An Empirical Framework for Sequential Assignment:The Allocation of Deceased Donor Kidneys∗†

Nikhil Agarwal, Itai Ashlagi, Michael Rees, Paulo Somaini, Daniel Waldinger‡

January 27, 2019

Abstract

An organ transplant can improve a patient’s life while also realizing substantial sav-ings in healthcare expenditures. Like many other scarce public resources, organs fromdeceased donors are rationed to patients on a waitlist via a sequential offer mechanism.The theoretical trade-offs in designing these rationing systems are not well understoodand depend on agent preferences. This paper establishes an empirical framework foranalyzing waitlist systems and applies it to study the allocation of deceased donorkidneys. We model the decision to accept an organ or wait for a more preferable or-gan as an optimal stopping problem, and develop techniques to compute equilibria ofcounterfactual mechanisms. Our estimates show that while some types of kidneys aredesirable for all patients, there is substantial match-specific heterogeneity in values. Wethen evaluate alternative mechanisms by comparing their effect on patient welfare toan equivalent change in donor supply. Past reforms to the kidney waitlist primarily re-sulted in redistribution, with similar welfare and organ discard rates as the benchmarkfirst come first served mechanism. These mechanisms and other commonly studiedtheoretical benchmarks remain far from optimal: we design a mechanism that increasespatient welfare by the equivalent of a 14.2 percent increase in donor supply.

∗We are grateful to Alex Garza, Sarah Taranto, and Jennifer Wainwright of UNOS for facilitating access tothe data. We thank Peter Arcidiacono, Lanier Benkard, Liran Einav, Gautam Gowrisankaran, Jacob Leshno,Anna Mikusheva, Michael Ostrovsky, Ariel Pakes, Parag Pathak, Peter Reiss, Andy Skrzypacz, GabrielWeintraub, Mike Whinston for helpful discussions. Layne Kirshon and Khizar Qureshi provided competentresearch assistance. The authors acknowledge support from the National Science Foundation (SES-1729090,SES-1254768), the National Institutes of Health (R21-DK113626), and the MIT SHASS Dean’s ResearchFund.†The data reported here have been supplied by UNOS as the contractor for the Organ Procurement and

Transplantation Network (OPTN). The interpretation and reporting of these data are the responsibility ofthe author(s) and in no way should be seen as an official policy of or interpretation by the OPTN or the U.S.Government.‡Agarwal: Department of Economics, MIT and NBER, email: agarwaln@mit.edu. Ashlagi: Management

Science and Engineering, Stanford University, email: iashlagi@stanford.edu. Rees: Department of Urol-ogy, University of Toledo Medical Center, email: michael.rees2@utoledo.edu. Somaini: Stanford GraduateSchool of Business and NBER, email: soma@stanford.edu. Waldinger: Furman Center and Department ofEconomics, New York University, email: dw120@nyu.edu.

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1 Introduction

As of January 7, 2019, there were 94,971 patients on the kidney waiting list in the UnitedStates, while only 13,483 deceased donor transplants were performed in 2018.1 Each trans-plant improves the expected quality and length of a transplanted patient’s life while savinghundreds of thousands of dollars in expected dialysis costs (Wolfe et al., 1999; Irwin et al.,2012; Held et al., 2016). Yet, approximately 20 percent of medically suitable organs ex-tracted for transplantation are discarded. The allocation of deceased donor kidneys doesnot use money because of ethical considerations and legal restrictions,2 making traditionalprice-based market-clearing mechanisms infeasible. Instead, available kidneys are allocatedthrough a centralized waitlist. Given these constraints, it is essential to find mechanismsthat efficiently allocate resources, minimize waste, and achieve equitable outcomes.3 Similarconsiderations motivate the use of waitlist systems to allocate other deceased donor organs,public housing, long-term care, child-care, and child-adoption.

Previous research and guidance on waitlist design either ignores dynamic incentives or as-sumes specific forms of market primitives. Theoretical approaches to designing dynamicassignment mechanisms have found that even qualitative trade-offs are sensitive to the prim-itives of the market.4 Absent clear recommendations from theory, many organ allocationagencies use simulations to predict the effects of alternative allocation rules. The simulations,including those used to design organ allocation rules,5 do not allow decisions to respond tochanges in the system. Moreover, recent empirical advances in analyzing allocation systemsare restricted to static choice settings.

This paper develops an empirical framework for analyzing waitlist mechanisms that sequen-tially assign objects to forward-looking agents, and applies these methods to study the de-ceased donor kidney allocation system in the U.S. We make several methodological andempirical contributions. First, we develop a method for estimating agent preferences usingtypical administrative data, and apply it to the kidney waitlist data from New York to es-

1Source: https://optn.transplant.hrsa.gov/data/view-data-reports/national-data/2The National Organ Transplantation Act (NOTA) makes it illegal to obtain human organs for transplan-

tation by compensating donors.3These goals are articulated by the Organ Procurement and Transplantation Network (OPTN), a contrac-

tor for the Health Resources & Services Administration (HRSA), in their policy document titled “Conceptsfor Kidney Allocation” (OPTN, 2011). A committee that was charged with reforming the allocation systemadopted a new mechanism in 2014. We discuss these reforms in greater detail below.

4Agarwal et al. (2018b) compare the results in Su and Zenios (2004), Leshno (2017), Arnosti and Shi(2017), and Bloch and Cantala (2017) and show by example that optimal design depends on the nature ofpreference.

5For example, Kidney Pancreas Simulated Acceptance Module (KPSAM) used by the kidney allocationcommittee to evaluate various proposed mechanisms prior to the reforms enacted in 2014 assumes thatacceptance decisions on the kidney waitlist do not depend on mechanism used, thereby ignoring differencesin dynamic incentives generated by various mechanisms. Similar methods are used by the organ allocationagencies in the U.K., Scandinavia and France.

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timate payoffs from various types of transplants. This step is based on an optimal stoppingproblem faced by a patient when she is offered a kidney. Second, we define a notion of steadystate equilibrium that is amenable to computation and counterfactual analysis of a broadclass of mechanisms. Finally, we use these techniques to compare alternative mechanisms onkey outcome measures such as efficiency, equity, and organ waste.

Our empirical results imply that the new kidney allocation system implemented in 2014leaves significant room for improvement on both average patient welfare and discard rates.This system, which was designed with the assistance of simulations that ignored dynamicincentives, mostly resulted in redistribution relative to the prior mechanism. In fact, ourresults suggest that these simulations yield biased and attenuated results, highlighting theimportance of dynamic incentives and their equilibrium effects. We use our estimates toderive a mechanism that can generate large welfare gains while also reducing waste.

Our empirical application studies deceased donor kidneys allocated in the New York Cityarea (henceforth, NYRT) between 2010 and 2013. The allocation mechanism used to matchdeceased donor kidneys with patients relies on a coarse point system based on donor andpatient characteristics and the patient’s waiting time. As soon as an organ becomes available,it is offered to patients on the waitlist in decreasing order of these priority points. The decisionof whether or not to accept an offer remains with the patient and the transplant surgeon.6Each organ is allocated to the highest-priority biologically compatible patient who acceptsit. The patient is removed from the waitlist once she is transplanted. Otherwise, she mayremain on the waitlist and may choose to accept the next organ she is offered. The prioritysystem does not depend on whether a patient has refused previous offers. Even though thetiming and quality of future offers are uncertain, it can be optimal to turn down an offer towait for a more suitable one.

We begin by documenting several key descriptive facts using rich administrative data onpatient and donor characteristics and acceptance decisions. Patients on the NYRT waitlistface extreme scarcity. While 1,400 patients join the waitlist each year, fewer than 400 de-ceased donor kidneys are recovered in NYRT. These donors vary widely in quality; some areaccepted immediately, while others are rejected by every patient and discarded. The chancea patient is high on the list for a particular type of organ, and therefore the chance of beingoffered desirable organs, increases with waiting time. As a result, patients have an incentiveto reject offers and wait for a better kidney. Indeed, Agarwal et al. (2018b) document thatacceptance rates are higher for patients who are less likely to receive offers. Therefore, con-sistent with dynamic considerations, patients with a higher option value of waiting are morelikely to refuse an offer for an organ.

Motivated by these descriptive facts, we model an agent’s decision to accept an offer as a6We refer to the decision-maker as the patient because our data does not directly identify cases in which

a surgeon makes a decision on behalf of her patient. This approach is reasonable if each surgeon acts in thebest interest of each of her patients.

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continuous-time optimal stopping problem. She accepts the current offer if the value fromthe object is higher than the expected value of continuing to wait. The distribution ofpotential future offers depends on the mechanism and the strategies of the other agents onthe list. Our empirical strategy combines acceptance probabilities with detailed knowledgeof the mechanism to recover the value of a transplant. Our technique adapts methods forinverting conditional choice probabilities in this continuous-time problem (Hotz and Miller,1993; Arcidiacono and Miller, 2011; Arcidiacono et al., 2016) to suit dynamic assignmentmechanisms. This approach eases the computation of the continuation values relative tofull solution methods (Pakes, 1986; Rust, 1987, for example) because the distribution offuture offers depends on all characteristics that influence priorities in the mechanism, and istherefore high-dimensional.

The estimated values from transplants are intuitive. All patients value certain characteristicsthat are correlated with organ quality – for instance, younger donors are preferred by allpatients, as are immunologically similar matches. We also estimate substantial donor-levelunobserved heterogeneity; including this feature significantly improves the fit of our estimatedchoice probabilities. In addition, there is significant match-specific heterogeneity in values.For example, we find that older patients place less value on younger donors as compared toyounger patients. This match-specific preference heterogeneity creates scope for the designof the allocation mechanism to improve match quality. Moreover, our descriptive resultsshow that the current mechanism produces significant mismatch on several dimensions. Forinstance, young patients are often allocated organs from older donors, while many old patientsreceive organs from young donors. Motivated by these facts, we conclude the paper byanalyzing equilibrium allocations of the pre- and post-2014 U.S. kidney allocation systems,benchmark mechanisms from the theoretical literature, and welfare-maximizing mechanismsthat exploit the rich heterogeneity across donors and patients.

Predicting assignments in these counterfactual mechanisms requires us to solve two technicalissues. First, we need to formulate an equilibrium notion that is both tractable and consistentwith our estimation procedure. Computing counterfactuals is challenging because it involvessolving a dynamic game with many players. An unrestricted state-space could include thecomposition of the entire waitlist, resulting in a system with an extremely high-dimensionalstationary distribution. To make progress, we develop a notion of a steady-state equilibriumin the spirit of previous approaches to simplifying this task (Hopenhayn, 1992; Weintraub etal., 2008; Fershtman and Pakes, 2012). Our approach computes a steady-state distributionof types and a steady-state queue length. This allows us to find a tractable computationalalgorithm that iterates between solving the value function using backwards induction andcalculating the steady-state composition of the queue by forward simulation.

Second, we need to ensure that assignments under counterfactuals of interest are indeed iden-tified. In dynamic models such as ours, counterfactuals may not be invariant to normalizingthe payoff of an action because they arbitrarily restrict payoffs across states (see Aguirre-

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gabiria and Suzuki, 2014; Kalouptsidi et al., 2015). Our estimates normalize the payoff ofnever receiving an assignment to zero. We formally show that our normalization is appropri-ate for the mechanism design counterfactuals we consider if the value of declining all offersremains fixed. In our empirical context, this assumption is satisfied if the value of remainingon dialysis until death, and the value and opportunity of receiving a living-donor transplantdo not change when the deceased donor allocation system is redesigned. We argue that boththese assumptions are reasonable.Our framework allows us to compare, for each patient, the welfare effect of a change in amechanism to an equivalent change in deceased donor supply (arrival rates) under the existingmechanism. This quantity has the advantage of being interpretable and independent of anarbitrary choice of units during estimation. We then aggregate these equivalent changes indonor supply across patients as a summary of the welfare effects.We start by comparing assignments generated by mechanisms that are used in practice totwo benchmarks recommended for the kidney allocation system in the theoretical literature.Specifically, we compare the pre- and post-2014 assignment systems with the first come firstserved (FCFS) (Bloch and Cantala, 2017) and last come first served (LCFS) (Su and Zenios,2004) waitlists. These benchmarks encode notions of procedural fairness and/or perform wellunder specific forms of agent preferences.The efficiency properties of various mechanisms turn on whether selective agents generatea net positive or negative externality on others. When an agent declines an object, sheallows others to receive an assignment earlier, generating a positive externality. However,by refusing an object and remaining on the list, she generates a negative externality as shetakes away future offers. Which force is more important depends on preferences and themechanism. Bloch and Cantala (2017) show that FCFS induces agents to be selective andproduces better assignments when preferences are highly heterogeneous. Selective agentsonly accept high idiosyncratic match value objects but allow others to receive an offer fora typical object earlier instead of later, generating a positive externality. However, FCFSperforms poorly when all agents value each object identically because selective agents passon only low quality objects onto those lower on the list. This fact creates strong incentives towait, resulting in organ waste. In fact, Su and Zenios (2004) show that in this environmentLCFS improves social surplus by reducing waste.7 It does so by providing incentives to acceptoffers quickly as waiting results in reduction in priority and therefore lower quality offers inthe future.Our results show that previously used mechanisms and commonly studied theoretical bench-marks can yield either high average patient welfare or low discards, but not both. Thereforms in 2014 resulted primarily in redistribution from older patients to younger patients,with little improvement in the welfare of the average patient. In fact, both the pre- and

7Observe that when all agents place an identical value on each object, social surplus is a function only ofwaste.

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post-2014 mechanisms yield patient welfare and organ discard rates within 1.9 percent of thebenchmark first come first served mechanism. Last come first served is able to dramaticallyreduce organ discard rates (27 percent), but at the cost of lowering patient welfare by 34percent due to poor match quality.To systematically find reallocation gains beyond those afforded by these previous recom-mendations, we calculate optimal assignments and optimal offer rates. In the former, thedesigner has full information about agent preferences and can force agents to accept organs;it is therefore a theoretical bound on the maximum welfare that can be achieved under anymechanism. In the latter, the designer uses only observable characteristics and cannot forceagents to accept offers. We use the solution to this problem to design an alternative to thecurrent mechanism. We also try a greedy approach to improving efficiency by offering organsto patients that are predicted to benefit from them the most.Our results show significant scope for improving assignments. The theoretical bound on themaximum possible gains from improved assignments is equivalent to a 28.1 percent increasein donor supply. While substantial, implementing this assignment requires full informationabout the value of an organ to all currently registered patients and the ability to dictateallocations. Even with full information, it may be politically infeasible to achieve these gainsin practice because such a mechanism would make take-it-or-leave-it organ offers to patientswith irreversible organ failure.We then design a waitlist mechanism aimed at increasing patient welfare. To do this, we solvefor offer rates as a function of agent and object observables that maximize patient welfare,but cannot dictate acceptance behavior. Using this solution, we design a scoring mechanismcan achieve an increase in the average patient’s welfare equivalent to a 14.2 percent increasein donor supply. This increase is approximately half of the potential gains from any possibleallocation reform. This mechanism also increases transplant rates by 7.8 percent, and as aresult, equilibrium queue lengths and waiting times are much shorter than under the pre-2014mechanism.We then consider mechanisms that aim to achieve welfare gains without substantially hurtingany patient type. Specifically, we solve of Pareto improving offer rates that maximize patientwelfare subject to making no patient type worse off. We then approximate this mechanismusing a scoring rule. The importance of the status quo and distributional constraint in thisexercise substantially influences the solution. Patient welfare increases by an equivalent ofonly 9.1 percent.Our solutions are able to reduce discards and increase patient values by simultaneously con-sidering heterogeneity in patients’ transplant values and dynamic incentives. They reallocateoffers to patients who not only have high values for specific organs, but also are very likelyto accept them. This allows the mechanism to increase the quantity of transplants withoutlowering match value. Explicitly considering incentives is crucial for these gains. A mecha-nism which naively prioritizes patients based on predicted transplant value only marginally

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improves on the pre-2014 mechanism, increasing patient welfare and transplant rates by 1.2and 0.3 percent, respectively.

These results point to the significant scope of using our empirical framework for improvingdynamic assignment mechanisms. Previously used empirical approaches for reforming organallocation systems are unable to identify these gains. Moreover, a naive comparison thatignores incentives and holds choice probabilities fixed, as done in KPSAM (SRTR, 2015),predicts much smaller differences between these mechanisms and therefore understates thetrade-off between match quality and organ discards. This finding suggests that recommenda-tions need to account for equilibrium responses, be specific to the primitives of the market,and incorporate the desired objectives of the policy maker.

Related Literature

The design of scoring rules for organ allocation has been an active area of research in theMedical and Operations Research communities (Zenios et al., 2000; Su et al., 2004; Zenios,2004; Kong et al., 2010; Su and Zenios, 2006; Bertsimas et al., 2013). However, this previousresearch, as well as the KPSAM acceptance module (see SRTR, 2015) used by the ScientificRegistry of Transplant Recipients (SRTR) to simulate the effects of various allocation sys-tems, does not empirically model patients’ dynamic incentives to accept or reject an organoffer. As a result, the current approach taken by SRTR assumes that acceptance decisions donot depend on the waitlist mechanism. Empirical evidence in Agarwal et al. (2018b) suggeststhat ignoring dynamic incentives may result in biased predictions.

Within economics, there is a large body of theoretical work (see Roth et al., 2004, 2007,for example) and a more recent empirical literature (see Agarwal et al., 2018a) studyingthe design of living donor kidney exchange markets. Despite the growth in the living donormarkets, deceased donors continue to be the primary source of kidneys for transplantation,enabling approximately 70 percent of all kidney transplants in the United States.8

Our paper is related to Zhang (2010), which uses a dynamic model to study how patientslearn about the quality of an organ. It shows that patients lower on the list are more likely torefuse an organ if patients that are higher have refused it. The paper argues that this patternis most consistent with a parametric model of observational learning. Our approach abstractsaway from learning,9 but allows for unobserved donor heterogeneity to capture correlationin acceptance behavior. We do this to focus on allocation issues and equilibrium responseswhen simulating changes to the offer system.

The methods in this paper contribute to the growing literature on empirical approaches foranalyzing centralized assignment systems (see Agarwal, 2015; Fack et al., 2015; Abdulka-

8In 2017, organs from deceased donors accounted for 14,038 out of 19,849 kidney transplants. Source:https://optn.transplant.hrsa.gov/data/view-data-reports/national-data/.

9Zhang (2010) uses data from 2002. Anecdotal evidence suggests that the information available to patientsabout donors was dramatically better and standardized during our sample period (2010-2013). This factsignificantly reduces the scope for observational learning.

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diroglu et al., 2017; Agarwal and Somaini, 2018, for example). These previous approacheshave focused on static assignment mechanisms. To our knowledge, the only exceptions areWaldinger (2017) and Reeling and Verdier (2018). Waldinger (2017) estimates a model ofpublic housing choice in which agents face a two-stage decision with a portfolio choice prob-lem in the first stage.10 Reeling and Verdier (2018) analyze a dynamic allocation mechanismin which agents can enter a lottery for a good each period, and apply it to study the alloca-tion of bear hunting licenses. In contrast, our methods pertain to an optimal stopping rulethat differs from these settings.

This distinction between static and dynamic assignment systems is important because thetheory of static allocation systems, e.g. mechanism design approaches to school choice (Ab-dulkadiroglu and Sönmez, 2003), is comparatively well-developed. Abdulkadiroglu et al.(2017) show that, at least in New York City, there is little difference between various well-coordinated school choice systems. In contrast, Leshno (2017), Bloch and Cantala (2017),Arnosti and Shi (2017), and Su and Zenios (2004) arrive at different conclusions about whichsequential offer system performs the best. Their results depend on the nature of preferenceheterogeneity. Therefore, estimating these primitives is essential when designing dynamicallocation mechanisms.

Our work builds on the estimation of dynamic discrete choice models (Wolpin, 1984; Pakes,1986; Rust, 1987; Hotz and Miller, 1993), particularly recent developments in continuous-time versions of these models (Arcidiacono et al., 2016), and the estimation of dynamicgames (Bajari et al., 2007; Pakes et al., 2007; Aguirregabiria and Mira, 2007; Pesendorferand Schmidt-Dengler, 2008). Additionally, we employ a model of beliefs and an equilibriumnotion that resembles concepts aimed at making the analysis of dynamic games tractable(Hopenhayn, 1992; Weintraub et al., 2008; Fershtman and Pakes, 2012). We discuss therelationship to this literature when we develop our approach.

Overview

Section 2 describes the kidney allocation system, the available data and documents key factsfrom the market. Sections 3 and 4 model the optimal stopping problem from the perspectiveof each agent and detail our estimation methods. Section 5 describes our parameter estimates.Section 6 defines a steady-state equilibrium and summarizes results on existence; presents ourapproach to welfare comparisons; and outlines the algorithm for computing an equilibrium.Section 7 describes the mechanisms we compare and presents the results. Section 8 concludes.

10This work is also related to a literature that estimates preferences for public housing to answer questionsabout how to design an allocation mechanism (see Geyer and Sieg, 2013; Sieg and Yoon, 2016; van Ommerenet al., 2016; Thakral, 2016). A key difference is that these approaches are based only on final assignmentsinstead of detailed choice data.

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2 Background, Data and Descriptive Evidence

This section starts with the basics of kidney transplation before describing the allocationsystem. Next, we detail our data sources and the information available for the study. Finally,we present key descriptive facts to motivate our model and empirical exercises.

2.1 Basics of Kidney Transplantation

As of December 31, 2016, there were 726,331 cases of End-Stage Renal Disease (ESRD)in the United States (USRDS, 2018). Medicare provides near universal coverage to ESRDpatients; unlike traditional Medicare that is age-specific, the Medicare ESRD program isthe only disease-specific entitlement program. This program cost the federal government$35.4 billion in 2016, accounting for 7.2 percent of overall Medicare paid claims (USRDS,2018). This program therefore accounts for approximately 1 percent of the federal budget.Transplantation is the best treatment for ESRD, as it improves both the quality and lengthof life while saving an estimated $270,000 over the life of a transplanted patient (Wolfe etal., 1999; Irwin et al., 2012; Held et al., 2016; USRDS, 2018).

A kidney from a deceased donor is considered transplantable to a patient if they are biologi-cally compatible. A donor’s organ is considered incompatible if the patient has a pre-existingimmune response to proteins on the organ’s cells. A biologically incompatible patient’s im-mune system will recognize and attack the transplanted organ, resulting in graft failure.11Following transplantation, medications allow transplant physicians to limit new immune re-sponses to foreign protein types, but pre-existing immune responses lead to immediate loss ofthe transplanted organ if not avoided. The specific form of incompatibility is not importantfor the purposes of this study.

11The immune system tags foreign objects (antigens) with antigen-specific antibodies so that white bloodcells (leukocytes) can defend against them. Each donor has blood-type antigens and up to 6 specific typesof human leukoctye antigen (HLA) proteins out of a set of hundreds of possible types that are relevant forkidney transplantation. Some patients have pre-existing antibodies to some subset of these antigens. Atransplant recipient’s immune system will immediately attack the donor’s kidney and reject the organ if therecipient has an antibody to any one of the donor antigens. A recipient is tissue-type compatible with adonor’s kidney if she has no pre-existing antibodies corresponding to the donor’s antigens Danovitch (2009),even if the donor and the recipient do not have the same antigens. A transplant between certain incompatiblepatient-donor pairs has become possible due to development of desensitization technologies (see Orandi et al.,2014), but compatible transplants are preferred. Following transplantation, the immune system will attackany foreign antigen if such an attack is not attenuated with immunosuppressive medications. Pre-existingdonor-specific antibodies cause immediate rejection of a transplanted organ and cannot be prevented usingimmunosuppression. Thus, transplant physicians measure pre-existing immune responses and avoid them,whereas they prevent future immune responses by treating a transplanted patient with immunosuppressivemedications that prevent an immune response to newly-identified foreign antigens. Also for this reason, anorgan from a donor that has some antigens in common with the recipient is more desirable because thepatient is less likely to develop a new immune response to the organ after transplantation.

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Because of biological compatibility, the transplantation possibilities available to patients differbased on their immune systems. In the United States, kidneys from deceased donors arelargely allocated to patients of the same blood type to maintain a balance in transplantationopportunities across blood types. However, some patients have immune systems that reactto a broader range of tissues, even from donors with the same blood type. These patientshave fewer transplantation options. A patient’s immune sensitization is commonly measuredby Calculated Panel Reactive Antibodies (CPRA), which is the percentage of donors in arepresentative sample with whom she is tissue-type incompatible. This measure is calculatedfrom blood tests conducted to determine the set of human protein types (antigens) to whicha patient has had a pre-existing immune response.

The benefits of a transplantation can substantially differ, even conditional on biological com-patibility. Organs from donors that are younger, died of brain death under controlled circum-stances, and did not suffer from diabetes, strokes or hypertension usually function longer.Measured kidney function, time from removal of the organ to anticipated transplantation,size and weight match between the patient and the donor is also considered important. Adonor with tissue proteins that are similar to the patient’s is considered preferable becausethe likelihood of the patient developing an immune response post-transplantation is reduced.In some cases, the donor may have an infectious disease that the patient is at risk of getting ifshe proceeds with a transplant. There are a number of other factors that influence the med-ical benefits from specific organs and to specific patients. We refer the reader to Danovitch(2009) for further details about kidney biology.

2.2 The Allocation of Deceased Donor Kidneys

Assignment of organs from a potential donor begins after death is declared (brain or cardiac)and necessary consent for donation has been obtained. The Organ Procurement Organization(OPO) in the donor’s area obtains information about the donor from tests and the donor’smedical history. This information is entered into a system, called UNet, that is used tocoordinate across transplant centers. The OPO staff use UNet to determine the order inwhich patients will be offered each of the donor’s organs, to transmit information about thedonor to the transplant centers, and to record accept/reject decisions. OPO staff usuallycontact the surgeons for several potential recipients simultaneously to solicit their decisions.This process can take place while the donor is on life-support and before the potential donor’sorgans have been extracted in order to maintain organ viability. Once a kidney has beenrecovered from the donor, transplant surgeons or patients that were potentially interested inreceiving that kidney may decline based on any new information discovered during medicaltesting or the physical examination of the kidney. These final decisions need to be madewithout delay, usually within an hour. A final compatibility blood test is then conductedusing samples from the donor and all patients that have accepted an organ. The donor’s

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kidneys are then allocated to the highest priority patients on the waitlist that were willingto accept the organs.

When offering organs, UNet first excludes patients who are not biologically compatible withthe donor. This exclusion is based on a detailed patient immunological profile that is submit-ted when the patient is registered on the waitlist. Specifically, the allocation system requirespatients to list unacceptable donor antigens, i.e. donor protein types with which the patient’simmune system is likely to react. The allocation system runs a “virtual crossmatch” withthese data, which we mimic in our analysis.

Next, UNet screens out patients that have listed pre-set exclusion criteria within the system.These criteria allow patients to automatically exclude kidneys that are transplantable butundesirable because of donor characteristics such as age, health conditions, and kidney func-tion. UNet then orders the remaining set of patients first by priority type, and then withinpriority type by priority points. Finally, it breaks ties in order of time waited.

The priority and points system is motivated by both equity and efficiency concerns. Thesystem in place during our sample period (2010 to 2013) first offers kidneys to patients witha perfect tissue-type match, then to patients from the local OPO in which the organs wererecovered, then regionally, and finally nationally. Within each priority group, the pointssystem is based on tissue type similarity, whether or not the patient is pediatric, patientsensitization, and waiting time. The detailed priority system is described in policy section 8in OPTN (2014).

New kidney allocation rules were implemented on December 4, 2014. This new system givesgreater priority to the healthiest patients for the most desirable donors, increases priority forextremely hard to match patients, and reduces emphasis on wait time. Israni et al. (2014)discusses this system and the rationale for the changes. We refer the reader to OPTN (2017)for a detailed description of the priorities and points used.

2.3 Data Sources

This study uses data from the Organ Procurement and Transplantation Network (OPTN).The OPTN data system includes data on all donors, wait-listed candidates, and transplantrecipients in the US, submitted by the members of the Organ Procurement and Transplan-tation Network (OPTN). The Health Resources and Services Administration (HRSA), U.S.Department of Health and Human Services provides oversight to the activities of the OPTNcontractor. For tractability, we restrict attention to data on the kidney waitlist and theacceptance decisions of all patients in the New York Organ Donor Network (NYRT) betweenJanuary 1st, 2010 and December 31st, 2013.12 NYRT is the largest donor service area (DSA),

12We end our sample in 2013 to rule out anticipatory effects and to avoid modeling transition dynamics asagents ancitipate the new system introduced in December 2014. Reports from the United Network for Organ

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in terms of number of patients, that used the standard allocation rules in the United Statesprior to 2014.13

The primary dataset on the waitlist, the Potential Transplant Recipient (PTR) dataset, con-tains the offers made and accept/reject decisions. This dataset is drawn from the recordsgenerated by UNet, which is the backbone software system used to coordinate offers anddecisions. In addition, we obtained detailed information on patient and donor characteris-tics from the Standard Transplantation Analysis and Research (STAR) dataset. Fields inthe STAR dataset are populated based on information gathered in UNet as well as formssubmitted by transplant centers after a transplant is performed.

2.4 Descriptive Analysis

We now describe our sample of patients and donors and document choice patterns. A strikingfeature of the waitlist is that even though there is extreme scarcity, some donors are rejectedby a very large number of patients. Choices suggest that large differences in donor qualitycombined with substantial priority for waiting time incentivize patients to reject low-qualitydonors and wait for more attractive offers.

Patients and Donors

Table 1 describes our patient sample. A total of 9,917 patients were registered with NYRTat some point between 2010 and 2013. Panel A shows the state of the waitlist on January1st of each year in our sample and summarizes a subset of important patient characteristics.Our dataset includes rich information on patient health status, including indicators of patienthealth (e.g. body mass index, age, total serum albumin), and medical history (e.g. diabetes,years on dialysis). The average patient on the list has waited for a little over two years, withthe average waiting time increasing over time. Recall that CPRA measures the probability oftissue-type mismatch with a randomly chosen donor. The average CPRA is about 12 percent,which indicates that there is more than a one-in-ten chance that a patient is tissue-typeincompatible with a randomly chosen donor. The standard deviation is large because thereare many patients with extremely low or extremely high CPRA. The allocation mechanismawards priority and points to high CPRA patients because of equity concerns.

Sharing (UNOS) that track transplantation rates after the adoption of the new system show the existenceof short-term transition dynamics (termed “bolus-effects” in these reports) immediately following the reform(Wilk et al., 2017).

13See Hart et al. (2017) for a map of DSAs in the United States. As mentioned above, except in cases ofperfect tissue-type matching, allocation takes place based on geography, with DSAs constituting the smallestunit. A little less than half of DSAs used rules that were different from the baseline rules. We identifiedthe DSAs that use non-standard rules via a special request for administrative documentation on the variousrules in use.

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Panel B describes the patients that enter and exit the waitlist during our sample. The list isgrowing; the number of new patients joining exceeds the number leaving. Panel C shows thereasons for departure from the list and the characteristics of patients by the stated reason fordeparture. The most common reason for depature is receiving a deceased donor transplant.The average patient has waited for 3.08 years before receiving a deceased donor. The secondmost common reason is that the patient either dies or becomes “too sick to transplant.” Thesepatients are 61.5 years old on average, compared to 54.3 years for patients who receiveddeceased donor transplants. The third most common departure reason is receiving a livedonor transplant, which is more likely for younger patients and often occurs within the firstyear on the waitlist. Finally, some patients leave for other or unknown reasons including amove outside the NYRT area.

Table 2 summarizes the rich set of donor characteristics used in our study. These includedonor age, cause of death, relevant medical history (diabetes, hypertension), and the leadingindicator of donor kidney function (donor creatinine). Panel A presents the statistics fordonors recovered within NYRT during our sample period. Just under 200 donors are recov-ered from the NYRT area each year, which is only one-seventh of the number of patientsjoining the waitlist in NYRT. Therefore, there is extreme scarcity in organ supply within theregion. The refusal rate remains high despite this scarcity. Across donors, the mean numberof biologically compatible offers that met the pre-set screening criteria is over 430, but themedian is much lower, at 27. This skewed distribution arises because undesirable kidneys arerejected by many patients, while desirable kidneys are accepted quickly. Indeed, over 20% ofdonors have at least one of their viable kidneys discarded. Organs from these donors wererefused by an average of almost 1,500 patients. The mean and median amongst the group ofdonors for which both of their viable kidneys were accepted are 123.1 and 15 respectively.14Our observable donor covariates which should predict organ quality – including age, cause ofdeath, and donor creatinine – are correlated with discards in the expected ways. The last twosets of columns show that a number of donors recovered in NYRT were ultimately offered topatients elsewhere. For these donors, only about 0.3 kidneys were transplanted per donor,indicating that most of these kidneys were discarded.

In addition to donors recovered within the local area, NYRT patients are also offered donorsfrom other parts of the country. Indeed, panel B shows that a total of 1,470 donors wereoffered to patients registered with NYRT in the average year. Because most of these donorswere recovered elsewhere in the country but offered to NYRT patients after a large numberof refusals, these donors are likely to be undesirable. It is therefore not surprising that theysee a very large number of offers and high discard rates. Again, the relatively poor qualityof these donors is captured by our observable characteristics. Compared to donors recoveredin NYRT, the average non-NYRT donor offered to NYRT patients is older, less likely to

14The number of transplanted kidneys amongst donors with no discards is less than two because somedonors have only one viable kidney for donation.

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have died of head trauma, more likely to be diabetic or hypertensive, more likely to be anundesirable donor (ECD) or be donating after cardiac death (DCD), and more likely to havea high creatinine level. The average donor offered to NYRT patients ultimately donates only0.75 kidneys.

Waitlist, Offers and Acceptance Rates

We now describe the offer and acceptance rates in the data. Throughout the paper, we onlyconsider blood and tissue type compatible offers. As a reminder, we refer to the decision-maker as the patient for simplicity of exposition, assuming that any decisions made by sur-geons are in the best interest of each individual patient. Overall, patients receive many offersand reject most of them. This is because desirable kidneys are accepted quickly, while lessdesirable kidneys are offered to many patients before being accepted or discarded. A pa-tient’s likelihood of receiving a high-quality offer rises as her waiting time increases, creatinga strong incentive for patients to wait for attractive offers.

While position on the list increases with waiting time, the kidney allocation system signif-icantly differs from a first come first served waitlist. Figure 1a plots waiting time againstposition on the list for offers where the donor met the patient’s screening criteria. The hor-izontal axis is the (donor-specific) position of a patient, and the vertical axis is the averagewaiting time for patients at that position. Mean waiting time falls quickly as we go down thelist. However, the system is not well approximated by a first come first served queue. Wecalculated the fraction of times that two patients who are offered the same donor are orderedidentically on the list for the next donor they are both offered. This fraction is 81.5 percent.It would be 100 percent in a first come first served system, and 50 percent in a system thatrandomly ordered patients for each donor. This summary shows the importance of the pointsand priorities other than for waiting time in determining position on the list.

Acceptance rates suggest that patients who are higher on the list have access to higher qualitydonors. Figure 1b shows the fraction of offers accepted by the position of the patient. Theacceptance rate in the first few positions is much higher than later positions, but still onlybetween 15 and 20 percent. Only about 1 percent of offers are accepted in lower positions,and this rate falls even farther below the top 20 positions. This sharp decline near the topoccurs for two reasons. First, for some donors, the first few offers are made to patients witha perfect tissue-type match. These offers have a high acceptance rate because perfect tissue-type matches are rare and extremely valuable. The second reason, which is the predominantone, is that lower positions see a negatively selected set of kidneys because desirable kidneysare likely to be accepted near the top of the list. Both reasons give patients a strong incentiveto reject offers and wait to receive a higher-quality donor in the future.

Because lower-quality kidneys are offered to more patients, patients receive many offers,but the overall acceptance rate, as a fraction of offers, is extremely low. Table 3 describes

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these overall patterns in offer and acceptance rates. Panel A considers all feasible offers,including offers that did not meet the patient’s pre-set criteria. A typical patient receivesabout 220 offers per year, but only 0.15 percent of offers are accepted. When interpretingthese low acceptance rates, it is important to remember that the majority of offers are fromvery undesirable donors. Offers from desirable donors are more likely to be accepted: kidneysrecovered in NYRT are accepted five times more often, and 10.6 percent of offers of a perfecttissue-type match are accepted. The panel also shows that patients with sensitized immunesystems – that is, patients with CPRA above 80 percent – receive many fewer offers fromtransplantable donors even though the system gives them higher priority, and are more likelyto accept. Panels B and C restrict to offers that are likely to be more attractive. Panel Bstudies offers that met patients’ pre-set criteria. The typical patient still gets such an offerevery three or four days. However, a typical patient only receives about two offers per monthfrom donors recovered within NYRT. Panel C restricts to offers in the first 10 positions,which are even more likely to be for organs from desirable donors. These offers are rare – thetypical patient can expect to receive less than one such offer each year – but they are verylikely to be accepted.Taken together, these statistics suggest that the supply of desirable donors in NYRT is scarceand that patients have to wait several years before receiving a transplant from a desirabledonor. Moreover, our dataset contains rich information predictive of the likelihood that adonor is refused.

Evidence on Mismatch

Table 4 provides suggestive evidence of mismatch between donors and transplanted patients.Panel A reports the rates at which patients receive any transplant. Pediatric patients arevery likely to be transplanted, either with a deceased donor kidney or through a livingdonor. The priority given to these patients is likely an important contributing factor. Adultpatients are less fortunate, but interestingly, among adults there is no signficant gradientin transplant probability with age. In contrast, the chance of receiving a live donor doesfall with age. Panel B describes transplanted donors by age for those patients who receivea kidney through the deceased donor waitlist. Pediatric patients are much more likely toreceive a transplant from a young donor as compared to older patients. Although thereis some assortative matching by age, signs of age mismatch remain. Many patients abovethe age of 65 continue to receive kidneys from young adults and middle-aged donors. Oneconcern in interpreting these numbers is that a kidney transplanted to an older patient may beundesirable for other reasons. Panel C focuses on a subset of donors with no clear medicallyundesirable characteristics such as diabetes, cardiac death, high creatinine levels or hepatitisC. The qualitative patterns of age mismatch persist. These patterns motivated some of the2014 OPTN reforms, which attempted to match healthier patients (typically young adults)to donors whose organs are predicted to last longest.

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Another patient characteristic that will be important in the improvements we ultimatelyidentify is whether the patient had already begun dialysis when they joined the waitlist.Some patients with end-stage renal disease who qualify for the waitlist still have marginalkidney function and can avoid dialysis. These patients are relatively healthy compared topatients already on dialysis at registration, but they are still likely to need dialysis in thefuture. The last two columns of Table 4 show that outcomes differ substantially for patientson and off dialysis. While patients not yet on dialysis at registration represent 32 percent ofour patient sample, they receive only 27 percent of deceased donor transplants. In contrast,they are much more likely to receive a living donor transplants (Panel A): 17 percent ofoff-dialysis patients received a live donor transplant during our sample, while only 7 percentof on-dialysis patients did. Consistent with being healthier, dialysis patients are much lesslikely to depart the waitlist because they died or were too sick to transplant. Despite thesedifferences, donors are nearly equally distributed by age among on- and off-dialysis patients(Panel B), though patients not on dialysis at registration are a bit more likely to receive avery young or very old donor.

Evidence on Response to Dynamic Incentives

A central assumption in our framework is that agents are forward-looking and respond todynamic incentives. One implication of this assumption is that patients for whom the optionvalue of waiting is lower should be less selective. Agarwal et al. (2018b) present descriptiveevidence consistent with dynamic incentives using data from all areas of the United States.They find that highly sensitized patients who are immunologically compatible with fewerdonors – and who can therefore expect to receive fewer offers in the future – are more likelythan less sensitized patients to accept an offer of a given quality. We replicated their researchstrategy using data from patients registered in NYRT and found similar patterns. We brieflydescribe the strategy and results below.

The ideal experiment would compare two identical populations of patients that face differentoption values for exogeneous reasons. However, we are not aware of such variation in thecontext of kidney allocations. Instead, Agarwal et al. (2018b) use the likelihood that a patientis biologically compatibile with a randomly chosen donor, as measured by the CalculatedPanel Reactive Antibody (CPRA), to study how variation in option values affect acceptancedecisions. A patient that is likely to be biologically compatible with a large number of donorsshould have a high option value of waiting, and therefore be more selective.

We replicate their findings for NYRT and show that, as predicted by the presence of dynamicincentives, CPRA is negatively correlated with offers for compatible organs and positivelycorrelated with acceptance rates (Figures B.1a and B.1b in the Appendix). This pattern isrobust to rich controls for patient priority and indicators of the value of an offer, for example,patient and donor characteristics, match characteristics, interactions of CPRA with tissue-

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type similarity (Table B.1 in the Appendix).The main concern is that immune sensitization also influences the value from a transplant.Patients develop sensitive immune systems primarily through blood transfusions and priortransplants. Therefore, these patients are more likely to be frail, making a transplant risky.This risk is less justified unless the donated organ is of high quality. By this argument,sensitized patients should be less likely to accept offers. The fact that we find the oppositecorrelation strengthens our argument for the importance of dynamic incentives.

3 A Model of Decisions in a Waitlist

This section presents a model of agents’ decisions in a waitlist mechanism that will formthe basis of our empirical strategy. We begin by defining a class of sequential assignmentmechanisms and the primitives governing agents’ decisions while on the waitlist. Agentsand objects arrive according to exogenous processes, and each object is offered to agents onthe waitlist in order of an agent-object-specific priority score until the object is acceptedby an agent or rejected by everyone. We then provide assumptions on agents’ payoffs andbeliefs, as well as the evolution of the state space, which lead to a tractable optimal stoppingproblem from the agent’s perspective. Though the model is motivated by the structure ofour application, it may be useful in other settings in which items are offered sequentially toagents, including other organ allocation settings.

3.1 Notation and Preliminaries

Consider a sequential assignment mechanism in which objects indexed by j are offered toagents indexed by i waiting on a list. Let xi denote observed characteristics of agent i; letzj and ηj respectively denote observed and unobserved characteristics of object j; and let tidenote the amount of time the agent has been waiting on the list. We assume that agentsobserve both ηj and zj. The model does not include unobserved agent heterogeneity. Wediscuss this restriction in Section 4.2.Objects may be incompatible with some agents. Let cij = 1 if object j is compatible withagent i, and 0 otherwise. Incompatibility can arise due to biological reasons in the organallocation context but they may arise due to other restrictions (e.g. legal) in other contexts.Time is continuous. Objects and agents arrive at poisson rates λ and γ, respectively. Thecharacteristics of each arriving agent x are independent and identically distributed (iid).Similarly, each object’s characteristics (z, η) are drawn iid from the cumulative distributionfunction (CDF) F upon arrival. We assume that each object must be assigned before thenext object is offered. The poisson arrival process and continuous time together imply thatsimultaneous arrivals are zero probability events.

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3.2 Mechanisms and Primitives

3.2.1 Mechanisms

We consider sequential assignment mechanisms that use a priority score. The mechanismallocates each object as it arrives:

• Step 1 (Ordering): The priority score sijt ≡ s (t;xi, zj) is calculated for all agents onthe waitlist. Ties in the score, if any, are broken using a known tie-breaking rule. Forexample, ties could be broken either uniformly at random or by waiting time.15

• Step 2 (Offers): Each agent may decide to accept or reject the object, with accep-tance denoted by aij = 1. The mechanism may solicit decisions from multiple agentssimultaneously. A mechanism does not make offers to agents that are known to beincompatible with the object.

• Step 3 (Assignment): The object(s) are allocated to agents with the highest qj prioritiesfor whom aij = 1, where qj is the number of copies of the object available. An objectcannot be allocated to an incompatible agent.

• Step 4 (Arrivals and Departures): An agent is removed from the waitlist once an objecthas been assigned to her. Other agents may exogenously join or leave the list.

Within the set of general offer-based waitlist systems, the primary restriction is on the orderin which offers are made. Specifically, we assume that an agent’s priority does not depend onthe other agents in the market. This restriction is adequate for estimation using the deceaseddonor kidney allocation system in place during our sample period. Moreover, such mecha-nisms are a natural class to consider because they are simple and transparent to implement.Indeed, all deceased-donor organ allocation mechanisms as well as systems considered by thekidney allocation committee during their deliberations prior to the 2014 reform were offermechanisms based on priority scores.16 In counterfactual analysis, we will compare assign-ments that result from various mechanisms that obey this structure to benchmark optimalassignments.

Typical administrative datasets from such assignment systems contain information on allcharacteristics used to determine the priority score because the characteristics are used tomake offers. This allows a researcher to calculate the order in which any object would beoffered. Our empirical exercises required us to develop computer code for this purpose, and

15If ties are broken by ti, it must be that no two agents have the same value. Since time is continuous andagent arrivals are governed by a poisson process, simultaneous arrivals will be zero-probability events, and tistrictly orders any two patients with probability one.

16Based on an examination of committee reports and public comments downloaded fromhttps://optn.transplant.hrsa.gov/members/committees/kidney-committee/.

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we were able to verify the output of our code using administrative records of the offers thatwere made during our sample period.

One complication in our setting is that organs must be allocated within a certain time framethat depends on the condition of the organ and various logistical factors. Limited manpowerat the Organ Procurement Organization (OPO) can limit the number of patients that canbe contacted and offered the organ. We treat the maximum number of offers that can bemade for each object as exogeneous.

3.2.2 Payoffs

There are three types of primitive payoffs in the model. The first is the (expected) netpresent value of agent i being assigned a compatible object j after waiting t periods, denotedΓij (t). In our application, this term captures the value placed by patients and surgeons ontransplants from various types of organs. The second is the expected net-present value fromfrom departure without an assigment, Di (t). In our application, departures occur due toliving donor transplants, death, or transfers to other listing centers (Table 1). We view Di (t)as incorporating any of those reasons.17 Finally, agents incur an expected flow payoff whilewaiting on the list, di (t). In our application, di (t) is best interpreted as the payoff fromliving without a functioning kidney, which includes dialysis for most patients.

Two economic implications of the payoffs in our model are worth noting. First, we abstractaway from costs of considering an offer. These costs are likely small relative to the value oftransplants and the flow costs of remaining on dialysis. Second, we assume that agents onlyvalue their own outcomes and not those of others. This assumption is commonly made intheoretical and empirical work on assignment mechanisms (e.g. Abdulkadiroglu and Sönmez,2003; Abdulkadiroglu et al., 2017). This restriction may be violated if surgeons value theoutcomes of other patients, especially those that they might be treating. NYRT has atotal of 10 transplant hospitals staffed with many more kidney transplant surgeons. Thislimits common agency problems that surgeons might face. The payoffs can be interpreted asaccruing to the patients if surgeons act in the best interest of each of their patients.

Our empirical framework makes the following assumptions on these payoffs:

Assumption 1. (i) The (expected) net present value of an assignment is additively separablein a payoff shock εijt:

Γij (t) ≡ Γ (t, xi, zj, ηj) + εijt. (1)17We can represent the value from a departure as a weighted average over the value from the various events,

i.e. Di (t) =∑k pik (t)Dik (t) where k denotes the type of depature (e.g. obtaining a live donor, death etc.)

and pik (t) is the probability of each type of depature conditional on a departure occuring. The formulationis agnostic about the sources of these payoffs. For example, the net present value of death can include anybequest motives.

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(ii) The random variables εijt are iid with a known, non-atomic distribution with cdf G.

(iii) The expected flow payoffs from waiting di (t) and the expected payoff from departingwithout an assignment Di (t) depend only on (xi, t) .

Restrictions on εijt imposed in Assumptions 1(i) and 1(ii) are common in the dynamic discretechoice literature. They will allow us to use an approach based on an inversion technique dueto Hotz and Miller (1993). The comparison with other methods and specific functional formassumptions on G, Γ (·), and the distribution of ηj are discussed in Section 4 below.

Assumption 1 implies that the model excludes agent-level unobserved heterogeneity. Wediscuss analytical challenges in relaxing this assumption once we have laid out our estimationapproach in Section 4.2.

3.2.3 Arrivals and Departures

Agents arrive stochastically, and may depart the list prior to assigment. We make the fol-lowing assumption on the arrival and departure processes:

Assumption 2. (i) Departures prior to assignment and arrivals are governed by Poissonprocesses that are independent of the waitlist composition and design.

(ii) The departure rate of agent i is given by δi (t) ≡ δ (t;xi). Further, each agent has aterminal date Ti <∞ at which departure occurs with probability 1.

In our application, we assume that patients die on or before their 100-th birthday. Ti thereforecorresponds to the waiting time for a patient on the day she turns 100 years of age.18

The primary economic restriction for our purposes is that departures prior to assignmentand arrivals do not depend on the design of the kidney waitlist. Table 1 shows that themost common reason for departure without a deceased donor transplant is death or patientsbecoming “too sick to transplant.” It seems safe to assume that these events are not responsiveto the design of the kidney waitlist. The second most common reason is receiving a livingdonor transplant. Departures due to this reason are exogeneous if the design of the kidneywaitlist does not affect the probability of finding a compatible living donor, and if patientsalways prefer a living donor to staying on the deceased donor waitlist. These conditionsare plausible in our setting because living donors are medically superior to deceased donors

18It is straightforward to extend the framework to allow for agents that could remain on the list forever,Ti = ∞. This generalization will primarily change computational techniques and require that the valuefunction for each patient approaches a constant. We restrict our attention to the finite time-horizon case forsimplicity of exposition.

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and produce better transplant outcomes in terms of patient and graft survival.19 Finally, aminority of patients depart for other reasons, in most cases for undisclosed reasons or becausethey move residences.

Similarly, we assume that agent arrivals do not depend on the design of the waitlist. Duringour sample period, patients could register as soon as kidney function is sufficiently poor.20Therefore, it is in a patient’s interest to join the waitlist as soon as possible. This feature ofthe priority system motivates our assumption that arrivals do not depend on the state of thewaitlist or the allocation mechanism. In counterfactuals, we consider priority systems thatdo not change how waiting time is calculated.

3.3 Individual Agent’s Problem

Agents on the waitlist who receive an offer of an object must decide whether to accept it orwait for a future offer. This results in an optimal stopping problem from the perspective ofthe agent (Pakes, 1986; Rust, 1987). We follow a common estimation strategy in dynamicgames by considering an agent’s optimal decision rule taking the distribution of actions ofother agents as observed in the data (Pakes et al., 2007; Bajari et al., 2007). Solving forcounterfactuals will require a notion of equilibrium, which we discuss in Section 6. Thissection starts by describing a general formulation of this single-agent problem before makingsimplifying restrictions.

3.3.1 Beliefs

To make an informed decision about whether to accept an offer, an agent must form beliefsabout the organs she may be able to obtain in the future if she declines the current offer.Recall that the kidney waitlist offers organs to patients in sequence of their priority scores

19Living donor superiority is partly driven by the higher medical quality of living donor kidneys, and alsoby the fact that living donation allows for a better planned transplant. For example, patients receiving aliving donor transplant can proactively start immunotherapy. OPTN and SRTR (2011) report the rate oflate graft failure for transplanted patients by donor type (living or deceased). This measures the time atwhich half of the transplanted patients are alive with the kidney still functioning. The rate of late graftfailure for adult patients transplanted in 1991 was 10.1 years for deceased donor kidneys, compared to 15.8years for living donor kidneys. Some of this difference may be due to selection into who receives each typeof transplant. Hart et al. (2017) compare outcomes following living and deceased donation by patient ageand primary diagnosis using the chances of graft failure 10 years after transplantation. This statistic foradults transplanted with a deceased donor kidney in 2005 is 52.8%, whereas it is only 37.3% for those thatreceived a living donor. They report that 5year patient survival differences for living donor and deceaseddonor recipients are large even when broken down by patient age or primary diagnosis (compare figures KI79 and KI 80 with figures KI 82 and KI 83 from Hart et al. (2017)).

20This feature is shared for all DSAs, including NYRT, that used standard allocation rules during oursample period. A patient was qualified for registration once they had begun dialysis or had a glomerularfiltration rate (GFR) below 20mL per minute.

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as long as the kidney remains viable for transplantation. The organs are allocated to thehighest priority agents that are compatible with and accept the organ. At the end of theallocation process, each organ j effectively has a cutoff priority s∗j , such that only an agentwith priority at least s∗j would have received the organ if she accepted it. This score dependson the decisions of all agents on the list at the time, their compatibility, and the number ofoffers that can be made for the object. Agent i can expect to obtain a compatible kidney aslong as her score, sijt, exceeds s∗j . Therefore, it is sufficient for an agent to form beliefs overthe probability distribution of s∗j in order to decide which organs are likely obtainable in thefuture.

The beliefs about the distribution of the cutoffs priority s∗j depend on the quality of the organand the information an agent may have about the competitive environment. Let

H (s;Fi,t, zj, ηj) = P(s∗j < s

∣∣∣Fi,t, zj, ηj) (2)

denote the belief for the CDF of s∗j given the information set Fi,t and the organ characteristicszj and ηj. Therefore, if agents know the scoring rule, then agent i believes that the probabilityof receiving an offer for organ j is given by H (s (ti;xi, zj) ;Fi,t, zj, ηj).

In principle, the information set can include the history of offers previously received (andrejected) by the agent as well as identities of the other agents on the waitlist at a given pointin time. However, there are several reasons, discussed below, why beliefs are unlikely to besensitive to such information. We therefore assume that an agent’s beliefs des not depend onsuch fine information:

Assumption 3. Each agent i’s believes that the probability that an object with characteristics(zj, ηj) is compatible and will be available to her after a waiting time of t is

π (t; zj, ηj, xi) = H (sijt; zj, ηj)× P (cij = 1| zj, xi) ,

where sijt = s (t; zj, xi) and H (·; zj, ηj) is the conditional distribution of the cutoff s∗j given(zj, ηj).

This assumption embeds three key restrictions. First, it assumes that beliefs are not sensitiveto short-term variation in the set of other agents currently on the waitlist. The primary threatto this restriction is that some surgeons may be treating multiple patients on the kidneywaitlist or may learn about other patients from their colleagues. This concern is limitedby the fact that the NYRT area has many transplant hospitals and surgeons. Second, itabstracts away from inference about the likelihood of receiving future offers based on pastoffers. This restriction is motivated by the institutional features and empirical observationsdiscussed below in Section 3.3.4. Finally, it assumes that the probability that an organ iscompatible depends only on observables and is independent of the cutoff. This restriction is

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appropriate in our context because surgeons list the blood- and tissue-types that are knownto be incompatible with the patient.

Our assumption on beliefs is a reasonable approximation if assessments about which organsare likely to be available are based primarily on a surgeon’s extensive experience treatingpatients. It also eases the analysis relative to beliefs of the form in equation (2) becauseit dramatically reduces the dimension of the information set, and therefore the state space,in the dynamic problem. It is well known that this curse of dimensionality can complicateanalysis and estimation in dynamic models (see Pakes and McGuire, 2001, for example).

The simplification of the state space in Assumption 3 is similar in spirit to equilibriumconcepts that have been introduced to make the estimation of dynamic games tractable.These concepts simplify the state space by abstracting away from aggregate uncertaintyin the composition of competitor types (Hopenhayn, 1992; Weintraub et al., 2008) or bymodeling beliefs as being based on past experience (Fershtman and Pakes, 2012). Despitethese simplifications, the state space continues to be quite rich because, in addition to aspectsthat influence payoffs, it contains all characteristics that influence priorities or determinewhether or not any given object j is compatible.

3.3.2 Value functions

We assume that agents make optimal accept/reject decisions by comparing the net presentvalue of an object to the value of waiting. Holding the strategies of other agents fixed, shedecides to remain on the list instead of accepting object j if the payoff from an assignmentΓij (t) is less than the value of continuing to wait conditional on the agent’s type xi andcurrent waiting time t, denoted Vi (t) ≡ V (t;xi). The Hamilton-Jacobi-Bellman differentialequation defining the value of waiting at time t is:

(ρ+ δi (t))Vi (t) = di (t) + δi (t)Di (t) +λ∫πij (t)Emax 0,Γij (t)− Vi (t) dF + Vi (t) , (3)

where the operator E takes expectations over the idiosyncratic payoff shocks εijt in equation(1), and, with a slight abuse of notation, πij (t) = π (t;xi, zj, ηj) defined in Assumption 3.

This expression can be derived by considering an agent’s value of waiting at time t for aninfinitesimal duration ∆t. In the event that no object arrives during this period, the agent in-curs flow payoffs from dialysis di (t) ∆t and may depart exogenously with probability δi (t) ∆t,incurring a payoff of Di (t) if such a departure offers. In the event that an object arrives usingthis period, which occurs with probability λ∆t, the object’s characteristics are drawn fromthe CDF F . The integral calculates the expected increment in the agent’s value function foreach arrival. Specifically, the agent receives an offer for this object with probability πij (t)and accepts it if Γij (t) > Vi (t), yielding an incremental value of Emax 0,Γij (t)− Vi (t)from an offer for object j. In the limit as ∆t→ 0, the probability that both departures and

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object arrivals occur within the interval ∆t tends to zero, yielding the differential equationabove.21

Equation (1) shows an intuitive result that values depend on the flow payoffs while waitingon the list, the possibility of and value from exogeneous departures, and the option valueof potential offers. The differential equation defining Vi (t) has a unique solution that isdetermined by the terminal condition Vi (Ti) = Di (Ti) because the probability of receivingadditional offers in the remaining time vanishes as t→ Ti.

3.3.3 Normalization and Simplifying the Value Function

A typical dataset from a sequential assigment mechanism such as ours only contains infor-mation about accept/reject decisions. As is well understood, data on actions alone may notbe sufficient for identifying all primitives of a dynamic discrete choice model, and the payofffrom one action must be normalized in each state (Magnac and Thesmar, 2002). However,Aguirregabiria and Suzuki (2014) and Kalouptsidi et al. (2015) point out that such normal-izations may not be innocuous because they arbitrarily restrict payoffs from specific actionsacross various states. These restrictions can affect the validity of answers to certain coun-terfactual predictions. This fact poses a potentially serious problem for empirical analysisif one is interested in answering questions that depend on primitives that are not identifiedfrom choice data.

Fortunately, the counterfactuals involving changes in the mechanism are identified in ourmodel. Intuitively, in any waitlist mechanism, the trade-offs between accepting an offer andwaiting should only depend on payoffs relative to the value of never receiving an assignment.Assumptions 1(iii) and 2(i) together imply that the value of never receiving an assignmentdoes not depend on the mechanism. This discussion suggests normalizing the value of refusingall offers, irrespective of the state.

21Specifically, the discretized version of the Hamilton-Jacobi-Bellman Equation defining the value of waitingat time t is:

Vi (t) = 11 + ρ∆t

[di (t) ∆t+ δi (t) ∆tDi (t) + λ∆t

∫πij (t)Emax Vi (t+ ∆t) ,Γij (t) dF

+ (1− (δi (t) + λi (t)) ∆t)Vi (t+ ∆t) + o (∆t)] ,

where λi (t) = λ∫πij (t) dF is the rate at which agent i expects to receive an offer at time t. The leading

fraction represents discounting due to time preferences. The first term is the flow payoff from remaining ondialysis. The second term is the expected probability of departure multiplied by its value. The third termrepresents the value for a kidney arriving. The fourth term denotes the value of waiting in period t+ ∆t inthe case when no offer arrives and an exogenous departure does not occur. The remainder term includes thepayoff in the event that multiple donors or objects arrive, or that a donor arrives and the patient departs,within ∆t. These events have probability of order o (∆t). Therefore, the remainder is of order o (∆t) as longas all expected payoffs are bounded. Taking the limit as ∆t→ 0 under mild continuity conditions yields thedifferential equation above.

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Formally, the value of refusing all offers, and therefore never being assigned, is defined bythe differential equation

(ρ+ δi (t))Oi (t) = di (t) + δi (t)Di (t) + Oi (t)

and the terminal condition O (Ti) = Di (Ti) . Under Assumptions 1(iii) and 2(i) this valuedoes not depend on the waitlist, and therefore future offer probabilities πij (t). MeasuringVi (t) and Γij (t) relative to Oi (t) suffices for analyzing decisions and differences in welfareunder the current and alternative mechanisms. Appendix B.1 formally shows that this is thecase.

With this in mind, we normalize the net present value of refusing all offers in the future,Oi (t), to zero at all t. This normalization implies that di (t) + δi (t)Di (t) = 0 for all t andthat Di (Ti) = 0. Equation (3) now simplifies to

(ρ+ δi (t))Vi (t) = λ∫πij (t)Emax 0,Γij (t)− Vi (t) dF + Vi (t) (4)

As can be seen, the advantage of this particular normalization is that the set of primitivesthat need to be estimated is greatly reduced. We no longer need to estimate the flow payoffsfrom remaining on the list or the net present value of departing. Going forward, we interpretΓij (t) and Vi (t) as values relative to never receiving an assignment.

The solution to differential equation above is

V (t;xi) =∫ Ti

texp (−ρ (τ − t)) p (τ |t;xi)

(λ∫π (τ ;xi, z, η) ψ (τ, xi, z, η) dFz,η

)dτ, (5)

wherep (τ |t;xi) ≡ exp

(−∫ τ

tδ (τ ′;xi) dτ ′

)is the probability that agent i does not exogenously depart before τ conditional on being onthe list at t and

ψ (τ, xi, z, η) ≡ Emax 0,Γ (τ, xi, z, η) + εijt − V (τ ;xi)

is the incremental value to agent i of receiving an offer of an object with characteristics (z, η)at time τ , with expectations taken over εijt. We have explicitly reintroduced agent and objectcharacteristics into the notation because this equation will form the basis of our empiricalstrategy. This solution is based on the boundary condition limt→Ti V (t;xi) = O (Ti;xi) = 0because the probability of receiving an offer after t vanishes as t → Ti. This equation alsoclarifies that the model and approach is readily extended to the case when Ti is infinite. Todo this, one would replace the condition that Vi (Ti) = 0 with the condition limt→∞ Vi (t) = 0.

26

3.3.4 Descriptive Evidence and Institutional Support for Assumption 3

In addition to the tractability provided by Assumption 3, the institutional and empirical fea-tures of our setting make it an appealing model of beliefs and acceptance behavior. To beginwith, there are several reasons why recent history should have limited predictive power forfuture offers. First, it is unrealistic that surgeons have detailed information about patients onthe waitlist that are not under their care. Privacy concerns preclude surgeons from obtainingsuch information. Second, the set and order of patients on the waitlist varies significantlyacross donors, limiting the ability to predict future offers based on recent experience. Wecalculated the fraction of times any two patients are prioritized in the same order for two ran-domly chosen donors. We find that the priorities do not preserve the order of patients 18.5%of the time. A random order would place this number at 50%. This calculation focuses onlyon cases where both patients are compatible with the two donors. Including incompatibilitywould indicate even less persistence. Last, but not least, we directly test for autocorrelationin priority-score cutoffs s∗j across organs ordered by the date on which they arrived. Onewould expect non-zero serial correlation across these cutoffs if the offers a patient observedcontained information about the likelihood of receiving future offers. We were not able rejectthe null hypothesis of zero rank autocorrelation across a range of partitions of donors (TableB.2 in the Appendix).22 Taken together, the evidence suggests that recent offers are notparticularly predictive of future cutoffs.

We also directly test whether recent offer history predicts current acceptance behavior, andfail to find evidence that it does. Suppose that a patient sees a string of unexpectedly frequentor high-quality offers. If the patient updates her beliefs based on recent history, she shouldinfer that she faces relatively little competition from other patients on the waitlist and canexpect a better offer set in the near future. As a result, the patient should become moreselective – less likely to accept an organ of a particular quality – if her recent offer history isbetter. In contrast, if the patient does not update her beliefs based on recent history, recentoffers should have no predictive power for current acceptance behavior.

We test this hypothesis by constructing an offer-specific variable for the number of years sincethe patient’s most recent offer. We include this variable as an additional predictor in theacceptance regressions presented in Section 2. Under the null hypothesis of no updating, thecoefficient on time since last offer should be zero. Under the alternative of updating based onrecent history, we expect a positive coefficient because patients who have waited a long timesince their last offer expect lower continuation values, and should therefore be more likely toaccept an offer of a given quality.

Table 5 presents coefficient estimates from several specifications that include measures ofthe patient’s recent offer history. The first two columns include time since last offer asa predictor in the standard acceptance regressions. Without controlling for current offer

22In fact, the p-values of the test statistic across relatively fine partitions are close to uniformly distributed.

27

characteristics, the estimated coefficient on time since last offer is positive and statisticallysignificant; however, with additional controls the coefficient becomes much smaller and sta-tistically insignificant. This finding suggests that recent offer rates do not predict currentacceptance behavior.

Of course, time since last offer may not fully capture the value of a patient’s recent offers,leading to a lack of power or omitted variables bias.23 For example, beliefs may be formednot only based on the most recent offer, but the last several, and a test based on the timesince the previous offer may be underpowered. Columns (3) and (4) show that when timesince last offer is averaged across a patient’s two and five most recent offers, the coefficientestimate on average time since last offer remains positive but statistically insignificant. Theremay also be other reasons why time since last offer is not the most relevant proxy for thevalue of a patient’s recent offers. Column (5) includes controls for donor characteristicsof the previous offer. This has almost no impact, and the coefficient estimates on donorcharacteristics are statistically insignificant. Finally, it is possible that our test does notconsider the relevant set of offers. Column (6) restricts the analysis to offers from idealdonors, and column (7) restricts to donors recovered in NYRT. In these two specifications,the time since last offer coefficient is statistically insignificant and in fact becomes negative.Taken together, there is no evidence that patients adjust their acceptance behavior based onthe recent offers they have experienced. We are therefore comfortable with the restrictionsembedded in Assumption 3.

4 Estimation

The key primitive needed to predict equilibrium allocations and welfare under alternativemechanisms are the transplant values, Γ (t, x, z, η). The challenge for estimation is thatacceptance decisions in our data depend on both the value of the offered organ and the valueof continuing to wait. The two leading techniques for estimating dynamic choice models ofthis type are the conditional choice probabilities (CCP) approach (Hotz and Miller, 1993;Aguirregabiria and Mira, 2007; Arcidiacono and Miller, 2011) and the full solution or nestedfixed point approach (Miller, 1984; Wolpin, 1984; Pakes, 1986; Rust, 1987). We employ theCCP approach because it affords a computationally tractable estimator that allows us touse detailed knowledge of the mechanism. This section begins by laying out our preferredapproach and the empirical specification before discussing alternatives in Section 4.2.

23Note that many sources of bias, such as patient unobserved heterogeneity and measurement error, wouldbias our estimates toward finding a positive coefficient on time since last offer. For example, suppose ourcontrols for patient priority were imperfect. In this case, some patients would be unobservably higher-priorityand, as a result, more selective. For these patients, we would observe lower times since last offer and loweracceptance rates for a given kidney, generating a positive omitted variables bias.

28

4.1 A CCP Approach for Sequential Assignments

Although equation (5) is an integral equation, it expresses the value function in terms of threesets of parameters. The first is the distribution of difference between the value of acceptingan offer and declining:

Γ (t, x, z, η) + ε− V (t;x) .

The second is the distribution of offers an agent receives in the future, which is a function ofthe object arrival rate γ, the distribution of object characteristics F , and the offer probabilitiesπ. The final set of parameters are the departure rates δ (t;x) and the parameter governingtime-preferences ρ. The value function can be recovered if these parameters can be estimated.The differences in the first set of parameters can then be used to obtain Γ (·) .

We estimate the model in four steps. First, we estimate departure rates using observed patientdepartures. Second, we estimate conditional choice probabilities from patient accept/rejectdecisions. We use these estimates and the Hotz-Miller inversion to solve for the differencesΓ (t, x, z, η)+ε−V (t;x) and ψ (τ, x, z, η) = Emax 0,Γ (t, x, z, η) + ε− V (t;x) in equation(5). Third, we compute the integral in equation (5) using the empirical distribution of donortypes and offer probabilities to estimate F and π. In the final step, we recover transplantvalues Γ (t, x, z, η) by solving for each patient’s value function at each date by evaluatingequation (5).

As is well known, time preferences are not identified from observed choices alone in dynamicdiscrete choice models (Magnac and Thesmar, 2002). We therefore set the discount rate ρ toa fixed value of 5 percent per year. Our results are robust to using an annual discount rateof 10 percent. For modest discount rates, most of the discounting of future offers is due tothe term δ (t;x), which is estimated at approximately 16% per year for the average patient.

Step 1: Estimating Departure Rates

A patient’s continuation value on the waiting list depends on how long she can expect tocontinue waiting before an exogenous departure. Our dataset contains information on howlong each patient is observed on the list without a transplant, and their reason for departure.We can therefore construct a censored measure of the length of time a patient would remain onthe list without a transplant. Censoring occurs if the patient is transplanted, or if she is stillon the list at the end of the sample period. These censored measures can be used to estimatedeparture rates independently of payoffs because Assumption 2 implies that, conditional onpatient characteristics, departure from the list prior to assignment is exogenous.

We estimate a censored Gompertz proportional hazards model in which the rate of departuretakes the form

δ (t;xi) = δ1 exp (δ2t) exp (xiβ) , (6)

29

where δ1 exp (δ2t) is the baseline hazard function for the Gompertz model and xi are observedpatient covariates. The parametric form for the baseline hazard function has the advantage ofallowing for a simple expression for the survival function p (τ |t;xi) ≡ exp (−

∫ τt δ (τ ′;xi) dτ ′).

It turns out that the estimated model yields a survival curve similar to the semi-parametricCox proportional hazards model.

Step 2: CCP Representation and Gibbs Sampling

Consider the probability that agent i refuses an offer of kidney j at time t. Assumption 1(i)implies that this probability is given by

Pijt = G (V (xi, t)− Γ (xi, zj, ηj, t)) ,

where G is the CDF of εijt. This quantity is referred to as the conditional choice probability(CCP) of refusing an offer, given xi, zj, ηj, and t. For now, assume that Pijt is known.Proposition 1 of Hotz and Miller (1993) shows that for any known distribution G that satisfiesAssumption 1(ii), there is a known function ψ such that

ψ (Pijt) = Emax 0,Γ (xi, zj, ηj, t) + εijt − V (xi, t) ,

where the dependence of ψ on G has been suppressed for simplicity of notation. Substitutingψ (Pijt) for ψ (τ, xi, zj, ηj) in equation (5), the value function can be re-written in terms ofthe CCPs as

V (t;xi) =∫ Ti

texp (−ρ (τ − t)) p (τ |t;xi)

(λ∫πij (τ)ψ (Pijt) dF

)dτ. (7)

Therefore, if Pijt can be estimated, the only remaining unknowns in this equation are πij (τ)and F . The value of a transplant Γ (xi, zj, ηj, t) = V (xi, t) − G−1 (Pijt) can therefore bewritten in terms of the remaining unknowns, Pijt, πi (·) and F , without directly solving theintegral equation (5).

In our application, we assume that εijt ∼ N (0, 1) .24 The variance of this term serves asour scale normalization. Further, we assume that Γ (·) is additively separable in ηj andapproximate

V (xi, t)− Γ (xi, zj, ηj, t) = χ (xi, zj, t) θ + ηj, (8)

where χ (·) is a flexible set of functions with interactions between its arguments. We includedummies in xi and zj for categorical variables and piecewise linear splines for their continuouselements, as well as piecewise linear splines in t. The bases in these categorical variables and

24Therefore, G = Φ is the CDF of the standard normal. This give us a simple expression for evaluatingψ (Pijt) because in this case ψ (P ) = φ

(Φ−1 (P )

)− (1− P ) Φ−1 (P ) .

30

splines are interacted with each other. Donor unobserved heterogeneity is parameterized asηj ∼ N (0, ση), with a variance to be estimated.

Because agents make an accept/reject decisions, they solve a binary choice problem. Weestimate the parameters (θ, ση) using a Gibbs’ sampler (McCulloch and Rossi, 1994; Gel-man et al., 2014), which yields a posterior distribution with a mean that is asymptoticallyequivalent to the maximum likelihood estimator (see van der Vaart, 2000, Theorem 10.1(Bernstein-von-Mises)).25

Identification of these parameters is intuitive. The parameter θ is identified by the rela-tionship between the covariates and the probability of acceptance. The variance, σ2

η, of thedonor-specific unobservable is identified because many donors have two kidneys offered topatients. The correlation between the number of offers before the acceptance of the firstand second kidneys reveals the importance of unobserved donor quality. If σ2

η is large, thenconditional on the observables xi, zj, and t, an early acceptance of the first kidney from adonor indicates that the second acceptance should soon follow. In constrast, if σ2

η is small,then the position of the first acceptance should have little information about the second. Theintuition is similar to those for results on the identification of measurement error models (seeKotlarski’s theorem in Rao, 1992; Hu and Schennach, 2008).

Step 3: Simulating the Mechanism

With estimates of the CCP parameters (θ, σ2η) and departure rates δi(t), our next objective is

to use equation (7) to calculate each patient’s value function and recover transplant values.To do this, we only need to estimate the inner integral in equation (7),

W (xi, t; θ0) =∫πj (t;xi)ψ (Pijt) dF,

= E[P (cij = 1|zj, xi) 1

sijt > s∗j

ψ (Pijt)

∣∣∣xi, t]because ρ is fixed and we have consistent estimates of p (τ ; t, xi), θ0 and λ0. Expectations inthis expression are taken over donor characteristics (zj, ηj) drawn from F ; and the priority-score cutoff s∗j drawn from the conditional distribution of cutoffsH (·; z, ηj) given (zj, ηj). Thesecond equality is implied by the definition of πij (t) given in Assumption 3. As a notationalreminder, cij = 1 if agent i is compatible with object j and sijt is the priority score of agenti for object j at time t.

We estimate this quantity by first determining the set of donors that patient i would havebeen offered had the donor arrived when the patient had waited for t periods. Recall from

25The Gibbs’ sampler obtains draws of θ and ση from a sequence of conditional posterior distributions usinga Markov chain given dispersed priors and an initial set of parameters

(θ0, σ0

η

). The invariant distribution

of the Markov chain is the posterior given the prior and the data. Details on the implementation, includingburn-in procedures and convergence diagnostics, are in Appendix B.2.

31

our discussion in Section 3.3.1 that an agent receives an offer for object j if she is compatibleand her priority score exceeds s∗j . Therefore, we constuct the sample analog

W(xi, t; θ

)= 1J

J∑j=1

P (cij = 1) 1sijt > s∗j

ψ(Pijt

)(9)

where the Pijt is replaced with Pijt = G(χ (xi, zj, t) θ + ηj

), and j indexes a donor in our

sample and the observed threshold priority for that donor, s∗j . Because the summand in thisexpression conditions on s∗j , we draw ηj from the posterior distribution given the observedaccept/reject decisions of all patients offered donor j. Knowledge of the mechanism allowsus to directly compare the patient’s priority score sijt with s∗j . Further, our dataset containsrich information on donor proteins and patient immune system characteristics that allow usto accurately estimate the probability with which cij = 1.26

It is worth emphasizing the importance of Assumption 3 in substituting the expectationwith the sample average. Under a richer information set Fi,t (as defined in equation 2) thatconditioned on the history of offers received by a patient or the configuration of the list, wewould only be able to use the subset of donors that were offered to a patient with exactlythe same information set when calculating the second approximation above. It is easy to seewhy this would restrict the sample size and limit our ability to accurately estimate W .

Theorem 1 in Appendix B.4 shows that, for each xi and t, W(xi, t; θ

)is a√J-consistent

estimator of W (xi, t; θ0) under conditions formalized in Assumption 4, also in AppendixB.4. The main requirement is on the serial dependence of the value of offers to a patient.Specifically, we require that the dependence of a potential future offer on the organ that hasarrived today diminishes with the time-horizon for the future offer. The other conditions forthe result are technical regularity conditions on the primitives, and the use of a well-behavedestimator for θ0.

Step 4: Estimating Γ

The final step recovers V (t;xi) and Γ (t, xi, zj, ηj) . We estimate V (t;xi) for time t and agenti by numerically integrating the expression in equation (7). To do this, we evaluate theintegrand at a large number of points. We substitute the sample analog for W (xi, τ ; θ0) inequation (9), the estimated departures model for p (τ |t;xi), and the observed donor arrivalrate for λ. Details of this procedure for computing V (t;xi) are provided in Appendix B.2.

26As mentioned in Section 2.2 a crossmatch is conducted using blood from the donor and patient in casethe virtual crossmatch yielded a false negative. We observe the rate of positive crossmatches in the datausing instances where a kidney was accepted because of a negative crossmatch, but the transplant did notoccur because the final crossmatch was positive. We use this conditional probability of a positive crossmatchfor P (cij = 1) in the expression above. Further details are provided in Appendix B.3.

32

Once V (t;xi) is calculated, we recover Γ (·) by inverting G :

Γ (t, xi, zj, ηj) = V (t;xi)−G−1(Pijt

).

This quantity can be calculated for any value of (t, xi, zj) .

4.2 Discussion

Unobserved Heterogeneity

The model omits patient unobserved heterogeneity. Arcidiacono and Miller (2011) developCCP approaches that allow for time-invariant unobserved heterogeneity with discrete types.There are three main complications in applying our methods. First, both the departure ratesδi (t) and choice probabilities would ideally depend on unobserved heterogeneity and wouldneed to be simultaneously estimated. Second, a solution to the well-known initial conditionsproblems needs to be developed for the agents already on the waiting list at the beginingof the sample. Third, one may further argue that unobserved heterogeneity may be timevarying if unobserved patient health is stochastic. This last extension may require significantrevision to the model.

We believe that abstracting away from unobserved heterogeneity still yields useful results be-cause our dataset contains a very rich set of patient characteristics. Nonetheless, we exploredspecifications in which unobserved heterogeneity was included in the CCP specification only.These specifications yielded qualitatively similar results for the counterfactual analysis (seeAppendix E for details).

Our model does include donor-level unobserved heterogeneity through η. One motivationfor doing this is the pattern of sharply declining acceptance rates by position documentedin Figure 1b. The observable characteristics included in the model do not explain all of thissharp decline or the composition of offers, especially at lower positions on the waitlist. Wewill compare model fit with and without such unobserved heterogeneity in the next section.

Comparison with Full Solution Approaches

An alternative to the CCP approach is to use a full solution or nested fixed point approach.The full solution approach would parametrize Γ (·) directly, say, as χ (x, z, t, η) θΓ, recoverthe value function by solving the fixed point in equation (5), and optimizing a statisticalloss function (e.g. maximimum likelihood) to obtain an estimate θΓ. Compared to the CCPapproach outlined above, the primary advantage of the full solution approach is to avoiddirectly parametrizing the (endogenous) difference in equation (8), Γ (·)− V (·), in terms of

33

χ (·). Aside from the appeal of using an implied functional form for V (·) that is consistentwith the primitives, the approach also uses the data more efficiently.However, this full solution approach is well known to be computationally burdensome whenthe state space is large (see Hotz and Miller, 1993; Arcidiacono and Miller, 2011). The highdimensionality of the state space remains a problem in the deceased donor allocation contextdespite the simplified model of beliefs. Specifically, the simulation in equation (9) computesthe compatibility and priority score for each patient and donor using all the variables thatenter the assignment mechanism and detailed information about the immune response of apatient to each potential donor. A full solution or nested fixed point method would requireus to separately solve for the value function using equation (5) for each patient at every singleoffer. Moreover, it would require such a solution to be computed for each guess of θΓ usedwithin an optimization routine.27

The CCP approach avoids these complications and makes it computationally feasible to re-spect the details of the mechanism. The main cost is loss in statistical efficiency because wedirectly estimate the difference in equation (8), and parametrize this difference in terms ofthe functions χ (·) . The latter cost is mitigated by using flexible functional forms. Nonethe-less, our implementation does impose some continuity across continuous states and limitsinteractions because a fully non-parametric approach is prohibitive given the dimensionalityof the state space.

5 Parameter Estimates

This section describes our estimates of patient departure rates, conditional choice probabili-ties, the value function, and the value of transplantation.

5.1 Departure Rates

Table 6 presents estimates from hazard models of departures from the kidney waitlist priorto transplantation. We estimated models under different parametric assumptions aboutthe baseline hazard functions. We used all of the patient-specific variables included in theCCP model.28 Across specifications, we estimate an increasing baseline hazard of departure,

27Essentially no two patients are identical because the mechanism awards points whenever a donor anda patient have overlapping antigens, and because immune responses to donors are idiosyncratic. There areabout 2.85 million offers in our dataset, making this problem extremely computationally expensive. Oneapproach to simplifying the problem, as done in our counterfactual analysis, is to evaluate the value functionfor each patient on a discrete grid of times. Because we have just under 10,000 patients, even a grid with100 points for each patient would require solving for approximately 1 million instances of the value functionat each guess of θΓ.

28The hazard model specifications include the CPRA variable from the CCP model, but omits a dummy forCPRA > 0.8. Because priority is discontinuous in CPRA at 0.8, it is important to allow acceptance behavior

34

consistent with patients becoming less healthy over time. Column (1) presents a Gompertzmodel, in which the estimated value of log (δ1) is -5.151 per day, indicating an increasingbaseline hazard. Other coefficient estimates reveal significant heterogeneity across patientsin their departure rates. For example, diabetic patients depart at higher rates. Patients withblood type A are more likely to depart than patients with blood type O, perhaps becauseof better living donor transplant opportunities. Among pediatric patients, older patientsare less likely to depart, perhaps because they are better able to tolerate dialysis. However,as patients get older, the departure rates begin to increase with age. Columns (2) and (3)estimate corresponding Weibull and Cox proportional hazards models with the same set ofpatient covariates. Point estimates for these coefficients remain stable across assumptions onthe baseline hazards, including the semi-parametric Cox proportional hazards model. FigureB.2 in the appendix compares the estimated survival curves from columns (1) and (3) andshows that the Gompertz hazards model yields a survival curve that is very similar to thenon-parametric Cox proportional hazards model. We use the Gompertz hazard model withpatient covariates reported in column (1) as our preferred specification.

5.2 Estimated CCPs

We estimated three specifications for the conditional choice probability of accepting an offer.All specifications include the rich set of patient and donor observed characteristics sum-marized in Tables 1 and 2. The first specification includes all of these baseline variables,but does not include donor unobserved heterogeneity (η) or the state variable time t. Thesecond specification adds donor unobserved heterogeneity, and the third specification addswaiting time interacted with a variety of characteristics. We explain the choice of baselinecharacteristics, and then describe the estimates.

The baseline characteristics common across specifications, as well as linear splines and inter-actions among these variables, were chosen by surveying the medical literature. Specifically,we use covariate and spline specifications from the KPSAM model, which was used by thekidney allocation committee to predict the outcomes of various allocation systems.29 We alsoinclude any covariates that were part of the survival models for kidney transplant patientsused in Wolfe et al. (2008). Following KPSAM and our earlier observation that donors fromother DSAs are less desirable, we include interactions of donor and patient characteristics

to be discontinuous at this point. Departures without a transplant, however, should not be discontinuous inCPRA. Moreover, specifications of the departures model that included this variable estimated a statisticallyinsignificant coefficient.

29We obtained the KPSAM module from the Scientific Registry of Transplant Recipients (SRTR),which contains the specification of the KPSAM acceptance model. Our dataset contained all butone of the variables used in that model. Visit https://www.srtr.org/requesting-srtr-data/simulated-allocation-models/ for a description of the various simulated allocation models and the pro-cedure to request these modules.

35

with whether the donor was recovered in NYRT. In addition, we include patient-donor spe-cific variables that capture match-specific heterogeneity, for example whether there are twoDR antigen mismatches. We specify piecewise linear splines for continuous covariates andinteract them with a variety of indicator variables.

Table 7 presents select parameter estimates from the three specifications (see Table B.3 for thefull specification). The estimated coefficients on observed donor characteristics are intuitiveand fairly robust across specifications. For example, offers from donors older than 50 yearsof age are less likely to be accepted than offers from 35 to 50 year old donors, and kidneysfrom younger donors are even more likely to be accepted. These differences are larger fordonors recovered in NYRT. A perfect tissue type match is very desirable, much more so thana young donor. Also intuitive are our estimates that offers of kidneys with more antigenmismatches (A, B or DR) are less desirable and that regional and national offers are lesslikely to be accepted.

We also estimate significant patient-level and match-specific heterogeneity in acceptancerates. Consistent with our discussion in Section 2, patients with more sensitive immunesystems (higher CPRA) are more likely to accept an offer. Acceptance rates also dependon patient age. Among pediatric patients, older patients are more likely to accept an offer,which is intuitive because size compatibility is important for younger pediatric patients.Adult patients of different ages are equally likely to accept a middle-aged donor, but olderpatients are more likely to accept a donor who is over 50 years old. This pattern is consistentwith the idea that it is more important for younger patients to obtain kidneys that are likelyto function for a long time, and suggests potential welfare gains from improved matching onpatient and donor age.

In the second specification, the estimated standard deviation of donor unobserved heterogene-ity is 1.03. The implied change in acceptance rate from a one standard deviation increase inη is therefore half the difference between a kidney recovered inside NYRT and one recoveredoutside NYRT. The third specification shows that acceptance rates fall rapidly with waitingtime in the first few years before starting to increase after year three. Most other parame-ter estimates, including the standard deviation of unobserved donor-level heterogeneity, aresimilar in the second and third specifications.

Figure 2 describes the fit of these models. The first panel mimics the fit of acceptance ratesby position described in Figure 1b, but includes offers that did not meet pre-set screening cri-teria. This panel shows that including donor unobserved heterogeneity much better capturesthe sharp decline in acceptance rates. The specification that excluded donor unobserved het-erogeneity does not even accurately capture the average acceptance rate across the first 20positions. Instead, it implies a more gradual decline as the composition of donor observableschanges moving down the list. The second panel of Figure 2 describes the fit of acceptancerates by time waited. Not surprisingly, the third specification does the best job of capturingchanges in acceptance rates over time.

36

Taken together, we feel comfortable with the fit of the CCPs in the third model and use thatas our preferred specification. All results that follow use estimates from this specification.

5.3 Estimated Value of Organ Offers

Below, we develop an interpretable measure for comparing values across patients in terms ofan equivalent change in donor arrival rates (supply). We then use these units to describe ourestimates.Sequential assignment mechanisms can re-assign offers from some patients to others. Moti-vated by this fact, consider the proportional increase in a patient’s value from a one-timeoffer for kidney j at the time of registration:

EVij ≡Emax 0,Γ (0, xi, zj, ηj) + εij − V (0;xi, λ)

V (0;xi, λ) ,

where the dependence of V on the object arrival rate λ is re-introduced for clarity. Thenumerator is the difference between the value with and without an additional one-time offerfor object j at time 0, and the denominator is the baseline value.Instead of a one-time offer, the same change in value can be generated by an increase in thedonor arrival rate. Specifically, let λij be defined such that

V (0;xi, λij) = V (0;xi, λ) + Emax 0,Γ (0, xi, zj, ηj) + εij − V (0;xi, λ) .

Using this expression, we can rewriteEVij as

EVij = V (0;xi, λij)− V (0;xi, λ)V (0;xi, λ)

≈ λij − λλ

,

where the approximation follows because, holding behavior and offer probabilities fixed,V (t;xi, λ) is approximately linear in λ (see equation 5). This approximation is appropriatefor small changes in λ, so that offer probabilities do not change substantially.The equivalent change in donor arrivals is, by definition, invariant to the scale of utility unitsacross agents and will be used to report welfare effects going forward. It equates an additionalone-time offer to the value of an alternative policy that is able to marginally increase theorgan supply. This feature makes this quantity similar in spirit to Equivalent Variation atthe time of registration, yielding a measure of payoffs that is interpretable across agents forsmall changes in the environment.Figures 3 – 5 describe our estimates in these units. The results are based on our preferredspecification, which includes donor unobserved heterogeneity and time waited. The plots

37

show how the value of an organ offer varies across specific patient and donor characteristics,holding all remaining characteristics fixed.

Healthier patients, whether measured by patient age or by dialysis status at registration,prefer younger donors. Figure 3 shows that patients across all age groups prefer youngeradult donors. However, the relative value of a young donor decreases with patient age. Fora 70 year old patient, a single offer from an 18 to 35 year old donor is as valuable as a 4 to 5percent increase in overall donor supply, while a donor over 50 years old is half as valuable.In contrast, for a 20 year old patient, a donor over 50 years old is only one twentieth asvaluable as an 18 to 35 year old donor.30 This pattern is consistent with the CCP estimatesand with the intuition that older donors should place less value on an organ that is likely tofunction for a very long period of time. Similarly, Figure 4 shows that patients off dialysisat registration place a relatively higher value for younger donors

In addition to donor age and patient health, a perfect tissue type match is especially impor-tant for patients (Figure 5). Such an offer from a young donor is worth a 32 percent increasein overall donor supply for a representative patient; from a donor over age 50, it is still wortha 17 percent increase in supply. This result is also intuitive because an organ with a perfecttissue type match is less likely to cause an adverse immune response, thereby increasing thelife-years afforded by the transplant. Offers which are not perfect tissue type matches are anorder of magnitude less valuable.

Detailed estimates for V and Γ (in εijt units) are presented in Tables B.3 and B.3 in theAppendix. The reported co efficients are obtained from a projection of these quantities onχ (·). The projection coefficients for both Γ and V are intuitive. For example, younger donorsare more valuable, while donors with tissue type mismatches are less valuable. Similarly,donors recovered outside NYRT are less desirable.

6 Steady State Equilibria and Welfare Comparisons

6.1 Equilibrium Concept

We now define an equilibrium concept for counterfactual analysis. The concept is intendedto capture a large pool of agents waiting for offers and making optimal decisions. Agentshave type x ∈ X , and objects have type z ∈ ζ, where we henceforth include the unobserveddonor characteristic η in z for notational simplicity. For computational reasons, we will treatX and ζ as finite sets.

30Older patients place higher values on all types of offers, in terms of equivalent supply increases. Thisis mainly because older patients exogenously depart more quickly, and therefore place a higher value on asingle current offer relative to a proportional increase in the stream of future offers.

38

To compactify notation, albeit with a slight abuse, the rest of the paper replaces subscriptsthat index individuals i and objects j with types x and z respectively. For instance, wewrite the value function as Vx (t) instead of V (t;xi) and the scoring rule as sxz (t) instead ofs (t;xi, zj) . The notation for other quantities such as π, Γ and δ is adapted analogously.Agents follow type-symmetric accept/reject strategies, σx : R × R+ → 0, 1, indexed byx ∈ X . The first element of the domain is the payoff of being assigned a particular object, Γ,and the second element is time waited, t ∈ R+. We exclude strategies that depend on richerinformation because beliefs are restricted to satisfy Assumption 3.We model the composition of the queue using a single steady state. Specifically, the queuecomposition will be governed by a probability density function, m, defined on the set X ×[0, T ], where T is the maximum wait time.31 This density governs the distribution of agentsof each type and how long they have waited. We write mx (t) to denote the density evaluatedat (x, t). The length of the queue is denoted by N .

Definition 1. A steady state equilibrium consists of an accept/reject strategy σ∗, beliefsπ∗, a queue size N∗, and a probability measure m∗ such that the following conditions hold:

1. Optimality: For each agent of type x ∈ X and an offer with net present value Γ,

σ∗x (Γ, t) = 1 Γ ≥ Vx (t; π∗) ,

where Vx (t; π∗) is the net present value for type x of declining the object and followingthe optimal strategy given π∗ after t.

2. Consistent beliefs: For each (t, x, z) , the beliefs π∗ (t;x, z) is consistent with equilibriumoffer probabilities. In particular, for mechanisms that uses the scoring rule s,

π∗xz (t) = H∗z (sxz (t))× P (c = 1|x, z) ,

where H∗z (s) is the probability that the object is available only to agents above thescore s if N∗ agents are drawn iid from m∗, and they follow strategy σ∗.32

3. Steady State: m∗ and N∗ satisfy the balance conditions

(a) For each x ∈ X , m∗x (t) and N∗ satisfy

mx (t) = −mx (t)κx (t) and mx (0) = γxN∗

,

31We assume that the density m is defined with respect to the Lebesgue measure on the Borel sets formedfrom X × [0, T ], where X is a finite set. Our counterfactuals continue to assume that no agent lives past onehundred years of age. Consequently, we set T to one hundred years and restrict mx (t) = 0 for all t such thata agent of type x would be more than one hundred years old.

32We do not restrict the queue length N∗to be an integer and therefore round it to the nearest integerwhen calculating π.

39

where γx is arrival rate of an agent of type x, and κx (t) is the equilibrium departurerate of an agent of type x at waiting time t.

(b) m∗ is a density: ∑x∈X∫ T

0 mx (τ) dτ = 1

The first condition states that each agent makes optimal decisions at each point in time givenher beliefs, assuming that she continues to make optimal decisions in the future. The valuefrom declining an offer is given by the Hamilton-Jacobi-Bellman equation defined in Section3.3. The second condition imposes that agents have correct beliefs. In the specific case ofa mechanism based on the scoring rule s, it writes agent beliefs about future offers as theproduct of the steady state distribution of cutoffs and exogenous compatibility realizations.The distribution H∗z governs the cutoffs that arise when agents use strategies σ∗ and N∗

agents are drawn from a distribution governed by m∗.33 The final condition determines thecomposition of agent characteristics on the list. The left-hand side in part (a) is the changein the density of agents of type x who have waiting time t. The right-hand side term is therate of departure for those agents. Departures occur for both exogenous reasons and becauseagents are removed from the waiting list once they are assigned; that is, κx (t) is the sumof δx (t) and the equilibrium rate at which agents of type x are assigned at time t. Thestrategy σ∗ and the offer rate of objects, given by π∗, determine the endogenous departures.The agent arrival rate γx is exogenous, and in the context of our application, it will only bepositive for agents with zero waiting time since patients begin to accumulate waiting timeonce they enter the queue. Part (b) ensures that m∗ integrates to one.

This equilibrium concept abstracts away from transitional dynamics in the size and compo-sition of the queue. For example, in our empirical context, it is possible that several patientsjoin the queue in close succession before any organs arrive. One approach may be to modelthese dynamics by assuming that the queue length and composition follow a Markov process.In principle, one could solve for the stationary distribution over the queue length and queuecompositions. However, this distribution is extremely high dimensional, and it would makecounterfactual exercises computationally intractable.

As in Assumption 3, we view our equilibrium concept as an approximation to the behaviorof the system. For example, although the mass of agents of type x is discrete, part 3(a)implies that the ratio mx (t) /mx (0) is given by the survival curve implied by the equilib-rium departure rates κx (t). Similarly, N∗ is assumed to be the expected queue length. Infact, Theorem 2 in Appendix C uses a concentration inequality to show that the stationarydistribution of the queue length concentrates mass on N∗. Our result implies that if N∗ is5000, the weight placed by the stationary distribution on queue lengths that deviate by morethan 5 percent is at most 0.5 percent. Hence, we expect that our equilibrium notion will bea good approximation for the behavior of the waitlist.

33In contrast with a direct continuum approximation with no aggregate uncertainty, this specificationallows for H∗z to be non-degenerate.

40

Finally, we prove the existence of a steady state equilibrium for sequential assignment mech-anisms that use a priority score in Theorem 3. The challenge in showing existence arisesbecause the strategies, beliefs and composition are a function of time, which is a continu-ous variable. We use the Brower-Schauder-Tychonoff fixed point theorem (Corollary 17.56,Aliprantis and Border, 2006) for general Banach spaces to prove existence. The primaryassumptions are technical regularity conditions imposing bounds and Lipschitz continuity ofprimitive objects. The main substantive condition is that the set of scores used in the mech-anism is finite. Our results do not rule out multiplicity of equilibria. However, we did notfind multiple equilibria for the set of counterfactuals and specifications that are consideredbelow.

6.2 Welfare Comparisons

Given a mechanismM and a donor arrival rate λ, let the steady-state value of an agentof type x be the integrated value function over current and future generations:

VMx (λ) = γxρ

∫ ∞0

exp (−ρτ)VMx (0;λ) dτ +∫ T

0N∗m∗x (τ)VMx (τ ;λ) dτ. (10)

The first term represents the discounted value of agents of type x that are expected to arrivein the future, and the second term represents the value of agents presently on the waitinglist. The first term is weighted by the net present value of the total mass of future arrivalsγxρ, and the second term is weighted by the equilibrium measure N∗mx (τ) that governs the

mass of agents of type x that have waited for τ periods. By considering the welfare of bothcurrent and future generations, we take the view of a social planner who is interested in thenet present value of payoffs generated by all future assignments. Our tables will also reportan alternative measure that only considers the value at the time of entry, VMx (0;λ). Thetheoretical discussion in this subsection applies to this alternative as well.

LetM0 be the baseline mechanism used during our sample period and let λ0 be the observeddonor arrival rate. For any mechanismM, define λx (M), the equivalent donor arrival ratefor agents of type x, as the solution to the equation VMx (λ0) = VM0

x (λx (M)) . As discussedin Section 5.3, we can express a change in the value function for type x as an equivalentchange in the donor arrival rate:

EVx (M) = VMx (λ0)− VM0x (λ0)

VM0x (λ0)

≈ λx (M)− λ0

λ0.

This measure describes the welfare effects for each patient in terms of an alternative policythat keeps the mechanism fixed, but is able to increase (or decrease) organ donation rates.

A common challenge in comparing the aggregate welfare effects of various mechanisms in

41

environments without transfers is the lack of a clear numeraire good that can be used tojustify a comparison based on a Kaldor-Hicks criterion. The measure EVx (M) is type-specific, and therefore does not make comparisons across patients of different types. Wewill summarize welfare effects by reporting equally weighted averages of EVx (M). Thisaggregation treats equivalent changes in donor arrivals for different patient types as equal.Of course, alternative welfare weights are easily entertained within this framework. Ourresults will also report distributional effects on key subgroups in order to allow the reader toindependently assess the welfare effects for alternative weights on these subgroups.

6.3 Computing Equilibria

We compute steady-state equilibria using an algorithm that iterates between computingthe value function and optimal decisions, and the steady-state composition of the waitlist.A detailed description with expressions for each step of the procedure and pseudocode isprovided in Appendix D. The following discussion provides a simplified description of thekey steps.

In addition to the primitives, the algorithm uses a discrete time grid t = t0, . . . , tl, tl+1, . . . , T ,abitrary initial beliefs π0, and a sample of patients and donors as inputs. An equilibrium iscomputed by iterating through the following steps for k ≥ 1:

1. Compute the value function V kx (tl), given beliefs πk−1, via backwards induction from

the value of waiting in the next period V kx (tl+1):

V kx (tl) =

∫ tl+1

tl

exp (−ρ (τ − tl)) pi (τ |tl)λ∫πk−1 (t;x, z)Emax

V kx (tl+1) ,Γ (τ ;x, z) + ε

dFdτ.

The inner integral in the above expression is approximated using sampled subset ofdonor types. This procedure starts with V k

x (T ) = 0 for all x and k, and workingbackwards to compute V k

x (t) for all t. This calculation also yields patient strategiesσkx (Γ, t) = 1

Γ ≥ V k

x (t)and departure rates κkx (t) .

2. Compute the queue composition mk via forward simulation:

mkx (tl) ∝ γx exp

(−∫ tl

0κkx (τ) dτ

).

3. Compute πk (t;x, z), which is the probability that an agent of type x is offered an objectof type z using the queue composition and the accept/reject strategies σkx (Γ, t).

4. For step k > 1: Terminate if the change in value functions and queue length/compositionsbetween iterations – supx,l

∣∣∣V kx (tl)− V k−1

x (tl)∣∣∣, supx,l

∣∣∣mkx (tl;x)−mk−1

x (tl)∣∣∣, Nk−Nk−1

42

– are uniformly below a chosen tolerance level. If these conditions are not satisfied,repeat steps 1-4.

If this algorithm terminates, the resulting accept/reject rules yield an equilibrium (up tothe threshold tolerance). Because the equilibrium we compute may not be unique, we trieddifferent starting values for π0. Our experiments at the estimated parameters do not indicatethat multiplicity is a concern in our setting.

To keep the computational burden manageable, the results we present below are based on atype space given by a random sample of 300 patients and 500 donors drawn from our dataset.Further, we discretize time into quarters for the first 15 years after registration, then every2 years until year 25, and every 25 years thereafter. As discussed in Appendix E, the resultsreported below are not sensitive to a larger set of types used to calculate equilibria of scoringmechanisms and to finer partitions after the first few years since the probability that a patientsurvives without a transplant falls dramatically.

7 Evaluating Design Trade-Offs

7.1 Alternative Mechanisms

A new deceased donor organ allocation system was adopted in December 2014. The mandateof the kidney committee, as laid out by the U.S. Department of Health and Human Services,was to find mechanisms that balanced the goals of providing equitable outcomes for patients,efficiently using available organs, and minimizing organ waste. This section compares mech-anisms aimed at achieving these goals to the two mechanisms used prior to and following the2014 re-design.34

We start by computing four benchmarks against which we compare simpler mechanismsmotivated by theory and practice.

• Optimal Assignments: This problem bounds the welfare gain achievable by anymechanism. To do this, we maximize the steady-state average welfare of all agentssubject only to feasibility constraints. The program solves for an assignment policya (ε;x, z, t) ∈ 0, 1 under full information about preferences and rational expecta-tions (but no foresight) about object arrival and agent departure processes. The policy

34Our calculations only change the allocation mechanism in NYRT. Evaluating a nationwide change wouldrequire us to use data on decisions made by all patients in the US, which is burdensome due to the patchworkof variants on points used in approximately half the states. To simplify this task, we keep the system usedto prioritize patients from the rest of the United States fixed to the pre-2014 system. We also assume thatthe policy function of patients from the rest of the US, which governs offers for non-local donors to patientsin NYRT, remains fixed.

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can dictate assignments to agents as a function of the agent type, the object type, timewaited, as well as the preference shocks ε for all agents.

We choose a to maximize the average equivalent donor supply increase, ∑xV ax (λ0)VM0x (λ0)

,where V a

x (λ0) is the steady-state value for type x under assignment policy a. Thefeasibility constraints ensure that the total rate of assignment does not exceed the totalarrival rate of objects. Specifically, for each z, we impose (a time-discretized versionof) the constraint

∑x

∫ T

0Nmx (t)P (c = 1, a = 1|x, z, t) dt ≤ qz.

The term P (c = 1, a = 1|x, z, t) is the probability that an object of type z is compatible(c = 1) and assigned (a = 1) to a randomly chosen agent of type x that has waited fortime t. In addition, we require that the composition of the waitlist be in steady state(Definition 1, part 3). These constraints are described in detail in Appendix D.2.1.

• Optimal Offer Rates: We solve for welfare maximizing offer rates in which thedesigner can choose offer rates, but cannot dictate assignments or condition on offerrates refused in the past. The offer rates πxz (t) depend on observable characteristics ofpatients, but not unobserved preference shocks ε. We solve for steady-state equilibria(Definition 1), implying that agents make optimal decisions given these offer rates. Jus-tifications for only not allowing the designer to dictate assignments include respect forpatient and doctor discretion and the concern that forcing assignments or penalizingterminally ill patients may be politically infeasible. In addition, a designer may nothave full information about idiosyncratic preferences, and may therefore be unable toimplement the optimal assignment solution even if she can dictate assignments.

Formally, we choose π to maximize ∑xV πx (λ0)VM0x (λ0)

, where, with a slight abuse of nota-tion, V π

x (λ0) is the equilibrium steady-state value for type x under offer rates π. Theoffer rates are subject to feasibility constraints, so that the steady-state rate of assign-ment implied by offer and acceptance rates does not exceed the arrival rate of objects.In this problem, the feasibility constraint from the optimal assignment problem is mod-ified by replacing the term P (c = 1, a = 1|x, z, t) with the probability πxzt of makingan offer to an agent of type x that has waited for time t multiplied by the probabilitythat this offer is the last offer than can be made. Ignoring, for the moment, the limiton the maximum number of offers that can be made, for each donor type z, we imposethe constraint that

∑x

∫ T

0Nmx (t) πxz (t)P (Γxzt + ε > Vx (t) |x, z) dt ≤ qz.

44

The left-hand side is the expected number of objects assigned under offer rates π andthe right-hand side is the total number of objects available. The mathematical problemwe solve is formally described in Appendix D.2.2.

Because this constraint is placed only on expected quantities, the offer rates may not beimplementable for any particular instance of objects of type z. The solution providesan upper bound on welfare under any offer mechanism that does not condition on pastbehavior or have information about idiosyncratic preferences.

• Approximately Optimal Priorities: Although the offer rates calculated above maynot be implementable, we can use the solution to implement very similar offer rates inequilibrium using a scoring rule. Specifically, we set sxz (t) to the values of πxz (t) thatsolve the optimal offer rates. Therefore, types with higher values of π are accordedhigher priority. This mechanism can be implemented and described using the existingorgan allocation infrastructure. We compute an equilibrium for these priorities usingthe algorithm outlined in Section 6.3.

• Approximately Optimal Pareto Improving Priorites: In practice, policy makersmay be constrained to reforms that do not systematically disadvantage specific groupsof patients. We design a priority system that aims to improve the welfare of the averagepatient without significantly hurting many patient types. To do this, we use a proceduresimilar to the one used to derive approximately optimal priorities. Specifically, we firstsolve for offer rates that are defined identically to the optimal offer rates except thatthey are constrained to make no agent type worse off than under pre-2014 priorities atthe time they join the waitlist.35 We then solve for the equilibrium allocation under ascoring function sxz (t) that is equal to the values of πxz (t) that are the solution to thisproblem.

It is worth noting that the resulting mechanism may not result in a strict Paretoimprovement in equilibrium. However, we expect that fewer agents will be significantlyworse off under these priorities relative to the approximately optimal priorities. Oursolutions will allow us to quantify these effects.

We compare these benchmarks to a set of waitlist priority rules motivated by theory andpractice. These mechanisms use scoring rules sxz (t) to order patients waiting for a transplantand break ties uniformly at random among patients with the same score. The focus on priorityrules is motivated by the design parameters considered by the kidney allocation committee.36

35During the 2014 reforms to the deceased donor kidney allocation mechanism, the kidney committeeconducted simulations to verify that there would be no adverse impacts by race, age group, geography, orCPRA.

36Our examination of the meetings of the kidney allocation committee indicate that they did not consider

45

Priority rules are simpler to describe and implement than mechanisms in which assignmentprobabilities are determined. Equilibria for this class of mechanisms can be computed usingthe algorithm described in Section 6.3.

• Post-2014: In December 2014, the kidney allocation mechanism switched to a sys-tem that awards greater priority to patients who are extremely difficult to match (highCPRA), and also prioritizes healthier patients for higher quality donors. The rationalefor the first change was that high CPRA patients have few opportunities for transplan-tation and are likely to accept most organs. Therefore, giving them additional prioritycould reduce organ waste and achieve more equitable outcomes for sensitized patients.The second change intended to offer high-quality kidneys to patients likely to benefitfrom them most, and in particular to reduce age mismatch.

• First Come First Served (FCFS): One concern of the kidney committee has beento maintain a transparent and procedurally fair offer system. FCFS is a procedurallyfair and commonly used mechanism which offers objects to agents in the order theyjoined the waiting list. FCFS also has attractive efficiency properties: Bloch andCantala (2017) show that it maximizes agent welfare within a broad class of primitiveswhen values for an object are drawn i.i.d. across agents. This result is driven by thefact that FCFS encourages agents to be selective and choose only objects with highmatch-specific values. We approximate this system by finely discretizing time on a gridt0, t1, . . . , tL and set sxz (t) = l if t ∈ [tl−1, tl).

• Last Come First Served (LCFS): Another goal of the kidney committee has beento minimize organ waste. Last come first served provides strong incentives to acceptorgans because agents that refuse an offer are demoted if other agents arrive in thefuture. Su and Zenios (2004) show that a last come first served system both maximizeswelfare and minimizes organ waste when there is agreement across agents on the valuesof various objects, that is, if preferences are vertical. This is because social welfaredepends not on who is assigned the object, but only on the fraction of objects allocated.We approximate this system by finely discretizing time on a grid t0, t1, . . . , tL and setsxz (t) = L− l if t ∈ [tl−1, tl).

• Greedy Priorities: This mechanism attempts to maximize welfare by using a greedyprocedure which offers organs to patients in order of predicted match value. For eachdonor, it divides patients into 10 equally sized bins based on our predicted welfare gainfrom assignment, measured in donor supply units (EVxz). sxz (t) is set to 10 for thosein the highest bin and to 1 for those in the lowest bin.

alternatives in which doctors were mandated to accept particular organs, or in which past rejections wereused to update priorities.

46

Some of the mechanisms we consider, particularly the approximately optimal and greedypriorities, condition finely on patient and donor characteristics. One concern with such fineconditioning is that agents may manipulate the characteristics they report to the mechanism.We therefore caution against interpreting our results as describing readily implementablemechanisms. However, it is worth noting that such manipulation would likely involve forgingmedical records. In addition, most patient characteristics in our model are already used bythe pre- or post-2014 mechanism.37 Another concern is that a finely tuned priority system iscomplicated to understand for market participants. Addressing this issue is beyond the scopeof this paper as it requires a well-articulated and practically relevant notion of mechanismcomplexity.

7.2 Comparing Mechanisms

We now describe the equilibrium outcomes under each mechanism described in the previoussection. Our analysis shows that previous theoretical and practical recommendations caneither increase patient welfare or organ discard rates, but not both. Fortunately, the optimalmechanisms described above can improve on both these goals while also ensuring that almostno patient type is significantly worse off.

The priority systems that have been in place in the U.S. since 2010 are very similar tothe benchmark first come first served mechanism in terms of average patient welfare andtransplant rates. In particular, the 2014 reforms did little to improve average patient welfare,and primary resulted in redistribution towards younger and more highly sensitized patients.Indeed, we find a small decline for patients above age 50 after the 2014 reforms and gains foryounger patients (Table 8). These welfare changes are small on average, and only a minority(16 percent) of patients, most of whom are young, benefit from the change. In addition,the pre- and post-2014 mechanisms yield patient welfare and organ discard rates within 1.9percent of the benchmark first come first served mechanism (Figure 6). Consistent with theseresults, waiting times, queue lengths, and the quality of the average donor transplanted aresimilar across the three mechanisms.

These results suggest that the priorities implemented in the pre- and post-2014 mechanismsprimarily result in redistribution relative to the FCFS benchmark. Most patients marginallyprefer FCFS to the two mechanisms that have been used in practice (Table 8). As mentionedearlier, Bloch and Cantala (2017) show that FCFS results in desirable equilibrium allocations

37In fact, all but two of the patient characteristics in our model were used to determine priority in eitherthe pre- or the post-2014 kidney allocation system. The patient characteristics in our model include thoseused to calculate the EPTS score in the post-2014 mechanism (see https://optn.transplant.hrsa.gov/resources/allocation-calculators/epts-calculator/), CPRA, total serum albumin, and body massindex. Only the last two characteristics are not used in the post-2014 mechanism. These characteristics wereimplicitly used in design proposals based on a model of estimated life-years from transplantation because thismodel includes these variables.

47

when preferences are heterogeneous. Patients that have a strong preference for a particulartype of kidney should be willing to wait for those organs. For example, even though thepre-2014 system did not explicitly prioritize young patients for kidneys from young donors,a significant degree of age matching was discernible (Table 4). For this reason, a relativelycoarse priority system may not be able to sufficiently improve welfare relative to FCFS.In contrast to the first three mechanisms, last come first served (LCFS) substantially reducesorgan discard rates at the cost of lower patient welfare. Discard rates fall by 26.6 percentunder LCFS relative to the pre-2014 mechanism, and the steady state queue length falls to2,694 from 4,992. This increase in transplant rates comes at a significant welfare cost becausepatients accept organs that are poorly matched to them: the welfare of the average patientfalls by an equivalent of a 33.6 percent reduction in donor supply (Figure 6). This result isalso reflected in donor characteristics, with patients accepting organs from donors that areolder, more likely to be hypertensive, and less likely to have died from head trauma (Table8).The stark difference between LCFS and the other systems occurs because LCFS dramaticallychanges agents’ incentives, penalizing them for rejecting poorly matched offers instead ofrewarding them with higher priority in the future. A model that ignores these incentivesfinds that the allocations are largely insensitive to the mechanism. Panel B in Table 8shows that changes in predicted discard rates, queue lengths, and donor characteristics aresimilar across the benchmark mechanisms when acceptance probabilities do not adjust to thenew equilibrium. Thus, it is essential to consider incentives when predicting the effects ofalternative allocation mechanisms.These results also highlight an important trade-off between organ discard rates and matchquality identified in the theoretical literature. Su and Zenios (2004) show, using a model inwhich agents have identical preferences over objects, that LCFS reduces selectivity becauseagents expect to receive lower quality offers in the future. This force encourages agentsto accept mismatched offers, which reduces organ discards but also lowers match quality.In contrast, FCFS increases selectivity as agents retain their priority when they decline anoffer, but also increases discard rates when there is a limit to the number of offers that canbe made. Bloch and Cantala (2017) use a model with highly heterogeneous agent preferencesto show that the benefits of improved match quality due to increased selectivity can makeFCFS a desirable mechanism. Our empirical findings weigh in favor of models that emphasizeheterogeneity in match value: in terms of patient welfare, the loss from lower match qualityunder LCFS substantially outweighs the gain from fewer discards.However, these theoretical benchmarks are far from optimal; a mechanism designed usingestimated preferences as inputs can substantially increase both patient welfare and transplantrates, and many of these gains are possible without significantly disadvantaging specificpatient groups.The upper bound on welfare implied by an optimal assignment is large: a designer who has

48

full information about preferences and can dictate assignments can improve patient welfareby an equivalent of a 28.1 percent increase in donor supply. At the same time, this designercan reduce discard rates by 13.6 percent. Table 8 shows that the vast majority (83 percent)of patient types are better off at registration under these assignments than under the pre-2014 mechanism. However, not all patients are better off even in this ideal case; there aretrade-offs between efficiency and distributional objectives.

The approximately optimal priorities achieve half of these potential gains. Patient welfareincreases by 14.2 percent, and discard rates fall by 7.8 percent (Table 8). Since patients aretransplanted at higher rates, they spend less time on the waiting list, and queue lengths fallfrom almost 5,000 to below 4,400. These results suggest that large improvements may bepossible by re-designing priority rules. In fact, the approximately optimal priorities mecha-nism achieves most of the gains possible under any offer mechanism. The optimal offer rates,which place an upper bound on welfare from any offer mechanism, performs only marginallybetter: it increases patient welfare by 17.4 percent and reduces discards by 7.5 percent.While these gains are large, one drawback of approximately optimal priorities is that gainsare concentrated amongst a minority of agents: only 42 percent of patient types are betteroff at registration than they were under the pre-2014 mechanism. The next section discusseshow the approximately optimal priorities mechanism is able to find these improvements, andin particular how gains are distributed across patients.

A significant fraction of these gains can be achieved while respecting the types of distri-butional constraints faced by policy makers. The approximately optimal Pareto improvingpriorities mechanism increases patient welfare by 9.1 percent and reduces discards by 4.5 per-cent. As under approximately optimal priorities, this mechanism increases welfare througha combination of higher transplant rates and improved match quality. Recall that becausethis mechanism approximates the optimal Pareto improving offer rates with a scoring rule,constraint that no type should be worse off may not be exactly satisfied. In fact, most patienttypes (85 percent) are slightly worse off at registration under our approximation than theywere under pre-2014 priorities. However,these patients experience small welfare losses: 99.7percent of patients are no more than 5 percent worse off at registration. Thus, the verystrong requirement that no patient type be significantly worse off ex-ante can be approxi-mately satisfied while achieving a substantial improvement in average patient welfare andtransplant rates.

The mechanism designer is only able to achieve these gains by considering patient incentives.The greedy priorities mechanism, which offers organs to patients based on predicted matchvalue alone, only marginally improves on the pre-2014 mechanism. It increases the averagepatient’s welfare by 1.2 percent and reduces organ discards by 0.3 percent (Figure 6). Thissmall improvement is surprising given that greedy priorities incorporates the rich estimatedheterogeneity in preference estimates, emphasizing the value of explicitly incorporating in-centive constraints into the mechanism design problem.

49

It is worth noting that our computed steady state queue length for the pre-2014 mechanismis 4,942 (Table 8, panel A), which is only slightly larger than the queue length of 4,632 onJanuary 1, 2013 (Table 1, panel A). This similarity is remarkable for two reasons. First,this moment of the data is not targeted in the CCP approach but matches our model.38 Aslightly longer steady state should be expected as the kidney waiting list was growing duringour sample period. Second, it suggests that our sample is likely close to the steady state.Indeed, panel A in Table 1 shows that the rate of growth in the length of the waiting list inNYRT declined during our sample period.

Finally, Appendix E assesses the sensitivity of our results to three variations. First, we esti-mate a model with a limited form of patient unobserved heterogeneity. Second, we evaluate amodel with a discount factor of 10 percent per year. Third, we re-compute the counterfactu-als using a sample with 1000 and 1500 types of patients and donors respectively instead of the300 and 500 types used in the baseline results. Our main qualitative conclusions are robustto these variations. For example, we robustly find that the pre- and post-2014 mechanismsare both similar in overall welfare and outcomes to the first come first served mechanism.

7.3 Optimal Offer Mechanism: Sources of Welfare Gains

Why is the approximately optimal priorities able to substantially outperform benchmarkmechanisms? The answer is closely tied to dynamic incentives. Compared to the pre-2014benchmark, approximately optimal priorities offers high quality organs to patients who notonly have high transplant values (in terms of equivalent increases in donor supply), but arealso are likely to accept the organs they are offered in equilibrium. This allows the mechanismto reduce discards while dramatically increasing the welfare of some patients.

In particular, the approximately optimal priorities reallocate desirable donors from patientson dialysis to patients off dialysis at registration. Table 10 compares transplant rates, offerprobabilities, and waiting times for an offer under the pre-2014, greedy, approximately opti-mal, and approximately optimal Pareto improving priorities, separately by patient age anddialysis status at registration.39 The first set of columns compares the number of transplantsper year from each donor type received by each patient group. Under baseline priorities,older off-dialysis patients (Panel B) receive about 11 transplants per year from young NYRTdonors, compared to more than 37 under approximately optimal priorities. Meanwhile, olderon-dialysis patients (Panel D) receive 30 kidneys each year from young healthy donors under

38The steady-state length of the waiting list N is approximately the ratio of the arrival rate of patients γto the average rate of departure (due to transplantation or otherwise) of each patient κ. This is because, forlarge N , the queue must satisfy the detail balance condition α = Nκ. The choice probabilities, our simulationof the mechanism and the estimated departure process without a transplant directly influence κ. That ourresults on queue lengths are close is reassuring for the methodology laid out in this paper.

39While we do not report reallocation patterns under the (infeasible) optimal offer rates mechanism, theyare very similar to the approximately optimal priorities.

50

baseline priorities, but only 25 under approximately optimal priorities. A similar realloca-tion occurs from young on-dialysis to young off-dialysis patients. In contrast, transplantrates from non-NYRT donors and low quality NYRT donors increase across all patient types,driving an overall increase in transplants. Approximately optimal Pareto improving prioritiesyield similar qualitative patterns to those in the unconstrained approximately optimal prior-ities, but the differences across patient groups are less stark. This is because the mechanismapproximates offer rates that are constrained to make each patient type better off ex-ante;therefore, it cannot reallocate as many desirable organs across groups.

This reallocation is achieved by offering high-quality donors to off-dialysis patients. The sec-ond group of columns in Table 10 reports the probability that a patient of each type receivesan offer from a donor of each type, conditional on the donor being offered to some NYRTpatients. These results capture the incentives faced by each patient type by quantifying thestream of offers they receive. The probability that a young off-dialysis patient receives anoffer from a young NYRT donor rises from 8.2 percent at baseline to 13.5 percent under ap-proximately optimal priorities; the same probability for a young on-dialysis patient falls from6.0 percent to 4.6 percent. On-dialysis patients see a uniform decrease in offer probabilityacross donor types under the approximately optimal priorities, and consistent with this, theyare worse off. In contrast, among off-dialysis patients, older patients see an increase in offerprobabilities for all donor types, while younger patients see a slight decrease for less desirableNYRT donors. Thus the approximately optimal priorities perform age/quality matching inaddition to reallocating organs to off-dialysis patients.

A natural question is why reallocation from on-dialysis to off-dialysis patients maximizespatient welfare. Table 11 shows that it is beneficial to reallocate organ offers to off-dialysispatients because they not only value each offer more, but are also more likely to accept. PanelA reports the average value of an offer from a randomly chosen donor to different groups ofpatients. To illustrate potential reallocation gains, patient types are weighted according totheir steady-state proportions under pre-2014 priorities. Off-dialysis patients benefit morethan on-dialysis patients from an offer from each donor group. An offer from a young, healthyNYRT donor is equivalent to a donor supply increase of 32.6 percent for an older off-dialysispatient, but only 18.1 percent for an older on-dialysis patient. The difference is even starkerfor non-NYRT donors, who are less desirable (equivalent to 0.9 and 0.4 percent increases indonor supply, respectively). This is in part because off-dialysis patients have low offer ratesin the baseline mechanism, so each additional offer is quite valuable. In addition, healthydonors are valued relatively more highly by young off-dialysis patients, which explains theage/quality matching under approximately optimal priorities. Panel B shows that off-dialysispatients are also more likely to accept an offer from a randomly chosen donor in the baselinemechanism. For example, off-dialysis patients would accept 1.2 percent of non-NYRT donors,while on-dialysis patients would accept 0.8 to 0.9 percent of them. Differences are similar forother donor types. Because of the limit on the number of offers than can be made, offering

51

the organs to patients more likely to accept increases transplant rates, and therefore expectedwelfare conditional on match quality. Approximately optimal priorities performs much betterthan greedy priorities in part because greedy priorities only considers transplant values andnot dynamic incentives, and therefore misses many of these potential gains.

Finally, one concern is that using dialysis status to determine priorities may result in agentstrying to game the system. While addressing this issue is beyond the scope of this paper,we make two observations. First, the post-2014 mechanism implicitly uses dialysis status todetermine priority through the Expected Post Transplant Survival model. Second, duringour sample period, patients were allowed to register on the list as soon as kidney function wassufficiently low (below a glomerular filteration rate of 20), but dialysis may not be necessaryfor all patients (for example, if the glomerular filteration rate is above 15). Therefore, ifgaming is of concern, then priority could be alternatively awarded based on measured kidneyfunction.

8 Conclusion

Previous reforms of organ allocation systems have been assisted by a simple simulation modelthat does not account for the dependence of accept/reject rules on agents’ incentives invarious mechanisms. This paper develops an empirically tractable method for estimatingthe value of various assignments in dynamic assignment systems, shows how to computeequilibria of counterfactual mechanisms, and computes optimal solutions. Our results showthat accounting for changes in agents’ incentives is important in predicting outcomes.

Moreover, we find that there is significant scope for further improving the deceased donorkidney allocation mechanism. There exist mechanisms that improve the average patient’swelfare by as much as the equivalent of a 14.2 percent increase in donor arrival rates whilealso reducing organ waste. An allocation system that also ensures that no patient type isworse off can realize the majority of these gains. In contrast, the reforms implemented in2014 were mostly redistributive, and both the pre- and the post-2014 systems are similarto a first come first served system. Alternatively, if our measure of patient welfare is of noconcern and the only goal is to reduce organ discards, it is possible to do so by 26.6 percentusing a last come first served system.

These findings suggest significant scope for using empirical approaches to improve dynamicassignment mechanisms. While the specific objective function for a designer may differ fromthose considered considered here, our methods can inform design once a well-defined objectiveand a set of acceptable mechanisms has been specified. Our approach can be used to bothevaluate outcomes from specific proposed mechanisms and to find optimal solutions.

Our approach makes several simplifying assumptions. First, and foremost, we assume thatagent beliefs are passive and we analyze steady state equilibria. Relaxing these assumptions is

52

an important direction for future work. Second, we do not allow for patient-level unobservedheterogeneity. Including this feature, potentially with time-varying components, may beimportant when applying these methods to other empirical contexts or data environments.Third, we assume that arrival rates of patients and objects are exogenous, an assumptionreasonable in our setting, but one that may be particularly important to relax in otherapplications. Finally, the outcomes we primarily focus on are organ waste and overall patientwelfare. Nonetheless, our framework can be used to evaluate the effects of various mechanismson other well-defined criteria that may be important for policy. These include equity or effectson predicted medical outcomes. Enriching this framework to incorporate these features isleft for future work.

Empirical approaches for evaluating dynamic assignment systems are particularly importantbecause previous theoretical approaches have not provided sharp guidance on these mecha-nisms. Models that emphasize heterogeneity in match value find that systems similar to firstcome first served are most efficient (Bloch and Cantala, 2017), while those that emphasizeheterogeneity in object quality suggest that last come first served be adopted (Su and Zenios,2004). Our results in this context find significant heterogeneity in match value and thereforefavor mechanisms of the former kind because they induce agent selectivity. Nonetheless,we find that targeting offers using observable characteristics and equilibrium considerationspromises additional improvements. These findings demonstrate the large scope for empiricalwork on understanding these designs more broadly than in the context of deceased donorkidney allocation.

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22.5

Num

ber

of K

idne

ys T

rans

plan

ted

per

Don

or1.

550.

780.

350.

481.

900.

421.

810.

510.

350.

73

Don

or A

ge43

.417

.856

.014

.239

.717

.141

.017

.054

.617

.4

Cau

se o

f D

eath

--

Hea

d T

raum

a26

.1%

44.0

%11

.7%

32.3

%30

.4%

46.0

%29

.3%

45.6

%11

.4%

31.9

%

Cau

se o

f D

eath

--

Str

oke

43.1

%49

.6%

60.9

%48

.9%

37.9

%48

.5%

39.6

%48

.9%

59.3

%49

.3%

Dia

betic

Don

or13

.8%

34.5

%25

.1%

43.5

%10

.4%

30.6

%10

.6%

30.8

%28

.6%

45.3

%

Hyp

erte

nsiv

e D

onor

37.2

%48

.4%

60.9

%48

.9%

30.2

%46

.0%

31.8

%46

.6%

62.1

%48

.7%

Exp

ande

d C

riter

ia D

onor

(E

CD

)29

.7%

45.7

%58

.1%

49.5

%21

.3%

41.0

%23

.3%

42.3

%59

.3%

49.3

%

Don

atio

n af

ter

Car

diac

Dea

th (

DC

D)

9.1%

28.7

%12

.3%

32.9

%8.

1%27

.3%

8.7%

28.2

%10

.7%

31.0

%

Don

or C

reat

inin

e1.

31.

51.

51.

21.

31.

61.

21.

41.

81.

9

Pan

el B

: All

Don

ors,

By

Num

ber

of O

rgan

s A

lloca

ted

or C

ateg

ory

of L

ast

Offe

r

Num

ber

of D

onor

s P

er Y

ear

1465

.75

906.

7555

920

1.25

1264

.5

Med

ian

Num

ber

of O

ffers

per

Don

or72

5.0

1200

.023

9.0

15.0

987.

5

Num

ber

of O

ffers

per

Don

or16

17.3

2560

.921

96.6

2990

.767

7.7

1123

.787

.616

0.6

1860

.826

77.0

Num

ber

of K

idne

ys T

rans

plan

ted

per

Don

or0.

750.

900.

210.

411.

640.

761.

760.

540.

590.

84

Don

or A

ge48

.018

.552

.615

.740

.620

.241

.117

.249

.118

.4

Cau

se o

f D

eath

--

Hea

d T

raum

a19

.2%

39.4

%15

.4%

36.1

%25

.4%

43.6

%28

.3%

45.1

%17

.8%

38.2

%

Cau

se o

f D

eath

--

Str

oke

48.1

%50

.0%

55.1

%49

.7%

36.8

%48

.2%

40.0

%49

.0%

49.4

%50

.0%

Dia

betic

Don

or20

.4%

40.3

%25

.0%

43.3

%12

.8%

33.5

%10

.3%

30.4

%22

.0%

41.4

%

Hyp

erte

nsiv

e D

onor

53.6

%49

.9%

63.1

%48

.3%

38.3

%48

.6%

33.2

%47

.1%

56.9

%49

.5%

Exp

ande

d C

riter

ia D

onor

(E

CD

)44

.7%

49.7

%54

.5%

49.8

%28

.8%

45.3

%23

.5%

42.4

%48

.0%

50.0

%

Don

atio

n af

ter

Car

diac

Dea

th (

DC

D)

12.6

%33

.2%

13.9

%34

.6%

10.4

%30

.6%

9.3%

29.1

%13

.1%

33.8

%

Don

or C

reat

inin

e1.

51.

31.

61.

21.

51.

51.

21.

31.

61.

3

Loca

l or

Per

fect

T

issu

e T

ype

Ma

tch

No

n-Lo

cal,

So

me

Tis

sue

Type

Mis

mat

ch

No

tes:

Pan

el A

con

sist

s o

f all

dece

ase

d ki

dney

don

ors

(7

84)

reco

vere

d in

NY

RT

and

offe

red

to N

YR

T p

atie

nts

betw

een

Jan

uary

1st

, 20

10 a

nd D

ece

mbe

r 31

st, 2

013

. Pa

nel B

in

clu

des

all d

ono

rs (

5,6

83)

offe

red

to N

YR

T p

atie

nts

dur

ing

the

sam

e pe

riod

, inc

ludi

ng d

onor

s re

cove

red

outs

ide

NY

RT.

Offe

rs e

xclu

de c

ases

in w

hic

h th

e do

nor

did

not m

eet

the

pa

tien

t's p

re-d

eter

min

ed c

rite

ria fo

r ac

cep

tabl

e do

nors

, or

in w

hich

the

patie

nt w

as

bypa

ssed

by

the

wai

tlist

sys

tem

due

to o

per

atio

nal

con

side

ratio

ns

tha

t did

not

invo

lve

an a

ctiv

e ch

oice

by

the

patie

nt o

r h

er s

urge

on.

60Ta

ble3:

Rates

ofReceiving

andAcceptin

gOffe

rs

Off

ers,

Pap

er

Pag

e 3

Offe

r &

Acc

ep

tan

ce R

ate

s

All

Do

no

rsN

YR

T D

on

ors

Pe

rfe

ct T

issu

e T

ype

Ma

tch

An

nu

al R

ate

% A

cce

pte

dA

nn

ua

l Ra

te%

Acc

ep

ted

An

nu

al R

ate

% A

cce

pte

d

Pa

ne

l A: A

ll O

ffers

All

99

17

21

8.4

0.1

5%

40

.10

.73

%0

.09

41

0.6

%

On

Dia

lysi

s a

t Re

gis

tra

tion

67

03

21

7.2

0.1

6%

42

.10

.76

%0

.09

01

1.3

%

No

t on

Dia

lysi

s a

t Re

gis

tra

tion

32

14

22

0.9

0.1

2%

36

.10

.67

%0

.10

49

.4%

Pe

ak

CP

RA

< 0

.80

88

64

23

4.5

0.1

0%

42

.80

.49

%0

.09

39

.3%

Pe

ak

CP

RA

>=

0.8

01

05

38

3.1

1.0

3%

17

.34

.39

%0

.10

31

5.8

%

Pa

ne

l B: O

ffers

tha

t Me

t Scr

ee

nin

g C

rite

ria

All

99

17

10

4.1

0.3

4%

23

.51

.35

%0

.05

12

1.8

%

On

Dia

lysi

s a

t Re

gis

tra

tion

67

03

10

2.7

0.3

7%

24

.81

.39

%0

.04

22

4.6

%

No

t on

Dia

lysi

s a

t Re

gis

tra

tion

32

14

10

7.1

0.2

7%

20

.61

.25

%0

.06

91

7.5

%

Pe

ak

CP

RA

< 0

.80

88

64

11

2.3

0.2

2%

25

.10

.89

%0

.04

91

9.7

%

Pe

ak

CP

RA

>=

0.8

01

05

33

5.8

2.5

1%

9.8

8.1

1%

0.0

68

29

.0%

Pa

ne

l C: O

ffers

With

in th

e F

irst 1

0 P

osi

tion

s th

at

Me

t S

cre

en

ing

Crit

eria

All

99

17

0.8

25

.38

%0

.82

6.6

8%

0.0

21

48

.3%

On

Dia

lysi

s a

t Re

gis

tra

tion

67

03

0.8

25

.42

%0

.82

6.6

8%

0.0

17

48

.5%

No

t on

Dia

lysi

s a

t Re

gis

tra

tion

32

14

0.8

25

.27

%0

.82

6.7

1%

0.0

29

47

.9%

Pe

ak

CP

RA

< 0

.80

88

64

0.8

15

.79

%0

.81

6.6

1%

0.0

19

51

.1%

Pe

ak

CP

RA

>=

0.8

01

05

30

.94

4.1

0%

0.9

46

.24

%0

.03

84

3.6

%

Nu

mb

er o

f P

atie

nts

No

tes:

the

re w

ere

2,8

50

,57

2 o

ffers

ma

de

to N

YR

T p

atie

nts

be

twe

en

Ja

nu

ary

1st

, 20

10

an

d D

ece

mb

er

31

st,

20

13

. P

an

el C

re

stric

ts t

o t

he

firs

t 1

0

NY

RT

pa

tien

ts in

ea

ch d

on

or's

offe

r se

qu

en

ce. A

n o

ffer

Me

t Scr

ee

nin

g C

rite

ria if

the

offe

r sa

tisfie

d a

pa

tien

t's p

re-d

ete

rmin

ed

crit

eria

fo

r a

cce

pta

ble

d

on

ors

. “A

nn

ua

l Ra

te”

colu

mn

s re

po

rt a

nn

ua

l offe

r ra

tes

com

pu

ted

by

pa

tien

t an

d th

en

ave

rag

ed

acr

oss

pa

tien

ts.

Pe

ak

CP

RA

is t

he

hig

he

st le

vel o

f C

alc

ula

ted

Pa

ne

l Re

act

ive

An

tibo

die

s (C

PR

A)

reco

rde

d fo

r e

ach

pa

tien

t du

ring

the

pe

riod

of s

tud

y.

61Ta

ble4:

Evidence

onMism

atch

Pa

tien

t Ag

eO

n D

ialy

sis

at

Re

gis

tra

tion

0-1

71

8-3

43

5-4

95

0-6

46

5+

Ye

sN

o

Sh

are

of P

atie

nt P

op

ula

tion

2.2

%1

1.1

%2

7.2

%4

1.5

%1

8.0

%6

7.7

%3

2.3

%

Sh

are

of D

ece

ase

d D

on

or

Tra

nsp

lan

ts4

.7%

9.6

%2

6.1

%4

1.9

%1

7.7

%7

2.6

%2

7.4

%

Sh

are

of I

de

al D

on

or

Tra

nsp

lan

ts6

.7%

9.8

%2

6.6

%3

9.8

%1

7.1

%7

3.1

%2

6.9

%

Pa

ne

l A: O

utc

om

es

of a

ll P

atie

nts

Re

ceiv

ed

De

cea

sed

Do

no

r T

ran

spla

nt

47

.9%

19

.2%

21

.3%

22

.4%

21

.9%

23

.8%

18

.9%

Re

ceiv

ed

Liv

e D

on

or

Tra

nsp

lan

t1

9.4

%1

8.6

%1

1.5

%8

.9%

6.5

%7

.2%

17

.2%

Stil

l Wa

itin

g2

8.9

%4

8.6

%5

0.4

%4

8.8

%4

4.0

%4

7.7

%4

8.3

%

Die

d o

r To

o S

ick

to T

ran

spla

nt

1.9

%4

.1%

6.8

%1

2.9

%2

0.0

%1

3.4

%6

.9%

De

pa

rte

d fo

r An

oth

er

Re

aso

n1

.9%

9.5

%1

0.1

%7

.0%

7.6

%7

.8%

8.7

%

Pa

ne

l B: D

on

or A

ge

an

d Q

ua

lity

Do

no

r is

Ide

al

97

.0%

70

.2%

69

.7%

64

.9%

66

.2%

68

.9%

67

.1%

Do

no

r Ag

e 0

-17

23

.8%

14

.6%

7.2

%4

.3%

4.5

%6

.6%

8.0

%

Do

no

r Ag

e 1

8-3

57

3.3

%3

7.6

%2

6.7

%1

7.3

%1

2.9

%2

2.8

%2

5.4

%

Do

no

r Ag

e 3

5-4

93

.0%

30

.2%

34

.9%

28

.7%

17

.7%

28

.1%

25

.2%

Do

no

r Ag

e 5

0-6

40

.0%

16

.6%

29

.2%

42

.1%

52

.0%

36

.7%

34

.4%

Do

no

r Ag

e >

= 6

50

.0%

1.0

%2

.0%

7.6

%1

2.9

%5

.7%

7.0

%

Pa

ne

l C: D

on

or A

ge

, Id

ea

l Do

no

rs

Do

no

r Ag

e 0

-17

23

.5%

17

.4%

9.5

%5

.8%

5.6

%8

.5%

10

.7%

Do

no

r Ag

e 1

8-3

57

3.5

%3

8.9

%2

3.7

%1

7.2

%1

1.2

%2

3.3

%2

4.9

%

Do

no

r Ag

e 3

5-4

93

.1%

26

.4%

33

.9%

22

.5%

14

.7%

24

.1%

21

.1%

Do

no

r Ag

e 5

0-6

40

.0%

16

.0%

30

.6%

45

.1%

51

.8%

37

.3%

34

.5%

Do

no

r Ag

e >

= 6

50

.0%

1.4

%2

.3%

9.4

%1

6.7

%6

.8%

8.9

%

Pa

ne

l A s

am

ple

is a

ll N

YR

T p

atie

nts

on

the

wa

itin

g li

st b

etw

ee

n J

an

ua

ry 1

st, 2

01

0 a

nd

De

cem

be

r 3

1st

, 20

13

. F

or

oth

er

pa

ne

ls,

the

sa

mp

le is

NY

RT

pa

tien

ts w

ho

re

ceiv

ed

de

cea

sed

do

no

r tr

an

spla

nts

be

twe

en

20

10

an

d 2

01

3. A

n id

ea

l do

no

r h

as

no

his

tory

of

dia

be

tes;

is

no

n-D

CD

; ha

s cr

ea

tinin

e b

elo

w 3

; an

d is

He

pa

titis

C n

eg

ativ

e.

62

Table 5: Acceptance Rate by Past Offer Rates

Dependent Variable: Current Offer Accepted

All Offers Ideal Donors NYRT Donors

(1) (2) (3) (4) (5) (6) (7)

Time Since Last Offer (Years) 0.0372 0.00227 0.00217 -0.000324 -0.00948

(0.00875) (0.00787) (0.00786) (0.00505) (0.00663)

Time Since Last Two Offers (Years) 0.00646

(0.00880)

Time Since Last Five Offers (Years) 0.0151

(0.0118)

Last Offer Donor Age -0.00000369

(0.00000304)

Last Offer from Diabetic Donor -0.0000854

(0.000116)

Last Offer from Expanded Criteria Donor -0.000174

(0.000116)

Last Offer from Donation after Cardiac Death 0.00000289

(0.000119)

Variables Affecting Priority X X X X X X

Patient Characteristics X X X X X X

Donor and Match Characteristics X X X X X X

Listing Center Fixed Effects X X X X X X

Observations 2793098 2831262 2831262 2821641 2831262 1083686 526233

R-squared 0.098 0.097 0.097 0.099 0.099 0.133 0.116

Mean Acceptance Rate 0.15% 0.15% 0.15% 0.15% 0.15% 0.25% 0.73%

Mean Time Since Last N Offers 0.005 0.005 0.005 0.005 0.005 0.012 0.023

S.D. Time Since Last N Offers 0.016 0.016 0.011 0.008 0.016 0.028 0.048

Notes: estimates from a linear probability model of offer acceptance as a function of the patient's recent offer history. Time Since Last N Offers measures the average number of years since the patient's previous offers, averaged over their last N offers. Time Since Last Offer, Including Inactive Periods counts inactive days as well as active days on the waitlist. Column (1) considers all offers and includes no controls for current offer characteristics. Columns (2) - (7) control for current patient, donor, and match characteristics. Column (5) includes controls for donor characteristics of the patient's previous offer. Column (6) restricts to offers from ideal donors, and Column (7) restricts to NYRT donors. Controls are as described in the notes for Appendix Table B.1. An ideal donor has no history of diabetes; is non-DCD; has creatinine below 3; and is Hepatitis C negative.

63

Table 6: Survival Model Estimates

Gompertz Weibull Cox

(1) (2) (3)

Diabetic Patient 0.112 0.104 0.115

(0.0334) (0.0334) (0.0334)

Bloodtype A Patient 0.147 0.115 0.154(0.0436) (0.0435) (0.0436)

Bloodtype O Patient 0.0146 0.0145 0.0132

(0.0390) (0.0390) (0.0391)

Calculated Panel Reactive Antibodies (CPRA) -0.000601 -0.000692 -0.000775

(0.00150) (0.00150) (0.00150)

CPRA = 0 0.182 0.171 0.173(0.0737) (0.0736) (0.0738)

CPRA - 80 if CPRA>=80 -0.0182 -0.0154 -0.0172

(0.00652) (0.00652) (0.00652)

Age (at Registration) -0.0261 -0.0213 -0.0210(0.0152) (0.0153) (0.0153)

Age - 18 if Age>=18 0.0230 0.0194 0.0186(0.0186) (0.0187) (0.0187)

Age - 35 if Age>=35 -0.00964 -0.0120 -0.0104

(0.00960) (0.00960) (0.00960)

Age - 50 if Age>=50 0.0232 0.0230 0.0239(0.00723) (0.00722) (0.00724)

Age - 65 if Age>=65 0.0243 0.0234 0.0241

(0.00923) (0.00922) (0.00925)

Prior Transplant 0.0205 0.0329 0.0289(0.0547) (0.0545) (0.0547)

Body Mass Index (BMI) -0.0131 -0.0123 -0.0131(0.00643) (0.00642) (0.00643)

Missing BMI 0.000107 0.132 -0.0477

(0.199) (0.199) (0.200)

BMI >= 18.5 -0.0653 -0.0743 -0.0675(0.105) (0.106) (0.106)

BMI >= 25 -0.0140 -0.0177 -0.0133(0.0492) (0.0492) (0.0492)

BMI >= 30 0.0285 0.0223 0.0269

(0.0595) (0.0594) (0.0595)

Total Serum Albumin -0.184 -0.180 -0.175(0.0543) (0.0543) (0.0543)

Missing Total Serum Albumin -0.675 -0.598 -0.621

(0.187) (0.187) (0.187)

Total Serum Albumin >= 3.7 -0.0836 -0.0817 -0.0871(0.0589) (0.0590) (0.0589)

Total Serum Albumin >= 4.4 0.0422 0.0308 0.0415(0.0507) (0.0506) (0.0507)

On Dialysis at Registration -1.087 -1.067 -1.086

(0.0659) (0.0660) (0.0659)

Log Years on Dialysis at Registration 0.108 0.109 0.107(0.0103) (0.0103) (0.0103)

Log Years on Dialysis at Registration * Over 5 Years -0.129 -0.123 -0.137

(0.105) (0.105) (0.105)

Constant -5.151 -4.702

(0.328) (0.337)

Constant (delta(2)) 0.000106(0.0000207)

Constant (delta) -0.0635

(0.0142)

Observations 9917 9917 9917

64

Table 7: Conditional Choice Probability of Acceptance (select co-efficients)

Base Specification Unobserved Heterog. Waiting Time + UH

(1) (2) (3)

Calculated Panel Reactive Antibody (CPRA) 0.60 (0.05) 0.67 (0.06) 0.54 (0.08)

Donor Age < 18 0.30 (0.10) -0.10 (0.18) -0.07 (0.19)

Donor Age 18-35 0.60 (0.12) 0.09 (0.18) 0.14 (0.18)

Donor Age 50+ -0.83 (0.15) -0.81 (0.20) -0.85 (0.21)

Expanded Criteria Donor (ECD) -0.15 (0.02) -0.51 (0.08) -0.54 (0.08)

Donation from Cardiac Death (DCD) -0.11 (0.02) -0.49 (0.08) -0.49 (0.07)

Perfect Tissue Type Match 2.28 (0.31) 2.79 (0.42) 2.75 (0.43)

2 A Mismatches -0.08 (0.01) -0.01 (0.02) -0.01 (0.02)

2 B Mismatches 0.06 (0.02) 0.03 (0.02) 0.02 (0.02)

2 DR Mismatches -0.07 (0.02) -0.06 (0.02) -0.06 (0.02)

Regional Offer -1.34 (0.05) -2.69 (0.17) -2.75 (0.17)

National Offer -1.50 (0.04) -2.92 (0.11) -2.95 (0.12)

Non-NYRT Donor, NYRT Match Run 1.21 (0.02) 1.97 (0.05) 2.01 (0.06)

Patient on Dialysis at Registration -0.04 (0.01) -0.12 (0.02) -0.11 (0.02)

Log Waiting Time (years) -0.01 (0.05)

Log Waiting Time * Over 1 Year -0.10 (0.06)

Log Waiting Time * Over 2 Years -0.17 (0.12)

Log Waiting Time * Over 3 Years 0.42 (0.11)

Log Dialysis Time at Registration (Years) 0.05 (0.00) 0.05 (0.00) 0.05 (0.01)

Log Dialysis Time at Registration * Over 5 years 0.48 (0.03) 0.42 (0.03) 0.41 (0.04)

NYRT Donor * Donor Age < 18 -0.05 (0.06) 0.22 (0.20) 0.23 (0.22)

NYRT Donor * Donor Age 18-35 0.10 (0.04) 0.11 (0.13) 0.14 (0.14)

NYRT Donor * Donor Age 50+ -0.42 (0.03) -0.62 (0.11) -0.62 (0.13)

Patient Age * Donor Age < 18 -0.01 (0.00) 0.00 (0.00) 0.00 (0.00)

Patient Age * Donor Age 18-35 -0.02 (0.00) 0.00 (0.01) 0.00 (0.01)

Patient Age * Donor Age 50+ 0.02 (0.00) 0.02 (0.01) 0.02 (0.01)

Patient Age - 35 if Age >= 35 * Donor Age 18-35 0.00 (0.01)

Patient Age - 35 if Age >= 35 * Donor Age 50+ 0.00 (0.01)

Donor Unobservable Std. Dev. 0.98 (0.21) 1.00 (0.23)

Idiosyncratic Shock Std. Dev. 1.00 1.00 1.00

Acceptance Rate 0.150% 0.150% 0.150%

Number of Offers 2850572 2850572 2850572

65

Table8:

Outcomes

inVa

rious

Mecha

nism

s

Pre

-201

4 P

riorit

ies

4992

.2--

2.52

43.8

25.6

%11

.6%

38.1

%--

----

--

Pos

t-201

4 P

riorit

ies

4913

.80.

5%2.

4943

.825

.5%

11.6

%38

.2%

-0.3

%-0

.9%

35.0

%98

.7%

Firs

t Com

e Fi

rst S

erve

d50

95.6

-1.4

%2.

5543

.725

.5%

11.6

%38

.0%

0.6%

1.2%

86.3

%95

.3%

Last

Com

e Fi

rst S

erve

d26

94.3

26.6

%2.

4545

.924

.3%

11.7

%44

.5%

-21.

8%-3

3.6%

2.0%

9.7%

Gre

edy

Prio

ritie

s49

17.4

0.3%

2.53

43.6

25.7

%11

.4%

37.9

%1.

7%1.

2%62

.0%

79.3

%

Opt

imal

Ass

ignm

ent

3815

.613

.6%

2.17

44.1

25.4

%11

.6%

40.6

%44

.4%

28.1

%83

.3%

96.3

%

Opt

imal

Offe

r Rat

es44

38.3

7.5%

2.40

44.1

25.5

%11

.6%

40.4

%27

.4%

17.4

%44

.3%

62.0

%

App

roxi

mat

ely

Opt

imal

Prio

ritie

s43

58.3

7.8%

2.38

44.2

25.3

%11

.5%

40.5

%24

.2%

14.2

%41

.7%

61.0

%

App

rox.

Opt

. Par

eto

Impr

ovin

g P

riorit

ies

4560

.94.

5%2.

4144

.025

.5%

11.5

%39

.5%

15.0

%9.

1%14

.7%

99.7

%

Pre

-201

4 P

riorit

ies

5387

.6--

2.86

43.3

25.8

%11

.5%

36.5

%--

----

--

Pos

t-201

4 P

riorit

ies

5337

.80.

0%2.

8643

.325

.8%

11.5

%36

.5%

----

----

Firs

t Com

e Fi

rst S

erve

d 54

60.6

-0.1

%2.

8443

.325

.8%

11.5

%36

.5%

----

----

Last

Com

e Fi

rst S

erve

d 51

75.0

3.6%

5.39

43.6

25.5

%11

.3%

37.7

%--

----

--

Gre

edy

Prio

ritie

s 52

64.2

0.9%

2.89

43.4

25.7

%11

.4%

36.8

%--

----

--

Que

ue

Leng

th

Red

uctio

n in

D

isca

rd

Rat

e

At L

istin

g V

x(0)

Cha

ract

eris

tics

of T

rans

plan

ted

Don

ors

Car

diac

D

eath

(D

CD

)

Ste

ady

Sta

teV

x

Cha

nge

in E

quiv

alen

t D

onor

Sup

ply

(Mea

n)Fr

actio

n w

ith Δ

Vx(0

)

> 0%

> -5

%

Pan

el A

: Ste

ady

Sta

te E

quili

briu

m, A

ll P

atie

nts

Pan

el B

: Est

imat

ed C

hoic

e P

roba

bilit

ies

from

Pre

-201

4 M

echa

nism

, All

Pat

ient

s

Age

Die

d of

H

ead

Trau

ma

Hyp

er-

tens

ive

Year

s on

W

aitli

st

Wai

tlist

66

Table 9: Outcomes in Various Mechanisms, by Patient Group

Pre-2014 Priorities 2.40 38.3 32.2% 13.4% 27.5% -- -- -- --

Post-2014 Priorities 2.37 39.3 33.8% 13.4% 28.5% 0.9% 0.3% 55.3% 97.4%

First Come First Served 2.43 41.0 29.2% 12.7% 35.2% -5.5% -4.6% 68.4% 84.2%

Last Come First Served 2.33 42.6 26.6% 13.0% 40.7% -32.0% -39.9% 0.0% 7.9%

Greedy Priorities 2.43 38.0 33.4% 11.6% 29.0% 1.2% 1.5% 71.1% 78.9%

Optimal Assignment 2.07 39.5 30.7% 11.9% 33.3% 43.9% 30.7% 81.6% 92.1%

Approximately Optimal Priorities 2.20 38.6 31.4% 11.3% 33.1% 20.4% 13.1% 68.4% 84.2%

Approx. Opt. Pareto Improving Priorities 2.34 37.7 30.9% 12.3% 33.7% 6.1% 4.6% 18.4% 100.0%

Pre-2014 Priorities 2.51 47.1 21.6% 12.0% 44.6% -- -- -- --

Post-2014 Priorities 2.48 46.6 21.8% 11.8% 44.1% -0.7% -1.3% 25.4% 100.0%

First Come First Served 2.52 45.6 23.4% 12.0% 40.9% 3.7% 4.1% 95.8% 100.0%

Last Come First Served 2.52 48.4 22.1% 12.3% 48.7% -30.3% -41.5% 0.0% 4.2%

Greedy Priorities 2.58 44.7 23.9% 11.8% 38.0% 5.2% 5.8% 59.2% 71.8%

Optimal Assignment 1.87 44.5 25.3% 12.4% 38.7% 96.9% 65.7% 95.8% 100.0%

Approximately Optimal Priorities 2.10 44.7 25.7% 12.7% 37.5% 70.0% 45.7% 74.6% 88.7%

Approx. Opt. Pareto Improving Priorities 2.33 46.1 23.9% 13.5% 39.3% 37.7% 25.9% 38.0% 100.0%

Pre-2014 Priorities 2.53 40.2 27.5% 11.5% 34.1% -- -- -- --

Post-2014 Priorities 2.50 40.3 27.2% 11.5% 33.7% 0.0% -0.6% 43.7% 98.6%

First Come First Served 2.57 40.9 27.0% 11.0% 35.8% -1.0% -0.2% 84.5% 93.0%

Last Come First Served 2.49 42.7 26.9% 11.3% 41.1% -20.3% -31.3% 1.4% 9.9%

Greedy Priorities 2.49 40.5 27.3% 10.7% 35.4% -0.7% -1.8% 70.4% 90.1%

Optimal Assignment 2.31 41.8 26.8% 11.0% 41.1% 6.5% 0.9% 71.8% 93.0%

Approximately Optimal Priorities 2.51 41.5 25.8% 10.9% 41.4% -7.4% -7.9% 14.1% 36.6%

Approx. Opt. Pareto Improving Priorities 2.46 39.1 28.1% 10.3% 36.2% -1.0% -2.4% 0.0% 98.6%

Pre-2014 Priorities 2.56 46.1 24.3% 10.9% 40.8% -- -- -- --

Post-2014 Priorities 2.53 46.1 23.6% 10.9% 41.1% -0.7% -1.3% 29.2% 98.3%

First Come First Served 2.59 45.3 24.5% 11.4% 38.7% 1.7% 2.3% 87.5% 97.5%

Last Come First Served 2.42 47.2 23.4% 11.3% 45.3% -14.4% -28.4% 4.2% 13.3%

Greedy Priorities 2.54 46.6 23.3% 11.5% 42.1% 1.1% 0.2% 55.8% 77.5%

Optimal Assignment 2.25 46.4 23.1% 11.3% 43.7% 35.8% 21.2% 83.3% 97.5%

Approximately Optimal Priorities 2.48 47.2 22.9% 11.2% 44.4% 17.1% 9.1% 30.0% 51.7%

Approx. Opt. Pareto Improving Priorities 2.45 47.6 23.2% 10.8% 43.2% 13.9% 7.4% 8.3% 100.0%

Change in Equivalent Donor Supply (Mean)

At Listing Vx(0)

Steady StateVx

Fraction with ΔVx(0)

> 0% > -5%

Waitlist

Cardiac Death (DCD)

Hyper-tensive

Characteristics of Transplanted Donors

AgeDied of Head

Trauma

Years on Waitlist

Panel A: Not on Dialysis at Registration, Age 0-49

Panel B: Not on Dialysis at Registration, Age 50+

Panel C: On Dialysis at Registration, Age 0-49

Panel D: On Dialysis at Registration, Age 50+

67

Table10:Fe

atures

ofVa

rious

Mecha

nism

s

Youn

gO

ldLo

w Q

ualit

y D

onor

Youn

gO

ldLo

w Q

ualit

y D

onor

Youn

gO

ldLo

w Q

ualit

y D

onor

Pre

-201

4 P

riorit

ies

(Bas

elin

e)21

.89.

77.

550

.28.

29.

831

.548

.02.

814.

543.

633.

52G

reed

y P

riorit

ies

19.2

11.5

7.9

48.5

6.8

9.6

28.5

41.8

3.63

4.04

3.21

2.96

App

roxi

mat

ely

Opt

imal

Prio

ritie

s20

.312

.28.

567

.113

.514

.329

.548

.53.

043.

392.

782.

86A

ppro

x. O

pt. P

aret

o Im

prov

ing

Prio

ritie

s18

.112

.36.

456

.69.

69.

723

.846

.62.

803.

432.

933.

01

Pre

-201

4 P

riorit

ies

(Bas

elin

e)10

.920

.318

.787

.65.

89.

930

.943

.84.

434.

933.

863.

82G

reed

y P

riorit

ies

22.3

24.0

16.3

72.6

4.2

6.7

22.8

31.7

4.64

4.80

3.62

3.31

App

roxi

mat

ely

Opt

imal

Prio

ritie

s37

.638

.328

.911

9.0

17.1

18.3

38.0

49.3

2.51

2.65

2.42

2.74

App

rox.

Opt

. Par

eto

Impr

ovin

g P

riorit

ies

23.4

29.0

25.4

98.5

8.2

9.7

27.9

43.4

3.50

3.55

2.84

3.01

Pre

-201

4 P

riorit

ies

(Bas

elin

e)30

.524

.417

.810

7.2

6.0

8.9

31.6

44.2

4.09

4.72

3.80

3.75

Gre

edy

Prio

ritie

s23

.725

.617

.511

3.5

7.3

11.7

35.9

53.0

3.90

4.15

3.33

3.17

App

roxi

mat

ely

Opt

imal

Prio

ritie

s7.

814

.514

.114

1.0

4.6

6.5

20.9

41.4

3.93

4.32

3.55

3.58

App

rox.

Opt

. Par

eto

Impr

ovin

g P

riorit

ies

23.7

22.2

14.2

125.

06.

47.

222

.542

.13.

694.

252.

983.

19

Pre

-201

4 P

riorit

ies

(Bas

elin

e)30

.041

.242

.117

2.1

5.1

9.8

31.4

46.4

4.40

4.86

3.83

3.80

Gre

edy

Prio

ritie

s28

.034

.344

.518

8.2

5.7

10.7

34.6

51.3

3.82

4.15

3.34

3.17

App

roxi

mat

ely

Opt

imal

Prio

ritie

s25

.132

.047

.223

0.7

5.6

8.8

26.5

46.5

3.77

3.97

3.40

3.46

App

rox.

Opt

. Par

eto

Impr

ovin

g P

riorit

ies

27.3

35.6

47.3

214.

75.

49.

833

.750

.03.

273.

552.

722.

90

Not

es: N

umbe

r of T

rans

plan

ts p

er Y

ear m

easu

res

the

num

ber o

f kid

neys

from

eac

h do

nor g

roup

tran

spla

nted

into

eac

h pa

tient

gro

up. O

ffer P

roba

bilit

y is

the

prob

abili

ty th

at a

pat

ient

rece

ives

an

offe

r fro

m a

do

nor i

n ea

ch g

roup

con

ditio

nal o

n th

e do

nor b

eing

offe

red

to s

ome

patie

nts.

Mea

n W

ait T

ime

at O

ffer r

epor

ts th

e m

ean

wai

t tim

e of

a p

atie

nt re

ceiv

ing

a ra

ndom

ly s

elec

ted

offe

r.

Num

ber o

f Tra

nspl

ants

per

Yea

rO

ffer P

roba

bilit

y (%

)M

ean

Wai

t Tim

e at

Offe

r (Ye

ars)

NY

RT

Don

orN

on N

YR

T D

onor

NY

RT

Don

orN

on N

YR

T D

onor

NY

RT

Don

or

Pan

el A

: Pat

ient

s no

t on

Dia

lysi

s at

Reg

istra

tion,

Age

0-4

9

Pan

el B

: Pat

ient

s no

t on

Dia

lysi

s at

Reg

istra

tion,

Age

50+

Pan

el C

: Pat

ient

s on

Dia

lysi

s at

Reg

istra

tion,

Age

0-4

9

Pan

el D

: Pat

ient

s on

Dia

lysi

s at

Reg

istra

tion,

Age

50+

Non

NY

RT

Don

or

68

Table 11: Sources of Welfare Gains

Young Old Low Quality Donor

Patients Not on Dialysis at Registration, Age 0-49 36.7 15.1 2.7 0.7Patients Not on Dialysis at Registration, Age 50+ 32.6 18.6 5.1 0.9Patients on Dialysis at Registration, Age 0-49 13.6 5.2 0.8 0.2Patients on Dialysis at Registration, Age 50+ 18.1 7.9 2.2 0.4

Patients Not on Dialysis at Registration, Age 0-49 34.7 16.4 5.3 1.2Patients Not on Dialysis at Registration, Age 50+ 26.7 18.5 6.2 1.2Patients on Dialysis at Registration, Age 0-49 26.4 14.2 3.2 0.8Patients on Dialysis at Registration, Age 50+ 25.6 15.7 5.3 0.9

NYRT Donor Non NYRT Donor

Panel B: Percentage Willing to Accept

Panel A: Value of an Offer

Notes: in each cell, quantities are averaged over randomly drawn donors and patients according to their proportions in the Pre-2014 Priorities (Baseline) mechanism. Panel A reports the average equivalent change in donor supply from an offer from each donor type to each patient type. All numbers are reported in percentage (%) units.

69

Figure 1: Offer Statistics by Position on Waiting List

(a) Waiting Time by Position

(b) Acceptance Rate by Position

Note: Sample is offers made to NYRT patients between 2010 and 2013, excluding offers that did not meet a patient’s pre-set

donor screening criteria. Positive crossmatches are counted as acceptances. In each figure, the black line plots the mean among

offered patients in each position group, and the shaded region represents pointwise 95 percent confidence intervals.

70

Figure 2: Model Fit

(a) Fit by position

(b) Fit by wait time

71

Figure 3: Value of an Organ Offer by Patient and Donor Age

Notes: Patient and donor characteristics other than those explicitly varied are set to median values from the NYRT sample.

Value functions are computed at registration.

72

Figure 4: Patient Age and Dialysis

Notes: Patient and donor characteristics other than those explicitly varied are set to median values from the NYRT sample.

Value functions are computed at registration.

73

Figure 5: Value of an Organ Offer by Match Quality

Notes: Patient and donor characteristics other than those explicitly varied are set to median values from the NYRT sample.

Value functions are computed at registration.

74

Figure 6: Welfare and Organ Utilization under Alternative Mechanisms

Pre‐2014 Priorities

Post‐2014 Priorities

First Come First Served

Last Come First Served

Greedy Priorities

Optimal Assignment

Optimal Offer RatesApproximately Optimal Priorities

Approx. Opt. Pareto Improving Priorities

‐40%

‐30%

‐20%

‐10%

0%

10%

20%

30%

40%

‐5% 0% 5% 10% 15% 20% 25% 30%

Equivalent Increase in

Don

or Arrival Rate

Reduction in Fraction Discarded

Notes: Results are reported relative to pre-2014 priorities and based on each mechanism’s steady state equilibrium. The

reduction in fraction discarded is defined as the change in the number of kidneys rejected by all NYRT patients to whom the

organ was offered.

75

Appendix for “An Empirical Framework for Sequential Assignment:The Allocation of Deceased Donor Kidneys”

Nikhil Agarwal, Itai Ashlagi, Michael Rees, Paulo Somaini, Daniel Waldinger.

A Data Appendix

A.1 Data Description

Our data on patients, donors, transplants, and offers are based on information submitted tothe Organ Procurement and Transplantation Network (OPTN) by its members. The maindataset on the waitlist is the Potential Transplant Recipient (PTR) dataset. It contains thesequences of offers made to patients on the deceased donor kidney waitlist, their decisions,and their reasons for refusal. Detailed information on patient characteristics, donor char-acteristics, and transplant outcomes come from the Standard Transplantation Analysis andResearch (STAR) dataset. UNOS also provided supplemental information for this study,including the ordering of distinct match runs conducted for the same deceased donor; thetransplant centers of donors and patients in our dataset; and dates of birth for pediatriccandidates, who joined the waitlist before turning 18 years of age.

The data contain unique identifiers that allow us to link the offer and acceptance data topatient and donor characteristics. Each deceased donor has a unique identifier. Similarly,each patient registration generates a unique patient waitlist identifier. Because patients maymove to different transplant centers or be registered in multiple centers simultaneously, someindividual patients have multiple waitlist id’s. Where appropriate, we de-duplicate offers sothat each patient can receive at most one offer from each donor. The patient history file alsocontains a unique patient record identifier corresponding to a particular state of the patienton the waitlist, including the patient’s CPRA, activity status, and pre-set screening criteria.Each offer in the PTR dataset contains the identifiers for the donor, the patient registration,and the patient history record that were used in the match run.

The PTR dataset contains all offers made to patients on the deceased donor kidney waitlist.Information include identifiers for the donor, patient, and patient history record that gener-ated the offer; the order in which the offers were made; each patient’s acceptance decision;and if the offer was not accepted, a reason for rejecting. Each offer record also containscertain characteristics of the match, including the number of tissue type mismatches.

The STAR dataset contains separate files on deceased donor characteristics, patient char-acteristics and transplant outcomes, and patient histories. The patient and donor charac-teristics from these tables are used to estimate our models of acceptance behavior, positivecrossmatch probabilities, and patient departure rates. We also use these characteristics to

76

replicate the mechanism and determine each patient’s compatibility with and priority scorefor each deceased donor in our sample.

A.2 Sample Selection

This section explains the selection of patients, donors, and offers used in our structural model.We consider patients who were registered in NYRT and actively waiting for a deceased donorkidney between January 1st, 2010 and December 31st, 2013. Our donor sample includes allU.S. deceased kidney donors whose organs were allocated according to the standard mech-anism. Our offer sample includes valid offers from all deceased donors to NYRT patientsduring this period recorded in the PTR data, as well as offers that were not made because ofpre-specified screening criteria. The next section discusses our replication of the offer mech-anism, which we use to determine offers that were refused through these screening criteria,as well as the waiting time each patient would need to have been offered each compatibledonor. The remainder of this section discusses the details of how we select our samples ofdonors, patients, and offers that met screening criteria.

Because NYRT patients may be offered donors from across the U.S., our procedure firstconstructs a nationwide sample of deceased donors, patients, and offers that meet our samplecriteria. We then restrict the sample to NYRT and omit certain donors and patients whoreceived non-standard treatment in the mechanism.

Our U.S. sample of deceased kidney donors comes from the intersection of donor identifiers inthe PTR and STAR deceased donor files. Patients in our sample were active on the deceaseddonor kidney waitlist after 2010 and were not jointly registered for a pancreas transplant.Patient registration date and activity status are determined from the patient history file. Wealso exclude patients who departed the waitlist for reasons which indicate that they did notultimately need a transplant. We exclude patients who were transplanted in another country,whose condition improved, or who could no longer be contacted. These departure reasonsare recorded in the STAR patient and transplant outcome dataset.

We then determine which offers were valid and could have been accepted by and transplantedinto the patient; patients’ acceptance decisions; and the resulting priority score cutoffs in eachmatch run.

We first exclude offers that are not valid. In certain cases, patients are bypassed when adonor is allocated to a specific recipient outside of the standard allocation rules. This canoccur if the donor is an armed service member; if the donor specified a particular recipient(directed donation); if there is a medical emergency or expedited placement attempt; or iforgan sharing among DSAs generates a “payback” in which one DSA allocates a kidney fromanother DSA as though it had been recovered in its own service area. There are also cases inwhich a patient is offered a tissue type incompatible donor, or a donor that did not meet the

77

patient’s pre-specified screening criteria. We identify these cases using a refusal reason codeprovided in the PTR data. In some cases, there is also text specifying specific circumstancesjustifying a rejection, which we parse to identify invalid offers in cases where the refusalcode does not provide a specific reason. Finally, some offers are refused due to technologicalconstraints if the patient needs a specific organ laterality or requires multiple simultaneousorgan transplants. We do not consider these cases to be genuine refusals, and omit themfrom the offer dataset.

Next, we created an algorithm to de-duplicate offers and acceptances within and across matchruns, and to determine the true priority score cutoff for each donor in each match run. Forsome donors, multiple match runs are conducted, and these match runs can include offers tooverlapping sets of patients. A specific kidney (e.g. the left kidney) may also be acceptedin multiple match runs. Finally, a patient can have multiple offers recorded from the samedonor, even in the same match run. Our algorithm assumes that later match runs takeprecedence over earlier ones (using the match run numbers provided by OPTN), and thatthe last observed match run in which an organ is placed takes that organ out of circulationfor subsequent match runs. After de-duplication, there were 11,428,540 offers nationwidethat met patient screening criteria. The U.S. sample contains 30,079 donors and 226,000patients.

We then implement the sample restrictions for NYRT. We consider all patients who wereregistered in NYRT and had active status sometime between January 1st, 2010 and December31st, 2013. At this stage, we exclude patients who received a transplant through non-standardallocation rules. This includes cases of medical urgency, an expedited placement attempt, amulti-organ transplant, or a military or directed donation. 68 patients were excluded becausethey received deceased donor kidney transplants for these reasons, leaving 9,917 patients inour NYRT sample. We also exclude the donors whose organs were placed according to thesereasons, even if the organs were allocated to non-NYRT patients. The NYRT offer samplecontains valid offers from the sample of deceased donors to the sample of NYRT patients.There are 1,281,024 such offers. These offers and patient acceptance decisions determine thepriority score cutoff in each match run for each donor’s available organs.

A.3 Replicating the Mechanism, Offer Dataset

Knowledge of the mechanism allows us to determine the set of offers that were declinedthrough pre-set screening criteria, as well as the waiting time required for a particular pa-tient to have access to a particular donor. These are essential for correctly modeling patientacceptance behavior and transplant opportunities under the current and counterfactual mech-anisms. We wrote compute code to replicate the standard deceased donor kidney allocationrules in place between January 1st, 2010 and December 31st, 2013.

78

For each deceased donor and match run, the algorithm begins with all concurrent patientwaitlist history records. It first determines which patients are incompatible with the donordue to their blood type and unacceptable human leukocyte antigen (HLA) antigens (if any).We use blood type and HLA equivalence tables followed by the OPTN, as well as the donor’sHLA antigens and the current unacceptable antigens listed by each patient. Next, we checkwhether the donor met each patient’s screening criteria. Finally, we determine the priorityscore of each patient given their CPRA, waiting time, geography, age, and number of HLAand DR mismatches with the donor. Given the priority score, we can calculate whether thepatient was above the priority score cutoff for the donor. We can also determine the amountof additional waiting time (which may be infinite) after which the patient’s priority scorewould exceed the donor’s cutoff.From the simulation, we obtain a set of offers predicted by our simulation of the mechanism.These are pairs of donors and patients where the patient met the priority score cutoff andwas blood and tissue type compatible with the donor. Some of these offers met the patient’sscreening criteria, while others did not. Those that did should appear in the PTR data.This provides a check on the performance of our mechanism code. Table A.3 tabulates offersappearing in our filtered PTR data and those predicted by simulation. The vast majorityof offers in the PTR data (93.1%) were predicted by our simulation. However, a substantialfraction of offers predicted by the simulation (29.6%) are not in our PTR offer sample.To estimate the patient acceptance model, we take as our offer sample the union of the PTRoffer dataset and the set of offers that the simulation predicts were filtered due to the patient’sscreening criteria but which would otherwise have appeared in the PTR data. In a final step,we de-duplicate offers at the patient level, since a patient registered at multiple centers willoccasionally receive multiple offers from the same donor. The final offer sample contains2,850,572 offers: 1,267,531 PTR offers, and 1,583,041 predicted offers that were “screenedout.” The PTR offers include the 88,366 offers that were not predicted by our simulation butwhich appear in the PTR data, and exclude the 502,239 offers predicted by the simulationbut which did not appear in PTR. Offers that were screened out are interpreted as rejectionssince the patient deemed the donor’s characteristics unacceptable.To calculate patient value functions, we store all compatible patient and donor pairs, includingpatients who did not meet the donor’s priority score cutoff.

A.4 Imputing Missing Donor DR Antigens

A donor’s HLA antigens are needed to determine tissue type compatibility with transplantcandidates as well as kidney points, which are in turn essential for replicating the mechanism.A limitation of our data is that we only observe a donor’s DR antigens if one of their kidneysor pancreas was transplanted into a patient. In this case, they appear in the KIDPAN(patient/transplant) dataset. If no organs were placed, a donor’s antigen information is

79

recorded in the deceased donor file for kidney/pancreas donors. The deceased donor file liststhe donor’s HLA antigens at the A and B loci, but not at the DR locus.

We either obtain or impute a donor’s missing DR antigens from two sources. First, somedeceased donors had a liver, lung, or part of their intestine transplanted even though theirkidneys and pancreas were not transplanted. The equivalent transplant files for these addi-tional organs are part of the STAR dataset, and we take the donor’s DR antigens directlyfrom those files.

Second, for deceased donors who had no organs transplanted, we use the reported numberof DR mismatches in the PTR offer dataset to impute the donor’s DR antigens. Because weobserve all patients’ HLA antigens, the number of DR mismatches between a donor and pa-tient is informative about the donor’s antigens. For example, if a donor-patient pair has zeroDR mismatches, the patient’s tissue type limits the donor’s antigens to a few possibilities.40A two DR mismatch pair also restricts the donor’s DR antigens, though less so than a zeromismatch. Since deceased donors whose organs are not transplanted are usually offered tomany patients, we can combine information across all offers to make an educated guess ofthe donor’s DR antigens.

We use the following imputation algorithm. For each donor without DR antigen information,we take all of the donor’s offers in the PTR data. Based on these offers, the recordednumber of DR mismatches, and the patient’s DR antigens and listed unacceptable antigens,we calculate a score for each DR antigen that the donor might have. We dock one point froma DR antigen’s score for each PTR offer it contradicts in the following cases:

• The offer has zero DR mismatches, and the antigen is not equivalent to one of thepatient’s DR antigens

• The offer has two DR mismatches, and the antigen is equivalent to one of the patient’sDR antigens

• The antigen was listed as unacceptable by the patient

For each donor, we take the two DR antigens with the highest scores. Ties are broken infavor of the antigens that appear most frequently among donors for whom DR antigens arerecorded.

40In the zero DR mismatch case, the donor may not share the patient’s exact DR antigens because ofHLA equivalences. Some distinct HLA proteins are equivalent in the sense that a patient with one DRantigen may desensitize it to several DR antigens. UNOS publishes HLA equivalence tables for measuringHLA mismatches, and a separate table for equivalent unacceptable antigens. Furthermore, even ignoringequivalences, a zero DR mismatch donor could be homozygous at the DR locus.

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B Estimation

B.1 Normalization

In this section, we show that the model described in Section 3.3.2 yields the same decision-rules as a model in which Oi(t) = 0 for all t (therefore, di (t) + δi (t)Di (t) = 0 for all t)and the net present value of a transplant is Γij(t) = Γij (t) − Oi (t) . To do this, the nextproposition establishes two results which hold for any assignment rule that does not affectthe value of being on dialysis or departing without a deceased donor transplant.

Proposition 1. Let pij (t) be the conditional probability that i is assigned object j given thatj arrives in period t. Let Vi (t; p) be i’s value of the assignment rule p. Then,

1. For any assignment rule p, Γij (t)−Vi (t; p) = Γij (t)−Vi (t; p), where Γij (t) = Γij (t)−Oi (t)and

(ρ+ δi (t)) Vi (t; p) = λ∫pij (t)

(Γij (t)− Vi (t; p)

)dF + Vi (t; p)

with boundary condition Vi (Ti; p) = 0.

2. For any two assignment rules p and p′, Vi (t; p)− Vi (t; p′) = Vi (t; p)− Vi (t; p′)

Proof. Part 1: First, we verify that Vi (t; p) = Vi (t; p)−Oi (t) satisfies the differential equationabove. Note that

(ρ+ δi (t))Vi (t; p) = di (t) + δi (t)Di (t) + λ∫pij (t) (Γij (t)− Vi (t; p)) dF + Vi (t; p) .

Therefore,

(ρ+ δi (t)) (Vi (t; p)−Oi (t)) = λ∫pij (t)

(Γij (t)− (Vi (t; p)−Oi (t))

)dF+ ∂

∂t(Vi (t; p)−Oi (t)) .

Hence, Vi (t; p) satisfies the necessary differential equation. It is straightforward to checkthat Vi (Ti; p) = Oi (Ti) = Di (Ti) showing that the solution with the boundary conditionVi (Ti; p) = 0 satisfies the requirements of the proposition.

Part 2: Observe that for any pair of assignment rules, p and p′,

Vi (t; p)− Vi (t; p′) = λ∫pij (t) (Γij (t)− Vi (t; p)) dF + Vi (t; p)− Vi (t; p′)

= λ∫pij (t)

((Γij (t)−Oi (t)

)−(Vi (t; p)−Oi (t)

))dF

+( ˙Vi (t; p)− Oi (t)

)−( ˙Vi (t; p′)− Oi (t)

)= Vi (t; p)− Vi (t; p′) .

81

Refer to the model with Oi (t) = 0 for all t and the related value function Vi (t; p) andthe payoffs Γij (t) as the normalized model. Part 1 shows that the normalized model alsoyields Γij (t) − Vi (t; p) as the difference in the value of acceping j relative to the valuewaiting if one expects assignments according to p. In particular, the result holds for pij (t) =πij (t) 1 Γij (t)− Vi (t) > 0 . Therefore, the normalized model yields an identical choice ruleand value function relative to no assignment.

Part 2 shows that the normalized model yields an identical difference in value functionsbetween any two assignment rules as the original model. This is useful for evaluating bothalternative mechanisms as well as for solving for equilibria. To see this, consider any actionspace At and strategy σi (t; j) → At. The action space need not be binary and may dependon time. For example, to consider a counterfactual with multiple wait-lists we can definethe action set as the choice of list at birth and an accept/reject decision thereafter. As thenotation indicates, each action can depend on the currently offered object, if any. As longas the analyst can then evaluate the assignment rule pij (t;σ) as a function of the strategyprofile σ = (σi, σ−i), the result says that the normalized model can be used to determine thedifference in values. To solve for equilibria, we would need to evaluate deviations (σ′i, σ−i)and compare

Vi (t; p (σ′i, σ−i))− Vi (t; p (σi, σ−i)) = Vi (t; p (σ′i, σ−i))− Vi (t; p (σi, σ−i)) .

To identify the value function relative to the current mechanism, we would need to compute

Vi (t; p (σ∗))− Vi (t; p) = Vi (t; p (σ∗))− Vi (t; p) ,

where p denotes the assignment probabilities under the factual mechanism and p (σ∗) denotesthe equilibrium assignment probabilities in an equilibrium of the counterfactual mechanism.

B.2 Details on the Estimator

Gibbs’ Sampler

Define yijt = V (xi, t) − Γ(xi, zj, ηj, t) − εijt = χ(xi, zj, ηj)θ − εijt and aijt = 1yijt < 0 .The sampler is initialized at any value of θ0, σ0

η and guesses for η0j and y0

ijt corresponding toobserved decisions such that y0

ijt ≥ 0 if and only if agent i rejected object j in period t. Wethen sample from the conditional posteriors and draws of y given the previous draws. The

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sampler iterates through the following sequence

ys+1ijt |θs, ηsj ; aijtηs+1j |ys+1

j , θs

θs+1|ys+1, ηs+1

σs+1η |ηs+1, (11)

where the conditioning on the priors and the observables is implicit, ys and ηs are vectorswith components ysijt and ηsj , and ys+1

j is a vector that stacks ys+1ijt across all i,t. The first

two steps involve data augmentation to simplify the sampling problem of the key parametersin the next step. Each of these steps involves draws from a closed-form distribution if theprior distribution on ση is specified as an inverse-Gamma distribution and the prior forθ ∼ N

(θ,Σθ

). With these priors, the first step involves sampling from a truncated normal,

the second and third steps involve sampling from a normal distribution, and the final stepinvolves sampling from an inverse-Gamma.

Computing the Value Function

Given t, for each patient i, the value of continuing is given by equation (7). Using equation(9), the sample analog of the value of continuing is given by

V i (t) = λ∫ Ti

texp (−ρ (τ − t)) p (τ |t;xi) W

(xi, τ ; θ

)dτ.

We numerically approximate this integral. First, we re-write V i (t) as follows:

V i (t) = λ∫ Ti

texp (−ρ (τ − t)) p (τ |t;xi)

1J

J∑j=1

1 cij = 1 1s (τ ;xi, zj) > s∗j

ψ(Pijτ

)dτ

= λ1J

J∑j=1

1 cij = 1∫ Ti

texp (−ρ (τ − t)) p (τ |t;xi) 1

s (τ ;xi, zj) > s∗j

ψ(Pijτ

)dτ

= λ1J

J∑j=1

1 cij = 1∫ Ti

τ ijt

exp (−ρ (τ − t)) p (τ |t;xi)ψ(Pijτ

)dτ,

where τ ijt = infτ > t : s (τ ;xi, zj) > s∗j

, with τ ijt = Ti if s (τ ;xi, zj) < s∗j for all τ ≤ Ti.

For each i and j, we approximate the integral above using B = 40 equally spaced pointsqb = b

B + 1 for b = 1, . . . , B on the unit interval. Let τ bijt = F−1(qb; ρ, τ ijt, Ti

)where

F(·; ρ, τ ijt, Ti

)is the cumulative distribution function of an exponential random variable

with parameter ρ that is truncated between τ ijt and Ti. We therefore compute the value

83

function asVi (t) = λ

ρ

1J

J∑j=1

1 cij = 1 1B

B∑b=1

p(τ bijt|t;xi

)ψ(Pijτbijt

).

This procedure ensures that there are B points of evaluation for each possible donor andpatient-time pair. The numerical performance is superior to an alternative that approximatesthe integral in equation (7) as a sum over a fixed set of draws because some patient, donor,time combinations may have a very small window of availability,

[τ ijt, Ti

].

B.3 Auxiliary Models

Positive Crossmatch Probability

Not all accepted offers result in transplantation. Even after a provisional acceptance, addi-tional testing may yield a positive crossmatch indicating that the patient is likely to developan immune response to the donor’s kidney. These transplants are not carried out, and ifpossible the organ is placed with another patient. To compute patient value functions andconduct counterfactual simulations, we must account for positive crossmatches. We thereforeestimate a probit model to predict the probability that a patient has a positive crossmatchwith an organ they have accepted. The specifications includes interactions between the pa-tient’s CPRA and the number of HLA mismatches with the donor, in addition to controls forpatient age and number of years on dialysis.41 We use a subset of the variables included inthe CCP model to avoid overfitting. Coefficient estimates and standard errors are displayedin Table B.4. Higher CPRA is associated with a higher positive crossmatch probability, asare more tissue-type dissimilarities (as measured by DR or HLA mismatches). In addition,CPRA and tissue type matches interact: high CPRA patients have an additional benefit fromfewer HLA mismatches. This is intuitive: patients with more sensitized immune systems maybe more likely to test positive against foreign antibodies, even if they have not tested positivein the past. Patients who have been on dialysis longer at registration have lower positivecrossmatch probabilities, but this pattern reverses above five years. This pattern may bedriven by blood transfusions to treat anemia, which is common amongst patients with kid-ney failure. Similarly, positive crossmatch probability increases with age for young patients,but then declines with age for older patients.

Maximum Number of Offers and Discards

Our data contain cases in which at least one of a donor’s organs is not offered to all compatiblepatients in NYRT. This usually occurs for two reasons. First, the organ may become unsuit-able for transplantation if it remains outside donor’s body for too long. Second, the organ

41We assume that test results are not manipulated. Therefore, crossmatch probabilities are identical incounterfactual mechanisms. Conversations with transplant surgeons support this assumption.

84

may be accepted by a patient in another OPO. We call these events “timeouts.” Timeoutsare driven by a combination of unobserved factors, including whether the organ remained inthe donor’s body during the offer process; the rate at which offers were made, which dependson patient/surgeon response times and the number of patients simultaneously contacted;the kidney’s rate of physical deterioration once outside the body; and decisions of patientsoutside NYRT.

We model the maximum number of offers that can be made for a given organ using a censoredexponential hazards model. Duration is the number of observed offers that met pre-specifiedscreening criteria. Censoring occurs if the organ is placed, or if it is discarded after beingoffered to all compatible patients. The hazard function is given by

λo (z) = λo exp (zβ) (12)

where z are characteristics of the donor, β is a vector of coefficients, and λ0 is the constantbaseline hazard rate. We allow the timeout hazard to depend on geography and indicatorsof donor quality. Specifically, we control for whether the donor is an expanded criteria donor(ECD); the donor’s cause of death (DCD); and whether the donor was recovered in NYRT,as well as interactions among these variables. The estimated timeout hazards are inputs tothe structural model.

In addition, we model the probability that a donor’s unallocated kidneys are discarded afterthe maximum number of offers has been reached using a probit model. Specifically, thisprobability is given by Φ

(zβ). The model is estimated by tracking whether the kidneys

were ultimately transplanted or not. With the remaining probability, the donor’s kidneysare allocated to a patient not registered in NYRT. This probit model includes the identicalset of covariates used to model the maximum number of offers that can be made. This partof the model does not influence allocation and incentives for patients in NYRT. It is used toproperly account for changes in discards for kidneys not allocated to patients in NYRT.

B.4 Consistency of W(xi, t; θ

)

We now show that W(xi, t; θ

)consistently estimates the quantity

W (xi, t; θ) =∫πij (t)ψ (Pijt) dF

=∫Hzj ,ηj (s (t;xi, Z))P (cij = 1|Z, xi; θ)ψ (xi, Z, η, t; θ) dFZ,η.

To do this, we need to introduce some notation. Define

gj (θ) = pc (zj, xi; θ)ψ (xi, zj, ηj, t; θ) 1s (t;xi, zj) > s∗j

,

85

where pc (zj, xi; θ) = P (cij = 1|zj, xi; θ) . For a vector x = (x1, . . . , xK) , we write |x| =(|x1| , . . . , |xK |) .

Finally, we index objects according to the order in which they arrive in our sample. Therefore,(zj, ηj) denotes the observed and unobserved characteristics of the j-th donor that arrived.Therefore, the data on

(zj, ηj, s

∗j

)generates a sequence of object arrivals.

We make the following assumptions on gj (θ):

Assumption 4. (i) (zj, ηj) is drawn i.i.d. with CDF F

(ii) gj (θ0) is weakly stationary42 with ∑∞k=−∞ γk <∞, where Cov (gj (θ0) , gj−k (θ0)) = γk.

(iii) θJ is√J-consistent, i.e.

(θJ − θ0

)= Op

(J−1/2

)(iv) There exists a function m (z), such that for each zj |g (zj; θ)− g (zj; θ′)| ≤ m (zj)·|θ − θ′|,and m (Z) has finite second moments.

Part (i), in our empirical context, assumes that the characteristics of the donor are drawnindependently each time a donor arrives. Part (ii) assumes that, at θ0, the offers and theirvalues for any given patient type xi, at any given time t, follows a weakly stationary process.That is, the covariance in these values across any two donors falls as they are further apartin the sequence. Given part (i), the only potential source of dependence between gj (θ0)and gk (θ0) is that the characteristics of donor j may affect the state of the waitlist fordonor k because of patient decisions. However, we expect that this dependence to fall asthese donors become further apart in their arrival sequence. Part (iii) assumes that θJ isconsistently estimated at a rate that is at least as fast as the square-root of the number ofdonors. These parameters govern the conditional choice probabilities and the probability ofa crossmatch failure at biological testing. Part (iv) is a regularity condition, assuming thatg (zj; θ) is Lipschitz continuous at each z, and imposes a bound on the second moment of thedistribution of Lipschitz constants. Proposition 2 shows that this property is satisfied undermore primitive conditions stated in Assumption 5.

We now show that for each xi, t, W = 1J

∑Jj=1 gj

(θ)

is a√J-consistent estimator of

W (xi, t; θ0) under this assumption.

Theorem 1. Fix xi, t. If Assumption 4 is satisfied, then∣∣∣∣∣∣W (xi, t; θ0)− 1J

J∑j=1

gj(θ)∣∣∣∣∣∣ = Op

(J−1/2

).

42The process gj is weakly stationary if (i) E [gj ] does not depend on j, and (ii) Cov (gj , gj−k) exists, isfinite and depends only on k, and not j.

86

Proof. For each xi, t, Assumption 4(i) implies that

E [gj (θ0)] = E[pc (zj, xi; θ0)ψ (xi, zj, ηj, t; θ0) 1

s (t;xi, zj) > s∗j

]= E

[pc (zj, xi; θ0)ψ (xi, zj, ηj, t; θ0)E

[1s (t;xi, zj) > s∗j

|zj, ηj

]]= E

[E[pc (zj, xi; θ0)ψ (xi, zj, ηj, t; θ0)Hzj ,ηj (s (t;xi, zj)) |zj, ηj

]]= W (xi, t; θ0) ,

where the equalities are a result of the law of iterated expectations, and the definitions ofHzj ,ηj (s) and W (xi, t; θ0) .

Because W (xi, t; θ0) = E [gj (θ0)], Chebychev’s inequality implies that,

P

J1/2

∣∣∣∣∣∣W (xi, t; θ0)− 1J

J∑j=1

gj (θ0)

∣∣∣∣∣∣ > ε

≤ V ar

(1√J

∑Jj=1 gj (θ0)

)ε2 .

Assumption 4(i) and Proposition 6.8 in Hayashi (2001) imply that

limJ→∞

V ar

1√J

J∑j=1

gj (θ0) = K.

Therefore, we have that

W (xi, t; θ0)− 1J

J∑j=1

gj (θ0) = Op

(J−1/2

). (13)

Lemma 1 implies that

W (xi, t; θ0)− 1J

J∑j=1

gj(θ)

= Op

(J−1/2

),

as requirements (i) and (ii) of Lemma 1 are part of Assumption 4(iv), requirement (iii) isequivalent to Assumption 4(iii), and requirement (iv) is proved in equation (13).

Lemma 1. Fix xi, t. Suppose that (i) g (zj; θ) is Lipschitz continuous for each zj with theLipschitz constant m (zj) ∈ RK i.e. |g (zj; θ)− g (zj; θ0)| ≤ m (zj) · |θ − θ0| , (ii) m (zj) hasfinite second moments, (iii)

∣∣∣θJ − θ0

∣∣∣ = Op

(J−1/2

), and (iv) 1

J

∑j g (zj; θ0)− E [g (zj; θ0)] =

Op

(J−1/2

). Then

1J

∑j

g(zj; θJ

)− E [g (zj; θ0)] = Op

(J−1/2

).

87

Proof. Because g (zj; θ) is Lipschitz continuous in θ, we have that for any θ ∈ Θ,

V ar (|g (zj; θ)− g (zj; θ0)|) ≤ |θ − θ0|T E(m (zj)mT (zj)

)|θ − θ0| ,

where xT is the transpose of the vector x. Therefore, because∣∣∣θJ − θ0

∣∣∣ = Op

(J−1/2

), with

probability approaching 1,

V ar(g(zj; θJ

)− g (zj; θ0)

)≤ VMKJ

−1,

for some finite constant K > 0, where VM = E(∑

k,k′mk (zj)mk′ (zj))and mk (zj) is the k-th

component of m (zj) . By the covariance inequality, with probability approaching 1,

Cov(g(zj; θJ

)− g (zj; θ0) , g

(zk; θJ

)− g (zk; θ0)

)≤ VMKJ

−1.

Therefore, with probability approaching 1, for all j, k ∈ 1, . . . , J,

V ar

1√J

∑j

g(zj; θJ

)− 1√

J

∑j

g (zj; θ0)

= 1J

J∑k=1

J∑j=1

Cov(g(zj; θJ

)− g (zj; θ0) , g

(zk; θJ

)− g (zk; θ0)

)≤ VMK.

By Chebychev’s inequality,

P

√J∣∣∣∣∣∣ 1J∑j

g(zj; θJ

)− E

[g(zj; θJ

)]− 1J

∑j

g (zj; θ0) + E [g (zj; θ0)]

∣∣∣∣∣∣ > ε

≤ VMK

ε2 .

Therefore,

AJ = 1J

∑j

g(zj; θJ

)− E

[g(zj; θJ

)]− 1J

∑j

g (zj; θ0) + E [g (zj; θ0)] = Op

(J−1/2

).

But, we know that 1J

∑j g (zj; θ0)− E [g (zj; θ0)] = Op

(J−1/2

). So, it must be that

1J

∑j

g(zj; θJ

)− E

[g(zj; θJ

)]= Op

(J−1/2

). (14)

Lipschitz continuity of g (zj; θ) and the Cauchy-Schwarz inequality imply that.

E[∣∣∣g (zj; θJ)− g (zj; θ0)

∣∣∣] ≤ E[m (zj) ·

∣∣∣θJ − θ0

∣∣∣]≤√VM

√E[∣∣∣θJ − θ0

∣∣∣ · ∣∣∣θJ − θ0

∣∣∣].

88

Because θJ belongs to the compact set Θ, and θJ−θ0 = Op

(J−1/2

), we have that

√E[∣∣∣θJ − θ0

∣∣∣ · ∣∣∣θJ − θ0

∣∣∣] =

Op

(J−1/2

). Together with the assumption that VM = E

(∑k,k′mk (zj)mk′ (zj)

)is finite, we

have thatE[∣∣∣g (zj; θJ)− g (zj; θ0)

∣∣∣] = Op

(J−1/2

). (15)

Finally, equation (14) and (15) together imply that

1J

∑j

g(zj; θJ

)− E

[g(zj; θJ

)]= 1J

∑j

g(zj; θJ

)− E [g (zj; θ0)] + E [g (zj; θ0)]− E

[g(zj; θJ

)]≤ 1J

∑j

g(zj; θJ

)− E [g (zj; θ0)] + E

[∣∣∣g (zj; θ0)− g(zj; θJ

)∣∣∣]=Op

(J−1/2

).

Lipschitz continuity of g (z; θ)

We now show primitive regularity conditions under which Assumption 4(iv) is satisfied.Recall that gj (θ) = P (cij = 1|zj, xi; θ)ψ (xi, zj, ηj, t; θ) 1

s (t;xi, zj) > s∗j

. Fix xi, t and

omit it from the notation for simplicity.

Assumption 5. (i) ψ (zj, ηj; θ) = E [max 0, χ (zj) · θ + ηj + ε] where ε has cdf Fε(ii) There exists a function m (z), such that m (Z) has finite fourth moments and for all θ, θ′,

a. |P (cij = 1|zj; θ)− P (cij = 1|zj; θ′)| ≤ m (zj) · |θ − θ′|b. |χ (z)| ≤ m (z) .

Part (i) follows from the definition of ψ (zj, ηj; θ) and the parametrization of the conditionalchoice probabilities in our model. It is repeated simply to keep this exercise self-contained.Part (ii) is a regularity condition that assumes lipschitz continuity of primitives, with suffi-ciently small lipschitz constants.

Proposition 2. If Assumption 5 is satisfied, then for all θ, θ′ ∈ Θ, |gj (θ)− gj (θ)| ≤ m (zj) ·|θ − θ′| , where m (zj) has finite second moments.

Proof. First, we show that for all θ, θ′ ∈ Θ |ψ (zj, ηj; θ)− ψ (zj, ηj; θ′)| ≤ m (zj) · |θ − θ′|.Define γ (x) = E [max 0, x+ ε] =

∫∞−x (x+ ε) dFε. Libniz’s rule implies that

γ′ (x) =∫ ∞−x−η

1dFε = 1− F (−x− η) ≤ 1.

89

Therefore, γ (x) is Lipschitz continuous with constant 1. Hence, Assumption 5(i) and (ii)b.,and Lemma 3 imply that

|ψ (zj, ηj; θ)− ψ (zj, ηj; θ′)| = |γ (χ (zj) θ + ηj)− γ (χ (zj) θ′ + ηj)|≤ |χ (zj) (θ − θ′)|≤ |χ (zj)| · |θ − θ′| ≤ m (zj) · |θ − θ′| , (16)

where the first inequality is due to the fact that γ (·) is Lipschitz continuous with constant 1.

Equation (16), Assumption 5(ii)a and Lemma 2 imply that |gj (θ)− gj (θ)| ≤ 2 (m (z) ∗m (z))·|θ − θ′|, where ∗ is the hadamard (or component-wise) product. Assumption 5(ii) implies thatm (z) = 2 (m (z) ∗m (z)) has finite second moments.

Lemma 2. Suppose that there exists a function m (z) such that (i) χ (z) ≤ m (z) and|χ (z, θ)− χ (z, θ′)| ≤ m (z) · |θ − θ′| , and (ii) supθ |χ (z, θ)| ≤ m (z) and supθ |χ (z, θ)| ≤m (z). Then,

|χ (z, θ)χ (z, θ)− χ (z, θ′)χ (z, θ′)| ≤ 2 (m (z) ∗m (z)) · |θ − θ′| ,

where ∗ is the hadamard (or component-wise) product.

Proof. Suppose that |χ (z, θ)− χ (z, θ′)| ≤ m (z) · |θ − θ′| and |χ (z, θ)− χ (z, θ′)| ≤ m (z) ·|θ − θ′| . Consider

|χ (z, θ)χ (z, θ)− χ (z, θ′)χ (z, θ′)|= |χ (z, θ)χ (z, θ)− χ (z, θ)χ (z, θ′) + χ (z, θ)χ (z, θ′)− χ (z, θ′)χ (z, θ′)|≤ |χ (z, θ) (χ (z, θ)− χ (z, θ′))|+ |(χ (z, θ)− χ (z, θ′))χ (z, θ′)|≤ |χ (z, θ)|m (z) · |θ − θ′|+ |χ (z, θ′)|m (z) · |θ − θ′|≤2 (m (z) ∗m (z)) · |θ − θ′| .

Lemma 3. Suppose that (i) ψ : R → R is Lipschitz continuous with constant K < ∞,and (ii) for each z, there exists m (z) ∈ RKθ such that |χ (z, θ)− χ (z, θ′)| ≤ m (z) · |θ − θ′|,then |ψ (χ (z, θ))− ψ (χ (z, θ′))| ≤ Km (z) · |θ − θ′| . In particular, if χ (z, θ) = χ (z) · θ whereχ (z) , θ ∈ RKθ , then |ψ (χ (z, θ))− ψ (χ (z, θ′))| ≤ K |χ (z)| · |θ − θ′| .

Proof. The first part follows definitionally because Lipschitz continuity of ψ implies that

|ψ (χ (z, θ))− ψ (χ (z, θ′))| ≤ K |χ (z, θ)− χ (z, θ′)| ≤ Km (z) · |θ − θ′| .

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For the second part, note that

χ (z) · (θ − θ′) ≤ |χ (z)| · |θ − θ′| .

C Equilibria: Existence and Approximations

C.1 Approximations in Long Queues

C.1.1 Queue Length

In this section, we show that the stationary distribution of the length of the waitlist concen-trates mass around the steady state value, N∗. This result suggests that using a deterministicqueue length is a good approximation for large values of N∗.

Theorem 2. Consider the continuous time Markov process Nτ , where τ denotes calendartime. For any given (π∗,m∗, σ∗), Nτ is an ergodic process with stationary distribution pn.Moreover, for any ε > 0,

limτ→∞

P (|Nτ −N∗| > εN∗) ≤ 2 exp(− N∗ε2

2 (1 + ε)

),

whereN∗ =

∫γxdFX∫ ∫

m∗x (t)κx (t) dtdFX.

Proof. Define κ =∫ ∫

m∗x (t)κx (t) dtdFX and γ =∫γxdFX . Observe that Nτ follows a birth-

death process with birth-rate γ and death rate Nτκ. Therefore, Nτ is an ergodic Markovprocess. The stationary distribution, pn, satisfies the detail balance conditions:

γp0 = κp1

(γ + nκ) pn = γpn−1 + (n+ 1)κpn+1 for n ≥ 1,

where the left-hand side denotes the exit rate from state n and the right hand side is theentry rate. Solving the system recursively and using the fact that ∑n pn = 1 yields

p0 =[1 +

∑ γn

n!κn]−1

= exp(−γκ

).

91

andpn = γn

n!κn exp(−γκ

).

Note that pn is the probability that a random variable N following a Poisson distributionwith parameter N∗ = γ

κtakes on the value n. Theorem 1, part 3 in Canone (2017) implies

the result.

C.1.2 Offer Probabilities

This section derives a computationally tractable approximation to offer probabilities given ascoring rule s, a large waitlist N∗ and an acceptance policy function.

In what follows, we fix a particular agent i with priority score s. Ties are broken randomly,and it is therefore without loss of generality to consider each agent’s tiebreaker to be drawnfrom a uniform distribution on the unit interval. Let 1− α be the tie-breaker for agent i.

There are two reasons why an offer for a kidney may be the last offer that can be made.The first reason is that the agent who received the offer accepts it. The second one is thatthe kidney expires after the offer is made. The first of these is governed by the acceptancebehavior of agents and the priority rule. The second is governed by the probability that theagent’s position on the list exceeds the maximum number of possible offers for the kidney.This model, specified in equation (12), is equivalent to an exponential hazards model withequal probability, p0 = λo (z) for an object of type z, of failure before the next offer is made.We fix z in what follows and drop if from the notation for simplicity.

An agent receives an offer if the total number of acceptances and expirations after offers toagents with a higher priority score than agent i is strictly less than the number of copies ofthe object available. We consider the probability of receiving an offer by considering waitliststhat are composed of N agents drawn randomly drawn with distribution governed by m.That is, we draw N agents sequentially and track the number of kidney acceptances andfailures. For each agent drawn from m, the probability that this agent is ordered above i andthat the kidney is either accepted by the agent or expires is:

p (s, α) = mH (s) pH (s) +mE (s)αpE (s) .

The first term represents the case when an agent with a higher priority (group H) is drawn.The probability of the kidney becoming unavailable conditional on an agent drawn from ahigher priority group is:

pH (s) = p0 + (1− p0) 1mH (s)

∑t,x

m (t;x) 1 s (t;x) > sP (Γ (t;x) + ε > Vx (t)) .

The second term represents the probability that an agent with priority score s is drawn. The

92

term pE (s) is defined analagously as pH (s).

The number of times that an kidney becomes unavailable after being offered to an agentordered above i is a binomial random variable X with parameters N and p (s, α). In thisnotation, an object is available to agent i if X < q, where q is the total number of copies ofthe object. Hence, the probability that i receives an offer is given by∫ 1

0P (X < q|s, α) dα, (17)

where we have integrated over the tie-breaker α, and explicit conditioning on N is subsumedfor simplicity.

For large N and small p (s, α), the distribution of X approaches the distribution of a Poissonrandom variable with parameter Np (s, α). Therefore, the expression in equation (17) yieldsthe following expression for πx (t):

πx (t) =∫ 1

0

∑q′<q

e−Np(s,α) (Np (s, α))q′

q′! dα,

where we use the Poisson approximation to re-write P (X < q|s, α). As a reminder, theobject type z is dropped from the notation for simplicity as it is fixed, although the offerprobabilities depend on it. This integral can be solved for in closed form. If q ∈ 1, 2 ,

πx (t) = e−Np(s,0) − e−Np(s,1)

N (p (s, 1)− p (s, 0)) + 1 q = 2 (1 +Np (s, 0))e−Np(s,0) − (1 +Np (s, 1))e−Np(s,1)

N (p (s, 1)− p (s, 0)) ,

(18)

C.2 Existence of Steady-State Equilibria

In this section, we prove that a steady-state equilibrium exists for sequential offer mechanismsthat use a scoring rule. Throughout, we assume that χ and ζ are finite sets, and that T isfinite. This assumption greatly simplifies the technical argument. Moreover, they are alsoused when computing steady-state equilibria.

We make the following regularly assumptions on the primitive objects:

Assumption 6. (i) The exogenous arrival and departure rates λ and γx are finite

(ii) The exogenous departure rate δ (τ ;x) is bounded below by δ > 0 and bounded above,unifomly for t ∈ [0, T ) and all x ∈ χ

(iii) The conditional probability density function fΓ|t,x,z exists, and is uniformly bounded

(iv) The conditional moment E [|Γ| |τ, x, z] =∫|Γ| dFΓ|τ,x,Z is uniformly bounded in t, x, z

93

(v) The family of functions g(t;x, z, Γ

)= FΓ|t,x,z

(Γ)indexed by Γ, x, z is Lipschitz contin-

uous in t with a common constant

(vi) The object arrival rate λ is strictly less than the total agent arrival rate ∑ γx

(vii) The set of scores S = s (t;x, z) : (t, x, z) ∈ [0, T ]× χ× ζ is finite.

Most empirical models will satisfy the continuity and boundedness assumptions above. Thetwo substantive assumptions that are worth discussing are parts (vi) and (vii). Part (vi)assumes that the objects that need to be assigned are scarce. This assumption ensures thatthe queue is unlikely to be empty. Part (vii) imposes a restriction on the mechanisms forwhich we prove existence. The assumption is used to ensure that the set of all functionsπxz (t) is sufficiently small (more precisely, compact). Other assumptions that yield thisconclusion would also suffice.

Our main result proves existence of a steady-state equilibrium.

Theorem 3. Suppose Assumption 6 is satisfied. Then a steady-state equilibrium for a se-quential offer mechanism with a scoring rule exists.

Proof. The proof proceeds by applying the Brower-Schauder-Tychonoff Fixed Point Theorem(Corollary 17.56, Aliprantis and Border, 2006). The proof proceeds in three parts. First, wedefine a set Ω to which the equilibrium objects belong. Second, we define a map A : Ω→ Ω.Finally, we show that A has fixed points, and that the fixed points of A yield steady-stateequilibria.

Part 1, Definition of Ω: The equilibrium objects are defined by five types of functions:

1. The conditional choice probability, given t and the agent and object characteristics xand z. We consider these choice probabilities as a function pσ : [0, T ]× χ× ζ → [0, 1].

2. The value function V : χ × [0, T ] → R+. It is convenient to define this function,although it is somewhat redundant with the choice probabilities above.

3. The offer probabilities π : [0, T ] × χ × ζ → [0, 1] where π (t;x, z) = Hz (sxz (t)) ×P (cij = 1|x, z).

4. The distribution of agent types m : χ× [0, T ]→ R+.

5. The queue length N ∈ R.

We denote the tuple of these objects with ω = (pσ, V, π,m,N) . We endow each of thefunctions in the first four objects with the supremum norm over their respective domains.The norm for ω is denoted ‖ω‖ = ‖pσ‖+ ‖V ‖+ ‖π‖+ ‖m‖+ |N |. Therefore, ω is an elementof a Banach space.

94

We further restrict ω to belong to a subset Ω of this Banach space. Specifically, we restrictthese components as follows:

1. The functions Vx (t) are uniformly bounded by λT supτ,X,z∫|Γ| dFΓ|τ,x,Z , and are Lips-

chitz continuous with a common constant λ supτ,x,z∫|Γ| dFΓ|τ,x,z.

2. The functions pσ (t;x, z) that are uniformly bounded by 1, and Lipschitz continuouswith a common constant K, where

K = λ supτ,x,z

(∫|Γ| dFΓ|τ,x,z sup

ΓfΓ|τ,x,z (Γ)

)+ sup

Γ,x,z,t,t′

∣∣∣FΓ|t,x,z(Γ)− FΓ|t′,x,z

(Γ)∣∣∣ / |t− t′| .

Note that Assumption 6 implies that K is finite.

3. The functions πx,z (t) such that πx,z (t) = πx,z (t′) if sxz (t) = sxz (t′) with range [0, 1].

4. The functions mx (t) are uniformly bounded by T supx γx and are Lipschitz continuouswith a common constant supx,τ γxδ (τ ;x).

5. The term N ∈[N, N

], where N = (∑x γx) infx

∫ T0 exp (−

∫ τ0 δ (τ ;x) dτ) /λ and N =

T∑

xγx

δ. Note that N > 0 because Assumption 6 requires that ∑x γx < λ and δ (τ ;x)

is uniformly bounded above.

Part 2, definition of A : Ω→ Ω: Denote AV [ω] as the V component of A [ω], where ω ∈ Ω.Likewise, define Aπ, Apσ , Am and AN . This map is defined as follows:

AV [ω] (x, t) =∫ T

texp (−ρ (τ − t)) p (τ |t;x)

(λ∫π (τ ;x, Z)

∫max 0,Γ− V (τ ;x) dFΓ|τ,x,ZdFZ

)dτ

Apσ [ω] (x, z, t) =∫

1 Γ ≥ AV [ω] (x, t) dFΓ|x,z,t

Am [ω] (x, t) = γx exp(−∫ t

0δ (τ ;x) + λ

∫π (τ ;x, Z) pσ (τ ;x, z) dFΓ|τ,x,Zdτ

)/N

AN [ω] = maxN,min

T∑x γx∑

x

∫ T0 mx (t)κx (t) dt

, N

Aπ [ω] (x, z, t) = Hz (sxz (t) ;Apσ [ω] , Am [ω] , AN [ω])× P (cij = 1|x, z)

wherep (τ |t;x) = exp

(−∫ τ

tδ (τ ′;x) dτ ′

)is the probability that agent of type x departs the list prior to τ conditional on being on thelist at t. It is clear that A is well defined. To ensure that the image is a subset of Ω, we needto show that A [ω] ∈ Ω for all ω ∈ Ω. We do this for each of the components separately:

95

1. AV : Since exp (−ρ (τ − t)), p (τ |t;x) and π (τ ;x, Z) are in [0, 1], and∫max 0,Γ− V (τ ;x) dFΓ|τ,x,Z ≤

∫|Γ| dFΓ|τ,x,Z ,

we have that AV [ω] is uniformly bounded by λT supτ,x,z∫|Γ| dFΓ|τ,x,z. Further, for any

t, t′ ∈ [0, T ], with t < t′, we have that

|AV [ω] (t)− AV [ω] (t′)|

=∣∣∣∣∣∫ t′

texp (−ρ (τ − t)) p (τ |t;x)

(λ∫π (τ ;x, Z)

∫max 0,Γ− V (τ ;x) dFΓ|τ,x,ZdFZ

)dτ∣∣∣∣∣

≤ λ |t′ − t| supτ,x,z

∫|Γ| dFΓ|τ,x,z.

Therefore, AV [ω] satisfies the necessary restrictions.

2. Apσ : Note that is Apσ [ω] uniformly bounded by 1. Moreover, for any x and z, andt, t′ ∈ [0, T ], we have that

|Apσ [ω] (t, x, z)− Apσ [ω] (t′, x, z)|

=∣∣∣∣∫ 1 Γ ≥ AV [ω] (x, t) dFΓ|x,z,t −

∫1 Γ ≥ AV [ω] (x, t′) dFΓ|x,z,t′

∣∣∣∣=∣∣∣∣∫ (1 Γ ≥ AV [ω] (x, t) − 1 Γ ≥ AV [ω] (x, t′)) dFΓ|x,z,t

∣∣∣∣+∣∣∣∣∫ 1 Γ ≥ AV [ω] (x, t′) d

(FΓ|x,z,t − FΓ|x,z,t′

)∣∣∣∣≤∣∣∣∣∣∫ maxAV [ω](x,t),AV [ω](x,t′)

minAV [ω](x,t),AV [ω](x,t′)1dFΓ|x,z,t

∣∣∣∣∣+∣∣∣FΓ|x,z,t′ (AV [ω] (x, t′))− FΓ|x,z,t (AV [ω] (x, t′))

∣∣∣≤λ |t′ − t| sup

τ,x,z

(∫|Γ| dFΓ|τ,x,z sup

ΓfΓ|τ,x,z (Γ)

)+ sup

Γ,x,z

(∣∣∣FΓ|t,x,z(Γ)− FΓ|t′,x,z

(Γ)∣∣∣ / |t− t′|) |t− t′|

≤[λ supτ,x,z

(∫|Γ| dFΓ|τ,x,z sup

ΓfΓ|τ,x,z (Γ)

)+ sup

Γ,x,z,t,t′

(∣∣∣FΓ|t,x,z(Γ)− FΓ|t′,x,z

(Γ)∣∣∣ / |t− t′|)] |t− t′|

Therefore, Apσ [ω] satisfies the necessary restrictions.

3. Aπ : Observe that Aπ [ω] (x, z, t) ∈ [0, 1] and Aπ [ω] (x, z, t) = Aπ [ω] (x, z, t′) if sxz (t) =sxz (t′) by construction.

4. Am : Since exp(−∫ t

0 δ (τ ;x) + λ∫π (τ ;x, Z) pσ (τ ;x, z) dFΓ|τ,x,Zdτ

)≤ 1 and N ≥ 1, we

have that Am [ω] is uniformly bounded by supx γx. Further, for any t, t′ ∈ [0, T ], with

96

t < t′, we have that

|Am [ω] (t)− Am [ω] (t′)|

≤γx exp(−∫ t′

tδ (τ ;x) + λ

∫π (τ ;x, Z) pσ (τ ;x, Z) dFZdτ

)

≤γx exp(−∫ t′

tδ (τ ;x) dτ

)≤ |t′ − t| sup

x,τγxδ (τ ;x) .

Therefore, Am [ω] satisfies the necessary restrictions.

5. AN : By construction, AN [ω] belongs to[1, T

∑xγx

δ

], satisfying the necessary restric-

tions.

Part 3, existence of equilibria: It is straightforward to verify that Ω is convex. Lemma4 imply that the components ΩV , Ωm and Ωpσ are compact sets. Lemma 5 shows that Ωπ

is compact. Assumption 6 (i), (ii) and (vi) imply that N > 0 and N is finite, implying thatΩN is compact. Therefore, Ω is compact because it is the cartesian product of compact sets.Lemma 6 shows that A is a continous map. Therefore, by the Brower-Schauder-TychonoffTheorem (Corollary 17.56, Aliprantis and Border, 2006) implies that there exists ω∗ ∈ Ωsuch that A [ω∗] = ω∗.

To complete the proof, we show that any fixed point ω∗ = (p∗σ, V ∗, π∗,m∗, N∗) correspondsto a steady-state equilibrium. Observe that for each x,

V ∗ (t;x) =∫ T

texp (−ρ (τ − t)) p (τ |t;x)

(λ∫π∗ (τ ;x, Z)

∫max 0,Γ− V ∗ (τ ;x) dFΓ|τ,x,ZdFZ

)dτ.

Therefore, V ∗ (t;x) is the value of declining an offer and following the optimal strategy giventhe offer rate π∗. Therefore,

p∗σ (x, z, t) = Apσ [ω∗] (x, z, t) =∫

1 Γ ≥ V ∗ (t;x) dFΓ|x,z,t.

For each x, z, t, the quantity F−1Γ|x,z,t (p∗σ (x, z, t)) = V ∗ (t;x). Therefore,

σ∗ (Γ, t) = 1

Γ ≥ F−1Γ|x,z,t (p∗σ (x, z, t))

is an optimal strategy, satisfying requirement 1 in Definition 1.

By construction, π∗ (x, z, t) = Aπ [ω∗] (x, z, t) = Hz (sxz (t) ; p∗σ,m∗, N∗)× P (cij = 1|x, z) sat-isfies requirement 2 of Definition 1 because p∗σ equals the acceptance probability of a type zobject by an agent of type x at time t.

97

Finally, m∗ = Am [ω∗] and N∗ = AN [ω∗] together satisfy requirement 3 in Definition 1. Therestriction of AN [ω∗] to

[N, N

]cannot strictly bind because N and N denote the smallest

and largest possible queue lengths given the exogenous arrival and departure rates.

C.3 Lemmata

Lemma 4. Suppose X ⊂ L∞ ([a, b]) is the set of all functions on the bounded interval [a, b]that are uniformly bounded by K1 and have a common Lipschitz constant K2. Then X iscompact.

Proof. Note that the set of functions X is uniformly equicontinuous. By the Arzela-Ascolitheorem, any sequence of functions xn ∈ X has a uniformly convergent subsequence xnk .Let the limit of this sequence by x∗, i.e. for each t, x∗ (t) = limk→∞ xnk (t). Therefore,supt |x∗ (t)| ≤ limk→∞ supt |xnk (t)| ≤ K1. Similarly, |x∗ (t)− x∗ (t′)| = limk→∞ |xnk (t)− xnk (t′)| ≤K2 |t− t′| . Hence, x∗ ∈ X. Consquently, we have that X is sequentially compact, which isequivalent to X being compact.

Lemma 5. Assumption 6(vii) implies that the set Ωπ consisting of functions π : [0, T ]×χ×ζ → [0, 1] endowed with the supremum norm such that πxz (t) = πxz (t′) if sxz (t) = sxz (t′) iscompact.

Proof. Assumption 6(vii) and finiteness of χ and ζ imply that Ωπ is finite dimensional.Further, Ωπ is closed and bounded by definition. By the Heine-Borel theorem, Ωπ is compact.

Lemma 6. Suppose Assumption 6 is satisfied. Then the map A : Ω→ Ω is continuous.

Proof. We do this for each component of A separately.

AV : Let Ω0 be an arbitrary subset of Ω. Consider ω ∈ Ω0, where Ω0 is the closure of Ω0.Since ω ∈ Ω0, there exists a sequence ωn ∈ Ω0 such that ‖ωn − ω‖ = εn → 0. Denote

98

Vn = AV [ωn] and drop x from the notation as it belongs to a finite set. Now, consider∣∣∣Vn (t)− V (t)∣∣∣

=∣∣∣∣∣∫ T

texp (−ρ (τ − t)) p (τ |t)λ

(∫πn (τ ;Z)

∫max 0,Γ− Vn (τ) dFΓ|τ,ZdFZ

)dτ

−∫ T

texp (−ρ (τ − t)) p (τ |t)λ

(∫π (τ ;Z)

∫max 0,Γ− V (τ) dFΓ|τ,ZdFZ

)dτ∣∣∣∣∣

≤Tλ supt,z

∣∣∣∣πn (t; z)∫

max 0,Γ− Vn (t) dFΓ|t,z − π (t; z)∫

max 0,Γ− V (t) dFΓ|t,z

∣∣∣∣≤Tλ sup

t,z

∣∣∣∣πn (t; z)∫|max 0,Γ− Vn (t) −max 0,Γ− V (t)| dFΓ|t,z

∣∣∣∣+ Tλ sup

t,z

∣∣∣∣|πn (t; z)− π (τ ; z)|∫

max 0,Γ− V (t) dFΓ|t,z

∣∣∣∣≤Tλ sup

t,z|Vn (t)− V (t)|+ Tλ sup

t,z

∫|Γ| dFΓ|t,z sup

t,z|πn (t; z)− π (τ ; z)|

≤Tλ(

1 + supt,z

∫|Γ| dFΓ|t,z

)εn.

Since εn → 0, Assumption 6(i) and (iv) imply that the right hand side converges to zero.Therefore, AV

[Ω0]⊂ AV [Ω0], implying that AV is continuous (Theorem 2.27, Aliprantis and

Border, 2006).

Apσ : Continuity follows by noting that AV is continuous in the sup-norm, and FΓ|t,x,z isabsolutely continuous with respect to Lebesgue measure for each t, x, z (Assumption 6(iii)).

Am : It is sufficient to fix x because χ is a finite set. Lemma 7 imply that the map definedby Aκ [ω] (t) = δ (t;x) + λ

∫π (t;x, Z) pσ (t;x) dFZ is continuous. Moreover, suptAκ [ω] (t) is

bounded above (Assumption 6(i)). Therefore, Aκ∗ [ω] (t) = −∫ t

0 δ (τ ;x)+λ∫π (τ ;x, Z)σx (Γ, t) dFZdτ

defines a continuous map from Ω to C ([0, T ]) . Because a composition of continuous func-tions is continuous, and g (a) = γx exp (a) /N is continuous for all N > 0, we have that Amis continuous.

AN : First we show that AN [ωn] is continuous. Lemma 7 implies that the map Aκ [ω] (t) =δ (t;x) + λ

∫π (t;x, Z) pσn (t;x, Z) dFZ is continuous for each x. A similar argument implies

that Aκ [ω] = ∑x

∫ T0 mx (t)κx (t) dt is continuous because mx (t) is bounded by γx. Further,

Aκ [ω] ∈ [δ,∞] since δ (t;x) is uniformly bounded below by δ (Assumption 6(ii)). Since thecomposition of real-valued continuous functions is continuous, and the reciprocal function iscontinuous for all arguments other than 0, we have that AN is a continuous map.

Aπ: Denote A [ω] = (Apσ [ω] , Am [ω] , AN [ω]) . We have shown that A is continuous andcompact. Note that for any sequence ωn,

supx,z,t|Aπ [ωn] (x, z, t)| ≤ sup

x,z,t

∣∣∣Hz

(sxz (t) ; A [ωn]

)∣∣∣ ≤ supz,s

∣∣∣Hz

(s; A [ωn]

)∣∣∣ ,

99

where the first inequality follows from the fact that P (cij = 1|x, z) ∈ [0, 1] and the secondinequality follows from set inclusion. Therefore, Lemma 8 and continuity of A implies thatfor each z, sups

∣∣∣Hz

(s; A [ωn]

)−Hz

(s; A [ω]

)∣∣∣→ 0 if ωn converges to ω. Since z belongs toa finite set, we therefore have that supx,z,t |Aπ [ωn] (x, z, t)− Aπ [ω] (x, z, t)| → 0. Hence, Aπis a continuous map. Finally, Lemma 8 and the fact that A [Ω] is compact imply that theimage of Aπ is compact.

Lemma 7. Fix x. The map Aκ : Ω→ L∞ ([0, T ]), where

Aκ [ω] (t) = δ (t;x) + λ∫π (t;x, Z) pσ (τ ;x, Z) dFZ

is continuous if λ is finite, and π and pσ are uniformly bounded by 1.

Proof. Let Ω0 be an arbitrary subset of Ω. Consider ω ∈ Ω0. Since ω ∈ Ω0, there ex-ists a sequence ωn ∈ Ω0 such that ‖ωn − ω‖ = εn → 0. Now, consider Aκ [ωn] (t) =λ∫πn (t;x, Z) pσn (τ ;x, Z) dFΓ|τ,x,Z , where we endow the range with the supremum norm.

‖Aκ [ωn]− Aκ [ω]‖ = λ∥∥∥∥∫ πn (t;x, Z) pσn (t;x, Z) dFZ −

∫π (t;x, Z) pσ (t;x, Z) dFZ

∥∥∥∥≤ λ sup

z,t|πn (t;x, z) pσn (t;x, z)− π (t;x, z) pσ (t;x, z)|

≤ λ supz,t|πn (t;x, z) (pσn (t;x, z)− pσ (t;x, z))|

+ λ supz,t|(πn (t;x, z)− π (t;x, z)) pσ (t;x, z)|

≤ λ supz,t|pσn (t;x, z)− pσ (t;x, z)|+ λ sup

z,t|πn (t;x, z)− π (t;x, z)|

≤ 2λεn.

Therefore, Aκ[Ω0]⊂ Aκ [Ω0], implying that Aκ is continuous (Theorem 2.27, Aliprantis and

Border, 2006).

Lemma 8. Fix z. The map AH : Ω → L∞ (R) defined by AH [ω] (s) = Hz (s; pσ,m,N) iscontinuous.

Proof. We omit z from the notation for simplicity as it is fixed. Equation (18) derives thefollowing expression for AH :

AH [ω] (t, x, z) =∫ 1

0

∑q′<q

e−Np(s,α) (Np (s, α))q′

q′! dα,

wherep (s, α) = mH (s) pH (s) +mE (s)αpE (s) ,

100

pH (s) = p0 + (1− p0) 1mH (s)

∑t,x

m (t;x) 1 s (t;x) > s pσ (t;x) ,

pE (s) is defined analagously as pH (s), and p0 is the probability of a failure ocurring. Here,we have P (Γ (t;x, z) + ε > Vx (t)) with the acceptance probabilities pσ (t;x, z). Recall that

mH (s) =∑t,x

m (t;x) 1 s (t;x) > s ,

mE (s) =∑t,x

m (t;x) 1 s (t;x) = s .

We are now ready to prove continuity of AH . We do this by first proving continuity of thecomponents mH , mE, pH and pE.

Continuity of mH and mE: Consider a sequence mn that converges in sup norm on x, t to mwe have that

|mn,H (s)−mH (s)| =∑x

∫ T

0|mn (t;x)−m (t;x)| 1 s (t;x) > s dt

≤ |χ|T supx,t|mn (t;x)−m (t;x)| .

Because this bound is independent of s, we have that sups |mn,H (s)−mH (s)| converges tozero. Therefore, AmH : Ω → L∞ (R) defined by AmH [ω] (s) = mH (s) is a continuous mapbecause AmH

(Ω0)

= AmH (Ω0) for any Ω0 ⊆ Ω (Theorem 2.27, Aliprantis and Border, 2006).An identical argument yields that AmE : Ω → L∞ (R) defined by AmE [ω] (s) = mE (s) is acontinuous map. We do this by first proving continuity of the various components.

Continuity of pH and pE: We show the argument only for pH because the argument for pEis identical. Consider a sequence of ωn that converges to ω, and the map ApH : Ω→ L∞ (R)defined by ApH [ω] (s) = p0 + (1− p0) 1

mH (s)∑t,xm (t;x) 1 s (t;x) > s pσ (t;x). Since p0 is

fixed, we only need to show continuity of the map from ω to 1mH (s)

∑t,xm (t;x) 1 s (t;x) > s pσ (t;x).

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For each s, we have that∣∣∣∣∣∣ 1mn,H (s)

∑t,x

mn (t;x) 1 s (t;x) > s pn,σ (t;x, z)

− 1mH (s)

∑t,x

m (t;x) 1 s (t;x) > s pσ (t;x)

∣∣∣∣∣∣≤

∣∣∣∣∣∣ 1mn,H (s)

∑t,x

mn (t;x) 1 s (t;x) > s pn,σ (t;x)

− 1mH (s)

∑t,x

m (t;x) 1 s (t;x) > s pn,σ (t;x)

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1mH (s)

∑t,x

m (t;x) 1 s (t;x) > s pn,σ (t;x)

− 1mH (s)

∑t,x

m (t;x) 1 s (t;x) > s pσ (t;x)

∣∣∣∣∣∣≤

∣∣∣∣∣∣ 1mn,H (s)

∑t,x

mn (t;x) 1 s (t;x) > s − 1mH (s)

∑t,x

m (t;x) 1 s (t;x) > s

∣∣∣∣∣∣ |pn,σ (t;x)|

+

∣∣∣∣∣∣ 1mH (s)

∑t,x

m (t;x) 1 s (t;x) > s

∣∣∣∣∣∣ |pn,σ (t;x)− pσ (t;x)|

≤

∣∣∣∣∣∣ 1mn,H (s)

∑t,x

mn (t;x) 1 s (t;x) > s − 1mH (s)

∑t,x

m (t;x) 1 s (t;x) > s

∣∣∣∣∣∣+ |pn,σ (t;x)− pσ (t;x)|

= |pn,σ (t;x)− pσ (t;x)|

The first inequality follows fromt he triangle inequality. The third inequality follows from thefact that |pn,σ (t;x)− pσ (t;x)| is bounded by 1 and mH (s) = ∑

t,xm (t;x) 1 s (t;x) > s bydefinition. The final inequality follows from the definition thatmn,H (s) = ∑

t,xmn (t;x) 1 s (t;x) > sand mH (s) = ∑

t,xm (t;x) 1 s (t;x) > s for all s. If ωn converges to ω, we have thatsupt,x |pn,σ (t;x)− pσ (t;x)| conveges to zero. Therefore, we have that sups |ApH [ωn] (s)− ApH [ω] (s)|also converges to zero. Hence, ApH is continuous because ApH

(Ω0)

= ApH (Ω0) for anyΩ0 ⊆ Ω (Theorem 2.27, Aliprantis and Border, 2006).

Continuity of p (s, α): The map ApH : Ω → L∞ (R× [0, 1]) defined by Ap [ω] (s, α) =mH (s) pH (s) +mE (s)αpE (s) is continuous because α is bounded by 1, the maps from ω tomH (s), pH (s), mE (s), pE (s) are continuous, and multiplication and addition are continuous.

Continuity of Aπ: The map from Ω to ∑q′<qe−Np(s,α)(Np(s,α))q

′

q′! is continuous beause the com-ponents are continuous, and the composition and multiplication operators are continuous.

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This term is bounded by 1. Therefore, the integral

∫ 1

0

∑q′<q

e−Np(s,α) (Np (s, α))q′

q′! dα

defines a continous map from Ω to the L∞ ([0, T ]) for each x.

D Computational Details

D.1 Counterfactual Scoring Mechanisms

We compute steady-state equilibria for counterfactual scoring mechanisms using the algo-rithm described in the pseudo-code below. It is based on three key steps:

Value Function Computation (Backwards Induction):

For a small h, the value function derived in equation (4) can be approximated as

(ρ+ δx (t))Vx (t) ≈ λ∫πx (t; z)Emax 0,Γ (t;x, z) + ε− Vx (t) dF + Vx (t+ h)− Vx (t)

h

Because the right-hand side is monotonically decreasing in Vx (t) , there is a unique value ofVx (t) that satisfies the equation. We will use this expression to obtain the value function bybackwards induction. At iteration k, given V k

x (tl+1)we use the bisection method to calculatethe value of v that solves:

(ρ+ δx (tl)) v = λ∫πkx (tl; z)Emax 0,Γ (tl;x, z) + ε− v dF + V k

x (tl+1)− vtl+1 − tl

(19)

Because this problem can be written as finding v = f (v) where f (·) is strictly decreasing, wecan take any initial guess v0and set the lower bound to min (v0, f (v0))and the upper boundto max (v0, f (v0)) . We use the initial guess v0 = V k

x (tl+1).

Offer Probabilities, πx,z (t):

The expression in equation (18) can be simplified and solved for analytically. We use thatsolution in our algorithm.

Waitlist Size/Composition (Forward Simulation), m,N :

We use κx (t) and γx to update the queue composition. Solving the ODE in Definition 1,part 3(a), we get that for any h > 0,

mx (t+ h) = mx (t) exp(−∫ h

0kx (t+ τ) dτ

),

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Algorithm 1 Steady State Equilibrium1: Inputs: Patient and donor characteristics, scoring rule s, parameters Γ, δ, ρ, and patient

age grid t0, . . . , tL = T. Let tl0x be the arrival time for patient of type x.2: Outputs: V ∗, π∗, m∗, N∗3: Initialize k = 0 and beliefs πkx(t) for all x and t ∈ t0, . . . , tL4: repeat5: V k ← Backwards Induction(πk)6: κkx(tl)← δx(tl) + λ

∑z π

kx,z(tl)P(Γ(tl;x, z) + ε > V k

x (tl))7: mk, Nk ← Forward Simulation(κk) . Waitlist Composition8: πk ← Compute Offer Probabilites(V k,mk, Nk) . Offer Probabilities9: k ← k + 110: until k > 1, ‖V k − V k−1‖∞ < ε, ‖mk −mk−1‖∞ < ε, and Nk = Nk−1 . Convergence11: V ∗ ← V k,m∗ ← mk, N∗ ← Nk, π∗ ← πk

12: function Backwards Induction(π)13: for all x do14: Set Vx(T ) = 015: for all x and tl = tL−1 to tl0x do16: Compute Vx(tl) by solving for v in equation (19)17: end for18: end for19: return Vx(tl) for all x and tl ∈ tl0x , . . . , T20: end function21: function Forward Simulation(κ)22: for all x do23: mx(tl0x)← λx24: for all tl = tl0x+1 to T do25: mx(tl+1)← mx(tl) exp(−κx(tl)(tl+1 − tl))26: end for27: end for28: Nk ← ∑

x,tlmkx(tl)κkx(tl) . Waitlist Size: Definition 1, part 3(b)

29: mx(tl)← mx(tl)/Nk for all x and tl30: return mx(tl) for tl ∈ tl0x , . . . , T and Nk

31: end function32: function Compute Offer Probabilities(m,V,N)33: pa(tl;x, z)← P(Γ(tl;x, z) + ε > Vx(tl)) for all x, tl34: for all s = max s(tl;x, z) to min s(tl;x, z) do35: Compute π using equation (18)36: end for37: return πk

38: end function

104

where mx (0) = λx. Appoximating κx (t+ τ) = κx (t+ h) for all τ ∈ (0, h), we have that

mx (tl+1) = mx (tl) exp (−κx (tl+1) (tl+1 − tl)) . (20)

Finally, we scale the output so that mx (tl) is a probability measure.

The size of the waitlist, N , is determined by part 3(b) of Definition 1.

105

D.2 Optimal Assignments and Optimal Offer Rates

The objective functions for these two problems are identical. It is given by:

∑ 1VM0x (λ0)

[γxρVx (0) +

∑l

Nmx (tl) (tl+1 − tl)Vx (tl)],

where VM0x (λ0) is defined in equation (10), m ≡ Nm and V are choice variables with inter-

pretations as in the rest of the paper. The variable m differs from m in only that it integratesacross χ × [0, T ] to the total queue length, instead of being a probability density functionthat integrates to 1. The constraints on the two problems differ and each has a separate,third choice variable. For the optimal assignment mechanism, we choose assignment policiesparametrized by µ. For the optimal offer mechanism, we choose offer rates parametrized byπ. We describe these variables and constraints below. The nonlinear problem is solved usingKNITRO optimizer interfaced with MATLAB. We supply analytic Jacobians and Hessiansto the solver.

D.2.1 Optimal Assignments

The allocation maximizes the objective function above by assigning an object of type z toagents currently on the list. The social planner has full information about the payoffs Γxzt aswell as the idiosyncratic shocks ε. She knows the steady-state distribution of agents waitingfor an assignment but not the future arrivals of objects or agents. The choice variable in thisproblem is the probability µzxt with which an object of type z is allocated to an agent of typex who has waited for t periods upon arrival. Given µ, the assignment is made to compatibleagents of type x that have waited for t periods and have the highest draws of ε. Choosing µis equivalent to choosing a cutoff εxzt such that µxzt = P (a (ε;x, z, t) = 1) = P (ε > εxzt).

There are three constraints:

1. Value Function: Finally, we chosen value to equal the agent’s net present value fromthe expected stream of assignments under the policy µzxt :(

1 +(ρ+ δx (tl) +

∑z

λzµxztl

)(tl+1 − tl)

)Vx (tl) = λ (tl+1 − tl)wx (tl) + Vx (tl+1) ,

where

wx (t) = 1∑z λzµxzt

∑z

λzµxztcxz [E [Γxzt + ε|ε > εxzt]P (ε > εxzt) + (Γxzt + εxzt)P (ε > εxzt)] .

This expression of V and w is obtained from solving the value function from followingthe policy of accepting offers with ε above εxzt, with offers made whenever an object

106

arrives. The term wx (t) denotes the expected value to an agent of type x for eachobject arrival. It is the sum of the values of assignment of each object type conditionalon the payoff shock exceeding εxzt, weighted by the probability of assignment.

2. Feasibility: The total mass of type z objects that are assigned upon arrival must notexceed the mass of objects that arrive. Specifically, for each z, we impose the constraint:

∑x,l

mx (tl) (tl+1 − tl) cxzµzxtl ≤ qz,

where cxz is the known (estimated) compatibility probability. The left hand side is thecumulative product of the (discretized) masses of each type of agent on the waitlist,mx (tl) (tl+1 − tl), multiplied by the assignment probabilities cxzµxztl for each agent.This quantity cannot exceed the right hand side, qz, which is the mass of objects thatarrive.

3. Steady-State Composition: The measure of agents of type x that have waited for t peri-ods is in steady state. This constraint is analogous to equation (20) above. Specifically,for each x and l > 0, we have that

mx (tl+1) = mx (tl) exp(−(δx (tl) +

∑z

λzcxzµxztl

)(tl+1 − tl)

)mx (t0) = γx.

The term ∑z λzcxzµxztl is the cumulative assignment rate across object for an agent of

type x at time tl. This, when added to δx (tl+1), yields the total departure rate.

In addition, we impose that each µxzt belongs to unit interval.

D.2.2 Optimal Offer Rates

The problem maximizes the objective function above by choosing a probability of offering anobject of type z to agents currently on the list. The social planner has full information aboutthe payoffs Γxzt, but does not know the idiosyncratic shocks ε. She knows the steady-statedistribution of agents waiting for an assignment but not the future arrivals of objects oragents. The choice variable in this problem is the probability πzxt with which an object oftype z is offered to an agent of type x who has waited for t periods upon arrival. Agentsmake optimal choices, given π, on which offers to accept.

As before, there are three constraints:

1. Value Function: Finally, we chosen value to equal the agent’s net present value from

107

the expected stream of assignments under the policy µzxt :

(1 + (ρ+ δx (tl)) (tl+1 − tl))Vx (tl) = λ (tl+1 − tl)wx (tl) + Vx (tl+1) ,

wherewx (t) = 1∑

z λzµxzt

∑z

λzµxztE [max 0,Γxzt + ε− Vx (t)]

andµxzt = πxztcxzP (Γxzt + ε > Vx (t)) .

As for the optimal assignment problem, wx (t) is the expected value to an agent of typex for each object arrival. However, in this problem, the agent makes optimal decisionsand offers do not depend on the payoff shocks. Therefore, an assignment occurs only ifthe agent is offered the object and the agent accepts. Acceptance occurs if the payoffshock exceeds Vx (t)− Γxzt.

2. Feasibility: The total mass of type z objects that are assigned upon arrival must notexceed the mass of objects that arrive. Specifically, for each z, we impose the constraint:

∑x,l

mx (tl) (tl+1 − tl)πzxtl [cxzP (Γxztl + ε > Vx (tl)) + p0,z] ≤ qz.

This constraint is also analogous to the feasibility constraint in the optimal assignmentproblem. The difference is that the assignment rate cxzµxzt is replaced by the term

πzxtl [cxzP (Γxztl + ε > Vx (tl)) + p0,z] .

The term πzxtl denotes the probability that an agent of type x receives an offer foran object of type z after she has waited for tl periods. The term in brackets is theprobability that any such offer is the last offer for the object that can be made. It isthe sum of the probability that object is compatible and transplanted

cxzP (Γxztl + ε > Vx (tl))

and the probability that no more offers can be made after this one. This term arisesfrom the technological constraint on the number of offers that can be made for an ob-ject. The model used to determine p0,z is described in Appendix B.3.

It is worth noting that this constraint only places a restriction on the expected numberof assignments. Therefore, the offer rates πxzt may not be implementable for specificdonor arrivals.

3. Steady-State Composition: The measure of agents of type x that have waited for t peri-

108

ods is in steady state. This constraint is analogous to equation (20) above. Specifically,for each x and l > 0, we have that

mx (tl+1) = mx (tl) exp(−(δx (tl+1) +

∑z

λzµxztl

)(tl+1 − tl)

)mx (tl) = γx,

whereµxzt = πzxtcxzP (Γxzt + ε > Vx (t)) .

The constraint is identical to the one used in the optimal assignment problem. However,in this problem, the assignment probability µxzt depends on the agents acceptancedecision.

In addition, we impose that each πxzt belongs to unit interval.

E Robustness

Finally, we assess sensitivity of the counterfactual results to three limitations of our analysis.We show that steady state comparisons of the queueing systems considered in Section 7 arerobust to persistent patient-level unobserved heterogeneity, alternative values of the discountfactor ρ, and a large sample of patients and donors.

E.1 Patient Unobserved Heterogeneity

Our baseline results abstracted away from patient unobserved heterogeneity to simplify esti-mation. This restriction could bias our results if patients differ in ways not captured by therich patient covariates in our data. For example, if some patients are systematically more se-lective than others and many patients are rarely willing to accept a kidney, we could overstatethe amount that changes to the priority system affect queue lengths and compositions.

To assess sensitivity to allowing for patient unobserved heterogeneity, we re-estimated themodel allowing for two unobserved patient types. Specifically, we introduced patient unob-served types αi and re-parametrized equation (8) as follows:

V (αi, xi, t)− Γ (αi, xi, zj, ηj, t) = αi + χ (xi, zj, t) θ + ηj,

where we allow αi ∈ α1, α2 with the parameters α1 and α2 to be estimated. In addition,we estimate the share π1 ∈ (0, 1) for unobserved type α1. The share of the other typeis 1 − π1. This parameterization allows patients to have systematically higher or lower

109

values of all transplants relative to their outside options.43 We assume that the exogenousdeparture rates do not depend on this unobserved preference type, and abstract away fromthe initial conditions problem, setting the proportion of each patient type in our sample tothe population average. This latter assumption is appropriate for patients that registered onthe waiting list during our sample period, but it abstracts away from selection that shouldarise for patients that registered prior to the sample of offers we consider.

We estimate this model by adding a data augmentation step to the Gibbs’ sampler. Thisstep draws αi for each agent i given their observed decisions and the parameters α1, α2, andπ1. Conditional on (α1, α2, π1), the posterior probability that αi = α1 is proportional tothe likelihood of observing the decisions made by agent i multiplied by π1. This likelihoodis simple to calculate as it is the product of the cumulative density functions of normaldistributions. With this data augmentation step, α1, α2, and π1 can be updated in theGibbs’ sampler using conjugate priors. We specify diffuse normal priors for α1 and α2 and aDirichlet prior for π1 (see Section 3.4, Gelman et al., 2014). As recommended in Gelman etal. (2014), we check for re-ordering and impose the restriction that α1 > α2.

Table B.5 presents the results for the steady-states of scoring mechanisms considered inthe main text. The results are both qualitatively and quantitatively similar to those fromour main specification. In particular, pre- and post-2014 priorities, first come first served,and greedy priorities have similar queue lengths (about 5,000) and total patient welfare.Last come first served causes both queue lengths and patient welfare to fall dramatically.Table B.6 shows that the distributional consequences are also similar to those from ourmain specification, though there are some small differences. Given that these differencesare few and small in magnitude, the results suggest that omitting patient-level unobservedheterogeneity does not substantially affect our conclusions about the welfare or distributionalimpacts of alternative assignment mechanisms.

E.2 Discount Factor

As discussed in Section 3, the discount faction ρ is not identified. The baseline results setthis parameter to 5 percent per year. Here, we evaluate sensitivity of our results to usingan annual discount rate of 10 percent. This change holds the estimated conditional choiceprobabilities fixed, but changes the calculation of V (t;xi) based on equation (7). Only Step4 in Section 4.1 must be revised to obtain estimates with an alternative discount rate.

Table B.7 presents the main counterfactuals and Table B.8 presents subgroup analysis. Equi-librium queue lengths and welfare comparisons are very close to those predicted by the base-

43A further relaxation would allow time-varying patient-level unobserved heterogeneity. This modificationwould raise additional identification challenges Connault (2016). It would also increase the computationalburden of our estimation procedure and counterfactuals; patients would have to form beliefs about theevolution of unobserved state variables governing their types, in addition to the distribution of future offers.

110

line specification; apart from last come first served, all queue length predictions are within100 of the baseline predictions, and welfare comparisons within a 1 percent change in equiva-lent donor supply. And as in the baseline specification, a 10 percent discount factor predictssmall subgroup effects, with somewhat larger effects for patients who are above 50 years oldand not on dialysis at registration.

One reason why the discount factor affects results minimally is that a large portion of dis-counting comes from patient departure rates, which are 16 percent annually for the averagepatient.

E.3 Larger Samples

Due to the computational burden of solving the optimal assignment and optimal offer mech-anisms, baseline estimates in the main text limit the number of types used in counterfactualcalculations to 300 patient types and 500 donor types. To assess whether the results aresensitive to the specific sample and number of types, we re-calculated the counterfactualsby drawing 1,000 patient types and 1,500 donor types. We recalculate equilibrium alloca-tions and welfare under the scoring mechanisms considered in the main text using the sameestimates of Γij(t) as in the baseline specification.

Table B.9 presents the overall results and Table B.10 presents the sub-group analyses. Onceagain, equilibrium allocations and changes in patient welfare are very similar to those fromthe smaller sample of patients and donors. The characteristics of transplanted donors areslightly different in the larger sample: transplanted donors are on average older, more likely tobe DCD, and less likely to be hypertensive. This difference is due in part to the compositionof donors in the larger sample. Distributional effects are also very similar in this largersample.

111

Table A.1: Patient Sample Restrictions

Number of Patients Registered

Kidney Candidates Registered in NYRT Between 2010 and 201 14499

Excluding Candidates Who Were Not Interested in a Transplant 13950

Excluding Inactive Candidates 9985

Excluding Candidates Receiving Non-Standard Allocations 9917

Notes: candidates who were not interested in a transplant include patients who departed the waitlist because they refused transplantation, received a transplant in another country, could not be contacted, or had an improved condition. Inactive candidates are patients who registered on the waitlist but never changed their status to "active," and therefore never received kidney offers, during the sample period. Candidates receiving non-standard allocations include placements from military or directed donations, expedited placement attempts, and medical emergencies.

Table A.2: Offer Sample Restrictions

Number of Offers

Offers in U.S. sample made on or before September 31st, 2015 61038882

Offers made to NYRT Patients 4516460

Offers made on or before December 31st, 2013 2883287

Excluding non-genuine refusals 1880672

Excluding offers after the donor's cutoff 1318961

Excluding patients and donors receiving non-standard allocations 1281024

Notes: non-genuine refusals include offers in the Potential Transplant Recipient (PTR) dataset which did not meet the patient's pre-set screening criteria; for which the patient or transplant surgeon was unavailable; or where the transplant could not occur for medical reasons. Non-standard allocations include placements from military or directed donations, expedited placement attempts, and medical emergencies. All offers were made on or after January 1st, 2010.

Table A.3: Fit of Mechanism Code: Predicted Offers

No Yes Total

No 14,907,971 88,366 14,996,337

Yes 502,239 1,192,658 1,694,897

Total 15,410,210 1,281,024 16,691,234

InPTRData

PredictedbySimulation

Notes:comparisonbetweenofferspredictedbysimulationofthemechanismandoffersappearinginthePTRdataset.Offersnotineitherdatasetsetincludecompatiblepatient-donorpairswherethepatientdidnotmeetthedonor'spriorityscorecutoff,orwherethedonordidnotmeetthepatient'spre-setscreeningcriteria.

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113

Table B.2: Cutoff Rank Autocorrelation Tests

Donor Type N p-value

Panel A: Donor Category

Standard Criteria, Age < 35 1065 2.01 0.55

Standard Criteria, Age >= 35 1598 1.99 0.46

Cardiac Death, Age < 35 178 2.02 0.56

Cardiac Death, Age >= 35 404 2.01 0.51

Expanded Criteria 2462 1.95 0.10

Expanded Criteria or Cardiac Death 156 2.12 0.78

Panel B: Donor Category and Origin

Standard Criteria, Age < 35; NYRT 234 2.09 0.74

Standard Criteria, Age < 35; non-NYRT 831 1.91 0.08

Standard Criteria, Age >= 35; NYRT 252 1.99 0.49

Standard Criteria, Age >= 35; non-NYRT 1346 2.01 0.60

Cardiac Death, Age < 35; NYRT 15 1.91 0.43

Cardiac Death, Age < 35; non-NYRT 163 2.25 0.95

Cardiac Death, Age >= 35; NYRT 50 2.51 0.97

Cardiac Death, Age >= 35; non-NYRT 354 2.08 0.77

Expanded Criteria; NYRT 227 2.16 0.88

Expanded Criteria; non-NYRT 2235 1.99 0.41

Expanded Criteria or Cardiac Death; NYRT 6 2.63 0.83

Expanded Criteria or Cardiac Death; non-NYRT 150 2.16 0.83

Panel C: Standard Criteria NYRT Donors, by Age and Blood Type

Standard Criteria, Age < 35; Blood Type O 117 1.98 0.45

Standard Criteria, Age < 35; Blood Type B 37 1.70 0.20

Standard Criteria, Age < 35; Blood Type AB 5 1.50 0.25

Standard Criteria, Age < 35; Blood Type A 75 1.74 0.11

Standard Criteria, Age >= 35; Blood Type O 131 1.87 0.24

Standard Criteria, Age >= 35; Blood Type B 32 2.00 0.49

Standard Criteria, Age >= 35; Blood Type AB 7 1.86 0.45

Standard Criteria, Age >= 35; Blood Type A 82 1.98 0.46

Autocorrelation Statistic

Notes: results from tests for autocorrelation of donor cutoffs in the NYRT sample based on the rank version of von Neumann's ratio statistic (Bartels, 1982). Each donor's cutoff is the priority score above which a patient would have received an offer from that donor, which is determined by the last patient in the donor's offer sequence. The rank of each donor's cutoff is its order statistic among the cutoffs of donors of the same type, with ties broken by a random number. The autocorrelation statistic for each donor type is computed for the observed sequence of donor cutoff ranks. Each p-value is the fraction of 1,000 randomly sampled permutations of donor arrival sequences for which the rank autocorrelation statistic is below that observed in sample.

114

Table B.3: Conditional Choice Probability of Acceptance (Detailed)

Base Specification Unobserved Heterog. Waiting Time + UH

(1) (2) (3)

Constant -3.69 (0.02) -4.44 (0.04) -4.46 (0.05)

Patient Diabetic -0.06 (0.01) -0.05 (0.02) -0.03 (0.02)

Calculated Panel Reactive Antibody (CPRA) 0.60 (0.05) 0.67 (0.06) 0.54 (0.08)

CPRA >= 0.8 0.27 (0.04) 0.13 (0.06) 0.17 (0.08)

CPRA = 0 -0.09 (0.02) -0.03 (0.03) -0.04 (0.03)

CPRA - 0.8 if CPRA >= 0.8 -0.41 (0.35) -0.39 (0.45) -0.58 (0.46)

Patient had Prior Transplant 0.38 (0.02) 0.36 (0.02) 0.16 (0.03)

Donor Age < 18 0.30 (0.10) -0.10 (0.18) -0.07 (0.19)

Donor Age 18-35 0.60 (0.12) 0.09 (0.18) 0.14 (0.18)

Donor Age 50+ -0.83 (0.15) -0.81 (0.20) -0.85 (0.21)

Donor Cause of Death Anoxia -0.03 (0.02) -0.09 (0.06) -0.09 (0.07)

Donor Cause of Death Stroke 0.01 (0.02) 0.02 (0.06) 0.04 (0.07)

Donor Cause of Death CNS 0.17 (0.08) -0.20 (0.29) -0.18 (0.29)

Donor Creatinine 0.5-1.0 -0.06 (0.03) -0.01 (0.10) -0.01 (0.10)

Donor Creatinine 1.0-1.5 0.02 (0.03) -0.02 (0.11) -0.01 (0.11)

Donor Creatinine >= 1.5 -0.13 (0.03) -0.23 (0.11) -0.22 (0.10)

Donor Pancreas Offered 0.35 (0.02) 0.52 (0.08) 0.51 (0.08)

Patient awaits Pancreas -2.49 (0.94) -8.62 (1.36) -14.56 (2.62)

Expanded Criteria Donor (ECD) -0.15 (0.02) -0.51 (0.08) -0.54 (0.08)

Donation from Cardiac Death (DCD) -0.11 (0.02) -0.49 (0.08) -0.49 (0.07)

Donor Male 0.00 (0.01) 0.07 (0.05) 0.05 (0.05)

Donor History of Hypertension 0.01 (0.01) -0.02 (0.05) -0.03 (0.05)

Perfect Tissue Type Match 2.28 (0.31) 2.79 (0.42) 2.75 (0.43)

2 A Mismatches -0.08 (0.01) -0.01 (0.02) -0.01 (0.02)

2 B Mismatches 0.06 (0.02) 0.03 (0.02) 0.02 (0.02)

2 DR Mismatches -0.07 (0.02) -0.06 (0.02) -0.06 (0.02)

ABO Compatible -0.36 (0.05) -0.43 (0.08) -0.46 (0.09)

Regional Offer -1.34 (0.05) -2.69 (0.17) -2.75 (0.17)

National Offer -1.50 (0.04) -2.92 (0.11) -2.95 (0.12)

Non-NYRT Donor, NYRT Match Run 1.21 (0.02) 1.97 (0.05) 2.01 (0.06)

Patient Blood Type A -0.17 (0.02) -0.29 (0.06) -0.29 (0.07)

Patient Blood Type O -0.31 (0.02) -0.40 (0.06) -0.39 (0.06)

Patient on Dialysis at Registration -0.04 (0.01) -0.12 (0.02) -0.11 (0.02)

Patient Age at Registration 0.03 (0.01) 0.09 (0.01) 0.09 (0.01)

Patient Age - 18 if Age >= 18 -0.04 (0.01) -0.10 (0.01) -0.10 (0.01)

Patient Age - 35 if Age >= 35 0.01 (0.00) 0.01 (0.01) 0.01 (0.01)

Patient Age - 50 if Age >= 50 0.00 (0.00) 0.00 (0.00) 0.00 (0.00)

Patient Age - 65 if Age >= 65 -0.01 (0.00) -0.01 (0.01) -0.01 (0.01)

Log Waiting Time (years) -0.01 (0.05)

Log Waiting Time * Over 1 Year -0.10 (0.06)

Log Waiting Time * Over 2 Years -0.17 (0.12)

Log Waiting Time * Over 3 Years 0.42 (0.11)

Patient BMI at Departure -0.04 (0.02) -0.08 (0.03) -0.07 (0.03)

Patient BMI - 18.5 if BMI >= 18.5 0.03 (0.02) 0.07 (0.03) 0.06 (0.03)

Patient BMI - 25 if BMI >= 25 0.02 (0.01) 0.02 (0.01) 0.02 (0.01)

Patient BMI - 30 if BMI >= 30 -0.01 (0.01) -0.02 (0.01) -0.02 (0.01)

Patient Serum Albumin -0.04 (0.03) -0.03 (0.03) -0.03 (0.03)

Serum Albumin - 3.7 if >= 3.7 -0.02 (0.05) -0.05 (0.05) -0.03 (0.06)

Serum Albumin - 4.4 if >= 4.4 0.11 (0.04) 0.16 (0.06) 0.15 (0.06)

Log Dialysis Time at Registration (Years) 0.05 (0.00) 0.05 (0.00) 0.05 (0.01)

Log Dialysis Time at Registration * Over 5 years 0.48 (0.03) 0.42 (0.03) 0.41 (0.04)

Perfect Tissue Type Match * Prior Transplant -0.43 (0.19) -0.37 (0.26) -0.28 (0.26)

Perfect Tissue Type Match * Diabetic Patient 0.02 (0.16) 0.04 (0.22) 0.04 (0.22)

Perfect Tissue Type Match * Patient Age -0.01 (0.01) -0.02 (0.01) -0.02 (0.01)

Perfect Tissue Type Match * CPRA 0.87 (0.34) 1.38 (0.47) 1.61 (0.47)

Perfect Tissue Type Match * CPRA above 80% -0.56 (0.30) -0.43 (0.40) -0.55 (0.41)

Perfect Tissue Type Match * ECD Donor -0.60 (0.16) -0.71 (0.23) -0.69 (0.23)

Perfect Tissue Type Match * DCD Donor -0.46 (0.33) -1.06 (0.46) -1.04 (0.46)

Perfect Tissue Type Match * NYRT Donor 0.50 (0.18) 0.10 (0.26) 0.12 (0.26)

Perfect Tissue Type Match * ABO Compatible 0.04 (0.17) 0.12 (0.23) 0.14 (0.23)

Donor Pancreas Offered * Patient awaits Pancrea 2.34 (1.07) 8.65 (1.46) 14.59 (2.68)

NYRT Donor * 2 A Mismatches 0.15 (0.02) 0.07 (0.04) 0.07 (0.04)

NYRT Donor * 2 B Mismatches -0.02 (0.03) -0.06 (0.04) -0.05 (0.04)

NYRT Donor * 2 DR Mismatches -0.02 (0.02) 0.01 (0.03) 0.01 (0.03)

115

NYRT Donor * Donor Age < 18 -0.05 (0.06) 0.22 (0.20) 0.23 (0.22)

NYRT Donor * Donor Age 18-35 0.10 (0.04) 0.11 (0.13) 0.14 (0.14)

NYRT Donor * Donor Age 50+ -0.42 (0.03) -0.62 (0.11) -0.62 (0.13)

Patient Age * Donor Age < 18 -0.01 (0.00) 0.00 (0.00) 0.00 (0.00)

Patient Age * Donor Age 18-35 -0.02 (0.00) 0.00 (0.01) 0.00 (0.01)

Patient Age * Donor Age 50+ 0.02 (0.00) 0.02 (0.01) 0.02 (0.01)

Patient Age - 35 if Age >= 35 * Donor Age 18-35 0.00 (0.01)

Patient Age - 35 if Age >= 35 * Donor Age 50+ 0.00 (0.01)

Log Waiting Time * Prior Transplant 0.22 (0.02)

Log Waiting Time * Patient Diabetic -0.03 (0.02)

Log Waiting Time * Patient Age 0.00 (0.00)

Log Waiting Time * CPRA 0.10 (0.05)

Log Waiting Time * CPRA >= 80 -0.01 (0.05)

Log Waiting Time * Patient Serum Albumin 0.00 (0.01)

Log Waiting Time * Patient BMI at Departure 0.00 (0.00)

Log Waiting Time * Patient Blood Type A 0.01 (0.03)

Log Waiting Time * Patient Blood Type O 0.01 (0.02)

Patient BMI Missing -0.81 (0.41) -1.47 (0.56) -1.27 (0.56)

Patient Serum Albumin Missing -0.01 (0.09) -0.02 (0.11) -0.10 (0.12)

Donor Unobservable Std. Dev. 0.98 (0.21) 1.00 (0.23)

Idiosyncratic Shock Std. Dev. 1.00 1.00 1.00

Acceptance Rate 0.150% 0.150% 0.150%

Number of Offers 2850572 2850572 2850572

Number of Donors 5863 5863 5863

Number of Patients 9788 9788 9788

116

Table B.3: Value of Transplant Estimates

Base Specification Unobserved Heterog. Waiting Time + UH

(1) (2) (3)

Constant -3.14 (0.00) -0.33 (0.00) -0.21 (0.00)

Patient Diabetic -0.16 (0.00) -0.89 (0.00) -0.98 (0.00)

Calculated Panel Reactive Antibody (CPRA) 1.98 (0.01) 2.09 (0.01) 1.11 (0.01)

CPRA >= 0.8 2.36 (0.01) 0.49 (0.02) 3.02 (0.02)

CPRA = 0 -0.69 (0.00) -4.04 (0.00) -3.87 (0.00)

CPRA - 0.8 if CPRA >= 0.8 -23.21 (0.08) -55.63 (0.15) -74.35 (0.15)

Patient had Prior Transplant 1.59 (0.00) 4.31 (0.00) 6.49 (0.00)

Donor Age < 18 0.34 (0.01) 0.31 (0.02) 0.32 (0.02)

Donor Age 18-35 0.69 (0.02) 0.39 (0.03) 0.82 (0.03)

Donor Age 50+ -0.87 (0.01) -0.90 (0.02) -1.17 (0.02)

Donor Cause of Death Anoxia -0.03 (0.00) -0.09 (0.00) -0.09 (0.00)

Donor Cause of Death Stroke 0.01 (0.00) 0.01 (0.00) 0.04 (0.00)

Donor Cause of Death CNS 0.19 (0.01) -0.12 (0.02) -0.16 (0.02)

Donor Creatinine 0.5-1.0 -0.06 (0.00) -0.03 (0.01) -0.01 (0.01)

Donor Creatinine 1.0-1.5 0.01 (0.00) -0.06 (0.01) -0.03 (0.01)

Donor Creatinine >= 1.5 -0.14 (0.00) -0.26 (0.01) -0.23 (0.01)

Donor Pancreas Offered 0.38 (0.00) 0.75 (0.01) 0.52 (0.01)

Patient awaits Pancreas -2.88 (0.01) -10.85 (0.02) -16.94 (0.02)

Expanded Criteria Donor (ECD) -0.15 (0.00) -0.50 (0.00) -0.53 (0.00)

Donation from Cardiac Death (DCD) -0.12 (0.00) -0.50 (0.00) -0.50 (0.00)

Donor Male 0.00 (0.00) 0.08 (0.00) 0.06 (0.00)

Donor History of Hypertension 0.01 (0.00) -0.03 (0.00) -0.03 (0.00)

Perfect Tissue Type Match 2.46 (0.11) 2.58 (0.23) 3.13 (0.23)

2 A Mismatches -0.08 (0.00) 0.06 (0.00) 0.04 (0.00)

2 B Mismatches 0.07 (0.00) 0.08 (0.00) 0.05 (0.00)

2 DR Mismatches -0.06 (0.00) -0.03 (0.00) -0.06 (0.00)

ABO Compatible -0.29 (0.00) 1.51 (0.01) 2.00 (0.01)

Regional Offer -1.36 (0.01) -2.99 (0.01) -2.75 (0.01)

National Offer -1.52 (0.01) -3.23 (0.01) -2.96 (0.01)

Non-NYRT Donor, NYRT Match Run 1.27 (0.00) 2.63 (0.00) 2.03 (0.00)

Patient Blood Type A -0.04 (0.00) 0.56 (0.00) 1.07 (0.00)

Patient Blood Type O -0.20 (0.00) 0.97 (0.00) 0.91 (0.00)

Patient on Dialysis at Registration 0.06 (0.00) 1.20 (0.00) 1.19 (0.00)

Patient Age at Registration 0.07 (0.00) 0.30 (0.00) 0.33 (0.00)

Patient Age - 18 if Age >= 18 -0.11 (0.00) -0.32 (0.00) -0.41 (0.00)

Patient Age - 35 if Age >= 35 0.03 (0.00) 0.06 (0.00) 0.14 (0.00)

Patient Age - 50 if Age >= 50 0.00 (0.00) -0.11 (0.00) -0.15 (0.00)

Patient Age - 65 if Age >= 65 -0.03 (0.00) -0.09 (0.00) -0.07 (0.00)

Log Waiting Time (years) -1.00 (0.01)

Log Waiting Time * Over 1 Year 0.77 (0.01)

Log Waiting Time * Over 2 Years 0.30 (0.01)

Log Waiting Time * Over 3 Years 5.05 (0.01)

Patient BMI at Departure -0.09 (0.00) -0.26 (0.00) -0.37 (0.00)

Patient BMI - 18.5 if BMI >= 18.5 0.07 (0.00) 0.28 (0.00) 0.44 (0.00)

Patient BMI - 25 if BMI >= 25 0.04 (0.00) 0.13 (0.00) 0.09 (0.00)

Patient BMI - 30 if BMI >= 30 -0.02 (0.00) -0.12 (0.00) -0.13 (0.00)

Patient Serum Albumin 0.02 (0.00) 1.07 (0.00) 0.96 (0.00)

Serum Albumin - 3.7 if >= 3.7 -0.09 (0.00) -1.19 (0.01) -0.96 (0.01)

Serum Albumin - 4.4 if >= 4.4 0.36 (0.00) 2.60 (0.01) 2.40 (0.01)

Log Dialysis Time at Registration (Years) 0.09 (0.00) 0.28 (0.00) 0.28 (0.00)

Log Dialysis Time at Registration * Over 5 years 3.46 (0.00) 7.94 (0.01) 8.39 (0.01)

Perfect Tissue Type Match * Prior Transplant -0.34 (0.07) -0.97 (0.14) -1.83 (0.14)

Perfect Tissue Type Match * Diabetic Patient -0.11 (0.05) 0.08 (0.11) -0.06 (0.11)

Perfect Tissue Type Match * Patient Age -0.01 (0.00) -0.02 (0.00) -0.02 (0.00)

Perfect Tissue Type Match * CPRA 0.53 (0.13) -0.39 (0.27) 1.19 (0.27)

Perfect Tissue Type Match * CPRA above 80% -0.75 (0.13) 1.39 (0.26) 0.12 (0.27)

Perfect Tissue Type Match * ECD Donor -0.57 (0.05) -0.59 (0.10) -0.66 (0.10)

117

Perfect Tissue Type Match * DCD Donor -0.54 (0.10) -1.25 (0.19) -1.33 (0.19)

Perfect Tissue Type Match * NYRT Donor 0.53 (0.09) 0.12 (0.17) 0.16 (0.18)

Perfect Tissue Type Match * ABO Compatible -0.05 (0.05) -1.55 (0.11) -2.01 (0.11)

Donor Pancreas Offered * Patient awaits Pancre 2.28 (0.06) 8.24 (0.12) 14.39 (0.12)

NYRT Donor * 2 A Mismatches 0.15 (0.00) 0.10 (0.01) 0.07 (0.01)

NYRT Donor * 2 B Mismatches -0.02 (0.00) -0.07 (0.01) -0.04 (0.01)

NYRT Donor * 2 DR Mismatches -0.02 (0.00) 0.00 (0.01) 0.01 (0.01)

NYRT Donor * Donor Age < 18 -0.07 (0.01) 0.07 (0.02) 0.22 (0.02)

NYRT Donor * Donor Age 18-35 0.11 (0.01) 0.13 (0.01) 0.19 (0.01)

NYRT Donor * Donor Age 50+ -0.43 (0.00) -0.77 (0.01) -0.62 (0.01)

Patient Age * Donor Age < 18 -0.01 (0.00) -0.01 (0.00) -0.01 (0.00)

Patient Age * Donor Age 18-35 -0.02 (0.00) -0.01 (0.00) -0.02 (0.00)

Patient Age * Donor Age 50+ 0.02 (0.00) 0.02 (0.00) 0.03 (0.00)

Patient Age - 35 if Age >= 35 * Donor Age 18-35 0.02 (0.00) 0.01 (0.00) 0.02 (0.00)

Patient Age - 35 if Age >= 35 * Donor Age 50+ -0.01 (0.00) -0.01 (0.00) -0.02 (0.00)

Log Waiting Time * Prior Transplant 2.05 (0.00)

Log Waiting Time * Patient Diabetic -0.29 (0.00)

Log Waiting Time * Patient Age 0.00 (0.00)

Log Waiting Time * CPRA 2.33 (0.01)

Log Waiting Time * CPRA >= 80 -3.20 (0.01)

Log Waiting Time * Patient Serum Albumin 0.06 (0.00)

Log Waiting Time * Patient BMI at Departure 0.02 (0.00)

Log Waiting Time * Patient Blood Type A 0.33 (0.00)

Log Waiting Time * Patient Blood Type O 0.29 (0.00)

Patient BMI Missing -1.79 (0.04) -5.95 (0.08) -8.64 (0.08)

Patient Serum Albumin Missing 0.39 (0.01) 4.31 (0.02) 3.41 (0.02)

Donor Unobservable Std. Dev. 0.98 (0.21) 1.00 (0.23)

Idiosyncratic Shock Std. Dev. 1.00 1.00 1.00

118

Table B.3: Net Present Value Estimates

Base Specification Unobserved Heterog. Waiting Time + UH

(1) (2) (3)

Constant 0.54 [0.00] 4.11 [0.00] 4.25 [0.00]

Patient Diabetic -0.11 [0.00] -0.84 [0.00] -0.95 [0.00]

Calculated Panel Reactive Antibody (CPRA) 1.38 [0.01] 1.40 [0.01] 0.54 [0.01]

CPRA >= 0.8 2.08 [0.01] 0.39 [0.02] 2.83 [0.02]

CPRA = 0 -0.59 [0.00] -4.03 [0.00] -3.86 [0.00]

CPRA - 0.8 if CPRA >= 0.8 -22.83 [0.08] -55.36 [0.15] -73.63 [0.15]

Patient had Prior Transplant 1.21 [0.00] 3.97 [0.00] 6.36 [0.00]

Patient awaits Pancreas -0.39 [0.01] -2.23 [0.01] -2.40 [0.02]

Patient Blood Type A 0.11 [0.00] 0.30 [0.00] 0.56 [0.00]

Patient Blood Type O 0.09 [0.00] 0.82 [0.00] 0.50 [0.00]

Patient on Dialysis at Registration 0.10 [0.00] 1.33 [0.00] 1.30 [0.00]

Patient Age at Registration 0.04 [0.00] 0.21 [0.00] 0.24 [0.00]

Patient Age - 18 if Age >= 18 -0.07 [0.00] -0.22 [0.00] -0.30 [0.00]

Patient Age - 35 if Age >= 35 0.02 [0.00] 0.05 [0.00] 0.12 [0.00]

Patient Age - 50 if Age >= 50 0.00 [0.00] -0.12 [0.00] -0.15 [0.00]

Patient Age - 65 if Age >= 65 -0.01 [0.00] -0.08 [0.00] -0.06 [0.00]

Log Waiting Time (years) -1.24 [0.01]

Log Waiting Time * Over 1 Year 0.84 [0.01]

Log Waiting Time * Over 2 Years 0.41 [0.01]

Log Waiting Time * Over 3 Years 4.67 [0.01]

Patient BMI at Departure -0.04 [0.00] -0.19 [0.00] -0.31 [0.00]

Patient BMI - 18.5 if BMI >= 18.5 0.04 [0.00] 0.21 [0.00] 0.38 [0.00]

Patient BMI - 25 if BMI >= 25 0.02 [0.00] 0.12 [0.00] 0.07 [0.00]

Patient BMI - 30 if BMI >= 30 -0.01 [0.00] -0.10 [0.00] -0.12 [0.00]

Patient Serum Albumin 0.06 [0.00] 1.10 [0.00] 1.00 [0.00]

Serum Albumin - 3.7 if >= 3.7 -0.07 [0.00] -1.17 [0.01] -0.96 [0.01]

Serum Albumin - 4.4 if >= 4.4 0.25 [0.00] 2.47 [0.01] 2.27 [0.01]

Log Dialysis Time at Registration (Years) 0.04 [0.00] 0.23 [0.00] 0.23 [0.00]

Log Dialysis Time at Registration * Over 5 years 2.98 [0.00] 7.50 [0.01] 7.94 [0.01]

Log Waiting Time * Prior Transplant 1.84 [0.00]

Log Waiting Time * Patient Diabetic -0.26 [0.00]

Log Waiting Time * Patient Age 0.00 [0.00]

Log Waiting Time * CPRA 2.25 [0.01]

Log Waiting Time * CPRA >= 80 -3.18 [0.01]

Log Waiting Time * Patient Serum Albumin 0.07 [0.00]

Log Waiting Time * Patient BMI at Departure 0.01 [0.00]

Log Waiting Time * Patient Blood Type A 0.59 [0.00]

Log Waiting Time * Patient Blood Type O 0.57 [0.00]

Patient BMI Missing -0.99 [0.04] -4.55 [0.08] -7.55 [0.08]

Patient Serum Albumin Missing 0.40 [0.01] 4.35 [0.02] 3.50 [0.02]

119

Table B.4: Positive Crossmatch Model

Dependent Variable: Positive Crossmatch

CPRA 1.039

(0.145)

0 or 1 HLA Mismatches -1.433

(0.477)

2 or 3 HLA Mismatches 0.181

(0.0820)

0 DR Mismatches -0.455

(0.0895)

CPRA * (0 or 1 HLA Mismatches) -0.337

(0.662)

CPRA * (2 or 3 HLA Mismatches) -0.449

(0.162)

CPRA 0 -0.592

(0.0787)

CPRA - 0.8 if CPRA > 0.8 -3.426

(0.785)

Log Dialysis Time at Registration (Years) -0.0300

(0.00797)

Log Dialysis Time at Registration * Over 5 Years 0.983

(0.0760)

Patient Age at Registration (Years) 0.0115

(0.00462)

Age at Registration - 35 if Age > 35 -0.0290

(0.00592)

Constant -0.239

(0.161)

Observations 4283

Notes: coefficient estimates from a probit regression of positive crossmatch on patient CPRA, the number of tissue type mismatches, patient age, and years on dialysis at registration. The sample is all offers accepted by NYRT patients between 2010 and 2013. Positive crossmatches are identified by the appropriate refusal code in the PTR data. CPRA is measured on a [0,1] scale.

120

TableB.5:

Rob

ustness:

Outcomes

inVa

rious

Mecha

nism

s,allowingforPa

tient

Uno

bservedHeterogeneity

Pre

-201

4 P

riorit

ies

5083

.5--

2.59

44.0

25.6

%11

.2%

38.6

%--

----

--

Pos

t-201

4 P

riorit

ies

5034

.50.

2%2.

5844

.025

.6%

11.3

%38

.5%

33.3

%18

.7%

-0.6

%-1

.0%

Firs

t Com

e Fi

rst S

erve

d51

95.2

-1.5

%2.

6344

.025

.7%

11.1

%38

.6%

83.7

%87

.3%

1.0%

1.7%

Last

Com

e Fi

rst S

erve

d30

53.3

22.7

%2.

7346

.223

.9%

10.1

%45

.9%

13.7

%0.

0%-2

6.7%

-37.

7%

Gre

edy

Prio

ritie

s50

36.3

0.1%

2.66

43.7

25.9

%11

.1%

37.7

%56

.3%

53.0

%0.

2%-0

.7%

Pre

-201

4 P

riorit

ies

5617

.3--

3.05

43.1

26.3

%11

.6%

35.2

%--

----

--

Pos

t-201

4 P

riorit

ies

5587

.20.

0%3.

0543

.126

.3%

11.6

%35

.1%

----

----

Firs

t Com

e Fi

rst S

erve

d 56

62.8

-0.1

%3.

0043

.226

.3%

11.5

%35

.3%

----

----

Last

Com

e Fi

rst S

erve

d 55

46.7

3.8%

5.46

43.4

25.8

%11

.5%

36.4

%--

----

--

Gre

edy

Prio

ritie

s 53

61.9

2.4%

3.22

43.4

26.1

%11

.3%

36.0

%--

----

--

Age

Ste

ady

Sta

teV

x

Die

d of

H

ead

Trau

ma

Pan

el B

: Est

imat

ed C

hoic

e P

roba

bilit

ies

from

Pre

-201

4 M

echa

nism

, All

Pat

ient

s

Wai

tlist

Cha

ract

eris

tics

of T

rans

plan

ted

Don

ors

Frac

tion

of P

atie

nts

Bet

ter O

ffC

hang

e in

Equ

ival

ent

Don

or S

uppl

y (M

ean)

Que

ue

Leng

th

Red

uctio

n in

D

isca

rd

Rat

e

Year

s on

W

aitli

stH

yper

-te

nsiv

eA

t Lis

ting

Vx(0

)S

tead

y S

tateV

x

At L

istin

g V

x(0)

Car

diac

D

eath

(D

CD

)

Pan

el A

: Ste

ady

Sta

te E

quili

briu

m, A

ll P

atie

nts

121

Table B.6: Robustness: Outcomes in Various Mechanisms, by Patient Group, allowing forPatient Unobserved Heterogeneity

Pre-2014 Priorities 2.49 38.6 35.9% 13.9% 27.2% -- -- -- --

Post-2014 Priorities 2.48 39.2 37.5% 14.0% 29.1% 44.7% 34.2% 0.0% -0.5%

First Come First Served 2.52 41.1 30.0% 12.6% 37.2% 63.2% 68.4% -6.5% -5.6%

Last Come First Served 2.56 42.6 26.7% 11.1% 41.8% 2.6% 0.0% -38.7% -45.5%

Greedy Priorities 2.56 38.5 33.0% 12.1% 31.1% 57.9% 60.5% -3.8% -3.8%

Pre-2014 Priorities 2.59 47.1 22.2% 11.1% 44.0% -- -- -- --

Post-2014 Priorities 2.58 46.6 22.5% 10.9% 43.0% 23.9% 14.1% -1.0% -1.3%

First Come First Served 2.61 45.6 25.1% 11.2% 39.8% 95.8% 98.6% 5.1% 5.6%

Last Come First Served 2.91 48.6 21.9% 10.4% 50.1% 12.7% 0.0% -37.8% -48.1%

Greedy Priorities 2.72 44.8 26.3% 10.3% 38.0% 53.5% 54.9% 2.2% 1.7%

Pre-2014 Priorities 2.60 40.4 27.3% 11.7% 35.0% -- -- -- --

Post-2014 Priorities 2.57 40.4 26.9% 11.9% 34.0% 42.3% 22.5% -0.5% -0.9%

First Come First Served 2.65 41.5 27.1% 10.6% 37.0% 80.3% 88.7% -0.9% 0.0%

Last Come First Served 2.81 43.2 26.6% 9.4% 42.6% 11.3% 0.0% -24.9% -35.0%

Greedy Priorities 2.61 40.5 27.1% 11.0% 35.0% 66.2% 60.6% -2.5% -3.3%

Pre-2014 Priorities 2.62 46.5 22.9% 10.1% 41.8% -- -- -- --

Post-2014 Priorities 2.61 46.5 22.5% 10.2% 42.0% 30.0% 14.2% -0.7% -1.1%

First Come First Served 2.66 45.6 23.9% 10.9% 39.4% 85.0% 85.8% 2.0% 2.6%

Last Come First Served 2.60 47.6 22.5% 10.2% 46.9% 19.2% 0.0% -17.4% -30.7%

Greedy Priorities 2.68 46.6 22.9% 11.3% 41.2% 51.7% 45.0% 1.8% 0.4%

Change in Equivalent Donor Supply (Mean)

At Listing Vx(0)

Steady StateVx

Fraction of Patients Better Off

At Listing Vx(0)

Steady StateVx

Waitlist Characteristics of Transplanted Donors

AgeDied of Head

Trauma

Cardiac Death (DCD)

Hyper-tensive

Panel A: Age < 50, Not on Dialysis at Registration

Panel B: Age 50+, Not on Dialysis at Registration

Panel C: Age < 50, on Dialysis at Registration

Panel D: Age 50+, on Dialysis at Registration

Years on Waitlist

122

TableB.7:

Rob

ustness:

Outcomes

inVa

rious

Mecha

nism

s,Ann

ualD

iscou

ntFa

ctor

0.9

Pre

-201

4 P

riorit

ies

5016

.8--

2.54

43.7

25.6

%11

.6%

38.0

%--

----

--

Pos

t-201

4 P

riorit

ies

4944

.00.

4%2.

5143

.825

.5%

11.6

%38

.0%

36.3

%14

.0%

-0.3

%-1

.5%

Firs

t Com

e Fi

rst S

erve

d51

14.2

-1.3

%2.

5643

.725

.5%

11.6

%37

.8%

86.3

%91

.3%

1.0%

2.1%

Last

Com

e Fi

rst S

erve

d29

54.2

23.2

%2.

6145

.824

.3%

11.7

%44

.1%

7.7%

0.0%

-19.

1%-4

1.9%

Gre

edy

Prio

ritie

s49

57.4

0.1%

2.57

43.6

25.7

%11

.4%

37.8

%59

.3%

50.3

%1.

1%0.

5%

Pre

-201

4 P

riorit

ies

5387

.6--

2.86

43.3

25.8

%11

.5%

36.5

%--

----

--

Pos

t-201

4 P

riorit

ies

5337

.80.

0%2.

8643

.325

.8%

11.5

%36

.5%

----

----

Firs

t Com

e Fi

rst S

erve

d 54

60.6

-0.1

%2.

8443

.325

.8%

11.5

%36

.5%

----

----

Last

Com

e Fi

rst S

erve

d 51

75.0

3.6%

5.39

43.6

25.5

%11

.3%

37.7

%--

----

--

Gre

edy

Prio

ritie

s 52

48.7

0.9%

2.88

43.4

25.7

%11

.4%

36.8

%--

----

--

Cha

ract

eris

tics

of T

rans

plan

ted

Don

ors

Pan

el B

: Est

imat

ed C

hoic

e P

roba

bilit

ies

from

Pre

-201

4 M

echa

nism

, All

Pat

ient

s

Car

diac

D

eath

(D

CD

)

Ste

ady

Sta

teV

x

At L

istin

g V

x(0)

Cha

nge

in E

quiv

alen

t D

onor

Sup

ply

(Mea

n)

Age

Pan

el A

: Ste

ady

Sta

te E

quili

briu

m, A

ll P

atie

nts

Hyp

er-

tens

ive

Frac

tion

Bet

ter O

ff

Year

s on

W

aitli

st

Wai

tlist

Que

ue

Leng

th

Red

uctio

n in

D

isca

rd

Rat

e

Ste

ady

Sta

teV

x

At L

istin

g V

x(0)

Die

d of

H

ead

Trau

ma

123

Table B.8: Robustness: Outcomes in Various Mechanisms, by Patient Group, Annual Dis-count Factor 0.9

Pre-2014 Priorities 2.42 38.2 32.3% 13.3% 27.3% -- -- -- --

Post-2014 Priorities 2.39 39.3 33.9% 13.4% 28.3% 57.9% 26.3% 1.2% -0.1%

First Come First Served 2.45 40.9 29.4% 12.7% 34.9% 68.4% 71.1% -5.7% -4.1%

Last Come First Served 2.46 42.4 26.7% 12.9% 40.4% 2.6% 0.0% -31.9% -46.8%

Greedy Priorities 2.47 38.0 33.6% 11.3% 29.4% 68.4% 63.2% 1.0% 2.0%

Pre-2014 Priorities 2.52 47.0 21.6% 12.0% 44.5% -- -- -- --

Post-2014 Priorities 2.49 46.6 21.9% 11.8% 44.0% 25.4% 8.5% -0.7% -1.8%

First Come First Served 2.52 45.6 23.4% 12.0% 40.8% 95.8% 98.6% 4.5% 5.1%

Last Come First Served 2.71 48.3 22.1% 12.3% 48.2% 2.8% 0.0% -25.6% -48.6%

Greedy Priorities 2.62 44.7 23.7% 11.8% 38.1% 54.9% 62.0% 3.0% 4.8%

Pre-2014 Priorities 2.55 40.2 27.4% 11.5% 33.9% -- -- -- --

Post-2014 Priorities 2.51 40.2 27.2% 11.5% 33.5% 45.1% 18.3% 0.1% -1.1%

First Come First Served 2.58 40.9 27.0% 11.0% 35.7% 84.5% 91.5% -0.8% 0.6%

Last Come First Served 2.66 42.6 26.9% 11.3% 40.8% 4.2% 0.0% -21.1% -40.7%

Greedy Priorities 2.52 40.5 27.3% 10.7% 35.3% 70.4% 52.1% -0.5% -2.5%

Pre-2014 Priorities 2.58 46.0 24.3% 10.9% 40.7% -- -- -- --

Post-2014 Priorities 2.56 46.0 23.6% 10.9% 41.0% 30.8% 10.8% -0.7% -2.0%

First Come First Served 2.60 45.2 24.4% 11.4% 38.5% 87.5% 93.3% 2.1% 3.2%

Last Come First Served 2.57 47.2 23.3% 11.4% 45.0% 14.2% 0.0% -10.1% -37.2%

Greedy Priorities 2.59 46.6 23.2% 11.6% 41.8% 52.5% 38.3% 0.9% -0.8%

Panel D: On Dialysis at Registration, Age 50+

Panel C: On Dialysis at Registration, Age 0-49

Panel B: Not on Dialysis at Registration, Age 50+

Panel A: Not on Dialysis at Registration, Age 0-49

Waitlist

Cardiac Death (DCD)

Hyper-tensive

Characteristics of Transplanted Donors

AgeDied of Head

Trauma

Years on Waitlist

Fraction Better Off

At Listing Vx(0)

Steady StateVx

Change in Equivalent Donor Supply (Mean)

At Listing Vx(0)

Steady StateVx

124

TableB.9:

Rob

ustness:

Outcomes

inVa

rious

Mecha

nism

s,La

rgeSa

mple

Pre

-201

4 P

riorit

ies

5049

.5--

2.71

45.4

26.7

%8.

3%45

.1%

----

----

Pos

t-201

4 P

riorit

ies

4964

.90.

6%2.

6845

.426

.6%

8.3%

45.1

%32

.3%

14.8

%0.

5%-0

.3%

Firs

t Com

e Fi

rst S

erve

d51

70.0

-1.5

%2.

7545

.326

.9%

8.3%

45.2

%90

.7%

90.7

%0.

7%1.

5%

Last

Com

e Fi

rst S

erve

d28

84.3

25.6

%2.

9846

.922

.8%

9.6%

49.5

%3.

7%0.

3%-2

1.2%

-33.

0%

Gre

edy

Prio

ritie

s49

64.7

0.6%

2.79

45.2

26.7

%8.

3%45

.0%

60.0

%53

.7%

4.6%

3.3%

Pre

-201

4 P

riorit

ies

5645

.1--

3.18

44.9

28.1

%7.

7%43

.8%

----

----

Pos

t-201

4 P

riorit

ies

5593

.70.

0%3.

1644

.928

.1%

7.7%

43.8

%--

----

--

Firs

t Com

e Fi

rst S

erve

d 57

21.8

-0.1

%3.

1744

.928

.2%

7.7%

43.8

%--

----

--

Last

Com

e Fi

rst S

erve

d 55

05.7

3.9%

5.88

45.3

27.1

%7.

9%44

.4%

----

----

Gre

edy

Prio

ritie

s 55

66.5

1.1%

3.44

44.9

27.9

%7.

7%43

.9%

----

----

Ste

ady

Sta

teV

xA

geD

ied

of

Hea

d Tr

aum

a

Hyp

er-

tens

ive

Frac

tion

Bet

ter O

ff

Year

s on

W

aitli

st

Wai

tlist

Que

ue

Leng

th

Red

uctio

n in

D

isca

rd

Rat

e

Pan

el A

: Ste

ady

Sta

te E

quili

briu

m, A

ll P

atie

nts

At L

istin

g V

x(0)

Cha

ract

eris

tics

of T

rans

plan

ted

Don

ors

Pan

el B

: Est

imat

ed C

hoic

e P

roba

bilit

ies

from

Pre

-201

4 M

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nism

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Pat

ient

s

Car

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eath

(D

CD

)

Ste

ady

Sta

teV

x

At L

istin

g V

x(0)

Cha

nge

in E

quiv

alen

t D

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Sup

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(Mea

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125

Table B.10: Robustness: Outcomes in Various Mechanisms, by Patient Group, Large Sample

Pre-2014 Priorities 2.47 40.7 32.8% 7.7% 36.3% -- -- -- --

Post-2014 Priorities 2.44 40.7 32.5% 7.7% 36.5% 29.9% 18.7% -0.3% -0.8%

First Come First Served 2.50 42.6 28.7% 9.0% 41.7% 86.6% 85.8% -4.1% -3.3%

Last Come First Served 2.41 43.9 24.2% 10.4% 45.8% 0.7% 0.0% -31.2% -39.5%

Greedy Priorities 2.48 41.1 31.4% 8.2% 37.0% 65.7% 64.9% 0.4% 0.3%

Pre-2014 Priorities 2.51 48.1 24.7% 8.2% 49.9% -- -- -- --

Post-2014 Priorities 2.49 47.8 24.7% 7.9% 49.3% 26.4% 13.7% -0.2% -0.9%

First Come First Served 2.54 47.2 26.5% 7.7% 48.4% 95.6% 96.0% 3.9% 4.5%

Last Come First Served 2.58 49.2 21.7% 9.2% 53.0% 1.3% 0.4% -27.3% -38.9%

Greedy Priorities 2.57 46.2 30.5% 7.5% 45.3% 59.0% 60.8% 15.4% 13.5%

Pre-2014 Priorities 2.99 41.8 29.9% 8.8% 39.5% -- -- -- --

Post-2014 Priorities 2.93 42.0 30.1% 8.9% 39.4% 41.6% 16.8% 1.6% 0.5%

First Come First Served 3.04 42.0 29.1% 9.0% 40.0% 86.0% 86.4% -0.9% 0.1%

Last Come First Served 3.91 43.5 24.8% 10.4% 44.8% 1.2% 0.0% -21.0% -32.0%

Greedy Priorities 3.23 41.9 27.4% 9.3% 40.5% 61.2% 49.6% -0.7% -2.2%

Pre-2014 Priorities 2.72 48.3 23.2% 8.1% 49.8% -- -- -- --

Post-2014 Priorities 2.68 48.3 23.0% 8.2% 50.1% 30.6% 12.9% 0.6% -0.3%

First Come First Served 2.76 47.5 25.0% 7.8% 48.2% 92.3% 92.0% 1.4% 2.3%

Last Come First Served 2.76 48.8 21.8% 9.1% 51.7% 7.7% 0.5% -14.3% -28.0%

Greedy Priorities 2.72 48.4 23.0% 7.9% 50.5% 57.8% 48.3% 3.0% 1.9%

Fraction Better Off

At Listing Vx(0)

Steady StateVx

Change in Equivalent Donor Supply (Mean)

At Listing Vx(0)

Steady StateVx

Waitlist

Cardiac Death (DCD)

Hyper-tensive

Characteristics of Transplanted Donors

AgeDied of Head

Trauma

Years on Waitlist

Panel D: On Dialysis at Registration, Age 50+

Panel C: On Dialysis at Registration, Age 0-49

Panel B: Not on Dialysis at Registration, Age 50+

Panel A: Not on Dialysis at Registration, Age 0-49

126

Figure B.1: Offer and Acceptance Rate by cPRA

(a) Offer Rate

(b) Acceptance Rate

Note: Sample includes all offers made to NYRT patients between 2010 and 2013, including offers that did not meet pre-set

donor screening criteria. Positive crossmatches are counted as acceptances. In each figure, the black line plots the mean among

offers to patients in each CPRA bin, and the shaded region represents pointwise 95 percent confidence intervals.

127

Figure B.2: Comparing Hazard Rates for Gompertz and Cox Proportional Hazards Models

01

23

45

Cum

ula

tive H

azard

0 1000 2000 3000 4000Days on List

Gompertz Hazards Model

01

23

45

Cum

ula

tive H

azard

0 1000 2000 3000 4000Days on List

Cox Proportional Hazards Model

Cumulative Hazard Rates