# From the Fair Distribution of Predictions to the Fair Distribution of Social
Goods: Evaluating the Impact of Fair Machine Learning on Long-Term
Unemployment

Sebastian Zezulka
University of Tübingen
<EMAIL_ADDRESS>
Konstantin Genin
University of Tübingen
<EMAIL_ADDRESS>

(January 2024)

###### Abstract

Algorithmic fairness focuses on the distribution of predictions at the time of
training, rather than the distribution of social goods that arises after
deploying the algorithm in a concrete social context. However, requiring a
‘fair’ distribution of predictions may undermine efforts at establishing a
fair distribution of social goods. Our first contribution is conceptual: we
argue that addressing the fundamental question that motivates algorithmic
fairness requires a notion of prospective fairness that anticipates the change
in the distribution of social goods after deployment. Our second contribution
is theoretical: we provide conditions under which this change is identified
from pre-deployment data. That requires distinguishing between, and accounting
for, different kinds of performative effects. In particular, we focus on the
way predictions change policy decisions and, therefore, the distribution of
social goods. Throughout, we are guided by an application from public
administration: the use of algorithms to (1) predict who among the recently
unemployed will remain unemployed in the long term and (2) target them with
labor market programs. Our final contribution is empirical: using
administrative data from the Swiss public employment service, we simulate how
such policies would affect gender inequalities in long-term unemployment. When
risk predictions are required to be ‘fair’, targeting decisions are less
effective, undermining efforts to lower overall levels of long-term
unemployment and to close the gender gap in long-term unemployment.

##  1 A fundamental question for fair machine learning

Research in algorithmic fairness is often motivated by the worry that machine
learning algorithms will reproduce or exacerbate the structural inequalities
reflected in their training data (Lum and Isaac, 2016, Tolbert and Diana,
2023). Indeed, whether an algorithm exacerbates an existing social inequality
is emerging as a central compliance criterion in EU non-discrimination law
(Weerts et al., 2023). However, the methodological solutions developed by
researchers in algorithmic fairness are, surprisingly, ill-suited for
addressing this fundamental question. At some level, the questions of
algorithmic fairness are ill-posed: often, it does not make sense to talk
about the fairness of a predictor, independent of the policy context in which
it is deployed. It is our policies and their effects that are just or unjust;
‘fair’ predictors can both support unjust policies and undermine just policy.
For example, public employment services use predictions of the risk of long-
term unemployment to decide who is given access to training programs. Policy
doves target those at the highest risk with training programs, while hawks,
considering those at the highest risk to be hopeless cases, withhold training
on grounds of ‘efficiency’. It is clear that the social consequences of errors
in prediction differ significantly depending on how these predictions will be
used. It would be surprising if we could say whether a predictor is fair
independent of this policy context. Therefore, rather than focusing on the
distribution of predictions at the time of training, we focus on the
distribution of social goods induced by deploying a predictive algorithm in a
policy context.

The field of algorithmic fairness has produced many mathematical
demonstrations of necessary trade-offs between different notions of
‘fairness’, and between ’fair’ and accurate prediction (Borsboom et al., 2008,
Kleinberg et al., 2016, Chouldechova, 2017, Menon and Williamson, 2018). This
lends the field an air of tragedy and makes the pursuit of fairness seem
fundamentally quixotic. But, while mathematical trade-offs exist between
predictive accuracy and the ‘fair’ distribution of predictions, predictive
accuracy does not necessarily trade-off against the fair distribution of
social goods (Tal, 2023a, Corbett-Davies et al., 2018). Indeed, we should
expect that accurate predictions help us to effectively implement policy aimed
at ameliorating unjust inequalities. In our empirical case study, we
demonstrate that (1) requiring risk predictions to be fair undermines efforts
to lower overall levels of long-term unemployment and to close the gender gap
in long-term unemployment, (2) that the hawkish policy of withholding training
programs from those at the highest risk is no more efficient than the dovish
policy of prioritizing those with the highest risk, and (3) that accurate
prediction of counterfactual treatment outcomes, rather than risk scores,
enables individualized targeting and therefore, a better and more equitable
distribution of social goods.

Of course, this shift in focus poses methodological challenges. To anticipate
the causal effects of embedding a predictive algorithm into a social process,
we must make some effort to, first, identify the contextually relevant
inequalities in the distribution of social goods; second, understand the
policy processes and decisions that partially give rise to, and could
conceivably ameliorate, these inequalities; and third, model how algorithmic
predictions might change these processes and, therefore, the distribution of
social goods. Standard algorithmic fairness methods neglect every part of this
process (Green, 2022, Selbst et al., 2019). All of these methods impose
constraints on predictions that hold in the (retrospective) training
distribution. By focusing on the distribution of predictions at the time of
training, they obscure substantive inequalities in real-world quantities and
neglect the changes in decision-making that arise from the deployment of
predictive algorithms. Consequently, these methods fail to anticipate the
effects of deploying these algorithms on the distribution of social goods. We
address these shortcomings in the following way:

  * •

We reconceptualize algorithmic fairness questions as policy problems.
Prospective fairness is a matter of anticipating the effect of deploying an
algorithmically informed policy on inequality in social goods.

  * •

We state formal conditions under which the effect of deploying an
algorithmically informed policy on context-relevant inequalities is identified
from pre-deployment data.

  * •

We illustrate our approach with a case study on the statistical profiling of
registered unemployed using a rich administrative dataset from Switzerland. We
study the likely effects of two algorithmic policy proposals on the gender gap
in the rates of long-term unemployment.

Our case study is based on administrative data from the Swiss Active Labor
Market Policy Evaluation Dataset. The original sample, collected in 2003,
contains roughly one hundred thousand observations of registered unemployed
aged $24$ to $55$. Although most unemployed were not assigned to any program,
we observe outcomes for six labor market programs. The Swiss labor market, as
outlined in Section 3, is characterized by an overall unemployment rate of
about $4\%$, a high rate of long-term unemployment (LTU), and a persistent
gender reemployment gap (2). In the administrative data, the LTU gender gap is
at $3.9\%$, with an LTU rate of $43.6\%$ among women and $39.7\%$ among men.
The gap between Swiss citizens and non-citizens is at $15.8\%$, with a rate of
$35.7\%$ among Swiss citizens and $51.5\%$ among non-citizens.

The plan of the paper is as follows: first, we argue for prospective fairness
as a conceptual framework and survey related work; section 3 introduces two
recently proposed algorithmic policies intended to support public employment
agencies in reducing long-term unemployment; we argue that, in this context,
the gender gap in long-term unemployment is a simple and intuitive measure of
systemic inequality; section 4 formalizes conditions under which the causal
effect of deploying an algorithmically informed policy on a measure of
systematic inequality is identified from pre-deployment data; in section 5 we
illustrate the method with an extended case study, simulating two proposed
profiling policies and their effects on the gender reemployment gap. Section 6
concludes and outlines directions for future work.

##  2 From retrospective to prospective fairness

In paradigmatic risk-assessment applications, machine learners are concerned
with learning a function that takes as input some features $X$ and a sensitive
attribute $A$ and outputs a score $R$ which is valuable for predicting an
outcome $Y$. The algorithmic score $R$ is meant to inform some important
decision $D$ that, typically, is causally relevant for the outcome $Y$. In the
application that concerns us in this paper, features such as the education and
employment history $(X)$ and gender $(A)$ of a recently unemployed person are
used to compute a risk score $(R)$ of long-term unemployment $(Y).$ This risk
score $R$ is meant to support a caseworker at a public employment agency in
making a plan $(D)$ about how to re-enter employment. This plan may be as
simple as requiring the client to apply to some minimum number of jobs every
month or referring them to one of a variety of job-training programs.

Formal fairness proposals require that some property is satisfied by either
the joint distribution $P(A,X,R,D,Y)$ or the causal structure $G$ giving rise
to it. Individual fairness proposals introduce a similarity metric $M$ on
$(A,X)$ and suggest that similar individuals should have similar risk scores.
In all these cases, the relevant fairness property is a function
$\varphi(P,G,M)$. Group-based fairness (Barocas et al., 2023) ignores all but
the first parameter; causal fairness (Kilbertus et al., 2017, Kusner et al.,
2017) ignores the last; and individual fairness (Dwork et al., 2012) ignores
the second. All these proposals agree that fairness is a function of the
distribution (and perhaps the causal structure) at the time when the
prediction algorithm has been trained, but before it has been deployed. We
claim that addressing the fundamental question of fair machine learning
requires comparing the status quo before deployment with the situation likely
to arise after deployment. In other words: prospective fairness is a matter of
anticipating the change from $\varphi(P_{\text{pre}},D_{\text{pre}},M)$ to
$\varphi(P_{\text{post}},D_{\text{post}},M)$. We do not claim that there is a
single correct inequality measure $\varphi(\cdot),$ nor even that there is an
all-things-considered way of trading off different candidates, only that we
must make a good faith effort to anticipate changes in the relevant measures
of inequality.

As shown in Figure 1, deploying a decision support algorithm introduces a
causal path from the predicted risk score $R$ to the decision $D$.
Importantly, the outcome variable $Y$ is causally downstream of this
intervention. The addition of a causal path is modeled as a structural
intervention (Malinsky, 2018, Bynum et al., 2022).

(a) Causal structure $G_{\text{pre}}$ before deploying an algorithmically
informed policy.

(b) Causal structure $G_{\text{post}}$ after deploying an algorithmically
informed policy.

Figure 1: The left hand side shows the pre-deployment causal graph
$G_{\text{pre}}$ inducing a joint probability distribution $P_{\text{pre}}$
over sensitive attributes $A$, features $X$, risk score $R$, decision $D$, and
outcome variable $Y$. The risk score $R$ is the output of a learned function
from $A$ and $X$. Since this graph represents the situation after training,
but before deployment, there is no arrow from the risk score $R$ to the
decision $D$. Retrospective fairness formulates constraints
$\varphi(G_{\text{pre}},P_{\text{pre}},M)$ on the pre-deployment arrangement
alone. The right-hand side represents the situation after the algorithmically
informed policy has been deployed, with predictions $R$ now affecting
decisions $D$. Prospective fairness requires comparing the consequences of
intervening on the structure of $G_{\text{pre}}$ and moving to
$G_{\text{post}}$. In other words, comparing
$\varphi(G_{\text{pre}},P_{\text{pre}},M)$ with
$\varphi(G_{\text{post}},P_{\text{post}},M)$.

From a dynamical perspective, static and retrospective fairness proposals go
wrong in two ways. In the worst case, they are self-undermining: satisfying
the fairness criteria at the time of training necessitates violating them
after implementation. For example, Mishler and Dalmasso (2022) show that
satisfying the fairness notions of sufficiency $(Y\perp A\leavevmode\nobreak\
|\leavevmode\nobreak\ R)$ or separation $(R\perp A\leavevmode\nobreak\
|\leavevmode\nobreak\ Y)$ at the time of training virtually ensures that they
will be violated after deployment. Illustrating the point in terms of
sufficiency, where $\perp$ denotes (conditional) statistical independence:

$Y\perp_{\text{pre}}A\leavevmode\nobreak\ |\leavevmode\nobreak\ R\hskip
4.0pt\text{ entails }\hskip
4.0ptY\not{\perp}_{\text{post}}A\leavevmode\nobreak\ |\leavevmode\nobreak\ R.$

Group-based notions of fairness like sufficiency and separation fall victim to
performativity: the tendency of an algorithmic policy intervention to shift
the distribution away from the one on which it was trained (Perdomo et al.,
2020). But as Mishler and Dalmasso (2022) show, they are undermined not by an
unintended and unforeseen performative effect, but by the intended, and
foreseen shift in distribution induced by algorithmic support, i.e.:

$P_{\text{pre}}(D\leavevmode\nobreak\ |\leavevmode\nobreak\ A,X,R)\neq
P_{\text{post}}(D\leavevmode\nobreak\ |\leavevmode\nobreak\ A,X,R).$

In other words, they are undermined by the fact that algorithmic support
changes decision-making, which, presumably, is the point of algorithmic
support in the first place. Since the distribution of the outcome $Y$ will
change after deployment, Berk et al. (2021) advises against group-based
metrics involving it, opting for statistical parity ($R\perp A$) instead. Of
course, independence requires a loss of predictive accuracy, which may
undermine even the most benevolent policies.

It is not likely that individual and causal fairness proposals are so
drastically self-undermining. So long as the similarity metric stays constant,
an algorithm that treats similar people similarly will continue to do so after
deployment. If, as Kilbertus et al. (2017) suggest, causal fairness is a
matter of making sure that all paths from the sensitive attribute $A$ to the
prediction $R$ are appropriately mediated, then causal fairness is safe from
performative effects so long as the qualitative causal structure upstream of
the prediction $R$ remains constant.

But even if causal and individual fairness proposals are not so dramatically
self-undermining, they are simply not testing whether the algorithm reproduces
or exacerbates inequalities in social goods, since the distribution of social
goods is causally downstream of algorithmic predictions. In particular, it is
customary to ignore the real-world dependence between $A$ and $Y$ induced by
the social status quo as the target of an intervention, since nothing can be
done about it at the time of training. Instead, fairness researchers focused
on whether the risk score itself is fair, whether in the group, individual, or
causal sense. However, from the dynamical perspective, it is perfectly
reasonable to ask whether the proposed algorithmic policy will exacerbate the
systemic inequality reflected in the dependence between gender $(A)$ and long-
term unemployment $(Y)$. Indeed, simple dynamical models and simulations
suggest that algorithms meeting static fairness notions at training may
exacerbate inequalities in outcomes in the long run (Liu et al., 2019, Zhang
and Liu, 2021). Streamlined dynamical models and simulations are a valuable
tool in evaluating the long-run effects of fairness-constrained algorithms.
The dual contributions of this paper are (1) a theoretical result giving
conditions under which the effect of deploying the algorithmic policy on the
joint distribution of $(Y,A)$ is identified and (2) a realistic case study
that forecasts, from administrative data, the effects of algorithmic policies
in public employment on the joint distribution of gender and long-term
unemployment.

###  2.1 Related Work

In machine learning, the fairness debate began with risk assessment tools for
decision- and policy-making (Angwin et al., 2016, Kleinberg et al., 2016,
Chouldechova, 2017, Mitchell et al., 2021). To this day, many standard case
studies e.g., lending, school admissions, and pretrial detention, fall within
this scope. See Berk et al. (2023) for a review on fairness in risk assessment
and Borsboom et al. (2008) and Hutchinson and Mitchell (2019) for predecessors
in psychometrics. Since then, researchers have stressed the importance of
explicitly differentiating policy decisions from the risk predictions that
inform them (Barabas et al., 2018, Kuppler et al., 2021, Beigang, 2022, Tal,
2023b) and of studying machine learning algorithms in their socio-
technological contexts (Selbst et al., 2019). We incorporate both of these
insights into the present work.

A central negative result emerging from recent fairness literature highlights
the dynamically self-undermining nature of group-based fairness constraints
that include the outcome variable $Y$. Mishler and Dalmasso (2022) show that a
classifier that is formally fair in the training distribution will violate the
respective fairness constraint in the post-deployment distribution. Coston et
al. (2020) suggests that the group-based fairness notion be formulated instead
in terms of the potential outcomes $Y^{d}.$ These alternative proposals are no
longer self-undermining, but they are still not testing the policy’s effect on
inequality in the distribution of social goods. This paper builds upon the
negative results of Berk et al. (2021) and Mishler and Dalmasso (2022): we
show how the post-interventional effect of an algorithmically informed policy
on the distribution of social goods can be identified from a combination of
(1) observational, pre-deployment data and (2) models of the policy proposal.

An emerging literature on long-term fairness focuses on the dynamic evolution
of systems under sequential-decision making, static fairness constraints, and
feedback loops; see Zhang and Liu (2021) for a survey. Ensign et al. (2018)
consider predictive feedback loops from selective data collection in
predictive policing. Hu and Chen (2018) propose short-term interventions in
the labor market to achieve long-term objectives. Using two-stage models, Liu
et al. (2019) and Kannan et al. (2019) show that retrospective fairness
constraints can, under some conditions, have negative effects on outcomes in
disadvantaged groups. With simulation studies, D’Amour et al. (2020) and Zhang
et al. (2020) confirm that imposing static fairness constraints does not
guarantee that these constraints are met over time and can, under some
conditions, exacerbate inequalities in social goods. Scher et al. (2023) model
long-term effects of statistical profiling for the allocation of unemployed
into labor market programs on skill levels. The picture emerging from this
literature is that post-interventional outcomes of algorithmic policies are a
relevant dimension for normative analysis that is not adequately captured by
retrospective fairness notions designed to hold in the training distribution.

##  3 Statistical profiling of the unemployed

Since the 1990s, participation in active labor market programs (ALMPs) has
been a condition for receiving unemployment benefits in many OECD countries
(Considine et al., 2017). ALMPs take many forms, but paradigmatic examples
include resume workshops, job-training programs, and placement services, see
Bonoli (2010) for a helpful taxonomy. Evaluations of ALMPs across OECD
countries find small but positive effects on labor market outcomes (Card et
al., 2018, Vooren et al., 2018, Lammers and Kok, 2019). Importantly, the
literature also reports large effect-size heterogeneity between programs and
demographics, as well as assignment strategies that are as good as random for
Switzerland (Knaus et al., 2022), Belgium (Cockx et al., 2023), and Germany
(Goller et al., 2021). This implies potential welfare gains from a more
targeted allocation into programs, especially when taking into account
opportunity costs—a compelling motivation for algorithmic support. Indeed, the
subsequent case study suggests that, if allocation decisions are made based on
data-driven estimates of individualized treatment effects, the gender
reemployment gap, as well as overall long-term unemployment, can be
significantly reduced.

Statistical profiling of the unemployed is current practice in various OECD
countries including Australia, the Netherlands, and Flanders, Belgium (Desiere
et al., 2019). Paradigmatically, supervised learning techniques are employed
to predict who is at risk of becoming long-term unemployed (LTU) (Mueller and
Spinnewijn, 2023). Such tools are regularly framed as introducing objectivity
and effectiveness in the provision of public goods and align with demands for
evidence-based policy and digitization in public administration. ALMPs target
supply-side problems by increasing human capital and matching problems by
supporting job search. Demand-side policies that focus on the creation of jobs
are not considered (Green, 2022).

Individual scores predicting the risk of long-term unemployment support a
variety of decisions. For example, the public employment service (PES) of
Flanders so far uses risk scores only to help caseworkers and line managers
decide who to contact first, prioritizing those at higher risk (Desiere and
Struyven, 2020). In contrast, the PES of Austria (plans to) use risk scores to
classify the recent unemployed into three groups: those with good prospects in
the next six months; those with bad prospects in the next two years; and
everyone else. The proposed policy of the Austrian PES is to focus support
measures on the third group while offering only limited support to the other
two. Advocates claim that, since ALMPs are expensive and would not
significantly improve the re-employment probabilities of individuals with very
good or very bad prospects, considerations of cost-effectiveness require a
focus on those with middling prospects (Allhutter et al., 2020a). However
intuitive this may seem, it is nowhere substantively argued that statistical
predictions of long-term unemployment from observational data are reliable
estimates for the effectiveness of administrative interventions. One worry is
that the unemployed who are labeled high-risk tend to be similar to those who,
historically, received ineffective programs. This is further complicated by
the presence of long-standing structural inequalities in the labor market,
which may be reproduced by algorithmic policies leaving those with “poor
prospects" to their own devices. In the subsequent simulation study, the
efficiency claims made in favor of Austrian-style policy are not corroborated.

Figure 2: Swiss Long-Term Unemployment Rates by Gender. Data for the period
2010-2022 are from Eurostat (European Commission, Eurostat, Accessed 17
January 2024), where the gender share of long-term unemployment is computed as
the share of all unemployed men/women ages 20-64 who are unemployed for more
than a year. Data for the period 1991-2007 are from the 2012 Swiss Social
Report (Bühlmann et al., 2013), where age information is not available. Data
for 2008-9 is not readily available.

Labor markets in OECD countries are structured by various inequalities. Gender
is a particularly long-standing and significant axis of inequality in labor
markets, with the gender pay gap and the child penalty being notorious
examples (Kleven et al., 2023, Bishu and Alkadry, 2016). On the other hand,
the gender gap in unemployment rates has largely disappeared over the last
decades (Albanesi and Şahin, 2018). Nevertheless, structural differences in
unemployment dynamics remain. For example, although women in Germany are less
likely to enter into unemployment, their exit probabilities are also lower
(Bundesagentur für Arbeit, 2023). Similarly, there is a longstanding gender
gap in long-term unemployment in Switzerland (see Figure 2). The obvious worry
is that prediction algorithms will pick up on these historical trends, as
demonstrated in Kern et al. (2021). The Austrian proposal for an LTU
prediction algorithm furnishes a particularly dramatic example. That algorithm
takes as input an explicitly gendered feature “obligation to care”, which has
a negative effect on the predicted re-employment probability and, by design,
is only active for women (Allhutter et al., 2020a). This controversial design
choice was justified as reflecting the “harsh reality" of the gendered
distribution of care responsibilities. Whatever the wisdom of this particular
variable definition, many other algorithms would pick up on the same
historical patterns. Moreover, if the intended use of these predictions is to
withhold support for individuals at high risk of long-term unemployment, it is
clear that such a policy might exacerbate the situation by further punishing
women for greater care obligations. Hopefully, the preceding motivates the
need for a prospective fairness methodology that assesses whether women’s re-
employment probability suffers under a proposed algorithmic policy. More
abstractly, what is needed is a way to predict how the pre-deployment
probability $P_{\text{pre}}(Y\leavevmode\nobreak\ |\leavevmode\nobreak\ A)$
will compare with the post-deployment probability
$P_{\text{post}}(Y\leavevmode\nobreak\ |\leavevmode\nobreak\ A)$. With these
estimates in hand, it would also be possible to predict whether the gender
reemployment gap is exacerbated, or ameliorated, under a proposed algorithmic
policy. The gender gap in reemployment probabilities is one particular choice
for a fairness notion $\varphi(\cdot).$ Variations on this simple metric could
be relevant in many other settings. For example, gender gaps in hiring, or
racial disparities in incarceration could be criteria that an algorithmically
informed policy should, minimally, not exacerbate (Kasy and Abebe, 2021). In
the following section, we give general conditions under which the post-
deployment change in the joint distribution of the outcome $(Y)$ and the
sensitive attribute $(A)$ is identified from pre-deployment data.

##  4 Identifiability of the Post-Deployment Distribution of Social Goods

Let $A,X,R,D,Y$ be discrete, observed random variables. In our example, $A$
represents gender; $X$ represents baseline covariates observed by the public
employment service for the registered unemployed; $R$ is an estimated risk of
becoming long-term unemployed; $D$ is an allocation decision made by the
public employment service and $Y$ is a binary random variable that is equal to
$1$ if an individual becomes long-term unemployed. For simplicity, we assume
that $R$ is a deterministic function of $A$ and $X$. We write
$\mathcal{A,X,R,D,Y}$ for the respective ranges of these random variables. For
$d\in\mathcal{D},$ let $Y^{d}$ be the potential outcome under policy $d,$ in
other words: $Y^{d}$ represents what the long-term unemployment status of an
individual would have been if they had received allocation decision $d.$
Naturally, $Y^{1},\ldots,Y^{|\mathcal{D}|}$ are not all observed. Our first
assumption is a rather mild one; we require that the observed outcome for
individuals allocated to $d$ is precisely $Y^{d}:$

$Y=\sum_{d\in\mathcal{D}}Y^{d}\mathbbm{1}[D=d].$ (Consistency)

Consistency is to be interpreted as holding both before and after the
algorithmic policy is implemented.

More substantially, we assume that the potential outcomes and decisions are
unconfounded given the observed features $(A,X)$ both before and after the
intervention:

$Y^{d}\perp_{t}D\leavevmode\nobreak\ |\leavevmode\nobreak\ A,X.$
(Unconfoundedness)

Unconfoundedness is a rather strong assumption that requires that the observed
features $A,X$ include all common causes of the decision and outcome. In the
case of a fully automated algorithmic policy, unconfoundedness holds by
design; but usually, risk assessment tools are employed to support human
decisions, not fully automate them (Levy et al., 2021). Although it is not
fated that all factors relevant to a human decision are available to the data
analyst, unconfoundedness is reasonable if rich administrative data sets
capture most of the information relevant to allocation decisions. For a case
in which this assumption fails, see Petersen et al. (2021).

We have argued that, in order to address the fundamental question of fair
machine learning, one must predict whether implementing the candidate
algorithmically informed policy leads to an improvement, or at least no
deterioration, in standards of justice. In the running example, this amounts
to comparing features of $P_{\text{pre}}(Y\leavevmode\nobreak\
|\leavevmode\nobreak\ A)$ with $P_{\text{post}}(Y\leavevmode\nobreak\
|\leavevmode\nobreak\ A)$. The first distribution is trivial to estimate, but
how to estimate $P_{\text{post}}(Y\leavevmode\nobreak\ |\leavevmode\nobreak\
A)$ from pre-deployment data? Here, the fundamental problem is performativity
(Perdomo et al., 2020). Our policy intervention will, in all likelihood,
change the process of allocation into labor market programs and, thus, change
the distribution of outcomes we are interested in. But not all kinds of
performativity are equal. Some performative effects are intended and
foreseeable. For example, the algorithmic effect is the intended change in
decision-making due to algorithmic support:

$P_{\text{pre}}\left(D=d\leavevmode\nobreak\ |\leavevmode\nobreak\
A=a,X=x\right)\neq P_{\text{post}}\left(D=d\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x\right).$ (Algorithmic Effect)

The first term in this inequality is the propensity score which can be
directly estimated from training data. The second term cannot be directly
estimated ex-ante. Nevertheless, it is possible to make reasonable conjectures
about the second term given a concrete proposal for how risk scores should
inform decisions. For example, if $D$ is binary, we could model the Austrian
proposal as providing support so long as the risk score is neither too high
nor low:

$P_{\text{post}}(D=1\leavevmode\nobreak\ |\leavevmode\nobreak\
A=a,X=x)=\mathbbm{1}\left[l<R(a,x)<h\right].$

More complex proposals for how risk scores should influence decisions require
more careful modeling. The subsequent empirical case study delivers a more
realistic model.

Although we allow for algorithmic effects, these cannot be too strong—the
policy cannot create allocation options that did not exist before. That is,
the risk assessment tools only change allocation probabilities into existing
programs. Moreover, we assume that the policy creates no unprecedented
allocation-demographic combinations:

$P_{\text{pre}}(D=d\leavevmode\nobreak\ |\leavevmode\nobreak\ A=a,X=x)>0\text{
if }P_{\text{post}}(D=d\leavevmode\nobreak\ |\leavevmode\nobreak\ A=a,X=x)>0.$
(No Unprecedented Decisions)

This would be violated if e.g., no women were allocated to some program before
the policy change.

Throughout this paper, we assume that no other forms of performativity occur.
In particular, we assume that the conditional average treatment effects
(CATEs) of the allocation on the outcome are stable across time:

$P_{\text{pre}}\left(Y^{d}\leavevmode\nobreak\ |\leavevmode\nobreak\
A=a,X=x\right)=P_{\text{post}}\left(Y^{d}\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x\right).$ (Stable CATE)

This amounts to assuming that the effectiveness of the programs (for people
with $A=a,X=x$) does not change, so long as all that has changed is the way we
allocate people to programs. In the case study, we assume that conditional
average treatment effects are stable under changes to allocation policies, as
well as to the total number of places available in (capacities of) each
program. This assumption could be violated if e.g., a program works primarily
by making some better off only at the expense of others—if everyone were to
receive such a program, it would have no effect (Crépon et al., 2013).

While algorithmic effects of deployment are intended and, to some degree,
foreseeable types of performativity, feedback effects that change the
covariates are more complicated to model.111In the classification of Pagan et
al. (2023), we focus on what they call “Outcome Feedback Loops”. In our
terminology, performativity is not exhausted by feedback effects. Following
Mishler and Dalmasso (2022) and Coston et al. (2020), we assume away the
possibility of feedback effects, leaving these for future research:

$P_{\text{pre}}\left(A=a,X=x\right)=P_{\text{post}}\left(A=a,X=x\right).$ (No
Feedback)

No Feedback amounts to assuming that the baseline covariates of the recently
employed are identically distributed pre- and post-deployment. Strictly
speaking, this is false, since the decisions of caseworkers will affect the
covariates of those who re-enter employment and some of them will, eventually,
become unemployed again. However, since the pool of employed is much larger
than the pool of unemployed, the policies of the employment service have much
larger effects on the latter than the former. For this reason, we may hope
that feedback effects are not too significant.

No Unprecedented Decisions, Stable CATE and No Feedback might fail
dramatically if e.g., the deployment of the policy coincided with a major
economic downturn. In a serious downturn, the employment service may have to
assist people from previously stable industries (violating No Unprecedented
Decisions and No Feedback), or employment prospects might deteriorate for
everyone (violating Stable CATE). However, the possibility of such exogenous
shocks is not a threat to our methodology. We are interested in the ceteris
paribus effect of the algorithmic policy on structural inequality, not an all-
thing-considered prediction of future economic conditions.

We are now in a position to show that, under the assumptions outlined above,
it is possible to predict $P_{\text{post}}(Y=y\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a)$ from pre-interventional data and a supposition
about $P_{\text{post}}(D=d\leavevmode\nobreak\ |\leavevmode\nobreak\
A=a,X=x)$. That means that we can also predict changes to the overall
reemployment probability $P_{\text{post}}(Y=0)$ as well as the gender
reemployment gap $P_{\text{post}}(Y=1\leavevmode\nobreak\
|\leavevmode\nobreak\ A=1)-P_{\text{post}}(Y=1\leavevmode\nobreak\
|\leavevmode\nobreak\ A=0).$ Each of these are natural and important instances
of $\varphi(\cdot).$ The proof is deferred to the supplementary material.

######  Theorem 1.

Suppose that Consistency, Unconfoundedness, No Unprecedented Decisions, Stable
CATE and No Feedback hold. Suppose also that $P_{\text{post}}(A=a)>0$. Then,
$P_{\text{post}}(Y=y\leavevmode\nobreak\ |\leavevmode\nobreak\ A=a)$ is given
by

$\sum_{(x,d)\in\Pi_{\text{post}}}P_{\text{pre}}(Y=y\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x,D=d)P_{\text{pre}}(X=x\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a)P_{\text{post}}(D=d\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x),$

where $\Pi_{t}=\left\\{(x,d)\in\mathcal{X\times
D}:P_{t}(X=x,D=d\leavevmode\nobreak\ |\leavevmode\nobreak\ A=a)>0\right\\}.$

Note that the first two terms in the product are identified from pre-
deployment data. Given a sufficiently precise proposal for how risk scores
influence decisions, it is also possible to model $\Pi_{\text{post}}$ and the
last term before deployment. This allows us to systematically compare
different (fairness-constrained) algorithms and decision procedures, and
arrive at a reasonable prediction of their combined effect on reemployment
probabilities (and the gender reemployment gap) before they are deployed. In
the following, we show how this approach works in a realistic case study.

##  5 Empirical Study: Long-term Unemployment in Switzerland

We are interested in forecasting the effect of using (fair) risk scores to
inform program allocation decisions on both the overall risk of long-term
unemployment and the gender reemployment gap. We present an extensive case
study built on Swiss administrative data to study three questions: do
fairness-constrained risk scores improve outcomes? are restrictive, Austrian-
style allocation policies more efficient than Flemish-style policies that
prioritize people at high risk? and can we improve outcomes with
individualized estimates of program effectiveness?

###  5.1 Methodology

Our analysis proceeds in the following stages: (1) Using double-robust machine
learning, we first estimate the effectiveness of each of the programs for all
individuals in our test sample. (2) We estimate risk scores for the
individuals in our test sample, using fairness-constrained and fairness-
unconstrained methods. We implement two fairness constraints: statistical
parity and equal opportunity. (3) For each of the risk scores from stage two,
we prioritize the individuals in the test sample. The Flanders-style policy
prioritizes those at the highest risk. The Austrian prioritization does the
same, but only for those in the $70-30$th risk percentiles; the rest go to the
end of the line. (4) For each priority list from stage three, we assign
unemployed to programs until program capacity is reached. We model two
assignment schemes. The first assigns individuals to programs randomly. The
second uses the results of stage one to assign individuals to the program with
the highest estimated effectiveness. Additionally, we consider the effect of
increasing program capacities. Finally, we summarize the effects of different
combinations of choices from steps (2-4) on overall rates of long-term
unemployment and the gender-reemployment gap.

####  5.1.1 Data

We exploit the administrative Swiss Active Labor Market Policy (ALMP)
Evaluation Dataset.222The data is available for scientific use at SWISSbase
(Lechner et al., 2020). The original sample contains observations on $100,120$
registered unemployed in $2003$, aged $24$ to $55$. Recently unemployed
received one of seven treatments: no program, vocational training, computer
programs, language courses, job search programs, employment programs, and
personality training. Among the seven treatment options, no program and job
search programs are by far the most common treatments. We restrict the
analysis to the German-speaking cantons as assignment strategies differ among
the three language regions (Knaus, 2022). To avoid overstating the
effectiveness of “no program”, we estimate pseudo program starting points for
individuals in this treatment arm and exclude those who are re-employed before
the pseudo starting point (Lechner, 1999, Knaus, 2022). The final data set
contains $64,296$ individuals, which we divide equally into training and test
sets. The simulation study is performed on the test set of $32,148$
individuals and all results are reported for this population. Descriptive
statistics for the simulation data are reported in Appendix B.1.

For all individuals, we observe employment status for $36$ months after
registration with the Swiss Public Employment Service (PES). Our target, long-
term unemployment, is defined as a binary variable indicating continuous
unemployment for $12$ months after the (pseudo) program start.333This is a
deviation from Körtner and Bach (2023), who define their target variable as
$12$ months after registration with the PES. The treatment variable is defined
as the first program assigned within six months after registering as
unemployed. The administrative data includes information on the individual
employment biographies, demographics, local labor market conditions as well as
information on the individual caseworker and their assessment of their
clients’ labor market outlook.

####  5.1.2 Individualized Average Potential Outcomes

We adopt double-robust machine learning for the estimation of individual
average potential outcomes (IAPOs) and treatment effects (IATEs) for the seven
treatment options (Chernozhukov et al., 2018, Alaa et al., 2023, Curth et al.,
2024). We follow Knaus (2022) and Körtner and Bach (2023) in their
identification strategy and use the R-package causalDML (Knaus, 2022). Inverse
probability weighting is used to account for non-random selection into the
programs under the identifying assumptions of Unconfoundedness (similar to our
Unconfoundedness), Common Support (No Unprecedented Decisions), and Stable
Unit Treatment Value (Consistency and Stable CATE). Especially important for
the plausibility of Unconfoundedness is the availability of information about
the individual caseworker. See Appendix B.3 for a more detailed discussion of
the estimation approach.

The resulting (individualized) average treatment effects are given in Figure 3
and Table 3(b). They are in line with the results reported in Knaus (2022) and
Körtner and Bonoli (2021). Vocational Training, Computer Programs, and
Language Courses have the strongest effects on reducing (long-term)
unemployment. We find that Job Search and Employment Programs on average
increase the risk of long-term unemployment by between $2$ to $3$ percentage
points and confirm the high effect heterogeneity in all treatments. The
reported treatment effects are the difference of the respective potential
outcome scores, where “no program” is the baseline program. IATEs broken down
by gender are given in Appendix B.3.

(a) Individualized Average Treatment Effects.

| ATE | SE | 95%-CI
---|---|---|---
Vocational | -11.12 | 0.06 | [-11.12, -11.12]
Computer | -11.37 | 0.05 | [-11.37, -11.37]
Language | -5.25 | 0.04 | [-5.26, -5.25]
Job Search | 3.43 | 0.03 | [3.43, 3.43]
Employment | 1.83 | 0.04 | [1.83, 1.83]
Personality | -1.84 | 0.04 | [-1.84, -1.84]
(b) Average Treatment Effects in percentage points, standard errors, and
$95\%$ confidence intervals. Negative treatment effects imply a lower risk of
becoming long-term unemployed.

Figure 3: Estimated (Individualized) Average Treatment Effects for the six
labor market programs. No program serves as the baseline.

####  5.1.3 Risk scores

In 2003, program assignment in the Swiss public employment service was made at
the discretion of the individual caseworker. This practice continues to this
day.444The canton of Freiburg had a pilot study from 2012-2014, providing
caseworkers with estimates of the expected length of the unemployment spell
(Arni and Schiprowski, 2015). For estimating the risk scores to determine a
prioritization, all caseworker information is excluded and only data that is
reasonably available at registration time is used: characteristics of the
unemployed person and the local labor market situation. The sensitive
attribute is included as a feature. The full list of features is given in
Appendix B.4.

To evaluate the impact of retrospective fairness on the distribution of social
goods, we estimate a fairness-unconstrained risk score and two risk scores
constrained to satisfy statistical parity555Also called demographic parity or
Independence of the predictions from the sensitive attribute (Barocas et al.,
2023). and equality of opportunity666The equality in true positive rates for
both groups. This is a relaxation of equalized odds, also called Separation
(Barocas et al., 2023)., respectively. Throughout, we use logistic ridge
regressions and the R-package fairml for estimation of fairness constrained
risk scores (Scutari et al., 2022). We do not require the fairness constraint
to be met perfectly.

All three methods, when applying a decision threshold of $.5$, achieve an
accuracy of about $64-65\%$. These results are in line with internationally
reported accuracy rates for the prediction of long-term unemployment (Desiere
et al., 2019). The unconstrained risk scores violate statistical parity, with
more women than men being predicted to become long-term unemployed (a
discrepancy of $0.116$). Further, the true (a discrepancy of $0.174$) and
false positive ($0.062$) rates are higher for women than for men. The fairness
constrained scores reduce these discrepancies. Details on the implementation
together with descriptive statistics for the risk scores can be found in
Appendix B.4.

####  5.1.4 Prioritization

For each of the three risk scores from the previous stage, we compile two
priority lists modeling the Belgian and Austrian proposals. The Belgian list
goes in order of decreasing risk (Desiere and Struyven, 2020). The Austrian
list does the same for those in the $30-70th$ risk percentiles. The others are
put at the end of the list, in random order (Allhutter et al., 2020a). This
yields six priority lists, one for each combination of risk score and
prioritization scheme.

####  5.1.5 Program Assignments

For each of the six lists from the previous stage, we assign individuals to
programs in order of priority. Individuals are assigned according to two
schemes: optimal and random. The first assigns each person to the program that
is most effective for them and not yet at capacity. This models the best-case
scenario in which caseworkers are very good at discerning which program is
best for each client. The second makes assignments by a uniform draw from the
available programs.777We run this scheme ten times per policy and average over
the resulting individual risks for long-term unemployment. These two
assignment schemes provide upper and lower bounds for what might happen when
caseworkers are informed by risk scores when making assignment decisions
instead of fully automating the decision. To model adjustments to the budget
constraint of the PES, we consider the effect of increasing program
capacities. As a baseline, we take the program sizes observed in the test set
(see Table 1). Then, we consider capacities that are $2-5x$ larger. Because
the most effective programs are also the smallest, increasing overall
capacities mainly influences outcomes by increasing the capacities of these
small but effective programs.

###  5.2 Results

####  5.2.1 Fair Prediction and the Fair Distribution of Social Goods.

Regardless of the notion of retrospective fairness and the choices made at
other stages, constraining risk predictions to be fair yields larger gender
reemployment gaps (Figure 4). This is because fairness constraints, by
shifting the distribution of risk scores among women to look more like the
distribution among men (Figure B.4), tend to underestimate their risk of long-
term unemployment. The effect of fairness constraints is to reserve a roughly
equal number of seats in effective training programs for men and women.
Therefore, fairness-constrained policies induce similar improvements in labor
market outcomes for both genders, which keeps the gender reemployment gap
relatively constant. On the other hand, fairness unconstrained risk scores
are, on average, higher for women. That means that more seats are reserved for
women in effective programs–the result is more aggressive reductions in rates
of long-term unemployment among women than among men. These effects are only
made more pronounced when budget constraints are relaxed and program
capacities are increased. For example, at baseline program sizes the
combination of Belgian prioritization and individualized treatment decisions
yields a $3.2\%$ gender gap in reemployment probabilities ($40.4\%$ vs
$37.2\%$) when risk scores are unconstrained and a $4.1\%$ gender gap
($40.9\%$ vs $36.8\%$) when risk scores are constrained to satisfy equal
opportunity. This means that, at baseline program sizes, the equal opportunity
constraint slightly exacerbated the ex-ante gender gap of $3.9\%$ ($43.6\%$
vs. $39.7\%$). If programs are made five times larger, the fairness
unconstrained policy reduces the gender gap to $.9\%$ ($35.1\%$ vs $34.2\%$)
whereas equal opportunity leaves the gender gap relatively unchanged at $3\%$
($36.2\%$ vs $33.2\%$). All results are given in Tables 5 for baseline
capacities and 6 for five-fold capacities. We observe similar patterns for
citizenship gaps, reported in Appendix B.6.

(a) Belgian Prioritization and Optimal Program

(b) Belgian Prioritization and Random Program

(c) Austrian Prioritization and Optimal Program

(d) Austrian Prioritization and Random Program

Figure 4: We plot the gender gap in long-term unemployment (LTU) against
program capacity for each combination of prioritization and assignment scheme.
The level of transparency shows the gender gap for the corresponding fairness
constraint: none, statistical parity, or equal opportunity. The unconstrained
risk scores (lowest transparency) result in the smallest gender gap. This
effect is especially pronounced as program capacity is increased and program
assignments are individualized (optimal).

####  5.2.2 Hawks and Doves.

Regardless of other choices, the Belgian policy is at least as efficient as
the Austrian policy, both in reducing overall rates of long-term unemployment
and reducing the gender reemployment gap (Figure 5). This holds both for the
optimal program assignment and the random assignment. For example: at baseline
program sizes, when the unemployed receive targeted assignment and risk scores
are not fairness constrained, the Belgian policy achieves an overall LTU rate
of $38.6\%$ and a gender reemployment gap of $3.2\%$ ($40.4\%$ vs. $37.2\%$)
whereas the Austrian policy induces an identical overall rate and a gap of
$3.4\%.$ If programs are made five times larger, the Belgian policy achieves
an overall rate of $34.6\%$ and a gender gap of $.9\%$ ($35.1\%$ vs $34.2\%$),
whereas the Austrian policy achieves an identical overall rate and a gender
gap of $1.2\%$ ($35.3\%$ vs $34.1\%$). Thus, targeting those at the highest
risk of long-term unemployment achieves improvements in gender equality
without any costs in overall efficiency. A more fine-grained analysis shows
that the Belgian prioritization closes the gender gap much more aggressively
among married non-citizens, who tend to have the worst labor market outcomes,
whereas the Austrian prioritization does slightly better among groups with
better average outcomes B.7. Similar effects are observed for citizenship gaps
B.6. Therefore we do not find any efficiency advantage for withholding
training from individuals at the highest risk of unemployment.

(a) Optimal Program

(b) Random Program

Figure 5: We plot overall long-term unemployment and the gender reemployment
gap against program capacity for each combination of prioritization and
assignment scheme. For clarity, results are shown only for fairness-
unconstrained risk scores. Regardless of the assignment scheme, the Belgian
prioritization results in slightly lower overall rates of long-term
unemployment (blue line) and a smaller gender gap. Individualized program
assignments (optimal) are markedly more effective.

####  5.2.3 Gains from Modeling Counterfactual Outcomes.

Regardless of other choices, assigning individuals to the program with the
highest estimated effectiveness reduces overall long-term unemployment and
reemployment gaps (Figures 4 and 5). This represents gains due to explicit
estimation of treatment effects rather than risk scores alone. For example: at
baseline program sizes, when risk scores are not fairness constrained,
targeting achieves a reduction of about $1.5$ percentage points in overall
long-term unemployment over random assignment, regardless of prioritization.
If programs are made five times larger, targeting achieves a reduction of
about $3.7$ percentage points over random assignment. Targeting is also much
more effective than random assignment at reducing gender gaps under both
prioritization regimes.

##  6 Conclusion and Future Work

We have argued that algorithmic fairness requires anticipating the causal
effects of deploying algorithms in concrete social settings on the
distribution of outcomes. We have shown that existing methods in algorithmic
fairness can have perverse distributive effects: requiring risk scores to be
fair may exacerbate inequalities in social goods. Moreover, contrary to the
accepted trade-offs between accurate and fair predictions, accurate prediction
of individualized counterfactual outcomes supports policy in reducing
inequality in the distribution of social goods.

Our approach has several limitations: we have not tried every fairness
constraint, nor accounted for uncertainty in the estimation of individualized
treatment effects. Realistic methods may have to make program assignments in
an online, rather than a batch, fashion. Next to anticipatory evaluations, the
design of algorithmically informed policies should also directly support the
ex-post identification and evaluation of the policy. We have also adopted a
rather paternalistic approach: future work should try to accommodate the
preferences of the unemployed. Finally, we have simulated a policy approach
that relies essentially on risk scores to facilitate prioritization. This
reflects the state of algorithmic policy. However, risk scores increasingly
seem like an unnecessary detour. We are inspired by the work of Körtner and
Bach (2023): future work might directly seek distributively optimal
allocations (perhaps with more sophisticated notions of optimality) without
recourse to risk scores (Kitagawa and Tetenov, 2019, Viviano and Bradic,
2023). This approach subjects claims of ‘efficiency’ to direct test and allows
the conceptual innovations of theorists of distributive justice like Rawls and
his interlocutors to flow directly into applications.

## Acknowledgements

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC number 2064/1 –
Project number 390727645. The authors thank the International Max Planck
Research School for Intelligent Systems (IMPRS-IS) for supporting Sebastian
Zezulka.

The authors thank Ruben Bach, Christoph Kern, Michael Knaus, and Thomas Grote
for helpful discussions and feedback.

## References

  * Alaa et al. (2023) Ahmed Alaa, Zaid Ahmad, and Mark van der Laan.  Conformal Meta-learners for Predictive Inference of Individual Treatment Effects, 2023.
  * Albanesi and Şahin (2018) Stefania Albanesi and Ayşegül Şahin.  The gender unemployment gap.  _Review of Economic Dynamics_ , 30:47–67, 2018.  doi: 10.1016/j.red.2017.12.005.
  * Allhutter et al. (2020a) Doris Allhutter, Florian Cech, Fabian Fischer, Gabriel Grill, and Astrid Mager.  Algorithmic Profiling of Job Seekers in Austria: How Austerity Politics Are Made Effective.  _Frontiers in Big Data_ , 3, feb 2020a.  doi: 10.3389/fdata.2020.00005.
  * Allhutter et al. (2020b) Doris Allhutter, Astrid Mager, Florian Cech, Fabian Fischer, and Gabriel Grill.  Der AMS-Algorithmus. Eine Soziotechnische Analyse des Arbeitsmarktchancen-Assistenz-Systems (AMAS). Endbericht.  Technical Report ITA-Projektbericht Nr. 2020-02, Institut für Technikfolgen-Abschätzung der Österreichischen Akademie der Wissenschaften, 2020b.
  * Angwin et al. (2016) Julia Angwin, Jeff Larson, Surya Mattu, and Lauren Kirchner.  Machine bias: There’s software used across the country to predict future criminals. and it’s biased against blacks, 2016.  URL https://www.propublica.org/article/machine-bias-risk-assessments-in-criminal-sentencing.
  * Arni and Schiprowski (2015) Patrick Arni and Amelie Schiprowski.  Die Rolle von Erwartungshaltungen in der Stellensuche und der RAV-Beratung - Teilprojekt 2: Pilotprojekt Jobchancen-Barometer.  _IZA Research Reports_ , (70), 2015.
  * Barabas et al. (2018) Chelsea Barabas, Madars Virza, Karthik Dinakar, Joichi Ito, and Jonathan Zittrain.  Interventions over Predictions: Reframing the Ethical Debate for Actuarial Risk Assessment.  In Sorelle A. Friedler and Christo Wilson, editors, _Proceedings of the 1st Conference on Fairness, Accountability and Transparency_ , volume 81 of _Proceedings of Machine Learning Research_ , pages 62–76. PMLR, 2018.
  * Barocas et al. (2023) Solon Barocas, Moritz Hardt, and Arvind Narayanan.  _Fairness and Machine Learning: Limitations and Opportunities_.  MIT Press, 2023.
  * Beigang (2022) Fabian Beigang.  On the Advantages of Distinguishing Between Predictive and Allocative Fairness in Algorithmic Decision-Making.  _Minds and Machines_ , 32(4):655–682, 2022.
  * Berk et al. (2021) Richard A Berk, Arun Kumar Kuchibhotla, and Eric Tchetgen Tchetgen.  Improving fairness in criminal justice algorithmic risk assessments using optimal transport and conformal prediction sets.  _Sociological Methods & Research_, 2021.
  * Berk et al. (2023) Richard A. Berk, Arun Kumar Kuchibhotla, and Eric Tchetgen Tchetgen.  Fair Risk Algorithms.  _Annual Review of Statistics and Its Application_ , 10(1):165–187, 2023.  doi: 10.1146/annurev-statistics-033021-120649.
  * Bishu and Alkadry (2016) Sebawit G. Bishu and Mohamad G. Alkadry.  A Systematic Review of the Gender Pay Gap and Factors That Predict It.  _Administration & Society_, 49(1):65–104, 2016.  doi: 10.1177/0095399716636928.
  * Bonoli (2010) Giuliano Bonoli.  The Political Economy of Active Labor-Market Policy.  _Politics & Society_, 38(4):435–457, 2010.  doi: 10.1177/0032329210381235.
  * Borsboom et al. (2008) Denny Borsboom, Jan-Willem Romeijn, and Jelte M. Wicherts.  Measurement invariance versus selection invariance: Is fair selection possible?  _Psychological Methods_ , 13(2):75–98, 2008.  doi: 10.1037/1082-989x.13.2.75.
  * Bühlmann et al. (2013) Felix Bühlmann, Céline Schmid Botkine, Peter Farago, François Höpflinger, Dominique Joye, René Levy, Pasqualina Perrig-Chiello, and Christian Suter.  _Swiss Social Report: Generations in Perspective_.  Seismo, 2013.  URL http://socialreport.ch/2012/first-level-page/long-term-unemployment/long-term-unemployment-in-switzerland-by-sex-1991-2010.html.
  * Bundesagentur für Arbeit (2023) Bundesagentur für Arbeit.  Statistik der Bundesagentur für Arbeit Berichte: Blickpunkt Arbeitsmarkt –Die Arbeitsmarktsituation von Frauen und Männern.  _Nürnberg, May_ , 2023.
  * Bynum et al. (2022) Lucius E. J. Bynum, Joshua R. Loftus, and Julia Stoyanovich.  Counterfactuals for the Future, 2022.
  * Card et al. (2018) David Card, Jochen Kluve, and Andrea Weber.  What Works? A Meta Analysis of Recent Active Labor Market Program Evaluations.  _Journal of the European Economic Association_ , 16(3):894–931, 2018.  doi: 10.1093/jeea/jvx028.
  * Chernozhukov et al. (2018) Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins.  Double/debiased machine learning for treatment and structural parameters.  _The Econometrics Journal_ , 21(1):C1–C68, jan 2018.  doi: 10.1111/ectj.12097.
  * Chouldechova (2017) Alexandra Chouldechova.  Fair Prediction with Disparate Impact: A Study of Bias in Recidivism Prediction Instruments.  _Big Data_ , 5(2):153–163, 2017.  doi: 10.1089/big.2016.0047.
  * Cockx et al. (2023) Bart Cockx, Michael Lechner, and Joost Bollens.  Priority to unemployed immigrants? A causal machine learning evaluation of training in Belgium.  _Labour Economics_ , 80:102306, 2023.  doi: 10.1016/j.labeco.2022.102306.
  * Considine et al. (2017) Mark Considine, Phuc Nguyen, and Siobhan O’Sullivan.  New public management and the rule of economic incentives: Australian welfare-to-work from job market signalling perspective.  _Public Management Review_ , 20(8):1186–1204, 2017.  doi: 10.1080/14719037.2017.1346140.
  * Corbett-Davies et al. (2018) Sam Corbett-Davies, Johann D. Gaebler, Hamed Nilforoshan, Ravi Shroff, and Sharad Goel.  The Measure and Mismeasure of Fairness.  2018\.  doi: 10.48550/ARXIV.1808.00023.
  * Coston et al. (2020) Amanda Coston, Alan Mishler, Edward H. Kennedy, and Alexandra Chouldechova.  Counterfactual risk assessments, evaluation, and fairness.  In _Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency_. ACM, 2020.  doi: 10.1145/3351095.3372851.
  * Crépon et al. (2013) Bruno Crépon, Esther Duflo, Marc Gurgand, Roland Rathelot, and Philippe Zamora.  Do Labor Market Policies have Displacement Effects? Evidence from a Clustered Randomized Experiment.  _The Quarterly Journal of Economics_ , 128(2):531–580, 2013.  doi: 10.1093/qje/qjt001.
  * Curth et al. (2024) Alicia Curth, Richard W. Peck, Eoin McKinney, James Weatherall, and Mihaela van der Schaar.  Using Machine Learning to Individualize Treatment Effect Estimation: Challenges and Opportunities.  _Clinical Pharmacology & Therapeutics_, January 2024.  ISSN 1532-6535.  doi: 10.1002/cpt.3159.
  * D’Amour et al. (2020) Alexander D’Amour, Hansa Srinivasan, James Atwood, Pallavi Baljekar, D. Sculley, and Yoni Halpern.  Fairness Is Not Static: Deeper Understanding of Long Term Fairness via Simulation Studies.  In _Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency_ , FAT* ’20, page 525–534. ACM, 2020.
  * Desiere et al. (2019) S. Desiere, K. Langenbucher, and L. Struyven.  Statistical profiling in public employment services.  _OECD Social, Employment and Migration Working Papers_ , (224), 2019.  doi: 10.1787/b5e5f16e-en.
  * Desiere and Struyven (2020) Sam Desiere and Ludo Struyven.  Using Artificial Intelligence to classify Jobseekers: The Accuracy-Equity Trade-off.  _Journal of Social Policy_ , 50(2):367–385, 2020.  doi: 10.1017/s0047279420000203.
  * Dwork et al. (2012) Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel.  Fairness through awareness.  In _Proceedings of the 3rd Innovations in Theoretical Computer Science Conference_. ACM, 2012.  doi: 10.1145/2090236.2090255.
  * Ensign et al. (2018) Danielle Ensign, Sorelle A. Friedler, Scott Neville, Carlos Scheidegger, and Suresh Venkatasubramanian.  Runaway Feedback Loops in Predictive Policing.  _Proceedings of Machine Learning Research_ , 81:1–12, 2018.
  * European Commission, Eurostat (Accessed 17 January 2024) European Commission, Eurostat.  Long-term unemployment by sex - annual data, Accessed 17 January 2024.  URL https://ec.europa.eu/eurostat/databrowser/view/une_ltu_a/default/table?lang=en.
  * Goller et al. (2021) Daniel Goller, Tamara Harrer, Michael Lechner, and Joachim Wolff.  Active labour market policies for the long-term unemployed: New evidence from causal machine learning, 2021.
  * Green (2022) Ben Green.  Escaping the Impossibility of Fairness: From Formal to Substantive Algorithmic Fairness.  _Philosophy & Technology_, 35(4), 2022.  doi: 10.1007/s13347-022-00584-6.
  * Hardt et al. (2016) Moritz Hardt, Eric Price, and Nathan Srebro.  Equality of Opportunity in Supervised Learning.  In _Advances in Neural Information Processing Systems_ , pages 3315–3323, 2016.  doi: 10.48550/ARXIV.1610.02413.
  * Hu and Chen (2018) Lily Hu and Yiling Chen.  A Short-term Intervention for Long-term Fairness in the Labor Market.  In _Proceedings of the 2018 World Wide Web Conference on World Wide Web_. ACM Press, 2018.  doi: 10.1145/3178876.3186044.
  * Hutchinson and Mitchell (2019) Ben Hutchinson and Margaret Mitchell.  50 Years of Test (Un)fairness.  In _Proceedings of the Conference on Fairness, Accountability, and Transparency_. ACM, 2019.  doi: 10.1145/3287560.3287600.
  * Kannan et al. (2019) Sampath Kannan, Aaron Roth, and Juba Ziani.  Downstream Effects of Affirmative Action.  In _Proceedings of the Conference on Fairness, Accountability, and Transparency_. ACM, 2019.  doi: 10.1145/3287560.3287578.
  * Kasy and Abebe (2021) Maximilian Kasy and Rediet Abebe.  Fairness, Equality, and Power in Algorithmic Decision-Making.  In _Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency_. ACM, mar 2021.  doi: 10.1145/3442188.3445919.
  * Kern et al. (2021) Christoph Kern, Ruben L. Bach, Hannah Mautner, and Frauke Kreuter.  Fairness in Algorithmic Profiling: A German Case Study, 2021.
  * Kilbertus et al. (2017) Niki Kilbertus, Mateo Rojas Carulla, Giambattista Parascandolo, Moritz Hardt, Dominik Janzing, and Bernhard Schölkopf.  Avoiding discrimination through causal reasoning.  In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, _Advances in Neural Information Processing Systems_ , volume 30. Curran Associates, Inc., 2017.
  * Kitagawa and Tetenov (2019) Toru Kitagawa and Aleksey Tetenov.  Equality-minded treatment choice.  _Journal of Business & Economic Statistics_, 39(2):561–574, 2019.  doi: 10.1080/07350015.2019.1688664.
  * Kleinberg et al. (2016) Jon Kleinberg, Sendhil Mullainathan, and Manish Raghavan.  Inherent Trade-Offs in the Fair Determination of Risk Scores, 2016.
  * Kleven et al. (2023) Henrik Kleven, Camille Landais, and Gabriel Leite-Mariante.  The Child Penalty Atlas.  _National Bureau of Economic Research_ , 2023.  doi: 10.3386/w31649.
  * Knaus (2022) Michael C. Knaus.  Double machine learning-based programme evaluation under unconfoundedness.  _The Econometrics Journal_ , 25(3):602–627, jun 2022.  doi: 10.1093/ectj/utac015.
  * Knaus et al. (2022) Michael C. Knaus, Michael Lechner, and Anthony Strittmatter.  Heterogeneous Employment Effects of Job Search Programs: A Machine Learning Approach.  _Journal of Human Resources_ , 57(2):597–636, 2022.  ISSN 1548-8004.  doi: 10.3368/jhr.57.2.0718-9615r1.
  * Körtner and Bonoli (2021) John Körtner and Giuliano Bonoli.  Predictive Algorithms in the Delivery of Public Employment Services.  _SocArXiv_ , 2021.
  * Kunaschk and Lang (2022) Max Kunaschk and Julia Lang.  Can Algorithms Reliably Predict Long-Term Unemployment in Times of Crisis? – Evidence from the COVID-19 Pandemic.  _IAB-Discussion Paper_ , 2022.  doi: 10.48720/IAB.DP.2208.
  * Kuppler et al. (2021) Matthias Kuppler, Christoph Kern, Ruben L. Bach, and Frauke Kreuter.  Distributive Justice and Fairness Metrics in Automated Decision-making: How Much Overlap Is There?, 2021.
  * Kusner et al. (2017) Matt J Kusner, Joshua Loftus, Chris Russell, and Ricardo Silva.  Counterfactual fairness.  _Advances in neural information processing systems_ , 30, 2017.
  * Körtner and Bach (2023) John Körtner and Ruben L. Bach.  Inequality-Averse Outcome-Based Matching.  2023\.  doi: 10.31219/osf.io/yrn4d.
  * Lammers and Kok (2019) Marloes Lammers and Lucy Kok.  Are active labor market policies (cost-)effective in the long run? Evidence from the Netherlands.  _Empirical Economics_ , 60(4):1719–1746, 2019.  doi: 10.1007/s00181-019-01812-3.
  * Lechner (1999) Michael Lechner.  Earnings and Employment Effects of Continuous Off-the-Job Training in East Germany After Unification.  _Journal of Business & Economic Statistics_, 17(1):74–90, January 1999.  ISSN 1537-2707.  doi: 10.1080/07350015.1999.10524798.
  * Lechner et al. (2020) Michael Lechner, Michael Knaus, Martin Huber, Markus Frölich, Stefanie Behncke, Giovanni Mellace, and Anthony Strittmatter.  Swiss active labor market policy evaluation [dataset].  _FORS, Lausanne, Switzerland_ , 2020.
  * Levy et al. (2021) Karen Levy, Kyla E. Chasalow, and Sarah Riley.  Algorithms and Decision-Making in the Public Sector.  _Annual Review of Law and Social Science_ , 17(1):309–334, 2021.  doi: 10.1146/annurev-lawsocsci-041221-023808.
  * Liu et al. (2019) Lydia T. Liu, Sarah Dean, Esther Rolf, Max Simchowitz, and Moritz Hardt.  Delayed Impact of Fair Machine Learning.  In _Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence_. International Joint Conferences on Artificial Intelligence Organization, 2019.  doi: 10.24963/ijcai.2019/862.
  * Lum and Isaac (2016) Kristian Lum and William Isaac.  To predict and serve?  _Significance_ , 13(5):14–19, 2016.
  * Malinsky (2018) Daniel Malinsky.  Intervening on structure.  _Synthese_ , 195(5):2295–2312, 2018.
  * Menon and Williamson (2018) Aditya Krishna Menon and Robert C. Williamson.  The cost of fairness in classification.  pages 107–118. PMLR, 2018.  doi: 10.48550/ARXIV.1705.09055.
  * Mishler and Dalmasso (2022) Alan Mishler and Niccolò Dalmasso.  Fair When Trained, Unfair When Deployed: Observable Fairness Measures are Unstable in Performative Prediction Settings, 2022.
  * Mitchell et al. (2021) Shira Mitchell, Eric Potash, Solon Barocas, Alexander D'Amour, and Kristian Lum.  Algorithmic Fairness: Choices, Assumptions, and Definitions.  _Annual Review of Statistics and Its Application_ , 8(1):141–163, 2021.  doi: 10.1146/annurev-statistics-042720-125902.
  * Mueller and Spinnewijn (2023) Andreas Mueller and Johannes Spinnewijn.  The Nature of Long-Term Unemployment: Predictability, Heterogeneity and Selection.  _National Bureau of Economic Research_ , 2023.  doi: 10.3386/w30979.
  * Pagan et al. (2023) Nicolò Pagan, Joachim Baumann, Ezzat Elokda, Giulia De Pasquale, Saverio Bolognani, and Anikó Hannák.  A Classification of Feedback Loops and Their Relation to Biases in Automated Decision-Making Systems.  In _Equity and Access in Algorithms, Mechanisms, and Optimization_. ACM, 2023.  doi: 10.1145/3617694.3623227.
  * Perdomo et al. (2020) Juan Perdomo, Tijana Zrnic, Celestine Mendler-Dünner, and Moritz Hardt.  Performative prediction.  In _International Conference on Machine Learning_ , pages 7599–7609. PMLR, 2020.
  * Petersen et al. (2021) Anette C. M. Petersen, Lars Rune Christensen, Richard Harper, and Thomas Hildebrandt.  "We Would Never Write That Down": Classifications of Unemployed and Data Challenges for AI.  _Proceedings of the ACM on Human-Computer Interaction_ , 5(CSCW1):1–26, 2021.  doi: 10.1145/3449176.
  * Scher et al. (2023) Sebastian Scher, Simone Kopeinik, Andreas Trügler, and Dominik Kowald.  Modelling the long-term fairness dynamics of data-driven targeted help on job seekers.  _Scientific Reports_ , 13(1), 2023.
  * Scutari et al. (2022) Marco Scutari, Francesca Panero, and Manuel Proissl.  Achieving fairness with a simple ridge penalty.  _Statistics and Computing_ , 32(5), September 2022.  ISSN 1573-1375.  doi: 10.1007/s11222-022-10143-w.
  * Selbst et al. (2019) Andrew D. Selbst, Danah Boyd, Sorelle A. Friedler, Suresh Venkatasubramanian, and Janet Vertesi.  Fairness and Abstraction in Sociotechnical Systems.  In _Proceedings of the Conference on Fairness, Accountability, and Transparency_. ACM, 2019.  doi: 10.1145/3287560.3287598.
  * Tal (2023a) Eran Tal.  Target specification bias, counterfactual prediction, and algorithmic fairness in healthcare.  In _Proceedings of the 2023 AAAI/ACM Conference on AI, Ethics, and Society_ , AIES ’23. ACM, August 2023a.  doi: 10.1145/3600211.3604678.
  * Tal (2023b) Eran Tal.  Target specification bias, counterfactual prediction, and algorithmic fairness in healthcare.  In _Proceedings of the 2023 AAAI/ACM Conference on AI, Ethics, and Society_ , pages 312–321, 2023b.
  * Tolbert and Diana (2023) Alexander Williams Tolbert and Emily Diana.  Correcting Underrepresentation and Intersectional Bias for Fair Classification, 2023.
  * Viviano and Bradic (2023) Davide Viviano and Jelena Bradic.  Fair Policy Targeting.  _Journal of the American Statistical Association_ , pages 1–14, 2023.  doi: 10.1080/01621459.2022.2142591.
  * Vooren et al. (2018) Melvin Vooren, Carla Haelermans, Wim Groot, and Henriëtte Maassen van den Brink.  The Effectivnesness of Active Labor Market Policies: A Meta-Analysis.  _Journal of Economic Surveys_ , 33(1):125–149, 2018.  doi: 10.1111/joes.12269.
  * Weerts et al. (2023) Hilde Weerts, Raphaële Xenidis, Fabien Tarissan, Henrik Palmer Olsen, and Mykola Pechenizkiy.  Algorithmic Unfairness through the Lens of EU Non-Discrimination Law.  In _2023 ACM Conference on Fairness, Accountability, and Transparency_. ACM, 2023.  doi: 10.1145/3593013.3594044.
  * Zhang and Liu (2021) Xueru Zhang and Mingyan Liu.  Fairness in Learning-Based Sequential Decision Algorithms: A Survey.  In _Handbook of Reinforcement Learning and Control_ , pages 525–555. Springer International Publishing, 2021.  doi: 10.1007/978-3-030-60990-0_18.
  * Zhang et al. (2020) Xueru Zhang, Ruibo Tu, Yang Liu, Mingyan Liu, Hedvig Kjellstrom, Kun Zhang, and Cheng Zhang.  How do fair decisions fare in long-term qualification?  _Advances in Neural Information Processing Systems_ , pages 1–13, 2020.

##  Appendix A Proof of Theorem 1

###### Proof of Theorem 1.

First, we need to show that all terms are well-defined. This amounts to
showing that $P_{\text{post}}(A=a,X=x)$, $P_{\text{pre}}(A=a)$ and
$P_{\text{pre}}(A=a,X=x,D=d)$ are strictly greater than zero for all
$(x,d)\in\Pi_{\text{post}}.$

We first show that $P_{\text{pre}}(A=a)>0.$ Note that

$\displaystyle P_{\text{pre}}(A=a)$
$\displaystyle=\sum_{x\in\mathcal{X}}P_{\text{pre}}(A=a,X=x)$
$\displaystyle=\sum_{x\in\mathcal{X}}P_{\text{post}}(A=a,X=x)$ (No Feedback)
$\displaystyle=P_{\text{post}}(A=a)>0.$

We now show that $P_{\text{post}}(A=a,X=x)>0$ for all
$(x,d)\in\Pi_{\text{post}}.$ Note that

$\displaystyle P_{\text{post}}(A=a,X=x)$
$\displaystyle=P_{\text{post}}(A=a)\sum_{e\in\mathcal{D}}P_{\text{post}}(X=x,D=e|A=a)$
$\displaystyle\geq P_{\text{post}}(A=a)P_{\text{post}}(X=x,D=d|A=a)>0.$

Finally, we show that $P_{\text{pre}}(A=a,X=x,D=d)>0$ for all
$(x,d)\in\Pi_{\text{post}}.$ Since $P_{\text{pre}}(A=a)>0$, it suffices to
show that $P_{\text{pre}}(X=x,D=d|A=a)>0$ for all $(x,d)\in\Pi_{\text{post}}.$
Accordingly, suppose that $(x,d)\in\Pi_{\text{post}}.$ Then

$P_{\text{post}}(X=x,D=d|A=a)=P_{\text{post}}(D=d|X=x,A=a)P_{\text{post}}(X=x|A=a)>0,$

which entails that both $P_{\text{post}}(D=d|X=x,A=a)>0$ and
$P_{\text{post}}(X=x|A=a)>0.$ By No Unprecedented Decisions,
$P_{\text{pre}}(D=x|X=x,A=a)>0$ and by No Feedback
$P_{\text{pre}}(X=x|A=a)>0.$ Therefore,

$P_{\text{pre}}(X=x,D=d|A=a)=P_{\text{pre}}(D=x|X=x,A=a)P_{\text{pre}}(X=x|A=a)>0;$

and the question of well-definedness is settled.

Next, note that: $P_{\text{post}}(Y=y\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a)=$

$\displaystyle=\sum_{(x,d)\in\Pi_{\text{post}}}P_{\text{post}}(Y=y\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x,D=d)P_{\text{post}}(X=x,D=d\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a)$ (Total Probability)
$\displaystyle=\sum_{(x,d)\in\Pi_{\text{post}}}P_{\text{post}}(Y=y\leavevmode\nobreak\
|\leavevmode\nobreak\
A=a,X=x,D=d)P_{\text{post}}(X=x|A=a)P_{\text{post}}(D=d\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x)$
$\displaystyle=\sum_{(x,d)\in\Pi_{\text{post}}}P_{\text{post}}(Y=y\leavevmode\nobreak\
|\leavevmode\nobreak\
A=a,X=x,D=d)P_{\text{pre}}(X=x|A=a)P_{\text{post}}(D=d\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x).$ (No Feedback)

Note that, whenever defined,

$\displaystyle P_{t}(Y=y\leavevmode\nobreak\ |\leavevmode\nobreak\
A=a,X=x,D=d)$
$\displaystyle=P_{t}\left(\sum_{e\in\mathcal{D}}Y^{e}\mathbbm{1}[D=e]=1\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x,D=d\right)$ (Consistency)
$\displaystyle=P_{t}\left(Y^{d}=y\leavevmode\nobreak\ |\leavevmode\nobreak\
A=a,X=x,D=d\right)$ $\displaystyle=P_{t}\left(Y^{d}=y\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x\right).$ (Unconfoundedness)

Therefore,

$\displaystyle P_{\text{post}}(Y=y\leavevmode\nobreak\ |\leavevmode\nobreak\
A=a,X=x,D=d)$ $\displaystyle=P_{\text{post}}\left(Y^{d}=y\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x\right)$
$\displaystyle=P_{\text{pre}}\left(Y^{d}=y\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x\right)$ (Stable CATE)
$\displaystyle=P_{\text{pre}}(Y=y\leavevmode\nobreak\ |\leavevmode\nobreak\
A=a,X=x,D=d);$

and therefore $P_{\text{post}}(Y=y\leavevmode\nobreak\ |\leavevmode\nobreak\
A=a)=$

$=\sum_{(x,d)\in\Pi_{\text{post}}}P_{\text{pre}}(Y=y\leavevmode\nobreak\
|\leavevmode\nobreak\
A=a,X=x,D=d)P_{\text{pre}}(X=x|A=a)P_{\text{post}}(D=d\leavevmode\nobreak\
|\leavevmode\nobreak\ A=a,X=x).$

∎

##  Appendix B Case Study

###  B.1 Descriptive Statistics: Simulation Data

| #Obs | LTU | Female | Age | Non-Citizen | Employability | Past Income
---|---|---|---|---|---|---|---
|  |  | (binary) | in years | (binary) |  | in CHF
Simulation Data | 32,148 | 0.41 | 0.44 | 36.8 | 0.36 | 1.93 | 43,461
No program | 23,785 | 0.41 | 0.43 | 36.6 | 0.37 | 1.92 | 42,557
Vocational | 423 | 0.28 | 0.32 | 37.5 | 0.32 | 1.91 | 49,349
Computer | 446 | 0.24 | 0.61 | 38.9 | 0.20 | 1.98 | 43,251
Language | 723 | 0.48 | 0.54 | 35.3 | 0.68 | 1.83 | 37,779
Job Search | 5,868 | 0.43 | 0.44 | 37.4 | 0.33 | 1.98 | 46,815
Employment | 321 | 0.46 | 0.43 | 35.3 | 0.39 | 1.84 | 36,902
Personality | 582 | 0.37 | 0.35 | 39.4 | 0.25 | 1.93 | 53,136
Table 1: Descriptive statistics for key demographic variables in the test and
simulation data and by observed treatment groups. Long-term unemployment
(LTU), Female, and Non-Citizen are given as shares. Age, Employability, and
Past Income are averages. Employability is an ordered variable from low (1) to
high (3), assigned by the caseworker. Knaus (2022) reports an exchange rate
USD/CHF of about $1.3$ for 2003.

###  B.2 Descriptive Statistics: Full sample

| #Obs | LTU | Female | Age | Non-Citizen | Employability | Past Income
---|---|---|---|---|---|---|---
|  |  | (binary) | in years | (binary) |  | in CHF
Full Sample | 64,296 | 0.41 | 0.44 | 36.8 | 0.36 | 1.93 | 43,391
No program | 47,631 | 0.41 | 0.44 | 36.6 | 0.37 | 1.93 | 42,529
Vocational | 858 | 0.29 | 0.33 | 37.5 | 0.30 | 1.93 | 48,654
Computer | 905 | 0.28 | 0.60 | 39.1 | 0.21 | 1.97 | 43,213
Language | 1,504 | 0.47 | 0.55 | 35.28 | 0.66 | 1.85 | 37,300
Job Search | 11,610 | 0.43 | 0.44 | 37.3 | 0.33 | 1.98 | 46,693
Employment | 611 | 0.43 | 0.41 | 35.3 | 0.38 | 1.83 | 37,084
Personality | 1,177 | 0.37 | 0.36 | 38.7 | 0.27 | 1.93 | 53,067
Table 2: Descriptive statistics for key demographic variables in the full
sample and by observed treatment groups. The simulation data is drawn from
this full sample. Long-term unemployment (LTU), Female, and Non-Citizen are
given as shares. Age, Employability, and Past Income are averages.
Employability is an ordered variable from low (1) to high (3), assigned by the
caseworker. Knaus (2022) reports an exchange rate USD/CHF of about $1.3$ for
2003.

###  B.3 Double-Robust Machine Learning for Estimating IAPOs

In Section 4, we have theoretically derived conditions under which the post-
interventional gender gap is identified. Two assumptions concern the internal
validity of our study. Unconfoundedness is the strongest assumption.
Replicating the work by Knaus (2022), Knaus et al. (2022) and Körtner and Bach
(2023), we rely on extensive information on caseworkers and their subjective
assessment of their clients in the estimation of treatment effects combined
with rich administrative data on the demographics and employment biographies
to support the assumption. No Unprecedented decisions requires that the
propensity scores are non-zero. The other two concern the external validity of
our simulation study. We presuppose that the treatment effects of the programs
are stable under different allocations and increased program capacities
(stable CATEs) and that the pool of unemployed stays the same (No Feedback on
the covariates).

First, we estimate the normalized conditional probability to be allocated into
each program (the propensity of treatment, $e_{d}(X_{i})$) and the conditional
outcome mean in the observed allocation (in short, conditional outcome,
$\mu(d,x)$). Given the small number of observations in most of the labor
market programs, we use the full data set and cross-validation for the
estimation of the nuisance parameters. The two nuisance parameters then allow
the estimation of the doubly robust score:

$\hat{\Gamma}_{i,d}=\hat{\mu}(d,X_{i})+\frac{D_{i}(d)(Y_{i}-\hat{\mu}(d,X_{i}))}{\hat{e}_{d}(X_{i})},$

where $D_{i}(d)$ indicates the treatment assignment for individual $i$ and
$Y_{i}$ the observed, pre-interventional outcome. This strategy is called
doubly robust because the functional form of either the propensity score or
the conditional outcome can be miss-specified without threatening the
identification (Chernozhukov et al., 2018, Knaus, 2022). In the last step, the
estimates of the debiased scores, $\hat{\Gamma}_{i,d}$, are used as pseudo
outcomes to estimate the conditional expected outcomes,
$\mathbf{E}[\hat{\Gamma}_{i,d}\leavevmode\nobreak\ |\leavevmode\nobreak\
X_{i}]$ using a regression forest. These estimates are the individualized
average potential outcomes for each treatment option under the outlined
identifying assumptions. For this step, the regression forest is trained only
on the training set.

We estimate individualized average treatment effects for each individual $i$
in the sample as differences between the respective individualized average
potential outcomes:

$\hat{\Delta}_{i,d,d^{\prime}}=\hat{\Gamma}_{i,d}-\hat{\Gamma}_{i,d^{\prime}}.$

In Table 6, we show the distribution of individualized average treatment
effects by gender. While the overall trends remain the same, all treatments
except job search programs on average are slightly more effective for women
than for men. Treatment effects are estimated against the baseline of no
program.

Figure 6: Individualized and Average Treatment Effects for all six labor
market programs by gender. Baseline is “no program”.

###  B.4 Risk Scores and Prioritization Policies

To determine the prioritization of registered unemployed in its Belgian or
Austrian variants we estimate risk scores for becoming long-term unemployed.
The full list of features is given in Table 3. For a discussion on the
predictability of long-term unemployment, see Mueller and Spinnewijn (2023).
Using administrative data from Germany, Kunaschk and Lang (2022) evaluate the
performance of risk scores under external shocks like the COVID-19 pandemic.
Kern et al. (2021) evaluate the violation of retrospective fairness criteria
when predicting long-term unemployment in the same context.

First, we estimate risk scores by a fairness-unconstrained logistic ridge
regression. The optimal regularization strength is chosen by cross-validation
at about $\lambda=0.049$. Second, we add a fairness constraint for statistical
parity and, third, a constraint for equal opportunity. In this case, the true
positive rates among the sensitive attribute are equalized, a relaxation of
Separation (Hardt et al., 2016). We make use of the the implementation by
(Scutari et al., 2022) for the estimation of fairness constrained risk scores.
To achieve statistical parity they use a ridge penalty to bound the variance
explained by the sensitive attribute (gender) over the total explained
variance. For equal opportunity, the risk score is regressed against the
sensitive attribute and the outcome variable with the ridge penalty bounding
the variance explained by the sensitive attribute over the total explained
variance. In both cases, we use a fairness penalty of $0.01$, where $0$
requires perfect fairness and $1$ corresponds to no fairness constraint.

Note some important differences between the Belgian and Austrian
implementations of our work. In Flanders, Belgium the probability of re-
employment within six months is estimated by a random forest model (Desiere
and Struyven, 2020). Sensitive attributes are no longer included due to
privacy regulations. In our simulation study, the definition of long-term
unemployment corresponds to the ILO definition with $12$ months of
uninterrupted unemployment.

In Austria, two different models are estimated (Allhutter et al., 2020b). The
first, short-term model, uses as a binary target at least $90$ days of
unsupported employment within seven months after the reference date. The
second, long-term model, uses at least $180$ days of unsupported employment
within $24$ months as the target. Those with a short-term probability of
employment of above $66\%$ are classified as low risk for LTU. Those with a
long-term probability of employment below $25\%$ are classified as high risk.
The middle group is built as a residual. That is, it includes all those not
classified as high or low risk. In difference to earlier reports (Allhutter et
al., 2020a), a stratification approach is applied, and logistic regressions
are used to evaluate the feature importance only (Allhutter et al., 2020b).
Sensitive attributes like gender and citizenship are included as features. In
difference to the Austrian proposal, we estimate one model and create the
prioritized middle group as those individuals falling in the $30-70th$
percentile of the respective risk distribution.

Features for the estimation of risk scores |
---|---
Age |
Mother tongue in canton’s language |
Lives in big city |
Lives in medium city |
Lives in no city |
Fraction of months employed in last 2 years |
Number of employment spells in last 5 years |
Female (binary) |
Foreigner with temporary permit |
Foreigner with permanent permit |
Cantonal GDP p.c. |
Married |
Mother tongue other than German, French, Italian |
Past income in CHF |
Previous job: Manager |
Previous job in missing sector |
Previous job in primary sector |
Previous job in secondary sector |
Previous job in tertiary sector |
Previous job: self-employed |
Previous job: skilled worker |
Previous job: unskilled worker |
Qualification: semiskilled |
Qualification: some degree |
Qualification: unskilled |
Qualification: skilled without degree |
Swiss citizenship |
Number of unemployment spells in last 2 years |
Cantonal unemployment rate in % |
Table 3: List of features used for the estimation of risk scores. All caseworker information is omitted from this estimation. See Lechner et al. (2020) for detailed information about the administrative data. | Reference | Ridge Regression | Statistical Parity | Equality of Opportunity
---|---|---|---|---
Accuracy | (1) | 0.644 | 0.644 | 0.645
Precision | (1) | 0.612 | 0.605 | 0.607
Recall | (1) | 0.384 | 0.404 | 0.404
Stat Parity | (0) | 0.116 | 0.041 | 0.019
Equal Opportunity | (0) | 0.173 | 0.07 | 0.044
False Positive Parity | (0) | 0.062 | 0.005 | -0.014
Positive Predictive Parity | (0) | 0.062 | 0.072 | 0.081
Negative Predictive Parity | (0) | 0.011 | 0.011 | 0.016
Table 4: Results for predicting long-term unemployment. To achieve predictions
of the binary target variable, a threshold of $0.5$ is applied to the risk
scores.

(a) No fairness constraint.

(b) Statistical parity constraint.

(c) Equal opportunity constraint.

Figure 7: Risk scores estimated by logistic ridge regression, with and without
fairness constraints. The vertical line at $.5$ gives the decision threshold
for binary predictions.

###  B.5 Results from the Simulation Study

| LTU | Women | Men | Gender gap | Non-Citizens | Citizen | Citizen Gap
---|---|---|---|---|---|---|---
Status quo | 0.414 | 0.436 | 0.397 | 0.039 | 0.515 | 0.357 | 0.158
Belgian, optimal |  |  |  |  |  |  |
Logistic Regression | 0.386 | 0.404 | 0.372 | 0.032 | 0.446 | 0.351 | 0.095
Stat. Parity | 0.386 | 0.408 | 0.368 | 0.039 | 0.448 | 0.35 | 0.097
Equal Opp. | 0.386 | 0.409 | 0.368 | 0.041 | 0.448 | 0.35 | 0.097
Belgian, random |  |  |  |  |  |  |
Logistic Regression | 0.4 | 0.421 | 0.385 | 0.036 | 0.473 | 0.359 | 0.114
Stat. Parity | 0.400 | 0.422 | 0.383 | 0.039 | 0.473 | 0.359 | 0.114
Equal Opp. | 0.400 | 0.423 | 0.383 | 0.04 | 0.474 | 0.359 | 0.115
Austrian, optimal |  |  |  |  |  |  |
Logistic Regression | 0.386 | 0.405 | 0.371 | 0.034 | 0.447 | 0.351 | 0.097
Stat. Parity | 0.386 | 0.408 | 0.369 | 0.038 | 0.451 | 0.349 | 0.101
Equal Opp. | 0.386 | 0.408 | 0.369 | 0.039 | 0.451 | 0.349 | 0.101
Austrian, random |  |  |  |  |  |  |
Logistic Regression | 0.402 | 0.423 | 0.385 | 0.037 | 0.476 | 0.359 | 0.117
Stat. Parity | 0.402 | 0.424 | 0.385 | 0.04 | 0.479 | 0.358 | 0.120
Equal Opp. | 0.402 | 0.425 | 0.385 | 0.04 | 0.479 | 0.359 | 0.120
Table 5: Rates of in long-term unemployment for the different algorithmically informed policies under baseline capacities. | LTU | Women | Men | Gender Gap | Citizens | Non-Citizen | Citizen Gap
---|---|---|---|---|---|---|---
Status quo | 0.414 | 0.436 | 0.397 | 0.039 | 0.515 | 0.357 | 0.158
Belgian, optimal |  |  |  |  |  |  |
Logistic Regression | 0.346 | 0.351 | 0.342 | 0.009 | 0.375 | 0.329 | 0.046
Stat. Parity | 0.345 | 0.36 | 0.333 | 0.026 | 0.378 | 0.326 | 0.051
Equal Opp. | 0.345 | 0.362 | 0.332 | 0.03 | 0.377 | 0.326 | 0.051
Belgian, random |  |  |  |  |  |  |
Logistic Regression | 0.383 | 0.395 | 0.373 | 0.022 | 0.44 | 0.350 | 0.09
Stat. Parity | 0.383 | 0.399 | 0.370 | 0.029 | 0.441 | 0.349 | 0.092
Equal Opp. | 0.383 | 0.400 | 0.369 | 0.031 | 0.442 | 0.349 | 0.092
Austrian, optimal |  |  |  |  |  |  |
Logistic Regression | 0.346 | 0.353 | 0.341 | 0.012 | 0.380 | 0.327 | 0.053
Stat. Parity | 0.346 | 0.361 | 0.334 | 0.026 | 0.388 | 0.322 | 0.066
Equal Opp. | 0.346 | 0.362 | 0.333 | 0.029 | 0.387 | 0.322 | 0.065
Austrian, random |  |  |  |  |  |  |
Logistic Regression | 0.383 | 0.397 | 0.373 | 0.024 | 0.444 | 0.349 | 0.095
Stat. Parity | 0.384 | 0.401 | 0.371 | 0.030 | 0.449 | 0.347 | 0.102
Equal Opp. | 0.384 | 0.402 | 0.370 | 0.032 | 0.449 | 0.347 | 0.102
Table 6: Rates in long-term unemployment for the different algorithmically
informed policies under five-fold capacities.

###  B.6 Citizen LTU Gap

(a) Belgian Prioritization and Optimal Program

(b) Belgian Prioritization and Random Program

(c) Austrian Prioritization and Optimal Program

(d) Austrian Prioritization and Random Program

Figure 8: We plot the citizen gap in long-term unemployment (LTU) against
program capacity for each combination of prioritization and assignment scheme.
The level of transparency shows the citizen gap for the corresponding fairness
constraint: none, statistical parity, or equal opportunity. All policy
combinations reduce the citizen gap. The unconstrained risk scores (lowest
transparency) result in the smallest citizen gap. This effect is especially
pronounced as program capacity is increased and program assignments are
individualized (optimal). Austrian prioritization compared to the Belgian
approach performs particularly poorly under fairness constraints for gender.

(a) Optimal Program

(b) Random Program

Figure 9: We plot overall long-term unemployment and the citizen reemployment
gap against program capacity for each combination of prioritization and
assignment scheme. For clarity, results are shown only for fairness-
unconstrained risk scores. Regardless of the assignment scheme, the Belgian
prioritization (blue line) results in the same long-term unemployment rate as
the Austrian and a slightly smaller citizen gap. Individualized program
assignments (optimal) are markedly more effective, especially under larger
program capacities.

###  B.7 Gender LTU Gaps for (un)married (non-)citizen

(a) Unmarried Non-Citizen

(b) Married Non-Citizen

(c) Unmarried Swiss Citizen

(d) Married Swiss Citizen

Figure 10: Long-term unemployment rates among the respective group (red and
blue line) and by gender for four sub-groups: unmarried non-citizen, unmarried
Swiss citizen, married non-citizen, and married Swiss citizen. Note the
different scales. The reduction in LTU rates and the gender gap is especially
pronounced for the group of married foreigners. For unmarried foreigners, the
gender gap even flips under both algorithmic policies at four- and five-fold
program capacities.