University of Southern California Dissertations and Theses

Patient choice and wastage in cadaveric kidney allocation

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Patient choice and wastage in cadaveric kidney allocation

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Content PATIENT CHOICE AND WASTAGE IN CADAVERIC KIDNEY ALLOCATION
by
Junxiong Yin
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(BUSINESS ADMINISTRATION)
May 2024
Copyright 2024 Junxiong Yin

Dedication
Dedicated to my grandmother, Zelan Yang,
for her unconditional love and care.
ii

Acknowledgements
Even when I am writing the acknowledgements section, the completion of the PhD thesis still
feels surreal. The past six years have been a rollercoaster ride, filled with bewildering moments,
excruciating times and fulfilling sparks. I would never have been able to get through this journey
without the guidance and support from many people.
First and foremost, I want to express my deepest gratitude to my advisor, Peng Shi, without
whom the thesis would not have been possible. As one of the most brilliant and inspiring individuals I have ever met, Peng has helped me tremendously both academically and personally.
Knowing that he is always there rooting for me and guiding me has enabled me to overcome
the immense struggles and challenges that I have encountered throughout the six years. I deeply
appreciate his wisdom and patience, and I am truly fortunate to become his first PhD student.
Moreover, I would like to thank Kimon Drakopoulos and Afshin Nikzard for serving on my
dissertation committee and providing invaluable feedback along the way. Besides, I would like to
thank Alvin Roth, Ramandeep Randhawa and Hamid Nazerzadeh for serving on my qualifying
committee and being so encouraging when I embarked on my research journey. I am really
grateful to have the precious opportunity to learn from these world-renowned scholars.
Furthermore, I would like to thank Vishal Gupta for being a diligent and supportive PhD
coordinator. Together with Andrew Daw and Lanore Larson, they showed me what it takes to
iii

become a top-notch and empathetic educator. Additionally, the excellence could not be achieved
without the help and support of Andy Rivera, Rebeca Gonzalez, Julie Phaneuf and Karla Mayorga
at the backstage.
The six years would have been much less fun and more isolated, especially during the pandemic, without the former and my fellow PhD students: Wilson Lin, Michael Huang, Simeng
Shao, Aikaterini Giannoutsou, Sebnem Manolya Demir, Justin Mulvany and Yiqiu Shen. I will
always cherish the sincere camaraderie and precious memories we have built. Beyond the department, I would like to thank my friends in Hong Kong, San Francisco and Los Angeles for
having shared my ups and downs despite sometimes the time zone difference.
Finally, I would like to thank my parents for their unwavering support and firm belief in my
completing the PhD. I would also like to thank Andy Xu for the companionship and support that
have shielded me from anxiety and loneliness. Six years ago, when I got out of the plane and
stepped out of LAX for the first time, I could never have imagined how my life would unfold. But
here I am, at the intersection, ready for the next chapter.
This work was supported in part by Health Resources and Services Administration contract 234-
2005-370011C. The content is the responsibility of the author(s) alone and does not necessarily reflect
the views or policies of the Department of Health and Human Services, nor does mention of trade
names, commercial products, or organizations imply endorsement by the U.S. Government.
The data reported here have been supplied by the United Network for Organ Sharing as the
contractor for the Organ Procurement and Transplantation Network (OPTN). The interpretation and
reporting of these data are the responsibility of the author(s) and in no way should be seen as an
official policy of or interpretation by the OPTN or the U.S. Government.
iv

Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Waste despite Scarcity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Existing Interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Targeting Over Reliance on Biopsy Results . . . . . . . . . . . . . . . . . . 5
1.2.2 Targeting Excessive Risk Aversion of Doctors . . . . . . . . . . . . . . . . 6
1.2.3 Targeting Congestion in Allocation Process . . . . . . . . . . . . . . . . . 7
1.3 Thesis Outline and Research Contributions . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Patient Choice: Predicting Offers for Each Patient . . . . . . . . . . . . . . 8
1.3.2 Wastage: Analyzing a Stylized Model of Organ Allocation . . . . . . . . . 10
1.3.3 Wastage: Developing an Allocation Mechanism for Practice . . . . . . . . 12
1.A Additional Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter 2: Patient Choice: Predicting Offers for Each Patient . . . . . . . . . . . . 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Suboptimality in Patients’ Kidney Acceptance . . . . . . . . . . . . . . . . 17
2.1.2 Source of Suboptimality: Offer Misprediction . . . . . . . . . . . . . . . . 19
2.2 Personalized Offer Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 Computing the Necessary Waiting Time from Data . . . . . . . . . . . . . 25
2.3 Doctors’ Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Computing the Offer Rate for a Patient Group from Data . . . . . . . . . . 29
2.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1 Data Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.2 Prediction Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
v

2.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.A Additional Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.B Heterogeneity in Patients’ Donor Access . . . . . . . . . . . . . . . . . . . . . . . 41
Chapter 3: Wastage: Analyzing a Stylized Model of Organ Allocation . . . . . . . 44
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 A Stylized Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.1 The Status Quo: the Waitlist with Choice . . . . . . . . . . . . . . . . . . . 50
3.2.2 Eliminating Waste: Running Lotteries for the Young Organs . . . . . . . . 53
3.2.3 Eliminating Waste: Randomizing the Patients’ Priorities . . . . . . . . . . 55
3.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.1 Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2 Allocation Mechanisms and Outcomes . . . . . . . . . . . . . . . . . . . . 58
3.3.2.1 Matching Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.2.2 Equilibrium Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.3 Evaluating Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.4 Allocating Everything upon Entry . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Eliminating Waste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 Cost of Eliminating Waste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.6.1 Heterogeneity in Beta and Unexpected Departures . . . . . . . . . . . . . 71
3.6.2 Approximation of the Allocation Mechanism in Practice . . . . . . . . . . 72
3.A Omitted Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.A.1 The Equilibrium Outcome of the Waitlist with Choice . . . . . . . . . . . . 72
3.A.2 The One-shot CEEI Never Exhibits Waste . . . . . . . . . . . . . . . . . . 74
3.A.3 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Chapter 4: Wastage: Developing an Allocation Mechanism for Practice . . . . . . 82
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 Periodic Boost Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Simulation Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.1 Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3.3 Cycling in the Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3.4 Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.4.1 Allocation Outcome and Waitlist Size . . . . . . . . . . . . . . . 98
4.3.4.2 Kidney Wastage . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.4.3 Welfare Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4.1 Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4.2 Calibrating the Utilities Using the Status Quo . . . . . . . . . . . . . . . . 104
4.4.3 Periodic Boost Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.4.3.1 Gap from the Optimal Allocation . . . . . . . . . . . . . . . . . . 112
vi

4.4.3.2 The Horizontal Differentiation Assumption . . . . . . . . . . . . 117
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.A Other Approaches Tried to Address the Cycling Issue . . . . . . . . . . . . . . . . 123
4.B Convergence of the Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Chapter 5: Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.1 Generalizable Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
vii

List of Tables
2.1 The table shows the performance of the doctors’ prediction on the training data,
based on grouping the patients by their blood type, allocation point range and
EPTS score range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 The table shows the performance of the doctors’ prediction on the training data,
based on grouping the patients by their blood type, allocation point range, EPTS
score range, prior liver transplant, age at registration (before or after 18 years
old), and willingness to accept both kidneys. . . . . . . . . . . . . . . . . . . . . . 32
2.3 The table shows the performance of the personalized offer prediction on the
training data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 The table shows the patients’ utilities for being matched to each organ type in
the stylized example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 The table shows the results of calibrating the utilities by varying the intercept. . . 106
4.2 The table shows the simulation results of the status quo, the regular periodic
boost policy and the boost upon entry with different parameter combinations. . . 109
4.3 The table shows the simulation results of the status quo with additional columns
showing the waste and welfare change calculated by removing the idiosyncratic
shock added to the utility of each patient-kidney type pair. . . . . . . . . . . . . . 119
4.4 The table shows the simulation results of the status quo and the boost upon entry
with different parameter combinations when there is no shock for the utility of
each patient-kidney type pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.5 The table shows the evolution of the convergence metrics and the waitlist size as
more iterations are simulated for the status quo with an intercept of 4.0. . . . . . 125
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List of Figures
1.1 The figure shows the annual numbers of new patients, kidneys recovered for
transplantation and kidneys transplanted, spanning the years from 1995 to 2022. . 3
1.2 The figure shows the kidney disposition by Kidney Donor Profile Index (KDPI)
range based on the recovered deceased donors in 2019. . . . . . . . . . . . . . . . 4
1.3 Distribution of the number of patient classifications used for each donor, based
on the 2019 kidney match-run data. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Distribution of the proportion of waiting time points in the total allocation
points for each patient-donor pair, based on the 2019 kidney match-run data. . . . 14
2.1 The figure shows the distribution of the expected waiting time for a better kidney
than the best rejected one for the patients over 60 years old, who account for
about 1/3 of the waitlist, based on the deceased donors recovered from 2016 to
2019. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 The panel shows the number of local donors within each KDPI range available
to the patients A, B and C as their waiting time increases, respectively, based on
the deceased donors recovered in 2019. . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 The figure shows the distribution of donor antigens at the DR locus, based on
the deceased donors recovered in 2019. To maintain clarity, only the antigens
with a frequency greater or equal to 100 are shown. . . . . . . . . . . . . . . . . . 22
2.4 The figure shows the actual offer numbers compared with the predicted offer
numbers by the best doctors’ prediction during the training phase, which uses
the match-run data from the past year and groups the patients by their blood
type, allocation point range and EPTS score range. . . . . . . . . . . . . . . . . . . 33
2.5 The figure shows the best personalized offer prediction during the training
phase, which uses the match-run data from the past year. . . . . . . . . . . . . . . 34
ix

2.6 The figure shows the mean offer numbers and MAE of each prediction approach
on the training data for the offer numbers from donors in each quality range for
each patient blood type group. The lower bound is obtained by calculating the
MAE under a Poisson distribution assumption for the offer numbers. . . . . . . . 34
2.7 The figure shows the percentage decrease in MAE by the personalized offer
prediction relative to the doctors’ prediction on the training data. . . . . . . . . . 35
2.8 The figure shows the doctors’ prediction on the testing data, which uses the
match-run data from the previous year and groups patients by their blood type,
allocation point range and EPTS score range. . . . . . . . . . . . . . . . . . . . . . 36
2.9 The figure shows the personalized offer prediction on the testing data, which
uses the match-run data from the previous year. . . . . . . . . . . . . . . . . . . . 36
2.10 The figure shows the mean offer numbers and MAE of each prediction approach
on the testing data for the offer numbers from donors in each quality range for
each patient blood type group. The lower bound is obtained by calculating the
MAE under a Poisson distribution assumption for the offer numbers. . . . . . . . 37
2.11 The figure shows the percentage decrease in MAE by the personalized offer
prediction relative to the doctors’ prediction on the testing data. . . . . . . . . . . 38
2.12 The figure shows the distribution of donor antigens at the A locus, based on the
deceased donors recovered in 2019. To maintain clarity, only the antigens with a
frequency greater or equal to 100 are shown. . . . . . . . . . . . . . . . . . . . . . 41
2.13 The figure shows the distribution of donor antigens at the B locus, based on the
deceased donors recovered in 2019. To maintain clarity, only the antigens with a
frequency greater or equal to 100 are shown. . . . . . . . . . . . . . . . . . . . . . 41
2.14 The panel shows the number of local donors within each KDPI range available
to the patients A’, B’ and C’ as their waiting time increases, respectively, based
on the donors recovered in 2019. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1 The figure shows that the set of organs available to a patient expands with her
waiting time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 The patients’ choice set under the waitlist with choice. . . . . . . . . . . . . . . . 52
3.3 The equilibrium outcome under the waitlist with choice. . . . . . . . . . . . . . . 52
3.4 The equilibrium outcome under the one-shot CEEI. . . . . . . . . . . . . . . . . . 54
3.5 The equilibrium outcome under the one-shot RSD. . . . . . . . . . . . . . . . . . . 55
x

3.6 The figure shows the geometric intuition for D(γ) being strictly increasing. . . . 79
4.1 The figure shows how a type i patient’s points for type j donors change with her
waiting time under the status quo allocation mechanism. . . . . . . . . . . . . . . 86
4.2 The figure shows how a type i patient’s points for type j donors change with her
waiting time under the periodic boost policy. . . . . . . . . . . . . . . . . . . . . . 87
4.3 The figure shows the distribution of the number of patients offered in the focal
OPO for the donors who were ever offered to the focal OPO, based on the 2019
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4 The figure shows the distribution of the update rates for patients’ acceptance
thresholds α, based on the 50000 simulated values. . . . . . . . . . . . . . . . . . . 104
4.5 The figure shows the waitlist evolution of the status quo in the simulation with
the intercept of 4.0 in the utility function. . . . . . . . . . . . . . . . . . . . . . . . 107
4.6 The figure shows the actual allocation matrix and the simulated allocation matrix
with the intercept of 4.0 in the utility function by patients’ age group. . . . . . . . 108
4.7 The figure shows how the proportion of acceptable kidneys out of the reasonable
kidneys changes with the waiting time for a blood type O patient under different
mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.8 The figure shows the upper bound allocation matrix and the allocation matrix
under the boost upon entry where the boost period is 1 month and the boost
points are 100000. The upper bound allocation matrix is obtained by solving
the optimization problem 4.13 with the objective of maximizing the equivalent
increase in kidney supply, where the patient types are grouped by their blood
types and age ranges, and with an additional constraint of Pareto improvement
over the status quo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.9 The figure shows the welfare change by patient blood type and age range under
each version of the boost upon entry relative to the status quo when there is no
shock for the utility of each patient-kidney type pair. . . . . . . . . . . . . . . . . 121
xi

Abstract
In most countries, there is a shortage of cadaveric kidneys, but many successfully procured and
medically tenable kidneys are being discarded. The wastage of cadaveric kidneys exacerbates
the shortage in kidney supply and the financial strains on healthcare systems. Although many
interventions have been proposed or are being implemented, the reduction in kidney wastage has
been limited. Therefore, in this thesis, we take on a futuristic perspective and study the research
question: what more is needed to eliminate waste?
First, after observing patients’ suboptimal choices regarding offer acceptance, we build a personalized offer prediction tool by leveraging the observed donor allocation cutoffs. This tool
enables the doctors and patients to assess their offer prospects more accurately and therefore
make better acceptance decisions.
Second, to provide theoretical insights on eliminating waste through mechanism design, we
analyze a stylized model to characterize the waste-eliminating mechanisms and identify the inevitable trade-offs of eliminating waste.
Finally, to connect the theoretical insights to practice, we propose a mechanism with a minor
departure from the status quo that reduces waste and improves patient welfare when the degree of
horizontal differentiation among the patients’ preferences is small. To examine the effectiveness,
xii

we develop a simulation engine and evaluate the mechanism using the actual kidney transplant
data.
xiii

Chapter 1
Introduction
In the United States, on average, patients are removed from the national kidney transplant waitlist
at a rate of about one per hour, either because they become too sick to transplant or because they
die while waiting for a transplant.∗ The patients on the waitlist typically suffer from end-stage
renal failure, for which dialysis and transplantation are the two primary treatment options. Compared to dialysis, transplantation offers better patient well-being and better long-term survival at
lower costs [35]. Currently, nearly 80% of the kidney transplants come from the deceased donors,
who usually died from strokes, heart attacks or car accidents and had generously consented to
donate their organs. Treated as precious common goods, these cadaveric kidneys are recovered
and allocated centrally by the Organ Procurement and Transplantation Network (OPTN) under
contract with the United Network for Organ Sharing (UNOS).
The kidney allocation system orchestrates the allocation process, which is a complex scoring
and ranking system [41]. Specifically, once procured, a deceased donor’s kidneys are offered to
all active patients on the waitlist. To stay active, patients must regularly provide their blood
samples and inform their listing centers of any updates, such as changes in the dialysis status,
∗For descriptive data on the national organ transplant waitlist, please refer to https://optn.transplant.hrsa.gov/
data/view-data-reports/national-data/.
1

address or insurance. The active patients are ranked along two dimensions. First, the patients are
categorized into different classifications determined by their physical conditions, compatibility
with the donor and proximity to the donor. For example, the patients who are difficult to match
yet perfectly compatible with the donor, and who live close to the donor will be prioritized.
Second, within each classification, the patients are ranked by their allocation points. Besides
physical conditions and compatibility with the donor, the patients who have been on the waitlist
longer accumulate more points and have higher priority.
To gain priority in the kidney allocation system, patients can strategically alter their physical
conditions and increase their hardness-to-match through medication. In comparison, accruing
longer waiting time is a more accessible lever, and patients are not penalized for rejecting offers.
Indeed, patients’ waiting times significantly influence their allocations. On average, for a donor,
only around 3 classifications are used, and more than 90% of patients’ allocation points come from
their waiting times (see Appendix 1.A for more details). Therefore, patients are incentivized to
wait for the good kidneys and bypass the bad ones.
The combination of the waiting incentive inherent in the kidney allocation system and various
logistical issues leads to a significant waste of cadaveric kidneys despite their scarcity.
1.1 Waste despite Scarcity
Currently, around 90,000 patients are on the waitlist with an expected waiting time of about 3
years, and roughly 40,000 new patients are added to the waitlist annually (see Figure 1.1). Although there was a significant decrease in the number of new patients in 2020 because of the
pandemic, the number of new patients has gradually returned to the pre-pandemic level. To
2

avoid the influence of the pandemic, the analysis in the thesis is based on the data prior to 2020.
In comparison, only around 25,000 transplants are performed each year, and more than 8,000
patients are removed from the waitlist annually due to deteriorating health conditions.
1995 2000 2005 2010 2015 2020
Year
0
10000
20000
30000
40000
50000
Count
Annual Kidney Demand, Supply and Transplants (1995-2022)
Number of new patients
Number of kidneys recovered for transplantation
Number of kidneys transplanted
Figure 1.1: The figure shows the annual numbers of new patients, kidneys recovered for transplantation and kidneys transplanted, spanning the years from 1995 to 2022.
Paradoxically, despite the shortage of cadaveric kidneys, a serious wastage problem is present
in the allocation process. Around 20% of the successfully procured cadaveric kidneys are discarded [19]. However, based on all observable characteristics, the transplanted and discarded
kidneys have a large overlap in quality [28]. Moreover, as shown in Figure 1.2, even the kidneys
of reasonable quality are discarded at non-trivial rates. The figure shows the disposition of the
kidneys recovered in 2019 by the KDPI range. The Kidney Donor Profile Index (KDPI) score combines various donor factors to summarize the risk of post-transplant graft failure in an average
adult recipient and is widely used in practice by the doctors to assess kidney quality. The lower
the KDPI, the better the kidney. Overall, we observe that as the KDPI score increases or kidney
3

quality declines, the discard rate increases. Especially for the kidneys with a KDPI score above
80, often referred to as the marginal kidneys, they are discarded at rates ranging from 40% to
70%. However, studies have shown that compared with remaining on dialysis and waiting for a
low-KDPI kidney, accepting these marginal kidneys offers significantly better long-term survival
and patient well-being. Even the kidneys of more reasonable quality, with a KDPI score of 20-80,
are discarded at non-trivial rates ranging from 5% to 30%. These observations indicate that many
of the discarded kidneys could have been utilized to save lives but were wasted.
[0,10) [10,20) [20,30) [30,40) [40,50) [50,60) [60,70) [70,80) [80,90) [90,100]
KDPI Range
0
500
1000
1500
2000
2500
3000
Number of kidenys
Transplanted
Discarded
0%
10%
20%
30%
40%
50%
60%
70%
80%
Discard rate
Kidney Disposition by KDPI Range in 2019
Discard rate
Figure 1.2: The figure shows the kidney disposition by Kidney Donor Profile Index (KDPI) range
based on the recovered deceased donors in 2019.
The wastage of cadaveric kidneys imposes an additional burden on the healthcare system.
The patients with end-stage renal failure constitute only 1% of the US Medicare population but
account for 7% of the total Medicare budget. In 2019, Medicare covered over $90,000 per year for
each dialysis patient, whereas only around $35,000 per year for each transplant recipient [48].
4

Therefore, reducing the wastage of cadaveric kidneys can simultaneously save lives, reduce
costs and ameliorate suffering.
1.2 Existing Interventions
Given the benefits of kidney transplantation and the scarcity of cadaveric kidneys, why are so
many kidneys wasted? The OPTN/SRTR 2019 Annual Data Report [19] highlighted that for the
cadaveric kidneys recovered for transplantation but ultimately discarded, the most prevalent reasons were “no recipient located, list exhausted” and “biopsy findings”. Given that tens of thousands of patients are on the waitlist, it is surprising to see “no recipient located, list exhausted”
as a common reason for discarding the recovered kidneys. A substantial medical and operations
literature has delved into different facets of the wastage problem. The proposed and implemented
interventions mainly target the following three aspects: over reliance on the biopsy results, excessive risk aversion of the doctors and congestion in the allocation process.
1.2.1 Targeting Over Reliance on Biopsy Results
15% of the cadaveric kidneys recovered for transplantation were discarded due to poor biopsy
findings [19]. However, recent medical literature has raised concerns about the reliability of the
biopsy results in determining kidney acceptance, prompting the recommendation to reexamine
the routine use of biopsy results. First, an extensive literature review found no consistent associations between the donor biopsy findings and post-transplant outcomes [64]. Second, the
on-call pathologists may not always possess the necessary expertise to accurately interpret the
5

biopsy results [6], and the general pathologists have been observed to overestimate the severity
of chronic lesions than the renal pathologists, leading to an overestimation of risks [12].
1.2.2 Targeting Excessive Risk Aversion of Doctors
Despite an extensive literature justifying and encouraging the utilization of kidneys with unfavorable characteristics, the doctors’ risk aversion †
still contributes to the kidney wastage [55].
The doctors tend to err on the side of caution with the marginal kidneys due to cognitive biases,
moral hazard, and monitoring pressure.
The cognitive biases against the marginal kidneys have been documented in the literature.
The use of the Expanded Criteria Donor (ECD) label itself has been found to lead to increased
hesitance to transplant such kidneys by the doctors and patients [15]. Sharif [53] pointed out
that the risk aversion among the doctors may be a manifestation of the prospect theory or stem
from the time-pressured nature of clinical decision-making. Besides, the patients further down
the waitlist may reject an offer if those at the top have already done so, inferring it to be of low
quality [66], although it has been found that the number of times a kidney has been rejected is
not independently and meaningfully indicative of the eventual patient survival or graft survival
outcomes [23]. To target these cognitive biases, decision support tools such as risk-adjusted posttransplant survival models have been developed to help the doctors and patients assess kidney
quality more objectively [54].
The opacity and misaligned incentive in the decision-making process create room for moral
hazard. Many deceased kidneys that were eventually transplanted were turned down many
†Alternatively one may interpret the risk aversion as risk seeking, as the doctors are waiting for the better
kidneys which demand longer waiting times. In the thesis, we consider this behavior to be risk aversion with respect
to the doctors’ caution with the marginal kidneys.
6

times by the transplant centers without the patients knowing [16]. As Montgomery [31] sharply
pointed out, “if a patient dies during or after transplantation, it’s the doctor’s responsibility; if
the patient dies from organ failure while awaiting a transplant, we can blame the indifference of
the universe”. To reduce the room for moral hazard, some centers have started involving patients
in the decision-making process so that patients’ preferences are better internalized.
Furthermore, in the US, the transplant centers were previously only evaluated based on their
post-transplant graft and patient survival rates, which had to be very high for the transplant
centers to receive Medicare funding and renew their OPTN membership. This has incentivized
the doctors to be extremely risk-averse and only accept the highest quality kidneys. Recognizing
this, many regulatory reforms have since been implemented to de-emphasize the post-transplant
graft and patient survival rates and account for the risks in the marginal kidneys when evaluating
the outcomes, so that the doctors who take higher risks are not penalized as harshly [38].
1.2.3 Targeting Congestion in Allocation Process
As the quality of a kidney deteriorates rapidly once it is stored on ice after procurement and
the maximum acceptable time between procurement and transplantation is 48 hours [37], the
allocation process is a race against time.
Nevertheless, over time, the congestion in the allocation process has gradually been ameliorated. Various IT infrastructures have been established to accelerate the offering process by
contacting patients in parallel and making offers in batches. Transplant centers can now utilize
data-driven offer filters to instantaneously turn down the offers they are unlikely to accept [26].
7

Policies such as the ECD policy and the Kidney Accelerated Placement (KAP) project are established to fast-track the marginal kidneys to the patients who have indicated willingness to
accept or to the transplant centers with a history of accepting such kidneys. Organ sharing
across regions also becomes easier now by the improved transplantation networks [65] and policy changes [63]. As a result, a procured cadaveric kidney can now be offered to thousands of
patients on the waitlist within the crucial 48-hour window.
1.3 Thesis Outline and Research Contributions
However, despite the wide recognition, extensive research and numerous interventions, the reduction in kidney wastage has been limited. Therefore, in this thesis, we take on a futuristic
perspective and ask the research question: what more is needed to eliminate waste?
To answer this research question, we take three steps, corresponding to the three main chapters of the thesis. The first chapter focuses on how to predict the offer prospects for each patient
so that the doctors and patients can make optimal choices regarding offer acceptance. The last
two chapters look at the wastage problem from the perspective of mechanism design and focus on
the theory and practice of designing a better allocation mechanism to reduce waste, respectively.
1.3.1 Patient Choice: Predicting Offers for Each Patient
In the kidney match-run data, we observe suboptimality in the patients’ kidney acceptance decisions. Around 30% of the patients over 60 years old, who account for about 1/3 of the waitlist,
are waiting for the good kidneys that require more than 5 years of waiting, a duration alarmingly
close to the life expectancy of these patients.
8

Through conversations with the doctors, we learned that the suboptimality in patients’ kidney
acceptance partly originates from offer misprediction, besides other logistical issues. Currently,
when deciding whether to accept a kidney offer within a short time frame, doctors rely on their
impressions of the received offers for other similar patients to gauge the continuation value of
waiting. However, this prediction approach does not fully account for the heterogeneity in patients’ offer prospects due to the subtle differences in patients’ physical conditions, leading to
inaccurate offer predictions and suboptimal acceptance decisions.
In this chapter, to address the offer misprediction, we develop an interpretable and personalized offer prediction tool that is more accurate than the doctors’ method. Specifically, to predict
the offer prospects for a patient, we examine the allocation cutoffs of the past donors and derive
the patient’s classification and allocation points for each donor according to the OPTN policy.
By doing so, we estimate the necessary waiting time for the patient to receive an offer from each
donor and hence construct the donor access trajectory for the patient as she waits longer on
the waitlist. This donor access trajectory allows us to provide a personalized prediction for the
patient based on her unique circumstances.
This chapter contributes to the literature by introducing an organ offer prediction approach
that does not require extrapolation across similar patients, fully accounts for the allocation system and continuously adapts to the changes in patients’ allocation points. In the organ offer
prediction literature, most work applies the widely used machine learning and statistical modeling techniques to predict the quantities and qualities of organ offers based on observational
data, such as random forest models [9], cox regressions [47] and Poisson regressions [22]. However, these methods are blind to the allocation system and require extrapolation among similar
patients without fully considering patients’ heterogeneity. Moreover, defining “similar patients”
9

requires a trade-off between patients’ heterogeneity and statistical power. In comparison, our
approach directly replicates the allocation system and predicts for each patient without the need
for extrapolation and loss of statistical power. In a work similar to ours, Jalbert et al. [20] provide
predictions at discrete time points, whereas our approach continuously adjusts for changes in
patients’ allocation points over time.
1.3.2 Wastage: Analyzing a Stylized Model of Organ Allocation
In this chapter, we study a stylized model that abstracts away from organ and patient characteristics to obtain theoretical insights on the wastage problem. A market is formalized as a set
of arriving organs with different types and arriving patients with heterogeneous preferences,
which specify patients’ utilities from getting each organ type. Due to deteriorating health conditions, patients become ineligible for transplantation after an exponentially distributed time. An
allocation mechanism specifies what patients can choose and what they will get. The current
allocation mechanism is approximated by a waitlist with choice, which prioritizes patients only
based on their waiting times and do not penalize patients for rejecting offers. The equilibrium
outcome of an allocation mechanism is characterized by three conditions: patients maximizing
utility, demand not exceeding supply and mechanism not withholding items. Moreover, an allocation mechanism exhibits waste if in the equilibrium outcome, there are organs being discarded
and patients leaving unmatched, although they could have benefited from the discarded organs.
By analyzing the stylized model, this chapter provides theoretical guidance on how to eliminate waste and makes the following contributions.
10

First, we characterize the necessary and sufficient condition for an allocation mechanism to
eliminate waste in all markets: Such a mechanism must reserve over-demanded organ types only
for the patients who recently joined the market, while keeping under-demanded types available
to everyone. If a mechanism does not satisfy this condition, then there exists a market in which
it exhibits waste. For example, in the current kidney allocation mechanism, patients have better
chances of getting higher quality kidneys by waiting longer. However, waiting is risky, since
patients’ health conditions can deteriorate at any moment, which makes them not able to receive
transplantation when the desired organs become available. Therefore, to eliminate waste, an
allocation mechanism needs to make waiting no longer attractive and make under-demanded
organs available to everyone at all times.
Second, we identify the necessary costs of moving to an allocation mechanism that eliminates
waste in all markets. Specifically, allocating over-demanded organs upon patients’ entry to make
waiting no longer attractive can be contentious, as it violates first-come first-served and requires
introducing other means to differentiate patients beyond their waiting times. Moreover, eliminating waste may not result in Pareto improvement over the status quo, as the patients who are
the most willing to wait may be worse off.
Finally, we derive the above insights in a model that allows rich preference heterogeneity. For
tractability reasons, previous theoretical studies of waitlist mechanisms have relied on restrictive
assumptions about agents’ preferences, such as completely horizontal preferences or completely
vertical preferences. In contrast, the modeling framework in this chapter remains tractable even
when patients’ preferences for various organ types are arbitrarily correlated, which makes the
theoretical insights more robust.
11

1.3.3 Wastage: Developing an Allocation Mechanism for Practice
To connect the theoretical insights to practice, in this chapter, we propose an allocation mechanism based on the status quo that reduces waste and improves the patient welfare when the
degree of horizontal differentiation among the patients’ preferences is small. Specifically, a cyclical boost is added to the current allocation system such that patients are boosted periodically to
the front of the waitlist, and this class of mechanism is referred to as the periodic boost policy.
Under the periodic boost policy, patients are incentivized to accept marginal kidneys when approaching the end of their boosted period, as the probability of getting good kidney offers is low,
and until approaching the next boost cycle, as patients expect to be boosted again and receive
good kidney offers. When patients are always boosted, the periodic boost policy recovers the
status quo; when patients are boosted only once upon entry, the periodic boost policy is referred
to as boost upon entry.
To evaluate the performance of the proposed mechanism compared to the status quo, we build
a simulation engine and conduct simulations with the actual kidney transplant data. Specifically,
the simulation process entails two steps. First, we simulate the allocation process using patients’
acceptance thresholds to obtain the donor cutoffs. Second, with the observed donor cutoffs, patients reassess their probabilities of getting offers from each donor type and solve the optimal
stopping problem to update their acceptance thresholds. We do these two steps iteratively until
the patients’ acceptance thresholds stabilize.
The first contribution of this chapter is introducing a practical mechanism with a minor
change from the status quo that reduces waste. Many mechanisms from the literature eliminate waste, such as the last-come first-served [56], one-shot competitive equilibrium with equal
12

incomes and one-shot random serial dictatorship. However, these mechanisms are contentious
and hard to be implemented, as they require significant changes from the status quo and may
appear unfair and harsh to the patients. The optimal mechanisms proposed by Agarwal et al.
[2] are complicated to be implemented and justified, as the precise knowledge of the patients’
preferences is hard to obtain and the priorities and allocation points are derived by replicating
the optimal offer rates rather than basing on biological compatibility and medical urgency, which
can be hard to be understood by the doctors and patients.
The second contribution of this chapter is building a simulation engine that accounts for patients’ incentives and has fewer assumptions. The simulation tool widely used by the organ transplant community to evaluate alternative allocation mechanisms does not capture the changes in
patients’ behavior under different mechanisms [18]. Agarwal et al. [2] built a more complex simulation engine that accounts for patients’ incentives, but some assumptions they made are hard
to defend: first, the long-run waitlist size is assumed to be a deterministic quantity; second, patients are assumed to believe that the set of other patients is drawn i.i.d. from the steady-state
long-run average density; third, the demand for kidneys from a particular donor is assumed to be
Poisson distributed with the mean based on integrating the steady-state density of patient types.
However, in reality, the waitlist size may fluctuate, and such fluctuations may generate certain
dependencies that violate the other two assumptions.
1.A Additional Figures
Figure 1.3 and Figure 1.4 highlight the importance of waiting times in the current kidney allocation system. For the marginal donors with KDPI equal to or greater than 85, patients’ allocation
13

points are only determined by their waiting times; for the other donors, on average, 90% of patients’ allocation points come from their waiting times.
2 4 6 8 10 12 14
Number of Classifications
0
1000
2000
3000
4000
Number of Donors
Distribution of Patient Classification Count
Figure 1.3: Distribution of the number of patient classifications used for each donor, based on the
2019 kidney match-run data.
0% 20% 40% 60% 80% 100%
Waiting Time Points / Total Allocation Points
0
10
20
30
40
50
60
% of Patient-Donor Pairs
Distribution of Waiting Time Points Proportion
Figure 1.4: Distribution of the proportion of waiting time points in the total allocation points for
each patient-donor pair, based on the 2019 kidney match-run data.
14

Chapter 2
Patient Choice: Predicting Offers for Each Patient
2.1 Introduction
Despite various decision support tools based on machine learning models and statistical modeling
techniques have been built to assist the doctors and patients make optimal choices, a considerable
number of kidneys of viable quality are still being discarded. Mohan et al. [28] found that while
the discarded kidneys were generally of inferior quality compared to those transplanted, such as
from relatively older donors, there was a substantial overlap of quality between the transplanted
and discarded kidneys.
Moreover, even the marginal kidneys with several unfavorable factors can provide the patients
with far superior outcomes than waiting on dialysis [36]. Ojo et al. [34] found that compared
with maintenance dialysis, transplantation of a marginal kidney is associated with a substantial
long-term survival benefit, with an average increase in life expectancy of 5 years. Moreover,
transplantation of a marginal kidney is particularly helpful for the patients above 50 years old,
whose life expectancy on dialysis is short [21]. Besides, Raslan et al. [45] found that compared
15

with the standard criteria deceased donors (SCD), transplantation from the expanded criteria
deceased donors (ECD) is associated with similar recipient and graft survival rates.
In addition to the survival benefits offered by these marginal kidneys, accepting them also
makes financial sense and contributes to the patients’ well-being. Through a cost-benefit analysis, Axelrod et al. [5] demonstrated that compared to maintenance dialysis, transplantation with
marginal kidneys is highly cost-effective with a considerably lower cost per quality-adjusted lifeyear (QALY). Hellemans et al. [14] further noted that even without the survival benefits, the
improved physical, mental, and social well-being associated with the ECD transplantation compared to dialysis might still justify pursuing the ECD transplantation for the patients.
The marginal kidneys have been used successfully in the European transplant programs.
Aubert et al. [4] compared the US and French kidney allocation systems and found that transplanting more marginal kidneys, mainly from the older deceased donors, would offer significant
survival benefits for the US patients. Ibrahim et al. [17] compared the US and UK kidney allocation systems using a suite of risk-adjusted metrics and highlighted the sharply higher utilization
of older and marginal kidney donors in the UK relative to the US.
Therefore, the discarded kidneys were not necessarily unsuitable or lacking value for the patients, and many of the discarded kidneys could have been utilized to save more lives. Given the
reality of thousands of patients leaving the waitlist without getting matched every year, it compels us to question whether the doctors and patients are waiting only because of the incentives
from the kidney allocation system, or also because of the suboptimality in their offer acceptance
decisions, such that we can propose tailored remedies.
16

2.1.1 Suboptimality in Patients’ Kidney Acceptance
Using the actual kidney transplant data, we observe significant suboptimality in the patients’
kidney acceptance decisions.
By examining each patient’s history of offers and identifying the best offer she rejected in
terms of donor proximity and quality, we gain insights into the patients’ acceptance thresholds for
the offers. Specifically, for each patient, her past rejected offers are ranked along two dimensions:
first by the distance between each donor’s recovered location and the patient’s listing center, and
second by the Kidney Donor Profile Index (KDPI) score of each donor. In the allocation process,
the distance is categorized into three groups: local, regional and national. The KDPI score is a
percentile score ranging from 0 to 100 derived from the measure developed by Rao et al. [44] and
combines various donors’ clinical and demographic factors to estimate the relative risk of posttransplant graft failure in an average adult recipient compared to a reference donor. In practice,
the KDPI score serves as an essential tool for the doctors and patients to evaluate the donor quality
and make informed decisions regarding offer acceptance. The lower the KDPI, the lower the risk
and the better the donor quality. For each patient, the best offer she rejected is the one among
her rejected offers that is the closest to the patient’s listing center and has the lowest KDPI score.
This provides an approximation of the highest acceptance threshold for each patient, because the
best offer is only among the observed rejected offers.
To understand how suboptimal the patients’ acceptance thresholds are, in Figure 2.1, we plot
the distribution of the expected waiting time for a donor that is better in terms of donor proximity
and quality than the best rejected one for the patients over 60 years old, who constitute 1/3 of
the waitlist. For each patient, the expected waiting time for a better kidney is calculated by first
17

deriving the waiting time for each donor that is not farther away and has an equal or lower KDPI
score than the patient’s best rejected offer and then averaging the waiting times for these donors.
How the necessary waiting time between each patient-donor pair is derived is discussed in more
detail in Section 2.2.
Remarkably, around 30% of these old patients are waiting for the kidneys that require more
than 5 years of waiting time, a duration that is comparable to the life expectancy of these patients [62]. This finding suggests that many patients are setting excessively high acceptance
thresholds and behaving suboptimally by waiting for the high-quality kidneys that are difficult
to obtain. As their health deteriorates over time, these patients may eventually have to settle with
a mediocre kidney or become too sick to transplant.
0 2 4 6 8 10
Years
0
2
4
6
8
10
12
% of Patients
Expected Waiting Time for a "Better Kidney" than the Best Rejected One for Old Patients
Figure 2.1: The figure shows the distribution of the expected waiting time for a better kidney than
the best rejected one for the patients over 60 years old, who account for about 1/3 of the waitlist,
based on the deceased donors recovered from 2016 to 2019.
18

2.1.2 Source of Suboptimality: Offer Misprediction
But why are patients behaving suboptimally? To understand how doctors make acceptance decisions in practice, as they often decide on behalf of their patients, we conducted extensive discussions with numerous doctors and identified a crucial source of suboptimality: offer misprediction.
For example, when an offer arrives to an old blood type O patient, the doctor in charge either
accepts or rejects and waits for future offers within a short time frame. In solving this optimal
stopping problem, the doctor assesses the continuation value of waiting by relying on her impressions of historical data and considering the offers received for other similar patients. She may
ask herself, “How quickly did other old blood type O patients receive offers in the last 6 months,
and what kind of offers did they receive?” Despite the small sample problem as the doctor bases
her prediction only on the data from her respective listing center, a fundamental question arises:
Are the offer prospects of other old blood type O patients even informative for this particular
patient?
We claim that the answer is maybe not, because each patient’s situation is unique, and even
among the similar patients, a considerable heterogeneity exists in their offer prospects due to factors such as the patients’ prior organ transplants, specific tissue types and other physical conditions. For a specific example, using the actual kidney transplant data, we reproduce the allocation
points of each patient for each donor according to the kidney allocation system. This allows us to
obtain a cutoff for each donor and subsequently estimate the patients’ donor access as their waiting times increase. Section 2.2 describes how the donor access trajectories are derived in more
detail. For instance, for the patient A who is over 60 years old and of blood type O, her access to
the local donors over time is plotted in Figure 2.2. Within the graph, each line represents how
19

the expected number of offers within a 6-month period for the donors of a specific quality range
(e.g., KDPI 0-20, 20-35, 35-85 and 85-100) changes with her waiting time. These four KDPI groups
are used because the kidney allocation system categorizes donors into these four groups and has
a set of classifications for each donor group. For two other patients B and C, who share the same
listing center, age range (above 60 years old), blood type and level of hardness to match as patient
A, we also graph out their donor access trajectories over time as in Figure 2.2. The patients’ level
of hardness to match is described by their Calculated Panel Reactive Antibodies (CPRA) scores,
which are percentile scores ranging from 0 to 100 derived from the percentage of donors with
whom the patients would be incompatible [39]. These three patients all have a CPRA score of 0,
which means that they are relatively easy to match.
For these three patients, the doctors may use the same acceptance threshold to decide whether
to accept an offer. However, the three figures reveal that these patients have different donor access
trajectories, therefore, they should adopt different acceptance strategies. Compared to patient B,
patient A has higher likelihood of receiving a high-quality local donor with KDPI 0-35 upon
entry and also higher likelihood of receiving a local donor with KDPI 20-85 as her waiting time
reaches 4-6 years. Therefore, patient A can benefit from waiting more than patient B and should
have a higher acceptance threshold. The better donor access for patient A is mainly attributed to
patient A having a more common combination of tissue types compared to patient B. In kidney
transplantation, blood type matching and tissue type matching are among the most important
factors to determine the compatibility between a patient-donor pair. Specifically, the tissue type
matching refers to the compatibility of human leukocyte antigens (HLAs), particularly the A, B
and DR loci, which affect both patients’ classifications and allocation points for each donor. Each
locus has two antigens, and there are hundreds of thousands of possible antigen combinations at
20

0 2 4 6 8 10
Waiting time (Years)
0
4
8
12
16
20
24
28
32
Expected # offers in 6 months
Local Donors
Deceased Donor Access of an Old Blood Type O Patient
KDPI Range
0-20
20-35
35-85
85-100
(a) Patient A
0 2 4 6 8 10
Waiting time (Years)
0
4
8
12
16
20
24
28
32
Expected # offers in 6 months
Local Donors
Deceased Donor Access of an Old Blood Type O Patient
KDPI Range
0-20
20-35
35-85
85-100
(b) Patient B
0 2 4 6 8 10
Waiting time (Years)
0
4
8
12
16
20
24
28
32
Expected # offers in 6 months
Local Donors
Deceased Donor Access of an Old Blood Type O Patient
KDPI Range
0-20
20-35
35-85
85-100
(c) Patient C
Figure 2.2: The panel shows the number of local donors within each KDPI range available to the
patients A, B and C as their waiting time increases, respectively, based on the deceased donors
recovered in 2019.
21

the A, B and DR loci. Among the A, B and DR loci, the number of antigen mismatches at the DR
locus has the greatest impact on the patients’ donor access, as the patients with fewer antigen
mismatches at the DR locus receive more allocation points equivalent to 1 or 2 years of waiting.
The distribution of donor antigens at the DR locus is shown in Figure 2.3 (see Section 2.A for the
distributions of donor antigens at the A and B loci, respectively). Patient A has antigens 4 and 1
at the DR locus, and patient B has antigens 7 and 16 at the DR locus. Therefore, patient A is more
likely to have fewer DR mismatches and has better donor access compared to patient B.
4 15 13 7 17 11 1 8 14 98 12 16 9 18 10 103
Antigen at the DR locus
0
1000
2000
3000
4000
Count
Distribution of Donor Antigens at the DR Locus
Figure 2.3: The figure shows the distribution of donor antigens at the DR locus, based on the deceased donors recovered in 2019. To maintain clarity, only the antigens with a frequency greater
or equal to 100 are shown.
Compared with patients A and B, patient C has significantly better donor access, which is
driven by that patient C has received a liver transplant before, which gives her higher priority
and classifications during the allocation process. The significant difference in the donor access
among the similar patients is not uncommon, as seen from the actual kidney transplant data (see
Section 2.B for another set of similar patients having different donor access trajectories).
22

Given the numerous patients to care for amid the hectic schedule and the hundreds of thousands of possible tissue type combinations, it is challenging for the doctors to fully account for the
heterogeneity among the patients when making the offer acceptance decisions. Moreover, when
doctors predict conditional on more factors, they encounter a severe small data problem. In light
of the importance of accounting for the patients’ heterogeneity, this chapter prescribes a personalized offer prediction tool based on the actual allocation cutoff of each donor. This approach fully
incorporates each patient’s unique circumstances and the rules of the kidney allocation system
to help the doctors make more tailored choice for each patient in the face of offer uncertainties.
Section 2.2 and Section 2.3 describe how the personalized offer prediction and the doctors’
prediction are constructed based on the actual kidney transplant data, respectively. Section 2.4
compares the two prediction approaches through first optimizing the two approaches on the
training data and then applying to the testing data.
2.2 Personalized Offer Prediction
In this section, we first describe the abstract idea of the personalized offer prediction approach
and then illustrate how to prepare the data for constructing the personalized offer prediction for
each patient.
2.2.1 Method
Formally, the goal of offer prediction is to predict the number of type j donor offers that patient
i will receive between time t and t + ∆, based on the historical offers observed between t
′ − δ
and t
′
, where ∆, δ > 0 and t
′ < t.
23

The personalized offer prediction approach involves two steps.
First, we plot the donor access trajectory of each donor type j for patient i. Specifically, for
each observed type j donor, based on the classification of the last patient offered and the classification of patient i according to the kidney allocation system, we can calculate the necessary
waiting time for patient i to receive an offer. There are four possible scenarios:
1. The donor is incompatible with patient i. The necessary waiting time is not applicable, as
patient i will be filtered out due to blood type incompatibility during the match-run.
2. The donor is last offered to a patient in a classification that precedes the classification of
patient i. The necessary waiting time is still not applicable, as patient i cannot move to a
prior classification by waiting [41].
3. The donor is last offered to a patient in patient i’s classification. We obtain the cutoff allocation points zj of the donor by examining the last offered patient. The necessary waiting
time is then
max{0, zj − [sj (i, t) − (t − ti)]},
where sj (i, t) is patient i’s allocation points for the donor at time t and ti
is patient i’s
waiting time qualification date. Here, t−ti
is patient i’s waiting time at time t and also her
waiting time points, because a patient obtains 1 allocation point for each year of waiting.
Therefore, sj (i, t) − (t − ti) is patient i’s allocation points from her physical conditions
other than the waiting time, and the points needed to reach zj on top of patient i’s physical
allocation points are her necessary waiting time to receive an offer from the donor.
24

4. The donor is last offered to a patient in a classification that succeeds the classification of
patient i. The necessary waiting time is 0, as patient i is in a higher class.
By doing so for every type j donor, we plot the type j donor access trajectory for patient i as
patient i remains on the waitlist.
Second, we predict the number of type j donor offers that patient i will receive between t and
t + ∆ by integrating the donor access trajectory over [t, t + ∆]. Specifically, let fij (τ ) denote the
type j donor access trajectory for patient i derived from the first step. The expected number of
type j offers that patient i will receive between t and t + ∆ is R t+∆
t
fij (τ )dτ · ∆/δ, where ∆/δ
is a scaling factor to adjust for the time length of the prediction period (∆) and the observational
data period (δ).
In summary, these two steps provide a personalized offer prediction that is intuitive and actionable for the doctors and patients. By visualizing the donor access trajectory, this approach
helps the doctors and patients make the optimal kidney acceptance choices.
2.2.2 Computing the Necessary Waiting Time from Data
To implement the personalized offer prediction using the actual kidney transplant data, the key
lies in how to plot the donor access trajectory for each patient, or essentially how to compute
the necessary waiting time for each patient to receive an offer from each donor. Following the
first step from above, to determine the necessary waiting time between each patient-donor pair,
we need to derive the cutoff for the donor and the classification and allocation points for each
donor-patient pair.
25

To derive the cutoff for each donor, we reproduced the allocation points for each patient-donor
pair according to the rules of the kidney allocation system [41], as the classifications already
exist in the match-run data obtained from the UNOS. For each donor, a patient’s allocation points
consist of her waiting time score, HLA matching score, CPRA score, pediatric score and donation
score for being a living donor. As we only have two observations of each patient’s CPRA score
(one when they joined the waitlist and the other when they left the waitlist or when the data
was obtained), we conducted linear interpolation for each patient’s CPRA score at the time of
allocation using the two observations and her waiting time on the waitlist. To ensure that the
calculated CPRA scores are close to what were used during the allocation, we further adjusted the
calculated CPRA score based on the classification that the patient was in during the allocation,
because some classifications specify the patients’ CPRA score range. For example, if a patient’s
CPRA score was calculated to be 85, but she was in a classification for the patients with CPRA
21-79, then her CPRA score used in the allocation was adjusted to be 79.
With the classification and allocation points between each patient-donor pair, the cutoff for
each donor is determined as follows. If the donor’s recovered kidneys were ever discarded, then
the offered patient from the match-run data with the lowest classification and allocation points
is taken as the cutoff patient. Otherwise, the transplanted patient with the lowest classification
and allocation points is taken as the cutoff patient.
To compute the necessary waiting time for a patient to be eligible for each donor given the
cutoff, we need to determine her classification and allocation points for the donor and then follow the rules described in Section 2.2.1 to compute the necessary waiting time. For a patient,
her allocation points for each donor can be computed as mentioned above, whereas determining
26

her classification for each donor requires reproducing a mapping according to the kidney allocation system [41]. Specifically, a patient’s classification for a donor depends on the following 10
features:
1. the patient’s blood compatibility with the donor;
2. the patient’s HLA compatibility at the A, B, DR loci with the donor;
3. the patient’s distance from the donor (local, regional or national);
4. whether the patient is a living donor;
5. whether the patient has received a liver transplant before;
6. whether the patient registered before 18 years old;
7. whether the patient is fine with receiving both kidneys from a donor;
8. whether the patient is an adult at the time of the match-run;
9. the patient’s CPRA score range;
10. and the patient’s Estimated Post-Transplant Survival (EPTS) score range.
The EPTS score is assigned to each waitlisted patient based on how long she is likely to live
with a kidney transplant compared to other patients, with a lower EPTS score indicating a longer
expected survival [40]. For a given donor, a patient may be qualified for several classifications
simultaneously. In such cases, the classification with the highest priority is selected. The constructed classification mapping table achieves an accuracy of 97.80% when applied to the actual
match-run data.
27

2.3 Doctors’ Prediction
In this section, similarly, we first describe the conceptual idea of the doctors’ prediction approach
and then illustrate how to prepare the data for constructing the doctors’ prediction for each patient.
2.3.1 Method
As mentioned in Section 2.1.2, doctors recall their experience with similar patients to predict the
offer prospects for a patient. Formally, to predict the number of type j donor offers that patient
i will receive between time t and t + ∆, the doctor’ prediction approach is approximated by the
following two steps:
First, the patients similar to patient i in the historical offer data observed between t
′ − δ and
t
′
are identified. The similarity is based on factors such as the listing center, blood type and
range of allocation points invariant to donors, which include the waiting time score, CPRA score,
pediatric score and donation score for being a living donor. The total number of type j donor
offers N and the total active time on the waitlist T are calculated for these similar patients. The
total active time is calculated to adjust the total offer counts such that the offer counts are not
deflated, because some patients may only arrive to the waitlist in the middle of the data period
or leave the waitlist before the end of the data period.
Second, the doctors’ prediction for the number of type j donor offers that patient i will receive
between t and t + ∆ is calculated as ∆N/T, where N/T represents the expected type j donor
offer rate for patient i.
28

2.3.2 Computing the Offer Rate for a Patient Group from Data
To construct the doctors’ prediction using the actual kidney transplant data, we only need to compute the total number of offers and total active time on the waitlist for a patient group. Specifically,
for a patient group, the total number of offers from a donor is obtained from the match-run data.
For each patient, her time on the waitlist during the data period is calculated by taking the latest
of the data start time and her registration time, and the earliest of the data end time and her time
of leaving the waitlist. Therefore, with the total number of offers and total active time, we can
compute the offer rate for each patient group.
2.4 Comparison
To compare the personalized offer prediction against the doctors’ prediction, we use the data
obtained from the UNOS, which contain comprehensive information on the deceased donors and
waitlisted patients and the match-run data for each deceased donor in the United States. The
match-run data describe the offers made of each donor and the accept/reject decisions made by
the patients.
Section 2.4.1 describes the data preparation process to prepare the data to implement the
personalized offer prediction approach and the doctors’ prediction approach. Section 2.4.2 details
the training phase of optimizing each approach using the training data. Finally, Section 2.4.3
presents the prediction results on the testing data.
29

2.4.1 Data Preparation
As seen from the sequence numbers, the generated patient list for each donor in the match-run
data is incomplete, due to the offer filters or patients’ prespecified preferences. However, as we do
not have access to the patients’ or listing centers’ offer filters, we reproduced the complete patient
list for each donor to fully account for the total number of offers that a patient is eligible for. By
doing so, we also avoid biasing the personalized offer prediction approach for not considering
these constraints of each patient, as it would have incorporated these filters if we had access. To
reproduce the complete patient list for each donor, the cutoff for each donor is first determined
following the approach described in Section 2.2.2. For each donor, all the patients who were active
when the donor was recovered and who would be either in a higher classification than the cutoff
patient or in the same classification as the cutoff patient but with higher allocation points are
considered as having been offered. Therefore, we obtain a more complete match-run data, and it
is used to compare the two prediction approaches.
We compare the performance of these two prediction approaches by the following task: the
objective is to predict the number of offers within each quality range a patient will receive in the
next 6 months, based on the previous year’s complete match-run data. Specifically, the donor
quality is grouped by the distance (local or regional) and the KDPI range (0-20, 20-35, 35-85 and
85-100), as this is also how the donors are grouped in the current kidney allocation system. The
national donors are not included in the prediction task, because they are generally less attractive
and less likely to be accepted by the doctors and patients, as the national donors are farther away
and associated with longer cold ischemic times.
30

Each prediction approach is first trained using the complete match-run data from Jun 30, 2017
to Jul 1, 2018 to predict the number of offers each patient will receive between Jul 2, 2018 and
Dec 31, 2018. Only the patients who appear between Jun 30, 2017 and Jul 1, 2018 are predicted.
Moreover, if a patient leaves the waitlist before Dec 31, 2018, then her offer numbers are adjusted
by the time that she stays on the waitlist, as patients’ departures cannot be captured by the offer
predictions. For example, if a patient left the market on Aug 2, 2018 and received 1 offer during
the time between Jul 2, 2018 and Aug 2, 2018, then her total number of offers between Jul 2, 2018
and Dec 31, 2018 would be adjusted to be 6. Both approaches are optimized to minimize the mean
absolute error (MAE) by varying the length of match-run data used, and the doctors’ prediction
approach is also optimized by the features used to group the patients. The MAE calculates the
average of the absolute differences between the predicted and actual offer numbers, and it provides a natural way to quantify the average prediction error, with lower values indicating better
performance.
After tuning, both approaches are applied to predict the number of offers each patient will
receive between Jan 1, 2019 and Jun 30, 2019 based on the complete match-run data in 2018.
Similarly, only the patients who appear in 2018 are predicted, and if a patient leaves the waitlist
before Jun 30, 2019, then her offer numbers are adjusted by the time that she stays on the waitlist.
2.4.2 Prediction Training
In practice, doctors predict the offer prospects for the patients mainly based on patients’ blood
types, age ranges and allocation point ranges. To approximate the doctors’ prediction, we assume
they use the following features to group the patients in their respective listing centers: blood
31

type, allocation point range at the prediction time, and EPTS score range. The allocation point
range includes a patient’s waiting time points at the prediction time and points from CPRA score,
pediatric score and living donation. The EPTS score range is also included, because the patients
with EPTS 0-20 are prioritized for the donors with KDPI 0-20. Other features that can affect
patients’ allocation classifications, such as previous liver transplants, registration before 18 and
acceptance of dual kidney transplants, are also considered in optimizing the doctors’ prediction.
During the training phase, we fine-tune the grouping and vary the length of match-run data
that doctors use to construct the predictions, which is more advanced than what the doctors are
doing in practice. We evaluate each combination using the MAE metric, and the training results
are shown in Table 2.1 and Table 2.2 below.
Length of Match-run Data Past 12 months Past 6 months Past 1 month
MAE 4.23 4.48 6.52
Table 2.1: The table shows the performance of the doctors’ prediction on the training data, based
on grouping the patients by their blood type, allocation point range and EPTS score range.
Length of Match-run Data Past 12 months Past 6 months Past 1 month
MAE 4.25 4.50 6.54
Table 2.2: The table shows the performance of the doctors’ prediction on the training data, based
on grouping the patients by their blood type, allocation point range, EPTS score range, prior
liver transplant, age at registration (before or after 18 years old), and willingness to accept both
kidneys.
Based on the training results, the doctors’ prediction performs best when using the matchrun data from the past year and grouping the patients based on their blood type, allocation point
range and EPTS score range. Figure 2.4 shows the actual offer numbers from the donors in each
32

quality range compared to the predicted offer numbers for each patient blood type group, with
the donor quality range characterized by the proximity and KDPI range.
0
2
4
6
8
10
12
14
# offers in 6 months
Local Donors | KDPI 0-20
0
2
4
6
8
10
12
14
Local Donors | KDPI 20-35
0
8
16
24
32
40
48
56
64
Local Donors | KDPI 35-85
0
1
2
3
4
5
Local Donors | KDPI 85-100
AB A B O
Patient Blood Type
0
3
6
9
12
15
18
21
24
# offers in 6 months
Regional Donors | KDPI 0-20
AB A B O
Patient Blood Type
0
3
6
9
12
15
18
21
24
Regional Donors | KDPI 20-35
AB A B O
Patient Blood Type
0
15
30
45
60
75
90
105
120
Regional Donors | KDPI 35-85
AB A B O
Patient Blood Type
0
15
30
45
60
75
90
105
120
135
150
Regional Donors | KDPI 85-100
Doctors' Prediction on Training Data: Actual vs Predicted
Actual
Predicted
Figure 2.4: The figure shows the actual offer numbers compared with the predicted offer numbers
by the best doctors’ prediction during the training phase, which uses the match-run data from
the past year and groups the patients by their blood type, allocation point range and EPTS score
range.
The personalized offer prediction is tuned by varying the length of match-run data used, with
the training results shown in Table 2.3.
Length of Match-run Data Two Years Ago Past Year Past 6 months
MAE 3.90 3.47 3.83
Table 2.3: The table shows the performance of the personalized offer prediction on the training
data.
Based on the training results, the personalized offer prediction performs the best when using
the match-run data from the past year. Figure 2.5 shows the actual offer numbers from the donors
in each quality range compared to the predicted offer numbers for each patient blood type group,
with the donor quality range characterized by the proximity and KDPI range.
To better visualize the comparison, Figure 2.6 shows the MAE for the offer numbers from
donors in each quality range for each patient blood type group, where the first light gray bar
33

0
2
4
6
8
10
12
14
# offers in 6 months
Local Donors | KDPI 0-20
0
2
4
6
8
10
12
14
Local Donors | KDPI 20-35
0
8
16
24
32
40
48
56
64
Local Donors | KDPI 35-85
0
1
2
3
4
5
Local Donors | KDPI 85-100
AB A B O
Patient Blood Type
0
3
6
9
12
15
18
21
24
# offers in 6 months
Regional Donors | KDPI 0-20
AB A B O
Patient Blood Type
0
3
6
9
12
15
18
21
24
Regional Donors | KDPI 20-35
AB A B O
Patient Blood Type
0
15
30
45
60
75
90
105
120
135
Regional Donors | KDPI 35-85
AB A B O
Patient Blood Type
0
15
30
45
60
75
90
105
120
135
150
Regional Donors | KDPI 85-100
Personalized Offer Prediction on Training Data: Actual vs Predicted
Actual
Predicted
Figure 2.5: The figure shows the best personalized offer prediction during the training phase,
which uses the match-run data from the past year.
represents the mean offer numbers in the next 6 months, the second red bar represents the MAE
of the best doctors’ prediction on the training data, and the yellow and green bars represent the
MAE from the personalized offer predictions using the match-run data from 2 years ago and the
last year, respectively. Additionally, to measure the inherent randomness, a lower bound is added
by calculating the MAE under a Poisson distribution assumption for the offer numbers [43].
0
1
2
3
4
5
6
7
8
9
10
# offers in 6 months
Local Donors | KDPI 0-20
0
1
2
3
4
5
6
7
8
9
10
Local Donors | KDPI 20-35
0
3
6
9
12
15
18
21
24
27
30
Local Donors | KDPI 35-85
0
1
2
3
4
5
Local Donors | KDPI 85-100
A AB B O
Patient Blood Type
0
1
2
3
4
5
6
7
8
9
10
# offers in 6 months
Regional Donors | KDPI 0-20
A AB B O
Patient Blood Type
0
2
4
6
8
10
12
14
Regional Donors | KDPI 20-35
A AB B O
Patient Blood Type
0
6
12
18
24
30
36
42
48
54
60
Regional Donors | KDPI 35-85
A AB B O
Patient Blood Type
0
6
12
18
24
30
36
42
48
54
60
Regional Donors | KDPI 85-100
Mean and MAE of Each Prediction Method on Training Data
Mean
Doctors' Prediction
Personalized: 2 Years Ago
Personalized: Last Year
Lower Bound
Figure 2.6: The figure shows the mean offer numbers and MAE of each prediction approach on
the training data for the offer numbers from donors in each quality range for each patient blood
type group. The lower bound is obtained by calculating the MAE under a Poisson distribution
assumption for the offer numbers.
Figure 2.6 shows that for the personalized offer prediction, using the last year’s match-run
data is close to the lower bound and outperforms the doctors’ prediction in almost all cases,
34

except for predicting the local donors with KDPI 85-100 whose offer numbers are already close to
0. This is because under the 2019 policy, there was no distinction between local or regional offers
when allocating the donors of KDPI 85-100, and the offers were all considered regional, except
for the perfectly matched local patients.
To better visualize the improvement by the personalized offer prediction, Figure 2.7 shows
the percentage decrease in MAE relative to the doctors’ prediction. On average, the personalized
offer prediction on the training data using the last year’s match-run data decreases the MAE by
18% relative to the doctors’ prediction.
0
2
4
6
8
10
12
14
16
18
20
% decrease in MAE
Local Donors | KDPI 0-20
9
6
3
0
3
6
9
12
15
Local Donors | KDPI 20-35
0
3
6
9
12
15
18
21
24
Local Donors | KDPI 35-85
60
54
48
42
36
30
24
18
12
6
0
Local Donors | KDPI 85-100
A AB B O
Patient Blood Type
0
5
10
15
20
25
30
35
40
45
50
% decrease in MAE
Regional Donors | KDPI 0-20
A AB B O
Patient Blood Type
0
4
8
12
16
20
24
28
32
36
40
Regional Donors | KDPI 20-35
A AB B O
Patient Blood Type
0
4
8
12
16
20
24
28
32
Regional Donors | KDPI 35-85
A AB B O
Patient Blood Type
50
40
30
20
10
0
10
20
30
Regional Donors | KDPI 85-100
Improvement in MAE Relative to Doctors' Prediction on Training Data
Personalized: 2 Years Ago
Personalized: Last Year
Figure 2.7: The figure shows the percentage decrease in MAE by the personalized offer prediction
relative to the doctors’ prediction on the training data.
2.4.3 Results
After training, the two prediction approaches are applied to predict the number of offers each
patient will receive between Jan 1, 2019 and Jun 30, 2019, using the match-run data from 2018.
Figure 2.8 and Figure 2.9 show the actual offer numbers compared to the predicted offer numbers
for the doctors’ prediction and the personalized offer prediction, respectively. For the doctors’
35

prediction, the doctors group the patients by their blood type, allocation point range and EPTS
score range.
0
2
4
6
8
10
12
14
# offers in 6 months
Local Donors | KDPI 0-20
0
2
4
6
8
10
12
14
16
18
20
Local Donors | KDPI 20-35
0
8
16
24
32
40
48
56
64
Local Donors | KDPI 35-85
0
1
2
3
4
5
Local Donors | KDPI 85-100
AB A B O
Patient Blood Type
0
3
6
9
12
15
18
21
24
# offers in 6 months
Regional Donors | KDPI 0-20
AB A B O
Patient Blood Type
0
3
6
9
12
15
18
21
24
Regional Donors | KDPI 20-35
AB A B O
Patient Blood Type
0
15
30
45
60
75
90
105
120
135
150
Regional Donors | KDPI 35-85
AB A B O
Patient Blood Type
0
20
40
60
80
100
120
140
160
Regional Donors | KDPI 85-100
Doctors' Prediction on Testing Data: Actual vs Predicted
Actual
Predicted
Figure 2.8: The figure shows the doctors’ prediction on the testing data, which uses the matchrun data from the previous year and groups patients by their blood type, allocation point range
and EPTS score range.
0
2
4
6
8
10
12
14
# offers in 6 months
Local Donors | KDPI 0-20
0
2
4
6
8
10
12
14
16
18
20
Local Donors | KDPI 20-35
0
8
16
24
32
40
48
56
64
Local Donors | KDPI 35-85
0
1
2
3
4
5
Local Donors | KDPI 85-100
AB A B O
Patient Blood Type
0
3
6
9
12
15
18
21
24
# offers in 6 months
Regional Donors | KDPI 0-20
AB A B O
Patient Blood Type
0
3
6
9
12
15
18
21
24
Regional Donors | KDPI 20-35
AB A B O
Patient Blood Type
0
15
30
45
60
75
90
105
120
135
150
Regional Donors | KDPI 35-85
AB A B O
Patient Blood Type
0
20
40
60
80
100
120
140
160
Regional Donors | KDPI 85-100
Personalized Offer Prediction on Testing Data: Actual vs Predicted
Actual
Predicted
Figure 2.9: The figure shows the personalized offer prediction on the testing data, which uses the
match-run data from the previous year.
In general, we observe that both prediction approaches tend to underestimate the offer numbers across all patient blood types and donor quality ranges. This is because the organ recovery
practice has become more aggressive over the years, with the number of donors recovered for
transplantation increasing by 13% in 2019 compared to 2018 (as seen from Figure 1.1).
36

Despite the time trend, the personalized offer prediction on the testing data using the last
year’s match-run data decreases the MAE by 20% relative to the doctors’ prediction, with the
MAE being 4.80 for the doctors’ prediction and 3.82 for the personalized offer prediction.
Figure 2.10 and Figure 2.11 visualize the comparison and improvement of the personalized
offer prediction relative to the doctors’ prediction for each patient blood type and donor quality
range. The results on the testing data are consistent with those observed from the training data,
with the personalized offer prediction showing substantial improvement over the doctors’ prediction across almost all patient blood types and donor quality ranges, except for predicting the
local donors with KDPI 85-100 whose offer numbers are already close to 0. The results highlight
the effectiveness of the personalized offer prediction, besides being transparent and interpretable.
0
1
2
3
4
5
6
7
8
9
10
# offers in 6 months
Local Donors | KDPI 0-20
0
1
2
3
4
5
6
7
8
9
10
Local Donors | KDPI 20-35
0
4
8
12
16
20
24
28
32
Local Donors | KDPI 35-85
0
1
2
3
4
5
Local Donors | KDPI 85-100
A AB B O
Patient Blood Type
0
2
4
6
8
10
12
14
# offers in 6 months
Regional Donors | KDPI 0-20
A AB B O
Patient Blood Type
0
2
4
6
8
10
12
14
Regional Donors | KDPI 20-35
A AB B O
Patient Blood Type
0
8
16
24
32
40
48
56
64
Regional Donors | KDPI 35-85
A AB B O
Patient Blood Type
0
8
16
24
32
40
48
56
64
Regional Donors | KDPI 85-100
Mean and MAE of Each Prediction Method on Testing Data
Mean
Doctors' Prediction
Personalized Prediction
Lower Bound
Figure 2.10: The figure shows the mean offer numbers and MAE of each prediction approach on
the testing data for the offer numbers from donors in each quality range for each patient blood
type group. The lower bound is obtained by calculating the MAE under a Poisson distribution
assumption for the offer numbers.
2.5 Discussion
In this chapter, motivated by the observation that the old patients are waiting for the hard-to-get
good kidneys considering their limited life expectancy, we uncover a source of suboptimality in
37

0
2
4
6
8
10
12
14
% decrease in MAE
Local Donors | KDPI 0-20
0
2
4
6
8
10
12
14
Local Donors | KDPI 20-35
0
3
6
9
12
15
18
21
24
Local Donors | KDPI 35-85
5
4
3
2
1
0
Local Donors | KDPI 85-100
A AB B O
Patient Blood Type
0
5
10
15
20
25
30
35
40
45
50
% decrease in MAE
Regional Donors | KDPI 0-20
A AB B O
Patient Blood Type
0
4
8
12
16
20
24
28
32
Regional Donors | KDPI 20-35
A AB B O
Patient Blood Type
0
5
10
15
20
25
30
35
40
45
Regional Donors | KDPI 35-85
A AB B O
Patient Blood Type
0
3
6
9
12
15
18
21
24
Regional Donors | KDPI 85-100
Improvement in MAE Relative to Doctors' Prediction on Testing Data
Personalized Prediction
Figure 2.11: The figure shows the percentage decrease in MAE by the personalized offer prediction
relative to the doctors’ prediction on the testing data.
patients’ kidney acceptance decisions: offer misprediction. We highlight that the doctors’ approach of predicting patients’ offer prospects by resorting to other similar patients overlooks the
subtle heterogeneity among the patients, such as the differences in their tissue types and their
physical conditions. This is suboptimal because it can lead to patients rejecting the reasonable
kidneys and ultimately leaving the waitlist without receiving a transplant, such as some old patients. Besides, it can also lead to patients accepting offers too soon, missing out on the better
kidneys they could have received by waiting slightly longer.
To mitigate this suboptimality, we develop a personalized offer prediction tool based on the
observed donor cutoffs that fully incorporates the patients’ heterogeneity and the rules of the
kidney allocation system. The personalized offer prediction shows significant improvement over
the doctors’ prediction, with a 20% decrease in mean absolute error. Moreover, the personalized
offer prediction does not suffer from the trade-off between the patient grouping and statistical
power, whereas in the doctors’ prediction, finer grouping of patients can lead to low statistical
power. In contrast, the personalized offer prediction is performed for each patient individually,
38

avoiding the issue of reduced statistical power on the patients’ side. Furthermore, the personalized offer prediction is easier to implement and understand by the doctors and patients, making
it accessible for the small listing centers that may lack the expertise or resources to effectively
train complicated prediction models.
We conclude by discussing the potential extensions of the personalized offer prediction approach.
1. Incorporating the patients’ other time-varying features. When constructing the donor access trajectory and computing the necessary waiting time for each donor for each patient,
we assume that the waiting time is the only feature that changes over time and increases the
patient’s allocation points. However, in practice, patients’ other features such as their EPTS
and CPRA scores may also increase over time, affecting patients’ classifications and allocation points, and consequently, their offer prospects. For example, an EPTS score higher than
20 decreases a patient’s priority for the donors with KDPI 0-20, and a higher CPRA score
increases a patient’s classification priority and allocation points. As we do not have data
on patients’ history of EPTS and CPRA scores, we are not able to model the progression of
these scores. Besides, we do not find any good model available. With access to more comprehensive data or good models to describe how patients’ EPTS and CPRA scores change
over time, one can easily incorporate these time-varying features into patients’ donor access graphs, leading to more accurate personalized offer predictions.
2. Leveraging machine learning approaches. In this chapter, we compare the personalized offer prediction with the doctors’ prediction but not with other machine learning approaches.
39

However, machine learning approaches based on the observational data can be seen as sophisticated ways of implementing the doctors’ prediction and still suffer from the tradeoff between the patient grouping and statistical power, as the fundamental idea is still to
smartly group patients and find similar patients to make predictions. In contrast, our approach does not require extrapolation across similar patients at all, but only the donor
cutoffs. While machine learning approaches can be trained on the donor access graph data
we constructed, the potential for improvement is limited, as the current personalized offer
prediction already shows performance close to the lower bound of assuming offers following a Poisson distribution. Moreover, incorporating machine learning may complicate the
intuition of the personalized offer prediction itself.
3. Building a personalized decision support tool with the offer predictions. In this chapter,
the developed tool only predicts the offer prospects for each patient but does not provide
direct suggestions on whether the patient should accept an offer. Providing such suggestions would require knowledge of the patient’s preferences to solve the optimal stopping
problem, which is not available in our data. However, once the data on patients’ preferences become available, the backward induction developed in Chapter 4 solves the optimal
stopping problem and can be devised into a personalized decision support tool to provide
concrete suggestions on offer acceptance decisions.
2.A Additional Figures
The distributions of donor antigens at the A and B loci are shown in Figure 2.12 and Figure 2.13.
40

2 1 3 24 68 11 30 23 29 32 98 31 26 33 25 74 34 66
Antigen at the A locus
0
1000
2000
3000
4000
5000
6000
Count
Distribution of Donor Antigens at the A Locus
Figure 2.12: The figure shows the distribution of donor antigens at the A locus, based on the deceased donors recovered in 2019. To maintain clarity, only the antigens with a frequency greater
or equal to 100 are shown.
44 7 35 8 62 51 60 18 57 27 39 65 53 58 61 13 49 98 45 55 38 72 52 37 42 50 41 64 63 71 56 48
Antigen at the B locus
0
500
1000
1500
2000
2500
Count
Distribution of Donor Antigens at the B Locus
Figure 2.13: The figure shows the distribution of donor antigens at the B locus, based on the deceased donors recovered in 2019. To maintain clarity, only the antigens with a frequency greater
or equal to 100 are shown.
2.B Heterogeneity in Patients’ Donor Access
Figure 2.14 shows another set of patients sharing the same listing center, age range (above 60
years old), blood type and level of hardness to match (CPRA score) but having different donor
access trajectories.
41

0 2 4 6 8 10
Waiting time (Years)
0
3
6
9
12
15
18
21
24
Expected # offers in 6 months
Local Donors
Deceased Donor Access of an Old Blood Type O Patient
KDPI Range
0-20
20-35
35-85
85-100
(a) Patient A’
0 2 4 6 8 10
Waiting time (Years)
0
3
6
9
12
15
18
21
24
Expected # offers in 6 months
Local Donors
Deceased Donor Access of an Old Blood Type O Patient
KDPI Range
0-20
20-35
35-85
85-100
(b) Patient B’
0 2 4 6 8 10
Waiting time (Years)
0
3
6
9
12
15
18
21
24
Expected # offers in 6 months
Local Donors
Deceased Donor Access of an Old Blood Type O Patient
KDPI Range
0-20
20-35
35-85
85-100
(c) Patient C’
Figure 2.14: The panel shows the number of local donors within each KDPI range available to the
patients A’, B’ and C’ as their waiting time increases, respectively, based on the donors recovered
in 2019.
42

Among the three patients, patient A’ has access to the high-quality local donors with KDPI
0-20, whereas the other two patients do not have access to such donors. This is due to the fact
that patient A’ has an Estimated Post-Transplant Survival (EPTS) score below 20, although these
three patients are all above 60 years old. The EPTS score is assigned to each waitlisted patient
based on how long she is likely to live with a kidney transplant compared to other patients, with
a lower EPTS score indicating a longer expected survival. Besides the patient age, the EPTS score
also depends on the patient time on dialysis, diagnosis of diabetes and prior solid organ transplants [40]. In the current kidney allocation system, the patients with EPTS 0-20 are prioritized for
the donors with KDPI 0-20 to maximize the transplanted patients’ QALYs for these high-quality
donors. Therefore, patient A’ is eligible for these high-quality donors whereas patients B’ and C’
do not.
Compared with patient C’, patient B’ has better access to the local donors with KDPI 20-85 as
patient B’ expects shorter waiting time to be eligible for these donors. This is driven by patient B’
having a more common combination of tissue types compared to patient C’. Specifically, patient
B’ has antigens 4 and 15 at the DR locus, and patient C’ has antigens 9 and 13 at the DR locus.
Therefore, patient B’ is more likely to have fewer mismatches at the DR locus and hence better
donor access.
43

Chapter 3
Wastage: Analyzing a Stylized Model of Organ Allocation
3.1 Introduction
In the United States, despite the shortage of organ supply, the wastage problem is prevalent across
a variety of cadaveric organs besides kidneys, such as livers, intestines and lungs, with discard
rates ranging between 5% and 10% [19]. The cadaveric organs are allocated through a prioritybased mechanism that ranks patients based on a combination of their waiting times, biological
compatibility and medical urgency. Currently, on average, patients are being removed from the
national organ transplant waitlist at a rate exceeding one per hour, either because they become
too sick to transplant or because they die. The patients on the waitlist generally suffer from
end-stage organ failure, for whom organ transplantation is the most viable option for better and
longer lives.
Given the benefits of organ transplantation and the scarcity of organs, why are so many
organs being wasted? As highlighted in Section 1.1, a substantial medical and operations literature has explored different aspects of the wastage problem, and the main findings point to
44

three primary factors: over reliance on uninformative quality indicators, excessive risk aversion
of doctors, and congestion in the allocation process.
Although these three factors have been analyzed in depth and promising solutions have been
proposed, the wastage of cadaveric organs persists. After helping doctors and patients make optimal organ acceptance decisions with the developed offer prediction tool, this chapter strives to
gain more theoretical understanding of the wastage problem and considers the following research
question from a futuristic perspective: suppose all of the problems mentioned in Section 1.1 were
to be resolved, what does it take to completely eliminate waste in cadaveric organ allocation?
To address this question, we analyze a stylized model that abstracts away from patient characteristics and approximates the current allocation mechanism as a waitlist with choice, which is a
first-come first-served queue in which patients can reject any offer and still maintain their positions in the waitlist. The phrase “with choice” differentiates this mechanism from the waitlists
currently being implemented to allocate public housing in the US, which penalizes participants
for rejecting offers.
3.1.1 Related Work
This chapter contributes to the literature on dynamic matching with private, one-sided preferences. Papers in this category are largely motivated by the allocation of cadaveric organs [56, 57,
58, 2] or public housing [61, 3]. One shortcoming of the existing literature is summarized in the
introduction of Agarwal et al. [2]:
Previous research and policy guidance on waitlist design is based on restrictive assumptions about agents’ preferences or behavior. The theoretical literature has found that
45

even qualitative trade-offs are sensitive to whether objects are vertically or horizontally
differentiated. Absent clear recommendations from theory, many organ allocation agencies use simulations to predict the effects of alternative allocation rules.
Agarwal et al. [2] address this by conducting an empirical analysis based on structural estimation. We comment on their work at the end of this section. This chapter strives to bridge
the gap in the theoretical literature described above by developing a tractable theoretical model
that gives clear recommendations without relying on restrictive assumptions about agents’ preferences: we allow preferences to be arbitrarily correlated across organ types. The proofs in this
chapter are generally not difficult, but the possibility for such simple analyses is not clear a priori,
given the above quote. Furthermore, the richness of the preference distributions allows the chapter to unearth new insights that have no analog in the literature, such as the finding that there
exist markets in which no mechanism can Pareto improve upon the status quo of the waitlist
with choice, even though the status quo exhibits waste (see Proposition 3 in Section 3.5). Without allowing for rich preference heterogeneity, Pareto dominance would not be as interesting of
a concept.
In terms of the other insights, a closely related series of papers is [56, 57, 58], all of which
analyze the tension between patient choice and aggregate welfare in cadaveric kidney allocation.
Su and Zenios [56] is the closest to this chapter, as they show that first-come first-served (FCFS),
which is equivalent to the waitlist with choice in this chapter, exhibits waste by incentivizing
patients to reject reasonable quality organs, but last-come first-served (LCFS) does not. We contribute by showing that besides LCFS, there are many other allocation mechanisms that eliminate
46

waste, and characterize what it takes for an allocation mechanism to do so in all markets. Moreover, we show that these allocation mechanisms eliminate waste under arbitrary heterogeneity
in preferences, whereas Su and Zenios [56] assume that patients have identical preferences. In
Su and Zenios [57, 58], the modeling assumptions eliminate waste by construction, so those papers are less relevant to the research question in this chapter. In Su and Zenios [57], the policy
designer knows each patient’s preference and can use this to assign each patient to a suitable
queue of arriving kidneys. They show that as long as the quality of kidneys in each queue is
sufficiently homogeneous, then patients will not turn down any offers. However, this allocation mechanism would not work if preferences are privately known, which is what we assume
in this chapter. In Su and Zenios [58], patients do have private information and can self-select
into various queues, but they cannot reject undesirable offers within each queue. However, the
current allocation mechanism allows patients to reject undesirable offers, and it is precisely this
cherry-picking behavior that we seek to ameliorate in this chapter.
The tension between allowing agent choice and reducing waste has also been highlighted in
several other theoretical papers. Bloch and Cantala [10] analyze mechanisms in which offers are
made to agents on the waitlist in a probabilistic manner, with the probability dependent on the
agent’s position in the waitlist. They find that FCFS results in the highest level of waste, but all
agents are better off under FCFS compared with any other probabilistic queuing discipline. One
shortcoming of their model is that they assume an exogenously given queue length, whereas in
practice the number of agents on the waitlist is determined by supply and demand. Another
shortcoming is that they are only able to fully solve their model with two agents. In contrast,
our model captures the endogeneity of queue length and can be fully solved when there is a
continuum of patients. One difference in the results is that in our model, not all agents are always
47

better off under FCFS, as will be illustrated in a stylized example in Section 3.2. Schummer [52]
shows that disallowing agents to reject offers can reduce waste but may hurt agents’ welfare. We
strengthen this result by showing in Section 3.5 that there are markets in which any deviation
from the current allocation mechanism will hurt some agents.
There is also a growing empirical literature on the waste problem. Zhang [66] shows that the
existence of observational learning contributes to waste, as when those at the top of the waitlist
reject an organ, the subsequent patients may infer that the organ must be of low quality and
reject it as well. This is referred to as herding and increases organ wastage. Our model does
not explicitly incorporate observational learning, and we show that even without this effect,
wastage is a problem whenever an allocation mechanism prioritizes agents based on waiting
times. Agarwal et al. [2] estimate a structural demand model of patient choice and compare
various alternative allocation mechanisms by simulation. They find that compared to the status
quo, alternative mechanisms can either yield a higher average welfare or a lower discard rate, but
not both. We corroborate this empirical observation by formally proving that in certain markets,
the status quo of waitlist with choice is not Pareto dominated by any other allocation mechanism,
even when it exhibits waste.
3.2 A Stylized Example
For clarity of exposition, we introduce the main ideas of the chapter in a stylized example with
two organ types, and defer the model in its full generality to Section 3.3. Although in practice,
clinicians or surgeons often make the decision to accept or reject an organ on behalf of patients,
48

they aim to keep the patients’ best interest in mind and consider each patient’s individual circumstances when making decisions. For simplicity of exposition, we will describe the decisions
as being made by patients, but one can understand it as physicians making decisions on their patients. This choice of language is consistent with the literature, which typically refers to patients
as decision makers [56, 57, 58, 2].
Suppose that there are two types of cadaveric organs: those from young donors and those
from old donors. For brevity, we refer to these as “young organs” and “old organs”. Suppose that
the young and old organs are procured for transplantation at rates of 0.45 and 0.3, respectively.
In comparison, patients request for transplantation at a rate normalized to 1. Hence, there is
an overall shortage of organ supply, which is the case in practice. For simplicity, the arrivals of
organs and patients are uniform over time, and the market is so large that each organ and each
patient can be thought of as being infinitesimally small.
In this example, all patients prefer the young organs to the old organs, but even the old organs
are beneficial to everyone. Patients vary in the intensity of their preference for young organs.
Concretely, there are two segments of patients of equal size: segment A patients have a strong
preference for young organs, whereas segment B patients only have a moderate preference. A
patient’s utility for staying unmatched is normalized to 0. The utilities that patients in each
segment obtain for being matched to each organ type are summarized in the following table:
Old organ Young organ
Segment A patients 1 8
Segment B patients 3 5
Table 3.1: The table shows the patients’ utilities for being matched to each organ type in the
stylized example.
49

Each patient leaves the market either after she accepts an organ or after she becomes ineligible
for transplantation due to worsening health conditions. While eligible, each patient exercises her
choice over whether to accept the organs offered by the allocation mechanism to maximize her
utility. We assume that patients’ lifetime of being eligible for transplantation is exponentially
distributed with a mean of 10 years.
3.2.1 The Status Quo: the Waitlist with Choice
Under the current allocation mechanism for cadaveric organs, patients’ waiting times play a significant role in determining their allocations, and patients can reject offers and maintain their
priority. Therefore, we approximate the current allocation mechanism as a waitlist that prioritizes patients only based on waiting times, and patients are not penalized for rejecting offers. We
refer to the mechanism as the waitlist with choice. This abstracts away from biological compatibility and medical urgency, but the same insights will hold under a richer setting as well.
In the waitlist with choice, patients gain better chances for popular organs by waiting longer.
Following this idea, the equilibrium of the waitlist with choice specifies the necessary waiting
time for patients to reach a good enough position and obtain an offer of each organ type. Given
the previous market primitives, we solve for the equilibrium waiting time for each organ type:
around 5 years for the young organs∗
and 0 for the old organs. The old organs are immediately
available to the patients, because the demand falls short of the supply in the equilibrium.
With the equilibrium waiting times, each patient decides which organ to aim for to maximize
her utility. Upon entry and before a patient’s waiting time reaches 5 years, she can only choose
∗More accurately, the waiting time for the young organs is 10 ln(5/3) years in the equilibrium of the waitlist
with choice.
50

to get an old organ and leave the market. After her waiting time reaches the cutoff for a young
organ, she can choose to get a young organ and leave the market. We plot how the set of organs
available to a patient changes with her waiting time in Figure 3.1.
Figure 3.1: The figure shows that the set of organs available to a patient expands with her waiting
time.
As in Figure 3.1, the set of organs available to patients expands with their waiting times.
Therefore, patients have incentives to turn down less desirable organs now to wait for better
offers in the future. However, since patients may become ineligible for transplantation during
waiting, their probability of surviving more than 5 years and getting the young organs is only
0.6. On the other hand, if they are okay with the old organs, then they can get one immediately
with probability 1. We define the set of probability bundles of old and young organs accessible
to patients as their choice set. This is a reduced form representation of the allocation mechanism
in equilibrium. For this numerical example, we plot the patients’ choice set in Figure 3.2, where
the axes denote the probabilities of getting the young and old organs, respectively.
The origin (0, 0) represents patients’ outside option, which is to reject everything. The point
(1, 0) represents accepting an old organ upon entry, whereas the point (0, 0.6) represents waiting
for a young organ. Patients are allowed to randomize over their choices, so their choice set is the
convex hull of these three points.
Every patient chooses a utility-maximizing probability bundle within the choice set. In the
equilibrium, segment A patients, who have a strong preference for the young organs, all choose to
51

Figure 3.2: The patients’ choice set under the waitlist with choice.
wait for them; segment B patients, who only have a moderate preference, play a mixed strategy.
In expectation, half of segment B patients choose to wait for the young organs and the other half
settle with the old ones. The equilibrium outcome is represented in Figure 3.3, where the patient
types are indicated by the valuation vectors (1, 8) and (3, 5), respectively.
Figure 3.3: The equilibrium outcome under the waitlist with choice.
The equilibrium outcome of the waitlist with choice exhibits waste. All segment A patients
and half of segment B patients leave the market without being matched with probability 0.4; in
the meantime, 1/6 of the positively valued old organs are discarded. This coincides with what
happens in practice: many patients pass on the less ideal cadaveric organs and are removed
from the waitlist due to worsening health conditions, and these less ideal organs end up being
discarded.
52

Moreover, waste is not unique to the waitlist with choice. The equilibrium outcome of the
waitlist with choice can be replicated by running independent lotteries for the young and old
organs with winning probabilities of 0.6 and 1, respectively. Besides, patients upon entry can
choose at most one of the two lotteries to play, and they can play only once in their lifetime.
The lottery implementation is similar to the one-shot version of the public housing allocation in
Singapore.†
Next, we describe two alternative allocation mechanisms that eliminate waste.
3.2.2 Eliminating Waste: Running Lotteries for the Young Organs
The fundamental source of waste in the waitlist with choice is the incentive to reject offers of
reasonable organs now to wait for better offers in the future. Patients are essentially playing a
high-stake gamble with their lives: if they win by outliving the waiting time needed for better
organs, then they obtain higher utilities; however, if they lose the gamble, then they leave the
market forever with nothing. To avoid such a risky gamble, we run an actual lottery for the young
organs upfront, so that a significant portion of the uncertainty in patients’ allocations is resolved
earlier and patients who lose the lottery have an opportunity to choose old organs as backup.
The “lottery + waitlist with choice” mechanism runs as follows. Each patient enters a lottery
for the young organs upon entry and gets offered one if she wins. Otherwise, she continues to
join the waitlist for the remaining organs. To eliminate waste, the parameters are set such that
each patient wins the lottery with probability 0.4, or she can wait for around 18 years to get a
†For more information about the public housing allocation in Singapore, see https://www.hdb.gov.sg/cs/infoweb/
homepage.
53

young organ‡
, whereas the old organs are always immediately available. The 18 years of wait
translates into a slim survival probability of 1/6.
With these parameters, the “lottery + waitlist with choice” gives rise to the choice set shown
in Figure 3.4. In the equilibrium, all patients who win the lottery get young organs immediately.
For the segment A patients who lose the lottery, their strong preference for the young organs
induces them to endure the long wait. In comparison, the segment B patients who lose the lottery
choose to settle with the old organs. Patients’ choices are shown graphically in Figure 3.4.
Figure 3.4: The equilibrium outcome under the one-shot CEEI.
The above equilibrium outcome would also arise if the policy designer implements a one-shot
version of competitive equilibrium with equal incomes (CEEI): Patients are given equal endowments of virtual currency and buy probability bundles of organs at market clearing prices. Moreover, patients can only participate once in their lifetime. If we normalize each patient’s virtual
endowment to be 1, then the market clearing prices are 2 per young organ and 1/3 per old organ.
Since all organs are utilized, the above equilibrium outcome does not exhibit waste. However,
eliminating waste does not come for free. The segment A patients now have a higher probability of leaving the market without getting matched and become worse off compared to being
‡More accurately, the waiting time for the young organs is 10 ln 6 years if a patient does not win the lottery.
54

under the waitlist with choice. Moreover, allocating the precious young organs through a lottery can be contentious, as those who lose the lottery can blame the policy designer, whereas
they have no one to blame but their own deteriorating health in the waitlist with choice. Both of
these downsides can impede the adoption of this waste eliminating mechanism and require hard
conversations within the organ transplantation community.
3.2.3 Eliminating Waste: Randomizing the Patients’ Priorities
If we are okay with allocating the young organs by lottery, then there are many possible means
of eliminating waste. For example, we can determine patients’ priorities entirely by lottery. The
mechanism, which we refer to as one-shot random serial dictatorship (RSD), assigns a lottery
number to each patient upon entry, and this lottery number determines her priority when making
offers.
Concretely, in equilibrium, a patient can choose between young and old organs if her lottery
number is among the top 45%, and she can get an old organ if her lottery number is among the top
75%. The remaining patients are not offered anything. Patients’ choices are shown in Figure 3.5.
Figure 3.5: The equilibrium outcome under the one-shot RSD.
55

Similarly, since all organs are utilized, the above equilibrium outcome does not exhibit waste.
However, similar to the “lottery + waitlist with choice” mechanism, segment A patients become
worse off compared to being under the waitlist with choice. Besides, ranking patients arbitrarily
can be hard to justify.
Summary of insights. The above discussion suggests the following insights:
1. To eliminate waste, we must disincentivize patients to gamble with their lives by turning
down acceptable offers now to wait for better offers in the future.
2. Moving to a waste eliminating mechanism may not be Pareto improving, as those most
willing to wait may be harmed.
The remainder of the chapter rigorously establishes these insights under a general model.
3.3 Model
In this section, we generalize the above stylized example by allowing for arbitrary many organ
types and an arbitrary distribution of patients’ preferences. Section 3.3.1 describes the market
primitives. Section 3.3.2 formalizes the concept of an allocation mechanism and an equilibrium
outcome. Section 3.3.3 defines waste and Pareto improvement. Section 3.3.4 discusses the property of differentiating agents’ only by their waiting times and introduces another extreme —
allocating everything upon entry.
56

3.3.1 Market
There are n types of items (i.e. organs), and type j items continuously arrive to the market at a rate
λj ≥ 0, j = 1, 2, . . . , n. Specifically, the “continuous arrival” follows from a fluid approximation
of the Poisson process. The interpretation of λj
is that for any time δ ≥ 0, a mass of λjδ type j
items arrive to the market.
Agents (i.e. patients) continuously arrive to the market at a rate normalized to 1. They leave
the market either when they accept items or when they become ineligible after an exponentially
distributed time with mean 1/β. As in the stylized example, patients may become ineligible for
transplantation due to worsening health conditions. The process that agents become ineligible is
exogenous and independent among the agents. In addition, it is independent from the item and
agent arrivals.
Each agent’s type is characterized by her valuation vector v = (v1, v2, . . . , vn) ∈ V ⊆ R
n
.
Agents’ valuation vectors are drawn independently and identically from a continuous probability
distribution on the set V with a cumulative distribution function F. To make the model general,
we do not impose any assumption on F, and agents’ preferences for different types of items
are allowed to be arbitrarily correlated. Therefore, agents do not necessarily have an agreed
ordering of the item types as in the stylized example. An agent with v obtains a one-shot utility
of vj and leaves the market if she accepts a type j item, and her payoff from staying unmatched
is normalized to 0.
The following definition encapsulates all the model primitives.
57

Definition 1 (Market). A market is defined as a tuple (n, λ, F, β), where n is the number of item
types, λ is the vector of items’ arrival rates, F is the cumulative distribution function of agents’
valuations, and 1/β is the agents’ mean time of being eligible for allocations.
3.3.2 Allocation Mechanisms and Outcomes
In this chapter, we focus on anonymous allocation mechanisms, which do not differentiate agents
by their types. As in the stylized example, the waitlist with choice treats both segment A patients
and segment B patients the same and offers them an identical choice set. Furthermore, in the
context of cadaveric organ allocation, anti-discrimination law is in place to prohibit differentiating patients based on their identities [13]. Despite that we only consider anonymous allocation
mechanisms, our derived insights carry over to non-anonymous allocation mechanisms.
Typically, in the market design literature, one defines a mechanism to be parameterized by
several parameters and an equilibrium of the mechanism to be a fixed point. For example, a
mechanism can be parameterized by prices and an equilibrium corresponds to a price vector that
clears the market. Under this approach, in order to find an equilibrium, one has to solve a fixed
point problem to determine the correct parameters.
In this chapter, we follow Arnosti and Shi [3] and adopt the following alternative construction,
which is mathematically equivalent to the above but avoids the need to solve the fixed point
problem. We define an allocation mechanismMto be a collection of matching rules parameterized
by a vector of parameters and an equilibrium outcome of M to be a matching rule that satisfies
certain feasibility conditions. Every such matching rule corresponds to an equilibrium outcome
58

of M, but we can analyze the equilibrium outcome without necessarily having to solve for the
parameters themselves in the matching rule.
We now formally define the matching rules and equilibrium outcomes.
3.3.2.1 Matching Rules
We revisit the waitlist with choice to motivate the definition of matching rules. As in the stylized
example, a vector z ∈ [0,∞)
n
specifies the necessary waiting time for each item type in the
waitlist with choice. Essentially, the waiting time vector z parameterizes a matching rule that
determines what agents get given their actions.
We first show that in the waitlist with choice, it suffices to think of each agent’s action as
picking her favorite item type. A general characterization of agents’ actions is to specify a tuple
(S, π) for each agent, where S ⊆ [n] := {1, 2, . . . , n} is a set of acceptable item types and π is
a permutation of S corresponding to her preference ranking. The interpretation is that when an
agent gets an offer, if the offer is acceptable, then she accepts the offer and leaves the market;
otherwise she rejects the offer. When more than one acceptable offers arrive, she breaks the tie
according to her preference ranking π and leaves the market with her favorite item. Given the
waiting time vector z, we denote j
∗
(S, π) as each agent’s favorite item type that arrives first, or
arg minj∈S zj
. Moreover, we observe that each agent always leaves with her j
∗
(S, π). Therefore,
it is equivalent to adopt a simpler construction and think of each agent’s action as picking her
j
∗
(S, π). Hence, the agents’ action set A reduces to {a0} ∪ [n], and agents can choose a = a0
to indicate that they reject everything or choose a = j to indicate that they only accept type
j items. If an agent chooses to only accept a type j item, then her probability of getting one
59

is exp(−βzj ), since she may become ineligible during waiting. Therefore, parameterized by a
waiting time vector z, the matching rule of the waitlist with choice is
rj (a) =



e
−βzj
, j = a
0, otherwise.
(3.1)
Following the idea above, we define the matching rules of general allocation mechanisms.
Denote the agents’ action set under an allocation mechanism M as A, and A is an arbitrary finite
set that includes the outside option a0. A matching rule r of M is a mapping from the agents’
actions to their allocations, r : A → Ω, where Ω = {x ∈ R
n
:
Pn
j=1 xj ≤ 1, xj ≥ 0 for all j} and
xj denotes the probability of receiving type j items.
As in the stylized example, a matching rule r gives rise to a choice set Cr for the agents. Since
agents are allowed to randomize over their actions, given a matching rule r, the generated choice
set Cr is the convex hull of {r(a) : a ∈ A}, and agents can pick any point from Cr ⊆ Ω.
3.3.2.2 Equilibrium Outcomes
To succinctly describe the final allocations, we define an outcome to be a mapping directly from
agents’ types to their allocations, µ : V → Ω. An equilibrium outcome of an allocation mechanism
is characterized by the following three conditions.
First, in the equilibrium outcome, agents are assigned to the probabilistic item bundles that
maximize their utilities. Concretely, given a matching rule r, agents’ allocations are solutions to
the following optimization problem,
60

max
x∈Cr
v · x. (3.2)
Second, the rate at which each item type is being allocated must not exceed its supply.
Third, any item type whose supply exceeds demand must be available to agents with probability 1. The last condition represents an implicit assumption that the allocation mechanism
cannot intentionally withhold items, since it is prohibited in cadaveric organ allocation [41] and
brings a trivial source of waste.
The following definition of equilibrium outcomes formalizes these three conditions.
Definition 2 (Equilibrium outcome). Given a market (n, λ, F, β), an outcome µ is an equilibrium
outcome of a mechanism M if
1. There exists a matching rule r ∈ Msuch that agents are assigned to allocations that maximize
their utilities under r: ∀v ∈ V, v · µ(v) = maxx∈Cr v · x;
2. The demand rates does not exceed the supply rates: ∀j ∈ [n], µ¯j
:= Ev∼F [µj (v)] ≤ λj
;
3. The items are not withheld intentionally: if µ¯j < λj
, then 1j ∈ C, where 1j
is a jth unit vector.
Following the definition, Proposition 1 shows that the waitlist with choice always has a unique
equilibrium outcome, and the proof is given in Section 3.A.1.
Proposition 1 (Existence and uniqueness). In any market, there exists a unique equilibrium outcome of the waitlist with choice.
61

3.3.3 Evaluating Outcomes
To compare different allocation mechanisms and evaluate equilibrium outcomes, we define waste
and Pareto dominance below.
One may simply define waste as discarding organs. However, if no one wants those organs,
then discarding them is not wasteful. To account for this, one may define waste as discarding
organs that some patients want, meaning that they prefer it over being unmatched. However, if
all of the patients who potentially want the discarded organs are already being matched with 100%
probability, then one can still argue that there is no waste, since utilizing the discarded organs
would not have increased the number of transplants. Hence, we define waste as discarding organs
that patients who are not fully matched want, so that offering those organs to those patients can
actually increase the number of transplants. In our model, the discarded organ types are those
for which the demand is less than the supply. Formally, given an outcome µ of an allocation
mechanism M, type j is said to be under-demanded if µ¯j < λj and over-demanded otherwise. A
formal definition of waste is given below.
Definition 3 (Waste). An outcome µ of an allocation mechanism M exhibits waste if there exists
an under-demanded type j, and it is possible for a positive measure of agents with vj > 0 to leave
the market without being matched. Otherwise, µ is said to eliminate waste.
The definition of waste implies that certain agents prefer gambling for better offers to accepting reasonably good offers. The gamble can be played with their lives as in the waitlist with
choice, or with their luck as in the one-shot lottery implementation. Either way, the gamble is
extremely risky as agents do not have backups and have to leave the market if they lose the bet.
62

Therefore, an allocation mechanism exhibits waste whenever it incentivizes agents to take on the
gamble despite already having reasonable offers.
The following definition of Pareto dominance is standard.
Definition 4 (Pareto dominance). An outcome µ Pareto dominates another outcome µ
′
if ∀v ∈
V, v · µ(v) ≥ v · µ
′
(v) with the inequality being strict for a positive measure of v ∈ V .
3.3.4 Allocating Everything upon Entry
Although the waitlist with choice incentivizes waiting and can lead to waste, it is used widely
because prioritizing agents by their waiting times is perceived to be acceptable to the stakeholders
and the general public [46]. Below we formally define the class of allocation mechanisms that
prioritize agents only by waiting times.
Definition 5 (Prioritize agents only by waiting times). An allocation mechanism M is said to
prioritize agents only by waiting times if for any matching rule r ∈ M, there exists a function C(t)
which specifies the set of item types available to an agent with waiting time t ≥ 0, and the generated
choice set Cr is the convex hull of the set of points {x ∈ Ω : x = exp(−βt)1j such that j ∈ C(t)},
where 1j
is the jth unit vector.
In other words, in such allocation mechanisms, agents’ actions are in the form of waiting for
certain time to gain enough priority and then choosing one of the available options. Following
the definition, the waitlist with choice prioritizes agents only by waiting times, despite at the cost
of potentially induced waste.
In the stylized example, we eliminate the waste by introducing lottery to make waiting less
attractive and allow for backups contingent on losing the lottery. To a further extreme, we define
63

a property of allocating everything upon entry. It implies that the agents who do not get their
ideal items can always choose any under-demanded items as backup.
Definition 6 (Allocating everything upon entry). An outcome µ is said to allocate everything upon
entry if ∀x ∈ Cr such that p :=
Pn
j=1 xj < 1 and type j
′
is under-demanded, then x+(1−p)1j
′ ∈ C.
3.4 Eliminating Waste
We now formalize what it takes to eliminate waste. From the stylized example, we observe that
introducing an external source of randomness, like lottery, in determining patients’ allocations
disincentivizes waiting and eliminates waste. However, not any kind of external randomness
serves this goal, as the equilibrium outcome of the waitlist with choice can always be replicated
by running independent lotteries.
There are several notable allocation mechanisms that eliminate waste in any market. In Section 3.A.2, we formally show that the one-shot CEEI always eliminates waste. In addition, it
is straightforward to show that for any ex-post Pareto optimal allocation mechanism from the
static setting, such as serial dictatorship, deferred acceptance or top trading cycles [1], if agents
are allowed to participate in the allocation mechanism for only once, then the equilibrium outcome eliminates waste. The results for serial dictatorship and deferred acceptance follow from
Azevedo and Leshno [7], and the result for top trading cycles follows from Leshno and Lo [25].
In the context of organ allocation, the analog of serial dictatorship would be to assign each patient a simple priority number upon entry, and using the same priority number to determine the
patient’s position in the queue for every organ, as illustrated in Section 3.2.2. Top trading cycles
would correspond to assigning each patient a lottery number for each organ type upon entry.
64

A patient is eligible for type j if and only if there exists a type j
′
for which her lottery number
exceeds a certain cutoff zjj′. The cutoff matrix z can be computed derived based on the market
primitives using the methodology derived in Leshno and Lo [25].
Motivated by these examples, we characterize the class of waste-eliminating allocation mechanisms in Theorem 1 with a minor regularity condition on the market.
Theorem 1 (Necessary and sufficient condition to eliminate waste). In any market where
the agents’ valuation distribution has positive density on the unit sphere§
, an equilibrium outcome
µ of a mechanism M eliminates waste if and only if µ allocates everything upon entry.
Proof. [Proof of Theorem 1] For one direction, suppose µ allocates everything upon entry. If on
the contrary, µ exhibits waste, then there exists an under-demanded type j
′
such that a positive
measure of agents with vj
′ > 0 may leave the market without being matched. Equivalently, for
these agents, their allocations satisfy Pn
j=1 xj < 1. Since µ allocates everything upon entry,
∀x ∈ Cr, x + (1 −
Pn
j=1 xj )1j
′ ∈ Cr. Therefore, each of these agents is strictly better off by
choosing x + (1 −
Pn
j=1 xj )1j
′, given her current allocation x. This contradicts to µ being an
equilibrium outcome.
For the other direction, suppose µ eliminates waste. When all item types are over-demanded,
µ allocates everything upon entry by definition. When all item types under-demanded, all items
are immediately available and µ also allocates everything upon entry by definition. For the other
cases, we reorder the item types such that under µ, type 1, 2, . . . , k are under-demanded and type
k + 1, k + 2, . . . , n are over-demanded, where 1 ≤ k ≤ n − 1.
§The regularity condition can be weakened to be that the agents’ valuation distribution has positive density on
an open set that contains the positive orthant, and the same proof works.
65

To show that µ allocates everything upon entry, it suffices to show that for any y ∈ Cr such
that p :=
Pn
j=1 yj < 1, y + (1 − p)1j
′ ∈ Cr for any j
′ ≤ k.
Since type k + 1 is over-demanded, we have that x¯k+1 := arg maxx∈Cr xk+1 < 1. Consider
the agents with v such that v1 > 0, vk+1 > 0, vj < 0 for all the other types and v1(1 − x¯k+1) +
vk+1x¯k+1 ≥ v1x1 + vk+1xk+1 for any x ∈ Cr. If (1 − x¯k+1, . . . , 0, x¯k+1, 0, . . . , 0) ̸∈ Cr, then a
positive portion of these agents may leave the market without getting matched and µ exhibits
waste. Contradiction. Therefore, (1 − x¯k+1, . . . , 0, x¯k+1, 0, . . . , 0) ∈ Cr. By symmetry, we have
that for all over-demanded item types j = k + 1, . . . , n, the point with x1 = 1 − x¯j and xj = ¯xj
is in Cr.
Besides, (1, 0, . . . , 0) ∈ Cr as type 1 is under-demanded. Therefore, by convexity of Cr, we
have that (1 −
Pn
j=k+1 yj
, 0, . . . , 0, yk+1, . . . , yn) ∈ Cr.
By symmetry, we have that (0, . . . , 0, yk+1, . . . , yn)+ (1−
Pn
j=k+1 yj )1j
′ ∈ Cr for any j
′ ≤ k.
By convexity of Cr again, we have that y + (1 − p)1j
′ ∈ Cr for any j
′ ≤ k.
As illustrated in the stylized example, waiting is inherently risky for patients because their
health conditions are deteriorating. Therefore, even a slight incentive offered by the allocation
mechanism to play the high stakes gamble may induce waste. Building on this intuition, Theorem 1 points out that when we are faced with a diverse group of agents, eliminating waste
essentially requires allocating everything upon entry. Because only by doing so, we eliminate
incentives for waiting and other high-stake games since agents do not gain by waiting and can
always have under-demanded items as backup.
Besides the aforementioned waste-eliminating allocation mechanisms, there are other ones
that allocate everything upon entry. For example, the LCFS allocates everything upon entry, since
66

agents’ priorities drop as they wait and agents can always leave with under-demanded items.
Although the lottery implementation of the waitlist with choice does not allocate everything
upon entry, if we allow the patients who lose the lottery to come back for under-demanded items,
then this modified lottery mechanism will allocate everything upon entry and always eliminate
waste.
3.5 Cost of Eliminating Waste
Although eliminating waste is an attractive goal, as we observe in the stylized example, it is not
free of concerns. First, allocating everything upon entry can be contentious and not as easily
justifiable as prioritizing patients by their waiting times. Second, the patients who are more
willing to wait become worse off after eliminating waste. We investigate both concerns in more
detail.
Eliminating waste requires moving away from differentiating agents only by their waiting
times. In the stylized example, we include the lottery component to differentiate agents by their
luck besides their lifetime. However, this can be a contentious reform as agents can affect their
waiting times but not their luck. Moreover, agents with bad luck may blame the allocation mechanism for creating artificial unfairness. But if we want to stick with prioritizing agents only by
waiting times, then Proposition 2 shows that the waitlist with choice is the only thing we can do,
and its unique equilibrium outcome given by Proposition 1 is the best we can hope for.
Proposition 2 (Implication of prioritizing only by waiting times). A mechanism prioritizes
agents only by their waiting times if and only if it is the waitlist with choice.
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Proof. [Proof of Proposition 2] By definition, the waitlist with choice prioritizes agents only by
their waiting times. For the other direction, if M is a mechanism that prioritizes agents only by
their waiting times, then for any matching rule r ∈ M, the generated choice set Cr is the convex
hull of the set of points {x ∈ Ω : x = exp(−β)1j such that j ∈ C(t)}. Let zj = mint{j ∈ C(t)},
which denotes the minimal time it takes to receive type j offers. Cr is then also the convex
hull of the set of points {x ∈ Ω : x = exp(−βzj )1j
, j = 1, . . . , n}. Notice that Cr is exactly
the generated choice set by the matching rule r of the waitlist with choice parameterized by z.
Therefore, M is the waitlist with choice.
Another challenge we may face when eliminating waste is that the resultant equilibrium outcome does not Pareto dominate the equilibrium outcome of the waitlist with choice. One may
think that in the waitlist with choice, given the discarded organs and the unmatched patients, we
can always find a better allocation mechanism to make everyone better off. However, Proposition 3 shows that this is not the case, as in some markets where the waitlist with choice exhibits
waste, it cannot be Pareto improved by any allocation mechanism.
Proposition 3 (Impossibility of Pareto improvement in certain markets). There are markets in which the equilibrium outcome of the waitlist with choice exhibits waste, but is not Pareto
dominated by any equilibrium outcome of any allocation mechanism.
The above result has no analog in the literature, largely because previous papers do not allow for rich preference heterogeneity, without which studying Pareto dominance is not interesting. The result is surprising because one might have thought that waste implies an opportunity
for Pareto improvement, as one can reallocate the wasted organs to the patients who are not
matched with 100% probability. For example, suppose a patient currently has a 50% chance of
68

getting a young kidney before her health deteriorates and 50% chance of departing without being
matched. The mechanism may better utilize the discarded old kidneys by giving her an additional
30% chance of getting an old kidney on top of her 50% chance of getting a young kidney. The
problem is that when this probabilistic bundle of (50%, 30%) is available, many people who were
previously opting for 100% chance of getting an old kidney may now choose this bundle instead,
thus increasing the demand for young kidneys, so those that only want young kidneys face more
competition and are strictly worse off.
The proof is based on this logic and given in Section 3.A.3. The main difficulty of the proof
is in showing that the above logic arises even under natural preference distributions, such as the
uniform distribution. One can show that the same proof holds whenever the density of preferences in polar coordinates is weakly increasing toward the under-demanded type.
Taken together, Proposition 2 and Proposition 3 suggest that eliminating waste may lead to
serious objections, as the allocation process becomes more contentious and some patients may be
harmed. Nevertheless, given the potential gains from eliminating waste, the policymakers need
to negotiate with the stakeholders to find a common ground between the benefits and costs of
eliminating waste.
3.6 Discussion
In this chapter, motivated by the serious wastage problem in the cadaveric organ allocation process, we uncover a source of waste inherent in the current allocation mechanism: prioritizing
patients by their waiting times. We highlight that it will still cause waste despite all the reforms
that are being undertaken, because it incentivizes patients to wait for better organs and leaving
69

the market during waiting is irreversible. Through studying a theoretical model, we show that
waste can only be eliminated by allocating over-demanded organ types upon entry while giving
patients under-demanded items as backup.
However, eliminating waste by allocating everything upon entry can be practically contentious and not Pareto improving. Therefore, our theoretical results suggest that hard conversations must be initiated within the organ transplantation community, so that the stakeholders
can weigh the benefits of eliminating waste against the costs, and a compromise may be found to
reduce waste. For example, we believe that adding a small extent of randomization to the current
allocation mechanism, such as giving random priority boosts to patients, can reduce waste while
being less contentious.
One practical consideration is that if over-demanded organs are reserved for those who recently enter the market, patients may strategize about when to enter the market. In our model,
the arrival rates of organs and patients are constant, so there is no better or worse time of entering
the market. In reality, there may be certain times periods in which there are more organs available relative to demand. The effect of strategic entry would be to stabilize the demand/supply
ratio over time, which may actually be good for the system. However, one has to rule out the
possibility of hearing about a good organ and entering immediately to snatch it. A simple rule to
accomplish this would be that organs are only offered to patients who are already in the market
before the donor’s death.
We conclude by discussing the robustness of our results when we generalize the model in
various ways.
70

3.6.1 Heterogeneity in β and unexpected departures
In the model, we assume that all agents share the same β and become ineligible unexpectedly at
the same rate. Although this assumption is widely used in the modeling literature [56, 58, 3], it
can be unrealistic since there is heterogeneity in patients’ health conditions when they join the
waitlist. The patients with worse health conditions may become ineligible faster and thus more
desperate in taking offers. Moreover, in practice, there are cases where surgeons are willing to
accept marginal organs when patients’ conditions deteriorate.
We can address the concern regarding heterogeneity in β by introducing a distribution on β.
However, this only complicates the analysis but does not alter our results. For the agents with
small enough β, they are still willing to gamble with their lives for better gains, which further
leads to waste. Following the same logic as in Theorem 1, the waste can only be eliminated by
allocating everything upon entry.
On the other hand, we may also interpret the heterogeneity in β as the heterogeneity in
agents’ valuation vectors. For example, the patients with worse health conditions tend to value
the more popular organs less than the other patients. We can think that the heterogeneity in β
has already been accounted for in the agents’ valuation vectors.
The assumption of unexpected departures is motivated by practical considerations. Large
transplant centers are usually faced with thousands of patients. Therefore, due to capacity constraints, it is impossible to keep track of all patients’ health conditions. But better waitlist management to enable decision-making based on patients’ most up-to-date health conditions can
indeed reduce waste [32].
71

3.6.2 Approximation of the Allocation Mechanism in Practice
In this chapter, we approximate the allocation mechanism for cadaveric organs as the waitlist with
choice, although other factors such as patients’ compatibility with organs and medical urgency
are also used to rank patients [41].
We can model a patient’s priority score given a cadaveric organ as a linear combination of
her waiting time, compatibility with the cadaveric organ under consideration and her personal
characteristics. The second component acts as a random boost in her priority score due to the
randomness in cadaveric organ supply, and the last component acts as a fixed boost. Therefore,
under this construction, a patient still gains by waiting in the sense that she increases her expected
priority score by increasing her waiting time.
Following the same logic as Theorem 1, our results still hold. Because with a diverse pool
of patients, certain patients may still find waiting profitable and this brings waste. Therefore, to
eliminate waste, more external randomness in patients’ priority scores should be introduced to
make waiting not attractive, and we arrive at allocating everything upon entry in the end.
3.A Omitted Proofs
3.A.1 The Equilibrium Outcome of the Waitlist with Choice
We first show the existence of an equilibrium outcome under the waitlist with choice and then
show its uniqueness.
72

To simplify the notation, for a given vector of waiting times z, we denote pj = e
−βzj
. Before
proving the proposition, we define the demand function, utilization function and equilibrium
waiting times of the waitlist with choice as below.
Definition 7 (Demand function and utilization function). For a given market (n, λ, F, β) and a
given vector of waiting times z ∈ [0,∞)
n
of the waitlist with choice, the demand function D(p) is
Dj (p) = Pv∼F

vjpj = max
j
′∈[n]
{vj
′pj
′}

, (3.3)
where pj = e
−βzj
, j = 1, . . . n. The corresponding utilization function U(p) is
Uj (p) = pjDj (p), (3.4)
Definition 8 (Equilibrium waiting time vector). For a given market (n, λ, F, β), a vector z ∈
[0,∞)
n
is an equilibrium wait-time vector of the waitlist with choice if its corresponding matching
rule gives rise to an equilibrium outcome.
To show the existence of an equilibrium, we define a mapping Γ(p) : [0, 1]n → [0, 1]n
and for
each component j, we have
Γj (p) = sup
x∈[0,1]
{Uj (h(p, j, x)) ≤ λj}, (3.5)
where
hk(p, j, x) =



x if k = j,
pk otherwise.
73

We show that Γ(p) is order preserving. Consider any two vectors p and p
′
such that p ≤ p
′
.
We have that for any fixed x ∈ [0, 1], Uj (h(p
′
, j, x)) ≤ Uj (h(p, j, x)) because offers of types other
than j are more attractive in p
′
. Therefore, Γj (p) ≤ Γj (p
′
) and Γ(p) ≤ Γ(p
′
).
By Tarski’s fixed point theorem, we have that there exists a fixed point such that Γ(p) = p.
The corresponding z such that zj =
1
β
ln 1
pj
is an equilibrium wait-time vector by construction of
Γ(p).
Now we prove by contradiction that the equilibrium wait-time vector z is unique.
Suppose that there are two different equilibrium waiting-time vectors z, z′
. Without loss
of generality, we assume that z and z
′ differ in the first component and z1 < z′
1
. Therefore,
z
′
1 > 0, p′
1 < 1 and U1(p
′
) = λ1.
Besides, we have D1(p) ≥ D1(p
′
) by construction of Γ(p) and p1 > p′
1
. Hence, U1(p) =
p1D1(p) > p′
1D1(p
′
) = U1(p
′
) = λ1, contradiction.
3.A.2 The One-shot CEEI Never Exhibits Waste
For a given market (n, λ, F, β) and a price vector q ∈ [0,∞)
n of the one-shot CEEI, the set of
possible allocations is Ω(q) = {x ∈ R
n
:
Pn
j=1 xj ≤ 1,
Pn
j=1 qjxj ≤ I, xj ≥ 0, j = 1, . . . , n},
where I ∈ [0,∞] is the finite income.
Since Ω(q) is a bounded polyhedron, Ω(q) has finite many vertices and we denote the set of its
vertices as A(q) = {x ∈ Ω(q) : x is a vertex of Ω(q)}. We define the corresponding continuous
utilization function and equilibrium price vector of the one-shot CEEI as below.
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Definition 9 (Utilization function). For a given market (n, λ, F, β) and a price vector q ∈ [0,∞)
n
of the one-shot CEEI, the utilization function U(q) is
Uj (q) = X
x∈A(q)
xjPv∼F

x · v = max
x′∈A(q)
x
′
· v

, j = 1, . . . , n. (3.6)
As U(q) is homogeneous of degree 0 in q, we normalize the space of price vectors such that
Pn
j=1 qj = 1.
Definition 10 (Equilibrium price vector). For a given market (n, λ, F, β), a price vector q ∈ Q =
{q ∈ [0,∞)
n
:
Pn
j=1 qj = 1} is an equilibrium price vector of the one-shot CEEI if its corresponding
matching rule gives rise to an equilibrium outcome.
To prove that the one-shot CEEI never exhibits waste, we first show the existence of an equilibrium outcome under the one-shot CEEI. We define a mapping Γ(q) : Q → Q, known as the
Gale-Nikaido mapping, and for each component j, we have
Γj (q) = qj + max{0, Uj (q) − λj}
1 + Pn
j=1 max{0, Uj (q) − λj}
. (3.7)
As Uj (q) is continuous for all j, we have that Γ(q) is continuous. Moreover, as Q is compact
and convex, following the Brouwer’s fixed point theorem, we have that there exists a fixed point
q
∗
such that Γ(q
∗
) = q
∗
.
Now we show that q
∗
is indeed an equilibrium price vector. The first condition holds by the
construction of U(q). The third condition follows from the Walras’ Law. It suffices to show that
under q
∗
, Uj (q
∗
) ≤ λj for all j, or equivalently, Pn
j=1 max{0, Uj (q
∗
) − λj} = 0.
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Suppose Pn
j=1 max{0, Uj (q
∗
) − λj} > 0, then there exists some j
′
such that Uj
′(q
∗
) > λj
′.
As q
∗
is a fixed point, we have that
∀j, q∗
j = Γj (q
∗
) =
q
∗
j + max{0, Uj (q
∗
) − λj}
1 + Pn
j=1 max{0, Uj (q
∗
) − λj}
, (3.8)
which implies that ∀j, q∗
j
Pn
j=1 max{0, Uj (q
∗
)−λj} = max{0, Uj (q
∗
)−λj}. Therefore, we have
that for all j, q
∗
j > 0 if and only if Uj (q
∗
) > λj
, as Pn
j=1 max{0, Uj (q
∗
) − λj} > 0. Hence, by the
existence of j
′
, we have that Pn
j=1 q
∗
j
(Uj (q
∗
) − λj ) > 0, which contradicts the Walras’ Law.
Thus, Pn
j=1 max{0, Uj (q
∗
) − λj} = 0, and q
∗
is an equilibrium price vector.
We now show that the equilibrium outcome µ of the one-shot CEEI arising from the equilibrium price vector q does not exhibit waste.
Suppose µ exhibits waste. By definition of waste, for an under-demanded type j, it is possible
for an agent with vj > 0 to leave the market without being matched. Since µ is an equilibrium
outcome and does not withhold items intentionally, we have that qj = 0. Therefore, the agents
with vj > 0 can be strictly better off by purchasing more probability share of type j, contradicting
that µ is an equilibrium outcome. Therefore, µ does not exhibit waste.
3.A.3 Proof of Proposition 3
We first construct a concrete market under which the proposition holds. Consider the market
(n, λ, F, β) where n = 2 and F is a uniform distribution over V = {(v1, v2) : v
2
1+v
2
2 < 1, v1, v2 >
0}. Below we show that whenever λ2 ∈ (0, 1/2) and λ1 is sufficiently large so that type 1 items are
under-demanded, then the equilibrium outcome of the waitlist with choice exhibits waste but is
76

not Pareto dominated by any equilibrium outcome of any mechanism. For technical convenience,
we parameterize V using polar coordinates, with (v1, v2) = (l cos θ, lsin θ), where l ∈ (0, 1) and
θ ∈ (0, π/2). Since l does not affect agent choice, we restrict attention to the valuation direction
θ, which is uniformly distributed within (0, π/2).
What does the outcome of the waitlist with choice look like in this market? For any λ2 ∈
(0, 1/2), there exists a unique ω ∈ (0, π/4) such that λ2 = 2ω tan(ω)/π, as the right hand side is
a strictly increasing function of ω within the domain ω ∈ (0, π/4). Define y := tan(ω) ∈ (0, 1).
The space of the allocations under the equilibrium outcome µ of the waitlist with choice with
type 1 items being under-demanded (λ1 > 1 − 2ω/π) is
∆ := Convex Hull({(0, 0),(1, 0),(0, y)}). (3.9)
Under µ, the agents with the valuation direction θ ∈ (0, π/2−ω) go for type 1 items, and the
agents with the valuation direction θ ∈ (π/2−ω, π/2) go for type 2 items. Given λ1 > 1−2ω/π,
µ exhibits waste of type 1 items, as those going for type 2 items have strictly positive probability
1 − y > 0 of leaving the market without being matched, whereas all agents have strictly positive
valuations for the under-demanded item type 1.
We now show that µ is not Pareto dominated by any equilibrium outcome of any alternative
mechanism. The main idea is that if an alternative mechanism offers an additional probabilistic
bundle outside the region ∆, agents who are choosing the under-demanded item type 1 would
substitute to the over-demanded type 2, which further increases the competition for type 2 items.
This hurts the patients who only desire type 2 items, as it makes the bundle (0, y) no longer
feasible. Precisely speaking, suppose on the contrary that there exists an equilibrium outcome µ
′
77

of any mechanism that Pareto dominates µ. Let the corresponding space of allocations be X. It
must be that X strictly contains ∆.
Define γmax := maxx∈X{yx1 + x2}. We have that γmax exists and γmax > y, since X is a
closed convex subset of the unit simplex Ω strictly containing ∆ by Pareto dominance. For any
γ ∈ [y, γmax], define the closed convex set
Xe(γ) := X
\
{(x1, x2) : yx1 + x2 ≤ γ} . (3.10)
Note that ∆ = Xe(y) and X = Xe(γmax). Hence, Xe(γ) defines a continuous interpolation between
the convex sets ∆ and X.
Define the total demand for type 2 items under the allocation space X˜(γ) as
D(γ) = 2
π
Z π/2
0
a2(γ, θ) dθ, where a(γ, θ) ∈ arg max
x∈Xe(γ)
{x1 cos θ + x2 sin θ} . (3.11)
a(γ, θ) is an optimal allocation for an agent with the valuation direction θ under the space of
allocations Xe(γ). Although there may be multiple optimal allocations for a fixed θ, this can only
occur for θ coming from a measure-zero set, so D(γ) is well defined. (This is because a compact
convex set has a unique tangent hyper-plane almost everywhere on its boundary.)
To complete the proof, we show that D(γ) is strictly increasing within the domain γ ∈
[y, γmax], from which it follows that D(γmax) > D(y). Here, D(γmax) is the total demand for
the over-demanded item type 2 under µ
′
, and D(y) is the total demand under µ. This inequality implies that the total demand for the over-demanded item type 2 is greater in X than in ∆,
so offering probabilistic bundles outside of ∆ increases the competition for the over-demanded
78

type, as described in the paragraph before the proof. However, D(y) = λ2, since type 2 is already
over-demanded under the waitlist with choice. This contradicts the feasibility of µ
′
.
The argument for showing the strict monotonicity of D(γ) is geometric. Given any γ ∈
(y, γmax), consider the difference D(γ)−D(γ −δ) for a small perturbation δ > 0. To a first order,
this difference is equal to the change in demand for item 2 among the agents who are choosing
points A(γ) and B(γ), which are the two points of the intersection between the boundary of X
and the line L(γ) := {(x1, x2) : yx1+x2 = γ}. Consider the normal cone to Xe(γ) at A(γ). Let the
absolute angle between the two extreme rays of the normal cone be α(γ). Similarly define β(γ)
for the normal cone to Xe(γ) at B(γ). The measure of agents choosing A(γ) in Xe(γ) is exactly
2α(γ)/π and the measure of agents choosing B(γ) is 2β(γ)/π. See Figure 3.6 for illustration.
Figure 3.6: The figure shows the geometric intuition for D(γ) being strictly increasing.
Note that as δ → 0, the difference
A2(γ) − A2(γ − δ) ≈
δ cos ω
sin α(γ)
sin(ω − α(γ)). (3.12)
This is because as γ decreases by a small perturbation, the point A(γ) approximately travels along
the tangent line which makes an angle of ω − α(γ) with the horizontal axis.
79

Hence, the left derivative,
∂−A2(γ) = −
cos ω
sin α(γ)
sin(ω − α(γ)). (3.13)
Similarly,
∂−B2(γ) = cos ω
sin β(γ)
sin(ω + β(γ)). (3.14)
Thus,
∂−D(γ) = 2
π
[α(γ)∂−A2(γ) + β(γ)∂−B2(γ)] (3.15)
=
2α(γ) cos ω
π

β(γ)
α(γ)
sin(ω + β(γ))
sin β(γ)
−
sin(ω − α(γ)
sin α(γ)

. (3.16)
To show that D(γ)is strictly increasing, it suffices to show that the left derivative ∂−D(γ) > 0
for all γ ∈ (y, γmax). Now, if ω − α(γ) < 0, then ∂−D(γ) > 0. So it suffices to consider the case
in which α(γ) ∈ (0, ω]. Furthermore, ω+β(γ) ≤ π/4, because Xe(γ) is a convex set that contains
(1, 0) and is contained in Ω.
For convenience, we write β instead of β(γ) and α instead of α(γ). However, the dependence
on γ is implicit. Note that for α, β ∈ (0, π/2), the ratio β/α is sandwiched between sin(β)/ sin(α)
and tan(β)/ tan(α). To see this, note thatsin(x)/x is strictly decreasing when x ∈ (0, π/2), while
tan(x)/x is strictly increasing when x ∈ (0, π/2). Hence, to show that Equation 3.16 is strictly
positive, it suffices to show that
sin β
sin α
sin(ω + β)
sin β
−
sin(ω − α)
sin α
> 0, (3.17)
80

and
tan β
tan α
sin(ω + β)
sin β
−
sin(ω − α)
sin α
> 0. (3.18)
After cancelling terms, Equation 3.17 follows from sin α > 0 and sin(ω + β) > sin(ω − α).
Moreover, Equation 3.18 follows from cos(α) = cos(−α), and the fact that the function g(x) =
sin(ω + x)/ cos(x) is monotonically increasing, as its derivative is
g
′
(x) = cos(ω + x) cos(x) + sin(ω + x) sin(x)
cos2
(x)
=
cos ω
cos2
(x)
> 0. (3.19)
81

Chapter 4
Wastage: Developing an Allocation Mechanism for Practice
4.1 Introduction
The kidney wastage has long been a pressing concern within the organ transplant community,
sparking ongoing discussions and efforts to address the problem. Among the proposals and interventions mentioned in Section 1.2, a significant focus has been put on increasing the utilization
of high-KDPI kidneys by expediting their placement, as about half of the procured high-KDPI
kidneys ended up being discarded. Specifically, it has been suggested that the high-KDPI kidneys could be preferentially offered to the transplant programs with a history of accepting such
offers [11] or to the patients with lower priority in exchange for reduced waiting times [8, 27].
Besides, fast-tracking the high-KDPI kidneys can reduce their time on ice, referred to as the cold
ischemic time, leading to better transplant outcomes and higher acceptance probabilities.
Two primary versions of the policies for fast-tracking the high-KDPI kidneys are: the Expanded Criteria Deceased Donor (ECD) policy and the Kidney Accelerated Placement (KAP)
project. The ECD policy enables patients to indicate their willingness to accept the high-KDPI
kidneys and bypasses the patients who are not willing to accept such kidneys during the allocation
82

process. By signing up for the ECD kidneys, the patients, who are usually old patients, can potentially receive a transplant earlier. A similar approach is employed in the European Senior Program, which allocates the kidneys from the donors aged 65 or older to the recipients of the same
age group. The KAP project is inspired by the observation that many kidneys with higher KDPI
scores and less desirable donor characteristics were allocated through OPO-initiated allocation
exceptions and transplanted into the patients significantly further down the match-run compared
to the non-exception transplants [24]. As these allocations were primarily based on the OPOs’
informal relationships with the transplant centers and the OPOs’ perceptions of the transplant
centers’ willingness to accept more challenging kidneys, this practice potentially exacerbates the
existing inequity in patients’ access to transplantation. Therefore, the OPTN implemented the
KAP project from July 18, 2019 to July 15, 2020 to institutionalize the idea of preferentially offering the hard-to-place kidneys to the programs with a history of accepting such kidneys. In the
KAP project, the “hard-to-place kidneys” are those from the adult donors with a KDPI score of 80
or higher and having reached the national allocation sequences, while the “qualified programs”
are those that have previously accepted and transplanted kidneys with similar or worse donor
characteristics than the donors being considered.
However, both the ECD policy and the KAP project do not address the kidney wastage problem. Studies have found that the discard rates of the ECD kidneys remained unchanged under
the ECD policy [59], although the policy led to a significant increase in the percentages of recovered kidneys from the ECD donors [60]. Schold et al. [49] found that after the ECD policy,
the older patients were more likely to receive the ECD kidneys, but there was no statistically
significant decrease in the cold ischemia times and waiting times overall. Moreover, there was
considerable heterogeneity in the transplant centers’ listing practices for the ECD kidneys: over
83

a quarter of centers listed fewer than 20% of their patients, and around a quarter of centers listed
90% of their patients. Only the centers with selective listing for the ECD kidneys observed reduced waiting times for the ECD recipients. Regarding the KAP project, Noreen et al. [33] found
that there was a notable 64% increase in the number of unique transplant programs that accepted
and transplanted the KAP-related kidneys, suggesting potential shifts in transplant programs’ behavior towards the hard-to-place kidneys. However, there was no statistically significant change
in the donor acceptance rate for the hard-to-place kidneys when compared to a similar period
and similar donor characteristics one year prior.
Numerous studies have been conducted to uncover the factors contributing to the limited
success of the ECD policy and the KAP project in addressing the kidney wastage problem. In
the ECD policy, the signals provided by the patients were noisy, as the patients could sign up
for the ECD kidneys without facing penalties for declining these offers later. Regarding the KAP
project, Noreen et al. [33] pointed out that the criteria for qualifying programs for the KAP project
might have been too lenient so that only the most selective programs were trimmed. Besides, the
accelerated placement was only triggered at the national level and came too late in the allocation
process, resulting in no significant difference in the cold ischemia times. Moreover, the definition
of “hard-to-place kidneys” might have been too strict such that not enough kidneys were included
to show the benefits of fast-tracking high-KDPI kidneys.
Besides these two reforms, many other attempts also failed to address the kidney wastage
problem and improve access for the vulnerable populations, despite the wide recognition and
extensive research of the kidney wastage problem [50]. This suggests a necessary departure
from exclusively focusing on expediting the allocation of high-KDPI kidneys and reducing the
cold ischemic times.
84

As suggested by the theoretical analyses in Chapter 3, what has been missing in these interventions is that the key in addressing the kidney wastage problem lies in how we allocate the
high-quality kidneys instead of the low-quality kidneys. Even if the perfect patient signals could
be obtained and the ideal KAP project could be implemented, there would still be waste, because
these policies do not change the prospects of waiting for the high-quality kidneys and the patients
are still incentivized to gamble with their life by rejecting the low-quality kidneys. Moreover, as
Mohan and Schold [30] cautioned, solely fast-tracking the low-quality kidneys might create a
permanent access advantage for a shrinking set of listing centers that receive and accept these
offers, potentially harming the access equity, a core value in the cadaveric kidney allocation.
In this chapter, leveraging the theoretical insights and building on the current kidney allocation system, we propose a practical mechanism and develop a simulation engine to evaluate it
with the actual kidney transplant data. We show that the proposed mechanism reduces kidney
wastage without significantly disadvantaging any patient group. Section 4.2 describes the intuition behind the proposed mechanism. Section 4.3 outlines the design of the simulation engine.
Section 4.4 presents the simulation results.
4.2 Periodic Boost Policy
The theory suggests that to eliminate waste, the good kidneys should be allocated only to the
newly arrived patients to reduce the incentive for waiting. However, such waste-eliminating
mechanisms may not be practical, as they significantly deviate from the current kidney allocation
mechanism, making them harder to be defended and implemented. Furthermore, such wasteeliminating mechanisms could hurt the patients who have waited for a long time.
85

Therefore, following the theoretical guidance, we propose a more practical mechanism called
the periodic boost policy, which requires a minor change to the status quo while reducing waste.
To understand the underlying intuition, a mechanism is described by a scoring rule sj (i, t) that
determines the allocation points for type j donors to type i patients. The status quo is represented
as t + αij , where t is a patient’s waiting time and αij describes the biological compatibility and
proximity between type i patients and type j donors. For example, the patients who have a
perfect tissue match with the donor, who live close to the donor and who are highly sensitized
have high αij s. Here, we assume that the patients’ waiting time is the only variable affected by
time. In practice, patients’ CPRA and EPTS scores can also change with time. However, as we
do not have access to the patients’ historical CPRA and EPTS scores and we do not find good
models describing how patients’ CPRA and EPTS scores change with time, we leave these for
future research. Therefore, for a type i patient, her allocation points for type j donors increase
linearly with time as shown in Figure 4.1.
Figure 4.1: The figure shows how a type i patient’s points for type j donors change with her
waiting time under the status quo allocation mechanism.
In comparison, the periodic boost policy is represented as t + αij + B1{t mod C < τ},
where B > 0 is the boost points, C > 0 is the length of the cycle time and τ > 0 is the length
of the boosted period. Therefore, on top of the allocation points under the status quo, patients
86

also receive the boost points B for time τ in every time C period. Hence, for a type i patient, her
allocation points for type j donors change periodically with time as shown in Figure 4.2.
Figure 4.2: The figure shows how a type i patient’s points for type j donors change with her
waiting time under the periodic boost policy.
Under the periodic boost policy, the patients are willing to accept the marginal kidneys when
they approach the end of the boosted period, because they expect that their probabilities of receiving the good kidneys are low; and until near the end of the unboosted period, because they
expect to be boosted again to receive the good kidneys. Moreover, by clearly specifying which
patients to be boosted and prioritized at each time, we help guide the listing centers in identifying
which patients to activate and get ready during each period. This can be a challenging problem
for the large transplant programs, as we learned from the doctors.
There are two special cases of the periodic boost policy that are worth noting. First, when
C = τ and the patients are always boosted, the periodic boost policy recovers the status quo.
Second, when C is very large and even greater than the patients’ maximum lifetime, the patients
are essentially boosted only once upon entry, and we refer to this subtype of the periodic boost
policy as the boost upon entry. By tuning the boost points and boost period, the boost upon entry
resembles the idea of allocating the good kidneys only to the newly arrived patients. One may
argue that the boost upon entry may incentivize the patients to game the system by picking the
87

optimal registration time such that when they join the waitlist and receive the boost, they would
just snatch the good kidneys. However, in equilibrium, this effect would be balanced out, as there
is a limited supply of the good kidneys and if there is such a better time to join the waitlist, then
there would be enough patients choosing that time to join to diminish its attractiveness.
4.3 Simulation Design
To evaluate the performance of the periodic boost policy compared to the status quo, we develop
a simulation engine that simulates the allocation process based on the inputs derived from the
actual kidney transplant data. Compared with the Kidney-Pancreas Simulated Allocation Model
(KPSAM) widely used by the organ transplant community [51], our simulation engine internalizes patients’ incentives and considers patients’ optimal behavior under alternative allocation
mechanisms. Besides, this simulation engine is different from the one in Agarwal et al. [2], because this simulation engine relies on fewer ad-hoc assumptions and can be better defended as
actually approximating a theoretical ideal.
For the simulation, the equilibrium concept is that of a mean-field equilibrium, or oblivious
equilibrium, with the two following conditions:
1. Optimality: Each agent is optimally responding based on the long-run average steady state
of the system, rather than to the current system state, which would be the case in a Markov
Perfect Equilibrium;
2. Consistency: The long-run average steady state is generated from the agents’ behaviors.
88

In contrast, the following ad-hoc assumptions were made in Agarwal et al. [2], which are hard
to be defended:
1. The long-run waitlist size N∗
is a deterministic quantity.
2. The agents believe that the set of other agents is drawn independently and identically from
the steady-state long-run average density.
3. The demand for the kidneys from a particular donor is Poisson distributed with the mean
based on integrating the steady-state density of the agent types.
In reality, the waitlist size may fluctuate, and such fluctuations may generate certain dependencies that violate the other two assumptions. It is possible that the above assumptions
from Agarwal et al. [2] are approximately correct, but without defining a ground-truth model, it
is hard to define what it means by “approximately correct”.
Specifically, our simulation engine contains two key steps. First, we simulate the allocation
using the patients’ acceptance thresholds to obtain the cutoffs for each kidney type. Second,
based on the cutoffs, patients reassess their chances of receiving offers, and we solve the optimal
stopping problem for each patient type using the cutoffs to obtain the new acceptance thresholds.
We do these two steps iteratively until patients’ acceptance thresholds stabilize.
Section 4.3.1 describes the primitives for the simulation. Section 4.3.2 describes the two steps
of the simulation in detail. Section 4.3.3 describes the challenge of cycling encountered in the
simulation due to the lack of randomness in patients’ kidney acceptance decisions and how we
address this challenge. Section 4.3.4 presents the metrics developed to evaluate and compare
different allocation mechanisms.
89

4.3.1 Primitives
There are n kidney types and m patient types. The type j kidneys arrive to the market according
to a Poisson process with a rate µj
, and the type i patients arrive according to a Poisson process
with a rate λi
. The utility of matching a type i patient to a type j kidney is equal to a deterministic quantity uij , whereas the patients’ utility of staying on dialysis or leaving without getting
matched is normalized to 0. Each patient type i exogenously departs from the market at a rate βi
,
and patients’ maximum lifetime in the market is capped at a constant L. In other words, patients’
lifetime is equal to the minimum of L and an exponentially distributed random variable with
mean 1/βi
.
An allocation system is described by a scoring rule sj (i, t), which describes the allocation
points of a type i patient who has been in the market with time t, with respect to the type j kidneys. The class of mechanisms being explored in the simulation are those that rank the patients
in the decreasing order of their allocation points and offer each kidney to the patients in such an
ordering until either the kidney is accepted by someone or discarded. This class of mechanisms
is rich enough to cover many mechanisms of interest, such as
1. The waitlist with choice (FIFO): sj (i, t) = t;
2. The last-come first-served (LIFO): sj (i, t) = −t, so that the newcomers are always prioritized;
3. The status quo: t + αij , where αij is a constant determined by the classification and allocation points of the type i patients for the type j donors based on the current kidney allocation system [41]. Specifically, in the kidney allocation system in 2019, there were at most 69
classifications for a donor, encoded with 1, 2, ..., 69, with a smaller classification indicating
90

a higher priority. Additionally, patients’ highest possible allocation points excluding the
waiting time were 217.1. Because the patients are first ordered by their classifications and
then by their allocation points, the αij is constructed by combining the classification and
allocation points for a patient-donor pair as (70 − classificationij ) ∗ 1000 + pointsij . In
this way, the original ordering in the kidney allocation system is preserved, with a higher
αij indicating a higher priority, and the patients cannot move to a higher classification by
waiting.
4. The periodic boost policy: t + αij + B1{t mod C < τ}, where B > 0 is the boost points,
C > 0 is the length of the cycle time and τ > 0 is the length of the boosted period.
Therefore, the patients receive the boost points on top of their original points under the
status quo periodically.
4.3.2 Estimation
Given the above simulation primitives, there are two objects that need to be estimated numerically:
1. V (i, t): The value function or acceptance threshold of a type i patient who has been in the
market for time t, which determines the patient’s acceptance decisions.
2. Fj (z): The cumulative distribution function of the cutoff allocation points z needed to
obtain a type j kidney, which determines the probability of receiving a type j kidney given
a patient’s current allocation points.
91

The estimation proceeds by an iterative procedure until it arrives at a certain fixed point
(V, F) that corresponds to an equilibrium. Specifically, the value function for each patient type i
is initialized to 0: ∀i, t, V 0
(i, t) = 0. In each iteration l ∈ {1, 2, · · · }, there are two steps:
1. Simulate the allocation to estimate the distribution of cutoff allocation points for
each kidney type j. To start, the kidney and patient arrivals are simulated, as well as the
patients’ exogenous departures according to the primitives. Each patient is represented by
a tuple (i, t0), which denotes her type i and arrival time t0.
When a type j kidney arrives at time t, it is offered to all the active patients according to
their allocation points sj (i, t − t0) in the decreasing order. The patient (i, t0) accepts the
kidney if and only if uij ≥ V
l−1
(i, t − t0). Once the kidney is accepted by a patient (i
′
, t′
0
),
the allocation is stopped, and the patient (i
′
, t′
0
) becomes inactive and gets removed from
the waitlist. The cutoff allocation points for this kidney are defined to be sj (i
′
, t − t
′
0
). If
the kidney is not accepted by any patient, then the cutoff allocation points for this kidney
are defined to be the lowest possible allocation points, for example, 0 under the waitlist
with choice, because it is accessible to any patient. At the beginning of the simulation,
as there are relatively fewer patients on the waitlist, the waitlist is not yet stabilized and
the observed cutoff allocation points can be an underestimate of the true cutoff allocation
points under a stabilized waitlist. Therefore, we only save the cutoff allocation points after
there are enough patients and the waitlist has stabilized, and we refer to the period when
the waitlist is still growing as the “burn-in period” of the simulation.
92

The allocation continues from time 0 until a large number T, such that after the allocation,
for each kidney type j we have a list of sample cutoff allocation points, using which we
compute the empirical CDF and denote it as F
l
j
(z).
2. Estimate the value function for each patient type i using the kidneys’ distributions
of cutoff allocation points. To compute the V (i, t), we discretize time into intervals of
δ periods, where δ > 0 is a sufficiently small constant. For each patient type i, the value
function at time L is initialized to be 0, i.e., ∀i, V l
(i, L) = 0, since all patients are assumed to
depart after hitting the time cap of L. At each grid point t ∈ {L−δ, L−2δ, L−3δ, · · · , 0},
the value function for each patient type i is estimated by the backward induction as below:
V
l
(i, t) = e
−βδV
l
(i, t + δ) +Xn
j=1
δµjF
l
j
(sj (i, t)) max
0, uij − e
−βδV
l
(i, t + δ)

. (4.1)
Here, e
−βδV
l
(i, t + δ) is the patient’s discounted continuation value when she has waited
for time t+δ, if the patient does not accept any offer during the period between t and t+δ.
As δ is sufficiently small, we assume that at most one kidney arrives within the period at
which the patient’s waiting time goes from t to t+δ, and the probability that a type j kidney
arrives is 1 − e
δµj ≈ δµj
. The probability that conditional on a type j kidney’s arrival, the
patient’s score sj (i, t) is greater than or equal to the cutoff points is equal to F
l
j
(sj (i, t)). If
the patient’s allocation points reach the cutoff, then the patient is offered the kidney, and
the additional value she gains compared to waiting is max(0, vj − e
−βδV
l
(i, t + δ)).
9

As δ is sufficiently small, we can estimate the value function between consecutive grid
points as follows: V
l
(i, t) = V
l
(i, L − ⌊(L − t)/δ⌋δ). In other words, we round up t to the
nearest grid point among {L, L − δ, L − 2δ, · · · , 0}.
After a sufficient number of iterations of the two steps above, the value function should have
sufficiently converged, such that λi supt{V
l
(i, t) − V
l−1
(i, t)}/
P
i
λi
is relatively small, which
is a weighted average of the maximum difference in the value function for each patient type
across the iterations by the respective arrival rate∗
. Once the value function has stabilized and
λi supt{V
l
(i, t) − V
l−1
(i, t)}/
P
i
λi
is smaller than a certain threshold, we terminate and conclude that we have reached an approximate equilibrium. With the obtained V (i, t), simulating
the allocation in Step 1 enables us to compute the metrics and compare between different mechanisms.
4.3.3 Cycling in the Simulation
Because the simulation does not allow the patients to play a mixed strategy, the simulation can
oscillate between different allocation outcomes. To illustrate, consider a simulation with two
types of kidneys: good kidneys and reasonable kidneys, and two types of patients: one type that
strongly prefers the good kidneys and the other type that slightly prefers the good kidneys. In
one iteration, we have that both patient types pursue the good kidneys, making the good kidneys
hard to get due to the long waiting time. In the next iteration, the patients who only slightly
prefer the good kidneys shift to the reasonable kidneys, because the long waiting time does not
∗Alternatively, one can also compute supi,t{V
l
(i, t) − V
l−1
(i, t)} as a more stringent measure of the convergence. However, during the simulation, we found that some rare patient types’ value functions are hard to converge,
but it does not affect the final allocation much because these patients types are rare. Therefore, we used this weighted
average as a measure of the convergence to ensure that most patients’ value functions sufficiently converge.
94

justify the additional utility for them. However, this shift can lead to a reduced competition for
the good kidneys, making them easy to get. Observing this, in the next iteration, the patients who
only slightly prefer the good kidneys come back for the good kidneys. Therefore, this dynamic
creates a cycling between the two allocation outcomes: one where every patient waits for the
good kidneys, and the other where only the patients who have strong preference for the good
kidneys wait for the good kidneys. In contrast, in a theoretical equilibrium, a proportion of
the patients who slightly prefer the good kidneys take the reasonable kidneys, whereas the rest
wait for the good kidneys. However, this theoretical equilibrium cannot be reproduced by the
simulation, as the simulation enforces all patients of the same type to adopt the same strategy by
following the same value function.
To address the cycling issue, two main approaches have been explored to allow the patients
of the same type to behave differently with respect to the same value function: adding random
shocks to the patients’ acceptance decisions, and randomizing the updating speed of the patients’
value functions. The first approach introduces randomness by making certain patients of the
same type behaving suboptimally: when faced with the same kidney, certain patients may reject
it even if the kidney is above the acceptance threshold or accept it even if the kidney is below
the acceptance threshold but still has a nonnegative value to them. The second approach introduces randomness by making the patients of the same type update their acceptance thresholds at
different rates, with some updating faster and others updating slower. The presented simulation
results are based on the second approach, as the suboptimality introduced in the first approach is
hard to interpret. For a more detailed discussion of the first approach, please refer to Section 4.A.
Below we focus on the second approach and its implementation.
95

To randomize the updating speed of the patients’ value functions, for each type i patient
who has been on the waitlist for time t in the iteration l, her acceptance threshold is a convex
combination of her current value V
l−1
(i, t) and her value from the previous iteration V
l−2
(i, t):
αV l−1
(i, t) + (1 − α)V
l−2
(i, t),
where 0 ≤ α ≤ 1. Different choices of α have been explored, such as:
1. α ∼ Uniform(0, 1). α is drawn independently and identically distributed (i.i.d.) from a
uniform distribution on the interval [0, 1].
2. α = min(X, 1), where X is drawn i.i.d. from an exponential distribution with mean a.
3. α ∼ Uniform(0, α¯), α¯ = σ

−A(|V
l−1
(i, t) − V
l−2
(i, t)| − B)

, where σ(x) = 1/(1 +
exp(−x)) is a sigmoid function, A > 0 adjusts the steepness of the sigmoid curve and
B > 0 adjusts the midpoint of the sigmoid curve. The interpretation is that for the patients
whose value functions update more across iterations, there is more heterogeneity in their
updating speeds to reduce the cycling brought by these patients. For the patients whose
value functions are already stable, less heterogeneity is needed.
In the actual simulation, the second and third approaches perform better compared to the first
approach in terms of addressing the cycling issue. For simplicity and clarity, the second approach
is implemented in the final simulation.
Therefore, we have the complete simulation algorithm as in Algorithm 1.
9

ALGORITHM 1: Simulation
Input: Kidney information: (n, µ). Patient information: (m, λ, u, β, L). Allocation
mechanism: sj (i, t). Simulation parameters: convergence tolerance ϵ, burn-in
period τ , time discretization δ, distribution of the update rates for patients’
thresholds a.
Output: Patient value function: V (i, t).
Initialize: Simulate kidney arrivals, patient arrivals and departures. ∀i, t, V 0
(i, t) = 0.
Iteration l = 1.
while l = 1 or λi supt{V
l
(i, t) − V
l−1
(i, t)}/
P
i
λi < ϵ do
// Simulate the allocation.
foreach kidney do
sort the active patients (i, t0) according to their scores sj (i, t − t0) in the
decreasing order;
foreach sorted active patient do
α = min(X, 1), X ∼ exp(a);
if uij ≥ αV l−1
(i, t − t0) + (1 − α)V
max(l−2,0)(i, t − t0) then
a type i patient receives the kidney and leaves the market;
if t > τ then
save the cutoff sj (i, t − t0)
end
break;
end
end
no patient accepts, and the kidney is discarded;
if t > τ then
save the cutoff 0
end
end
// Backward induction using the observed cutoffs.
foreach patient type i do
V
l
(i, L) = 0;
for t ∈ {L − δ, L − 2δ, L − 3δ, · · · , 0} do
V
l
(i, t) =
e
−βδV
l
(i, t + δ) + Pm
j=1 δµjF
l
j
(sj (i, t)) max
0, uij − e
−βδV
l
(i, t + δ)

end
end
end
9

4.3.4 Evaluations
Using the V (i, t) obtained from the simulation, we can simulate a round of allocation to derive the
allocation outcome and compute various metrics, such as the waitlist size and kidney wastage,
to compare different allocation mechanisms. All allocation outcomes and metrics are computed
using the simulation results after the burn-in period.
4.3.4.1 Allocation Outcome and Waitlist Size
The allocation outcomes are summarized in matrices where each row corresponds to the allocation vector, or the probability of being matched with each kidney type, for each patient type.
Specifically, to construct the allocation matrix under each allocation mechanism, for each patient
type, we count the number of patients who either gets matched with a kidney or who leave the
market without getting matched after the burn-in period. We then normalize the counts to obtain
a probability vector for each patient type. As there are many patient types in the simulation, we
also aggregate the patient types by grouping them into broader categories, such as only by their
age range. For each aggregated patient group, we compute an aggregated allocation vector by
taking an average of the allocation vectors of the constituent subtypes weighted by the arrival
rate of each subtype.
For each allocation mechanism, the waitlist size is calculated by averaging the number of
patients on the waitlist across the time points of kidney arrivals after the burn-in period.
4.3.4.2 Kidney Wastage
Following the definition from the theory (see Section 3.3.3), a kidney is considered wasted if
offering it to the patients who are not matched increases the total number of kidney transplants.
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Therefore, given an allocation matrix, the number of kidneys wasted is calculated by solving a
maximum matching problem between the discarded kidneys and the patients who leave without
getting matched. Let xij denote the number of type j kidneys that are discarded and can be
matched to the type i patients who leave the market without getting matched, and pij denote the
probability that a type i patient is matched to a type j kidney from the allocation matrix. The
maximum matching problem is formalized as the following linear optimization problem 4.2.
max X
i,j
xij (4.2)
s.t. X
j
xij ≤ λi

1 −
X
j
pij!
∀i (Demand constraint) (4.3)
X
i
xij ≤ µj

1 −
X
i
pij!
∀j (Supply constraint) (4.4)
xij = 0 ∀i, j s.t. uij < 0 (Nonnegative utility) (4.5)
xij ≥ 0 ∀i, j. (4.6)
Here, the demand constraint denotes that the number of kidneys matched to the type i patients
does not exceed the number of type i patients who leave the market being unmatched. Similarly,
the supply constraint denotes that the number of patients matched with the type j kidneys does
not exceed the number of type j kidneys that are discarded. Finally, the nonnegative utility
constraint ensures that a type i patient only accepts a type j kidney if she obtains a nonnegative
utility for that kidney type. The optimal objective value to the optimization problem can be
viewed as an upper bound of the actual kidney wastage, as the precise timing of patient and
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kidney arrivals is not considered. However, given the density of the market, the approximation
should be reasonably accurate.
4.3.4.3 Welfare Comparison
To compare the patient welfare across different allocation mechanisms, we follow the approach
in Agarwal et al. [2] and measure the welfare change of an alternative allocation mechanism
relative to the status quo as an equivalent change in the kidney arrival rates under the status
quo. Specifically, let M0 denote the status quo mechanism and M denote an alternative mechanism. Following Agarwal et al. [2], the welfare change EVi(M) for the type i patients under the
alternative mechanism is expressed as a relative change in the expected utility as below:
EVi(M) =
P
j
pM
ij uij −
P
j
pM0
ij uij
P
j
pM0
ij uij
, (4.7)
where pM
ij denotes the probability of a type i patient being matched with a type j kidney under
the alternative mechanism M, and uij denotes the utility of a type i patient being matched with
a type j kidney.
By averaging EVi(M) across different patient types, we obtain a measure of average welfare
change of the alternative mechanism M relative to the status quo.
4.4 Simulation Results
To evaluate how the periodic boost policy performs compared to the status quo, we conduct simulations using the primitives estimated from the kidney transplant data obtained from the UNOS
for a major OPO in Southern California in 2019. The data contain comprehensive information
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on the waitlisted patients in the OPO and the deceased donors offered to the OPO. We focus our
simulations on a single OPO instead of the entire national kidney allocation system, both because
of the computational constraints and the sufficiency to demonstrate the key findings.
Section 4.4.1 describes the primitives of the simulation estimated from the data. Section 4.4.2
describes how the utility between each patient-donor type pair is calibrated based on the estimates from Agarwal et al. [2]. Section 4.4.3 discusses the simulation results of the periodic boost
policy.
4.4.1 Primitives
On the demand side, based on the number of new patients registered to the waitlist in the OPO in
2019, the simulation assumes that the patients arrive to the market following a Poisson process
with a rate of 2276 per year. The patients are discretized into 412 types, based on their listing
center, blood type, age range, CPRA score range and EPTS score range. The life expectancy of
each patient type staying on dialysis is estimated using the data from the United States Renal Data
System [62] given the age upon entry, and patients’ lifetimes follow an exponential distribution
with their respective life expectancy. Moreover, the time that a patient stays on the waitlist is
capped at 25 years, according to the data.
On the supply side, all the donors ever offered to the OPO may be considered as supply. However, this would inflate the supply in the simulation, because we only simulate for a single OPO
and as shown in Figure 4.3, many donors were only offered to a few patients in the OPO. To have
a reasonable supply level for the simulation, the 25th percentile of the number of patients offered
is used as a threshold to determine whether a donor is considered as supply for the simulation,
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which is 4 †
. Besides, for the donors considered as national donors for the OPO, we only consider
the ones that have been accepted or provisionally accepted by the OPO as the simulation supply,
because most national donors are deemed undesirable due to their prolonged cold ischemia time.
Hence, we consider all the local donors, regional donors and accepted or provisionally accepted
national donors that have been offered to at least 4 patients in the OPO as the donor supply for the
simulation, which gives rise to 1017 donors arriving to the market every year following a Poisson
process. As for 93% donors, both of their kidneys were recovered, the simulation assumes that
both kidneys are recovered for every donor and the total kidney supply amounts to 2034 per year.
The kidneys are discretized into 229 types, based on the OPO that initiated the match run, the
blood type and the KDPI range of the kidney.
0 25 50 75 100 125 150 175 200
Number of Patients in the OPO Offered
0
100
200
300
400
Number of Donors
Distribution of Number of Patients Offered Figure 4.3: The figure shows the distribution of the number of patients offered in the focal OPO
for the donors who were ever offered to the focal OPO, based on the 2019 data.
For each patient-kidney type pair, the classification and allocation points are determined according to the rules of the kidney allocation system in 2019. The utility for each pair is calculated
using the coefficients from the conditional choice probabilities estimated in Agarwal et al. [2].
†This may be further tuned to have a reasonable supply level for the simulation.
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As directly applying the coefficients makes the patients unwilling to accept any kidneys and we
are using data from a different OPO than the one used in Agarwal et al. [2], the utilities are further calibrated by varying the intercept to match the local kidney discard rate observed in the
simulation with the reality. ‡ The calibration process is discussed in more detail in Section 4.4.2.
To account for the random shocks and horizontal differentiation in the patients’ preferences, two
types of normal shocks are added to the utility matrix following the estimates from Agarwal et al.
[2]: a normal shock with a mean of 0 and a standard deviation of 1.02 §
is added for each kidney type and a normal shock with a mean of 0 and a standard deviation of 1 is added to each
patient-kidney type pair.
In the simulation, each kidney is offered to all the active patients who are on the waitlist and
have not yet been matched or departed without a transplant, until either being accepted by a
patient or discarded. Besides, each patient can only be matched with at most one kidney, as dual
kidney transplants are rare in reality.
For the simulation parameters, each year is discretized into time periods of length δ = 1/3000,
and 52 years of allocation are simulated. In each iteration, the first half of the simulation is
considered as the burn-in period to allow the waitlist to reach a steady state, and the patients’
thresholds are derived based on the kidney cutoffs observed after the burn-in period. To address
the cycling issue mentioned in Section 4.3.3, we use α = min(X, 1), where X is drawn i.i.d.
from an exponential distribution with mean a = 0.2. As shown in Figure 4.4, the majority of
the patients update their acceptance thresholds relatively slowly, preventing them from cycling
‡Other metrics such as the local and regional kidney discard rate may be used to calibrate the utilities. However,
the utilities calibrated with the local and regional kidney discard rate produces an allocation outcome that is farther
away from the actual allocation. Therefore, we choose the local kidney discard rate as the calibration metric during
the simulation.
§To be more accurate, the standard deviation used is 1.0177.
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rapidly among different strategies. Besides, instead of choosing a fixed ϵ as the convergence
tolerance to determine when to stop the simulation, we run the simulations for each intercept
or each mechanism for 30 or 40 iterations, respectively, and we use the last 10 iterations for our
analysis and results presented below. In Section 4.B, we discuss in more detail that generally the
simulations have stabilized after 20 iterations, and therefore averaging these iterations gives a
robust approach to derive the approximate equilibrium for each intercept and mechanism.
0.0 0.2 0.4 0.6 0.8 1.0
0
1000
2000
3000
4000
Count
Distribution of Update Rates for Acceptance Thresholds
Figure 4.4: The figure shows the distribution of the update rates for patients’ acceptance thresholds α, based on the 50000 simulated values.
4.4.2 Calibrating the Utilities Using the Status Quo
With the primitives, the utility between each patient-kidney type pair is calibrated by varying
the intercept with the coefficients estimated from Agarwal et al. [2]. The intercepts ranging from
2.5 to 6 in increments of 0.1 are examined. For each constant, the status quo is simulated for 30
iterations, and the average of the last 10 iterations is used to compute the kidney wastage, local
kidney discard rate and waitlist size.
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The calibration is done by minimizing the absolute difference between the local kidney discard
rates observed in the simulation and in reality. Using the kidney transplant data, the local kidney
discard rate is calculated as the proportion of the local kidneys offered to the OPO but ultimately
discarded, which is 21.17%. The calibration results are shown in Table 4.1.
As shown in the “Discard Rate Diff” column, the intercept of 4.0 minimizes the absolute difference between the local discard rates in the simulation and in reality. Hence, 4.0 is picked as
the intercept in the utility function.
With the intercept of 4.0, we compare both the waitlist size and the allocation matrix from
the simulation with those observed from the data. As shown in Figure 4.5, the waitlist size in
the simulation stabilizes at around 9749 after the burn-in period, which is longer than the actual
waitlist size of 8515 at the OPO. Figure 4.6 presents a side-by-side comparison of the allocation
matrices by patients’ age group from the simulation and the reality. In each allocation matrix,
each row represents the average allocation vector for a patient age group, and each column represents a kidney KDPI group or departure without transplant. Note that the younger patients
have a significantly higher probability of receiving the good kidneys with KDPI 0-20. This is
because the status quo prioritizes the younger patients for the low-KDPI kidneys to maximize
the post-transplant life expectancy for these high-quality kidneys, an idea known as longevity
matching.
The comparison of the waitlist size and allocation matrix between the simulation and the
reality shows that in the simulation, the waitlist is longer, and the departure rates of different
patient age groups are slightly higher. Besides, the younger patients are less likely to be matched
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Intercept Simulated Waste Simulated Discard Rate Waitlist Size Discard Rate Diff
2.5 6.19% 24.19% 10 860.36 3.02%
2.6 6.88% 23.96% 10 804.47 2.78%
2.7 6.84% 23.66% 10 723.37 2.49%
2.8 6.68% 23.44% 10 625.37 2.27%
2.9 6.88% 23.20% 10 497.27 2.02%
3.0 7.37% 23.27% 10 443.55 2.10%
3.1 7.84% 23.00% 10 409.67 1.83%
3.2 8.06% 22.85% 10 356.38 1.68%
3.3 8.53% 22.74% 10 318.79 1.56%
3.4 9.12% 22.40% 10 222.76 1.23%
3.5 10.46% 22.30% 10 163.05 1.13%
3.6 11.40% 22.03% 10 096.17 0.86%
3.7 13.55% 22.11% 10 002.82 0.94%
3.8 15.24% 21.85% 10 017.05 0.68%
3.9 16.26% 21.60% 9951.00 0.43%
4.0 16.12% 21.41% 9749.10 0.24%
4.1 15.65% 20.80% 9552.12 0.38%
4.2 15.21% 20.37% 9423.29 0.80%
4.3 16.00% 20.07% 9237.73 1.11%
4.4 16.08% 19.77% 9080.17 1.41%
4.5 16.14% 19.59% 9026.97 1.58%
4.6 19.18% 19.62% 8826.07 1.56%
4.7 19.04% 19.60% 8746.63 1.57%
4.8 19.49% 19.69% 8673.61 1.48%
4.9 19.63% 16.92% 8413.10 4.25%
5.0 18.42% 14.75% 8205.35 6.42%
5.1 18.27% 14.79% 8133.48 6.38%
5.2 18.52% 13.52% 8007.21 7.66%
5.3 20.39% 13.51% 7922.65 7.67%
5.4 19.86% 13.45% 7823.02 7.72%
5.5 18.72% 13.38% 7647.88 7.80%
5.6 18.57% 13.32% 7606.96 7.86%
5.7 18.57% 13.25% 7569.53 7.92%
5.8 18.85% 13.25% 7566.29 7.92%
5.9 18.99% 13.33% 7548.34 7.84%
6.0 17.85% 13.22% 7347.90 7.96%
Table 4.1: The table shows the results of calibrating the utilities by varying the intercept.
with the kidneys with KDPI 20-40 but more likely to be matched with the kidneys with KDPI 40-
60, whereas the older patients are less likely to be matched with the kidneys with KDPI 80-100.
These discrepancies can be attributed to several reasons as below.
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0 20000 40000 60000 80000 100000 120000
# of Kidneys Simulated
0
2000
4000
6000
8000
10000
# of Patients on the Waitlist
Stats Quo (Simulation): Waitlist Evolution
Figure 4.5: The figure shows the waitlist evolution of the status quo in the simulation with the
intercept of 4.0 in the utility function.
First, the simulation assumes that the patients are behaving optimally and that the system
reaches an equilibrium, which may not be the case in reality. As illustrated in Chapter 2, the
doctors and patients often make suboptimal decisions due to offer mispredictions and other behavioral biases, which can lead to patients’ accepting offers too soon or waiting for the good
kidneys that are hard to get. Moreover, the kidney allocation system in reality may not have
reached an equilibrium yet, as the kidney recovery practices and patient arrivals are changing
over the years.
Second, the utility estimates from Agarwal et al. [2] may not accurately reflect the preferences
of the patients in this OPO. The estimates from Agarwal et al. [2] are based on an OPO in New
York back in 2013, whereas our data is based on an OPO in Southern California in 2019. The
differences in location and time may lead to differences in patient preferences and choices.
Third, the kidney supply in the simulation may underestimate the actual kidney supply in
reality. As mentioned in Section 4.4.1, only the local donors, regional donors and accepted or
provisionally accepted national donors that have been offered to at least 4 patients in the OPO
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0-20 20-40 40-60 60-80 80-100 departure
Donor KDPI Range
0-40
40-50
50-60
60-100
Patient Age Range
0.23 0.21 0.13 0.07 0.01 0.34
0.07 0.16 0.08 0.16 0.09 0.44
0.05 0.09 0.08 0.12 0.16 0.50
0.03 0.06 0.10 0.10 0.16 0.55
Status Quo (Reality): Allocation Vectors by Patient Age Range
0.1
0.2
0.3
0.4
0.5
(a) Actual Allocation Matrix by Patient Age Group
0-20 20-40 40-60 60-80 80-100 departure
Donor KDPI Range
0-40
40-50
50-60
60-100
Patient Age Range
0.18 0.11 0.18 0.09 0.02 0.43
0.11 0.06 0.19 0.12 0.06 0.47
0.05 0.15 0.10 0.17 0.03 0.51
0.05 0.08 0.10 0.18 0.03 0.56
Status Quo (Simulation): Allocation Vectors by Patient Age Range
0.1
0.2
0.3
0.4
0.5
(b) Simulated Allocation Matrix by Patient Age Group
Figure 4.6: The figure shows the actual allocation matrix and the simulated allocation matrix with
the intercept of 4.0 in the utility function by patients’ age group.
are considered as the supply for the simulation, which can be stringent. However, identifying the
precise conditions for determining which kidneys should be counted as the supply is challenging,
as the simulation only focuses on a single OPO and does not consider the inter-OPO competition.
To more accurately reproduce the allocation outcome in the practice, one may re-estimate the
utility function and simulate the national kidney allocation system, which is beyond the scope of
this thesis. As the simulation with the intercept of 4.0 in the utility function reasonably approximates the status quo, we proceed to evaluate the periodic boost policy.
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4.4.3 Periodic Boost Policy
Two types of the periodic boost policy are examined: the regular periodic boost policy where the
boost points are sufficiently large (100000) to boost the patients to the front of the waitlist for a
month, with cycle times of 1, 3 and 5 years; and the boost upon entry where the patients receive
the boost points of 4, 6, 10 with a boost period of 1 month or the boost points of 100000 with a
boost period of 1 week, 2 weeks or 1 month. For the status quo and each parameter combination
for the periodic boost policy, 40 iterations are simulated, and the average of the last 10 iterations
is used to compute the allocation matrix, kidney wastage, waitlist size and patient welfare. The
simulation results are shown in Table 4.2.
Mechanism Waste Waitlist Size Welfare Change
Status Quo 16.28% 9799.13 0
Regular Periodic Boost Policy
1 month per year with 100000 points 14.58% 9626.24 −0.70%
1 month per 3 years with 100000 points 13.86% 9552.81 −2.39%
1 month per 5 years with 100000 points 12.31% 9341.04 −3.58%
Boost upon Entry
1 month with 4 points 14.70% 9749.25 −1.34%
1 month with 6 points 11.81% 9368.05 −3.68%
1 month with 10 points 8.23% 8938.85 −7.80%
1 week with 100000 points 2.95% 7154.19 −5.63%
2 weeks with 100000 points 2.34% 7114.42 −8.82%
1 month with 100000 points 2.02% 7087.62 −9.74%
Table 4.2: The table shows the simulation results of the status quo, the regular periodic boost
policy and the boost upon entry with different parameter combinations.
Compared with the status quo, both the regular periodic boost policy and the boost upon
entry reduce waste and shorten the waitlist, resulting in a shorter time on the waitlist for the
patients. Under the status quo, the patients gamble with their lifetimes to wait for the good
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kidneys, which is costly and leads to significant kidney wastage. In contrast, under the regular
periodic boost policy and boost upon entry, much of the uncertainty on the good kidneys is
resolved by whether the patients are boosted instead of their waiting times, making the patients
less incentivized to wait and less picky. For a specific example, Figure 4.7 shows for a blood
type O patient, how the proportion of acceptable kidneys out of the kidneys she finds reasonable
changes with her waiting time under different mechanisms. A kidney is considered “reasonable”
if the patient obtains a nonnegative utility from being matched with the kidney and “acceptable”
if the patient’s utility for the kidney exceeds her acceptance threshold at that time. Therefore,
the higher the ratio between the number of acceptable kidneys and the number of reasonable
kidneys is, the less picky the patient is.
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
Waiting Time (Years)
0
20
40
60
80
100
Acceptable / Reasonable (%)
Kidney Acceptance Threshold for a Blood Type O Patient
Mechanism
Status Quo
Periodic Boost (1 mon per 3 yrs with 100000 pts)
Boost upon Entry (1 mon with 10 pts)
Boost upon Entry (1 mon with 100000 pts)
Figure 4.7: The figure shows how the proportion of acceptable kidneys out of the reasonable
kidneys changes with the waiting time for a blood type O patient under different mechanisms.
The patient is generally less picky under the periodic boost policy, especially the boost upon
entry, than the status quo. Under the periodic boost policy, the patient does not have a chance
for the good kidneys during the first few boosts, as there are insufficient good kidneys available
and obtaining a good kidney requires a combination of boost points and sufficient waiting time.
With enough waiting time, as expected, the patient has a lower acceptance threshold when she is
not boosted and a higher acceptance threshold when she approaches the boost period or becomes
110

boosted. As patients are still incentivized to wait for the boosts, the periodic boost policy reduces
waste but not by much. In comparison, the boost upon entry makes the patient much less picky
and significantly reduces the wastage, because if the patient does not receive a good kidney during
the initial boost, then she is much less likely to get one by waiting. With a fixed boost period
of 1 month, when the boost points are small, the patient expects to wait for an extended time if
she does not get a good kidney during the boosted period; when the boost points are large, the
patient does not expect to get the good kidneys by waiting and she becomes much more willing
to settle with a reasonable kidney if she does not receive a good kidney during the boost period.
However, although the periodic boost policy and the boost upon entry reduce waste, they hurt
the patient welfare as shown in the “Welfare Change” column. The welfare change is measured
by the equivalent change in kidney supply. Specifically, as directly calculating the relative utility
change for each patient type yields a noisy estimate of the welfare change, we first calculate
the mean utility for each patient group by blood type and age range, and the rest follows from
the Section 4.3.4.3. Under the status quo, the patients are free to wait for the kidneys that they
prefer. However, under the periodic boost and boost upon entry policies, the introduction of the
boost points increases the role of luck and reduces patients’ choices. Although the patients are
more likely to get matched, their utilities conditional on being matched become lower.
The results highlight the tension between reducing kidney wastage and improving patient
welfare, as suggested in Chapter 3. To delve deeper, we further examine whether it is impossible
to both reduce kidney wastage and improve patient welfare in this market. Besides, we discuss
the assumption of the horizontal differentiation in the patients’ preferences.
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4.4.3.1 Gap from the Optimal Allocation
To examine whether it is possible to both reduce kidney wastage and improve patient welfare
with any mechanism, we develop a linear optimization program to solve an LP relaxation of the
allocation problem.
Given the primitives from Section 4.3.1, the following linear program can be used to solve for
the optimal allocation when we do not consider the differences in the patients’ life expectancy and
the specific timing of patient and kidney arrivals. The decision variables are the xij s, where xi
is
an n-dimensional vector that encodes the probability that a type i patient is eventually matched
with a type j kidney. Let vi denote an n-dimensional vector that encodes the utility that a type i
patient gets from being matched with a type j kidney, and u
0
i denote the type i patients’ expected
utility under the status quo. The welfare change measured by the equivalent increase in kidney
supply is then represented by P
i
λixi
· vi/(u
0
i
P
i
λi). The parameter α ∈ [0, 1] is the relative
weight of the welfare change compared to the kidney utilization. When α = 1, the objective
function is the equivalent increase in kidney supply; when α = 0, the objective function is the
total match rate, which is equivalent to minimizing the kidney wastage.
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max X
i
λixi
·

α
vi
u
0
i
+ (1 − α)1

(4.8)
s.t. X
j
xij ≤ 1 ∀i (Feasibility constraint) (4.9)
X
i
λixij ≤ µj ∀j (Supply constraint) (4.10)
vi(xi − xi
′) ≥ 0 ∀i, i′
satisfying certain conditions (Envy-free constraint) (4.11)
xi ≥ 0 ∀i. (4.12)
As the status quo prioritizes the patients with low EPTS score and high CPRA score, the envyfree constraint is such that a patient of certain type does not envy any patient type who has the
same blood type, lower EPTS score and higher CPRA score.
However, as patients of different age range upon entry can have different life expectancy, the
above linear program is further revised to account for the differences in patients’ life expectancy.
Suppose that there are l possible different departure rates of the patients: β1 < β2 < · · · < βl
.
Define x
k
ij as the probability that a type i patient would obtain a type j kidney if her departure
rate were βk. For each patient type i, let k(i) denote their true departure rate. The following
linear program is equivalent to the above but allowing for different patients’ departure rates.
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max X
i
λixi
·

α
vi
u
0
i
+ (1 − α)1

(4.13)
s.t. X
j
x
1
ij ≤ 1 ∀i (Feasibility constraint) (4.14)
x
k+1
ij ≤ x
k
ij ∀i, j, 1 ≤ k ≤ l − 1 (Weakly increasing) (4.15)
(βk+1 − βk−1)x
k
ij ≤ (βk+1 − βk)x
k−1
ij (4.16)
+ (βk − βk−1)x
k+1
ij ∀i, j, 2 ≤ k ≤ l − 1 (Convexity) (4.17)
X
i
λix
k(i)
ij ≤ µj ∀j (Supply constraint) (4.18)
vi(x
k
i − x
k
i
′) ≥ 0 ∀i, i′
, k satisfying certain conditions (Envy-free constraint) (4.19)
x
k
i ≥ 0 ∀i, k. (4.20)
The weakly increasing constraint follows from that a patient’s set of allocations would stay
the same or expand if her life expectancy were longer. The convexity constraint follows from
that each patient’s allocation vector is a sum of exponential terms as each patient’s lifetime is
assumed to be exponentially distributed. Because it is challenging to precisely account for the
exponential lifetimes in the linear optimization problem, we adopt this relaxation to provide an
upper bound on the optimum.
To examine whether it is possible to reduce kidney wastage while Pareto improving upon the
status quo, we add the following Pareto improvement constraint
vixi ≥ u
0
i
, ∀i,
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to the optimization problem 4.13 and solve it with the objective of maximizing the equivalent
increase in kidney supply where the patient types are grouped by their blood type and age range,
as in Table 4.2. This leads to the allocation matrix in panel A of Figure 4.8, with an equivalent
increase in kidney supply of 15.00% and waste of 0.06%.
0-20 20-40 40-60 60-80 80-100 departure
Donor KDPI Range
0-40
40-50
50-60
60-100
Patient Age Range
0.14 0.14 0.18 0.16 0.14 0.24
0.11 0.08 0.18 0.20 0.11 0.32
0.07 0.16 0.06 0.15 0.12 0.45
0.07 0.08 0.11 0.17 0.09 0.48
Upper Bound: Allocation Vectors by Patient Age Range
0.1
0.2
0.3
0.4
(a) The upper bound allocation matrix solved from optimization.
0-20 20-40 40-60 60-80 80-100 departure
Donor KDPI Range
0-40
40-50
50-60
60-100
Patient Age Range
0.21 0.11 0.12 0.15 0.13 0.28
0.10 0.09 0.14 0.26 0.08 0.33
0.05 0.17 0.10 0.16 0.10 0.43
0.04 0.09 0.15 0.14 0.08 0.50
Boost upon Entry (1 mon with 100000 pts): Allocation Vectors by Patient Age Range
0.1
0.2
0.3
0.4
(b) The allocation matrix under the Boost upon Entry (1 mon with 100000 pts)
Figure 4.8: The figure shows the upper bound allocation matrix and the allocation matrix under the boost upon entry where the boost period is 1 month and the boost points are 100000.
The upper bound allocation matrix is obtained by solving the optimization problem 4.13 with
the objective of maximizing the equivalent increase in kidney supply, where the patient types
are grouped by their blood types and age ranges, and with an additional constraint of Pareto improvement over the status quo.
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Interestingly, as shown in panel B of Figure 4.8, the allocation matrix under the boost upon
entry with the boost period of 1 month and boots points of 100000 closely resembles the upper
bound allocation. However, although the boost upon entry reduces the kidney wastage to 2.02%,
it does not improve patient welfare.
This discrepancy arises because the boost upon entry does not differentiate the patients by
their preferences, whereas the upper bound allocation requires knowing the preference of each
patient to offer the right kidney. Although the boost upon entry may similarly personalize the
boost points or boost duration for each patient type, it is hard to achieve in reality, as the precise
knowledge of the patients’ preferences is difficult to collect and the rules used to differentiate the
patients in the upper bound allocation or personalized boost upon entry may be hard to justify. In
comparison, it is more justifiable to differentiate and rank the patients based on their compatibility
with the donor and waiting times, as in the status quo. However, the lack of differentiating the
patients by their preferences makes it challenging to both reduce kidney wastage and improve
patient welfare.
To approximate how the lack of differentiating the patients by their preferences can limit the
improvement on patient welfare, we add the following constraint to the optimization problem 4.13
to mimic the setting where we do not have the precise knowledge of the patients’ preferences:
xij = xij′, ∀i, j, j′
s.t. j and j
′
are similar kidney types.
Therefore, the mechanism does not perfectly capture the patients’ horizontal differentiation of
the similar kidneys. Specifically, two kidney types are considered “similar” if their match runs
were initiated by the same OPO and they are of the same blood type and in the same KDPI range
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with increments of 20 rather than 10 for the original kidney types. After adding this constraint,
Pareto improving upon the status quo becomes infeasible, and a patient group by blood type
and age range becomes 37% worse off at best in terms of the equivalent change in kidney supply.
Moreover, even reducing the kidney wastage becomes challenging with the lowest possible waste
being 14.02%.
4.4.3.2 The Horizontal Differentiation Assumption
The value of knowing each patient’s preference and offering the right kidney partly stems from
the horizontal differentiation in the patients’ preferences. Recall that following Agarwal et al. [2],
to capture the horizontal differentiation in the patients’ preferences, a normal shock with a mean
of 0 and a standard deviation of 1 is added to each patient-kidney type pair when constructing
the utility matrix.
However, given that the magnitude of the patients’ utilities is about 4.5, it remains an open
question whether there exists this much of horizontal differentiation among the patients for the
kidneys of the same blood type and recovery location and similar KDPI range. The variance
observed in the patients’ choices could be a result of issues other than rational utilities, such as
the weekend effects, which describe that the kidneys are more likely to be rejected and discarded
during the weekends due to the limited resources compared to the weekdays [29].
To examine the impact of the horizontal differentiation assumption, we first remove the normal shock added to the utility of each patient-kidney type pair and recalculate the welfare change
for each mechanism in Table 4.2 using the same simulation results. The interpretation is that the
utilities without the idiosyncratic shocks represent the true preferences of patients and the idiosyncratic shocks represent patients’ mistakes instead of horizontal differentiation. The result
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is shown in Table 4.3, and among the policies examined, the welfare change for the boost upon
entry with the boost period of 1 week and boost points of 100000 changes from -5.63% to 3.97%.
Hence, when there is no idiosyncratic shock for each patient-kidney type pair and not much horizontal differentiation among the patients’ preferences, the boost upon entry with a large boost
for a short time significantly reduces waste and improves overall patient welfare.
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Mechanism Waitlist Size Local Kidney Discard Rate Waste w/ Idio Shocks Welfare Change w/ Idio Shocks Waste w/o Idio Shocks Welfare Change w/o Idio Shocks
Status Quo 9799.13 21.29% 16.28% 0 3.11% 0
Regular Periodic Boost Policy
1 month per year w/ 100000 9626.24 20.07% 14.58% -0.70% 2.61% -7.01%
1 month per 3 years w/ 100000 9552.81 19.07% 13.86% -2.39% 1.81% -2.45%
1 month per 5 years w/ 100000 9341.04 18.39% 12.31% -3.58% 1.02% -3.46%
Boost upon Entry
1 month with 4 points 9749.25 21.15% 14.70% -1.34% 2.47% -1.03%
1 month with 6 points 9368.05 19.74% 11.81% -3.68% 1.53% -14.87%
1 month with 10 points 8938.85 19.25% 8.23% -7.80% 0.86% -21.79%
1 week with 100000 points 7154.19 15.91% 2.95% -5.63% 0.62% 3.97%
2 weeks with 100000 points 7114.42 17.42% 2.34% -8.82% 0.58% -9.16%
1 month with 100000 points 7087.62 18.15% 2.02% -9.74% 0.56% -12.46%
Table 4.3: The table shows the simulation results of the status quo with additional columns showing the waste and welfare change
calculated by removing the idiosyncratic shock added to the utility of each patient-kidney type pair.
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Moreover, in another set of simulations, we maintain the same setting as in Table 4.2 but only
remove the normal shock added to the utility for each patient-kidney type pair. As the boost upon
entry reduces the kidney wastage more significantly, we only simulate the boost upon entry with
different parameter combinations. The simulation results are shown in Table 4.4.
Mechanism Waste Waitlist Size Welfare Change
Status Quo 9.11% 11 596.31 0
Boost upon Entry (1 mon with 4 pts) 5.82% 11 304.94 0.44%
Boost upon Entry (1 mon with 6 pts) 5.22% 11 209.52 0.60%
Boost upon Entry (1 mon with 10 pts) 4.72% 11 082.18 0.48%
Boost upon Entry (1 wk with 100000 pts) 1.01% 9765.66 6.29%
Boost upon Entry (2 wk with 100000 pts) 0.89% 9753.51 3.33%
Boost upon Entry (1 mon with 100000 pts) 0.89% 9809.55 1.29%
Table 4.4: The table shows the simulation results of the status quo and the boost upon entry with
different parameter combinations when there is no shock for the utility of each patient-kidney
type pair.
Similarly, compared to the status quo, the boost upon entry significantly reduces the kidney
wastage and shortens the waitlist. Moreover, after removing the normal shock and decreasing
the level of horizontal differentiation among the patients, the boost upon entry also improves
the patient welfare, with an equivalent increase of up to 6.29% in kidney supply when the boost
points are large. However, as shown in Figure 4.9, for the versions of the boost upon entry tested,
they do not Pareto dominate the status quo when the patients are group by blood type and age
range. The blood type A patients are generally worse off under the boost upon entry, as the blood
type AB patients may take away the blood type A donors from the blood type A patients when
boosted. Despite this, the boost upon entry with the boost period of 1 month and boost points of
4 reduces the kidney wastage by 36% without significantly disadvantaging any patient group.
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A 0-40
A 40-50
A 50-60
A 60-100
AB 0-40
AB 40-50
AB 50-60
AB 60-100
B 0-40
B 40-50
B 50-60
B 60-100
O 0-40
O 40-50
O 50-60
O 60-100
Patient Blood Type and Age Range
20
0
20
40
Welfare Improvement (%)
Welfare Improvement by Patient Blood Type and Age Range
Mechanism
Boost upon Entry (1 mon with 4 pts)
Boost upon Entry (1 mon with 10 pts)
Boost upon Entry (1 wk with 100000 pts)
Boost upon Entry (1 mon with 100000 pts)
Figure 4.9: The figure shows the welfare change by patient blood type and age range under each
version of the boost upon entry relative to the status quo when there is no shock for the utility
of each patient-kidney type pair.
In summary, the periodic boost policy, especially the boost upon entry, effectively reduces
the kidney wastage, but whether it improves the patient welfare depends on the level of horizontal differentiation among the patients’ preferences, which requires further investigation of the
patients.
4.5 Discussion
In this chapter, motivated by the observation that many reforms aimed at addressing the kidney
wastage problem have focused exclusively on expediting the allocation of the marginal kidneys
but have failed to reduce the kidney wastage, we take a departure from these practices and propose a minor change on the status quo allocation mechanism. Following the theoretical guidance,
we introduce adding a periodic boost on top of the patients’ allocation points to break the monotonicity between the patients’ waiting times and their chances of receiving the good kidneys. A
variety of the periodic boost policy is the boost upon entry, in which patients only receive the
boost points once upon entry. By introducing this exogenous shock, the periodic boost policy
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resolves the uncertainty of receiving the good kidneys earlier and deters patients from gambling
with their lives to see whether they obtain the good kidneys.
To evaluate how the periodic boost policy performs compared to the status quo, we develop
a simulation engine to simulate the periodic boost policy with the actual kidney transplant data.
Because as suggested by the theoretical analysis, reducing kidney wastage may not result in a
Pareto improvement over the status quo in patient welfare. This could make a waste-reducing
mechanism hard to be adopted, as changing an allocation system requires buy-in from different
stakeholders. Compared to other existing simulation engines, our simulation engine incorporates
patients’ incentives and relies on fewer ad-hoc assumptions.
The simulation results show that the periodic boost policy, especially the boost upon entry,
effectively reduces the kidney wastage. However, whether the boost upon entry improves the
patient welfare in terms of the equivalent increase in kidney supply depends on how much horizontal differentiation there is among the patients, which may be further estimated and evaluated
through questionnaires or focus groups with the patients. Despite this, given the simplicity and
effectiveness in reducing the kidney wastage, the periodic boost policy, especially the boost upon
entry, is promising to be pursued.
We conclude this chapter by discussing the potential extensions of the simulation engine.
1. Simulate the national kidney transplant system and incorporate the recent modifications.
In this chapter, the simulation is conducted based on the kidney and patient arrivals from a
single Organ Procurement Organization (OPO), which does not capture the effect from the
inter-OPO competition. Nevertheless, with a reasonable utility measure for each patientkidney pair across the U.S., this simulation engine is able to simulate a national kidney
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allocation system. Besides, the tiered proximity system (local, regional and national) has
been replaced with a 250 nautical mile distribution circle [42]. However, incorporating
this policy change would require more information on the exact location of each kidney
recovery location and each patient’s listing center, which is not available in our current
data. But with such information, one can easily adjust the input and simulate the new
kidney allocation system using our simulation engine.
2. Incorporate the random shocks on patients’ acceptance decisions and adjust the metrics
accordingly. In the simulation, we did not explicitly model the exogenous shocks to the
patients’ acceptance decisions during the allocation process. To incorporate the exogenous
shocks, the backward induction step would need to be revised by utilizing a truncated
normal distribution for the patients’ expected utilities. Besides, the waste calculation would
need to be adjusted as well, as the patients may accept a kidney that they value negatively
on average due to the random shocks. Although these changes are feasible, in this chapter,
to still allow for the heterogeneity in patients’ horizontal preferences, we chose to add the
random shocks to the input utility matrix rather than during the allocation process to have
a cleaner simulation. Moreover, to better approximate the patients’ decisions, one may
need to refit the choice model for each OPO and the magnitude of the random shocks may
vary across different regions.
4.A Other Approaches Tried to Address the Cycling Issue
The cycling in the simulation emerges as a result of lack of randomness in the patients’ kidney
acceptance decisions. To address the cycling issue, two main approaches have been tried: adding
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random shocks to the patients’ acceptance thresholds, and randomizing the updating speed of
the patients’ acceptance thresholds. The latter approach is discussed in more detail in the main
body of the thesis, and the appendix focuses on the former approach.
The intuition of adding random shocks to the patients’ acceptance thresholds is that even
with the same value function, some patients of the same type may behave suboptimally due to
the exogenous shocks. Specifically, for a type i patient who has been on the waitlist for time t in
the iteration l, her acceptance threshold is max{V
l−1
(i, t) + ϵ, 0}, where ϵ ∼ Uniform(−∆, ∆)
and ∆ = |V
l−1
(i, t)−V
l−2
(i, t)|. The uniform distribution can be replaced with other reasonable
probability distributions. The interpretation is that for the patients who update more across
iterations, there is a greater shock to create enough heterogeneity to break the cycling. For the
patients whose value functions are stable, there is less shock such that these patients behave more
optimally.
This approach is not implemented in the final simulation, because adding random shocks
make the results harder to interpret as the suboptimality is not estimated from real data but only
added to address the cycling issue. Besides, adding random shocks in the simulation complicates the waste calculation, because the random shocks affect whether a patient finds a kidney
acceptable and can greatly reduce the waste.
4.B Convergence of the Simulation
The simulation of each mechanism generally stabilizes after 20 iterations. For an example, we
show the convergence metrics and waitlist evolution when simulating the status quo with an
intercept of 4.0 in Table 4.5. The convergence is a weighted average of the maximum difference
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in the value function for each patient type across the iterations by the respective arrival rate,
whereas the mean convergence is a simple average of the difference in the value functions for
different patient types across the iterations. As seen, both convergence metrics tail down after
a few iterations. Although the convergence does not become 0, the allocation is fairly stable, as
shown by the stability of the waitlist size.
Iteration Convergence Mean Convergence Waitlist Size
1 3.76 3.96 6324
2 0.18 0.02 6540
3 1.00 0.33 11 388
4 0.71 0.16 10 809
5 0.80 0.16 10 346
6 0.62 0.09 10 112
7 0.70 0.13 10 065
8 0.33 0.06 9876
9 0.79 0.16 9708
10 0.48 0.08 9699
11 0.52 0.09 9791
12 0.42 0.07 9641
13 0.67 0.13 9759
14 0.44 0.08 9626
15 0.76 0.14 9824
16 0.52 0.09 9861
17 0.75 0.15 9815
18 0.37 0.08 9635
19 0.87 0.16 9810
20 0.60 0.11 9933
21 0.76 0.14 9763
22 0.37 0.08 9582
23 0.86 0.16 9767
24 0.61 0.10 9897
25 0.76 0.14 9818
26 0.68 0.13 9684
27 0.44 0.08 9773
28 0.72 0.14 9758
29 0.47 0.08 9670
30 0.80 0.16 9778
Table 4.5: The table shows the evolution of the convergence metrics and the waitlist size as more
iterations are simulated for the status quo with an intercept of 4.0.
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Chapter 5
Concluding Remarks
Motivated by the significant wastage despite the scarcity in cadaveric kidney allocation, this thesis presents both theoretical guidance and actionable solutions to ameliorate the wastage issue.
The theoretical guidance sheds lights on how to eliminate the wastage and highlights the associated trade-offs. The actionable solutions include a personalized offer prediction tool and a
practical kidney allocation mechanism based on the status quo.
In the concluding chapter, we summarize the insights that could be generalized to other contexts, such as the allocation of other cadaveric organs and public housing, and discuss the future
work needed to effectively implement these tools.
5.1 Generalizable Insights
First, leveraging state-of-the-art machine learning models and statistical techniques to make predictions based on observational data might be a convenient default approach. However, this fails
to capture the intricacies of the allocation process and faces a trade-off between accounting for
126

the participants’ heterogeneity and maintaining statistical power. In comparison, directly incorporating the rules of the allocation process can yield more personalized, accurate and intuitive
predictions without compromising the statistical power.
Second, simply expediting the allocation of the wasted items may not necessarily reduce their
wastage, and the allocation of the more desirable items plays a crucial role. To reduce wastage, a
mechanism needs to consider the participants’ preferences and resolve the uncertainty of getting
the more desirable items early.
Third, reducing wastage not necessarily makes everyone happier, and any effort to reduce
the wastage should be carefully evaluated in terms of the impact on the participants’ welfare.
Moreover, if there is significant horizontal differentiation among the participants, then with an
anonymous mechanism, it is challenging to reduce the wastage without hurting some participants. Because reducing the wastage requires making the participants less picky, while maintaining their welfare requires allowing them to choose. In such cases, incorporating the knowledge
of the differences in the participants’ preferences may help, but it might lead to prioritizing the
participants based on some rules that are hard to explain or justify.
5.2 Future Work
First, the personalized offer prediction tool shows the doctors what the patients’ donor access
trajectories are like as the patients remain on the waitlist. However, it does not directly suggest
whether the doctors should accept or reject each offer. As solving the optimal stopping problem
to obtain the acceptance threshold for each patient requires a thorough understanding of the
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patient’s preference, one may obtain the patients’ preferences through questionnaires in practice
and solve the optimal stopping problem using part of the simulation algorithm in Chapter 4.
Second, to conduct the simulation, we leveraged the utility estimates from Agarwal et al.
[2]. However, the utility estimates may vary across different areas and times. To enhance the
robustness of the simulation results, one may re-estimate the utility coefficients for each area
and re-run the simulations. Alternative approaches to address the cycling issue in the simulation
may be explored and tested.
Third, recent modifications to the kidney allocation system, such as the removal of the donation service area (DSA) and OPTN region in the allocation process [42], should be incorporated
into the prediction and simulation tools to make these tools more relevant to the present.
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Bibliography
[1] Atila Abdulkadiroğlu and Tayfun Sönmez. “School Choice: A Mechanism Design
Approach”. In: American Economic Review 93.3 (2003), pp. 729–747.
[2] Nikhil Agarwal, Itai Ashlagi, Michael A Rees, Paulo Somaini, and Daniel Waldinger.
“Equilibrium Allocations under Alternative Waitlist Designs: Evidence from Deceased
Donor Kidneys”. In: Econometrica 89.1 (2021), pp. 37–76.
[3] Nick Arnosti and Peng Shi. “Design of Lotteries and Wait-lists for Affordable Housing
Allocation”. In: Management Science 66.6 (2020), pp. 2291–2307.
[4] Olivier Aubert, Peter P Reese, Benoit Audry, Yassine Bouatou, Marc Raynaud,
Denis Viglietti, Christophe Legendre, Denis Glotz, Jean-Phillipe Empana, Xavier Jouven,
et al. “Disparities in Acceptance of Deceased Donor Kidneys Between the United States
and France and Estimated Effects of Increased US Acceptance”. In: JAMA Internal
Medicine 179.10 (2019), pp. 1365–1374.
[5] David A Axelrod, Mark A Schnitzler, Huiling Xiao, William Irish,
Elizabeth Tuttle-Newhall, Su-Hsin Chang, Bertram L Kasiske, Tarek Alhamad, and
Krista L Lentine. “An Economic Assessment of Contemporary Kidney Transplant
Practice”. In: American Journal of Transplantation 18.5 (2018), pp. 1168–1176.
[6] M Antonieta Azancot, Francesc Moreso, Maite Salcedo, Carme Cantarell, Manel Perello,
Irina B Torres, Angeles Montero, Enric Trilla, Joana Sellarés, Joan Morote, et al. “The
Reproducibility and Predictive Value on Outcome of Renal Biopsies from Expanded
Criteria Donors”. In: Kidney International 85.5 (2014), pp. 1161–1168.
[7] Eduardo M Azevedo and Jacob D Leshno. “A Supply and Demand Framework for
Two-sided Matching Markets”. In: Journal of Political Economy 124.5 (2016),
pp. 1235–1268.
[8] Masoud Barah, Vikram Kilambi, John J Friedewald, and Sanjay Mehrotra. “Implications of
Accumulated Cold Time for US Kidney Transplantation Offer Acceptance”. In: Clinical
Journal of the American Society of Nephrology 17.9 (2022), pp. 1353–1362.
129

[9] Dimitris Bertsimas, Jerry Kung, Nikolaos Trichakis, David Wojciechowski, and
Parsia A Vagefi. “Accept or Decline? An Analytics-based Decision Tool for Kidney Offer
Evaluation”. In: Transplantation 101.12 (2017), pp. 2898–2904.
[10] Francis Bloch and David Cantala. “Dynamic Assignment of Objects to Queuing Agents”.
In: American Economic Journal: Microeconomics 9.1 (2017), pp. 88–122.
[11] Matthew Cooper, Richard Formica, John Friedewald, Ryutaro Hirose, Kevin O’Connor,
Sumit Mohan, Jesse Schold, David Axelrod, and Stephen Pastan. “Report of National
Kidney Foundation Consensus Conference to Decrease Kidney Discards”. In: Clinical
transplantation 33.1 (2019), e13419.
[12] Mark Haas. “Donor Kidney Biopsies: Pathology Matters, and So Does the Pathologist”. In:
Kidney International 85.5 (2014), pp. 1016–1019.
[13] US Department of Health and Human Services. OPTN Policy. Accessed: 2024-03-20. 2000.
url: https://www.ecfr.gov/current/title-42/chapter-I/subchapter-K/part-121.
[14] Rachel Hellemans, Anneke Kramer, Johan De Meester, Frederic Collart, Dirk Kuypers,
Michel Jadoul, Steven Van Laecke, Alain Le Moine, Jean-Marie Krzesinski,
Karl Martin Wissing, et al. “Does Kidney Transplantation with a Standard or Expanded
Criteria Donor Improve Patient Survival? Results from a Belgian Cohort”. In: Nephrology
Dialysis Transplantation 36.5 (2021), pp. 918–926.
[15] Richard A Hirth, Qing Pan, Douglas E Schaubel, and Robert M Merion. “Efficient
Utilization of the Expanded Criteria Donor (ECD) Deceased Donor Kidney Pool: an
Analysis of the Effect of Labeling”. In: American Journal of Transplantation 10.2 (2010),
pp. 304–309.
[16] S Ali Husain, Kristen L King, Stephen Pastan, Rachel E Patzer, David J Cohen,
Jai Radhakrishnan, and Sumit Mohan. “Association Between Declined Offers of Deceased
Donor Kidney Allograft and Outcomes in Kidney Transplant Candidates”. In: JAMA
Network Open 2.8 (2019), e1910312–e1910312.
[17] Maria Ibrahim, Gabe Vece, Jenny Mehew, Rachel Johnson, John Forsythe, David Klassen,
Chris Callaghan, and Darren Stewart. “An International Comparison of Deceased Donor
Kidney Utilization: What Can the United States and the United Kingdom Learn from Each
Other?” In: American Journal of Transplantation 20.5 (2020), pp. 1309–1322.
[18] Ajay K Israni, Nicholas Salkowski, Sally Gustafson, Jon J Snyder, John J Friedewald,
Richard N Formica, Xinyue Wang, Eugene Shteyn, Wida Cherikh, Darren Stewart, et al.
“New National Allocation Policy for Deceased Donor Kidneys in the United States and
Possible Effect on Patient Outcomes”. In: Journal of the American Society of Nephrology
25.8 (2014), pp. 1842–1848.
130

[19] Ajay K Israni, David Zaun, John D Rosendale, C Schaffhausen, W McKinney, and
Jon J Snyder. “OPTN/SRTR 2019 Annual Data Report: Deceased Organ Donors”. In:
American Journal of Transplantation 21 (2021), pp. 567–604.
[20] Jonathan Jalbert, Jean-Noel Weller, Pierre-Luc Boivin, Sylvain Lavigne, Mehdi Taobane,
Mike Pieper, Andrea Lodi, and Héloise Cardinal. “Predicting Time to and Average Quality
of Future Offers for Kidney Transplant Candidates Declining a Current Deceased Donor
Kidney Offer: A Retrospective Cohort Study”. In: Canadian Journal of Kidney Health and
Disease 10 (2023), p. 20543581231177844.
[21] Colleen L Jay, Kenneth Washburn, Patrick G Dean, Ryan A Helmick, Jacqueline A Pugh,
and Mark D Stegall. “Survival Benefit in Older Patients Associated with Earlier
Transplant with High KDPI Kidneys”. In: Transplantation 101.4 (2017), pp. 867–872.
[22] Vikram Kilambi, Kevin Bui, Gordon B Hazen, John J Friedewald, Daniela P Ladner,
Bruce Kaplan, and Sanjay Mehrotra. “Evaluation of Accepting Kidneys of Varying Quality
for Transplantation or Expedited Placement with Decision Trees”. In: Transplantation
103.5 (2019), pp. 980–989.
[23] Kristen L King, Sulemon G Chaudhry, Lloyd E Ratner, David J Cohen, S Ali Husain, and
Sumit Mohan. “Declined Offers for Deceased Donor Kidneys Are Not an Independent
Reflection of Organ Quality”. In: Kidney360 2.11 (2021), pp. 1807–1818.
[24] Kristen L King, S Ali Husain, Adler Perotte, Joel T Adler, Jesse D Schold, and
Sumit Mohan. “Deceased Donor Kidneys Allocated Out of Sequence by Organ
Procurement Organizations”. In: American Journal of Transplantation 22.5 (2022),
pp. 1372–1381.
[25] Jacob D Leshno and Irene Lo. “The Cutoff Structure of Top Trading Cycles in School
Choice”. In: The Review of Economic Studies 88.4 (2021), pp. 1582–1623.
[26] Amanda Macchi. Better Screening Accelerates Organ Offers. Accessed: 2024-03-20. 2020.
url: https://unos.org/news/innovation/better-screening-accelerates-organ-offers/.
[27] Sanjay Mehrotra, Juan Marcos Gonzalez, Karolina Schantz, Jui-Chen Yang,
John J Friedewald, and Richard Knight. “Patient Preferences for Waiting Time and Kidney
Quality”. In: Clinical Journal of the American Society of Nephrology 17.9 (2022),
pp. 1363–1371.
[28] Sumit Mohan, Mariana C Chiles, Rachel E Patzer, Stephen O Pastan, S Ali Husain,
Dustin J Carpenter, Geoffrey K Dube, R John Crew, Lloyd E Ratner, and David J Cohen.
“Factors Leading to the Discard of Deceased Donor Kidneys in the United States”. In:
Kidney International 94.1 (2018), pp. 187–198.
131

[29] Sumit Mohan, Karl Foley, Mariana C Chiles, Geoffrey K Dube, Rachel E Patzer,
Stephen O Pastan, R John Crew, David J Cohen, and Lloyd E Ratner. “The Weekend Effect
Alters the Procurement and Discard Rates of Deceased Donor Kidneys in the United
States”. In: Kidney International 90.1 (2016), pp. 157–163.
[30] Sumit Mohan and Jesse D Schold. Accelerating Deceased Donor Kidney Utilization Requires
More Than Accelerating Placement. 2022.
[31] Robert A Montgomery. “Getting Comfortable with Risk”. In: New England Journal of
Medicine 381.17 (2019), pp. 1606–1607.
[32] Sara Moriarty. Achieving a Shorter, More Active Kidney Waitlist. Accessed: 2024-03-20.
2019. url:
https://unos.org/news/improvement/achieving-a-shorter-more-active-waitlist/.
[33] Samantha M Noreen, David Klassen, Roger Brown, Yolanda Becker, Kevin O’Connor,
Jennifer Prinz, and Matthew Cooper. “Kidney Accelerated Placement Project: Outcomes
and Lessons Learned”. In: American Journal of Transplantation 22.1 (2022), pp. 210–221.
[34] Akinlolu O Ojo, Julie A Hanson, Herwig-Ulf Meier-Kriesche, Chike N Okechukwu,
Robert A Wolfe, Alan B Leichtman, Lawrence Y Agodoa, Bruce Kaplan, and
Friedrich K Port. “Survival in Recipients of Marginal Cadaveric Donor Kidneys Compared
with Other Recipients and Wait-listed Transplant Candidates”. In: Journal of the American
Society of Nephrology 12.3 (2001), pp. 589–597.
[35] Gabriel C Oniscu, Helen Brown, and John LR Forsythe. “Impact of Cadaveric Renal
Transplantation on Survival in Patients Listed for Transplantation”. In: Journal of the
American Society of Nephrology 16.6 (2005), pp. 1859–1865.
[36] María José Pérez-Sáez, Núria Montero, Dolores Redondo-Pachón, Marta Crespo, and
Julio Pascual. “Strategies for an Expanded Use of Kidneys from Elderly Donors”. In:
Transplantation 101.4 (2017), pp. 727–745.
[37] Hessel Peters-Sengers, Julia HE Houtzager, Mirza M Idu, Martin BA Heemskerk,
Ernst LW van Heurn, Jaap J Homan van der Heide, Jesper Kers, Stefan P Berger,
Thomas M van Gulik, and Frederike J Bemelman. “Impact of Cold Ischemia Time on
Outcomes of Deceased Donor Kidney Transplantation: An Analysis of a National
Registry”. In: Transplantation Direct 5.5 (2019), e448.
[38] Organ Procurement and Transplantation Network. Enhance Transplant Program
Performance Monitoring. Accessed: 2024-03-20. 2022. url:
https://optn.transplant.hrsa.gov/policies-bylaws/enhance-transplant-programperformance-monitoring/.
132

[39] Organ Procurement and Transplantation Network. How Calculated Panel Reactive
Antibodies Score Is Calculated and Used. Accessed: 2024-03-20. 2024. url: https:
//optn.transplant.hrsa.gov/data/allocation-calculators/cpra-calculator/learn-about-cpra/.
[40] Organ Procurement and Transplantation Network. How Estimated Post Transplant
Survival Score Is Calculated and Used. Accessed: 2024-03-20. 2024. url: https:
//optn.transplant.hrsa.gov/data/allocation-calculators/epts-calculator/learn-about-epts/.
[41] Organ Procurement and Transplantation Network. OPTN Policy. Accessed: 2024-03-20.
2024. url: https://optn.transplant.hrsa.gov/policies-bylaws/policies/.
[42] Organ Procurement and Transplantation Network. Removal of DSA and Region from
Kidney Allocation Policy. Accessed: 2024-03-20. 2024. url:
https://optn.transplant.hrsa.gov/professionals/by-organ/kidney-pancreas/kidneyallocation-system/removal-of-dsa-and-region-from-kidney-allocation-policy/.
[43] TA Ramasubban. “The Mean Difference and the Mean Deviation of Some Discontinuous
Distributions”. In: Biometrika 45.3/4 (1958), pp. 549–556.
[44] Panduranga S Rao, Douglas E Schaubel, Mary K Guidinger, Kenneth A Andreoni,
Robert A Wolfe, Robert M Merion, Friedrich K Port, and Randall S Sung. “A
Comprehensive Risk Quantification Score for Deceased Donor Kidneys: the Kidney
Donor Risk Index”. In: Transplantation 88.2 (2009), pp. 231–236.
[45] Mohamed Raslan, Mahmoud Samhan, Eyad Halabi, Jellan Shanab, Bobby Matheow,
Wael Al Shomali, and Moustafa Al Mousawi. “Outcome of Kidney Transplantation from
Expanded Criteria Deceased Donors”. In: Transplantation 101.5S-3 (2017), S46.
[46] Kevin Sack. In Discarding of Kidneys, System Reveals Its Flaws. Accessed: 2024-03-20. 2012.
url: https://www.nytimes.com/2012/09/20/health/transplant-experts-blame-allocationsystem-for-discarding-kidneys.html.
[47] Juliana Feiman Sapiertein Silva, Gustavo Fernandes Ferreira, Marcelo Perosa,
Hong Si Nga, and Luis Gustavo Modelli de Andrade. “A Machine Learning Prediction
Model for Waiting Time to Kidney Transplant”. In: PLoS One 16.5 (2021), e0252069.
[48] Rajiv Saran, Bruce Robinson, Kevin C Abbott, Jennifer Bragg-Gresham, Xiaoying Chen,
Debbie Gipson, Haoyu Gu, Richard A Hirth, David Hutton, Yan Jin, et al. US Renal Data
System 2019 Annual Data Report: Epidemiology of Kidney Disease in the United States. 2020.
[49] JD Schold, RJ Howard, MJ Scicchitano, and H-U Meier-Kriesche. “The Expanded Criteria
Donor Policy: An Evaluation of Program Objectives and Indirect Ramifications”. In:
American Journal of Transplantation 6.7 (2006), pp. 1689–1695.
133

[50] Jesse D Schold, Sumit Mohan, Anne Huml, Laura D Buccini, John R Sedor,
Joshua J Augustine, and Emilio D Poggio. “Failure to Advance Access to Kidney
Transplantation Over Two Decades in the United States”. In: Journal of the American
Society of Nephrology 32.4 (2021), pp. 913–926.
[51] Jesse D Schold and Peter P Reese. “Simulating the New Kidney Allocation Policy in the
United States: Modest Gains and Many Unknowns”. In: Journal of the American Society of
Nephrology 25.8 (2014), pp. 1617–1619.
[52] James Schummer. “Influencing Waiting Lists”. In: Journal of Economic Theory 195 (2021),
p. 105263.
[53] Adnan Sharif. “Risk Aversion, Organ Utilization and Changing Behavior”. In: Transplant
International 35 (2022), p. 10339.
[54] Jon J Snyder, Nicholas Salkowski, S Joseph Kim, David Zaun, Hui Xiong, Ajay K Israni,
and Bertram L Kasiske. “Developing Statistical Models to Assess Transplant Outcomes
Using National Registries: the Process in the United States”. In: Transplantation 100.2
(2016), pp. 288–294.
[55] Darren E Stewart, Victoria C Garcia, John D Rosendale, David K Klassen, and
Bob J Carrico. “Diagnosing the Decades-long Rise in the Deceased Donor Kidney Discard
Rate in the United States”. In: Transplantation 101.3 (2017), pp. 575–587.
[56] Xuanming Su and Stefanos Zenios. “Patient Choice in Kidney Allocation: The Role of the
Queueing Discipline”. In: Manufacturing & Service Operations Management 6.4 (2004),
pp. 280–301.
[57] Xuanming Su and Stefanos A Zenios. “Patient Choice in Kidney Allocation: A Sequential
Stochastic Assignment Model”. In: Operations Research 53.3 (2005), pp. 443–455.
[58] Xuanming Su and Stefanos A Zenios. “Recipient Choice Can Address the
Efficiency-equity Trade-off in Kidney Transplantation: A Mechanism Design Model”. In:
Management Science 52.11 (2006), pp. 1647–1660.
[59] Randall S Sung, Laura L Christensen, Alan B Leichtman, SM Greenstein, Dale A Distant,
James J Wynn, Mark D Stegall, Francis L Delmonico, and Friedrich K Port. “Determinants
of Discard of Expanded Criteria Donor Kidneys: Impact of Biopsy and Machine
Perfusion”. In: American Journal of Transplantation 8.4 (2008), pp. 783–792.
[60] Randall S Sung, Mary K Guidinger, Craig D Lake, Maureen A McBride,
Stuart M Greenstein, Francis L Delmonico, Friedrich K Port, Robert M Merion, and
Alan B Leichtman. “Impact of the Expanded Criteria Donor Allocation System on the Use
of Expanded Criteria Donor Kidneys”. In: Transplantation 79.9 (2005), pp. 1257–1261.
134

[61] Neil Thakral. The Public-housing Allocation Problem. Tech. rep. Technical Report, Harvard
University, 2016.
[62] United States Renal Data System. 2023 USRDS Annual Data Report: Epidemiology of Kidney
Disease in the United States. Bethesda, MD: National Institutes of Health, National Institute
of Diabetes, Digestive, and Kidney Diseases, 2023. url: https://adr.usrds.org/2023.
[63] UNOS. Kidney Monitoring Data Shows 16% Increase in Transplant. Accessed: 2024-03-20.
2022. url: https://unos.org/news/in-focus/kidney-monitoring-data-shows-16-increasein-transplant/.
[64] CJ Wang, JB Wetmore, GS Crary, and BL Kasiske. “The Donor Kidney Biopsy and Its
Implications in Predicting Graft Outcomes: a Systematic Review”. In: American Journal of
Transplantation 15.7 (2015), pp. 1903–1914.
[65] Guihua Wang, Ronghuo Zheng, and Tinglong Dai. “Does Transportation Mean
Transplantation? Impact of New Airline Routes on Sharing of Cadaveric Kidneys”. In:
Management Science 68.5 (2022), pp. 3660–3679.
[66] Juanjuan Zhang. “The Sound of Silence: Observational Learning in the US Kidney
Market”. In: Marketing Science 29.2 (2010), pp. 315–335.
135

Abstract (if available)

Abstract In most countries, there is a shortage of cadaveric kidneys, but many successfully procured and medically tenable kidneys are being discarded. The wastage of cadaveric kidneys exacerbates the shortage in kidney supply and the financial strains on healthcare systems. Although many interventions have been proposed or are being implemented, the reduction in kidney wastage has been limited. Therefore, in this thesis, we take on a futuristic perspective and study the research question: what more is needed to eliminate waste?

First, after observing patients' suboptimal choices regarding offer acceptance, we build a personalized offer prediction tool by leveraging the observed donor allocation cutoffs. This tool enables the doctors and patients to assess their offer prospects more accurately and therefore make better acceptance decisions.

Second, to provide theoretical insights on eliminating waste through mechanism design, we analyze a stylized model to characterize the waste-eliminating mechanisms and identify the inevitable trade-offs of eliminating waste.

Finally, to connect the theoretical insights to practice, we propose a mechanism with a minor departure from the status quo that reduces waste and improves patient welfare when the degree of horizontal differentiation among the patients' preferences is small. To examine the effectiveness, we develop a simulation engine and evaluate the mechanism using the actual kidney transplant data.

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Asset Metadata

Creator Yin, Junxiong (author)

Core Title Patient choice and wastage in cadaveric kidney allocation

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Degree Conferral Date 2024-05

Publication Date 05/17/2024

Defense Date 04/24/2024

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag market design,matching markets,OAI-PMH Harvest,organ allocation,wait-list

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Shi, Peng (committee chair), Drakopoulos, Kimon (committee member), Nikzad, Afshin (committee member)

Creator Email junxiong.yin@gmail.com,junxiong.yin@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC113939946

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Legacy Identifier etd-YinJunxion-12930

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Format theses (aat)

Rights Yin, Junxiong

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