#  Payday loans - blessing or growth suppressor? - Machine Learning Analysis

Rohith Mahadevan
<EMAIL_ADDRESS>Sam Richard
<EMAIL_ADDRESS>Kishore Harshan Kumar
<EMAIL_ADDRESS>Jeevitha Murugan
<EMAIL_ADDRESS>Santhosh Kannan
<EMAIL_ADDRESS>Saaisri
<EMAIL_ADDRESS>Tarun
<EMAIL_ADDRESS>Raja CSP Raman
<EMAIL_ADDRESS>
###### Abstract

The upsurge of real estate involves a variety of factors that have got
influenced by many domains. Indeed, the unrecognized sector that would affect
the economy for which regulatory proposals are being drafted to keep this in
control is the payday loans. This research paper revolves around the impact of
payday loans in the real estate market. The research paper draws a first-hand
experience of obtaining the index for the concentration of real estate in an
area of reference by virtue of payday loans in Toronto, Ontario in particular,
which sets out an ideology to create, evaluate and demonstrate the scenario
through research analysis. The purpose of this indexing via payday loans is
the basic - debt: income ratio which states that when the income of the person
bound to pay the interest of payday loans increases, his debt goes down
marginally which hence infers that the person invests in fixed assets like
real estate which hikes up its growth.

Keywords: Real estate,Domains,Unrecognized
sector,Economy,Paydayloans,Toronto,Ontario, Indexing,Debt:Income ratio,
Interest,Invests,Assests

##  1 Introduction

The most underrated factor that comes into play in the history of the real
estate market in Toronto seems to be the payday loans whose APR (annual
percentage rate) is much higher than other loans in Canada. In addition to the
high-interest rates charged, the late payment fee and the NSF (non-sufficient
fund - return check charge) fee are charged by the lender to the borrower in
some cases of need. The eligibility for a person to borrow is at least 18
years old or the age of majority in a province or territory, have proof of an
income and be a Canadian citizen or permanent resident. One must also need to
have a valid bank account. Some lenders may ask for access to the borrower’s
bank account which often creates a hassle of the overdraft, which holds an
ultimatum to the debt cycle, further the income of the person drains which
results in the worst scenario of investment in fixed assets like real estate.
In addition to it, the verge of increasing indebtedness causes a dramatic
decline in the economic status of the citizens and for their upliftment and
the social status, The Government of Canada offers 3 programs to help first-
time homebuyers – the first-time homebuyer incentive, the first-time home
Buyer’s amount tax credit, and Home Buyers’ Plan. However, this would help the
citizens sufficiently. Still, a further increase in the interest turns the
situation to be alarming and would decline the growth factor of the real
estate market. Hence, the paper has undertaken the theme of indexing the
upsurge of real estate via the reduction of the concentration of payday loans.
Unknowingly, this domain of analysis stays farther from the views of the
common man, who turns out to be the prey to the huge interest amount further
expanding the debt cycle. Hitherto, there have been no signs of drastic growth
of real estate in the area of payday loans concentration which turns out to be
harsh both on the economy as well as the citizens. Thereby, the upcoming parts
of this paper explain in detail the analysis made using the datasets,
unsupervised machine learning, data analytics, etc to come to a conclusion of
the indexing satisfying the motto of the research paper.

##  2  Literature Survey:

[1]The highest level of financial literacy among Canadians has caused unusual
growth in Canada over the years. The Canadian government implemented First-
Time Home Buyer Incentive(FTHBI) to increase in Canadian ownership; as a
result, people faced many financial, unemployment, and intersectional shifts
in labour among various sectors. [2]According to observations, the Canadian
labour market has experienced rapid change in the last decade. Unemployment
can result in intersectoral shifts in labour demand and the process of labour
reallocation across sectors. The government has increased the payroll type of
tax in recent years. Wages and salaries increased by approximately 10 percent,
while supplement labour increased by 40 percent, increasing people’s salaries
by approximately 3 percent. This leads to inconsistency in consumption and
borrowing, forcing these people to rely on micro-credit. [3] Furthermore, if
borrowers are ”naive” about their current focus, overly optimistic about their
future financial situation, or do not anticipate their high likelihood of
repeat borrowing for any other reason, they may underestimate the costs of
repaying a loan. In this case, restricting credit access may benefit.[4]
Credit constraints and financial exclusion are used to achieve this. Income,
assets or wealth, age, household size, education, debt, gender, and region of
residence are used as predictors of financial exclusion. [5]This information
is obtained using credit reports which contain a wealth of information about
consumers’ financial habits. Hundreds of consumer-related variables are
evidently collected in these reports from a variety of sources. [6] Although
most customers were unaware that being denied a loan had an impact on their
credit history, learning this increased their interest in providing an
indication of loan eligibility on the site. Loans were compared based on both
total cost and Annual Percentage Rate(APR). While APR was not widely
understood, it was perceived as simple to compare and served as a proxy for
price when the total repayment amount was not specified. [7] Because of
geographic variation in access to loans caused by regulation, the effects of
payday loans on consumers are very mixed. There are more prisoners in the area
as a result of the crisis. Regressions were used to examine the lender’s
credit history. This provides adequate lending and lender information.[8]
Network data adds value by increasing the predictive accuracy of these models.
Predictors are intended to inform decisions that affect the very outcomes that
are the target of prediction, such as when the predictor is intended to
estimate the risk of an adverse outcome so that a decision-maker can intervene
to prevent that outcome. A predictor trained to be fair in one population may
be unfair in another, but performativity can induce unfairness even when the
populations in which the predictor is trained and deployed are the same, i.e.
there is no covariate shift. [9] Thus, two major conclusions can be reached.
To begin, future research should concentrate on addressing each stage of the
payday loan process, beginning with understanding the needs of payday loan
users. The findings indicate that borrowers’ decisions are close to optimal in
light of their liquidity needs; however, the initial liquidity needs that
drive people to seek payday loans may be the result of suboptimal consumption
and savings decisions.

##  3 DATA ARCHITECTURE:

Figure 1: Flow chart on data architecture

##  4 Methodology

###  4.1  Data collection:

Initially, the data is gathered from numerous sources on the internet to
determine the number of payday loan lenders in each location based on facts
such as 2 bedrooms, 3 bedrooms, and 4 bedrooms in 2010 and 2020 among various
regions of Canada. For a high level of accuracy, the data was cross-verified
numerous times. such that this data is more adequate for further data
preparation

###  4.2 Data preparation:

Before removing null values from structured data, the data was properly
cleansed. The clustered data was then structured into a single data frame for
easier identification, and an average analysis was returned for 2010 and 2020.
For a better comprehension of data, numerous visualizations are used. The data
was further divided into several number of subsets and visualized accordingly.
Further, it is proceeded with various algorithms to reach better accuracy.

###  4.3 Inference from analytics:

Analytics is the process that consists of data cleaning, inspecting,
modelling, and transforming the data to obtain useful insights from which
conclusions can be drawn. Analytics is invaluable to the process of decision-
making and optimization. Modelling and visualizing are two major aspects of
analytics that help us to better understand the data and utilize resources
effectively.

Initially, the data is divided into sub-sections for analysis purposes. After,
cleaning the data and preparing it, statistical analysis is done. Initial
observation of how many payday loan lenders are present in each city in the
Ontario province is seen.

Figure 2: Graph indicates payday loan lenders with reference to various
regions of Canada

###  4.4 Statistical inference:

The value in the real estate price is calculated by using the percentage
increase formula. The highest percentage increase of 861 percent is observed
in the 4-bedroom apartment. The real estate price increased from 2010 to 2020
drastically in specific areas. An average increase of 128 percent is observed
in the overall data.deep analysis, it is inferred that the mean increase was
very less in 2 bedroom apartment.

Figure 3: Analysis on dataset

Even though the dataset is cleaned, some data from 2010 are approximated
values. The visualization is done with the present data that are taken from
multiple resources. The data is taken from up-to-date websites that are found
online. A result comparison of real estate prices in 2010 and 2020 is done.
The chart shows that there is an overall increase but in some areas, there is
a high increase in prices. In some areas, the prices are increased moderately
with an increase in time. For further drill-down analysis, the data is cleaned
in such a way as to model the efficiency based on the available data.

Figure 4: House prices in 2010 and 2020

The median of the price increase is inclined towards the first quartile
region. This means that there is a huge increase observed over the period

Figure 5: Box plot analysis on Median of increase in price

####  4.4.1 Regression analysis

The data from the dataset is preprocessed and a simple plot is made. Different
combination of average increase vs payday loan lenders, 2 bedroom price
percentage increase vs payday loan lenders, 3 bedroom price percentage
increase vs payday loan lenders, 4 bedroom price percentage increase vs payday
loan lenders is made. Considering all the combinations, 4 bedroom houses show
that the percentage increase was in receding trend. The trend was normal and
predicted a downfall

Figure 6: Analysis of payday loan lenders vs average increase in pricing

The chart analysis interprets that as the payday loan lender’s percentage
increases the price variation in real estate price decreases. The downtrend
graph proves the statement. For further analysis, k means clustering is done
to find whether there are any cluster of groups formed with respect to payday
loan lender’s and the average percentage increases in price.

##  5 Data Modeling

###  5.1 Percent increase formula:

percent increase =$\frac{final-initial}{|initial|}{\cdot 100}$

###  5.2 K-Means Clustering:

K-Means Clustering is a simple unsupervised learning algorithm that is used to
divide the dataset into K number of distinct, non-overlapping clusters. It
divides the data into clusters based on the centroid of the cluster. A
cluster’s centroid is the arithmetic mean of the values in the cluster. A data
point is allocated to a cluster if the squared distance between the data point
and the cluster’s centroid is a minimum. The value of K is the hyperparameter
in this algorithm that is generally found using the elbow and the silhouette
methods.

The K means clustering algorithm is implemented in the dataset. The elbow
method is used to find the number of clusters required to train the dataset .
In the Elbow method, we are varying the number of clusters ( K ) from 1 – 10.
For each value of K, we are calculating WCSS ( Within-Cluster Sum of Square ).
WCSS is the sum of the squared distance between each point and the centroid in
a cluster. When we plot the WCSS with the K value, the plot looks like an
Elbow. As the number of clusters increases, the WCSS value will start to
decrease. WCSS value is largest when K = 1. When the graph is analyzed it is
evident that the graph will rapidly change at a point and thus creating an
elbow shape. From this point, the graph starts to move almost parallel to the
X-axis. The K value corresponding to this point is the optimal K value or an
optimal number of clusters.

###  5.3 Implementation of K-Means clustering:

The algorithm is implemented using the python sklearn library. The elbow
method determines the number of clusters that are actually needed for the
analysis. The n clusters that are required are 3 and the data is trained with
the algorithm. After training, the data points are split into three different
groups. As discussed, a cluster is formed where the payday loan lenders are
less and the average increase in percentage is increasing.

Figure 7: Analysis plot on three different subsets

This supports our previous regression analysis. The average trend tends to be
in a state of descending.

##  6 Result analysis:

The interpretation of the graphical representation starts with the
normalization of the data before visualizing it which plays a pivotal role in
the data analysis. The dataset involves the pricing of 2,3,4 Bedroom houses in
the Ontario province and is estimated. A percentage over the difference of the
range is obtained and plotted in the graph. The graph is now visualized using
a regression plot .

Overall, this analysis shows that there is a gradual decrease in the growth
factor of real estate by virtue of payday loans. However, the graph is plotted
against the payday loan lenders and the average increase of the houses which
leads to a marginally decreasing slope which is welcoming to support our
primary thesis.

Further, anomaly has to be checked wherein there is one single anomaly which
is inferred that is the shift of the interquartile range which states that all
the outcomes of the cluster blobs come under the position and have one anomaly
data in 4Br. To figure this out and to give out a precise outcome we take into
consideration the k means clustering from the unsupervised machine learning
which involves the division of the entire graphical representation into small
parts of subset ; say over a range of 0-30. The quartile 3 of the graph has
more outcome markings as compared to the quartile 1 because of that the region
of q3 is more as compared to q1. More the cluster in the graph less is the
payday loans the median shifts in the q1 region.

The elbow chart developed from the k means cluster analysis shows the kirks/
bents in the decreasing slope that is interpreted, thereby the random state is
declared,if k increases, the payday loan lenders will decrease, and each
cluster will now have fewer constituent instances and the instances occur
closer to the respective centroids. However, the increase in payday loans will
decline the growth of real estate where k is taken into consideration.

##  7 Limitations of the study:

The hypothesis that exists in the real world is that payday loan lenders
indirectly affect the real estate growth of a particular area. The research
paper correlates all the data that are available on the internet and tries to
possibly prove the hypothesis. The research study is concentred only on the
Ontario province. The data analytics and modelling are done for the dataset
which is concentrated towards the specific region thus resulting in very
minimal insights. The problem with the dataset is all the real estate prices
from 2010 are not available as there were fewer data collected at that time. A
crucial takeaway from this result is that we can’t fully foresee the impact of
real estate growth on payday loans because the land or surroundings may not
meet the expectations of consumers, causing real estate growth to spike.

##  8 Conclusion :

This research paper mainly focuses on Ontario province and the cities in it
the above study concludes that amongst many factors influencing the growth
rate of real estate; payday loan lenders too play a significant role. The
research analysis concludes that the increase in payday loans would marginally
decrease the real estate growth in particular; which in general may involve a
variety of factors too. However, the study stated above involves likely the
views of the small group of researchers while the participants of the paper do
not claim or may not be representative of larger segments of populations;
which may challenge traditional findings, implications, generalizability, and
measures of validity.

##  9 Future work:

The future work of this research paper is to concentrate on all the provinces
in Canada and to find more insights into the impact of payday loans on people.
Thus, this will strengthen the study and will help us to find more about the
indirect impact of the payday loan concept and how it affects the real estate
growth in the country. The study will also concentrate on the factors which
have an indirect impact on the economic growth of the country and possibly try
to increase the overall economic status of the common people living across the
country.

##  10 Acknowledgements:

We’d like to express our gratitude to our Featurepreneur team for collecting
data to improve this model’s accuracy. Thank you for mentoring the paper
Snekha Suresh.

## References

  * [1] Sanjana Bhatnagar.  An assessment of the first-time home buyer incentive in canada.  2019\.
  * [2] Khan Islam and Wayne Simpson.  Payday lending and microcredit: Two faces of the same problem?: Payday lending and microcredit.  Journal of International Development, 30, 12 2017.
  * [3] Hunt Allcott, Joshua J Kim, Dmitry Taubinsky, and Jonathan Zinman.  Are high-interest loans predatory? theory and evidence from payday lending.  Working Paper 28799, National Bureau of Economic Research, May 2021.
  * [4] Gabriel F. R. Vasconcelos Giacomo De Giorgi, Matthew Harding.  Predicting mortality from credit reports.  2021\.
  * [5] John Gathergood, Benedict Guttman-Kenney, and Stefan Hunt.  How Do Payday Loans Affect Borrowers? Evidence from the U.K. Market.  The Review of Financial Studies, 32(2):496–523, 08 2018.
  * [6] John Gathergood, Benedict Guttman-Kenney, and Stefan Hunt.  How Do Payday Loans Affect Borrowers? Evidence from the U.K. Market.  The Review of Financial Studies, 32(2):496–523, 08 2018.
  * [7] Fair When Trained Alan Mishler, Niccolò Dalmasso.  Unfair when deployed:observable fairness measures are unstable in performativeprediction settings.  2022\.
  * [8] Brian Distelberg.  Exploratory study of payday lending: A human ecology evaluation.  2009\.
  * [9] Stephen S. Poloz.  The causes of unemployment in canada:.  1994\.


# Modeling Access Differences to Reduce Disparity in Resource Allocation

Kenya S. Andrews
University of Illinois at Chicago
<EMAIL_ADDRESS>
&Mesrob I. Ohannessian
University of Illinois at Chicago
<EMAIL_ADDRESS>
&Tanya Berger-Wolf
Ohio State University
<EMAIL_ADDRESS>

###### Abstract

Motivated by COVID-19 vaccine allocation, where vulnerable subpopulations are
simultaneously more impacted in terms of health and more disadvantaged in
terms of access to the vaccine, we formalize and study the problem of resource
allocation when there are inherent access differences that correlate with
advantage and disadvantage. We identify reducing resource disparity as a key
goal in this context and show its role as a proxy to more nuanced downstream
impacts. We develop a concrete access model that helps quantify how a given
allocation translates to resource flow for the advantaged vs. the
disadvantaged, based on the access gap between them. We then provide a
methodology for access-aware allocation. Intuitively, the resulting allocation
leverages more vaccines in locations with higher vulnerable populations to
mitigate the access gap and reduce overall disparity. Surprisingly, knowledge
of the access gap is often not needed to perform access-aware allocation. To
support this formalism, we provide empirical evidence for our access model and
show that access-aware allocation can significantly reduce resource disparity
and thus improve downstream outcomes. We demonstrate this at various scales,
including at county, state, national, and global levels.

##  1 Introduction

All common resources must be allocated in some way to members of a society.
During such an allocation, the goal should be to optimize the benefits of the
resources for that society. To be transformational, the way we allocate needs
to be not only superficially fair but also deeply just and willfully acting to
mitigate unfairness. We learned a great deal about what happens when we are
not intentional in our allocation efforts through the COVID-19 virus health
crisis. Yet, amidst the rise in monkeypox cases, we find ourselves in familiar
territory with shortages and allocation issues for the monkeypox virus
vaccines [LaFraniere et al.(2020), for Preparedness and (ASPR)(2022)]. For
most of the COVID-19 pandemic, government officials and healthcare workers
have been rapidly seeking ways to minimize loss caused by the COVID-19 virus.
In this search, they found the virus affects members of the vulnerable
population more severely; namely people with underlying conditions, members of
racial minority groups, those who lack personal transportation, those of lower
economic status, and older people [for Immunization et al.(2022)]. Vulnerable
people experience worse short and long term health side effects and higher
mortality rates [CDC(2022b)]. Additionally, the impact on cities and states
when members of the vulnerable populations get sick from COVID-19 leads to
higher hospitalization rates, lack of healthcare staff availability, and less
hospital space to care for other conditions. Though these realities were
recognized, many of the plans that were developed to minimize loss, including
vaccination, did not fully take this into consideration. Instead, when
vaccines began to be distributed, they were distributed according to age
(oldest) or career (most essential workers) [Jean-Jacques and Bauchner(2021)].
Additionally, points of distribution for vaccines were receiving proportional
amounts of vaccines [Cronk(2020)]. It is seemingly ideal to have such an
allocation, but there are several examples where this logic breaks. For
example, age is not the only good measure because we see Black Americans have
a shorter life expectancy than their White counterparts due to health
inequities [Kirby and Kaneda(2010)]. Thus, only focusing on age exacerbates
these issues for Black Americans.

Motivated by COVID-19 vaccine allocation, we formalize the problem of fair
resource allocation when there are inherent access differences that correlate
with advantage and disadvantage. We make and support three key premises:

  * (A)

The population is stratified into disadvantaged and advantaged subpopulations.

  * (B)

The disadvantaged subpopulation is more vulnerable to the lack of resources.

  * (C)

The disadvantaged subpopulation has obstacles in acquiring allocated
resources.

In Sec. 3, we describe the formal problem by closely following premises (A)
and (B). Our first contribution there is to highlight reducing resource
disparity between advantaged and disadvantaged subpopulations as a key goal of
resource allocation. Then, in Sec. 4, our second contribution is a concrete
access model, which we develop through premise (C). This allows us to
determine how resources would flow to the disadvantaged vs. advantaged
subpopulations, upon performing a given allocation. In Sec. 5, we show that
resource disparity can be effectively thought of as a proxy to more complex
impact disparities downstream. In Sec. 6, we combine the formal problem with
the access model to obtain our third contribution, an access-aware allocation
algorithm that reduces resource disparity while maintaining approximate
geographic proportionality. A key finding here is that access-aware allocation
can often be performed without explicit knowledge of the access gap between
the subpopulations. We acknowledge that there are many contributors to access
disparities other than disadvantage. That said, in Sec. 7.1 we give empirical
evidence in support of our access model. Then, in Sec. 7.2, we revisit the
COVID-19 vaccine allocation problem as a use case. Our fourth contribution is,
we show significant reduction in resource disparity and thus downstream impact
can be achieved with minimal deviation from proportionality. In Sec. 8, we
conclude the paper with discussion points. We start with a survey of related
work in Sec. 2.

##  2 Related Work

The importance of considering equity and justice when allocating resources has
not been lost on the scientific community, especially for the COVID-19 vaccine
distribution problem, the primary case study here. The role of access
differences and its interplay with vulnerability and disadvantage is often
alluded to, but not directly formalized as we do.

For example, in [Jean-Jacques and Bauchner(2021)], the authors note that those
65 and older account for 80% of deaths from COVID-19; however, they warn that
“if this is done without explicit attention to promoting health equity, it
will, once again, exacerbate major health disparities.” They attribute this to
the fact that: making a vaccination appointment requires substantial time,
reliable internet access and technology, ability to travel to a vaccination
site, and trust in the safety of the vaccine. Note that inability to do any of
these and the existence of pre-exisiting health conditions correlates with a
person being more vulnerable. Several solutions are offered in [Jean-Jacques
and Bauchner(2021)], including prioritizing hardest hit and lowest economic
ZIP Codes, which is very much in line with the access-aware allocation
solutions that come out of our explicit model. Other suggestions are: local
vaccine education, decreasing transportation barriers, and simplifying
registration procedures. Though we do not discuss these, we do acknowledge
that they could aid in grassroots efforts of vaccine administration.

In [Wedlund and Kvedar(2021)], the authors develop a tool to identify people
who will benefit most from receiving the vaccine at a specific time period.
They argue that those at most risk of contracting should be prioritized.
However, access differences are not made explicit, and this focus does not
necessarily address minimizing the harshest effects of contracting COVID-19.

In [Estiri et al.(2021)], data from electronic hospital records (EHRs) is used
to train a model which can determine the likelihood of a person to die from
contracting COVID-19. While we do not depend on health records to determine
whether a person would die from COVID-19, we do use the CDC’s full definition
of what it means to be vulnerable. The findings of [Estiri et al.(2021)] (see
also [Garg et al.(2020a)] for early variant findings) clearly show affect the
level of hospitalization and death risk. It is also worth noting that health
records for racial minorities, lower-income people, and those who are in more
rural areas tend to not be complete profiles due to access barriers [Norris
and Nissenson(2008)].

Many data-driven methods to guide vaccine allocation have also been proposed.
In [Davahli et al.(2021)], the authors cluster states with similarly reported
effective reproduction numbers111an epidemiological property that describes
the expected number of new cases due directly to one case to look for an
optimal distribution. Their focus does not include equity and is more on
preventing spread, which we do not address. By concentrating on resource
equity between vulnerable and non-vulnerable subpopulations, we focus on who
would benefit most from receiving enough vaccines.

In [Chen et al.(2020)], allocation under limited resources is addressed but
without considerations of equity, disparity, or access. There, the importance
of dynamic allocation is emphasized, which we do not address. In [Perakis et
al.(2021)], the authors consider overall COVID-19 impacts (death, job loss,
and changes in the economy). They build a linear optimization method to
predict testing allocation, based on predicting the influx of cases, over
different time periods to maximize detection of the virus amongst different
population groups (symptomatic, asymptomatic, and contact tracing). They
incorporate a basic geographic notion of equity, by allocating a minimum
number of resources at each location, but do not model access disparity. In a
different relevant line of work, resources are allocated to increase direct
(spatial) access. For example, in [Zhang et al.(2021)], the placement of
hospital beds is optimized using a model of spatial accessibility. This does
not involve an explicit relationship between vulnerability and resource
acquisition but is relevant nonetheless in that it centers on issues of access
by the disadvantaged and findings include resulting improvements in equity
also.

In the theoretical CS and OR communities, models of equity and allocation have
also been proposed. Recently, [Donahue and Kleinberg(2020)] studied resource
allocation across groups with varying demand distributions and showed
tradeoffs between utilization and fairness across groups. The varying demand
distributions could be interpreted as access differences, but it is important
to note that upon instantiation of the demand, there is no inherent difference
in the ease with which the resource is acquired. More importantly, in that
model, individual allocations can be made for each group, which may not
ideally describe shared public resource allocation, save for assistance models
that occur through an application process. Though we focus on access
differences in physical resource acquisition, we acknowledge the importance of
differences in access to information and their impact, e.g., as is done in
[Garg et al.(2020b)] for standardized testing. Of course, there are many
dimensions beyond access, such as dynamics and correlations, that further
complicate resource acquisition. For example, this was explored in depth by
[Manshadi et al.(2021)] for rationing medical supplies in the COVID-19
backdrop.

A key theme in the current paper is that the disadvantaged lack ease of access
to resources and, simultaneously, are disproportionally affected by the burden
of lack of resource [CDC(2021b), of Health and Services(2021)]. Sometimes they
can be targeted directly (e.g., mobile clinics, government-supplied PPE, etc).
Though these are helpful solutions, these efforts are expensive and resources
often run out before having had sufficient impact [Simmons-Duffin(2022)].
Policies that place more of these resources where they are needed, such as the
access-aware allocation methods proposed here, can boost resource acquisition
indirectly. Reaching vulnerable populations and mitigating the obstacles they
face are critical in making optimization and data-driven methods achieve their
true clinical potential [Zhou et al.(2021), Marwaha and Kvedar(2022)]. We
believe explicitly modeling access differences is an essential tool for
achieving this potential.

##  3 Problem Description

##### Allocation Task and Notation

We are tasked with allocating an amount $N_{\mbox{$\Sigma$}}$ of resources,
assumed to be arbitrarily divisible, across $k$ locations indexed by
$j\in[k]:=\\{1,\cdots,k\\}$. The decision variables are the amounts to be
allocated to each location $j$, denoted by $N_{j}$, or equivalently the
fraction of the whole that these represent, denoted by
$n_{j}:=N_{j}/N_{\mbox{$\Sigma$}}$. It is desirable to entirely allocate the
resources, so that $\sum_{j}N_{j}=N_{\mbox{$\Sigma$}}$ and $\sum_{j}n_{j}=1$.
We write $\mathbf{n}:=(n_{j})_{j\in[k]}$. The implicit target for this
allocation are the population sizes $P_{j}$ at each location. Using parallel
notation to $N_{\mbox{$\Sigma$}}$, let $P_{\mbox{$\Sigma$}}=\sum_{j}P_{j}$ be
the total population size and $p_{j}:=P_{j}/P_{\mbox{$\Sigma$}}$ be the
fraction of the population at location $j$, so that $\sum_{j}p_{j}=1$. We
write $\mathbf{p}:=(p_{j})_{j\in[k]}$. Resources are assumed to be limited,
$N_{\mbox{$\Sigma$}}<P_{\mbox{$\Sigma$}}$, and we parametrize this explicitly
as $\alpha:=N_{\mbox{$\Sigma$}}/P_{\mbox{$\Sigma$}}\in(0,1)$.

##### Disadvantage vs. Advantage

A central premise of this work, Premise (A), is that the population is
stratified by disadvantage and that this stratification varies across
locations. The _disadvantaged_ should be regarded as particularly vulnerable
in the absence of a resource. We adopt the simplest model that encapsulates
this, by using a parameter $\beta_{j}\in[0,1]$, which describes the fraction
at location $j$ who is a particularly disadvantaged subpopulation. While the
remainder $1-\beta_{j}$ of the population may also have some vulnerability, we
will describe them as _advantaged_. This stratification influences the
allocation task through a difference in how allocated resources are acquired
by each subpopulation. At any given location $j$, given $N_{j}$ resources, we
assume there exists an acquisition function $\rho$, that determines the
fraction of $N_{j}$ acquired by the disadvantaged, while the remainder
$1-\rho$ is acquired by the advantaged.

How can we characterize $\rho$? We can only do this through a clear model of
access that describes the process by which allocated resources are acquired by
each subpopulation and the difference between them. This is the motivation of
Sec. 4, which elaborates our proposed access model theoretically — this is
later supported empirically in Sec. 7.1. The resulting $\rho$ is found to
depend on the local quantities $N_{j}$, $P_{j}$, and $\beta_{j}$, as well as a
global parameter $\eta$ that describes the _access gap_ between disadvantaged
and advantaged subpopulations. In what follows, for conciseness, we refer to
the value that $\rho$ takes at location $j$ as $\rho_{j}$. For reference,
please refer to Table 1 for a glossary of the main notations.

$P_{\mbox{$\Sigma$}}$ | total population size |  | $\mathbf{n}$ | vector of all $n_{j}$
---|---|---|---|---
$N_{\mbox{$\Sigma$}}$ | total resources available |  | $\beta_{j}$ | percentage of disadvantaged at location $j$
$\alpha$ | resource availability, $N_{\mbox{$\Sigma$}}/P_{\mbox{$\Sigma$}}$ |  | $\rho_{j}$ | percentage of $N_{j}$ acquired by the disadvantaged
$k$ | number of locations |  | $\eta$ | access gap between disadvantaged and advantaged
$P_{j}$ | population size at location $j\in\\{1,\cdots,k\\}$ |  | $\mathsf{d}_{1}$ |  $\ell_{1}$ distance
$N_{j}$ | resources allocated to location $j$ |  | $\mathsf{d}_{\infty}$ | relative $\ell_{\infty}$ distance
$p_{j}$ | fraction of total population at location $j$, $P_{j}/P_{\mbox{$\Sigma$}}$ |  | $\varepsilon$ | tolerance in distance from proportionality
$n_{j}$ | fraction of total resources allocated to $j$, $N_{j}/N_{\mbox{$\Sigma$}}$ |  | $\mathbf{n}^{\star}_{\cdot}$ | optimal allocation when distance $\mathsf{d}_{\cdot}$ is used
$\mathbf{p}$ | vector of all $p_{j}$ |  | $\mathsf{RD}$ | resource disparity, Eq. (1)
Table 1: Glossary of main notations

##### Resource Rate Disparity

Upon an allocation $\mathbf{n}$, each subpopulation acquires part of the
$N_{\mbox{$\Sigma$}}$ resources. The disadvantaged acquire
$\sum_{j}\rho_{j}N_{j}=N_{\mbox{$\Sigma$}}\sum_{j}\rho_{j}n_{j}$, while the
advantaged acquire the rest
$\sum_{j}(1-\rho_{j})N_{j}=N_{\mbox{$\Sigma$}}\sum_{j}(1-\rho_{j})n_{j}$. As a
result, resource acquisition rate may vary across the two subpopulations,
which we quantify through _rate disparity_ :

$\mathsf{RD}=\frac{\sum_{j}(1-\rho_{j})N_{j}}{\sum_{j}(1-\beta_{j})P_{j}}-\frac{\sum_{j}\rho_{j}N_{j}}{\sum_{j}\beta_{j}P_{j}}=\alpha\frac{\sum_{j}(1-\rho_{j})n_{j}}{\sum_{j}(1-\beta_{j})p_{j}}-\alpha\frac{\sum_{j}\rho_{j}n_{j}}{\sum_{j}\beta_{j}p_{j}}.$
(1)

If the notion of disadvantage is true to its name, then $\mathsf{RD}$ will
typically be non-negative, as the resource acquisition rate will be higher
within the advantaged group. As we see in Sec. 7, this is indeed the case for
the vaccine allocation problem. Moreover, it is desirable not only to have
$\mathsf{RD}$ close to zero, but also to make it negative, as downstream
outcomes may improve with higher allocation rates among the disadvantaged. We
expand on this in Sec. 5. This codifies Premise (B) and is the reason why the
definition in Eq. (1) does not enforce positivity. That said, $|\mathsf{RD}|$
may also be handled with minimal modifications within the present methodology.

##### Proportional Allocation and Geographic Disparity

The baseline allocation is proportional to the population, that is
$n_{j}=p_{j}$, for all $j$. This is fundamentally a notion of geographic
equity that is an expected and desirable property of a potential allocation in
many policies, including some recent resource allocation work, e.g. in
COVID-19 testing allocation [Perakis et al.(2021)]. To allow ourselves to
remain close to this property, we relax the strict equality of $n_{j}=p_{j}$
for all $j$ to a distance on the $k$-simplex between $\mathbf{n}$ and
$\mathbf{p}$. One can think of this as a form of headroom for social
intervention, up to a limit that could be potentially determined by law or by
agreement. We primarily consider the absolute $\ell_{1}$ and relative
$\ell_{\infty}$ distances:

$\mathsf{d}_{1}(\mathbf{n},\mathbf{p})=\sum_{j}|n_{j}-p_{j}|\qquad\qquad\mathsf{d}_{\infty}(\mathbf{n},\mathbf{p})=\max_{j}\left|\frac{n_{j}}{p_{j}}-1\right|$
(2)

Note that $\mathsf{d}_{\infty}$ is the more stringent of the two distances as
$\mathsf{d}_{1}=\sum_{j}p_{j}\left|\frac{n_{j}}{p_{j}}-1\right|\leq\mathsf{d}_{\infty}$.
(The methodology could also handle absolute $\ell_{\infty}$ and relative
$\ell_{1}$ distances, given by $\max_{j}|n_{j}-p_{j}|$ and
$\sum_{j}\left|\frac{n_{j}}{p_{j}}-1\right|$ respectively, and weighted
variants.)

##### Allocation Objective

It is worth remarking that if $\rho_{j}=\beta_{j}$, that is if disadvantaged
and advantaged subpopulations get their proportional shares of the allocated
resources at each location, then proportional allocation yields zero rate
disparity, $\mathsf{RD}=0$. Intuitively, this is a scenario where there is no
access gap between the disadvantaged and advantaged. When there is a gap, we
expect proportional allocation to not yield the best rate disparity. How can
we allocate resources to improve disparity across subpopulations while
maintaining a reasonable geographic parity? This trade-off leads to the
central objective of this paper, which we can express at a high level as:

$\begin{array}[]{lcrcl}\mathbf{n}^{\star}_{\cdot}&=&\operatorname*{arg\,min}&&\mathsf{RD}(\mathbf{n},\mathbf{p},N_{\mbox{$\Sigma$}},P_{\mbox{$\Sigma$}},\beta,\eta)\\\
&&\textrm{subject
to}&&\mathsf{d}_{\cdot}(\mathbf{n},\mathbf{p})\leq\varepsilon\\\
&&&&\alpha\mathbf{n}\leq\mathbf{p},\end{array}$ (3)

where the last inequality is a _no-waste constraint_ , which prohibits the
allocation of more resources than people at an location $j$: $N_{j}\leq
P_{j}$. The $\cdot$ can be either $1$ or $\infty$, indicating the distance
used.

##  4 Modeling Access

In this section, our scope is a single location $j$. As such, for clarity, we
drop all $j$ subscripts.

###  4.1 Resource Acquisition

To model access differences across disadvantaged and advantaged populations,
we need to model how resources are acquired upon allocation. For this, we
adopt a simple yet effective model: _Poisson acquisition until saturation or
exhaustion_. Each subpopulation acquires a unit of resource according to an
independent Poisson arrival process, which stops when one of two things
happen: (1) an amount of resource equal to the subpopulation size is acquired
by the subpopulation (saturation) or (2) a total amount of resource equal to
the allocated size is acquired (exhaustion). This model allows us to account
for access differences via the relative rate of each Poisson process. In what
follows, we focus on a specific location $j$ with population $P$ where $\beta$
fraction of the population is disadvantaged and $N$ resources have been
already allocated. Our goals are to specify the acquisition model, describe
access gaps, and derive the shape of the acquisition function $\rho$. We
assume that resources and subpopulation sizes are integers.

##### Base Model

Let $\lambda^{-}$ and $\lambda^{+}$ be the rates of acquisition of the
disadvantaged and advantaged respectively. When there are no access gaps,
these rates reflect only the relative sizes of the subpopulations. We set the
expected time for a single resource acquisition per person in this case as our
unit of time, and treat the two populations as a split. This gives us the base
rates of $\lambda^{-}=\beta P$ and $\lambda^{+}=(1-\beta)P$.

##### Access Gap

Premise (C) states that the disadvantaged face obstacles in acquiring
resources. We can model this as a _slowdown_ relative to the base rate. Let
$\eta$ be the (relative) access gap between the disadvantaged and advantaged
populations, where $\eta=1$ signifies no gap and $\eta=0$ is the limit of
infinite gap. Under this model, the modified rate for the disadvantaged
becomes $\lambda^{-}=\eta\beta P$ while the rate of the advantaged remains
unchanged $\lambda^{+}=(1-\beta)P$.

###  4.2 Naïve Acquisition Function

With this model, we can now concretely define the acquisition function $\rho$
to be the expected fraction of the total allocated $N$ resources that will be
acquired by the disadvantaged. To develop intuition, let us start with a naïve
characterization that ignores saturation. This can be thought of as the limit
of very scarce resources ($\alpha\rightarrow 0$), so much so that no
subpopulation in any location can be saturated. In this case, subpopulations
acquire resources until exhaustion. If $\ddot{U}_{t}$ represents the
disadvantaged process and $\ddot{V}_{t}$ represents the advantaged process,
then the exhaustion time can be defined as
$\ddot{S}=\min\\{t~{}:~{}\ddot{U}_{t}+\ddot{V}_{t}\geq N\\}$. With this, the
acquisition function can be defined as
$\ddot{\rho}:=\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}}\right]/N$.
The $\ddot{\phantom{\rho}}$ notation is used to evoke _“naïve”_ , and not
second derivative. We have (the proof is given in the supplementary material):

###### Proposition 4.1

If saturation is ignored, the naïve acquisition function $\ddot{\rho}$ depends
only on $\beta$ and $\eta$, and is of the form:

$\ddot{\rho}=\frac{\lambda^{-}}{\lambda^{-}+\lambda^{+}}=\frac{\eta\beta}{\eta\beta+1-\beta}$
(4)

###  4.3 Exact Acquisition Function

The model above is unrealistic, because it has a positive probability for
$\ddot{U}_{\ddot{S}}>\beta P$ or $\ddot{V}_{\ddot{S}}>(1-\beta)P$. As the
availability of the resource increases, i.e., for larger values of $\alpha$,
this probability is non-negligible as subpopulations may saturate. We thus
need an exact characterization of $\rho$ that reflects this. Let the
disadvantaged and advantaged processes in this case be $U_{t}$ and $V_{t}$
respectively. These are now _stopped_ versions of the Poisson processes,
$\ddot{U}_{t}$ and $\ddot{V}_{t}$, of the naïve case. More precisely, if we
define saturation times of these processes as
$T_{U}=\min\\{t~{}:~{}\ddot{U}_{t}\geq\beta P\\}$ and
$T_{V}=\min\\{t~{}:~{}\ddot{V}_{t}\geq(1-\beta)P\\}$ respectively, i.e., the
times when they hit their respective population sizes, then

$U_{t}=\left\\{\begin{array}[]{lcl}\ddot{U}_{t}&;&\textrm{if }t\leq T_{U}\\\
\beta P\phantom{(1-)}&;&\textrm{if }t>T_{U}\end{array}\right.\qquad\qquad
V_{t}=\left\\{\begin{array}[]{lcl}\ddot{V}_{t}&;&\textrm{if }t\leq T_{V}\\\
(1-\beta)P&;&\textrm{if }t>T_{V}\end{array}\right.$

The exhaustion time is now modified to be: $S=\min\\{t~{}:~{}U_{t}+V_{t}\geq
N\\}$, which is well-defined assuming that no wasteful allocations are
allowed, so that $N\leq P$ always. The exact characterization of the
acquisition function is now
$\rho:=\operatorname*{\mathbf{E}}\left[{U_{S}}\right]/N$. We have (the proof
is given in the supplementary material):

###### Proposition 4.2

When both saturation and exhaustion occur, the exact acquisition function
$\rho$ depends not only on $\beta$ and $\eta$, but also on $N$ and $P$. It can
be written in terms of the naïve acquisition function $\ddot{\rho}$ of Eq. (4)
as follows:

$\displaystyle\rho=\ddot{\rho}\quad$ $\displaystyle-$
$\displaystyle\ddot{\rho}\overline{F}\left(\beta
P-1;N-1,\ddot{\rho}\right)+\frac{\beta P}{N}\overline{F}\left(\beta
P;N,\ddot{\rho}\right)$ (5) $\displaystyle+$
$\displaystyle(1-\ddot{\rho})\overline{F}\left((1-\beta)P-1;N-1,1-\ddot{\rho}\right)-\frac{(1-\beta)P}{N}\overline{F}\left((1-\beta)P,1-\ddot{\rho}\right)$

####  4.3.1 Tractable Approximation

The acquisition function directly affects the allocation objective expressed
in Eq. 3 through the rate disparity, as seen in Eq. 1. While the
characterization of $\rho$ in Prop. 4.2 is exact, it is not directly amenable
to reduction to a convex optimization problem. Fortunately, we can develop a
tractable approximation to $\rho$ that also helps develop intuition about its
behavior. One can think of the acquisition process in two stages: (1) both
subpopulations acquire simultaneously, (2) one subpopulation saturates, while
the other continues to acquire.

To quantify these stages, first note that with unlimited resources, the
advantaged saturation time is $t=1$, on average. This is because the expected
$m^{\textrm{th}}$ arrival time of $\ddot{V}$ is $m/\lambda^{+}$, with
$m=\lambda^{+}=(1-\beta)P$. Similarly, the disadvantaged saturation time is
$t=1/\eta$, on average. As a result, it is highly unlikely for the
disadvantaged to saturate prior to the advantaged. This sets the stage for the
approximation, by taking these times to be equal to their expectations. The
strength of such an approximation stems from the fact that for large
parameters, a Poisson random variable concentrates very sharply around its
mean. The approximate model is:

  * •

Before $t=1$, $\operatorname*{\mathbf{E}}\left[{U_{t}}\right]$ and
$\operatorname*{\mathbf{E}}\left[{V_{t}}\right]$ grow linearly at the rate of
$\eta\beta P$ and $(1-\beta)P$ respectively. This stage thus reaches a maximal
acquisition of $(\eta\beta+1-\beta)P$.

  * •

After $t=1$, $\operatorname*{\mathbf{E}}\left[{V_{t}}\right]$ saturates at
$(1-\beta)P$, while $\operatorname*{\mathbf{E}}\left[{U_{t}}\right]$ continues
to grow linearly at the rate of $\eta\beta P$.

  * •

Exhaustion happens when
$\operatorname*{\mathbf{E}}\left[{U_{t}}\right]+\operatorname*{\mathbf{E}}\left[{V_{t}}\right]=N$.

In this model, if $N<(\eta\beta+1-\beta)P$ we have
$S\approx(N/P)/(\eta\beta+1-\beta)$ and thus
$\rho=\operatorname*{\mathbf{E}}\left[{U_{S}}\right]/N\approx(\eta\beta
PS)/N=\ddot{\rho}$. In other words, if exhaustion occurs during stage (1), the
naïve model’s acquisition function applies. Otherwise, the acquisition
function is fully determined by the size of the saturated subpopulation, since
all that is left for the other is the remainder of the resources. More
precisely, in that case we have $\rho\approx[N-(1-\beta)P]/N$. We can write:

$\rho\approx\left\\{\begin{array}[]{lcl}\frac{\eta\beta}{\eta\beta+1-\beta}&;&\textrm{if
}N<\frac{1}{\alpha}P(\eta\beta+1-\beta)\\\
1-\frac{(1-\beta)P}{N}&;&\textrm{otherwise}.\end{array}\right.$

Further, using the fact that $N/P=\alpha n/p$ and observing that
$\frac{\eta\beta}{\eta\beta+1-\beta}>1-\frac{(1-\beta)p}{\alpha n}$ is
equivalent to $n<\frac{1}{\alpha}p(\eta\beta+1-\beta)$, we can write the
approximate acquisition function as:

$\tilde{\rho}:=\max\left\\{\ddot{\rho},1-\frac{(1-\beta)p}{\alpha n}\right\\}$
(6)

Eq. (6) clearly demarcates the two possible behaviors of $\tilde{\rho}$: the
first dominated by exhaustion ($\ddot{\rho}$) and the second dominated by
saturation. Note that the absolute sizes of $N$ and $P$ do not affect this
approximation, only their relative size $N/P$ matters. In fact, we can think
of this approximation as the limit of $\rho$ for large $N$ and $P$. Fig. 1(a)
illustrates this visually.

(a) Acquisition Behavior  (b) Robustness to $\eta$

Figure 1: Visualizations — (1(a)) Acquisition behavior versus time (with
$\eta=0.5$, $\beta=0.5$). Blue and red plots are resources acquired by all and
by the advantaged, respectively for $P=10,100,1000,10000$ (faintest to
darkest), for the maximal expected time. For a specific $N$, resources are
exhausted sooner (the orange example). For each case, the average resource is
the solid line. The error bars represent the $5$ to $95$ percentile range. The
limits (dotted red and blue) correspond to the approximation model in Section
4.3.1. These are indistinguishable from the average $P=1000$ case and from the
entire confidence range for $P=10000$. The acquisition function is always the
expected fraction of resources acquired by the disadvantaged upon exhaustion.
This can have two distinct behaviors, based on whether only exhaustion occurs
or both saturation and exhaustion occur. (1(b)) Constraint set of the example
in Sec. 6.3 (interior of the blue region). Direction of the vector $c_{j}$ of
$\mathsf{RD}$ in Eq. (7), as $\eta$ varies (in pink). The narrow fan implies
that there exists a single discrepancy-optimizing allocation that is optimal
for all $\eta$, and thus robust to lack of specification of $\eta$.

##  5 Resource Disparity and Downstream Impact

Reducing resource disparity may be thought of as a goal in its own right, as a
form of virtue or deontological ethics. However, in the present context one
may argue that this goes hand in hand with a more consequentialist notion.
Assume that, in the absence of the resource, the advantaged are likely to
incur an adverse outcome with probability $x$ and that the disadvantage are
more likely to incur such an outcome, say with probability $(1+\delta)x$.
These outcomes represent an abstraction of potential downstream impact. In the
presence of the resource, say the adverse outcome’s likelihood drops by a
factor $q\in(0,1)$ for the advantaged and by a factor of $q^{\prime}$ for the
disadvantaged. Thus, with the resource, the advantaged and disadvantaged have
a probability $qx$ and $q^{\prime}(1+\delta)x$ of adverse outcomes
respectively. We can then make the following observation.

###### Proposition 5.1

In the adverse outcome scenario described above, the expected number of
adverse outcomes is a linear function of $\mathsf{RD}$. Furthermore, the slope
of the relationship is positive whenever
$\delta>\frac{q^{\prime}-q}{1-q^{\prime}}$.

One can see this by noting that $\mathsf{RD}$ in Eq. (1) is a linear function
of the number of vaccines that go to the disadvantaged and that the same is
also true for the expected number of adverse outcomes. The condition of Prop.
5.1 then follows from a direct calculation. We omit the proof, as it is
straightforward. Thus, if the disadvantage is large enough at first (large
enough $\delta$) and if the resource allows a decent improvement for the
disadvantaged ($q^{\prime}$ not much larger than $q$) then reducing
$\mathsf{RD}$ also reduces the expected number of adverse outcomes. The first
facet is intuitive. To see why the second is also necessary, imagine that the
resource did nothing to the disadvantaged, i.e., $q^{\prime}=1$. In that case,
it doesn’t help to channel more resources to them, and it is in fact more
advantageous (for downstream impact) to channel more to the advantaged.
However, these are arguably satisfied in practice, especially if the resource
is even more effective for the disadvantaged ($q^{\prime}<q$).

##### Example

Using the CDC estimates [CDC(2022a)] in the period 10/2021 to 2/2022, the
average rate of COVID19 hospitalization per 100,000 people in the unvaccinated
(advantaged) 18-64 age group is $99$ and the (disadvantage) 65+ group is
$415$. For the fully vaccinated, the respective rates are $18$ and $78$. This
gives $q=18/99\approx 0.182$ and $q^{\prime}=78/415\approx 0.188$,
corresponding to a minuscule threshold of $0.007$ that $\delta=415/99-1\approx
3.19$ readily exceeds. If one considers the 50+ group as advantaged instead,
then one gets $q^{\prime}<q$ and the condition holds trivially. $\square$

##  6 Access-Aware Allocation

We are now in position to instantiate concrete version of Eq. (3). To develop
intuition, we first do this for the naïve acquisition model and show that it
reduces to a linear program. We then show how the resulting solution can be
used iteratively, to heuristically solve allocation under the approximate
access model. As for the exact access model, one could in principle use
nonlinear optimization with Eq. 5 directly, but we do not address this
currently. Before proceeding, let us rewrite resource disparity in Eq. (1) as
follows:

$\mathsf{RD}=\alpha\frac{\sum_{j}(1-\rho_{j})n_{j}}{\sum_{j}(1-\beta_{j})p_{j}}-\alpha\frac{\sum_{j}\rho_{j}n_{j}}{\sum_{j}\beta_{j}p_{j}}=\sum_{j}\underbrace{\alpha\left(\frac{1-\rho_{j}}{\sum_{j^{\prime}}(1-\beta_{j^{\prime}})p_{j^{\prime}}}-\frac{\rho_{j}}{\sum_{j^{\prime}}\beta_{j^{\prime}}p_{j^{\prime}}}\right)}_{\textstyle
c_{j}}n_{j}.$ (7)

###  6.1 Naïve Allocation

In the naïve model, resource disparity becomes a linear function of
$\mathbf{n}$, because plugging in $\ddot{\rho}$ instead of $\rho$ in the
expression for $c_{j}$ in Eq. (7) gives:

$c_{j}=\frac{\alpha}{\eta\beta_{j}+1-\beta_{j}}\left(\frac{1-\beta_{j}}{\sum_{j^{\prime}}(1-\beta_{j^{\prime}})p_{j^{\prime}}}-\frac{\eta\beta}{\sum_{j^{\prime}}\beta_{j^{\prime}}p_{j^{\prime}}}\right),$

which does not depend on $\mathbf{n}$. Furthermore, the no-waste constraint is
already linear and the constraints with both
$\mathsf{d}_{1}(\mathbf{n},\mathbf{p})$ and
$\mathsf{d}_{\infty}(\mathbf{n},\mathbf{p})$ of Eq. (2) can be expanded into a
set of linear constraints, via slack variables. As such, in this case the
allocation problem (3) can be written as a linear program. Here are the
instances that result for each distance:

$\begin{array}[]{lcrclcll}\mathbf{n}^{\star}_{{\color[rgb]{1,0,0}1}}&=&\\!\\!\\!\\!\\!\\!\operatorname*{arg\,min}&&\sum_{j}c_{j}n_{j}&&&\\\
&&\\!\\!\\!\\!\\!\\!\textrm{subject
to}&&n_{j},s_{j}&\geq&0&\textrm{\hbox{\multirowsetup$\left.\rule{0.0pt}{28.45274pt}\right\\}\,\forall
j$}}\\\ &&&&n_{j}-s_{j}&\leq&p_{j}&\\\ &&&&-n_{j}-s_{j}&\leq&-p_{j}&\\\
&&&&\alpha n_{j}&\leq&p_{j}&\\\
&&&&{\color[rgb]{1,0,0}\sum_{j}s_{j}}&{\color[rgb]{1,0,0}\leq}&{\color[rgb]{1,0,0}\varepsilon}&\\\
&&&&\sum_{j}n_{j}&=&1&\end{array}\begin{array}[]{lcrclcll}\textrm{or}\quad\mathbf{n}^{\star}_{{\color[rgb]{1,0,0}\infty}}&=&\\!\\!\\!\\!\\!\\!\operatorname*{arg\,min}&&\sum_{j}c_{j}n_{j}&&&\\\
&&\\!\\!\\!\\!\\!\\!\textrm{subject
to}&&n_{j},s_{j}&\geq&0&\textrm{\hbox{\multirowsetup$\left.\rule{0.0pt}{31.29802pt}\right\\}\,\forall
j$}}\\\ &&&&n_{j}-s_{j}&\leq&p_{j}&\\\ &&&&-n_{j}-s_{j}&\leq&-p_{j}&\\\
&&&&\alpha n_{j}&\leq&p_{j}&\\\
&&&&{\color[rgb]{1,0,0}s_{j}}&{\color[rgb]{1,0,0}\leq}&{\color[rgb]{1,0,0}\varepsilon
p_{j}}&\\\ &&&&\sum_{j}n_{j}&=&1&.\end{array}$ (8)

###  6.2 Better Allocation

Eq. (8) demonstrates concretely how the access model influences the allocation
decision. Access determines resource disparity, which in turn can be traded
off with geographic disparity. While conveniently in linear program form,
these do not capture the phenomenon of saturation that will occur whenever
sufficient resources are available at a given location. The exact acquisition
function in Eq. (5) does address this, but is highly nonlinear. Let us
consider instead the approximate acquisition function in Eq. (6). The
resulting $\mathsf{RD}$ is still nonlinear, because $\tilde{\rho}_{j}$ depends
on $n_{j}$ and $p_{j}$, and thus so will $c_{j}$. Moreover, at each location,
Eq. (7) multiplies Eq. (6) by $-n$, yielding the minimum of two linear
functions, which is thus concave. Since $\mathsf{RD}$ is the sum of these
functions, the result is a concave minimization problem, which can generally
be non-tractable. That said, this informs us that the solution remains on the
boundary of the constraint set. In addition, the two distinct behaviors of
$\tilde{\rho}$ suggest that one could start with a naïve model and then
“correct” the model and repeat. This gives the following iterative heuristic:

  1. 1.

Solve the naïve allocation problem (8), assuming that exhaustion does not
occur.

  2. 2.

Use the current allocation to detect exhaustion locations $j$ and correct
$\tilde{\rho}$ according to (6) at these locations.

  3. 3.

Re-solve the naïve allocation problem (8), but now with the $c_{j}$ weights in
(7) using the updated $\tilde{\rho}_{j}$.

  4. 4.

Repeat Steps (2) and (3) until convergence.

This is a heuristic both because $\tilde{\rho}$ is an approximation, even if
an accurate one, and also because a priori the iterations could either not
converge or converge to a suboptimal stable point. Nevertheless, in practice
the iterations converge extremely quickly and to good solutions, i.e.,
solutions that significantly reduce resource disparity, as we illustrate
experimentally in Sec. 7. In the supplementary material, we empirically verify
the quality of these using exhaustive search.

###  6.3 Robustness to the Access Gap

To consider the allocation scheme of Sec. 6.2 as a viable alternative to the
baseline proportional allocation, one should be able to readily implement it.
Apart from design choices, such as the type of allowable deviation from
proportionality, all variables are known but one: the access gap $\eta$. How
should this be addressed? One approach is to fit the model to historical data,
e.g., using a similar approach to the one we use to provide empirical evidence
for the access model in Sec. 7.1. While viable, this requires careful
elimination of confounding variables. Surprisingly, we experimentally observe
that $n_{1}^{\star}$ and $n_{\infty}^{\star}$ are _robust_ to the choice of
$\eta$, in that the solution remains stable for a large range of values and in
some cases, for _all_ $\eta$. We give here an intuitive understanding of this
robustness, through a simple example. In the supplementary material, we
suggest more direct means to address not knowing $\eta$.

##### Example

Consider a case with only three locations, $j=1,2,3$. Let the population be
evenly distributed,
$\mathbf{p}=\left(\tfrac{1}{3},\tfrac{1}{3},\tfrac{1}{3}\right)$. Say
locations have varying proportions of disadvantaged: low $\beta_{1}=0.2$,
medium $\beta_{2}=0.5$, and high $\beta_{3}=0.8$. Assume resources are limited
to $70\%$ of the population, $\alpha=0.7$. We are willing to deviate our
allocation as much as $\varepsilon=0.4$ away from $\mathbf{n}$ in $\ell_{1}$,
to reduce the resource disparity $\mathsf{RD}$ between the disadvantaged and
advantaged. Fig. 1(b) displays the constraint set centered around the interior
point $\mathbf{p}$. The fan shape represents the direction of the objective
function vector $c_{j}$ in $\mathsf{RD}$ of Eq. (7) at the optimal allocation,
as $\eta$ varies. Note that $\eta$ does not affect the constraint set. Even if
$\mathsf{RD}$ depends nonlinearly on $\mathbf{n}$, this informs us that the
optimal allocation is indeed at the upper left corner
$n^{\star}_{1}=\left(\tfrac{2}{15},\tfrac{41}{105},\tfrac{10}{21}\right)$ of
the constraint polytope, since otherwise an infinitesimal gradient descent
would move away. This implies that the solution is completely unaffected by
$\eta$! Interestingly, this phenomenon also occurs with most of the real data
cases that we cover in Sec. 7.2. Pragmatically, this means that we have an
alternative to proportional allocation that _does not require knowledge of the
access gaps_ and yet results in a much more equitable distribution of
resources. Note that the resulting allocation is intuitive: it boosts the
highly disadvantaged location’s allocation, reduces the low’s allocation, and
keeps the medium’s roughly the same. $\square$

##  7 Empirical Validation

###  7.1 Evidence for the Access Model

To determine whether the proposed model of access has merit in practice, we
look at how COVID-19 vaccine acquisition varies across locations. We cannot
directly observe the number of vaccines acquired by the advantaged. We can,
however, observe the overall vaccination rate. According to the the naïve
model, i.e., assuming no saturation for simplicity, at any time $t$ the
overall vaccination rate is
$\operatorname*{\mathbf{E}}\left[{\ddot{U}_{t}+\ddot{V}_{t}}\right]/(Pt)=(\eta\beta+1-\beta)=-(1-\eta)\beta+1$.
In what follows, we empirically show that we indeed observe this linear
dependence on $\beta$, despite the potential presence of confounding
variables.

##### Data

For the rates of COVID-19 vaccines, we use publicly available timeseries data
from the CDC at the county level [CDC(2021a)]. This data is provided to the
CDC by state or territory health departments by healthcare providers. A few
counties have no reported vaccinations, we discard these. The features we
captured were the dates (Date), county name (Recip_County), state name
(Recip_State), number of people who received 1 dose of a COVID-19 vaccine
(Administered_Dose1_Recip), and their percentage at the county level
(Administered_Dose1_Pop_Pct). This data reflects recipients over the age of 5.
To characterize each county’s overall vulnerability, we use publicly available
data from Surgo Ventures’ Precision for Community Vulnerability to COVID-19:
Explore the U.S. Data Tool [Ventures(2021)]. They look at many of the factors
the CDC uses to judge vulnerability for COVID-19 such as socioeconomic status,
household composition, racial status, existence of pre-exiting conditions, and
accessibility to healthcare and transportation. They use these factors to
calculate overall vulnerability at the county-level. From Surgo, we captured
county name, state name, and vulnerability at $(\textsf{very
high},\textsf{high},\textsf{moderate},\textsf{low},\textsf{very low})$ levels.
To explore the applicability of our observations at every scale, we also look
at the whole world, using Gallup numbers for vulnerability [Esipova et
al.(2020)] and vaccination numbers from One World in Data [Mathieu et
al.(2021)]. We include only a subset of $142$ countries that have clean
representation across data sets.

##### Experiments

For each county in California, Illinois, Ohio, and Pennsylvania, we look at
the percentage of the population that has acquired at least the first-dose of
the COVID-19 vaccine on December 30, 2021. We do the same at the global level,
per country. Let’s call these numbers $y_{j}$. At the state level, we consider
the proportion of the vulnerable at each county to be
$\beta_{j}=\textsf{moderate}_{j}+\textsf{high}_{j}+0.5\times\textsf{very
high}$, where the very high vulnerability range is slightly weighted down
because it is often noisy and incomplete. For each of the 4 states, Fig. 2(a)
shows a scatter plot of $y$ (the bubbles have areas proportional to population
size), along with its soft nearest-neighbor interpolation (with $\lambda=20$):
$\hat{y}(\beta)=\frac{\sum_{j}y_{j}\exp(-\lambda|\beta-\beta_{j}|)}{\sum_{j}\exp(-\lambda|\beta-\beta_{j}|)}.$
In the global case, we set
$\beta_{j}=\textsf{moderate}_{j}+\textsf{high}_{j}$, the 2 categories of
Gallup, and illustrate the parallel results in Fig. 2(b).

##### Results

We observe that, despite noise and confounding variables (e.g., differences
between urban/suburban/rural regions, the deliberately ignored saturation
which affects the low-$\beta$ range, noise and outliers, etc.), the general
trend of $y_{j}$ versus beta is not only monotonically decreasing, but is
indeed roughly linear. In particular, we do see clear segments of linear
behavior for primary clusters of counties. We conclude that, though highly
idealized, the presented access model has the potential to capture a realistic
correlation between the disadvantaged’s lack of resource and access obstacles.
Note also the common behavior across states, highlighted with the red dotted
line in Fig. 2(a). We conjecture that this may be formalized as a methodology
to learn $\eta$ from the data. More strikingly, the same behavior extends all
the way to the global scale, showing the ubiquity of the phenomenon
anticipated by our model.

(a) U.S. States (b) Global

Figure 2: Vaccination rate vs. vulnerability — (2(a)) Each bubble represents a
county, with area proportional to its population. The $x$-axis is the
estimated percentage of vulnerable in the county ($\beta$). The $y$-axis is
the percentage of the overall population with their first COVID-19 vaccination
dose as of 12/30/2021. The orange curve is a soft nearest-neighbor
interpolation $\hat{y}(\beta)$ of the relationship between $\beta$ and $y$.
The dotted red is a visual guide illustrating the common dominant behavior
across states. (2(b)) Global manifestation of the same phenomenon as in Fig.
2(a) — A roughly linear inverse relationship between vulnerability numbers and
vaccination rates across countries.

###  7.2 Behavior and Outcomes of Access-Aware Allocation

We now explore, numerically, the behavior and outcomes to expect when
employing the proposed access-aware allocation, in comparison to proportional
allocation. The latter has a simple characterization: we always have
$\frac{n_{j}}{p_{j}}=\alpha$. To assess how access-aware allocation deviates
from proportional allocation, we visualize the behavior of
$\frac{n_{j}}{p_{j}}$ in this case, for locations with varying $\beta_{j}$. In
order to demonstrate the possible gains achieved versus proportional
allocation, we simultaneously visualize the latter’s discrepancy $\mathsf{RD}$
along with the discrepancy achieved by access-aware allocation.

##### Data

In addition to the county-level CDC data, we also used data privately
communicated by the Ohio Health Department, including tract-level maps with
information about the number of vulnerable people in each tract, for 22
counties in Ohio[of Health(2020)]. The criterion for vulnerability in this
case differs from Surgo’s, but is based on similar metrics. Using tract-level
data from the Census Bureau [Bureau(2020)], we could then determine the
population size at each tract, and thus the percentage vulnerable ($\beta$) in
each. This allows us to explore the applicability of the present methodology
at such finer granularity. We also explore a much coarser granularity, with
allocation of vaccines across all $50$ states in the US. The data for this is
an aggregation of that obtained from CDC and Surgo, as described in Sec. 7.1.
The vulnerable percentage in this section is taken to be everyone categorized
as very high, high, or moderate.

##### Experiments

We fix the value of $\varepsilon$ to $0.1$ throughout. The influence of
$\varepsilon$ on the allocation is apparent nonetheless, and we comment on it
below. We look at the states of California, Illinois, Georgia, Ohio, and
Pennsylvania (locations = counties), in addition to Franklin County, OH
(locations = Census tracts), and the US (locations = states). We consider
$\alpha=0.1$ (low), $0.5$ (medium), and $0.9$ (high) levels of resource
availability. We vary $\eta$ from $0$ to $1$. In each scenario, we solve for
$n_{1}^{\star}$ and $n_{2}^{\star}$ using the iterative approach in Sec 7.2,
by solving Eq. (8) and alternating with updates of $\rho$.

##### Results — Behavior

Fig. 3(a) illustrates the behavior of the $n_{1}^{\star}$ allocation through
the ratio $\frac{n_{j}}{p_{j}}$. The dotted line corresponds to $\alpha$, the
proportional allocation case. We observe that in all cases the access-aware
allocation has roughly the same form: no allocation $\frac{n}{p}=0$ below a
lower threshold, full allocation $\frac{n}{p}=1$ above an upper threshold, and
proportional allocation $\frac{n}{p}=\alpha$ in between. The value of
$\varepsilon$ influences how close the upper and lower thresholds are for
fixed $\alpha$. This is a harsh policy and is due to the lax nature of
$\mathsf{d}_{1}$ that allows such deviation from proportionality. It is worth
noting that this is not always the form of $n_{1}^{\star}$, as we have seen in
the example of Sec. 6.3.

Fig. 3(b) illustrates the behavior of the $n_{\infty}^{\star}$ allocation,
which is much simpler: slightly less allocation below a threshold
$\frac{n}{p}<\alpha$ and slightly more allocation above a threshold
$\frac{n}{p}>\alpha$. The value of $\varepsilon$ determines the amount of
change from $\alpha$. It is also interesting to note that in this case, unlike
the thresholds of $n_{1}^{\star}$, this threshold does not depend on $\alpha$.
This is a lax policy, giving a much gentler deviation from proportionality,
due to the harsh nature of $\mathsf{d}_{\infty}$.

Lastly, we note that we do not display multiple policies for each $\eta$,
because under each scenario the policy is unaffected by the choice of $\eta$,
thus supporting the claims of robustness in Sec. 6.3.

(a) Behavior of $n^{\star}_{1}$ (b) Behavior of $n^{\star}_{\infty}$

Figure 3: Behavior of Allocations — Each point is a location $j$. The $x$-axis
is the percentage vulnerable $\beta$ at location $j$ and the $y$-axis is the
ratio of allocation to its population $\frac{n}{p}$. The allocation for
$\alpha=0.1$, $\alpha=0.5$, and $\alpha=0.9$ are also indicated with dotted
lines. Main observations — (3(a)) $n^{\star}_{1}$ specifies two thresholds,
does not allocate below the lower, fully allocates above the higher, and
maintains $\alpha$ in between. (3(b)) $n^{\star}_{\infty}$ fixes a threshold
and alters the proportional allocation slightly lower below it and slightly
higher above it.

##### Results — Outcomes

Fig. 4(a) and Fig. 4(b) illustrate the improvement in resource disparity that
$n_{1}^{\star}$ and $n_{\infty}^{\star}$ afford. Each constraint results in
specific characteristics. As we could expect from their harsh and lax
respective behaviors, $n_{1}^{\star}$ can deliver a more marked improvement to
resource disparity than $n_{\infty}^{\star}$ can.

More interestingly, under $n_{1}^{\star}$, access-awareness is generally
(though not always) more impactful at higher access gaps (small $\eta$) and
less so at lower gaps (large $\eta$). There isn’t such a monotonic
relationship for $n_{\infty}^{\star}$, with $\mathsf{RD}$ often peaking in
$\eta$ in that case. This seems to suggest that when too much deviation from
proportionality isn’t desirable (so one would best use $\mathsf{d}_{\infty}$),
then allocation, to be effective, needs to be compounded with enough social
effort to bring the disadvantaged above a certain level of access beyond which
it is able to mitigate any remaining difference. However, if more bold
redistribution is accepted (thus one could use $\mathsf{d}_{1}$) then
allocation on its own can go a long way to mitigate disparity. Thus this work
reveals an interesting tradeoff. To borrow some political jargon, if one is
“conservative” in resource allocation, they ought to be more “progressive” in
providing social safety nets. Conversely, if one is “socialist” in resource
allocation, they can afford to be less regulated and more “libertarian” when
it comes to society.

Both instances do however share a common yet counterintuitive pattern, in that
the disparity reduction is relatively higher when resources are more available
($\alpha=0.9$), especially in moderate access gaps (mid to high $\eta$). This
is due to saturation effects, as without access-awareness, the advantaged
subpopulation can quickly acquire all the needed resources, leaving scraps to
the disadvantaged.

What is practically most significant, however, is that at moderate
availability ($\alpha=0.5$) the improvement in $\mathsf{RD}$ is consistent in
both instances, no matter what the value of $\eta$ is. We see this visually as
the $\mathsf{RD}$ plots having shifted down. This highlights the potential
impact that access awareness could have on distributing resources more
equitably, no matter what the gap is between the advantaged and disadvantaged.

(a) Outcomes of $n^{\star}_{1}$ (b) Outcomes of $n^{\star}_{\infty}$

Figure 4: Outcomes of Allocations — (Left) Resource (vaccination) disparity
($y$-axis, $\mathsf{RD}$) of both the access-aware allocation
$n^{\star}_{\infty}$ offers and the proportional allocation (faint plots), for
$\alpha=0.1,0.5,0.9$. (Right) Difference between the two, which captures the
improvement in disparity by using the access-aware allocation. The $x$-axis is
the access gap, $\eta$. Main observations — (4(a)) For $n^{\star}_{1}$,
improvements are more marked when the access gap is larger, i.e. when $\eta$
is smaller. Counterintuitively, more availability (larger $\alpha$) offers
relatively more opportunity to mitigate disparity, as explained in the text.
(4(b)) For $n^{\star}_{\infty}$, the general behavior is comparable to that of
$n^{\star}_{1}$. However, in this case we are more constrained not to deviate
from proportionality, and the gains are thus less.

A final phenomenon that is worth clarifying is the following. Imagine
$\eta=0$, meaning that no matter how resources are allocated, the
disadvantaged acquire _nothing_. In that case, one cannot hope to improve on
proportional allocation (or for that matter on any non-wasteful allocation),
since all resources will then flow to the advantaged, and no reallocation can
change it. One would then expect, near $\eta=0$, for the $\mathsf{RD}$ of the
access-aware allocations to approach that of the proportional allocation. Yet,
we observe a clear separation between the two. How can we explain this? This
is due to the fact that our access model operates with an infinite time
horizon, which creates a discontinuity. While the disadvantaged acquire
nothing at $\eta=0$, even a modicum of access $\eta>0$ no matter how small,
the disadvantaged can acquire whatever is leftover after the advantaged
saturate. They can do this, even if it takes them an arbitrarily long time.
And this offers an opportunity for access-aware allocation to make a
difference. This also opens up an avenue to explore finite time horizon
effects by appropriately modifying the access model.

##  8 Discussions

Throughout this work, we have considered technical means by which to address
issues with access to limited resources for members of vulnerable groups. We
acknowledge that this is an oversimplification of society and that there are
many factors, both endogenous and exogenous, that can hamper a person’s
acquisition of a limited resource.

##### Other Factors of Slowed Access

Often, members of vulnerable groups have been marginalized and mistreated by
systems offering aid and assistance to illness or societal issues (e.g.
Tuskegee Experiments) [Wasserman et al.(2007)]. This mistreatment has lead to
mistrust and reluctance to accepting proposed solutions amongst some
subpopulations. In addition, there is often a wave of misinformation
distributed in mass on public concerns [Naeem et al.(2021)]. Without the
correct information on a resource, people are robbed of the opportunity to
decide whether obtaining a resource will truly be beneficial for them.
Educating people on the true benefits and disadvantages of obtaining a
resource could help in their decision-making. Even with a trustworthy system,
true education on obtaining a resource, and the solution of proximity issues,
more barriers exist, e.g., lack of internet, inflexible jobs, mobility issues,
or no health insurance [Jean-Jacques and Bauchner(2021)].

##### Calls for Justice

Our approach to mitigating slowed down access through increased proximity
could be interpreted as a form of justice, in that those a part of the
disadvantaged population have often been marginalized, creating lack of access
and exacerbating the effects of this deficit. This marginalization has been
caused by multiple factors and reared in various ways, some include: redlining
which hurt access to jobs and healthcare [Garber(2021)], social exclusion
which can lead to lack of education, and inability to gather those resources
[Alston and Kent(2003)], etc. Such history has shifted the distribution of
power [Kasy and Abebe(2021)]. Though these effects are deeply rooted and ring
throughout the lives of the disadvantaged, the hope is that taking such steps
will encourage appropriate visibility of these members of society and, by
doing so, work toward reversing injustice.

##  9 Acknowledgements

This paper is based upon work supported in part by the NSF LSAMP Bridge to the
Doctorate Program under Award No. HRD-1906075 and the NSF-Amazon Program on
Fairness in AI in Collaboration with Amazon under Award No. IIS-1939743,
titled FAI: Addressing the 3D Challenges for Data-Driven Fairness: Deficiency,
Dynamics, and Disagreement. Any opinion, findings, and conclusions or
recommendations expressed in this paper are those of the authors and do not
necessarily reflect the views of the National Science Foundation or Amazon.

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##  Appendix A Verification of the Heuristic

Let us take a closer look at the behavior of the heuristic optimization via
successive iteration, as described in Sec. 6.2 and used throughout our
experiments. Recall that the underlying problem is a minimization of a
piecewise-linear concave objective over a bounded convex polytope. This means
that the solution lies on a vertex of this polytope. To empirically verify the
heuristic, we can exhaustively test all these vertices. Because such an
approach is feasible only at a small scale, we restrict this verification to
small state-level examples: states CT, MA, ME, and VT, which have 8, 14, 16,
and 14 counties respectively. Vulnerability ($\beta$) is based on moderate to
very high percentages from the Surgo data. We fix
$\varepsilon=0.1,\alpha=0.5,$ and $\eta=0.3$ as representative values. We
employ the Avis–Fukuda algorithm [Avis and Fukuda(1992)] to enumerate the
vertices of the constraint polytope in each case. We then evaluate the
objective at each vertex. Fig. 5 illustrates the sorted objective values
across all vertices in each state. In all cases, the heuristic converges. In
most cases, it is approximately (ME) or exactly (MA and CT) at the optimal
value. However, in some cases it can remain sub-optimal (NH). Note that even
then, the access-aware heuristic improves on proportional allocation (the
$\times$ mark). We conclude that, awaiting further investigation into better
optimization approaches, the access model and the corresponding heuristic
allocation presented in this paper are a pragmatic approach to tackle resource
disparity. That said, to explore potential avenues to remedy suboptimality, we
conjecture that initializing the iterations with a noisy version of the naïve
acquisition function can dislodge from local minima. Such restarts are much
faster than vertex enumeration, which becomes prohibitive with $20+$
locations. In initial experiments, choosing the best out of $100$ random noisy
restarts (additive standard normal per coordinate) yielded the optimal
solution every time. Completely random restarts performed worse.

Figure 5: Sorted $\mathsf{RD}$ values at every vertex of the constraint
polytope for allocation across counties for $4$ states, CT, MA, ME, and VT.
The orange $\times$ indicates the $\mathsf{RD}$ of proportional allocation.
The dots represent the $\mathsf{RD}$ of the access-aware allocation, either in
the heuristic form of the paper, or augmented via noisy restarts.

##  Appendix B No Knowledge of the Access Gap

We include her some discussion on potential approaches that could be taken in
situations where the methodology of the paper is applied, but where robustness
to $\eta$ is not inherent. Let us assume that the range $(0,1)$ of $\eta$ can
be partitioned to finitely many, $L$, regions with a representative $\eta_{i}$
in each (the piecewise linear nature of the problem facilitates this, though
does not always guarantee it, cf. the adversarial version in the next
paragraph). If a Bayesian estimate of the distribution of the $\eta_{i}$ is
available, then one could also minimize the expected discrepancy
$\sum_{i=1}^{L}\operatorname*{\mathbb{P}}\left({\eta=\eta_{i}}\right)\mathsf{RD}(\mathbf{n},\eta_{i})$
with the same methodology as in Sec. 6.2, as it is still a concave function
admitting a minimum on the boundary of the convex constraint set. More
precisely, the iterations would maintain a separate updated
$\tilde{\rho}_{j}(i)$ and corresponding $c_{j}(i)$ for each $i$ in Steps (1)
and (2), and use the effective $c_{j}$ to be
$\sum_{i=1}^{L}\operatorname*{\mathbb{P}}\left({\eta=\eta_{i}}\right)c_{j}(i)$
in Steps (1) and (3).

One may also want to find an adversarially robust allocation via
$\min_{\mathbf{n}}\max_{i}\mathsf{RD}(\mathbf{n},\eta_{i})$. The objective
function here can become irregular as a max of concave functions is not even
concave anymore, and the the optimal allocation could fall in the interior of
the constraint set. In this case, one could relax this minimax problem into
its maximin dual, by solving
$\max_{\mathbb{P}}\min_{\mathbf{n}}\sum_{i=1}^{L}\operatorname*{\mathbb{P}}\left({\eta=\eta_{i}}\right)\mathsf{RD}(\mathbf{n},\eta_{i})$
instead. Then, the earlier Bayesian solution can be used along with a
numerically differentiated interior point method to optimize over
$\mathbb{P}$. We can then guarantee that
$\min_{\mathbf{n}}\max_{i}\mathsf{RD}(\mathbf{n},\eta_{i})\in[1,L]\times\max_{\mathbb{P}}\min_{\mathbf{n}}\sum_{i=1}^{L}\operatorname*{\mathbb{P}}\left({\eta=\eta_{i}}\right)\mathsf{RD}(\mathbf{n},\eta_{i})$.

##  Appendix C Proofs

#### Proof of Prop. 4.1

Conditionally on $\ddot{S}=s$, $\ddot{U}_{\ddot{S}}$ and $\ddot{V}_{S}$ are
equivalent to independent Poisson random variables $X$ and $Y$ with respective
rates $\mu:=\lambda^{-}s$ and $\nu:=\lambda^{+}s$, conditioned on $X+Y=N$.
Note that $X+Y$ itself is then Poisson with parameter $\mu+\nu$. To calculate
$\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}|\ddot{S}=s}\right]$, we
can equivalently calculate:

$\displaystyle\operatorname*{\mathbf{E}}\left[{X|X+Y=N}\right]$
$\displaystyle=$
$\displaystyle\sum_{x=0}^{N}x\operatorname*{\mathbb{P}}\left({X=x|X+Y=N}\right)=\sum_{x=0}^{N}x\frac{\operatorname*{\mathbb{P}}\left({X=x,Y=N-x}\right)}{\operatorname*{\mathbb{P}}\left({X+Y=N}\right)}=\frac{\sum_{x=0}^{N}x\frac{\mu^{x}\mathrm{e}^{-\mu}}{x!}\frac{\nu^{N-x}\mathrm{e}^{-\nu}}{(N-x)!}}{\frac{(\mu+\nu)^{N}\mathrm{e}^{-(\mu+\nu)}}{N!}}$
$\displaystyle=$
$\displaystyle\frac{N\mu}{(\mu+\nu)}\sum_{x=1}^{N-1}\tfrac{(N-1)!}{(x-1)!(N-x)!}\left(\tfrac{\mu}{\mu+\nu}\right)^{x-1}\left(\tfrac{\nu}{\mu+\nu}\right)^{N-x}=\frac{N\mu}{(\mu+\nu)},$

where we used the binomial identity. By substituting for $\mu$ and $\nu$, we
get that
$\operatorname*{\mathbf{E}}\left[{U_{\ddot{S}}|\ddot{S}=s}\right]/N=\frac{\lambda^{-}}{\lambda^{+}+\lambda^{-}}$
regardless of $s$. Taking iterated expectation over $\ddot{S}$ completes the
proof. $\square$

#### Proof of Prop. 4.2

Consider the two events $\mathcal{A}=\\{\ddot{U}_{\ddot{S}}>\beta P\\}$ (the
disadvantaged have saturated, $T_{U}<\ddot{S}$) and
$\mathcal{B}=\\{\ddot{V}_{\ddot{S}}>(1-\beta)P\\}$ (the advantaged have
saturated $T_{V}<\ddot{S}$). These are disjoint, since otherwise
$\ddot{U}_{\ddot{S}}+\ddot{V}_{\ddot{S}}>\ddot{P}$, which contradicts the
definition of $\ddot{S}$. Along with the complement
$\mathcal{C}=(\operatorname*{\mathbb{P}}\left({\mathcal{A}}\right)\cup\operatorname*{\mathbb{P}}\left({\mathcal{B}}\right))^{\mathsf{c}}$
of their union, these events thus form a partition of the sample space. In
$\operatorname*{\mathbb{P}}\left({\mathcal{C}}\right)$, we have $S=\ddot{S}$
(since no saturation occurs) and $U_{S}=\ddot{U}_{\ddot{S}}$. Now let us write
the following two total expectations:

$\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}}\right]=\operatorname*{\mathbb{P}}\left({\mathcal{A}}\right)\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}|\mathcal{A}}\right]+\operatorname*{\mathbb{P}}\left({\mathcal{B}}\right)\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}|\mathcal{B}}\right]+\operatorname*{\mathbb{P}}\left({\mathcal{C}}\right)\underbrace{\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}|\mathcal{C}}\right]}_{\operatorname*{\mathbf{E}}\left[{U_{S}|\mathcal{C}}\right]}$

and

$\operatorname*{\mathbf{E}}\left[{U_{S}}\right]=\operatorname*{\mathbb{P}}\left({\mathcal{A}}\right)\underbrace{\operatorname*{\mathbf{E}}\left[{U_{S}|\mathcal{A}}\right]}_{\beta
P}+\operatorname*{\mathbb{P}}\left({\mathcal{B}}\right)\underbrace{\operatorname*{\mathbf{E}}\left[{U_{S}|\mathcal{B}}\right]}_{N-(1-\beta)P}+\operatorname*{\mathbb{P}}\left({\mathcal{C}}\right)\operatorname*{\mathbf{E}}\left[{U_{S}|\mathcal{C}}\right],$

where the latter two expressions are due to the saturation events of
$\mathcal{A}$ and $\mathcal{B}$. Subtracting the first from the second:

$\operatorname*{\mathbf{E}}\left[{U_{S}}\right]=\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}}\right]+\operatorname*{\mathbb{P}}\left({\mathcal{A}}\right)\left(\beta
P-\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}|\mathcal{A}}\right]\right)+\operatorname*{\mathbb{P}}\left({\mathcal{B}}\right)\left(N-(1-\beta)P-\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}|\mathcal{B}}\right]\right).$

This is convenient because we have now expressed the saturated process’s
expectation completely in terms of the unsaturated process. All the quantities
to be found depend only on $\ddot{S}$, $\ddot{U}_{t}$, and $\ddot{V}_{t}$.
Using a similar approach as in the proof of Prop. 4.1 (conditioning on
$\ddot{S}=s$, using the equivalent Poisson random variables $X$ and $Y$, then
performing iterated expectation), we get:

$\begin{array}[]{rclcrcl}\operatorname*{\mathbb{P}}\left({\mathcal{A}}\right)&=&1-F(\beta
P;N,\ddot{\rho})&&\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}|\mathcal{A}}\right]&=&N\ddot{\rho}\frac{1-F(\beta
P-1;N-1,\ddot{\rho})}{\operatorname*{\mathbb{P}}\left({\mathcal{A}}\right)}\\\
\operatorname*{\mathbb{P}}\left({\mathcal{B}}\right)&=&1-F((1-\beta)P;N,1-\ddot{\rho})&&\operatorname*{\mathbf{E}}\left[{\ddot{V}_{\ddot{S}}|\mathcal{B}}\right]&=&N(1-\ddot{\rho})\frac{1-F((1-\beta)P-1;N-1,1-\ddot{\rho})}{\operatorname*{\mathbb{P}}\left({\mathcal{B}}\right)}\par\end{array}$

Note that
$\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}|\mathcal{B}}\right]=N-\operatorname*{\mathbf{E}}\left[{\ddot{V}_{\ddot{S}}|\mathcal{B}}\right]$,
and we have already determined that
$\operatorname*{\mathbf{E}}\left[{\ddot{U}_{\ddot{S}}}\right]=N\ddot{\rho}$.
Combining these expressions completes the proof. $\square$