triage scenarios. This view is also strongly endorsed by medical ethicists based on thebackward-looking principle of reciprocity and the forward-looking principle of instrumental value (Emanuel et al., 2020). Nevertheless, states such as New York State and Minnesota had to giveup on this consideration, largely due to concerns about the extreme scenarios where no unitsmay remain for the rest of the society. One of the reasons offered in Vawter et al. (2010) by theMinnesota Pandemic Working group is as follows: As of July 2020, the only exception is Michigan which recommends a priority system with a lexicographichierarchy between several tiers of patients. .. it is possible that they [key workers] would use most, if not all, of the short supply ofventilators; other groups systematically would be deprived access. The New York State Task Force also struggled with this topic, recognizing the need to provide“insurance” for frontline health workers (Zucker et al., 2015). However, they ultimately decidedagainst it, stating:
Expanding the category of privilege to include all the workers listed above may meanthat only health care workers obtain access to ventilators in certain communities. Thisapproach may leave no ventilators for community members, including children; this al-ternative was unacceptable to the Task Force.
For both states, a limitation of priority systems resulted in completely giving up these ethicalvalues. It is important to emphasize that these concerns are direct consequences of the mechanicsof a priority system where providing preferential access to any group for any portion of theresources means giving preferential treatment for all of them.Fourth, a single priority order struggles to integrate the principle of non-exclusion. Thisprinciple is the idea that every patient, no matter his or her circumstances, should have somehope of obtaining a life-saving resource. In the March 2020 Alabama rationing plan, individualswith severe or profound mental disabilities were considered “unlikely candidates for ventilatorsupport.” Washington state guidelines recommend that hospital patients with “loss of reservesin energy, physical ability, cognition and general health” be switched to outpatient or palliativecare (Fink, 2020). In a priority system coupled with excess demand for available resources bythe higher-priority groups – even without any explicit exclusion of certain types of individuals– there will be some patients in a lower-priority group who would never be treated during ashortage.
These conceptual challenges with a priority point system are reflected in actual design challengesin several guidelines. We briefly describe three examples.Massachusetts revised their critical care guidelines on April 7, 2020 for the COVID-19 pan-demic. Like many guidelines, this document was the result of a committee consisting of medicalexperts and ethicists. The guidelines provided an adaption of the system described in Table 1without the life-cycle consideration. However, after precisely spelling out the priority order witha table of numbers for each dimension, the document states:
Individuals who perform tasks that are vital to the public health response, including allthose whose work directly to support the provision of care to others, should be givenheightened priority.
This clause provides no further description on how heightened priority is to be implemented.This lack of transparency contrasts with the level of precision regarding other ethical principles,and may reflect their inability to arrive at consensus given the underlying priority point system. After widespread backlash, this plan was withdrawn on April 9, 2020. Carter (2020) warns that plans whichdiscriminate against the disabled may violate the Americans with Disabilities Act.
In the event that there are ties in priority scores between patients, life-cycle considerationswill be used as a tiebreaker, with priority going to younger patients, who have had lessopportunity to live through life’s stages. In addition, individuals who perform tasks thatare vital to the public health response – specifically, those whose work directly supportsthe provision of acute care to others – will also be given heightened priority (e.g., as atiebreaker between identical priority scores)
In their adaptation of Table 1, the designers saw saving the most lives as more justified thaneither the life-cycle principle or the instrumental value principle. However, the guidelines didnot choose between these two latter ethical values in the event of tie-breaking.The third example is from Arizona’s June 6, 2020 update to their allocation framework(DHS, 2020). This document also offers a table prioritizing patients based on SOFA scores andwhether a patient is expected to live or die within one or five years despite successful treatmentof illness. It then warns that “a situation could arise where limited resources are needed bytwo or more patients with the same triage priority scores” in which case “additional factors may be considered as priorities.” Among the list of additional factors are whether patientsare pediatric patients, first responders or health care workers, single caretakers for minors ordependent adults, pregnant, or have not had an opportunity to experience life stages. There isno further detail on how multiple tie-breakers would be implemented.Beyond these specific updates to guidelines during the COVID-19 pandemic, there are alsoconcerns that incomplete descriptions have rendered such guidelines ineffective in other settings.During the 2004 shortage of the influenza vaccine, Schoch-Spana et al. (2005) state that CDCguidelines were too general and broad. Specifically,
Local providers thus faced gaps in the local supply of inactivated vaccine as well as theabsence of a priori prioritization standards relevant to initial and evolving local conditions.Practitioners and local and state health authorities throughout the U.S. faced a similarpredicament.
Despite these vagaries, some state departments of health penalized clinicians if protocols are notfollowed. For example, Lee (2004) describes that the Massachusetts threatened a fine or prisontime for whoever violates the CDC order on distribution during the 2004 flu-shot shortage. Therequirement to follow an incompletely specified system places clinicians in a difficult position.
Many of the challenges presented in Section 2.2 stem from one simple but limiting feature of apriority point system: it relies on a single priority ranking of patients that is identical for all units.A reserve system eliminates this feature of the mechanism because it allows for heterogeneity ofpatient claims over different units.A reserve system has three main parameters. They are:10. a division of all units into multiple segments referred to as reserve categories ,
2. number of units in each of these categories, and3. specification of a priority order of the patients in each of these categories.If reserve categories are processed sequentially, as in the case in several real-life applications ofreserve systems, processing order of reserve categories can also be an additional parameter. Forsome (or all) of the reserve categories, there can also be exclusion criteria , based on the natureof the medical resource that is being rationed along with the patient’s clinical assessment. Thepriority order of patients for each category also incorporates this information. Reserve categoriescan differ either based on the groups to receive higher priority or the combination of ethicalprinciples to be invoked. The main idea is to use the associated priority order – which embedsethical principles – when allocating units in each reserve category. Importantly, the priorityorder need not be the same between reserve categories.A reserve system provides a natural resolution to the representation and implementationissues discussed in Sections 2.2. However, adoption of a reserve system without a clear un-derstanding of one important aspect of its implementation can have unintended distributionalconsequences. When a patient qualifies for a unit through multiple reserve categories, she doesnot care through which one she receives a unit. And yet this choice influences the outcome forother patients. We therefore next present a general theory of reserve systems.
While our primary application is rationing of scarce medical resources, in this section we providea general model of reserve systems which has several other applications including affirmativeaction in school choice (Pathak and S¨onmez, 2013; Dur et al., 2018; Correa et al., 2019), collegeadmissions (Ayg¨un and Bo, 2020; Baswana et al., 2019), assignment of government positions(S¨onmez and Yenmez, 2019 a , b ), and skill diversity in immigration visa allocation (Pathak, Rees-Jones and S¨onmez, 2020). Despite this generality, the terminology is tailored to our mainapplication.There is a set of patients I and q identical medical units to allocate. There is a set of reserve categories C . For every category c ∈ C , r c units are reserved so that (cid:80) c ∈C r c = q .It is important to emphasize that individual units are not associated with the categories inour model. The phrase “ r c units are reserved” does not mean specific units are set aside forcategory c . Rather, it means that for the purposes of accounting, a total of unspecified r c unitsare attached to category c .For every category c ∈ C , there is a linear priority order π c over the set of patients I and ∅ . This priority order represents the relative claims of the patients on units in category c aswell as their eligibility for those units. For every category c ∈ C and patient i ∈ I , we say that This division is for accounting purposes only, and it does not attach a specific unit to a category. See Dur et al. (2018) and Pathak, Rees-Jones and S¨onmez (2020) for real-life examples of this phenomenonfrom school choice and H1-B visa allocation. i is eligible for category c if i π c ∅ . Given priority order π c , we represent its weak order by π c . That is, for any x, y ∈ I ∪ {∅} , x π c y ⇐⇒ x = y or x π c y. For our main application of pandemic rationing, π c orders patients based on the balance ofethical principles guiding the allocation of units in category c .A matching µ : I → C ∪ {∅} is a function that maps each patient either to a category orto ∅ such that (cid:12)(cid:12) µ − ( c ) (cid:12)(cid:12) ≤ r c for every category c ∈ C . For any patient i ∈ I , µ ( i ) = ∅ meansthat the patient does not receive a unit and µ ( i ) = c ∈ C means that the patient receives a unitreserved for category c . Let M denote the set of matchings.For any matching µ ∈ M and any subset of patients I (cid:48) ⊆ I , let µ ( I (cid:48) ) denote the set ofpatients in I (cid:48) who are matched with a category under matching µ . More formally, µ ( I (cid:48) ) = (cid:8) i ∈ I (cid:48) : µ ( i ) ∈ C (cid:9) . These are the patients in I (cid:48) who are matched (or equivalently who are assigned units) undermatching µ .In real-life applications of our model, it is important to allocate the units to qualified in-dividuals without wasting any and abiding by the priorities governing the allocation of theseunits. We next formulate this idea through three axioms: Definition 1.
A matching µ ∈ M complies with eligibility requirements if, for any i ∈ I and c ∈ C , µ ( i ) = c = ⇒ i π c ∅ . Our first axiom formulates the idea that units should be awarded only to eligible individuals.For rationing of vital medical resources, any patient who is eligible for one category must alsobe eligible for any category. And if a patient is ineligible for all categories, then this patientcan be dropped from the set of patients. Hence, compliance with eligibility requirements alwaysholds for our main application.
Definition 2.
A matching µ ∈ M is non-wasteful if, for any i ∈ I and c ∈ C , i π c ∅ and µ ( i ) = ∅ = ⇒ (cid:12)(cid:12) µ − ( c ) (cid:12)(cid:12) = r c . Our second axiom formulates the idea that no unit should be wasted. That is, if a unitremains idle, then there should not be any unmatched individual who is eligible for the unit.For rationing vital resources, each patient is eligible for all units, and therefore non-wastefulnessin this context corresponds to either matching all the units or all the patients.
Definition 3.
A matching µ ∈ M respects priorities if, for any i, i (cid:48) ∈ I and c ∈ C , µ ( i ) = c and µ ( i (cid:48) ) = ∅ = ⇒ i π c i (cid:48) . In many real-life applications of reserve systems, the outcome is often publicized through asystem that identifies the lowest priority individual that qualifies for admission for each category.This representation makes it straightforward to verify that an allocation was computed followingthe announced policy because an individual can compare her priority to the announced cutoffs.Often a cardinal representation of the priority order, such as a merit score or a lottery number,is used to identify these “cutoff” individuals. (See, for example, Appendix Figure A1.) Thisobservation motivates the following equilibrium notion.For any category c ∈ C , a cutoff f c is an element of I ∪ {∅} such that f c π c ∅ . Cutoffsin our model play the role of prices for exchange or production economies, where the lastcondition corresponds to prices being non-negative in those economies. We refer to a list ofcutoffs f = ( f c ) c ∈C as a cutoff vector . Let F be the set of cutoff vectors.Given a cutoff vector f ∈ F , for any patient i ∈ I , define the budget set of patient i atcutoff vector f as B i ( f ) = { c ∈ C : i π c f c } . A cutoff equilibrium is a pair consisting of a cutoff vector and a matching ( f, µ ) ∈ F × M such that1. For every patient i ∈ I ,(a) µ ( i ) ∈ B i ( f ) ∪ {∅} , and(b) B i ( f ) (cid:54) = ∅ = ⇒ µ ( i ) ∈ B i ( f ) .
2. For every category c ∈ C , | µ − ( c ) | < r c = ⇒ f c = ∅ . A cutoff equilibrium is an analogue of a competitive equilibrium for a reserve system. A cutoffvector-matching pair is a cutoff equilibrium if1. each patient who has a non-empty budget set is matched with a category in her budgetset, and each patient who has an empty budget set remains unmatched, and2. each category which has not filled its quota under this matching has cutoff ∅ .13he first condition corresponds to “preference maximization within budget set” and the secondcondition corresponds to the “market clearing condition.”Our first characterization result gives an equivalence between cutoff equilibrium matchingsand matchings that satisfy our basic axioms. Theorem 1.
For any matching µ ∈ M that complies with eligibility requirements, is non-wasteful, and respects priorities, there exists a cutoff vector f ∈ F that supports the pair ( f, µ ) as a cutoff equilibrium. Conversely, for any cutoff equilibrium ( f, µ ) ∈ F × M , matching µ complies with eligibility requirements, is non-wasteful, and respects priorities. There can be multiple equilibrium cutoff vectors that supports a matching at cutoff equilibria.Next, we explore the structure of equilibrium cutoff vectors.For any matching µ ∈ M and category c ∈ C , define f µc = (cid:40) min π c µ − ( c ) if | µ − ( c ) | = r c ∅ otherwise and, (1) f µc = (cid:40) min π c (cid:110) i ∈ µ ( I ) : i π c max π c (cid:0) I \ µ ( I ) (cid:1) ∪ {∅} (cid:111) if max π c (cid:0) I \ µ ( I ) (cid:1) ∪ {∅} (cid:54) = ∅∅ otherwise . (2)Here, • f µc identifies – the lowest π c -priority patient who is matched with category c if all category- c unitsare exhausted under µ , and – ∅ if there are some idle category- c units under µ ,whereas • f µc identifies – the lowest π c -priority patient with the property that every weakly higher π c -prioritypatient than her is matched under µ if some category- c eligible patient is unmatchedunder µ , and – ∅ if all category- c eligible patients are matched under µ .Let µ be any matching that respect priorities. By construction, f µc π c f µc for any c ∈ C . Our next result characterizes the set of cutoff vectors.
Lemma 1.
Let µ ∈ M be a matching that complies with eligibility requirements, is non-wasteful,and respects priorities. Then the pair ( g, µ ) is a cutoff equilibrium if, and only if, f µc π c g c π c f µc for any c ∈ C .
14n immediate corollary to Lemma 1 is that for each cutoff equilibrium matching µ , f µ =( f µc ) c ∈C is a maximum equilibrium cutoff vector and f µ = ( f µc ) c ∈C is a minimum equi-librium cutoff vector .Of these equilibrium cutoff vectors, the first one has a clear economic interpretation. Themaximum equilibrium cutoff of a category indicates the selectivity of this particular category.The higher the maximum cutoff is the more competitive it becomes to receive a unit through thiscategory. This is also the cutoff that is typically announced in real-life applications of reservesystems due to its clear interpretation. The interpretation of the minimum equilibrium cutoffof a category is more about the entire matching than the category itself, and in some sense it isartificially lower than the maximum equilibrium cutoff due to individuals who are matched withother categories. All other equilibrium cutoffs between the two are also artificial in a similarsense. Therefore, for much of our analysis we focus on the maximum equilibrium cutoff vector. Although Theorem 1 gives a full characterization of matchings that satisfy our three axioms, itleaves the question of how to find such a matching open. In this section, we present a procedureto construct all such matchings utilizing the celebrated deferred-acceptance algorithm by Galeand Shapley (1962).Consider the following hypothetical many-to-one matching market. The two sides of themarket are the set of patients I and the set of categories C . Each patient i ∈ I can be matchedwith at most one category, whereas each category c ∈ C can be matched with as many as r c patients. Category c is endowed with the linear order π c that is specified in the primitives ofthe original rationing problem.Observe that in our hypothetical market, all the primitives introduced so far naturally followsfrom the primitives of the original problem. The only primitive of the hypothetical market thatis somewhat “artificial” is the next one:Each patient i ∈ I has a strict preference relation (cid:31) i over the set C ∪ {∅} , such that, for eachpatient i ∈ I , c (cid:31) i ∅ ⇐⇒ patient i is eligible for category c. While in the original problem a patient is indifferent between all units (and therefore all cat-egories as well), in the hypothetical market she has strict preferences between the categories.This “flexibility” in the construction of the hypothetical market is the basis of our main char-acterization.For each patient i ∈ I , let P i be the set of all preferences constructed in this way, and let P = × i ∈ I P i .Given a preference profile (cid:31) = ( (cid:31) i ) i ∈ I , the individual-proposing deferred-acceptance algo-rithm (DA) produces a matching as follows. Individual Proposing Deferred Acceptance Algorithm (DA)Step 1:
Each patient in I applies to her most preferred category among categoriesfor which she is eligible. Suppose that I c is the set of patients who apply to category15 . Category c tentatively assigns applicants with the highest priority according to π c until all patients in I c are chosen or all r c units are allocated, whichever comesfirst, and permanently rejects the rest. If there are no rejections, then stop. Step k:
Each patient who was rejected in Step k-1 applies to her next preferredcategory among categories for which she is eligible, if such a category exists. Supposethat I kc is the union of the set of patients who were tentatively assigned to category c in Step k-1 and the set of patients who just proposed to category c . Category c tentatively assigns patients in I kc with the highest priority according to π c until allpatients in I kc are chosen or all r c units are allocated, whichever comes first, andpermanently rejects the rest. If there are no rejections, then stop.The algorithm terminates when there are no rejections, at which point all tentativeassignments are finalized.A matching µ ∈ M is called DA-induced if it is the outcome of DA for some preferenceprofile (cid:31) ∈ P .We are ready to present our next result:
Theorem 2.
A matching complies with eligibility requirements, is non-wasteful, and it respectspriorities if, and only if, it is DA-induced.
Not only is this result a second characterization of matchings that satisfy our three basicaxioms, it also provides a concrete procedure to calculate all such matchings. Equivalently,Theorem 2 provides us with a procedure to derive all cutoff equilibria. This latter interpretationof our characterization leads us to a refinement of cutoff equilibrium matchings explored in ournext section.
An interpretation of the DA-induced matchings is helpful to motivate in focusing a subset of thesematchings. Recall that the hypothetical two-sided matching market constructed above relies onan artificial preference profile ( (cid:31) i ) i ∈ I of patients over categories. What this corresponds tounder the DA algorithm is that any patient i is considered for categories that deem her eligiblein sequence, following the ranking of these categories under her artificial preferences (cid:31) i . Unlessthere is a systematic way to construct these preferences, it may be difficult to motivate adoptingthis methodology for real-life applications. For example, if a patient i is considered first foran unreserved category and then for an essential personnel category, whereas another patient j with similar characteristics is considered for them in the reverse order, it may be difficult tojustify this practice. That is, while there is a potentially large set of matchings that satisfyour three axioms, not all are necessarily obtained through an intuitive procedure. This may bea challenge especially in the context of medical rationing, since procedural fairness is also animportant ethical consideration in this context. Procedural fairness is the main motivation forour focus in a subset of these matchings. 16n many real-life applications of reserve systems, institutions process reserve categories se-quentially and allocate units associated with each category one at a time using its category-specific priority order. We next formulate matchings obtained in this way and relate them toour characterization in Theorem 2.An order of precedence (cid:46) is a linear order over the set of categories C . For any twocategories c, c (cid:48) ∈ C , c (cid:46) c (cid:48) means that category- c units are to be allocated before category- c (cid:48) units. In this case, we saycategory c has higher precedence than category c (cid:48) . Let ∆ be the set of all orders of precedence.For a given order of precedence (cid:46) ∈ ∆, the induced sequential reserve matching ϕ (cid:46) , is amatching that is constructed as follows:Suppose categories are ordered under (cid:46) as c (cid:46) c (cid:46) . . . (cid:46) c |C| . Matching ϕ (cid:46) is found sequentially in |C| steps: Step 1:
Following their priority order under π c , the highest priority r c category- c -eligible patients in I are matched with category c . If there are less than r c eligiblepatients in I than all of these eligible patients are matched with category c . Let I be the set of patients matched in Step 1. Step k:
Following their priority order under π c k , the highest priority r c k category- c k -eligible patients in I \ ∪ k − k (cid:48) =1 I k (cid:48) are matched with category c k . If there are lessthan r c k eligible patients in I \ ∪ k − k (cid:48) =1 I k (cid:48) then all of these eligible patients are matchedwith category c k . Let I k be the set of patients matched in Step k.Given an order of precedence (cid:46) ∈ ∆, the induced sequential reserve matching complies witheligibility requirements, is non-wasteful, and it respect priorities. Thus, it is DA-induced byTheorem 2. Indeed it corresponds to a very specific DA-induced matching. Proposition 1.
Fix an order of precedence (cid:46) ∈ ∆ . Let the preference profile (cid:31) (cid:46) ∈ P be suchthat, for each patient i ∈ I and pair of categories c, c (cid:48) ∈ C , c (cid:31) (cid:46)i c (cid:48) ⇐⇒ c (cid:46) c (cid:48) . Then the sequential reserve matching ϕ (cid:46) is DA-induced from the preference profile (cid:31) (cid:46) . We conclude this section with a comparative static result regarding the maximum equilibriumcutoff vectors supporting sequential reserve matchings:
Proposition 2.
Fix a preferential treatment category c ∈ C , another category c (cid:48) ∈ C \ { c } , anda pair of orders of precedence (cid:46), (cid:46) (cid:48) ∈ ∆ such that: • c (cid:48) (cid:46) c , • c (cid:46) (cid:48) c (cid:48) , and for any ˆ c ∈ C and c ∗ ∈ C \ { c, c (cid:48) } ˆ c (cid:46) c ∗ ⇐⇒ ˆ c (cid:46) (cid:48) c ∗ . That is, (cid:46) (cid:48) is obtained from (cid:46) by only changing the order of c with its immediate predecessor c (cid:48) .Then, f ϕ (cid:46) (cid:48) c π c f ϕ (cid:46) c . Recall that the maximum equilibrium cutoff for a category is indicative of how selective thecategory is. Therefore, the earlier a category is processed under a sequential reserve matchingthe more selective it becomes by Proposition 2. This result is intuitive because the earlier acategory is processed, the larger is the set of patients who compete for these units in a setwiseinclusion sense.
In many real-life applications of reserve systems, there is a baseline priority order π of individuals.Starting with Hafalir, Yenmez and Yildirim (2013), the earlier market design literature onreserve systems exclusively considered this environment. This priority order may depend onscores in a standardized exam, a random lottery, or arrival time of application. In our mainapplication of pandemic resource allocation, it may depend on SOFA scores described in Section2.1. This baseline priority order is used to construct the priority order for each of the reservecategories, although each category except one gives preferential treatment to a specific subset ofindividuals. For example, in our main application these could be essential personnel or personsfrom disadvantaged communities. In this section, we focus on this subclass of reserve systemsand present an analysis of reserve matching on this class.To formulate this subclass, we designate a beneficiary group I c ⊆ I for each category c ∈ C . It is assumed that all patients in its beneficiary group are eligible for a category. Thatis, for any c ∈ C and i ∈ I c , i π c ∅ . There is an all-inclusive category u ∈ C , called the unreserved category, which has allpatients as its set of beneficiaries and endowed with the baseline priority order. That is, I u = I and π u = π. Any other category c ∈ C \ { u } , referred to as a preferential treatment category, has amore exclusive set I c ⊂ I of beneficiaries and it is endowed with a priority order π c with thefollowing structure: for any pair of patients i, i (cid:48) ∈ I , i ∈ I c and i (cid:48) ∈ I \ I c = ⇒ i π c i (cid:48) ,i, i (cid:48) ∈ I c and i π i (cid:48) = ⇒ i π c i (cid:48) , and i, i (cid:48) ∈ I \ I c and i π i (cid:48) = ⇒ i π c i (cid:48) . Including Hafalir, Yenmez and Yildirim (2013), several prior studies consider the allocation of heterogeneousobjects and the baseline priority often depends on the object. π c , beneficiaries of category c are prioritized over patients who are not, but otherwisetheir relative priority order is induced by the baseline priority order π .Let I g , referred to as the set of general-community patients , be the set of patients whoare each a beneficiary of the unreserved category only: I g = I \ ∪ c ∈C\{ u } I c . In particular two types of such problems have widespread applications.We say that a priority profile ( π c ) c ∈C has soft reserves if, for any category c ∈ C and anypatient i ∈ I , i π c ∅ . Under a soft reserve system all individuals are eligible for all categories. This is the case, forexample, in our main application of pandemic resource allocation.We say that a priority profile ( π c ) c ∈C has hard reserves if, for any preferential treatmentcategory c ∈ C \ { u } ,1. i π c ∅ for any of its beneficiaries i ∈ I c , whereas2. ∅ π c i for any patient i ∈ I \ I c who is not a beneficiary.Under a hard reserve system, only the beneficiaries of a preferred treatment category are eligiblefor units in this category. This is the case, for example, in H1-B visa allocation in the US. Allocation rules based on sequential reserve matching are used in a wide range of practicalapplications. While an aspect that is often ignored in practical applications, it is important topay attention to the choice of the order of precedence in these problems, for it has potentiallysignificant distributional implications. In this subsection we focus on sequential reserve matchingunder soft reserves, as that is the relevant case for our main application of pandemic rationing.We already know from Proposition 2 that the later a category is processed, the less compet-itive it becomes. A natural follow up question is whether this also means that the beneficiariesof this category necessarily benefits from this comparative static exercise. The answer would beof course straightforward, if each patient was a beneficiary of a single category. But this is notthe case in our model, because even if each patient is a beneficiary of at most one preferentialtreatment category, they are also each a beneficiary of the unreserved category. Indeed, even ifthat was not the case, unless the reserves are hard non-beneficiaries may still be matched withunits from preferential treatment categories. So the answer to this question is not an immediateimplication of Proposition 2. Under some assumptions such as when there is only one preferen-tial treatment category (Dur et al., 2018), this question is already answered in the affirmative.However, as we present in the next example, this is not always the case.
Example 1.
Suppose there are q = 6 medical units to be allocated in total. There are sixcategories: the unreserved category u and five preferential treatment categories c, c (cid:48) , c ∗ , ˆ c, ˜ c andeach category has a single unit capacity. uppose there are seven patients. All patients are beneficiaries of the unreserved category u : I u = { i , i , i , i , i , i , i } . The beneficiaries of preferential treatment categories c , c ∗ , and ˜ c are given as I c = { i , i , i } , I c ∗ = { i , i } , I ˜ c = { i , i } , while there are no beneficiaries of preferential treatment categories c (cid:48) and ˆ c : I c (cid:48) = ∅ and I ˆ c = ∅ .There are also no general-community patients: I g = ∅ . Suppose π , the baseline priority order ofpatients, is given as i π i π i π i π i π i π i . Also assume that all patients are eligible for all preferential treatment categories besides theunreserved category u .We consider two sequential reserve matchings based on the following two orders of precedence: (cid:46) : c (cid:48) (cid:46) c (cid:46) c ∗ (cid:46) ˆ c (cid:46) ˜ c (cid:46) u, and (cid:46) (cid:48) : c (cid:46) (cid:48) c (cid:48) (cid:46) (cid:48) c ∗ (cid:46) (cid:48) ˆ c (cid:46) (cid:48) ˜ c (cid:46) (cid:48) u. In the following table, we demonstrate the construction of the two induced sequential reservematchings by processing their mechanics in parallel:Order of Precedence (cid:46)
Order of Precedence (cid:46) (cid:48)
Step Category Patient Category Patient c (cid:48) i c i c i c (cid:48) i c ∗ i c ∗ i c i ˆ c i c i ˜ c i u i u i Thus the two sequential reserve matchings match the patients ϕ (cid:46) = { i , i , i , i , i , i } and ϕ (cid:46) (cid:48) = { i , i , i , i , i , i } . In this problem category- c (cid:48) and ˆ c units are treated as if they are of unreserved category u , asthese two categories do not have any beneficiaries in the problem. We use the baseline priorityorder π to match them.Under the first order of precedence (cid:46) , the highest π -priority patient i , who is also a category- c beneficiary, receives the first unit, which is reserved for category c (cid:48) . As a result, i , who is thenext category- c beneficiary, receives the only category- c unit. In the end, units associated withcategories c ∗ and ˜ c are matched with their highest and lowest priority beneficiaries i and i ,respectively. The highest priority beneficiary of ˜ c , patient i , receives the category- ˆ c unit, whichis processed like the unreserved category and before ˜ c . Hence, the lowest priority beneficiary of ategory c , i remains unmatched as the last unit, which is reserved for the unreserved category,goes to i . Thus, ϕ (cid:46) ( I c ) = { i , i } is the set of matched category- c beneficiaries.Under the second order of precedence (cid:46) (cid:48) that switches the order of c and c (cid:48) , the selectivity ofcategory c increases as it is processed earlier: the highest priority category- c patient i receivesits unit instead of i . This leads to the units associated with categories c ∗ and ˜ c being matchedwith their lowest and highest priority beneficiaries i and i , respectively – this is a switch ofroles for these categories with respect to (cid:46) . This is because the highest-priority beneficiary of c ∗ ,patient i , is now matched with category c (cid:48) , which is processed like the unreserved category before c ∗ . This enables the lowest priority beneficiary of category c , patient i , to be matched with theunreserved category as she is prioritized higher than i under the baseline priority order. Hence, ϕ (cid:46) (cid:48) ( I c ) = { i , i , i } is the set of matched category- c beneficiaries.Thus, ϕ (cid:46) ( I c ) (cid:40) ϕ (cid:46) (cid:48) ( I c ) although category c is processed earlier under (cid:46) (cid:48) than under (cid:46) . Observe that our negative example holds even though each patient is a beneficiary of atmost one preferential treatment category. Nevertheless, a positive result holds for our mainapplication of soft reserves provided that there are at most five categories and each patient is abeneficiary of at most one preferential treatment category.
Proposition 3.
Assuming1. there are at most five categories, and2. each patient is a beneficiary of at most one preferential treatment category,consider a soft reserve system induced by a baseline priority order. Fix a preferential treatmentcategory c ∈ C \ { u } , another category c (cid:48) ∈ C \ { c } , and a pair of orders of precedence (cid:46), (cid:46) (cid:48) ∈ ∆ such that: • c (cid:48) (cid:46) c , • c (cid:46) (cid:48) c (cid:48) , and • for any ˆ c ∈ C and c ∗ ∈ C \ { c, c (cid:48) } , ˆ c (cid:46) c ∗ ⇐⇒ ˆ c (cid:46) (cid:48) c ∗ . That is, (cid:46) (cid:48) is obtained from (cid:46) by only changing the order of c with its immediate predecessor c (cid:48) .Then, ϕ (cid:46) (cid:48) ( I c ) ⊆ ϕ (cid:46) ( I c ) . .2 Smart Reserve Matching Although virtually all practical applications of reserve systems are implemented through sequen-tial reserve matching, this class of mechanisms may suffer from one important shortcoming: theymay lead to Pareto inefficient outcomes, due to myopic processing of reserves. The followingexample illustrates both how this may happen, and also motivates a possible refinement basedon smart processing of reserves.
Example 2.
Consider a hard reserve system induced by baseline priority order π . There aretwo patients I = { i , i } who are priority ordered as i π i under the baseline priority order π . There are two categories; an unreserved category u withan all-inclusive beneficiary set of I = { i , i } , and a preferential treatment category c witha beneficiary set I c = { i } of a single preferential treatment patient. Both categories have acapacity of one unit each (i.e., r c = r u = 1 ). Since the reserves are hard, the resulting category-specific priority orders are given as follows: i π u i π u ∅ and i π c ∅ π c i . Consider the sequential reserve matching ϕ (cid:46) induced by the order of precedence (cid:46) , where u (cid:46) c. Under matching ϕ (cid:46) , first patient i is matched with the unreserved category u and subsequentlythe category- c unit is left idle since no remaining patient is eligible for this preferential treatmentcategory. Therefore, ϕ (cid:46) = (cid:32) i i u ∅ (cid:33) , resulting in the set of matched patients ϕ (cid:46) ( I ) = { i } .Next consider the sequential reserve matching ϕ (cid:46) (cid:48) induced by the order of precedence (cid:46) (cid:48) , where c (cid:46) (cid:48) u. Under matching ϕ (cid:46) (cid:48) , first patient i is matched with the preferential treatment category c andsubsequently patient i is matched with the unreserved category u . Therefore, ϕ (cid:46) (cid:48) = (cid:32) i i c u (cid:33) , resulting in the set of matched patients ϕ (cid:46) (cid:48) ( I ) = { i , i } . Since ϕ (cid:46) ( I ) (cid:40) ϕ (cid:46) (cid:48) ( I ) , matching ϕ (cid:46) isPareto dominated by matching ϕ (cid:46) (cid:48) . Observe that Example 2 also illustrates that the cause of Pareto inefficiency is the myopicallocation of categories under sequential reserve matchings. Under matching ϕ (cid:46) , the more flexibleunreserved unit is allocated to patient i who is the only beneficiary of category- c . This results22n a suboptimal utilizations of reserves, which can be avoided with the concept of “smart reservematching” we introduce below.To this end, we first introduce a new axiom, which together with non-wastefulness implyPareto efficiency. Definition 4.
A matching µ ∈ M is maximal in beneficiary assignment if µ ∈ arg max ν ∈M (cid:12)(cid:12) ∪ c ∈C\{ u } ( ν − ( c ) ∩ I c ) (cid:12)(cid:12) . This axiom simply requires that the reserves should be maximally assigned to target bene-ficiaries to the extent it is feasible. It precludes the myopic assignment of categories to patientssince the desirability of a matching depends on the structure of the matching as a whole ratherthan the individual assignments it prescribes for each category.It is worth noting that the inefficiency observed in Example 2 is specific to the case of hardreserves and cannot happen for soft reserves, as in our main application of pandemic rationing.Nonetheless, maximality in beneficiary assignment is a desirable axiom for soft reserves becausesub-optimal utilization of reserves may receive heightened scrutiny. For example, consider ascenario with two preferential treatment categories, essential personnel and disadvantaged, eachwith one reserve. Suppose patient A is both essential personnel and disadvantaged, patientB is disadvantaged, and there are several other patients who are neither. One possible wayto use these reserves is to assign patient A to the disadvantaged reserve, leaving no otherpreferential treatment patients available for the essential personnel reserve. In this case, theessential personnel reserve would be opened up to patients who are neither essential personnelnor disadvantaged. This could in turn mean only one of the reserves is assigned to members of thetarget beneficiary groups. This outcome could be seen problematic since an alternative, whichassigns patient A to the essential personnel reserve (instead of the disadvantaged reserve) andpatient B to the disadvantaged reserve, accommodates both reserves. By imposing maximalityin beneficiary assignment, we avoid this shortcoming through a “smart” utilization of reserves.Building on this insight, we next present an algorithm that generates smart cutoff matchings:
Smart Reserve Matching Algorithm
Fix a parameter n ∈ { , , . . . , r u } that represents the number of unreserved units tobe processed in the beginning of the algorithm. The remaining unreserved unitsare to be processed at the end of the algorithm.Fix a baseline priority order π , and for the ease of description relabel patients sothat i π i π . . . π i | I | . Proceed in two steps.
Step 1:
Iteratively construct two sequences of patient sets J u ⊆ J u ⊆ . . . ⊆ J u | I | ,which determine patients to be matched with the unreserved category u in thisstep, and J ⊆ J ⊆ . . . ⊆ J | I | , which determine the patients to be matched with For n = 0, this algorithm is equivalent to the horizontal envelope algorithm in S¨onmez and Yenmez (2020). C \ { u } that they are beneficiaries of, and asequence of sets of matchings M ⊇ M ⊇ . . . ⊇ M | I | in | I | substeps.Define Step 1.( k ) for any k ∈ { , , . . . , | I |} as follows:If k = 1, let J u = ∅ , J = ∅ , and M be the set of all matchings that are maximal in beneficiary assignment; thatis M = arg max ν ∈M (cid:12)(cid:12) ∪ c ∈C\{ u } ( ν − ( c ) ∩ I c ) (cid:12)(cid:12) . If k >
1, then sets of patients J uk − and J k − and set of matchings M k − are con-structed in the previous substep, Step 1.( k − Step 1.( k ): Process patient i k . Three cases are possible: • If | J uk − | < n and there exists a matching in M k − that matchespatient i k with the unreserved category u , then define J uk = J uk − ∪{ i k } , J k = J k − , and M k = (cid:8) µ ∈ M k − : µ ( i k ) = u (cid:9) . • Otherwise, if there exists a matching in M k − that matches patient i k with a preferential treatment category c ∈ C \ { u } that she is abeneficiary of, that is i k ∈ I c , then define J uk = J uk − , J k = J k − ∪ { i k } , and M k = (cid:8) µ ∈ M k − : µ ( i k ) (cid:54)∈ {∅ , u } and i k ∈ I µ ( i k ) (cid:9) . • Otherwise, define J uk = J uk − , J k = J k − , and M k = M k − . Step 2:
For any matching µ ∈ M | I | , construct a matching σ ∈ M as follows: • Assign µ ( i ) to every patient i ∈ J | I | ∪ J u | I | . • One at a time iteratively assign the remaining units to the remaining highestpriority patient in I \ ( J u | I | ∪ J | I | ) who is eligible for the category of the assignedunit in the following order:1. the remaining units of the preferential treatment categories in an arbitraryorder, and2. the remaining units of the unreserved category u .Every matching σ constructed in this manner is referred to as a smart reserve matching induced by assigning n unreserved units subsequently at the beginning of the algorithm. Let M nS be the set of all reserve matchings for a given n .We have the following result about the sets of patients matched under smart reserve match-ings: 24 emma 2. Consider either a soft reserve system or a hard reserve system induced by a baselinepriority order π . For any n ∈ { , , . . . , r u } and any two smart reserve matchings σ, ν ∈ M nS , σ − ( u ) = ν − ( u ) , and ∪ c ∈C\{ u } ( σ − ( c ) ∩ I c ) = ∪ c ∈C\{ u } ( ν − ( c ) ∩ I c ) , and moreover, σ ( I ) = ν ( I ) . Lemma 2 states that every smart reserve matching for a given n matches the same set ofpatients with the unreserved category u , the same set of patients with preferential treatmentcategories in C \ { u } that they are beneficiaries of, and the same set of patients overall.In a soft reserve system or a hard reserve system, for a given n , we denote the set patientsmatched in every smart reserve matching with n unreserved units processed first as I nS . ByLemma 2, for any σ ∈ M nS , I nS = σ ( I ) . Our second result on smart reserve matchings is as follows:
Proposition 4.
Consider either a soft reserve system or a hard reserve system induced by abaseline priority order π . For any n ∈ { , , . . . , r u } , any smart reserve matching in M nS complieswith eligibility requirements, is non-wasteful, respects priorities, and is maximal in beneficiaryassignment. The choice of parameter n is not without a consequence. In particular, matchings producedby the algorithm with the lowest parameter n = 0 and the highest parameter n = r u both havedistinctive distributional consequences. Theorem 3.
Consider either a soft reserve system or a hard reserve system induced by a baselinepriority order π . Let σ r u ∈ M r u S be a smart reserve matching when all unreserved units areassigned first (i.e., n = r u ) and σ ∈ M S be a smart reserve matching when all unreservedunits are assigned last (i.e., n = 0 ). Then, for any cutoff equilibrium matching µ ∈ M that ismaximal in beneficiary assignment, f σ ru u π f µu π f σ u . Theorem 3 states that among all maximum equilibrium cutoff vectors that support maximalmatchings in beneficiary assignment, the selectivity of the unreserved category is • the most competitive for smart reserve matchings with n = r u , that is, when all unreservedcategory units are assigned in the beginning of the algorithm, and • the least competitive for smart reserve matchings with n = 0, that is, when all unreservedcategory units are assigned at the end of the algorithm. We describe a polynomial-time algorithm to construct I nS and a smart reserve matching in Appendix A.4. n ),respectively. However, there can be multiple ways of assigning these patients to different pref-erential treatment categories. Thus, the maximum equilibrium cutoffs of preferential treatmentcategories are not uniquely determined for these matchings. Our study of reserve systems contributes to literature in matching market design focused ondistributional issues. Our main results of Theorems 1-3 as well as Propositions 1 and 2 have noantecedents in the literature, and they are novel to this paper. In contrast, Propositions 3 and4 extend previously known ideas to our model.While Theorem 1 is novel and our paper is the first one to formally introduce the notionof cutoff equilibrium for reserve systems, due to its intuitive appeal the use of this notion iswidespread in real-life applications of reserve systems. In particular, the outcomes of reservesystems are often announced together with the cutoffs that support them. Examples includeadmission to exam high schools in Chicago (Dur, Pathak and S¨onmez, 2020), assignment of gov-ernment positions in India (S¨onmez and Yenmez, 2019 a , b ), college admissions in Brazil (Ayg¨unand Bo, 2020), and H1-B visa allocation in the US for years 2006-2008 (Pathak, Rees-Jones andS¨onmez, 2020). For the first three of these applications the cutoffs are given in terms of examor merit scores, whereas for the last application the cutoffs are given in terms of the date ofvisa application receipt. While the concept of cutoff equilibrium for reserve systems is novel toour paper, cutoffs are used in simpler matching environments in the absence of distributionalconsiderations (see, e.g., Balinski and S¨onmez (1999), Azevedo and Leshno (2016), and Leshnoand Lo (2020)).In addition to presenting a characterization of outcomes that satisfy three basic axioms,Theorem 2 also provides a procedure to calculate all cutoff matchings. While our characterizationitself is novel, the use of the deferred acceptance algorithm to derive a specific cutoff matchingis not. This technique to obtain cutoff levels for reserve systems has been employed in variousreal-life applications, including in school choice algorithms of Boston Public Schools (Dur et al.,2018) and Chile (Correa et al., 2019), and college admissions algorithm for Engineering Collegesin India (Baswana et al., 2019).Our characterizations in Theorems 1 and 2 use three simple axioms. As such, the resultingmatchings can fail to be Pareto efficient in some applications. This failure has to do with rathermechanical and inflexible assignment of agents to categories as presented by Example 2. Thiscan be mitigated by filling reserves in a “smart” way. In Section 4.2 we introduce a class ofsmart reserve matching algorithms which does precisely this and through maximal utilization ofreserves always generates a Pareto efficient matching. The smart reserve matching algorithm isa generalization of the horizontal envelope algorithm introduced by S¨onmez and Yenmez (2020).under exclusively “minimum guarantee” type reserves. Moreover, for n = 0 the smart reservematching algorithm becomes equivalent to the horizontal envelope algorithm.26f the two analytical results that extend previously known ideas in the literature, Proposition3 extends the comparative static results of Dur et al. (2018); Dur, Pathak and S¨onmez (2020);Pathak, Rees-Jones and S¨onmez (2020). What is perhaps more novel is Example 1, which showsthat the comparative static exercise in Proposition 3 fails to hold once there are more than fivecategories. Another implication of this situation is that the proof of Proposition 3 is considerablymore involved than its predecessors. Proposition 4, on the other hand, extends the analysis onhorizontal envelope algorithm in S¨onmez and Yenmez (2020) to the more general smart reservematching algorithm.In addition to above described papers which directly relate to our analysis, there are alsoseveral others that have examined allocation in the presence of various distributional constraintssuch as minimum-guarantee reserves (or lower quotas), upper quotas, and regional quotas. Someof the most related ones include Abdulkadiro˘glu (2005), Biro et al. (2010), Kojima (2012),Budish et al. (2013), Hafalir, Yenmez and Yildirim (2013), Westkamp (2013), Ehlers et al.(2014), Echenique and Yenmez (2015), Kamada and Kojima (2015), Kamada and Kojima (2017)Kamada and Kojima (2018), Ayg¨un and Turhan (2020), Bo (2016), Dogan (2016), Kominersand S¨onmez (2016), and Fragiadakis and Troyan (2017).Our paper also introduces the medical rationing into the market design literature. By con-sidering a real-world resource allocation problem, we contribute to the study of formal propertiesof specific allocation processes in the field and the study of alternative mechanisms. Studies inthis vein include those on entry-level labor markets (Roth, 1984; Roth and Peranson, 1999),school choice (Balinski and S¨onmez, 1999; Abdulkadiro˘glu and S¨onmez, 2003), spectrum auc-tions (Milgrom, 2000), kidney exchange (Roth, S¨onmez and ¨Unver, 2004), internet auctions(Edelman, Ostrovsky and Schwarz, 2007; Varian, 2007), course allocation (S¨onmez and ¨Unver,2010; Budish, 2011), cadet-branch matching (S¨onmez and Switzer, 2013; S¨onmez, 2013), assign-ment of airport arrival slots (Schummer and Vohra, 2013; Schummer and Abizada, 2017), andrefugee resettlement (Jones and Teytelboym, 2017; Delacr´etaz, Kominers and Teytelboym, 2019;Andersson, 2019). In Sections 3 and 4, we model pandemic rationing as a one-shot static reserve system. Severalvital resources, however, must be rationed during a pandemic as patients in need present. Hence,it is important to formulate how our static model can be operationalized in an application wherepatient arrival and allocation are both dynamic. The adequate formulation depends on thespecific characteristics of the rationed resource. Most notably, answers to the following twoquestions factor in the implementation details:1.
Is the resource fully consumed upon allocation or is it durable, utilized over a period, andcan it be re-allocated? Is there immediate urgency for allocation?
A unit of a vaccine is consumed upon allocation and reallocation of the unit is not possible.Moreover, there is no immediate urgency to allocate a vaccine. Hence, a large number of unitscan be allocated simultaneously. Therefore, vaccine allocation is an application of our modelwhere our proposed reserve system can be implemented on a static basis as vaccines becomeavailable.This is, however, not the only reasonable way a reserve system can be operationalized forvaccine allocation. In the United States, there is a tradition of distributing influenza vaccinesat local pharmacies or healthcare providers on a first-come-first-serve basis. This practice canbe interpreted as a single-category special case of a reserve system where the priorities arebased on the time of arrival. This practice can easily be extended to any sequential reservematching system with multiple categories where the baseline priorities are determined by thetime of arrival. Under this dynamic implementation of a reserve system, as a patient arrivesto a healthcare provider she is allocated a vaccine as long as there is availability in a categoryfor which she is a beneficiary. If there are multiple such categories, the patient is assigned aunit from the category that has the highest precedence under the sequential reserve matching.While many have criticized first-come-first-serve allocation because of biases it induces basedon access to health care (e.g., Kinlaw and Levine (2007)), reserve categories can be designed tomitigate these biases, even if priority is first-come-first-serve within each category. For example,there can be a reserve category for patients from rural areas. There is an important precedentfor using a reserve system in this dynamic form. Between 2005-2008, H-1B immigration visasin the US were allocated through a reserve system with general and advanced-degree reservecategories where priority for each category was based on the application arrival time (Pathak,Rees-Jones and S¨onmez, 2020).
Since the relevant characteristics of ventilators and ICU beds are identical in relation to ourmodel, the implementation of reserve systems for these resources will be similar. Therefore, wepresent the details of their implementation together. For simplicity in this subsection, we referthe resource in short supply as a ventilator.A ventilator is durable and can be reassigned once its use by its former occupant is com-pleted. Moreover, there is always urgency in allocation of this vital resource. These two featuresmake direct static implementation of a reserve system impractical; implementation always hasto be dynamic. One important observation on ventilator allocation is key to formulate the im-plementation: since a ventilator is durable and assigned to a patient for a period, it can beinterpreted as a good which is allocated at each instant. During the course of using a ventilator,a patient’s clinical situation and her priority for one or more categories may change. Therefore,with the arrival of each new patient, the allocation of all units has to be reevaluated. As such,28he following additional ethical and legal consideration has an important bearing on the designof a reserve system:3.
Can a patient be removed from a ventilator once she is assigned?
There is widespread debate on this issue in the United States. Piscitello et al. (2020) describes25 states with protocols that discuss the ethical basis of re-assigning ventilators. As of June2020, the majority of guidelines support ventilator withdrawal. If a ventilator can be withdrawn,the design is simpler (and effectively identical to static implementation with each new arrival).While patient data needs to be updated through the duration of ventilator use, no fundamentaladjustment is needed for the design of the main parameters of the reserve system. Of course inthis scenario, it is possible that the category of the unit occupied by the patient may changeover time. For example, a patient may initially be assigned a unit from the general categoryeven though she has sufficiently high priority for multiple categories such as the general categoryand essential personnel category. At a later time, she may only have high enough priority forthe latter category. In this case, the patient will continue using the ventilator although foraccounting purposes she will start consuming a unit from a different category.If a ventilator cannot be withdrawn, a reserve system can still be applied with a grandfa-thering structure to reflect the property rights of patients who are already assigned. In thiscase, the priority system has to give highest priority to occupants of the units from any categoryfor as long as they can hold these units despite a change in their clinical situation or arrival ofpatients who otherwise would have higher priority for these units.
For anti-virals drugs and treatments, the vital resource is consumed upon allocation (as invaccine allocation) but there is typically urgency and allocation decisions will need to be madeas patients arrive (as in ventilator allocation). One possible dynamic implementation is basedon first-come-first-serve arrival within reserve categories. This would be akin to the dynamicallocation scenario for ventilators with a baseline priority structure that depends on patientarrival time as described in Section 5.1.1. Alternatively, drug assignment can be batched withinpre-specified time-windows. Drugs can then be assigned based on expectations of the numberof patients in each category over this time window. Since drugs would be administered by aclinician, the relationship between a reserve system and cutoffs can be particularly valuable. Aclinician can simply assign the treatment to a patient if she clears the cutoff for any reservefor which she is eligible. In fact, after the first version of our paper was circulated, some ofthe authors assisted with the design of the system used at the University of Pittsburgh MedicalCenter to allocate the anti-viral drug remdesivir in May 2020 with this implementation. Thesystem had special provisions for hardest hit and essential personnel and used lotteries forprioritization (see White et al. (2020) for more details on this system.)
A common theme of several medical rationing guidelines is the importance of the followingprocedural dimensions: transparency and community engagement; accountability and responsi-29ility; and adaptability and flexibility (WHO, 2007; Prehn and Vawter, 2008). We next brieflydescribe some virtues of a reserve system along these dimensions.Task forces which may include medical personnel, ethicists, and lawyers often decide onprioritization. There is a widespread consensus about the need to accommodate these differentperspectives. For example, a John Hopkins study on community perceptions of medical rationingstates (Biddison et al., 2013):
Both groups felt strongly that no single principle could adequately balance the competingaims and values triggered by allocation decisions. Some felt that a combination ofprinciples should be used.
We think that a clear link between a reserve category and a particular ethical principle mayhelp with community engagement and facilitate compromises between different stakeholders.For example, suppose that a task force values reciprocity and instrumental value, but severalother goals as well. Then they could establish a reserve category for essential personnel for afraction of resources and use other reserve categories to balance these other goals.The transparency associated with reserve categories may be one reason systems are often usedin settings that involve community engagement. For example, following debates between the pro-neighborhood and pro-choice factions, Boston’s school assignment system established a reservewhere half of each school’s seats prioritize applicants from the walk zone in 1999. Likewise,reserves were developed as part of India’s affirmative action system after more than a decade ofcommunity involvement summarized in the 1979 Mandal Commission Report and formulated inthe landmark 1992
Indra Sahwney
Supreme Court case. Similarly, after widespread communityoutreach, OPTN introduced a reserve category for kidney allocation in 2014. This categorygives preferential treatment to patients with the highest expected benefit for 20% of the highestquality deceased-donor kidneys. These reserve-system precedents suggest that stakeholders findreserve categories easy to interpret.The salience of reserve categories also helps with accountability and fostering public trust.In a priority point system, by contrast, it may be more difficult to explain exactly how anallocation rule reflects a particular value when points are aggregated across several dimensions.Moreover, our result relating reserve systems to cutoff equilibrium also simplify the process ofexplanation in situations where a patient is denied a resource. For example, if a patient is deniedan anti-viral, a clinician can show that the patient’s score was below the cutoffs for any categoryin which the patient is eligible.Reserve systems also allow for adaptability and flexibility because of the way the alloca-tion problem is divided into smaller pieces. Within each piece, a prioritization decision canbe adjusted without the need to alter the categories. Emanuel et al. (2020) emphasizes that“prioritization guidelines should differ by intervention and should respond to changing scientificevidence.” In particular, vaccines and anti-viral drugs may rely on different ethical principlesthan ventilators and ICU beds. We see this dimension of flexibility as particularly valuable giventhe need for guidelines to evolve during different stages of a public health emergency. Moreover,for certain resources, criteria may evolve as clinical information becomes available. For example,when there is no information on clinical effectiveness of an anti-viral, it will be difficult chose a30riority to maximize expected health outcomes because those are unknown; this may motivatea lottery within the category. As clinical data emerge, however, these priorities can change,without requiring a change in the category. We next briefly describe some possible reserve categories and their relationship to some currentdebates.
As we discussed in Section 2.2.1, there is widespread agreement on the desirability of prioritizingessential personnel. Yet, Michigan is the only state to provide clear priority for essential person-nel. Several states struggled to find a way to prioritize essential personnel, which led some tocompletely abandon it, others to be vague, and still others to use essential personnel status onlyas a tie-breaker. Adoption of a reserve system that includes a category giving preferential treat-ment to essential personnel provides a tool to accommodate the underlying ethical principles ofreciprocity and instrumental value.
Public health emergencies can have differential impact across communities, and there are callsfor rationing guidelines to respond to differential incidence. For example, growing evidencesuggests that COVID-19 is hitting communities of color harder than other groups (see, e.g.,Price-Haywood et al. (2020) and Sequist (2020)). These concerns have motivated criticisms ofexisting rationing guidelines. Indeed, shortly after Massachusetts released their revised crisisstandards of care in April 2020, Manchanda, Couillard and Sivashanker (2020) criticize themfor exacerbating biases in expected health outcomes driven by discrimination in access to healthcare or other social inequalities. For vaccines, Schmidt (2020) argues that rationing guidelinesshould give priority to groups that have been structurally and historically disadvantaged. Infact, Melinda Gates, a major benefactor of the Global Alliance for Vaccines and Immunization,emphasized that after health care workers, there should be tiered vaccine allocation to “blackpeople next, quite honestly, and many other people of color” (Ducharme, 2020). A reservesystem offers flexibility to accommodate these concerns. A portion of scarce resources couldbe set aside in the form of a disadvantaged category based on legally-permissible measures ofdisadvantage. With either lottery or non-lottery based prioritization, it is also possible to use a reserve system to measurethe effectiveness of the rationed medical resource (see Abdulkadiroglu et al. (2017) and Abdulkadiroglu et al.(2019)). In fact, following circulation of the first draft of this paper, Schmidt (2020) advocated for a reserve systemfor vaccines. .3.3 Disabled Reserve Categories Disabilities rights advocates have opposed rationing plans based on expected health outcomesusing survival probabilities because such criteria are inherently discriminatory. Ne’eman (2020)argues that provisions that exclude certain groups can undermine overall trust in the medicalsystem “based on a well-founded fear of being sacrificed for the greater good.” Persad (2020)recounts that several prefer either random selection or minimal triage that completely ignoresany differences in likelihood or magnitude of benefit, or the likely quantity of resources requiredfor benefit. A reserve system allows for a resolution of this dispute. In particular, a disabledcategory can be established for disabled patients reserving some of the units for these groups. Ifthe representatives of these groups reach a decision to implement random lottery within disabledpatients for these units, this can be implemented under a reserve system without interfering withthe priority order for units in other categories.
Another possible category is a
Good Samaritan reciprocity category , which reserves a fraction ofresources for those who have verified Good Samaritan acts. In such a category, a small fractionof resources are reserved for those who have saved lives through their past Good Samaritan acts.These could be participants for clinical trials on vaccine or treatment development (Emanuelet al., 2020), altruistic donors who have donated their kidneys to a stranger, or people whohave donated large quantities of blood over the years. Good Samaritan status can also beprovided for compatible patient-donor pairs who voluntarily participate in kidney exchangeeven though they do not have to, and save another patient’s life who was incompatible withhis/her donor. This type of incentive could save a large number of lives. S¨onmez, ¨Unver andYenmez (2020) estimate 180 additional kidney patients could receive living donor transplants forevery 10 percent of compatible pairs who can be incentivized to participate in kidney exchange.Community engagement exercises can determine which acts “deserve” a Good Samaritan status.In addition to the ethical value of reciprocity, this category can also be motivated from incentives.The mere existence of a modest reserve of this nature may mitigate more persistent and ongoingcrises in other healthcare domains through these incentives.
While our analysis pertains to the rationing problem of a single entity, like a hospital system, itcan be extended to multiple entities. This extension would allow for considerations that can alsoreduce waste in the system. For example, hospitals in the system can “loan” their unused unitsto the system, say to a virtual hospital that consists of excess units loaned to the system, andthey can earn credit from the system for future use of the units at the virtual hospital when theyhave a shortage. Hospitals can be incentivized to loan their unused units to the virtual hospital, The Disability Rights Education and Defense Fund argues that the probability of survival should not beconsidered as long as it is positive (DREDF, 2020). That is, they claim that the “use of a disability to denyor limit an individual’s access to health care or to provide them a lower priority in accessing scarce resources orsupplies constitutes a clear violation of disability nondiscrimination law.”
32f their patients receive some priority for some of the units in the virtual hospital. There canbe a specific reserve category where priorities may depend on credits earned by the hospitals,while for another category priorities may be determined by clinical criteria only.
Because of the anticipated and ongoing shortage of key medical resources during public healthemergencies, several leaders in the medical ethics community have made important recommenda-tions regarding medical rationing. These recommendations reflect compromises between severalethical principles – maximizing lives, maximizing life-years, life-cycle considerations, instrumen-tal values, reciprocity, protecting to the sickest, and non-exclusion. We have argued that areserve system offers additional flexibility to balance competing objectives.The theory of reserve systems we develop is based on three simple criteria. Our formalresults relate these abstract principles to a cutoff equilibrium concept, and we provide a completecharacterization of all cutoff equilibria through a hypothetical two-sided matching market. Wethen focus on an intuitive class of reserve matchings and analyze comparative static propertieswithin this class. Finally, we develop several formal results for scenarios where there is a baselinepriority order.Aside from our main motivation of pandemic rationing, our results can be directly applied toseveral other resource allocation problems with reserves. These include immigration visa alloca-tion in the United States (Pathak, Rees-Jones and S¨onmez, 2020), affirmative action in schoolchoice systems in Boston, Chicago, New York City and Chile (Dur et al., 2018; Dur, Pathak andS¨onmez, 2020; Correa et al., 2019), affirmative action for public school and government positionsin India (Ayg¨un and Turhan, 2017; Baswana et al., 2019; S¨onmez and Yenmez, 2019 a , b ), anddiversity plans for college admissions in Brazil (Ayg¨un and Bo, 2020). We leave explorations ofthese connections to future research.We hope that the reserve system we have analyzed will rarely be deployed. However, evenif rationing guidelines are rarely applied, their mere existence reflects a statement of values.Several aspects of the design, including those related to essential personnel, disadvantaged anddisabled communities, adults and children run the risk of upsetting social harmony. For example,Fink (2020) describes a risk with poorly designed systems excluding certain principles: “at theend you have got a society at war with itself. Some people are going to be told they don’t matterenough.” We have shown that a reserve system provides policy makers with an additional toolto navigate these complex challenges. Although confronting scarcity in life-and-death situationsis dire, we also hope some of our ideas, such as creating a Good Samaritan category, hold thepotential to make progress on longstanding medical problems beyond immediate public healthemergencies. 33 ppendix A Proofs (Online Appendix) A.1 Proofs of Results in Section 3
Proof of Theorem 1.Part 1.
Suppose matching µ ∈ M complies with eligibility requirements, is non-wasteful, andrespects priorities. We construct a cutoff vector f ∈ F as follows: For each category c ∈ C ,define f c = (cid:40) min π c µ − ( c ) if | µ − ( c ) | = r c , ∅ otherwise . Fix a category c ∈ C . If | µ − ( c ) | = r c then f c ∈ µ − ( c ) by construction. Since µ complieswith eligibility requirements, then f c π c ∅ . On the other hand, if | µ − ( c ) | < r c , then f c = ∅ .Therefore, in either case f c π c ∅ . We showed that f ∈ F , i.e., it is a well-defined cutoff vector.Next, we show that ( f, µ ) is a cutoff equilibrium. Condition 2 in cutoff equilibrium definitionis immediately satisfied because if for any c ∈ C , | µ − ( c ) | < r c , then f c = ∅ by construction.We next show that Condition 1 in cutoff equilibrium definition is also satisfied in two parts.Let i ∈ I .(a) We show that µ ( i ) ∈ B i ( f ) ∪{∅} . If µ ( i ) = ∅ then we are done. Therefore, suppose µ ( i ) = c for some c ∈ C . Two cases are possible: – If | µ − ( c ) | = r c , then f c = min π c µ − ( c ), and hence i π c f c . Thus, c ∈ B i ( f ). – If | µ − ( c ) | < r c , then f c = ∅ by construction. Since µ complies with eligibilityrequirements, i π c f c = ∅ . Thus, µ ( i ) ∈ B i ( f ).(b) We show that B i ( f ) (cid:54) = ∅ = ⇒ µ ( i ) ∈ B i ( f ). Suppose B i ( f ) (cid:54) = ∅ ; but to the contraryof the claim, suppose that µ ( i ) (cid:54)∈ B i ( f ). By Condition 1(a) in the definition of a cutoffequilibrium, µ ( i ) = ∅ . Let c ∈ B i ( f ). Since µ respects priorities, then for every j ∈ µ − ( c )we have j π c i . If | µ − ( c ) | = r c , then by construction, f c ∈ µ − ( c ), and hence, f c π c i ,contradicting c ∈ B i ( f ). We conclude that | µ − ( c ) | < r c . Then by construction, f c = ∅ .Since c ∈ B i ( f ), i π c f c = ∅ . These two statements together with µ ( i ) = ∅ contradictnon-wastefulness of µ . Thus, µ ( i ) ∈ B i ( f ).Hence, we showed that ( f, µ ) is a cutoff equilibrium. Part 2.
Conversely, suppose pair ( f, µ ) ∈ F × M is a cutoff equilibrium. We will show thatmatching µ complies with eligibility requirements, is non-wastefulness, and respects priorities. Compliance with eligibility requirements:
Consider a patient i ∈ I . Since by Condition 1(a) ofcutoff equilibrium definition µ ( i ) (cid:54) = ∅ implies µ ( i ) ∈ B i ( f ), we have i π c f c . Since the cutoffsatisfies f c π c ∅ , by transitivity of π c , i π c ∅ . Therefore, µ complies with eligibility requirements. Non-wastefulness:
Let i ∈ I be such that µ ( i ) = ∅ and i π c ∅ for some c ∈ C . We show that | µ − ( c ) | = r c . Then by Condition 1(a) of the definition of a cutoff equilibrium for ( f, µ ), we have34 i ( f ) = ∅ . In particular c / ∈ B i ( f ). Then f c π c i , implying that f c π c ∅ and hence | µ − ( c ) | = r c .Thus, µ is non-wasteful. Respect of Priorities:
Let patient i ∈ I be such that for some category c ∈ C , µ ( i ) = c whilefor some patient j ∈ I , µ ( j ) = ∅ . We show that i π c j , which will conclude that matching µ respects priorities. By Condition 1(b) of cutoff equilibrium definition, B j ( f ) = ∅ . In particular, f c π c j . Since µ ( i ) = c , by Condition 1(a) of cutoff equilibrium definition, c ∈ B i ( f ), implyingthat i π c f c . By transitivity of π c , i π c j . Proof of Lemma 1.
We prove the lemma in three claims. Let matching µ ∈ M comply witheligibility requirement, be non-wasteful, and respect priorities. Claim 1. f µ is the maximum equilibrium cutoff vector supporting µ , i.e., ( f µ , µ ) is a cutoffequilibrium and for every cutoff equilibrium ( f, µ ), f µc π c f c for every c ∈ C . Proof.
We prove the claim in two parts.Part 1.We show that ( f µ , µ ) is a cutoff equilibrium:We restate the definition of f µ given in Equation (1) in the main text: For every c ∈ C , f µc = (cid:40) min π c µ − ( c ) if | µ − ( c ) | = r c ∅ otherwise . By this definition f µ ∈ F . Moreover, Condition 2 in the definition of a cutoff equilibrium istrivially satisfied.We show that Condition 1(a) holds next. Let i ∈ I . If µ ( i ) = ∅ then Condition 1(a) issatisfied for i . Suppose µ ( i ) = c for some c ∈ C . We have i π c min π c µ − ( c ). Moreover i π c ∅ ,as µ complies with eligibility requirements. Thus, i π c f µc ∈ {∅ , min π c µ − ( c ) } , and hence, µ ( i ) ∈ B i ( f µ ), showing Condition 1(a) is satisfied.Finally, we show that Condition 1(b) is satisfied. We prove its contrapositive. Let i ∈ I besuch that µ ( i ) (cid:54)∈ B i ( f µ ). Thus, µ ( i ) = ∅ by Condition 1(a). Let c ∈ C . If | µ − ( c ) | < r c , then f µc = ∅ π c i by non-wastefulness of µ . If | µ − ( c ) | = r c , then j π c i for every j ∈ µ − ( c ) as µ respects priorities; thus, f µc = min π c µ − ( c ) π c i . In either case, we have c (cid:54)∈ B i ( f µ ). Thus, weget B i ( f µ ) = ∅ , showing that Condition 1(b) also holds for ( f µ , µ ), and hence, completing theproof that ( f µ , µ ) is a cutoff equilibrium.Part 2. Let ( f, µ ) be a cutoff equilibrium. We prove that f µc π c f c for every c ∈ C :Suppose, for contradiction, that there exists some category c ∈ C such that f c π c f µc . Then | µ − ( c ) | = r c as ( f, µ ) is a cutoff equilibrium and f c π c ∅ , which follows from the fact that f µc π c ∅ .Thus, f µc = min π c µ − ( c ) π c ∅ by definition and µ complying with eligibility requirements. Thenfor the patient i = f µc , µ ( i ) = c (cid:54)∈ B i ( f ) contradicting that ( f, µ ) is a cutoff equilibrium.Thus, such a category c does not exist, and hence, f µ is the maximum cutoff equilibrium vectorsupporting matching µ . (cid:5) Claim 2. f µ is the minimum equilibrium cutoff vector supporting µ , i.e., ( f µ , µ ) is a cutoffequilibrium and for every cutoff equilibrium ( f, µ ), f c π c f µc for every c ∈ C . Proof.
We prove the claim in two parts. 35art 1. We show that ( f µ , µ ) is a cutoff equilibrium:We restate f µ using its definition in Equation (2): for every c ∈ C , f µc = (cid:40) min π c (cid:110) i ∈ µ ( I ) : i π c x c (cid:111) if x c (cid:54) = ∅∅ otherwisewhere x c is defined as x c = max π c (cid:0) I \ µ ( I ) (cid:1) ∪ {∅} . For every c ∈ C , since x c π c ∅ , we have min π c (cid:8) i ∈ µ ( I ) : i π c x c (cid:9) π c ∅ , if x c (cid:54) = ∅ . Hence, f µc π c ∅ showing that f µ ∈ F .We show that the conditions in the definition of a cutoff equilibrium are satisfied by ( f µ , µ ). Condition 2.
Suppose | µ − ( c ) | < r c for some c ∈ C . For any i ∈ I \ µ ( I ) we have ∅ π c i bynon-wastefulness of µ . Thus, x c = ∅ . This implies f µc = ∅ by its definition. Hence, Condition 2is satisfied. Condition 1(a).
Let i ∈ I . If µ ( i ) = ∅ then Condition 1(a) is satisfied for i . Suppose µ ( i ) = c for some c ∈ C . We have i π c f µc , since we showed in Claim 1 that ( f µ , µ ) is a cutoff equilibrium.Two cases are possible about µ − ( c ): • If | µ − ( c ) | < r c : we showed in proving Condition 2 that x c = ∅ , thus, f µc = f µc = ∅ . Since i π c ∅ , c = µ ( i ) ∈ B i ( f µ ) showing that Condition 1(a) holds for i . • If | µ − ( c ) | = r c : Then f µc = min π c µ − ( c ) π c x c : as otherwise – if x c ∈ I , then µ ( x c ) = ∅ (by definition of x c ) and yet c ∈ B x c ( f µ ), a contradiction to( f µ , µ ) being a cutoff equilibrium; – if x c = ∅ , then (i) x c π c f µc ∈ I contradicts f µ being a cutoff vector, and (ii) x c = f µc contradicts | µ − ( c ) | = r c . Thus we cannot have x c π c f µc in this case either.Thus, f µc π c min π c { i ∈ I : i π c x c } = f µc . Then i π c f µc π c f µc , implying c = µ ( i ) ∈ B i ( f µ ) and showing that Condition 1(a) holds for i . Condition 1(b).
Let i ∈ I be such that B i ( f µ ) (cid:54) = ∅ . Let c ∈ B i ( f µ ). • if x c = ∅ : Then i π c ∅ = f µc . By definition of x c , i ∈ µ ( I ), i.e., µ ( i ) (cid:54) = ∅ . • if x c (cid:54) = ∅ : Then i π c f µc π c x c , which in turn implies that µ ( i ) (cid:54) = ∅ by definition of f µc and x c . 36n either case, by Condition 1(a), µ ( i ) ∈ B i ( f µ ). Thus, Condition 1(b) is satisfied for i .These conclude proving that ( f µ , µ ) is a cutoff equilibrium.Part 2. Let ( f, µ ) be a cutoff equilibrium. We prove that f c π c f µc for every c ∈ C :Suppose to the contrary of the claim f µc π c f c for some c ∈ C . Now, f µc is a patient,because f c π c ∅ by the definition of a cutoff vector. By definition of f µc , x c (cid:54) = ∅ and f µc π c x c .We cannot have x c π c f c , as otherwise, we have c ∈ B x c ( f ); however, by definition of x c , µ ( x c ) = ∅ , contradicting ( f, µ ) is a cutoff equilibrium. Thus f c π c x c . Since x c is eligible for c , f c ∈ I . Furthermore, f c ∈ µ ( I ), since c ∈ B f c ( f ) and ( f, µ ) is a cutoff equilibrium. Therefore, f c ∈ { j ∈ µ ( I ) : j π c x c } . Since f µc = min π c { j ∈ µ ( I ) : j π c x c } , we have f c π c f µc , contradicting f µc π c f c . Therefore, such a category c cannot exist, and hence, f µ is the minimum equilibriumcutoff vector supporting µ . (cid:5) Claim 3.
For any given two cutoff equilibria ( f, µ ) and ( g, µ ) such that f c π c g c for every c ∈ C ,the pair ( h, µ ) is also a cutoff equilibrium where h ∈ F satisfies for every c ∈ C , f c π c h c π c g c . Proof.
We can obtain cutoff vector h from g after a sequence of repeated applications of thefollowing operation: Change the cutoffs of one of the categories c ∈ C of an input vector f (cid:48) ∈ F so that its cutoff increases by one patient and gets closer to h c than f (cid:48) c . We start with f (cid:48) = g tothe sequence. We show that each iteration of this operation results with a new equilibrium cutoffvector g (cid:48) supporting µ and we use this g (cid:48) as the input of the next iteration of the operation.Since the outcome vector gets closer to h at each step, the last cutoff vector of the sequence is h by finiteness of the patient set, and thus, ( h, µ ) is a cutoff equilibrium:Suppose c (cid:48) ∈ C is such that h c (cid:48) π c (cid:48) g c (cid:48) . We prove that for cutoff vector g (cid:48) ∈ F such that g (cid:48) c = g c for every c ∈ C \ { c (cid:48) } and g (cid:48) c (cid:48) = min π c (cid:48) { i ∈ I : i π c (cid:48) g c (cid:48) } , ( g (cid:48) , µ ) is a cutoff equilibrium. Itis straightforward to show that g (cid:48) ∈ F . Observe also that B i ( g (cid:48) ) = B i ( g ) for every i ∈ I \ { j } where j = g c (cid:48) . Three cases are possible regarding j : • If j (cid:54)∈ I : j = ∅ . • If j ∈ I and µ ( j ) = c (cid:48) : Category c (cid:48) ∈ B j ( f ) as ( f, µ ) is a cutoff equilibrium. However, f c (cid:48) π c (cid:48) h c (cid:48) π c (cid:48) j = g c (cid:48) , contradicting that c (cid:48) ∈ B j ( f ). Therefore, this case cannot happen. • If j ∈ I and µ ( j ) (cid:54) = c (cid:48) : Observe that µ ( j ) (cid:54) = ∅ , as c (cid:48) ∈ B j ( g ) and ( g, µ ) is a cutoffequilibrium. Moreover, we have µ ( j ) ∈ B j ( g ), in turn together with µ ( j ) (cid:54) = c (cid:48) implyingthat µ ( j ) ∈ B j ( g (cid:48) ) as g (cid:48) µ ( j ) = g µ ( j ) .These and the fact that ( g, µ ) is a cutoff equilibrium (specifically its Condition 1(b)) show that µ ( i ) ∈ B i ( g (cid:48) ) for every i ∈ I such that B i ( g (cid:48) ) (cid:54) = ∅ , proving Condition 1(b) holds in the definitionof cutoff equilibrium for ( g (cid:48) , µ ).Since ( g, µ ) is a cutoff equilibrium (specifically Conditions 1(a) and 1(b) of the definition)imply that for every i ∈ I , µ ( i ) = ∅ ⇐⇒ B i ( g ) = ∅ . Therefore, we have µ ( i ) = ∅ for every i ∈ I such that B i ( g (cid:48) ) = ∅ , because B i ( g (cid:48) ) ⊆ B i ( g ) = ∅ , where the set inclusion follows from thefact that the cutoffs have weakly increased for each category under g (cid:48) . This and Condition 1(b)that we showed above imply that for all i ∈ I , µ ( i ) ∈ B i ( g (cid:48) ) ∪ {∅} . Thus, Condition 1(a) in thedefinition of a cutoff equilibrium is also satisfied by ( g (cid:48) , µ ).37e show Condition 2 is also satisfied proving that for every c ∈ C , g (cid:48) c = ∅ if | µ − ( c ) | < r c toconclude that ( g (cid:48) , µ ) is a cutoff equilibrium. Suppose | µ − ( c ) | < r c for some c ∈ C . If c (cid:54) = c (cid:48) , then g (cid:48) c = g c = ∅ , where the latter equality follows from ( g, µ ) being a cutoff equilibrium (specificallyits Condition 2). If c = c (cid:48) , f c (cid:48) = ∅ π c (cid:48) g c (cid:48) , where the first equality follows from ( f, µ ) being acutoff equilibrium (specifically its Condition 2). This contradicts. g ∈ F . So c (cid:54) = c (cid:48) , completingthe proof. (cid:5) Proof of Theorem 2.
Sufficiency : We first prove that any DA-induced matching complies with eligibility require-ments, is non-wasteful, and respects priorities. Let (cid:31)∈ P be a preference profile of patients overcategories and ∅ . Suppose µ ∈ M is DA-induced from this preference profile. Compliance with eligibility requirements:
Suppose that µ ( i ) = c for some c ∈ C . Then i mustapply to c in a step of the DA algorithm, and hence, c (cid:31) i ∅ . By construction of (cid:31) i , this means i π c ∅ . Therefore, matching µ complies with eligibility requirements. Non-wastefulness:
Suppose that i π c ∅ and µ ( i ) = ∅ for some category c ∈ C and patient i ∈ I .By construction of (cid:31) i , c (cid:31) i ∅ because i is eligible for c . As patient i is unmatched in µ , sheapplies to c in some step of the DA algorithm. However, c rejects i at this or a later step. Thismeans, c should have been holding at least r c offers from eligible patients at this step. Fromthis step on, c always holds r c offers and eventually all of its units are assigned: (cid:12)(cid:12) µ − ( c ) (cid:12)(cid:12) = r c .Hence, matching µ is non-wasteful. Respecting priorities:
Suppose that µ ( i ) = c and µ ( i (cid:48) ) = ∅ for two patients i, i (cid:48) ∈ I and acategory c ∈ C . For every category c (cid:48) ∈ C , π c (cid:48) is used to choose eligible patients at every stepof the DA algorithm. Therefore, µ ( i ) = c implies i π c ∅ . Since µ ( i (cid:48) ) = ∅ , then it must be eitherbecause ∅ π c i (cid:48) or because i π c i (cid:48) . In the first case, we get i π c i (cid:48) as well because π c is transitive.Therefore, matching µ respects priorities. Necessity:
We now prove that any matching µ ∈ M with the three stated properties is DA-induced from some preference profile. We construct a candidate preference profile (cid:31)∈ P asfollows: • Consider a patient i ∈ µ − ( c ) where c ∈ C . Since µ complies with eligibility requirements, i must be eligible for category c . Let i rank category c first in (cid:31) i . The rest of the rankingin (cid:31) i is arbitrary as long as all eligible categories are ranked above the empty set. • Consider an unmatched patient i ∈ µ − ( ∅ ). Let i rank categories in any order in (cid:31) i suchthat only eligible categories are ranked above the empty set.We now show that µ is DA-induced from preference profile (cid:31) . In the induced DA algorithmunder (cid:31) , for every category c (cid:48) ∈ C , patients in µ − ( c (cid:48) ) apply to category c (cid:48) first. Every unmatchedpatient j ∈ µ − ( ∅ ) applies to her first-ranked eligible category according to (cid:31) j , if there is any.Suppose c ∈ C is this category. Since µ respects priorities, j has a lower priority than any patientin µ − ( c ), who also applied to c in Step 1. Furthermore, since µ is non-wasteful, (cid:12)(cid:12) µ − ( c ) (cid:12)(cid:12) = r c (asthere are unmatched eligible patients for this category, for example j ). Therefore, all unmatched38atients in µ are rejected at the first step of the DA algorithm. Moreover, for every category c (cid:48) ∈ C , all patients in µ − ( c (cid:48) ) are tentatively accepted by category c (cid:48) at the end of Step 1.Each unmatched patient in j ∈ µ − ( ∅ ) continues to apply according to (cid:31) j to the othercategories at which she is eligible. Since µ respects priorities and is non-wasteful, she is rejectedfrom all categories for which she is eligible one at a time, because each of these categories c ∈ C continues to tentatively hold patients µ − ( c ) from Step 1 who all have higher priority than j according to π c , as µ respects priorities. Moreover, by non-wastefulness of µ , (cid:12)(cid:12) µ − ( c ) (cid:12)(cid:12) = r c , asthere are unmatched eligible patients (for example j ) under µ .As a result, when the algorithm stops, the outcome is such that, for every category c (cid:48) ∈ C , allpatients in µ − ( c (cid:48) ) are matched with c (cid:48) . Moreover, every patient in µ − ( ∅ ) remains unmatchedat the end. Therefore, µ is DA-induced from the constructed patient preferences (cid:31) . Proof of Proposition 1.
Let (cid:46) ∈ ∆ be a precedence order and ϕ (cid:46) be the associated sequentialreserve matching. We show that ϕ (cid:46) is DA-induced from preference profile (cid:31) (cid:46) = ( (cid:31) (cid:46)i ) i ∈ I .For every patient i ∈ I , consider another strict preference relation (cid:31) (cid:48) i such that all categoriesare ranked above the empty set and, furthermore, for any c, c (cid:48) ∈ C , c (cid:31) (cid:48) i c (cid:48) ⇐⇒ c (cid:46) c (cid:48) . Note that the relative ranking of two categories for which i is eligible is the same between (cid:31) (cid:46)i and (cid:31) (cid:48) i .We use an equivalent version of the DA algorithm as the one given in the text. Consider aStep k : Each patient i who is not tentatively accepted currently by a category applies to thebest category that has not rejected her yet according to (cid:31) (cid:48) i . Suppose that I kc is the union of theset of patients who were tentatively assigned to category c in Step k − c . Category c tentatively assigns eligible patients in I kc with thehighest priority according to π c until all patients in I kc are chosen or all r c units are allocated,whichever comes first, and permanently rejects the rest.Since for any category c ∈ C and any patient i ∈ I who is ineligible for category c , ∅ π c c , theoutcome of the DA algorithm when the preference profile is (cid:31) (cid:46) and (cid:31) (cid:48) = ( (cid:31) (cid:48) i ) i ∈ I are the same.Furthermore, when the preference profile is (cid:31) (cid:48) , the DA algorithm works exactly like thesequential reserve procedure that is used to construct ϕ (cid:46) . We show this by induction. Suppose (cid:46) orders categories as c (cid:46) c (cid:46) . . . (cid:46) c |C| . As the inductive assumption, for k >
0, supposefor categories c , . . . , c k − , the tentative matches at the end of Step k − (cid:31) (cid:48) are identical to their matches in sequential reservematching ϕ (cid:46) .We next consider Step k of the DA algorithm from (cid:31) (cid:48) . Only patients who are rejected fromcategory c k − apply in Step k of the DA algorithm and they all apply to category c k . Then c k uses its priority order π c k to tentatively accept the r c k highest-priority eligible applicants (andif there are less than r c k eligible applicants, all eligible applicants), and rejects the rest. Observethat since every patient who is not tentatively accepted by a category c , . . . , c k − applied tothis category in Step k , none of these patients will ever apply to it again; and by the inductiveassumption no patient who is tentatively accepted in categories c , . . . , c k − will ever be rejected,39nd thus, they will never apply to c k , either. Thus, the tentative acceptances by c k will becomepermanent at the end of the DA algorithm. Moreover, this step is identical to Step k of thesequential reserve procedure under precedence order (cid:46) and the same patients are matched withcategory c k in ϕ (cid:46) . This ends the induction.Therefore, we conclude that ϕ (cid:46) is DA-induced from patient preference profile (cid:31) (cid:46) . Proof of Proposition 2.
Let J (cid:46) , J (cid:46) (cid:48) ⊆ I be the sets of patients remaining just before category c is processed under the sequential reserve matching procedure induced by (cid:46) and (cid:46) (cid:48) , respectively.Since c is processed earlier under (cid:46) (cid:48) and every other category preceding c and c (cid:48) under (cid:46) and (cid:46) (cid:48) are ordered in the same manner order, J (cid:46) ⊆ J (cid:46) (cid:48) . Two cases are possible:1. If | ϕ − (cid:46) (cid:48) ( c ) | < r c : Then J (cid:46) ⊆ J (cid:46) (cid:48) implies | ϕ − (cid:46) ( c ) | < r c . Therefore, by Equation (1), f ϕ (cid:46) (cid:48) c = ∅ = f ϕ (cid:46) c .2. If | ϕ − (cid:46) (cid:48) ( c ) | = r c : Then J (cid:46) ⊆ J (cid:46) (cid:48) implies, f ϕ (cid:46) (cid:48) c = min π c ϕ − (cid:46) (cid:48) ( c ) π c min π c ϕ − (cid:46) ( c ) , where the first equality follows by Equation (1). By the same equation, f ϕ (cid:46) c ∈ {∅ , min π c ϕ − (cid:46) ( c ) } and by the definition of a cutoff vector, f ϕ (cid:46) (cid:48) c π c ∅ . Thus, f ϕ (cid:46) (cid:48) c π c f ϕ (cid:46) c . A.2 Proofs of Lemma 2, Proposition 4, and Theorem 3 in Section 4
Proof of Lemma 2.
By the definition of the smart reserve matching algorithm inducedby assigning n unreserved units subsequently at the beginning, in Step 1.( k ) for every k ∈{ , , . . . , | I |} , and for every every matching µ ∈ M k , • µ ( i ) = u for every i ∈ J uk , and • µ ( i ) (cid:54)∈ { u, ∅} and i ∈ I µ ( i ) for every i ∈ J k .We show that for any i ∈ I \ ( J u | I | ∪ J | I | ) there is no matching µ ∈ M | I | such that µ ( i ) (cid:54)∈ { u, ∅} and i ∈ I µ ( i ) . Suppose contrary to the claim that such a patient i and matching µ exist. Patient i is processed in some Step 1.( k ). We have i / ∈ J uk ∪ J k ⊆ J u | I | ∪ J | I | . We have µ ∈ M k − as M k − ⊇ M | I | . Then | J uk − | = n , as otherwise we can always match i with u even if we cannotmatch her with a preferential treatment category that she is a beneficiary of when she is processedunder a matching that is maximal in beneficiary assignment, contradicting i / ∈ J u | I | ∪ J | I | . Butthen as µ ( i ) (cid:54)∈ { u, ∅} and i ∈ I µ ( i ) , we have µ ∈ M k and i ∈ J k , contradicting again i / ∈ J u | I | ∪ J | I | .Thus, in Step 2 no patient is matched with a preferential treatment categories that she isa beneficiary of. These prove ∪ c ∈C\{ u } ( σ − ( c ) ∩ I c ) is the same set regardless of the matching σ ∈ M nS we choose.To prove that σ − ( u ) is the same for every σ ∈ M nS , we consider two cases (for Step 2):40 If we have a soft reserve system: Then all patients are eligible for the remaining max { , q −| J u | I | ∪ J | I | |} units. Since we assign the patients in I \ ( J u | I | ∪ J | I | ) based on priority accordingto π to the remaining units, we have σ ( I ) is the same patient set regardless of the matching σ ∈ M nS we choose. Since the remaining r u − n unreserved units are assigned to the lowestpriority patients that are matched in any σ ∈ M nS , if any, and µ − ( u ) is a fixed setregardless of which matching µ ∈ M | I | we choose (as we proved above), then σ − ( u ) isthe same set regardless of which σ ∈ M nS we choose. • If we have a hard reserve system: Then in Step 2 no patient is matched with a unit of acategory that she is not a beneficiary of. Thus, the r u − n remaining unmatched units areassigned to the highest π -priority patients in I \ ( J u | I | ∪ J | I | ). This concludes proving thateach of σ ( I ) and σ − ( u ) is the same patient set regardless of which σ ∈ M nS we choose. Proof of Proposition 4.
For any n ∈ { , , . . . , r u } , we prove that every smart reservematching in M nS complies with eligibility requirements, is non-wasteful, respects priorities, andis maximal in beneficiary assignment. Compliance with eligibility requirements:
By construction, no patient is ever matched with acategory for which she is not eligible during the procedure.
Non-wastefulness:
Suppose to the contrary of the claim that there exists some σ ∈ M nS that iswasteful. Thus, there exists some category c ∈ C and a patient i ∈ I such that σ ( i ) = ∅ , i π c ∅ ,and | µ − ( c ) | < r c . Then in Step 2 patient i or another patient should have been matched with c as we assign all remaining units to eligible patients, which is a contradiction. Respect for Priorities:
Let σ ∈ M nS be a smart reserve matching. Suppose patients i, j ∈ I aresuch that i π j and σ ( i ) = ∅ . We need to show either (i) σ ( j ) = ∅ or (ii) i (cid:54)∈ I σ ( j ) and j ∈ I σ ( j ) ,which is equivalent to j π σ ( j ) i . Suppose σ ( j ) (cid:54) = ∅ . Suppose to the contrary that i ∈ I σ ( j ) and j ∈ I σ ( j ) . Consider the smart reserve matching procedure with n . Two cases are possible: j ∈ J u | I | ∪ J | I | or not. We show that either case leads to a contradiction, showing that σ respectspriorities. • If j ∈ J u | I | ∪ J | I | : Consider the matching ˆ σ obtained from σ as follows: ˆ σ ( i ) = σ ( j ), ˆ σ ( j ) = ∅ ,and ˆ σ ( i (cid:48) ) = σ ( i (cid:48) ) for every i (cid:48) ∈ I \ { i, j } . Since i, j ∈ I σ ( j ) , and we match i instead of j with σ ( j ), ˆ σ is a matching that is maximal in beneficiary assignment as well. Since i π j , i is processed before j in Step 1. Let i be processed in some Step 1.( k ). Since σ ( i ) = ∅ , i (cid:54)∈ J k ∪ J uk . Then ˆ σ ∈ M k − as σ ∈ M k − . Two cases are possible: – if σ ( j ) = u : As j is matched with an unreserved unit in Step 1, then an unreservedunit is still to be allocated in the procedure when i is to be processed in Step 1.( k )before j . We try to match i with an unreserved unit first. Since ˆ σ ∈ M k − andˆ σ ( i ) = u this implies i ∈ J uk . This contradicts i (cid:54)∈ J uk ∪ J k . – if σ ( j ) (cid:54) = u : Then when i is to be processed in Step 1.( k ), we are trying to matchher (i) if it is possible, with unreserved category u first and if not, with a preferential41reatment category that she is a beneficiary of, or (ii) directly with a preferentialtreatment category that she is a beneficiary of without sacrificing the maximality inbeneficiary assignment. However, as σ ( i ) = ∅ we failed in doing either. Since i ∈ I ˆ σ ( i ) and ˆ σ ( i ) (cid:54) = u , at least there exists a matching in M k − that would match i with apreferential treatment category that she is a beneficiary of. Hence, this contradicts i (cid:54)∈ J uk ∪ J k . • If j (cid:54)∈ J u | I | ∪ J | I | : Therefore, j is matched in Step 2 of the smart reserve matching algorithmwith n unreserved units processed first. Since we match patients in Step 2 either with thepreferential treatment categories that they are eligible but not beneficiary of or with theunreserved category u , and we assumed j ∈ I σ ( j ) then σ ( j ) = u . Since i is also availablewhen j is matched, and i π j , patient i or another patient who has higher π -priority than j should have been matched instead of j , which is a contradiction. Maximality in Beneficiary Assignment:
By construction M nS ⊆ M , which is the set of match-ings that are maximal in beneficiary assignment in the smart reserve matching procedure with n . The following lemma and concepts from graph theory will be useful in our next proof. Westate the lemma as follows: Lemma 3 (Mendelsohn and Dulmage Theorem, 1958) . Let M b be the set of matchings thatmatch patients with only preferential treatment categories that they are beneficiaries of andotherwise leave them unmatched. If there is a matching in M b that matches patients in some J ⊆ I and there is another matching ν ∈ M b then there exists a matching in M b that matchesall patients in J and at least | ν − ( c ) | units of each category c ∈ C \ { u } . See for example page 266 of Schrijver (2003) for a proof of this result.Let us define I ∅ = ∅ for notational convenience.We define a beneficiary alternating path from µ to ν for two matchings in µ, ν ∈ M asa non-empty list A = ( i , . . . , i m ) of patients such that (cid:2) i (cid:54)∈ I µ ( i ) or µ ( i ) = u (cid:3) & (cid:2) i ∈ I ν ( i ) and ν ( i ) (cid:54) = u (cid:3) , (cid:2) i m ∈ I µ ( i m ) and µ ( i m ) = ν ( i m − ) (cid:3) & (cid:2) i m ∈ I ν ( i m ) and ν ( i m ) (cid:54) = u (cid:3) for every m ∈ { , , . . . , m − } , (cid:2) i m ∈ I µ ( i m ) and µ ( i m ) = ν ( i m − ) (cid:3) & (cid:2) i m (cid:54)∈ I ν ( i m ) or ν ( i m ) = u (cid:3) . A beneficiary alternating path begins with a patient i who is not matched with a preferentialtreatment category that she is a beneficiary of under µ and ends with a patient i m who is notmatched with a preferential treatment category that she is a beneficiary of under ν . Everybodyelse in the path is matched under both matchings with a preferential treatment category that sheis a beneficiary of. We state the following observation, which directly follows from the finitenessof categories and patients. 42 bservation 1. If µ and ν ∈ M are two matchings such that for every c ∈ C\{ u } , | µ − ( c ) ∩ I c | = | ν − ( c ) ∩ I c | and there exists some i ∈ (cid:0) ∪ c ∈C\{ u } ν − ( c ) ∩ I c (cid:1) \ (cid:0) ∪ c ∈C\{ u } µ − ( c ) ∩ I c (cid:1) , then thereexists a beneficiary alternating path from µ to ν beginning with patient i . We are ready to prove our last theorem:
Proof of Theorem 3.
By Proposition 4 and Theorem 1, any σ ∈ M S and σ r u ∈ M r u S are cutoff equilibrium matchings. Let µ ∈ M be any other cutoff equilibrium matching that ismaximal in beneficiary assignment.We extend the definitions of our concepts to smaller economies: given any I ∗ ⊆ I and r ∗ = ( r ∗ c ) c ∈C such that r ∗ c ≤ r c for every c ∈ C , all properties and algorithms are redefined forthis smaller economy ( I ∗ , r ∗ ) by taking the restriction of the baseline priority order π on I ∗ , anddenoted using the argument ( I ∗ , r ∗ ) at the end of the notation. For example M ( I ∗ , r ∗ ) denotesthe set of matchings for ( I ∗ , r ∗ ). Proof of f σ ru u π f µu :We prove the following claim first. Claim 1.
For any set of patients I ∗ ⊆ I and any capacity vector r ∗ ≤ r , suppose matching ν ∈ M ( I ∗ , r ∗ ) is maximal in beneficiary assignment for ( I ∗ , r ∗ ). Let σ ∈ M r ∗ u S ( I ∗ , r ∗ ) be a smartreserve matching with all unreserved units processed first. Then | σ − ( u ) | ≥ | ν − ( u ) | . Moreover, according to the baseline priority order π , for any k ∈ (cid:110) , . . . , (cid:12)(cid:12) ν − ( u ) \ σ − ( u ) (cid:12)(cid:12)(cid:111) ,let j k be the k th highest priority patient in σ − ( u ) \ ν − ( u ) and j (cid:48) k be the k th highest prioritypatient in ν − ( u ) \ σ − ( u ), then j k π j (cid:48) k . Proof.
Suppose to the contrary of the first statement | σ − ( u ) | < | ν − ( u ) | . Since both σ and ν are maximal in beneficiary assignment for ( I ∗ , r ∗ ), then the exists some patient i ∈ ( ∪ c ∈C ν − ( c ) ∩ I c ) \ ( ∪ c ∈C σ − ( c ) ∩ I c ). This patient is not committed to be matched in Step 1 of the smartreserve matching algorithm with all unreserved units first, despite the fact that there exists atleast one available unreserved unit when she was processed, which is a contradiction. Thus, | σ − ( u ) | ≥ | ν − ( u ) | .For the rest of the proof, we use induction on the cardinality of I ∗ and on the magnitude ofvector of category capacity vector r ∗ : • For the base case when I ∗ = ∅ and r ∗ c = 0 for every c ∈ C , the claim holds trivially. • As the inductive assumption, suppose that for all capacity vectors of categories boundedabove by vector r ∗ and all subsets of I bounded above by cardinality k ∗ , the claim holds.43 Consider a set of patients I ∗ ⊆ I such that | I ∗ | = k ∗ and capacity vector of categories r ∗ = ( r ∗ c ) c ∈C . Let σ ∈ M r ∗ u S ( I ∗ , r ∗ ) be a smart reserve matching for ( I ∗ , r ∗ ) with allunreserved units processed first and ν ∈ M ( I ∗ , r ∗ ) be maximal in beneficiary assignmentfor ( I ∗ , r ∗ ). If σ − ( u ) ⊇ ν − ( u ) then the claim for ( I ∗ , r ∗ ) is trivially true. Thus, supposenot. Then, there exists j ∈ ν − ( u ) \ σ − ( u ). Moreover, let j be the highest π -prioritypatient in ν − ( u ) \ σ − ( u ). We have two cases that we consider separately:Case 1. There is no patient i ∈ I ∗ such that i π j and i ∈ σ − ( u ) \ ν − ( u ):We show that this case leads to a contradiction, and hence, it cannot hold.When j is processed in Step 1 of the smart reserve matching algorithm with allunreserved units processed first, since σ ( j ) (cid:54) = u , either(i) all units of the unreserved category are assigned under σ to patients with higher π -priority than j , or(ii) some unreserved category units are still available when j is processed.Observe that (i) cannot hold, because it contradicts Case 1. Thus, (ii) holds.Since j is the highest π -priority patient in ν − ( u ) \ σ − ( u ) and since we are in Case1, for every i ∈ I ∗ such that i π j , we have ν ( i ) = u ⇐⇒ σ ( i ) = u. (3)We construct a new matching ˆ ν ∈ M ( I ∗ , r ∗ ) from ν and fix a patient i ∈ I ∗ as follows.We check whether there exists a patient i ∈ I ∗ such that i π j, σ ( i ) (cid:54) = u, and i (cid:54)∈ I ν ( i ) , (4)(a) If such a patient i does not exist, then let ˆ ν = ν and i = j .(b) If such a patient i exists, then let her be the highest π -priority patient with theproperty in Equation 4.We show that i ∈ I σ ( i ) . Consider the smart reserve matching algorithm. Since ν ( j ) = u and i π j , just before i is processed in Step 1, there is still at least oneunreserved unit available by Equation 3. Since we are processing all unreservedunits first and since σ ( i ) (cid:54) = u , it should be the case that we had to match i witha preferential treatment category that she is a beneficiary of. Thus, i ∈ I σ ( i ) .We create a new matching for ( I ∗ , r ∗ ) from ν , which we refer to as ˆ ν , such thatˆ ν matches every patient in I ∗ exactly as under ν except that ˆ ν leaves patient j unmatched and matches i with category u instead. Since ν is maximal inbeneficiary assignment for ( I ∗ , r ∗ ), so is ˆ ν .So far, we have for every i (cid:48) ∈ I ∗ such that i (cid:48) π i ,1. σ ( i (cid:48) ) = u ⇐⇒ ν ( i (cid:48) ) = u (by Equation 3 and i π j ),2. σ ( i (cid:48) ) ∈ I σ ( i (cid:48) ) (an unreserved unit is available before i is processed in Step 1 ofthe smart reserve matching algorithm with all unreserved units processed first;thus, every patient processed before i is matched if possible, with u , and if notpossible, with a preferential treatment category that she is a beneficiary of underthe restriction of maximality in beneficiary assignment), and44. ˆ ν ( i (cid:48) ) ∈ I ˆ ν ( i (cid:48) ) (by definition of i as the highest π -priority patient satisfying Equa-tion 4).We also have ˆ ν ( i ) = u and σ ( i ) (cid:54) = u .Patient i is processed in some Step 1.( k ) in the smart reserve matching algorithmwith all unreserved units processed first. As σ ( i ) (cid:54) = u we have i / ∈ J uk ( I ∗ , r ∗ ). Onthe other hand, since σ ∈ M k − ( I ∗ , r ∗ ), by Statements 1, 2, and 3 above, we haveˆ ν ∈ M k − ( I ∗ , r ∗ ) as well and it matches i with u , contradicting i / ∈ J uk ( I ∗ , r ∗ ).Therefore, Case 1 (ii) cannot hold either.Case 2. There is some i ∈ σ − ( u ) \ ν − ( u ) such that i π j :Construct a matching ˆ σ from σ that it leaves every patient who is matched in Step2 of the smart reserve matching algorithm with all unreserved units processed first:for every i ∗ ∈ I ∗ , ˆ σ ( i ∗ ) = σ ( i ∗ ) if i ∗ ∈ I σ ( i ∗ ) and ˆ σ ( i ∗ ) = ∅ otherwise. Clearlyˆ σ ∈ M ( I ∗ , r ∗ ) and is maximal in beneficiary assignment for ( I ∗ , r ∗ ), since σ is. ByLemma 3, there exists a matching ˜ ν ∈ M b ( I ∗ , r ∗ ) such that under ˜ ν all patients in ∪ c ∈C\{ u } ˜ ν − ( c ) ∩ I c are matched with the preferential treatment categories in C \ { u } that they are beneficiaries of, and for every c ∈ C \ { u } , | ˜ ν − ( c ) ∩ I c | = | ˆ σ − ( c ) ∩ I c | (equality follows rather than ≥ as dictated by the lemma, because ˆ σ is maximalin beneficiary assignment for ( I ∗ , r ∗ )). For every i ∗ ∈ I ∗ , we have ν ( i ∗ ) = u = ⇒ ˜ ν ( i ∗ ) = ∅ as ˆ ν ∈ M b ( I ∗ , r ∗ ). We modify ˜ ν to obtain ˆ ν : For every i ∗ ∈ ν − ( u ), we setˆ ν ( i ∗ ) = u and for every i ∗ ∈ I ∗ \ ν − ( u ), we set ˆ ν ( i ∗ ) = ˜ ν ( i ∗ ). Clearly, ˆ ν ∈ M ( I ∗ , r ∗ )and is maximal in beneficiary assignment for ( I ∗ , r ∗ ), since ν is. We will work withˆ σ and ˆ ν instead of σ and ν from now on.Recall that ˆ σ ( i ) = u and ˆ ν ( i ) (cid:54) = u . Two cases are possible: i ∈ I ˆ ν ( i ) or ˆ ν ( i ) = ∅ .1. If i ∈ I ˆ ν ( i ) : Then by Observation 1, there exists a beneficiary alternating path A from ˆ σ to ˆ ν beginning with i and ending with some i (cid:48) ∈ I ∗ such that (i)ˆ σ ( i (cid:48) ) ∈ I ˆ σ ( i (cid:48) ) and ˆ σ ( i (cid:48) ) (cid:54) = u , and (ii) ˆ ν ( i (cid:48) ) = u or ˆ ν ( i (cid:48) ) = ∅ .By the existence of the beneficiary alternating path, it is possible to match either i or i (cid:48) with a preferential treatment category that she is a beneficiary of andmatch the other one with u without changing the type of match of any otherpatient i ∗ ∈ I ∗ \ { i, i (cid:48) } has, i.e., either i ∗ is matched with a preferential treatmentcategory under both matchings or not. Yet, when i is processed in Step 1 of thesmart reserve matching algorithm with all unreserved units processed first, wechose i to be matched with u and i (cid:48) with a preferential treatment category. Thismeans i π i (cid:48) . Let ˆ c (cid:54) = u be the category that i is matched with under ˆ ν .If ˆ σ ( i (cid:48) ) (cid:54) = u , then modify ˆ ν by assigning an unreserved unit to i (cid:48) instead of j :ˆ ν ( j ) = ∅ and ˆ ν ( i (cid:48) ) = u . Otherwise, do not modify ˆ ν any further. As defined in the hypothesis of the lemma, ˜ ν ∈ M b ( I ∗ , r ∗ ) means that for every i ∗ ∈ I ∗ and c ∈ C , i ∗ ∈ ˜ ν − ( c )implies c (cid:54) = u and i ∗ ∈ I c . I (cid:48) , r (cid:48) ) such that I (cid:48) = I ∗ \ { i, i (cid:48) } and for every c ∈ C , r (cid:48) c = r ∗ c − c ∈ { ˆ c, u } and r (cid:48) c = r ∗ c , otherwise.We show that a smart reserve matching σ (cid:48) ∈ M r (cid:48) u S ( I (cid:48) , r (cid:48) ) can be obtained from theoriginal smart reserve matching σ ∈ M r ∗ u S ( I ∗ , r ∗ ) and ˆ ν . Consider the beneficiaryalternating path A we discovered above starting with patient i and ending withpatient i (cid:48) from ˆ σ to ˆ ν : Suppose A = ( i, i , . . . , i m − , i (cid:48) ). Define σ (cid:48) ( i ∗ ) = σ ( i ∗ ) for every i ∗ ∈ I (cid:48) \ { i , . . . , i m − } and σ (cid:48) ( i ∗ ) = ˆ ν ( i ∗ ) for every i ∗ ∈ { i , . . . , i m − } . Observe that σ (cid:48) ∈ M ( I (cid:48) , r (cid:48) ). The existence of σ (cid:48) shows that it is feasible to matchevery patient in J u | I ∗ | ( I ∗ , r ∗ ) \{ i } with u and it is feasible to match every patient in J | I ∗ | ( I ∗ , r ∗ ) \ { i (cid:48) } with a preferential treatment category that she is a beneficiaryof in ( I (cid:48) , r (cid:48) ). Thus, the smart reserve matching algorithm with all unreservedunits processed first proceeds exactly in the same manner as it does for ( I ∗ , r ∗ )with the exception that it skips patients i and i (cid:48) in the smaller economy ( I (cid:48) , r (cid:48) ).Hence, σ (cid:48) ∈ M r (cid:48) u S ( I (cid:48) , r (cid:48) ).Let the restriction of matching ˆ ν to ( I (cid:48) , r (cid:48) ) be ν (cid:48) . Observe that ν (cid:48) is a matchingfor ( I (cid:48) , r (cid:48) ). Moreover, it is maximal in beneficiary assignment for ( I (cid:48) , r (cid:48) ), since ˆ ν is maximal in beneficiary assignment for ( I ∗ , r ∗ ).Now one of the two following cases holds for ν :(a) If ν ( i (cid:48) ) (cid:54) = u : Recall that while ν ( j ) = u , we updated ˆ ν so that ˆ ν ( i (cid:48) ) = u and ˆ ν ( j ) = ∅ . Thus, ν (cid:48) ( j ) = ∅ as well. Since i π j , this together with theinductive assumption that the claim holds for ( I (cid:48) , r (cid:48) ) imply that the claimalso holds for ( I ∗ , r ∗ ), completing the induction.(b) If ν ( i (cid:48) ) = u : Since i π i (cid:48) , this together with the inductive assumption that theclaim holds for ( I (cid:48) , r (cid:48) ) imply that the claim also holds for ( I ∗ , r ∗ ), completingthe induction.2. If ˆ ν ( i ) = ∅ : Recall that ˆ ν ( j ) = u . We modify ˆ ν further that ˆ ν ( i ) = u andˆ ν ( j ) = ∅ . Consider the smaller economy ( I (cid:48) , r (cid:48) ) where I (cid:48) = I ∗ \ { i } , r (cid:48) u = r ∗ u − r (cid:48) c = r ∗ c for every c ∈ C \ { u } .Let σ (cid:48) and ν (cid:48) be the restrictions of σ and ˆ ν to ( I (cid:48) , r (cid:48) ), respectively. Since, σ ( i ) =ˆ ν ( i ) = u both σ (cid:48) and ν (cid:48) are matchings for ( I (cid:48) , r (cid:48) ). Since the capacity of category u is decreased by one, σ (cid:48) is a smart reserve matching with all unreserved unitsprocessed first for ( I (cid:48) , r (cid:48) ). To see this observe that the algorithm proceeds as itdoes for ( I ∗ , r ∗ ) with the exception that it skips i . Matching ν (cid:48) is maximal inbeneficiary assignment for ( I (cid:48) , r (cid:48) ). Therefore, by the inductive assumption, theclaim holds for ( I (cid:48) , r (cid:48) ). This together with the fact that i π j imply the claimholds for ( I ∗ , r ∗ ), completing the induction. (cid:5) If | µ − ( u ) | < r u then f σ ru u π f µu = ∅ . On the other hand, if | µ − ( u ) | = r u , Claim 1 impliesthat f σ ru u = min π σ − r u ( u ) π f µu = min π µ − ( u ).46 roof of f µu π f σ u : We prove the following claim first.
Claim 2.
For any set of patients I ∗ ⊆ I and any capacity vector r ∗ ≤ r , suppose ν ∈ M ( I ∗ , r ∗ )is a matching that is maximal in beneficiary assignment for ( I ∗ , r ∗ ). Let σ ∈ M S ( I ∗ , r ∗ ) be asmart reserve matching with all unreserved units processed last, J = ∪ c ∈C\{ u } ( σ − ( c ) ∩ I c ) , and J (cid:48) = ∪ c ∈C\{ u } ( ν − ( c ) ∩ I c ) . According to the baseline priority order π , for any k ∈ { , . . . , | J (cid:48) \ J |} , let j k be the k th highestpriority patient in J \ J (cid:48) and j (cid:48) k be the k th highest priority patient in J (cid:48) \ J , then j k π j (cid:48) k . Proof. We use induction on the cardinality of I ∗ and on the magnitude of vector of capacitiesof categories r ∗ : • For the base case when I ∗ = ∅ and r ∗ c = 0 for every c ∈ C , the claim holds trivially. • In the inductive step, suppose for every capacity of categories bounded above by vector r ∗ and subsets of patients in I bounded above by cardinality k ∗ the claim holds. • Consider set of patients I ∗ ⊆ I such that | I ∗ | = k ∗ and capacity vector for categories r ∗ = ( r ∗ c ) c ∈C . If J \ J (cid:48) = ∅ then the claim holds trivially. Suppose J \ J (cid:48) (cid:54) = ∅ . Let i ∈ J \ J (cid:48) be the highest priority patient in J \ J (cid:48) according to π .We have | J | = | J (cid:48) | by maximality of σ and ν in beneficiary assignment for I ∗ . Thus, | J \ J (cid:48) | = | J (cid:48) \ J | , which implies J (cid:48) \ J (cid:54) = ∅ .By Lemma, 3 there exists a matching ˆ ν ∈ M b ( I ∗ , r ∗ ) that matches patients only withpreferential treatment categories that they are beneficiaries of such that ˆ ν matches patientsin J (cid:48) and | σ − ( c ) ∩ I c | units reserved for every preferential treatment category c ∈ C \{ u } . Since both ν and σ are maximal in beneficiary assignment for ( I ∗ , r ∗ ), then only patientsin J (cid:48) should be matched under ˆ ν and no other patients (as otherwise ν would not bemaximal in beneficiary assignment for ( I ∗ , r ∗ )).Since ˆ ν ( i ) = ∅ , i ∈ I σ ( i ) , and σ ( i ) (cid:54) = u , by Observation 1, there exists a beneficiaryalternating path A starting with i from ˆ ν to σ and ending with a patient i (cid:48) ∈ I ˆ ν ( i (cid:48) ) (andˆ ν ( i (cid:48) ) (cid:54) = u by its construction), and yet i (cid:48) / ∈ I σ ( i ) or σ ( i (cid:48) ) = u .Existence of the beneficiary alternating path shows that it is possible to match i or i (cid:48) (but not both) with preferential treatment categories that they are beneficiaries of withoutaffecting anybody else’s status as committed or uncommitted in Step 1 of the smart reserve This claim’s proof is similar to the proof of Proposition 1 Part 2 in S¨onmez and Yenmez (2020). Although both σ and ν may be matching some patients with categories that they are not beneficiaries of orwith the unreserved category, we can simply leave those patients unmatched in σ and ν and apply Lemma 3 tosee such a matching ˆ ν exists. σ matches i with apreferential treatment category that she is a beneficiary of at the cost of patient i (cid:48) , wehave i π i (cid:48) . Next consider the smaller economy ( I (cid:48) , r (cid:48) ) in which we remove (i) i and i (cid:48) and set I (cid:48) = I ∗ \ { i, i (cid:48) } , (ii) one of the units associated with preferential treatment category ˆ c = ˆ ν ( i (cid:48) )and set r (cid:48) ˆ c = r ∗ ˆ c −
1, and (iii) keep the capacity of every other category c ∈ C \ { ˆ c } the sameand set r (cid:48) c = r ∗ c .Let ν (cid:48) be the restriction of ˆ ν to ( I (cid:48) , r (cid:48) ). As ˆ ν ( i ) = ∅ , ˆ ν ( i (cid:48) ) = ˆ c (cid:54) = u such that i ∈ I ˆ c andthe capacity of ˆ c is reduced by 1 in the smaller economy, ν (cid:48) ∈ M ( I (cid:48) , r (cid:48) ), and furthermore,it is maximal in beneficiary assignment for ( I (cid:48) , r (cid:48) ).We form a matching σ (cid:48) ∈ M ( I (cid:48) , r (cid:48) ) by modifying σ and ˆ ν using the beneficiary alternatingpath A we found before. Recall that A is the beneficiary alternating path from ˆ ν to σ beginning with i and ending with i (cid:48) . Suppose A = ( i, i , . . . , i m − , i (cid:48) ). Define σ (cid:48) ( i ∗ ) = σ ( i ∗ ) for every i ∗ ∈ I (cid:48) \ { i , . . . , i m − } and σ (cid:48) ( i ∗ ) = ˆ ν ( i ∗ ) for every i ∗ ∈ { i , . . . , i m − } . Observe that σ (cid:48) ∈ M ( I (cid:48) , r (cid:48) ). The existence of σ (cid:48) shows that it is possible to match everypatient in J | I ∗ | ( I ∗ , r ∗ ) \{ i } with a preferential treatment category that she is a beneficiary ofin ( I (cid:48) , r (cid:48) ). Thus, the smart reserve matching algorithm with all unreserved units processedlast proceeds as it does for ( I ∗ , r ∗ ) with the exception that it skips patients i and i (cid:48) .Therefore, σ (cid:48) ∈ M S ( I (cid:48) , r (cid:48) ).By the inductive assumption, the claim holds for σ (cid:48) and ν (cid:48) for ( I (cid:48) , r (cid:48) ). This completes theinduction, as we already showed i π i (cid:48) . (cid:5) Thus, we showed that at the end of Step 1 of the smart matching algorithm with all un-reserved units processed last, weakly lower priority patients have remained uncommitted in J ∗ = I \ J | I | = I \ ∪ c ∈C\{ u } ( σ − ( c ) ∩ I c ) than in ˆ J = I \ ∪ c ∈C\{ u } ( µ − ( c ) ∩ I c ).Two cases are possible: • If we have a soft reserves system: As both σ and µ are maximal in beneficiary assignment,an equal number of units are assigned to the highest π -priority patients in ˆ J (by Step 2 ofthe smart reserve matching algorithm) and J ∗ (as by Theorem 1, µ respects priorities andis non-wasteful). Under σ , the unreserved units are assigned last in order in Step 2 ofthe algorithm. On the other hand, the remainder of µ , i.e., the units assigned to the non-beneficiaries of preferential treatment categories and beneficiaries of u , can be constructedby assigning the rest of the units sequentially to the highest priority patients in J ∗ one byone when unreserved units are not necessarily processed last.Therefore, if | µ − ( u ) | < r u then | σ − ( u ) | < r u , in turn implying f µu = f σ u = ∅ . If | µ − ( u ) | = r u then | σ − ( u ) | ≤ r u and f µu = min π µ − ( u ) π f σ u ∈ {∅ , min π σ − ( u ) } .48 If we have a hard reserves system: The proof is identical as the above case with theexception that now only unreserved units are assigned as both σ and µ comply witheligibility requirements. A.3 Proof of Proposition 3 in Section 4
In this subsection, we first show some lemmas that we will use in the proof of Proposition 3.Fix a soft reserve system induced by the baseline priority order π . Suppose each patient isa beneficiary of at most one preferential treatment category.First, we introduce some concepts.We introduce function τ : I → ( C \ { u } ) ∪ {∅} to denote the preferential treatment categorythat a patient is beneficiary of, if there is such a category. That is, for any patient i ∈ I , if i ∈ I c for some c ∈ C \ { u } , then τ ( i ) = c , and if i ∈ I g , i.e., i is a general-community patient, then τ ( i ) = ∅ .For a category c ∗ ∈ C , a set of patients ˜ I ⊆ I , and a patient i ∈ ˜ I , let rank( i ; ˜ I, π c ∗ ) denotethe rank of i among patients in ˜ I according to π c ∗ .We consider incomplete orders of precedence. For a given subset of categories C ∗ ⊆ C , wedefine an order of precedence on C ∗ as a linear order on C ∗ . Let ∆( C ∗ ) be the set of ordersof precedence on C ∗ .We extend the definition of sequential reserve matchings to cover incomplete precedenceorders and match a subset of patients ˜ I ⊆ I as follows: A sequential reserve matchinginduced by (cid:46) ∈ ∆( C ∗ ) over ˜ I is the outcome of the sequential reserve procedure whichprocesses only the categories in C ∗ in the order of (cid:46) to match only the patients in ˜ I and leavesall categories in C \ C ∗ unmatched and patients in I \ ˜ I unmatched. Let ϕ ˜ I(cid:46) denote this matching.
Lemma 4.
Suppose that ˜ I ⊆ I , c ∈ C \ { u } , and (cid:46), (cid:46) (cid:48) ∈ ∆( { u, c } ) are such that • (cid:46) is given as u (cid:46) c , • (cid:46) (cid:48) is given as c (cid:46) (cid:48) u , • I (2) = ˜ I \ µ ( ˜ I ) where µ = ϕ ˜ I(cid:46) , • I (cid:48) (2) = ˜ I \ µ (cid:48) ( ˜ I ) where µ (cid:48) = ϕ ˜ I(cid:46) (cid:48) , and • µ ( ˜ I c ) (cid:40) ˜ I c .Then the following results hold:1. | I (2) \ I (cid:48) (2) | = | I (cid:48) (2) \ I (2) | ,2. I (cid:48) (2) \ I (2) ⊆ ˜ I c ,3. I (2) \ I (cid:48) (2) ⊆ ˜ I \ ˜ I c , . if i ∈ I (2) \ I (cid:48) (2) and i (cid:48) ∈ I (cid:48) (2) , then i π i (cid:48) , and5. if i (cid:48) ∈ I (cid:48) (2) \ I (2) and i ∈ I (2) ∩ I c , then i (cid:48) π i . Proof of Lemma 4.
The first statement in Lemma 4 holds because under soft reservesevery patient is eligible for every category, which implies that | µ ( ˜ I ) | = | µ (cid:48) ( ˜ I ) | . As a result, | µ ( ˜ I ) \ µ (cid:48) ( ˜ I ) | = | µ (cid:48) ( ˜ I ) \ µ ( ˜ I ) | , which is equivalent to | I (cid:48) (2) \ I (2) | = | I (2) \ I (cid:48) (2) | since µ ( ˜ I ) \ µ (cid:48) ( ˜ I ) = I (cid:48) (2) \ I (2) and µ (cid:48) ( ˜ I ) \ µ ( ˜ I ) = I (2) \ I (cid:48) (2).The second statement in Lemma 4 holds because if i ∈ µ − ( u ), then rank( i ; ˜ I, π ) ≤ r u .Therefore, i ∈ µ (cid:48) ( ˜ I ). Furthermore, every i ∈ µ − ( c ) is a category- c patient since there exists j ∈ ˜ I c such that j / ∈ µ ( ˜ I ). As a result, we get˜ I c ⊇ µ − ( c ) ⊇ µ ( ˜ I ) \ µ (cid:48) ( ˜ I ) = I (cid:48) (2) \ I (2) . To prove the third statement in Lemma 4, suppose for contradiction that there exists i ∈ I (2) \ I (cid:48) (2) such that i ∈ ˜ I c . Therefore, i ∈ µ (cid:48) ( ˜ I ) \ µ ( ˜ I ) = I (2) \ I (cid:48) (2). By the first statementin Lemma 4, | I (cid:48) (2) \ I (2) | = | I (2) \ I (cid:48) (2) | ≥ I (2) \ I (cid:48) (2) has at least one patient. Bythe second statement in Lemma 4, I (cid:48) (2) \ I (2) ⊆ ˜ I c . Therefore, there exists i (cid:48) ∈ I (cid:48) (2) \ I (2) = µ ( ˜ I ) \ µ (cid:48) ( ˜ I ) such that i (cid:48) ∈ ˜ I c . Since i ∈ µ (cid:48) ( ˜ I ), i (cid:48) / ∈ µ (cid:48) ( ˜ I ), and τ ( i ) = τ ( i (cid:48) ), we get i π i (cid:48) . Likewise, i (cid:48) ∈ µ ( ˜ I ), i / ∈ µ ( ˜ I ), and τ ( i ) = τ ( i (cid:48) ) imply i (cid:48) π i. The two displayed relations above contradict each other.The fourth statement in Lemma 4 is true because for every i ∈ I (2) \ I (cid:48) (2) = µ (cid:48) ( ˜ I ) \ µ ( ˜ I )we know that i / ∈ ˜ I c by the third statement in Lemma 4. Since µ ( ˜ I c ) (cid:40) ˜ I c , there are at least r c patients in ˜ I c . Therefore, µ (cid:48)− ( c ) ⊆ ˜ I c , which implies that i ∈ µ (cid:48)− ( u ). Since i (cid:48) ∈ I (cid:48) (2) isequivalent to i (cid:48) / ∈ µ (cid:48) ( ˜ I ), we get i π i (cid:48) .The fifth statement in Lemma 4 follows from i, i (cid:48) ∈ ˜ I c , i (cid:48) ∈ µ ( ˜ I ), and i / ∈ µ ( ˜ I ). Lemma 5.
Suppose that c, c (cid:48) ∈ C \ { u } are different categories. Let ˜ I ⊆ I and (cid:46), (cid:46) (cid:48) ∈ ∆( { c, c (cid:48) } ) be such that • (cid:46) is given as c (cid:48) (cid:46) c , • (cid:46) (cid:48) is given as c (cid:46) (cid:48) c (cid:48) , • I (2) = ˜ I \ µ ( ˜ I ) where µ = ϕ ˜ I(cid:46) , • I (cid:48) (2) = ˜ I \ µ (cid:48) ( ˜ I ) where µ (cid:48) = ϕ ˜ I(cid:46) (cid:48) , and • µ ( ˜ I c ) (cid:40) ˜ I c .Then the following results hold:1. | I (2) \ I (cid:48) (2) | = | I (cid:48) (2) \ I (2) | , . I (cid:48) (2) \ I (2) ⊆ ˜ I c ,3. I (2) \ I (cid:48) (2) ⊆ ˜ I \ ˜ I c ,4. if i ∈ I (2) \ I (cid:48) (2) and i (cid:48) ∈ I (cid:48) (2) , then i π i (cid:48) , and5. if i (cid:48) ∈ I (cid:48) (2) \ I (2) and i ∈ I (2) ∩ ˜ I c , then i (cid:48) π i . Proof of Lemma 5. If | ˜ I c (cid:48) | ≥ r c (cid:48) , then µ ( ˜ I ) = µ (cid:48) ( ˜ I ) and, therefore, I (2) = I (cid:48) (2). Then allthe statements in Lemma 5 hold trivially. Suppose that | ˜ I c (cid:48) | < r c (cid:48) for the rest of the proof.The first statement in Lemma 5 follows as in the proof of the first statement in Lemma 4.The second statement in Lemma 5 holds because if i ∈ µ − ( c (cid:48) ) and i ∈ ˜ I c (cid:48) , then i ∈ µ (cid:48) ( ˜ I )since | ˜ I c (cid:48) | < r c (cid:48) . If i ∈ µ − ( c (cid:48) ) and i / ∈ ˜ I c (cid:48) , then rank( i ; ˜ I \ ˜ I c (cid:48) , π ) ≤ r c (cid:48) − | ˜ I c (cid:48) | . As a result i ∈ µ (cid:48) ( ˜ I ). These two statements imply that µ − ( c (cid:48) ) ⊆ µ (cid:48) ( ˜ I ). Furthermore, every i ∈ µ − ( c ) is acategory- c patient since there exists i ∈ ˜ I c such that i / ∈ µ ( ˜ I ). As a result, we get that I (cid:48) (2) \ I (2) = µ ( ˜ I ) \ µ (cid:48) ( ˜ I ) = µ − ( c ) \ µ (cid:48) ( ˜ I ) ⊆ µ − ( c ) ⊆ ˜ I c . The proof of the third statement in Lemma 5 is the same as the proof of the third statementin Lemma 4.The fourth statement in Lemma 5 is true because for every i ∈ I (2) \ I (cid:48) (2) = µ (cid:48) ( ˜ I ) \ µ ( ˜ I ) weknow that i / ∈ ˜ I c by the third statement in Lemma 5. Moreover, µ (cid:48)− ( c ) ⊆ ˜ I c , as there exists j ∈ ˜ I c such that j (cid:54)∈ µ ( ˜ I ), which implies that there are at least r c category- c patients. Thisimplies i ∈ µ (cid:48)− ( c (cid:48) ). Furthermore, i / ∈ ˜ I c (cid:48) because ˜ I c (cid:48) ⊆ µ ( ˜ I ), which follows from | ˜ I c (cid:48) | < r c (cid:48) .Consider i (cid:48) ∈ I (cid:48) (2). Then i (cid:48) / ∈ µ (cid:48) ( ˜ I ), which implies that i π i (cid:48) because µ (cid:48) ( i ) = c (cid:48) , τ ( i ) (cid:54) = c (cid:48) , and µ (cid:48) ( i (cid:48) ) = ∅ .The proof of the fifth statement in Lemma 5 is the same as the proof of the fifth statementin Lemma 4. Lemma 6.
Suppose that c ∈ C \ { u } and c (cid:48) , c ∗ ∈ C \ { c } are different categories. Let ˜ I ⊆ I and (cid:46), (cid:46) (cid:48) ∈ ∆( { c, c (cid:48) , c ∗ } ) be such that • (cid:46) is given as c (cid:48) (cid:46) c (cid:46) c ∗ , • (cid:46) (cid:48) is given as c (cid:46) (cid:48) c (cid:48) (cid:46) (cid:48) c ∗ , • I (3) = ˜ I \ µ ( ˜ I ) where µ = ϕ ˜ I(cid:46) , • I (cid:48) (3) = ˜ I \ µ (cid:48) ( ˜ I ) where µ (cid:48) = ϕ ˜ I(cid:46) (cid:48) , and • µ ( ˜ I c ) (cid:40) ˜ I c .Then the following results hold:1. | I (3) \ I (cid:48) (3) | = | I (cid:48) (3) \ I (3) | ,2. I (cid:48) (3) \ I (3) ⊆ ˜ I c ,3. I (3) \ I (cid:48) (3) ⊆ ˜ I \ ˜ I c , and . If i (cid:48) ∈ I (cid:48) (3) \ I (3) and i ∈ I (3) ∩ ˜ I c , then i (cid:48) π i . Proof of Lemma 6.
The first statement in Lemma 6 follows as in the proof of the firststatement in Lemma 4. Likewise, the fourth statement in Lemma 6 follows as in the proof ofthe fifth statement in Lemma 4.To show the other two statements, we use Lemma 4 and Lemma 5. Let ˆ (cid:46), ˆ (cid:46) (cid:48) ∈ ∆( { c, c (cid:48) } ) besuch that ˆ (cid:46) : c (cid:48) ˆ (cid:46) c and ˆ (cid:46) (cid:48) : c ˆ (cid:46) (cid:48) c (cid:48) . Let I (2) = ˜ I \ ϕ ˜ I ˆ (cid:46) ( ˜ I ) and I (cid:48) (2) = ˜ I \ ϕ ˜ I ˆ (cid:46) (cid:48) ( ˜ I ). Then I (3) = I (2) \ µ − ( c ∗ ) and I (cid:48) (3) = I (cid:48) (2) \ µ (cid:48)− ( c ∗ ).For both precedence orders (cid:46) and (cid:46) (cid:48) under the sequential reserve matching procedure, con-sider the beginning of the third step, at which category c ∗ is processed. For (cid:46) , the set ofavailable patients is I (2). For (cid:46) (cid:48) , the set of available patients is I (cid:48) (2). If I (2) = I (cid:48) (2), then allthe statements hold trivially because in this case we get I (3) = I (cid:48) (3). Therefore, assume that I (2) (cid:54) = I (cid:48) (2). For every precedence order, r c ∗ patients with the highest priority with respect to π c ∗ are chosen.We consider each patient chosen under (cid:46) and (cid:46) (cid:48) for category c ∗ one at a time in sequencewith respect to the priority order π c ∗ . For both precedence orders there are r c ∗ patients matchedwith c ∗ because µ ( ˜ I c ) (cid:40) ˜ I c . Let i k be the k th patient chosen under (cid:46) for c ∗ and i (cid:48) k be the k th patient chosen under (cid:46) (cid:48) for c ∗ where k = 1 , . . . , r c ∗ . Let J k be the set of patients available whenwe process (cid:46) for the k th patient and J (cid:48) k be the set of patients available when we process (cid:46) (cid:48) forthe k th patient where k = 1 , . . . , r c ∗ . For k = 1, J k = I (2) and J (cid:48) k = I (cid:48) (2). By definition, J k +1 = J k \ { i k } and J (cid:48) k +1 = J (cid:48) k \ { i (cid:48) k } . We show that(a) J (cid:48) k \ J k ⊆ ˜ I c ,(b) J k \ J (cid:48) k ⊆ ˜ I \ ˜ I c , and(c) if i ∈ J k \ J (cid:48) k and i (cid:48) ∈ ( J k ∩ J (cid:48) k ) \ ˜ I c ∗ , then i π c ∗ i (cid:48) .by mathematical induction on k . These three claims trivially hold for k = 1 by Statements 2,3, and 4 in Claims 1 and 2.Fix k . In the inductive step, assume that Statements (a), (b), and (c) hold for k . Consider k + 1. If J k +1 = J (cid:48) k +1 , then the statements trivially hold. Assume that J k +1 (cid:54) = J (cid:48) k +1 whichimplies that J (cid:96) (cid:54) = J (cid:48) (cid:96) for (cid:96) = 1 , . . . , k . There are four cases depending on which sets i k and i (cid:48) k belong to. We consider each case separately. Case 1: i k ∈ J k \ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k ∩ J k . Then J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = J (cid:48) k \ J k and J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = (cid:0) ( J k \ J (cid:48) k ) \ { i k } (cid:1) ∪ { i (cid:48) k } . By Statement (a) of the inductive assumption for k , J (cid:48) k +1 \ J k +1 = J (cid:48) k \ J k ⊆ ˜ I c . Therefore,Statement (a) holds for k + 1. 52s J (cid:48) k (cid:54) = J k and | J k | = | J (cid:48) k | because of the soft-reserves condition, J (cid:48) k \ J k has at least onecategory- c patient. Moreover, this patient has higher priority with respect to π c ∗ than any othercategory- c patient in J (cid:48) k ∩ J k because the former is chosen under (cid:46) while the later is not chosenunder (cid:46) . Therefore, i (cid:48) k cannot be a category- c patient. As a result, J k +1 \ J (cid:48) k +1 ⊆ ˜ I \ ˜ I c , soStatement (b) holds for k + 1.To show Statement (c) for k + 1, observe that J k +1 \ J (cid:48) k +1 = (( J k \ J (cid:48) k ) \ { i k } ) ∪ { i (cid:48) k } and J k +1 ∩ J (cid:48) k +1 = ( J k ∩ J (cid:48) k ) \ { i (cid:48) k } ). Therefore, Statement (c) for k + 1 follows from Statement (c)for k and the fact that i (cid:48) k π c ∗ i for any i ∈ J k +1 ∩ J (cid:48) k +1 . Case 2: i k ∈ J k \ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k \ J k . Then, J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = ( J (cid:48) k \ J k ) \ { i (cid:48) k } and J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = ( J k \ J (cid:48) k ) \ { i k } . Therefore, J (cid:48) k +1 \ J k +1 ⊆ ˜ I c and J k +1 \ J (cid:48) k +1 ⊆ ˜ I \ ˜ I c by Statements (a) and (b) for k , respectively,implying Statements (a) and (b) for k + 1.To show Statement (c) for k + 1, observe that J k +1 \ J (cid:48) k +1 = ( J k \ J (cid:48) k ) \ { i k } and J k +1 ∩ J (cid:48) k +1 = J k ∩ J (cid:48) k . Therefore, Statement (c) for k + 1 follows from Statement (c) for k trivially. Case 3: i k ∈ J k ∩ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k ∩ J k . In this case, i k = i (cid:48) k , then J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = J (cid:48) k \ J k and J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = J k \ J (cid:48) k . Therefore, Statements (a) and (b) for k + 1 follows from the respective statements for k .To show Statement (c) for k + 1, observe that J k +1 \ J (cid:48) k +1 = J k \ J (cid:48) k and J k +1 ∩ J (cid:48) k +1 =( J k ∩ J (cid:48) k ) \ { i k } . Therefore, Statement (c) for k + 1 follows from Statement (c) for k trivially. Case 4: i k ∈ J k ∩ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k \ J k . We argue that this case is not possible. Since i (cid:48) k ∈ J (cid:48) k \ J k , i (cid:48) k must be a category- c patient by Statement (a) for k . If c ∗ = u , then every patient in J k \ J (cid:48) k has a higher priority with respect to π than every patient in J k ∩ J (cid:48) k , which cannot happen since i k ∈ J k ∩ J (cid:48) k . Therefore, c ∗ (cid:54) = u . Since i (cid:48) k is a category- c patient, there must not be a category- c ∗ patient in J (cid:48) k . By Statement (c) for k , we know that every patient in J k \ J (cid:48) k has a higher prioritywith respect to π c ∗ than every patient in ( J k ∩ J (cid:48) k ) \ I c ∗ = J k ∩ J (cid:48) k . This is a contradiction to i k ∈ J k ∩ J (cid:48) k . Therefore, Case 4 is not possible.Since I (3) = J r c ∗ and I (cid:48) (3) = J (cid:48) r c ∗ , Statements 2 and 3 in Lemma 6 follow from Statements(a) and (b) above, respectively. Lemma 7.
Suppose that c ∈ C \ { u } and c (cid:48) , c ∗ , ˜ c ∈ C \ { c } are different categories. Let ˜ I ⊆ I and (cid:46), (cid:46) (cid:48) ∈ ∆( { c, c (cid:48) , c ∗ , ˜ c } ) be such that • (cid:46) is given as c (cid:48) (cid:46) c (cid:46) c ∗ (cid:46) ˜ c , • (cid:46) (cid:48) is given as c (cid:46) (cid:48) c (cid:48) (cid:46) (cid:48) c ∗ (cid:46) (cid:48) ˜ c , I (4) = ˜ I \ µ ( ˜ I ) where µ = ϕ ˜ I(cid:46) , • I (cid:48) (4) = ˜ I \ µ (cid:48) ( ˜ I ) where µ (cid:48) = ϕ ˜ I(cid:46) (cid:48) , and • µ ( ˜ I c ) (cid:40) ˜ I c .Then the following results hold:1. | I (4) \ I (cid:48) (4) | = | I (cid:48) (4) \ I (4) | ,2. I (4) \ I (cid:48) (4) ⊆ ˜ I \ ˜ I c ,3. if i (cid:48) ∈ I (cid:48) (4) \ I (4) , i (cid:48) / ∈ ˜ I c , and i ∈ I (4) , then i (cid:48) π i , and4. if i (cid:48) ∈ I (cid:48) (4) \ I (4) , i (cid:48) ∈ ˜ I c , and i ∈ I (4) ∩ ˜ I c , then i (cid:48) π i . Proof of Lemma 7.
The first statement in Lemma 7 follows as in the proof of the firststatement in Lemma 4. Likewise, the fourth statement in Lemma 7 follows as in the proof ofthe fifth statement in Lemma 4.To prove the other two statements, we use Lemma 6. Let ˆ (cid:46), ˆ (cid:46) (cid:48) ∈ ∆( { c, c (cid:48) , c ∗ } ) be such thatˆ (cid:46) : c (cid:48) ˆ (cid:46) c ˆ (cid:46) c ∗ and ˆ (cid:46) (cid:48) : c ˆ (cid:46) (cid:48) c (cid:48) ˆ (cid:46) (cid:48) c ∗ . Let I (3) = ˜ I \ ϕ ˜ I ˆ (cid:46) ( ˜ I ) and I (cid:48) (3) = ˜ I \ ϕ ˜ I ˆ (cid:46) (cid:48) ( ˜ I ). Then I (4) = I (3) \ µ − (˜ c ) and I (cid:48) (4) = I (cid:48) (3) \ µ (cid:48)− (˜ c ).For both precedence orders (cid:46) and (cid:46) (cid:48) under the sequential reserve matching procedure, con-sider the beginning of the fourth step, at which category ˜ c is processed. For (cid:46) , the set of availablepatients is I (3). For (cid:46) (cid:48) , the set of available patients is I (cid:48) (3). If I (3) = I (cid:48) (3), then I (4) = I (cid:48) (4)which implies all the statements in Lemma 7. Therefore, assume that I (3) (cid:54) = I (cid:48) (3). For everyprecedence order, r ˜ c patients with the highest priority with respect to π ˜ c are chosen.We consider each patient chosen under (cid:46) and (cid:46) (cid:48) for category ˜ c one at a time in sequence withrespect to the priority order π ˜ c . Since µ ( ˜ I c ) (cid:40) ˜ I c , r ˜ c patients are matched with ˜ c under bothprecedence orders. Let i k be the k th patient chosen under (cid:46) for ˜ c and i (cid:48) k be the k th patient chosenunder (cid:46) (cid:48) for ˜ c where k = 1 , . . . , r ˜ c . Let J k be the set of patients available when we process (cid:46) forthe k th patient and J (cid:48) k be the set of patients available when we process (cid:46) (cid:48) for the k th patientwhere k = 1 , . . . , r ˜ c . For k = 1, J k = I (3) and J (cid:48) k = I (cid:48) (3). By definition, J k +1 = J k \ { i k } and J (cid:48) k +1 = J (cid:48) k \ { i (cid:48) k } .We show that(a) J k \ J (cid:48) k ⊆ ˜ I \ ˜ I c ,(b) if ˜ c = u , i (cid:48) ∈ J (cid:48) k \ J k , τ ( i (cid:48) ) (cid:54) = c , and i ∈ J k , then i (cid:48) π i ,(c) if ˜ c (cid:54) = u , i (cid:48) ∈ J (cid:48) k \ J k , τ ( i (cid:48) ) (cid:54) = c , and i ∈ J k , then ( J k ∪ J (cid:48) k ) ∩ I ˜ c = ∅ and i (cid:48) π i (d) if ˜ c (cid:54) = u , then ( J (cid:48) k \ J k ) ∩ I ˜ c = ∅ . 54y mathematical induction on k . These three claims trivially hold for k = 1 by Statements 2and 3 in Lemma 6.Fix k . In the inductive step, assume that Statements (a), (b), (c), and (d) hold for k .Consider k + 1. If J k +1 = J (cid:48) k +1 , then the statements trivially hold. Assume that J k +1 (cid:54) = J (cid:48) k +1 which implies that J (cid:96) (cid:54) = J (cid:48) (cid:96) for (cid:96) = 1 , . . . , k . There are four cases depending on which sets i k and i (cid:48) k belong to. We consider each case separately. Case 1: i k ∈ J k \ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k ∩ J k . When |{ i (cid:48) ∈ J (cid:48) k \ J k : τ ( i (cid:48) ) (cid:54) = c }| ≥ i (cid:48) k ∈ J (cid:48) k \ J k byStatements (b) and (c) for k . Therefore, |{ i (cid:48) ∈ J (cid:48) k \ J k : τ ( i (cid:48) ) (cid:54) = c }| = 0. Furthermore, J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = J (cid:48) k \ J k and J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = (( J k \ J (cid:48) k ) \ { i k } ) ∪ { i (cid:48) k } . Since | J (cid:48) k | = | J k | and J (cid:48) k (cid:54) = J k , we get | J (cid:48) k \ J k | ≥
1. Therefore, J (cid:48) k \ J k has at least onecategory- c patient because |{ i (cid:48) ∈ J (cid:48) k \ J k : τ ( i (cid:48) ) (cid:54) = c }| = 0. Moreover, this patient has higherpriority with respect to π ˜ c than any other category- c patient in J (cid:48) k ∩ J k because the formerpatient is chosen under (cid:46) and the latter is not, so i (cid:48) k cannot be category c . Therefore, Statement(a) for k + 1 follows from Statement (a) for k . Statements (b) and (c) trivially hold for k + 1 aswell because { i (cid:48) ∈ J (cid:48) k +1 \ J k +1 : τ ( i (cid:48) ) (cid:54) = c } = { i (cid:48) ∈ J (cid:48) k \ J k : τ ( i (cid:48) ) (cid:54) = c } = ∅ . Statement (d) for k + 1 follows from Statement (d) for k . Case 2: i k ∈ J k \ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k \ J k . Then J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = ( J (cid:48) k \ J k ) \ { i (cid:48) k } and J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = ( J k \ J (cid:48) k ) \ { i k } . Since J k +1 \ J (cid:48) k +1 ⊆ J k \ J (cid:48) k , Statement (a) for k + 1 follows from Statement (a) for k . Likewise,Statements (b), (c), and (d) for k + 1 follow from the corresponding statements for k because J (cid:48) k +1 \ J k +1 ⊆ J (cid:48) k \ J k , J (cid:48) k +1 ∪ J k +1 ⊆ J (cid:48) k ∪ J k , and J k +1 ⊆ J k . Case 3: i k ∈ J k ∩ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k ∩ J k . In this case, i k = i (cid:48) k , then J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = J (cid:48) k \ J k and J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = J k \ J (cid:48) k . In this case, Statements (a), (b), (c), and (d) for k + 1 follow from their respective statementsfor k . Case 4: i k ∈ J k ∩ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k \ J k . In this case, J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = (( J (cid:48) k \ J k ) \ { i (cid:48) k } ) ∪ { i k } J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = J k \ J (cid:48) k . Statement (a) for k + 1 follows from Statement (a) for k trivially.Statement (b) for k + 1 follows from i k π i for any i ∈ J k +1 and also from Statement (b) for k . To show Statement (d) for k + 1, suppose that ˜ c (cid:54) = u . Then by Statement (d) for k , J (cid:48) k \ J k does not have a category-˜ c patient. Since i (cid:48) k ∈ J (cid:48) k \ J k , this implies that there are no category-˜ c patients in J (cid:48) k . Therefore, i k does not have category ˜ c . We conclude that J (cid:48) k +1 \ J k +1 =(( J (cid:48) k \ J k ) \ { i (cid:48) k } ) ∪ { i k } does not have a category ˜ c patient, which is the Statement (d) for k + 1.To show Statement (c) for k + 1, suppose that ˜ c (cid:54) = u , i (cid:48) ∈ J (cid:48) k +1 \ J k +1 , τ ( i (cid:48) ) (cid:54) = c , and i ∈ J k +1 . If i (cid:48) (cid:54) = i k , then i (cid:48) ∈ J (cid:48) k \ J k since J (cid:48) k +1 \ J k +1 = (( J (cid:48) k \ J k ) \ { i (cid:48) k } ) ∪ { i k } and Statement(c) for k + 1 follows from Statement (c) for k because i ∈ J k +1 ⊆ J k . Otherwise, supposethat i (cid:48) = i k . By Statement (d) for k + 1, i (cid:48) does not have category ˜ c , which implies that thereare no category-˜ c patients in J k ; this in turn implies there are no category-˜ c patients in J k +1 since J k +1 ⊆ J k . Furthermore, by Statement (d) for k + 1, there are no category-˜ c patients in J (cid:48) k +1 \ J k +1 . We conclude that there are no category-˜ c patients in J (cid:48) k +1 ∪ J k +1 . Finally, i (cid:48) π ˜ c i for any i ∈ J k +1 = J k \ { i (cid:48) } and since there are no category-˜ c patients in J (cid:48) k +1 ∪ J k +1 we get i (cid:48) π i .Since I (4) = J r ˜ c and I (cid:48) (4) = J (cid:48) r ˜ c , Statement 2 in Lemma 7 follows from Statement (a) andStatement 3 in Lemma 7 follows from Statements (b) and (c). Lemma 8.
Suppose that c ∈ C \ { u } and c (cid:48) , c ∗ , ˜ c, ˆ c ∈ C \ { c } are different categories. Let ˜ I ⊆ I and (cid:46), (cid:46) (cid:48) ∈ ∆( { c, c (cid:48) , c ∗ , ˜ c, ˆ c } ) be such that • (cid:46) be such that c (cid:48) (cid:46) c (cid:46) c ∗ (cid:46) ˜ c (cid:46) ˆ c , • (cid:46) (cid:48) be such that c (cid:46) (cid:48) c (cid:48) (cid:46) (cid:48) c ∗ (cid:46) (cid:48) ˜ c (cid:46) (cid:48) ˆ c , • I (5) = ˜ I \ µ ( ˜ I ) where µ = ϕ ˜ I(cid:46) , • I (cid:48) (5) = ˜ I \ µ (cid:48) ( ˜ I ) where µ (cid:48) = ϕ ˜ I(cid:46) (cid:48) , and • µ ( ˜ I c ) (cid:40) ˜ I c .Then the following results hold:1. | I (5) \ I (cid:48) (5) | = | I (cid:48) (5) \ I (5) | and2. I (5) \ I (cid:48) (5) ⊆ ˜ I \ ˜ I c . Proof.
The first statement in Lemma 8 follows as in the proof of the first statement in Lemma4. To prove the second statement, we use Lemma 7. Let ˆ (cid:46), ˆ (cid:46) (cid:48) ∈ ∆( { c, c (cid:48) , c ∗ , ˜ c } ) be such thatˆ (cid:46) : c (cid:48) ˆ (cid:46) c ˆ (cid:46) c ∗ ˆ (cid:46) ˜ c and ˆ (cid:46) (cid:48) : c ˆ (cid:46) (cid:48) c (cid:48) ˆ (cid:46) (cid:48) c ∗ ˆ (cid:46) ˜ c. Let I (4) = ˜ I \ ϕ ˜ I ˆ (cid:46) ( ˜ I ) and I (cid:48) (4) = ˜ I \ ϕ ˜ I ˆ (cid:46) (cid:48) ( ˜ I ). Then I (5) = I (4) \ µ − (˜ c ) and I (cid:48) (5) = I (cid:48) (4) \ µ (cid:48)− (˜ c ).56or both precedence orders (cid:46) and (cid:46) (cid:48) under the sequential reserve matching procedure, con-sider the beginning of the fifth step, at which category ˆ c is processed. For (cid:46) , the set of availablepatients is I (4). For (cid:46) (cid:48) , the set of available patients is I (cid:48) (4). If I (4) = I (cid:48) (4), then we get I (5) = I (cid:48) (5), which implies all the statements. Therefore, assume that I (4) (cid:54) = I (cid:48) (4). For everyprecedence order, r ˆ c patients with the highest priority with respect to π ˆ c are chosen.We consider each patient chosen under (cid:46) and (cid:46) (cid:48) for category ˆ c one at a time in sequence withrespect to the priority order π ˆ c . Since µ ( ˜ I c ) (cid:40) ˜ I c , r ˆ c patients are matched with ˆ c under bothprecedence orders. Let i k be the k th patient chosen under (cid:46) for ˆ c and i (cid:48) k be the k th patient chosenunder (cid:46) (cid:48) for ˆ c where k = 1 , . . . , r ˆ c . Let J k be the set of patients available when we process (cid:46) forthe k th patient and J (cid:48) k be the set of patients available when we process (cid:46) (cid:48) for the k th patientwhere k = 1 , . . . , r ˆ c . For k = 1, J k = I (4) and J (cid:48) k = I (cid:48) (4). By definition, J k +1 = J k \ { i k } and J (cid:48) k +1 = J (cid:48) k \ { i (cid:48) k } .We show that(a) J k \ J (cid:48) k ⊆ ˜ I \ ˜ I c and(b) if i (cid:48) ∈ J (cid:48) k \ J k and i ∈ J k ∩ ˜ I c , then i (cid:48) π ˆ c i by mathematical induction on k . These results trivially hold for k = 1 by Statements 2, 3, and4 in Lemma 7.Fix k . In the inductive step, assume that Statements (a) and (b) hold for k . Consider k + 1.If J k +1 = J (cid:48) k +1 , then the statements trivially hold. Assume that J k +1 (cid:54) = J (cid:48) k +1 which implies that J (cid:96) (cid:54) = J (cid:48) (cid:96) for (cid:96) = 1 , . . . , k . There are four cases depending on which sets i k and i (cid:48) k belong to. Weconsider each case separately. Case 1: i k ∈ J k \ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k ∩ J k . Then J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = J (cid:48) k \ J k and J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = (cid:0) ( J k \ J (cid:48) k ) \ { i k } (cid:1) ∪ { i (cid:48) k } . If i (cid:48) k ∈ ˜ I c , then we get a contradiction to Statement (b) for k . Therefore, i (cid:48) k / ∈ ˜ I c , which impliesthat Statement (a) holds for k + 1 by Statement (a) for k and the second displayed equation.Statement (b) for k + 1 follows trivially from Statement (b) for k , the first displayed equation,and J k ⊇ J k +1 . Case 2: i k ∈ J k \ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k \ J k . Then J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = ( J (cid:48) k \ J k ) \ { i (cid:48) k } and J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = ( J k \ J (cid:48) k ) \ { i k } . In this case, Statements (a) and (b) for k + 1 follow trivially from the corresponding statementsfor k . 57 ase 3: i k ∈ J k ∩ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k ∩ J k . In this case, i k = i (cid:48) k , then J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = J (cid:48) k \ J k and J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = J k \ J (cid:48) k . In this case, Statements (a) and (b) for k + 1 follow trivially from the corresponding statementfor k . Case 4: i k ∈ J k ∩ J (cid:48) k and i (cid:48) k ∈ J (cid:48) k \ J k . In this case, J (cid:48) k +1 \ J k +1 = ( J (cid:48) k \ { i (cid:48) k } ) \ ( J k \ { i k } ) = (( J (cid:48) k \ J k ) \ { i (cid:48) k } ) ∪ { i k } and J k +1 \ J (cid:48) k +1 = ( J k \ { i k } ) \ ( J (cid:48) k \ { i (cid:48) k } ) = J k \ J (cid:48) k . Then Statement (a) for k + 1 follows from Statement (a) for k . Furthermore, i k π ˆ c i for any i ∈ J k +1 , which together with Statement (b) for k imply Statement (b) for k + 1.Since I (5) = J r ˆ c and I (cid:48) (5) = J (cid:48) r ˆ c , Statement 2 in Lemma 8 follows from Statement (a). Proof of Proposition 3.
Let |C| ≤
5. Let C ∗ = { c ∗ ∈ C : c ∗ (cid:46) c (cid:48) } be the set of categoriesprocessed before c (cid:48) under (cid:46) and before c under (cid:46) (cid:48) . The orders of categories in C ∗ are the samewith respect to (cid:46) and (cid:46) (cid:48) . Thus, just before category c (cid:48) is processed under (cid:46) and c is processedunder (cid:46) (cid:48) , the same patients are matched in both sequential reserve matching procedures. Let ˜ I be the set of patients that are available at this point in either procedure.Let ˆ (cid:46) be the incomplete precedence order on C \C ∗ that processes categories in the same orderas in (cid:46) . Likewise, let ˆ (cid:46) (cid:48) be the incomplete precedence order on C \ C ∗ that processes categoriesin the same order as in (cid:46) (cid:48) .If ϕ ˜ I ˆ (cid:46) ( ˜ I c ) = ˜ I c then the result is proven. Therefore, assume that ϕ ˜ I ˆ (cid:46) ( ˜ I c ) (cid:40) ˜ I c in the rest ofthe proof. Let k = |C \ C ∗ | be the number of remaining categories. • If k = 2, then by Lemmas 4 and 5, we obtain ϕ ˜ I ˆ (cid:46) (cid:48) ( ˜ I c ) ⊆ ϕ ˜ I ˆ (cid:46) ( ˜ I c ). • If k = 3, then by Lemma 6, we obtain ϕ ˜ I ˆ (cid:46) (cid:48) ( ˜ I c ) ⊆ ϕ ˜ I ˆ (cid:46) ( ˜ I c ). • If k = 4, then by Lemma 7, we obtain ϕ ˜ I ˆ (cid:46) (cid:48) ( ˜ I c ) ⊆ ϕ ˜ I ˆ (cid:46) ( ˜ I c ). • If k = 5, then by Lemma 8, we obtain ϕ ˜ I ˆ (cid:46) (cid:48) ( ˜ I c ) ⊆ ϕ ˜ I ˆ (cid:46) ( ˜ I c ).These imply that ϕ (cid:46) (cid:48) ( I c ) ⊆ ϕ (cid:46) ( I c )completing the proof. 58 .4 A Polynomial Time Method for Smart Reserve Matching Procedure Consider the following algorithm for any n :Step 0. Find a matching that is maximal and complies with eligibility requirements by temporarilydeeming that a patient i ∈ I is eligible for a category c ∈ C \ { u } if and only if i ∈ I c , and no patient is eligible for unreserved category u . This is known as a bipartitemaximum cardinality matching problem in graph theory and many augmenting alternatingpath algorithms (such as those by Hopcroft and Karp, 1973; Karzanov, 1973) can solveit in polynomial time. The solution finds the maximum number of patients who can bematched with a preferential treatment category that they are a beneficiaries of. Denotethe number of patients matched by this matching as n b .Step 1. Let J u = ∅ , J = ∅ . Fix parameters κ (cid:29) (cid:15) > (cid:15) < κ > | I | would work.For k = 1 , . . . , | I | we repeat the following substep given J uk − , J k − :Step 1.( k ). Suppose i k is the patient who is prioritized k th in I according to π .i. if | J uk − | < n continue with (i.A) and otherwise continue with (ii).A. Temporarily deem all patients in J uk − ∪ { i k } eligible only for category u andall other patients eligible only for the categories in C \ { u } that they arebeneficiaries of.B. for every pair ( i, x ) ∈ I × C ∪ {∅} define a weight W i,c ∈ R as follows: • If x ∈ C and i is temporarily eligible for x as explained in (i.A), – if i ∈ J uk − ∪ J k − , then define W i,x := κ , – otherwise, define W i,x := (cid:15) . • If x ∈ C and i is not temporarily eligible for x as explained in (i.A), define W i,x := − (cid:15) . • If x = ∅ , define W i,x := 0.C. Solve the following assignment problem to find a matching σ ∈ arg max µ ∈M (cid:88) i ∈ I W i,µ ( i ) using a polynomial algorithm such as the Hungarian algorithm (Kuhn, 1955).D. If | σ ( I ) | = n b + | J uk − | + 1 then define J uk := J uk − ∪ { i k } and J k := J k − , and go to Step 1.( k + 1) if k < | I | and Step 2 if k = | I | .E. Otherwise, go to (ii).ii. Repeat (i) with the exception that i k is temporarily deemed eligible only for thecategories in C \ { u } that she is a beneficiary of in Part (ii.A). Parts (ii.B) and(ii.C) are the same as Parts (i.B) and (i.C), respectively, with the exception thatweights are constructed with respect to the eligibility construction in (ii.A). Parts(ii.D) and (ii.E) are as follows: 59. If | σ ( I ) | = n b + | J uk − | and σ ( i ) (cid:54) = ∅ for all i ∈ J k − ∪ { i k } , then define J uk := J uk − and J k := J k − ∪ { i k } , and go to Step 1.( k + 1) if k < | I | and Step 2 if k = | I | .E. Otherwise, J uk := J uk − and J k := J k − , and go to Step 1.( k + 1) if k < | I | and Step 2 if k = | I | .Step 2. (a) Find a matching σ as follows:i. Temporarily deem all patients in J u | I | eligible only for category u , all patients in J | I | eligible only for the categories in C \ { u } that they are beneficiaries of, andall other patients ineligible for all categories.ii. Find a maximal matching σ among all matchings that comply with the temporaryeligibility requirements defined in (i) using a polynomial augmenting alternatingpaths algorithm (for example see Hopcroft and Karp, 1973; Karzanov, 1973).(b) Modify σ as follows:One at a time assign the remaining units unmatched in σ to the remaining highestpriority patient in I \ ( J u | I | ∪ J | I | ) who is eligible for the category of the assigned unitin the real problem in the following order:i. the remaining units of the preferential treatment categories in C \ { u } in anarbitrary order, andii. the remaining units of the unreserved category u .Step 3. Define I nS := σ ( I ) . Matching σ is a smart reserve matching with n unreserved category units processed first.The difference between this algorithm and the procedure we gave in the text is that we do nothave to construct the matching sets M k in every Step 1( k ), as this is an NP-complete problemto solve. 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