Causal Fairness for Outcome Control
Authors: Drago Plecko
Elias Bareinboim
Venue: NeurIPS 2023 poster (2023)
Keywords: Fair Machine Learning
Causal Inference
Decision-Making
TL;DR: We introduce a causal framework for assessing whether a decision-making process is fair, and whether the conditional treatment effect of the decision on the outcome is affected by the protected attribute.

Abstract: As society transitions towards an AI-based decision-making infrastructure, an ever-increasing number of decisions once under control of humans are now delegated to automated systems. Even though such developments make various parts of society more efficient, a large body of evidence suggests that a great deal of care needs to be taken to make such automated decision-making systems fair and equitable, namely, taking into account sensitive attributes such as gender, race, and religion. In this paper, we study a specific decision-making task called outcome control in which an automated system aims to optimize an outcome variable $Y$ while being fair and equitable. The interest in such a setting ranges from interventions related to criminal justice and welfare, all the way to clinical decision-making and public health. In this paper, we first analyze through causal lenses the notion of benefit, which captures how much a specific individual would benefit from a positive decision, counterfactually speaking, when contrasted with an alternative, negative one. We introduce the notion of benefit fairness, which can be seen as the minimal fairness requirement in decision-making, and develop an algorithm for satisfying it. We then note that the benefit itself may be influenced by the protected attribute, and propose causal tools which can be used to analyze this. Finally, if some of the variations of the protected attribute in the benefit are considered as discriminatory, the notion of benefit fairness may need to be strengthened, which leads us to articulating a notion of causal benefit fairness. Using this notion, we develop a new optimization procedure capable of maximizing $Y$ while ascertaining causal fairness in the decision process.

Causal Fairness for Outcome Control
Drago Plecko and Elias Bareinboim
Department of Computer Science
Columbia University
àp3144@collmbia.eer ebåcs. cides.columbia. cowwmbia..cooomocmooooom.......a edu
Abstract
As society transitions towards an Al-based decision-making infrastructure, an ever- increasing number of decisions once under control of humans now delegated automated systems. Even though such developments make various parts of society more efficient, large body of evidence suggests that great deal of care needs to be taken to make such automated decision-making systems fair and equitable, namely, taking into account sensitive attributes such gender, race, and religion. In this paper, we study a specific pocision-making task called outcome control in which an automated system aims optimize an outcome variable Y while being fair and equitable. The interest in such a setting ranges from interventions related to criminal justice and welfare, all the way to clinical decision-makinn and public health. In this paper, we first analyze through causal lenses the notion of benefit, which captures how much specific individual would benefit from a positive decision, counterfactually speaking, when contrasted with an alternative, negative one. We introduce the notion of benefit fairness, which can be seen as the minimal fairness requirement in decision-making and develop an algorithm for satisfying it. We then note that benefit itself may be influenced by the protected attribute, and propose causal tools which can be used to analyze this. Finally, if some of the variations of the protected attribute in the benefit are considered as discriminatory, the notion of benefit fairness may need to be strengthened, which leads us to articulating a notion causal benefit fairness. Using this notion, we develop optimization procedure capable of maximizing Y while ascertaining causal fairness in the decision process.
Introduction
Decision-makine systems based on artificial intelligence and machine learning are being increasingly deployed in real-world settings where they have life-changing consequences on individuals and on society more broadly, including hiring decisions, university admissions, law enforcement, credit lending, health care access, and finance 19 pervasive in those settings when decisions are being made by humans, and remain (or are potentially amplified) when decisions are made using machines with little transparency or accountability. Exam- ples include reports such biases in decision support systems for sentencing []. face-detection [7]. online advertising (37 10), and authentication [35]. A large part of the underlying issue is that AI systems designed to make decisions are trained with data that contains various historical biases and past discriminatory decisions against certain protected groups, and such systems may potentially lead to more discriminatory process, unless they have degree of faimess and transparency.
26 G
Issues of unfairness and discrimination are
In this paper, we focus on the specific task of outcome control, characterized by a decision which precedes the outcome of interest Y. The setting of outcome control appears across a broad range of applications, from clinical decision-making [13] and public health [15], to criminal justice (3] and various welfare interventions 9 In general, outcome control will cover settings in which an
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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institution (such as a hospital or a social service) may attempt to maximize an outcome (such as survival or well-being) using a known control (such as surgery or a welfare intervention ' Importantly due to historical biases, certain demographic groups may differ in their distribution of covariates and therefore their benefit from treatment (e.g., from surgery or intervention) may lower than expected, leading to a lower allocation of overall resources. We next discuss two lines of related literature.
Firstly, a large body of literature in reinforcement learning 36 38 and policy learning I 33 21
18
2] analyzes the task of optimal pacision-making Often, these works consider the conditional average treatment effect (CATE) that measures how much probabilistic gain there is from a positive versus a negative decision for specific group of individuals when experimental data is available. Subsequent policy decisions are then based on the CATE, a quantity that will be important in our approach as well. The focus of this literature is often on developing efficient procedures with desirable statistical properties, and issues of fairness have not traditionally been explored in this context.
On the other hand, there is also a growing literature in fair machine learning, that includes various different settings. One can distinguish three specific and different tasks, namely (1) bias detection and quantification for currently deployed policies; (2) construction of fair predictions of an outcome; (3) construction of fair decision-making policies. Most of the work in fair ML falls under tasks (1) and (2), whereas our setting of outcome control falls under (3). Our work also falls under
the growing literature that explores fairness through causal lens [22 20 28
32
For concreteness, consider the causal diagram in Fig. [ that represents the setting of outcome control, with X the protected attribute, 2 possibly multidimensional set of confounders, W set of mediators. Decision D based on the variables X, Z, and W, and the outcome Y depends on all other variables in the model. In this setting, we also assume that the decision-maker is operating under budget constraints.
Y
D
W
Previous work introduces a fairesss definition that conditions on the potential outcomes of the decision D, written Ydo Yd,, and ensures that the decision D is independent of the protected attribute X for any fixed value of Yd,, Ys, [1]. Another related work, in a slightly different setting of risk assessment (]] proposes conditioning on the potential outcome under a negative decision, Yd,- and focuses on equalizing counterfactual error rates. Finally, the work 25 is also related to our setting, and proposes that fairness should be assessed with respect to three causal effects of interest: (i) total effect X + Y: (ii) total effect D - Y; and (iii) total effect X + D. Our framework, however, integrates considerations of the above effects more coherently, and brings forward a new definition of fairness based on first principles.
Figure 1: Standard Fairness
Model (SFM) [31 extended
for outcome control.
However, the causal approaches mentioned above take a different perspective from the policy learning literature, in which policies are built based on the CATE of the decision D, written E[[2, Yd, x, z, w/. which we will refer to as benefi ? Focusing exclusively on the benefit, though, will provide no fairness guarantees apriori. In particular, as can be seen from Fig. [ the protected attribute X may influence the effect of D on Y in three very different ways: (i) along the direct pathway X - Y; (ii) along the indirect pathway X -> W- - Y; (iii) along the spurious pathway X <--> 2 - Y. Often, the decision-make may view these causal effects differently, and may consider only some of them as discriminatory. Currently, no approach in the literature allows for a principled way of detecting and removing discrimination based on the notion of benefit, while accounting for different underlying causal mechanisms that may lead to disparities.
In light of the above, the goal of this paper is to analyze the foundations of outcome control from a causal perspective of the pocision-maker. Specifically, we develop a causal- based decision-making framework for modeling fairness with the following contributions:
(i) We introduce Benefit. Fairness (BF, Def. 2 to ensure that at equal levels of the benefit, the protected attribute X is independent of the decision D. We then develop an algorithm for achieving BF (Alg. T and prove optimality guarantees (Thm. 2
(ii) We develop Alg. 2 that determines which causal mechanisms from X to the benefit (direct, indirect, spurious) explain the difference in the benefit between groups The decision-maker
can then decide which causal pathways are considered as discriminatory.
I (Our setting is not related to the concept of outcome control in the procedural justice literature We remark that the notion of benefit discussed in this work different from that in (34).
24

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(iii) We define the notion of causal benefit faimess (CBF, Def. B). which models discriminatory pathways in a fine-graine fine-grained way. We further develop an algorithm (Alg. 3 that ensures the removal of such discriminatory effects, and show some theoretical guarantees (Thm. 4
1.1 Preliminaries
We use the language of structural causal models (SCMs) as our basic semantical framework (30] A structural causal model (SCM) is a tuple M == (V, U, F, P(w)) where V. U are sets of endogenous (observable) and exogenous (latent) variables respectively, F is set of functions fv; one for each V E V. where Vi f- fv. (pa Uv)) for some VV) S V and Uv, S U. P(u) is a strictly positive probability measure over U. Each SCM M is associated to causal diagram over the node set V, where V -> V if V; is an argument of /v.. and V; a--+ Vj if the corresponding Uv. Uv, are not independent. Throughout, our discussion will be based the specific causal diagram known as the standard fairness model (SFM, see Fig. [T. Further, an instantiation of the exogenous variables U is called a unit. By (u) we denote the potential response of Y when setting X x for the unit u, which is the solution for Y (u) to the set of equations obtained by evaluating the unit in the submodel Ma. in which all equations in F associated with X are replaced by X I.
2 Foundations of Outcome Control
In the setting of outcome control, we are interested in the following decision decision-making task: Definition 1 (Decision-Making Optimization). Let M be an SCM compatible with the SFM We define the optimal decision problem as finding the (possibly stochastic) solution to the following optimization problem given a fixed budget b:
D* arg max
D(a,2,w)
subject to
P(d) < b.
(2)
The budget b constraint is relevant for scenarios when resources are scarce, in which not all patients possibly requiring treatment can be given treatment. In such setting, the goal is treat patients who are most likely to benefit from the treatment, as formalized later in the text. We next discuss two different perspectives on solving the above problem. First, we discuss the problem solution under perfect knowledge, assuming that the underlying SCM and the unobserved variables are available to us (we call this the oracle's perspective). Then, we move on to solving the problem from the point of view of the decision-maker, who only has access to the observed variables in the model and a dataset generated from the true SCM.
2.1 Oracle's Perspective
The following example, which will be used throughout the paper, is accompanied by a vignette that performs inference using finite sample data for the different computations described in the sequel. We introduce the example by illustrating the intuition of outcome control through the perspective of an all- all-knowing oracle:
Example (Cancer Surgery- continued). A clinical team has access to information about the sex of cancer patients (X- = Zn male, X = xı female) and their degree of illness severity determined from tissue hiopsy (W E [0, 1)). They wish to optimize the 2-year survival of each patient (Y), and the decision D indicates whether to perform surgery. The following SCM M describes the data generating mechanisms (unknown to the team):
X X+ Ux
(3)
UUW if I0:
W
(4)
11 VT Uw if X I1
F*,P"(U)
D+ + fD(X, W)
(5)
1 + 10U + 3 -WD
W > 0.5).
(6)
5
Ux f0, 1}, P(Ux =1 0.5,
(7)
Uw, Uy ~ Unif(0, 1].
(8)
3

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Units ạ
X
Y
W D
Units in the oracle example.
Causal diagram for the running example.
Figure 2: (a) Units in the Oracle example; (b) Standard Fairness Model (SFM) for the cancer surgery
example, corresponding to W (W) and = the SFM in Fig.
where the fo mechanism is constructed by the team.
The clinical team has access to an oracle that is capable of predicting the future perfectly. In particular, the oracle lells the team how each individual would respond to surgery. That is, for each unit U = u (of the 500 units), the oracle returns the values of
Yás (u), Yd. (u).
(9)
Having access to this information, the clinicians quickly realize how to use their resources. In particular, they notice that for units for whom (Yap (u), Ya, (u)) equals (0,0) (1, 1), there is effect of surgery, since they will (or will not) survive regardless of the decision. They also notice that
surgery is harmful for individuals for whom (Ya, (w), Yd, (u)) (1, (1. These individuals would survive if given surgery, but would survive otherwise. Therefore, they ultimately decide to treat 100
individuals who satisfy
Ya), (0,
(10)
since these individuals are precisely those whose death can be prevented by surgery. They learn there are 100 males and 100 females in the (0, group, and thus, to be fair with respect to sex, they decide to treat 50 males and 50 females.
The space of units corresponding to the above example is represented in Fig. 2a The groups described by different values of Ya, (u), Ys, (u) in the example are known as canonical types (3) or principal strata [12. Two groups cannot be influenced by the treatment decision (which will be called "Safe" and "Doomed", see Fig. 2a The third group represents those who are harmed by treatment (called "Harmed"). In fact, the decision perform surgery for this subset of individuals is harmful. Finally, the last group represents exactly those for whom the surgery is life-saving, which is the main goal of the clinicians (this group is called "Helped").
This example illustrates how, in presence of perfect knowledge, the team can allocate resources efficiently. In particular, the consideration of fairness comes into play when deciding which of the individuals corresponding to the (0, 1) principal stratum will be treated. Since the number of males and females in this group is equal, the team decides that half of those treated should be female. The approach described above can be seen as appealing in many applications unrelated to to the medical setting, and motivated the definition of principal fairness [14]. As show next, however, this viewpoint is often incompatible with the decision-make perspective.
2.2 Decision-Maker* Perspective
We next discuss the policy construction from the perspective of the decision-maker
Example (Cancer Surgery continued). The team of clinicians constructs the causal diagram associated with the decision-making process, shown in Fig, 2b Using data from their electronic health records (EHR), they estimate the benefit of the treatment based on x, w:
EY,, Yáo w. w. E[Ya, Ya, w.
(11)
In words, at each level of illness severity W w, the proportion of patients who benefi from the surgery is the same, regardless of sex. In light of this information, the clinicians decide construct

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(4)
(b)
U-space Oracle
U-space for Decision-Msker
(resilience)
Canonical Type:
Benelit
Figure 3: Difference in perspective between perfect knowledge of an oracle (left) and imperfect knowledge of decision-maker in practical application (right).
the decision policy fo such that fD(W) 1(W > 22. In words, if a patient's 's illness severity W is above 1. the patient will receive treatment.
After implementing the policy and waiting for the 2-year follow-up period, clinicians estimate the probabilities of treatment within the stratum of those helped by surgery, and compute that
7
P(d yd. = =0, yd, L, 20)
(12)
8"
1
P(d] ydo ydh = 1, 21)
(13)
2"
indicating that the allocation of the decision is not independent of sex. That is. within the group of those who are helped by surgery, males are more likely to be selected for treatment than females. O
The decision-making policy introduced by the clinicians, somewhat counter-intuitively does not allocate the treatment equally within the principal stratum of those who are helped, even though each level of illness severity, the proportion of patients who benefit from the treatment is equal between the sexes. What is the issue at hand here?
To answer this question, the perspective under perfect knowledge is shown in Fig. 3.. In particular, on the horizontal axis the noise variable uy is available, which summarizes the patients' unobserved resilience. Together with the value of illness severity (on the vertical axis) and the knowledge of the structural causal model, we can perfectly pick apart the different groups (i.c., principal strata) according to their potential outcomes (groups indicated by color). In this case, it is clear that our policy should treat patients in the green area since those are the ones who benefit from treatment.
In Fig. 3p. however, we see the perspective of the decision makers under imperfect knowledge. Firstly, the decision-maken have no knowledge about the values on the horizontal axis, since this represents variables that outside their model. From their point of view, the key quantity of interest the conditional average treatment effect (CATE), which we call benefit in this contex defined as
A 2, w) P P(yd, x2, 2, w) P(ya, x, 2, w).
(14)
Importantly, in setting, the benefit A can be uniquely computed (identified) from observational data, implying that we may use the notion of benefit for practical data analyses:
Proposition (Benefit Identifiability). Let M be an SCM compatible with the SFM in Fig. Then the benefit Axx, z, w) is identifiable from observational data and equals:
A z, w) = P④y d1, x. z, w) PPy do. x, w)
(15)
for any choice of x, 2, W.
'In practice, however, clinical decision making may not be based on the actual value of the benefit A but biased, imperfect version of it, corresponding to the intuition of the clinician.
5

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Algorithm Decision-Making with Benefit Faimess
Inputs: Distribution P(V), Budget b
Compute A (xx, 2, w) = Y Yo x, 2, w| for (z,
2: If P(A 0) S b, set D 1 A x. 2, w) > 0) and RETURN(D").
3: Find ò, > such that
P(A > 66) b, P(A 66) > b.
(17)
4: Otherwise, define
= ((x, 2) A (z, 161. B = (zz 2, w) A 2, [6}
(18)
(19)
5: Construct the policy D such that:
1 for (z, 2, w) E I,
D =
with prob. b-P
P(B)
for (No: 2, w) E B,
(20)
with prob. 6-P(2) for (21, 2, w) E B.
P(B)
The benefit is simply the increase in survival associated with treatment. After computing the benefit (shown in Eq. []) the decision-makers visualize the male and female groups (lighter color indicates larger increase survival associated with surgery) according to it. is visible from the figure that the estimated benefit from surgery is higher for the group than for Therefore, to the best of their knowledge, the decision deckeion-makers makers decide to treat more patients from the To group.
The example illustrates why the oracle's perspective may be mislcading for the decision rocision-makers. makers. The clinicians never determine exactly which patients belong to the group
Ya (u) 0, Yd. (u) = 1,
that is, who benefits from the treatment. Instead, they have to rely on illness severity (W) as proxy for treatment benefit (A). In other words, understanding of treatment benefit will always be probabilistic, and we need to account for this when considering the decision-maker 's task. A further
discussion the relation to principal fairness 44 is given in Appendix A
2.3 Benefit Fairness
To remedy the above issue, we propose an alternative definition, which takes the viewpoint of the decision-maker:
Definition 2 (Benefit Fairness). Let A denote the benefit of treatment in Eq. 14 and let a fixed value of A. We say that the pair (Y, D) satisfies benefit fairness (BF, for short) if
P(d A 6, xo) P(d A 5, V8.
(16)
The above definition pertains the population-level setting, but without oracle knowledge. In this setting, the objective function in Eq. iis expected to be lower than in the oracle case. Further, in finite samples, the constraint from Def. 2 ii expected hold approximately. Notice that BF takes the perspective of the Socision-makee who only has access to the umit's attributes (z, 2, w), opposed to the exogenous
f In particular, the benefit A 2, w) estimable from the data, i.e., attainable by the decision- maker. BF then requires that at each level of the benefit, A 2, w) = ō, the rate of the decision does not depend on the protected attribute. The benefit is closely related to the canonical types discussed earlier, namely
A (x, 2, w) c(x, 2, w) d(x, 2.)
(21)
The values of c, d in Eq. are covariate-specinn and indicate the proportions of patients helped and harmed by the treatment, respectively, among all patients coinciding with covariate values x, 2, w (i.e.,
FFrmmally having access to the exogenous instantiation of U = implies knowing which principal stratum from Fig. the unit belongs to, since U/ determines all of the variations of the model.

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Algorithm 2 Benefit Fairness Causal Explanation
Inputs:
Distribution P(V), Benefit A(r, 2, w), Decision policy D
1: Compute the causal decomposition of the resource allocation disparity into its direct, indirect, and spurious contributions
44 31
P(d x1) - P(d| mо) - DE + IE + SE.
(22)
2: Compare the distributions P(A x1) and P(A) zo).
3: Compute the causal decomposition of the benefit disparity
E(A x1) - E(A | mo) = DE + IE + SE.
(23)
4: Compute the counterfactual distribution P(Ac x) for specific interventions that remove the direct, indirect, total effect of X the benefit A.
all s.t. (x, 7, ()) = (a,, 2, w)). The proof of the claim Eq. 21 is given in Appendix B Using this connection of canonical types and the notion of benefit, we can formulate a solution to the problem in Def. T given in Alg. [ particular, Alg. | takes as input the observational distribution P(V), but its adaptation inference from finite samples follows easily. In Step 2 check whether we are operating under resource scarcity, and if not, the optimal policy simply treats everyone who stands to benefit from the treatment. Otherwise, we find the ō, > 0 which uses the entire budget ( and separate the interior I of those with the highest benefit (all of whom are treated), and the boundary B (those who are to be randomized) in Step 4 The budget remaining to be spent the boundary is 6 PT)), and thus individuals on the boundary are treated with probability b- P(2) P(S)
Importantly, male and female groups are treated separately in this random selection process (Eq. 20
reflecting BF. The BF criterion can be seen as minimal fairness requirement in decision-makan and is often aligned with maximizing utility, as seen from the following theorem:
Theorem 2 (Alg. Optimality). Among all feasible policies D for the optimization problem in Eqs. 2 the result of Alg. i is optimal and satisfies benefit fairness.
A key extension discuss relates to the cases in which the benefit itself may be deemed as discriminatory towards a protected group
3 Fairness of the Benefit
As discussed above, benefit fairness guarantees that at each fixed level of the benefit A 6, the protected attribute plays no role in the treatment assignment. However, benefit fairness does not guarantee that treatment probability equal between groups, i.e., that P(d x1) P(d 2o)- In fact, benefit fairness implies equal treatment allocation in cases where the distribution of the benefit is equal across groups (shown formally in Appendix F. which may not always be the case:
Example (Cancer Surgery- continued). After applying benefi and implementing the optimal policy D 1(W > 1)) the clinicians compute that P(d x1) - P(d xo) = -50%, that is, females are 50% less likely be treated than males.
In our example benefit fairess results in a disparity in resource allocation. Whenever this is the case, implies that the benefit A differs between groups. In Alg. 2 we describe formal procedure that helps the decision-maker to obtain a causal understanding of why that is, i.e., which underlying causal mechanisms (direct, indirect, spurious) lead to the difference in the benefit. In Appendix G we discuss in detail how the terms appearing in the decompositions in Eqs. 22] and 23 are defined. The key subtlety here is how to define counterfactuals with respect to the random variable A. In the
appendix we define this formally and show that the variable A can be considered as an auxillary variable the structural causal model, Furthermore, we show that the notions of direct, indirect, and spurious effects are identifiable under the SFM in Fig. [ad provide the identification expressions for them, allowing the data analyst to compute the decompositions in Alg. in practice (see Appendix G We next ground the idea behind Alg. 2 in our example:
7

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Digtnbutton Denety
Distribuiton Denally
22
Benefit
Benefit a
Distributin
X/ (Male)
4/x; (Female)
Distribution
4/Xo
Au, Way
(a) Density of benefit A by group.
(b) Density of counterfactual benefit Ac.
Figure 4: Elements of analytical tools from Alg. 2
in Ex.
Example (Decomposing the disparity). Following Alg. 2 the clinicians first decompose the observed disparities into their direct, indirect, and spurious components:
P(d| 21) P(d| xo)
0%
50%
0%
(24)
DE
18
SE
EE 11) [FA xo)
0
(25)
DE
SE
IE
showing that the difference between groups is entirely explained by the levels of illness severity, that is, male patients are on average more severely ill than female patients (see Fig. 4a). Direct and spurious effects. in this example, do not explain the difference in benefit between the groups.
Based these findings, the clinicians realize that the main driver of the disparity in the benefit A the indirect effect. Thus, they decide to compute the distribution of the benefit P P(Ar,WW Wag T1).
which corresponds the distribution of the benefit had X been equal Do along the indirect effect. The comparison of this distribution, with the distribution P(A xo) is shown in Fig. 4b indicating that the distributions are in fact equal.
In the above example, the difference between groups is driven by the indirect effect, although generally, the situation may more complex, with combination of effects driving the disparity. Still, the tools of Alg. 2 equip the reader for analyzing such more complex cases. The key takeaway here is that the first step analyzing disparity treatment allocation obtain causal understanding of why the benefit differs between groups. Based on this understanding, the decision maker may decide that benefit A is unfair, which is what we discuss next.
3.1 Controlling the Gap
A causal approach. The first approach for controlling the gap in resource allocation takes counterfactual perspective. We first define what means for the benefit A to be causally fair:
Definition 3 (Causal Benefit Faimess).. Suppose C (Cu, C1) describes causal pathway from X to Y which is deemed unfair, with Co, C1 representing possibly counterfactual interventions. The pair (Y, D) satisfies counterfactual benefit fairness (CBF) if
E(UCi,d -Ci.do x. w) = EVVCo,d UCo.da Ix, w) Vx, 2,
(26)
P(d A, zo) Pdd A, 21).
(27)
To account for discrimination along a specific causal pathway (after using Alg. 2). the decisionnmakke needs to compute an adjusted version the benefit such that the protected attribute has no effect along the intended causal pathway C. For instance, C = ( ([2o}: }. {ai } describes the total causal effect, whereas C = ([xo): ; {#1, Wn. describes the direct effect. In words, CBF requires that treatment benefit A should not depend on the effect of X on Y along the causal pathway C, and this condition covariate- speecfi, i.e., holds for any choice of covariates J, z, w. Additionally, the decision policy D should satisfy BF, meaning that at each degree of benefit A = 8, the protected attribute plays

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5. El In Step [ the factual benefit values A, together with the adjusted, counterfactual benefit values Ac (that satisfy Def. 3 are computed. Then, åcr is chosen to match the budget b, and all patients with a counterfactual benefit above PCF are treated as demonstrated in the following example:
role in deciding whether the individual is treated or not. This statement is for fixed value of A and possibly considers individuals with different values of z, 2, w. CBF can be satisfied using Alg.
Example (Cancer Surgery- Counterfactual Approach), The clinicians realize that the difference in illness severity comes from the fact that female patients are subject to regular screening tests, and are therefore diagnosed earlier. The clinicians want to compute the adjusted benefit, by computing the counterfactual values of the benefit AR,. A.,, (u) for all u such that X(u) 21. For the computation. they assume that the relative order of the illness severity for females the counterfactual world would have stayed the same (which holds true in the underlying SCM). Therefore, they compute that
1
W., (u)
3 3V1
(n))²,
(28)
for each unit with X X(u) F1. After applying Alg. with the counterfactual benefit values Ac. the resulting policy DCF = 1(Ac > 1)) has a resource allocation disparity of
The above example illustrates the core of the causal counterfactual approach to discrimination removal. BF was not appropriate itself, since the clinicians are aware that the benefit of the treatment depends on sex in way they deemed unfair. Therefore, to solve the problem, they first remove the undesired effect from the benefit A, by computing the counterfactual benefit Ac. After this, they apply Alg. B with the counterfactual method (CF) to construct a fair decision policy. Notably, the causal approach to reducing the disparity relies on the counterfactual values of the benefit ( as opposed to the factual benefit values A (u)). As it turns out, measuring (or removing) the dovariate-specinn direct effect of X on the benefit A relatively simple in general, while the indirect effect may be more challenging to handle:
Remark 3 (Covariate- Speciifc Direct Effect of X on Benefit A Computable). Under the assump- tions of the SFM. the potential outcome Woo (u) is identifiable for any unit with X = zo for which the attributes (()) = z, W (u) ) = w are observed. However, the same is not true for the indirect effect. While the counterfactual distribution of the benefit when the indirect effect is manipulated, written P(Azo, zzo). is identifiable under the SFM, the covariate-level values An Wr, (zo, 2, 2, are not identifable without further assumptions.
A utilitarian/factuaa approach. An alternative, utilitarian (or factual) approach to reduce the disparity in resource allocation uses the factual benefit A Auu), instead of the counterfactual benefit Ac(u) used in the causal approach. This approach is also described in Alg. 3 with the utilitarian (UT) method. Firstly, in Step 5 the counterfactual values Ag are used to compute the disparity that would arise from the optimal policy in the hypothetical, counterfactual world:
(29) The idea then is to introduce different thresholds ((70)_8(x)) for Iu and æ1 groups such that they introduce a disparity of at most M. In Step 6 we check whether the optimal policy introduces a disparity bounded by M. If the disparity is larger than M by an e, in Step 7 we determine how much slack the disadvantaged group requires, by finding thresholds 8 g(20), gTer) that either treat everyone in the disadvantaged group, or achieve a disparity bounded by M. The counterfactual (CF) approach focused on the counterfactual benefit values Ac and used single threshold. The utilitarian (UT) approach focuses on the factual benefit values A, but uses different thresholds within groups However, the utilitarian approach uses the counterfactual values Ae to determine the maximum allowed disparity. Alternatively, this disparity can be pre- specified, as shown in the following example:
M == IP(Ac 2 åCF [x)) - P(Ac 2 8CF [0)]--
Example (Cancer Surgery- Utilitarian Approach). Due to regulatory purposes, clinicians decide that M 20% is the maximum allowed disparity that can be introduced by the new policy D. Using Alg. they construct DUT and find that,or 8( 0.21, 8(21) = 0.12,
P(A > 8(00) (x0) Ri 60%, P(A > g(01) x1) R 40%,
(33)
which yields PT xe 50% and P(d"T x1) - P(d UT mb) z 20%, which is in line with the hospital resources and the maximum disparity allowed by the regulators.
In this section, for clarity of exposition we assume that distribution of the benefit admits density, although the methods are easily adapted to the case when this does not hold.
9

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Algorithm 3 Causal Discrimination Removal for Outcome Control
Inputs: Distribution P(V), Budget b, Intervention C, Max. Disparity M, Method E {CF,UT) }T) Compute A (xx, 2, w), Ac(s, 2, w) for all (x, 2, w:)
2: If P(A 0) S b, set D = 1 (A (2, 2, w) > 0) and RETURN(D).
3: Find dCF > such that
P(Ac 2 bcF) =
(30)
4: If Method is CF, set DCF 1( 1(Ac(1, 2, w) 2 bcF) and RETURN DCF).
5: If M not pre-specified, compute the disparity
M = P(Ac 2 scr 21) - P(Ac 2 scr 20).
(31)
6: Find GUT such that P(A 2 SUT) = b. If
PPAZ dUT (2)) P(A 2 SUT mn)l S M,
set DUT ((,(,, w) 2 dUT) and RETURN(DUT).
7: Otherwise, suppose w.l.o.g. that P(A 2 6n x1) P(A 86 x0) M + fore 0. Define
P(a1) and let g(20) be such that
:: P(za)
P
(20) 1th
20) P(A 2 66
20) +
1
1+1
Set (( max(02 ((ar)) 0), and g(Z)) s.t. P(A 2 8(m))
[1) p(z)) - P 2 g(ar)) xp).
b.
8: Construct and RETURN the policy DUT;
for (#1, 2, w) s.t. A( w) 2 ((1))
DUT for (zo, 2, w) s.t. A 2, w) 2 (10)
(32)
otherwise.
Finally, we describe the theoretical guarantees for the methods in Alg. (proof given in Appendix
Theorem 4 (Alg. Guarantees). The policy DCF is optimal among all policies with budget S b that in the counterfactual world described by intervention C. The policy DUT is optimal among all policies with a budget S that either introduce bounded disparity in resource allocation [P(d] xn) - P(d] wo)l S M or treat everyone with a positive benefi in the disadvantaged group.
We remark that policies DCF and DUT do not necessarily treat the same individuals in general. In Appendix we discuss a formal condition called counterfactual crossing that ensures that DCF and DUT treat the same individuals, therefore explaining when the causal and utilitarian approaches are equivalent [29]. In Appendix E we provide an additional application of our outcome control framework to the problem of allocating respirators
[5] in intensive care units (ICUs), using the
MIMIC-IV dataset [7].
4 Conclusion
In this paper we developed causal tools for understanding fairness in the task of outcome control. We introduced the notion of benefit fairness (Def. 2). and developed a procedure for achieving (Alg. [ Further, we develop a procedure for determining which causal mechanisms (direct, indirect, spurious) explain the difference in benefit between groups (Alg. 2). Finally, we developed two approaches that allow the removal of discrimination from the decision process along undesired causal pathways (Alg. 3.. The proposed framework was demonstrated through a hypothetical cancer surgery example (see vignette) and a real-world respirator allocation example (see Appendix EE.. We leave for future work the extensions of the methods to the setting of continuous decisions D, and the setting of performing decision decision-making makiig under uncertainty imperfect causal knowledge.
10

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Acknowledgements This research was supported in part by the NSF, ONR, AFOSR, DoE, Amazon, JP Morgan, and The Alfred Sloan Foundation.
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Supplementary Material for Causal Fairness for Outcome Control
The source code for reproducing all the experiments can be found in our our code repository Furthermore, the vignette accompanying the main text can be found here
A Principal Fairness
We start with the definition of principal fairness:
Definition (Principal Fairness [4]. Let D be a decision that possibly affects the outcome Y. The pair (Y, D) is said to satisfy principal fairness if
P(d| ydo Yd, x1) = P(d ydo yd, Fu),
(34)
for each principal stratum (ydo: Udy), which can also be written as D X Ydo: Ya,- Furthermore, define the principal fairness measure (PFM) as:
PFM (d) ydo yd,) = P(d| ydo: Md, ; - P(d| yd Ud z0)
(35)
The above notion of principal fairness aims to capture the intuition described in oracle example in Sec. 2. However, unlike in the example, the definition needs to be evaluated under imperfect knowledge, when only the collected data is available An immediate cause for concern, in this context, is the joint appearance of the potential outcomes Ydg Yd, in the definition of principal fairness. As is well-known in the literature, the joint distribution of the potential outcomes Y Yd. is in general impossible to obtain, which leads to the lack of identifiability of the principal fairness criterion:
Proposition 5 (Principal Fairness is Not Identifiable). The Principal Fairness (PF) criterion from Eg. 34 is not identifiable from observational or experimental data.
The implication of the proposition that principal fairness, in general, cannot be evaluated, even an unlimited amount of data was available. One way to see why PF is not identifiable is the following construction. Consider an SCM consisting of two binary variables D, E {0. 1} and the simple graph D + Y. Suppose that we observe P(d)- Pa. and P(y d1) = m1. P(y do) = for some constants Pa, m1, my (additionally assume mg S my w.l.o.g.). It is easy to show that these three values determine all of the observational and interventional distributions of the SCM. However,
notice that for any [0, - [mm] the SCM given by
D - UD
(36)
Y + 1(U E mo - Al) + D1 E [mo - X, mi])+
(37)
(1 ()-)) [m],m] + A)),
Uy ~ Unif(0, 1, ~ Bernoullippa),
(38)
satisfies P(d) = Pa, Ply d,) = m, and P(y | do) mg. but the joint distribution P(ydo = 0, ydd = mi mm + X depends on the À parameter and is therefore non-identifiable.
A.1
Monotonicity Assumption
To remedy the problem of non-identifiability of principal faimess, 14 proposes the monotonicity assumption:
Definition 5 (Monotonicity). We say that an outcome Y satisfies monotonicity with respect to a decision D if
Ya, (u) > Yáo (u).
(39)
In words, monotonicity says that for every unit, the outcome with the positive decision (D D) would not be worse than with the negative decision (D 0). We now demonstrate how monotonicity aids the identifiability of principal faimess.
As implied by the definition of the SCM, we almost never have access to the unobserved sources of variation (w) that determine the identity of each unit.
14

---

Under the monotonicity assumption (Eq. 39 the principal fairness criterion is identifiable under the Standard Fairness Model (SFM).
Proposition 6.
Proof. The main challenge in PF is to obtain the joint distribution P(Udo Vd,), which is non- identifiable in general. Under monotonicity, however, we have that
Yáp (w) 0 A Ya, (w) =0 Yá (u) = A Ya, (w) =1 Ydo (u) = 1.
Ya, (u) 0,
(40)
(41)
Therefore, follows from monotonicity that
P(yd, 1, yd, 0) 0,
(42)
P(yd = 0, yd. = 0) = P(yd, = 0),
(43)
P(yd, 1, Md. 1) P(ydo 1),
(44)
P(yd, = O. yd, = =1)= = 1- -(Ud, = 0) P(ya, = =1),
(45)
thereby identifying the joint distribution whenever the interventional distributions P(ya.). P ((y..)
are identifiable.
In the cancer surgery example, the monotonicity assumption would require that the patients have strictly better survival outcomes when the surgery is performed, compared to when it is not. Given the known risks of surgical procedures, the assumption may be rightfully challenged in such a setting. In the sequel, we argue that the assumption of monotonicity is not really necessary, and often does not help the deciision-maken even if it holds true. To fix this issue, in the main text we discuss a relaxation of the PF criterion that suffers from neither of the above two problems but still captures the essential intuition that motivated PF.
B Canonical Types & Bounds
Definition 6 (Canonical Types for Decision-Making! Let Y be the outcome of interest, and D binary decision. We then consider four canonical rypes of units:
(i) units u such that Yd, (u) 1, Ya, (u) 1, called safe, (ii) units u such that Ydo (u) Y4, (u) 0, called harmed,
(iii) units u such that Yd (u) Ya, (u) 1, called helped, (iv) units u such that Yd, (u) 0, Ya, (u) 0, called doomed.
In decision-making, the goal is to treat as many units who are helped by the treatment, and as few who are harmed by it. As we demonstrate next, the potential outcomes Yd, (u), Ya, (u) depend precisely on the canonical types described above.
Proposition (Canonical Types Decomposition). Let M be an SCM computible with the SFM. Let D be binary decision that possibly affects the outcome Y. Denote by (8, d, u)(x, z, w) the proportion of each of the canonical types from Def. 6 respectively, for a fixed set of covariates (z, 2, w). It then holds that
P(ya, x (,,2,w) w) = c(x, x, w) + ss w), P(yd, [x,2,w) d(0, 2, w) + 5Xx, w.) w).
(46)
(47)
Therefore, we have thut
(x, 2, w) = P(yd, x., 1. w) - P(ydo Ix, w)
(48)
c(x, z, w) d(I, z, w).
(49)
Proof. Notice that we can write:
P(ya x, 2,2,,, w) P(yd, = 1, Ndo = 1 1 + P(ya, = 1, yds = 0 xr 22 w)
(50)
s(x, z, w) + (1
(51)
15

---

where the first line follows from the law of total probability, and the second by definition. Similarly, we have that
P(ydo 2,,,,) P = 14, xz w) + P(ydo = 1, yd, = 0 x 2, 2, w) s(z, 2, w) + 2,
(52)
(53)
thereby completing the proof.
The proposition shows us that the degree of benefit (xx, 2, w) captures exactly the difference between the proportion of those helped by the treatment, versus those who are harmed by it. From the point of
view of the decision-makes this is very valuable information since higher A(x, 2, w) values indicate higher utility of treating the group corresponding to covariates (zz 2, w). This insight can be used to prove Thm. which states that the policy D* obtained by Alg. is optimal:
Proof. Note that the objective in Eq. 1 can be written
E E[YD]
P(YD= 1)
(54)
= Σ
P(YD = 1 2, 2, P(z,,, 2,
(55)
2, 2,30
Σ
P(Y 1, 1= 1,,,, P(Y = x 2, 2,,,w P(ar, 2, w). (56)
3,2,00
Eq. 55 follows from the law of total probability, and Eq. 56 the consistency axiom. Now, note that Yág, Yd, LDD X, Z, W, from which it follows that
E B
Σ
P((a. 2,2, (. = x, x, w) + P(yd, 1z,,w) = 0 x z, m)
(57)
2,5,00
P(x, w).
By noting that P(D (1,,,,2) = 1 PPD = 1 2,2, w), we can rewrite the objective
Σ 2,2,M
(s(,,2,0) c1, 2, P(d w)
(58)
+(s(2,2, + d(x, w) (1 - P(d x, 2, w)) P(x, w)
(0,,2,0)) w) P(d x, 2, w)) [c(3, (,, w) d(xr, 2, wi)] P P(rr, z,
(59)
P(ydo 1,8d, 1) Σ P(d x, P(r, (
(60)
Only the second term in Eq. 60 can be influenced by the decision-maker, and optimizing the term is subject the budget constraint:
Σ 2, (() <
(61)
Such an optimization problem is a simple linear programming exercise, for which the policy D* from Alg. is a (possibly non-unique) optimal solution.
Finally, as the next proposition shows, the values of P(yd, xr, 2.), P(ydo x., 2, w) can be used to bound the proportion of different canonical types: Proposition 8 (Canonical Types Bounds and Tightness). Ler (s, d, c, u) 2, w) denote the propor- tion of each the canonical types from from Def. 6or a fixed set of covariates (z, 2, w). Let m2 2, w) P(ya, x, z, w) and mo(x, z, w) = P(yá, x, 2, w) and suppose that mi (I, w) 2 mo 2, w). We then have that (dropping (x, 2, w) from the notation):
d E [0. min(mo, mn)) )}:
my
(62)
e [mi ma, mn).
(63)
In particular, the above bounds are tight, meaning that there exists an SCM M, computible with the observed data, that attains each of the values within the interval. Under monotonicity, the bounds collapse to single points, with d 0 and mm mp-
16

---

d( 2. (0)
<(x, c(,,2,w) 2,
5xx 2, w)
Figure 5: Canonical types solution space. The unit simplex is shown in yellow, the + mi plane in blue, and the + me plane in red. The solution space for the possible values of (s(2, w), c(z, 2, w), d(x, 2, w)) lies at the intersection of the red and blue planes, indicated by the dashed black line.
Proof. There are three linear relations that the values 8, d, c, obey:
m1,
(64)
8 + d = m0.
(65)
d + c=
(66)
On top of this, we know that 8, d, are all non-negative. Based on the linear relations, we know that the following parametrization the vector (s, d, c, u) holds
(s, d, c, u) (mo
+ m1 mo, m1 - d):
(67)
which represents line in the 3-dimensional space (s, d, c). In particular, know that the values of (s, d, c) have to lie below the unit simplex Fig. (in yellow). In particular, the red and the
blue planes represent the linear constraints from Eq. 64
65 The line parametrized in Eq. 67 lies
m1)] since each of the elements in Eq. 67 is positive. This bound d also implies that e [m] mo, mi). Finally, need to construct an fy mechanism that achieves any value within the bounds. To this end, define
the intersection of the red and blue planes. Notice that d [0, min(mo.
fy (x, 2, w. d, uny =1 (u, E [0, s)) + d 1u +
(68)
(1
1uu, t,
(69)
U Unif(O, 1].
(70)
which is both feasible and satisfies the proportion of canonical types to be (s, d, c, u).
C Proof of Thm.
Proof. The first part of the theorem states the optimality of the DCF policy in the counterfactual world. Given that the policy uses the true benefit values from the counterfactual world, we apply the argument of Thm. to to prove its optimality.
We next prove the optimality of the DUT policy from Alg. In Step we check whether all individuals with a positive benefit can be treated. If yes, then the policy Doy is the overall optimal policy. If not, in Steplo we check whether the overall optimal policy has disparity bounded by M. If this is the case, D the overall optimal policy for a budget uses the entire budget b (since we are
and cannot be strictly improved.
For the remainder of the proof, we may suppose that operating under scareity), and that DUT has introduces disparity 2 M. We also assume that the benefit A admits density, and that probability P(A [a, b] x) for any (a, b] [0, 1] and
17

---

Let 8(#0), 8(m1) be the two thresholds used by the DUT policy. Suppose that DUT is policy that has higher expected utility and introduces a disparity bounded by M, or treats everyone in the disadvantaged group. Then there exists an alternative policy D De with higher or equal utility that takes the form
if A (x1, 2, w) 8(01)
D UT
if A 2, 8(x0)
(71)
0 otherwise.
with 8(=0)', 8(x1)' non-negative (otherwise, the policy can be trivially improved). words, for any policy D D there is threshold based policy that is no worse. The policy D is also threshold based policy. Now, if we had
glaaY < g(xx)
(72)
g < 5(00)
(73)
it would mean policy D"T is using larger budget than DUT However, DUT uses a budget of b, making D" infeasible. Therefore, we must have that
slarY < <5(e)),8 5lw0) > g(2u)
(74)
glar)' 5(zz), glao8 < 5(z0)
(75)
We first handle the case in Eq. 74 In this case, the policy D UUT introduces larger disparity than
DUT, Since the disparity of DUT is at least M, the disparity of D" is strictly greater than M.
Further, note that 8(ma)' 8(mo) 2 0. showing that D does not treat all individuals with a positive benefit in the disadvantaged group. Combined with a disparity of > M, this makes the policy D UT infeasible.
For the second case in Eq. 75 let (8o, 61) denote the utility of threshold based policy: (60, 61) = EE (A > 6x) = =o)] + EJA1(A > 61)1(X = =1)]
(76)
Thus, we have that
U(8(m0), ((z1)) U((=0o) s(an))) = -E E[A E ( 1)1(X = x1)
(
(77)
EIA 181Z0), 8(00) 1)1(X zo)]
(78)
2 Slm)) [I E (8(m1), (1(m))
11{XX = æ1)
(79)
8(10) (OEE)
5(20)
1x zo)]
(80)
2 S(0o) (TE(N(A E (8(01) 8(01))
1) =x1]]
(81)
EEI(AA (8(20) ,0(x0)))1(X = z0))
(82)
(00) (P(A 18(11)) 5(())) 1,21)
(83)
P(A E (o(oo)", 5(r0)),
(84)
20,
(85)
where the last line follows from the fact that D UT also gives a contradiction.
has a budget no higher than DUT, Thus, this case
Therefore, we conclude that policy DUT is optimal among all policies with a budget S b that either introduce bounded disparity in resource allocation [P(d] mn) P(d xo) M or treat everyone with a positive benefit in the disadvantaged group.
D Equivalence of CF and UT Methods in Alg. 3
A natural question to ask is whether the two methods in Alg. 3 yield the same decision policy in terms of the individuals that are selected for treatment. To examine this issue, we first define the notion of counterfactual crossing:
18

---

Definition 7 (Counterfactual crossing). We say that two units of the population u1, U2 satisfy counterfactual crossing with respect to an intervention C if
(i) u1, U2 belong the same protected group, X(u)) ((2)).
(ii) unil un has a higher factual benefit than u2, A(ui) > A(u2).
(iii) unit un has a lower counterfactall benefit than U2 under the intervention C, Ac(u)) <
(c(
In words, two units satisfy counterfactual crossing if in has a higher benefit than U2 in the factual world, while in the counterfactual world the benefit is larger for the unit U2. Based on this notion, we can give condition under which the causal and utilitarian approaches
equivalent:
Proposition 9 (Causal and Utilitarian Equivalence). Suppose that no two units of the population satisfy counterfactual crossing with respect to an intervention C, and suppose that the distribution of the benefis A admits a density. Then, the causal approach based on applying Alg. Tg.Thith counterfactual benefit Ac. and the utilitarian approach based on factual benefit A and the disparity M defined in will select the same set of units for treatment.
Eq. 29
Proof. The policy DUT treats individuals who have the highest benefit A in each group. The DCF policy treats individuals with the highest counterfactual benefit Ac. Importantly, the policies treat the same number of individuals in the zg and æ1 groups. Note that, in the absence of counterfactual crossing, the relative ordering of the values of A, Ac does not change, since
Aun)) > A (u2) = Ac(w)) > (u2).
(86)
Thus, since both policies pick the same number of individuals, and the relative order of A, Ac is the same, DUT and DCF will treat the same individuals.
E Experiment
We apply the causal framework of outcome control to the problem of allocating mechanical ventilation in intensive care units (ICUs), which is recognized as an important task when resources are scarce [5]. such as during the COVID-19 pandemic [40 42] An increasing amount of evidence indicates that a sex-specific bias in the process of allocating mechanical ventilation may exist (27). and thus
the protected attribute X will be sex (wr for females, for males).
To investigate this issue using the tools developed in this paper, we use the data from the MIMIC-IV dataset [17 [16 that originates from the Beth Israel Deaconess Medical Center in Boston, Mas- sachusetts. In particular, we consider the cohort of all patients in the database admitted the ICU, Patients who are mechanically ventilated immediately upon entering the ICU are subsequently re- moved. By focusing on the time window of the first 48 hours from admission to ICU, for each patient we determine the earliest time of mechanical ventilation, labeled tMv. Since mechanical ventilation used to stabilize the respiratory profile of patients, for each patient we determine the average oxygen saturation in the three-hour period [L 3, [MV] prior to mechanical ventilation, labeled Oz-pre. We also determine the oxygen saturation in the three-hour period following ventilation ['MV, + 3). labeled Oy-post. For controls (patient not ventilated at any point in the first 48 hours), we take the reference point as 12 hours after ICU admission, and calculate the values in three hours before and after this time. Patients" respiratory stability, which represents the outcome of interest Y, is measured as follows:
2 97, (O Y==1-00-P0s-- 97)² otherwise.
Y 0 if
O2-post Of
(87)
Values of oxygen saturation above 97 are considered stable, and the larger the distance from this stability value, the higher the risk for the patient. We also collect other important patient characteristics before intervention that are the key predictors of outcome, including the SOFA score 39 respiratory rate, and partial oxygen pressure (PaOz). The data loading is performed using the ricu R R-package 4. and the source code for reproducing the entire experiment can be found here.
19

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0.03
Ses
0.02
Parrade
Value
Ma.
Piimschoiccal
0.01
2:15
0.10
0.00
0.05
02
03
D4
06
06
07
Causal Fairness Measure
Bencttt Dociio
(a) Disparity in Deurr decomposed.
Benefit fairness criterion Deurr.
Figure 6: Analyzing the existing policy Dow
Step 1: Obtain the SFM. After obtaining the data, the first step of the modeling is to obtain the
standard fairness model (SFM). The SFM specification is the following:
X sex,
(88)
Z age,
(89)
W (SOFA score, respiratory rate, PaO2):
(90)
D mechanical ventilation,
(91)
Y respiratory stability.
(92)
Step 2: Analyze the current policy using Alg. Then, perform analysis of the currently implemented policy Deurr, by computing the disparity in resource allocation, P(dowrr xi) P(d zo). and also the causal decomposition of the disparity into its direct, indirect, and spurious contributions:
Pl x1)) - P(d (00) = 1.6%
0.3% 0.3%
(93)
DE
IF.
SE
The results are shown in Fig. 6a
with vertical bars indicating 95% confidence intervals obtained using bootstrap. The decomposition demonstrates that the decision mechanically ventilate patient has a large direct effect of the protected attribute X, while the indirect and spurious effects explain smaller portion of the disparity in resource allocation. We then compute
p(d
A 8, - P(d A 8,
(94)
across the deciles of the benefit A. In order to do so, we need to estimate the conditional potential outcomes Yá,: Yd, and in particular their difference [E[a, Yd, I, w, w.. We fit an xgboost model which regresses Y on D, X, Z, and W, to obtain the fit Y. The learning rate was fixed at 77 0.1, and the optimal number of rounds chosen via 10-fold validaanii We then use the obtained model to generate predictions
Pa (I. 2, w), Ddo (,,,,,))
(95)
from which we can estimate the benefit A. The results for the probability of treatment given fixed decile shown in Fig. [bb] Interestingly, each decile, women are less likely to be mechanically ventilated, indicating a possible bias.
Step 3: Apply Alg. to obtain D Our next step is to introduce optimal policy that satisfies benefit fairness. To do so, we make use of the benefit values. In our cohort 50,827 patients, a total of 5,404 (10.6%) mechanically ventilated. We assume that the policy D* needs achieve the same budget. Therefore, we bin patients according to the percentile of their estimated benefit A. For the percentiles [90%, 100%). all of the patients treated. In the 89-90 percentile, only 60% of the patients can be treated. We thus make sure that
P(d" A E [ds9%, 0y0%) x1) P(d" A 1009%, 890%)] xi) 0.6.
(96)
Due to the construction, the policy D* satisfies the benefit faimess criterion from Def.
20

---

0.4
0.010
0.3
Value
Value
0.005
0.2
0.1
0.000
0.0
Causal Fairness Measure
Causal Fairness Measure
(a) Disparity in D* decomposed.
(b) Benefit disparity decomposition.
Figure 7: Causal analysis of D and A using Alg.
Step 4: Apply Alg. to analyze D We next decompose the disparity of the new policy D and also decompose the disparity in the benefit A. We obtain the following results:
P(d [11)=
P(d* (o))
0.1%
0.6%
0.2%
(97)
DE
18
E(A x1) E(A xo) = 0.08 + 0.21 + 0.04
(98)
DE
IE
SE
The two decompositions are also visualized in Fig. Therefore, even after applying benefit fairness, some disparity between the sexes remains. The causal analysis reveals that males require more mechanical ventilation because they are more severely (indirect effect). They also require more mechanical ventilation since they are older (spurious effect), although this effect is not significant. Finally, males also seem to benefit more from treatment when all other variables are kept the same (direct effect, see Fig. 7b). We note that using Alg. has reduced the disparity in resource allocation, with substantial reduction of the direct effect (see Eq. 93 vs. Eq. 97).
Step 5: Apply Alg. to create DCF, In the final step, we wish to remove the direct effect of sex on the benefit A. To construct the new policy DCF we will make use of Alg. Firstly, we need to compute the counterfactual benefit values, in the world where X = along the direct pathway, while W attains its natural value Wx(w) (u). That is, we wish estimate Ar,Wxi., for all patients in the cohort. For the computation of the counterfactual values, we make use of the xgboost model developed above. In particular, we use the fitted model to estimate the potential outcomes
Ydo Wx(us> Yd "xes
(99)
The adjusted potential outcomes allow us to estimate Az, Wxoo after which we obtain the policy DCF that satisfies the CBF criterion from Def.
After constructing DCF, we have a final look the dis- parity introduced by this policy. By another application of Alg. 2
we obtain that
Volure
OF 1 OF zo) 0%
0.5% + 0.2%
1.25
DE
IE
SE
Therefore, we can see that the removal of the direct effect from the benefit A resulted in a further decrease in the overall disparity. The comparison of the causal decompositions for the original policy Dearr, optimal policy D* obtained from Alg. T and the causally fair
IE
SE
Faimess Measure
Policy
D
Figure 8: Causal comparison of policies
policy DCF is shown in Fig. g
Dourr D*, and DCF
F Connection of Benefit Fairness and Demographic Parity
In this appendix, we diseuss the connection of the benefit fairness (BF) criterion with demographic parity (DP). The connection is given in the following formal result.
21

---

Proposition 10 (Benefit Faimesss & Demographic Parity). Suppose that the distribution of the benefit A is equal between the groups Tn, Ti, that is
A
A
I
A
zo
(100)
Then any decision policy satisfying benefit fairness also satisfies demographic parity.
Proof.
P(d x1)
Σ P(d T1, A x1)
(101)
Σ P(d To. A = P - (11)
using BF
(102)
Σ P(d To. To. A= 8) x
using 21
d
(103)
P(d za);
(104)
implying demographic parity.
G Practical Considerations for Decompositions in Alg.
In this appendix, we discuss some practical details about how to perform the decompositions of the treatment disparity and benefit disparity as described in Alg. 2 particular, we begin by showing that the benefit A can be considered as variable in the causal diagram.
Proposition 11 (Connection of Benefit with Structural Causal Models). Let M be an SCM compatible with the SFM in Fig. [ Let the benefit A be defined as in Eq. 14 Let fy be the structural mechanism of the outcome Y, taking X, Z, W, D as inputs, together with the noise variable Uy. The structural mechanism fo the benefit A is then given by:
fa 2, w) = E., [fr (x, w, d1, uy)) - Eu, (Ix 2, W. do uy)l
(105)
where Fm, integrates over the randomness Uy Therefore, the benefit is a cicermmmiistcffuctttii of X, 7, W. and we can add it the causal diagram as follows:
Z
A
W
Therefore, after adding to the causal diagram as in the above proposition, we can view it as another outcome variable. In particular, it also allows to define various counterfactuals with respect to it, via the standard definitions for counterfactuals used in structural causal models. For instance, the potential outcome Azo (which is random variable) is simply given by S' (æo, Z, W), where Z, W are random variables but X = To is fixed. Other, possibly nested counterfactuals are also defined analogously.
Equipped with structural understanding how benefit relates to the original causal model, we can now explain in more detail how the decompositions in Alg. 2 are performed. In particular, the
decompositions follow the previously proposed approaches of 44 31:
Proposition 12 (Computing Decompositions in Alg. 2). Let M be an SCM compatible with the SFM
Fig. [ Let the benefi be defined as in E4. 14 and suppose is added the causal diagram as in Prop. 11 The decompositions of the disparity in treatment allocation, and the disparity in expected
22

---

benefit, can be computed as follows:
P(d z1) - P(d zo) = P(derwa, xo) P(ddo x00)
(106)
direct effect DE(X -+D)
P(drr,Weo x0)) P(dz) 200)
(107)
indirect effectt -E()
P(dz) (D) - P(dr) x
(108)
spuriions effect SE(X+--01)
E(A 21) E(A xτ)
E E( W Wep 20 TE( 200 (00)
(109)
directt effecr DE(X -+))
E ( An,.Wa xo) - 1E
20
(110)
indirect effect 1E(XX -+)
E E(Ar. TD) E(A 11 11
(111)
spurious effect SE X
Furthermore, under the assumptions of the SFM in Fig, both of the decompositions are identifiable and their identification expressions are given by
DE(X F) Σ
E(F 31,
E(F ro, 2, w) P(w| za, za, P( xo);
(112)
IE(X - F) Σ
[(FF 11, 2, 1) P(w x0, - P(wl xi, x1, P( mo),
(113)
2,30
SE(X +--+ F) Σ (F 21, 2) P(z (P(2120) xo) P(2 P(=1z1).
(114)
where the random variable F is either replaced by the treatment decision D to obtain the direct, indirect, and spurious terms Eqs. 706
T06 108 or by the benefit A obtain the terms in Eqs. T09 117
23


Reviews:
[{"review_id": "jWbbiqrZoT", "reviewer": "Reviewer_9Jxk", "summary": "The paper focuses on outcome control from a causal perspective of the decision-maker. The authors introduce benefit fairness taking the perspective of the decision maker and provide theoretical guarantee that the algorithmic result is optimal and satisfies benefit fairness. To support the decision maker in indicating potential discrimination early on, algorithm 2 evaluates the difference in benefit to demographic groups. And the paper defines causal benefit fairness with an accompanying algorithm and a guarantee of optimality.", "questions": "The authors do not actually provide a clear definition of “outcome control” as used in this paper. Based on the content, I assumed they are using the high level definition put forth by [Procedural Justice in Algorithmic Fairness, CSCW 2019], which defines outcome control as enabling the correctability and possible recourse against an individual decision. But this paper is not cited in the present paper. The authors reference [Causal Fairness Analysis, 2022], which is an ICML tutorial where section 5.3.3 contains exact wording and sentences from the present paper lines 35-38. The tutorial includes more insight clarifying that outcome control setting requires that the institution and the individual have the same utility function. It would be helpful to clarify this in the main text of the present paper. What is the definition of “outcome control” being used in this paper?\n", "limitations": "It is not clear how generalizable this framework is given the demonstration on clinical examples.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1 - In fairness research, the need for more causal perspectives has been a growing discussion. This research is significant in contributing to an underdeveloped area.\n\nS2 - The research problem is well developed and simple guarantees provided with proofs in the appendix.\n", "weaknesses": " Exposition clarity needs minor improvement. For instance, what is the relationship between the first and second paragraph of the introduction? What do the authors see as the connection between outcome control and historical biases?\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time and effort in reviewing the paper. We are encouraged by the fact that the reviewer appreciated the main strengths of the paper. We provide further clarifications to the questions raised in the sequel. \n\n[W1: Historical biases] We appreciate this insightful and fundamental question, thank you. Following historical biases, certain demographic groups differ in their distribution of covariates (either confounder Z, or mediators W). The overall benefit from treatment may then _be lower in the protected group_, due to a difference in the covariates that originates historically. Examples of this are numerous (we mention in passing the example of kidney malfunction, more common in some minority groups, which in turn reduces the possible benefit of heart surgery – this example falls under outcome control). We have now added this remark to the Intro, to make the transition more smooth. We thank the reviewer for this suggestion!\n\n[Q1: Outcome Control definition] Thank you for pointing this out. We have now made clear in the Introduction that outcome control in the context of the paper refers to the specific setting described by the causal diagram in Figure 1, in which the standard fairness model is extended to include a treatment $D$ that precedes the outcome $Y$. More formally, in outcome control, the goal is to maximize a specific outcome using a known control (e.g., maximizing survival using surgery). Furthermore, it is true that this setting is often characterized by the fact that individuals and institutions have the same utility function. This is now clarified in the text. Furthermore, we also add the following explanation and citation relating to the work that you mentioned:\n\n“Therefore, in the setting of outcome control, an institution may attempt to maximize an outcome (such as survival or well-begin) using a known control (such as surgery or a welfare intervention). This is different from the concept of outcome control in the procedural justice literature [1], which refers to the settings when individuals have some influence over assigning their own outcomes.” \n\n[1] Lee, Min Kyung, et al. \"Procedural justice in algorithmic fairness: Leveraging transparency and outcome control for fair algorithmic mediation.\" Proceedings of the ACM on Human-Computer Interaction 3.CSCW (2019): 1-26.\n\n[L1] Here, we remark that the examples of outcome control are numerous, and go beyond just the clinical setting. For instance, consider a setting of outcome control in the context of criminal justice. A judge may wish to minimize the number of individuals who recidivate, that is, minimize the outcome $Y$. The judge can make the decision $D$ to detain the individual. This example, therefore, falls under outcome control, since the judge would attempt to use the decision $D$ to minimize the outcome $Y$. The COMPAS dataset is collected from a relatively similar setting to this one. In a further example, (Coston et al., 2020) analyzed a specific setting of a child hotline. There, the intervention is to send a social welfare team (decision $D$), while the intent is to maximize welfare and minimize harm (the outcome $Y$). The main considerations of the approach, for these settings, would remain largely the same as for the clinical examples in the paper. That is, the judge would attempt to detain those individuals for whom the detention decision reduces the probability of re-offending the most; a hotline responder would intervene to calls that are most urgent, and for which there is the greatest reduction in harm; both of these are analogous to treating patients who benefit the most from treatment in a clinical setting. We reflected this discussion in the manuscript, but, please, let us know if the answer is satisfactory or further elaboration would be desirable; thank you."}, {"review_id": "NaApDi7OcG", "reviewer": "Reviewer_pGG2", "summary": "This paper studies the problem where a decision maker must allocate a treatment $D$ to optimize an outcome variable $Y$ while ensuring that the decision is fair, formulating fairness as the protected attribute $X$ not having an influence on $D$. It uses a clinical decision-making process (a running example on Cancer Surgery, Fig. 2(b)) to showcase its proposed method.\n \nThe paper proposes the notion of benefit based on the potential outcomes (PO) framework. $Y_{d_0}$ denotes the outcome of a patient that didn’t undergo surgery while $Y_{d_1}$ denotes the outcome of a patient that did undergo surgery. Under PO, only one of the two potential outcomes are observed per patient. Benefit is defined as $Y_{d_1} - Y_{d_0}$. Hence, a decision maker will derive a $D^*$ that maximizes overall benefit: i.e., those patients for which $Y_{d_0}=0$ (died without surgery) and $Y_{d_1}$ (survived with surgery).\n \nThrough the running example, the paper shows how a decision maker (under imperfect information) can be unfair via $D$ despite optimizing for the benefit. It then introduces a set of algorithms to reach fair benefit. \n", "questions": "See my questions in the Weaknesses section. I’m willing to increase my score if these questions are addressed.", "limitations": "Further discussion on $\\Delta$ is missing. It’s a considerable limitation for the application of this definition and it’s not evaluated here theoretically.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "[S1] The topic is relevant and novel within (causal) fairness. Under a heterogenous population of individuals, it is reasonable to expect that not all individuals will benefit from a treatment (captured by $D$ and its causal effect on $Y$ in the paper). The notion of benefit becomes relevant as (i) it formalizes which individuals need $D$ and which do not; (ii) it raises the potential issue of what happens when those that benefit under $D$ don’t embody societal representational fairness goals.\n\n[S2] The use of the running example was very useful for understanding the paper’s goal.\n\n[S3] Fig. 3 illustrates well the problem of allocating a treatment without knowing the type of patient w.r.t. to $D$.\n\n[S4] Def. 2 (Benefit Fairness), putting aside the estimation issues, is a nice extension to other group-level fairness definitions (like Equal Opportunity). It states that individuals across the protected attribute $X$ that have the same (potential) benefit $\\delta$ to gain from treatment should have the same probability of getting treated.\n", "weaknesses": "[W1] PO limitations should be discussed further: Given the use of the PO framework, there’s a lack of treatment on how the benefit ($\\Delta = Y_{d_1} - Y_{d_0}$) can be estimated in a consistent and unbiased way, at least, at the individual level. For each individual, we can only observe $Y_{d_0}$ or $Y_{d_1}$: for each individual, the one observed is the factual and the other one the counterfactual. However, the paper lacks a meaningful discussion on how we can estimate the counterfactuals in the non-oracle setting, which is the one of practical interest. Even in the running example, isn’t there a risk that the clinicians are using a biased $\\Delta$? If so, how would that affect the proposed algorithms? The limitations should be discussed explicitly.\n\nIn particular, the derivation and justification of Eq. (16), which is central to the algorithm(s) is not clear. Is it a statement on the same tuple $(x, z, w)$: if so, how can the same individual be benefited and harmed at the same time by the treatment? Or is it a “match” between two individuals? Also, what is $z$ in the running example (this random variable is only presented and used in Fig. 1)?\n\nCounterfactual generation/estimation is not exploited in the example. For instance, if we have access to the SCM model, couldn’t we generate the counterfactual distribution via Pearl’s abduction, action, and prediction steps for $Y$. We could then identify the “doomed” and “safe”? This is important as Algorithm 1 can only operate meaningfully if it can identify the borders of the types of individuals (w.r.t. the notion of benefit). Line 1 in Algorithm 1 needs to be discussed further.\n\n[W2] Stronger Section 2: This section could be tighter. For instance, what’s the role of the budget $b$ from Def. 1? Even in the oracle setting, it is possible that we can’t treat all 100 patients if the budget is low enough, no? Conversely, if the budget is high enough and $\\Delta \\geq 0$, we could treat all patients, no?\n\nFurther, as a follow up to W1, if there is the pair \\{$Y_{d_0}$, $Y_{d_1}$\\} for each individual then there is also a $\\Delta$ parameter. Now, under the oracle, we can see through the future and allocate $D$ based on $\\Delta$. In the non-oracle setting, there’s a brief discussion about using $W$ as a proxy that leads to unfairness: how does that translate into estimating $\\Delta$? Otherwise, if we can estimate it, then why are we even considering the proxy $W$? What I’m hinting at here – do correct me if I’m wrong – is that Def. 2 needs to have some measure of uncertainty to highlight that the non-oracle allocation will have some error w.r.t. to the oracle allocation (for a fixed $b$). Otherwise, it seems like we are still in the oracle setting.\n\n[W3] Limited evaluation: The running example is essentially the only use case. Although it shows the proposed methodology, it would’ve helped to, e.g., test the algorithms under different parameters for the same synthetic data. Similarly, could the algorithms handle other variables on top $W$ or under a second protected attribute like race? The evaluation, even for the single use case, could’ve been pushed further to show the robustness of the approach.\n\n[W4] Some definitions are unclear or not fully explained. For instance, \nIn Def. 1, the role of the budget $b$ is not explained. It would’ve also helped to formulate it under the PO framework explicitly. \nIn Def. 2, please define $\\Delta$ and $\\delta$ within the definition itself. These are presented later.\nIn Def. 3, is it at the individual level? Under the SCM presented (Fig. 2, but also Fig. 2), how are the conditionals the same under the interventions captured by the causal pathways. Don’t we need to update for downstream effects under $\\mathcal{C}$?\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "In the weaknesses section, the questions were good and we believe to have addressed them in a factual and robust way. Please let us know if there are any further issues!\n\n--- \n\n[W1a: PO Limitations] Thanks for pointing this out. See global comment [G1] for a detailed response.\n\n[W1b: Biased $\\Delta$] This is indeed possible and is likely to be the case in practice. In some sense, using $\\Delta$ is “optimal”, while clinicians probably use a less-than-perfect version of $\\Delta$. We’ve included a specific remark to acknowledge this. It’s worth noting that our goal is to understand the conditions under which decision-making should be made, rather than describing how currently stakeholders make decisions – which, as you pointed out, may be suboptimal. We hope our work can serve as a normative guide to enhance decision-making in practice.\n\n[W1c: Eq. (16) justification] Thanks for the opportunity to clarify this issue, which is central to our contributions. The key distinction here is between ‘covariate-specific’ and ‘unit-level’. By ‘covariate specific’, we mean all units $u$ of the SCM that are compatible with the observed values $(x, z, w)$. The value of $\\Delta$ is computed for a group of units with fixed covariates. However, for each of these units, the outcome can only be 0 or 1, as you rightly pointed out. $\\Delta$ represents an average over these outcomes. Unfortunately, both ‘covariate-specific’ and ‘unit-level’ approaches are called “individual” in the literature, leading to much confusion. To address this, we have dded the following for clarity:\n\n“The values of $c, d$ are covariate-specific, and indicate the proportions of patients helped and harmed by the treatment, respectively, among all patients coinciding with the observed event $X=x, Z=z, W=w$ (i.e., all $u \\in U$ s.t. $(X, Z, W)(u) = (x,z,w)$). “\n\n[W1d: What is Z?] Generally, the $Z$ can be an arbitrary set of observed confounders, whereas in our running example $Z = \\emptyset$. Thanks for noting this. We have added this explicitly in the description of Figure 2. \n\n[W1e: $\\Delta$ estimation] Agreed! Please see suggested improvements (i)-(iii) we mention in global answer [G1]. However, we remark that identifying doomed and safe groups would not be possible in the absence of the SCM. However, identifying the difference of the proportions $c(x,z,w) - d(x,z,w)$ is possible from observational data.\n\n--- \n\n[W2: Section 2] Thanks – both assertions are indeed correct. Firstly, we make the following addition to the text, just below Definition 1 :\n\n“The budget b is relevant for scenarios when resources are scarce, in which not all patients possibly requiring treatment can be given treatment. In such a setting, the goal is to treat patients who are most likely to benefit from the treatment, as formalized next in the text.”\n\nFurther, note that if b is large enough, only the $\\Delta > 0$ condition needs to be checked. This is reflected in Line 2 of Algorithm 1, and also Line 2 of Algorithm 3 (and corresponds exactly with the intuition you described). So, the importance of having a budget comes for cases when not all patients who benefit from treatment can be treated. In this case, the reasoning becomes more involved, and we need to estimate $\\Delta(x,z,w)$ for all $x,z,w$ values, and treat patients with the largest benefit $\\Delta$, as described in the Algorithms 1 and 3.\n\n---\n\n[W3a: Limited Eval] Please see global response [G2].\n\n[W3b: Other Ws and second attribute] We appreciate the question, thank you. Once again, we can provide some positive answers. Firstly, the set of variables $W$ can be multidimensional (see Appendix E for a specific real-world example). Furthermore, the example in Appendix E also shows how we can have a set of confounders $Z$. Therefore, the framework is compatible with datasets of a larger dimension.\n\nRegarding the extension of the protected attribute – this is indeed possible. The simplest way of seeing this is to define the protected groups as all elements of the Cartesian product of the domains of the two attributes (e.g., multiple (sex, race) combinations). Then, for example, Eq. (15) could be an equality relating more than two different groups, but would in principle remain the same.\n\n---\n\n[W4a: Def. Explanations] We hope the discussion on the budget provided above helped to clarify this issue. Still, let us know if you think this is sufficient. Definitions of $\\Delta$ and $\\delta$ are now put in Definition 2. \n\n[W4b: Individual Level Def. 3?] Thanks for asking; Yes, the first part of the definition is covariate-specific (Eq. (25)), and the second part is specific to a fixed $\\delta$ level (Eq. (26)), but across covariate tuples. We now updated the text to make this transparent:\n\n“CBF requires that treatment benefit $\\Delta$ should not depend on the effect of $X$ on $Y$ along the causal pathway $\\mathcal{C}$, and this condition is covariate-specific, i.e., holds for any choice of covariates $x, z, w$. Additionally, the decision policy $D$ should satisfy BF, meaning that at each degree of benefit $\\Delta = \\delta$, the protected attribute plays no role in deciding whether the individual is treated or not. This statement is for a fixed value of $\\Delta = \\delta$, and possibly considers individuals with different values of $x, z, w$.“\n\n[W4c: Same conditionals?] The question pertains to Figures 4(a,b), right? (there may be a typo in the question). But you are correct – in Figure 4(a), we plot the densities of $\\Delta \\mid x_0$, $\\Delta \\mid x_1$ which correspond to Males and Females, respectively. Then, after adjusting for the indirect effect, the two densities become the same, as noted in your comment. We have now updated the Figure 4(a) label to have labels $\\Delta \\mid x_0$, $\\Delta \\mid x_1$ instead of Male, Female, to make things clearer. Furthermore, we increase the label size for Figure 4(b), so it is more clearly legible that the indirect effect is removed, i.e., the subscript is $W_{x_0}, x_1$."}, {"review_id": "9UtLvr4ESb", "reviewer": "Reviewer_H5h5", "summary": "The paper proposes a new fairness notion called benefit fairness that considers fairness in the outcomes of the decisions. In order to avoid the unidentifiability issue of the principle fairness, this paper proposes to condition on the conditional average treatment effect (CATE). The paper provides an algorithm that is proven to be optimal and satisfy benefit fairness. The paper in addition provides an algorithm to satisfy both benefit fairness and counterfactual benefit fairness.", "questions": "Although I didn’t check the correctness of the proof, Theorem 2 seems quite strong. Are there any assumptions about the policy or the causal model in Theorem 2?\n\nMy understanding of benefit fairness is that, it considers the causal effect of the link X->Y. If we remove this link, then $\\Delta(x_0,z,w)$ is always equal to $\\Delta(x_1,z,w)$, and benefit fairness degrades to the conditional demographic parity. Thus, the meaning of benefit fairness is to make the causal effect of the link X->D match the causal effect of the link X->Y? Is this understanding correct? \n\nIf my above understanding is correct, does that mean that CBF cannot be satisfied if the link X->Y exists, because CBF generally means that  the decision should not depend on X?\n", "limitations": "None.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The proposed new fairness notion is quite interesting. It is different from traditional fairness notions, yet it is well-motivated and -explained using a real-world example. \n\nThe proposed algorithms come with theoretical analysis that shows their optimality and fairness satisfaction. \n\nA budget is involved in the problem formulation that makes the algorithms more generally applicable. \n", "weaknesses": "The connection between benefit fairness and traditional fairness notions other than principle fairness, such as demographic parity, equal opportunity, etc., is not clear. Are they consistent or do they conflict to some extent? A rigorous study may be out of the scope of this paper but some discussions will be helpful.\n\nThe measure of benefit fairness depends on the observation of the outcome Y. Can we generally assume that Y is available when we construct the decision policy? In the running example, Y is observed after a 2-year period. How can we estimate Y when we construct the decision policy before the 2-year period?\n\nSome statements and notations are not very clear. For example, in Definition 3, $y_C$ is not defined and the definition of the pathway is not clear. My understanding is that the paper attempts to formulate path-specific counterfactual fairness, but the notations are quite confusing. In addition, $\\Delta_{C}$ is not defined either. The two terms CBF and BFC may easily confuse the readers.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the provided comments, it was nice to see that the main strengths were appreciated! We respond point-by-point in the sequel.\n\n[W1: Connection with traditional notions] Great question, the manuscript indeed does not touch on this. We make the following observations and add the following paragraph to the text: \n\n“If the benefit distributions between groups are substantially different, a decision satisfying BF will not achieve demographic parity. However, for cases in which all causal pathways (direct, indirect, and spurious) between the protected attribute and the benefit are either zero or removed by adjustment, the Causal BF criterion implies demographic parity (i.e., equal allocation of resources by $D$)”.\n\nFurthermore, we also add a formal statement on this, which is added to the appendix due to considerations of space:\n\n“Suppose that the distribution of the benefit $\\Delta$ is equal between the groups $x_0, x_1$. Then, benefit fairness implies demographic parity.\"\n\nThe simple proof is also quite insightful for showing what is going on.\n\nProof: \n\\begin{align}\n    P(d \\mid x_1) = \\sum_\\delta P(d \\mid x_1, \\Delta = \\delta) P(\\Delta \\mid x_1) \n    \\overset{\\text{using BF}}{=} \\sum_\\delta P(d \\mid x_0, \\Delta = \\delta) P(\\Delta = \\delta \\mid x_1)\n\\end{align}\n\\begin{align}\n\\overset{\\Delta \\mid x_1 \\overset{d}{=} \\Delta \\mid x_0}{=} \\sum_\\delta P(d \\mid x_0, \\Delta = \\delta)P(\\Delta = \\delta \\mid x_0) = P(d \\mid x_0),\n\\end{align}\nimplying parity. We thank the reviewer for raising this point; it indeed adds an important connection to the existing literature.\n\n[W2: Dependence on outcome Y] This is a good point. The label may indeed not be available for the cohort at hand in some settings. However, the upside is that it may be possible to use retrospective data, under the assumptions encoded in Figure 1. Therefore, one may use past data to assess the $\\Delta$ quantity, possibly adjust it, and design a new and fair policy. On the other hand, if no data at all is available on the label $Y$, then designing a policy with such fairness guarantees is not possible, at least, with the level of assumptions currently considered. \n\n[W3: Notation Clarity] Thanks. We have added a clarification of what a clause $C$ is; further, we also define a pathway. The terms CBF and BFC are indeed too similar. We for that reason, throughout, use CBF for Causal Benefit Fairness, and BF for just Benefit Fairness. We hope this distinction between BF and Causal BF will be easier to follow.\n\n[Q1: Theorem 2 proof]   Good question; the main assumptions are the lack of confounding assumptions required for performing the identification of the effects necessary for constructing the policy. The second part of the answer is that the statement considers an infinite-sample, population-level case. We remark there is an additional level of difficulty for providing finite sample guarantees, which is not covered by our theorem statement. We feel these are intriguing challenges for future work. \n\n[Q2: X->D, X->Y effects] These are both great questions, thank you. There are two distinct parts of the CBF definition. \n\n(i) The first part ascertains the fairness of the $\\Delta$ quantity,\n(ii) The second part ascertains that at equal levels of $\\Delta$, the probability of treatment does not depend on the protected attribute. \n\nYour question seem to relate to part (i), if we understood it correctly. If there is no link X->Y, then $\\Delta(x_1, z, w) = \\Delta(x_0, z, w)$, which is indeed accurate. Put differently, as you noted, the absence of a direct X -> Y link implies conditional demographic parity for the $\\Delta$ quantity. Note, however, that this does not imply anything about the parity for decision D, related to part (ii). \n\nNow, more generally, if the link X -> Y exists and it is considered to be discriminatory, our proposal is to first _adjust the_ $\\Delta$ quantity accordingly. In other words, CBF cannot be satisfied unless we _remove the discriminatory effect from_ $\\Delta$. After having this condition ascertained, benefit fairness from part (ii) can be applied to ensure a notion of equity. Thus, the fairness requirement can be thought of as a “two-step” process.   We hope this makes sense but happy to provide further elaboration. \nThe note on balancing the effects X->Y and X->D is very interesting but we haven’t thought about doing that explicitly. Stll, in order to achieve the described notion of fairness, both of these pathways are affected by our procedure. Please let us know if you have any further questions, or if you see the balancing of the effects from a different perspective. "}, {"review_id": "1tkQLJa7qz", "reviewer": "Reviewer_BFhw", "summary": "The authors analyze a new dimension of decision making: outcome control and benefit from decisions. Specifically, this work uses causal analysis to introduce new definitions of fairness on benefits and outcome control. The authors also propose new algorithms to study fairness and design fair algorithms for outcome fair decisions. ", "questions": "* Two works [1, 2] seem somewhat related to the authors’ notions and framework. How does the framework of decision outcome control relate to the thinking of fairness of “treatments” in [1]? Similarly, how does the notion of “benefit” the authors introduce relate to the notion of “harm” in [2]? \n\n[1] Madras, David, et al. \"Fairness through causal awareness: Learning causal latent-variable models for biased data.\" Proceedings of the conference on fairness, accountability, and transparency. 2019. [2] Richens, Jonathan, Rory Beard, and Daniel H. Thompson. \"Counterfactual harm.\" Advances in Neural Information Processing Systems 35 (2022): 36350-36365.\n", "limitations": "* While the paper’s primary focus is on introducing a new theoretical framework of fairness in decision making related to outcome control and benefit, more analysis and results on different experimental settings might have been an added benefit.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "* The paper proposes new fairness notions for outcome control, that is an important aspect of many decision making settings\n* The authors use a simple example to walk through all the different aspects and notions, that greatly helps the readability of the paper.\n* Through the synthetic use case, the authors show when and how benefit fairness is important, and how causal path analysis could also become critical for decision making while accounting for fairer outcomes.\n", "weaknesses": "* While the appendix provides additional results on a real-world COVID dataset, the main paper has limited results on various different settings.\n* While the authors assume causal knowledge, the paper does not discuss limitations explicitly for when computing counterfactuals and especially path-specific counterfactuals might become challenging in practice.\n* Section 3 should have better explained which counterfactuals we compute to look at the path-specific factors and how the 3 different components are found out by decomposition.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the provided comments, which were encouraging. We provide further clarifications regarding the points raised below. \n\n---\n\n[W1: Limited Evaluation] Please see global comment [G1].\n\n---\n\n[W2: Computing Counterfactuals] Please see global comment [G2]. Also, we have added the following:\n\n\n- (i) We add a Proposition that states the identification property of $\\Delta$. In particular, the statement is the following:\n“Under the assumptions of the extended Standard Fairness Model from Fig. 1, the benefit $\\Delta(x,z,w)$ can be identified from observational data.”\n- (ii) Further we also added a comment on the fact that direct, indirect, and spurious effects of $X$ on $\\Delta$ are identifiable in this settings.\n- (iii) We also remark that the $\\Delta(x, z, w)$ itself may be easily adjusted to remove the direct effect of $X$. However, indirect effect counterfactuals at a covariate-specific level $x, z, w$ are more difficult to compute and may require further assumptions such as additivity or monotonicity of the response.\n\nWe hope the above three changes adequately address the concern. Please let us know. \n\n[W3: Terms in the decomposition] Thanks for noting this. We have implemented this suggestion. We now spell out the expression for the direct, indirect, and spurious effects, appearing in the decomposition, to increase transparency. \n\n---\n\n[Q1: Two related works] Thanks for pointing us to these references. You are indeed correct – the setting considered in [1] is related to ours. However, the approach taken therein differs from our approach. In particular, [1] computes three causal effects of interest: (i) total effect X -> Y; (ii) total effect D -> Y;  and (iii) total effect X -> D. Then, the paper suggests that fairness should be assessed based on these three quantities. Our framework, however, integrates the above effects more coherently, and brings forward a new definition of fairness based on first principles. We now also added a citation for [1].\n\nRegarding [2], we also found it quite interesting. However, the concept of harm / benefit defined in [2] is qualitatively different from our approach, due to the max operator used in Eq. (6) in [2]. The notion of harm / benefit in [2] considers what is in the literature known as canonical types [3] or principal strata [4]. However, our notion of benefit corresponds to the conditional average treatment effect (CATE), which measures the proportion who benefit minus the proportion of those harmed (in the language of [2]). We now clarify in the text that our notion of benefit differs from that in [2], to better contextualize our approach. Thanks for mentioning these works!\n\n[3] A. Balke and J. Pearl. Counterfactual probabilities: Computational methods, bounds and applications. In Uncertainty Proceedings 1994, pages 46–54. Elsevier, 1994\n[4] C. E. Frangakis and D. B. Rubin. Principal stratification in causal inference. Biometrics, 58(1): 21–29, 2002.\n"}]

Author rebuttal:
The authors would like to sincerely thank all the reviewers for this paper. The main strengths and novelty were clearly appreciated, and the questions raised were quite useful for us to revise and improve the paper.

In our response, we index all weaknesses with W, questions Q, and limitations L. We do not fully cite reviewer's questions (due to character limit), but try our best to provide a caption for each W/Q/L. 

Here, we would like to provide two global responses, which are then cited in the individual responses as well.

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[G1: Limited Evaluation] Unfortunately, during the submission process, the external links of the document have stopped working. In particular, the submission included the following links (the anonymized has now been shared with the Area Chair):

- (i) In a vignette that accompanies the paper, we performed inference for the running example on finite samples, but this has not been highlighted due to the missing link. Please, have a look at Supplementary Material, source-code/vignette.html. Therein, we perform the inferences described in the paper, based on the data itself (and not the SCM).
- (ii) Furthermore, in Appendix E, we performed a real-world experiment based on the MIMIC-IV dataset. This application, as the citations in the Appendix also show, comes from the literature in intensive care medicine, and is a well-known issue (sex-specific bias in allocating respirators). We hope the reviewers will find this real application compelling. 

In the current version of the writing, we decided to prioritize theoretical underpinnings rather than practical demonstrations. Of course, this is a matter of taste. We thought this type of exposition would maintain the most clarity in the concepts, while we deferred the empirical studies to the supplemental materials. However, we now suggest moving the real-world experimental results from Supplement E to the main text. We are happy to hear further suggestions on how to improve the presentation from the reviewers.

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[G2, Identification of Counterfactuals] The paper leaves the issue of identification of $\Delta$ implicit but we agree that making it explicit could add better transparency and improve the flow. We changed the manuscript to add the following points more explicitly. 

- (i) We added a Proposition that states the identification property of $\Delta$. In particular, the statement is the following: “Under the assumptions of the extended Standard Fairness Model (Fig. 1), the benefit $\Delta(x,z,w)$ is identifiable from observational data.”
- (ii) We further added an Appendix with the proof of identification using the counterfactual axioms (Ch. 7, Pearl, 2000). In the Appendix, we also provide the identification expression for $\Delta(x,z, w)$.
- (iii) Below the proposition, we pointed to more general identification strategies. Interestingly, these are also applicable to cases with both observational and experimental data.
“More generally, the approach of [1] can be used for testing identifiability of $\Delta(x,z,w)$, and this approach also handles cases when both experimental and observational data are available.”
- (iv) Further we also added a comment on the fact that direct, indirect, and spurious effects of $X$ on $\Delta$ are identifiable in this settings. A proof of this claim, together with identification expressions, is now added to the appendix.
- (v) We also remark that the $\Delta(x, z, w)$ itself may be easily adjusted to remove the direct effect of $X$. However, indirect effect counterfactuals at a covariate-specific level $x, z, w$ are more difficult to compute and may require further assumptions such as additivity or monotonicity of the response.

Thus, we hope this clearly demonstrates that our approach is applicable to real data; that is, our approach does not assume the knowledge of the SCM itself.


Meta-review: Novel use of causal analysis to introduce new definitions of fairness on benefits and outcome control.
Decision: Accept (poster)