# Wealth Dynamics Over Generations:
Analysis and Interventions

Krishna Acharya  Georgia Tech ISyE  Eshwar Ram Arunachaleswaran  University of
Pennsylvania Department of Computer and Information Sciences  Sampath Kannan
University of Pennsylvania Department of Computer and Information Sciences
Aaron Roth  University of Pennsylvania Department of Computer and Information
Sciences  Juba Ziani  Georgia Tech ISyE

###### Abstract

We present a stylized model with feedback loops for the evolution of a
population’s wealth over generations. Individuals have both talent and wealth:
talent is a random variable distributed identically for everyone, but wealth
is a random variable that is dependent on the population one is born into.
Individuals then apply to a downstream agent, which we treat as a university
throughout the paper (but could also represent an employer) who makes a
decision about whether to admit them or not. The university does not directly
observe talent or wealth, but rather a signal (representing e.g. a
standardized test) that is a convex combination of both. The university knows
the distributions from which an individual’s type and wealth are drawn, and
makes its decisions based on the posterior distribution of the applicant’s
characteristics conditional on their population and signal. Each population’s
wealth distribution at the next round then depends on the fraction of that
population that was admitted by the university at the previous round.

We study wealth dynamics in this model, and give conditions under which the
dynamics have a single attracting fixed point (which implies population wealth
inequality is transitory), and conditions under which it can have multiple
attracting fixed points (which implies that population wealth inequality can
be persistent). In the case in which there are multiple attracting fixed
points, we study interventions aimed at eliminating or mitigating inequality,
including increasing the capacity of the university to admit more people,
aligning the signal generated by individuals with the preferences of the
university, and making direct monetary transfers to the less wealthy
population.

##  1 Introduction

The wealth of a population evolves over generations as a function of the
opportunities available to it. Opportunities available to a generation depend
not only on their talent, but also on the wealth of the previous generation.
In such a dynamical system, the initial wealth of a population determines how
wealth evolves and what it will be in the limit. Understanding this system can
help illuminate when and why inequalities can arise and persist.

In this paper we define and analyze a simple, mathematically-tractable model
for this feedback system, before considering possible interventions to make
its behavior more equitable. To discuss the main conclusions of our paper, we
first need to provide a sketch of our model. Individuals are divided across
multiple _populations_ , and have both a _type_ (an abstraction of talent) and
a _wealth_. Within a single population, the distribution of wealth and types
are given by Gaussians with known means and variances. Types are distributed
identically across populations, but each population has its own distribution
of wealth. An individual from a particular population is sampled by sampling
their type $T$ from the (universal) type distribution, and their wealth $W$
from the wealth distribution particular to their population. An individual
then generates a _signal_ $S=\beta T+(1-\beta)W$, i.e., some convex
combination of their wealth and type. This signal could represent e.g. an
individual’s score on a standardized test, or the rating that results from an
interview. Here we allow that the signal might have a dependence on wealth
rather than just type because of the indirect effects it can have on
evaluations: for example, the ability to engage in additional test
preparation. Downstream, a university111Throughout this paper we describe the
downstream agent as a university admitting students. However we could also
view the downstream agent as an employer hiring employees, or any other agent
allocating opportunities based on evidence that conflates talent and wealth
that have effects on the long-term wealth of the selected individuals. who has
no direct knowledge of applicant’s types or wealth (but with knowledge of the
distributions from which they were drawn) observes the signal, and forms a
posterior belief about the applicant’s type and wealth. The university seeks
to select individuals for whom another convex combination $\alpha
T+(1-\alpha)W$ exceeds some threshold $\tau$, and so selects exactly those
applicants for whom $\mathbb{E}[\alpha T+(1-\alpha)W|S]\geq\tau$. Here again
we allow that the university might have an explicit preference for wealth (and
not purely for type). This might represent e.g. a desire for full tuition
payments or future alumni donations, or a more nebulous desire for “culture
fit” or for skills associated with wealth (e.g. students who can walk on to
the sailing or squash team). For each population we then let the mean wealth
of the next generation be a non-decreasing function of the fraction of people
admitted to the university at the previous round. We also assume that the
distribution of types (or talent) remains unchanged over generations and is
identical for all individuals, independent of their population.

First, we consider the fixed points of these wealth dynamics. If there is only
a single fixed point (and the dynamics converge to it), this implies that
wealth inequality across population groups is transitory, and that over time
it will equalize (as the mean wealth of all populations moves to the single
attracting fixed point). On the other hand, if there are multiple fixed points
of the wealth dynamics, then wealth inequality can persist, with different
populations “stuck” at different fixed points.

We give conditions under which the dynamics correspond to a contraction map
and have a single fixed point (implying that wealth inequality is transitory).
These conditions in particular include the case when $\alpha=1$ — i.e. when
the university is selecting entirely based on inferred type. On the other
hand, there are other situations (in which, necessarily the university places
some weight $(1-\alpha)>0$ on wealth) in which case there can be two
attracting fixed points (and a third unstable fixed point), which can result
in persistent inequality absent intervention: one population can be “trapped”
in the less wealthy fixed point, while the other one is in the more wealthy
fixed point.

We then turn our attention to interventions. We focus our study of
interventions on ways to move a population’s wealth from the lower fixed point
to the higher fixed point, or to modify the dynamics so that there is a single
attracting fixed point (which leads to wealth equality). We consider three
types of interventions:

  1. 1.

Increasing The Capacity of the University: We consider what happens when the
university is able to admit more applicants (by lowering its threshold
$\tau$). We show that doing this has positive effects: either it shifts the
dynamics from the regime in which there are multiple fixed points to the
regime in which there is a single fixed point (thus leading to long-term
wealth equality), or it _raises the wealth of both attracting fixed points_.

  2. 2.

Changing the Design of the Signal $S$: We consider what happens if we are able
to better align the signal the university receives with the university’s
objective function (by shifting $\beta$ closer to $\alpha$ — i.e. by having
the signal weight type and wealth more similarly to how they are weighted in
the university’s objective function). We show that as $\beta$ is moved closer
to $\alpha$ the disparity between the two fixed points is reduced. Notably,
and perhaps counter-intuitively, making the signal depend more on type (by
increasing $\beta$) is _not_ always the way to reduce disparities (despite the
fact that type is distributed identically across populations).

  3. 3.

Direct Subsidies to the Disadvantaged Population: Finally we consider making
direct financial subsidies to the disadvantaged population, to shift them from
the lower wealth attracting fixed point to the basin of attraction of the
higher wealth fixed point (from which they will naturally proceed to the
higher wealth fixed point without further intervention). We consider a
parameterized family of objective functions that the designer might have, that
differ in how they relatively weight the _cost_ of the subsidy with the wealth
of the disadvantaged population, and in how they discount time. Within this
class of interventions, we focus on two options: the most aggressive “1-shot”
option makes a large 1-shot payment to directly increase the wealth of the
disadvantaged population to move them to the basin of attraction of the
wealthier fixed point. The least aggressive “limiting” option makes the
minimal payment per round that is guaranteed to cause eventual convergence to
the wealthier fixed point. We derive conditions under which the “1-shot”
option is preferred by the designer over the “limiting” option and vice versa.

###  1.1 Discussion and Limitations

For mathematical tractability, we study a simple stylized model, which should
be viewed as a first cut at attempting to model wealth inequality rather than
a faithful description of the full problem. For example, we have assumed that
the university has access to an applicant’s wealth only indirectly via
inferences that can be drawn from their test score and population. In
practice, a university has a number of other signals at their disposal. One
should interpret the wealth populations in our model as equivalence classes
induced by the information available to them at admissions time. Similarly, we
have modeled individual talent via a static “type” distribution, when in fact
talent is multi-dimensional and not static (and might _depend_ on
opportunities that different populations might have different access to prior
to university admissions). We have not modelled university capacity
constraints, and this allows us to treat each population independently of the
others.

Nevertheless, several qualitative takeaways emerge from our modelling that we
think are interesting: for example, in our model, the persistence of
inequality (multiple attracting fixed points) depends on the university using
a selection rule that intentionally takes into account wealth, rather than
just talent. We show that if the university puts $\alpha=1$ weight on type in
our model, there is only a single fixed point. This suggests that changes in
admission policies that reduce the focus on wealth (for example, switching to
need-blind admissions and reducing or eliminating legacy admissions) might
have beneficial long-term effects. As another takeaway, we find that
aggressive interventions (in our model, that aim to lift the lower-wealth
population in one shot to the basin of attraction of the higher-wealth fixed
point) are often the most cost effective in the long run, compared to more
modest interventions that would accomplish the same goal after $k>1$ rounds.
On the other hand, incremental interventions become optimal when society
heavily discounts the future. This suggests that institutions that are able to
formulate longer-term goals, but have the resources to act on them immediately
(for example, non-profit universities with large endowments) may be able to
combat wealth inequality more effectively.

###  1.2 Related Work

Our paper is related to economic models of inequality, which date back to
Arrow (1973) and Phelps (1972). For example, Coate and Loury (1993) and Foster
and Vohra (1992) study two stage models in which the existence of self-
confirming equilibria can cause inequality to be persistent even when
populations are ex-ante identical.

More recently, the computer science community has begun studying dynamic
models of fairness. Jabbari et al. (2017) study the costs of imposing fairness
constraints on learners in general Markov decision processes. Hu and Chen
(2018) study a dynamic model of the labor market similar to that of Coate and
Loury (1993); Foster and Vohra (1992) in which two populations are symmetric,
but can choose to exert costly effort in order to improve their value to an
employer. They study a two-stage model of a labor market in which
interventions in a “temporary” labor market can lead to high-welfare symmetric
equilibrium in the long run. Liu et al. (2018) study a two-round model of
lending in which lending decisions in the first round can change the type
distribution of applicants in the 2nd round, according to a known, exogenously
specified function. Liu et al. (2020) study a dynamic model where in each
round, strategic individuals decide whether to invest in qualifications and
the decision-maker updates his classifier that decides which individuals are
qualified; they characterize the equilibria of such dynamics and develop
interventions that lead to better long-term outcomes. Arunachaleswaran et al.
(2020) study a model in which decisions over individuals and populations are
made along a multi-layered pipeline, where each layer corresponds to a
different stage of life. They consider the algorithmic problem faced by a
budgeted centralized designer who aims to intervene on the transitions between
layers to obtain optimally fair outcomes, when such modifications are costly.
Kannan et al. (2019) study a two stage model of affirmative action in which a
college may set different admissions policies for an advantaged and
disadvantaged group, but a downstream employer makes hiring decisions that
maximize their expected objective given their posterior belief on student
qualifications (that depend on the college’s policies). Jung et al. (2020)
study an equilibrium model of criminal justice in which two populations with
different outside option distributions make rational decisions as a function
of criminal justice policy; they show that policies that have been proposed
with equity considerations in mind (equalizing false positive and negative
rates) actually emerge as optimal solutions to a social planner’s optimization
problem even without an explicit equity goal.

We highlight two closely related papers. Heidari and Kleinberg (2021) also
study a model of inter-generational wealth dynamics across many rounds, in
which both wealth (which they model as a binary) and talent play a role in
success, as a function of opportunities that can be allocated to a limited
portion of the population. Like us, Heidari and Kleinberg (2021) use college
admissions as a running example of an institution allocating the
opportunities, and like us, study a model in which admissions to college plays
the role of determining wealth increase or decrease from one generation to the
next. Our models differ in a number of specifics, but the primary difference
between these two works is that Heidari and Kleinberg (2021) study the optimal
policy for a very patient institution interested in maximizing its long-run
payoff, and show that it recovers a form of affirmative action, preferentially
offering opportunities to the less wealthy population so that it can reap the
benefits of their resulting increased wealth in future generations. In
contrast, we study institutions that are myopic, and optimize only for their
short-term reward. In this setting, we study conditions under which such
institutions do or do not perpetuate inequality, and study interventions in
the settings in which they do. Mouzannar et al. (2019) study a continuous time
dynamic in which two populations with binary, fully observable type are
selected by a college with a myopic objective, whose selection rate within
each population changes their type distribution at the next round. They study
conditions under which imposing an “affirmative action” constraint (having the
same selection rate within each population) can lead to equal or improved
outcomes in the long run. In addition to various differences in the models
(they study a setting with fully observed, binary types) and the class of
interventions considered (we study changing what is observable to the
university and offering direct subsidies), the most salient difference is that
our model seeks to understand how wealth evolves over the long run, and has
the crucial feature that wealth can be partially conflated with type in the
signal observed by the university. We also focus on intervening directly on
wealth via monetary subsidies. In contrast, Mouzannar et al. (2019) do not
model wealth, and in their dynamic, the type distribution directly evolves and
is fully observable. Hence their myopic college does not need to do any
inference as in our model.

##  2 Preliminaries

######  Definition 1 (Attracting fixed points).

Let $f:~{}\mathbb{R}\to\mathbb{R}$ be a real-valued function and let $x^{*}$
be such that $f(x^{*})=x^{*}$. We call $x^{*}$ a fixed point of $f$. Further,
let $a_{t}(x)$ be the sequence defined by $a_{0}=x$ and $a_{t+1}=f(a_{t})$; we
say that $x^{*}$ is attracting for $x$ if and only if $a_{t}(x)$ converges to
$x^{*}$.

######  Claim 1 (Attracting fixed points).

Let $f$ be a real-valued, continuous, non-decreasing function such that
$x^{*}$ is a fixed point of $f$. If $f(x)>x$ for all $x\in[a,x^{*})$, $x^{*}$
is attracting on $[a,x^{*})$. Similarly, if $f(x)<x$ for all $x\in(x^{*},b]$,
$x^{*}$ is attracting on $(x^{*},b]$.

###### Proof.

Let $x\in[a,x^{*})$. Let $a_{t}(x)$ be the sequence defined by $a_{0}=x$ and
$a_{t+1}=f(a_{t})$. $a_{1}=f(a_{0})>a_{0}$, Since $f$ is non-decreasing
$a_{2}=f(f(a_{0}))\geq f(a_{0})$, but $f(x)>x$ $\forall x\in[a,x^{*})$, so
this inequality is in fact strict, i.e $a_{2}>a_{1}$. Note that by induction,
we have for all $t$ that $x^{*}=f(x^{*})\geq
a_{t+1}(x)=f(a_{t}(x))>a_{t}(x)\ldots>a_{0}$; hence $a_{t}$ is increasing and
$a_{t}\in[a,x^{*}]$ for all $t$. In particular, $a_{t}$ is a convergent
sequence with a finite limit in $[a,x^{*}]$. Now, since $f(x)>x$ for all
$x<x^{*}$, $x^{*}$ is $f$’s unique fixed point on $[a,x^{*}]$. Because $f$ is
continuous, we must have
$\lim_{t\to+\infty}a_{t+1}=\lim_{t\to+\infty}f(a_{t})=f\left(\lim_{t\to+\infty}a_{t}\right)$,
i.e. the limit $l$ must satisfy $f(l)=l$. The only point on $[a,x^{*}]$ that
satisfies this condition is $x^{*}$, yielding the first part of the result. A
similar argument holds for the second part of the proof. ∎

##  3 Model

We consider a university that has a non-atomic set of applicants from two
different sub-populations (or groups), denoted $1$ and $2$, and must decide
which applicants to admit. Each applicant has a type $T$, where the types are
random variables drawn i.i.d. from a known distribution $\mathcal{D}$; we
assume that the distribution of types is the same for both groups. Further,
each applicant also has a wealth $W_{i}$; wealth is drawn i.i.d. from a known
distribution $\mathcal{W}_{i}$ which may depend on the applicant’s group $i$.
We assume that wealth and types are drawn independently of each other.

The university aims to make decisions based on each applicant’s type and
wealth, and would like to admit individuals for whom

$\alpha T+(1-\alpha)W\geq\tau,$

for some parameter $\alpha\in[0,1]$ and some threshold $\tau$. Here, $\alpha$
represents how much the university values type versus wealth. However, the
university cannot directly observe this quantity. Instead, we assume that it
can only see a score $S$ for each applicant (for example in the form of a
standardized test result), where the score is a convex combination of both an
applicant’s type $T$ and wealth $W$. We write

$S=\beta T+(1-\beta)W,$

for some known $\beta\in[0,1]$. The university then performs a Bayesian update
and admits individuals that satisfy

$\mathbb{E}_{T,W}\left[\alpha T+(1-\alpha)W|S\right]\geq\tau.$

We are interested in understanding the long-term dynamics of a process where
the university’s decisions (made as described above) affect the individuals’
future attributes. We consider a discrete time horizon, in which at each time
step $t\in\mathbb{Z}^{+}$, the university’s decisions shapes the distribution
of wealth in each group in time step $t+1$. In particular, we assume that the
expected wealth $\mu_{i}^{t+1}$ of group $i$ in step $t+1$ is the fraction of
group $i$ that is admitted by the university at time step $t$. I.e., we write

$\displaystyle\mu_{i}^{t+1}=\mathbb{P}_{S}\left[\mathbb{E}_{T,W}\left[\alpha
T+(1-\alpha)W|S\right]\geq\tau\right]$ (1)

This is motivated by the fact that students that are admitted to competitive
universities are expected to reach better life outcomes and accumulate more
wealth.

In the rest of the paper, we make the following assumptions on the functional
form of the type and wealth distributions:

######  Assumption 1.

The type $T$ at any time instant is drawn from the distribution
$T\sim\mathcal{N}\left(0,\gamma^{2}\right)$. The initial wealth at time $0$
for population $i$ satisfies $\mu^{0}\in[0,1]$, and
$W_{i}\sim\mathcal{N}\left(\mu_{i}^{t},\sigma^{2}\right)$ at time step $t$ for
a fixed constant $\sigma$.

Note that the type can be centered around $0$ without loss of generality, by
changing the value of $\tau$ used by the employer by the corresponding amount.
The assumption $\mu^{0}\in[0,1]$ is also without loss of generality, and
simply renormalizes the average wealth of a group to be between $[0,1]$, so
long as we consider populations with bounded wealth. Note that with our
assumptions the mean wealth of each group always stays in the range $[0,1]$
although the sampled wealth of individuals can fall outside this interval.

##  4 Wealth Dynamics and Properties

We note that the dynamics of each group only depend on the decisions made by
the university within that group. Therefore, we can treat groups
independently. In this section, we focus on a single group at a time, and drop
the dependencies on $i$ in our notations for simplicity. We show that several
attracting fixed points can arise from our dynamics; in particular, there are
regimes of parameters under which there is a low wealth fixed point that
groups with initially low wealth converge to, and a high wealth fixed point
that groups with initially high wealth converge to. In Section 5.3, we
consider interventions that apply to more general update functions that the
ones described in this Section, so long as they have similar fixed point
properties.

###  4.1 Computing the Wealth Update Rule

We start by characterizing the joint distributions of the type $T$, the wealth
$W$, and the score $S$.

######  Claim 2.

Let $\mu\triangleq\mathbb{E}[W]$. We have that $(T,S)$ forms a bivariate
Gaussian distribution with mean $(0,(1-\beta)\mu)$ and covariance matrix

$\begin{bmatrix}\gamma^{2}&\beta\gamma^{2}\\\
\beta\gamma^{2}&\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}.\end{bmatrix}$

Similarly, $(W,S)$ forms a bivariate Gaussian distribution with mean
$(\mu,(1-\beta)\mu)$ and covariance matrix

$\begin{bmatrix}\sigma^{2}&(1-\beta)\sigma^{2}\\\
(1-\beta)\sigma^{2}&\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}.\end{bmatrix}$

The proof is provided in Appendix A.1. This allows us to compute the update
function that maps the wealth of a group in the current round, $\mu^{t}$, to
the wealth of that same group in the next round, $\mu^{t+1}$:

######  Lemma 1.

Recall that $\tau$ is the threshold used by the university to decide
admission. At every time step $t$, we have

$\displaystyle\mu^{t+1}=1-\Phi\left(K\left(\alpha,\beta,\gamma,\sigma\right)\left(\tau-(1-\alpha)\mu^{t}\right)\right),$

where
$K\left(\alpha,\beta,\gamma,\sigma)\right)\triangleq\frac{\sqrt{\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}}}{\alpha\beta\gamma^{2}+(1-\alpha)(1-\beta)\sigma^{2}}$
and $\Phi$ is the cumulative density function of a standard Gaussian. We
denote the update rule function

$\displaystyle f(x)\triangleq
1-\Phi\left(K\left(\alpha,\beta,\gamma,\sigma\right)\left(\tau-(1-\alpha)x\right)\right).$
(2)

For simplicity of notation, we omit the dependency of $f$ in the parameters of
the problem when clear from context. When not, we explicitly write the
dependency of $f$ in the parameters of interest. The proof of Lemma 1 is
mostly algebraic, and is provided in Appendix A.2.

###  4.2 Fixed Points and Convergence of the Dynamics

We can now use the closed-form expression for the update rule to study the
properties of the wealth dynamics. In this section, we bound the number of
fixed points of our dynamics, provide properties of these fixed points, and
characterize which fixed point each initial wealth converges to. We start by
noting that the update rule has a simple shape. Indeed:

######  Claim 3.

$f(x)$ is continuous and increasing in $x$. Further, $f$ is convex on
$[0,x^{*}]$ and concave on $[x^{*},1]$ where

$\displaystyle x^{*}=\begin{cases}0&\text{if}~{}\tau\leq 0,\\\
\frac{\tau}{1-\alpha}&\text{if}~{}0<\tau<1-\alpha,\\\ 1~{}\text{if}&\tau\geq
1-\alpha.\end{cases}$

The proof of the above claim is given in Appendix A.3. We now use the above
properties on the shape of $f$ to derive properties of its fixed point. First
we remark that $f$ has at least one fixed point, since $f(0)>0$ and $f(1)<1$,
and $f$ is continuous. Now, note that the number of fixed points of $f$ is
also upper-bounded:

######  Lemma 2.

Suppose $0<\tau<1-\alpha$, then $f(x)=x$ has at most $3$ solutions for
$x\in[0,1]$. If $f$ has $3$ fixed points $z_{1}<z_{2}<z_{3}$, it must be that
$z_{1}<\frac{\tau}{1-\alpha}<z_{3}$. If $\tau\leq 0$ or $\tau\geq 1-\alpha$,
$f(x)=x$ only has a single solution for $x\in[0,1]$.

The proof is provided in Appendix A.4. Lemma 2 has direct implications for
disparities across groups with different starting expected wealth. In
particular, the number of fixed points of $f$ determines whether different
groups must converge to equal wealth (the case in which there is only a single
fixed point) in the long-run or whether there are cases in which wealth
inequality is persistent (the case in which there are multiple fixed points).
We discuss these implications in more details in the rest of this section.

#### The Case of a Single Fixed Point

We now study conditions under which $f$ has single vs. multiple fixed points.
We first consider the case of a single fixed point. In this case, we remark
that the single fixed point has the following property:

######  Claim 4.

If $z$ is the single fixed point of $f$, then $z$ is attracting on $[0,1]$.

###### Proof.

Since $f(0)>0$, $f(1)<1$, and $f$ is continuous and has a single fixed point
$z$, it must be that $f(x)>f(z)=z$ for $x<z$ and $f(x)<f(z)=z$ for $x>z$.
Applying Claim 1 concludes the proof. ∎

This implies in particular that when $f$ has a single fixed point $z$, wealth
dynamics converge to this fixed point no matter what the starting wealth was.
This means in particular that there are no long-term disparities between
populations of different initial socio-economic statuses (though they may take
different amounts of time to reach the same wealth), i.e. the dynamics self
correct for initial wealth disparities. Figure 1 shows an instantiation of a
wealth update functions with a single fixed point and the corresponding wealth
dynamics (in green); the plots illustrate convergence of the dynamics to the
single fixed point starting both from an initially low wealth (Figure 1 (a))
and from an initially high wealth (Figure 1 (b)).

(a) Low initial wealth (b) High initial wealth

Figure 1: A wealth update function with a single fixed point, for
$\alpha=0.1,\beta=0.6,\gamma=0.4,\sigma=1.1,\tau=0.2$. The update function is
plotted in blue, its single fixed point in red, and the wealth dynamics
induced by the update function in green. Sub-figure (a) considers dynamics
starting at an initial wealth of $0.2$ while sub-figure (b) considers dynamics
starting at wealth $1.0$.

We note that Lemma 2 already implies that there exist interesting situations
in which the dynamics have a single attracting fixed point and wealth dynamics
are self-correcting. The first one is when $\tau$ is small ($\tau<0$); i.e.,
the university is not very selective in its admissions. Intuitively, this
leads to most individuals from any group being admitted (almost) independently
of their starting wealth, which allows even economically disadvantaged groups
to build wealth over time. The other situation deriving from Lemma 2 arises
when $\tau>1-\alpha$. This can arise for two reasons: first, is the university
is very selective and sets high values of $\tau$, wealth becomes insufficient
to qualify an individual for admission (as then $(1-\alpha)E[W|S]\leq
1-\alpha<\tau$); an agent must have sufficiently high (inferred) type to be
admitted, which helps reduce disparities due to wealth. This can also arise
when $\alpha$ is large and the university is mostly interested in type over
wealth. Intuitively, in this case, the university pays significant attention
to their posterior belief on the type of an individual, which facilitates
equalizing the treatment of groups of different wealth since they have the
same type distributions; while the university cannot observe type directly,
they discount for average wealth more (by a factor of $(1-\alpha)\mu$) hence
correct for wealth disparities more as $\alpha$ is smaller.

Below, we provide an additional condition under which $f$ has a single fixed
point:

######  Claim 5.

If $K(\alpha,\beta,\gamma,\sigma)\leq\frac{\sqrt{2\pi}}{1-\alpha}$, $f$ is a
contraction mapping and has a unique attracting fixed point.

###### Proof.

This immediately follows from
$f^{\prime}(x)=\frac{K(1-\alpha)}{\sqrt{2\pi}}\exp\left(-K^{2}(\tau-(1-\alpha)x)^{2}/2\right)$
and from
$\exp\left(-K^{2}(\tau-(1-\alpha)x)^{2}/2\right)\leq 1$ (with equality at
$x=\tau/(1-\alpha)$). Note that $f(0)>0$ and $f(1)<1$ so the fixed point $z$
must satisfy $f(x)<z$ if and only if $x<z$ and must be attracting. ∎

We note that $K(\alpha,\beta,\gamma,\sigma)=\frac{\sqrt{Var(S)}}{Cov(D,S)}$
where $D=\alpha T+(1-\alpha)W$ . This implies that, holding the college’s
objective function (i.e., $\alpha$) constant, the better the scoring rule
aligns with the university’s admissions criteria (i.e. as the covariance
between $D$ and $S$ increases), the smaller $K$ becomes. This makes the
condition that $f$ is a contraction mapping with a single fixed point easier
to satisfy, which in turn causes wealth dynamics to self-correct for initial
inequality. When $\alpha\to 1$, the condition is always satisfied, and $f$ has
a single fixed point. This may not be surprising in that in this case the
university only cares about type in admissions, and the university requires a
higher threshold on scores for wealthier populations; this helps reduce
disparities across populations with disparate wealth.

#### The Case of Multiple Fixed Points

We first characterize which fixed points are attracting when multiple points
arise, and which regime of initial wealth lead to which fixed points. We focus
on the case of three fixed points, as the case of two fixed points is a corner
case than can only arise if $f(x)$ is tangent to $Id(x)=x$ at one of the fixed
points.222Suppose this is not the case. $f(0)>0$ hence $f(x)>x$ before the
first fixed point. Because it is not tangent to the identity line, it must
then be that $f(x)<x$ between the first and the second fixed point. Similarly,
it must then be that $f(x)>x$ after the second, last fixed point. This
contradicts $f(1)<1$.

######  Claim 6.

Suppose $f$ has $3$ fixed points, denoted $z_{1}<z_{2}<z_{3}$. Then $z_{1}$ is
attracting for $[0,z_{2})$ and $z_{3}$ is attracting for $(z_{2},1]$.

###### Proof.

This follows from the proof of lemma 2. Indeed, let $g(x)=f(x)-x$, we have
that $g(0)=f(0)>0$, then $g$ must decrease below $0$, increase above $0$, and
decreases below $0$ again as $g(1)=f(1)-1<0$. This implies that $f(x)>x$ for
$x<z_{1}$ and $x\in(z_{2},z_{3})$, while $f(x)<x$ for $x\in(z_{1},z_{2})$ and
$x>z_{3}$. ∎

In particular, when there are three fixed points, a population that starts
with low wealth will converge to the first fixed point, while a group with
large initial wealth will converge to the third fixed point. In this case,
initial disparities in wealth persist in the long term, and interventions are
needed for different populations to obtain equitable long-term wealth
outcomes. Figure 2 shows a wealth update function with three fixed points and
the corresponding dynamics for two different starting points. We note that
starting at low wealth leads to convergence to the first and lowest fixed
point, while starting at relatively high health leads to convergence to the
highest fixed point. The figure illustrates how wealth disparities can
propagate and amplify over time.

(a) Low initial wealth (b) High initial wealth

Figure 2: A wealth update function with 3 fixed points, for
$\alpha=0.1,\beta=0.95,\gamma=1.4,\sigma=1.1,\tau=0.5$. the update function is
plotted in blue, its fixed points in red, and the wealth dynamics induced by
the update function in green. Sub-figure (a) considers dynamics starting at an
initial wealth of $0.5$ while sub-figure (b) considers dynamics starting at an
initial wealth of $0.7$.

We note that such situations can only arise in the regime in which
$0<\tau<(1-\alpha)$. In particular:

######  Claim 7.

Suppose $K(\alpha,\beta,\gamma,\sigma)>\frac{\sqrt{2\pi}}{1-\alpha}$ and
$\tau=\frac{1-\alpha}{2}$. Then $f$ has $3$ fixed points.

###### Proof.

In this case, note that $x^{*}=\frac{\tau}{1-\alpha}=\frac{1}{2}$. Further, we
know that

$\displaystyle f(x^{*})$
$\displaystyle=1-\Phi\left(K\left(\alpha,\beta,\gamma,\sigma\right)\cdot\left(\tau-(1-\alpha)x^{*}\right)\right)$
$\displaystyle=1-\Phi(0)$ $\displaystyle=1/2,$

implying that $x^{*}=1/2$ is a fixed point of $f$. Further,

$\displaystyle f^{\prime}(x^{*})=\frac{K(1-\alpha)}{\sqrt{2\pi}}>1,$

hence $f(x)<x$ in a small neighborhood $(x^{*}-\varepsilon,x^{*})$ and
$f(x)>x$ in a small neighborhood $(x^{*},x^{*}+\varepsilon)$. By continuity of
$x$ and the fact that $f(0)>0$, $f(x)=x$ must have a solution on $[0,x^{*})$.
Similarly, since $f(1)<1$, $f(x)=x$ must have a solution on $(x^{*},1]$. ∎

Note that because $f$ is continuous in $\tau$, $f$ must have three fixed
points for any $\tau$ in a neighborhood of $\frac{1-\alpha}{2}$. I.e., there
exists a continuous range of values of $\tau$ for which $f$ has three fixed
points, showing that such situations are not a corner case of our framework,
unlike when $f$ has two fixed points.

######  Remark 1.

There is a gap between the conditions given in Claim 5 and Lemma 2 under which
a single fixed point arises, and the condition given in Claim 7. In
particular, we remark that even if $f$ is not a contraction mapping and
$K>\frac{\sqrt{2\pi}}{1-\alpha}$, or when $0\leq\tau\leq 1-\alpha$, it may
still have only a single fixed point. To investigate how often $3$ fixed
points can arise, we picked a uniform grid of parameter values
$(\alpha,\beta,\gamma,\sigma,\tau)\in[0,1]^{5}$ and investigated what fraction
of the parameters that satisfy either $0\leq\tau\leq 1-\alpha$ or
$K>\frac{\sqrt{2\pi}}{1-\alpha}$ actually lead to an update rule with three
fixed points. We found that this was the case for roughly $30$ percent of the
values we explored, implying the existence of a significant range of
parameters for which there are disparities in the long term wealth of
different populations.

##  5 Interventions To Improve Long Term Population Wealth

In this section, we consider different types of intervention aiming at
improving and equalizing population wealth, when the wealth dynamics have
multiple fixed points (and so are not necessarily self correcting). We
consider three types of interventions: i) changing the design of the admission
rule used by the university, ii) changing the design of the standardized test
or scoring rule that the university relies on, and iii) providing subsidies to
disadvantaged groups.

###  5.1 Changing the Admission Rule

Since in our model, it is admission to university that confers a wealth
advantage to the next generation, a natural intervention is to increase the
capacity of the university, thereby admitting more people. Rather than
changing the objective function of the university $(\alpha)$, we model this
kind of intervention by decreasing the university’s admissions threshold
$\tau$. We note that although we do not explicitly quantify it, increasing the
capacity of a university will come at some financial cost, and so this kind of
intervention is not necessarily incomparable to the direct subsidies we
consider later.

The claim below characterizes how the fixed points of $f$ change when we
change the value of $\tau$. For the sake of notation, we let $f(.,\tau)$ be
the update rule when the chosen threshold is $\tau$, and omit the dependencies
of $f$ in the other parameters of the problem.

######  Theorem 1.

Fix $\alpha,\beta,\gamma,\sigma$. For a given admission threshold $\tau$, let
$z_{1}(\tau)<z_{2}(\tau)<z_{3}(\tau)$ be the fixed points of $f$ when all $3$
exist. Let $\tau^{\prime}<\tau$ be such that $f(.,\tau^{\prime})$ has $3$
fixed points, we have

$z_{1}(\tau^{\prime})>z_{1}(\tau)~{}~{}\text{and}~{}~{}z_{3}(\tau^{\prime})>z_{3}(\tau),$

but

$z_{2}(\tau^{\prime})<z_{2}(\tau).$

###### Proof Sketch.

The proof follows simply by showing that the update function is decreasing in
$\tau$. In turn, decreasing $\tau$ moves the update rule $f$ “up”, which
increases attracting fixed points and decreases unstable ones. The full proof
is given in Appendix B.1. ∎

In interpreting Theorem 1, we recall that only the first and third fixed
points $z_{1}$ and $z_{3}$ are attracting, and that $z_{2}$ is unstable (has
no points $x$ for which it is attracting for $x\neq z_{2}$). Hence, if we are
in a situation in which there is persistent wealth inequality (multiple fixed
points), we find that if we can decrease the admissions threshold $\tau$ of
the university, then _either_ :

  1. 1.

We increase the wealth of both of the attracting fixed points (and hence the
wealth of both populations, irrespective of which attractive fixed point they
are at). By decreasing $z_{2}$ we also reduce the size of the attracting
region $[0,z_{2})$ of the lower wealth fixed point, thus enabling poorer
populations to converge to the most desirable fixed point. Or

  2. 2.

We move the dynamics to one in which there is only a single fixed point, and
hence eliminate wealth inequality.

###  5.2 Changing the design of the scoring rule $S$

The college has to engage in inference about an applicant’s type and wealth
when $\beta\neq\alpha$, because the signal it receives does not align with its
objective function. What if we can modify the signal (by e.g. changing the
design of a standardized test) to more closely align the signal with the
college’s objective?

In this section we characterize how the fixed points of $f$ change when we
change the value of $\beta$. We denote $f(.,\beta)$ the update rule when the
scoring rule uses parameter $\beta$, while the other parameters of the problem
remain fixed.

######  Theorem 2.

Fix $\alpha,\tau,\gamma,\sigma$. For a given $\beta$, let
$z_{1}(\beta)<z_{2}(\beta)<z_{3}(\beta)$ be the fixed points of $f$ when all
$3$ exist. Suppose $\beta<\alpha$ and let $\beta^{\prime}\in(\beta,\alpha)$ be
such that $f(.,\beta^{\prime})$ has $3$ fixed points, then

$z_{1}(\beta^{\prime})>z_{1}(\beta)~{}~{}\text{and}~{}~{}z_{3}(\beta^{\prime})<z_{3}(\beta).$

Similarly, if $\beta>\alpha$ and $\beta^{\prime}\in(\alpha,\beta)$, we have

$z_{1}(\beta^{\prime})>z_{1}(\beta)~{}~{}\text{and}~{}~{}z_{3}(\beta^{\prime})<z_{3}(\beta).$

###### Proof Sketch.

The first part of the proof follows simply by showing that the update function
is increasing in $\beta$ for $x<\tau/(1-\alpha)$ (where the first fixed point
lies) and decreasing in $\beta$ as for $x>\tau/(1-\alpha)$ (where the third
fixed point lies). In turn, increasing $\beta<\alpha$ towards $\alpha$ move
the update rule $f$ “up” around the first fixed point and “down” around the
third fixed point, which increases $z_{1}$ and decreases $z_{3}$. A similar
argument holds for $\beta>\alpha$. The full proof is given in Appendix B.2. ∎

Intuitively, one might suppose that to reduce wealth disparities, we should
redesign tests so as to make them reflect type more strongly and wealth less
strongly (since types are distributed identically across groups). But Theorem
2 shows that counter-intuitively, this need not be the case333Even when
$\beta\to 1$, $f$ may have three fixed points: by Claim 7, this arises for
example when
$K(\alpha,1,\gamma,\sigma)=\frac{1}{\alpha\gamma}>\frac{1-\alpha}{\sqrt{2\pi}}$
and $\tau=\frac{1-\alpha}{2}$. In this case, setting $\beta=\alpha$
surprisingly leads to better outcomes than $\beta=1$.. Instead, what Theorem 2
shows is that in order to reduce inequality, we want to move $\beta$ towards
$\alpha$, causing the signal to better reflect the objective function of the
college —- even when this results in reducing the extent to which the signal
reflects type444Of course, if we can, we would prefer to increase the extent
to which the college _values_ type rather than wealth, but to the extent that
we cannot do this, then we want to align the test with the college’s
objective.. Theorem 2 shows that moving $\beta$ towards $\alpha$ always has
the effect of reducing wealth disparities. It either:

  1. 1.

Increases the wealth of the less wealthy attracting fixed point, and decreases
the wealth of the more wealthy attracting fixed point, thereby decreasing the
long term wealth disparity, or it

  2. 2.

Shifts the dynamic to one that has only a single fixed point, thereby
eliminating long term wealth disparities.

###  5.3 Direct Subsidies

We have thus far considered interventions that can be applied by the college
(admitting more students) or a testing body (changing the design of the
signal). In this section, we take the point of view of a funding body or
governmental agency that can provide direct monetary subsidies to populations.
We generalize the class of functions we study to include any function $f$
satisfying the following properties:

######  Assumption 2.

$f$ is continuous and increasing. Further, $f$ has three fixed points
$z_{1}<z_{2}<z_{3}$ with $f(x)>x$ on $[0,z_{1}]$ and $[z_{2},z_{3}]$ and
$f(x)<x$ on $[z_{1},z_{2}]$ and $[z_{3},1]$.

The above assumption captures the main properties of our function
$f=1-\Phi\left(K\left(\alpha,\beta,\gamma,\sigma\right)\left(\tau-(1-\alpha)x\right)\right)$
when it has three fixed points and implies the same attracting properties we
established for $z_{1}$, $z_{2}$, and $z_{3}$, but also encompasses more
general update rules that need not result from the Gaussian inference process
we have studied thus far. We note that this allows us to study general
S-shaped function with diminishing returns at both ends of the socio-economic
spectrum. Such functions model situation in which people of very low income or
very high income see little upward mobility (in the first case because of a
lack of access to opportunities, and in the second case due to the fact that
individuals of higher income are rare), whereas middle income individuals have
significant opportunities to improve their wealth.

We denote by $C(\mu,t)$ the subsidy given to a population with wealth $\mu$ at
time step $t$. The wealth of a population $t+1$ then depends of the wealth in
time $t$ as

$\mu^{t+1}=f\left(\mu^{t}+C(\mu^{t},t)\right).$

In this setting, we consider interventions that allow a population to reach
beyond the second fixed point $z_{2}$. Once a population reaches wealth (even
slightly) over $z_{2}$, their wealth naturally evolves to the highest
attracting fixed point $z_{3}$ over time; i.e., wealth dynamics self-correct
for disparities with no intervention needed. For the same reason, we only
consider $\mu^{0}\in[z_{1},z_{2}]$; this is because populations with
$\mu^{0}<z_{1}$ will converge to $z_{1}$ without intervention, and we can
start intervening once $\mu^{0}$ reaches $z_{1}$, while a population with
$\mu^{0}>z_{2}$ will reach the best long-term outcome (the highest fixed
point, $z_{3}$) on its own. Therefore, from now on, we assume $C(\mu)=0$ for
all $\mu\notin[z_{1},z_{2}]$. We can now formulate our centralized designer’s
objective, which is to minimize the following loss function:

$L(C)=\lambda\sum_{t=0}^{T(C)-1}\rho^{t}C(\mu^{t},t)+(1-\lambda)\sum_{t=0}^{T(C)-1}\rho^{t}(z_{2}-\mu^{t}),$

where $\rho,\lambda\in[0,1)$, and
$T(C)=\min\\{t~{}~{}\text{s.t.}~{}\mu^{t}\geq z_{2}\\}$ is the first time step
such that $\mu^{t}\geq z_{2}$. Here $\rho$ is a discounting factor; the lower
$\rho$ is, the less the designer cares about future as opposed to immediate
outcomes. The objective is a convex combination of two terms, with weights
controlled by $\lambda$. The first term consists of the discounted monetary
cost of the subsidies (the sum goes up to time $T(C)-1$, since after the
wealth of the population crosses $z_{2}$, the subsidies cease. This term
represents a preference to spend less money on direct subsidies. The second
term consists of the sum discounted difference between the target wealth
$z_{2}$ that the intervention is aiming at, and the wealth of the population
at the current round. This term represents a preference to quickly increase
the wealth of the lower wealth population. $\lambda$ represents the relative
strength of these two preferences.

Note that $z_{2}-\mu^{0}$ is a constant term that does not depend on the
designer’s interventions, hence we will equivalently aim to minimize

$L(C)=\lambda\sum_{t=0}^{T(C)-1}\rho^{t}C(\mu^{t},t)+(1-\lambda)\sum_{t=1}^{T(C)-1}\rho^{t}(z_{2}-\mu^{t}),$

where we drop the discounted difference between $z_{2}$ and the initial wealth
$\mu^{0}$ at $t=0$.

#### Algorithmically finding a near-optimal subsidy function $C(.)$

We note that in our setting, one may discretize the space of possible costs
and use dynamic programming to find optimal interventions from each possible
starting point. However, doing so requires carefully understanding the wealth
update function $f$. In practice, detailed knowledge of $f$ will be hard to
come by. For this reason, in the rest of this section, we will aim for a
“detail free” solution and consider a simple class of constant subsidies and
study how they can be applied with minimal information about the wealth update
$f$.

#### Constant Subsidies

In the rest of this section, we consider the case in which $C(\mu)$ is
constant in $\mu$ for $z_{1}\leq\mu\leq z_{2}$. I.e. there exists $C\in[0,1]$
such that $C(\mu)=C$ for all $\mu\in[z_{1},z_{2}]$ and $C(\mu)=0$ otherwise.
We call these $C$-subsidy interventions. We qualify a $C$-subsidy intervention
as a $k$-shot intervention if it takes $k$ time steps under the subsidy to
reach wealth (at least) $z_{2}$ when starting at wealth $z_{1}$, i.e. if
$T(C)=k$. Note that different values of $C$ may lead to the same number of
steps $k$ such that $\mu^{k}\geq z_{2}$, i.e. there may be several values of
$C$ that qualify as a $k$-shot intervention for a given value of $k$.

Our aim is to give guidelines on how to choose $C$ while using minimal
information about the function $f$. Here, we will encode this minimal
information as a single, real parameter $\Delta$, defined as

$\displaystyle\Delta=\max_{x\in[z_{1},z_{2}]}~{}~{}x-f(x).$ (3)

Intuitively, $\Delta$ measures how difficult it is for a subsidy to have an
effect on wealth that propogates in the next round. When $\Delta\to 0$, we
have that $f(x)\to x$ on $x\in[z_{1},z_{2}]$, and investing a subsidy of $C$
increases the population wealth by $C$, since $f(\mu^{t}+C)\to\mu^{t}+C$.
However, we have that for at least one value of $\mu^{t}$,
$f(\mu^{t}+C)=\mu^{t}+C-\Delta$, implying that when $\Delta$ is large, a large
amount of the subsidy is lost in the next round, and so its overall effect is
small. If we want to guarantee that our subsidies will eventually lift the
lower wealth population to the higher wealth fixed point independently of its
starting point, we need to consider subsidies in which $C>\Delta$.

######  Claim 8.

Suppose $C\leq\Delta$. Then there exists a starting wealth
$\mu^{0}\in[z_{1},z_{2})$ such that $\mu^{t}<z_{2}$ for all $t$; i.e.,
$\mu_{t}$ never reaches $z_{2}$. On the other hand, if $C>\Delta$, there
exists $t$ such that $\mu^{t}\geq z_{2}$.

###### Proof.

Let $x_{\Delta}\in(z_{1},z_{2})$ be any value of $x$ such that $f(x)=x-\Delta$
(note that $x_{\Delta}\neq z_{1},~{}z_{2}$ where $f(x)-x=0$, since $f(x)<x$ on
$[z_{1},z_{2}]$ if $f$ has three fixed points). Suppose
$\mu^{t}<x_{\Delta}-\Delta$ and $C\leq\Delta$, then
$\mu^{t+1}=f\left(\mu_{t}+C\right)<f\left(x_{\Delta}-\Delta+C\right)\leq
f(x_{\Delta})=x_{\Delta}-\Delta$. I.e., $\mu_{t}<x_{\Delta}-\Delta<z_{2}$ for
all $t$ so long as $\mu^{0}\in[z_{1},x_{\Delta}-\Delta)$; note that the
interval is not empty as $x_{\Delta}-\Delta=f(x_{\Delta})>z_{1}$. For the
second part of the proof, note that by definition of $\Delta$, for all $t$,
$\mu^{t+1}=f\left(\mu^{t}+C\right)\geq\mu^{t}+C-\Delta$, hence the group
wealth increases by at least a constant amount $C-\Delta$ at each time step. ∎

Intuitively, this holds because if $C$ is smaller than $\Delta$, it becomes
insufficient to compensate the fact that the wealth of a group can decrease by
an amount up to $\Delta$ at each round. In the rest of this section, we aim to
understand how different interventions for different values of $C$ compare to
each other, and when to choose low-cost versus high-cost interventions. Before
doing so, we note that there is always a single, optimal $1$-shot intervention
among all such $1$-shot interventions:

######  Fact 1.

The $1$-shot intervention with cost $C=z_{2}-\mu^{0}$ has smaller cost than
any other $1$-shot intervention. This immediately follows from the fact that
any $1$-shot intervention with cost $C$ has loss $\lambda C$, and that no
intervention with $C<z_{2}-\mu^{0}$ can reach $z_{2}$ in one shot, as
$\mu^{1}=f(\mu^{0}+C)<f(z_{2})=z_{2}$.

We now provide a sufficient condition under which the $1$-shot intervention is
guaranteed to be optimal.

######  Theorem 3.

Suppose $\rho\geq\lambda$. Then, any $k$-shot intervention has higher loss
than the $1$-shot, $\left(z_{2}-\mu^{0}\right)$-subsidy intervention. I.e. the
$\left(z_{2}-\mu^{0}\right)$-subsidy intervention is optimal.

###### Proof.

The proof follows by induction on $k$. First, let us consider the base case
when $k=2$, and let $C$ be any cost that leads to convergence in two shots.
Consider any starting point $\mu^{0}\in[z_{1},z_{2}]$. Note that the sequence
of wealth $\mu^{0}\to\mu^{1}\to\mu^{2}$ must satisfy $\mu^{1}<z_{2}$ and
$\mu^{2}\geq z_{2}$. Further, note that because
$\mu^{t+1}=f\left(\mu^{t}+C\right)\leq\mu^{t}+C$ we must have
$C\geq\mu^{t+1}-\mu^{t}$. We then have that the loss $L$ satisfies

$\displaystyle L(C)$ $\displaystyle=\lambda
C+(1-\lambda)\rho(z_{2}-\mu^{1})+\rho(\lambda C)$
$\displaystyle\geq\lambda(\mu^{1}-\mu^{0})+(1-\lambda)\rho(z_{2}-\mu^{1})+\rho[\lambda(z_{2}-\mu^{1})]$
$\displaystyle=\lambda(\mu^{1}-\mu^{0})+\rho\left(z_{2}-\mu^{1}\right)$
$\displaystyle\geq\lambda(\mu^{1}-\mu^{0})+\lambda\left(z_{2}-\mu^{1}\right)=\lambda(z_{2}-\mu^{0}).$

This concludes the case of $k=2$.

For $k>2$, note that we have $\mu^{k-1}<z_{2}$ and $\mu^{k}\geq z_{2}$.
Letting $C$ be any cost that leads to reaching $z_{2}$ in $k$ shots, we have
that the loss function is given by

$\displaystyle L(C)$
$\displaystyle\geq\lambda(\mu^{1}-\mu^{0})+(1-\lambda)\rho(z_{2}-\mu^{1})+\left[\lambda\sum_{t=1}^{k-1}\rho^{t}C+(1-\lambda)\sum_{t=2}^{k-1}\rho^{t}(z_{2}-\mu^{t})\right]$
$\displaystyle=\lambda(\mu^{1}-\mu^{0})+(1-\lambda)\rho(z_{2}-\mu^{1})+\rho\left[\lambda\sum_{t=1}^{k-1}\rho^{t-1}C+(1-\lambda)\sum_{t=2}^{k-1}\rho^{t-1}(z_{2}-\mu^{t})\right]$
$\displaystyle=\lambda(\mu^{1}-\mu^{0})+(1-\lambda)\rho(z_{2}-\mu^{1})+\rho\left[\lambda\sum_{t=0}^{k-2}\rho^{t}C+(1-\lambda)\sum_{t=1}^{k-2}\rho^{t}(z_{2}-\mu^{t+1})\right].$

The second term in the last line of the inequality is the loss function when
starting at $\mu^{1}\in[z_{1},z_{2}]$ instead of $\mu^{0}$. Indeed, write
$\nu^{t}=\mu^{t+1}$ the sequence that starts at $\mu^{1}$ and satisfies
$\nu^{k}=\mu^{k-1}<z_{2}$ but $\nu^{k-1}=\mu^{k}\geq z_{2}$ (hence this new
sequence converges in $k-1$ rather than $k$ steps); the loss of this sequence
is given by:

$\displaystyle\lambda\sum_{t=0}^{k-2}\rho^{t}C+(1-\lambda)\sum_{t=1}^{k-2}\rho^{t}(z_{2}-\nu^{t})=\lambda\sum_{t=0}^{k-2}\rho^{t}C+(1-\lambda)\sum_{t=1}^{k-2}\rho^{t}(z_{2}-\mu^{t+1}).$

By the induction hypothesis, since the cost of a one-shot intervention is
lower than that of any $k-1$-shot intervention, we have that

$\lambda\sum_{t=0}^{k-2}\rho^{t}C+(1-\lambda)\sum_{t=1}^{k-2}\rho^{t}(z_{2}-\mu^{t+1})\geq\lambda\left(z_{2}-\mu^{1}\right).$

Therefore, $L(C)$ is lower bounded by

$\displaystyle\lambda(\mu^{1}-\mu^{0})+(1-\lambda)\rho(z_{2}-\mu^{1})+\rho[\lambda(z_{2}-\mu^{1})]$
$\displaystyle\geq\lambda(\mu^{1}-\mu^{0})+\lambda(z_{2}-\mu^{1})\geq\lambda(z_{2}-\mu^{0}).$

∎

In particular, $1$-shot interventions become optimal when the discounting
factor $\rho$ is relatively large, or when $\lambda$ is relatively small. The
first result intuitively arises because when $\rho$ becomes large, the
centralized designer cares about cost and wealth of the group at each time
step; a $1$-shot intervention allows the designer to incur a single up-front
cost for intervening (instead of inefficiently investing a smaller cost per
round over more rounds, and losing some of this invested cost, up to $\Delta$,
at each time step) while immediately reaching high wealth outcomes. On the
other hand, no matter what $\rho$ is, when $\lambda$ becomes small, the
designer only cares about reaching high wealth as soon as possible, hence
prefers faster interventions. We now provide sufficient conditions under which
$1$-shot is not optimal:

######  Theorem 4.

If $\rho<\lambda\left(1-\frac{C}{z_{2}-\mu^{0}}\right)$, the $C$-subsidy
intervention has lower loss than the $1$-shot,
$\left(z_{2}-\mu^{0}\right)$-subsidy intervention.

###### Proof.

Consider any intervention with cost $C$ such that $\mu^{k}\geq z_{2}$, i.e. we
reach $z_{2}$ after at most $k$ time steps. First, remember that the loss for
this intervention is given by

$\lambda\sum_{t=0}^{k-1}\rho^{t}C+(1-\lambda)\sum_{t=1}^{k-1}\rho^{t}(z_{2}-\mu^{t}).$

Noting that $\mu^{t}\geq\mu^{0}$ for all $t$, hence $z_{2}-\mu^{t}\leq
z_{2}-\mu^{0}$, we can upper bound the loss by

$\displaystyle\lambda\sum_{t=0}^{+\infty}\rho^{t}C+(1-\lambda)\sum_{t=1}^{+\infty}\rho^{t}(z_{2}-\mu^{0})$
$\displaystyle=\frac{\lambda
C}{1-\rho}+(1-\lambda)\frac{\rho}{1-\rho}(z_{2}-\mu^{0})$
$\displaystyle\leq\frac{\lambda
C}{1-\rho}+(1-\lambda)\frac{\rho}{1-\rho}(z_{2}-\mu^{0}).$

In turn, we have that a sufficient condition for $C$-subsidy to have a lower
loss than one-shot is given by

$\frac{1}{1-\rho}\left(\lambda
C+(1-\lambda)\rho(z_{2}-\mu^{0})\right)<\lambda(z_{2}-\mu^{0}).$

This can be rewritten as

$\lambda
C+(z_{2}-\mu^{0})\rho-\lambda(z_{2}-\mu^{0})\rho<\lambda(z_{2}-\mu^{0})-\lambda(z_{2}-\mu^{0})\rho,$

i.e.

$(z_{2}-\mu^{0})\rho<\lambda(z_{2}-\mu^{0})-\lambda C,$

which immediately leads to the theorem statement. ∎

Theorem 4 gives conditions under which the minimal 1-shot intervention has
higher cost than the $C$-subsidy intervention. But recall that we can take $C$
as small as $\Delta+\epsilon$ (for arbitrarily small $\epsilon$) and still get
an intervention that reaches the region of attraction for the highest wealth
fixed point. Thus we have the following corollary, which gives a necessary
condition for the 1-shot intervention to be optimal:

######  Corollary 1.

If $\rho<\lambda\left(1-\frac{\Delta}{z_{2}-\mu^{0}}\right)$, then the
$(\Delta+\varepsilon)$-subsidy intervention has lower loss than the $1$-shot,
$(z_{2}-\mu^{0})$-subsidy intervention as $\varepsilon\to 0$.

The above corollary provides the most stringent condition that we can derive
from Theorem 4 for $1$-shot not to be optimal. In particular, we note that the
cheapest intervention we can use, the $(\Delta+\varepsilon)$-subsidy one, is
better than the $1$-shot intervention so long as
$\rho<\lambda\left(1-\frac{\Delta}{z_{2}-\mu^{0}}\right)$. We note that the
combination of Theorem 3 and Corollary 1 show that when $\Delta$ becomes small
and subsidy interventions are efficient, the condition that $\rho\geq\lambda$
becomes nearly tight for optimality of the $1$-shot, $(z_{2}-\mu^{0})$-subsidy
intervention. When $\Delta$ is large, there are still situations in which the
condition of Corollary 1 is essentially necessary and sufficient for the
$(\Delta+\varepsilon)$-subsidy to be better than the 1-shot intervention, as
evidenced by the example below:.

######  Example 1.

Suppose $f$ is continuous, but such that it is linear on interval
$(a,b)\subset(z_{1},z_{2})$ with $f(x)=x-\Delta$ within said interval. We have
immediately that $\mu^{t+1}=\mu^{t}+C-\Delta$ hence
$\mu^{t}=\mu^{0}+t(C-\Delta)$ so long as $t$ is such that $\mu^{t}\in(a,b)$.
Here, the one-shot intervention still has loss $\lambda(z_{2}-\mu^{0})$.
However, the $(\Delta+\epsilon)$-subsidy intervention reaches
$z_{2}-\varepsilon$, hence $z_{2}$, after no less than
$T_{\varepsilon}=\frac{b-\mu^{0}}{\varepsilon}\to_{\varepsilon\to 0}+\infty$
time steps. In turn, it has loss at least

$\displaystyle L(\Delta+\varepsilon)$
$\displaystyle\geq\lambda\sum_{t=0}^{T_{\varepsilon}-1}\rho^{t}(\Delta+\varepsilon)+(1-\lambda)\sum_{t=1}^{T_{\varepsilon}-1}\rho^{t}(z_{2}-\mu^{0}-t\varepsilon)$
$\displaystyle\to_{\varepsilon\to
0}~{}~{}\frac{\lambda}{1-\rho}\Delta+\frac{\rho(1-\lambda)}{1-\rho}(z_{2}-\mu^{0}).$

The proof of Theorem 4 shows that the loss is also upper-bounded by

$L(\Delta+\varepsilon)\leq\frac{\lambda}{1-\rho}\left(\Delta+\varepsilon\right)+\frac{\rho(1-\lambda)}{1-\rho}(z_{2}-\mu^{0}).$

Hence, it must be that this bound is essentially tight, i.e. that

$L(\Delta+\varepsilon)\to_{\varepsilon\to
0}\frac{\lambda}{1-\rho}\Delta+\frac{\rho(1-\lambda)}{1-\rho}(z_{2}-\mu^{0}).$

In particular, in this case, the condition of Theorem 4 and Corollary 1 is not
only sufficient but also necessary for the
$\left(\Delta+\varepsilon\right)$-subsidy intervention to have better loss
than the one-shot intervention.

### Acknowledgements

This work was supported in part by NSF grants AF-1763307 and FAI-2147212 and a
grant from the Simons Foundation.

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##  Appendix A Omitted Proofs for Section 1: Wealth Dynamics

###  A.1 Proof of Claim 2

Because $S$ is a convex combination of $W$ and $T$, both $(T,S)$ and $(W,S)$
are multivariate Gaussians. The covariances are given by

$Cov(T,S)=\beta Cov(T,T)+(1-\beta)Cov(T,W)=\beta\gamma^{2},$ $\displaystyle
Cov(W,S)$ $\displaystyle=\beta Cov(W,T)+(1-\beta)Cov(W,W)=(1-\beta)\sigma^{2}$

and

$\displaystyle Cov(S,S)$
$\displaystyle=\beta^{2}Cov(T,T)+(1-\beta)^{2}Cov(W,W)+2\beta(1-\beta)Cov(T,W)=\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}.$

###  A.2 Proof of Lemma 1

Using Claim 2, we have that

$\displaystyle E[T|S=s]$
$\displaystyle=\frac{Cov(T,S)}{Var(S)}(s-(1-\beta)\mu)=\frac{\beta\gamma^{2}}{\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}}(s-(1-\beta)\mu),$

and

$\displaystyle E[W|S=s]$
$\displaystyle=\mu+\frac{Cov(W,S)}{Var(S)}(s-(1-\beta)\mu)=\mu+\frac{(1-\beta)\sigma^{2}}{\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}}(s-(1-\beta)\mu).$

Therefore, the university admits a student with score $s$ if and only if

$\displaystyle\left(\frac{\alpha\beta\gamma^{2}+(1-\alpha)(1-\beta)\sigma^{2}}{\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}}\right)(s-(1-\beta)\mu)\geq\tau-(1-\alpha)\mu,$

which can be rewritten as

$\displaystyle\frac{s-(1-\beta)\mu}{\sqrt{\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}}}\geq\frac{\sqrt{\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}}}{\alpha\beta\gamma^{2}+(1-\alpha)(1-\beta)\sigma^{2}}\cdot\left(\tau-(1-\alpha)\mu\right).$

Noting that by Claim 2,
$\frac{S-(1-\beta)\mu}{\sqrt{\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}}}$
follows a normal distribution with mean $0$ and variance $1$; the expression
for $\mu^{t+1}$ (hence the update rule) is then given by

$\displaystyle
1-\Phi\left(\frac{\sqrt{\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}}}{\alpha\beta\gamma^{2}+(1-\alpha)(1-\beta)\sigma^{2}}\cdot\left(\tau-(1-\alpha)\mu\right)\right)=1-\Phi\left(K\left(\alpha,\beta,\gamma,\sigma\right)\cdot\left(\tau-(1-\alpha)\mu\right)\right)$

This concludes the proof.

###  A.3 Proof of Claim 3

For simplicity of notations, let us write $K$ instead of
$K(\alpha,\beta,\gamma,\sigma)$. Continuity is immediate from $f$ being the
composition of a linear (hence continuous) function and the continuous
function $\Phi$. Now, we have

$f^{\prime}(x)=\frac{K(1-\alpha)}{\sqrt{2\pi}}\exp\left(-K^{2}(\tau-(1-\alpha)x)^{2}/2\right)\geq
0,$

showing $f$ is increasing. Finally, the second order derivative of the update
rule $f^{\prime\prime}(x)$ is given by

$\frac{K^{3}(1-\alpha)^{2}(\tau-(1-\alpha)x)}{\sqrt{2\pi}}\cdot
e^{-K^{2}(\tau-(1-\alpha)x)^{2}/2}.$

The result immediately follows, as $f^{\prime\prime}(x)\geq 0$ if and only if
$x\leq\frac{\tau}{1-\alpha}$.

###  A.4 Proof of Lemma 2

Let us write $g(x)=f(x)-x$. Note that $f(x)$ has a fixed point if and only if
$g(x)=0$.

  1. 1.

$\tau\geq 1-\alpha$ and $f$ is convex on $[0,1]$. Then
$g^{\prime}(x)=f^{\prime}(x)-1$, $g^{\prime\prime}(x)=f^{\prime\prime}(x)$,
and $g$ is also convex. Further, note that $g(0)=f(0)-0>0$ and
$g(1)=f(1)-1<0$. Therefore, $g(x)=0$ can only have one solution at most.
Indeed, let $x^{*}$ be the smallest value in $[0,1]$ for which $g(x^{*})=0$;
we have that for all $x\in(x^{*},1]$, we can write $x^{*}=\lambda
x+(1-\lambda)1$ for some $\lambda\in(0,1]$, Then, we have $g(x)\leq\lambda
g(x^{*})+(1-\lambda)g(1)<0$ by convexity.

  2. 2.

$\tau\leq 0$ and $f$ is concave on $[0,1]$. Then $f$ can have at most $1$
fixed point by the same argument as above.

  3. 3.

Otherwise, note that $g^{\prime}(x)=f^{\prime}(x)-1$ is first increasing up
until $x^{*}=\tau/(1-\alpha)$ then decreasing in $x$. Therefore, $g^{\prime}$
has at most two zeros $x^{-}$ and $x^{+}$. If $g^{\prime}$ has two zeros, they
must satisfy $x^{-}<x^{*}$ and $x^{+}>x^{*}$, and that $g^{\prime}(x)<0$ for
$x<x^{-}$, $g^{\prime}(x)\geq 0$ for $x\in[x^{-},x^{+}]$, and
$g^{\prime}(x)<0$ for $x>x_{+}$. $g$ then has at most three intersection with
$0$, with the first intersection on $[0,x^{-}]$, the second on
$[x^{-},x^{+}]$, and the third on $[x^{+},1]$. When $g^{\prime}$ has at most
one zero, $g$ can only have at most $2$ zeros

This concludes the proof.

##  Appendix B Omitted Proofs for Section 5: Interventions for Long-term
Fairness

###  B.1 Proof of Theorem 1

This follows from the fact that $f(x,\tau)$ is decreasing in $\tau$ for all
$x\in[0,1]$. First, this implies that $f(x,\tau^{\prime})>f(x,\tau)\geq x$ for
all $x\in[0,z_{1}(\tau)]$. Hence $z_{1}(\tau^{\prime})>z_{1}(\tau)$. For the
third fixed point, note that
$f(z_{3}(\tau),\tau^{\prime})>f(z_{3}(\tau),\tau)=z_{3}(\tau)$; because $f$ is
continuous and $f(1)<1$, this immediately implies that $f$ has a fixed point
on $(z_{3}(\tau),1]$, hence $z_{3}(\tau^{\prime})>z_{3}(\tau)$.

Finally, let us consider the case of the second fixed point. First, we note
that it must be that $z_{1}(\tau^{\prime})<z_{2}(\tau)$. Suppose this is not
the case, it must be that $f(x,\tau^{\prime})>x$ for all $x<z_{2}(\tau)$.
Further, for all $x\in[z_{2}(\tau),z_{3}(\tau)]$, we must have
$f(x,\tau^{\prime})>f(x,\tau)\geq x$, hence $f(x,\tau^{\prime})>x$ for all
$x<z_{3}(\tau)$. This implies $z_{1}(\tau^{\prime})>z_{3}(\tau)$. However, we
must have $z_{3}(\tau)\geq\tau/(1-\alpha)$ while
$z_{1}(\tau^{\prime})\leq\tau^{\prime}/(1-\alpha)<\tau/(1-\alpha)$, which is a
contradiction.
Now that we have $z_{1}(\tau^{\prime})<z_{2}(\tau)$, note that it must be that
$f(x,\tau^{\prime})<x$ on a small neighborhood
$(z_{1}(\tau^{\prime}),z_{1}(\tau^{\prime})+\varepsilon)$ by our
characterization of the fixed points of $f$. Since $f$ is continuous and
$f(z_{2}(\tau),\tau^{\prime})>f(z_{2}(\tau),\tau)=z_{2}(\tau)$, there exists a
fixed point on $(z_{1}(\tau^{\prime}),z_{2}(\tau)$. Since
$z_{3}(\tau^{\prime})>z_{3}(\tau)>z_{2}(\tau)$, this must be the second fixed
point $z_{2}(\tau^{\prime})$.

###  B.2 Proof of Theorem 2

The partial derivative of $f$ with respect to $\beta$ is given by

$\displaystyle\frac{\partial}{\partial\beta}f(x,\beta)=[\tau-(1-\alpha)x]~{}\cdot~{}\phi\left(K(\alpha,\beta,\gamma,\sigma)(\tau-(1-\alpha)x)\right)$
$\displaystyle\times\frac{(\alpha-\beta)\gamma^{2}\sigma^{2}}{\sqrt{\beta^{2}\gamma^{2}+(1-\beta)^{2}\sigma^{2}}(\alpha\beta\gamma^{2}+(1-\alpha)(1-\beta)\sigma^{2})^{2}}$

where $\phi$ is the probability density function of a standard Gaussian. Note
that for $\alpha<\beta$, $\frac{\partial}{\partial\beta}f(x,\beta)<0$ when
$x<\tau/(1-\alpha)$ and $\frac{\partial}{\partial\beta}f(x,\beta)>0$ when
$x>\tau/(1-\alpha)$. In particular, $f(x,\beta^{\prime})>f(x,\beta)\geq x$ for
all $x\leq z_{1}(\beta)(<\tau/(1-\alpha))$, hence $f(.\beta^{\prime})$ has no
fixed point on $[0,z_{1}(\beta)]$. This means that
$z_{1}(\beta^{\prime})>z_{1}(\beta)$. Similarly,
$f(x,\beta^{\prime})<f(x,\beta)\leq x$ for all $x\geq
z_{3}(\beta)(>\tau/(1-\alpha))$, hence $f$ has no fixed point on
$[z_{3}(\beta),1]$ and $z_{3}(\beta^{\prime})<z_{3}(\beta)$. A similar proof
follows for $\alpha\geq\beta^{\prime}>\beta$.


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