# Fairness-Accuracy Trade-Offs: A Causal Perspective

Drago Plecko and Elias Bareinboim
Department of Computer Science
Columbia University
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

With the widespread adoption of AI systems, many of the decisions once made by
humans are now delegated to automated systems. Recent works in the literature
demonstrate that these automated systems, when used in socially sensitive
domains, may exhibit discriminatory behavior based on sensitive
characteristics such as gender, sex, religion, or race. In light of this,
various notions of fairness and methods to quantify discrimination have been
proposed, also leading to the development of numerous approaches for
constructing fair predictors. At the same time, imposing fairness constraints
may decrease the utility of the decision-maker, highlighting a tension between
fairness and utility. This tension is also recognized in legal frameworks, for
instance in the disparate impact doctrine of Title VII of the Civil Rights Act
of 1964 – in which specific attention is given to considerations of business
necessity – possibly allowing the usage of proxy variables associated with the
sensitive attribute in case a high-enough utility cannot be achieved without
them. In this work, we analyze the tension between fairness and accuracy from
a causal lens for the first time. We introduce the notion of a path-specific
excess loss (PSEL) that captures how much the predictor’s loss increases when
a causal fairness constraint is enforced. We then show that the total excess
loss (TEL), defined as the difference between the loss of predictor fair along
all causal pathways vs. an unconstrained predictor, can be decomposed into a
sum of more local PSELs. At the same time, enforcing a causal constraint often
reduces the disparity between demographic groups. Thus, we introduce a
quantity that summarizes the fairness-utility trade-off, called the causal
fairness/utility ratio, defined as the ratio of the reduction in
discrimination vs. the excess in the loss from constraining a causal pathway.
This quantity is particularly suitable for comparing the fairness-utility
trade-off across different causal pathways. Finally, as our approach requires
causally-constrained fair predictors, we introduce a new neural approach for
causally-constrained fair learning. Our approach is evaluated across multiple
real-world datasets, providing new insights into the tension between fairness
and accuracy.

##  1 Introduction

Automated decision-making systems based on machine learning and artificial
intelligence are now commonly implemented in various critical sectors of
society such as hiring, university admissions, law enforcement, credit
assessments, and health care (Khandani et al., 2010; Mahoney and Mohen, 2007;
Brennan et al., 2009). These technologies now significantly influence the
lives of individuals and are frequently used in high-stakes settings (Topol,
2019; Berk et al., 2021; Taddeo and Floridi, 2018). As these systems replace
or augment human decision-making processes, concerns about fairness and bias
based on protected attributes such as race, gender, or religion have become a
prominent consideration in the ML literature. The available data used to train
automated systems may contain past and present societal biases as an imprint
and therefore has the potential to perpetuate or even exacerbate
discrimination against protected groups. This is highlighted by reports on
biases in systems for sentencing Angwin et al. (2016), facial recognition
Buolamwini and Gebru (2018), online ads Sweeney (2013); Datta et al. (2015),
and system authentication Sanburn (2015), among many others. Despite the
promise of AI to enhance human decision-making, the reality is that these
technologies can also reflect or worsen societal inequalities. As alluded to
before, the issue does not arise uniquely from the usage of automated systems;
human-driven decision-making has long been analyzed in a similar fashion.
Evidence of bias in human decision-making is abundant, including studies on
the gender wage gap (Blau and Kahn, 1992, 2017) and racial disparities in
legal outcomes (Sweeney and Haney, 1992; Pager, 2003). Therefore, without
proper care about fairness and transparency of the new generation of AI
systems, it is unclear what its impact will be on the historically
discriminated groups, and issues of inequity more broadly.

Within the growing literature on fair machine learning, a plethora of fairness
definitions have been proposed. Commonly considered statistical criteria,
among others, include demographic parity (independence (Darlington, 1971)),
equalized odds (separation (Hardt et al., 2016)), and calibration (sufficiency
(Chouldechova, 2017)). These definitions, however, have been shown as mutually
incompatible (Barocas and Selbst, 2016; Kleinberg et al., 2016). Despite a
number of proposals, there is still a lack of consensus on what the
appropriate measures of fairness are, and how statistical notions of fairness
could incorporate moral values of the society at large. For this reason, a
number of works explored the causal approaches to fair machine learning
(Kusner et al., 2017; Kilbertus et al., 2017; Nabi and Shpitser, 2018; Zhang
and Bareinboim, 2018b, a; Wu et al., 2019; Chiappa, 2019; Plečko and
Meinshausen, 2020), and an in-depth discussion can be found in (Plečko and
Bareinboim, 2024). The main motivation for doing so is that the causal
approach may allow the system designers to attribute the observed disparities
between demographic groups to the causal mechanisms that underlie and generate
them in the first place. In this way, by isolating disparities transmitted
along different causal pathways, one obtains a more fine-grained analysis, and
the capability to decide which causal pathways are deemed as unfair or
discriminatory. More fundamentally, such considerations also form the basis of
the legal frameworks for assessing discrimination in the United States and
Europe. For instance, in the context of employment law, the disparate impact
doctrine within the Title VII of the Civil Rights Act of 1964 (Act, 1964)
disallows any form of discrimination that results in a too large of a
disparity between groups of interest. A core aspect of this doctrine, however,
is the notion of business necessity (BN) or job-relatedness. Considerations of
business necessity may allow variables correlated with the protected attribute
to act as a proxy, and the law does not necessarily prohibit their usage due
to their relevance to the business itself (or more broadly the utility of the
decision-maker). Often, the wording that is used is that to argue business
necessity in front of a court of law, the plaintiff needs to demonstrate that
“there is no practice that is less discriminatory and achieves the same
utility” (Els, 1993). This concept illustrates the tension between fairness
and utility, and demonstrates that we cannot be oblivious to considerations of
utility from a legal standpoint.

The question of fairness-utility trade-offs has been explored in the fairness
literature (Corbett-Davies et al., 2017), with the essential argument that an
unconstrained predictor always achieves a greater or equal utility than a
constrained one. Despite this, the literature seems to be divided on this
issue. For instance, some works argue that fairness and utility trade-offs are
negligible in practice (Rodolfa et al., 2021), while others argue that such
trade-offs need not even exist (Maity et al., 2020; Dutta et al., 2020). In
this paper, we directly refute such arguments and demonstrate that, from a
causal viewpoint, fairness and utility are always in a trade-off. The key
subtlety in the works that find no trade-off (Rodolfa et al., 2021; Maity et
al., 2020; Dutta et al., 2020) are the fairness metrics used to impose
constraints. Naturally, the implications on the predictor’s utility will
strongly depend on the exact type of the fairness constraint that is
enforced111Works that discuss how fairness-accuracy trade-offs do not exist or
are insignificant in practice often focus on equality of odds (Hardt et al.,
2016) ($\widehat{Y}{\perp\\!\\!\\!\perp}X\mid Y$). Notably, this metric always
allows for the perfect predictor $\widehat{Y}=Y$, and thus in settings with
good predictive power, the cost of enforcing this constraint may indeed be
negligible., and we demonstrate that imposing causal fairness constraints
along different causal pathways almost always reduces predictive power.

Interestingly, the tension between fairness and utility has been largely
unexplored in the causal fairness literature (with some exceptions such as
(Nilforoshan et al., 2022; Plecko and Bareinboim, 2024a; Plečko and
Bareinboim, 2024)). Our main aim of this paper is to fill in this gap, and
provide a systematic way of analyzing the fairness-accuracy trade-off from a
causal lens. We illustrate our approach in a simple linear setting:

######  Example 1 (Linear Fairness-Accuracy Causal Trade-Offs).

Consider variables $X,W,Y$ behaving according to the following system of
equations:

$\displaystyle X$ $\displaystyle\leftarrow\text{Bernoulli}(0.5)$ (1)
$\displaystyle W$ $\displaystyle\leftarrow\beta X+\epsilon_{w}$ (2)
$\displaystyle Y$ $\displaystyle\leftarrow\alpha X+\gamma W+\epsilon_{y},$ (3)

where $\epsilon_{w}\sim N(0,\sigma_{w}^{2}),\epsilon_{y}\sim
N(0,\sigma_{y}^{2})$. Variable $X$ is the protected attribute, and $Y$ is the
outcome of interest. The causal diagram of Eqs. 1 -3 is shown in Fig. 3 (with
the $Z$ set empty). Attribute $X$ can influence $Y$ along two different
pathways: the direct path $X\to Y$, and the indirect path $X\to W\to Y$.
Therefore, we consider fair predictors $\widehat{Y}^{S}$ of the form
$\widehat{Y}^{S}=\hat{\alpha}_{S}X+\hat{\gamma}_{S}W,$ where the predictor
$\widehat{Y}^{S}$ removes effects in the set $S$, with $S$ ranging in
$\\{\emptyset,\text{DE},\text{IE},\\{\text{DE, IE}\\}\\}$ (DE, IE stand for
direct and indirect effects). For instance, the optimal predictor
$\widehat{Y}^{\emptyset}$ has
$\hat{\alpha}_{\emptyset}=\alpha,\hat{\gamma}_{\emptyset}=\gamma$, and
therefore its mean-squared error (MSE) equals
$\mathbbm{E}[Y-\widehat{Y}^{\emptyset}]^{2}=\sigma_{y}^{2}$. The DE-fair
predictor has $\hat{\alpha}_{\text{DE}}=0,\hat{\gamma}_{\text{DE}}=\gamma$,
and the fully-fair predictor $\widehat{Y}^{\\{\text{DE, IE}\\}}$ has
$\hat{\alpha}_{\\{\text{DE, IE}\\}}=0,\hat{\gamma}_{\\{\text{DE, IE}\\}}=0$.
Thus, the corresponding MSE values can be written as (see Appendix A for
details):

$\displaystyle\mathbbm{E}[Y-\widehat{Y}^{\text{DE}}]^{2}=\sigma_{y}^{2}+\frac{\alpha^{2}}{2},\quad\mathbbm{E}[Y-\widehat{Y}^{\\{\text{DE,
IE}\\}}]^{2}$
$\displaystyle=\sigma_{y}^{2}+\frac{\alpha^{2}+\gamma^{2}\beta^{2}+\alpha\gamma\beta}{2}+\gamma^{2}\sigma_{w}^{2}.$
(4)

Our goal is to decompose the total excess loss originating from imposing the
fairness constraints into the path-specific contributions:

$\displaystyle\underbrace{\mathbbm{E}[Y-\widehat{Y}^{\\{\text{DE,
IE}\\}}]^{2}}_{\text{fully-fair predictor's
loss}}-\underbrace{\mathbbm{E}[Y-\widehat{Y}^{\emptyset}]^{2}}_{\text{unconstrained
loss}}$ $\displaystyle=\underbrace{\frac{\alpha^{2}}{2}}_{\text{Term
I}}+\underbrace{\frac{\gamma^{2}\beta^{2}+\alpha\gamma\beta}{2}+\gamma^{2}\sigma_{w}^{2}}_{\text{Term
II}}$ (5)
$\displaystyle=\underbrace{\mathbbm{E}[Y-\widehat{Y}^{\text{DE}}]^{2}-\mathbbm{E}[Y-\widehat{Y}^{\emptyset}]^{2}}_{\text{Term
I = excess DE loss}}$ (6)
$\displaystyle+\underbrace{\mathbbm{E}[Y-\widehat{Y}^{\\{\text{DE,
IE}\\}}]^{2}-\mathbbm{E}[Y-\widehat{Y}^{\text{DE}}]^{2}}_{\text{Term II =
excess IE loss}}.$ (7)

At the same time, for each pathway, we can also observe the decrease in group
disparity associated with removing the effect along the path $S^{\prime}$, by
computing

$\displaystyle\underbrace{(\mathbbm{E}[\widehat{Y}^{\\{\text{DE, IE}\\}}\mid
x_{1}]-\mathbbm{E}[\widehat{Y}^{\\{\text{DE, IE}\\}}\mid
x_{0}])}_{\text{disparity after removing DE,
IE}}-\underbrace{(\mathbbm{E}[Y^{\emptyset}\mid
x_{0}]-\mathbbm{E}[Y^{\emptyset}\mid x_{0}])}_{\text{disparity before removing
DE, IE}}=-\alpha-\beta\gamma,$ (8)

which again decomposes into the contributions along direct and indirect
effects:

$\displaystyle(\mathbbm{E}[\widehat{Y}^{\text{DE}}\mid
x_{0}]-\mathbbm{E}[\widehat{Y}^{\text{DE}}\mid
x_{0}])-(\mathbbm{E}[\widehat{Y}^{\emptyset}\mid
x_{0}]-\mathbbm{E}[\widehat{Y}^{\emptyset}\mid x_{0}])$
$\displaystyle=-\alpha$ (9)
$\displaystyle(\mathbbm{E}[\widehat{Y}^{\\{\text{DE, IE}\\}}\mid
x_{1}]-\mathbbm{E}[\widehat{Y}^{\\{\text{DE, IE}\\}}\mid
x_{0}])-(\mathbbm{E}[\widehat{Y}^{\text{DE}}\mid
x_{0}]-\mathbbm{E}[\widehat{Y}^{\text{DE}}\mid x_{0}])$
$\displaystyle=-\beta\gamma.$ (10)

Prototypical fairness/utility trade-offs for predictors $\widehat{Y}^{S}$ are
visualized in Fig. 1, for three randomly drawn triples
$(\alpha,\beta,\gamma)$. Predictor $\widehat{Y}^{\emptyset}$ is optimal and
thus always has $0$ excess loss (and lies on the vertical axis). Fully-fair
predictor $\widehat{Y}^{\\{\text{DE, IE}\\}}$ removes both direct and indirect
effects, and thus always has $0$ TV measure (and lies on the vertical axis).
The slopes in the plot between $\widehat{Y}^{\emptyset}$,
$\widehat{Y}^{\text{DE}}$, and $\widehat{Y}^{\text{DE, IE}}$ (indicated by
arrows), geometrically capture the tension between excess loss and reducing
discrimination upon imposing a constraint. These slopes are computed as
$\frac{\text{TV difference}}{\text{Excess Loss}}$, a quantity that we call the
causal fairness-utility ratio (CFUR). Based on Eqs. 6-7 and 9-10 we can
compute that

$\displaystyle\text{CFUR}(\text{DE})=-\frac{1}{\alpha},\quad\text{CFUR}(\text{IE})=-\frac{2\beta\gamma}{\gamma^{2}\beta^{2}+\alpha\gamma\beta+2\gamma^{2}\sigma_{w}^{2}},$
(11)

summarizing the fairness-utility trade-off for each pathway.

Figure 1: TV vs. Excess Loss.

The above example illustrates how in the simple linear case we can attribute
the increased loss from imposing fairness constraints to the specific causal
pathway in question. It also shows we can compute the associated change in the
disparity between groups, measured by the so-called TV measure
$\mathbbm{E}[\widehat{Y}\mid x_{1}]-\mathbbm{E}[\widehat{Y}\mid x_{0}]$ (also
known as the parity gap), a result known under the rubric of causal
decompositions (Zhang and Bareinboim, 2018b; Plečko and Bareinboim, 2024)
(this connection is explored explicitly in Appendix C). With the above example
in mind, we list below the key contributions of this manuscript:

  1. (i)

We introduce the notion of a path-specific excess loss (PSEL) associated with
imposing a fairness constraint along a causal path (Def. 3), and we prove how
the total excess loss (TEL) can be decomposed into a sum of path-specific
excess losses (Thm. 1),

  2. (ii)

We develop an algorithm for attributing path-specific excess losses to
different causal paths (Alg. 1), allowing the system designer to explain how
the total excess loss is affected by different fairness constraints. In this
context, we show the equivalence of Alg. 1 with a Shapley value (Shapley et
al., 1953) approach (Prop 3),

  3. (iii)

For purposes of applying Alg. 1, a key requirement is the construction of
causally-fair predictors $\widehat{Y}^{S}$ that remove effects along pathways
in $S$. We introduce a novel Lagrangian formulation of the optimization
problem for such $\widehat{Y}^{S}$ (Def. 5) accompanied with a training
procedure for learning the predictor (Alg. 2),

  4. (iv)

We introduce the causal fairness/utility ratio (CFUR, Def. 4) that summarizes
how much the group disparity can be reduced per fixed cost in terms of excess
loss. We compute CFURs on a range of real-world datasets, and demonstrate that
from a causal viewpoint fairness and utility are almost always in tension.

###  1.1 Preliminaries

We use the language of structural causal models (SCMs) (Pearl, 2000). An SCM
is a tuple $\mathcal{M}:=\langle V,U,\mathcal{F},P(u)\rangle$ , where $V$, $U$
are sets of endogenous (observable) and exogenous (latent) variables,
respectively, $\mathcal{F}$ is a set of functions $f_{V_{i}}$, one for each
$V_{i}\in V$, where $V_{i}\leftarrow f_{V_{i}}(\mathrm{pa}(V_{i}),U_{V_{i}})$
for some $\mathrm{pa}(V_{i})\subseteq V$ and $U_{V_{i}}\subseteq U$. $P(u)$ is
a strictly positive probability measure over $U$. Each SCM $\mathcal{M}$ is
associated to a causal diagram $\mathcal{G}$ (Bareinboim et al., 2022) over
the node set $V$ where $V_{i}\rightarrow V_{j}$ if $V_{i}$ is an argument of
$f_{V_{j}}$, and $V_{i}\dashleftarrow\\!\\!\\!\\!\\!\\!\dashrightarrow V_{j}$
if the corresponding $U_{V_{i}},U_{V_{j}}$ are not independent. An
instantiation of the exogenous variables $U=u$ is called a unit. By $Y_{x}(u)$
we denote the potential response of $Y$ when setting $X=x$ for the unit $u$,
which is the solution for $Y(u)$ to the set of equations obtained by
evaluating the unit $u$ in the submodel $\mathcal{M}_{x}$, in which all
equations in $\mathcal{F}$ associated with $X$ are replaced by $X=x$.
Throughout the paper, we assume a specific cluster causal diagram
$\mathcal{G}_{\text{SFM}}$ known as the standard fairness model (SFM) (Plečko
and Bareinboim, 2024) over endogenous variables $\\{X,Z,W,Y,\widehat{Y}\\}$
shown in Fig. 3. The SFM consists of the following: protected attribute,
labeled $X$ (e.g., gender, race, religion), assumed to be binary; the set of
confounding variables $Z$, which are not causally influenced by the attribute
$X$ (e.g., demographic information, zip code); the set of mediator variables
$W$ that are possibly causally influenced by the attribute (e.g., educational
level or other job-related information); the outcome variable $Y$ (e.g., GPA,
salary); the predictor of the outcome $\widehat{Y}$ (e.g., predicted GPA,
predicted salary). The SFM encodes the assumptions typically used in the
causal inference literature about the lack of hidden confounding. The
availability of the SFM and the implied assumptions are a possible limitation
of the paper, while we note that partial identification techniques for
bounding effects can be used for relaxing them (Zhang et al., 2022).

$Z$$X$$W$$Y$$\widehat{Y}$

Figure 2: Standard Fairness Model.

$\emptyset$$\\{I\\}$$\\{D\\}$$\\{S\\}$$\\{D,I\\}$$\\{I,S\\}$$\\{D,S\\}$$\\{D,I,S\\}$

Figure 3: Graphical representation $\mathcal{G}_{\text{PSEL}}$.

Based on the SFM, we will use the following causal fairness measures:

######  Definition 1 (Population-level Causal Fairness Measures (Pearl, 2001;
Plečko and Bareinboim, 2024)).

The natural direct, indirect, and spurious effects are defined as

$\displaystyle\text{NDE}_{x_{0},x_{1}}(y)$
$\displaystyle=P(y_{x_{1},W_{x_{0}}})-P(y_{x_{0}})$ (12)
$\displaystyle\text{NIE}_{x_{1},x_{0}}(y)$
$\displaystyle=P(y_{x_{1},W_{x_{0}}})-P(y_{x_{1}})$ (13)
$\displaystyle\text{NSE}_{x}(y)$ $\displaystyle=P(y\mid x)-P(y_{x}).$ (14)

A causally-fair predictor for a subset $S$ of the above measures is defined
as:

######  Definition 2 (Causally Fair Predictor (Plecko and Bareinboim, 2024b)).

The causally $S$-fair predictor $\widehat{Y}^{S}$ with respect to a loss
function $L$ and pathways in $S$ is the solution to the following optimization
problem:

$\displaystyle\widehat{Y}^{S}=$ $\displaystyle\operatorname*{arg\,min}_{f}$
$\displaystyle\qquad\mathbbm{E}\;$ $\displaystyle L(Y,f(X,Z,W))$ (15) subject
to $\displaystyle\;\;\;\text{NDE}_{x_{0},x_{1}}(f)$
$\displaystyle=\text{NDE}_{x_{0},x_{1}}(y)\cdot\mathbb{1}(\text{DE}\notin S)$
(16) $\displaystyle\text{NIE}_{x_{1},x_{0}}(f)$
$\displaystyle=\text{NIE}_{x_{1},x_{0}}(y)\cdot\mathbb{1}(\text{IE}\notin S)$
(17) $\displaystyle\text{NSE}_{x_{0}}(f)$
$\displaystyle=\text{NSE}_{x_{0}}(y)\cdot\mathbb{1}(\text{SE}\notin S)$ (18)
$\displaystyle\text{NSE}_{x_{1}}(f)$
$\displaystyle=\text{NSE}_{x_{1}}(y)\cdot\mathbb{1}(\text{SE}\notin S).$ (19)

The definition of $\widehat{Y}^{S}$ has a straightforward interpretation. For
any pathway in the set $S$, the corresponding causal effect should be $0$, as
proposed in the path-specific causal fairness literature (Nabi and Shpitser,
2018; Chiappa, 2019). However, importantly, pathways that are not in $S$ also
need to be constrained – the effect of $X$ on $\widehat{Y}$ along these paths
should not change compared to the true outcome $Y$ (Plecko and Bareinboim,
2024b). For instance, if the direct path is not in $S$, then we expect to have
$\text{NDE}_{x_{0},x_{1}}(\widehat{y})=\text{NDE}_{x_{0},x_{1}}(y)$ (and
similarly for IE, SE).

##  2 Path-Specific Excess Loss

In this section, we introduce the concept of a path-specific excess loss, and
then demonstrate how the total excess loss can be decomposed into path-
specific excess losses.

######  Definition 3 (Path-Specific Excess Loss).

Let $L(\widehat{Y},Y)$ be a loss function and $\widehat{Y}^{S}$ the optimal
$S$-fair predictor with respect to $L$. Define the path-specific excess loss
(PSEL) of a pair $S,S^{\prime}$ as:

$\displaystyle\text{PSEL}(S\to
S^{\prime})=\mathbbm{E}[L(\widehat{Y}^{S^{\prime}},Y)]-\mathbbm{E}[L(\widehat{Y}^{S},Y)].$
(20)

The quantity $\text{PSEL}(\emptyset\to\\{D,I,S\\})$ is called the total excess
loss (TEL).

The total excess loss computes the increase in the loss for the totally
constrained predictor $\widehat{Y}^{\\{D,I,S\\}}$ with direct, indirect, and
spurious effects removed compared to the unconstrained predictor
$\widehat{Y}^{\emptyset}$. 222Generally, other classifiers that remove only
subsets of the causal paths between $X$ and $Y$ may be considered fair,
depending on considerations of business necessity (Plečko and Bareinboim,
2024). Still, the rationale developed in this paper can be easily adapted to
such settings. The total excess loss can be decomposed as a sum of path-
specific excess losses (all proofs are provided in Appendix B):

######  Theorem 1 (Total Excess Loss Decomposition).

The total excess loss $\text{PSEL}(\emptyset\to\\{D,I,S\\})$ can be decomposed
into a sum of path-specific excess losses as follows:

$\displaystyle\text{PSEL}(\emptyset\to\\{D,I,S\\})$
$\displaystyle=\text{PSEL}(\emptyset\to\\{D\\})+\text{PSEL}(\\{D\\}\to\\{D,I\\})$
(21) $\displaystyle\quad+\text{PSEL}(\\{D,I\\}\to\\{D,I,S\\}).$ (22)

######  Remark 2 (Non-Uniquness of Decomposition).

The decomposition in Thm. 1 is not unique. In particular, the
$\text{PSEL}(\emptyset\to\\{D,I,S\\})$ can be decomposed as

$\displaystyle\text{PSEL}(\emptyset\to\\{S_{1}\\})+\text{PSEL}(\\{S_{1}\\}\to\\{S_{1},S_{2}\\})+\text{PSEL}(\\{S_{1},S_{2}\\}\to\\{D,I,S\\})$
(23)

for any choice of $S_{1},S_{2}\in\\{D,I,S\\}$ with $S_{1}\neq S_{2}$.
Therefore, six different decompositions exist (three choices for $S_{1}$, two
for $S_{2}$).

Fig. 3 provides a graphical overview of all the possible path-specific excess
loses. In the left side, we start with $S=\emptyset$ and the predictor
$\widehat{Y}^{\emptyset}$. Then, we can add any of $\\{D,I,S\\}$ to the
$S$-set, to obtain the predictors $\widehat{Y}^{D},\widehat{Y}^{I}$, or
$\widehat{Y}^{S}$, and so on. The graph representing all the possible states
$\widehat{Y}^{S}$ and transitions between pairs
$(\widehat{Y}^{S},\widehat{Y}^{S^{\prime}})$ shown in Fig. 3 is labeled
$\mathcal{G}_{\text{PSEL}}$. There are six paths starting from $\emptyset$ and
ending in $\\{D,I,S\\}$. In Alg. 1, we introduce a procedure that sweeps over
all the edges and paths in $\mathcal{G}_{\text{PSEL}}$ to compute path-
specific excess losses, while also computing the change in the TV measure
between groups in order to track the reduction in discrimination.

Input: data $\mathcal{D}$, predictors $\widehat{Y}^{S}$ for $S$-sets
$\subseteq\\{D,I,S\\}$

1 foreach  _edge $(S,S^{\prime})\in\mathcal{G}_{\text{PSEL}}$_ do

2 compute the path-specific excess loss of $S^{\prime}\setminus S$, given by
$\text{PSEL}(S\to S^{\prime})$

3 compute the TV measure difference of $S^{\prime}\setminus S$, written
$\text{TVD}(S\to S^{\prime})$ given by
$\text{TV}_{x_{0},x_{1}}(\widehat{Y}^{S^{\prime}})-\text{TV}_{x_{0},x_{1}}(\widehat{Y}^{S})$

4

5foreach  _causal path $S_{i}$ $\in\\{D,I,S\\}$_ do

6 compute the average path-specific excess loss and TV difference across all
paths $\emptyset\to\dots\to\\{D,I,S\\}$,  $\displaystyle\text{APSEL}(S_{i})$
$\displaystyle=\frac{1}{3!}\sum_{\begin{subarray}{c}\text{paths
}\pi\in\mathcal{G}_{\text{PSEL}}:\\\ \emptyset\text{ to
}\\{D,I,S\\}\end{subarray}}\text{PSEL}(\pi^{<S_{i}}\to\pi^{<S_{i}}\cup S_{i})$
(24) $\displaystyle\text{ATVD}(S_{i})$
$\displaystyle=\frac{1}{3!}\sum_{\begin{subarray}{c}\text{paths
}\pi\in\mathcal{G}_{\text{PSEL}}:\\\ \emptyset\text{ to
}\\{D,I,S\\}\end{subarray}}\text{TVD}(\pi^{<S_{i}}\to\pi^{<S_{i}}\cup S_{i}),$
(25) where $\pi^{<S_{i}}$ denotes the set of causal paths that precede $S_{i}$
in $\pi$.

return _set of $\text{PSEL}(S\to S^{\prime})$, $\text{TVD}(S\to S^{\prime})$,
attributions APSEL($S_{i}$), ATVD($S_{i}$)_

Algorithm 1 Path-Specific Excess Loss Attributions

Formally, for any edge $(S,S^{\prime})$ in $\mathcal{G}_{\text{PSEL}}$ the
value of $\text{PSEL}(S\to S^{\prime})$ is computed, together with the
difference in the TV measure (TVD) from the transition $S\to S^{\prime}$,
defined by

$\displaystyle\text{TVD}(S\to
S^{\prime})=\underbrace{\mathbbm{E}[Y^{S^{\prime}}\mid
x_{1}]-\mathbbm{E}[Y^{S^{\prime}}\mid x_{0}]}_{\text{TV after removing
}S^{\prime}\setminus S}-\underbrace{\mathbbm{E}[Y^{S}\mid
x_{1}]-\mathbbm{E}[Y^{S}\mid x_{0}]}_{\text{TV before removing
}S^{\prime}\setminus S}.$ (26)

The quantities $\text{PSEL}(S\to S^{\prime})$ and $\text{TVD}(S\to
S^{\prime})$ are naturally associated with the effect that was removed, i.e.,
$S^{\prime}\setminus S$. As there are multiple ways of reaching the set
$\\{D,I,S\\}$ from $\emptyset$ in $\mathcal{G}_{\text{PSEL}}$, each of the
corresponding effects (direct, indirect, spurious) will be associated with a
number of different PSELs and TVDs (generally, note that the complexity is
exponential in the number of causal paths included). In Eqs. 24-25 inside the
algorithm, we compute the average PSEL and TVD across all the edges that are
associated with a specific effect $S_{i}$. This simple intuition,
corresponding to taking an average across all of the possible decompositions
of the total excess loss (Eq. 23), turns out to be equivalent to a Shapley
value (Shapley et al., 1953) of a suitably chosen value function:

######  Proposition 3 (PSEL Attribution as Shapley Values).

Let the functions $f_{1}(S),f_{2}(S)$ be defined as:

$\displaystyle f_{1}(S)$ $\displaystyle=\text{PSEL}(\emptyset\to
S),f_{2}(S)=\text{TVD}(\emptyset\to S).$ (27)

The Shapley value for the effect $S_{i}\in\\{D,I,S\\}$ and function $f_{k}$,
is computed as

$\displaystyle\phi^{k}(S_{i})=\sum_{S\subseteq\\{D,I,S\\}\setminus\\{S_{i}\\}}\frac{1}{n\binom{n-1}{|S|}}\left(f_{k}(S\cup\\{S_{i}\\})-f_{k}(S)\right).$
(28)

where $n=3$ for the choice $\\{D,I,S\\}$. The averaged path-specific excess
loss of $S_{i}$ and the averaged TV difference of $S_{i}$ are equal to the
Shapley values of $S_{i}$ associated with functions $f_{1},f_{2}$,
respectively,

$\displaystyle\phi^{1}(S_{i})$
$\displaystyle=\text{APSEL}(S_{i}),\phi^{2}(S_{i})=\text{ATVD}(S_{i}),$ (29)

with $\text{APSEL}(S_{i}),\text{ATVD}(S_{i})$ defined in Eqs. 24, 25,
respectively.

The above proposition illustrates how averaging the influence of removing a
causal effect over all possible ways of reaching $\\{D,I,S\\}$ from
$\emptyset$ is equivalent to computing the Shapley values of an appropriate
value function $f$. We next introduce the notion of a causal fairness/utility
ratios:

######  Definition 4 (Causal Fairness/Utility Ratio (CFUR)).

The causal fairness/utility ratio (CFUR) for a causal path $S_{i}$ is defined
as

$\displaystyle\text{CFUR}(S_{i})=\frac{\text{ATVD}(S_{i})}{\text{APSEL}(S_{i})}.$
(30)

The CFUR quantity may be particularly useful for comparing different causal
effects, and the connection of the CFUR with local TVD and PSEL values is
described in Appendix D. The intuition behind the quantity is simple – for
removing a causal effect $S_{i}$ from our predictor $\widehat{Y}$, we want to
compute how much of a reduction in the disparity that results in (measured in
terms of the ATVD measure) per unit change in the incurred excess loss. This
quantity attempts to assign a single number to a causal path that succinctly
summarizes how much fairness is gained vs. how much predictive power is lost
by imposing such a causal constraint.

##  3 Causally-Fair Constrained Learning

In the preceding section, we developed an approach for quantifying the tension
between fairness and accuracy from a causal viewpoint. The results were
contingent on finding the optimal causally-fair predictors $\widehat{Y}^{S}$
following Def. 2. However, computing the predictors $\widehat{Y}^{S}$ is quite
challenging in practice, due to several complex causal constraints in the
optimization problem. In this sectionm we develop a practical approach for
solving this problem. We begin by introducing a Lagrangian form of the optimal
causally-fair predictor:

Input: training data $\mathcal{D}_{t}$, evaluation data $\mathcal{D}_{e}$,
precision $\epsilon$

1 Set $\lambda_{\text{low}}=0$, $\lambda_{\text{high}}=$ large

2 while  _$|\lambda_{\text{high}}-\lambda_{\text{low}}| >\epsilon$_ do

3 set
$\lambda_{\text{mid}}=\frac{1}{2}(\lambda_{\text{low}}+\lambda_{\text{high}})$

4 fit a neural network to solves the optimization problem in Eqs. 32-34 with
$\lambda=\lambda_{\text{mid}}$ on $\mathcal{D}_{t}$ to obtain the predictor
$\widehat{Y}^{S}(\lambda_{\text{mid}})$

5 compute the causal measures of fairness NDE, NIE, NSE of
$\widehat{Y}^{S}(\lambda_{\text{mid}})$ on evaluation data $\mathcal{D}_{e}$

6 test the hypothesis  $\displaystyle
H_{0}^{CE}:\text{NCE}(\widehat{y}^{S}(\lambda_{\text{mid}}))$
$\displaystyle=\text{NCE}(y)\cdot\mathbb{1}(\text{CE}\notin S)$ (31) where NCE
ranges in
$\text{NDE}_{x_{0},x_{1}},\text{NIE}_{x_{1},x_{0}},\text{NSE}_{x_{0}},\text{NSE}_{x_{1}}$

7 if any of $H_{0}^{DE},H_{0}^{IE},H_{0}^{SE_{0}},H_{0}^{SE_{1}}$ rejected
then $\lambda_{\text{low}}=\lambda_{\text{mid}}$ else
$\lambda_{\text{high}}=\lambda_{\text{mid}}$

return _predictor $\widehat{Y}^{S}(\lambda_{\text{mid}})$_

Algorithm 2 Causally-Fair Constrained Learning (CFCL)

######  Definition 5 (Causal Lagrange Predictor $\widehat{Y}^{S}$).

The causally $S$-fair $\lambda$-optimal predictor $\widehat{Y}^{S}(\lambda)$
with respect to pathways in $S$ and the loss function $L$ is the solution to
the following optimization problem:

$\displaystyle\operatorname*{arg\,min}_{f}$
$\displaystyle\;\mathbbm{E}\;L(Y,f(X,Z,W))+\lambda\big{(}\text{NDE}_{x_{0},x_{1}}(f)-\text{NDE}_{x_{0},x_{1}}(y)\cdot\mathbb{1}(\text{DE}\notin
S)\big{)}^{2}+$ (32)
$\displaystyle\lambda\big{(}\text{NIE}_{x_{1},x_{0}}(f)-\text{NIE}_{x_{1},x_{0}}(y)\cdot\mathbb{1}(\text{IE}\notin
S)\big{)}^{2}+$ (33)
$\displaystyle\lambda\big{(}\text{NSE}_{x_{0}}(f)-\text{NSE}_{x_{0}}(y)\cdot\mathbb{1}(\text{SE}\notin
S)\big{)}^{2}+\lambda\big{(}\text{NSE}_{x_{1}}(f)-\text{NSE}_{x_{1}}(y)\cdot\mathbb{1}(\text{SE}\notin
S)\big{)}^{2}$ (34)

The above definition reformulates the problem of finding $\widehat{Y}^{S}$ to
a Lagrangian form. This makes the problem amenable to standard gradient
descent methods, and we propose a procedure for finding a suitable predictor
$\widehat{Y}^{S}$ in Alg. 2 (CFCL). CFCL performs a binary search to find an
appropriate value of the $\lambda$ parameter. For an interval
$[\lambda_{\text{low}},\lambda_{\text{high}}]$ it takes the midpoint
$\lambda_{\text{mid}}$. For this parameter value, it computes the optimal
predictor $\widehat{Y}^{S}(\lambda_{\text{mid}})$ for the optimization problem
in Eqs. 32-34 by fitting a feed-forward neural network with $n_{h}$ hidden
layers and $n_{v}$ nodes in each layer. After this, the causal measures of
fairness are computed for the predictor
$\widehat{Y}^{S}(\lambda_{\text{mid}})$ on an evaluation set, and these
measures are compared to the causal measures for the true outcome $Y$ (CFCL
performs the appropriate hypothesis tests as in Eq. 31 ). If none of the
hypotheses are rejected, it means that $\lambda_{\text{mid}}$ is large enough
to enforce the causal fairness constraints, and the algorithm moves to the
interval $[\lambda_{\text{low}},\lambda_{\text{mid}}]$. If a hypothesis is
rejected, it indicates that $\lambda_{\text{mid}}$ is not large enough, and
the algorithm moves to the interval
$[\lambda_{\text{mid}},\lambda_{\text{high}}]$. Thus, CFCL leads to a
systematic search for the parameter $\lambda$.

##  4 Experiments

In this section, we perform the causal fairness-accuracy analysis described in
Sec. 2. We analyze the Census 2018 (Ex. 2), COMPAS (Ex. 3), and UCI Credit
(Ex. 4 in Appendix E) datasets.

(a) (A)PSEL and (A)TVD values.

(b) CFUR values.

(c) Fairness/Utility of $\widehat{Y}^{S}$ predictors.

$\emptyset$$\\{I\\}$$\\{D\\}$$\\{S\\}$$\\{D,I\\}$$\\{I,S\\}$$\\{D,S\\}$$\\{D,I,S\\}$(0,
0.2)(0.06, 0.14)(0.03, -0.07)(0.06, 0.15)(0.07, 0.15)(0.02, 0.2)(0.03,
-0.06)(0.02, -0.06)(-0.01, 0.2)(0.02, -0.06)(0, 0.2)(0.07, 0.15)

(d) $\mathcal{G}_{\text{PSEL}}$ with PSEL (blue) and TVD values (red).

Figure 4: Application of Alg. 1 on the Census 2018 dataset.

######  Example 2 (Salary Increase of Government Employees (Plečko and
Bareinboim, 2024)).

The US government is building a tool for automated allocation of salaries for
new employees. For developing the tool, they use the data collected by the
United States Census Bureau in 2018, including demographic information $Z$
($Z_{1}$ for age, $Z_{2}$ for race, $Z_{3}$ for nationality), gender $X$
($x_{0}$ female, $x_{1}$ male), marital and family status $M$, education $L$,
and work-related information $R$. The government wants to predict the outcome
$Y$, the yearly salary of the employees (transformed to a log-scale). The
standard fairness model (Fig. 3) is constructed as
$\\{X=X,Z=\\{Z_{1},Z_{2},Z_{3}\\},W=\\{M,L,R\\},Y=Y\\}$.

The team developing the ML predictor is also concerned with the fairness of
the allocated salaries. In particular, they wish to understand how the
different causal effects from the protected attribute $X$ to the predictor
$\widehat{Y}$ affect the prediction, and how much the salary predictions would
have to deviate from the optimal prediction to remove an effect along a
specific pathway (in particular, they focus on the root mean squared error
(RMSE) loss). For analyzing this, they utilize the tools from Alg. 1, and
build causally fair predictors $\widehat{Y}^{S}$ (for different choices of
$S$-sets) using Alg. 2.

The analysis results are shown in Fig. 4, with uncertainty bars indicating
standard deviations over 10 bootstrap repetitions. In the analysis of PSEL and
TVD values (Fig. 4(a)), the team notices that imposing fairness constraints
does not reduce RMSE, with the largest excess loss for the indirect effect,
and smaller excess losses for direct and spurious effects. When looking at TVD
values, they find that removing direct and indirect effects reduces the group
differences substantially. In terms of causal fairness-utility ratios (Fig.
4(b)), the team finds that removing the direct effect has the best value in
terms of reducing the disparity between groups vs. increasing the loss. The TV
measure vs. excess loss dependence for different predictors $\widehat{Y}^{S}$
is shown in Fig. 4(c) (binary labels (D, I, S) in the figure indicate which
effects were removed), and the graph $\mathcal{G}_{\text{PSEL}}$ with the
values of PSEL and TVD for each transition is shown in Fig. 4(d). Based on the
analysis, the team decides to use the predictor with the direct effect removed
$\widehat{Y}^{D}$.

(a) (A)PSEL and (A)TVD values.

(b) CFUR values.

(c) Fairness/Utility of $\widehat{Y}^{S}$ predictors.

$\emptyset$$\\{I\\}$$\\{D\\}$$\\{S\\}$$\\{D,I\\}$$\\{I,S\\}$$\\{D,S\\}$$\\{D,I,S\\}$(0,
0.01)(0.1, -0.05)(0.04, -0.03)(0.15, -0.05)(0.09, -0.05)(-0.01, 0.01)(0.09,
-0.03)(0.03, -0.03)(0, 0.01)(0.06, -0.03)(-0.04, 0.01)(0.12, -0.05)

(d) $\mathcal{G}_{\text{PSEL}}$ with PSEL (blue) and TVD values (red).

Figure 5: Application of Alg. 1 on the COMPAS dataset.

######  Example 3 (Recidivism Prediction on COMPAS (Larson et al., 2016)).

Courts in Broward County, Florida use machine learning algorithms, developed
by a private company, to predict whether individuals released on parole are at
high risk of re-offending within 2 years ($Y$). The algorithm is based on the
demographic information $Z$ ($Z_{1}$ for gender, $Z_{2}$ for age), race $X$
($x_{0}$ denoting White, $x_{1}$ Non-White), juvenile offense counts $J$,
prior offense count $P$, and degree of charge $D$. The grouping
$\\{X=X,Z=\\{Z_{1},Z_{2}\\},W=\\{J,P,D\\},Y=Y\\}$ constructs the standard
fairness model.

The team from ProPublica wishes to understand the tension between fairness and
accuracy, since the latter is an important concern of the district court. For
the analysis, they use the complement of the area under the receiver operator
characteristic curve ($1-$AUROC). They apply Alg. 1, and find that the APSEL
value is the largest for the indirect effect (around 12% of AUROC), while it
is smaller for the spurious effect (6%), and negligible for the direct effect,
implying that the direct effect does not play a significant role in the
prediction (see Fig. 5(a)). Regarding TVD values, the team finds that the
removal of the spurious effect results in a 3% decrease in the TV measure,
while the removal of the indirect effect decreases the disparity by 5%. Based
on this, they find that the spurious effect is the best for reducing the
disparity between groups (see Fig. 5(b) for CFUR values). Finally, the team
also visualizes the TV and excess loss dependence (Fig. 5(c)), and the graph
$\mathcal{G}_{\text{PSEL}}$ with PSEL and TVD values associated with each
transition and effect removal (Fig. 5(a)). In the upcoming court hearing, the
team proposes the usage of the causally-constrained predictor
$\widehat{Y}^{\\{D,S\\}}$ that removes direct and spurious effects, and they
use the above analysis to demonstrate the impact of this choice on the
predictor’s accuracy.

##  5 Conclusion

The tension between fairness and accuracy is a fundamental concern in the
modern applications of machine learning. The importance of this tension is
also recognized in the legal frameworks of anti-discrimination, such as the
disparate impact doctrine, which may allow for the usage of covariates
correlated with the protected attribute if they are sufficiently important for
the decision-maker’s utility (this concept is known as business necessity). In
this work, we developed tools for analyzing the fairness-accuracy trade-off
from a causal standpoint. Our approach allows the system designer to quantify
how much excess loss is incurred when removing a path-specific causal effect
(Def. 3). We also showed how the total excess loss, defined as the difference
between the loss of the predictor fair along all causal pathways vs. an
unconstrained predictor, can be decomposed into a sum of path-specific excess
losses (Thm. 1). Based on this, we developed an algorithm for attributing
excess loss to different causal pathways (Alg. 1), and introduced the notion
of a causal fairness-utility ratio that captures the $\frac{\text{fairness
gain}}{\text{excess loss}}$ ratio and in this way summarizes the trade-off for
each causal path. Since our approach is dependent on causally-fair predictors
(Def. 2), we introduced a new neural approach for constructing such predictors
(Def. 5, Alg. 2). Finally, we demonstrated empirically that from causal
perspective fairness and accuracy are almost always in tension (Exs. 2-4).

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## Technical Appendices for Fairness-Accuracy Trade-Offs: A Causal Perspective

The source code for reproducing all the experiments can be found in the
anonymized code repository. The code is also included with the supplementary
materials, in the folder source-code. All experiments were performed on a
MacBook Pro, with the M3 Pro chip and 36 GB RAM on macOS 14.1 (Sonoma). Each
experiment can be run with less than 1 hour of compute on the above-described
machine or equivalent. Within each experiment, calls to Alg. 2 are included,
which includes optimizing neural networks (using pytorch). Here, we use feed-
forward networks with $n_{l}=2$ hidden layers, and $n_{\text{hidden}}=16$
nodes in each layer. Training is carried out using the Adam optimizer [Kingma
and Ba, 2014], with the learning rate $\eta=0.001$ fixed, for 500 epochs with
batch size 512, and early stopping regularization (20 epochs of patience), and
5 random initial restarts.

##  Appendix A Linear Path-Specific Excess Losses

Consider the SCM in Eqs. 1-3. The unconstrained, DE-fair, IE-fair, and fully-
fair predictors are given by

$\displaystyle\widehat{Y}^{\emptyset}$ $\displaystyle=\alpha X+\gamma W$ (35)
$\displaystyle\widehat{Y}^{\text{DE}}$ $\displaystyle=\gamma W$ (36)
$\displaystyle\widehat{Y}^{\text{IE}}$ $\displaystyle=\alpha X$ (37)
$\displaystyle\widehat{Y}^{\\{\text{DE, IE}\\}}$ $\displaystyle=0.$ (38)

Note that we can compute:

$\displaystyle\mathbbm{E}[Y-\widehat{Y}^{\\{\text{DE, IE}\\}}]^{2}$
$\displaystyle=\mathbbm{E}Y^{2}$ (39) $\displaystyle=\mathbbm{E}[\alpha
X+\gamma W+\epsilon_{y}]^{2}$ (40)
$\displaystyle=\mathbbm{E}[\epsilon_{y}^{2}]+\alpha^{2}\mathbbm{E}X^{2}+\gamma^{2}\mathbbm{E}W^{2}+2\mathbbm{E}\epsilon_{w}\mathbbm{E}\alpha
X+\mathbbm{E}\epsilon_{w}\mathbbm{E}\gamma W$ (41)
$\displaystyle\quad+2\mathbbm{E}\alpha X\mathbbm{E}\gamma W$ (42)
$\displaystyle=\sigma_{y}^{2}+\frac{\alpha^{2}}{2}+\gamma^{2}\mathbbm{E}[\beta
X+\epsilon_{w}]^{2}+2\frac{\alpha}{2}\mathbbm{E}\gamma[\beta X+\epsilon_{w}]$
(43)
$\displaystyle=\sigma_{y}^{2}+\frac{\alpha^{2}}{2}+\gamma^{2}\sigma_{w}^{2}+\frac{\gamma^{2}\beta^{2}}{2}+\frac{\alpha\gamma\beta}{2}$
(44)
$\displaystyle=\sigma_{y}^{2}+\frac{\alpha^{2}+\gamma^{2}\beta^{2}+\alpha\gamma\beta}{2}+\gamma^{2}\sigma_{w}^{2}.$
(45)

Similarly, we have that

$\displaystyle\mathbbm{E}[Y-\widehat{Y}^{\text{DE}}]^{2}$
$\displaystyle=\mathbbm{E}[\alpha X+\gamma W+\epsilon_{y}-\gamma W]^{2}$ (46)
$\displaystyle=\sigma_{y}^{2}+\alpha^{2}\mathbbm{E}X^{2}$ (47)
$\displaystyle=\sigma_{y}^{2}+\frac{\alpha^{2}}{2}$ (48)
$\displaystyle\mathbbm{E}[Y-\widehat{Y}^{\text{IE}}]^{2}$
$\displaystyle=\mathbbm{E}[\alpha X+\gamma W+\epsilon_{y}-\alpha X]^{2}$ (49)
$\displaystyle=\sigma_{y}^{2}+\gamma^{2}\mathbbm{E}W^{2}$ (50)
$\displaystyle=\sigma_{y}^{2}+\frac{\gamma^{2}\beta^{2}}{2}+\gamma^{2}\sigma_{w}^{2}$
(51) $\displaystyle\mathbbm{E}[Y-\widehat{Y}^{\emptyset}]^{2}$
$\displaystyle=\mathbbm{E}[\alpha X+\gamma W+\epsilon_{y}-\alpha X-\gamma
W]^{2}$ (52) $\displaystyle=\mathbbm{E}\epsilon_{y}^{2}=\sigma_{y}^{2}.$ (53)

Therefore, we have that

$\displaystyle\mathbbm{E}[Y-\widehat{Y}^{\\{\text{DE,
IE}\\}}]^{2}-\mathbbm{E}[Y-\widehat{Y}^{\emptyset}]^{2}$
$\displaystyle=\frac{\alpha^{2}+\gamma^{2}\beta^{2}+\alpha\gamma\beta}{2}+\gamma^{2}\sigma_{w}^{2}$
(54)
$\displaystyle=\frac{\alpha^{2}}{2}+\frac{\gamma^{2}\beta^{2}+\alpha\gamma\beta}{2}+\gamma^{2}\sigma_{w}^{2}$
(55)
$\displaystyle=\mathbbm{E}[Y-\widehat{Y}^{\text{DE}}]^{2}-\mathbbm{E}[Y-\widehat{Y}^{\emptyset}]^{2}$
(56) $\displaystyle\quad+\mathbbm{E}[Y-\widehat{Y}^{\\{\text{DE,
IE}\\}}]^{2}-\mathbbm{E}[Y-\widehat{Y}^{\text{DE}}]^{2}.$ (57)

##  Appendix B Theorem Proofs

###### Thm. 1 Proof:.

The total excess loss (TEL) is defined as:

$\displaystyle\text{PSEL}(\emptyset\to\\{D,I,S\\})=\mathbbm{E}[L(\widehat{Y}^{\\{D,I,S\\}},Y)]-\mathbbm{E}[L(\widehat{Y}^{\emptyset},Y)].$
(58)

Let $S_{1},S_{2}\subset\\{D,I,S\\}$ such that $S_{1}\neq S_{2}$. The quantity
can be expanded as follows:

$\displaystyle\text{PSEL}(\emptyset\to\\{D,I,S\\})$
$\displaystyle=\mathbbm{E}[L(\widehat{Y}^{\\{D,I,S\\}},Y)]-\mathbbm{E}[L(\widehat{Y}^{\emptyset},Y)]$
(59)
$\displaystyle=\mathbbm{E}[L(\widehat{Y}^{\\{D,I,S\\}},Y)]-\mathbbm{E}[L(\widehat{Y}^{S_{1}},Y)]$
(60)
$\displaystyle\quad+\mathbbm{E}[L(\widehat{Y}^{S_{1}},Y)]-\mathbbm{E}[L(\widehat{Y}^{\emptyset},Y)]$
(61)
$\displaystyle=\underbrace{\mathbbm{E}[L(\widehat{Y}^{\\{D,I,S\\}},Y)]-\mathbbm{E}[L(\widehat{Y}^{\\{S_{1},S_{2}\\}},Y)]}_{\text{PSEL}(\\{S_{1},S_{2}\\}\to\\{D,I,S\\})}$
(62)
$\displaystyle\quad+\underbrace{\mathbbm{E}[L(\widehat{Y}^{\\{S_{1},S_{2}\\}},Y)]-\mathbbm{E}[L(\widehat{Y}^{S_{1}},Y)]}_{\text{PSEL}(S_{1}\to\\{S_{1},S_{2}\\})}$
(63)
$\displaystyle\quad+\underbrace{\mathbbm{E}[L(\widehat{Y}^{S_{1}},Y)]-\mathbbm{E}[L(\widehat{Y}^{\emptyset},Y)]}_{\text{PSEL}(\emptyset\to
S_{1})}$ (64) $\displaystyle=\text{PSEL}(\emptyset\to
S_{1})+\text{PSEL}(S_{1}\to\\{S_{1},S_{2}\\})$ (65)
$\displaystyle\quad+\text{PSEL}(\\{S_{1},S_{2}\\}\to\\{D,I,S\\}),$ (66)

completing the theorem’s proof for the choice $S_{1}=D,S_{2}=I$. The proof
also implies other decompositions of the total excess loss
$\text{PSEL}(\emptyset\to\\{D,I,S\\})$ as mentioned in Rem. 2. ∎

###### Prop. 3 Proof:.

Let $\mathcal{S}=S_{1},\dots,S_{m}$ be a set of causal pathways between the
attribute $X$ and the outcome $Y$ (such as direct, indirect, and spurious
effects in the context of the Standard Fairness Model). Consider the value
function

$\displaystyle f(S)=\text{PSEL}(\emptyset\to S).$ (67)

The Shapley value of pathway $S_{i}$ associated with $f$ is given by:

$\displaystyle\phi(S_{i})=\sum_{S\subseteq\mathcal{S}\setminus\\{S_{i}\\}}\frac{1}{m\binom{m-1}{|S|}}\left(f(S\cup\\{S_{i}\\})-f(S)\right).$
(68)

Consider now the generalized $\mathcal{G}_{\text{PSEL}}$ graph for $m$
pathways shown in Fig. 6. The graph is split into columns, and in column
$\ell$ all of the sets $S$ of size $\ell$ appear. The quantity
$\text{APSEL}(S_{i})$ considers all pathw $\pi$ that start in $\emptyset$ and
end in $\\{S_{1},\dots,S_{m}\\}$, and takes the average of excess losses
associated with $S_{i}$ along each path $\pi$, written
$\text{PSEL}(\pi^{<S_{i}}\to\pi^{\leq S_{i}})$. The APSEL quantity is written
as:

$\displaystyle\frac{1}{n!}\sum_{\begin{subarray}{c}\text{paths
}\pi\in\mathcal{G}_{\text{PSEL}}:\\\ \emptyset\text{ to
}\\{D,I,S\\}\end{subarray}}\text{PSEL}(\pi^{<S_{i}}\to\pi^{\leq S_{i}}).$ (69)

We now split the expression for the Shapley value by conditioning on the size
of the set $S$:

$\displaystyle\phi(S_{i})$
$\displaystyle=\sum_{\ell=0}^{m-1}\sum_{S\subseteq\mathcal{S}\setminus\\{S_{i}\\}:|S|=\ell}\frac{1}{m\binom{m-1}{\ell}}\left(f(S\cup\\{S_{i}\\})-f(S)\right)$
(70)
$\displaystyle=\sum_{\ell=0}^{m-1}\sum_{S\subseteq\mathcal{S}\setminus\\{S_{i}\\}:|S|=\ell}\frac{1}{m\binom{m-1}{\ell}}\left(\text{PSEL}(\emptyset\to\\{S\cup
S_{i}\\})-\text{PSEL}(\emptyset\to S)\right)$ (71)
$\displaystyle=\sum_{\ell=0}^{m-1}\sum_{S\subseteq\mathcal{S}\setminus\\{S_{i}\\}:|S|=\ell}\frac{1}{m\binom{m-1}{\ell}}\text{PSEL}(S\to\\{S\cup
S_{i}\\}).$ (72)

Now, fix a set $S$ of size $|S|=\ell$. There are exactly $\ell!$ paths leading
from the $\emptyset$ to $S$. This is followed by a fixed transition $S\to
S\cup S_{i}$ is chosen. The remaining steps from $S\cup S_{i}$ to
$S_{1},\dots,S_{m}$ along the path $\pi$ can be chosen in $(m-\ell-1)!$ ways.
Therefore, across all paths $\pi$, the contribution $\text{PSEL}(S\to\\{S\cup
S_{i}\\}$ appears $\ell!(m-\ell-1)!$ times. Therefore, the APSEL quantity can
be re-written as:

$\displaystyle\text{APSEL}(S_{i})$
$\displaystyle=\frac{1}{m!}\sum_{\ell=0}^{m-1}\sum_{S\subseteq\mathcal{S}\setminus\\{S_{i}\\}:|S|=\ell}{\ell!(n-\ell-1)!}\text{PSEL}(S\to\\{S\cup
S_{i}\\})$ (73)
$\displaystyle=\sum_{\ell=0}^{m-1}\sum_{S\subseteq\mathcal{S}\setminus\\{S_{i}\\}:|S|=\ell}\frac{\ell!(m-\ell-1)!}{m!}\text{PSEL}(S\to\\{S\cup
S_{i}\\})$ (74)
$\displaystyle=\sum_{\ell=0}^{m-1}\sum_{S\subseteq\mathcal{S}\setminus\\{S_{i}\\}:|S|=\ell}\frac{1}{m\binom{m-1}{\ell}}\text{PSEL}(S\to\\{S\cup
S_{i}\\})$ (75)
$\displaystyle=\sum_{S\subseteq\mathcal{S}\setminus\\{S_{i}\\}}\frac{1}{m\binom{m-1}{|S|}}\text{PSEL}(S\to\\{S\cup
S_{i}\\})$ (76) $\displaystyle=\phi(S_{i}).$ (77)

The proof for the ATVD quantity follows the same steps. ∎

$\emptyset$$\vdots$$\\{S_{1}\\}$$\\{S_{m}\\}$$\\{S_{1},\dots,S_{\ell}\\}$$\\{S_{m-\ell+1},\dots,S_{m}\\}$$\vdots$$\\{S_{1},\dots,S_{m}\\}$Column
0Column 1Column $\ell$Column
$m$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$
Figure 6: Generalized $\mathcal{G}_{\text{PSEL}}$ graph for $m$ causal
pathways.

##  Appendix C Connection to Causal Decompositions

In this appendix, we explore the connection of the fairness/accuracy trade-off
described in the main text with the literature on causal decompositions [Zhang
and Bareinboim, 2018b, Plečko and Bareinboim, 2024]. As described in the main
text and Alg. 1, along with tracking the reduction in predictive power from
imposing a causal fairness constraint, we were also interested in the change
in the TV${}_{x_{0},x_{1}}(\widehat{y}^{S})$, captured by the TV difference
(TVD) quantity defined as:

$\displaystyle\text{TVD}(S\to
S^{\prime})=\underbrace{\mathbbm{E}[Y^{S^{\prime}}\mid
x_{1}]-\mathbbm{E}[Y^{S^{\prime}}\mid x_{0}]}_{\text{TV after removing
}S^{\prime}\setminus S}-\underbrace{\mathbbm{E}[Y^{S}\mid
x_{1}]-\mathbbm{E}[Y^{S}\mid x_{0}]}_{\text{TV before removing
}S^{\prime}\setminus S}.$ (78)

In words, the quantity $\text{TVD}(S\to S^{\prime})$ measures how much the TV
measure changes once the effect in $S^{\prime}\setminus S$ is constrained to
$0$. Put differently, the $\text{TVD}(S\to S^{\prime})$ is a quantification of
the amount of disparity that is transmitted along the path
$S^{\prime}\setminus S$. Other notions for quantifying the size of the
disparity transmitted along a specific causal pathway have also been explored
in the literature on causal decompositions [Zhang and Bareinboim, 2018b,
Plečko and Bareinboim, 2024]. An important result from [Plečko and Bareinboim,
2024] we draw a connection to is the decomposition of the TV measure into the
NDE, NIE, and NSE contributions:

######  Proposition 4 (TV Decomposition [Plečko and Bareinboim, 2024]).

The TV measure can be decomposed into the natural direct, indirect, and
spurious contributions:

$\displaystyle\text{TV}_{x_{0},x_{1}}(y)=\text{NDE}_{x_{0},x_{1}}(y)-\text{NIE}_{x_{1},x_{0}}(y)+\text{NSE}_{x_{1}}(y)-\text{NSE}_{x_{0}}(y).$
(79)

Based on the decomposition in Eq. 79, we can try to quantify how much the
disparity would be reduced by if we were to remove an effect. For instance, if
the direct effect was removed, the reduction in the disparity would be

$\displaystyle\text{TV}_{x_{0},x_{1}}(y^{\text{DE}})-\text{TV}_{x_{0},x_{1}}(y)=-\text{NDE}_{x_{0},x_{1}}(y)$
(80)

where $y^{\text{DE}}$ is the outcome with the direct effect removed, which we
assume satisfies:

$\displaystyle\text{NDE}_{x_{0},x_{1}}(y^{\text{DE}})$ $\displaystyle=0$ (81)
$\displaystyle\text{NIE}_{x_{1},x_{0}}(y^{\text{DE}})$
$\displaystyle=\text{NIE}_{x_{1},x_{0}}(y)$ (82)
$\displaystyle\text{NSE}_{x_{0}}(y^{\text{DE}})$
$\displaystyle=\text{NSE}_{x_{0}}(y)$ (83)
$\displaystyle\text{NSE}_{x_{1}}(y^{\text{DE}})$
$\displaystyle=\text{NSE}_{x_{1}}(y).$ (84)

A similar reasoning could be applied to understand the reduction in the TV
resulting from removing indirect, and spurious effects. Therefore, in this way
we can obtain a different quantification of the TVD measure, which we call TV
reduction (TVR), satisfying:

$\displaystyle\text{TVR}(S\to S\cup\text{DE})$
$\displaystyle=-\text{NDE}_{x_{0},x_{1}}(y)$ (86)
$\displaystyle\text{TVR}(S\to S\cup\text{IE})$
$\displaystyle=\text{NIE}_{x_{1},x_{0}}(y)$ (87) $\displaystyle\text{TVR}(S\to
S\cup\text{SE})$ $\displaystyle=-\text{NSE}_{x_{1}}(y)+\text{NSE}_{x_{0}}(y),$
(88)

where in each line we assume the set $S$ does not contain DE, IE, and SE,
respectively. To establish a connection between our approach with the
literature on causal decompositions, we empirically compare the TVR and TVD
measures. We again use the same datasets as in the main text, namely the
Census 2018 dataset from Ex. 2 and the COMPAS dataset from Ex. 3.

(a) TVD and TVR comparison on Census 2018 data.

(b) TVD and TVR comparison on COMPAS data.

Figure 7: Comparison of alternative causal disparity quantifications TVD, TVR.

The results of the empirical comparisons of TVR and TVD values are shown in
Fig. 7, where the indicate bars are standard deviations obtained from 10
bootstrap samples of the data. The figure illustrates that on both COMPAS and
Census data, TVD and TVR values are not statistically different, across
direct, indirect, and spurious effects. This finding adds further validity to
our analysis in the main text, meaning that the TVD measure corresponds
closely with existing notions for quantifying discrimination in the
literature.

##  Appendix D CFUR Quantity Representation

The CFUR quantity (Def. 4) of a causal path $S_{i}$ is given by

$\displaystyle\frac{\text{ATVD}(S_{i})}{\text{APSEL}(S_{i})}.$ (89)

The ATVD and APSEL quantities are defined as the average values of TVD, PSEL
quantities, defined through Eqs. 24-25. In this appendix, we discuss the
connection of the CFUR value with the local fairness-utility ratios, defined
via:

$\displaystyle\text{LCFUR}(S,S_{i})=\frac{\text{TVD}(S\to S\cup
S_{i})}{\text{PSEL}(S\to S\cup S_{i})}.$ (90)

From Eqs. 24-25, we see that

$\displaystyle\text{CFUR}(S_{i})=\frac{\frac{1}{m!}\sum_{S\in\mathcal{I}}\text{TVD}(S\to
S\cup S_{i})}{\frac{1}{m!}\sum_{S\in\mathcal{I}}\text{PSEL}(S\to S\cup
S_{i})},$ (91)

where $m$ is the number of causal paths and in the summations the set $S$
ranges within $\mathcal{I}$, the set of all path prefixes of $S_{i}$ along all
paths connecting $\emptyset$ and $\\{S_{1},\dots,S_{m}\\}$ in the
$\mathcal{G}_{\text{PSEL}}$ graph. Now, we want to connect the global CFUR
with the local LCFUR values. This can be done through the following
proposition:

######  Proposition 5 (LCFUR and CFUR).

The CFUR$(S_{i})$ value of a causal path $S_{i}$ can be written as:

$\displaystyle\text{CFUR}(S_{i})=\sum_{S\in\mathcal{I}}\lambda(S_{i})\times\text{LCFUR}(S,S_{i}),$
(92)

where the weights $\lambda(S,S_{i})$ are defined as:

$\displaystyle\lambda(S,S_{i})=\frac{\text{PSEL}(S\to S\cup
S_{i})}{\sum_{S^{\prime}\in\mathcal{I}}\text{PSEL}(S^{\prime}\to
S^{\prime}\cup S_{i})}.$ (93)

The above proposition shows that the global CFUR value is a weighted average
of all the local LCFUR values. The weight assigned to each local LCFUR is
proportional to the size of its path-specific excess loss $\text{PSEL}(S\to
S\cup S_{i})$ among the sum of all path-specific excess losses that are
considered $\sum_{S^{\prime}\in\mathcal{I}}\text{PSEL}(S^{\prime}\to
S^{\prime}\cup S_{i})$. Therefore, we see that the CFUR value places larger
weights on $S\to S\cup S_{i}$ transitions that result in a larger path-
specific excess loss.

##  Appendix E UCI Credit Example

(a) (A)PSEL and (A)TVD values.

(b) CFUR values.

(c) Fairness/Utility of $\widehat{Y}^{S}$ predictors.

$\emptyset$$\\{I\\}$$\\{D\\}$$\\{S\\}$$\\{D,I\\}$$\\{I,S\\}$$\\{D,S\\}$$\\{D,I,S\\}$(0.01,
0)(0.05, 0.01)(0, 0)(0.04, 0.01)(0.04, 0.01)(0.01, 0)(0, 0)(0, 0)(0, 0)(0,
0)(0.01, 0)(0.05, 0.01)

(d) $\mathcal{G}_{\text{PSEL}}$ with PSEL (blue) and TVD values (red).

Figure 8: Application of Alg. 1 on the UCI Credit dataset.

######  Example 4 (UCI Credit [Yeh, 2016]).

A commercial bank is developing an automated payment default scoring system
based on the UCI Credit Dataset, collected in 2018. This dataset includes
demographic variables $Z$ ($Z_{1}$ for age), the protected attribute gender
$X$ ($x_{0}$ female, $x_{1}$ male), marital status $M$, education level $L$,
employment status $E$, and financial history $F$. The institution aims to
predict the binary outcome $Y$ that indicates whether the individual defaulted
on a credit payment ($Y=1$ for defaulting, $Y=0$ otherwise). The grouping
$\\{X={X},Z=\\{Z_{1}\\},W=\\{L,M,E,F\\},Y=Y\\}$ induces the standard fairness
model [Plečko and Bareinboim, 2024] for this analysis.

The development team is focused on ensuring fairness in the risk of default
predictions. They examine the causal effects from the protected attribute $X$
to the predicted default probability $\widehat{Y}^{S}$, evaluating how much
scores need to be adjusted to mitigate specific causal pathways, using the
complement of AUROC ($1-$AUROC) as the loss metric. They employ Alg.1 for this
purpose and develop fair scoring models $\widehat{Y}^{S}$ using different
$S$-sets as per Alg.2.

The analysis results are depicted in Fig.8, with uncertainty bars indicating
standard deviations over 10 bootstrap repetitions. Fig.8(a)) shows the TVD and
PSEL values. The team notes that for direct and spurious effects the APSEL and
ATVD values are not significantly different from $0$. In words, they find that
imposing a fairness constraint along these causal pathways does not
significantly reduce the predictive power (in terms of reducing AUROC) of the
predictor $\widehat{Y}^{S}$. For the indirect effect, they find an APSEL value
of 5% of AUROC, and an ATVD of 1.5%, showing that removing the indirect effect
may reduce discrimination.

The team then looks at CFUR values, finding that DE and SE values within one
standard deviation contain $0$, again compatible no trade-off between fairness
and accuracy. This is because the causal effects along DE and SE paths are
already very close to $0$, and constraining such pathways has little effect on
the predictor $\widehat{Y}^{S}$. The team then plots the TV values and the
excess reduction in AUROC in Fig. 8(c), for all predictors $\widehat{Y}^{S}$
for the set $S$ ranging in the powerset of $\\{DE,IE,SE\\}$. The figure
illustrates that the predictors $\widehat{Y}^{S}$ are statistically
indistinguishable in performance for two groups, namely:

$\displaystyle\widehat{Y}^{\emptyset},\widehat{Y}^{\\{\text{D}\\}},\widehat{Y}^{\\{\text{S}\\}},\widehat{Y}^{\\{\text{D,
S}\\}},\text{ and}$ (94)
$\displaystyle\widehat{Y}^{\\{I\\}},\widehat{Y}^{\\{\text{D,
I}\\}},\widehat{Y}^{\\{\text{S, I}\\}},\widehat{Y}^{\\{\text{D, I, S}\\}}.$
(95)

Once again, the removal of direct and spurious effects has little effect on
the predictor, while the removal of the indirect effect seems to reduce
discrimination. Based on these findings, the team considers whether to remove
the indirect effect from the predictions. However, after a detailed discussion
with a regulatory consultant, they decide that the usage of mediators $W$ is
justifiable since $W$ reflects the spending behavior of the customers.
Therefore, the team decides to use the $\widehat{Y}^{\emptyset}$ predictor.