Fairness Aware Counterfactuals for Subgroups
Authors: Loukas Kavouras
Konstantinos Tsopelas
Giorgos Giannopoulos
Dimitris Sacharidis
Eleni Psaroudaki
Nikolaos Theologitis
Dimitrios Rontogiannis
Dimitris Fotakis
Ioannis Emiris
Venue: NeurIPS 2023 poster (2023)
Keywords: subgroup fairness
recourse
counterfactual explanations
TL;DR: FACTS is an auditing mechanism for fairness of recourse, i.e., how difficult is it to achieve the desired outcome

Abstract: In this work, we present Fairness Aware Counterfactuals for Subgroups (FACTS), a framework for auditing subgroup fairness through counterfactual explanations. We start with revisiting (and generalizing) existing notions and introducing new, more refined notions of subgroup fairness. We aim to (a) formulate different aspects of the difficulty of individuals in certain subgroups to achieve recourse, i.e. receive the desired outcome, either at the micro level, considering members of the subgroup individually, or at the macro level, considering the subgroup as a whole, and (b) introduce notions of subgroup fairness that are robust, if not totally oblivious, to the cost of achieving recourse. We accompany these notions with an efficient, model-agnostic, highly parameterizable, and explainable framework for evaluating subgroup fairness. We demonstrate the advantages, the wide applicability, and the efficiency of our approach through a thorough experimental evaluation on different benchmark datasets.

Fairness Aware Counterfactuals for Subgroups
Loukas Kavouras
Konstantinos Tsopelas IMSI / Athena RC
Giorgos Giannopoulos
IMSI/ / Athena RC
IMSI Athena RC
kavouras@athenarc..u
k. taaopelas@athemarc..gg
giann@athenarc. gr
Dimitris Sacharidis
Eleni Psaroudaki
ULB dimitris. hta.rrisssacarrddssuu.....
NTUA & IMSI / Athena RC
epsaroudaki @mail. .ntua.gr
Nikolaos Theologitis
Dimitrios Rontogiannis
IMST / Athena RC
IMSI Athena RC dronto@gmail.com
n. theollogttiis@@ttheerr..rr
Dimitris Fotakis
Ioannis Emiris NKUA & IMSI / Athena RC emiris@athenarc..rr
NTUA & Archimedes / Athena RC fotakis@cs ntua.gr
Abstract
In this work, we present Fairness Aware Counterfactuals for Subgroups (FACTS), a framework for auditing subgroup faimess through counterfactual explanations. We start with revisiting (and generalizing) existing notions and introducing new, more refined notions of subgroup fairness. We aim (a) formulate different aspects of the difficulty of individuals certain subgroups to achieve recourse, i.e., receive the desired outcome, either the micro level, considering members of the subgroup individually, or at the macro level, considering the subgroup as whole, and (b) introduce notions of subgroup fairness that are robust, the cost of achieving recourse. We accompany these notions with an efficient, model-agnostic, highly parameterizable, and explainable framework for evaluating subgroup fairness. We demonstrate the advantages, the wide applicability, and the efficiency our approach through thorough experimental evaluation of different benchmark datasets.
totally oblivious,
1 Introduction
Machine learning is now integral part of decision-making processes across various domains, e.g.. medical applications [22], employment [5, 7J. recommender systems [29], education [26], credit assessment [4]. Its decisions affect everyday life directly, and, if unjust discriminative, could potentially harm our society [28]. Multiple examples of discrimination or bias towards specific population subgroups in such applications [33] create the need not only for explainable and interpretable machine learning that more trustworthy [6], but also for auditing models in order to detect hidden bias for subgroups [30].
Bias towards protected subgroups most often detected by various notions of fairness of prediction, e.g., statistical parity, where all subgroups defined by protected attribute should have the same probability of being assigned the positive (favorable) predicted class. These definitions capture the explicit bias reflected in the model's predictions. Nevertheless, implicit form of bias is the
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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difficulty for, or the burden [32, 20] of, an individual (or group thereof) to achieve recourse, i.e., perform the necessary actions change their features so as to obtain the favorable outcome [10, 35]. Recourse provides explainability (i.e., a counterfactual explanation [38]) and actionability to an affected individual, and is legal necessity in various domains, e-g., the Equal Credit Opportunity Act mandates that an individual can demand to learn the reasons for a loan denial. Fairness of recourse captures the notion that the protected subgroups should bear equal burden [10, 14, 37, 20].
To illustrate these notions, consider a company that supports its promotion decisions with an Al system that classifies employees as good candidates for promotion, the favorable positive class, based on various performance metrics, including their cycle time efficiency (CTE) and the annual contract value (ACV) for the projects they lead. Figure la draws ten employees from the negative predicted class as points in the CTE-ACV plane and also depicts the decision boundary of the classifier. Race is the protected attribute, and there are two protected subgroups with five employees each, depicted as circles and triangles. For each employee, the arrow depicts the best action to achieve recourse, i.e., to cross the decision boundary, and the number indicates the cost of the action, here simply computed as the distance to the boundary [10]. For example, I1 may increase their chances for promotion mostly by acquiring more high-valued projects, while z2 mostly by increasing their efficiency. Burden is defined the mean cost for protected subgroup [32]. For the protected race O, the burden is 2, while it is 2.2 for race 1, indicating thus unfairness of recourse against race 1. In contrast, assuming there is an equal number of employees of each race in the company, the classifier satisfies fairness of prediction in terms of statistical parity (equal positive rate in the subgroups).
Cycle Time Efficincy (CTE)
Ad
Cyole Time Elficiancy (CT5)
ay JJCV CTR (-99)
222
Effertionnss
0.8
a2-14C7
10056
X1
0.4
0.2
race
arganie
Coo
eegustiv
Annual Contrart Pister (ACV)
Anmull Contrnct Valor (ACV)
(b)
Figure 1: (a) An example of affected individuals, the decision boundary, actions, and a subpopulation (in the shaded region), depicted in the feature space: (b) Cumulative distribution of cost of recourse for the individuals in (a); (c) Comparison of two actions achieve recourse for two individuals.
While fairness of recourse is an important concept that captures distinct notion of algorithmic bias, as also explained in [14, 37J, we argue that is much more nuanced than the mean cost of recourse (aka burden) considered in all prior work [10, 32, 37, 20], and raise three issues.
First, the mean cost, which comprises a micro viewpoint of the problem, does not provide the complete picture of how the cost of recourse varies among individuals and may lead to contrasting conclusions. Consider again Figure la, and observe that for race I all but one employee can achieve recourse with cost at most 2, while an outlier achieves recourse with cost 6. It is this outlier that raises the mean cost for race above that of race O. For the two race subgroups, Figure 1b shows the cumulative distribution of cost, termed the effectiivenss-coot distribution (ecd) in Section 2. These distributions allow the fairness auditor to inspect the tradeoff between cost and recourse, and define the appropriate fairness of recourse notion. For example, they may consider that actions with cost of more than 2 are unrealistic (c.g., because they cannot be realized within some timeframe). and thus investigate how many employees can achieve recourse under this constraint; within budger in Section 2.3. Under this notion, there is unfairness against race 0, as only 60% of race 0 employees (compared to 80% of race 1) can realistically achieve recourse.
refer to this equal effectiveness
There are several options to go beyond the mean cost. One is to consider fairness of recourse at the individual level, and compare an individual with their counterfactual counterpart had their protected attribute changed value [37]. However, this approach is impractical as, similar to other causal-based definitions of fairness, c.g., [21], it requires strong assumptions about the causal structure in the

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domain [15]. In contrast, we argue that it's preferable to investigate faimess in subpopulations and inspect the trade-off between cost and recourse.
Second, there are many cases where the aforementioned micro-level aggregation of individuals' costs is not meaningful in auditing real-world systems. To account for this, we introduce a macro viewpoint where a group of individuals is considered as a whole, and an action is applied to and assessed collectively for all individuals in the group. An action represents an external horizontal intervention, such as an affirmative action in society, or focused measures in an organization that would change the attributes of some subpopulation (e.g., decrease tax or loan interest rates, increase productivity skills). In the macro viewpoint, the cost of recourse does not burden the individuals, but the external third party, e.g., the society or an organization. Moreover, the macro viewpoint offers a more intuitive way to audit a system for faimess of recourse, as it seeks to uncover systemic biases that apply to large number of individuals.
To illustrate the macro viewpoint, consider the group within the shaded region in Figure la. In the micro viewpoint, each employee seeks recourse individually, and both race subgroups have the same distribution of costs. However, we can observe that race 0 employees, like achieve recourse by actions in the ACV direction, while race I employees, like z2 the orthogonal CTE direction. This becomes apparent when we take the macro viewpoint, and investigate the effect of action 01. depicted on the border of the shaded region, discovering that it disproportionally effective on the race subgroups (leads to recourse for two-thirds of one subgroup but for none in the other). In this example, action a might represent the effect of a training program enhance productivity skills, and the macro viewpoint finds that it would perpetuate the existing burden of race O employees.
Third, existing notions of faimess of recourse have important practical limitation: they require cost function that captures one's ability to modify one's 's attributes, whose definition may involve a learning process [31], or an adaptation of off-the-shelf functions by practitioners [35]. Even the idea of which attributes are actionable, in the sense that one can change (e.g., one cannot get younger), or "ethical" to suggest as actionable (e.g., change of marital status from married divorced) hides many complications [36].
Conclusions drawn about fairess of recourse crucially depend on the cost definition. Consider individuals Ii, T2, and actions d1, 42, shown in Figure Ic. Observe that is hard to say which action is cheaper, as one needs compare changes within and across very dissimilar attributes. Suppose the cost function indicates action 01 is cheaper; both individuals achieve recourse with the same cost. However, if action 02 is cheaper, only 22 achieves recourse. Is the classifier fair?
To address this limitation, we propose definitions that are oblivious to the cost function. The idea is to compare the effectiveness of actions to the protected subgroups, rather than the cost of recourse for the subgroups. One way to define a cost-oblivious notion of fairness of recourse is the equal choice for recourse (see Section 2.3), which we illustrate using the example in Figure Ic. According to it, the classifier is unfair against Z1, as Ij has only one option, while 22 has two options to achieve recourse among the set of actions (01,a2):
Contribution Our aim is to showcase that fairness of recourse is an important and distinct notion of algorithmic fairness with several facets not previously explored. We make series of conceptual and technical contributions.
Conceptually, we distinguish between two different viewpoints. The micro viewpoin/ follows literature on recourse fairness [10, 14, 37, 20] in that each individual chooses the action that cheaper for them, but revisits existing notions. It considers the trade-off between cost and recourse and defines several novel notions that capture different aspects of it. The macro viewpoint considers how an action collectively affects group of individuals, and quantifies its effectiivness. It allows the formulation of cost-oblivions notions of recourse fairness. It also leads to an alternative trade-off between cost and recourse that may reveal systemic forms of bias.
Technically, we propose an efficient, interpretable, agnnsti,,,, highly parametrizable framework termed FACTS (Fairess--Aware Countefactuals for Subgroups) to audit for faimess of recourse. FACTS efficient in computing the effectiveness-cost distribution, which captures the trade-off between cost and recourse, in both the micro or macro viewpoint. The key idea is that instead of determining the best action for each individual independently (i.e., finding their nearest counterfactual explanation [38]), it enumerates the space of actions and determines how many and which individuals
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achieve recourse through each action. Furthermore, FACTS employs systematic way to explore the feature space and discover any subspace such that recourse within. FACTS ranks the subspaces in decreasing order of recourse bias it detects, and for each provides an interpretable summary of its findings.
bias exists among the protected subgroups
Related Work We distinguish between fairness of predictions and fairness of recourse. The former aims to capture and quantify unfairness by comparing directly the model's predictions [27, 2] at: the individual level, e.g., individual fairness [9], counterfactual/caasal-biee the group level, c.g., demographic parity [39], equal odd/opportunity [12].
fairness [21, 19, 26]: and
Fairness of recourse is more recent notion, related to counterfactual explanations [38], which explains a prediction for an individual (the factual) by presenting the "best" counterfactual that would result in the opposite prediction, offering thus recourse the individual. Best, typically means the nearest counterfactual in terms of a distance metric in the feature space. Another perspective, which we adopt here, is to consider the action that transforms factual into counterfactual, and specify cost function to quantify the effort required by an individual to perform an action. In the simplest case, the cost function can be the distance between factual and counterfactual, but it can also encode the feasibility of an action (c.g., it is impossible to decrease age) and the plausibility of a counterfactual (e.g., it act on structural causal model capturing cause-effect relationships among attributes [15]. Hereafter, we adopt the most general definition, where a cost function is available, and assume that the best counterfactual explanation is the one that comes from the minimum cost action. Counterfactual explanations have been suggested as a mechanism to detect possible bias against protected subgroups. e.g., when they require changes in protected attributes [13].
of-distribution). It is also possible view actions as interventions that
Fairness of recourse, first introduced in [35] and formalized in [10], is defined the group level as the disparity of the mean cost to achieve recourse (called burden in subsequent works) among the protected subgroups. Fairness of recourse for an individual when they require the same cost to achieve recourse in the actual world and in an imaginary world where they would have a different value in the protected attribute [37]. This definition however only applies when structural causal model of the world is available. Our work expands on these ideas and proposes alternate definitions that capture a macro and micro viewpoint of fairness of recourse.
There is a line of work on auditing models for fairness of predictions at the subpopulation level [17, 18]. For example, [34] identifies subpopulations that show dependence between a performance measure and the protected attribute. [3] determines whether people are harmed due to their membership in a specific group by examining a ranking of features that are most associated with the model's behavior. There is no equivalent work for fairness of recourse, although the need to consider the subpopulation recognized in [16] due to uncertainty in assumptions or intentionally study fairness. Our work is the first that audits for fairness of recourse at the subpopulation level.
A final related line of work is global explainability. For example, recourse summaries [31, 24, 25] summarizes individual counterfactual explanations globally, and as the authors in [31, 25] suggest can be used manually audit for unfairness in subgroups of interest. [23] aims explain how a model behaves in subspaces characterized by certain features of interest.
[8] uses counterfactuals to unveil whether black-box model, that already complies with the regulations that demand the omission of sensitive attributes, is still biased or not, by trying to find a relation between proxy features and bias. Our work is related to these methods in that we also compute counterfactual explanations for all instances, albeit in a more efficient manner, and with the goal to audit fairness of recourse on subpopulation level.
2 Fairness of Recourse for Subgroups
2.1 Preliminaries
We consider feature space X = X1 x... Xn, where X,, denotes the protected feature, which, for ease of presentation, takes two protected values {0, 1}. For an instance x E X, we use the notation x.X; refer its value in feature X,.
We consider a binary classifier h : X + 1) where the positive outcome the favorable one. For given h, we are concerned with dataset D of adversely affected individuals, i.c., those who

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receive the unfavorable outcome. We prefer the term instance to refer to any point in the feature space X, and the term individual to refer to an instance from the dataset D.
We define an action a as a set of changes to feature values, e.g., a = {country US, education-num + 12). We denote A the set of possible actions. An action when ap- plied to an individual (a factual instance) I results in counterfactual instance x' = a(r). If the individual a was adversely affected (h(x) -1) and the action results in counterfactual that receives the desired outcome (h(a(z)) - 1), we say that action a offers recourse to the individual x and is thus effective. In line with the literature, we also refer to an effective action counterfactual explanation for individual [38].
An action incurs cost to individual x, which denote cost(a, x). The cost function captures both how feasible the action a is for the individual x, and how plausible the counterfactual a(r) is [14].
Given a set of actions A, we define the recourse cost rc(AA, .) of an individual as the minimum cost among effective actions if there is one, otherwise some maximum cost represented as Coci
[ist/ aa A: h(a(x)) = 1}, if 3a A: h(a(x))
rc(A, 2)
Coos
otherwise.
An effective action of minimum cost is also called nearest counterfactual explanation [14).
We define subspace X₂ S X using a predicate p. which is a conjunction of feature-level predicates
of the form "feature-opereee operator-value"" e.g., the predicate p = (country = US) A (education 2 9) defines instances from the US that have more than years of education.
Given a predicate we define the subpopulation group Gp S D as the set of affected individuals that satisfy p. i.c., Gp {a E Dlp(z)). We further distinguish between the protected subgroups Gp.1 = {ir E Dlp(z) A x.X,, = 1} and Gp.0- = {i E Dlp(z) A x.X,, = 0}. When the predicate understood, we may omit in the designation of group simplify notation.
2.2 Effectiveness-Coss Trade-Off
For a specific action a, we naturally define its effectiveness (eff) for a group G, as the proportion of individuals from G that achieve recourse through
1 G
eff G)
(+ GIM((()) -]]--
Note that effectiveness termed correctness in [31] and coverage in [25]. We want examine how recourse is achieved for the group G through a set of possible actions A. We define the aggregate effectiveness (aeff) of A for G in two distinct ways.
In the micro viewpoint, the individuals in the group are considered independently, and each may choose the action that benefits itself the most. Concretely, we define the enirr-esfeecivveness of set of actions A for group G as the proportion of individuals in G that can achieve recourse through some action in A, i.e..:
aeff, (A.
|G| Hx G| A, e.f x) 1-/-
In the macro viewpoint, the group is considered as a whole, and an action is applied collectively to all individuals in the group. Concretely, we define the macro-effectivennss of set of actions A for group G as the largest proportion of individuals in G that can achieve recourse through the same action in A, i.e.:
aeffM (A, G)
max A
( Gl (f, 1}}-
G
For a group G, actions A, and a cost budget c, we define the in-budget actions as the set of actions that cost at most e for any individual in G:
A. = {a E AlVa E G, cost(a,, c).
We define the effectiveness-cost distribution (ecd) as the function that for a cost budget e returns the aggregate effectiveness possible with in-budget actions:
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ecd A, A,G) G) aeff(Ac: G).
We use ecdy, ecdM to refer to the micro, macro viewpoints of aggregate effectiveness. A similar concept, termed the coverage-cost profile, appears in [25].
The value ecd(c; A, G) the proportion of individuals in G that can achieve recourse through actions A with cost at most c. Therefore, the ecd function has an intuitive probabilistic interpretation. Consider the subspace Xp determined by predicate p. and define the random variable C' as the cost required by an instance I E Xp to achieve recourse. The function ecd(c; A, Gp) is the empirical cumulative distribution function of using sample Gp-
The inverse effectiveness-cost distribution function ecd- A. takes input effectiveness level o E [0, 1] and returns the minimum cost required so that o|G] individuals achieve recourse.
2.3 Definitions of Subgroup Recourse Fairness
We define recourse fairness of classifier h for group G by comparing the ecd functions of the protected subgroups Go. G in different ways.
The first two definitions are cost-oblivious, and apply whenever we can ignore the cost function. Specifically, given set of actions and group G, assume that all actions in considered equivalent, and that the cost of any action is the same for all individuals in the group; i.e., cast(a. x) cost(a', n²), Va * a' E A, x + x' E G. The definitions simply compare the aggregate effectiveness of set of actions the protected subgroups.
Equal Effectiveness This definition has micro and macro interpretation, and says that the classifier fair the same proportion of individuals the protected subgroups can achieve recourse: aef Go) (A G1).
Equal Choice for Recourse This definition has only macro interpretation and claims that the classifier fair if the protected subgroups can choose among the same number of sufficiently effective actions achieve recourse, where sufficiently effective means the actions should work for least 1000% (for & E [0, I)) of the subgroup:
la E eff(a, Go) 2 oH = Ia E A| (f, G1) 2 o
The next three definitions assume the cost function is specified, and have both micro and a maero interpretation.
Equal Effectiveness within Budget The classifier is fair if the same proportion of individuals in the protected subgroups can achieve recourse with a cost at most o:
ecd(c; A, Go) ecd Gz).
Equal Cost of Effectiveness The classifier is fair if the minimum cost achieve aggregate effective-
ness of E [0, 1] in the protected subgroups is equal:
ecd ( (0; A, Go) ecd (d;
Fair Effectiveness- Cost Trade-Off The classifier is fair if the protected subgroups have the same effectiveness-con distribution, or equivalently for each cost budget their aggregate effectiveness is cqual:
max ecd A, Go) ecd(c; A, G1) 0.
The left-hand side represents the two-sample Kolmogorov- Smirnov statistic for the empirical cumula- tive distributions (ecd) of the protected subgroups. We say that the classifier is fair with confidence
if this statistic is less than |-In(aa)) GG
2|G,,ol|G,.11
The last definition takes a micro viewpoint and extends the notion of burden [32] from literature to the case where not all individuals may achieve recourse. The mean recourse cost of a group G,
rc(A,)
rc(A,
G

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considers individuals that cannot achieve recourse through A and have recourse cost of Coe. To exclude them, we denote as G the set of individuals of G that can achieve recourse through an action in A, i.e., G {- E G| E A, h(a(x)) 1). Then the conditional mean recourse cost is the mean recourse cost among those that can achieve recourse:
rc' (A, G) = G
Σ rc(AAx)
DEG
If G = G the definitions coincide with burden.
Equal (Conditional) Mean Recourse The classifier is fair if the (conditional) mean recourse cost for the protected subgroups is the same:
re' Go: =C
Note that when the group G the entire dataset of affected individuals, and all individuals can achieve recourse through A, this fairness notion coincides with fairness of burden [35, 32].
3 Fairness-aware Counterfactuab for Subgroups
This section presents FACTS (Fairness-aware Counterfactuals for Subgroups), framework that implements both the miero and the macro viewpoint, and all respective fairness definitions provided in Section 2.3 to support auditing of the "difficulty to achieve recourse" in subgroups. The output of FACTS comprises population groups that are assigned (a) an unfairness score that captures the disparity between protected subgroups according to different fairness definitions and allows us to rank groups, and (b) istrrintuitive, easily explainable counterfactual summary, which we term Comparative Subgroup Counterfactuals (CSC).
Figure 2 presents an example result of FACTS derived from the adult dataset [1]. The "if clause" represents the subgroup Gp, which contains all the affected individuals that satisfy the predicate p (hours-per-week FullTime) A (marital-status Married-civ-spouse) (occupation Adm-clerical). The information below the predicate refers to the protected subgroups Gp.o and Gp.1 which are the female and male individuals of G, respectively. With blue color, we highlight the percentage cov(Gp.i) = Gp.sl/ | which serves as an indicator of the size of the protected subgroup. The most important part the representation below each protected subgroup and are evaluated in terms of fairness metrics. In this example, the metric the Equal Cost of Effectiveness with effectiveness threshold = 0.7. For Gp.0 there no action surpassing the threshold @ = 0.7, therefore we display message accordingly. On the contrary. the action a (hours-per-week + Overtime, occupation + Exec-managerial) has effectiveness 0.72 > 6 for Gp. thus allowing a respective 72% of the male individuals of Gp to achieve recourse. The unfairness score is "inf", since no recourse is achieved for the female subgroup.
the actions applied that appear
If hours werrs--pprweeeeellll marital sattl-ttata -spouse www.ppttionstttr-lleee clerica:: covered
Protected Subgroup 'Male'. secupatios=Exe-mmnaee with effectiveness 72.00% Make
Protected Subgroup = "Female 1.80% covered
against this subgroup.
Equal Effectiveness (1hreshold=0.7), Unfairness score inf.
Figure 2: CSC for highly biased subgroup in terms Equal Cost of Effectiveness with Φ = 0.7.
Method overview Deploying FACTS comprises three main steps: (a) Subgroup and action space generation that creates the sets of groups & and actions to examine; (b) Counterfactual summaries generation that applies appropriate actions to each group G E G; and (c) CSC construction and fairness ranking that applies the definitions of Section 2.3. Next, we describe these steps in detail.
(a) Subgroup and action space generation Subgroups are generated by executing the fp-growth [11] frequent itemset mining algorithm on D, and on D, resulting to the sets of subgroups Go and 9i and then by computing the intersection g Go n1 In our setting, an item is feature-level predicate of the form "feature-operator-valeee and, consequently, an itemset is predicate p defining subgroup Gp- This step guarantees that the evaluation in terms of fairness will be performed
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between the common subgroups G of the protected populations. The set of all actions A is generated by executing fp-growth on the unaffected population increase the chance of more effective actions and to reduce the computational complexity. The above process is parameterizable w.r.t. the selection of the protected attribute(s) and the minimum frequency threshold for obtaining candidate subgroups. We emphasize that alternate mechanisms to generate the sets A and & are possible.
(b) Counterfactual summaries generation For each subgroup Gp E G, the following steps are performed: (i) Find valid actions, i.e., the actions in A that contain subset of the features appearing in p and at least one different value in these features; (ii) For each valid action a compute eff(a, G,.o) and eff Gp,1). The aforementioned process extracts, for each subgroup Gp, subset V of the actions A, with each action having exactly the same cost for all individuals of Gp. Therefore, individuals of GoDD and Gp.1 evaluated in terms of subgroup-leve actions, with fixed cost for all individuals of the subgroup, in contrast to methods that rely on aggregating the cost of individual counterfactuals. This approach provides a key advantage our method in cases where the definition of the exact costs for actions is either difficult or ambiguous: a misguided or even completely erroneous attribution of cost to the extent that the respective fairness definition allows it. In the setting of individual counterfactual cost aggregation, changes in the same sets of features could lead highly varying action costs for different individuals within the same subgroup.
action will equally affect all individuals of the subgroup and only
(c) CSC construction and fairness ranking FACTS evaluates all definitions of Section 2.3 on all subgroups, producing an unfairness score per definition, per subgroup. In particular, each definition of Section 2.3 quantifies a different aspect of the difficulty for a protected subgroup to achieve recourse. This quantification directly translates to difficulty scores for the protected subgroups Gp.0 and G,.11 of each subgroup Gy. which we compare accordingly (computing the absolute difference between them) to arrive at the unfairness score of each Gp based on the particular fairness metric.
The outcome of this process is the generation, for each fairness definition, of ranked list of CSC representations, in decreasing order of their unfairness score. Apart from unfairness ranking, the CSC representations will allow users to intuitively understand unfairness by directly comparing differences in actions between the protected populations within a subgroup.
4 Experiments
This section presents the experimental evaluation of FACTS on the Adult dataset [1]: more information about the datasets, experimental setting, and additional results can be found in the appendix. The code is available at: https: / ///itttt. com/ wuttoaarrrtttnaa///a the experimental setting and then present and discuss Comparative Subgroup Counterfactuals for subgroups ranked as the most unfair according to various definitions of Section 2.3.
First, we briefly describe
Table 1: Unfair subgroups identified in the Adult dataset.
Subgroup
Subgroup 2
Suhgroup 3
runk
Dias aguiisit
unfairness scote
Fank
biss against
unlaimess score
rank
bias spainst
untaimess score
Equal Edfecciveness
2950
Male
0.11
10063
Female
0,0004
275
Female
0.32
Equal Choice for Recourse (6 - 01.3
Fuir
0
12
Female
2
Fair
0
Equal Choice for Recourse (6 - 0.7)
6
Male
1
Pemale
6
Fair
0
Equal Effasivanss within Budget
B)
Fair
0
2806
Female
03005
70
Famile
01.3
Equal Effectiveness within Budget
110)
2350
Male
0.11
8518
Femule
0,0004
226
Femile
11.
Equal Effectiveness within Budget 18)
2675
Male
0.11
9222
Female
0,0004
272
Femile
0.3
Equal Cos of Fifeeciieenes 69- 0.5)
Fair
0
Fiir
Femile
inf
Equal Cosl of Effletiveenss
(1.7)
Male
inf
12
Femule
2.
Fair
Foir Lifectiveness Cost Trade Off
4065
Male
0.11
3579
Female
0.13
305
Female
0.32
Equal (Conditionll Meon Recoume
Fair
3145
Female
0.35
Fai
Experimental Setting The first step was the dataset cleanup (e.g., removing missing values and duplicate features, creating bins for continuous features like age). The resulting dataset was split randomly with 70:30 split ratio and was used to train and test respectively a logistic regression model (consequently used as the black-box model to audit). For the generation of the subgroups and the set of actions we used fp-growth with 1% support threshold on the test set. We also implemented various cost functions, depending on the type of feature, i.e., categorical, ordinal, and numerical. A detailed description of the experimental setting, the models used, and the processes of our framework can be found in the supplementary material.
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Unfair subgroups Table I presents three subgroups which were ranked at position according to three different definitions: Equal Cost of Effectiveness
0.7), Equal Choice for Recourse (ø 0.7) and Equal Cost of Effectiveness (Φ 0.3), meaning that these subgroups were detected to have the highest unfairness according to the respective definitions. For each subgroup, its rank, bias against, and unfairness score are provided for all definitions presented in the left-most column. When the unfairness score is O, we display the value "Fair" in the rank column. Note that subgroups with exactly the same score w.r.t. definition will receive the same rank. The CSC representations for the fairness metric that ranked the three subgroups of Table at the first position are shown in Figure
Subgroup is ranked first (highly unfair) based on Equal Cost of Effectiveness with Φ = 0.7. while it ranked much lower (see the values of the "rank" column for Subgroup 1). The same pattern is observed for Subgroup 2 and Subgroup they are ranked first based on Equal Choice for Recourse with Φ = 0.7 and Equal Cost of Efffectiveness with ø 0.3 accordingly, but much lower according to the remaining definitions. This finding provides a strong indication of the utility of the different fairness definitions, i.e., the fact that they are able to capture different aspects of the difficulty in achieving recourse.'
is even considered as fair according most of the remaining definitions
Subgroup
age=(41.0,
race=White. relatiinship--uut----aaiii sht
Protected Subgroup 'Male", 1.34% covered
No recourses for this subgroup.
Protected Subgroup "Female" ", 1.77% covered
wwwaaiiossiip-aarreee with effectivaness 70.49% Make marital Marrtue-Mrrid----- -sw--cisssssseee.eeeee
Bias against "Male" due to Equal Cost Effectiveness (thresheld 0.7), Unfairness score inf.
Suhgroup 2
If Dorklass-Private poous--pr--eck-aulTTaa marital Marries -ivecci-sssspeee spouse wwwppttion--tttteeeeeeeeeoo clerical rrce
White:
Protected Subgroup " 1.04% covered
Mike weu--ppr-eed--Derriii occupatios--Bee- managerial with effactiveness
Prof-ssciitt with effectiveness 70.00%
Make hours-per-weeke wer--weeBBBBaiDDaa occupation-Fxee -itriiion---e-emmaaeiiii with effeetivenesé 70.005
peus-per-eeed-Braidd
occspation-Pot--ppeee with effectiveness 70.008
effectiveness 70.00%
Werkchss-Sllf-mme
Mike Workelas--Slf--epp- comp-in.hours... hours pouss-per-eed--Deee weck-Dre/Timee bte./riimeeeccuuatiiieee sccupatios-Exee- Exee--manngeeii with effectiveness 80.00%
Make Sorkclass=SSlff-ee hours-
occupatioosSalee with effectiveness 70.00%
Warkelas--Sef--am-- hours-per-woot- Baans--nr-eddd-aaiiiniiinnn oeespattion--eee- managerial with
70.00%
Protected Subgroup "Fomale". 3.51 covered
Mike WorkclassSelf-omp- hearrrinnhorr---e hours-per-wedd weo---ww--vveiimmm seupptiossSee-mmaggoo with effectiveness 74.51%
Make Sorkelas--Slff-om--- pours-per-wekk- sert--ee-aBaaDDoooi
Catatiion--eeemmaaae with effectiveness 74.51%
against "Fomale" due
Choice
(thresheld Unfairness
Sabgroup
age-(41.0, 50.0], accupation-Sales
Protected Subgroup "Male". 1.18% covered
Make occupation Craft-repair with effectiveness 0.00%
Make oceapation-Amn-cle
0.00%
Make ocapatiaa-tebb-uupprt
effectiveness
Make peappaiion-hot--pecilll with effectiveness 28.218
Make Scapption-Exce-msaagg
with effectiveness 39.74%
Protected Subgroup "Female". 1.56% effectiveness
Make ocappttio-CCatt-reee
Make ocapation-kkm-clcriaal with effeetiveness 0.00%
Make ocuppttion-tthh-spppr with effectiveness 0.00%
with Make acapatian-Paa--seeidti with effectiveness 6.94%
Make occppttin--fee-mmangg
Make age=(30.0.90.0) with effectiveness 6.94%
Make age-(50.0.900 .000..)..cccpptiior pcccppttion-Poo--- Prof-specialty with effecriveness 6.94%
Make oge-(50.0.90) 20000..cceppation ascepaation-Crat: Crapaatioo-Catt--epppae
effectiveness 6.94%
Make age=i50.0 90.0 .ecoppationsAmmcleriii
with effectiveness 6.94%
Make age-(50.0, (0c-(50..990.1 sccapation-Esee-- manaaaaaaallaa with effeetiveness 6.94%
Bias
agoinst "Pemale" due Equal Cost of Effectiveness (threshold 0.3). Unfairmess seare inf.
Figure 3: Comparative Subgroup Counterfactuals for the subgroups of Table 1.
What more, these different aspects can easily be motivated by real-world auditing needs. For example, out of the aforementioned definitions, Equal Cost of Effectiveness with ø 0.7 would be suitable in scenario where a horizontal intervention to support subpopulation needs be performed, but a limited number of actions is affordable. In this case, the macro viewpoint demonstrated in the CSC Subgroup 1 (top result in Figure 3) serves exactly this purpose: one can easily derive single, group-level actions that can effectively achieve recourse for a desired percentage of the unfavored
Additional examples, well as statistics measuring this pattern a significantly larger sample, are included in the supplementary material to further support this finding.

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subpopulation. On the other hand, Equal Choice for Recourse (6 0.3), for which the CSC of a top result is shown in the middle of Figure 3. is mainly suitable for cases where assigning costs to actions might be cumbersome or even dangerous/unethical. This definition is oblivious to costs and measures bias based on the difference in the number of sufficiently effective actions to achieve recourse between the protected subgroups.
Another important observation comes again from Subgroup 1, where bias against the Male protected subgroup is detected, contrary to empirical findings from works employing more statistical-orient approaches (e.g., [34]. where only subgroups with bias against the Female protected subgroup are reported. We deem this finding important, since it directly links to the problem of gerymannering [17], which consists in partitioning attributes in such way and granularity to mask bias in subgroups. Our framework demonstrates robustness to this phenomenon, given that it can be properly configured to examine sufficiently small subgroups via the minimum frequency threshold described in Section 3.
5 Limitations
There exist numerous challenges and constraints when attempting to determine costs associated with specific counterfactuals. The cost functions involved are intricate, dataset-specific, and frequently necessitate the expertise of a domain specialist. These cost functions may depend individual characteristies or human interpretation of the perceived difficulty of specific changes, potentially giving rise to concerns like breaches of user privacy (as the expert may require access to all individual user characteristics [25]) or the risk of malicious specialists manipulating cost functions to conceal existing biases.
We acknowledge the difficulties of finding the dataset-dependen cost functions and proceed to imple- ment various "natural" cost functions tailored to different feature types (e.g., L1 norm, exponential distances). These are only suggestions to the expert who is going to use our framework, are suscepti- ble to change, and can be tailored for the specific datasets. Our primary focus is to identify potential biases towards specifie subpopulations and recognizing the aforementioned inherent difficulties in defining costs, introduce fairness metrics that remain independent of cost considerations. We recognize that a single metric, whether cost-dependent or not, cannot identify all forms of bias that may be present within model's predictions. We do not aim to replace the existing statistical measures for bias detection, rather than complement them by focusing on the bias of achieving recourse.
It is essential to note that FACTS, like other resource generation algorithms, may be susceptible to errors, as demonstrated in prior research [31]. Outcomes produced by FACTS can vary based on the chosen configuration, including hyperparameters and metric selection. These variations can either obscure biases present within a classifier or indicate their absence. since FACTS is explainable framework for bias assessment within subgroups, its primary purpose is to serve as a guidance tool for auditors to identify subgroups requiring in-depth analysis of fairness criteria violations. Responsible usage of our framework necessitates transparency regarding the chosen configuration. Therefore, we recommend running FACTS with different hyperparameters and utilizing many metrics as possible. Appendix G provides a comprehensive discussion of FACTS limitations and responsible disclosure guidelines.
crucial to emphasize that
6 Conclusion
In this work, we delve deeper into the difficulty (or burden) of achieving recourse, an implicit and less studied type of bias. In particular, we go beyond existing works, by introducing a framework of fairness notions and definitions that, among others, conceptualizes the distinction between the micro and macro viewpoint of the problem, allows the consideration of a subgroup as a whole when exploring recourses, and supports oblivious to action cost fairness auditing. We complement this framework with an efficient implementation that allows the detection and ranking of subgroups according to the introduced fairness definitions and produces intuitive, explainable subgroup repre- sentations in the form of counterfactual summaries. Our next steps include the extension of the set of fairness definitions, focusing on the comparison of the effectiveness-cor distribution; improvements on the exploration, filtering, and ranking of the subgroup representations, via considering hierarchical relations or high-coverage of subgroups; user study to evaluate the interpretability of fairness judgments, and the development of fully-fiedged auditing tool.
10

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Acknowledgments and Disclosure of Funding
This work has been funded by the European Union's Horizon Europe research and innovation programme under Grant Agreement No. 101070568 (AutoFair).
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Appendix
This is the appendix the main paper, discussing how numerical attributes are handled (Appendix A), describing in detail the experimental setting (Appendix B), presenting the datasets (Appendix C). providing additional results (Appendix D) and discussion (Appndix E), quantitatively comparing the various fairness notions (Appendix F) and describing in detail limitations and usage guidelines of our framework avoid malicious usage (Appendix G).
A Discussion about Numerical Attributes
FACTS works for datasets and models with numerical or continuous features; in fact it works with any mixture of categorical and numerical features. The core idea is that numerical attributes are binned. These bins used to define subpopulations, e.g., people in the salary range [40K, 50K], and actions, e.g., make salary [50K, 60K). So the action "if salary is [40K, 50K]. make salary [50K, 60K]" means that all individuals within that salary range are mapped to their counterfactuals where their salaries are increased by 10K.
Binning is necessary when defining subpopulations, and present conclusions a manner that is easily understandable by humans, e.g., "there unfairness against females when married=no, salary in [40K, 50K]".
want explore the entire feature space
Binning also necessary when considering actions over numerical attributes. As explained. binning of granularity 10K for salary means that we consider actions that change salary by =10K, =20/, etc. We emphasize that because the number of actions/counterfactuuls infinite, all methods that work with model-agnostic counterfactual-batt explore only a small set of actions.
explanations (e.g.. FACTS, and [31, 25]) have to
Our goal is to approximate the effectiveness-cos distribution, which means we should consider as many actions possible. Therefore, on the one hand, binning for actions should rather be fine- grained, while on the other hand, binning for subpopulations should be moderately coarse-grained so draw conclusions that concern many affected individuals.
With that said, the bimning gramularity for actions and subpopulations can differ. For example consider bins of length 10K for subpopulations and 5K for actions. An action, "if salary is [40K, 50K], make salary [55K, 60K means that individuals with salary in [40K, 45 increase their salary by 15K, and individuals with salary in [45K, 50K] increase their salary by 10K.
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B Experimental Setting
Models To conduct our experiments, have used the Logistic Regression" classification model, where we use the default implementation of the python package scikit-learn', This model corresponds to the black box one that our framework audits in terms of fairness of recourse.
Train-Test Split For our experiments, all datasets are split into training and test sets with proportions 70% and 30%, respectively. Both shuffling of the data and stratification based on the labels were employed. Our results can be reproduced using the random seed value 131313 in the data split function (train_test berii_test__plle splitt from the python package scikit-learph FACTS is deployed solely on the test set.
Frequent Itemset Mining The set of subgroups and the set of actions are generated by executing the fp-growthh algorithm for frequent itemset mining. We used the implementation in the Python package mlxtend" We deploy fp- fp-growth with support threshold 1%, i.e., require the return of subgroups and actions with at least I % frequency the respective populations. Recall that subgroups derived from the affected populations Do and D1 and actions are derived from the unaffected population.
Effectiveness and Budgets As we have stated in Section 2 of our main paper, the metrics Equal Choice for Recourse and Equal Cost of Effectiveness require the definition of a target effectiveness level o, while the metric Equal Effectiveness within Budgel requires the definition of target cost level (or budget) c.
Regarding the metrics that require the definition of an effectiveness level ø, we used two different val- ues arbitrarily, i.e., relatively low effectiveness level of 30% and relatively high effectiveness level of Φ = 70%.
For the estimation of budget-level values we followed more elaborate procedure. Specifically,
Compute the Equal Cost of Effectivenesss (micro definition) with a target effectiveness level of 6 = 50% to calculate, for all subgroups G, the minimum cost required to flip the prediction for at least 50% of both Go and G1.
2. Gather all such minimum costs of step in array.
3. Choose budget values as percentiles of this set of cost values. We have chosen the 30%, 60% % and 90% percentiles arbitrarily.
Cost Functions Our implementation allows the user to define any cost function based on their domain knowledge and requirements. For evaluation and demonstration purposes, we implement an indicative set of cost functions, according to which, the cost of a change of a feature value to the value s' defined as follows:
Numerical features: [norm(v) - norm(v'),, where norm is a function that normalizes
values to [0, 1].
2. Categorical features: if + v', and otherwise.
3. Ordinal features: (pos(v) - (1)) where pos a function that provides the order for each value.
Additionally to the above costs, the user is able define feature-specific weight that indicates the difficulty of changing the given feature through an action. Thus, for each dataset, the cost of actions be simply determined by specifying the numerical, categorical, and ordinal features, well as the weights for each feature.
LogistiiRRegression..ff html
http:///ww....anngggggtt/tldtutt test_ hast_split.hmm httt::://gttab...co///tt//ttittt
train_
15

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Feasibility Apart from the cost of actions, we also take care of some obvious unfeasible actions such as that the age and education features can be reduced and actions should not lead unknown or missing values.
Compute resources Experiments were run on commodity hardware (AMD Ryzen 5600H proces- sor, 8GB RAM). On the software side, all experiments were run in an isolated conda environment using Python 3.9.16.
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C Datasets Description
We have used five datasets in our experimental evaluation; the main paper presented results only on the first. For each dataset, we provide details about the preprocessing procedure, specify feature types, and list the cost feature weights applied.
C.1 Adult
We have generated CSCs in the Adult dataset? using two different features as protected attributes, i.e., 'sex', and 'race". The assessment of bias for each protected attribute is done separately. The results for 'sex' as the protected attribute are presented in the main paper. Before we present our results for race as the protected attribute, we briefly discuss the preprocessing procedures and feature weights used for the adult dataset.
Preprocessing We removed the features 'fnlwgt' and 'education' and any with unknown values. The hours-per-weekk and "age" features have been discretized into 5 bins each.
Features All features have been treated as categorical, except for 'capital-gain' and 'capital-loss' which are numeric, and "education-num" and 'hours-per-week which we treat as ordinal. The feature weights that used for the cost function presented Table 2. We need to remind here that this comprises only an indicative weight assignment to serve our experimentation: the weight below try to capture the notion of how feasible/actionable it is to perform a change specific feature.
Table 2: Cost Feature Weights for Adult
feature name
weight value
feature name
weight value
native-country
Workelass
marital-status
5
hours-per-week
2
relationship
capital-gain
age occupation
10
capital-loss
1
education-nie
C.2 COMPAS
We have generated CSCs in the COMPAS dataset for race the protected attribute. Apart from our results, we provide some brief information regarding preprocessing procedures and the cost feature weights for the COMPAS dataset.
Preprocessing We discard the features 'age' and 'c_ t_charge_deese seseee__ese_____ The 'priors_couue feature has been discretized into bins: [-0.1,1), [1, 5) [5, 10) [10, 15) and [15, 38), while trying keep the frequencies of each bin approximately equal (the distribution of values is highly asymmetric so this not possible with the direct use of e.g., pandas qcut")).
Features We treat the features 'juv fel___uuttt,,,, 'juv_misd miidd_ccunt' juu__other_ove count' as numerical and the rest as categorical. The feature weights used for the cost function are shown in Table
C.3 SSL
We have generated CSCs the SSL ! for race as the protected attribute. Before we move to our results, we discuss briefly preprocessing procedures and feature weights applied in the SSL dataset.
datasets. html
http:////////.//o//////////////a chicago-sslle saaeeoo-ss-cleen.cov
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Table 3: Cost Feature Weights for COMPAS
feature name
weight value
age_cat
10
juv juv_fel_ cou_t
juv uu__el_ count
juv_ ____tther____ other_ccunt
I
priors_coure
__chargg_degree
Preprocessing We remove all rows with missing values ('U' or 'X') from the dataset. We also discretize feature 'PREDICTOR RAT TREND IN CRIMINAL ACTIVITY" into bins. Finally, since the target labels are values between and 500, we 'binarize" them by assuming values above 344 to be positively impacted and below 345 negatively impacted (following the principles used in [3].
Features In this dataset, treat all features numerical (apart from the protected race feature).
Table 4 presents the feature weights used for the cost function.
Table Cost Feature Weights for SSL
feature name
weight value
PREDICTOR RAT AGE AT LATEST ARREST
PREDICTOR RAT VICTIM SHOOTING INCIDENTS
PREDICTOR RAT VICTIM BATTERY OR ASSAULT
PREDICTOR RAT ARRESTS VIOLENT OFFENSES
PREDICTOR RAT GANG AFFILIATION PREDICTOR RAT NARCOTIC ARRESTS
PREDICTOR RAT TREND IN CRIMINAL ACTIVITY PREDICTOR RAT UUW ARRESTS
C.4 Ad Campaign
We have generated CSCs in the Ad Campaign dataset for gender as the protected attribute.
We decided not remove missing values, since they represent the vast majority of values all features. However, we did not allow actions that lead to missing values in the CSCs representation.
Preprocessing
Table 5: The assigned weights for each feature used in the calculation of the cost function in the Ad Campaign dataset
feature name
weight value
religion
5
politics
2
parents
3
age
10
income
3
area
2
college_educater
3
homeowner
1
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Features In this dataset, we treat all features, apart from the protected one, categorical. The feature weights used for the cost function shown Table 5.
C.5
Credit
We have generated CSCs in the Statlog (German Credit) dataset for sex as the protected attribute.
Preprocessing We performed only minimal preprocessing. This involved adding column names, based on the provided documentation, since they are not automatically included in the dataset. We also simplified the values of the 'sex' feature into 'Male' and 'Female', from their original, more complex form (which originally was included in a coded version of the combination of personal status and sex). Finally, changed the labels from {1, 2] to {0, 1].
Features The features 'duration', 'credit', 'rate', 'residence'. "age". 'cards', and 'liables' are treated as numerical, while the remaining features are regarded as categorical. We assigned the same weight value, specifically 1, for all features in the dataset when calculating the cost function, as depicted in Table 6.
Table 6: The assigned weights for each feature used in the calculation of the cost function in the German Credit dataset
feature name
weight value
status
duration
hist
purpose
credit
savings
empl
rate
sex
others
residence
property
age
plans
housing
cards
job
liables
telephone
foreign
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D Additional Results
This section repeats the experiment described in the main paper, concerning the Adult dataset with 'gender' as the protected attribute (Section 4), in three other cases. Specifically, we provide three subgroups that were ranked first in terms of unfairness according to a metric, highlight why they were marked as unfair by our framework, and summarize their unfairness scores according to the rest of the metrics.
D.I Results for Adult with race as the protected attribute
We showcase three prevalent subgroups for which the rankings assigned by different fairness defini- tions truly yield different kinds of information. This is showcased in Table 7. We once again note that the results presented here for 'race' protected attribute, while the corresponding results for 'gender' are presented in Section 4 of the main paper.
Table 7: Example of three unfair subgroups in Adult (protected attribute race)
Sulgroup
Subgroup 2
Sulgooup 3
nack
his againdt
unfairmno scord
mikk
Niw sprindt
unfairmes sccec
rink
biss agvinsi
unfaimess soore
Equal Bilectiveness
Fair
3047.0
Non
White
Equal Choice for Recousse 0.3)
Non White
Non White
Equal Cholce Recourse 0.7)
Fair
0.0
Eggal lztectiveness within Budget
10.01)
White
0.242
Nen White
4035.
White
Bllativeaess willin Buigett
Fair
0.0
Nen White
1663.0
White
Cost Eflectivenzss
Non White
0.15
Non White
Equa Cost Elfectiveness
Fair
0.0
Fair
Far Hilectieenss--Coo Trslo-(ff
900.0
Non- Whote
0.242
4807
WhiteWite
0.115
2644D
Non-WWitte WhitWWite
0, 0,162
Equi (Cuiliioon)) Mocr Recmure
5897.0
White
0.021
$309.0
White
0.047
In Figure 4 we present the Comparative Subgroup Counterfactual representation for the subgroups of Table 7 that corresponds to the fairness metric for which each subgroup presents the minimum rank.
These results are line with the findings reported in the main paper (Section 4), on the same dataset (Adult), but different protected attribute (race instead of gender). Subgroups ranked first (highly unfair) with respect to a specific definition, are ranked much lower or even considered as fair according to most of the remaining definitions. This serves as an indication of the utility of the different fairness definitions, which is further strengthened by the diversity of the respective CSCs of Table 7. For example, the Subgroup 1 CSC (ranked first Equal Choice for Recourse (6 0.3)), demonstrates unfairness by contradicting a plethora of actions for the "White" protected subgroup. as opposed to much fewer actions for the the "Non-Whh White" protected subgroup. For Subgroup 2, much more concise representation is provided, tied to the respective definition (Equal Cost of Effectiveness (ó 0.3)): recourses are identified for the desired percentage of the "Non-White" unfavored population, as opposed to the "White" unfavored population.
D.2 Results for COMPAS
We present some ranking statistics for three interesting subgroups for all fairness definitions (Table 8). The Comparative Subgroup Counterfactuals for the same three subgroups are shown in Figure 5.
Table 8: Example of three unfair subgroups in COMPAS
Subgroup
Subgroup 2
Subggrup3 3
egainst
itftimes scoce
rask
agaiest
unfuimes score
Ios ogainst
unfaiitess KOTE
Exuol Rifactioonns
Fhir
Foir
30
116.0
0.151
200.0
Minran-Ammiin
(L071
Figuil Choiee for Rossens (t.)
Fiir
For
300
5.0
Aftiom-Amriiaan
1.0
Fair
Fair
B0
Equal Choice for Recounse (sl 0.7)
Africas Aunerionn
3.0
Fazr
Fiir
0.0
Fair
Fair
0.0
Exual Bilectoonns within Budga (o-m
Alfican-Ammric
0.167
74.0
Aliion-Amriian American
0U071
Badga
Effectivnees
Far
Adikaai Antefiann
0.0
Eqni Cost of Ellosivmna (9 e 6.7)
Fair
Far
00
Far
Fair
0.0
Fatr
Far
0.0
Fax Ellectomos- Oad Trade-Off
3,0
Amiican-Amarinn
0.5
214.0
Alricn- American
0.151
$76.0
Afrisn-mmiiin
(L071
Equil iC (Coddtoraa) Meas Recousse
5910
Affican Affican-Ameriio
1,667
Far
Fiir
0.0
Afficen American
Hff
D.3 Results for SSL
In Table we present a summary of the ranking statisties for three interesting subgroups. and their respective Comparative Subgroup Counterfactuals in Figure 6.
20

---

Sabgrup1
If Porkelass-Private age-(34.0. 41.0). eapital -gain-D, eapital- marital -tertsst native- CiatiinssiisNoo-in--tae
Cosntry-Unitedd
Protected Subgroup 'Noe-White" 1.095 covered
Make Workela--Fderall -eeee gov, ago-(41.0, 50.0), marital -tatus- -ar---Marrddd---d-eeee sposes,, relatiosship-Mariee
with effectiveness 36.84%. Make -gain----i--1 Marital-status-Meriidd -pposs, relationship-Marial with effeativeness
Make age=(41.0 o. s0.0], capitl-gain= marital -ataa----aaaaa-rd------------ relationship=Mariee with effectiveness 180.00%.
Protected Subgroup "White",
cuvered
Make age=/41.0, 50.0], marital -statss--ttttt-Maarraa -poesse relationthip=Mariie effectiveness
45.14%.
Make marital-statuseMmrre Marree-Mdddddd---sooooee relationshipsMMriied with effectiveness 40.00%
Make age=(50.0. 90.0],
Married-elv spouss, relationship=Mariiel
effectiveness
428.
Make Workclas=LLeell-o age=(41.0, 50.0],
maritul mariaal-tunooeeaMaaeed---------- relationshipsMariin
with effectivesess 44.008.
WorkelawsLLeal--or age=(41.0, 50.0], partidl--bblussaaaiie-------pppo relationship-Mariied
with effectiveness. 44.00%.
Make
Workelas-Self -emp-ine, Som------ll------nn,, age=(41.0. 50.0], maiial--oatts--Mridddd-------- relationship-Mariiee
with effectiveness 52.57%.
Make
Serkelaw=Sllf-epp-ae
a5e=(50.0, 90.0].
mariall-ttttue22arr-----------p- relationship-Mariid
effeetivenesa 49.71%.
Make Vorkdlas=Lussal-oor maw.hhh---taaasaarrrrddd--ee-ee- oag. with effectiveness
36 008.
Workelas==Selfeemp-ti mariaal-tatuu--aarii------------ relatiosship-Mariidd
effectivenexs 45.14%.
Make Workelasse Federal-gov -0 50.0]. marital mariaa--ttatussaaiiid------ relationship-Mariid
with effectiveness 62.29%
Make
Workelas=statee
agez(41.0, 50.0], marital wariiaal-tttttteeaaaddded------- relationship=Morrid
with effectiveness 40.57%.
Make capital -qain=15444 mariuul-stattss-Maridddd--- bttp::/---puuuueoaaaaaaaaoaaaiaa relationssip=MMrried with effectiveness 99.43%.
Workelass-Lecal-por, age=(50.0, 90.01, satiaal-tttuuu--Maeedd---------- relationsshp-Marriee
with effectiveaess 40.00%.
Make age=(4 0. 50.0]. capital -gin==103 marinl--sausssmmiddii------- -ppouss relationship=Mariie
effectiveaess 100.00%.
against White
Equal for (threshuld 0.3). Unfairness score 10.
Subgroup 2
hours ppers--pe-week FullTime. native -country United-States, occupation Adn-clerical, relationship
Married:
Protected Subgroup
Roe-White: 1.66% covered
No recourses for this subgroup-
Protected Subgroup "White's 1.66% covered Make heurs -iss-ppee week BrainDrain. occupation Buss-managerial effectiveness
Bias against "Noe-While White due to Equal Cost of Effectiveness (threshold = 0.3). Unfairness score inf.
Sabgroup
houts-per-weck PariTime.
Divorced, native-countey United-States: Protected Subgroup = Woit-Witte " 1.15% covered
Make marital
With effectiveaess 0.00%.
Make hours-per-wead-bee
marital-ssata--- -stata-Marriad -spuse with effectiveness 0.005..
with effeativeness
Make hours Maurs-per-weeb-allTTie
marital -trrtus-
Make heurs hours-per--ee perr--pe-ee--uriiiieee marital Marree -clv
-iv----ousss -speuse with effectiveness 0.00s.
Make hours-pes houri-per-wwd---e per-wedd-tverrinee
Make hears-per- tours-per-weok-BBie -er----rr----iiiaariiii marital-statuss AtatussNNvad-mrrde with effactiveness 0.00..
siv-se
effectiveaess 0.00%.
Protected Subgroup
1.66 covered
Make marital stataal-stttt-Mariie -arriedd-iiv--- -ividd-----pouse with effectiveneas 101S..
Make hours ponr--ppr-eed--ii/iii marital statt--arrieddll----a--
with effectiveness 1.01%.
Make hourt-per-wol--ullTiie
marital -statts--aariiddddii
effectivenest 7.07%..
Make hours-per-ee Saars--pr-wcad-Ovvriine marital- -status-Marrid-civ -iy-eee with effectiveness 7.07%..
Make hours poors--er-ee -er----eee---rellllee
marital Neatus-Never-mrriia with effectiveness 15.15%.
Make hours-per ppere-ppeewweeeeee Bradd-BraisDaiis marital Marr------i-----eee with effectiveness 16.168.
agoinst "Noa-White due Equal Conditionsl Mean Recourse. Unfairness
Figure 4: Example of three Comparative Subgroup Counterfactuals in Adult (protected attribute race); ref. Table 7
D.4 Results for Ad Campaign
In Table 10 we present, we did for the other datasets, the ranking results for 3 interesting subgroups, while in Figure 7, we show the respective Comparative Subgroup Counterfactuals for these subgroups.
21

---

Sabgrup
If sge cat 45, c_charge degree M. juy misd coast 9, priors count (10.0, 15.0]:
Protected Subgroup 'Caucosian" 1.36 covered
Make e_charge deggeeee,. priors countst--.1.
effectiveness 100.00%.
Make srioors_ountt(-a0)
1.0]
effeetivenesA 100.009.
Make silors_coutt(110. 5.0] with effectiveness 100.00S..
Make cat=Grester than 45, priors count=(-0.1, 1.0] with effectiveness 100.00%.
Make age_eat-Greeter
c_sharge_degreen
priers_couet-(-D0
effectiveness 100.00%.
Make age CatzGreater than 45, degrresr, priors_cuunt=(( (1.0. 5.0] with effectiveness 100.003.
Make age catzGrester than 45, priors cosst=(1.0. (1.0, 5.0] with effectiveness 100.00S.
Protected Subgroup - 'African American" 1.16 covered
Make mrtors_countt((11
1.0]
83.33%
Make age cat=Greater than 45, priors posst=(-0.1; 1.0] effectiveness
Make age_eat eat--atrrrrerr thas 45,
priors_count-i __count-(-B-1,, 1.0] with effectiveness 100.00%.
Make cat_eat=rreatter than 45, priors_ count=((.0, 5.0] with effectiveness 83.33% African-Aman
Equal Choice
(threshold 0.7). Unfairness
Subgroup 2
If c_charge degree M. other count
Protected Subgroup 'Caucasian 3.59%
Make jay e__other_coope Protected Subgroup = "African-Amerrce" AAmeriann " 3.48% covered
with effeetiveness 42.86%.
No revourses this subgroup.
against African
Equal Effectiveness (threshold 0.3), Unfairmess score
Sabgrop
cat_ cat
45, c_cbarge_degres jav_ffii_count juv_misd_count 0, juy_other_coun o,
sex Male:
Protected Subgroup = 'Caucasian' 7.18% covered
Make e_charge_degreesth
effectiveness 7.14%
Protected Subgroup
Make "_charge_degrees with effectiveness 0.008.
Bias agaiast African American due Equal Conditional Mean Recourse. Unfairmess score int.
Figure 5: Example of three Comparative Subgroup Counterfactuals in COMPAS; ref. Table 8
Table 9: Example of three unfair subgroups in SSL
Subgroup
Subgrmmp
Sabgroup
Eflssiivesss
Black
0.077
11.0
Black
for
12.0
Black
Choice for Recourse
Black
Fiir
Fiir
Effectiveeess Budget
195.0
1692.0
While
Effectiveness within Budget
2427.0
Black
0.111
126.0
3636.0
White
Lffectiveness within Budget
2557.0
Black
0,076
13.0
1496.0
Black
Coe Eifeectiveess (6 0.3)
0.0
Block
Black
Equal Coe of Fiffectiveess (6 0.7)
Black
inf
Fair
Fair
For
Fair
0.0
For FiforrieeeeeeCCCo Cod Tradk-Off
1901.0
Black
0.111
443.0
Black
0.663
2605.
Black
0.151
Eipal (Condtoonl) Mean Recourse
3496.0
Black
0,033
Black
1374.
White
0.95
Table 10: Example of three unfair subgroups in Ad Campaign
Subgroup
Subgroup
Subgroup
biss unfaimess
bizs
unfairmess
rank
against
unfairness
Effectiveness
319.0
Female
0,286
Fair
467.0
(1059
Choics Recourse
5.0
Female
Female
Fair
0.0
Choic:
Fair
Funtle
Fair
Effectienees within Rudget
Fair
0.3)
Difectiveness 0.7)
Fii Hlfetivenes-CCo
Fonale
547.0
Equal (Condiioaa) Recourse
22

---

Subgroup 1
If PREDICTOR RAT ARRESTS VIOLENT OFFENSES 1, PREDICTOR RAT NARCOTIC ARRESTS 1. PREDICTOR RAT VICTIM BATTERY OR ASSAULI
Protected Subgroup 'Black' 1.04% covered No recourses for this subgroup.
Protected Subgroup 'White" 1.0% covered
Make PREDICTOR RAT ARRESTS VIOLENT OFFENSES PREDICTOR RAT NARCOTIC ARRESTS O. PREDICTOR RAT VICTIM BATTERY OR ASSAULT
with effectiveness 72.73%
Make PREDICTOR RAT ARRESTS VIOLENT OPPENSES d. PREDICTOR RAY NARCOTIC ARRESTS 1. PREDICTOR RAY VICTIM
BATTERY OR ASSAULT = O with effectiveness 72.73%
Make PREDICTOR BATTERY OR ASSAUTT
ARRESTS VIOLENT OPRENSES
PREDICTOR RAT NARCOTIC ARRESTS PREDICTOR RAT VICTIM
with effectiveness 72.73%
Bias against "Black" due to Equal Cost of Effectiveness (threshold - 0.7). Unfairness score - int.
Subgroup
PREDICTOR RAT NARCOTIC ARRESTS PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (-0.2, -0.1], PREDICTOR RAT LEW ARRESTS O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS O:
Protected Subgroup "Black" 2.51% covered
Make PREDECTOR RAT NARCOTIC ARRESTS
PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (-8. 20099999999999
-0.31. PREDICTOR RAT LEW ARRESTS O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS with effectiveness
0.00%
Mike PREDICTOR RAT NARCOTIC ARRESTS O. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (-0.1, O. PREDICTOR LLW ARRESTS
VICTIM SHOOTING INCIDENTS Mike PREDICTOR RAT NARCOTIC ARRESTS PREDICTOR RAT TREND IN CRIMINAL ACTIVITY = (0.1, 0.3], PREDICTOR
PREDICTOR
effectiveness 0.00%
RAT LUW ARRESTS - O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS - O with effestiveness 0.00%
Make PREDECTOR RAT NARCOTIC ARRESTS PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (-8. 200999999999999
-0.31, PREDICTOR RAT
ARRESTS
PREDICTOR
VICTIM SHOOTING INCIDENTS
Make PREDECTOR RAT NARCOTIC ARRESTS O. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY - (0.3, 7.31, PREDICTOR
RAT LLW ARRESTS O. PREDICTOO RAT VICTIM SHOOTING INCIDENTS O with effectiveness 0.00%
Mike PREDECTOR RAT NARCOTIC ARRESTS PREDICTOR RAT LUW ARRESTS O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS
PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (-0.3. -0.21
with effectiveness 0.00%
Mike PREDICTOR RAT NARCOTIC ARRESTS 1. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY = (-B.1, 0.1 PREDICTOR
RAT LEW ARRESTS - O, PREDICTOR RAT VICTIM SHOOTIIN INCIDENTS - O with effectiveness 0.00% PREDICTOR
NARCOTIC ARRESTS O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS
PREDICTOR RAT
IN CRIMINAL ACTIVITY (-0.2. -0.11.
PREDICTOR RAT
with effectiveness 0.00%
Mike PREDECTOR RAT NARCOTIC ARRESTS I. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (0.1, 0.31, PREDICTOR RAT LUW ARRESTS O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS
O with effectiveness 0.00%
Mike PREDICTOR RAT NARCOTIC ARRESTS
PREDICTOR RAT TREND IN CRIMINAL ACTIVITY
200990099999000
-0.31. PREDICTOR RAT UUW ARRESTS O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS 0 with effectiveness
0.00%
Mike PREDECTOR RAT NARCOTIC ARRESTS 1. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (0.3, 7.31. PREDICTOR UUW ARRESTS
PREDICTOR
VICTIM SHOOTING INCIDENTS
effectiveness 0.00%
Protected Subgroup "White" 2.878 covered
Mike PREDICTOR RAT NARCITIC ARRESTS O. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (-8. 200999999999999
57. -0.31, PREDICTOR RAT UOW ARRESTS O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS 0 with effectiveness
Make PREDICTOR RAT NARCOTIC ARRESTS O. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY = (-0.1, 0.11. PREDICTOR
RAT ULW ARRESTS = O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS = O with effectiveness 0.00%
Mike PREDICTOR RAT NARCOTIC ARRESTS O. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (0.1. 0.3], PREDICTOR LLW ARRESTS
SHOOTING INCIDENTS = Make PREDICTOR RAT NARCOTIC ARRESTS PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (-8. 20099999999999
PREDICTOR
effectiveness 0.00%
-0.3]. PREDICTOR RAT LOW ARRESTS O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS 0 with offectiveness
0.00%
PREDICTOR NARCOTIC ARRESTS
PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (0.3, 7.31, PREDICTOR
RAT LUW ARRESTS = O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS = O with effectiveness 0.00%
Make PREDICTOR RAT NARCOTIC ARRESTS O. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY = (-0.3, -0.21.
PREDICTOR RAT UUW ARRESTS O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS
with effectiveness 0.00%
PREDICTOR RAT NARCOTIC ARRESTS
PREDICTOR RAT
CRIMINAL ACTIVITY (-0.1. 0.11 PREDICTOR
RAT ULW ARRESTS = O. PREDICTOR RAT VICTIM SHOKTING INCIDENTS = with effectiveness 0.00%
Make PREDICTOR RAT NARCOTIC ARRESTS J. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY = (-0.2, II.
PREDICTOR RAT LUW ARRESTS O. PREDICTOR RAI VICTIM SHOOTING INCIDENTS PREDICTOR
with effectiveness 0.00%
NARCOTIC
PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (0.1, 0.31, PREDICTOR
RAT LUW ARRESTS = O. PREDICTOR RAT VICTIM SHOOTING INCIDENTS = with effectiveness 0.00%
Make PREDICTOR RAT NARCOTIC ARRESTS = 2. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY = (-8.2009 200999999999999 -0.31. PREDICTOR RAT UUW ARRESTS
PREDICTOR RAT VICTIM SHOOTING INCIDENTS
with effectiveness
Make PREDICTOR RAT NARCOTIC ARRESTS 1. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY (0.3, 7.31. PREDICTOR RAT LUW ARRESTS = 0. PREDICTOR RAT VICTIM SHOOTING INCIDENTS = 0 with effectiveness 0.00%
Bias
"Black" due Egual Conditional) Mean Recourse. Unfairness score inf.
Subgroup 3
1f PREDICTOR RAI GANG APFILIATION 1. PREDICTOR RAT NARCOTIC ARRESTS 2. PREDICTOR RAT TREND IN CRIMINAL ACTIVITY 1-8.20099998999999 -0.31. PREDICTOR RAT VICTIM SHOOTING INCIDENTS O: Protected Subgroup "Black" " 1.88% covered
No recourses for this sabgroup.
Protected Subgroup "White 1.00% covered
Make PREDICTOR RAT GANG AFFILIATION O. PREDICTOR RAT NARCOTIC ARRESTS PREDICTOR RAT TREND CRIMINAL ACTIVITY offectiveness 31.82%
9.92.8999999999999 -0.31, PREDICTOR RAF VICTIM SHODLING INCIDENTS
Bias against "Back"" due to Equal Cost of Effectiveness (threshold 0.3). Unfairaess score
inf.
Figure 6: Example of three Comparative Subgroup Counterfactuals in SSL; ref. Table 9
23

---

Subgroup 1
45-54, Unknown.
Protected Subgroup
Make age=55-64, area=Rural with effectiveness 30.77%. Protected Subgroup 'Female" 1.15 covered
No resourses this against
Erfectiveness (threshold=0.3) Unfairness
Subgroup 2
55-64,
Unknown,
polities Unknown, religion
Unknown:
Protected Subgroup 'Male", 2.53% govered
Make nomeowner-0, parents-l with effectiveness 100.00%.
Make homeowner=0, parents=l, veligios-Caristianiyy with effectiveness 100.00%
homeowneraD,
parentszl. relizion=Other
effectiveness
Make area=Urban.
parents-]. religion=Chiistiasiyy with effectiveness 93.91%.
Make area=Urban,
parents=]. religion=Other with effectiveness 93.91%.
Make homeowner-0, income-<100K, parents-l
effectiveness 100.005.
area-Kural,
parentse]. religion=Uther
effectiveness 100.00%.
Make area=Rural, parents=]. religionsChrissiasig with effectiveness 100.00%.
Make homcowner-D, income=cl000, parenti-l, sellgiou-Christianii with effactiveness 100.00%.
Make homeowner-l)
incom=<cl00,, parents=l, religion-Other with effectiveness 100.005.
Subgroup 'Female' Make homeowner-D, parents-l with effectiveness 100.00%.
covered
Make homcowner=D) parents-l, teligios-Caristianii with effectiveness 100.00S. Make homeowner=0),
religion=Other with effectiveness 100.00%.
Make homeowaer=D), income<<ll0x, parenti-l effectiveness
Make homeowneral), incom=<cI00K, parentsel,
with effectiveness 100.00%.
Make homeowner-D, income<c100K, parenti-l, religion-Other with effectivencas 100.00%.
Choice Recourse (thresheld Unfaireess
Sabgrup.3
If ages 55-64, income <100K. religion Unkaows:
Protected Subgroup
1.02%
Make religion-Christan
effectiveness
Make rellgion-Other with effectiveness 0.00%.
Protected Subgroup 'Fomale". 1.08% covered
Make Daligion-Christianii
9.868.
Make religion-Oe
effectiveness 9.86%.
Bias against Male due ta Equal Conditional
Mon Recourse. Unfairness score inf.
Figure 7: Example of three Comparative Subgroup Counterfactuals in Ad Campaign; ref. Table 10
24

---

E Further Discussion on Experiments
E.1 Model Agnostic Results
We note that our method is model-agnostic and does not depend on the model class. Nonetheless, we have considered two additional models, XGBoost and a Neural Network, for the Adult dataset. We provide an example CSC for Subgroup of Figure 3, one for the output of each model (XGBoost in Figure and NN in Figure 9)
age (41.0, 30.01. -evet otw.aaaiee............aaaeeeeeee relaatiosstipp-oo-----t-
Protected Subgroup "Male" 1.13% covered No requerses for this subgroup.
Protected Subgroup "Female' 1.14% covered
No requerses No blas
subgroup.
Figure 8: Output of FACTS on an XGBoost model, for the same subgroup as Subgroup 1 of Figure 3.
Sabgropp
age=(41.0,
pariial-staluusaNeer---rr- race=White. adlatioshhip-No--ii-fmmii. Protected Subgroup Male 1.21% covered
No requerses for this subgroup.
Protected Subgroup 'Female" ", 1.488 covered
Make marital sartsssMarried-ee -ipoossses.. relationship=Mariied effectiveness
Bias against "Male" due to Equal Cost of Effectiveness (threshold 0.7). Unfairness score inf.
Figure 9: Output of FACTS on a simple NN model, for the same subgroup as Subgroup 1 of Figure 3.
E.2 Generalization to multiclass demographic groups
The generalization of the algorithm to multiple protected groups (comprising multiclass demographic groups) straightforward process. For instance, we can compare the burden of each protected group against all other groups using a one-vs-all approach. An example of this methodology applied the Adult dataset and its multi-valued race attribute is illustrated in Figure 10.
1f capital -gann=ggnnnn---- marital-sistnn-Divvree
Protected Subgroup 'Asian-Pic- 12.95% covered
capital -gain-15024. Marid -spose effectiveneas
Make -gann=7888 marital -stitsl--ttttssaMariii -eir -eiv---osss with effectiveness 75.00%.
Make capital -aaiill-ga--778 maritaalla---ttts------ ----Marridd-i----pe -iiv--soee with effectiveness 75.00%.
Protected Subgroup 'Amer-ladian
covered
Make capital - -ipooo with effectiveness
Make capital-gaan- -gain maritat mar.mararaaasstttaa Maristd--Maiiieee----- with effectiveness 81,829..
Make
with effectiveness 72.73%
Subgroap Black 15.15%
Make capital -aiinal-3 saaiii--0222.mmisttttttttt. www.mstl-ttttt--aarddddd..
effectiveness
Protected Subgroup = 'Other" 12.975 covered
Make capital -apinal--ainn150000 -gain=15044
with effectiveneas 100.00%.
Make capital -gain-7688, httaa-/88..arinll-sttt----------
76.929.
Make -aiin=--g.nn7298... -writal-sttttsselll
-iiriidddi---spsss with effectiveness 69.23%.
Protected Subgroup (iia-1102,,maritttt sattll---tttt-Mariee sorrieddeiv-spouue with effectiveneas 99.418.
'White 16.96% covered
Make sapital -apital-gain- -gain-15024
Make capital -gain=7688, maaa -aras----oiadll---sspse
Make capital -gain
aarariaa--ttttes -Mtrris-Marridd -eiv--passe with effectiveness 70.43%.
Bias against "Black" due to Equal Choice of Revourse. Unfairness score = 2.
Figure 10: Output of FACTS for Adult for multi- valued protected attribute (race); all protected subgroups other than 'Black' have three options (actions) achieve recourse with an effectiveness of at least 0.7, while 'Black' has a single option. Therefore we conclude bias against the 'Black" using the Equal Choice of Recourse.
E.3 Runtime and statisties
In Table we present the runtime of FACTS for all datasets. We also provide statistics on the output generated by fp-growth for all these datasets, offering preliminary insights into scalability.
25

---

Additionally, we include the runtime for the Adult dataset, with sex serving as the protected attribute. for various combinations of minimum support threshold and the number of bins in the numerical attributes (see Table 13). These results encompass both the total runtime of FACTS and the time taken by FACTS the assessment of fairness step. In Table 14 we report the corresponding statistics from fp-growth for the same combinations as presented in 13. It is worth noting that our primary contribution is conceptual nature, focusing on the formalization of recourse unfairness notions, rather than technical, i.e. how to efficiently determine good set of actions to explore.
Table 11: Runtime for all datasets (min-support threshold: 0.01 for Adult, COMPAS, SSL. Ad campaign: 0.04 for German credit)
Adult (sex)
Adult (race)
COMPAS
SSL
Ad campaign
German credit
Runtime (scconds)
1,875
2,208
11
1,807
860
689
Table 12: Statistics on the output of fp- growtt for all datasets
Adult (sex)
Adult (race)
COMPAS
SSL.
Ad campaign
German credit
# male (white) subgroups (affected)
27,510
27,767
1431
7,836
1.658
244,777
# female (non-white) subgroups (affected)
27,
24,664
3,984
7,801
1,700
241,287
# actions (unaffected)
$7,008
56,394
2,895
17,270
3,476
83,631
# subgroups
13,300
18,692
95
6,552
1,651
36,970
# valid actions
97,413
112,879
188
86,782
4,274
117.143
# subgroups
31,509
Table 13: Runtime for Adult (sex) while varying min-support threshold and number of bins
(min-support threshold, # bins)
(0.1, 10)
(0.05, 10)
(0.01, 10)
(0.05. 5)
(0.05, 20)
Runtime (seconds)
35
178
3,539
265
155
Runtime to assess fairness (seconds)
8
10
17
10
10
Table Statistics on the output of fp-growth for Adult (sex) while varying min-support threshold and number of bins
(min-support threshold, # bins)
(0.1,10)
(0.05,10)
(0.01,10)
(0.05.5)
(0.05,20)
# of male subgroups (affected)
997
3,017
25,569
4,017
2,309
# of female subgroups (affected)
933
3,024
26,173
4,021
2,397
# of common subgroups
453
1,583
12,093
1,969
1,126
of actions (unaffected)
2,828
7,777
60,006
10,709
5,961
# of all valid actions
351
3,120
87,687
4,348
2,538
of subgroups
283
1,235
10,975
1,706
887
26

---

F
Comparison of Fairness Metrics
The goal of this section is to answer the question: "How different are the fairness of recourse metrics". To answer this, we consider all subgroups and compare how they rank in terms of unfairness according to 12 distinct metrics. The results justify our claim the main paper that the fairness metrics capture different aspects of recourse unfaimess. For each dataset and protected attribute, we provide (a) the ranking analysis table, and (b) the aggregated rankings table.
The first column of the ranking analysis table shows the number of the most unfair subgroups per metric, i.e., how many ties are in rank 1. Depending on the unit of the unfairness score being compared between the protected subgroups (namely: cost, effectiveness, or number of actions), the number of ties can vary greatly. Therefore, we expect to have virtually no ties when comparing effectiveness percentages and to have many ties when comparing costs. The second and third columns show the number of subgroups where we observe bias in one direction (e.g., against males) and the opposite (e.g., against females) among the top 10% most unfair subgroups.
The aggregated rankings table used as evidence that different fairness metries capture different types of recourse unfairness, Each row concerns the subgroups that are the most unfair (i.e.. tied at rank 1) according to each fairness metrie. The values in the row indicate the average percentile ranks of these subgroups (i.e., what percentage of subgroups are more unfair) when ranked according to the other fairness metrics, shown as columns. Concretely, the value v of each cell i, j of this table computed as follows:
1. We collect all subgroups of the fairness metric appearing in row i that are ranked first (the most biased) due to this metric.
2. We compute the average ranking of these subgroups in the faimess metric appearing in column j.
3. We divide with the largest ranking tier of the fairness metric of column j to arrive at e.
Each non non-diagonal value of this table represents the relative ranking based on the specific metric of the column for all the subgroups that are ranked first in the metric of the respective row (all diagonal values of this table are left empty). A relative ranking of v in a specific metric m means that the most unfair subgroups of another metric are ranked lower on average (thus are fairer) for metric m.
F. Comparison on Adult for protected attribute gender
The number of affected individuals in the test set for the adult dataset is 10,205. We first split the affected individuals the set of affected males D, and the set of affected females Do. The number of subgroups formed by running fp-growth with support threshold 1 1% on D, and on Do and computing their intersection is 12,880. Our faimess metrics will evaluate and rank these subgroups based on the actions applied.
Tables 15 and 16 present the ranking analysis and the aggregated rankings respectively the gender attribute, on the Adult dataset. Next, we briefly discuss the findings from these two tables; similar findings stand for the respective tables of the other datasets, thus we omit the respective discussion.
is evident from Table 15 that the different ways to produce ranking scores by different definitions can lead to considerable differences in ties, i.e., the number of subgroups receiving the same rank (here only rank is depicted). The "Top 10%" columns demonstrate interesting statistics on the protected subgroup for which bias is identified: while it is expected that mostly bias against "Female" will be identified, subgroups with reverse bias (bias against "Male") "" are identified, indicating robustness to gerrym verrymandeinn, as hinted in Section 4 of the main paper.
Table 16 is produced provide stronger evidence on the unique utility of the various presented definitions (see footnote 1 of the main paper: "Additional examples, as well as statistics measuring this pattern on a significantly larger sample, are included the supplementary material to further support this finding.") "). In particular, in this table, for all subgroups that are ranked first in definition, we calculate their average relative (normalized in [0. 1) ranking the remaining definitions. Given this, a value close to 1 means very low average rank and a value close to 0 means very high rank. Consequently, values away from 0 indicate the uniqueness and non-triviality of the different definitions and this becomes evident from the majority of the values of the table.
27

---

Table 15: Ranking Analysis in Adult (protected attribute gender)
Most Unfaic Subgroups
Subgroups Bias against Males Top 10% Unfair Subgroups)
SSubgroup Bias against Femalcs Top Unfair Subgroups)
(Equal Cost of Effeccivvnesst Macre))) 0.3)
1673
56
206
(Equal Cost EfficciivenessMMacre 0.7)
301
26
37
(Fquua Choice for Recourse, 0.3)
54
286
(Equal Choice for Recourse, 0.7)
31
30
Equal Effectiveness
1040
(Equal Effectiveness within Budget, 5.0
41
616
(Equal Effectiveness within Budget, 10.0)
904
(Equal Effectiveness within Budget, 18.0)
22
964
(Equal Cost of Effectivenness 0.3)
1523
10
226
(Equal Cost Effectiveness
0.7)
290
38
27
Equali Condiioootl Mean Recourse)
764
540
565
(Fair Effectiveness Cost Trade Off, value)
61
1156
Table 16: Aggregated Rankings in Adult (protected attribute gender)
8-100 11.1
1o-go 1441
(Eeal Decid Erponhaoo Narvocy
0:31
113.3
1.3
0.214
0.505
8.352
1.355
8.5
120
0.40
0357
(Egial Chils Acsuess 321
0.23
13.80
0.511
0130
8:913
0181
1.0
0.83
0180
0.3
(Eeal Diockaac widle brdos 13
802.6
13
1.34
1.36
1.03
3017
1.0
038
030
020
(Real Gecd Kiichhae Marza All
8338
0351
1.
09266
0453
8.30
3215
10
046
615
0.0
0224
030
050
F2
Comparison on Adult for protected attribute race
The number of affected individuals in the test set for the adult dataset is 10,205. We first split the affected individuals on the set of affected whites Dj and the set of affected non-whites Do. The number of subgroups formed by running fp-growth with support threshold 1% on D, and on Do and computing their intersection is 16,621. Our fairness metrics will evaluate and rank these subgroups based on the actions applied.
Table 17: Ranking Analysis in Adult (protected attribute race)
Subgroups against Whites
Subgroups against (in Top 10% Unfair Subgroups)
Subgroups
(in Top 10% Unfair Subgroups)
(Equal Cost of Eflectiveness(Macca 0.3)
1731
0
295
(Equal Cost of FRlectiveness(Maern)) 0.7)
325
51
(Equal Choice for Recourse, 0.3)
391
(Equal Choice for Recourse, 0.7)
2
10
60
Equal Eflectiveness
6
1433
(Fiuul Effectiveness within Budget, 1.15
1
50
24
(Equal Effectiveness within Budget, 10.0
(Equal Effectivencss within Budget, 21.0)
0
1423
(Equal Cost of Elfectiveness/ (Mcro)), 0.3)
1720
294
(Equal Cost of Effectiveness 0.7)
325
7
Equal Ponua(CCndittonal Mean Recourse)
2545
1316
(Fair Effectiveness-Cor Tradc-Off, value)
2
F.3 Comparison on COMPAS
The number of affected individuals in the test set for the COMPAS dataset 745. We first split the affected individuals on the set of affected Caucasians Dj and the set of affected African Americans Do- The number subgroups formed by running fp-growth with support threshold 1% on Dj and on Do and computing their intersection is 995. Our faimess metrics will evaluate and rank these subgroups based on the actions applied.
F.4 Comparison on SSL
The number of affected individuals in the test set for the SSL dataset is 11,343. We first split the affected individuals on the set of affected blacks Dj and the set of affected whites Dg based on the race attribute (appears with the name RACE CODE CD in the dataset). The number of subgroups
28

---

Table 18: Aggregated Rankings in Adult (protected attribute race)
(Raal Cerrd Alflloomm (ar
0.5
0.539
3.403
02713
0.313
01000
1.08
0700
0142)
0.13
11100
1.8
1.050
1121
0111
1.62
ah Michore widi let
4331
1000
0331
8.311
1181
130
0300
1385
1008
0311
130
(Sea becel Moms
0000
04
3.20
3140
0000
2.00
148
0411
00
la Cal Iod valaa
6012
600
Table 19: Ranking Analysis in COMPAS
# Most Unfair Suhgroups
SShhgruups w. Bias against Coneasians (in Top 10% Unfair Suhproups)
Sabgroupe w. Biss agrins AfricunAAmrriaas (in Top 10% Unfaii Subproups)
(Equal Cost of Elfectiveness Macro), 0.3)
51
0
(Equal Cost of Bffectivaness(Macro)) 0.7)
(Equal Chosce for Recourse, 0,3)
13
12
8
(Equal Choice Recourse, 0.7)
8
6
Equal Effactivenees
1
(Equal Effectiveness within Budget, 1.0)
4
16
30
(Equal Effectiveness
Budget,
(Equal Cost of (enetiivenes(MMicn) 0.3)
0
(Equal Cost of Bffeetivcness(Mico 0.7)
46
0
6
EnnaatConditional Mean Recourse)
(Fair Effectivenes--Coo Trade-Off, valuc)
5
18
62
Table 20: Aggregated Rankings in COMPAS
Coc Mous
Cod MMrrn
Crals
Eig Clace
Epuil
Eigel
Sical
Stcaal Cast
Goad Car.
Ssloen
Eedgen 2005
(Maxx.7)
Dilechmow (.3)
131
$1300
032
02301
(lyn Claice Rarein by
0:33
6.76
0.115
0.333
1.0
01170
18.33
0.50
0.301
0.16
Sqpll Dill Eimnn
013
1.0
0.20
1.3
1.0
01005
033
1.0
2.911
0101
(igul Bitadiane will Bulecs 36.0
162
1.0
0.22
1.3
9.00
10
02
18
2.00
0200
dog Us tihechooo coooooaann
6.9
012
1922
1225
1123
031
03300
100
0022
011.0
(ou iovee Tade OI. salet
1.64
140
030
6931
068
formed by running fp-growth with support threshold 1% on D1 and on Du and computing their intersection is 6,551. Our fairness metrics will evaluate and rank these subgroups based the actions applied.
Table 21: Ranking Analysis in SSL
Most Unfair Subgroups
# Subgroups w. Bias against Whites (in Top 10% Unfair Subgroups)
# Subgroups w. Bias against Blacks (in Top 10% Unfair Subgroups)
(Equal Cost Eifeetiiveness(Marro), 0.3)
107
(Equal Cost of Fifeetiveness Macro), 0.7)
627
26
124
(Equal Choice for Recourse, 0.3)
108
184
(Equal Choice Recourse, 0.7)
Equal Effectiveness
15
389
(Equal Effectiveness within Budget, 1.0)
18
18
436
(Equal Effectivencss within Budget. 2.0)
532
(Equal Effectiveness within Budget, 10.0)
15
548
(Equal Cost of ElffectivenessMMic Micro)), 0.3)
458
5
135
(Equal Cost Effectiivnees MMcro), 0.7)
671
23
130
ConditCCo Mean Recourse)
100
41
434
(Fair Effectiveness- Cost Trade-Off, value)
80
544
Table 22: Aggregated Rankings in SSL
Sigal
Sigul
Sia
TEgal Dorcel ElycPrroo
1.33
Pilm 24 aval
2401
03477
11125
13:.00
01171
3.102
(375)
(3211
13:00
6.2%
(Paal Chere ln Eeeeeee
1145
1.10
1790
015.9
12045
0782
3177
19000
0110
617%
130
(Izal Difcctas vidl Dudga 21)
133
8.30
0825
0250
0002
0.5
0286
0286
130
Iqtal Docol trscionoo Mart.
1442
036
1391
0110
11315
0180
3789
1376
1386
6016
Pa Kilukw Tude valar
690
1367
135
0557
8770
3907
646
0991
0011
680
29

---

E55 Comparison on Ad Campaign
The number affected individuals in the test for the Ad campaign dataset is 273,773. We first split the affected individuals on the set of affected males D1 and the set of affected females Do based on the gender attribute. The number of subgroups formed by running fp-growth with support threshold 1 1% on Dj and on Do and computing their intersection is 1,432. Our fairness metrics will evaluate and rank these subgroups based on the actions applied.
Table 23: Ranking Analysis in Ad Campaign
Most Unfair Subgroups
Subgroups Bias Males
*Sbbgroups Bias against Females (in Top 10% Unfair Subgroups)
úin Top 10% Unfair Subgroups)
(Equal Cost Effecciiveness Macro), 0.3)
427
44
(Equal Cost of (Mfeciveness(MMaro) 0.7)
264
26
(Equal Choice for Recourse, 0,3)
(Equal Choice for Recourse, 0.7)
384
39
Equal Effectiveness
15
123
(Equal Effectiveness within Budget, L.0)
42
(Equal Effectiveness within Budget, 5.0)
10
114
(Equal Cost Effectivveeess Micm),, 0.3)
427
44
(Equal Cost Effectiveness Micro), 0.7)
264
26
FquualConditiona Mean Recourse)
108
74
(Fair Effectiveness Cost Trade OffeeOOff, value)
15
Table 24: Aggregated Rankings in Ad Campaign
Car
Tiga Can
a
Tical Chibe
Teot
Cae
Etal Car
Drdgi
Drdge 501
(Saul Bllechrppe 3:3)
0.351
0.2%
0863
9.35
1.5
1.0
13.55
1.0
1.0
0.55
1.0
0111
013
Sqaal Bilicvaan
0.5
0.5
0.333
3.333
1.0
0.355
0.3
833
0.00
0631
(ig Rinddmee wikke hales S.f.
835
8.5
0.33
0335
2.001
100
035
85
0.00
0601
Suul Bileetrroooeeeee E.)
835
83
005
233
135
130
0.1
0.56
dex Flw toeass Toals DE. sole
075
05
599
10
6325
030
30

---

G Discussion of Limitations & Responsible Usage Guidelines
Determining costs for specific counterfactuals poses numerous challenges and constraints. The asso- ciated cost functions are intricate, dataset-specific and frequently require the expertise of a domain specialist. These cost functions can depend on individual characteristics or human interpretation of the perceived difficulty of specific changes, potentially giving rise to concerns like breaches of user privacy (as the expert may require access to all individual user characteristics [25]) or potential manipulation of cost functions by malicious specialists to conceal existing biases.
Recognizing the difficulties of finding the dataset-dependent cost functions, implement various "natural" cost functions tailored to different feature types (c.g., L1 norm, exponential distances, etc.). These serve only as suggestions to the expert using our framework, are susceptible change, and can be customized for specific datasets. While the creation of precise cost models is an interesting research area, it is not our primary focus. Instead, our primary goal is to identify a specific type of bias (difficulty to achieve recourse) towards specific subpopulations. Therefore, we have introduced fairness metries that are independent of cost considerations, recognizing the aforementioned inherent difficulties in defining costs. We recognize that a single metric, whether cost-dependent not, cannot identify all forms of bias that may be present within a model's 's predictions. We do not aim to replace the existing statistical measures for bias detection, rather than complement them and capture possible gaps, as described in the introduction of our paper and Section 5.
Our framework evaluates subgroups based on range of fairness metries, identifying 'candidate' subgroups that may experience unfair treatment under scrutiny. While the number of such 'candidate' subgroups can be substantial across different metrics, this abundance can serve as a safeguard against malicious users, as cost-oblivious rules are immune to user influence. We emphasize our efforts to produce comprehensive summaries that are easily understandable for the end-users and help them make informative decisions regarding the existence of the groups that are unfairly treated.
The ultimate decision regarding which of the 'candidate' subgroups are considered unfairly treated rests with the expert. Our comprehensive summaries highlight the top-scoring 'candidate" subgroups across multiple metrics, with the ultimate goal of helping the end users and deterring malicious users from deliberately ignoring the framework's suggestions. Explanations accompany FACTS' suggestions, requiring users to justify their rejection and serving as a barrier against deliberate malicious actions. While our summaries are interpretable and customizable, we plan to enhance our framework with visualizations in future work to further assist end-users. A user study would also be helpful assess FACTS' interpretability to the end-user.
It is important to note that FACTS, like other resource generation algorithms, may be susceptible to errors, as demonstrated in prior research [31]. The outcomes produced by FACTS can vary based on the chosen configuration, including hyperparameters and metric selection. These variations can either obscure biases present within classifier or indicate their absence. is crucial emphasize that since FACTS is an explainable framework for bias assessment within subgroups, its primary purpose is serve as a guidance tool for auditors to identify subgroups requiring in-depth analysis of fairness criteria violations. Responsible usage of our framework necessitates transparency regarding the chosen configuration. Therefore, we recommend running FACTS with different hyperparameters and utilizing as many metrics as possible.
Another potential limitation relates GDPR and similar regulations, which may restrict access protected attributes (confidential information, etc.) even in bias detection applications. It's important to clarify that our framework maintains privacy and avoids personal information leakage. Only summaries of results are presented to the end-users, limited to subgroups that impose the minimum size requirement. These features could be further enhanced by adding restrictions the size requirements if legislation or standardized requirements existed. In essence, our approach addresses complex challenges bias detection, while upholding fairness, respecting privacy requirements, and ensuring interpretability for end-users.
An acknowledged constraint of our framework pertains to how we generate actions, explained in Section 3. Both actions and subgroups are derived using a frequent itemset mining algorithm. We have consciously opted against altering our approach to action generation, minimize computational complexity and enhance action effectiveness. Importantly, the same actions are applied to all groups sharing common predicate during the computation of fairness metrics.
31


Reviews:
[{"review_id": "rAvNgcaXXd", "reviewer": "Reviewer_m3rQ", "summary": "The paper proposes a framework (termed as FACTS) for analysing the recourse fairness of a machine learning model. The work introduces multiple metrics for quantifying recourse fairness - both at micro level (individuals considered separate) and macro level (individuals considered together). Some proposed recourse fairness metrics are argued to be unaffected by the underlying cost metric. Finally, the work demonstrates the proposed framework for a logistic regression model by training it on several datasets. All except one dataset results are present in the supplementary material.", "questions": "1. It seems to create counterfactual summaries, FACTS mines conjunctions of predicates for all positive samples and then uses them to find counterfactuals. Hwoever, doesn't this approach limit the coverage (effectiveness) because if the positive samples are not diverse (i.e., the feature space is not well covered), we may not find counterfactuals for many individuals, resulting in poor coverage.\n\n2. I think FACTS have many similarities in terms of metrics with [2]. For example, micro and macro vs finding a global direction and scaling it locally. Similarly, ECD vs coverage-cost profiles. I would like to hear authors comment on how their work differs from GLOBE-CE.\n\n3. Some points needs clarification/correction\n    1. What is \"horizontal intervention\". lines 342 and line 70\n    2. I think the term \"effectiveness\" (Line 193) is same as the term \"coverage\" used in the context of GCEs? See Ref[3]\n    3. Further, set of actions A can be considered as diverse CFs. ECD profiles same as in GCEs? Micro vs Macro too in G~LOBE-CE\n    4. Line 154: I don’t think this claim is correct. AReS[3], GLOBE-CE[2] and Gupta et al. have tried to address this problem before using global (group-level) counterfactuals.\n    6. Line 161: It would be helpful to understand authors comments on how FACTS compares to other global explanations framework in terms of efficiency.\n\n[2] A Translation-Based Approach for Global Counterfactual Explanations. Dan Ley, Saumitra Mishra, Daniele Magazzeni. GLOBE-CE: CoRR abs/2305.17021 (2023)\n\n[3] Beyond Individualized Recourse: Interpretable and Interactive Summaries of Actionable Recourses. Kaivalya Rawal and Himabindu Lakkaraju. NeurIPS 2020", "limitations": "The authors do not discuss any limitations of their work. However, it is important to note, given that FACTS is proposed as a tool to investigate recourse fairness, detailed experimentation and analysis would be needed before using the tool in the real-world applications.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 4, "contribution": 2, "strengths": "1. Analysing model fairness from the perspective of recourse explanations is an important research problem. There exists much work in analysing the fairness of model predictions, but less on analysing if the generated recourses are fair.\n2. The paper is very well-written. The motivation, definitions, and metrics are all clearly explained. The authors cover the current metrics clearly, highlight their weaknesses, and then introduce their proposed metrics.\n3. Bibliography is extensive and covers most of the recent papers on the related topics.", "weaknesses": "1. Experiments section seems insufficient. Please see below for some examples\n   \n   a) The experiments section and the supplementary material presents outputs of the FACTS framework for different datasets, however, it is unclear how to evaluate the efficacy of the proposed framework. It would have been great if the authors had included experiments using toy datasets to create a biased model and then demonstrated that FACTS can uncover the recourse bias while the existing methods don’t/do it with poor coverage.\n\n  b) I think the current experiments could have been more detailed for example by covering other model classes (e.g., tree-based, neural networks) to understand if the proposed metrics generalise to non-linear and non-differentiable models.\n\n  c) FACTS is argued to be scalable, interpretable and highly parametrizable (Line 113). However, there are no experiments to support this. It would be great to understand interpretability through a user study for example. Similarly, analysing how FACTS scales to continuous datasets would be very helpful because as it uses an itemset miming algorithms which generally don't scale well with continuous datasets as feature binning creates a large search space [1]. \n\n2. Line 119 - FACTS can explore systematically the feature space. I wonder if there are any guarantees about this? If the feature dimension is high and the dataset has continuous features then the search space becomes too large to scale well.\n\n[1] Dan Ley, Saumitra Mishra, Daniele Magazzeni. Global Counterfactual Explanations: Investigations, Implementations and Improvements. ICLR Workshop 2022.\n\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**W1a \"experiments using toy datasets to create a biased model and then demonstrated that FACTS can uncover the recourse bias\"**\n\nPlease note that we have provided a toy example in Fig, 1 where we show how burden is not nuanced enough to capture other notions of recourse bias.\n\nMost importantly, we have experimented with real datasets with known statistical bias, including Adult, COMPAS and IBM Ad Campaign. Tables 1, 6, 7, 8, 9 demonstrate that burden fails to uncover other forms of recourse bias. A more extensive assessment is presented in Tables 11, 13, 15, 17 and 19.\n\n**W1b \"covering other model classes (e.g., tree-based, neural networks)\"**\n\nWe note that our method is model-agnostic and does depend on the model class. Nonetheless, we have considered two additional models, XGBoost and a NN, for the Adult dataset. An example CSC is in Figures 1 and 2 in the pdf attached in the author rebuttal.\n\n**W1c-i \"interpretability through a user study\"**\n\nThis is outside the scope of this work, but we acknowledge the utility of a standardized study to evaluate the interpretability of fairness judgements. We recognize that GLOBE-CE has taken steps towards that direction.\n\n**W1c-ii \"analysing how FACTS scales to continuous datasets\"**\n\nWe present results reporting the runtime as we vary the number of bins per numerical attribute and the minimum support threshold for frequent itemsets. Refer to Tables 3 and 4 in the pdf.\n\n**W2 \"\"FACTS can explore systematically the feature space.\" I wonder if there are any guarantees about this. ... the search space becomes too large to scale well.\"**\n\nFACTS is an offline auditing method. Indeed the search space can become large and this is a known limitation of all algorithms that examine subpopulations. We note that a similar approach is used by FairTest for auditing prediction fairness of subpopulations and by AReS to generate a summary of counterfactual explanations. We also note that GLOBE-CE does not examine subpopulations.\n\n\n**Q1-i \"It seems to create counterfactual summaries, FACTS mines conjunctions of predicates for all positive samples\"**\n\nWe emphasize that FACTS does not aim to create counterfactual summaries. It uses all counterfactuals discovered to audit for fairness.\n\n**Q1-ii \"we may not find counterfactuals for many individuals, resulting in poor coverage\"**\n\nIt is indeed possible that we may not find recourse for some individuals. Note however that this holds for any model-agnostic method that examines a finite set of actions and does not explore all, possibly infinite, actions. The more actions we examine the better the approximation of the effectiveness-cost distribution will be.\nIf we observe poor effectiveness (coverage in [A]) in some subpopulations, there are some simple remedies: we may lower the minimum support threshold when mining for actions; we may additionally examine actions generated by a different process, like uniformly sampling the space of possible actions.\n\n**Q2 \"I think FACTS have many similarities in terms of metrics with [2]. For example, micro and macro vs finding a global direction and scaling it locally. Similarly, ECD vs coverage-cost profiles. I would like to hear authors comment on how their work differs from GLOBE-CE.\"**\n\nWe summarize the differences:\n- FACTS is a method designed to audit for fairness of recourse. GLOBE-CE aims to find the best counterfactual summary,  which can be used to audit for fairness at the population level.\n- FACTS examines subpopulations and may uncover instances of unfairness hidden at the population level. GLOBE-CE can only summarize the recourses of the entire population.\n- When GLOBE-CE is used to audit for fairness it compares the average cost, aka *burden* in the literature, among the protected subgroups. FACTS is more nuanced, studying in depth the notion of fairness of recourse and making several novel contributions; most notably, it motivates and formalizes a series of fairness metrics, some of which are based on effectiveness and thus oblivious to the cost function, and it defines the micro and macro viewpoints.\n- When GLOBE-CE is used to audit for fairness it draws conclusions based on the best recourse direction discovered, i.e., it forces all individuals to achieve recourse through actions along one direction. In contrast, FACTS draws conclusions by considering all examined actions.\n- \"Finding a global direction and scaling it locally\" is neither the micro (it restricts the actions available to individuals) nor the macro view (it allows for different actions to individuals). \n- The coverage-cost profile in GLOBE-CE is a *constrained* version of the effectiveness-cost distribution (ECD) in FACTS. Specifically, the coverage-cost profile is an ECD where only actions along a *single* recourse direction are aggregated in the micro viewpoint of FACTS.\n \n**Q3 “Some points needs clarification/correction [...]”**\n\n1. An intervention for all individuals.\n2. Correct\n3. Refer to answer to *Q2*\n4. Note that the term subgroup/subpopulation refers to a group of individuals defined by more than one attributes, following the terminology in [17]. Regarding the references:\n-AReS [30] is a global explainability method. Quoting “It is thus important to be cognizant of the fact that AReS is finally an explainable algorithm (as opposed to being a fairness technique) that is meant to guide decision makers.” Fairness auditing is merely discussed as a potential merit of the method and not systematically treated.\n-GLOBE-CE [A] is also primarily a global explainability method that can be used for fairness auditing. What is important, is that it does not audit fairness on the subpopulation level. \n- Gupta et al. [10] similarly considers groups defined by a single protected attribute and not subpopulations. Further, the paper focuses on fairness correction and not auditing.\n5. FACTS is not a global explainability method."}, {"review_id": "VMUhF5PRPB", "reviewer": "Reviewer_CDVq", "summary": "In this paper, the authors explore the fairness of recourse in detail and distinguish between the micro and macro viewpoints. Moreover, they propose an efficient, interpretable, model-agnostic, highly parameterizable framework, called FACTS, to audit for fairness of recourse and provide an interpretable summary of its finding.", "questions": "- In general, how realistic is it to be oblivious to the cost function and have an equal choice for recourse? \n", "limitations": "- The experimental section provides limited insight as it only contains one dataset.\n", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The authors claim that their work is the first that audits for fairness of recourse at the subpopulation level.\n- The results of their work are explainable and interpretable, which makes it more practical and easier to take action in order to mitigate the bias for different subgroups.\n- The authors provide a thorough analysis of various fairness metrics and their advantages and limitations to motivate their work. They also define various subgroup recourse fairness metrics and produce separate subgroup rankings per definition.", "weaknesses": "- The experiments section could be improved by including more datasets and comparisons with other fairness metrics and definitions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**W \"The experiments section could be improved by including more datasets and comparisons with other fairness metrics and definitions.\"**\n\nWe would like to point out that due to lack of space in the main text, we have only included a single dataset Adult with sex as the protected attribute. In the supplementary material, we have additional experiments using Adult (race is protected), SSL, COMPAS, and Ad campaign datasets. Moreover, in the pdf attached to the author rebuttal, we include results on German credit.\n\nSince fairness of recourse is a quite recent notion of algorithmic fairness, there exist no other fairness metrics beside the average recourse cost or \"burden\". In results shown in Table 1, and in additional results in the supplementary material, we showcase that the various fairness metrics we introduce capture distinct interpretations of recourse fairness.\n\n\n**Q \"In general, how realistic is it to be oblivious to the cost function and have an equal choice for recourse?\"**\n\nEqual choice for recourse is one of our proposed definitions of recourse fairness and may find application in some settings. To recall, it says that if you have a set of actions such that each action is considered equal in terms of cost (this can be when actions have equal or comparable cost, or when cost is not an issue or irrelevant or cannot be well quantified) you can be *cost oblivious* and define fairness just in terms of *effectiveness* with respect to this set of actions. \n\nA cost-oblivious setting where equal choice for recourse makes sense can be seen in Figure 1a in our main text. Suppose that a company makes available to its employees two training programs. Action $a_1$ refers to a training program to enhance productivity skills (and thus affect the CTE aspect of individuals), while action $a_2$ refers to a similar training program to enhance project acquisition skills (and thus affect the ACV aspect of individuals). It can be argued that the cost of actions $a_1$ and $a_2$ is irrelevant and that we can be cost oblivious for them. Equal choice for recourse would investigate if individuals from race 0 and race 1 can equally benefit from these actions.\n"}, {"review_id": "FUatFUHihw", "reviewer": "Reviewer_fCWq", "summary": "This paper considers the problem of fairness of machine learning based decisions. The setting is as follows: there is a set of features X in R^n where X_n denotes the demographic group, and a classifier h: X to {0,1} and we have access to a dataset D of individuals with h(X)=0. Each individual can take actions in an action space A to potentially change their classification decision to h(a(X))=1: this is recourse, and a is the counterfactual explanation. The paper considers how different actions to achieve recourse can be fair on either a micro or macro level. \nThey first define multiple notions for effectiveness of actions to achieve recourse, that depend on a group, a cost budget or an effectiveness level. They then define and recall multiple definitions to define fairness of recourse (6 such definitions in section 2.3). \nThey propose an algorithm \"FACTS\" that enables one to check for subgroups (based on features X) where there exist unfairness based on the 6 definitions and allows one to find the interventions to achieve needed effectiveness rates. They evaluate and showcase their algorithm on four different datasets.", "questions": "- can you address the point in the weaknesses section?\n\n- does the algorithm generalize to multiclass demographic groups and multiclass outcomes? \n\n- how can one find recourse for continuous features based on the provided algorithm? \n\n- the current algorithm lacks statistical tests to check if the subgroup found and the violations are statistically significant, how can one ensure that the violations found are in fact significant?\n\n- what is the runtime of the algorithm and how does it scale when features have very large spaces (i.e. suppose feature X_2 can take 10000 values)\n\ncomments:\n\n- I really liked the introduction, but best to separate related work from the introduction. I would also encourage the authors to make the intro more succinct to avoid repeating the explanations in later sections.\n\n- the figures to show the recourse are not easy to read or attractive, I would suggest the authors spend some time improving the design of the CSC (figure 2 and figure 3)\n\n- sections 2.2 and 2.3 now read as a list of definitions without much continuity or story between the definitions and constraints.", "limitations": "limitations are adequately addressed.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Originality: I think the way this paper thinks about recourse is novel in terms of the trade offs between effectiveness, cost and subgroups. Furthermore their algorithm is novel and intuitive way to find violations of fairness of recourse constraints. \n\nQuality: the experimental results seem sound (code is provided) and definitions are well discussed.\n\nClarity: I enjoyed reading this paper, in particular, the introduction. However, there are some minor changes that can improve the exposition. \n\nSignificance: I think the FACTS tool can serve as a nice way to audit algorithms for fairness of recourse. Moreover, the definitions in section 2.3 and the exposition in the introduction is very insightful for the community in algorithmic recourse.", "weaknesses": "- there are two limitations of the algorithm: the form of the predicates (subgroups) and the form of the actions. In particular the subgroups are conjunctions of multiple features (does not allow arbitrary subgroups) and second actions are also conjunctions of feature values. This will lead to finding too many subgroups and too many actions, that could have instead been grouped under one subgroup or one action. Second, the algorithm only applies to categorical features and actions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**W-i \"There are two limitations of the algorithm: the form of the predicates (subgroups) and the form of the actions. In particular the subgroups are conjunctions of multiple features (does not allow arbitrary subgroups) and second actions are also conjunctions of feature values. This will lead to finding too many subgroups and too many actions, that could have instead been grouped under one subgroup or one action.\"**\n\nWe argue that conjunctions of multiple features is the most natural and interpretable way to define and refer to subpopulations. Investigating algorithmic fairness for clearly defined and understood subpopulations is very important. In contrast, arbitrary definitions of subgroups is also susceptible to gerrymandering, where subgroups may be maliciously defined so as to hide unfairness.\n\nGenerating many actions is actually desirable. When we audit for fairness, we consider all of the generated actions, aggregate their effectiveness and cost, and define the effectiveness-cost distribution on which all our fairness metrics are based.\n\n\n**W-ii \"Second, the algorithm only applies to categorical features and actions.\"**\n\nThis is not true. There seems to be a misconception about how FACTS handles numerical attributes, which we try to clarify in the author rebuttal. Please refer to the section on *Numerical Attributes and Binning*.\n\n**Q1 \"Does the algorithm generalize to multiclass demographic groups and multiclass outcomes\"**\n\nThe generalization to multiple protected groups (multiclass demographic groups) is straightforward. For example, we can simply compare, say the burden, of each protected group to all others, e.g., one-vs-all. An example of this approach for the Adult dataset and the multi-values race attribute is presented in Figure 3 in the pdf attached to the author response.\n\nRegarding multiclass outcomes, we note that the notion of recourse (and by extension the notion of fairness of recourse) assumes there is a favorable and an unfavorable class, i.e. implies a binary classification setting. This also means that these notions can transfer to a multiclass setting when one of the classes can be considered as the favorable and all others as unfavorable; we note however that we have not seen such a case in the literature of algorithmic recourse.\n\n\n**Q2 \"How can one find recourse for continuous features based on the provided algorithm?\"**\n\nPlease refer to the global rebuttal for a detailed explanation. Briefly the idea is to perform binning on the continuous features and define actions based on these bins.\n\n\n**Q3 \"The current algorithm lacks statistical tests to check if the subgroup found and the violations are statistically significant, how can one ensure that the violations found are in fact significant?\"**\n\nRegarding the statistical significance of a subgroup/subpopulation, we note that this task should be application and context specific, as when a subgroup should be considered representative might differ in different applications, and even in the same application when different policies are followed. Our framework enables a simple parameterization that lets the auditor decide the sizes of the subpopulations examined (set to 1% of the dataset size in our experiments).\n\nRegarding the statistical significance of fairness violations, we note that for some fairness metrics it is easy to compute the statistical significance. For example, for mean recourse cost, we can simply use the t-test to assess how significant the difference between the means of the two sub-populations is; for fair effectiveness-cost trade-off, we can report the result of the Kolmogorov-Smirnov test, as discussed in the main text.\n\nNonetheless, we argue that the *importance* of any fairness violation should be best judged by an expert auditor with domain knowledge and assisted by tools such as FACTS. Note that FACTS depicts alongside the unfairness metric (e.g., difference of mean recourse cost), the sub-population sizes (the coverage percentages depicted as blue in Figs. 2 and 3 in the main text).\n\n\n**Q4 \"What is the runtime of the algorithm and how does it scale when features have very large spaces (i.e. suppose feature X_2 can take 10000 values)\"**\n\nTables 1 and 3 in the pdf attached to the author rebuttal report the runtime of our algorithm for all datasets, and also includes results while varying the number of bins in the numerical attributes.\n"}, {"review_id": "NgisTjDJCQ", "reviewer": "Reviewer_EeTT", "summary": "The paper presents a framework called FACTS (Fairness Aware Counterfactuals for Subgroups) for auditing subgroup fairness through counterfactual explanations. The authors aim to formulate different aspects of the difficulty individuals face in achieving recourse, either at the micro level (individuals within subgroups) or at the macro level (subgroups as a whole). They introduce new notions of subgroup fairness that are robust and provide an efficient, model-agnostic, and explainable framework for evaluating subgroup fairness. The authors demonstrate the advantages and wide applicability of their approach through an experimental evaluation using benchmark datasets.", "questions": "Please refer to the weakness section for main questions. In addition to points raised above, there are a few questions remained:\n\n- How does the proposed framework handle the issue of defining a cost function for recourse?\n- Are there any computational or scalability limitations of the FACTS framework?\n- How does the performance of the FACTS framework compare to other state-of-the-art approaches for auditing subgroup fairness?\n- The proposed framework relies on the assumption that counterfactual explanations can accurately capture the underlying causes of bias and provide actionable insights.\n- The effectiveness of the FACTS framework may vary depending on the specific dataset and domain.\n- The framework may face challenges in cases where the protected attributes are not well-defined or are subject to interpretation.", "limitations": "The main paper does not have a limitation statement, although the authors mentioned a few in the conclusion. There's no broader societal impact statement about the potential negative societal impact.\n\n", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "+ The paper addresses an important and timely topic in machine learning, namely fairness in decision-making processes.\n+ The authors propose a novel framework, FACTS, for auditing subgroup fairness, which is model-agnostic and highly parameterizable.\n+ The paper provides a thorough explanation of the different notions of subgroup fairness and their implications.\n+ The experimental evaluation demonstrates the effectiveness and efficiency of the proposed approach on benchmark datasets.\n+ The paper is well written and organized.", "weaknesses": "There are a few weaknesses:\n\n- The paper could benefit from a more detailed description of the methodology used in the experimental evaluation. While the authors have provided an overview of the experimental setup and the data collection process, a more comprehensive explanation of the specific steps taken would greatly enhance the clarity and reproducibility of the study. For example, providing information about the specific criteria used to select the sources or the process of extracting and preprocessing the data would be useful for identifying the potential data generating bias.\n\n- In addition, providing more information on the statistical analysis performed on the experimental results would enhance the credibility of the findings. The authors mainly use Comparative Subgroup Counterfactuals for evaluating. However, it remains unclear whether the observations are significant, and how do they generalize to high-dimensional data as well.\n\n- The framework may face challenges in cases where the protected attributes are not well-defined or are subject to interpretation. How does FACTS perform when the underlying sensitive attributes are unknown (which is esp. true for many applications)?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**W1 \"more detailed description of the methodology used\"**\n\nWe acknowledge that some details about the experimental methodology description are missing in the main text, and probably not adequately covered in the supplementary material. We intend to remedy this should the paper be accepted. We should note however, that standard preprocessing and train-test practices were applied on all datasets, which are mostly widely used datasets for benchmarking fairness methods. Finally, we note that we have included more detailed information for the additional experiments we have conducted during this rebuttal phase, and which we present in the pdf attached to the author rebuttal.\n\n\n**W2 \"more information on the statistical analysis performed on the experimental results\" \"whether the observations are significant, and how do they generalize to high-dimensional data\"**\n\nWe note that the experimental results are derived by applying the steps outlined in Section 3 and then applying the definitions of Section 2.3 in a straightforward manner. There is no statistical post-processing to include or exclude data points.\n\nHigh-dimensional data may lead to subgroups with few individuals. Please refer to our response to comment *Q3* of *Reviewer fCWq* on significance.\n\n\n**W3 \"How does FACTS perform when the underlying sensitive attributes are unknown?\"**\n\nSimilar to fairness literature (see Fliptest, Fairtest, AReS, Globe-CE), the problem formulation that FACTS adopts is that the protected attributes are known in advance. Techniques that discover protected attributes is a very interesting but orthogonal research direction; i.e., such techniques can be applied prior to running FACTS. \n\nNonetheless, we wish to draw attention to two important aspects of FACTS.\n\nFirst, FACTS can investigate whether recourse bias manifests in intersectional subgroups, e.g. black men of certain age groups, despite being absent when comparing all black with all white men, for example.\n\nSecond, FACTS is not restricted to apply solely on protected attributes. Any attribute (and even a multiclass/multivalued attribute) can be treated as \"protected\" and given as input to FACTS. FACTS then would assess any disparities in recourse across subpopulations defined by the selected attribute.\n\n\n**Q1 \"How does the proposed framework handle the issue of defining a cost function for recourse?\"**\n\nWe consider the definition of cost as an orthogonal, highly application/domain dependent and task, which is out of scope of our work; we refer the reader to the discussion in [30,34] and [A] about the challenges in defining cost functions. \n\nOur framework is built so that it can support arbitrarily defined cost functions on top of the available attributes. Upon this, our framework utilizes the produced recourse scores in different ways (or not at all) depending on the selected definition. In particular though, since we recognize the difficulty of defining costs, two of our definitions actually are cost-oblivious, i.e., they rely on measuring effectiveness of actions and not their costs (making the assumption that the same action will have the same cost for all individuals of the examined subgroup). We note that in our experiments, we select a straightforward cost configuration that is used solely to demonstrate the results of our method, as described in \"Experimental Setting\" of the supplementary material.\n\n\n**Q2 \"Are there any computational or scalability limitations?\"**\n\nWe note that as reported in the pdf attached to the global rebuttal, the runtime of FACTS is in the order of a few minutes for the datasets examined. \n\n\n**Q3 \"How does the performance of the FACTS framework compare to other state-of-the-art approaches for auditing subgroup fairness?\"**\n\nIn our work, we audit for a specific type of algorithmic fairness, fairness of recourse. All methods of the literature (e.g. Fliptest, Fairtest, \"Preventing Fairness Gerrymandering\") audit for predictive fairness (like equal opportunity), and thus are not comparable. \n\nNote that mean cost of recourse or \"burden\" [20,31,36] is the only known fairness of recourse metric in the literature. We also note that AReS [30] and follow-up work are global explainability methods, which can be used for auditing for the burden metric. Such methods however are not designed with auditing as their main goal. In particular, AReS only provides a toy example with qualitative results of how their framework can be used for fairness auditing and states that: \"It is thus important to be cognizant of the fact that AReS is finally an explainable algorithm (as opposed to being a fairness technique) that is meant to guide decision makers.\"\n\nExperiments with real datasets, presented in Tables 1, 6, 7, 8, 9 and also in Tables 11, 13, 15, 17, 19 demonstrate that burden fails to uncover other forms of recourse bias.\n\n**Q4 \"the assumption that counterfactual explanations can accurately capture the underlying causes of bias and provide actionable insights.\"**\n\nIndeed we operate on this assumption. We believe that algorithmic recourse (via counterfactual explanations) is an important aspect of the behavior of a model that captures how difficult it is for an individual to have agency over algorithmic decisions that concern them. In this sense, fairness of recourse mandates that individuals are not discriminated against in their capacity to receive desirable outcomes.\n\n\n**Q5 \"The effectiveness of the FACTS framework may vary depending on the specific dataset and domain.\"**\n\nWe note that we have evaluated our method on four widely-used datasets in the fairness literature and we report interesting/meaningful findings for all of them (in the main text and in the supplementary material). We also show that the various fairness of recourse definitions we have introduced are quite distinct notions, and it is thus up to auditors and domain experts to decide how best to apply the FACTS framework.\n"}, {"review_id": "xjXQlCt37L", "reviewer": "Reviewer_ZFdX", "summary": "The authors identify various aspects of fairness that require consideration when assessing recourse bias between subgroups. They propose FACTS (Fairness Aware Counterfactuals for Subgroups), in an attempt to lay foundations for a framework that can be used to audit subgroup fairness through counterfactual explanations (CEs). Experimental evaluation is relatively well thought out and thorough.", "questions": "The bins chosen for numerical features are rather large, and so may not effectively capture the minimum cost recourses between subgroups if the numerical features are influential, which might commonly be the case. This is one of the issues with frequent itemset mining, and something that [3] originally suffered with- do the authors have any proposed solutions? From my experience in the area, this is a fairly serious shortcoming, and something that prevented me from awarding this paper an even higher score.", "limitations": "Described above. Limitations associated with choosing parameters for fairness metrics and determining appropriate action spaces should be discussed in the revision.\n\nThe research contributes valuable insights and provides a robust foundation for future work, though I would suggest to keep in mind the inherent complexity of the fairness problem, which is indeed emphasised in the paper. Overconfidence in the reliability of the proposed framework may not encourage an appropriately critical approach to this multifaceted issue i.e. conclusions drawn should be cautiously optimistic and mindful of the above limitations.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "1. The paper is easy to read, with a logical structure and natural flow. In most cases, relevant examples are provided where necessary.\n\n2. I believe the issue being addressed is complex, and hard conclusions regarding the effectiveness of fairness tools should be made sparingly. However, the paper does an excellent job of analysing the problem at a deeper level than that which currently exists in the literature [1]. Explicit separation between the micro and macro viewpoint is well positioned.\n\n3. While there are still gaps to be filled, this paper lays a strong foundation for future research into the use of counterfactuals in assessing recourse bias/unfairness. It is worth noting that [2] conducts similar analyses of the potential pitfalls of bias assessment via subgroup comparison, arriving at a similar conclusion to this paper that recourses should be compared 1 to 1 for more reliable results.\n\n[1] Michael Kearns, Seth Neel, Aaron Roth, and Zhiwei Steven Wu. Preventing fairness gerrymandering: Auditing and learning for subgroup fairness. ICML 2018.\n\n[2] Dan Ley, Saumitra Mishra, Daniele Magazzeni. GLOBE-CE: A Translation-Based Approach for Global Counterfactual Explanations. ICML 2023.\n", "weaknesses": "I appreciate the thoroughness of the authors in tackling an issue as complex as this. There are a few important gaps that I believe may have evaded their attention, which I will detail below.\n\n1. The assumption that while cost is not exactly quantifiable, a given action's cost is uniform across the input space, and can thus be compared between instances, does not always hold. One simple example would be changing salary by amount X. Individuals commanding higher initial salaries are likely to find this action easier. I would thus modify the \"oblivious to the cost function\" claim appropriately (though it is still a more than reasonable starting point for now). \n\n2. Evaluation of fairness appears quite sensitive to the set of actions A chosen (specifically the fixed cost used/described in lines 290-302). Such a budget can often handle poorly the asymmetries between subgroup cost distributions for a given action. That said, multiple cost budgets are evaluated in the experiments, to provide a more complete overview. It's also worth considering the idea of scaling the cost of certain action directions, as in [2], and evaluating fairness accordingly.\n\n3. In a similar vein, frequent itemsets are prone to resolution issues with numerical features. Moving beyond the apriori approach in [3] is wise, though frequent itemsets often do not uncover the minimal cost directions to flip predictions. Additionally, if the motivation behind fp-growth is primarily efficiency, it would be useful to have some analysis of this.\n\n4. Discussion of when and how the fairness metrics can be manipulated (as above) is needed. Additionally, in practical settings, individuals are each in control of diverse actions, often unique to them. The effect of such actions (not included in but confounding with the feature space) is worth pointing out.\n\n5. A large quantity of metrics are introduced. Table 1 caption could do with more clarification so readers don't have to study the text to understand the highlighted values.\n\n[3] Kaivalya Rawal and Himabindu Lakkaraju. Beyond individualized recourse: Interpretable and interactive summaries of actionable recourses. NeurIPS 2020.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**W1 \"The assumption that while cost is not exactly quantifiable, a given action's cost is uniform across the input space, and can thus be compared between instances, does not always hold. One simple example would be changing salary by amount X. Individuals commanding higher initial salaries are likely to find this action easier.\"**\n\nWe thank the reviewer for raising this point, which only concerns the cost-oblivious fairness metrics. There, we will make this assumption explicit: the cost of a given action $a$ (e.g., increase salary by amount $\\delta$) is the same for all individuals $x \\neq x'$, i.e., $cost(a, x) = cost(a, x')$.\nWhen this is not the case, we can think of two alternatives that are straightforward to implement in FACTS:\n- We can keep the subpopulation bins for that attribute small (or adjust them accordingly) so that the assumption approximately holds for each subpopulation.\n- Adapt the definition of actions so as to satisfy the assumption. E.g., for salary, we consider actions that increase salary by $\\gamma$%.\n\n\n**W2 \"Evaluation of fairness appears quite sensitive to the set of actions A chosen (specifically the fixed cost used/described in lines 290-302). Such a budget can often handle poorly the asymmetries between subgroup cost distributions for a given action. That said, multiple cost budgets are evaluated in the experiments, to provide a more complete overview. It's also worth considering the idea of scaling the cost of certain action directions, as in [A], and evaluating fairness accordingly\"**\n\nWe emphasize that our goal is to accurately capture the real effectiveness-cost distribution (ecd) within a subpopulation. We achieve this by considering the cost and effectiveness of all valid actions among a set $A$. Augmenting this set with additional actions can help improve the accuracy of the ecd. As suggested, for each action $a \\in A$ we may also consider other actions derived from $a$ by scaling $a$'s numerical part; e.g., if $a$ says change income from [40K,50K] to [50K,60K], i.e., add 10K, we can also consider actions that change the income by 5K, 10K, 15K, for example. This can also be achieved by applying a different numerical binning for the actions as we discuss in the author rebuttal. Another approach would be to sample the space of possible actions uniformly at random.\n\n\n**W3 \"In a similar vein, frequent itemsets are prone to resolution issues with numerical features. Moving beyond the apriori approach in [3] is wise, though frequent itemsets often do not uncover the minimal cost directions to flip predictions.\"**\n\nWe want to emphasize that all model-agnostic methods that explore a finite set of actions $A$ have an inherent limitation to how well they can discover the minimum recourse cost. Nonetheless, as discussed in the previous comment, we can foresee several ways to augment the set $A$ so as to most accurately represent the ecd.\n\n\n**W3 \"Additionally, if the motivation behind fp-growth is primarily efficiency, it would be useful to have some analysis of this.\"**\n\nTables 1-4 in the global response show the runtime of our fp-growth-based generation of subpopulations and actions, and also provides some scalability results. We note however that our main contribution is a conceptual one, i.e., how to formalize notions of recourse unfairness, rather that a technical one, i.e., how to efficiently determine a good set of actions to explore.\n\n\n**W4 \"Discussion of when and how the fairness metrics can be manipulated (as above) is needed. Additionally, in practical settings, individuals are each in control of diverse actions, often unique to them. The effect of such actions (not included in but confounding with the feature space) is worth pointing out.\"**\n\nWe thank the reviewer for raising these issues; we intend to discuss them in greater detail in a limitation section. The manipulability of the method and the robustness of the conclusions drawn is an important issue. We note that while local/individual counterfactuals have been found to be prone to manipulability, our fairness auditing method is more robust as it considers multiple counterfactuals for the same individual and draws conclusions at a subpopulation level. Clearly, more research in this direction is called for.\n\nRegarding the last observation, this boils down to how well the cost function models the world; e.g., it could capture causal structures in the domain, and personalized/unique actions/costs. \n\n\n**Q1 \"The bins chosen for numerical features are rather large, and so may not effectively capture the minimum cost recourses between subgroups if the numerical features are influential, which might commonly be the case. This is one of the issues with frequent itemset mining, and something that [30] originally suffered with- do the authors have any proposed solutions? From my experience in the area, this is a fairly serious shortcoming, and something that prevented me from awarding this paper an even higher score.\"**\n\nWe note the distinction between binning for subpopulations and binning for actions, which can be done separately as discussed in the global response, and has the potential to define a more fine-grained action set $A$. Additional ways to augment or to define a better set $A$ without relying on frequent itemset mining are also outlined in the response to **W2**, but are somewhat orthogonal to our main contribution. Certainly, the implementation of our auditing framework can benefit from recent advances in the field like [A].\n"}]

Author rebuttal:
We are thankful to the reviewers for their insightful and constructive comments. In this global response, we would like to address some misunderstandings about the focus of our work and how we handle numerical attributes, and also discuss limitations.

### Global Explainability vs Auditing for Fairness

FACTS is a method to audit a model for fairness of recourse in subpopulations. AReS [30, 24] and GLOBE-CE [A] are methods to generate summaries for global counterfactual-based explanations. As such the papers and methods have distinct focus and objectives.

Please note that GLOBE-CE appears in ICML 2023 and was first published on May 26th in arXiv, *after* the NeurIPS 2023 submission deadline of May 8th.

AReS and GLOBE-CE consider global counterfactual explanations (GCEs); a GCE is an explanation that applies to a group of instances collectively (akin to our macro viewpoint). Their methods differ in the GCE definitions. A GCE in AReS is a translation in the feature space (akin to an action in our terminology). A GCE in GLOBE-CE is a direction in the feature space along which instances can achieve recourse with differing translation magnitudes. Nonetheless, AReS and GLOBE-CE have the same goal: find a small set of GCEs to "best" explain a group of instances (according to some objectives). The motivation is to construct an accurate, easily understandable *summary* of counterfactuals; the main contribution of GLOBE-CE is that it constructs a more accurate summary than AReS. Note also the analogy with local explainability, where the goal is to find the best (typically the nearest) counterfactual for a single instance.

In contrast, auditing for fairness is *not* an optimization problem. We do not generate counterfactuals and then select a *few* good among them. In fact, we take the opposite approach: we generate as many counterfactuals as we can afford to, and use *all* of them to approximate the effectiveness-cost distribution.

Global explainability methods can be used to audit for recourse fairness, but only as an afterthought. In contrast, our main objective is to formalize the notion of recourse fairness. We are the first to motivate and define different variants going beyond the mean recourse cost or “burden” in literature, and consider micro and macro viewpoints when examining subpopulations.

Finally, we note that model auditing is typically an offline process, and the runtime is often not an issue. Nonetheless, the runtime of FACTS is in the order of minutes for all datasets and configurations tested, as we report in the pdf attached (Tables 1 and 3).

### Numerical Attributes and Binning

We would like to clarify that FACTS indeed works for datasets/models with numerical/continuous features; in fact it works with any mixture of categorical and numerical features. The core idea is that numerical attributes are binned. These bins are used to define subpopulations, e.g., people in the salary range [40K,50K], and actions, e.g., make salary [50K,60K]. So the action "if salary is [40K,50K], make salary [50K,60K]" means that all individuals within that salary range are mapped to their counterfactuals where their salaries are increased by 10K.

Binning is *necessary* when defining subpopulations, as we want to explore the entire feature space and present conclusions in a manner that is easily understandable by humans. E.g. compare the interpretability of "there is unfairness against females when married=no, salary in [40K,50K]" with that of "... when married=no, salary=39K or 42K or 45.5K".

Binning is also *necessary* when considering actions over numerical attributes. As explained, binning of granularity 10K for salary means that we consider actions that change salary by ±10K, ±20K, etc. We emphasize that because the number of actions/counterfactuals is infinite, *all* methods that work with model-agnostic counterfactual-based explanations (e.g., FACTS, AReS, GLOBE-CE) have to explore only a small set of actions. Note again the distinction between global explainability and auditing for fairness. In global explainability only a few of the explored actions will be selected, whereas we argue that all of them should be used to reliably audit for fairness.

Recall that our goal is to approximate the effectiveness-cost distribution, which means we should consider as many actions as possible. Therefore, on the one hand, binning for actions should rather be fine-grained, while on the other hand, binning for subpopulations should be moderately coarse-grained so as to draw conclusions that concern many affected individuals.

With that said, the binning granularity for actions and subpopulations can *differ*. For example consider bins of length 10K for subpopulations and 5K for actions. An action, "if salary is [40K,50K], make salary [55K,60K]" means that individuals with salary in [40K,45K] increase their salary by 15K, and individuals within [45K,50K] increase their salary by 10K.

In all our experiments, we have shown results using the same binning for actions and subpopulations. Adapting our algorithm to handle differing binning granularities is straightforward. Note that Tables 3 and 4 in the attached pdf present results as we vary the binning granularity.

### Limitations and Potential Societal Impact

We would like to thank the reviewers for raising awareness on several aspects, which we intend to discuss in the suppl. material and include in the guidelines for ethical usage of FACTS:
- the definition of costs to actions
- transparency about the subpopulations explored
- how to interpret fairness metrics and auditing results (incl. importance and statistical significance)
- need for a user study on interpretability
- compliance with GDPR and similar regulations in auditing using protected/sensitive attributes
- discussion and study on robustness and sensitivity to malicious actors.


[A] GLOBE-CE: A Translation-Based Approach for Global Counterfactual Explanations. Ley et al. ICML 2023.


Meta-review: After some discussion, the reviewing team felt that this work had enough merits to accept it as a poster.
Decision: Accept (poster)