queue

MANUFACTURING&SERVICE

OPERATIONS MANAGEMENT Vol.6,No.4,Fall2004,pp.280–301

issn1523-4614 eissn1526-5498 04 0604 0280

inf orms

?doi10.1287/msom.1040.0056

?2004INFORMS

Patient Choice in Kidney Allocation:

The Role of the Queueing Discipline

Xuanming Su

Walter A.Haas School of Business,University of California,Berkeley,California94720,

xuanming@https://www.wendangku.net/doc/174783509.html,

Stefanos Zenios

Graduate School of Business,Stanford University,Stanford,California94305,stefzen@https://www.wendangku.net/doc/174783509.html,

T his paper develops and analyzes a queueing model to examine the role of patient choice on the high rate of organ refusals in the kidney transplant waiting system.The model is an M/M/1queue with homoge-neous patients and exponential reneging.Patients join the waiting system and organ transplants are re?ected by the service process.In addition,unlike the standard M/M/1model,each service instance is associated with a variable reward that re?ects the quality of the transplant organ,and patients have the option to refuse an organ(service)offer if they expect future offers to be better.Under an assumption of perfect and complete information,it is demonstrated that the queueing discipline is a potent instrument that can be used to maximize social welfare.In particular,?rst-come-?rst-serve(FCFS)ampli?es patients’desire to refuse offers of marginal quality,and generates excessive organ wastage.By contrast,last-come-?rst-serve(LCFS)contains the inef?cien-cies engendered by patient choice and achieves optimal organ utilization.A numerical example calibrated using data from the U.S.transplantation system demonstrates that the welfare improvements possible from a better control of patient choice are equivalent to a25%increase in the supply of organs.

Key words:kidney allocation;queues;priority;last-come-last-served;stochastic games;ef?ciency-equity trade-off

History:Received:June1,2002;accepted:July15,2004.This paper was with the authors7months for

3revisions.

1.Introduction

Kidney transplantation is the preferred treatment for most patients diagnosed with chronic kidney failure. However,there is a signi?cant shortage of organs for transplantation as demonstrated by the expand-ing transplant waiting list.In the10-year period between1992and2002,the national waiting list grew from22,063to51,144,and the median waiting time increased from624days to1,144days(UNOS2002). Despite these alarming trends,more than12%of all organs recovered from a donor are eventually dis-carded,because they are repeatedly refused for trans-plantation by patients on the waiting list and by their surgeons.The mostcommon reason for t hese refusals is that these organs are of marginal quality,and hence the patient expects to bene?t by waiting for a better organ(UNOS2002).

Recognizing the inef?ciency created by these refus-als,the United Network of Organ Sharing(UNOS),

which is charged with the management of the trans-plant waiting system has recently introduced the fol-lowing modi?cation to the transplant allocation sys-tem:Organs from marginal donors are now reserved for patients who declare in advance their willingness to accept such organs.While these patients remain eligible for all other organs,they are likely to be offered an organ from a marginal donor several months or even years before they would expect an offer of a“regular”donor organ.Consequently,UNOS expects to place more of these“marginal”organs and to diminish their wastage by accelerating patient’s access to them.Yet,this pro-posed modi?cation is not free of problems.Speci?cally, while increasing patient involvement in the transplant allocation system is desirable,there are no guarantees that this will improve the overall system.In fact,read-ers of this journal are familiar with the classical result in Naor(1969),where consumer choice in a waiting system increases congestion and degrades performance.Is it 280

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possible that patient choice in the transplant waiting list can lead to more wastage and longer waiting times? This paper aims to provide an answer to this question by focusing on the interaction between patient choice and the queueing discipline used to rank patients on the transplant waiting list.

In our view,the effect of patient choice on the transplant allocation system is dictated by two com-plementary forces.First,depending on their intrinsic characteristics,some patients may be more likely to accept organs of marginal quality than others.Second, the rule used to prioritize patients on the transplant waiting list can encourage patients to be either more or less stringent in their choices.While understanding the interaction between these two forces is desirable, our approach is based on the principle of“divide and conquer.”In two companion papers(Su and Zenios 2004a,b)we focus on the role of patient heterogene-ity,while suppressing the dynamics of the prioritiza-tion rule.By contrast,the focus here is on the effect of the prioritization rule with patient heterogeneity suppressed.The emerging picture is incomplete,but it can be argued that some of the?ndings obtained from the simpli?ed setting are universally valid.

A queueing model provides the natural abstrac-tion for our investigation.The model incorporates two independent arrival streams,one for patients and a second one for organs,and“service”re?ects the transplant operation when a patient and an organ are merged and depart the system.Because organs cannot be stored,and thus cannot be placed in a queue,their arrival can be conveniently captured by the“service”process with the service time equal to the time between organ arrivals.This implies that the basic model is an M/M/1queue,but with two unique features that deviate from the standard assumptions and re?ect the reality of the transplant waiting sys-tem.First,each“service”offer is associated with a reward that captures the quality of the organ.And second,patients may refuse an organ offer if they perceive its quality to be unacceptable and expect a future offer to be of better quality.Although the refused organ will then be offered to the next patient in line,it is theoretically possible for the resulting queueing model to“idle”even when the queue is not empty,because the quality of some organs may make them unattractive for all patients on the trans-plant waiting list.The model also captures patient death through exponential patient reneging,and it assumes that all patients are homogeneous despite their well-documented heterogeneity.As explained above,the latter is a conscious choice made primarily for tractability’s sake.To re?ect the difference in med-ical outcomes between waiting and transplantation, the model also assumes that each patient on the wait-ing listreceives a reward per unitt ime t hatre?ect s their quality of life prior to a transplant,and when the patients accept an organ offer,they also receive a reward given by the patient’s expected discounted quality-adjusted life years(QALYs)after a transplant. Patients in this system behave as rational economic agents and determine whether to accept or decline each offer based on the offer’s quality.Similarly,the medical planner overseeing the system determines who should be offered each organ.Each patient’s objective is to maximize his or her own total expected discounted QALYs,and hence the patient solves an optimal stopping time problem:when to accept an organ offer.On the other hand,the medical plan-ner wishes to maximize the sum of the rewards for all patients and has two policy levers at its disposal: organ rationing and patient prioritization.That is,the planner can in?uence system outcomes by limiting access to certain organs(rationing)and by dynami-cally prioritizing the candidates on the waiting list. Of course,in reality,there is not a single planner,but rather a community of stakeholders that collectively determine the rationing and prioritization rule.The monolithic medical planner in our model re?ects the collective actions of this community.

Our analysis starts with a frictionless ideal,where the medical planner is a benevolent dictator and the patients do not exercise choice,but instead accept any offer.This identi?es the socially ef?cient(or Pareto optimal)outcome and provides a comparison bench-mark for the effects of patient choice.In the context of this system,patient prioritization is not effective because patients are identical,but rationing is effec-tive because there is a tradeoff between waiting and transplanting a not-so-good organ.In fact,when the queue length is small,the planner only offers organs of the best quality.But as the queue length increases,

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organs of lower quality are also offered to counterbal-ance the increased wait.

The next step is to consider the competitive equi-libria that will emerge when patients exercise their choice.We assume?rst that patients are ranked according to the FCFS discipline,and investigate whether,given this priority discipline,rationing increases the planner’s objective.It turns out that rationing is ineffective because it is confounded by patient choice.Speci?cally,in the absence of explicit rationing by the planner,“implicit”rationing will emerge as the equilibrium outcome:Organs of low quality will be discarded because no one on the waiting list would accept them.Further,the quality threshold that separates the acceptable organs from the ones that are declined is decreasing in the queue length.Imposing rationing that raises these thresholds beyond their naturally occurring equilibrium does not bene?tt he planner in any way.These equilibrium thresholds are higher than the ones obtained in the socially ef?cient case,mirroring the results in Naor (1969),where customer autonomy in a queueing sys-tem caused performance degradation.In our context, patients’quality requirements in the FCFS system are more stringent than is socially optimal,and their behavior causes excessive organ wastage.

Next,we turn to investigate whether the ineffec-tiveness of rationing is an artifact of the FCFS pri-ority system.Our analysis demonstrates that it is not:treatment rationing is always ineffective,and hence,the only meaningful lever for the medical plan-ner to use is the priority discipline.This motivates the following question:What is the priority disci-pline that maximizes social welfare when patients exercise choice?An examination of the FCFS disci-pline demonstrates that it is ineffective because future arrivals do not affect the patients already in the sys-tem,hence existing patients do not consider the con-gestion externalities they impose when they decline an organ offer.Motivated by this observation,we then consider the extreme opposite priority rule—LCFS—and demonstrate that,in this rule,patients internalize the externalities of their own decisions,and system performance achieves the socially optimal ideal. This brings to the forefront the equity-ef?ciency trade-off that underlies any medical waiting system. Speci?cally,itis a fundament al premise in medicine that FCFS is fair,and in that respect,LCFS can be blatantly unfair.It is therefore natural to consider a priorit y rule t hatis notas inef?cientas CF S but not as unfair as LCFS.To do that,we introduce a

family of prioritization schemes,with both FCFS and LCFS belonging to that family,and examine the effect of these schemes.The main?nding formalizes the intuition that in systems judged fair according to the FCFS criterion,patients internalize only a small frac-tion of the externalities caused by their autonomy, but in systems that are unfair according to the same criterion,patients internalize most of the external-ities.This casts serious doubts on the validity of the premise that FCFS is a gold standard for fair-ness.When patients exercise choice,FCFS is the most “unfair”of all policies because the externalities of patient autonomy are borne by everyone except the person exercising this autonomy.

The root of the inef?ciency identi?ed by this anal-ysis lies in the well-known divergence between indi-vidual and social optima in queueing systems.This was?rstst udied in Naor’s(1969)seminal paper, which demonstrated that an admission toll is nec-essary to induce the social optimum.Yechiali(1971) considers the perspective of a pro?t-maximizing?rm, and Lippman and Stidham(1977)and Stidham(1978) study the structural properties of the optimal con-gestion toll.Mendelson(1985)embeds the queue-ing model into an economic framework that directly considers the effect of such externalities.The situa-tion becomes more complex when there are multi-ple customer types,because customers now have the added incentive to misrepresent themselves to obtain a better service level.Mendelson and Whang(1990) derive incentive-compatible priority pricing policies that simultaneously induce truthtelling and socially optimal behavior,and Van Mieghem(2000)com-bines this with dynamic scheduling policies.Another instance of similar incentive problems arises in the context of multiserver systems.Bell and Stidham (1983)show that when arriving customers are free to choose which server to join,low-cost/high-speed servers will become overcongested.

Compared with all these papers,our work exhibits important differences in modeling methodology,pol-icy implications,and solution techniques.First,while most existing models are concerned with customers’

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decisions at arrival epochs(e.g.,whether to join the queue,what priority level to purchase,and which server to choose),we are interested in patients’deci-sions to accept transplantation at service epochs

(which correspond to the times when organs become available).Kaplan(1987)studies a similar queue-ing model in the context of public housing pro-grams,in which registrants may refuse to move into a public housing project if they would rather wait for a more attractive accommodation option.How-ever,while he models tenant choice as an exogenous acceptance probability,the queueing dynamics in our model account for patients’utility-maximizing deci-sions.Next,the current work also yields entirely dif-ferent policy implications.While intelligent pricing is essential in previous work,we show that priori-tization alone is suf?cient to coordinate the system studied here.Bell and Stidham(1983)mention pric-ing under the LCFS discipline,and Hassin(1985)pro-vides remarks on the optimality of LCFS.However, to the best of our knowledge,no other work has rig-orously showed that the socially optimal ideal can be attained by merely changing the priority rule to LCFS.In fact,the paper by Hassin(1985)provides an informal argumentfavoring LCF S butnota for-mal proof.Finally,while most papers in the literature assume that queue lengths are not observable and focus on equilibrium analysis,we allow patients to observe queue lengths and analyze this less tractable case using stochastic game techniques.The only other methodologically similar paper that we know of is Altman and Shimkin(1998),which analyzes individ-ual equilibrium decisions to join a processor-sharing system.

In the health economics literature,most work on medical queueing systems is based on economic mod-els t hatuse costbene?tanalysis t o quant ify a socially optimal waiting list con?guration(for a comprehen-sive survey,see Cullis etal.2000).Incent ive consid-erations in waiting lists focus mainly on the attitudes of physicians.Among others,Yates(1995)expresses concern for the possibility that the pursuit of pri-vate practice by consultants in the United Kingdom’s National Health Services(NHS)may create a con-?ict of interest,and Weinstein(2001)contemplates the dual role of physicians as gatekeepers.The incen-tives of hospital management have also been studied,such as in Feldman and Lobo(1997),although to a lesser extent.The papers by Goddard et al.(1995)and Iversen(1997)examine pat ientpreferences,butdiffer from the current study in that they consider patients’

choices between seeking treatment from the waiting list or the private sector,whereas we are concerned with the behavior of patients while on the waiting list.Furthermore,the majority of papers in this area focus on static equilibrium models.Some exceptions include Goddard and Tavakoli(1994)who presenta queueing analysis of the impact of various prioritiza-tion regimes,and Van Ackere and Smith(1999)and Smith and Van Ackere(2002)who model the NHS waiting lists,using system dynamics methodology. Most of these papers assume that patients’choices to seek treatment are exogenously given(with the excep-tion of Van Ackere and Smith1999,where patient arrival rates depend on perceived waiting times). In contrast,we examine patients’utility-maximizing decisions by explicitly modeling their utility func-tions.We hope that our dynamic analysis of patient choice in the context of kidney transplantation will shed some light on the trade-offs that have not been previously studied.

The kidney allocation problem creates some of modern medicine’s mostvexing policy dilemmas,and its study would bene?t from the rigor provided by queueing models.Extensive simulation studies have been pursued in an effort to clarify the role of dif-ferent allocation policies on queueing outcomes such as waiting time(see Zenios et al.2002,Howard2001, Votruba2002),and Zenios(1999)provides some ana-lytical results using a multiclass queueing model with reneging.However,these papers suppress patient choice and fail to recognize its critical role.

We now provide an outline for the remainder of this paper.In§2,we provide an overview of kidney transplantation in the United States.A description of the model is presented in§3.Section4analyzes the socially optimal outcome when patients do not exercise choice.Section5considers the competitive equilibrium that emerges under the FCFS priority dis-cipline.A comparison of the socially optimal outcome to the outcomes from the competitive equilibrium ver-i?es a welfare loss,whose source is examined in detail in§6.The subsequent two sections verify that pri-oritization is a potent policy instrument.Speci?cally,

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§7shows that the socially ef?cient outcome can be recovered by using LCFS,and§8discusses a fam-ily of prioritization schemes that capture the essence of the equity-ef?ciency trade-off underlying the selec-tion of different priority schemes.Section9presents a numerical example that investigates the magnitude of the welfare losses under FCFS and the model parameters that contribute to these losses.Concluding remarks are presented in§10.All proofs are relegated to the appendix.

2.Background on Kidney

Transplantation

We now presentan overview of kidney t ransplan-tation in the United States focusing on its aspects most related to the questions addressed in this paper. Readers who wish to obtain more details should con-sultUNOS(2002)and Organ Procurementand Trans-portation Network(OPTN)(2003).

End-Stage Renal Disease(ESRD)or chronic kid-ney failure is a condition affecting more than400,000 patients in the United States as of2002.Most of these patients undergo dialysis therapy,which involves vis-iting a dialysis center for at least12hours each week. The other treatment alternative is kidney transplan-tation,which is often preferred because it enables patients to resume regular activities.However,to minimize the risk of acute graft rejection(graft is the medical term for the transplant organ),patients who have received a t ransplantmustt ake immunosup-pressive drugs inde?nitely.Although some patients are able to obtain a kidney from a living relative, mostmustrely on cadaveric donors(t hatis,deceased donors)and continue to receive dialysis treatment while waiting for a suitable organ.

In2002,there were23,328new ESRD patients but only11,860cadaveric organs were procured in the same year.UNOS oversees the allocation of organs to transplant candidates by coordinating the activ-ities of59organ procurement organizations(OPO) that operate in distinct geographical regions.Individ-ual OPOs are responsible for procuring all organs donated in their region,and patients join the waiting listatt heir local t ransplantcent ers.Typically,when a kidney becomes available,itis?rstallocat ed t o patients at the local level,then at the regional level,and then at the national level.Furthermore,because post-transplant survival relies heavily on clinical vari-ables such as age and tissue types,UNOS attempts to promote favorable transplant outcomes by providing the OPOs with speci?c organ allocation guidelines. This includes a point system that prioritizes potential transplant recipients based on points that re?ect the quality of the tissue match between the donor and the candidate,as well as points that re?ect the candi-date’s waiting time.The continued shortage of organs and the associated explosion in waiting times has con-tributed to a convergence of this point system to a system that resembles FCFS.

Despite the apparent supply shortage,nearly half of the procured organs are refused by the?rst-offered transplant candidate.Placement of these refused organs is not an easy task because a substantial num-ber of patients may prefer to remain on dialysis and wait for a better organ offer,rather than obtain an organ of marginal quality.During the search for a recipient,organs accumulate cold ischemia time (during which they are kept frozen),which causes transplant outcomes to deteriorate,and organs are usually discarded if they are not transplanted within 48hours.As a result,in2001,about12%of all recov-ered kidneys were not transplanted,thus exacerbating the supply shortage.The modi?ed policy described in the introduction was developed in an effort to increase the utilization of these marginal organs;this policy is formally known as the Expanded Criteria Donor program and came into effect in October2002. However,the effectiveness of this new policy and other policies that encourage patient choice has yet to be examined rigorously.

3.Model Description

We now introduce the queueing model for the local transplant waiting list.Patients arrive according to a time-homogeneous Poisson process with rate ,and cadaveric organs arrive according to an independent Poisson process with rate .Patients can depart from the waiting list either when they receive and accept an organ offer,or when they die after an exponentially distributed amount of time with mean1/ .The death process is independent from both patient and organ arrivals.For brevity of notation,it is convenient to normalize the organ arrival rate ≡1.

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285

The reward structure for the model assumes that patients are assigned a quality of life score that depends on whether they are waiting,they have received a transplant,or they have died.The qual-ity of life score re?ects the desirability of each of the three states (dialysis,post-transplant,death)and patient preferences are homogeneous in the sense that all patients have the same quality of life score in each state.Patients on dialysis receive a continuous payoff ata rat e h per unit time.Patients who die receive an instantaneous payoff d .In addition,patients receive a payoff from transplantation.To re?ect variability in organ types,we assume that there exists a measure of “organ quality”captured by a continuous random

variable X ,which takes values in x ˉx

and has proba-bility density f .When a donor organ arrives,its qual-ity X =x is realized and publicly observed,and this re?ects the post-transplant total expected discounted QALYs for the patient receiving the organ.All rewards are continuously discounted at rate .For future refer-ence,itis also convenientt o de?ne t he reverse cumula-tive probability distribution F x = ˉ

x x f x dx and also the function g x = ˉ

x x xf x dx/ F x ,which gives the

expected conditional reward from organs with quality that exceeds a threshold x .

Patients have the option to decline an organ offer in anticipation of a potentially better future offer,and their objective is to exercise their “decline”option carefully to maximize their total quality-adjusted life expectancy.A policy for a patient speci?es the range of donor organs that are acceptable to the patient ateach pointin t ime.Similarly,t he medical planner (referred to as “she”)must decide how to allocate the organs that become available and her objective is to maximize the total quality-adjusted expected life years of all patients.A policy for the medical plan-ner speci?es,ateach pointin t ime,t he range of donor organs that will be assigned to patients waiting and the rank of all patients on the waiting list.An organ is ?rst assigned to the top-ranked candidate.If she or he declines it,the organ will be offered to the second-ranked candidate.The process will be repeated until the organ is either accepted by someone on the wait-ing listor itis refused by everyone.The organ is t hen discarded.

Throughout the analysis,the focus will be on sta-tionary Markovian strategies for all parties.It will also

be assumed that each patient and the medical planner have perfect information about the queue length and the quality of organs.All model primitives are known to all parties.

The current speci?cation is quite general and does not require additional assumptions on the param-eters.There are no restrictions on the signs of h or d ,

although ?niteness on the bounds x ˉx

is required.For ease of exposition,we will develop our analy-sis assuming that the quality random variable X has a continuous density f ,but this assumption is not necessary;for example,X could be a discrete ran-dom variable.Furthermore,the analysis can be be car-ried over to the undiscounted case by taking the limit →0.

Before we proceed further,it is worthwhile to describe the modeling assumptions that deviate from the real-life kidney allocation system reviewed in §2.While it is virtually impossible to develop a tractable model that captures every single aspect of reality,it is nonetheless important to discuss how the simpli?-cations made in the model affect the validity of our ?ndings.The following listis notexhaust ive,butit attempts to cover the assumptions that represent the mostsigni?cantdeviat ions from realit y.3.1.Homogeneous Patients

As explained in the introduction,heterogeneity in patient characteristics is suppressed from our model.The ESRD patient population exhibits wide differ-ences in various clinical attributes (such as age and tissue types)and the outcome from a transplant depends both on the attributes of the organ (the so-called organ type)and the clinical attributes of the patient (the so-called patient type).By suppress-ing patient heterogeneity,our model assumes that the organ type is a much more important predic-tor for the outcome of a transplant than the patient type.In fact,survival analysis performed using trans-plant data from the United States Renal Data Sys-tem (USRDS)partially supports this assumption:donor age is the factor that most signi?cantly in?u-ences post-transplant survival (see Su 2004).There-fore,incorporating only the variability in organ types and not in patient types captures the most important driver of transplant outcomes.This is not to say that variability in patient attributes is any less important.In fact,its interaction with patient choice is studied

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in Su and Zenios(2004a),where itis shown t hat organ types can be partitioned into domains such that organs in each domain are offered to patients of a given type or types.Further,appropriately designing the partition and the assignment of organ domains to patient types minimizes the wastage caused by patient choice.

Beyond the suppressed patient heterogeneity, patients may also be better informed about unob-servable risk factors that in?uence transplant out-comes,and aboutt heir preferences among different health states.This information asymmetry introduces another important dimension of strategic behavior that may cause wastage:patients may provide mis-leading information about their true preferences and health conditions to improve their individual out-comes.Such behavior may exacerbate the problems of patient choice,and its study is left as a topic for future research.

3.2.Perfect Information

The second unrealistic assumption made in our model is that patients have perfect information about the size of the waiting list.Although patients(or at least their physicians)are kept updated about their posi-tion on the waiting list,?uctuations in the waiting list because of addit ions and removals make itdif?cultt o continuously monitor the state of the system.How-ever,as comprehensive information systems are made publicly available over the Internet,the assumption of perfectinformat ion will become increasingly rele-vant.In fact,the website maintained by UNOS pro-vides regular updates about the size of the waiting list,new additions,and any departures.Patients can use this information to intelligently monitor changes in t heir own waitlistposit ion.

3.3.No Organ Deterioration

As mentioned in§2,donor organs accumulate cold ischemia time during the search for a recipient and this causes a degradation in outcomes that is not cap-tured in our model.Theoretically,this is not an impor-tant issue because our model permits us to identify the patient(if any)who would accept the organ,and thus placement of the organ can be almost instanta-neous.However,contacting that particular individual imposes logistical challenges,and moreover,that indi-vidual may not be perfectly rational as assumed in our model,and hence may refuse the offer.There-fore,the reader should assume that our analysis will understate the welfare loss caused by patient choice, and will potentially overestimate the improvements generated by any of our proposals.

3.4.Geographical Factors

In reality,when the local waiting list is exhausted, UNOS will attempt to allocate the kidney to patients on the regional or even national waiting list.How-ever,this incurs substantial time delays(e.g.,when transporting the kidney across long distances),which result in organ deterioration as described above.In our model,we simply assume that the kidney is dis-carded after being refused by all local patients.This captures the negative effects of organ refusal in an imperfect way,and may overstate the bene?ts of our proposals.

3.5.Social Welfare Function

Our analysis assumes that the medical planner is interested in maximizing the total welfare of all patients participating in the organ allocation system, but is not concerned with other sources of social costs such as the?nancial cost of treatment.Incorporating health care costs into the analysis provides a different perspective not pursued here.

4.Socially Optimal Outcome

Having described our modeling framework,we now consider the problem of a benevolent medical plan-ner;here,patients do not exercise choice and accept all organ offers.For each queue length n,the plan-ner’s policy speci?es an organ acceptance threshold b n such that only organs with quality exceeding b n are offered to a transplant patient.Let V n denote the medical planner’s optimal total expected discounted reward.Bellman’s equation of optimality states that

V 0 =

+

V 1 (1)

V n =sup

b n ∈ x ˉx

1

+1+ + n

· hn+ F b n V n?1 +g b n

+ 1? F b n V n + V n+1

+ n V n?1 +d (2)

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To interpret(2),notice that the total waiting payoff rate for all patients is hn when there are n patients on the waiting list.This is earned continuously until the next transition,which occurs with rate1+ + n. Then,there are four possible transitions.

(1)With probability F b n / 1+ + n an organ with quality that exceeds the threshold b n arrives,in which case a patient receives that organ and obtains an instantaneous payoff of g b n ,and the waiting listdecreases by one.

(2)With probability 1? F b n / 1+ + n an organ with quality below the decision threshold arrives,but this does not affect the state of the system because itis discarded.

(3)A new arrival occurs with probability / 1+ + n and this increases the queue length by one.

(4)A patient dies with probability n/ 1+ + n , in which case the queue length decreases and a payoff d is accrued.

Equation(1)is a boundary condition,which states that there is no waiting payoff when the system is empty and that the only possible transition is a new pat ientarrival occurring ata rat e .

The optimality equation has two important corol-laries.First,it can be shown that the optimal decision thresholds b? n are related to the optimal value func-tion as follows(assuming that the constraint b? n ≥x is notbinding):

b? n =V n ?V n?1 (3) This expression can be derived from the?rst-order conditions for the maximization problem in the right-hand side of(2),and it is valid because the function in the right-hand side of(2)is concave in b n —it has a nonpositive second derivative.More intuitively, this expression follows because the medical planner determines each threshold by weighing two alterna-tives:(1)approve a transplant and earn a payoff of atleast b n +V n?1 or(2)deny the transplant and maintain the continuation payoff V n .Expression(3) states that the medical planner is indifferent between these two choices on the margin.

The second corollary is that the decision thresholds b n are nonincreasing in the queue length n.Thatis, the medical planner is selective in the choice of organs when the queue length is small,but as the queue length increases,lower quality organs also become acceptable.

Proposition1.The socially optimal decision thresh-olds b? n are nonincreasing in n.

The resultis proven in t he appendix using an argu-ment presented in Bertsekas(1995)and extended by George and Harrison(2001).

https://www.wendangku.net/doc/174783509.html,petitive Equilibrium

Under FCFS

We now extend the analysis to explore the competi-tive equilibrium when patients retain their autonomy and the medical planner uses the FCFS priority dis-cipline.We provide?rst a precise de?nition of the medical planner’s and of each patient’s strategy,intro-duce the appropriate equilibrium concept and?nally provide an algorithm that derives the equilibrium. Consider?rst the medical planner.Her strategy consists of the FCFS priority rule and a queue-length-dependentrat ioning rule b= b n n=1 . Under this rule,when the queue length is n,only organs with a value no less than b n are assigned to patients on the transplant waiting list on a FCFS order.The rationing rule b is common knowledge to all patients.When a patient declines an organ offer, the organ is then offered to the next candidate on the waiting list.

Next,consider the patients on the waiting list. Rather than characterize the strategy of each patient, we de?ne a strategy pro?le that characterizes the behavior of all patients.Because the underlying sys-tem dynamics are exponential,it is natural to focus on Markovian strategies,where each patient’s decision on the range of acceptable organs depends only on the current queue length and his position on the wait-ing list.We also restrict attention to threshold-based acceptance policies,because each patient will accept all organs that yield a higher reward than his con-tinuation payoff from remaining on dialysis.Hence, a strategy pro?le is characterized by a set of decision thresholds a k n n≥1 n≥k≥1 with the follow-ing interpretation:When the queue length is n,

t

he patient in position k will only acceptorgans when their quality is no less than the decision threshold a k n ;otherwise the patient will retain his position in line and pass on the organ to the next patient.For brevity of notation,we let a= a k n n≥1 n≥k≥1 . In addition,given the medical planner’s threshold

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strategy b,it follows that a strategy pro?le must be such that

a1 n ≥a2 n ≥···≥a n n ≥b n (4) that is,the threshold for the k th position cannot be less than the threshold for the k+1stposit ion,and all thresholds are no less than the medical planner’s threshold.This condition follows from the FCFS pri-ority rule,which implies that violations of this con-dition are not implementable:the decision thresholds for patients that are higher on the priority list deter-mine the feasible thresholds for all patients behind them,and the medical planner’s decision threshold places a lower bound on everyone’s thresholds. Embedded in our strategy pro?le is the assumption that patient strategies are symmetric:although differ-ent patients will use different acceptance thresholds at any given time because they will be at different positions along the queue,these thresholds are sym-metric in the sense that all patients who are ever in position k when the queue length is n will use the same threshold a k n .The assumption of a symmetric equilibrium is common when there are multiple equi-libria(see Fudenberg and Tirole1990,p.160),because there is no reason for homogeneous patients to behave differently when faced with the same set of circum-stances.And even though this assumption does not exclude the existence of other equilibria,it streamlines the analysis considerably.

In this setting,the patients are involved in an in?-nite horizon dynamic game and the relevant equilib-rium conceptis t hatof subgame perfect ion(Gibbons 1992).In this concept,a strategy pro?le is a subgame perfect Nash equilibrium if patients cannot gain by unilateral one-stage deviations from the equilibrium strategy.That is,there is no single state,de?ned by the pair n k of the queue length and patient position, where the k th patient may gain by deviating from the actions prescribed by the strategy pro?le at this one state.This equilibrium concept involves two assump-tions that are worth discussing:(1)each patient can only change one threshold at a time and(2)patients cannot collectively choose their threshold.The latter is a standard assumption because it is practically impos-sible for patients to“collude”and set their thresholds jointly.The former re?ects the assumption that even if a particular patient chooses to change his whole strat-egy pro?le,these changes should satisfy Bellman’s principle of optimality,and thus,it is suf?cient to con-sider unilateral changes that involve deviations in one threshold at a time.

We will now provide an algorithm that identi?es a subgame perfect strategy pro?le for this game.The ?rst step is to derive expressions for each patient’s continuation payoffs(or value function)for any given strategy pro?le a.Speci?cally,assuming that all patients comply with strategy pro?le a,let V a k n denote the total expected discounted payoff for the patient in position k when the queue length is n,and let V a denote the collection V a k n n≥1 n≥k≥1 . The symmetric assumption is re?ected in the fact that V a k n is the same for all patients who will ever be in position k when the queue length is n.Then,V a is derived as follows:

V a1 1 =

1

+1+ +

· h+ F a1 1 g a1 1 + 1? F a1 1 V a1 1

+ V a1 2 + d (5) V a1 n =

1

+1+ + n

· h+ F a1 n g a1 n + F a n n ? F a1 n

·V a1 n?1 + 1? F a n n V a1 n

+ V a1 n+1 + d+ n?1 V a1 n?1 (6) V a n n =

1

+1+ + n

· h+ F a n?1 n V a n?1 n?1 + F a n n g a n n

? F a n?1 n g a n?1 n + 1? F a n n V a n n

+ V a n n+1 + d+ n?1 V a n?1 n?1 (7) V a k n =

1

+1+ + n

· h+ F a k?1 n V a k?1 n?1 + F a k n g a k n

? F a k?1 n g a k?1 n + F a n n ? F a k n

·V a k n?1 + 1? F a n n V a k n + V a k n+1

+ k?1 V a k?1 n?1 + d

+ n?k V a k n?1 (8)

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For brevity of notation,it is convenient to express (5)–(8)using the shorthand notation V a=T a V a,where the operator T a is de?ned by the right-hand side of (5)–(8).

The expressions in(5)–(8)are structurally similar to the expressions in the single-party,dynamic decision-making problem(1)–(2).The differences arise because (5)–(8)capture the objective of a single patient and not the objective of the medical planner.We will now interpret these expressions by examining(8),which gives the most general case;the interpretation for Equations(5)–(7)follows a similar line of thinking. To explore the individual components of(8)notice that h/ +1+ + n is the expected waiting pay-off until the next transition,which occurs at a rate 1+ + n.Then,there are three possible transitions: organ arrival(scaled to rate1),patient arrival(rate ), and patient death(rate n).We shall systematically consider each possible type of transition,its proba-bility of occurrence,and its associated continuation payoff.The following are the possible cases.

(1)Organ Arrival.When an organ arrives,there are four possibilities from the perspective of the k th patient:(a)Its value exceeds a k?1 n ,which implies that it will be accepted by one of the?rst k?1patients leading into a reduction in the queue length and a reduction in the k th patient’s position.(b)Its value will be between a k?1 n and a k n ,implying that it will be accepted by the k th patient who will receive the reward and depart.(c)Its value will exceed a n n butwill be less t han a k n ,implying t hatitwill be accepted by one of the patients behind the k th patient leading into a reduction in the queue length.(d)Its value will be less than the threshold a n n ,implying that the organ is discarded and the state of the system is unchanged.

(2)Patient Arrival.This transition occurs with prob-ability / 1+ + n ,it leaves the position of the k th patient unchanged,and it increases the queue length by one.

(3)Patient Death.This occurs with rate n and there are three distinct possibilities:(a)One of the k?1 patients in front of the k th patient dies,advancing the position of the k th patient and decreasing the queue length by one(this transition occurs with probabil-ity k?1 / 1+ + n .(b)The k th patient dies with probability / 1+ + n and leaves the system with instantaneous payoff d.(c)One of the n?k patients behind the k th patient dies,decreasing the queue length by one,but leaving the k th patient’s position unchanged(this transition occurs with prob-ability n?k / 1+ + n ).

Having completed the derivation of the value func-tion V a and the operator T a,we are in a position to formally de?ne the equilibrium concept as follows: A strategy pro?le a F is a subgame perfectNash equi-librium if it attains the following supremum(where the supremum is taken over one component of the strategy pro?le at a time)

V F=sup

a k n a1 n ≥a2 n ≥···≥a n n ≥

b n

T a V F (9)

The?xed point V F is the continuation payoff for the equilibrium strategy a F the superscript F here rep-resents an abuse of notation and indicates that the priority rule is FCFS.The notation in(9)represents in

a shorthand notation the following substitutions into

(5)–(8).First,the value function V a is now replaced by V F.Second,the right-hand sides of(5)–(8)are max-imized as follows:(5)is maximized over a1 1 ,(6)is maximized over a1 n ,(7)is maximized over a n n , and(8)is maximized over a k n .The reader may verify that these right-hand expressions are quasi-concave,and thus each has a unique maximizer. For example,differentiating(5)with respect to a1 1 yields f a1 1 V a1 1 ?a1 1 ,demonstrating that(5) increases with a1 1 for a1 1 V a1 1 .Therefore,the theory of dynamic programming guarantees existence and uniqueness of the equilibrium value function V F,and quasi-concavity ensures that the corresponding equi-librium strategy a F is also unique.

In equilibrium,when the queue length is n,

t

he patient in position k will acceptorgans wit h qual-ity no less than a F k n .Further,that patient is free to change the decision threshold unilaterally,but such changes will notimprove his expect ed payoff.He is also free to unilaterally change his policy in every subsequent state,but again,this will not provide any improvement as long as the strategies of all other patients are unchanged.An improvement can be real-ized only when several patients agree to change their decision thresholds simultaneously.

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Having completed the characterization of the equi-librium strategy pro?le,we are now in a position to derive a simple expression relating the strategy pro-?le to the continuation payoffs.Speci?cally,as in the socially optimal case,it is straightforward to con?rm that the?rst-order optimality conditions are neces-sary and suf?cient(ignoring momentarily the bound-ary constraints a k n ≥a k+1 n ≥b n ),which implies that the following conditions hold:

a F k n =V F k n?1 (10)

a F n n =V F n n (11) Intuitively,these conditions state that a patient is indif-ferent between either accepting the marginal organ a F k n or passing it along to the next patient on the waiting list.If he passes it to the next patient,then that patient will accept it,and hence the queue length decreases by one leaving him with a continuation pay-off V F k n?1 .Similar interpretations are valid in the boundary cases,where the patient is either the only one waiting or the last one waiting.In these boundary cases,the patient’s position and queue length remain unchanged if the marginal organ is refused. Having characterized the equilibrium strategy pro-?le that emerges as a response to the medical plan-ner’s organ allocation policy,we can now take a step back and return to the following question:Given our prediction for the equilibrium strategy pro?le, what is the medical planner’s optimal treatment rationing strategy

b that would maximize the plan-ner’s objective?

The following proposition,proven in the appendix, provides the answer.

Proposition2.It is optimal to impose no treatment rationing by choosing b n ≡x for all n.

The main observation employed in the proof is that the medical planner’s decision thresholds b n appear in the equilibrium characterization(9)only as a constraint on the feasible decision thresholds for the patients.Therefore,explicitly restricting the range of available organs is ineffective,because its only impact would be to discard kidneys that would not have been otherwise discarded.Interpreted more negatively,this result states that rationing is not a use-ful policy instrument when patients are autonomous. Rationing corresponds to a form of service rate con-trol,because restricting the range of acceptable organs for transplant effectively modi?es the“service rate”of the organ allocation system.In conventional queue-ing models with adjustable service rates,it is optimal to dynamically increase the service rate(at a cost) when excessive work load builds up.Even in the sys-tem studied here“service rate”controls are effective when patients are not autonomous(see the results in§4).However,these controls becomes ineffective when patients are autonomous.The implicit thresh-olds implemented by the patients’equilibrium strat-egy are as effective as any explicit thresholds that can be imposed by the medical planner.

From a computational standpoint,Proposition2 simpli?es the derivation of the equilibrium strat-egy pro?le.Speci?cally,when the medical planner imposes no control on the range of feasible donor organs,patients no longer have to consider the size of the waiting list in their decision problem.The only relevant information is their position on the wait-ing list.This is in stark contrast to the case where the medical planner imposes controls,because then queue length matters because of its effect on the range of organs available to the patients.This implies that V F k n is independent of the queue length and depends only on the patient’s position k,thus we let V F k n =V F k and a F k n =a F k .With this simpli?ca-tion,it follows that the equilibrium strategy pro?le is derived as follows:

V F 1 =sup

a 1 ∈ x ˉx

1

· h+ F a 1 g a 1 + 1? F a 1 V F 1 + d

(12) V F k =sup

a k ∈ x a k?1

1

+1+ k

· h+ F a k?1 V F k?1 + F a k g a k

? F a k?1 g a k?1 + 1? F a k V F k

+ d+ k?1 V F k?1 (13) This characterization of the equilibrium strategy pro?le also reveals the main shortcoming of FCFS. First,note that absent any explicit control by the plan-ner,the system with autonomous patients achieves an implicit decision threshold such that only organs with quality greater than a F n are accepted when the queue length is n.However,these equilibrium

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291

thresholds obtained by (12)–(13)will deviate from the socially optimal thresholds b ? n .A more detailed examination of this ?nding will be pursued in the nextsect ion.

6.

Welfare Loss from Patient Autonomy

We will now compare the socially optimal deci-sion thresholds b ? n to the competitive equilibrium thresholds a F k identi?ed in §5.It will be shown that the latter can be obtained by solving a dynamic pro-gramming recursion analogous to that solved in the socially optimal case (1)–(2),but with patient arrivals ignored.Hence,the competitive thresholds are inef-?cient.The argument proceeds by considering the

aggregate value function V F k = k i =1

V F i ,and it shows that V

F k and a F k are obtained by solving (1)–(2),butwit h =0.This statement is made precise in the following proposition,which is proven in the appendix.

Proposition 3.Under FCFS,the solution a F

k to the optimality Equations (12)–(13)also solves (1)–(2),but with the arrival rate =0.

This result suggests that if the social planner is faced with a hypothetical system with an arrival rate of zero,the optimal controls she would choose for this hypothetical system would coincide with the thresholds that arise in competitive equilibrium under FCFS.That is,the welfare loss in the competitive sys-tem arises because the self-serving behavior of cur-rent patients ignores the welfare of future patients.It can also be shown that the competitive thresholds are higher than the socially optimal ones

a F n ≥

b ? n

(14)

implying that the competitive outcome is inef?cient because patients are too stringent in their choices,and thus generate congestion externalities.A formal proof for this result will be presented in §8,where a general class of priority rules and their corresponding equi-libria will be analyzed.

In summary,our analysis has shown that treat-ment rationing is an ineffective way to control patient behavior,and that the system is inef?cient under the commonly used FCFS rule.However,we shall show in the next section that social ef?ciency can be achieved if the priority rule is LCFS.

7.The Role of Prioritization

The analysis for the LCFS discipline follows the steps developed in §5with the exception that each new patient now arrives at the top of the line,shifting back the position of all other patients by https://www.wendangku.net/doc/174783509.html,ing the approach of §5,we can show that the medical planner will never restrict the range of organs offered to the patients,and hence,the value function for a patient in position k does not depend on the queue length.If we now let V L k denote the value function for a patient in position k and a L k denote that patient’s equilibrium acceptance threshold,then the dynamic programming recursion for the competitive equilib-rium is as follows:

V L 1 =sup

a 1 ∈ x ˉx

1

+1+ + · h + F

a 1 g a 1 + 1? F a 1 V L 1 + V L 2 + d

(15)

V L k =

sup

a k ∈ x a k ?1 1

+1+ + k

· h + F

a k ?1 V L k ?1 + F a k g a k ? F

a k ?1 g a k ?1 + 1? F a k V L k + V L k +1 + d + k ?1 V L k ?1 (16)

Unlike the FCFS system,in the LCFS system,arrivals

are taken into account because they cause each patient’s position to increase by one.One can proceed by considering the aggregate value functions and comparing the competitive acceptance thresholds under LCFS to the socially optimal thresholds.The following result shows that the aggregate value func-tion is obtained by solving the dynamic programming recursion (1)–(2),and hence,with the LCFS priority rule,the medical waiting system with autonomous patients is socially ef?cient.

Proposition 4.Under LCFS,the optimal strategy for the medical planner is to exercise no treatment rationing.The competitive equilibrium that emerges, a L k ,is socially optimal.

We shall defer the proof of this proposition until a more general version of this result is presented in §8.This result establishes that with LCFS the external-ities caused by patient choice are completely inter-nalized,because a patient who refuses an offer will

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drop on the waiting list when a new patient arrives. The threat of this reduction in position is suf?cient to align the patient’s behavior with the behavior desired by the medical planner,and thus the socially optimal

ideal can be attained.

Most of the existing literature on queueing mod-els with hidden information relies on monetary pay-ments to distinguish between customers of different (unobserved)types.The observation that prioritiza-tion alone is suf?cient to coordinate such queueing systems was?rst made informally by Hassin(1985), and to the best of our knowledge,Proposition4is the ?rstcase in which t his insightis rigorously proved. Further,previous work has focussed mainly on static equilibrium analysis in which individual decisions are made based on long-run average quantities(such as expected delay).Because these quantities are indepen-dent of the queueing discipline by virtue of Little’s Law,static models do not fully capture the power of priorities(within a single customer class).In this regard,our dynamic analysis provides the additional insightt hatan appropriat ely chosen queueing disci-pline can indeed completely eradicate incentive prob-lems among identical customers,as long as real-time system information is readily available and intelli-gently utilized.

Our analysis of the two polar extremes of LCFS and FCFS allows us to conclude that the former is ef?-cient,whereas the latter is not.However,the social ef?ciency of LCFS should be treated with caution because of strategic dif?culties associated with its implementation:without any form of monitoring,any person in line has the motivation to balk and reenter the system at the top of the line(see Hassin1985). Although we shall assume that this is not permitted, we also acknowledge the massive administrative costs involved with preventing such behavior,which could partly explain why LCFS systems are rarely observed in practice.

On the other hand,the result that FCFS is inef?-cient appears incompatible with the observation that it is commonly used in practice.In fact,apart from minor provisions made for exceptional cases,it is the primary prioritization scheme being used.While itis recognized in Larson(1987)t hatF CF S priorit i-zation,being a symbol of justice and equity,enjoys many advantages that go beyond economic welfare,our model highlights the inherent inef?ciency caused by the inability of FCFS to contain the externalities generated by patients’self-serving behavior.

8.The Equity-Ef?ciency Trade-off

The resultt hatLCF S achieves a socially ef?cientcom-petitive equilibrium,while FCFS suffers deadweight losses caused by externality problems brings to the forefront the trade-off between ef?ciency and equity: The priority rule that maximizes system ef?ciency is the one that deviates the most from FCFS,the acceptable standard for equity.While this tradeoff has attracted considerable attention,previous studies have focused on patient heterogeneity as the main driver behind it.Speci?cally,while an ef?cient policy would allocate organs to patients most likely to bene-?t from transplantation,this would be unjust because it would create disparities in access to transplantation between different ethnic and racial groups(see Zenios et al.2000).By contrast,the results developed in this paper demonstrate that a trade-off between ef?ciency and equity can exist even when patients are homo-geneous,because of patient choice and its interaction with the queueing priority rule.

To quantify this trade-off,we now consider a con-tinuum of priority rules called randomized absolute priority rules.In these rules,new patients will either receive absolute priority and skip to the head of the line or join the end of the line.Each new patient is granted absolute priority independently with proba-bility p∈ 0 1 ;we let a p k denote the equilibrium decision thresholds under this priority system.The case of p=0corresponds to FCFS and p=1cor-responds to LCFS.These prioritization schemes are probabilistic hybrids of the FCFS and LCFS rules and are thus natural candidates for analysis.

The reader could almostpredictour nextresult, which states that under the randomized absolute pri-ority rule with parameter p,the arrival rate when patients exercise choice is effectively p.The proof, presented in the appendix,relies on the aggregation argumentdescribed in§6.

Proposition5.Under randomized absolute priority with parameter p,the competitive equilibrium thresholds

a p k can be obtained by solving the optimality Equations

(1)–(2)with the arrival rate set at p.

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This proposition presents a continuum of cases covering both FCFS and LCFS prioritization,and con?rms the antithetical relationship between abso-lute priority and externalities.Absolute priority

embodies nuances of social injustice,while negative externalities,manifested through distortions in offer acceptance rates,lead to economic inef?ciency.The relationship between the absolute priority parame-ter p and the effective arrival rate p can thus be interpreted as a quantitative representation of the equity-ef?ciency trade-off,because absolute priorities provide a mechanism to minimize the impact of the negative externalities.

The effect of the priority parameter p on controlling the externalities can also be con?rmed in the follow-ing result,which states the impact of the parameter on the equilibrium thresholds.

Proposition6.Let0≤p≤p ≤1.Then,for each queue length n,

a p n ≥a p n (17) This proposition states that patients become less selective as the priority parameter increases and,con-sequently,as the threat of a reduction in their prior-ity following an organ refusal becomes more severe. Therefore,the social planner is able to mitigate the externality effects with randomized absolute prior-ity rules.Such regimes can vary in intensity accord-ing to the parameter p,and more importantly,can be justi?ed in several practical cases.(For example, emergency cases and pediatric patients may arguably deserve absolute priority.)On a broader level,our interpretations also suggest that these externality problems can be keptunder cont rol using a more gen-eral class of preemptive regimes that go beyond ran-domized absolute priority rules—regimes in which patients who decline an organ offer would expect a decrease in their priority position.

9.Numerical Study

In this section,we present results from a numerical study that has two main objectives.The?rst one is to illustrate that patient choice can signi?cantly degrade the performance of the transplant waiting list as mea-sured by average waiting time and expected patient reward,and to identify key system parameters that either exacerbate or alleviate such performance degra-dation.The second objective is to study the effect of absolute priorities on system performance.In particu-lar,it will be demonstrated that a system where a small fraction of the patients are granted absolute priorities, while the rests are prioritized according to FCFS,can recover most of the losses caused by the FCFS system. The parameters for the study are estimated from kidney transplant data obtained from UNOS(2002). The arrival rate is set at =200patients per year and the service rate is set at =100organs per year. These were obtained by scaling down(and rounding to the closest hundred)the corresponding?gures for the national waiting list by a factor of100,so they represent the waiting list maintained by a small OPO (there are59OPOs in the United States).Patient death rate is =0 124,calculated from mortality statistics of transplant candidates as reported in USRDS(2002) and in Wolfe etal.(1999).The following reward st ruc-ture is used as a basis for decision making.Following Zenios etal.(2000),a reward of h=0 6per year is assigned to patients on the waiting list.This implies that the relative quality of life on the waiting list is 60%of that for healthy individuals.The corresponding reward per unit time for patients who have received a transplant is taken to be0.75.Patient reward after death is d=0,measured in QALYs.

We next proceed to estimate the payoffs obtained from receiving a https://www.wendangku.net/doc/174783509.html,ing a USRDS data set of37,756kidney transplants performed between November1994and December1999,we?tan expo-nential survival model with covariates that include, for example,patient and donor tissue types,age, sex,and race.(See Venables and Ripley(1997)for an overview of the relevant survival analysis tech-niques.)For the average patient,we use this model to estimate survival under two extreme scenarios(best-possible kidney and worst-possible kidney)and then obtain the quality-adjusted life expectancy assuming a quality of life with transplant of0.75as above. Then,the best-case estimate is8.98years and the worst-case estimate is4.05years.Given these esti-mates,we assume that the random variable X rep-resenting the reward from a random organ offer is uniformly distributed between x=4 00andˉx=9 00. All rewards are discounted at a continuous rate = 3%.Notice that with this reward structure,a patient’s expected discounted QALYs from waiting inde?nitely

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is h/ + =3 89,hence even the worst kidney offer is attractive.

First,we address the effect of patient choice by comparing the performance of the waiting system under both FCFS and LCFS.FCFS is a proxy for the allocation system that now prevails in the United States.Our analysis allows us to compute system transition rates,stationary queue-length distributions,and value functions for both systems,and these quan-tities can then be used to compute the average waiting time,average queue length,average life expectancy,and welfare loss de?ned as the percentage differ-ence in QALY between FCFS and the socially optimal LCFS.Table 1summarizes the key performance met-rics.The results demonstrate that patient choice in the FCFS system increases the average queue length and average waiting time until treatment relative to LCFS by 17.8%and 20.4%,respectively.This is because a fraction of the organs allocated under LCFS are dis-carded under FCFS because they are refused by the patients on the waiting list.This organ wastage cul-minates into a 5.9%decrease in each patient’s quality-adjusted life expectancy.This seemingly small welfare loss translates into an annual loss of more than 7,000patient life years (because more than 23,328patients join the waiting list each year and the life expectancy of each patient is reduced by 0.3years).The eco-nomic consequences are also signi?cant:Because each of the approximately 9,636patients receiving cadav-eric transplants each year has to spend an additional 1.1years on dialysis with FCFS and because the cost of treatment on the waiting list is $45,000per patient per year,this translates into an approximate increase in health care costs of more than $476million per year.These ?gures are calculated based on waiting list statistics for 2002reported in OPTN (2003)and dialy-sis costs reported in Pastan and Bailey (1998),which are all expected to increase in the future.

Table 2

Dependence of Welfare Loss on System Parameters

Welfare loss

LCFS

FCFS (%)

Baseline scenario: =200 =100X ～U 4 9 5.20 4.89 5.90Increased scale: =2,000 =1,000X ～U 4 9 5.32 4.897.98Increased variability: =200 =100X ～U 4 10 5.45 5.08 6.79Increased mean: =200 =100X ～U 4 5 9 5 5.45 5.11 6.28Increased supply: =200 =125X ～U 4 9 5.52

5.16

6.57

Table 1Welfare Loss in Terms of Performance Metrics

Mean Waiting time Expected queue until transplant

Discarded survival length

(years)

kidneys (%)

(QALY)First-best LCFS 8065 4805 20FCFS

9506 6015 84 89%change under FCFS +17.8

+20 4

—

?5 90

Next,we examine the impact of different system parameters on welfare losses.We obtain the quality-adjusted life expectancy under four additional sce-narios,representing:(1)a bigger OPO,(2)increased variability in organ quality,(3)improved overall

organ quality,and (4)increased organ supply.The results are summarized in Table 2.

The following four observations can be made.

(1)A 10-fold increase in arrival and service rates is associated with an increase in welfare loss from 5.90%to 7.98%.Further,increased scale is associated with improved overall performance in LCFS (QALYs increase from 5.20t o 5.32),butnotin C S.This suggests that the statistical economies of scale that are frequently present in standard queueing systems and that explain the improved performance of the LCFS system in this scenario,also cause an increase in welfare losses under the FCFS system.Increased scale with FCFS makes organ refusals more appeal-ing for the patients on the top of the waiting list.The increased arrival rate elevates these patients’likeli-hood of receiving a better future offer and drives their decision to refuse organs of low quality.

(2)In the scenario representing increased vari-ability,the organ quality distribution is changed to U 4 10 ;an alternative is to use U 3 10 ,which has the same mean as the baseline case and higher vari-ance,but this creates boundary problems when the value of the organ is less than 3.89—the value of wait-ing in perpetuity.We will compare this case to the

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scenario representing increased mean quality(with organ quality distribution of U 4 5 9 5 ),because the mean organ quality is the same for both scenarios. We see that the welfare loss is slightly higher in an

environment of higher variability(6.79%compared to 6.28%),because the patients’option to wait becomes more valuable.

(3)An improvementin overall organ qualit y by increasing post-transplant QALYs uniformly by0.5 years improves quality-adjusted life expectancy in F CF S from4.89years t o5.11years,butitalso increases the total welfare loss.Further,the increased QALY under FCFS(5.11years)falls short of the socially opti-mal performance of the baseline system(5.20years). This implies that remedying the externality problems in the original system would have a larger welfare impactcompared t o an improvementin average organ quality.

(4)In the same vein,increasing the organ supply by a practically impossible25%leads to an increase in quality-adjusted life expectancy from4.89years to 5.16years in the FCFS system.This improvement has the same order of magnitude as the welfare gain from LCFS in the baseline system.That is,the welfare gain caused by adopting a system that aligns patient choice with social ef?ciency is equivalent to a welfare gain that could be achieved by a25%increase in organ supply.

In summary,these results suggest that the kidney transplantation system suffers from substantial wel-fare losses caused by pat ientchoice.While mostef-forts to improve the performance focus on supply-side strategies,our analysis suggests that the effectiveness of supply-side interventions pales in comparison to that of the demand-based priority rules proposed here. How much absolute priority should be granted?While the previous analysis suggests that the welfare gains obtained by switching from FCFS to LCFS are sub-stantial,there are several hurdles that would prevent the adoption of LCFS.The analysis in§8suggests that randomized absolute priority systems can recover part of the welfare losses,because patients internal-ize some of the externalities caused by their decision to decline an organ offer.We will now investigate the relationship between the fraction of patients receiv-ing absolute priorities,and the fraction of welfare loss that can be recovered.Figure1System Performance Under Various Degrees of

Absolute Priority

00.10.20.30.40.50.60.70.80.9 1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Randomized Absolute Priority Parameter, p

F

r

a

c

t

i

o

n

o

f

W

e

l

f

a

r

e

L

o

s

s

R

e

c

o

v

e

r

e

d

λ = 200,μ = 100

λ = 500,μ = 100

λ = 100,μ = 100

We start with the baseline scenario and investigate how the performance of the system(measured by the quality-adjusted life expectancy)changes as the frac-tion p of patients who receive absolute priority also changes.The solid curve in Figure1summarizes our results:the parameter p is re?ected on the horizon-tal axis and the fraction of welfare losses recovered is plotted on the vertical axis.As expected,the frac-tion of welfare loss recovered increases from0to1as we increase p from0(FCFS)to1(LCFS).Further,this curve is concave,indicating that the?rst few units of absolute priority are more effective than subsequent units.We note that90%of the welfare loss is recov-ered with p=0 41,and the socially ef?cient outcome is attained with p=0 69.

Next,we repeat the experiment and vary the supply-to-demand ratio.We consider the following two cases( =100 =100and =500 =100),and represent the results using dotted lines in Figure1. When demand is?ve times as much as supply,90%of the welfare loss is already recovered with p=0 10,and the socially ef?cient outcome is attained with p=0 17. When demand and supply are balanced,a random absolute priority parameter of p=0 72is required to recover90%of the welfare loss,and the socially ef?-cient outcome is attained only when p=0 98.This comparison demonstrates that as the supply-demand imbalance increases,the initial units of absolute pri-ority become more effective.Therefore,the limited

Su and Zenios:Patient Choice in Kidney Allocation:The Role of the Queueing Discipline 296Manufacturing&Service Operations Management6(4),pp.280–301,?2004INFORMS

use of randomized priority can improve social wel-fare in overloaded organ allocation systems,which are expected to become more common in the future.

10.Concluding Remarks

Despite the continued shortage of organs for trans-plantation,approximately12%of all kidneys pro-cured from deceased donors are routinely discarded. This wastage contributes to the already long waiting times and to the high cost of treatment for patients on the waiting list.It is also widely recognized that patient choice is a main driver behind this wastage. In fact,a recent report by UNOS(2002)states that “a subsetof deceased organs of marginal qualit y has a high discard rate after procurement because their placementis arduous and prolonged.”This observa-tion has motivated the development of an Expanded Criteria Donor policy that aims to expedite the place-ment of marginal organs by exploiting patient choice. This paper attempts to clarify the effect of patient choice on the organ allocation system by examining a stylized queueing model.It demonstrates that the pri-ority discipline used to rank patients on the transplant waiting list can exacerbate the wastage of organs.In fact,FCFS,an established standard of equity,increases patients’desire to refuse organs of marginal quality and aggravates the organ discard rate.By contrast, LCFS eliminates all externality problems and achieves optimal organ utilization.This?nding highlights a new dimension in the equity-ef?ciency dilemma.The mostequit able CF S policy is inef?cientbecause it exaggerates the externalities of patient choice.This ef?ciency loss is over and above the well-studied inef-?ciencies caused by FCFS’s failure to allocate organs to the patients more likely to bene?t from them(see Zenios etal.(2000)and references t herein). However,our?ndings are still several steps away from providing immediately applicable practical rec-ommendations beyond the general warning that “FCFS exacerbates the organ wastage caused by patient choice.”Speci?cally,LCFS is practically infea-sible,and the absence of patient heterogeneity in our model may create suspicion about the validity of our?ndings.A companion paper(Su and Zenios 2004a)explores the role of heterogeneity in more detail and shows that the inef?ciency of the FCFS rule remains relevant in that context.In another work (Su and Zenios2004b),we study a variant of the “multiserver FCFS”queueing discipline:kidneys are classi?ed into quality grades,and there is a separate FCFS queue for each grade of kidneys.The grades are designed so that better kidneys involve longer waits, and patients balance the trade-off between the qual-ity of the kidney and the waiting time by declaring which grade of kidney they wish to receive.Theoret-ical analysis demonstrates that this system mitigates the inef?ciencies of FCFS,and a simulation model that captures the realistic complexities of the kidney allo-cation environment(see Su et al.2004),demonstrates that the same system accelerates the placement of kid-neys and reduces organ wastage.

One of the objectives of this paper is to determine the impact of the priority discipline on the number of organs discarded because of patient choice.The numerical results demonstrate that under a FCFS dis-cipline,as many as15%of organs may be wasted because of patient refusals.This estimate compares favorably to the actual percentage of12%reported in clinical studies,and supports the hypothesis that a signi?cantnumber of organs are refused because of the prioritization system.However,the actual num-ber of organs discarded cannot be estimated precisely by our model.This is because the local waiting sys-tems that make up the national waiting list share organs,and thus the U.S.waiting system looks like a massive single waiting list—much different that the smaller waiting systems analyzed in§9.One may then argue that organ refusals will be rare in such a massive system,because the effects of the priority discipline will be dwarfed by the scale of the waiting system.While compelling,this argument is?awed: In a national waiting system,refusals cause an organ to be transported across regions,incurring transporta-tion delays that cause a deterioration in the quality of the organ.So even if the organ is not discarded fol-lowing refusals,it will be substantially inferior.How-ever even if one ignores the transportation delays, the numerical results in§9suggest that the welfare losses because of organ refusals increase with the scale of the system,and thus,they may be more pro-found in a massive national system compared to a smaller local system.Nevertheless,we do not wish to argue that our?ndings make a convincing case that much of the currently refused organs are because of

Su and Zenios:Patient Choice in Kidney Allocation:The Role of the Queueing Discipline

Manufacturing&Service Operations Management6(4),pp.280–301,?2004INFORMS297

the priority discipline.While our informal discussions

with clinicians in two transplant centers suggest that

patients frequently refuse organs because they recog-

nize that once they reach the top of the waiting list

they can afford to be selective,one of the reviewers

shared with us a different conversation with a trans-

plant team that contradicted our informal discussions.

Absent any primary unbiased data,the results in this

paper support the hypothesis postulated above,but

do notconvincingly prove it.

In summary,while it is desirable to expand the

role of patient choice in the organ allocation sys-

tem,poorly designed initiatives can have the oppo-

site of the intended effect.This paper highlights these

“unintended consequences”and identi?es contain-

ment mechanisms that help improve the utilization of

scarce organ resources.

Acknowledgments

The authors have bene?ted from stimulating discussions

with Haim Mendelson and Korok Ray.Partial?nancial

support was provided by the National Science Foundation

Grant SBER-9982446.This paper has bene?ted substantially

from the comments and suggestions of the Senior Editor

and three anonymous referees.

Appendix.Proofs

Proof of Proposition1.The proof shall proceed in three

steps.

Step1.In this step,we use value iteration to establish

that the relative value function de?ned as n ≡V n ?

V n?1 is nonincreasing in n.Let V k n denote the k th

iterate for the value function,with the supremums attained

by b k n ;the superscript here represents an abuse of nota-

tion because in the main part of this paper,superscripts in

the value function are used to represent different priority

rules.That is,starting arbitrarily with V0 n ≡0,we have,

for every n≥0

V k+1 0 =

+

V k 1

V k+1 n =sup

b n ∈ x ˉx

1

· hn+ F b n V k n?1 +g b n

+ 1? F b n V k n + V k n+1

+ n V k n?1 +d (18) Next,de?ne for k≥0and n≥1,

k n ≡V k n ?V k n?1 which,by the convergence of the value iteration algorithm, converges to n as k→ for every n≥1.

Now,to establish that n is nonincreasing in n,itsuf-?ces to show that k n is nonincreasing in n for every k≥0.We shall show this by induction.Notice that this holds trivially for k=0.For k≥0and n>1,elementary algebra shows that

k+1 n+1 =V k+1 n+1 ?V k+1 n

≤

1

+1+ + n+1

· h+ F b k n+1 k n

+ 1? F b k n+1 k n+1

+ k n+2 + n k n + d (19) k+1 n =V k+1 n ?V k+1 n?1

≥

1

+1+ + n+1

· h+ F b k n?1 k n?1

+ 1? F b k n?1 k n

+ k n+1 + n?1 k n?1

+ k n + d (20) To obtain the inequalities in(19)–(20),we use the fact that b k n+1 and b k n?1 are suboptimal in state n,and we uni-formize the transition rates.Inequalities(19)–(20),together with the inductive hypothesis,imply

k+1 n+1 ? k+1 n ≤

1

· 1? F b k n+1 k n+1 ? k n

+ F b k n?1 k n ? k n?1

+ k n+2 ? k n+1

+ n?1 k n ? k n?1

≤0 (21) Similarly,for the boundary terms,we have,for k≥0, k+1 2 ≤

1

+1+ +2

· h+ F b k 2 k 1 + 1? F b k 2 k 2

+ k 3 + k 1 + d (22) k+1 1 ≥

1

+1+ +2

h+ k 1 + k 2 + k 1 + d

(23) where(22)follows from taking n=1in(19),and transition rates are appropriately uniformized.Inequalities(22)–(23),

Su and Zenios:Patient Choice in Kidney Allocation:The Role of the Queueing Discipline 298Manufacturing&Service Operations Management6(4),pp.280–301,?2004INFORMS

together with the inductive hypothesis,imply

k+1 2 ? k+1 1 ≤

1

+1+ +2

· 1?F b k 2 k 2 ? k 1 + k 3 ? k 2

≤0 (24) Combining(21)and(24),our inductive proof is complete, and we have shown that n is nonincreasing in n.

Step2.Next,after removing terms that do not depend on b n ,observe that for each n≥1,b? n is the maximizer of:

V b n n ≡ F b n g b n ? n (25) This function satis?es the following increasing differences (ID)property:

V b ?V b is strictly increasing in ∈

for any b b ∈ x ˉx s.t.b

Step3.Finally,consider some arbitrary n n such that n

Case1. n = n

Here,b? n and b? n are maximizing the same objective function(25).If the solution is unique,then b? n =b? n ; otherwise,we can choose the same solution for both and we still have b? n =b? n .

Case2. n > n

Suppose for sake of contradiction that b? n

V b? n n ?V b? n n

>V b? n n ?V b? n n

but by the optimality of b? n and b? n in(25),we know that

V b? n n ?V b? n n

≤0≤V b? n n ?V b? n n which gives the desired contradiction.Therefore,b? n ≥b? n .

Therefore,we can always?nd an optimal policy b? n that is nonincreasing in n.The proof is complete. Proof of Proposition2.Consider an arbitrary control policy b n ,let V k n and a k n denote the equilibrium value function and patient decision thresholds.We shall begin by showing that b n cannot be optimal if there is some n for which

a n n >V n n and a n n =x (26)

The proof proceeds by contradiction.Consider the value of n for which(26)holds.In this case,we must have a n n =b n ;otherwise,the equilibrium cannot be sustained because a smaller a n n is feasible and will be chosen instead since a n n denotes the lowest acceptable quality, and V n n is the continuation payoff from waiting.This sim-ilarly implies that if b n is reduced to b n ? ,the equilib-rium value of a n n decrease and V n n will increase.Hence, the policy b n is therefore suboptimal.

This establishes that if b n is optimal,then a n n ≤V n n or a n n =x for every n.Since a n n

Proof of Proposition3.This follows as a corollary to Proposition5by using p=0.

Proof of Proposition4.This follows as a corollary to Proposition5by using p=1.

Proof of Proposition5.We shall begin by writing down the optimality equations under randomized absolute priority with parameter p.Let a p k denote the decision threshold that attains the supremum and V p k the optimal value function for a patient in position k,we can express the optimality equations as

V p 1 =

1

+1+ +

· h+ F a p 1 g a p 1 + 1? F a p 1 V p 1

+ pV p 2 + 1?p V p 1 + d (27) V p k =

1

+1+ + k

· h+ F a p k?1 V p k?1 + F a p k g a p k

? F a p k?1 g a p k?1 + 1? F a p k V p k

+ pV p k+1 + 1?p V p k

+ d+ k?1 V p k?1 (28) Now,let V p k =

k

i=1

V p i .Then,retaining the supremum only for the last term in each summation and substituting a p k into the other terms,we have

V p 1 =sup

a 1 ∈ x ˉx

1

· h+ F a 1 g a 1 + 1? F a 1 V p 1

+ p V p 2 ?V p 1 + 1?p V p 1 + d (29)

Su and Zenios:Patient Choice in Kidney Allocation:The Role of the Queueing Discipline

Manufacturing&Service Operations Management6(4),pp.280–301,?2004INFORMS299

V p k =1

+1+ +

· h+ F a p 1 g a p 1 + 1? F a p 1 V p 1

+ pV p 2 + 1?p V p 1 + d

+k?1

i=2

1

+1+ +i

h+ F a p i?1 V p i?1 + F a p i g a p i ? F a p i?1 g a p i?1 + 1? F a p i V p i + pV p i+1

+ 1?p V p i + d+ i?1 V p i?1

+sup

a k ∈ x a p k?1

1

+1+ +k

h+ F a p k?1

·V p k?1 + F a k g a k

? F a p k?1 g a p k?1

+ 1? F a k V p k + pV p k+1 + 1?p V p k + d

+ k?1 V p k?1

(30)

=

1

+1+ +k

· h+ F a p 1 g a p 1 + 1? F a p 1 V p 1

+ pV p 2 + 1?p V p 1 + d+ k?1 V p 1

+

k?1

i=2

1

+1+ +k

h+ F a p i?1 V p i?1

+ F a p i g a p i ? F a p i?1 g a p i?1

+ 1? F a p i V p i + pV p i+1

+ 1?p V p i + d+ i?1 V p i?1

+ k?i V p i

+sup

a k ∈ x a p k?1

1

+1+ +k

h+ F a p k?1

·V p k?1 + F a k g a k

? F a p k?1 g a p k?1

+ 1? F a k V p k + pV p k+1

+ 1?p V p k + d

+ k?1 V p k?1

=sup

a k ∈ x a p k?1

1

+1+ + k

· hq+ F a k g a k + V p k?1 + 1? F a k V p k

+ p V p k+1 ?V p 1 + 1?p V p k + kd

+

q

i=1

k?i V p i +

q

i=1

i?1 V p i?1 (31)

=sup

a k ∈ x a k?1

1

+1+ + k

· hk+ F a k g a k + F a k V p k?1

+ 1? F a k V p k + p V p k+1 ?V p 1

+ 1?p V p k + kd+ k V p k?1 (32)

Next,let J k =z+ V p k ,where z= p/ V p 1 .Then,from

the expression for V p 1 in(29),and after some algebra we

have

J 1 =sup

a 1 ∈ x ˉx

1

+1+ +

·

h+ F a 1 g a 1 +

p

+ p

J 1 + 1? F a 1 J 1

+ pJ 2 + 1?p J 1 +

d+

p

+ p

J 1

(33)

Similarly,using the expression for V p k in(32),we have

J k =sup

a k ∈ x a k?1

1

+1+ + k

· hk+ F a k g a k + F a k J k?1

+ 1? F a k J k + pJ k+1 + 1?p J k

+ kd+ kJ k?1 (34)

After removing“dummy”transitions,which were intro-

duced purely for purposes of uniformization,(33)–(34)

become

J 1 =sup

a 1 ∈ x ˉx

1

·

h+ F a 1

g a 1 +

p

+ p

J 1

+ 1? F a 1 J 1 + pJ 2 +

d+

p

+ p

J 1

(35)

J k =sup

a k ∈ x a k?1

1

+1+ p+ k

· hk+ F a k g a k + F a k J k?1

+ 1? F a k J k + pJ k+1 + kd

+ kJ k?1 (36)