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Fair Healthcare Rationing to Maximize Dynamic
Utilities
Aadityan Ganesh1∗, Pratik Ghosal1∗, Vishwa Prakash HV1∗, and Prajakta
Nimbhorkar1,2∗
1Chennai Mathematical Institute, India
2UMI ReLaX
{aadityanganesh,pratik,vishwa,prajakta}@cmi.ac.in
Abstract. Allocation of scarce healthcare resources under limited logis-
tic and infrastructural facilities is a major issue in the modern society.
We consider the problem of allocation of healthcare resources like vac-
cines to people or hospital beds to patients in an online manner. Our
model takes into account the arrival of resources on a day-to-day basis,
different categories of agents, the possible unavailability of agents on cer-
tain days, and the utility associated with each allotment as well as its
variation over time.
We propose a model where priorities for various categories are modelled
in terms of utilities of agents. We give online and offline algorithms to
compute an allocation that respects eligibility of agents into different
categories, and incentivizes agents not to hide their eligibility for some
category. The offline algorithm gives an optimal allocation while the on-
line algorithm gives an approximation to the optimal allocation in terms
of total utility. Our algorithms are efficient, and maintain fairness among
different categories of agents. Our models have applications in other areas
like refugee settlement and visa allocation. We evaluate the performance
of our algorithms on real-life and synthetic datasets. The experimental
results show that the online algorithm is fast and performs better than
the given theoretical bound in terms of total utility. Moreover, the exper-
imental results confirm that our utility-based model correctly captures
the priorities of categories.
1 Introduction
Healthcare rationing has become an important issue in the world amidst the
COVID-19 pandemic. At certain times the scarcity of medical resources like
vaccines, hospital beds, ventilators, medicines especially in developing countries
raised the question of fair and efficient distribution of these resources. One natu-
ral approach is to define priority groups. For example, for vaccination, the main
priority groups considered include health care workers, workers in other essen-
tial services, and people with vulnerable medical conditions [36,44]. Racial equity
has been another concern [9]. Having made the priority groups, it still remains a
*The authors contributed equally to this work and are listed in alphabetical order
arXiv:2303.11053v1 [cs.MA] 20 Mar 2023
2 Authors Suppressed Due to Excessive Length
challenge to allocate resources within the groups in a transparent manner [15,45].
A New York Times article has mentioned this as one of the hardest decisions for
health organizations [16]. In light of this, it is a major problem to decide how
to allocate medical resources fairly and efficiently while respecting the priority
groups and other ethical concerns.
The healthcare rationing problem has been recently addressed by market
designers. In [35], the problem was framed as a two-sided matching problem
(see e.g. [39]). Their model has reserve categories each with its own priority
ordering of people. This ordering is based on the policy decisions made accord-
ing to various ethical guidelines. It is shown in [35] that running the Deferred
Acceptance algorithm of Gale and Shapley [18] has desired properties like eligi-
bility compliance, non-wastefulness and respect to priorities. This approach of
[35] has been recommended or adopted by organizations like the NASEM (Na-
tional Academies of Sciences, Engineering, and Medicine) [19]. It has also been
recognized in medical literature [36,43], and is mentioned by the Washington
Post [12]. The Smart Reserves algorithm of [35] gives a maximum matching sat-
isfying the desired properties mentioned earlier. However, it assumes a global
priority ordering on people. In a follow-up work, [5] generalize this to the case
where categories are allowed to have heterogeneous priorities. Their Reverse Re-
jecting (REV) rule, and its extension to Smart Reverse Rejecting (S-REV) rule
are shown to satisfy the goals like eligibility compliance, respect to priorities,
maximum size, non-wastefulness, and strategyproofness.
However, the allocation of healthcare resources is an ongoing process. On a
day-to-day basis, new units arrive in the market and they need to be allocated
to people. The variation in the availability of medical resources over a period of
time, and the possible unavailability of recipients on certain days is an important
factor in making allocation decisions. For example, while allocating vaccines,
the unavailability of people on certain days might lead to wastage of vaccines,
especially if the units are reserved for categories a priori. The previous models
do not encompass this dynamic nature of resources. Moreover, the urgency with
which a resource needs to be allocated to an individual also changes over time.
While priority groups or categories aim to model this by defining a priority order
on people, defining a strict ordering is not practically possible. While dealing
with a large population, defining a strict ordering on people is not desirable.
For instance, in the category of old people, it is neither clear nor desirable to
define a strict order on people of the same age and same vulnerabilities. Even if
categories are allowed to have ties in their ordering, the ordering still provides
only an ordinal ranking.
Our model provides the flexibility to have cardinal rankings in terms of pri-
oritizing people by associating a utility value for each individual. Thus, in our
work, categories do not define an ordering over people, instead, there is a util-
ity value associated with allocation of the resource to each person. The goal
is to find an allocation with maximum total utility while respecting category
quotas. However, utilities can change over time. For instance, the advantage of
allotting a ventilator to a person today might be far more beneficial than al-
Fair Healthcare Rationing to Maximize Dynamic Utilities 3
lotting it tomorrow. Similarly, vaccinating the vulnerable population as early as
possible is much more desirable from a social perspective than delaying it to a
later day. We model this through dynamic utilities. Thus, we consider utilities
that diminish over time. The discounting factor 0< δ < 1is multiplicative.
Such exponential discounting is commonly used in economics literature [38,40].
Our utility maximization objective can thus be seen as maximization of social
welfare. Another advantage is that the division of available units into various
categories is not static. It is dynamically decided depending on the supply of
units and availability of people on each day.
Our algorithms to find a maximum utility allocation are based on network
flows. They adhere to the following important ethical principles which were in-
troduced by Aziz et al in [5]:
1. complies with the eligibility requirements
2. is strategyproof (does not incentivize agents to under-report the categories
they qualify for or days what they are available on),
3. is non-wasteful (no unit is unused but could be used by some eligible agent)
Additionally our algorithms give an approximate maximum weight match-
ings, where the weights denote the utility value of a matching. We note that
the current state-of-practice algorithms such as first-come first-serve or random
ordering do not guarantee non-wastefullness as the matching returned by them
may not be of maximum size. Furthermore, matchings returned by these algo-
rithms could be of arbitrarily low total utility. Using category quotas and utility
values, we provide a framework in which more vulnerable populations can be
prioritized while maintaining a balance among the people vaccinated through
each category on a day-to-day basis.
1.1 Related Work
The topic of constrained matching problems has been an active area or research
and it has been studied in the context of school choice and hospital residents
problem apart from healthcare rationing [30,5,27,28,8,21,41]. The setting with
two-sided preferences has been considered in [22,29,23,17]. The fairness and wel-
fare objectives have been covered in a comprehensive manner in [32].
Another application of the constrained matching problem is in the refugee
resettlement problem. Refugee resettlement is a pressing matter in the twenty-
first century where people have been forced to leave their country in order to
escape war, persecution, or natural disaster. In the refugee resettlement pro-
cess the refugee families are settled from asylum countries to the host countries
where the families are given permanent residentship. The reallocation is done
keeping in mind the necessities of the families as well as the infrastructure ca-
pacities of the host countries. Delacrétaz et al. [13] formalized refugee allocation
as a centralized matching market design problem. The refugee allocation prob-
lems have been studied both in terms of matching problems with preferences
[3,14,6,25,26,34,42] and without preferences[7,13]. In the matching problem with
4 Authors Suppressed Due to Excessive Length
preferences, the goal is to match the refugees to localities based on the preference
of either one or both sides, while satisfying the multidimensional resettlement
constraints. Delacretaz et al. considered the problem both in terms of with and
without preferences. The problem without preference can be reduced to mul-
tiple multidimensional knapsack problems [13]. The branch-and-bound method
can be used to find the exact solution. Bansak et al. [7] used a combination of
supervised machine learning and optimal matching to obtain a refugee matching
that satisfies the constraints of refugees and localities. The dynamic version of
the refugee resettlement problem [4,1,11] has also been considered in literature.
1.2 Our Models
We define our model below and then define its extension. Throughout this pa-
per, we consider vaccines as the medical resource to be allocated. People are
referred to as agents. Note that although the discussion assumes perishability of
resources, it can easily be extended to non-perishable resources.
Model 1: Our model consists of a set of agents A, a set of categories C, and a set
of days D. For day dj∈D, there is a daily supply sjdenoting the number of vac-
cine shots available for that day. For each category ci∈C, and each day dj∈D,
we define a daily quota qij . Here qij denotes the maximum number of vaccines
that can be allocated for cion day dj. There is a priority factor αkassociated
with an agent ak. Let αmax = maxi{αi|αiis the priority factor of agent ai}
and αmin = mini{αi|αiis the priority factor of agent ai}. Utilities have
adiscount factor δ∈(0,1) denoting the multiplicative factor with which the
utilities for vaccinating agents reduce with each passing day. Thus if akis vacci-
nated on day dj, the utility obtained is αk·δj. Each agent akhas an availability
vector vk∈ {0,1}|D|. The jth entry of vkis 1if and only if akis available for
vaccination on day dj.
Model 2: Model 2 is an extension of Model 1 in the following way. The sets
A, C, D and the daily supply and daily quotas are the same as those in model
1. Apart from the daily quota, each category cialso has an overall quota qi
that denotes the maximum total number of vaccines that can be allocated for
category ciover all the days. Note that overall quota is also an essential quantity
in applications like visa allocation and refugee settlement.
In both the models, a matching M:A→(C×D)∪{∅} is a function denoting
the day on which a person is vaccinated and the category through which it is
done, such that the category quota(s) and daily supply values do not exceed on
any day. Thus if we define variables xijk such that xijk = 1 if M(ak)=(ci, dj)
and xijk = 0 if M(ak) = ∅, then we have Pi,j xijk ≤1for each k,Pk,j xijk ≤qi
for each i,Pkxijk ≤qij for each i, j, and Pi,k xij k ≤sjfor each j. Here
1≤i≤ |C|,1≤j≤ |D|,1≤k≤ |A|. If M(ak) = ∅for some ak∈A, it means
the person could not be vaccinated through our algorithm within |D|days.
In both the models, the utility associated with akis αk·δj−1where M(ak) =
(ci, dj). The goal is to find a matching that maximizes the total utility.
Fair Healthcare Rationing to Maximize Dynamic Utilities 5
1.3 Our Contributions
The utilities αkand discounting factor δhave some desirable properties. If agent
akis to be given a higher priority over agent a`, then we set αk> α`. On any
day dj,αk·δj> α`·δj. Moreover, the difference in the utilities of the two agents
diminishes over time i.e. if j < j0then (αk−α`)δj>(αk−α`)δj0. Thus the
utility maximization objective across all days vaccinates akearlier than a`.
We consider both online and offline settings. The offline setting relies on the
knowledge about availability of agents on all days. This works well in a system
where agents are required to fill up their availability in advance e.g. in case of
planned surgeries, and visa allocations. The online setting involves knowing the
availability of all the agents only on the current day as in a walk-in setting. Thus
the availability of an agent on a day in future is not known.
We give an optimal algorithm for Model 1in the offline setting..
Theorem 1. There is a polynomial-time algorithm that computes an optimal
solution for any instance of Model 1in the offline setting.
We also give algorithms for both Model 1and Model 2in the online setting.
We give theoretical guarantees on the performance of online algorithms in terms
of their competitive ratio in comparison with the utility of an offline optimal
solution.
Theorem 2. There is an online algorithm (Algorithm 1) that gives a competi-
tive ratio of (i) 1 + δfor Model 1and (ii) of 1 + δ+ (αmax/αmin)δfor Model 2
when δis the common discounting factor for all agents. The algorithm runs in
polynomial time.
We prove part (i)of Theorem 2 in Section 3.2 whereas part (ii)is proved in
Appendix.
Strategy-proofness: It is a natural question whether agents benefit by hiding their
availability on some days. We show that the online algorithm is strategy-proof. In
this context, we analyze our online algorithm for Model 1from a game theoretic
perspective. We exhibit that the offline setting has a pure Nash equilibrium that
corresponds to the solution output by the online algorithm. For this, we assume
that the tie-breaking among agents is done according to an arbitrary permutation
πof agents.
Theorem 3. Let an offline optimal solution that breaks ties according to a ran-
dom permutation πmatch agent aion day di. Then for each agent ai, reporting
availability exactly on day di(unmatched agents mark all days as unavailable)
is a pure Nash equilibrium. Moreover, the Nash equilibrium corresponds to a
solution output by the online algorithm.
Experimental Results: We also give experimental results in Section 6 using real-
world datasets. Apart from maximization of utilities, we also consider the number
of days taken by the online algorithm for vaccinating high priority people. Our
experiments show that the online algorithm almost matches the offline algorithm
in terms of both of these criteria.
6 Authors Suppressed Due to Excessive Length
Selection of utility values: An important aspect of our model is that the choice
of utility values does not affect the outcome as long as the utility values have the
same numerical order as the order of priorities among agents. Thus the output of
online as well as offline algorithm remains the same as long as αk> α`whenever
agent akhas a higher priority over agent a`.
2 Optimal Offline Algorithm for Model 1
The problem can be modelled as an instance of the minimum cost flow network.
We define the minimum cost flow problem here for completeness.
The minimum cost flow problem: The input is a flow network G= (V, E)as a
directed graph with node set V, edge set E, capacities ce>0and cost ue∈R
on each edge e∈E, and a source sand sink t. A flow f:E→Ris a valid flow
in Gif f(e)≤c(e), and the incoming flow at any node except sand tequals the
outgoing flow. The cost of a flow f(e)along an edge eis ue·f(e). A minimum
cost flow in the network is the one that minimizes the sum of costs of the flow
along all edges.
There are polynomial-time algorithms known for the minimum cost flow prob-
lem. Also, it is known that if all the capacities are integers, then the optimum
flow is an integer. We refer the reader to [2] for the details of minimum cost flow.
Reduction: The construction of the flow network is shown in Figure 2. The flow
network consists of a source s, a sink t, nodes for each day dj, each agent akand
nodes cij for each (ci, dj)∈C×D. Each edge (s, dj)has capacity sjdenoting
the daily supply for day dj, each edge (dj, cij)has capacity equal to qij , and
all other edges have capacity 1. All the edges are directed. Additionally, each
(cij , ak)edge has cost −uk·δj
kwhereas other edges have cost 0.
Proof. (of Theorem 1) We show that a minimum cost flow fin the flow network
corresponds to a maximum utility matching in the given instance. The integrality
of minimum cost flow implies that each edge incident on tcan have a flow of
either 0or 1. For each k, if f(ak, t)=1, then there is exactly one cij such
that f(cij , ak) = 1. Set M(ak) = (ci, dj)in the corresponding matching M.
Similarly, for any matching M, A corresponding flow can be shown as follows.
If M(ak) = (ci, dj)then set f(ak, t) = f(cij , ak) = 1, and set f(s, dj)equal to
the number of agents vaccinated on day dj, and f(dj, cij )equal to the number
of agents vaccinated on day djthrough category ci. It is clear that this is a valid
flow in the network, and the negation of the cost of the flow is the same as the
utility of the corresponding matching.
3 Algorithms for Model 1
We give a flow based polynomial-time optimal offline algorithm for Model 1
in Appendix. Here, we give an online algorithm for the same which achieves a
competitive ratio of 1 + δ, where δis the discounting factor of the agents.
Fair Healthcare Rationing to Maximize Dynamic Utilities 7
s
d1
d2
c11
c12
c13
c21
c22
c23
a1
a2
a3
t
Fig. 1: Flow network for finding a maximum utility matching in Model 1
3.1 Online Algorithm for Model 1
We present an online algorithm which greedily maximizes utility on each day.
We show that this algorithm indeed achieves a competitive ratio of 1 + δ.
Outline of the Algorithm: On each day di, starting from day d1, we construct
a bipartite graph Hi= (Ai∪C, Ei, wi)where Aiis the set of agents who are
available on day diand are not vaccinated earlier than day di. Let the weight
of the edge (aj, ck)∈Eibe wi(aj, ck) = αj.δi−1. We define capacity of the
category ck∈Cas b0
i,k. In this graph, our algorithm finds a maximum weighted
b-matching of size not more than the daily supply value si.
The following lemma shows that the maximum weight b-matching computed
in Algorithm 1 is also a maximum size b-matching of size at most si.
Lemma 1. The maximum weight b-matching in Hiof size at most siis also a
maximum size b-matching of size at most si.
Proof. We prove that applying an augmenting path in Hiincreases the weight
of the matching. Consider a matching Miin Hisuch that Miis not of maximum
size and |Mi|< si. Let ρ= (a1, c1, a2, c2,· · · , ak, ck)be an Mi-augmenting path
in Hi. We know that every edge incident to an agent has the same weight in Hi.
If we apply the augmenting path ρ, the weight of the matching increases by the
8 Authors Suppressed Due to Excessive Length
Algorithm 1 Online Algorithm for Vaccine Allocation
Input: An instance Iof Model 1
Output: A matching M:A→(C×D)∪ {∅}
1: Let D, A, C be the set of Days, Agents and Categories respectively.
2: M(aj)←∅for each aj∈A
3: for day diin Ddo
4: Ai← {aj∈A|ajis available on diand ajis not vaccinated}
5: Ei← {(aj, ck)∈Ai×C|ajis eligible to be vaccinated under category ck}
6: for (aj, ck)in Eido
7: Let wi(aj, ck)←αjδi−1
8: end for
9: Construct weighted bipartite graph Hi= (Ai∪C, Ei, wi).
10: for ckin Cdo
11: b0
i,k ←qik {Where qik is the daily quota}
12: end for
13: Find maximum weight b-matching Miin Hiof size at most si. {Where siis the
daily supply}
14: for each edge (aj, ck)in Mido
15: M(aj)←(ck, di){Mark ajas vaccinated on day diunder category ck}
16: end for
17: end for
18: return M
weight of the edge (a1, c1). This proves that a maximum weight matching in Hi
of size at most siis also a maximum size b-matching of size at most si.
3.2 Charging scheme
We compare the solution obtained by Algorithm 1 with the optimal offline so-
lution to get the worst-case competitive ratio for Algorithm 1. Let Mbe the
output of Algorithm 1 and Nbe an optimal offline solution. To compare Mand
N, we devise a charging scheme by which, each agent apmatched in Ncharges a
unique agent aqmatched in M. The amount charged, referred to as the charging
factor here is the ratio of utilities obtained by matching apand aqin Mand N
respectively.
Properties of the charging scheme:
1. Each agent matched in Ncharges exactly one agent matched in M,
2. Each agent aqmatched in Mis charged by at most two agents matched in
N, with charging factors at most 1and δ. This implies that the utility of N
is at most (1 + δ)times the utility of M.
We divide the agents matched in Ninto two types. Type 1agents are those
which are matched in Mon an earlier day compared to that in N. Thus ap∈A
is a Type 1agent if apis matched on day diin Mand on day djin N, such that
i < j. The remaining agents are called Type 2agents. Our charging scheme is
as follows:
Fair Healthcare Rationing to Maximize Dynamic Utilities 9
1. Each Type 1agent apcharges themselves with a charging factor δ, since the
utility associated with them in Nis at most δtimes that in M.
2. Here onwards, we consider only Type 2agents and discuss the charging
scheme associated with them.
Let Xibe the set of Type 2agents matched on day diin N, and let Yi
be the set of agents matched on day diin M. Since Algorithm 1 greedily
finds a maximum size b-matching of size at most si, and as each edge in
the b-matching corresponds to a unique agent, we show the following lemma
holds:
Lemma 2. For each di∈D, the set |Xi| ≤ |Yi|.
Proof. Since Xicontains only Type 2agents matched in Ni, the agents in Xi
are not matched by Muntil day i−1. Therefore Xi⊆Ai, where Aiis defined
in Algorithm 1. The daily quota and the daily supply available for computation
of Niand Miis the same i.e. qi,k, and sirespectively. By construction, Miis a
matching that matches maximum number of agents in Ai, up to an upper limit
of si,|Xi|≤|Yi|.
To obtain the desired competitive ratio we design an injective mapping according
to which, each agent apin Xican uniquely charge an agent aqin Yisuch that
αp≤αq. The following lemma shows that such an injective mapping always
exists.
Lemma 3. There exists an injective mapping f:Xi→Yisuch that if f(ap) =
aq, then αp≤αq.
Proof. Let Niand Mirespectively be the restrictions of Nand Mto day di.
We construct an auxiliary bipartite graph Giwhere Xi∪Yiform one bipartition
and categories form another bipartition. The edge set is Ni∪Mi. Then we set
the capacity of ckin Gito be bi,k =qi,k.
The charging scheme is as follows. Consider the symmetric difference Mi⊕Ni.
It is known that Mi⊕Nican be decomposed into edge disjoint paths and even
cycles [33].
Consider a component Cwhich is an even cycle as shown in Fig 2a. Since
each agent apin Chas both Miedge and Niedge indecent on it, agent apin Xi
charges her own image in Yiwith a charging factor of 1.
Now, Consider a component which is a path ρ. There are two cases.
1. Case 1: The path ρhas an even length: If ρstarts and ends at a cat-
egory node, then each agent along the path is matched in both Niand
Mi. Hence, all such agents can charge themselves with a charging factor
of 1. Suppose ρstarts and ends at an agent as shown in Fig 2b i.e. ρ=
(a1, c1, a2, c2,· · · , ak−1, ck−1, ak). Let a1be matched in Miand akis matched
in Ni. Then, α1must be greater than or equal to αk. Otherwise from
Lemma 1, Mi⊕ρis a matching of higher weight - which contradicts the
fact that Miis the maximum weight matching. Now, every agent in ρexcept
a1and akcharge themselves with a charging factor of 1and akcharges a1
with a charging factor of αk/α1.
10 Authors Suppressed Due to Excessive Length
a1c1
a2
c2
a3
c3
a4
c4
1
1
1
1
(a) Agents in cycles charge themselves
with a charging factor or 1
a1
c1
a2
c2
a3
c3ck−2
ak−1
ck−1
ak
1 1 1
αk/α1
(b) Agents who are matched in both
Niand Micharge themselves. Agent ak
charges a1with a factor or αk/α1. Red
edges represent Niand blue edges repre-
sent Mi
Fig. 2: Charging schemes
2. Case 2: The path ρhas an odd length: Then either ρbegins and ends with
an Miedge or with an Niedge. If ρstarts and ends with an Miedge, then
every agent along the path who is matched in Niis also matched in Mi.
Therefore all the agents on ρcharge themselves.
Consider the case when ρstarts with an Niedge. Since ckis an end-point of
ρwith an Ni-edge, ckmust have more agents matched to it in Nithan that
in Mi. So ckcannot be saturated in Mi.
As Miis a maximum size matching [1], we cannot augment Mito Mi⊕ρin Gi
even though both endpoints are unsaturated. This can happen only because
the daily supply is met. That is |Mi|=si. As a1is vaccinated in category c1
in Ni, we claim that the weight w(a1, c1)is less than every other edge in Mi.
This is because if there exists an edge e∈Misuch that w(e)< w(a1, c1),
we can remove the edge efrom Miand apply the augmenting path ρto
get a matching with a higher weight, which is a contradiction. Therefore, as
w(a1, c1)is less than every other edge in Mi, agent a1can safely charge any
agent aqwho is matched in Mi. Since |Mi|≥|Ni|, we are guaranteed to have
sufficient agents in Nifor charging.
Order of charging among Type 2 agents: First, every agent who has both
Miand Niedges indecent on it, charges herself. Next every agent who is an
end-point of an even-length path charges the agent represented by the other
end-point. The rest of the agents are end-points of an odd-length path matched
in Ni. We proved that the edges incident on these agents have a weight smaller
than every edge in Mi. They can charge any agent of Miwho has not been
charged yet by any agent of Ni, as stated above.
Proof (of Theorem 2 (i)). Let aqbe an agent who is vaccinated by the online
matching Mon day i. Then aqcan be charged by at most two agents matched
in N. Suppose aqis vaccinated by the optimal matching Non some day i0> i.
Assume that the agent apof type 2 who also charges aq. If the priority factor of
Fair Healthcare Rationing to Maximize Dynamic Utilities 11
aqand apare αqand αprespectively, then
αp.δi+αq.δi0
αq.δi=αp
αqi
+δi0−i≤1 + δ.
The last inequality follows as 0< αp≤αq<1,and i0> i. Therefore the utility
obtained by apand aqin Miis atmost 1 + δtimes the the utility of aqin Mi.
Therefore the competitive ratio of Algorithm 1 is at most 1 + δ.
In the Appendix, we show a tight example which achieves this compititve
ratio.
Since the daily supply of day d1is 1, vaccinating a1maximizes the utility
gained on the first day. Hence there exists a run of Algorithm 1 where a1is vac-
cinated under category c1on day d1. In this run, agent a2cannot be vaccinated
on day d2as she is unavailable on that day. Hence, total utility gained by the
online allocation is α1. Whereas in a optimal allocation scheme all the agents
can be vaccinated. We vaccinate agent a2on day d1under category c2, agent a1
on day d2under category c1. This sums to a total utility of α1+α1δ. Therefore
the competitive ratio is α1+α1δ
α1= 1 + δ.
3.3 Tight example for the Online Algorithm
a1α1α1δ
α1-
a2
d1d2
Fig. 3: A tight example with competitive ratio 1 + δ. Online allocation indicated
in red, Optimal allocation indicated in green and arrows indicate charging
The following example shows that the competitive ratio of Algorithm 1 is
tight. Let the set of agents A={a1, a2}and categories C={c1, c2}. Agent a1is
eligible under {c1, c2}and agent a2is eligible only under {c2}. The daily supply:
s1= 1 and s2= 1. The daily quota of each category on each day is set to 1. The
priority factor for both the agents is α1. Assume that a1is available on both the
days whereas the agent a2is available only on the first day. Figure 3 depicts this
example.
4 Online Algorithm for Model 2
We present an online algorithm which greedily maximizes utility on each day. We
assume that the discounting factor of the agents is δ. Moreover each agent akhas
12 Authors Suppressed Due to Excessive Length
a priority factor αk. Let αmax = maxi{αi|αiis the priority factor of agent ai}
and αmin = mini{αi|αiis the priority factor of agent ai}. We show that this
algorithm indeed achieves a competitive ratio of 1 + δ+αmax
αmin δ.
Outline of the Algorithm: On each day di, starting from day d1, we construct
a bipartite graph Hi= (Ai∪C, Ei, wi)where set Aiis the set of agents who are
available on day diand are not vaccinated earlier than day di. Let the weight
of the edge (aj, ck)∈Eibe wi(aj, ck) = αj.δi−1. Let b0
i,k represent the capacity
of ck∈Cin Hi. In this graph, our algorithm finds a maximum weighted b-
matching of size not more than the daily supply value si. This can be found in
polynomial time [31]. Lemma 1 proves that the maximum weight b-matching is
also a maximum cardinality b-matching of Hi.
Algorithm 2 Online Algorithm for Vaccine Allocation
Input: An instance Iof Model 2
Output: An allocation M:A→(C×D)∪ {∅}
1: Let D, A, C be the set of Days, Agents and Categories respectively.
2: M(aj)←∅for each agent aj∈A
3: rk←qkfor each category ck∈C
4: for day diin Ddo
5: Ai← {aj∈A|ajis available on diand ajis not vaccinated}
6: Ei={(aj, ck)∈Ai×C|ajis eligible to be vaccinated under category ck}
7: Construct bipartite graph Hi= (Ai∪C, Ei).
8: for ckin Cdo
9: b0
i,k ←min(qik, rk){Capacity for each ckin Hi}
10: end for
11: Find maximum weight b-matching Niin Hiof size at most si.
12: for each edge (aj, ck)in Mido
13: M(aj)←(ck, di){Mark ajas vaccinated on day diunder category ck}
14: rk←rk−1{Update remaining overall quota}
15: end for
16: end for
17: return M
4.1 Outline of the charging scheme
We compare the solution obtained by Algorithm 2 with the optimal offline so-
lution to get the worst-case competitive ratio for Algorithm 2. Let Mbe the
output of Algorithm 2 and Nbe an optimal offline solution. To compare Mand
N, we devise a charging scheme similar to that in Section 3.2, by which each
agent amatched in Ncharges a unique agent a0matched in M. The amount
charged, referred to as the charging factor here is the ratio of utilities obtained
by matching aand a0in Mand Nrespectively.
Properties of the charging scheme:
1. Each agent matched in Ncharges exactly one agent matched in M,
Fair Healthcare Rationing to Maximize Dynamic Utilities 13
2. Each agent matched in Mis charged by at most three agents matched in N,
with charging factors at most 1, δ and αmax
αmin δ. This implies that the utility of
Nis at most (1 + δ+αmax
αmin δ)times the utility of M.
We divide the agents matched in Ninto two types. Type 1agents are those
which are matched in Mon an earlier day compared to that in N. Thus a∈A
is a Type 1agent if ais matched on day diin Mand on day djin N, such that
i < j. The remaining agents are called Type 2agents. Our charging scheme is
as follows:
1. Type 1agents charge themselves with a charging factor δ, since the utility
associated with them in Nis at most δtimes that in M.
2. Here onwards, we consider only Type 2agents and discuss the charging
scheme associated with them.
Let Xibe the set of Type 2agents matched on day diin N, and let Yibe
the set of agents matched on day diin M.
(a) Case 1: |Xi| ≤ |Yi|:From Lemma 3 we claim that each agent ap∈Xi
charges an agent in aq∈Yiwith αp≤αq. Therefore the agents in Xi
charge the agents in Yiwith a charging factor of 1.
(b) Case 2: |Xi|=|Yi|+z, z > 0:Let Niand Mirespectively be the
restrictions of Nand Mto day di. We construct an auxiliary bipartite
graph Giwhere Xi∪Yiform one bipartition and categories form another
bipartition. The edge set is Ni∪Mi. For a category ck, let nj,k and
mj,k be the number of agents matched in Nand Mrespectively, under
category ckon day dj. Then we set the quota of ckin Gito be bi,k =
min{qi,k,max{qk−Pi−1
j=1 nj,k, qk−Pi−1
j=1 mj,k}}. This is the maximum
of the quotas of ckthat were available for computation of Niand Mi
respectively.
The charging scheme is given by the following. Consider the symmetric
difference Mi⊕Ni. Since |Ni|=|Mi|+z, there are exactly zedge-disjoint
alternating paths in Mi⊕Nithat start and end with an edge of N[33].
Let ρ=ha1, c1, a2, . . . , ak, ckibe one such path. Then a2, . . . , ak−1are
matched in both Miand Ni, so they charge themselves with a charging
factor of 1. From Lemma 3, the agent a1charges akwith charging factor
of at most 1. It remains to decide whom akcharges.
Since ρterminates at ckwith an Ni-edge, the number of agents matched
to ckin Niis more than those matched to ckin Mi. In Lemma 4, we show
that this can happen only because of exhaustion of qkin Algorithm 2 on
or before day di. So agent akcan charge some agent almatched to ckin
Mon an earlier day, with charging factor αk
αlδ≤αmax
αmin δ.
Lemma 4. If node ckis an end-point of a path ρin Gi, then qkis exhausted in
Algorithm 2 on or before day di.
Proof. Suppose ckbe an endpoint of ρin Gi. The number of agents matched to
ckin Niis more than those matched to ckin Mi. We know that the daily supply
siof the day diis an upperbound for both |Mi|and |Ni|. Since |Ni|=|Mi|+z,
14 Authors Suppressed Due to Excessive Length
we have |Mi|< si. From Algorithm 2 we know that Miis a maximum-size b-
matching in Hiof size at most si. If the capacity of ckis not saturated in Hi,
then we can augment the path ρcontradicting the maximality of Mi. Since ck
has more edges of Mithan Niincident to it, from the definition of bi,k, category
ckmust have exhausted the overall quota qkin Algorithm 2 on or before day di.
4.2 Tight Example
The following example shows that the competitive ratio of Algorithm 2 is tight.
Let set of agents A={a1, a2, a3}and categories C={c1, c2}. Agent a1is
eligible under {c1, c2}. Agent a2is eligible only under {c1}and agent a3is
eligible only under {c2}. The daily supply: s1= 1 and s2= 2. Overall quotas:
q1= 1 and q2= 2. The daily quota of each category on each day is set to 1. The
utility discounting factor for each agent is δ. The priority factor of the agent ai
is αifor i= 1,2,3. We assume that 0≤α1=α3< α2≤1. Agent a1is available
on both the days. Agent a3is available only on the first day, whereas agent a2
is available only on the second day. Figure 4 depicts this example.
a1
a2
a3
α1α1δ
-α2δ
α3-
d1d2
Fig. 4: A tight example with competitive ratio 1+δ+α2
α1δ. Online allocation indi-
cated in red, Optimal allocation indicated in green and arrows indicate charging
Since the daily supply of day d1is 1, vaccinating a1maximizes the utility
gained on the first day. Hence there exists a run of Algorithm 2 where a1is vacci-
nated under category c1on day d1. In this run, agent a2cannot be vaccinated on
day d2as she is eligible only under category c1and overall quota of category c1
is exhausted. Hence, total utility gained by the online allocation is α1. Whereas
in a optimal allocation scheme all the agents can be vaccinated. Vaccinate agent
a3on day d1under category c2, agent a1and a2on day d2under categories c2, c1
respectively. This sums to a total utility of α3+α1δ+α2δ. Therefore the compet-
itive ratio of the online algorithm is α3+α1δ+α2δ
α1=α1+α1δ+α2δ
α1= 1 + δ+αmax
αmin δ.
The first equality holds as α1=α3. The second equality holds as αmax =α2
and αmin =α1.
Fair Healthcare Rationing to Maximize Dynamic Utilities 15
5 Strategy-proofness of the online algorithm
We give the details of the Pure Nash Equilibrium here.
5.1 Pure Nash equilibrium
The offline algorithm might choose any arbitrary matching that maximizes the
utility. We present a deterministic tie-breaking rule similar to the one used in [5]
to force the algorithm to pick a unique matching. For this, we fix an ordering πon
agents. We show the existence of a pure Nash equilibrium under the deterministic
tie-breaking. We cast our problem as a linear program as given in Fig 5.
maximize: X
i∈A,j∈C,
k∈D
uik.xij k
subject to: X
i∈A,j∈C
xijk ≤sk,∀k∈D
X
i∈A
xijk ≤qjk ,∀(j, k)∈C×D
X
j∈C,k∈D
xijk ∈[0,1],∀i∈A
xijk ∈[0,1],∀(i, j, k)∈A×C×D
Fig. 5: Here uik is the utility value of agent ion day k, and sk&qjk are the daily
supply and daily quotas respectively.
It can be seen that this LP models the network flow formulation of our
problem stated in Section 2. It is known ([31]) that the polytope arising from
the network flow problem is integral. To impose the deterministic tie breaking,
we modify the objective function as follows.
maximize X
i∈A,j∈C,
k∈D
uik.xij k +λ×REG, where
REG =X
i∈APk∈D,j∈Cxij k
2π(i)
For a sufficiently small λ(λ<δ|D|+1), the difference between utilities of any
two allocations is greater than REG. Therefore, the linear program in Figure 5
maximizes the objective function in Fig 5, but breaks ties to maximize REG.
16 Authors Suppressed Due to Excessive Length
Let Adibe defined as the set of agents matched on a day di∈Dand A∞be
the set of unmatched agents at the end of a run of the Algorithm 1. Let agent
apbe matched on di(WLOG, assume all unmatched agents are matched on day
∞. Now, we present a proof of Theorem 3.
Proof. (of Theorem 3) Suppose the agent apis matched on day di, and deviates
to reporting a subset of the actual available days.
If agent apgets matched on a day dj,j < i, because of misreporting her
available days, then some agent aqon day djwill remain unmatched. This follows,
since on any given day, the matching computed by algorithm 1 is of maximum size
and all agents other than apturn up on at most one day. The rest of the matching
will remain unchanged. But, agent aqis prioritized by πover agent ap. Otherwise,
algorithm 1 would have matched apand not aqon day dj. Hence, agent apcannot
replace agent aqon day djeven after misreporting her availability.
Therefore agent aphas no advantage in deviating from the strategy. Hence,
the above matching is a pure Nash equilibrium.
6 Experimental Evaluation
In Section 3 we prove worst-case guarantees for the online algorithm. We also
give a tight example instance achieving a competitive ratio of 1+2δ. Here, we
experimentally evaluate the performance of the online algorithm and compare
it with the worst-case guarantees on a real-life dataset. For finding the optimal
allocation that maximizes utility, we solve the networkflow linear program with
the additional constraint for overall quota Pi∈A,k∈Dxij k ≤qj∀cj∈C. This
LP is described in the Appendix. The code and datasets for the experiments can
be found at [24]
6.1 Methodology
All experiments run on a 64-bit Ubuntu 20.04 desktop of 2.10GHz * 4 Intel Core
i3 CPU with 8GB memory.
The proposed online approximation algorithm runs in polynomial time. In
contrast, the optimal offline algorithm solves an integer linear program which
might take exponential time depending on the integrality of the polytope. We
relax the integrality constraints to achieve an upper-bound on the optimal allo-
cation. For comparing the performance of the online Algorithm 1 and the offline
Algorithm, we use vaccination data of 24 hospitals in Chennai, India for the
month of May 2022. We use small data-sets with varying instance sizes for eval-
uating the running times of the algorithms. We use large data-sets of smaller
instance sizes for evaluating competitive ratios.
All the programs used for the simulation are written in Python language.
For solving LP, ILP, and LPR, we use the general mathematical programming
solver COIN-OR Branch and Cut solver MILP (Version: 2.10.3)[10] on PuLP
(Version 2.6) framework[37]. When measuring the running time, we consider the
time taken to solve the LP.
Fair Healthcare Rationing to Maximize Dynamic Utilities 17
6.2 Datasets
Our dataset can be divided into two parts.
Supply: We consider vaccination data of twenty four hospitals of Chennai,
India for the month of May 2022. This data is obtained from the official COVID
portal of India using the API’s provided. The data-set consists of details such
as daily vaccination availability, type of vaccines, age limit, hospital ID, hospital
zip code, etc. for each hospital.
Demand: Using the Google Maps API [20], we consider the road network for
these 24 hospitals in our data-set. From this data we construct a complete graph
with hospitals as vertices and edge weights as the shortest distance between any
two hospitals. For each hospital h∈H, we consider the cluster C(h)as the set
of hospitals which are at most five kilo meters away from h. We consider these
clusters as our categories. Now, we consider 10000 agents who are to be vacci-
nated. For each agent a, we pick a hospital huniformly at random. The agent
abelongs to every hospital in the cluster C(h). Each agent’s availability over 30
days is independently sampled from the uniform distribution. Now, we consider
the age wise population distribution of the city. For each agent we assign an age
sampled from this distribution. Now, we partition the set of agents as agents
of age 18-45years, 45-60years and 60+. We assign α-values 0.96,0.97 and 0.99
respectively. We also consider the same dataset with α-values 0.1,0.5and 0.9
respectively. We set the discounting factor δto be 0.95.
For analyzing the running time of our algorithms, we use synthetically gen-
erated datasets with varying number of instance sizes ranging from 100 agents
to 20000 agents. Each agent’s availability and categories are chosen randomly
from a uniform distribution.
6.3 Results and Discussions
We show that the online algorithm runs significantly faster than the offline al-
gorithm while achieving almost similar results. We give a detailed emperical
evaluation of the running times in the Appendix.
To compare the performance of the online Algorithm 1 against the offline algo-
rithm we define a notion of remaining fraction of un-vaccinated agents. That is,
on a given day di, we take the set of agents Pdiwho satisfy both of the following
conditions:
1. Agent ais available on some day djon or before day di.
2. Agent abelongs to some hospital hand hhas non-zero capacity on day dj
Pdiis the set of agents who could have been vaccinated without violating
any constraints. Let γi=|Pdi|.
Let Vdibe the set of agents who are vaccinated by the algorithm on or before
day di. Let ηi=|Vdi|. Now, 1−ηi/γirepresents the fraction of unvaccinated
agents. In Figure 6 we compare the age-wise 1−ηi/γiof both of our online and
offline algorithms. We note that the vaccination priorities given to vulnerable
18 Authors Suppressed Due to Excessive Length
groups by the online approximation algorithm is very close to that of the offline
optimal algorithm. In both the algorithms, By the end of day 2, 50% of 1−ηi/γi
was achieved for agents of 60+ age group. By the end of day 8, only 10% of the
most vulnerable group remained unvaccinated.
Fig. 6: The 1−ηi/γivalue achieved by the online algorithm is very similar to
that of the offline algorithm across age groups. Both algorithm vaccinate achieves
vaccinate 90% of the most vulnerable group within 8 days.
6.4 Running Time Analysis
In Table 1 we compare the performance of the online algorithm and the offline
algorithm against the same dataset. We consider alpha values (0.96,0.97,0.99)
and (0.1,0.5,0.9). In both the cases, the online algorithm vaccinates almost the
same number of agents as that of the offline while algorithm achieving similar
total utility. The competitive ratio is 0.99. The online algorithm runs significantly
faster than the offline algorithm.
Fair Healthcare Rationing to Maximize Dynamic Utilities 19
Online
Algorithm
Offline
Algorithm
αvalue α1α2α1α2
δ0.95 0.95 0.95 0.95
Running time
(in sec) 319.04 336.55 888.90 806.65
Total
no. of agents
vaccinated
7154 7145 7192 7192
Total Utility 3567.95 1550.23 3580.68 1573.95
Table 1: The vector α1=
(0.96,0.97,0.99) and vector α2=
(0.1,0.5,0.9) represent the alpha
values for the three age groups . The
average competitive ratio is 0.99.
The average running time of the
online and the offline algorithms are
327.79 seconds and 847.77 seconds
respectively.
2 4 6 8 10 12 14 16
100
500
1000
2000
5000
10000
20000
1.95 ·10−2
0.12
0.14
0.36
0.91
1.69
5.68
1.91 ·10−2
0.14
0.23
0.59
1.81
4.43
13.76
Running time (in 50 sec)
Size of Instance (number of agents)
Online Algorithm Offline Algorithm
Fig. 7: Time taken by offline and online
algorithms (on synthetic datasets) vs
instance size
Comparing the running time of Algorithm 1 and the offline algorithm, Fig-
ure 7 shows that the online algorithm runs significantly faster than the offline
algorithm for all input sizes.
6.5 Performance Analysis
In Figure 8, we plot the number of agents of age group 18-45 getting vaccinated
by the online algorithm 1 on each day for alpha values 0.96 and 0.1. It is clear
that the vaccination follows almost identical pattern as long as the order of
alpha values remain the same. Figure 9 shows similar results for the optimal
offline algorithm. The independence on cardinal values shows that the algorithm
is practically useful as ordering the vulnerable groups is much more feasible than
assigning a particular value. Similar plots for other age groups are given in the
appendix.
In Figure 10, we plot the number of agents of age group 45-60 getting vacci-
nated by the online algorithm 1 on each day for alpha values 0.97 and 0.5. It is
clear that the vaccination follows almost identical pattern as long as the order
of alpha values remain the same. Figure 11 shows similar results for the opti-
mal offline algorithm. Figure 12 and Figure 13 plot similar results for the 60+
age group population. We note that in both online and the offline algorithm,
allocations of vaccines for the age group 60+ are higher in the initial days and
decreases with days. Most of the agents from this group are vaccinated by the
end of 10th day.
20 Authors Suppressed Due to Excessive Length
Fig. 8: Number of agents in the 60+ age group vaccinated by the online algorithm
for alpha-values 0.96 and 0.1respectively.
Fig. 9: Number of agents in the 60+ age group vaccinated by the offline algorithm
for alpha-values 0.96 and 0.1respectively.
Fair Healthcare Rationing to Maximize Dynamic Utilities 21
Fig. 10: Number of agents in the 45-60 age group vaccinated by the online algo-
rithm for alpha-values 0.97 and 0.5respectively.
Fig. 11: Number of agents in the 45-60 age group vaccinated by the offline algo-
rithm for alpha-values 0.97 and 0.5respectively.
22 Authors Suppressed Due to Excessive Length
Fig. 12: Number of agents in the 60+ age group vaccinated by the online algo-
rithm for alpha-values 0.99 and 0.9respectively.
Fig. 13: Number of agents in the 60+ age group vaccinated by the offline algo-
rithm for alpha-values 0.99 and 0.9respectively.
Fair Healthcare Rationing to Maximize Dynamic Utilities 23
7 Conclusion
We investigate the problem of dynamically allocating perishable healthcare goods
to agents arriving over a period of time. We capture various constraints while
allocating a scarce resource to a large population, like production constraint on
the resource, infrastructure and constraints. While we give an offline optimal
algorithm for Model 1, getting one for Model 2or showing NP-hardness remains
open. We also propose an online algorithm approximating welfare that elicits
information every day and makes an immediate decision. The online algorithm
does not require a foresight and hence has a practical appeal.Our experiments
show that the online algorithm generates a utility roughly equal to the utility of
the offline algorithm while achieving very little to no wastage.
References
1. Ahani, N., Gölz, P., Procaccia, A.D., Teytelboym, A., Trapp, A.C.: Dynamic place-
ment in refugee resettlement. arXiv preprint arXiv:2105.14388 (2021)
2. Ahuja, R., Magnanti, T., Orlin, J.: Network Flows: Theory, Algorithms, and Ap-
plications. Prentice Hall (1993)
3. Andersson, T., Ehlers, L.: Assigning refugees to landlords in sweden: Stable max-
imum matchings. Cahier de recherche 122(3), 937–965 (2016)
4. Andersson, T., Ehlers, L., Martinello, A.: Dynamic Refugee Matching. Working
Papers 2018:7, Lund University, Department of Economics (Mar 2018)
5. Aziz, H., Brandl, F.: Efficient, Fair, and Incentive-Compatible Healthcare Ra-
tioning, p. 103–104. Association for Computing Machinery, New York, NY, USA
(2021)
6. Aziz, H., Chen, J., Gaspers, S., Sun, Z.: Stability and pareto optimality in refugee
allocation matchings. In: Proceedings of the 17th International Conference on Au-
tonomous Agents and MultiAgent Systems. pp. 964–972 (2018)
7. Bansak, K., Ferwerda, J., Hainmueller, J., Dillon, A., Hangartner, D., Lawrence,
D., Weinstein, J.: Improving refugee integration through data-driven algorithmic
assignment. Science 359(6373), 325–329 (2018)
8. Biró, P., Fleiner, T., Irving, R.W., Manlove, D.F.: The college admissions problem
with lower and common quotas. Theoretical Computer Science 411(34), 3136–3153
(2010)
9. Bruce, L., Tallman, R.: Promoting racial equity in covid-19 resource allocation.
Journal of Medical Ethics 47(4), 208–212 (2021)
10. COIN-OR branch and cut solver. https://coin-or.github.io/Cbc/intro.html
(2022), accessed: 2022-08-15
11. Cilali, B., Barker, K., González, A.D.: A location optimization approach to refugee
resettlement decision-making. Sustainable Cities and Society 74, 103153 (2021)
12. https://www.covid19reservesystem.org/media (2022), accessed: 2022-01-14
13. Delacrétaz, D., Kominers, S.D., Teytelboym, A.: Refugee resettlement. University
of Oxford Department of Economics Working Paper (2016)
14. Delacrétaz, D., Kominers, S.D., Teytelboym, A.: Matching Mechanisms for Refugee
Resettlement. Working Papers 2019-078, Human Capital and Economic Opportu-
nity Working Group (Dec 2019)
24 Authors Suppressed Due to Excessive Length
15. Emanuel, E.J., Persad, G., Upshur, R., Thome, B., Parker, M., Glickman, A.,
Zhang, C., Boyle, C., Smith, M., Phillips, J.P.: Fair allocation of scarce medical
resources in the time of covid-19. New England Journal of Medicine 382(21), 2049–
2055 (2020)
16. Fink, S.: The hardest questions doctors may face: Who will be saved? who won’t?
The New York Times (Mar 2020)
17. Fleiner, T., Kamiyama, N.: A matroid approach to stable matchings with lower
quotas. Math. Oper. Res. 41(2), 734–744 (2016)
18. Gale, D., Shapley, L.S.: College admissions and the stability of marriage. The
American Mathematical Monthly 69(1), 9–15 (1962)
19. Gayle, H., Foege, W., Brown, L., Kahn, B. (eds.): Framework for Equitable Al-
location of COVID-19 Vaccine. National Academies of Sciences Engineering and
Medicine (2020)
20. Google maps platform. https: // developers. google. com/ maps (2022), ac-
cessed: 2022-08-15
21. Goto, M., Iwasaki, A., Kawasaki, Y., Kurata, R., Yasuda, Y., Yokoo, M.: Strat-
egyproof matching with regional minimum and maximum quotas. Artificial Intelli-
gence 235, 40–57 (2016)
22. Hamada, K., Iwama, K., Miyazaki, S.: The hospitals/residents problem with lower
quotas. Algorithmica 74(1), 440–465 (jan 2016)
23. Huang, C.: Classified stable matching. In: Proceedings of the Twenty-First An-
nual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010. pp. 1235–1253
(2010)
24. HV, Vishwa Prakash: Fair_healthcare_rationing, https: // github. com/
severus-tux/ Fair_ Healthcare_ Rationing
25. Jones, W., Teytelboym, A.: The international refugee match: A system that respects
refugees’ preferences and the priorities of states. Refugee Survey Quarterly 36(2),
84–109 (2017)
26. Jones, W., Teytelboym, A.: The local refugee match: Aligning refugees’ preferences
with the capacities and priorities of localities. Journal of Refugee Studies 31(2),
152–178 (2018)
27. Kamada, Y., Kojima, F.: Efficient matching under distributional constraints: The-
ory and applications. American Economic Review 105(1), 67–99 (Jan 2015)
28. Kamada, Y., Kojima, F.: Recent developments in matching with constraints. Amer-
ican Economic Review 107(5), 200–204 (May 2017)
29. Kavitha, T.: A size-popularity tradeoff in the stable marriage problem. SIAM Jour-
nal on Computing 43(1), 52–71 (2014)
30. Kojima, F.: New Directions of Study in Matching with Constraints, pp. 479–482.
Future of Economic Design (Nov 2019)
31. Lawler, E.L.: Combinatorial optimization: networks and matroids. Courier Cor-
poration (2001)
32. Moulin, H.: Fair division and collective welfare. MIT press (2004)
33. Nasre, M., Rawat, A.: Popularity in the generalized hospital residents setting. In:
Computer Science - Theory and Applications - 12th International Computer Sci-
ence Symposium in Russia. pp. 245–259 (2017)
34. Nguyen, H., Nguyen, T., Teytelboym, A.: Stability in matching markets with com-
plex constraints. Management Science 67(12), 7438–7454 (2021)
35. Pathak, P.A., Sönmez, T., Ünver, M.U., Yenmez, M.B.: Fair Allocation of Vac-
cines, Ventilators and Antiviral Treatments: Leaving No Ethical Value Behind in
Health Care Rationing. Boston College Working Papers in Economics 1015, Boston
College Department of Economics (Jul 2020)
Fair Healthcare Rationing to Maximize Dynamic Utilities 25
36. Persad, G., Peek, M.E., Emanuel, E.J.: Fairly Prioritizing Groups for Access to
COVID-19 Vaccines. JAMA 324(16), 1601–1602 (Oct 2020)
37. PuLP optimization. https: // coin-or. github. io/ pulp/ (2022), accessed:
2022-01-14
38. Ramsey, F.P.: A Mathematical Theory of Saving. The Economic Journal 38(152),
543–559 (12 1928)
39. Roth, A.E., Sotomayor, M.A.O.: Two-Sided Matching: A Study in Game-Theoretic
Modeling and Analysis. Econometric Society Monographs, Cambridge University
Press (1990)
40. Samuelson, P.A.: A Note on Measurement of Utility. The Review of Economic
Studies 4(2), 155–161 (02 1937)
41. Sankar, G.S., Louis, A., Nasre, M., Nimbhorkar, P.: Matchings with group fair-
ness constraints: Online and offline algorithms. In: Proceedings of the Thirtieth
International Joint Conference on Artificial Intelligence, IJCAI 2021,. pp. 377–
383 (2021)
42. Sayedahmed, D.: Refugee Settlement and Other Matching Problems with Prior-
ity Classes and Reserves: A Market Design Perspective. Ph.D. thesis, Concordia
University (2020)
43. Sönmez, T., Pathak, P.A., Ünver, M.U., Persad, G., Truog, R.D., White, D.B.:
Categorized priority systems: A new tool for fairly allocating scarce medical re-
sources in the face of profound social inequities. Chest 159(3), 1294–1299 (2021)
44. Truog, R.D., Mitchell, C., Daley, G.Q.: The toughest triage – allocating ventilators
in a pandemic. New England Journal of Medicine 382(21), 1973–1975 (2020)
45. WHO: A global framework to ensure equitable and fair allocation of covid-19 prod-
ucts and potential implications for covid-19 vaccines. Tech. rep., World Health
Organization (2020)
