# Allocating Opportunities in a Dynamic Model of Intergenerational Mobility

Hoda Heidari
Carnegie Mellon University
<EMAIL_ADDRESS>
Jon Kleinberg
Cornell University
<EMAIL_ADDRESS>

###### Abstract

Opportunities such as higher education can promote intergenerational mobility,
leading individuals to achieve levels of socioeconomic status above that of
their parents. We develop a dynamic model for allocating such opportunities in
a society that exhibits bottlenecks in mobility; the problem of optimal
allocation reflects a trade-off between the benefits conferred by the
opportunities in the current generation and the potential to elevate the
socioeconomic status of recipients, shaping the composition of future
generations in ways that can benefit further from the opportunities. We show
how optimal allocations in our model arise as solutions to continuous
optimization problems over multiple generations, and we find in general that
these optimal solutions can favor recipients of low socioeconomic status over
slightly higher-performing individuals of high socioeconomic status — a form
of socioeconomic affirmative action that the society in our model discovers in
the pursuit of purely payoff-maximizing goals. We characterize how the
structure of the model can lead to either temporary or persistent affirmative
action, and we consider extensions of the model with more complex processes
modulating the movement between different levels of socioeconomic status.

##  1 Introduction

Intergenerational mobility — the extent to which an individual’s socioeconomic
status differs from the status of their prior generations of family members —
has emerged as a central notion in our understanding of inequality. A large
amount of empirical work has gone into estimating the extent of mobility for
different subsets of society; while many of the effects are complex and
challenging to measure, two broad and fairly robust principles emerge from
this work. First, socioeconomic status is persistent across generations: an
individual’s socioeconomic status is strongly dependent on parental status. As
Lee and Solon (2009) write in the opening to their survey of this topic, “Over
the past two decades, a large body of research has documented that the
intergenerational transmission of economic status in the United States is much
stronger than earlier sociological and economic analyses had suggested”.
Second, certain types of opportunities can serve as strong catalysts for
socioeconomic mobility; a canonical example is higher education, which has the
potential to raise an individual’s socioeconomic status (and, by the previous
principle, that of their current or future children as well). As Chetty et al.
(2014) write, “The fact that the college attendance is a good proxy for income
mobility is intuitive given the strong association between higher education
and subsequent earnings”.

An important question from a social planning perspective is thus the choice of
policy for allocating opportunities to people of different levels of
socioeconomic status. (Again, we can think of access to higher education as a
running example in this discussion.) Many goals can motivate the choice of
policy, including the reduction of socioeconomic inequality and the
prioritization of opportunities to those most in need. Such goals are often
viewed as operating in tension with the aim of maximizing the achievable
payoff from the available opportunities, which would seem to suggest targeting
the opportunities based only on the anticipated performance of the recipient,
not their socioeconomic status. In this view, society is implicitly being
asked to choose between these goals; this consideration forms a central
ingredient in the informal discourse and debate around the allocation of
opportunity. But through all of this, a challenging question remains: to what
extent is the tension between these goals genuine, and to what extent can they
be viewed as at least partially in alignment?

A large body of work in economics compares various allocation policies in
terms of the above seemingly-competing criteria — typically in simplified
settings in which only two generations are considered. The literature includes
seminal work by Nobel Laureate Garry Becker with Nigel Tomes (Becker and
Tomes, 1986) and by Glenn Loury (Loury, 1981). In multigenerational settings,
however, deriving the optimal policy becomes exceedingly challenging, and it
has been highlighted as a class of open questions in this literature. For
example, in his work on models of college admissions and intergenerational
mobility, Durlauf (2008) notes: “ A college admissions rule has
intergenerational effects because it not only influences the human capital of
the next generation of adults, but also affects the initial human capital of
the generation after next. […] Efficiency in student allocation [in this case]
is far more complicated than before. I am unaware of any simple way of
describing efficiency conditions for college assignment rules analogous to
[the above setting].” In this work, we address this challenge and the
associated open questions concerning the behavior of multigenerational models.
A key ingredient in our progress on these questions is the development of
methods for working with a class of Markov Decision Processes that operate
over continuous states and continuous actions. Our analysis of
multigenerational models enables us to investigate the apparent tension
between efficiency and fairness considerations in allocating opportunities.

####  Allocating Opportunities in a Payoff-Maximizing Society.

We work with a simple mathematical model representing a purely payoff-
maximizing society, operating over multiple generations. As we discuss briefly
in Section 1 and at more length in Appendix A, our model is grounded in the
types of models proposed in economic theory work on these problems. The
society must decide how to allocate opportunities in each generation across a
population heterogeneous in its socioeconomic status. The payoff to the
society is the total performance of everyone who receives the opportunities,
summed (with discounting) over all generations. Although the set-up of the
model is highly streamlined, the analysis of the model becomes quite subtle
since society must solve a continuous-valued dynamic programming problem over
multiple generations.

What we find from the model is that the optimal solution will in general tend
to offer opportunities to individuals of lower socioeconomic status over
comparable individuals of higher socioeconomic status, even when these
competing individuals are predicted to have a slightly better performance from
receiving the opportunity. This is not arising because the optimal solution
has any a priori interest in reducing socioeconomic inequality (although such
goals are important in their own right (Forde-Mazrui, 2004)); rather it is
strictly trying to maximize payoff over multiple generations. But given two
individuals of equal predicted performance, the one with lower socioeconomic
status confers an added benefit to the payoff function: their success would
grow the size of the socioeconomically advantaged class, resulting in higher
payoffs in future generations. Because the difference in payoff contributions
between these two individuals is strictly positive, the same decision would be
optimal even if the individual of lower socioeconomic status had a slightly
lower predicted performance from receiving the opportunity. The optimal
solution should still favor the candidate with lower status in this case.

In other words, the society in this model discovers a form of socioeconomic
affirmative action in allocating opportunities, based purely on payoff-
maximizing motives. The model thus offers a view of a system in which reducing
inequality is compatible with direct payoff maximization. In this sense, our
results belong to a genre of analyses (popularized by Page (Page, 2008) and
others) asserting that policies and interventions that we think of as
motivated by equity concerns, can also be motivated by purely performance-
maximizing considerations: even if society only cares about performance, not
equity, it should still (at least in the underlying models) undertake these
policies. In addition to providing a purely utilitarian motivation for
socioeconomic affirmative action, our model provides novel insights regarding
the shape and extent of effective affirmative action policies by specifying
the way in which criteria for receiving the opportunity should be adjusted
based on socioeconomic status to maximize society’s performance across
multiple generations.

We now give a rough overview of the model and results; a complete description
of the model is provided in the following section.

####  A Model for Allocating Opportunities.

We consider a population that is partitioned into two groups of different
socioeconomic status: $D$ (disadvantaged), consisting of a $\phi_{0}$ fraction
of the population, and $A$ (advantaged), consisting of a $\phi_{1}=1-\phi_{0}$
fraction of the population. Each agent $i$ (from either group) has an ability
$a_{i}$ drawn uniformly at random from the interval $[0,1]$,

Society has the ability to offer an opportunity to an $\alpha$ fraction of the
population. Note that the parameter $\alpha$ specifies the inherent limitation
on the amount of opportunities available. Since opportunities are limited, the
society has to wrestle with the question of how to allocate them. An
individual $i$ in group $D$ who is offered the opportunity has a probability
$\sigma a_{i}$ of succeeding at it, for a parameter $0<\sigma<1$. An
individual $i$ in group $A$ who is offered the opportunity has a probability
$\sigma a_{i}+\tau$ of succeeding at it, for the same $\sigma$ and an
additional parameter $0<\tau\leq 1-\sigma$ reflecting the advantage. We will
refer to the above quantities as the success probabilities of the agents.
Success probabilities reflect various levels of performance when agents are
offered the opportunity.

Anyone in group $D$ who is offered the opportunity and succeeds at it moves up
to group $A$. Each individual is then replaced by one offspring of the same
socioeconomic status and the process continues to the next generation. In the
general form of the model, there is also some probability that an individual’s
offspring does not perfectly inherit their socioeconomic status. The payoff to
society is the number of individuals who succeed at the opportunity summed
over all generations, with the generation $t$ steps into future multiplied by
$\gamma^{t}$ for a discount factor $0<\gamma<1$.

####  Summary of Results.

In any given generation, society’s policy will consist of a threshold for
group $D$ and a (possibly different) threshold for group $A$: the opportunity
is given to every individual whose success probability is above the threshold
for their group. The optimal policy is given by a dynamic program over the
continuous set of all possible choices for the population composition
$(\phi_{0},\phi_{1})$ as state variables. We solve the most basic version of
the model analytically. We computationally solve more complex versions of the
model by discretizing the state space, then applying standard dynamic
programming solutions for finite decision processes.

If the problem of allocating the opportunity only spanned a single generation,
then the payoff-maximizing policy would use the same threshold for both
groups. But given the discounting sum over multiple generations, we find that
society’s optimal policy can, in general, use a lower threshold for group $D$
than for group $A$. The difference in thresholds is a form of socioeconomic
affirmative action, and it arises due to the intuition discussed above:
boosting the number of individuals from group $D$ who receive the opportunity
will increase the number of available candidates from group $A$ in future
generations, each of whom provides a (discounted) payoff in future generations
via their enhanced performance. Finding the correct trade-off in allocating
opportunity thus involves a delicate balance between immediate and future
utility.

(a)

(b)

(c)

Figure 1:  Visualizing the triples $(\alpha,\tau,\gamma)$ for which the
optimal policy uses persistent affirmative actions. (We set $\sigma=1-\tau$
for these plots.) Points that lie above the surfaces in panels (a) and (c),
and below the surface in panel (b), correspond to parameter values yielding
persistent affirmative action. (1(a)) When $1-\frac{\alpha\sigma}{\tau}<0$,
any $\gamma>0$ suffices for persistent affirmative action and eventually
moving the entire population to group $A$. When
$1-\frac{\alpha\sigma}{\tau}>0$, there exists some $\gamma<1$ (and hence a
finite level of patience $\left(\frac{\gamma}{1-\gamma}\right))$ that suffices
for persistent affirmative action. (1(b)) When $\tau$ is sufficiently large,
the optimal policy does not use persistent affirmative action; this is because
for a large $\tau$, the extent of affirmative action required to pick up the
best performing members of $D$ is large—which in turn significantly reduces
the immediate payoff. For any given value of $\alpha$, there exists a
sufficiently small $\tau$ that guarantees persistent affirmative action.
(1(c)) When $\alpha$ is small relative to $\tau$, the optimal policy does not
use persistent affirmative action; this is because the cost of picking the
best performing members of $D$ is very high and a small $A$ group suffices for
filling the available opportunities. Note that for some values of $\tau$, no
matter how large $\alpha$ is, the optimal policy never employs persistent
affirmative action.

Whether socioeconomic affirmative action is employed by the optimal solution —
and the extent to which it is employed — depends on the fraction $\phi_{0}$ of
individuals from group $D$; in the most basic model, the amount of affirmative
action decreases monotonically as $\phi_{0}$ is reduced. The extent of
affirmative action is also determined by the amount of opportunity available
($\alpha$), the dependence of success on ability and socioeconomic status
($\sigma$ and $\tau$), and society’s patience in trading off immediate payoff
in return for payoff from future generations ($\gamma$). We characterize the
optimal solution in this respect as a function of these parameters, finding
that for some regions of the parameter space, the society employs temporary
affirmative action, reducing the size of group $D$ to a given level before
equalizing thresholds in subsequent generations; in other parts of the
parameter space, the society employs persistent affirmative action, in which
the threshold for group $D$ is strictly lower in every generation and the size
of group $D$ converges to 0 over time.

Figure 1 provides some ways of describing the regions of parameter space in
which the optimal solution uses persistent affirmative action. As the
partitions of the space there make apparent, the interactions among the key
parameters is fairly subtle. First, persistent affirmative action is promoted
by large values of $\alpha$ and small values of $\tau$, since these make it
easier to include high-performing members of group $D$ without a large
difference in thresholds; and it is promoted by larger values of $\gamma$,
indicating greater concern for the payoffs in future generations. One might
have suspected that persistent affirmative action would only be realized in
the optimal solution in the limit as society’s patience (essentially
$\gamma/(1-\gamma)$) goes to infinity; but in fact, a sufficiently large
finite amount of patience is sufficient for the optimal policy to use
persistent affirmative action.

In our model, we include a probabilistic background process by which
individuals can also move between groups $A$ and $D$; this reflects the idea
that there are many mechanisms operating simultaneously for socioeconomic
mobility, and we are studying only one of these mechanisms via the opportunity
under consideration. The most basic version posits a single probability $p$
that each individual independently loses their group membership and re-samples
it from the current distribution of group sizes. We also consider a version of
the model in which this probability of loss of group membership is different
for groups $A$ and $D$; in this case, we are only able to solve the model
computationally, and these computational results reveal interesting non-
monotonicities in the amount of affirmative action employed as a function of
the relative size of group $D$ ($\phi_{0}$).

####  Utilitarianism, Prioritarianism, and the Desert Principle.

Our simple mathematical model allows us to represent and distinguish among
several distinct worldviews toward allocation policies (see, e.g., (Arneson,
2013) for further discussion of these views): (1) a _utilitarian_ view, which
generally favors slightly lower-ability members of $A$ to comparable, but
slightly higher ability members of $D$ in pursuit of maximizing social utility
and productivity (recall that membership in $A$ confers a boost in success
probability); (2) a _prioritarian_ view, which evaluates a policy according to
its impact on the well-being of the worse-off members of society. Our model
can capture the priority view through large discount factors (recall that as
the society’s patience increases, it effectively increases the priority
assigned to the disadvantaged group members), or by adjusting the welfare
function; (3) a _desert-principle_ view, which advocates for allocating
opportunities based on some notion of deservingness. Deservingness in this
view is often defined in terms of the contributions people make to the social
utility. Hence success probability in our model is arguably the closest match
to individual desert. With that definition for desert, desert-based principles
would allocate opportunities myopically in each generation. As our analysis
illustrates, such policies often fail to maximize the social utility in the
long-run.

####  Limitations and Interpretations.

Our model is designed to incorporate the basic points we just mentioned in as
simplified a fashion as possible; as such, it is important to note some of its
key limitations. First, it is intended to model the effect of a single
opportunity, and it treats other forms of mobility probabilistically in the
background. It also assumes that the fundamental parameters
($\alpha,\sigma,\tau,\gamma$) are constant over all generations as well as
over individuals within one generation. It treats an individual’s group
membership ($A$ and $D$) and ability as a complete description of their
performance, rather than including any dependence on the group membership of
the individual’s parent. (That is, an individual in group $A$ performs the
same in the model regardless of whether their parent belonged to group $A$ or
$D$.) All of these would be interesting restrictions to relax in an extension
of the model. Second, much of the past theoretical work on intergenerational
mobility focuses on an issue that we do not consider here: the strategic
considerations faced by parents as they decide how much to consume in the
present generation and how much to pass on to their children. Our interest
instead has been in the optimization problem faced by a social planner in
allocating opportunities, treating the behavior of the agents as fixed and
simple. Here too, it would be interesting to explore models that address these
issues in combination. Finally, because our focus is on intergenerational
mobility in a socioeconomic sense, we do not model discrimination based on
race, ethnicity, or gender, and the role of race- or gender-based affirmative
action in combatting these effects. The model is instead concerned with
_socio-economic_ or _class-based_ (Malamud, 1995; Kahlenberg, 1996)
affirmative action. That said, the ingredients here could be combined with
models of statistical or taste-based discrimination on these attributes to
better understand their interaction (as outlined in Section 5).

The simplicity of our model, however, does allow us to make a correspondingly
fundamental point: that even a purely payoff-maximizing society can discover
affirmative action policies from first principles as it seeks to optimize the
allocation of opportunities over multiple generations. Moreover, the optimal
allocation policy is deeply connected to dynamic programming over the
generations; the society is essentially attempting to “steer” the balance of
group $A$ and group $D$ over time, making sure not to turn things too abruptly
(giving up present benefit) or too gradually (giving up future benefit). This
idea that society is searching for a way to turn optimally toward a better
outcome is not specific to our model; it is an image that has arisen in
qualitative discourse over several centuries. It can be seen in a quote
popularized by Martin Luther King, that “the arc of the moral universe is
long, but it bends toward justice” (Cohen, 2006). Interestingly, the original
form of this quote, by the American minister Theodore Parker in 1853, has an
even more abstractly mathematical flavor: “I do not pretend to understand the
moral universe; the arc is a long one, my eye reaches but little ways. I
cannot calculate the curve and complete the figure by the experience of sight;
I can divine it by conscience. And from what I see I am sure it bends towards
justice” (Parker, 1853). It is a curiously apt image for the way in which our
optimal solutions gradually turn through the state space to reshape the
distribution of socioeconomic groups, and it can be seen as added motivation
for the issues at the heart of the model.

### Related Work

Here, we briefly mention several lines of scholarship that are closely related
to our work. See Appendix A for a more in-depth discussion.

####  Long-term Implications of Fair ML.

Several recent articles study the long-term impact of ML-based decision-making
and fairness interventions on society, including the enforcement of
statistical parity in hiring (Hu and Chen, 2018), and responses by individuals
and populations to an ML-based decision rule (Liu et al., 2018; Mouzannar et
al., 2019; Kannan et al., 2019). Liu et al. (2018), for example, study the
conditions under which the choices of a myopic profit-maximizing institution
(e.g., a bank lending money to individuals) work in the interest of the
disadvantaged group. Dong et al. (2018); Hu et al. (2019); Milli et al. (2019)
address _strategic classification_ where the goal is to design classifiers
robust to strategic manipulation. This body of research focuses on strategic
responses by agents being evaluated; in contrast, we assume idealized
prediction so as to focus on the perspective of a social planner who seeks
optimal allocation of opportunities over time.

####  Intergenerational Income Mobility.

A substantial literature in economics studies how higher inequality results in
lower income mobility across generations (see, e.g., (Becker and Tomes, 1979;
Maoz and Moav, 1999; Corak, 2013)). The precise measurements of inequality and
mobility significantly influence the strength of this effect (see, e.g.,
(Solon, 1992; Piketty, 2000)). Theoretical models of income mobility have
studied utility-maximizing parents deciding how much of their capital to
consume and how much of it to invest in their offspring (Becker and Tomes,
1986; Becker and Tomes, 1979; Loury, 1981; Solon, 1999). We deliberately set
aside parental strategic considerations and focus instead on deriving the
optimal policy that maximizes the discounted payoff over generations.

####  Affirmative Action Policies.

A rich body of work in economics investigates statistical discrimination
(Arrow, 1973) and the role of affirmative action in redressing it (Fang and
Moro, 2011). Outcome-based policies including affirmative action targets have
long been proposed and implemented as temporary remedies to eliminate group-
level inequalities. Race-based affirmative action and socioeconomic
affirmative action can be viewed as distinct categories of intervenions
(Kahlenberg, 1996; Carnevale and Rose, 2013), with the former addressing long-
term effects of racial bias and the latter facilitating access for
economically disadvantaged individuals (Reardon et al., 2017). While the
relationship between them is complex and contested in the literature (Kane,
1998; Reardon et al., 2006; Gaertner and Hart, 2013), they can co-exist
without necessarily competing.

####  Affirmative Action in College Admissions and Comparison with (Durlauf,
2008).

Durlauf (2008) provides a model to compare the equality and efficiency of
affirmative action policies with those of meritocratic policies in the context
of admission rules to public universities. In his concluding remarks, Durlauf
poses the key question our work sets out to answer: how do multigenerational
considerations impact the optimal allocation policy? As we address this open
question that he poses, we follow his model in many respects, although we
depart from it in a few key areas. 111For a more detailed comparison, see
Section A.4. Similar to our work, Durlauf provides a condition under which
efficiency and equality considerations are aligned, but he focuses on settings
where a diverse body of students on college campuses improves the human
capital development for all of them. Durlauf assumes the policymaker aims to
maximize the level of human capital among _the next generation_ of adult
citizens. He restricts attention to two generations only, and instead of
solving for the optimal policy, he compares the meritocratic policy with
affirmative action in terms of the average human capital of the next
generation. In contrast, we model the dynamics of human-capital development
across _multiple generations_ and derive the _optimal allocation policy_.
Moreover, groups in model corresponds to _socio-economic tiers_ , whereas
Durlauf defines them in terms of _race_.

In both models, generations are represented as follows: in each time step, a
new member is born into each family/dynasty, and he/she replaces the current
member of the family in the next generation. The _initial human capital_ in
Durlauf’s model corresponds to our notion of _success probability_. _Adult
human capital_ in his model is determined by college attendance, and it
roughly maps to our notion of _success_ (i.e., whether the individual succeeds
if given the opportunity.) In both models, an admission rule maps a student’s
human capital and group membership into a binary outcome indicating whether
the student is given the opportunity. In both models, the admission rule may
vary across time and generations; the level of state expenditures on education
is assumed constant across generations ($\alpha$ is fixed in our model); the
only output of universities is human capital (our objective function is made
up of the percentage of the population who succeed in each generation); and
finally, the initial human capital is accurately measured for every student
(we assume ability and success probability are perfectly observable.)

##  2 A Dynamic Model

### Agents, Circumstances, Abilities, & Generations

We consider a model of a society that consists of a continuum of agents in two
different sets of socioeconomic circumstances — a disadvantaged circumstance
$D$ and an advantaged circumstance $A$. These circumstances are
(probabilistically) inherited from one generation to the next, but we can try
to increase the number of agents in the advantaged circumstance in future
generations by offering opportunities to disadvantaged agents in the current
generation. This comes with a trade-off, however, since a competing option is
to offer these opportunities to advantaged agents in the current generation.
Our goal is to model this trade-off. We say that an agent $i$ has circumstance
$c_{i}=0$ if they are disadvantaged ($i\in D$), and circumstance $c_{i}=1$ if
they are advantaged ($i\in A$). Each agent $i$ also has an ability $a_{i}$,
which is a real number in $[0,1]$.

Time advances in discrete periods, beginning with period $t=0$. We think of
these as generations. Consider an agent $i$ who has circumstance
$c^{\text{init}}_{i}$ at the beginning of time $t$. Depending on whether $i$
receives the opportunity, his/her circumstance may change to
$c_{i}^{\text{post}}$. At the end of time step $t$, $i$ produces a new agent
$i^{\prime}$ in generation $t+1$. This new agent $i^{\prime}$ has an ability
$a_{i^{\prime}}$ drawn uniformly at random from $[0,1]$. With some fixed
probability (specified below) $i^{\prime}$ inherits the ex-post circumstance
of $i$ (so $c_{i^{\prime}}=c^{\text{post}}_{i}$), otherwise, it takes on a
circumstance randomly selected from the background distribution of
circumstances within the population in generation $t$. More specifically, in a
given period $t$, let $\phi_{j}(t)$ denote the fraction of agents who have
circumstance $j$, for $j=0,1$. If $c^{\text{post}}_{i}=0$, then with a fixed
probability $1-p_{D}$, $i^{\prime}$ inherits circumstance $D$, and with fixed
probability $p_{D}$, it receives a circumstance randomly selected from the
background distribution $(\phi_{0}(t),\phi_{1}(t))$. Similarly, If
$c^{\text{post}}_{i}=1$, then with a fixed probability $1-p_{A}$ ,
$i^{\prime}$ inherits circumstance $A$, and with probability $p_{A}$, it
receives a circumstance randomly selected from the background distribution
$(\phi_{0}(t),\phi_{1}(t))$.

The movement probabilities, $p_{A}$ and $p_{D}$, capture all processes—other
than the opportunity we seek to allocate optimally—through which individuals
can change their circumstance from their parental inheritance. For example, in
the college admissions example, while our model focuses on how admission
decisions can reshape circumstances over generations, there are many other
forces and processes that impact the evolution of circumstances within society
(e.g., number of jobs in the economy, training opportunities outside college,
or pure luck). The movement probabilities summarize and capture all these
alternative upward or downward movement possibilities.

### Opportunities and Payoffs

We consider the problem of performing an intervention in this society, which
consists of offering an opportunity to a subset of the population. We only
have the resources to offer the opportunity to an $\alpha$ fraction of the
population. An agent who is offered the opportunity has some probability of
succeeding at it, as a function of their ability and circumstances that we
specify below. Succeeding at the opportunity confers two benefits on society:

  1. (i)

it produces an immediate payoff/reward to the society in the form of
productivity;

  2. (ii)

if the agent is disadvantaged, it moves them (and subsequently their future
generations) into the advantaged group.

The central problem to be solved in the model, as we will see below, is how to
balance the immediate gains from (i) against the long-term gains from (ii)
over multiple generations.

In particular, if an agent of ability $a_{i}\in[0,1]$ and circumstance
$c_{i}\in\\{0,1\\}$ is offered the opportunity, their probability of
succeeding at it is $a_{i}\sigma+c_{i}\tau,$ where $\sigma,\tau>0$ and
$\sigma+\tau\leq 1$. Note that since $c_{i}\in\\{0,1\\}$, this simply means
that $\tau$ gets added to the success probability of all agents whose
circumstance is equal to $1$.

Our payoff (or reward) $r(t)$ in period $t$ is simply the fraction of the
population that both receives the opportunity and succeeds at it. Our total
payoff is a discounted sum of payoffs over all periods, with discount factor
$0<\gamma<1$; that is, the total payoff $r$ is equal to
$\sum_{t=0}^{\infty}\gamma^{t}r(t)$. As noted earlier, agents with
circumstance $0$ who receive the opportunity and succeed at it will produce
offspring who (are more likely to) have circumstance $1$; this matters for the
payoff because the total payoff $r=\sum_{t=0}^{\infty}\gamma^{t}r(t)$ depends
on the fraction of agents with each type of circumstance in all time periods.

### Thresholds and Interventions

The way we allocate the opportunity at time $t$ is to set a threshold
$\theta_{j}(t)$ for agents with circumstance $j$, for $j\in\\{0,1\\}$, and to
offer the opportunity to all agents $i$ with circumstance $j$ whose success
probability, $a_{i}\sigma+c_{i}\tau$, is at least $\theta_{j}(t)$. That is,

$a_{i}\sigma+c_{i}\tau\geq\theta_{j}.$

We will sometimes write the threshold $\theta_{j}$ and the population fraction
with each circumstance $\phi_{j}$ without the explicit dependence “$(t)$” when
we are considering a single fixed time period.

Agents of circumstance $0$ make up a $\phi_{0}$ fraction of the population,
and a $1-\frac{\theta_{0}}{\sigma}$ fraction of them receive the opportunity,
for a total fraction of the population equal to
$\phi_{0}\times(1-\frac{\theta_{0}}{\sigma})$. Similarly, agents of
circumstance $1$ make up a $\phi_{1}$ fraction of the population, and a
$1-\frac{\theta_{1}-\tau}{\sigma}$ fraction of them receive the opportunity,
for a total fraction of the population equal to
$\phi_{1}\times(1-\frac{\theta_{1}-\tau}{\sigma})$. The sum of these two
fractions must add up to $\alpha$ — the portion of the population to whom we
can offer the opportunity:

$\displaystyle\forall 0\leq\theta_{0}\leq\sigma\text{ and
}\tau\leq\theta_{1}\leq\sigma+\tau:\text{
}\phi_{0}\times\left(1-\frac{\theta_{0}}{\sigma}\right)+\phi_{1}\times\left(1-\frac{\theta_{1}-\tau}{\sigma}\right)=\alpha$
(1) $\displaystyle\Leftrightarrow$ $\displaystyle\forall
0\leq\theta_{0}\leq\sigma\text{ and }\tau\leq\theta_{1}\leq\sigma+\tau:\text{
}\phi_{0}\theta_{0}+\phi_{1}\theta_{1}=\sigma(1-\alpha)+\phi_{1}\tau.$

This also shows how our choice of thresholds is a one-dimensional problem in
the single variable $\theta_{0}$ (or equivalent in the single variable
$\theta_{1}$), since after setting one of the two thresholds, the other is
determined by this equation. More precisely, we have that:

$\begin{cases}\theta_{1}=\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\quad\forall\phi_{0}<1\\\
\theta_{1}\in[\tau,\sigma+\tau]\quad\text{ for }\phi_{0}=1.\end{cases}$ (2)

Note that if $\phi_{0}=1$, $\theta_{1}$ can take on any value in
$[\tau,\sigma+\tau]$, the threshold will not affect how opportunities are
allocated. The same holds for $\phi_{0}=0$ and any $\phi_{0}\in[0,\sigma]$.

### Dynamics in a Single Period

Recall that our payoff $r(t)$ in period $t$ is simply the fraction of the
population that both receives the opportunity and succeeds at it. We can
decompose this as follows.

  * •

Agents of circumstance $0$ make up a $\phi_{0}$ fraction of the population,
and a $1-\frac{\theta_{0}}{\sigma}$ fraction of them receive the opportunity,
for a total fraction of the population equal to
$\phi_{0}\left(1-\frac{\theta_{0}}{\sigma}\right)$. Not all of these agents
succeed at the opportunity; the average success probability in this group is
$\frac{1}{2}(\sigma+\theta_{0})$, so the expected quantity that succeeds is

$\phi_{0}\left(1-\frac{\theta_{0}}{\sigma}\right)\frac{(\sigma+\theta_{0})}{2}=\frac{\phi_{0}}{2\sigma}\left(\sigma^{2}-\theta_{0}^{2}\right).$

  * •

Agents of circumstance $1$ make up a $\phi_{1}$ fraction of the population,
and a $1-\frac{\theta_{1}-\tau}{\sigma}$ fraction of them receive the
opportunity, for a total fraction of the population equal to
$\phi_{1}\left(1-\frac{\theta_{1}-\tau}{\sigma}\right)$. Again, not all of
these agents succeed at the opportunity; the average success probability in
this group is $\frac{1}{2}(\sigma+\tau+\theta_{1})$, so the expected quantity
that succeeds is

$\phi_{1}\left(1-\frac{\theta_{1}-\tau}{\sigma}\right)\frac{(\sigma+\tau+\theta_{1})}{2}=\frac{\phi_{1}}{2\sigma}\left((\sigma+\tau)^{2}-\theta_{1}^{2}\right).$

The total payoff in period $t$ is the sum of these two terms:

$r(t)=\frac{\phi_{0}(t)}{2\sigma}\left(\sigma^{2}-\theta_{0}(t)^{2}\right)+\frac{\phi_{1}(t)}{2\sigma}\left((\sigma+\tau)^{2}-\theta_{1}(t)^{2}\right).$
(3)

### Dynamics over Multiple Periods

If we were just optimizing the payoff in this single time period, then we’d
have a single-variable optimization problem in the variable $\theta_{0}$ (or
equivalently in $\theta_{1}$), with the objective function given by (3). But
since there is also the set of discounted payoffs in future time periods, we
also need to look at the effect of our decisions on the quantities
$\phi_{0}(.)$ and $\phi_{1}(.)$ in future periods.

If $p_{A}=p_{D}=0$, $\phi_{1}(t+1)$ grows relative to $\phi_{i}(t)$ depending
on the fraction of the population that transitions from circumstance $0$ to
circumstance $1$ by succeeding at the opportunity. Thus we have

$\phi_{0}(t+1)=\phi_{0}(t)-\frac{\phi_{0}(t)}{2\sigma}\left(\sigma^{2}-\theta_{0}(t)^{2}\right).$
(4)

More generally when $p_{A}$ or $p_{D}$ are non-zero, let’s define
$\phi^{\text{post}}_{0}(t)=\phi_{0}(t)-\frac{\phi_{0}(t)}{2\sigma}\left(\sigma^{2}-\theta_{0}(t)^{2}\right)$.
(For simplicity, we drop “$(t)$” and simply use $\phi^{\text{post}}_{0}$ in
the remainder of this section). We have:

$\phi_{0}(t+1)=\phi^{\text{post}}_{0}(1-p_{D})+\phi^{\text{post}}_{0}p_{D}\phi^{\text{post}}_{0}+\left(1-\phi^{\text{post}}_{0}\right)p_{A}\phi^{\text{post}}_{0},$
(5)

It is easy to see that:

###### Proposition 1

If $p_{A}=p_{D}$, then
$\phi_{0}(t+1)=\phi_{0}(t)-\frac{\phi_{0}(t)}{2\sigma}\left(\sigma^{2}-\theta_{0}(t)^{2}\right)$.

Proof  Suppose $p_{A}=p_{D}=p$. Then we can re-write (5) as follows:

$\displaystyle\phi_{0}(t+1)$ $\displaystyle=$
$\displaystyle\phi^{\text{post}}_{0}(1-p)+\phi^{\text{post}}_{0}p\phi^{\text{post}}_{0}+\left(1-\phi^{\text{post}}_{0}\right)p\phi^{\text{post}}_{0}$
$\displaystyle=$
$\displaystyle\phi^{\text{post}}_{0}(1-p)+\phi^{\text{post}}_{0}p\left(\phi^{\text{post}}_{0}+\left(1-\phi^{\text{post}}_{0}\right)\right)$
$\displaystyle=$
$\displaystyle\phi^{\text{post}}_{0}(1-p)+\phi^{\text{post}}_{0}p$
$\displaystyle=$
$\displaystyle\phi^{\text{post}}_{0}=\phi_{0}(t)-\frac{\phi_{0}(t)}{2\sigma}\left(\sigma^{2}-\theta_{0}(t)^{2}\right)$

where in the last line, we replace $\phi^{\text{post}}_{0}$ with its
definition.

The above proposition shows that with respect to dynamics and optimal policy,
settings in which $p_{A}=p_{D}$, are essentially equivalent to settings in
which $p_{A}=p_{D}=0$.

In summary, the full problem is to choose thresholds
$\theta_{0}(t),\theta_{1}(t)$ for each time period $t$ so as to maximize the
infinite sum $r=\sum_{t=0}^{\infty}\gamma^{t}r(t)$. Each term $r(t)$ depends
not just on the chosen thresholds but also on the fractions of agents with
each type of circumstance $\phi_{0}(t),\phi_{1}(t)$, which evolve according to
the recurrence in Equation (5). Note that the intuitive trade-off between
$\theta_{0}$ and $\theta_{1}$ shows up in the formulation of (3) and (4):
lowering $\theta_{0}$ and lowering $\theta_{1}$ have different effects, both
in period $t$ and in future time periods.

##  3 Theoretical Analysis for $p_{D}\geq p_{A}$

In this section, we focus on settings in which $p_{D}\geq p_{A}$ and
characterize the optimal policy to maximize the discounted payoff over
generations. (We only provide the analysis for settings of $p_{D}=p_{A}$ but
the extension to $p_{D}>p_{A}$ is straightforward). We cast the problem as
deriving the infinite-time-horizon optimal policy in a continuous state- and
action-space (Markov) decision process. We characterize the optimal threshold
and value function for every state $\phi_{0}\in[0,1]$. Importantly, we show
that there exists a tipping point $\phi^{*}_{0}$ that splits the state space
into two distinct regions: states at which the optimal threshold uses strict
affirmative action, and states at which the optimal policy consists of
imposing equally high thresholds on both $A$ and $D$ groups.

### The Decision Process

Given $\alpha,\sigma,\tau$, and $\gamma$, we define a decision process
$\mathcal{D}_{\alpha,\sigma,\tau,\gamma}=(\Phi,\Theta,S,R)$ (or $\mathcal{D}$
for short) with a continuous state space $\Phi=[0,1]$, action space
$\Theta=[0,\sigma]$, state transition $S:\Phi\times\Theta\rightarrow\Phi$, and
reward function $R:\Phi\times\Theta\rightarrow[0,1]$. Each state
$\phi_{0}\in\Phi$ corresponds to a particular fraction of disadvantaged
individuals within the population. For instance, the states $0$ (or $1$)
represents a society in which no one (or everyone) belongs to $D$.

The set of thresholds admissible in each state $\phi_{0}$ is denoted by
$\Theta_{\phi_{0}}$. $\Theta_{\phi_{0}}$ consist of all thresholds
$0\leq\theta_{0}\leq\sigma$ that can satisfy the capacity constraint,
$\alpha$. In other words, for any $\in\theta_{0}\Theta_{\phi_{0}}$ if we
impose the threshold $\theta_{0}\in[0,\sigma]$ on group $D$, we can find a
threshold $\theta_{1}\in[0,\sigma+\tau]$ for group $A$ such that exactly
$\alpha$ fraction of the overall population receives the opportunity. This
capacity constraint translates into two conditions on $\Theta_{\phi_{0}}$:

  1. 1.

A threshold $\theta_{0}\in\Theta_{\phi_{0}}$ should not give the opportunity
to _more than_ $\alpha$ fraction of the population. Formally,
$\forall\theta_{0}\in\Theta_{\phi_{0}}:\phi_{0}\left(1-\frac{\theta_{0}}{\sigma}\right)\leq\alpha$,
which is equivalent to:

$\forall\theta_{0}\in\Theta_{\phi_{0}}:\quad\theta_{0}\geq\sigma\left(1-\frac{\alpha}{\phi_{0}}\right).$
(6)

  2. 2.

A threshold $\theta_{0}\in\Theta_{\phi_{0}}$ should not waste opportunities,
that is, it should give the opportunity to _at least_ $\alpha$ fraction of the
overall population. Formally,
$\forall\theta_{0}\in\Theta_{\phi_{0}}:\phi_{0}\left(1-\frac{\theta_{0}}{\sigma}\right)+\phi_{1}\geq\alpha.$
Replacing $\phi_{1}$ with $1-\phi_{0}$ and rearranging terms, the above is
equivalent to

$\forall\theta_{0}\in\Theta_{\phi_{0}}:\quad\theta_{0}\leq\frac{\sigma(1-\alpha)}{\phi_{0}}.$
(7)

Figure 2 illustrates the actions satisfying conditions (6) and (7) for every
state $\phi_{0}\in[0,1]$.

Figure 2:  The admissible thresholds for every state of the decision process,
$\mathcal{D}$. The x-axis specifies the state $\phi_{0}$, and the y-axis
highlights (in light red) the admissible thresholds, $\Theta_{\phi_{0}}$, at
every state, $\phi_{0}$.

The state transition function, $S:\Phi\times\Theta\rightarrow\Phi$, specifies
the state transitioned to for every (state, admissible threshold) pair.
Formally,

$\forall\phi_{0}\in\Phi,\forall\theta_{0}\in\Theta_{\phi_{0}}:\quad
S(\phi_{0},\theta_{0})=\phi_{0}-\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2}).$
(8)

The reward function, $R:\Phi\times\Theta\rightarrow[0,1]$, is defined as
follows: $R(\phi_{0},\theta_{0})$ denotes the immediate reward/payoff of
imposing threshold $\theta_{0}$ at state $\phi_{0}$. Formally,

$\forall\phi_{0}\in\Phi,\forall\theta_{0}\in\Theta_{\phi_{0}}:\quad
R(\phi_{0},\theta_{0})=\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2})+\frac{\phi_{1}}{2\sigma}\left((\sigma+\tau)^{2}-\theta_{1}^{2}\right).$

Replacing $\phi_{1}$ with $1-\phi_{0}$ and $\theta_{1}$ with the right hand
side of (2), we obtain the following equivalent expression for $R$:

$R(\phi_{0},\theta_{0})=\begin{cases}\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})\quad\text{
for }\phi_{0}=1\text{, otherwise: }\\\
\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2})+\frac{1-\phi_{0}}{2\sigma}\left((\sigma+\tau)^{2}-\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)^{2}\right)\end{cases}$
(9)

### Characterization of the Optimal Policy

Next, we illustrate and characterize the optimal policy for the decision
process $\mathcal{D}$ defined above. (For further information and references
on continuous-state decision processes, see Appendix A.5.)

A deterministic policy $\pi$ for a decision process $\mathcal{D}$ is a mapping
$\pi:\Phi\rightarrow\Theta$ such that $\pi(\phi_{0})$ prescribes the threshold
at state $\phi_{0}$. The value $V_{\pi}(\phi_{0})$ of a state $\phi_{0}$ under
policy $\pi$ is the discounted reward of executing policy $\pi$ on
$\mathcal{D}$ starting with initial state $\phi_{0}$. A policy $\pi$ is
optimal if its value function $V_{\pi}$ satisfies Bellman Optimality—defined
recursively as follows:

$V_{\pi}(\phi_{0})=\max_{\theta_{0}}R(\phi_{0},\theta_{0})+\gamma
V_{\pi}(S(\phi_{0},\theta_{0})).$ (10)

We establish in Appendix C that for our decision process $\mathcal{D}$, the
value function satisfying the above functional equation is _unique_ ,
_continuous_ , and _differentiable_. (We prove these facts utilizing tools
from recursive analysis and dynamic programming (Stokey, 1989; Cotter and
Park, 2006)). For simplicity, from this point on we refer to this unique
optimal value function as $V(.)$ and drop the subscript $\pi$.

Let the correspondence $\Pi^{*}_{0}:\Phi\rightarrow 2^{\Theta}$ denote all
optimal policies for $\mathcal{D}$. More precisely, for any $0\leq\phi_{0}\leq
1$, the set $\Pi^{*}_{0}(\phi_{0})$ contains all optimal threshold values at
$\phi_{0}$. Figure 3(a) illustrates $\Pi^{*}_{0}$ for a sample setting of the
parameters $\alpha,\sigma,\tau,\delta$. (See Figure 6 in the Appendix for more
instances.) Figure 7 in the Appendix illustrates the value function $V$. Note
that the value function is consistently decreasing and concave.

(a) The optimal $\theta_{0}$ at every state $0\leq\phi_{0}\leq 1$. The optimal
threshold decreases with $\phi_{0}$. The dashed lines indicate the tipping
points, $\phi^{*}_{0}$, below which $\sigma$ is the only optimal threshold,
and above it $\sigma$ is not optimal.

(b) The difference between $\theta_{0}$ and $\theta_{1}$ at every state
$0\leq\phi_{0}\leq 1$. The dashed lines specify the tipping points. Strict
affirmative action is employed beyond $\phi^{*}_{0}$ only and the extent of
affirmative action is increasing in $\phi_{0}$.

(c) The state to which the optimal policy converges given the initial state
$\phi_{0}$. The dashed lines specify the tipping point, $\phi^{*}_{0}$. Note
that the optimal policy never shrinks the size of group $D$ to a value less
than $\phi^{*}_{0}$.

Figure 3: Illustration of the (a) optimal policy, (b) extent of affirmative
action, and (c) absorbing state.

We say that the optimal policy, $\Pi^{*}_{0}$, uses affirmative action at a
state $\phi_{0}$ if at $\phi_{0}$, it imposes a lower threshold on group $D$
compared to $A$.

######  Definition 1 ((Strict) Affirmative Action)

The optimal policy uses (strict) affirmative action at state $\phi_{0}$ if for
all $\theta_{0}\in\Pi^{*}_{0}(\phi_{0})$ and all
$\theta_{1}\in\Pi^{*}_{1}(\phi_{0})$, $\theta_{0}<\theta_{1}$.

If the inequality above is not strict (i.e., $\theta_{0}\leq\theta_{1}$), we
say the optimal policy uses _weak_ affirmative action. Figure 3(b) shows the
extent of affirmative action for a sample setting of the parameters
$\alpha,\sigma,\tau,\delta$. (See Figure 8 in the Appendix for more
instances).

Our main result (Theorem 1) characterizes the states at which the optimal
policy employs affirmative action. In particular, it establishes the existence
of a tipping point $0\leq\phi^{*}_{0}\leq 1-\alpha$ that designate the region
of affirmative action: At any state $\phi_{0}\leq\phi^{*}_{0}$, the optimal
policy assigns similar high threshold to both the advantaged and disadvantaged
groups. In contract, at any state $\phi_{0}\leq\phi^{*}_{0}$ the optimal
policy imposes a strictly lower threshold on group $D$ compared to $A$.

###### Theorem 1

Given $\alpha,\sigma,\tau,\gamma$ and the decision process $\mathcal{D}$, let

$\phi^{*}_{0}=\max\left\\{0,\min\left\\{1-\alpha,1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)\right\\}\right\\}.$
(11)

  * •

For any state $\phi_{0}\leq\phi^{*}_{0}$, there exists
$\theta_{0}\in\Pi^{*}_{0}(\phi_{0})$ and $\theta_{1}\in\Pi^{*}_{1}(\phi_{0})$
such that $\theta_{0}=\theta_{1}$.

  * •

For any state $\phi_{0}>\phi^{*}_{0}$, for all
$\theta_{0}\in\Pi^{*}_{0}(\phi_{0})$ and all
$\theta_{1}\in\Pi^{*}_{1}(\phi_{0})$ such that $\theta_{0}<\theta_{1}$.

We prove Theorem 1 by establishing a series of Lemmas. Lemma 1 determines the
largest state $\phi^{*}_{0}\in[0,1]$ below which the optimal policy consists
of applying the high threshold of $\sigma$ to group $D$. Clearly, below such
point, the optimal policy does not use affirmative action. Next, we
investigate states larger than $\phi^{*}_{0}$. For every state
$\phi_{0}>\phi^{*}_{0}$, Lemma 2 identifies the set of thresholds that exhibit
affirmative action. Lemma 3 establishes that the optimal policy uses weak
affirmative action beyond $\phi^{*}_{0}$. That is, it shows that for any state
$\phi_{0}>\phi^{*}_{0}$, it is never optimal to impose a strictly higher
threshold on D compared to A. Proposition 2 shows that beyond $\phi^{*}_{0}$,
the optimal policy in fact uses _strict_ affirmative action. That is, at every
state $\phi_{0}>\phi^{*}_{0}$, the optimal policy imposes a strictly lower
threshold on D compared to A.

Lemma 1 determines the state $\phi^{*}_{0}\in[0,1]$ up to which $\sigma$ is an
optimal threshold. Note that if $\sigma$ is an optimal threshold at a state
$\phi_{0}$, the optimal policy does not use affirmative action at $\phi_{0}$.
To see this, note that when $\sigma$ is optimal, any action
$\theta_{0}>\sigma$ is also optimal (they all pick a 0-fraction of group $D$
and are, therefore, effectively equivalent). (All omitted proof can be found
in Appendix C.1.)

######  Lemma 1 (The Tipping Point)

For any state $\phi_{0}<\phi^{*}_{0}$, $\sigma\in\Pi^{*}_{0}(\phi_{0})$.

The following Lemma characterizes the region of affirmative action in the
state-action space.

######  Lemma 2 (Region of Affirmative Action)

For a state $\phi_{0}^{*}<\phi_{0}<1$, the threshold
$\theta_{0}\in\Pi^{*}_{0}(\phi_{0})$ uses affirmative action if and only if
$\theta_{0}\leq\sigma(1-\alpha)+\tau(1-\phi_{0})$.

Proof  Let $\theta$ be the threshold that if applied to both D and A at
$\phi_{0}$, allocates opportunities exactly. We have that

$\phi_{0}\left(1-\frac{\theta}{\sigma}\right)+(1-\phi_{0})\left(1-\frac{\theta-\tau}{\sigma}\right)=\alpha$

or equivalently,

$\theta=\sigma(1-\alpha)+\tau(1-\phi_{0}).$

Note that $\theta_{0}<\theta$ if and only if $\theta_{1}>\theta$—otherwise the
capacity constraints would not be maintained. Therefore, for any
$\theta_{0}<\theta$, $\theta_{1}>\theta_{0}$, which implies affirmative
action.

The following Lemma shows that beyond $\phi^{*}_{0}$, the optimal threshold
for the disadvantaged is never higher than that for the advantaged.

######  Lemma 3 (Weak Affirmative Action)

Consider a state $\phi_{0}>\phi^{*}_{0}$. For all
$\theta_{0}\in\Pi^{*}_{0}(\phi_{0})$ and all
$\theta_{1}\in\Pi^{*}_{1}(\phi_{0})$, $\theta_{0}\leq\theta_{1}$.

Proof  Suppose not and there exists $\theta_{0}\in\Pi^{*}_{0}(\phi_{0})$ and
$\theta_{1}\in\Pi^{*}_{1}(\phi_{0})$ such that $\theta_{0}>\theta_{1}$. If we
lower $\theta_{0}$ down to $\sigma(1-\alpha)+\tau(1-\phi_{0})$ and increase
$\theta_{1}$ up to $\sigma(1-\alpha)+\tau(1-\phi_{0})$, we maintain the
capacity constraints and at the same time achieve the following:

  * (a)

We improve the immediate reward of the current time step— because we replace
advantaged agents with low success probabilities in the range of
$[\theta_{1},\sigma(1-\alpha)+\tau(1-\phi_{0})]$ with disadvantaged agents
with higher success probabilities in
$[\sigma(1-\alpha)+\tau(1-\phi_{0}),\theta_{0}]$.

  * (b)

We move to a state with a relatively smaller size of group $D$ (simply because
we gave the opportunity to more disadvantaged agents). The value function is
strictly decreasing in $\phi_{0}$. Therefore, this new next state has a higher
value compared to the previous one.

The fact that we can improve the value contradicts the optimality of
$\theta_{0},\theta_{1}$. Therefore, $\theta_{0}>\theta_{1}$ cannot hold

The following Proposition establishes that beyond $\phi^{*}_{0}$, the optimal
policy uses strict affirmative action. (The proofs establishing the
proposition can be found in Appendix C.1.)

######  Proposition 2 (Strict Affirmative Action)

Consider a state $\phi_{0}>\phi^{*}_{0}$, and let $V^{\prime}(\phi_{0})$
denote the derivative of the value function $V$ evaluated at $\phi_{0}$. If
$V^{\prime}(\phi_{0})<0$, then for all $\theta_{0}\in\Pi^{*}_{0}(\phi_{0})$
and all $\theta_{1}\in\Pi^{*}_{1}(\phi_{0})$, $\theta_{0}<\theta_{1}$.

####  Insights from the Analysis.

We end this section by making several observations about the optimal policy:
First, note that the optimal policy never shrinks the size of $D$ to a value
less than $\phi^{*}_{0}$. For every initial state $\phi_{0}\in[0,1]$, Figure 9
shows the state one converges to when the optimal policy is simulated for
$1000$ steps on $\mathcal{D}$. In other words, affirmative action is optimal
from a utilitarian point of view as long as group $D$ is sufficiently large.
Second, the precise derivation of $\phi_{0}^{*}$, as specified in (11), allows
us to gain new insights into how the interaction between the parameters of our
model can give rise to or avert affirmative action. Figure 1 depicts the
status of persistent affirmative action (i.e., $\phi_{0}^{*}\leq 0$) for
$\alpha,\tau,\gamma$ in settings where $\sigma=1-\tau$. (The derivation behind
the plots can be found in Appendix C.2.) Notice that persistent affirmative
action is promoted by large values of $\alpha$ and small values of $\tau$,
since these make it easier to include high-performing members of group $D$
without a large difference in thresholds. Persistent affirmative action is
also promoted by larger values of $\gamma$, indicating greater concern for the
rewards in future generations. Note, however, that a finite level of patience
often suffices for persistent affirmative action to be optimal.  Finally, a
frequent objection to affirmative action polices is their potential for
reverse discrimination (see, e.g., Hopwood v. Texas, 78 F.3d 932 (5th Cir.
1996)). Translating these concerns into the terminology of our model, one may
object that “if the highest-ability member of $D$ has ability below the
lowest-ability member of $A$, then _any_ affirmative action in this scenario
will violate the desert principle”. Note, however, that in our model,
abilities for both $A$ and $D$ group members are uniform in the $[0,1]$
interval. So the optimal policy will never favor a lower-ability member of $D$
to a higher ability member of $A$.

##  4 Computational Analysis for $p_{D}<p_{A}$

The focus of our theoretical analysis was on the settings in which $p_{A}\leq
p_{D}$. Next, we computationally investigate the optimal policy for cases
where $p_{A}>p_{D}$.

Recall that when $p_{A}$ and $p_{D}$ are non-zero, the circumstance of an
offspring is not deterministically specified by that of their parent. Instead,
$p_{A}$ and $p_{D}$ specify the offspring’s probability of spontaneous
movement to the background distribution of circumstances in society, that is,
$(\phi_{0},\phi_{1})$. When $p_{A}\leq p_{D}$, both the spontaneous movement
dynamics and the allocation of opportunities work toward reducing the relative
size of group $D$ over time. This fact allowed us to utilize backward
induction to characterize the optimal policy theoretically. When
$p_{A}>p_{D}$, however, the spontaneous movement work in the opposite
direction of opportunity allocation: with no opportunity allocated, the
spontaneous movement dynamics gradually shift the entire population to group
$D$. In such settings, the role of the allocation policy consists of
combatting the natural flow of the population to $D$.

Although in settings with $p_{A}>p_{D}$, it is significantly more challenging
to characterize the optimal allocation policy, we can still approximately
compute this policy via discretization followed by solution methods for finite
decision problems. More specifically, we discretize the state- and action-
space of the decision process, $\mathcal{D}$, then compute the optimal policy
of the resulting finite decision process using methods such as policy
iteration. In what follows, we report the result of our simulations when the
state- and action-space are approximated by 1000 equidistant states and
actions (i.e., $\tilde{\Phi}=(0,0.001,0.002,\cdots,1)$ and
$\tilde{\Theta}=\sigma(0,0.001,0.002,\cdots,1)$) and the reward and transition
functions are approximated accordingly.

Figure 4: The optimal policy $\theta_{0}$ at every state $0\leq\phi_{0}\leq 1$
for various $p_{A}$ values. Figure 5: The absorbing state for every initial
state $0\leq\phi_{0}\leq 1$. The optimal policy is non-monotone and exhibits
sharp drops when better absorbing states become cost-effective.

Figure 4 shows how the optimal policy varies with $p_{A}$ for a fixed value of
$p_{D}$. Compared to the optimal policy for settings of $p_{A}\leq p_{D}$, we
observe two significant differences here:

  * •

The optimal policy can be non-monotone in $\phi_{0}$.

  * •

There exists a sharp drop in the optimal threshold at some intermediate state
$\phi_{0}\in[0,1]$.

Next, we provide some justifications for both the intuitive and potentially
counter-intuitive aspects of these phenomena.

For every initial state $\phi_{0}\in[0,1]$, Figure 5 illustrates the state one
converges to (i.e., the absorbing state) if the optimal policy is simulated
for 1000 steps. Note that in all cases, the optimal policy successfully
reduces the absorbing state to a more desirable value less than 1 (recall that
with no intervention, we always converge to $\phi_{0}=1$ through the
spontaneous movements).

As illustrated in Figure 5, the sharp drop in the optimal threshold coincides
with an abrupt change in the absorbing state. When the initial state is close
to 1, the allocation policy must employ extensive affirmative action to reduce
and maintain a better absorbing state. This feat, however, comes at the cost
of immediate reward. The optimal policy, therefore, settles for reaching and
obtaining a slightly more desirable absorbing state (a value between 0.6 and
0.8 in Figure 5). For smaller initial states, the cost of employing extensive
affirmative action goes down, and at some point, it becomes viable to reach
and maintain a much better absorbing state (a value between 0 to 0.3 in Figure
5). The sharp drop in the optimal threshold happens precisely at the state
where the benefit of extensive affirmative action outweighs its cost.

In summary, when $p_{A}>p_{D}$, we need persistent (albeit non-monotone)
affirmative action to reach and maintain more desirable absorbing states. The
optimal extent of affirmative action crucially depends on the initial state of
the population: If the disadvantaged group is large to begin with, the optimal
policy has to forgo extensive affirmative action to maintain sufficient short-
term return. If the advantaged group initially has a sufficient mass,
extensive affirmative action becomes optimal, and in the long-run, manages to
significantly increases the fraction of advantaged individuals in the
population. These findings are in sharp contrast with settings of $p_{A}\leq
p_{D}$. In those settings, affirmative action is optimal under the much more
straightforward condition that the disadvantaged group exceeds a
$\phi_{0}^{*}$ fraction of the population, for a constant $\phi_{0}^{*}$ that
we can specify precisely, and it ceases to be optimal as soon as society gains
at least $1-\phi_{0}^{*}$ mass in the advantaged group.

##  5 Discussion and Future Directions

In this paper, we developed and analyzed a model for allocating opportunities
in a society with limited intergenerational mobility. These opportunities
produce benefits in the present generation, but they can also raise the
socioeconomic status of the recipients and their descendants. This creates a
trade-off: whether to maximize the current value achievable from the
opportunities or to increase the value achievable in future generations. We
have shown how to resolve this trade-off by solving a continuous optimization
problem over multiple generations, and we have seen how optimal solutions to
this problem can exhibit a form of socioeconomic affirmative action, favoring
individuals of low socioeconomic status over slightly higher-performing
individuals of high socioeconomic status. Characterizing the conditions under
which this type of affirmative action occurs in the model provides insights
into the interaction between the amount of opportunity available, the
magnitude of the gap between different socioeconomic classes, and the
“patience”’ of the society in evaluating the benefits achievable from future
generations.

####  Insights and Implications.

Our work provides a purely utilitarian account of a society–with no intrinsic
interest in reducing inequality– which nonetheless chooses to employ
affirmative action along socioeconomic lines. In that sense, our work responds
to concerns around affirmative action hurting social utility. In addition, our
analysis present new insights on the shape and extent of effective
socioeconomic affirmative action policies across multiple generations. Our
findings offer several important insights to the Fair-ML literature: (a) If
temporal dynamics are taken into account, there are important cases where
fairness interventions can be fully aligned with performance-maximizing goals.
(b) Effective fairness interventions should often adapt to population changes
over time. (c) For a comprehensive assessment of fairness for an allocation
policy/algorithm, we may need to look beyond its immediate impact on its
direct subjects. Our work, for instance, characterizes optimal allocations
when decisions made for individuals today impact their future generations.

####  Toward a Broader Class of Models Leading to Affirmative Action.

Our work’s main focus was on _intergenerational dynamics_ and their impact on
the effectiveness of _socioeconomic affirmative action policies_. But we hope
that our work also serves as a stepping stone toward a broader class of models
characterizing conditions under which affirmative action comes out of optimal
policy derivations. In particular, a generalized version of our decision
process can capture _race-dependent_ dynamics of movement between various
socioeconomic levels. Race-dependent dynamics are motivated by the observation
that advantaged Black people are more likely to be downwardly mobile than
their advantaged white counterparts (Chetty et al., 2020). A more general
version of our decision process would consist of race-dependent $\sigma$’s and
$\tau$’s. Analyzing the resulting dynamics can shed light on the tradeoffs
between race/ethnicity-based and socioeconomic affirmative action policies. We
leave this analysis as a crucial direction for future work.

There are a number of interesting further research directions suggested by the
results in the paper, and several of them address the limits of the model
noted in the introduction. In particular, it would be interesting to consider
extensions of the model in which basic parameters of the society varied over
time rather than being fixed and could only be estimated with delayed feedback
(e.g., after several years); the ability distributions were non-uniform; the
dynamics would amplify the interaction between ability and socioeconomic
advantage; or there were multiple opportunities being allocated concurrently.
It would be interesting to incorporate strategic considerations as in other
work on intergenerational mobility. And finally, there is a natural
opportunity to try integrating the models here with work on the nature of
statistical discrimination and the design of interventions to alleviate it.

####  On the Scope and Limitations of Mathematical Models.

Our work addresses one potential role that affirmative action policies can
have on future generations and mitigating socioeconomic inequality. Our work
follows a long tradition in the mathematical social sciences, where a stylized
model is proposed to capture certain aspects of a complex societal issue; the
model is then rigorously analyzed with the hope that the formal analysis
provides new insights about alternative policy choices and their
counterfactual effects. These insights can in turn inform policymakers at a
qualitative level. We conclude this article by acknowledging that all
mathematical models are by definition highly simplified representations of the
phenomena at hand, and as such it is important to understand and interpret
them keeping their limitations and scope of applicability in mind. We have
grounded our work in a broad literature from economics so as to draw on the
insights that earlier modelers have brought to this setting. But we emphasize
that theoretical models and their implications should never be taken as exact
representations of the way complex societal processes operate and evolve. As
such, it is important to not draw policy interpretations on the basis of such
models alone.

## Acknowledgments

This work was supported in part by a Simons Investigator Award, a Vannevar
Bush Faculty Fellowship, a MURI grant, AFOSR grant FA9550-19-1-0183, and
grants from the ARO and the MacArthur Foundation. We are grateful to Lawrence
E. Blume, Alexandra Chouldechova, Zachary C. Lipton, Rediet Abebe, Manish
Raghavan, Kate Donahue, the AI, Policy, and Practice (AIPP) group at Cornell,
and the FEAT reading group at Carnegie Mellon for invaluable discussions.
Finally, we would like to thank the anonymous reviewers of our work for their
insightful and constructive feedback.

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##  Appendix A Expanded Related Work

###  A.1 Long-term Implications of Fair-ML

Several recent papers study the long-term impact of decision-making models and
fairness interventions on society and individuals. Liu et al. [2018] and
Kannan et al. [2019] study how a utility-maximizing _decision-maker_ may
respond to the predictions made by the model. For instance, the decision-maker
may interpret and use the predictions in a certain way, or update the model
entirely. Dong et al. [2018]; Hu et al. [2019]; Milli et al. [2019] address
_strategic classification_ —a setting in which decision subjects are assumed
to respond _strategically_ and potentially _untruthfully_ to the choice of the
classification model, and the goal is to design classifiers that are robust to
strategic manipulation.

[Hu and Chen, 2018] and [Mouzannar et al., 2019] take inspiration from
existing models of statistical discrimination and affirmative action. Hu and
Chen [2018] study the impact of enforcing statistical parity on hiring
decisions made in a temporary labor market that precedes a permanent labor
market. They show that under certain conditions, statistical parity can result
in an equilibrium that Pareto-dominates the one that would emerge in an
unconstrained labor market.

Mouzannar et al. [2019] model the dynamics of how a population reacts to a
selection rule by changing their qualifications. The qualification of an
individual is assumed to be estimated via some classification function and
this estimate is assumed to be the same as their true qualification/label of
the individual. They model a selection rule by its selection rates across the
qualified and unqualified members of two socially salient groups. In other
words, the rule is modeled by four non-negative numbers. They assume that
there exist continuously differentiable functions $f_{1}$ and $f_{2}$ that map
the selection rates in each group to the percentage of qualified individuals
in that group in the next round. These functions model how qualifications
change over time in response to the selection rule. Within this model,
Mouzannar et al. [2019] study two types of myopic utility-maximizing policies:
an affirmative action policy which forces the selection rates to be equal
across the two groups, and an unconstrained policy that simply picks the
qualified in each group. They then provide several conditions under which each
of these policies lead to social equality (i.e., groups that are
indistinguishable in terms of their qualifications). As for affirmative action
policies, under-acceptance of qualified individuals in one group (to guarantee
equal selection rates) is shown to be inefficient (both in terms of the
decision-maker’s utility and average qualifications). Over-acceptance of the
unqualified, on the other hand, can lead to a more qualified population
(although it may fail to guarantee social equality).

###  A.2 Intergenerational Income Mobility

A large economic literature addresses the relationship between income
inequality and intergenerational mobility. At a high-level, the literature
suggests that higher inequality results in lower income mobility across
generations (see e.g., [Maoz and Moav, 1999; Corak, 2013]). The precise
measurements of inequality and mobility significantly influence the strength
of this effect (see, e.g., [Solon, 1992; Piketty, 2000]). The literature
recognizes two main mechanisms through which a parent may affect its offspring
income: (1) by transmitting endowments (e.g., innate abilities, connections,
etc.) to the next generation; (2) by investing in the human capital of the
next generation [Becker and Tomes, 1979; Solon, 1999]. In the existing
theoretical models of income mobility, utility-maximizing parents decide how
much of their income/human capital to consume and how much of it to invest in
their offspring (depending on their level of altruism) [Becker and Tomes,
1986; Becker and Tomes, 1979; Loury, 1981; Solon, 1999]. This decision,
combined with the social institutions and the technology that converts
parental investment to the next generation’s human capital, determine the fate
of the next generation.

For example, Loury [1981] provides a dynamic model of how the earnings
distributions of the next generation depends on the parent’s earnings and
his/her investment decision for his/her offspring. The earnings of the
offspring is determined by their innate ability/endowment $\alpha$222Similar
to ours, Loury’s model the ability of each individual is drawn randomly from
some fixed distribution. The ability of the offspring is only observed after
the parent makes their investment decision. and how much the parent invests in
their training $e$ through a function $h(\alpha,e)$. Unlike our model, Loury
assumes the the parents are expected-utility maximizing when making their
investment decisions for their offspring. In particular, he assumes the
utility $u(.,.)$ of a parent with earnings $y$ is a function of his/her own
consumption $c$ and the expected utility of the offspring as the result of the
investment he/she makes in their training, which is $y-c$. This recursive
definition of utility specifies how the utility of one generation depends on
that of all their subsequent generations (not just the immediate offspring).
Loury defines the indirect utility associated with an earning $y$ to be the
function the parent uses to estimate their offspring’s utility. If this
function is the same as the one obtained by solving the parent’s utility
maximization problem, the indirect utility function is called _consistent_. He
goes on to show that under certain (weak) assumptions, there is a unique
consistent indirect utility function. He also defines an _equilibrium_
earnings distribution as one that if it characterizes the earnings
distribution for one generation, it continues to do so for all subsequent
generations. He then provides several comparative statics results for various
policies (e.g., “education specific tax policies”). For example, he shows that
under certain conditions, egalitarian policies that redistribute earnings of
the next generation have insurance effects that make every member of society
today better off.

Other factors that have been shown to causally affect income mobility are
neighborhood [Chetty and Hendren, 2018, 2018], parental education [Torche,
2011], and family background characteristics (e.g., welfare receipt, health,
attitudes and social behavior [Black and Devereux, 2010]).

###  A.3 Affirmative Action Policies

A rich body of work in economics investigates sources of statistical
discrimination333According to [Fang and Moro, 2011] “Statistical
discrimination generally refers to the phenomenon of a decision-maker using
observable characteristics of individuals as a proxy for unobservable, but
outcome-relevant, characteristics.” and the role of affirmative action
policies in redressing it. For an excellent survey of this literature, see
[Fang and Moro, 2011].

In contrast to taste-based theories of discrimination—which attribute group
inequality to racial or gender preference against members of certain
groups—statistical discrimination theories cast group inequality as a
consequence of interaction between two _rational_ parties:

  1. 1.

A utility maximizing decision-maker (e.g., an employer) who has imperfect
information about a decision subject’s characteristics and uses his/her group
membership as a signal for his/her outcome-relevant unobservable
characteristics (e.g., employee’s productivity);

  2. 2.

Several groups of individuals (e.g., racial or gender groups) that are ex-ante
identical in terms of the distribution of qualifications. Individuals are
utility maximizing and best-respond to the decision maker’s strategy by
adjusting their investment in qualifications.

In a seminal work, Arrow [1973] argues that differences between groups can be
explained as a form of _coordination failure_ : In equilibrium, the decision
maker holds asymmetric beliefs about group qualifications, and this serves as
a _self-fulfilling stereotype_. Because of this belief, members of the
disadvantaged group don’t have enough incentive to invest in skills and
qualifications—precisely because they know that the decision maker will treat
them unfavorably). This in turn rationalizes the decision maker’s
belief—members of the disadvantaged group indeed end up being less qualified
(on average) than the advantage group.

Outcome-based policies, such as affirmative action quotas or the application
of disparate impact tests, have long been proposed and implemented as a
_temporary_ remedy to eliminate group-level inequalities. Such policies may
seem particularly effective when self-fulfilling stereotypes are the primary
cause of group inequalities. Imposing quota constraints can lead the players
to coordinate on a symmetric outcome, that is, and equilibrium in which the
decision maker holds symmetric beliefs about different groups. This, however,
is not the only possible consequence of imposing quota constraints. Coate and
Loury [1993] have shown that quota constraints may reduce the disadvantaged
group’s incentives to invest in skills, and subsequently, result in an even
worse asymmetric the equilibrium. Coate and Loury call this phenomenon
_patronization_. (This is a potential consequence of algorithmic fairness-
enhancing interventions.)

Advocates of affirmative action have often argued that the larger
representation of minorities in higher social positions can generate _role
models_ that can positively influence future generations of minorities in
their investment decisions. Chung [2000] formalizes these arguments, by
allowing for groups to differ in their costs of investment.

While the existing literature on affirmative action largely focuses on race-
based interventions and policies, some scholars have advocated for _socio-
economic_ or _class-based_ affirmative action as a race-neutral alternative
(see e.g., [Kahlenberg, 1996; Carnevale and Rose, 2013]). Race-neutral
alternatives become particularly important to understand when race-sensitive
interventions are constitutionally challenged (for example, see the case of
Fisher v. University of Texas at Austin (2013)). The extent to which
substituting socio-economic status for race in college admissions is effective
in improving minority enrollment is contested (see, e.g., [Kane, 1998; Reardon
et al., 2006; Gaertner and Hart, 2013; Reardon et al., 2017]). However, socio-
economic affirmative action can certainly facilitate access to college for
economically disadvantaged students. The resulting socioeconomic diversity is
a desirable outcome in and of itself [Reardon et al., 2017] and in essence, it
does not have to compete with or replace race-sensitive admission policies.

###  A.4 Comparison with [Durlauf, 2008]

[Durlauf, 2008] is one of the closest papers to our work—in terms of the
motivation and modeling choices. In his concluding remarks, Durlauf poses the
key question our work sets out to answer: how do multigenerational
considerations impact the optimal allocation policy? Next, we provide a brief
summary of [Durlauf, 2008] and compare our work with it in terms of modeling
choices and results.

Durlauf focuses on a condition under which efficiency and equality
considerations are aligned, and that is when a diverse body of students on
college campuses improves the human capital development for all of them.
Durlauf [2008] provides a model to compare the equality and efficiency of
affirmative action policies with those of meritocratic policies in the context
of admission rules to public universities. An admission rule maps the initial
human capital of students (and possibly their demographic information, such as
race) to their admission decisions. If a student is admitted, their human
capital is then improved by the quality of the college they attend and the
average human capital of their classmates in that college. The state wishes to
employ an admission policy that maximizes the aggregate human capital. He
argues that while the meritocratic policy may be efficient under certain
conditions, the same holds for affirmative action policies (e.g., when
“diversity improves education for all students.”).

In many respects, our model is similar to [Durlauf, 2008], although we depart
from his model and analysis in a few key areas. Most importantly, we model the
dynamics of human-capital development across _multiple generations_ and derive
the _optimal allocation policy_.

Similar to our work, Durlauf studies a multi-generational model of a society,
wherein each generation, a new member is born into every family/dynasty, and
he/she replaces the current adult member of the family in the next generation.
Even though our model is not expressed in terms of _overlapping generations_ ,
one can equivalently cast it in those terms.

The _initial human capital_ in Durlauf’s model corresponds to our notion of
_success probability_. _Adult human capital_ in his model is determined by
college attendance, and it roughly maps to our notion of _success_ (i.e.,
whether the individual succeeds if given the opportunity.)

In Durlauf’s model, students always prefer classmates with higher human
capital and colleges with higher fundamental quality. Unlike his model, we do
not capture the _spillover effects_ of students attending the same college on
each other. We also don’t allow for various levels of college quality. (It may
be worth noting that while in his general model, Durlauf allows for various
college qualities, in his analysis he focuses on at most two quality
levels—one college of high quality, the other of low quality.)

In both models, an admission rule maps a student’s human capital and group
membership into a binary outcome indicating whether the student is admitted.
In our work, groups corresponds to _socio-economic tiers_ , whereas in
Durlauf’s, they correspond to _racial groups_. In both models, the admission
rule may vary across time and generations.

Other simplifying assumptions in Durlaufs model are:

  * •

The level of state expenditures on education is fixed across generations.
Similarly, alpha is fixed in our model.

  * •

The only output of universities is human capital. Similarly, we only care
about the percentage of the population who succeed.

  * •

The initial human capital is accurately measured for every student. Similarly,
we assume ability and success probability are perfectly observable.

As for the objective function, Durlauf assumes the policymaker wants to
maximize the level of human capital among _the next generation_ of adult
citizens. He restricts attention to two generations only, but he notes that
“When one moves to this dynamic perspective, one also needs to consider an
_inter-temporal_ objective function; in parallel to my simplest specification,
which is based on the level of human capital at one point in time, it is
appropriate to consider a _weighted average_ of the levels across time. A
standard way to make this move involves working with […] a _discounted sum_ of
adult human capital across two generations.” The latter is precisely the
objective function we aim to optimize. Unlike our work, Durlauf does not solve
for the optimal policy. He merely compares the meritocratic policy with
affirmative action in terms of the average human capital of the next
generation. (Similar to our work, he defines a meritocratic admissions policy
to mean that each student is admitted to college exclusively on the basis of
their initial human capital. An affirmative action policy in his view is one
that takes group membership into account as well.)

Both models take the perspective of a policymaker who needs to choose between
affirmative action and meritocratic rules and aim to understand how efficiency
and equity interact. Similar to our findings, he finds that depending on the
parameters of the model, affirmative action can be more efficient than myopic
meritocratic admission rules. In our model, this happens because the human
capital of a student is directly impacted by that of his/her parent. In
Durlauf’s work, in contrast, this can happen because of the spillover effects
of students on each other and the impact of college quality on developing
adult human capital. In more concrete terms, Durlauf’s model can capture the
competing claims that ”diversity improves education for all students on
campus”, or ”stronger students going to the same college leads to higher
social capital for all of them”. Our model does not capture such _spillover_
effects. We instead focus on the _intertemporal_ or intergenerational effects
of admission rules. As mentioned earlier, Durlauf emphasizes the importance of
intergenerational factors in Section 6.2 of ”Affirmative action, meritocracy,
and efficiency”.

###  A.5 MDPs with Continuous State and Action Spaces

Finding the optimal policy for a continuous state- and action-space MDP is
often a difficult task [Marecki et al., 2006]. Two main categories of
algorithms have been proposed to (approximately) calculate the optimal policy:
(1) A typical approach is to discretize the state and action space, then solve
the resulting MDP using standard methods, such policy or value iteration. This
approach suffers from the well-known “curse of dimensionality”. (2) Another
approach approximates the optimal value function with a parametric form, then
sets out to fit those parameters such that they satisfy the Bellman equation
(see, e.g., [Li and Littman, 2005; Hauskrecht and Kveton, 2004].

##  Appendix B Limitations and Interpretations

Our model is designed to incorporate the basic points we mentioned in the
introduction in as simplified a fashion as possible; as such, it is important
to note some of its key limitations. First, it is intended to model the effect
of a single opportunity, and it treats other forms of mobility
probabilistically in the background. It also assumes that its fundamental
parameters ($\alpha,\sigma,\tau,\gamma$) as constant over all generations. It
treats an individual’s group membership ($A$ and $D$) and ability as a
complete description of their performance, rather than including any
dependence on the group membership of the individual’s parent. (That is, an
individual in group $A$ performs the same in the model regardless of whether
their parent belonged to group $A$ or $D$.) All of these would be interesting
restrictions to relax in an extension of the model.

Much of the past theoretical work on intergenerational mobility focuses on an
issue that we do not consider here: the strategic considerations faced by
parents as they decide how much to consume in the present generation and how
much to pass on to their children. Our interest instead has been in the
optimization problem faced by a social planner in allocating opportunities,
treating the behavior of the agents as fixed and simple. Here too, it would be
interesting to explore models that address these issues in combination.

Finally, because our focus is on intergenerational mobility in a socioeconomic
sense, we do not model discrimination based on race, ethnicity, or gender, and
the role of race-based and gender-based affirmative action in combatting these
effects. The model is instead concerned with _socio-economic_ or _class-based_
[Malamud, 1995; Kahlenberg, 1996] affirmative action. That said, the
ingredients here could be combined with models of statistical or taste-based
discrimination on these attributes to better understand their interaction.

The simplicity of our model, however, does allow us to make a correspondingly
fundamental point: that even a purely payoff-maximizing society can discover
affirmative action policies from first principles as it seeks to optimize the
allocation of opportunities over multiple generations. Moreover, the optimal
allocation policy is deeply connected to dynamic programming over the
generations; the society is essentially attempting to “steer” the balance of
group $A$ and group $D$ over time, making sure not to turn things too abruptly
(giving up present benefit) or too gradually (giving up future benefit).

##  Appendix C Properties of the Value Function

In this section, we show that the value function $V(.)$ for the decision
process $\mathcal{D}$ is unique, continuous, differentiable, and monotonically
decreasing. We begin by offering an alternative formulation of the decision
process that has the exact same optimal policy and value function, but is more
conducive to recursive analysis. Next, we prove several properties of the
state space $\Phi$ and the reward function $R^{\prime}$ for this alternative
decision process. Then we apply previously-established theorems from dynamic
programming and recursive analysis [Stokey, 1989] to establish the
aforementioned properties of $V(.)$.

#### A Decision Process Equivalent to $\mathcal{D}$

Given the decision process $\mathcal{D}=(\Phi,\Theta,S,R)$ (as defined in
Section 3), we first provide an alternative formulation, called
$\mathcal{D}^{\prime}$, that fits the standard representation of decision
processes in the Dynamic Programming literature. This standard formulation
allows us to import tools and theorems from Dynamic Programming with minor
modifications.

We construct $\mathcal{D}^{\prime}$ from $\mathcal{D}$ by re-defining the
action space—not in terms of thresholds, but in terms of the states reachable
through an admissible threshold from a given state. The state transitions in
this formulation become trivial, but we need to re-write the reward function
in terms of the new parameters (i.e., current and the next state).

Formally, given $\mathcal{D}$, we define
$\mathcal{D}^{\prime}=(\Phi,\Gamma,I,R^{\prime})$ as follows:

  * •

The correspondence $\Gamma:\Phi\rightarrow\Phi$ specifies the states reachable
by one admissible $\theta_{0}$ from any given state $\phi_{0}\in\Phi$. More
precisely

$\Gamma(\phi_{0})=\\{\omega\in\Phi|\exists\theta_{0}\in\Theta(\phi_{0})\text{
s.t. }\omega=S(\phi_{0},\theta_{0})\\}.$

  * •

$I:\Phi_{0}\times\Phi_{0}\rightarrow\Phi_{0}$ simply returns its second
argument, that is, for all $\phi_{0}\in[0,1]$, $I(\phi_{0},\omega)=\omega$.

  * •

To recast the reward function in terms of $(\phi_{0},\omega)$, we first write
$\theta_{0}$ as a function of $\phi_{0},\omega$:

$\displaystyle\omega=\phi_{0}-\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta^{2}_{0})$
$\displaystyle\Leftrightarrow$ $\displaystyle
2\sigma\frac{\omega}{\phi_{0}}+\sigma^{2}-2\sigma=\theta^{2}_{0}$
$\displaystyle\Leftrightarrow$
$\displaystyle\theta_{0}=\sqrt{2\sigma\frac{\omega}{\phi_{0}}+\sigma^{2}-2\sigma}$

Given the above change of variables, we can write:

$R^{\prime}(\phi_{0},\omega_{0})=\begin{cases}R\left(\phi_{0},\sqrt{2\sigma\frac{\omega}{\phi_{0}}+\sigma^{2}-2\sigma}\right)\quad\text{
if }\phi_{0}>0\\\ R(0,\sigma)\quad\text{ if }\phi_{0}=0\end{cases}$

###### Property 1

$\Phi$ is a convex subset of $\mathbb{R}$, and the correspondence $\Gamma$ is
nonempty, compact-valued, and continuous.

Proof  $\Phi=[0,1]$, which is clearly a convex subset of $\mathbb{R}$. The
correspondence $\Gamma$ can be characterized as follows:

$\displaystyle\Gamma(\phi_{0})$ $\displaystyle=$
$\displaystyle\\{\omega\in\Phi|\exists\theta_{0}\in\Theta(\phi_{0})\text{ s.t.
}\omega=S(\phi_{0},\theta_{0})\\}$ $\displaystyle=$
$\displaystyle\left\\{\omega\in\Phi|\omega=\phi_{0}-\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2})\text{
where
}\sigma\left(1-\frac{\alpha}{\phi_{0}}\right)\leq\theta_{0}\leq\frac{\sigma(1-\alpha)}{\phi_{0}}\text{
and }0\leq\theta_{0}\leq 1\right\\}$ $\displaystyle=$
$\displaystyle\begin{cases}\omega\in\left[\phi_{0}\left(1-\frac{1}{2\sigma}\right),\phi_{0}\right]\quad\text{
if }\phi_{0}\leq\alpha\\\
\omega\in\left[\phi_{0}-\sigma\alpha\left(1+\frac{\alpha}{2\phi_{0}}\right),\phi_{0}\right]\quad\text{
if }\alpha<\phi_{0}\leq 1-\alpha\\\
\omega\in\left[\phi_{0}-\sigma\alpha\left(1+\frac{\alpha}{2\phi_{0}}\right),\phi_{0}-\frac{\phi_{0}}{2\sigma}\left(1-\frac{1-\alpha}{\phi_{0}}\right)\left(1+\frac{1-\alpha}{\phi_{0}}\right)\right]\quad\text{
if }\phi_{0}\geq 1-\alpha\end{cases}$ $\displaystyle=$
$\displaystyle\left[\max\left\\{\phi_{0}\left(1-\frac{1}{2\sigma}\right),\phi_{0}-\sigma\alpha\left(1+\frac{\alpha}{2\phi_{0}}\right)\right\\},\min\left\\{\phi_{0},\phi_{0}-\frac{\phi_{0}}{2\sigma}\left(1-\frac{1-\alpha}{\phi_{0}}\right)\left(1+\frac{1-\alpha}{\phi_{0}}\right)\right\\}\right]$

Given the above definition, it is trivial to verify that $\Gamma$ is indeed
nonempty, compact-valued, and continuous.

###### Property 2

The reward function $R^{\prime}$ is bounded and continuous.

Proof  The reward function $R$ specifies the fraction of the population who
succeed if given the opportunity. So clearly $0\leq R(.,.)\leq 1$ is bounded.
As a result $R^{\prime}$ is also bounded. To establish continuity of
$R^{\prime}$, we first establish the continuity of $R$. It is trivial to see
that $R(.,.)$ (defined in (9)) is continuous at any $\phi_{0}<1$. At
$\phi_{0}=1$, we have

$\displaystyle\lim_{\phi_{0}\rightarrow
1}\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2})+\frac{1-\phi_{0}}{2\sigma}\left((\sigma+\tau)^{2}-\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-\lim_{\phi_{0}\rightarrow
1}\frac{1-\phi_{0}}{2\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-\lim_{\phi_{0}\rightarrow
1}\frac{1}{2\sigma}\frac{\left(\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}\right)^{2}}{1-\phi_{0}}$
($\Gamma(1)=\\{\sigma(1-\alpha)\\}$) $\displaystyle=$
$\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-\lim_{\phi_{0}\rightarrow
1}\frac{1}{2\sigma}\frac{\left(\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\sigma(1-\alpha)\right)^{2}}{1-\phi_{0}}$
(L’Hospital’s rule) $\displaystyle=$
$\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-\lim_{\phi_{0}\rightarrow
1}\frac{1}{\sigma}\frac{(\tau+\sigma(1-\alpha))\left(\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\sigma(1-\alpha)\right)}{1}$
$\displaystyle=$ $\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-0$
$\displaystyle=$ $\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})$
$\displaystyle=$ $\displaystyle R(1,\theta_{0}).$

So $R$ is continuous at $\phi_{0}=1$, as well. Finally, note that
$\sqrt{2\sigma\frac{\omega}{\phi_{0}}+\sigma^{2}-2\sigma}$ is continuous
function of $(\phi_{0},\omega)\in\Gamma(\phi_{0})$, so $R^{\prime}$ is also
continuous.

Let $C(\Phi)$ be the space of bounded continuous functions
$f:\Phi_{0}\rightarrow\mathbb{R}$ with the sup norm
$\|f\|=\max_{\phi_{0}\in\Phi}|f(\phi_{0})|$. We define the operator $T$ on the
space $C(\Phi)$ as follows:

$(Tf)(\phi_{0})=\max_{\omega\in\Gamma(\phi_{0})}R^{\prime}(\phi_{0},\omega)+\gamma
f(\omega).$

######  Theorem 2 (Adapted version of Theorem 4.6 in [Stokey, 1989])

Let $\Phi$, $\Gamma$, and $R^{\prime}$ satisfy Properties 1 and 2. Then the
operator $T$ maps $C(\Phi)$ into itself, $T:C(\Phi)\rightarrow C(\Phi)$, and
$T$ has a unique fixed point $V\in C(\Phi)$. Moreover, given $V$, the optimal
policy correspondence $\Pi^{*}:\Phi\rightarrow\Phi$ is compact-valued and
upper hemi continuous (u.h.c).

According to Cotter and Park [2006], if the reward function $R^{\prime}$ is
differentiable, the value function $V$ is differentiable on any interior point
that is an optimal “next state” for some current state. More precisely,

######  Theorem 3 (Adapted version of Theorem 2 in [Cotter and Park, 2006])

Suppose $\omega\in\Pi^{*}(\phi_{0})\cap(0,1)$ for some $\phi_{0}\in[0,1]$. If
$R^{\prime}$ is continuously differentiable, then $V$ is differentiable at
$\omega$, with
$\frac{\partial}{\partial\phi_{0}}V|_{\omega}=\frac{\partial}{\partial\phi_{0}}R^{\prime}|_{(\omega,\omega^{\prime})}$
for any $\omega^{\prime}\in\Pi^{*}(\omega)$.

It only remains to show that:

###### Property 3

$R^{\prime}$ is continuously differentiable on the interior of its domain.

Proof  To establish continuous differentiablity of $R^{\prime}$ with respect
to $\omega$, note that:

$\frac{\partial}{\partial\omega}R^{\prime}(\phi_{0},\omega)=\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})\frac{\partial}{\partial\omega}\theta_{0}.$

and both terms on the right hand side of the above equation are continuous.

$\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})=\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\theta_{0}}{1-\phi_{0}}\right)$
$\frac{\partial}{\partial\omega}\theta_{0}=\frac{\sigma}{\sqrt{2\sigma\omega\phi_{0}+(\sigma^{2}-2\sigma)\phi_{0}^{2}}}$

Therefore, $\frac{\partial}{\partial\omega}R^{\prime}(\phi_{0},\omega)$ is
trivially continuous at any $(\phi_{0},\omega)\in(0,1)^{2}$.

To establish continuous differentiability of $R^{\prime}$ with respect to
$\phi_{0}$, note that:

$\frac{\partial}{\partial\phi_{0}}R^{\prime}(\phi_{0},\omega)=\frac{\partial}{\partial\phi_{0}}R(\phi_{0},\theta_{0})+\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})\frac{\partial}{\partial\phi_{0}}\theta_{0}.$

It is easy to see that all three terms in the right hand side of the above are
continuous:

$\displaystyle\frac{\partial}{\partial\phi_{0}}R(\phi_{0},\theta_{0})$
$\displaystyle=$
$\displaystyle\frac{1}{2\sigma}(\sigma^{2}-\theta_{0}^{2})-\frac{1}{2\sigma}\left((\sigma+\tau)^{2}-\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)^{2}\right)$
$\displaystyle-\frac{1}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)\left(\frac{\sigma(1-\alpha)-\theta_{0}}{1-\phi_{0}}\right)$
$\frac{\partial}{\partial\phi_{0}}\theta_{0}=\frac{\sigma\omega}{\phi_{0}^{2}\sqrt{2\sigma\omega\phi_{0}+(\sigma^{2}-2\sigma)\phi_{0}^{2}}}$

Therefore, $\frac{\partial}{\partial\phi_{0}}R^{\prime}(\phi_{0},\omega)$ is
continuous at any $(\phi_{0},\omega)$ in the interior of $R^{\prime}$s domain.

###### Proposition 3

$V(\phi_{0})$ is monotonically decreasing at all $\phi_{0}\in(0,1)$.

Proof  For any $\phi_{0}\leq\phi^{*}_{0}$, the statement is easy to verify
given the closed form expression for the value function in (12). For any
$\phi_{0}>\phi^{*}_{0}$ we show that $V$ is decreasing in an open neighborhood
on the left side of $\phi_{0}$. Since $V$ is differentiable at $\phi_{0}$,
this implies $V^{\prime}(\phi_{0})\leq 0$.

Let $\phi^{\prime}_{0}$ be the state we get to if we apply the optimal
threshold $\theta_{0}$ at $\phi_{0}$. We have that
$\phi^{\prime}_{0}<\phi_{0}$ (note that at $\phi_{0}>\phi^{*}_{0}$,
$\theta_{0}<\sigma$, which implies $\phi^{\prime}_{0}<\phi_{0}$). Now for any
state $\phi^{\prime}_{0}<\phi^{\prime\prime}_{0}<\phi_{0}$, we can show that
$V(\phi^{\prime\prime}_{0})>V(\phi_{0})$. This is simply because we can reach
$\phi^{\prime}_{0}$ from $\phi^{\prime\prime}_{0}$ using a threshold
$\theta^{\prime\prime}_{0}>\theta_{0}$. To see this, note that:

$\displaystyle
S(\phi_{0},\theta_{0})=\phi^{\prime}_{0}=S(\phi^{\prime\prime}_{0},\theta^{\prime\prime}_{0})$
$\displaystyle\Leftrightarrow$
$\displaystyle\phi_{0}-\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2})=\phi^{\prime\prime}_{0}-\frac{\phi^{\prime\prime}_{0}}{2\sigma}(\sigma^{2}-{\theta^{\prime\prime}_{0}}^{2})$
$\displaystyle\Leftrightarrow$
$\displaystyle\phi_{0}(1-\frac{\sigma}{2}+\frac{\theta_{0}^{2}}{2\sigma})=\phi^{\prime\prime}_{0}(1-\frac{\sigma}{2}+\frac{{\theta^{\prime\prime}_{0}}^{2}}{2\sigma})$

Since $\phi_{0}>\phi^{\prime\prime}_{0}$, it must be the case that
${\theta^{\prime\prime}_{0}}>\theta_{0}$ for the above equation to hold.

Next, observe that compared to applying $\theta_{0}$ at $\phi_{0}$, using the
higher threshold of $\theta^{\prime\prime}_{0}$ at $\phi^{\prime\prime}_{0}$
leads to a higher immediate reward and the same next state value at
$\phi^{\prime\prime}_{0}$. From this observation, we can conclude that the
value of $\phi^{\prime\prime}_{0}$ is higher than that of $\phi_{0}$, because:

$\displaystyle V(\phi^{\prime\prime}_{0})$ $\displaystyle\geq$ $\displaystyle
r(\phi^{\prime\prime}_{0},\theta^{\prime\prime}_{0})+\gamma
V(\phi^{\prime}_{0})$ $\displaystyle\geq$ $\displaystyle
r(\phi_{0},\theta_{0})+\gamma V(\phi^{\prime}_{0})$ $\displaystyle=$
$\displaystyle V(\phi_{0}).$

Figure 6: The optimal policy $\theta_{0}$ at every state $0\leq\phi_{0}\leq 1$
for various settings of $\alpha,\sigma,\tau$, and $\gamma$. Note that in all
cases, the optimal threshold is monotonically decreasing with $\phi_{0}$.
Moreover, there exists a point below which $\sigma$ is the only optimal
threshold, and above which $\sigma$ is no longer optimal. Observe that this
point coincides with the tipping point, $\phi^{*}_{0}$, established in Theorem
1 (depicted by dashed lines). The dotted red line illustrates the line of
affirmative action, derived in Lemma 2. Notice that beyond $\phi^{*}_{0}$, the
optimal policy is always below the line of affirmative action.

Figure 7: The scaled value function at every state $0\leq\phi_{0}\leq 1$ for
various settings of $\alpha,\sigma,\tau$, and $\gamma$. The dashed lines
specify the tipping point, $\phi^{*}_{0}$. Note that in all cases, the value
function is continuous, concave, and decreasing.

Figure 8: The difference between $\theta_{0}$ and $\theta_{1}$ at every state
$0\leq\phi_{0}\leq 1$ for various settings of $\alpha,\sigma,\tau$, and
$\gamma$. The dashed lines specify the tipping point, $\phi^{*}_{0}$. Note
that strict affirmative action is only employed beyond $\phi^{*}_{0}$. Also
note that the extent of affirmative action is monotonically increasing in
$\phi_{0}$.

Figure 9: The state the optimal policy converges to with the initial state
$\phi_{0}$. The dashed lines specify the tipping point, $\phi^{*}_{0}$. Note
that the optimal policy never shrinks the size of group $D$ to a value less
than $\phi^{*}_{0}$.

###  C.1 Omitted Proofs

####  Proof of Lemma 1.

Proof  We utilize _backward induction_ to pinpoint the largest state at which
$\sigma$ is an optimal threshold. First, observe that if $\phi_{0}$ if
sufficiently small, $\sigma\in\Pi^{*}_{0}(\phi_{0})$. To see this, note that
at least for $\phi_{0}=0$, $\sigma\in\Pi^{*}_{0}(\phi_{0})$. Second, note that
if $\sigma\in\Pi^{*}_{0}(\phi_{0})$), Bellman optimality (10) must hold:

$\displaystyle V(\phi_{0})$ $\displaystyle=$ $\displaystyle
R(\phi_{0},\sigma)+\gamma V(S(\phi_{0},\sigma))$ $\displaystyle=$
$\displaystyle R(\phi_{0},\sigma)+\gamma V(\phi_{0}),$

where in the second line we utilized the fact that
$S(\phi_{0},\sigma)=\phi_{0}$ for all $\phi_{0}\in[0,1]$. This fact can be
readily verified through (8). Rearranging the above equation, we obtain:

$V(\phi_{0})=\frac{1}{1-\gamma}R(\phi_{0},\sigma).$

Replacing $R$ above with its definition (9) and plugging $\sigma$ in place of
$\theta_{0}$, we obtain:

$\displaystyle V(\phi_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{1-\gamma}\frac{1-\phi_{0}}{2\sigma}\left((\sigma+\tau)^{2}-\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\sigma}{1-\phi_{0}}\right)^{2}\right)$
(12) $\displaystyle=$
$\displaystyle\frac{1}{1-\gamma}\frac{1-\phi_{0}}{2\sigma}\left((\sigma+\tau)^{2}-\left(\frac{-\sigma\alpha+(1-\phi_{0})\tau+(1-\phi_{0})\sigma}{1-\phi_{0}}\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{1-\gamma}\frac{1-\phi_{0}}{2\sigma}\left((\sigma+\tau)^{2}-\left(\sigma+\tau-\frac{\sigma\alpha}{1-\phi_{0}}\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{1-\gamma}\frac{1-\phi_{0}}{2\sigma}\frac{\sigma\alpha}{1-\phi_{0}}\left(2(\sigma+\tau)-\frac{\sigma\alpha}{1-\phi_{0}}\right)$
$\displaystyle=$
$\displaystyle\frac{\alpha}{(1-\gamma)}\left((\sigma+\tau)-\frac{\sigma\alpha}{2(1-\phi_{0})}\right)$

A corollary of the above is that

$V(0)=\frac{\alpha}{(1-\gamma)}\left((\sigma+\tau)-\frac{\sigma\alpha}{2}\right).$

Next, we derive the largest $\phi_{0}$ at which $\sigma$ remains an optimal
threshold. Let’s denote this point by $\tilde{\phi}_{0}$. From Bellman
optimality, we know that an action is optimal if and only if it is optimal
with respect to the (optimal) value function. So taking (12) as the value
function $V$ up to $\tilde{\phi}_{0}$, any optimal threshold at
$\phi_{0}\leq\tilde{\phi}_{0}$ must maximize $R(\phi_{0},\theta_{0})+\gamma
V(S(\phi_{0},\theta_{0}))$. Since $R(\phi_{0},\theta_{0})+\gamma
V(S(\phi_{0},\theta_{0}))$ is differentiable and $\tilde{\phi}_{0}$ is the
largest state at which $\sigma$ is an optimal policy, $\sigma$ must satisfy
the first order condition at $\tilde{\phi}_{0}$—that is, the partial
derivative of $R(\phi_{0},\theta_{0})+\gamma V(S(\phi_{0},\theta_{0}))$
(w.r.t. $\theta_{0}$) must be 0 at $(\tilde{\phi}_{0},\sigma)$.

We can derive the derivative of $R(\phi_{0},\theta_{0})+\gamma
V(S(\phi_{0},\theta_{0}))$ as follows:

$\frac{\partial}{\partial\theta_{0}}\left\\{R(\phi_{0},\theta_{0})+\gamma
V(S(\phi_{0},\theta_{0}))\right\\}=\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})+\gamma\frac{\partial}{\partial\theta_{0}}\
S(\phi_{0},\theta_{0})\frac{\partial}{\partial\phi_{0}}V(S(\phi_{0},\theta_{0})).$
(13)

We can calculate each term in (13) as follows:

$\displaystyle\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})$
$\displaystyle=$
$\displaystyle-\frac{\phi_{0}\theta_{0}}{\sigma}+\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}\right)$
(14) $\displaystyle=$
$\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\phi_{0}\theta_{0}}{1-\phi_{0}}-\theta_{0}\right)$
$\displaystyle=$
$\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\theta_{0}}{1-\phi_{0}}\right).$
$\frac{\partial}{\partial\theta_{0}}S(\phi_{0},\theta_{0})=\frac{\phi_{0}\theta_{0}}{\sigma}.$
(15)
$\frac{\partial}{\partial\phi_{0}}V(\phi_{0})=-\frac{\sigma\alpha^{2}}{2(1-\gamma)(1-\phi_{0})^{2}}.$
(16)

Plugging (8), (16), (14), and (15) into (13), we obtain that:

$\displaystyle\frac{\partial}{\partial\theta_{0}}\left\\{R(\phi_{0},\theta_{0})+\gamma
V(S(\phi_{0},\theta_{0}))\right\\}=$ (17)
$\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\theta_{0}}{1-\phi_{0}}\right)-\gamma\frac{\phi_{0}\theta_{0}}{\sigma}\frac{\sigma\alpha^{2}}{2(1-\gamma)(1-\phi_{0}+\frac{\phi_{0}}{2\sigma}(\sigma^{2}-\theta_{0}^{2}))^{2}}.$

As mentioned earlier, at $(\tilde{\phi}_{0},\sigma)$, the derivative (17) must
amount to 0. Therefore, to find $\tilde{\phi}_{0}$, we must solve the
following equation (obtained by replacing $\theta_{0}$ in (17) with $\sigma$):

$\displaystyle
0=\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\sigma}{1-\phi_{0}}\right)-\gamma\phi_{0}\frac{\sigma\alpha^{2}}{2(1-\gamma)(1-\phi_{0})^{2}}$
$\displaystyle\Rightarrow$ $\displaystyle
0=(-\alpha\sigma+(1-\phi_{0})\tau)(1-\phi_{0})-\gamma\frac{\sigma^{2}\alpha^{2}}{2(1-\gamma)}$
$\displaystyle\Rightarrow$ $\displaystyle
0=\tau(1-\phi_{0})^{2}-\alpha\sigma(1-\phi_{0})-\frac{\gamma\sigma^{2}\alpha^{2}}{2(1-\gamma)}$

(Note that in the second line of the derivation above, we multiplied both
sides by $\sigma(1-\phi_{0})^{2}/\phi_{0})$.) Solving the above quadratic
equation for $(1-\phi_{0})$, we have:

$\tilde{\phi}_{0}\in
1-\frac{\alpha\sigma}{2\tau}\left(1\pm\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$

To obtain the tightest bound, we pick the smaller value among the above two
possibilities:

$\tilde{\phi}_{0}=1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$

Note that $\tilde{\phi}_{0}$ must always be between $0$ and $(1-\alpha)$ (due
to the budget constraints illustrated in Figure 2). So the above derivation
only goes through if $\tilde{\phi}_{0}\leq(1-\alpha)$. Therefore, we have:
$\phi^{*}_{0}=\max\left\\{0,\min\left\\{1-\alpha,\tilde{\phi}_{0}\right\\}\right\\}.$

####  Proof of Proposition 2.

Proof  Note that according to Lemma 3, for all
$\theta_{0}\in\Pi^{*}_{0}(\phi_{0})$ and all
$\theta_{1}\in\Pi^{*}_{1}(\phi_{0})$, $\theta_{0}\leq\theta_{1}$. It only
remains to show that the inequality is strict. That is, for all
$\theta_{0}\in\Pi^{*}_{0}(\phi_{0})$ and all
$\theta_{1}\in\Pi^{*}_{1}(\phi_{0})$, $\theta_{0}\neq\theta_{1}$.

Suppose not and there exists $\theta_{0}\in\Pi^{*}_{0}(\phi_{0})$ and
$\theta_{1}\in\Pi^{*}_{1}(\phi_{0})$, such that $\theta_{0}=\theta_{1}$.
According to Lemma 2, this implies that
$\theta_{0}=\sigma(1-\alpha)+\tau(1-\phi_{0})$. Next we show that
$\theta_{0}=\sigma(1-\alpha)+\tau(1-\phi_{0})$ cannot be an optimal threshold
at $\phi_{0}$.

Recall that

$\Pi^{*}_{0}(\phi_{0})=\arg\max_{\theta_{0}}R(\phi_{0},\theta_{0})+\gamma
V(S(\phi_{0},\theta_{0})).$

So if $\sigma(1-\alpha)+\tau(1-\phi_{0})\in\Pi^{*}_{0}(\phi_{0})$, it must
satisfy the following first-order condition:

$\displaystyle\frac{\partial}{\partial\theta_{0}}\left(R(\phi_{0},\theta_{0})+\gamma
V(S(\phi_{0},\theta_{0}))\right)=\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})+\gamma\frac{\partial}{\partial\theta_{0}}\
S(\phi_{0},\theta_{0})\frac{\partial}{\partial\phi_{0}}V(S(\phi_{0},\theta_{0}))=0$

But note that $\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})$ at
$\theta_{0}=\sigma(1-\alpha)+\tau(1-\phi_{0})$ is 0:

$\displaystyle\frac{\partial}{\partial\theta_{0}}R(\phi_{0},\theta_{0})$
$\displaystyle=$
$\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\theta_{0}}{1-\phi_{0}}\right)$
$\displaystyle=$
$\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{\sigma(1-\alpha)+(1-\phi_{0})\tau-\left(\sigma(1-\alpha)+\tau(1-\phi_{0})\right)}{1-\phi_{0}}\right)$
$\displaystyle=$
$\displaystyle\frac{\phi_{0}}{\sigma}\left(\frac{0}{1-\phi_{0}}\right)=0$

So $\theta_{0}=\sigma(1-\alpha)+\tau(1-\phi_{0})$ can only be optimal if

$\frac{\partial}{\partial\theta_{0}}\
S(\phi_{0},\theta_{0})\frac{\partial}{\partial\phi_{0}}V(S(\phi_{0},\theta_{0}))=0.$

But this equality cannot hold because
$\frac{\partial}{\partial\theta_{0}}S(\phi_{0},\theta_{0})=\frac{\phi_{0}\theta_{0}}{\sigma}>0,$
and $\frac{\partial}{\partial\phi_{0}}V(S(\phi_{0},\theta_{0}))<0$. So
$\theta_{0}=\sigma(1-\alpha)+\tau(1-\phi_{0})$ cannot be the optimal threshold
at $\phi_{0}$.

###  C.2 Derivation of $\gamma^{*}$, $\tau^{*}$, and $\alpha^{*}$

The precise derivation of $\phi_{0}^{*}$ in Equation 11 allows us to gain
insight into how the interaction between the primitive parameters of our model
can promote or avert affirmative action. In Figure 1, we focus on the
interactions among $\alpha,\tau,\gamma$ (for simplicity assuming that
$\sigma=1-\tau$) and illustrate the regimes of persistent affirmative action
(i.e., $\phi_{0}^{*}\leq 0$). We define and investigate the following
quantities:

  * •

Given $\tau$ and $\alpha$, $\gamma^{*}$ specifies the minimum discount factor
required for $\phi_{0}^{*}\leq 0$.

  * •

Given $\gamma$ and $\alpha$, $\tau^{*}$ specifies the maximum level of $\tau$
that can maintain $\phi_{0}^{*}\leq 0$.

  * •

Given $\tau$ and $\gamma$, $\alpha^{*}$ specifies the minimum level
opportunities required for $\phi_{0}^{*}\leq 0$.

Next, we derive the above quantities using Equation 11. In what follows, we
assume $\tau>\alpha\sigma$, because we have

$\displaystyle
1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$
$\displaystyle\leq$ $\displaystyle
1-\frac{\alpha\sigma}{2\tau}\left(1+1\right)$ $\displaystyle\leq$
$\displaystyle 1-\frac{\alpha\sigma}{\tau}$

and if $\tau<\alpha\sigma$, then $1-\frac{\alpha\sigma}{\tau}<1-1=0$ and
$\phi^{*}_{0}=0$.

#### Derivation of $\tau^{*}$

Assuming that $\alpha>0$, $\tau,\gamma\in(0,1)$, $\sigma=1-\tau$ and
$2\tau>\sigma\alpha$,

$\displaystyle
1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)\leq
0$ $\displaystyle\Leftrightarrow$ $\displaystyle
1\leq\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$
(assuming $\alpha>0$ and $0<\tau<1$) $\displaystyle\Leftrightarrow$
$\displaystyle\frac{2\tau}{\alpha\sigma}\leq
1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}$ $\displaystyle\Leftrightarrow$
$\displaystyle\frac{2\tau}{\alpha\sigma}-1\leq\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}$
(assuming $2\tau>\alpha(1-\tau)$) $\displaystyle\Leftrightarrow$
$\displaystyle\left(\frac{2\tau}{\alpha\sigma}-1\right)^{2}\leq
1+\frac{2\tau\gamma}{1-\gamma}$ $\displaystyle\Leftrightarrow$
$\displaystyle\frac{4\tau^{2}}{\alpha^{2}\sigma^{2}}-\frac{4\tau}{\alpha\sigma}+1\leq
1+\frac{2\tau\gamma}{1-\gamma}$ $\displaystyle\Leftrightarrow$
$\displaystyle\frac{4\tau^{2}}{\alpha^{2}\sigma^{2}}-\frac{4\tau}{\alpha\sigma}\leq\frac{2\tau\gamma}{1-\gamma}$
(divide by $\tau$ assuming $\tau>0$) $\displaystyle\Leftrightarrow$
$\displaystyle\frac{2\tau}{\alpha^{2}\sigma^{2}}-\frac{2}{\alpha\sigma}\leq\frac{\gamma}{1-\gamma}$
$\displaystyle\Leftrightarrow$ $\displaystyle
2\tau-2\alpha\sigma\leq\frac{\gamma\alpha^{2}\sigma^{2}}{1-\gamma}$ (18)
$\displaystyle\Leftrightarrow$ $\displaystyle
2\tau-2\alpha(1-\tau)\leq\frac{\gamma\alpha^{2}}{1-\gamma}(1-\tau)^{2}$
$\displaystyle\Leftrightarrow$ $\displaystyle
2-2+2\tau-2\alpha(1-\tau)\leq\frac{\gamma\alpha^{2}}{1-\gamma}(1-\tau)^{2}$
$\displaystyle\Leftrightarrow$ $\displaystyle
2-2(1-\tau)-2\alpha(1-\tau)\leq\frac{\gamma\alpha^{2}}{1-\gamma}(1-\tau)^{2}$
$\displaystyle\Leftrightarrow$ $\displaystyle
0\leq\frac{\gamma\alpha^{2}}{1-\gamma}(1-\tau)^{2}+2(1+\alpha)(1-\tau)-2$
$\displaystyle\Leftrightarrow$ $\displaystyle
0\leq\frac{\gamma\alpha^{2}}{2(1-\gamma)}(1-\tau)^{2}+(1+\alpha)(1-\tau)-1$
$\displaystyle\Leftrightarrow$ $\displaystyle\tau\leq
1-\frac{-(1+\alpha)+\sqrt{(1+\alpha)^{2}+\frac{2\gamma\alpha^{2}}{(1-\gamma)}}}{\frac{\gamma\alpha^{2}}{(1-\gamma)}}$

where the last line is derived by obtaining the roots of the quadratic
function $q(x)=\frac{\gamma\alpha^{2}}{2(1-\gamma)}x^{2}+(1+\alpha)x-1$, as
follows:

$x^{*}_{1}=\frac{-(1+\alpha)-\sqrt{(1+\alpha)^{2}+\frac{2\gamma\alpha^{2}}{(1-\gamma)}}}{\frac{\gamma\alpha^{2}}{(1-\gamma)}}\text{
,
}x^{*}_{2}=\frac{-(1+\alpha)+\sqrt{(1+\alpha)^{2}+\frac{2\gamma\alpha^{2}}{(1-\gamma)}}}{\frac{\gamma\alpha^{2}}{(1-\gamma)}}.$

Note that $\frac{\gamma\alpha^{2}}{2(1-\gamma)}>0$, so $q(x)\geq 0$ if and
only if $x<x^{*}_{1}$ or $x>x^{*}_{2}$. Note that $x^{*}_{1}<0$ so if we know
$x$ is positive, then $q(x)\geq 0$ if and only if $x>x^{*}_{2}$. Replacing $x$
with $(1-\tau)$, gives us equation (18).

#### Derivation of $\alpha^{*}$

Assuming that $\tau\in(0,1)$ and $\gamma<1$,

$\displaystyle
1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)\leq
0$ $\displaystyle\Leftrightarrow$ $\displaystyle
1\leq\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{2\tau}{\sigma}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)^{-1}\leq\alpha$

#### Derivation of $\gamma^{*}$

Assuming that $\alpha,\tau,\sigma,\gamma\in(0,1)$ and $\tau>\sigma\alpha$,

$\displaystyle
1-\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)\leq
0$ $\displaystyle\Leftrightarrow$ $\displaystyle
1\leq\frac{\alpha\sigma}{2\tau}\left(1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}\right)$
$\displaystyle\Leftrightarrow$ $\displaystyle\frac{2\tau}{\alpha\sigma}\leq
1+\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}$ $\displaystyle\Leftrightarrow$
$\displaystyle\frac{2\tau}{\alpha\sigma}-1\leq\sqrt{1+\frac{2\tau\gamma}{1-\gamma}}$
$\displaystyle\Leftrightarrow$
$\displaystyle\left(\frac{2\tau}{\alpha\sigma}-1\right)^{2}\leq
1+\frac{2\tau\gamma}{1-\gamma}$ $\displaystyle\Leftrightarrow$
$\displaystyle\left(\frac{2\tau}{\alpha\sigma}-1\right)^{2}-1\leq\frac{2\tau\gamma}{1-\gamma}$
$\displaystyle\Leftrightarrow$
$\displaystyle\frac{1}{2\tau}\left(\frac{2\tau}{\alpha\sigma}-1\right)^{2}-\frac{1}{2\tau}\leq\frac{\gamma}{1-\gamma}$
$\displaystyle\Leftrightarrow$ $\displaystyle
1-\left(\frac{1}{2\tau}\left(\frac{2\tau}{\alpha\sigma}-1\right)^{2}-\frac{1}{2\tau}+1\right)^{-1}\leq\gamma$