Priorities vs. Quotas

Oguzhan Celebi∗

MIT

Joel P. Flynn†

MIT

August 18, 2021

Abstract

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: ocelebi@mit.edu†MIT Department of Economics, 50 Memorial Drive, Cambridge, MA 02142. Email: jpflynn@mit.eduWe are grateful to Daron Acemoglu, George-Marios Angeletos, Jonathan Cohen, Roberto Corrao,

Stephen Morris, Anh Nguyen, Parag Pathak, Karthik Sastry, Tayfun Sonmez, Alexander Wolitzky, andparticipants 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 measureof already admitted agents of the same attribute. Thus, our proposed mechanism requires only that theauthority is able to rank all students within each group at the point of assignment, and requires no knowledgeof 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 ofstable matchings.

4The slot-specific priority model of Kominers and Sonmez (2016) embeds these previous models. Furtherrelated 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 ∈ Ω, which the authority believes has distribution Λ ∈ ∆(Ω). 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 × Ω→ 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:

supφ

∫Ω

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) =

∫Θ

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) =

∫[0,1]

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:

Assumption 2. The authority’s utility function is separable:

ξ (sh, x) ≡ g

(sh +

∑m∈M

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 = Amm∈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., |Ω| = 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 xtarm 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

(x− βx

2x2

)(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 minκ, q > 1+γ−ω1κ

+γβx+κ(ω−ω) and κ(1−ω) < 1+γ−ω

1κ

+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,minκ, 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 − Qplaces 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 ∆ = κ2q

(1− κ

q γβx

)V[ω].

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

2 x2)

(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

2(1− κγβx)V[ω] (7)

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:

∆∗ ≡ minV ∗ − VP , V ∗ − VQ =

12

(κγβx)2 κV[ω]

1+κγβx, κγβx ≤ 1,

12

κV[ω]1+κγβx

, 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 κ2

(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

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

the authority is sufficiently risk-averse and 1κ< 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), whichcorresponds 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 selectioneffect is equivalent to the effect that price regulation gives rise to the greatest production in states wherethe firm’s marginal cost is lowest. Moreover, the guarantee effect is equivalent to the ability of quantityregulation 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 1κ≥ 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

2

(E[κ]− (V[κ] + E2[κ])γβx

)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 ∈ Ω and show that the claimed

subsidy schedule implements the same allocation. Fix an ω ∈ 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 ∈ Mthat 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 = xmm∈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 = xmm∈M as:

maxxm∈[0,xm], ∀m∈M

∑m∈M

∫ h(1)

sm(xm)

sfm(s)ds+∑m∈M

um(xm) s.t.∑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

considering the Lagrangian:

L(x, λ, κ, κ) =∑m∈M

∫ h(1)

sm(xm)

sfm(s)ds+∑m∈M

um(xm)

+ λ

(q −

∑m∈M

xm

)+∑m∈M

κm(xm − xm) +∑m∈M

κmxm

(13)

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

∂L∂xm

= −s′m(xm)sm(xm)fm(sm(xm)) + u′m(xm)− λ− κm + κm = 0 (14)

λ∂L∂λ

= λ

(q −

∑m∈M

xm

)= 0 (15)

κm∂L∂κm

= κm(xm − xm) = 0 (16)

κm∂L∂κm

= κmxm = 0 (17)

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

s′m(xm) = − 1

fm(sm(xm))(18)

Thus, we can simplify Equation 14 to:

∂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

∫ h(1)

maxminsC−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)

which is trivially satisfied for λ = sC . As q =∑

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:

κm = h(0) + u′m(xm)− λ (23)

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 = ymm∈M and the state ω:

Am(y, s;ω) ≡ h−1

(h(s) +

ξxm (sh(y, ω), y)

ξsh (sh(y, ω), y)

)(24)

where sh(y, ω) is the match quality index in state ω when the highest scoring y = ymm∈Magents 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 |Ω| = 1 and let x∗m denote the measure of attribute m agents in the

optimal allocation, with x∗ = x∗mm∈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 ≤ xtarm and u′m(xm) = 0 for xm > xtar

m where km is sufficiently large

for all m ∈ M and∑

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

m

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

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

∑m∈M xtar

m , with

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

ssds. Moreover, when we admit measure x minority

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

sds = κ(1− s).

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

all score ω. Thus, the total score is given by s = qω+ (1−ω)x− 12κx2 for 0 ≤ x ≤ minκ, 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

κ+ γβx

)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 = maxQ, 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 minκ, q and floored by 0.

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

15

minκ, q and:

UQ(Q) = qE[ω] + (1 + γ − E[ω]) minκ, q − 1

2

(1

κ+ γβx

)minκ, q2 (26)

Second, Q ∈ [κ(1− ω),minκ, q) and:12

UQ(Q) = qE[ω] + (1 + γ − E[ω])Q− 1

2

(1

κ+ γβx

)Q2 (27)

Third, Q ∈ (κ(1− ω), κ(1− ω)) and:

UQ(Q) = qE[ω] +

∫ ω

1−Qκ

((1 + γ − ω)Q− 1

2

(1

κ+ γβx

)Q2

)dΛ(ω)

+

∫ 1−Qκ

ω

((1 + γ − ω)κ(1− ω)− 1

2

(1

κ+ γβx

)(κ(1− ω))2

)dΛ(ω)

(28)

Finally, Q ≤ κ(1− ω) and:

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

2

(1

κ+ γβx

)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]1κ

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

1+γ−ω1κ

+γβx+ κ(ω − ω) and κ(1− ω) < 1+γ−ω

1κ

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

payoff is given by:

U ′Q(Q) =

∫ ω

1−Qκ

((1 + γ − ω)−

(1

κ+ γβx

)Q

)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]

1κ

+ γβx(31)

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

κ(1− ω) <1 + γ − ω

1κ

+ γβx(32)

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

16

which we have assumed. We therefore have that:

VQ = qE[ω] + (1 + γ − E[ω])Q∗ − 1

2

(1

κ+ γβx

)Q∗2 (33)

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

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

UP (α) = qE[ω] + (1 + γ − E[ω]) minκ, q − 1

2

(1

κ+ γβx

)minκ, q2 (34)

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

UP (α) = qE[ω] +

∫ 1+α−min qκ,1

ω

((1 + γ − ω) minκ, q − 1

2

(1

κ+ γβx

)minκ, q2

)dΛ(ω)

+

∫ ω

1+α−min qκ,1

((1 + γ − ω)κ(1 + α− ω)− 1

2

(1

κ+ γβx

)[κ(1 + α− ω)]2

)dΛ(ω)

(35)

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

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

2

(1

κ+ γβx

)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

κ+ γβx

)E[(x(α) + ε)2]

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

2

(1

κ+ γβx

)x(α)2 − 1

2

(1

κ+ γβx

)E[ε2]

(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

κ+ γβx

)x(α)2 +

κ

2(1− κγβx)V[ω] (38)

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

17

to check that this optimal α∗ indeed lies within this case, or equivalently that κ(1+α∗−ω) ≤minκ, 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 − ω1κ

+ γβx+ κ(ω − ω) < minκ, q

(39)

where the final inequality follows by our assumption that minκ, q > 1+γ−ω1κ

+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) =

∫ ω

1+α−min qκ,1

d

dα

((1 + γ − ω)κ(1 + α− ω)− 1

2

(1

κ+ γβx

)[κ(1 + α− ω)]2

)dΛ(ω)

= κ

∫ ω

1+α−min qκ,1

((1 + γ − ω)−

(1

κ+ γβx

)(m(α) + ε(ω))

)dΛ(ω)

(40)

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

x(α) >1 + γ − E[ω|ω ≥ 1 + α−min q

κ, 1]

1κ

+ γβx− E

[ε(ω)|ω ≥ 1 + α−min q

κ, 1]

(41)

for all x(α) ∈ [minκ, q−κ(E[ω]−ω),minκ, 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:

minκ, q − κ(E[ω]− ω) >1 + γ − ω

1κ

+ γβx− κ(E[ω]− ω) (42)

Which holds as we assumed that minκ, q > 1+γ−ω1κ

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

that:

∆ = VP − VQ =κ

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

κ+ γβx

)x(ω)2 (44)

18

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

x(ω) =κ(1 + γ − ω)

1 + κγβx(45)

Under our maintained assumptions, we have that:

x(ω) =κ(1 + γ − ω)

1 + κγβx≤ 1 + γ − ω

1κ

+ γβx+ κ(ω − ω) < minκ, q (46)

and:

x(ω) =κ(1 + γ − ω)

1 + κγβx≥ 1 + γ − ω

1κ

+ γβx> κ(1− ω) (47)

Thus, this policy is feasible. Substituting, we have that:

V ∗ = qE[ω] +1

2

E[κ(1 + γ − ω)2]

1 + κγβx(48)

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

VQ = qE[ω] +1

2

κ(1 + γ − E[ω])2

1 + κγβx, VP = VQ +

κ

2(1− κγβx)V[ω] (49)

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

LQ =1

2

κV[ω]

1 + κγβx(50)

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

LP = LQ −∆ =1

2(κγβx)

2 κV[ω]

1 + κγβx(51)

Enveloping over these losses yields the claimed formula.

4.4. Proof of Corollary 2

Proof. We show that a quota-second policy Q is equivalent to a priority subsidy of α(Q) = Qκ

.

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

by zero and capped by minκ, q. A priority policy α(Q) = Qκ

admits the highest-scoring

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

minκ, q. Thus, state-by-state, quota-second policy Q and priority subsidy α(Q) = Qκ

yield

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

19

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