How should authorities concerned with match quality, fairness, and
diversity, but un-

certain over the distribution of agents’ characteristics, allocate
a resource? We show

that, when preferences over these dimensions are separable, a new
monotone subsidy

schedule (MSS) mechanism requires no knowledge of the state and is
ex post optimal.

We rationalize the common priority and quota mechanisms as limits
of MSS when risk

aversion over diversity is low and high, respectively. Echoing
lessons from price vs.

quantity regulation (Weitzman, 1974), priorities positively select
agents over states,

while quotas guarantee a level of diversity, but MSS achieve
both.

∗MIT Department of Economics, 50 Memorial Drive, Cambridge, MA
02142. Email:

[email protected] †MIT Department of Economics, 50
Memorial Drive, Cambridge, MA 02142. Email:

[email protected] We are
grateful to Daron Acemoglu, George-Marios Angeletos, Jonathan
Cohen, Roberto Corrao,

Stephen Morris, Anh Nguyen, Parag Pathak, Karthik Sastry, Tayfun
Sonmez, Alexander Wolitzky, and participants in the MIT Theory
Lunch for helpful comments. First posted version: March 27,
2020.

1. Introduction

Authorities in charge of resource allocation in institutions often
face conflicting objectives.

On the one hand, they want to allocate resources to the most
suitable individuals to maximize

match quality or respect some notion of fairness. On the other
hand, they want to ensure that

those who receive the resources are diverse according to a variety
of characteristics such as

socio-economic background, religion, race, age, or gender. To this
end, such authorities have

broadly used two classes of policies: quotas,1 where a certain
portion of the resource is set

aside for given groups; and priority subsidies (or simply
priorities), where such individuals

are given higher scores than an underlying index. This motivates
two related questions:

What constitutes optimal policy in this setting? And when, if ever,
are priorities or quotas

optimal?

These questions are important because priorities and quotas are the
primary policies that

have been used for resource allocation in non-price contexts.
Quotas have been introduced

across a range of markets, for example: Chicago Public Schools
employs reserves for students

from different socio-economic groups at its competitive exam
schools; Boston Public Schools

used reserves for walk-zone students at all schools; and
universities such as University of

California, Davis instituted a quota system for minority students.
Priority subsidies have

also been widely employed, for example: the widely used New York
State Task Force on

Ventilator Allocation guidelines give differential priority to
agents with differing mortality

risk; church-run schools in the UK give explicit admissions points
to students from various

religious groups; the University of Michigan and the University of
Texas have used different

admissions scales for minority students; and the Vietnamese
university entrance exam has

given explicit exam points boosts to students from certain
disadvantaged groups. However,

there is currently no formal understanding of whether these
policies are optimal, how these

policies are different, or which policy an authority should
pursue.

These settings all have four features in common. First, the
individuals have an underlying

score (e.g., exam score, index of clinical need) that allocates
them property rights over the

resource. Second, the authority is endowed with some power to
affect these property rights by

designing a set of rules that transform the scores of individuals
with certain characteristics

(e.g., quotas or priority subsidies for certain groups). Third,
this set of rules is usually

designed when there is substantial uncertainty over the economy
(e.g., the score distribution

of the students, number of individuals and doctors who need
treatment during a pandemic).

Fourth, once the rules are designed, the authority implements an
outcome that is fair with

respect to the (transformed) scores in the sense that they never
allocate the resource to a

1We use quota as a general term that includes the widely used
reserve policies (see Definition 3).

1

lower score individual when individuals with higher score are not
allocated the resource.

In this paper, we therefore formulate and solve the optimal
mechanism design problem of

an authority who allocates a resource to agents who are
heterogeneous in their suitability for

the resource and other attributes. The authority cares separably
about an index of match

quality and the numbers of agents of different attributes who are
allocated the resource.

However, they are uncertain about the distribution of scores and
attributes in the population,

which varies arbitrarily across states of the world.

We introduce a new class of mechanisms, monotone subsidy schedules,
which proceed in

two steps. First, monotonically transform an agent’s score s when
they have attribute m into

a new score Am(ym, s) that depends on the measure of agents who
have the same attribute

and a higher score ym. Second, allocate the resource in order of
transformed scores until the

resource is exhausted. This mechanism is fair in the sense that,
for any set of agents with

the same attributes, the highest-scoring agents are allocated the
resource.2

Addressing our first question, we derive in closed-form a monotone
subsidy schedule

mechanism that implements the ex post optimal allocation in every
state of the world while

requiring only knowledge of the authority’s own preferences
(Theorem 1). We argue that

policymakers could use the class of fair mechanisms we propose to
improve outcomes relative

to priority and quota mechanisms which are, by contrast, generally
suboptimal.

This result not only suggests an improvement for policymakers
relative to priority and

quota mechanisms, but also allows us to answer our second question
and provide natural

sufficient conditions under which priorities and quotas are
optimal. This allows us to offer

the following simple rationalizations of priority and quota
policies (Corollary 1): first, if an

authority is certain of the composition of the population, then
both priorities and quotas

are optimal. Second, if an authority is risk-neutral or highly
risk-averse over the number of

assigned agents of different attributes, they can optimally use
priority or quota mechanisms,

respectively.

To both illustrate and develop the intuition behind these results,
we study a detailed

example that allows for a closed-form comparison of priorities,
quotas, and the optimal

subsidy schedule mechanism. We do this in the spirit of the seminal
analysis of Weitzman

(1974), who compares price and quantity regulation in product
markets. In the example,

the resource corresponds to seats at a school and there are two
groups of students (minority

and non-minority students). The authority is uncertain over the
relative scores of minority

and non-minority students, and has linear-quadratic preferences
over the scores of admitted

2This is equivalent to assigning agents in order of their
transformed scores evaluated at the measure of already admitted
agents of the same attribute. Thus, our proposed mechanism requires
only that the authority is able to rank all students within each
group at the point of assignment, and requires no knowledge of the
state.

2

students and the number of minority students admitted to the
school.

The preference of the authority between priority and quota
mechanisms is governed by

its risk aversion over the number of admitted minority students:
there is a cutoff value

such that quotas are preferred when risk-aversion exceeds this
threshold and priorities are

otherwise preferred (Proposition 1). On the one hand, quotas
guarantee a level of diversity by

mandating a minimal level of minority admissions. On the other
hand, priorities positively

select minority students across states of the world as relatively
more minority students

receive the resource in the states in which minority students have
relatively higher scores,

improving match quality. Monotone subsidy schedules optimally
exploit the guarantee effects

of quotas and the positive selection effects of priorities, and are
always optimal. Thus, our

paper shows how standard price-theoretic lessons regarding
instrument choice carry over to

markets without an explicit price mechanism.3 Finally, we leverage
the example to provide

insights into optimal precedence order design and a recent debate
regarding the allocation

of medical resources (Pathak, Sonmez, Unver, and Yenmez,
2021).

Related Literature Of most relevance to our analysis are the
studies of quotas by Ko-

jima (2012), who shows how affirmative action policies that place
an upper bound on the

enrollment of non-minority students may hurt all students, Hafalir,
Yenmez, and Yildirim

(2013) who introduce the alternative and more efficient minority
reserve policies, and Ehlers,

Hafalir, Yenmez, and Yildirim (2014) who generalize reserves to
accommodate policies that

have floors and ceilings for minority admissions.4 The issue of
priority design has also been

studied. Erdil and Kumano (2019) and Echenique and Yenmez (2015)
study the effect of a

certain class of substitutable priorities, while Celebi and Flynn
(2021) study how to opti-

mally coarsen underlying scores into priorities. Our focus on
comparing priorities, quotas,

and optimal mechanisms distinguishes our analysis from this
literature which considers the

properties of each policy in isolation and without an explicit
treatment of uncertainty.

Finally, our result regarding the optimal order in which to process
quotas (Corollary 2)

contributes to the literature that studies the effects of changes
in precedence order (Dur,

Kominers, Pathak, and Sonmez, 2018; Dur, Pathak, and Sonmez, 2020;
Pathak, Rees-Jones,

and Sonmez, 2020a,b). The difference between our paper and these is
that we analyze the

optimal precedence order under uncertainty when the level of quotas
is also under the control

of the authority.

3Spiritually, this builds on Azevedo and Leshno (2016) who
introduced the price-theoretic analysis of stable matchings.

4The slot-specific priority model of Kominers and Sonmez (2016)
embeds these previous models. Further related papers study quota
policies in university admissions in India (Sonmez and Yenmez,
2021, 2020a,b), in Germany (Westkamp, 2013) and in Brazil (Aygun
and Bo, 2021), and simultaneous processing of quotas (Delacretaz,
2020).

3

2. Optimal Mechanisms

An authority allocates a single resource of measure q ∈ (0, 1) to a
unit measure of agents.

Agents differ in their type θ ∈ Θ = [0, 1]×M comprising their
scores s ∈ [0, 1] and personal

attributes m ∈ M, where their score denotes their suitability for
the resource and M is

a finite set comprising potential attributes such as race, gender,
or socioeconomic status.

The true distribution of types is unknown to the authority. The
authority’s uncertainty

is paramaterized by ω ∈ , which the authority believes has
distribution Λ ∈ (). In

state of the world ω, the type distribution is Fω ∈ (Θ) with
density fω. An assignment

µ : Θ → {0, 1} specifies for any type θ ∈ Θ whether they are
assigned to the resource.

The set of possible assignments is U . An assignment is feasible if
it allocates no more than

measure q of the resource. A mechanism is an ω−measurable function
φ : (Θ) → U that

returns a feasible assignment for any possible distribution of
types. The authority is an

expected utility maximizer with Bernoulli utility ξ : U × → R.
Given a mechanism φ, let

µφ(ω) be the assignment in state of the world ω. A first-best
mechanism is any mechanism

that attains the value:

ξ(µφ(ω), ω)dΛ(ω) (1)

Is there a set of rules the authority can design without knowledge
of the state of the world

that yields the same value as a first-best mechanism? There is, of
course, no guarantee that

this is possible. Nevertheless, whenever the authority’s payoff
derives from match quality

and diversity, and is separable in these desiderata, we derive an
explicit first-best mechanism

that is fair,5 and can be implemented with no knowledge of ω on the
part of the authority.

To place some structure on preferences, we first assume that the
authority cares only

about (i) an index of match quality

sh(µ, ω) =

µ(s,m)h(s)dFω(s,m) (2)

for some continuous, strictly increasing function h : [0, 1]→ R+,
which determines the extent

to which the authority values agents with higher scores, and (ii)
the measure of agents of

each attribute allocated the resource x(µ, ω) = {xm(µ, ω)}m∈M

xm(µ, ω) =

µ(s,m)fω(s,m)ds (3)

5Agents with higher scores are allocated before agents with lower
scores and the same characteristics.

4

Assumption 1. There exists some ξ : R|M|+1 → R such that:

ξ(µ, ω) ≡ ξ (sh(µ, ω), x(µ, ω)) (4)

We next assume that the authority’s preferences are separable in
match quality and

diversity:

ξ (sh, x) ≡ g

um(xm)

) (5)

for some continuous, strictly increasing function g : R→ R and
differentiable, concave, and

weakly increasing functions um : R→ R for all m ∈M.

Here, um determines their preference for assigned agents of
attribute m, and g determines

their risk preferences over their utility over scores and diversity
across states of the world.

We maintain these assumptions throughout our analysis.

2.1. Monotone Subsidy Schedule Mechanisms Are Optimal

We now introduce a new and simple class of subsidy schedule
mechanisms that we will

demonstrate are first-best optimal.

Definition 1 (Subsidy Schedule Mechanisms). A subsidy schedule
mechanism A = {Am}m∈M,

where Am : R× [0, 1]→ R, transforms the score s of agents with
attribute m who have mea-

sure ym higher scoring agents of the same attribute into Am(ym, s).
Agents are allocated the

resource in order of their transformed scores Am(ym, s) until it
reaches capacity.

We will say that a subsidy schedule mechanism A is monotone when
Am(·, s) is a de-

creasing function for all m ∈ M, s ∈ [0, 1] and Am(ym, ·) is an
increasing function for all

m ∈M, ym ∈ R. Observe that monotone subsidy schedule mechanisms are
fair in the sense

that they preserve the ranking of agents within any attribute.
Moreover, they can be im-

plemented in the following “greedy” fashion: within each attribute
m, rank all agents in

order of their score and assign agents in order of their
transformed scores evaluated at the

measure of already admitted agents of the same attribute. Thus, A
can be specified ex ante

without any contingency on the unknown state, and all that is
required in the interim to

implement it is knowledge of the scores that individual agents have
– a necessary condition

for performing any form of prioritized assignment.

Theorem 1. The subsidy schedule mechanism Am(ym, s) ≡
h−1(h(s)+u′m(ym)) is monotone

and a first-best mechanism.

5

The proofs of all results are provided in Section 4. Observe that A
requires only that the

authority knows its preferences over match quality h and diversity
um.6 To gain intuition

for the form of this mechanism, suppose that the authority has
linear utility over scores

h(s) ≡ s. In this case, Am(ym, s) = s + u′m(ym), so an agent with
attribute m is awarded a

subsidy of u′m(ym) when there are ym higher scoring agents of the
same attribute, their direct

marginal contribution to the diversity preferences of the
authority. This is optimal, because

this subsidy precisely trades off the marginal benefit of
additional diversity with the marginal

costs of reduced match quality, which are constant. To generalize
this beyond linear utility

of scores, consider the following observation: we can map agents’
scores from s to h(s), and

consider the optimal subsidy mechanism in this space. As h is
monotone, this preserves

the ordinal structure of the optimal allocation, and the authority
has linear preferences over

h(s). Thus, in this transformed space, the optimal subsidy remains
additive and given by

u′m(ym). To find the optimal transformed score in the original
space, we simply invert the

transformation h and apply it to the optimal score in the
transformed space, yielding the

formula for the optimal mechanism in Theorem 1.

2.2. Rationalizing Priority and Quota Mechanisms

As we have discussed, the primary classes of mechanisms that have
been used in practice

are priority and quota mechanisms. Priority mechanisms give each
agent a priority based

on their score and personal attributes, and allocate the resource
in order of the priority.7

Quota mechanisms reserve some portion of the resource for agents
with different attributes

and allocate each portion in order of the score. Formally, we
define these mechanisms as:

Definition 2 (Priority Mechanisms). A priority mechanism P : Θ→ R
awards each student

θ ∈ Θ a priority P (θ), and then allocates the resource in order of
priority until measure q

has been allocated.

Definition 3 (Quota Mechanisms). A quota mechanism (Q,D) reserves
Qm measure of the

capacity for agents of each attribute m ∈ M such that ∑

m∈MQm ≤ q, with the remaining

capacity QR = q− ∑

m∈MQm allocated to a merit slot R in which all student types are
eligible.

The mechanism arranges these slots via a bijection D :M∪{R} → {1,
2, . . . , |M|+ 1} (the

precedence order). The mechanism then proceeds by allocating the
measure QD−1(k) agents of

attribute D−1(k) to the resource in ascending order of k until
measure q has been allocated.

In general, as Theorem 1 makes clear and the example in the next
section will demon-

strate, neither priority or quota mechanisms are optimal. This is
because they fail to adapt

6At the end of Section 4.1, we show how separability (Assumption 2)
is necessary for this conclusion. 7We allow priority mechanisms to
reverse the scores of agents with the same attribute, but this is
never

optimal as they always prefer to allocate to individuals with
higher scores, all else equal.

6

to the state of the world: P and (Q,D) are both fixed ex ante and
depend only on individual

characteristics. The subsidy schedule circumvents this issue by
using a rank-dependent score

adjustment which allows the mechanism to adapt to the state of the
world without needing

to know it.

Nevertheless, there are simple sufficient conditions on the
uncertainty and diversity prefer-

ences of the authority that allow us to rationalize priority and
quota mechanisms as optimal.

Corollary 1. The following statements are true:

1. If there is no uncertainty (i.e., || = 1), then there exist
first-best priority and quota

mechanisms.

2. If the authority is risk-neutral over the measures of assigned
agents of different at-

tributes (i.e., um(xm) is linear for all m ∈ M), then there exists
a first-best priority

mechanism given by P (s,m) = h−1(h(s) + u′m).

3. If the authority is highly risk-averse over the measures of
assigned agents of different

attributes around a diversity target (i.e., u′m(xm) ≥ km for xm ≤
xtarm and u′m(xm) = 0

for xm > xtarm where km is sufficiently large for all m ∈ M and
∑

m∈M xtarm < q), then

there exists a first-best quota mechanism in which Qm = xtarm and
D(R) = |M|+ 1.

This result formalizes the idea that the suboptimality of priority
and quota mechanisms

stems from their inability to adapt to the state. However, it also
provides conditions on

preferences such that this inability is not problematic. On the one
hand, if the authority

is risk-neutral over the measure of agents of different attributes,
then they can perfectly

balance their match quality and diversity goals without regard for
the state of the world as

there is a constant “exchange rate” between the two, so priorities
are optimal. On the other

hand, if the authority is highly risk-averse as to the prospect of
failing to assign xtar m agents

of attribute m, then a quota allows them to always achieve this
target level of assignment in

all states of the world while minimally sacrificing match
quality.

This offers the following simple rationalizations of priority and
quota policies. First, if

an authority is certain of the composition of the population, then
both priorities and quotas

are optimal. Second, if an authority is risk-neutral or highly
risk-averse over the measure of

assigned agents of different attributes, they can optimally use
priority or quota mechanisms,

respectively.

3. Priorities vs. Quotas: A Closed-Form Example

We now illustrate and clarify the intuition underlying these
results in a simple example

in which the welfare gains and losses from using priorities or
quotas can be derived in closed

form. To do so, we follow an intellectual approach similar to that
of Weitzman (1974) in

7

his seminal comparison of price and quantity mechanisms in product
markets. We use the

example to study optimal precedence order design and medical
resource allocation.

3.1. The Setting of the Example

A single school has capacity q. Students are of unit total measure
and either minority or

majority students. The authority has linear-quadratic preferences ξ
: R2 → R over students’

total scores s and the number of admitted minority students
x:8,9

ξ(s, x) = s+ γ

) (6)

where γ ≥ 0 indexes their general concern for admitting minority
students relative to en-

suring high scores and βx ≥ 0 indexes the degree of risk-aversion
regarding the measure of

admitted minority students. The minority students are of measure κ
and have a distribution

of underlying scores that is uniform over [0, 1]. The majority
students are of the residual

measure and all have common underlying score ω ∈ [ω, ω] ⊆ [0, 1].
Finally, we assume that

the affirmative action preference is neither too small nor too
large with the following two

conditions min{κ, q} > 1+γ−ω 1 κ

+γβx +κ(ω−ω) and κ(1−ω) < 1+γ−ω

1 κ

+γβx . These conditions ensure that

optimal affirmative action policies will neither be so large as to
award all slots to minority

students in some states nor so small that there is no affirmative
action in some states.

The authority can either implement a subsidy schedule mechanism
(which will be op-

timal), an additive priority subsidy mechanism, or a quota
mechanism. In this setting, a

subsidy schedule mechanism awards an additive score subsidy of A(y)
to a minority student

when measure y other minority students have higher scores, and then
allocates the school to

students in order of their transformed scores. An additive priority
subsidy α ∈ R+ increases

uniformly the scores of minority students for the purposes of
gaining admission: the score

used in admissions becomes uniform over [α, 1 + α]. The authority
then admits the highest

scoring measure q students. A quota policy Q ∈ [0,min{κ, q}] sets
aside measure Q of the

capacity for the minority students. The measure Q highest scoring
minority students are

first allocated to quota slots, and all other agents are then
admitted to the residual q − Q places according to the underlying
score.10

8Alternatively, if the authority cares about both the average score
and the proportion of minority students,

all of the analysis goes through. For example, Equation 7 becomes =
κ 2q

( 1− κ

q γβx

) V[ω].

9To nest this in our more general setting, set h(s) = s,
uminority(x) = γ ( x− βx

2 x 2 )

(which is weakly

increasing over the relevant region given our parametric
assumptions) and umajority ≡ 0. 10This corresponds to a precedence
order that processes quota slots first. We discuss the importance
of

precedence orders in section 3.3 together with how our model can
produce insights about their design.

8

3.2. Comparing Mechanisms

Let the authority’s expected utility be V ∗ under any first-best
optimal mechanism, VS

under an optimal subsidy schedule mechanism, VP under an optimal
priority mechanism,

and VQ under an optimal quota mechanism. The following proposition
characterizes the

relationships between these mechanisms:

Proposition 1. The following statements are true:

1. The comparative advantage of priorities over quotas is given
by:

≡ VP − VQ = κ

Thus, priorities are preferred to quotas if and only if:

1

κ ≥ γβx (8)

2. The monotone subsidy schedule mechanism A(y) = γ(1 − βxy) is
first-best optimal,

V ∗ = VS. The comparative advantage of subsidy schedule mechanisms
over priorities

and quotas is given by:

∗ ≡ min{V ∗ − VP , V ∗ − VQ} =

1 2

, κγβx > 1. (9)

Which is increasing in κγβx for κγβx ≤ 1, decreasing in κγβx for
κγβx > 1, and equals

zero when κγβx = 0.

To develop intuition for the comparative advantage of priorities
over quotas, observe

the following. First, a quota of Q admits measure Q minority
students in all states of

the world under our assumptions. However, a priority policy induces
variability in the

measure of admitted minority students across states of the world.
This costs a priority

policy κ 2

(1 + κγβx)V[ω] in payoff terms. Second, a priority policy
positively selects minority

admissions across states of the world. In the proof of the result,
we show that minority

admissions in state ω under the optimal priority policy are x(α, ω)
= x(α) + ε(ω) where

x(α) = κ(1 + α − E[ω]) and ε(ω) = κ (E[ω]− ω). Thus, the optimal
priority policy admits

more minority students when minority students score relatively well
and fewer when minority

students score relatively poorly. This benefits a priority policy
by −C[ω, ε(ω)] = κV[ω] in

payoff terms. Which is preferred then depends on the risk
preferences of the authority over

the measure of admitted minorities. If the authority is close
enough to risk-neutral and

9

1 κ > γβx, then priorities are strictly preferred as positive
selection dominates guarantees. If

the authority is sufficiently risk-averse and 1 κ < γβx, then
quotas are strictly preferred as the

guarantee effects dominate positive selection. Finally, the extent
of uncertainty V[ω] may

intensify an underlying preference but never determines which
regime is preferred.11

In this example, the optimal subsidy schedule is linear in the
minority students’ ranks,

with slope given by the authority’s risk aversion over minority
admissions. This allows the

subsidy schedule to optimally balance the positive selection and
guarantee effects, and imple-

ment the first-best allocation in every state. From this, we learn
that the loss from priority

and quota policies relative to the optimum is greatest when the
authority is indifferent be-

tween the two regimes. Echoing our rationalization from earlier,
the loss from restricting to

priority or quota policies is zero when the authority is
risk-neutral or there is no uncertainty

regarding relative scores, and decreases as the authority becomes
highly risk averse.

3.3. Optimal Precedence Orders

In this example so far, we modelled quotas by first allocating
minority students to quota

slots and then allocating all remaining students according to the
underlying score. However,

we could have instead allocated q−Q places to all agents according
to the underlying score

and then allocated the remaining Q places to minority students. The
order in which quotas

are processed is called the precedence order in the matching
literature and their importance

for driving outcomes has been the subject of a large and growing
literature (see e.g., Dur

et al., 2018, 2020; Pathak et al., 2020a). Our framework can be
used to understand which

precedence order is optimal, a question that has not yet been
addressed.

In this example, the same factors that determine whether one should
prefer priorities or

quotas determine whether one should prefer processing quotas second
or first. By virtue of

uniformity of scores, it can be shown in the relevant parameter
range that a priority subsidy

of α is equivalent to a quota policy of κα when the quota slots are
processed second. Thus,

the comparative advantage of priorities over quotas is exactly
equal to the comparative

advantage of processing quotas second over first. The intuition is
analogous: processing

quotas second allows for positive selection while processing quotas
first fixes the number of

admitted minority students. Thus, on the one hand, when the
authority is more risk averse,

they should process quota slots first to reduce the variability in
the admitted measure of

minority students. On the other hand, when they are less risk
averse, they should process

11There is in fact a formal mapping between in our setting and that
of Weitzman (1974), which corresponds to the comparative advantage
of prices over quantities. Mapping Weitzman’s C ′′−1 7→ κ, B′′ 7→
−γβm, V[α(θ)] 7→ V[ω], we have that Weitzman’s coincides with our
own. The positive selection effect is equivalent to the effect that
price regulation gives rise to the greatest production in states
where the firm’s marginal cost is lowest. Moreover, the guarantee
effect is equivalent to the ability of quantity regulation to
stabilize the level of production.

10

quotas second to take advantage of the positive selection effect
such policies induce. These

results are summarized in the following corollary:

Corollary 2. The optimal quota-second policy achieves the same
value as the optimal pri-

ority policy; quota-second policies are preferred to quota-first
policies if and only if 1 κ ≥ γβx.

3.4. Beyond Affirmative Action: Medical Resource Allocation

The lessons of this paper apply not only to affirmative action in
academic admissions, but

much more broadly to other settings in which centralized
authorities must allocate resources

to various groups. One prominent such context is the allocation of
medical resources during

the COVID-19 pandemic. An important issue faced by hospitals is how
to prioritize frontline

health workers (doctors, nurses and other staff) in the receipt of
scarce medical resources:

hospitals wish to both treat patients according to clinical need
and ensure the health of

the frontline workers needed to fight the pandemic. To map this
setting to our example,

suppose that the score s is an index of clinical need for a scarce
medical resource available

in amount q, the measure of frontline health workers is κ, and ω
indexes the level of clinical

need in the patients currently (or soon to be) treated by the
hospital, which is unknown.

The risk aversion of the authority γβx corresponds to both a fear
of not treating sufficiently

many frontline workers and excluding too many clinically needy
members of the general

population.

In practice, both priority systems and quota policies have been
used, as detailed exten-

sively by Pathak et al. (2021). The primary concern that has been
voiced is that if a priority

system is used, some attributes (or characteristics) may be
completely shut out of allocation

of the scarce resource and that this is unethical, so quotas should
be preferred. Our frame-

work can be used to understand this argument; if there is an
unusually high draw of ω, a

priority system would lead to the allocation of very few resources
to frontline workers, and

vice-versa. Our Proposition 1 implies that if the authority is very
averse to such outcomes

(γβx is high), quotas will be preferred and for exactly the reasons
suggested. However, we

also highlight a fundamental benefit of priority systems in
inducing positive selection in

allocation: when ω is high, it is beneficial that fewer resources
go to the less sick medical

workers and more to the relatively sicker general population. More
generally, as per Theorem

1, we argue that a subsidy schedule mechanism that awards frontline
workers a score subsidy

that depends on the number of more clinically needy frontline
workers could further improve

outcomes.

Finally, an important additional consideration in this context
arises if the hospital or

authority must select a regime (priorities or quotas) before it
understands the clinical need

of its frontline workers κ, after which it can decide exactly how
to prioritize these workers

11

or set quotas, but before ultimate demand for medical resources ω
is known. It follows from

Proposition 1 that the comparative advantage of priorities over
quotas is:

E[] = 1

) V[ω] (10)

Thus, an increase in uncertainty V[κ] regarding the need of
frontline workers leads to a greater

preference for quotas. This is for the reason that the volatility
in the number of frontline

workers is convex in κ, the sensitivity of the measure allocated to
medical workers to the

underlying demand for medical resources ω. This highlights a
further advantage of quotas in

settings where a clinical framework must be adopted in the face of
uncertainty regarding the

clinical needs of frontline workers, as was the case at the onset
of the COVID-19 pandemic.

4. Proofs

4.1. Proof of Theorem 1

Proof. We characterize the optimal allocation for each ω ∈ and show
that the claimed

subsidy schedule implements the same allocation. Fix an ω ∈ and
suppress the dependence

of Fω and fω thereon, and define the utility index of a score as s
= h(s) with induced densities

over s given by fm for all m ∈ M. Let the measure of agents with
any attribute m ∈ M that is allocated the resource be xm ∈ [0, xm]
where xm =

∫ h(1)

h(0) fm(s)ds. Observe that,

conditional on fixing the measures of agents of each attribute that
are allocated the resource

x = {xm}m∈M, there is a unique optimal allocation (i.e., ξ-maximal
µ). In particular, as g

and h are continuous and strictly increasing, the optimal
allocation conditional on x satisfies

µ∗(s, m;x) = 1 ⇐⇒ s ≥ sm(xm) for some thresholds {sm(xm)}m∈M that
solve:∫ h(1)

sm(xm)

fm(s)ds = xm (11)

We can then express the problem of choosing the optimal x = {xm}m∈M
as:

max xm∈[0,xm], ∀m∈M

∑ m∈M

xm ≤ q (12)

where a solution exists by compactness of the constraint sets and
continuity of the objective.

We can derive necessary and sufficient conditions on the
solution(s) to this problem by

12

∫ h(1)

sm(xm)

The first-order necessary conditions to this program are given
by:

∂L ∂xm

λ ∂L ∂λ

κm ∂L ∂κm

= κmxm = 0 (17)

for all m ∈M. By implicitly differentiating Equation 11, we obtain
that:

s′m(xm) = − 1

∂L ∂xm

= sm(xm) + u′m(xm)− λ− κm + κm = 0 (19)

Observe that all constraints are linear. Thus, if the objective
function is concave, the

first-order conditions are also sufficient. To this end, as all
cross-partial derivatives of the

objective function are zero, it suffices to check that ∂L ∂xm

is a decreasing function of xm for all

m ∈M. Observe by Equation 18 that sm(xm) is a decreasing function
of xm. Moreover u′m

is a decreasing function of xm by virtue of the assumption that um
is concave for all m ∈M.

Thus, the objective function is concave.

Thus, to verify that our claimed subsidy schedule is a first-best
mechanism, it suffices to

show that the allocation it implements satisfies Equations 14 to
17. The subsidy schedule

Am(ym, s) = h−1 (h(s) + u′m(ym))) in the transformed score space
yields transformed scores

h (Am(ym, s)) = s+ u′m(ym). Define xm as the admitted measure of
students of attribute m

under this mechanism. Agents of attribute m ∈ M are allocated the
resource if and only if

13

s+ u′m(xm) ≥ sC for some threshold sC that solves:

∑ m∈M

max{min{sC−u′m(xm),h(1)},h(0)} fm(s)ds = q (20)

We can therefore partition M into three sets that are uniquely
defined: (i) interior MI =

{m ∈ M|sC − u′m(xm) ∈ (h(0), h(1))}; (ii) no allocation M0 = {m ∈
M|sC − u′m(xm) ≥ h(1)}; (iii) full allocation M1 = {m ∈ M|sC −
u′m(xm) ≤ h(0)}. For all m ∈ M0, we

implement xm = 0. For all m ∈M1, we implement xm = xm. For all m
∈MI , we implement

xm ∈ (0, xm). For any m ∈MI , the allocation threshold is sm(xm) =
sC − u′m(xm). For any

m ∈M0, the allocation threshold is h(1). For any m ∈M1, the
allocation threshold is h(0).

We now verify that this outcome satisfies the established necessary
and sufficient condi-

tions. For all m ∈MI , by the complementary slackness conditions we
have that κm = κm =

0. Substituting the above into Equation 14 for all m ∈MI we obtain
that:

sC − λ = 0 (21)

m∈M xm, the complementary slackness

condition for λ is then satisfied. For all m ∈M0, by complementary
slackness we have that

κm = 0 and Equation 14 is satisfied by:

κm = λ− h(1)− u′m(0) (22)

For all m ∈ M1, by complementary slackness we have that κm = 0 and
Equation 14 is

satisfied by:

This completes the proof.

In Footnote 6, we comment that Assumption 2 is necessary for this
result. Following

the same steps as above but without Assumption 2 (while assuming
that ξ is differentiable

and weakly increasing), an optimal generalized subsidy schedule
necessarily depends on the

entire vector y = {ym}m∈M and the state ω:

Am(y, s;ω) ≡ h−1

) (24)

where sh(y, ω) is the match quality index in state ω when the
highest scoring y = {ym}m∈M agents of each attribute are allocated.
Observe that this generally depends on ω via the joint

14

distribution of attributes and scores and is therefore not
implementable without knowledge

of the state. Observe further that this collapses to the optimal
subsidy schedule we derive

when Assumption 2 is imposed.

4.2. Proof of Corollary 1

Proof. Part (i): Suppose || = 1 and let x∗m denote the measure of
attribute m agents in the

optimal allocation, with x∗ = {x∗m}m∈M. A priority policy P (s,m) =
h−1(h(s) + u′m(x∗m)) =

Am(x∗m, s) implements the same allocation as the optimal subsidy
schedule mechanism and

by Theorem 1, is optimal. A quota mechanism with (Q,D) where Qm =
x∗m implements x∗

for all D. Part (ii): When um is linear, u′m is constant and the
first-best optimal subsidy

schedule mechanism is a priority mechanism P (s,m) = h−1(h(s) +
u′m). Part (iii): When

u′m(xm) ≥ km for xm ≤ xtar m and u′m(xm) = 0 for xm > xtar

m where km is sufficiently large

for all m ∈ M and ∑

m∈M xtar m < q, observe that the optimal mechanism admits xm ≥
xtar

m

for all m ∈ M in all states of the world, but conditional on xm ≥
xtar m for all m ∈ M

admits the highest scoring set of agents. A quota Qm = xtar m and
QR = q−

∑ m∈M xtar

D(R) = |M|+ 1 implements this allocation and is first-best
optimal.

4.3. Proof of Proposition 1

Proof. Part (i): First, if we admit all minority students over some
threshold s, the total score

of admitted minority students is κ ∫ 1

s sds. Moreover, when we admit measure x minority

students where x ≤ min{κ, q}, this admissions threshold is defined
by x = κ ∫ 1

s ds = κ(1− s).

Thus, we have that s = 1− x κ . Finally, the residual measure q−x
admitted majority students

all score ω. Thus, the total score is given by s = qω+ (1−ω)x− 1 2κ
x2 for 0 ≤ x ≤ min{κ, q}.

As both quota and priority policies always admit the
highest-scoring minority students, the

authority’s utility is given by:

U = qE[ω] + E[(1 + γ − ω)x]− 1

2

( 1

) E[x2] (25)

We now derive the admitted measure of minority students. In the
absence of a priority

or quota policy, α = 0 or Q = 0, we have that x = κ(1 − ω) measure
minority students

are admitted. Thus, under a quota policy Q, measure x = max{Q, κ(1
− ω)} minority

students are admitted. Under a priority policy, the measure of
admitted minority students

is x = κ ∫ 1

ω−α dx = κ(1 + α− ω). In each case x is capped by min{κ, q} and
floored by 0.

The expected utility function over quotas is given by one of four
cases. First, Q >

15

min{κ, q} and:

UQ(Q) = qE[ω] + (1 + γ − E[ω]) min{κ, q} − 1

2

( 1

Second, Q ∈ [κ(1− ω),min{κ, q}) and:12

UQ(Q) = qE[ω] + (1 + γ − E[ω])Q− 1

2

( 1

UQ(Q) = qE[ω] +

2

( 1

UQ(Q) = qE[ω] + E [(1 + γ − ω)κ(1− ω)]− 1

2

( 1

) E [ (κ(1− ω))2] (29)

We claim that the optimum lies in the second case. See that in case
two the strict maximum

is attained at Q∗ = 1+γ−E[ω] 1 κ

+γβx ∈ (κ(1 − ω),min{κ, q}), by our assumptions that min{κ, q}
>

1+γ−ω 1 κ

+γβx + κ(ω − ω) and κ(1− ω) < 1+γ−ω

1 κ

+γβx . Moreover, in case three, the first derivative of the

payoff is given by:

) dΛ(ω) (30)

Thus, checking that the sign of this is positive amounts to
verifying that for all Q ∈ (κ(1− ω), κ(1− ω)), we have that:

Q < 1 + γ − E[ω|ω ≥ 1− Q

κ ]

1 κ

+ γβx (31)

As the RHS is an increasing function of Q, it suffices to show
that:

κ(1− ω) < 1 + γ − ω

1 κ

+ γβx (32)

12By our maintained assumptions we have that this interval has
non-empty interior.

16

VQ = qE[ω] + (1 + γ − E[ω])Q∗ − 1

2

( 1

) Q∗2 (33)

We now turn to characterizing the value of priorities. There are
three cases to consider.

First, when κ(1 + α− ω) ≥ min{κ, q} we have that x = min{κ, q}
and:

UP (α) = qE[ω] + (1 + γ − E[ω]) min{κ, q} − 1

2

( 1

) min{κ, q}2 (34)

Second, when κ(1 + α− ω) ≥ min{κ, q} ≥ κ(1 + α− ω) we have
that:

UP (α) = qE[ω] +

ω

2

( 1

2

( 1

) dΛ(ω)

(35)

Finally, when min{κ, q} ≥ κ(1 + α− ω), we have that:

UP (α) = qE[ω] + E[(1 + γ − ω)κ(1 + α− ω)]− 1

2

( 1

) E[(κ(1 + α− ω))2] (36)

We claim that the optimum under our assumptions lies only the third
case. First, we argue

that there is a unique local maximum in the third case. Second, we
show the value in the

second case is decreasing in α. By continuity, the unique optimum
then lies in the third case.

First, it is helpful to write x(α) = κ(1 + α − E[ω]) and ε = κ
(E[ω]− ω). The value in

the third case can then be re-expressed as:

UP (α) = qE[ω] + E[(1 + γ − ω) (x(α) + ε)]− 1

2

( 1

= qE[ω] + (1 + γ − E[ω])x(α)− E[ωε]− 1

2

( 1

(37)

Finally, we have that E[ε2] = κ2V[ω] and E[ωε] = C[ω, ε] = −κV[ω].
Thus:

UP (α) = qE[ω] + (1 + γ − E[ω])x(α)− 1

2

( 1

2 (1− κγβx)V[ω] (38)

We then see that the optimal α∗ in this range sets x(α∗) = Q∗ <
min{κ, q}. It remains only

17

to check that this optimal α∗ indeed lies within this case, or
equivalently that κ(1+α∗−ω) ≤ min{κ, q}. To this end, see that κ(1
+ α∗ − E[ω]) = Q∗, and:

κ(1 + α∗ − ω) = Q∗ + κ(E[ω]− ω) ≤ Q∗ + κ(ω − ω)

≤ 1 + γ − ω 1 κ

+ γβx + κ(ω − ω) < min{κ, q}

(39)

where the final inequality follows by our assumption that min{κ, q}
> 1+γ−ω 1 κ

+γβx + κ(ω − ω).

Second, in the second case we have that the first derivative of the
payoff in α is given by:

U ′P (α) =

d

dα

2

( 1

( (1 + γ − ω)−

) dΛ(ω)

(40)

Checking that the sign of this is negative for all α such that κ(1
+ α − ω) ≥ min{κ, q} ≥ κ(1 + α− ω) then amounts to checking
that:

x(α) > 1 + γ − E[ω|ω ≥ 1 + α−min{ q

κ , 1}]

1 κ

+ γβx − E

κ , 1} ]

(41)

for all x(α) ∈ [min{κ, q}−κ(E[ω]−ω),min{κ, q}−κ(E[ω]−ω)]. So it
suffices to check that

the minimal possible value of the LHS exceeds the maximal possible
value of the RHS. A

sufficient condition for this is that:

min{κ, q} − κ(E[ω]− ω) > 1 + γ − ω

1 κ

+ γβx − κ(E[ω]− ω) (42)

Which holds as we assumed that min{κ, q} > 1+γ−ω 1 κ

+γβx + κ(ω − ω). We have now established

that:

2 (1− κγβx)V[ω] (43)

Part (ii): By Theorem 1, we have that A(y) = u′(y) = γ(1 − βxy) is
first-best optimal.

See in state ω that the payoff from admitting x(ω) minority
students is given by:

qω + (1 + γ − ω)x(ω)− 1

2

( 1

Thus, the x(ω) that solves the FOC is given by:

x(ω) = κ(1 + γ − ω)

x(ω) = κ(1 + γ − ω)

1 κ

and:

1 κ

V ∗ = qE[ω] + 1

1 + κγβx (48)

We have already shown that the value functions of priorities and
quotas are given by:

VQ = qE[ω] + 1

1 + κγβx , VP = VQ +

2 (1− κγβx)V[ω] (49)

We can therefore compute the loss from restricting to quota
policies:

LQ = 1

1 + κγβx (50)

To find the loss from restricting to priority policies, we
compute:

LP = LQ − = 1

4.4. Proof of Corollary 2

.

A quota-second policy admits the highest-scoring x = κ(1−ω)+Q
minority students, floored

by zero and capped by min{κ, q}. A priority policy α(Q) = Q κ

admits the highest-scoring

x = κ(1 + α(Q) − ω) = κ(1 − ω) + Q minority students, floored by
zero and capped by

min{κ, q}. Thus, state-by-state, quota-second policy Q and priority
subsidy α(Q) = Q κ

yield

the same allocation. The claims then follow from Proposition
1.

19

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Rationalizing Priority and Quota Mechanisms

Priorities vs. Quotas: A Closed-Form Example

The Setting of the Example

Comparing Mechanisms

Proofs