Performative Validity of Recourse Explanations
Authors: Gunnar König
Hidde Fokkema
Timo Freiesleben
Celestine Mendler-Dünner
Ulrike von Luxburg
Venue: NeurIPS 2025 poster ()
Keywords: recourse
explainable machine learning
XAI
performativity
causality
distribution shift
robustness
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: Recourse algorithms can induce a distribution shift that makes their own recommendations invalid.

Abstract: When applicants get rejected by a high-stakes algorithmic decision system, recourse explanations provide actionable suggestions for applicants on how to change their input features to get a positive evaluation. A crucial yet overlooked phenomenon is that recourse explanations are *performative*: When many applicants act according to their recommendations, their collective behavior may shift the data distribution and, once the model is refitted, also the decision boundary. Consequently, the recourse algorithm may render its own recommendations *invalid*, such that applicants who make the effort of implementing their recommendations may be rejected again when they reapply.
In this work, we formally characterize the conditions under which recourse explanations remain valid under their own performative effects. In particular, we prove that recourse actions may become invalid if they are influenced by or if they intervene on non-causal variables. Based on this analysis, we caution against the use of standard counterfactual explanations and causal recourse methods, and instead advocate for recourse methods that recommend actions exclusively on causal variables.

Performative Validity of Recourse Explanations
Gunnar Königl. Hidde Fokkema', Timo Freiesleben
Celestine Mendler-Dünner'' Ulrike von Luxburg
Tübingen AI Center, Uniiersity of Tübingen Korteweg-de Vries Institute for Mathematics, University of Amsterdam
*LMU Munich. 'Munich Center for Machine Learning (MCML)
ELLII Institute Tübingen, Max Planck Institute for Intelligent Systems, Tübingen g koenig edu@ edu@pm.me
Abstract
When applicants get rejected by high-stakes algorithmic decision system, recourse explanations provide actionable suggestions for applicants on how to change their input features to get a positive evaluation. crucial yet overlooked phenomenon is that recourse explanations are performative: When many applicants act according to their recommendations, their collective behavior may shift the data distribution and, once the model is refitted, also the decision boundary. Consequently, the recourse algorithm may render its own recommendations invalid, such that applicants who make the effort of implementing their recommendations may be rejected again when they reapply. this work, we formally characterize the conditions under which recourse explanations remain valid under their own performative effects. In particular, we prove that recourse actions may become invalid they are influenced by if they intervene non-causal variables. Based this analysis, caution against the use of standard counterfactual explanation and causal recourse methods, and instead advocate for recourse methods that recommend actions exclusively on causal variables.
1 Introduction
Modern machine learning systems can significantly impact people's lives. Automated systems may determine whether someone receives a loan, is admitted to graduate program, gets invited to job interview. such high-stakes scenarios, applicants who receive an unfavorable decision - such as being rejected for loan or job interview often seek guidance how improve their chances the future. To address this need, recourse explanations are employed: they inform applicants of concrete changes they could implement to achieve the desired outcome from the system [Karimi cet al. 2022). For example, rejected loan applicants might be recommended to reduce their credit card utilization, and rejected job applicants might be advised to obtain a master's degree and reapply.
Ideally, recourse explanations are valid meaning that applicants follow the recommended actions (e.g., obtaining master's degree), they will indeed receive the desired outcome when reevaluated by the machine learning model (e.g.. be invited to the interview). In practice, however, deployed prediction models are rarely static. Model owners routinely monitor for distribution shifts and retrain their models to maintain performance. Recent work has shown that model updates can undermine the validity of recourse (Rawal et al. 2020 Upadhyay et al.) [021 Nguyen et al. 2023) These studies examine how different types of distribution shift such as data corrections, temporal drift, geospatial variability recommendations that remain robust under such shifts.
affect recourse validity, and propose methods for generating
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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What these works overlook, however, is that by recommending certain actions, recourse itself can cause a distribution shift. We refer to this phenomenon as the performativity of recourse explanations. As running example, consider a company using machine learning model to screen candidates for interviews for software engineering job (see Figure D. Based on historical data, the model learns that applicants invited to interviews often hold a master's 's degree in software engineering or, alternatively, show regular activity on GitHub. Two recourse actions might therefore be recommended to rejected applicants: 1. Earn a master's degree in software engineering. 2. Increase GitHub activity by making regular commits. Both actions have performative effects: when implemented. they induce a distributional shift. However, they differ critically in their causal impact qualifica- tion: While completing& a master's 's degree imparts substantial knowledge in software engineering, GitHub activity is merely correlate of coding expe- rience. Many tools exist that automatically generate daily commits, inflating an applicant's ' profile with- out improving their actual skills. If many applicants adopt such tools, the model will learn that GitHub ac- tivity no longer correlates with programming ability. As result, future versions of the model may ignore the feature that applicants were advised to pursue.
SE muster's 's degree
Y: qualified SF.
Li gets interviewed
GitHub activity
Figure 1: Simplified causal graph of features used to predict if applicants should be invited for a software engineering (SE) job interview.
the very recourse
In this work, we investigate whether recourse explanations remain valid under their own performative effects. Unrewarded recourse can impose significant burdens; thus, this question is particularly press- ing in light of the social impact of recourse practices. When recourse fails, applicants may waste time and resources pursuing actions that ultimately do not improve their outcomes /Nenaatasubaamanian and Alfano 2020).
Contributions. In Section 4 we introduce the two core concepts of our analysis. First, formalize the performative effect of recourse explanations as causal interventions in response to recourse recommendations on acted-upon features in the data- gett-generating process. Second, introduce the notion of performative validity: a recourse explanation is performatively valid if, after applicants act upon it, they receive the desired outcome under the optimal post-recourse prediction model. In Section 5) we prove that recourse becomes performatively invalid if the implemented action carries predictive information about the post-recourse outcome. Building this insight, we identify two mechanisms that can induce such invalidity: actions that are influenced by effect variables (Section [5.1). and actions that intervene on effect variables (Section 5.2)). Then, in Section 6 we empirically study the performative validity of three recourse methods: counterfactual explanations |Wachter et al., 2017], causal recourse [Karimi et al. 2021], and wwwovveeeent-oosuseeeeee causal recourse [König et al.) [003]]. Our findings show that counterfactual explanations and causal recourse, by targeting non-causal variables. often result in performatively invalid recourse! In contrast, improvement-focuseed causal recourse remains valid across a wide range of data-generating processes.
2 Related Work
We refer to Karimi al. 2022 for a comprehensive review on the literature of algorithmic recourse, and Mishra et al. [2021 for an overview on robust recourse. Most relevant for our work studies that consider the validity of recourse when the underlying predictive model changes over time. Rawal et al. 2220)) demonstrated empirically that recourse recommendations become invalid in the face of model shifts resulting from natural dataset shifts, including tempo- ral changes, or geospatial variation. This has motivated the design of recourse that is robust to
Robust recourse:
distributional shifts (Upadhyay al. 202T, Nguyen et al. 2023 Tiang al. 2024] and temporal
factors (De Toni et al.) 2024]. Similarly, Pawelezyk et al. [220 Black et al. [202T]. König et al.
(2023) find that natural variations in model retraining even when performed on the same data can render previously recommended recourse invalid. In response, approaches have been designed that optimize for robustness to model multiplicity, hyperparameter choices, and stochasticity, among others (Dutta et al.] 2022 Hamman et al. 2023 [024]. We also focus on the validation of recourse under distributional shift. However, our approach is novel in examining performative distribution thifts-that is, endogenous shifts that are induced by the very provision of recourse itself.

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Performativity: Predictive models may influence the very data they are later evaluated on phenomenon known as performativity [MacKenzie and Millo 2003). Implications for risk minimiza- tion have been studied under the umbrella of performative prediction [Perdomo et al. 2020a]. The framework formalizes two foundational concepts: performative optimality and performative stability. A model is performatively optimal if it is the best response to the data distribution induced by its predecessor. It is performatively stable if it remains optimal even after the distribution shift it itself induces. Building optimal or stable models; We refer to Hardt and Mendler-Dümme of related work on performative prediction. Our work extends this line of research by developing conceptual framework for performative recourse, analogous to performative prediction. Recourse that induces no change in P(Y X) corresponds to natural analogue of a performatively stable point. We generalize this notion to recourse explanations that are rewarded even if the optimal model changes in response to the recourse-induced distribution shift.
these ideas, subsequent work has proposed algorithms that yield performatively
[2025] for a comprehensive review
Strategic classification:
Strategic classification is a specific form of performativity in which agents deliberately manipulate their inputs to receive a favorable prediction from a model (Hardt
al. 2016]. A growing body of work investigates how build models to guard against such strategic
manipulations [Levanon and Rosenfeld [021 Chen et al.) 2020 Zmic et al. 2021 Chen et al. [023
Haghtalab et al. al.] 2021]. Thereby, a causal perspective has proven especially fruitful: In particular, the
literature distinguishes between two types of user actions: gaming and improvement [Kleinberg and
Raghavan 2020, Miller et al.) 2020, Ghalme et al., [021] Tsirtsis et al., 2024 Efthymiou et al. 2025].
Gaming refers to actions on non-causal variables that change the model prediction Y without altering the true target Y. while improvement refers to actions causal variables that affect Y itself. While improvement affects only the marginal distribution P(X), gaming may also alter the conditional distribution P(Y | X) [fisterr et al. 2021 König et al., 2021 Horowitz and Rosenfeld 2023). We find that standard recourse methods since they can inadvertently encourage users to game the system (König et al.) 2023 commonly induce shifts similar to those described in [Pfister et al., 2021 Horowitz and Rosenfeld [023]. By modelling the applicant's 's action as dependent on their own characteristics, we identify an additional source of performative invalidity: Recourse actions may leak otherwise unknown information about the applicant into the post-recourse distribution- recourse avoids gaming. As consequence, further assumptions about the data-generating process are required to guarantee performative validity.
3 Preliminaries
We consider supervised learning setting where the goal is to predict some target variable Y, such as an applicant's qualification
software engineer. The prediction is based on a vector of observed features X = .X., Xp) in some space fo- instance, university degrees or GitHuub activity. Companies may only want to invite applicants above a certain qualification threshold. To formalize this, we binarize the target variable Y using a threshold, formally L := 1[Y 2 0] 10. We denote the conditional probability of a positive label given features h(x)) P X z). and the corresponding binary classifier as Ī(x) [[(()) 2 tc], with decision threshold te E [0. 1].
Causal model. Algorithmic recourse concerns causal interventions that applicants perform in the world KKrimi et al. 2 König et al.) 2023). To estimate these effects, causal model is required. Throughout this work, we assume that the sittaggenerating process can be modeled using an acyclic Structural Causal Model (SCM) M = (x, U. J' [Pearl] 2009]: X includes all variables in the system, including the features X; and the target Y. U consists of independent noise variables U, capturing unobserved causal influences (for example, coding enthusiasm underlying GitHub activity), and ff is a set of structural equations that specify how each variable in X is generated from its causal parents and its corresponding noise. SCMs enable us to analyze the effects of interventions and counterfactual scenarios [Parr| 2009]. For example, completing a master's 's degree can be modeled as intervention. Formally, an intervention a set of features I, is represented using Pearl's s do-operator: do(a) = do((X; = 0 her.) = do(X, = ).
Each acyelic SCM induces Directed Acyelic Graph (DAG) G that visualizes the causal relationships between the features. An example of such a graph can be found in Figure For variable E X, we denote its direct causes as pa(Z) (parents), its direct effects as ch(Z) (children), its full set of
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direct and indirect causes as an Z) (ancestors), its full set of direct and indirect effects as de( 2) (descendants). For example, Figure master's degree is direct cause of qualification and thus parent in the graph, and GitHub activity is direct effect of qualification and hence a child in the graph. We abbreviate the direct effects of the target Y as F. == ch(Y), ) its causes as == an(Y), and the direct parents of the direct effects as S == pa( X E). also referred to as the spouses.
Recourse methods. Over the course of this paper, we will refer to three conceptually different recourse methods: Counterfactual Explanations (CE) [Wachter et al., 2017]. Causal Recourse (CR)
KKaimi et al. 2020b 202T), and cettovemmen--oousseee Causal Recourse (ICR) [König et al. [022]
Let x be an applicant's 's current feature vector and f, E [0. the targeted success probability for the recourse recommendation. Further, let be a cost function that captures the effort required to
implement the recommendation. The cost function takes two arguments: the applicant's 's observation and the suggested change, which, depending on the method, is either causal intervention or new datapoint x'. The methods are defined as follows:
(CE) A Counterfactual Explanations is defined as
(x) arg min c(x, s.l. Max)=11 =1.
CEs identify the closest input x' to the current instance x that flips the predicted outcome.
They are widely used and many extensions have been proposed (Ustun et 2019 et al. 2020a, Dandl et al., 2020].
Karimi
(CR) Causal Recourse is defined a
CR (x) = arg min (x, s.t. P([(XP)= = x, do(a)) 2 tr,
where X denotes the individual's observation after implementing the action do(a). In- tuitively, CR identifies the least costly action that leads positive prediction with high probabiliiyy. Unlike CEs, CR takes the causal relationships between features into account.
(CCR) improvement-focusee Causal Recourse is defined
a ICR (x) = arg min c(1, a) s.t. P(L-= = =1 = do(a)) 2 tr
where LP is the applicant's 's label after implementing the action. Intuitively, ICR recom- mends the least costly action that leads positive outcome with high probability. That is, while CE and CR directly focus on changing the prediction, ICR guides the applicant to become qualified. When the decision model is accurate, ICR reverts the decision with high probability, as shown in Proposition of König et al. 2023
The approaches differ fundamentally in the types of interventions they suggest: Where CE and CR may suggest interventions on effects that change the prediction without altering the applicant's qualification, ICR exclusively intervenes on causes and changes the qualification.
We note that for CR and ICR, versions exist: The individualized version (ind. CR or ind. ICR) which is based on individualized causal effect estimates but requires knowledge of the SCM. And the subpopulation-basse based version (sub. CR ind. ICR) that only requires the causal graph but resorts less accurate average causal effect estimates (details in Appendix B). All theoretical results hold for both versions.
4 Performativity of Recourse Explanations
When rejected individuals implement recourse, they modify their characteristics and reapply. As such, recourse itself can shift the applicant distribution and, once the model is updated, also the decision boundary. We refer to this phenomenon as the performativity of recourse.
Distribution Shift and Model Retraining. To formalize the shift caused by recourse, we distin- guish between the pre-recourse and post-recourse states of applicant, denoting the post-recourse

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state with the superseript p. Assuming that an i.i.d. sample of rejected individuals implements recourse and reapplies, the new applicant distribution can be written as the mixture
P(L" = =,, =)) == aP(L = L, = x) (1 a) = 1,X"= = -) (1)
where superscript m denotes mixture variables, P(L,X) is the pre-recourse applicant distribution, P(LP, == 0) the distribution of previously rejected applicants after implementing recourse, and E (0.1) the mixing weight that determines the proportion of pre- versus post-recourse applicants. The optimal predictor for this updated applicant distribution becomes:
(='(:)) [P("" 1|x = x) Le]-
4.1 Performative validity
For individuals implementing recourse, it is fundamental that the validity of recourse is not affected by its performative effects. We refer to this requirement as performative validity.
Definition 4.1 (Performative Validity). Given predictive model L, recourse method is called performatively valid with respect to a set of individuals X' X. for all x E X" and a E [o, 1] the optimal updated model _ satisfies [=(()= 2 L(x).
Intuitively, the definition requires that whenever recourse is effective for an individual E 2' with respect to the original model, the updated model must also accept the applicant. Thereby, it should not matter how much data is collected, and thus the guarantee must hold irrespective of the mixing weight a. Note that performative validity is related to the notion of performative stability [Perdomo et al., 2020b), which requires a model to remain optimal for the distribution it entails. As such, if the model is performatively stable w.r.t. the distribution map implied by recourse, it is also performatively valid. However, in general the converse does not hold.
4.2 A model of the post-recourse distribution
To obtain the post-recourse state of applicants, we assume that rejected individuals follow the actions recommended by the recourse algorithm while all other factors remain constant. To determine the causal effects of these actions on downstream variables, we assume access the underlying structural causal model (SCM). In the SCM, each variable is function of its observed and unobserved causes, formally 2; := (Fpa(j): My). Causal interventions do(a) = do(X,, oI,) are implemented by replacing this function with fixed assignment x == fj.
Since recourse actions differ across applicants, we introduce dedicated action variable A and obtain the post-recourse state by replacing each intervened-upon structural equation Jj with a function l that switches between the normal assignment and the intervention based on A, that is
a(x)) if [(x)= =0 and 1 = = (3 u5.0) do(2) otherwise
S -
e otherwise
A
(
The action A causally depends on the pre-recourse observation x: first, via the decision model f.(2)). which decides whether a recommendation is made, and second, via the recommendation a(x). In short, we say that the action is influenced by æ. To capture this influence, we include both the pre-recourse variables X and post-recourse variables X in the model. The causal relationships among pre-recourse variables are governed by the original structural equations f. while those among post-recourse variables follow the modified structural equations f8.
Our model induces the causal graph in Figure 2 a). Notably, the pre- and post-recourse states connected not only via A, but also via the unobserved causes U and UP. We typically assume that unobserved causes remain unchanged between the pre- and post-recourse states, though in some settings it is more realistic to allow them to change (we discuss this in Section 5.1
Recourse May Invalidate Itself if Relies on Effect Variables
After introducing the setup, we now formally characterize conditions under which performative validity holds. The following theorem shows that performative validity guaranteed if the proposed action A is conditionally independent of L given XP.
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X
D
R
7
x
M
X
Figure 2: Causal graphs. In each graph, color open paths between action A and post-recourse target YP given post-recourse observations XP via the incoming and outgoing edges, and highlight the decisive edges Xg + and + X using thicker arrows. (a): We illustrate the relationships between the target Y and its causes Xc, its direct effects Xg, and its spouses Xs. as well as their relationships with their post-recourse counterparts Yr, PP, In general, the pre- and post-recourse variables are connected both via the action A and the unobserved causal influences U, U (Section 4.2). and there are open paths via X - A and A + X (Theorem 5.2p. (b): The causal graph for Example 5.3 where Xc = D indicates having master's degree and X g G indicates the GitHub open. (c): The causal graph for Example 5.8 where Xc R is the general risk score and X M indicates the mood. Since the unobserved causal influences are resampled, only the path via + M is open.
activity. Since only interventions on causes are suggested, only the path via G
Proposition 5.1 (Uninformative actions imply performative validity). Consider any recourse method and all points that would have been accepted by the original pre-recourse model, that is, [(x)=1. Then the optimal original and post-recourse models h and he are the same on all such points x if and only if observing whether an intervention was performed 1[A A & do(Ø)) does not help predict post-recours:
[A + do(2)) IL LP XP
M(x) = 1P(xa) Vx [[xx==1)) =1
As result, we can guarantee performative validity if observing the intervention does not help to predict the post-recourse outcome, that is
A L XP
Yx 1 (x) L(x).
To evaluate whether A is conditionally independent of L given X", we consider the causal graph in aa). According to standard results causal inference (see Appendix A two variables are conditionally independent in the data if they are d-separated in the graph. From the graph, we can identify two potential reasons why A may not be d-separated from L given XP:
Theorem 5.2 (Two Sources of Invalidity). Consider the retrained post-recourss model L of Eq. ① and assume there performative invalidity:
unobserved confounders. Then there exist only two potential sources of
1. Xaety) + A. that is, recourse actions A are influenced by effect variables, meaning that the decision model L(X) or recourse method a(X) causally depends on effects Xae(Y)-
2. XXP that is, recourse actions A intervene on effect variables, meaning that the recourse method suggests interventions on direct effects of the prediction target.
Proof. (Sketch) A d-separated from Yp given XP in the graph, then IL YP|X" in the distribution induced by the corresponding SCM, and performative validity can be guaranteed (Propo- sition 5.Ip. We observe that the general graph A is d-connected with YP given XP, but becomes d-separated if we remove two edges: (1) the incoming edge X Xde(Y) A which represents that the recommended action may influenced by the pre-recourse effects, and (2) the outgoing edge
+ XP. which represents that the actions may intervene the post-recourse direct effects.
D

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Theorem 5.2
shows that performative validity can be guaranteed if we avoid using effect variables. Applied to our running example we can guarantee validity if we avoid using the effect variable GitHub activity, that is, (1) neither the decision model nor the recourse algorithm are influenced by information about GitHub activity to arrive their outputs, and (2), recourse does not suggest to change the GitHub activity. Conversely, if we rely on the effect variable GitHub activity, it is unclear whether recourse is performatively valid. To better understand whether and under which assumptions relying on effects leads to performative invalidity, we now study each of the two sources in detail.
5.1 Performative invalidity due to actions that are influenced by effect variables
In this section, we find that actions that are influenced by effect variables may indeed cause performa- tive invalidity. Intuitively, the reason is that when actions are influenced by effect variables, they may contain relevant information about the unobserved causal influences U that otherwise could not be obtained from the post-recourse observation. When the actions are implemented, this information leaks into the post-recourse variables and their relationship with the target may change.
Example 5.3 (Interventions that are influenced by effects may cause performative invalidity). Let Xe ~ (0.5) be whether someone has a degree, L the applicant's qualification, XE whether someone has significant GitHub activity, and let Le 0.5 be the decision threshold. Let furthermore U2 ~ Uni 1) capture the applicant's autonomy, an unobserved cause of qualification. Further- more, suppose that GitHub activity is only predictive of autonomy the subpopulation of applicants without a degree.
[1U > 0.55] L:=
Xc 0 Xc =
X = { L Xc = Xx=
[1U 0.45]
The causal graph is visualized in Figure (b). In this setting, the action A is influenced by the effect variable (GitHub activity Xxx) and captures information about the unobserved cause (autonomy Uy): ): Applicants rejected if and only if they have no GitHuub activity, which only happens if they have low autonomy "y E [0, 0.55]. When those low-autonomy applicants reapply after getting a degree, resulting X (1, 1). the relationship between having a degree and qualification changes: Pre-recourse, it is unclear whether individuals with a degree X = (1, 1) have high or low autonomy; Post-recourse it is more likely that applicants with degree have low autonomy and are thus less qualified. This can indeed lead to invalidity: If the model is updated on mixture with only 20 percent post-recourse applicants, the updated model would already reject everyone with observation (1, 1). All details are reported in Appendix D.I
The result is concerning, since it proves that there are settings where performative validity cannot be guaranteed, even if only interventions on causes are suggested. To avoid performative invalidity in such settings, the only remedy is to change the decision model such that does not rely on effect variables (this follows from Theorem 5.2). However, the model may not be in the explanation provider's 's control, and abstaining from using effect variables may result in a significant drop in predictive accuracy.
As follows, we present two assumptions that allow us to partially recover from this sobering result. We first observe that in Example 5.3 the action could only leak information because the unobserved causal influence, the applicant's 's autonomy, remains the same pre- and post-recourse. This may not always be the case: Think of scenario where patient in a closed psychiatric facility applies to spend a day outside, and the decision is made based on the person's 's risk to harm themselves Y. which is predicted based on their average risk score X and their score in a quick examination Xx that captures the daily mood. Here, the unobserved causes could be the weather Uy and the selection of questions for the examination Ug factors that are resampled every day. More formally, the pre- and post-recourse unobserved causal influences (Uy, Ug) and (U&, U%) are independent (Assumption 5.4). When this assumption is met, no relevant information about the post-recourse target Y' can leak via the suggested action, and thus recourse methods that only intervene on causes can be guaranteed to be performatively valid (Proposition 5.5).
Assumption 5,4 (Unobserved Noise Changes Post-Recourse). The unobserved causal influences of Y and Xg are resampled post-recourse, that is Uy and UY, as well as Ug and UE are i.i.d.
Proposition 5.5 (Assumption 5.4 restores performative validity). If IfAssumption 5.4 holds, recourse methods that avoid interventions on effects performatively valid.
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Second, we observe that the action in Example 5.3 can only change the post-recourse conditional distribution because it has access to information that otherwise cannot be obtained from the post- recourse observation. Specifically, we recall that in the example GitHub activity is only predictive of autonomy when the person has no degree, which is only the case pre-recourse. However, depending on the functional form of the causal relationships, the pre- and post-recourse observation may provide similar information about the unobserved causal influences. Specifically, we introduce Assumption and show that it implies performative validity when the recourse actions do not intervene on effects (Theorem 5.7). Intuitively, the assumption ensures that any two observations that were generated with the same structural equations and unobserved influences provide the same information about U. Furthermore, it covers many naturally occurring types of structural equations, including linear additive and multiplicative noise models.
Assumption 5.6 (Marginalization with Invertible Aggregated Noise). For an SCM with structural equations fE and fr, we assume that
there exists aggregation function Bagg and structural equation fagy such that we can rewrite the composition fÆ(fy (IC, uy), 2g up) as faeg Ts. ( (tyy, ug)), and (iI) fags is invertible, that is, there exists fagg such that fagg (DC, Fg, xg) = buge [ug. uy).
Theorem 5.7 (Assumption 6.6 performative validity.). Let Assumption 5.6 be satisfied. Further assume that the unobserved causal influences stay the same post-recourse, that is (Uy, Ug) (Ug, Ug. Then, recourse methods that abstain from interventions on effects are performatively valid.
5.2 Performative invalidity due to actions that intervene on effect variables
To show that interventions on effects may indeed cause performative invalidity, we show that invalidity may occur in a setting where we can rule out that actions that are influenced by effects are responsible. This is tricky, since we cannot enforce that recourse actions are not influenced by effects but intervene on effects at the same time: To ensure that interventions are not influenced by effects the decision model must be invariant to changes in the effect variables- and as result, no actions that intervene on effects would be suggested anyway. Instead, to show that interventions on effects are responsible focus setting where actions that influenced by effects not for performative invalidity, problematic. In Section we introduced two such settings, captured in Assumptions B..mptttonsssaaaaaaa 5nd [56] As follows, consider formal version of the psychiatric facility setting where the unobserved causal influences are resampled post-recourse and thus Assumption 5.4 is met.
Example 5.8 (Interventions effects cause performative invalidity). Let Y capture a patient's s daily risk of harming themselves, Xe the patient's general risk score, and X a quick examination that aims to capture the daily mood. More formally, let X ~ ~N 1), Uy ~ (0. 1), Ug ~ 10) and their causal relationships be governed by the following structural equations:
Y == Xc + Uy,
X = Y +UB
L= 1| 0j.
(2)
The causal graph is visualized in Figure (c). We suppose that recourse suggests intervening on Xg, that is, to give more favorable answers in the quick examination. Since giving more favorable answers per se does not improve the patient's 's daily mood, the action only makes the patient's mood appear favorable. When many applicants start to game the examination, the variable becomes less predictive of patient risk, and updated models will reduce their reliance on the feature. Formally, recourse changes zg to move rejected individuals onto the decision boundary .e = -rg. The updated decision boundary moves towards TC = 0, such that recourse- soourse-implmmmnting applicants with aC < 0 get rejected when they reapply (confer Appendix F. for details). Since Assumption 5.6 holds too, the phenomenon occurs irrespective of whether the noise is resampled (Assumption 5.4).
The example demonstrates that interventions on effects may indeed cause performative invalidity. More generally, we can prove that recourse via interventions on effects always leads to a shift in the conditional distribution (Appendix [E). Whether the shift is severe enough to lead to performative invalidity depends on the concrete setting. These results are concerning, since the two most widely used recourse methods, namely CE and CR, may suggest intervening on effects.
Corollary 5.9. In a setting where the first source of invalidity (influence on A) can be excluded, recourse algorithms that abstain from interventions on
that is if either Assumption direct effects are performatively valid but other recourse algorithmss may not. Without additional assumptions, performative validity can only be guaranteed for ICR, but not for CR and CE.
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1.0
ACC. (relit originall
0.5
Ind. ICR
0.5
sub, ICR
0.0
sub. CR
-1
-0.5
-0.
-1.0
1.0
LAdd
UMult
NLAdd
NLMutt
LCubic
LAdd LMult Ni Ada NiMult Gubic GPA
Credit
SCM
SCM
Figure 3: Experimental results. (QI, left): The pointwise differences between pre- and post- recourse conditional distribution aggregated using the mean, the lines indicate the range. All values averages over 10 runs. (Q2, right): The difference in acceptance rate (refit minus original), average (+) and standard deviation (lines) over 10 runs. While CE and CR lead to unfavorable shifts and performative invalidity, ICR is performatively valid in all settings.
6 Experiments Confirm That Interventions on Effects Must Be Avoided
Our theory suggests that the performative validity of recourse critically depends on the recourse method and the functional form of the underlying causal relationships. To investigate this empirically, we study the performative effects of CE, and the different versions of CR and ICR in synthetic and real-world settings. Specifically, study the following classes of structural equations:
(i) Additive noise (LAdd): [5(x, u) I(x) + u where I is linear
(ii) Multiplicative noise (LMulr)): [z, u) m(x)u where m is linear in each dimension
(iii) Nonlinear relations and additive noise (NLAdd): fa u) = g(x) + u where g nonlinear
(iv) Nonlinear relations and multiplicative noise (NLMul()) f2 u) g(x)u where g nonlinear
(v) Polynomial noise (LCubic); f,(x, = (I(x)) +))
We note that the first two settings (LAdd and LMult) satisfy Assumption 5.6 but the remaining settings do not. To enable pointwise comparisons of the conditional distributions, we rely on discrete noise distributions with finite support.
In addition, we include two real-world settings: College admission (GPA) and credit scoring (Credit).
For GPA we assume the causal graph in [Harris al.. 2022 and fit linear SCM with additive
Gaussian noise on the dataset [OpenIntro) 2020 For Credit we rely on the graph by Chen et al. (2033 and fit random-foree forest based, nonlinear SCM with additive Gaussian noise (Yeh] 2009]] The Cred setting has eleven features; all others include one cause and one effect variable. To highlight the differences between methods, we choose the costs such that interventions
effects are more
lucrative. Consequently, CE and CR intervene on the effect, while ICR only intervenes on the cause.
To allow both sources of invalidity come into effect, we always model the noise stay the same post-recourse. We provide a detailed description of setup and results in Appendix F
Conditional distributions (Q1): Does the conditional distribution that is modeled by the predictor change? More formally, are P(L 1|x = x) and P(L 1|X x²) the same?
For the synthetic settings with discrete support we compare the pre- and post-recourse conditional distributions computing the pointwise differences P(Y" X x²) P(Y 1|X x') and aggregating them using the minimum, the maximum, and the mean. When computing the mean, we weight the different points x' according to their post-recourse probability in the population of rejected applicants, formally P(XP x' L = 0). The results are reported in Figure (left). We find that methods that intervene on effects (CE and CR) significantly decrease the conditional probability of favorable outcome across all settings. For example, in the nonlinear additive setting, the conditional probability of a favorable outcome decreases by between 70% and 100%. In contrast, for ICR, the conditional probability remains the same nearly every setting, including but not limited to the two settings covered by Assumption 5.6] The exception is the setting Lin LunCi,,,, where for ind. ICR the conditional probability of favorable outcome increases by between 0% and 60%. This is consistent with our theory.

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Acceptance rates for the updated model (Q2): Does performative validity hold? Which percent- age of the excours--ipplememeee applicants gets accepted by the original vs the updated models?
To compare the pre- and post-refit acceptance rates, we compute the percentage of the resource- implementing applicants who get accepted by the original model an updated model that is trained on mixture with a = 1/3 post-recourse data. The results are reported in Figure (right). We observe that the interventions on effects recommended by CE and CR not only change the conditional distribution, but also lead to a dramatic drop in the post-recourse acceptance rates. For example, in the GPA setting, the acceptance rate drops by nearly 80% for CE and CR. In contrast, the acceptance rates for ICR remain unchanged across all settings, extending beyond our theoretical results.
In conclusion, the empirical results confirm our theory: Even recourse methods that only intervene on causes (ICR) may lead to a shift when they are influenced by effects but only methods that suggest to intervene effects (CE and CR) cause shift in settings covered by Assumption [.6 Extending beyond our theoretical analysis, we observe that ICR seldom leads to a shift, and when it does, the shift is not problematic. In contrast, CE and CR shift the distribution across all settings and cause severe performative invalidity. Our findings consistently confirm that interventions on effects must be avoided.
7 Discussion and Future Work
In this paper, we show that improvement-focusee causal recourse (ICR), unlike counterfactual explanations (CEs) and causal recourse (CR), can maintain performative validity across a broad range of data-generating processes, including those involving resampling additive noise. Empirically, ICR appears to remain valid even under more general conditions. key direction for future research is to formally characterize the full class of performative validity. Additionally, because full causal knowledge is rarely available in practice, would be interesting to extend our framework to settings with incomplete causal knowledge, including cases that violate causal sufficiency.
processes under which ICR guarantees
This paper has examined the performative effects of recourse explanations from the perspective of the applicant. A promising direction for future work is to complement this view with the perspective of the model authority (Fokkema et al. 2024]. Model authorities may use recourse explanations to strategically steer applicants toward regions of the input space where model performance improves or institutional goals are better met. In this setting, our notion of recourse validity could be extended to actions that are valid post-recourse but not pre-recourse, offering a conceptual analogue to performative optimality [Perdomo et al., 2020a]. However, such steering raises important ethical concerns- particularry when it overlooks or confficts with the applicant's own goals and values [Ki
and Perdomo 2022
Hardt al. 2022 Zezulka and Genin 2024].
We focused on single-step recourse where applicants act on a one-time recommendation and then face the updated model. However, many real-world scenarios involve repeated decision-making such as applicants reapplying for loans multiple times [Verma et al., 2022 Fonseca et al. 2003)]. ]. Building on insights from the performative prediction literature future research could investigate whether different recourse strategies converge to stable equilibria, and if so, analyze their properties and desirability.
Practical Insight. Counterfactual explanations are widely used in practice to guide applicants toward their desired outcomes. However, our findings show that, due to the performative effects of their actions, applicants may still fail achieve these outcomes even even after following the recommended recourse. To prevent applicants from taking costly yet ineffective actions, model authorities should only give recourse recommendations that improve the qualification of the applicant. In particular, we advise against relying on standard counterfactual or causal recourse explanations. Thereby, we strengthen the position taken in previous work (König et al. 2023), which demonstrates the benefits of focuued recourse from the perspective of the model authority.
recommendation
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Acknowledgments and Disclosure of Funding
The authors are grateful to Anna Vollweiter for running preliminary, yet different, experiments the early stages of the project. This work has been supported by the German Research Foundation through the Cluster Excellence "Machine Learning New Perspectives for Science" (EXC 2064/1 number 390727645) and the Carl Zeiss Foundation through the CZS Center for Al and Law. Freiesleben was additionally supported by the Carl Zeiss Foundation through the project "Certification and Foundations of Safe Machine Learning Systems in Healthcare".
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A
Causality Preliminaries
Causal graphs cannot only be used visualize causal relationships but, under certain assumptions, also to reason about conditional independencies in the data. To do so, we first introduce the so-called d-separation criterion.
Definition A.1 (d-separation). Two variable sets X, Y are d-separated given a variable set Z, denoted Z, if and only if, for every path p from X Y one of the following holds
as X L Y
i p contains a chain + m + j or fork i <- m + where m E Z. 4 such that and all of its descendants are not in Z.
p contains collider
Under certain assumptions, d-separation in the graph can be linked to conditional (in)dependencies in the data: First, if the so-called Markov property holds, d-separation in the graph to implies independence in the data. Second, if faithfulness holds, independence in the data implies d-separation in the graph.
Definition A.2 (Markov Property). Given a DAG G and a joint distribution P over the nodes, this distribution is said to satisfy the Markov property with respect to the DAG g if
X G Y Z = X Y Z.
for all disjoint vertex sets X, Z Definition A.3 (Faithfulness), Given a DAG & and joint distribution P over the nodes, this distribution is said to be faitful to the DAG g if
X Z - x X
Throughout the paper, we only use the graph to read off conditional independence relationships, For instance, in Theorem [52] we only say that there may be a dependence via the open paths but make no claim that it must be present in the data. As such, we only need the Markov property for our results. The Markov property is met if the data is induced by SCMs with independent noise terms, more generally, if there are no unobserved confounders.
Proposition A.4. Assume that P is induced by an SCM with graph G. Then, P is Markovian with respect to G.
Proof. See for example Proposition 6.31 in Peters et al. 2017)
B Individualized and Subpopulation-based Causal Recourse Methods
Karimi et al. 2020b propose two versions of Causal Recourse (CR): an individualized and subpopulation-busee version. The individualized version leverages counterfactuals to make causal effect estimates, which take the information about the individual obtained from the observed features X into account. Therefore, so-called rung 3 causal knowledge is required, which is, for example, given by an SCM [Pearl 2009].
However, the SCM is rarely readily available in practice, and thus the subpopulation- based version was introduced as a fallback that only requires access to the causal graph. The causal graph is easier to obtain, since it only captures whether nodes are causally related, but does not detail the functional form of their relationship. However, causal graphs do not enable the estimation of individualized treatment effects (rung 3 on Pearl's ladder of causation [Paar 2009). Instead, causal graphs can be used to compute conditional average treatment effects, that is, interventional distributions for a subpopulation or similar individuals (rung 2). Specifically, causal graph-based treatment effect estimates can account for characteristics that are assumed be unaffected by the intervention. In the graph, these are the non-descendants of the intervened-upon variables Xad(I.).
In the background sections, we introduced the individualized versions of the causal recourse methods. The definitions of the subpopulation based versions of CR and ICR only differ in the conditioning set (highlighted red):
CR (x:) arg min c(z, a) s.t. P(L(X*)= mna(T,) do(a)) tr.
21

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and
ICR (z) arg min s.t. P(I = Tdd( do(a)) 2 4.
C Proofs of Section
Let us make two general remarks, before we start with the proofs. First, when the context allows, shorten the notation of conditional distributions. Specifically, sometimes drop the random variable and only write the corresponding observation. For example,
P(L = X x², Uc uc) P(L = x², uc).
Second, of our results prove that the conditional distribution of the underlying potentially continuous target Y remains the same. These results imply performative validity.
Lemma C.I. If P(Y* < y X² x) = P(Y S y X 2) for all (2, E x y, then P(TP = 1 X = x) P(T. =1 X = for all x E X.
Proof. The following string equalities proves the result,
P(L" = XP = P(Y s X x) = P(Y 50 PLL= 1 x).
As follows, we repeat and prove the results from Section
Proposition 5.1 (Uninformative actions imply performative validity). Consider any recourse method and all points that would have been accepted by the original pre-recourse model, that is, L(x)=1. Then the optimal original and post-recourse models h and he are the same all such points x if and only if observing whether an intervention was performed 1[A + do(E)) does not help predict post-recourse:
[[A + do(2)) I1 L' |
M(x)) = hP(x)) Vr [(x)= =1
As result, guarantee performative validity if observving the intervention does not help to predict the post-recourse outcome, that is
A IL IP X
Vx L. (x) L(x).
Proof. In the first two steps, we prove the equivalence between the conditional independence and the equivalence to show that conditional
the stability of the predictor. In the last step, independence of the action implies performative validity.
Step (LP IL 1[A = do(@)] X + P(L" X²) = P(L" X", = do(2))). The first step holds by the definition conditional independence.
P(LP, A = do ()) XP)
P(P X", (P,A=do(2))
(Bayes' rule)
P(A do( XP)
P(L XP)P do(Ø) XP)
(L' 1l 1[A do(2)) XP)
P(A = do XP)
= P(TP XP).
Step 2 (P(IP) X2 x, A -d(()) X x)).
In the subpopulation of individuals where no intervention is performed the structural equations are the same pre- and post-recourse. Since we assume that all other factors remain the same well, the pre- and post-recourse distribution are the same. More formally,
P(L X A do(2)) P(L X, = do(
Furthermore, the action A is function of X and result
P(L X, A = do(e)) P(L X).
22

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Note: Only accepted individuals [(x) = 1 are recommended to change nothing A = do(Ø). ) and thus P(Y X do( only defined for accepted z. As result, step and 2 only apply where [(x) = 1, and our result is restricted to x where L(x) = 1.
We recall that h(z) == P(L 1|X =x) and h+(x)= := P(L 1|X x). So taken together, the conditional independence is equalivalent to equal pre- and post-recourse model in the acceptance region.
Step 3 (4 11 1' X"
T. (z) 2 ((x)).
To obtain the final implication, we observe that when h(x))
h*(xx) for all where T((r)) = 1, it also holds that [*)( 2 [xx): Any a that the original classifier L is also accepted by the updated classifier T. and for any x that rejected by T. the classification can only improve. Overall we get
X
1[A = do(2)] IL L' |
h(x) = (x) for all x E where [(x) =
h(x) h" (x) for all x E X where L(x) 1
[x)x)) 2 [(x).
Theorem 5.2 (Two Sources of Invalidity). Consider the retrained post- post-recourse model L of Eq. and assume there are no unobserved confounders. Then there exist only two potential sources of performative invalidity:
X + A, that is, recourse actions A are influenced by effect variables, meaning that the decision model ((X) or recourse method aXX) causally depends effects Xaety)-
2. A + X that is, recourse actions A intervene effect variables, meaning that the recourse method suggests interventions on direct effects of the prediction target.
Proof. If there are unobserved confounders, which is for example the case if the data generating mechanism can be written as a SCM with independent noise terms, then the Markov property holds, and d- d-separation in the causal graph implies independence in the data. If A were be d-separated from Y X", it would thus hold that A IL Y" XP, and as a consequence of Proposition 5.1 recourse would be performatively valid. However, if there open paths between A and Y given XP, then A may be dependent on YP given X and recourse may be performatively invalid.
Given X there are only few possible open paths between A and XP. To follow the proof, we recommend inspecting Figure first.
An arrow with asterisks, * indicates that the arrow could go both ways. We observe that the following paths are guaranteed to be closed:
All paths via causes of X are blocked: All causes X are observed, and as a result any is closed for any
path of the form y + X E C.
A or YP t XP
All paths via effects effects are blocked: All effects X are observed, and as a result any path of the form YP X ++ A is closed for any i E E.
All paths via spouses blocked: Spouses observed and thus any path YP - X. + xy + ++ or YP + X <- Xp * A is closed for any E and j E S.
All remaining paths either include the segment A ..... Up
Yp or A
yp
with E. Whenever the action causally influences an effect variable the action d-connected with YP, since the edge A + X² + YP is open whenever X"" is observed. Paths including
A ++ Up Y are only open if the effect variables X causally influence A:
23

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Any path that involves a collider structure A # + Y t Uy or A+ -, +
Y f- Uy closed if X A A.
All other paths include an edge A t X; where i E de(Y).
To summarize: Unless there is edge A - X Xee(r) A, A is d-separated from y given
X?, and thus A 1L Y X", and as consequence recourse performatively valid.
D Proofs of Section 5.1
D.1 Details of Example 5.3
Interventions that are influenced by effects may cause performative invalidity We start by deriving the conditional probabilities P(L X). The variable X = (Xc, can take three different values: (0, (), (0, 1) (1, 1). In the first two cases, X perfectly predicts the label L, that is
P(L = 1 X (0,0)) =0 P(L 1 X (0) 1.
In the third case, Xc 1. When Xc 1, the value of Xg is determined by Xc (by definition of the SCM). Thus, the conditional probability reduces to
P Xc 1, XE 1) P(L Xc 1)
= 0.55.
(by definition)
Assuming that the predictor is Bayes optimal, that is T.(x) | x) 2 0.5), only applicants with observation X = (0, 0) are rejected. To move these rejected individuals across the decision boundary, the only possible action is to intervene on X by setting do(Xc = 1), yielding a post-recourse observation x" = 1).
To derive the updated model, we need to determine the post-recourse probability of a favorable label for people with x² 11, Therefore, we first derive the unobserved noise distribution for the individuals who implement recourse. By definition of the SCM, the unobserved noise for the rejected
individuals is distributed according to
P(UL 0) = P(UL X (0, Unif 0.55).
Thus we get that P(L = X" x P(L" = - 0) P(U, > 0.45) = 0.1/0.55.
To obtain the prediction by the updated model i. we derive the mixture pm with = 0.2 and observe that the observation X 11 now gets conditional probability of
0.1
P(L." x (1, 1)) = 0.2 * 0.55
+ 0.8 0.55 0.4844 < le-
As such, the recourse outcome (1, 1) is not valid for (rejected by) the updated model.
D.2 Independent noise assumption and proofs
Proposition 5.5 (Assumption 5.4 restores performative validity). If Assumption 5.4 holds, recourse methods that avoid interventions on effects are performatively valid.
Proof. Note that there are no connections between U and UP in the graph depicted in Figure 2 a), whenever UP is independently resampled. As the probability of intervening on effects is 0, the arrow from A to X is not present and we observe the d-seperation of A 1g LP XP and thus also the statistical conditional independence A Ill L2 Xp. This also gives us that [[4 Ø] Il L2 XP, as the indicator is function of A. Proposition 5. then tells us immediately that the conditional
distribution stays the same for [(xx) = 1, i.e.
h²(x) = P(L = X" = x) P(L =
h(x).
24

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D.3 Invertible aggregated noise assumption and proofs
We restate the assumption and stability results again for completeness sake. Assumption 5.6 (Marginalization with Invertible Aggregated Noise). For an SCM with structural equations fE and fy, we assume that
(i) there exists an aggregation function bugg and a structural equation fagy such that we can rewrite the composition fa(fr (xc, uy s ug) as ( FS, Dugy (yy, ug)), and iii fags is invertible, that is, there exists fig such that fagg ( C.C. 2E, bage (ug, uy
Linear Additive and Multiplicative SCMs satisfy Assumption 5.6 In the Section D we mention that Linear Additive models and Multiplicative SCMs satisfy Assumption [36] Here we will quickly show that this is the case. general, the structural equations for the effect variables Xg and Y given by the following structural equations:
Y (X +y
Xg = LE(Y, Xc, + Ug..
where ly and Eg are linear functions (for example, (Y,Y, Xc, Xs) By substituting the expression Y into the equation for Xg we get
X = (Iy (Xc) + Nx Xs) + UE =1 Xc, Xs) BUy y UE-
We can rewrite the structural equations
Uage (x) BUy + Ue
f Jugg (rc, FS, Magy) Iæ(ly (#c), (C,, Uags
agg (C Ts TE) JE LE(ly (
and thus Assumption 5.6 is satisfied.
In multiplicative settings with multilinear aggregation functions m (linear in each argument, for me (X Y, Xs) = X-XXXXXXxx Xs33,, we get
Y (Xc)Ux
X mx(Xo, X,
Substituting the expression for Y we get
X = m=(Xc, Y, XU)URR
mE((c, (Y
m (Xc), Uy
We can rewrite the structural equations as
Uags buge(u) (u) == (UyUg)
Sagg (IC, Ug Uagg) mg(rc: my (Ic)),, 2 gg)
gg
f (EC, US, mg(xc, my (Ic), xs))
and thus Assumption 5.6 satisfied.
Theorem 5.7 (Assumption 5.6 restores performative validity.). Let Assumption 5.6 be satisfied.
Further assume that the unobserved causal influences stay the same post-recoure that (U (Uy, Ug) (UR, U.). Then, recourse methods that abstain from interventions effects are performatively valid.
Proof. We will again use the characterization given in Proposition 5.I show that the required conditional independence is present. Let TA be the index set which intervention A is performed. We write,
Using Assumption 5.6 we
P(I = Ø X = v²) = P(L IA = 2, P(I = Ø X = x²)x")
25

---

We can rewrite the first probability as
P(L LA = 0 =
uy, IA = 2)p(ux L0,1A = Ø) duy P(LP =1 11/2p.uy
ur
P(L= =1 uy xx, Ø) duy.
Lyy
Where the second equality follows from the fact that L is completely determined by X and Uy. For conditional probability on the noise we have the following sequence of equalities, because we can condition on no intervention being performed
p(uy X x", =====0 Ø) p(uy X x., I 2) [ p(uy
The first equality here follows from the fact that X and X have the same functional equations, when no intervention is performed. The last equality follows as TA is deterministic function of X.
What rests to show is that p(uy X2 xd) p(uy | X 2²). The basis for this proof that Xc, Xg and X constrain = (Uy, Ug) in the same way, whenever no interven- tions the effects are performed. That is, for p(x, x²) > 0 it holds that either the conditionals p(rg IC, Ig, uy, ug) and p(=² | xez zgg. Uy Up) are exactly zero or exactly one. The reason is Assumption on the structural equations. We know that
p(ag TS, uy, up) 1 | [s(F xs ((u,,ur))
p(a T r'g uy, ug) 1 1 fa( r''. (uy, ug))].
We reformulate the constraints as
(uy, ((uy,ug) f (xc, FE 2s)) b, (uy, uF) = f (x x'y x's).
We recall that
p(uy, ug X = x)) pluy, UE x)) p(x)
and that
pluy, ug ug. xx))
p(wy, Wg XP= =P=)p)
p(xP)
The joint distribution p(uy, ug, x), can be written out p(uy, ug, p(uy hww.euueencccccccsshhne IC, uy, ug)
giving a conditional distribution of
p(uy buwy)pluu)pp(cc)))ssss) re. æs; uy ug)
pluy, ur = x)
Ju pluly pluy (ow())((((p))) #c, uy, ug)
IC, xs, uy: u'g) du
(3)
Ju plu,y ((()))(( TC, dly, w'y du
Now, for the second conditional can write as p(uy, ugg x²)
/ pluy,ug,re N2, x2,)) dac
Xe
/ pluy, ug, Fc)p(c 1.c. uy, uE)p(r x uy, ug) dxe
= =((x)) = d' I uy, ug)
p(oc)p(10 xc. uy, ug) drc
8o = =(lur)p(ur)pppfp.e dg sy uy, ur) /
p(xc)plo% xo, us = (ur, dre
Xe
26

---

Because of the assumption, we know that p(a r's =c. uy, ug) is zero unless buy = f (2 r'g. Thus we can replace p(zc | Ic, Us = 6,(u,, uE)) with placo 2c, =
E g)) and get
ug, plwy))((u))pleeeeeeee d's oe uy ug) / p(ac)p(ab Ic,u, us = fo (x²)) drc-
(0,
Remark that the final integral only depends on x?, we will shorten this integral by setting
Z(x") = / p(zc)p(20 0c,4 = fo (x")) dxc.
Ne
Plugging everything together we get
pluy (luu)p((' d' ze. uy ugy 2(r")
p(uy, ug X x²)
Ju p(l,
d's
Lo uy (xx)) du
pluy plu ((1 plle 2 uy ug ((x))
Su pluly P(u pl( p Jg. uyy E Z xv) du
pluy pluu)p(('' Le r'z uY ug) pluyy (p(p))(('
(4)
Te J u u du
If we now substitute x" into Equation are equal.
3). we observe that the expressions in Equations
3 and
Theorem D.I. Consider the causal graph in Figure Aa). Assume that U = U and that Assumption i. satisfied. the recourse method suggests interventions on direct effects with non-zero probability, then there exist x with [(x) = for which (2(x) + h(x) and the post-recourse conditional probability is described by
1²(x) a<=)h()) + (())(((()))((()))))))))))))))))) +
where g(z) S te. r(x) E [0, and o., B, are all functions bounded in [0, 1].
Compared to Lemma E.T one of the components will always be smaller than the decision threshold, but there will also be component, for which it is impossible to tell if it is higher or lower than the decision threshold in general.
Proof. We will prove this result by direct calculation. First, we split the post-recourse condi- tional probability by conditioning on where the intervention are performed. Let be all possible combinations of the sets C, E, S and the empty We can rewrite the probability as,
P(L = =1-1111 x²) Σ
P(LA- = X" x') X IA i).
SET
We immediately have that P(L = XP = x,
-
= h(x)) and for any h(x) by the proof of
i without E we have P(LP- = XP x, 2,LA = P( Theorem 5.7 This gives us that a = Σiet, Egi =
P(A==i| x
As the next step, we will focus on the terms with E i, but C This will be the g function and we set
g(x)= = Σ
P(YP =1 XP = IA =)
iEI: BEL,C
B(x) = Σ
P(IA =i X" = =x).
VEI: BELCE
We show by direct calculation that each term can be bounded by the pre-recourse probability. We make use of several conditional independencies in the graph depicted in Figure bb) and the fact that
27

---

the structural equation of Y is the same as the structural equation of Y".
P(LP = 1 2², = /
P(L = 1 x., x, i)p(æ x², () dr
x
/
P(L = -1 ,,,,,, uy, i)p(uy x², 2, t)p(a x", i) duy dæ
Ut
-11
|PLL P(L 1 = = uy uplrr xx, ()(( I xf duy da
x Uty
-f
p(uy ,,, 1)p(z) x², duy dæ.
{uy: fr t 1200
Now, we use that Uy Lg X2, A X Uy 11 XP, X and that 2ec Ic, as only the effects intervened upon, to write
/ p(x x²,i)
= -W'=11X' x==
p(uy I duy dæ
x
Huy: (x(e,,u))20)
f
p(ar x. (x, ()) x .)) dr
x
Ve / p(( |x?;
x
le-
Where the last inequality is consequence of only suggesting intervention whenever h(x) < to and thus p(z x", i) will only be non-zero for points x with h(x) < te- The inequality will be strict if P(LA ni Ø X2 2²) > 0.
The final term will be the r-function, which defined as
r(x)
I 1). Σ P
iEI: BEL
We can repeat the previous derivation, but we do not use the step x IC. This gives
r(xp)= = [p(x p(x)
x², (C,
p(uy | x)) duy dæ.
X'
[uy: (x )20)
Now that we cannot change x'C TC, this expression cannot be simplified further. The relation between uy and could have positive influence negative influence on the overall probability. Putting everything together gives the required expression.
E
Proofs of Section 5.2
Now, we will perform the calculation to get the excact form of the conditional distribution when the noise is resampled.
Proposition F.1. Consider the causal graph in Figure aa) Let Assumption 5.4 satisfied. Further assume that the probability of interventions on direct effects is non-zero. Then, there exist x with L(x) = for which hP hP(x)) # h(x)) and the post-recourse distribution takes the following form,
1P(x) (1 ((z))h(z) ((z) Xc xC) 3(x) E (0,
28

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Proof. To calculate the exact form of the conditional, we write LA for the index set of the recourse recommendation and we condition on which interventions are performed and splitting up the proba- bilities. As before, I will denote the set of all combinations of C, E, S and the empty set. We write the conditional probability as
= P(L = X = =) Σ P(IA- = x = = =
(5)
LEI
Whenever there was no intervention was performed, the conditional distribution does not change as all the noise distributions are the same and the structural equations remain unchanged. This means that
= = --
=
Whenever an intervention on the cause or one of the spouses is performed is performed. no change will be observed. So without loss generality we can focus only on the case where LA E.
Now that we assume that an intervention is performed on the effects of Y only, we see that the arrow from Y X and the arrow from X3 to X area broken. Coincidentally, the variables y and L only depend on X2.
P(I==1 = E, X" x) P(L" = 1| LA E, X =cc
P = X xc)
= P X = xc).
Combining everything together find that Equation 5
reduces to
P(L== = X =))
Σ
P(I - x²=
X=x)
iEI: Egi
E = x x²== 1 = =xc)
IE:: EEi
(1 B(x)) || x x) P(=) =1 x Xc = xc).
Where we set
B(x) = Σ
İEI: BE
and hence B e (0, 1].
All the results in this section and the previous section now give Corollary 5.9
Corollary 5.9. In a setting where the first source of invalidity (influence on A) can be excluded, that is if either Assumption 5.4 3.4 holds, recourse algorithms that abstain from interventions on direct effects are performatively valid but other recourse algorithms may not. Without additional assumptions, performative validity can only be guaranteed for ICR, but not for CR and CE.
Proof. Theorem 5.7 gives us that ICR performatively stable when the noise stays the same pre- and post-recourse and Proposition tells tells us the same when the noise is independently resampled.
Propositions D.] and E. tell us that CR and CE may not be performatively valid.
E.1 Details of Example 5.8
Interventions on effects cause performative invalidity First, we will calculate the explicit form of the conditional distribution. That is, h(z) = 1| = x) & ((Ic + xze/v// where the cumulative distribution function of the standard normal distribution. We start by rewriting the conditional distribution as
P(L= 1 x) P(Y 2 0 X x)
plu x) dy.
29

---

An explicit expression for the conditional density p y x)) can be obtained through a Markov decomposition using the structure of the graph and an application of the Bayes' rule
ply x) p(Zc,,E,
2
(Markov decomposition) p(zc)p(y #c)p(TE y)
p(zc)p(ZE xc)
ply IC p v)
(Bayes' rule)
p( xC))
Because we know the exact relations between the variables through the structural equations, we can infer the conditional densities needed in Equation (6). Remark that the noise variables Uy and Uy are distributed according to independent standard normals. The structural equation for Y is defined Jy(Xc: ) Xc + Uy, und since Uy and Xc are independent when we condition on Xc Te, meaning that P()] ~ (z 1) which is the distribution of Uy shifted by rc. We can repeat this derivation for the other variables conclude that
get that Y xc. + Uy
I Xc =g ((cc,)),
X Y y (w,,
X X ~ (xc.2).
Substituting the densities of these random variables into Equation gives e (wv--))
I
ply x) V
25
(w0-
4n
24 +x.N 2ot t2g
1 == - 102
π
y² (1e (10 tu? - Jrezo
e
π
(y =atte)
The last expression we recognize the density of N (zoteR 1) distribution. Finally, we use that we can translate and rescale the edf of a normal distribution to the edf of the standard normal. Let T o, be the edf of normal distribution, then holds that
11
(x) = ①b
Using μ ECtPE and 0² 2, we get
P(L - 1 x P(Y 2 0 X xx)
P(V 5 X x
2ct-g
11/2
+ (re + zE)/V²)
The final equality follows from the symmetry of the standard normal distribution around 0. The Bayes optimal classifier is given by i(z) [P(L X = 2) 2 2]- In this case, have the threshold t. and the decision boundary is those points where
+ (Do + z
=
1t)==e)/V22 + FC -2g-
So, in the pre-recourse setting, the decision boundary is at 2C -I
30

---

Now we apply Proposition E.1 to get the expression for the post post-recourse conditional distribution, which is
- +
Xc =rc)
Similarly as before, we can calculate that P = X =c)=+(aa (. The property to note
that (xx) < whenever Fc O. Now, for any point with sC <0, but #c = E we observe
a 3 x.)
[²(x)= = ((o
2
Here, the 3(x) indicates the proportion of the post-recourse individuals that intervened on the effect variables, which will be non-zero amount in this example. The fact that h²(x) 2. but h(x) shows that all points on the decision boundary with IC < O, are invalidated post-recourse. Indeed, the final mixed distribution will be given by
1 1
1
hm (=) ah(x) (1 a)hp(z)
2
2
where a indicates the mixing parameter.
Finally, the probability mass of the people that get rejected that were originally accepted is non-zero, because the recourse recommendation moves non-zero mass towards the decision boundary.
F
Experiments
All code is publicly available via GitHut]
F1
Settings
In our experiments, report the results for five synthetic and one real-world dataset. In the synthetic settings we focus on varying the type of functional relationship while ensuring that the data generating process has finite support. Furthermore, corresponding model can always be changed by modifying only one of the features.
chose the parameters such that the prediction of the
To obtain data with finite support, we rely on the discrete binomial distribution. The binomial distribution normally takes two parameters, n and p, where is the number of trials and p the probability of positive outcome in each trial. The support of the binomial are integers in [0. and the mean np.
For our settings, we sometimes shift the distribution by an offset, e.g., to ensure positive values. We refer this shifted binomial as Sh p, μ4) where μ is the new mean. To sample from ShBin(n, P, n), we sample values from Bin(n, p) and shift them by the offset mp.
Now are ready to introduce all synthetic DGPs in detail. Setting F.1 (Additive Noise, LinAdd).
Xc Uc
Uc ShBin 0.5,
Y = Xc + Uy
Uy ~ S2Bim(2,0.5,00
Xg Y + Xc Up
Ug 0.5
Setting F22 (Multiplicativve Noise, LinMult).
Xc Uc
Uc ShBin 0,5,5// + 1)
Y = Xelly
Uy ~ ShBim(1,0,,,0,.
X == YXCUR
UE S.B
31

---

Setting F.3 (Nonlinear Additive, NlinAdd).
Xc = =-Cc Uc
Uc ShBin (8, 0.5,0) 0)
Y = (Xc)² +Uy
Uy S2 0.5,
X = (Y + xx)) + UE
Ug S2 0.5, 0)
Setting F.4 (Nonlinear Multiplicative, NlinMult).
Xc Uc
Uc 0.5ShBim(2, 0.5, 0.3ShBin(4, 0.5,, Uy Sh 0.5,
Y XaUr
Xg (Y + UE
Ug S2 0.5,,
Setting F.5 (Polynomial Noise, LinCubic).
Xc == Uc
Uc ~ ShBin (4 0.5, -1) Uy ~ S2 0.5, 1)
== (Xc + Uy)
Xg == (Xo+ UU+ YB-
Ug ~ ShBin 0.5, 0.5)
For the GPA example we obtain data from OpenIntro 2020] and assume the causal graph suggested in [Harris al., 2022). Then, we fit linear Gaussian SCM, that is, for each node we fit linear model to predict the node from its parents, and fit a normal distribution to the residual to obtain the corresponding noise distribution. For the Credit example we obtain data from Chen et al. (2023). and assume the causal graph suggested by (Ych) 2009]. Then, we fit random forest based additive Gaussian SCM, meaning that we random forest to predict each node from parents, and fit normal distribution the residuals to obtain the noise distributions.
Setting F.6 (College Admission, SAT).
Xc := Uc Y ay Xc + Uy
Uc N(O,1) 1) Uy oY) UF ~ NN
X == YFY + UF
In settings, the target variable is binarized using the median as treshold. That is,
:: 1 2 median
Costs In our experiments we want to compare the different recourse methods, and to reveal their differences chose the cost such that interventions effects intervene on the effects, and ICR intervenes on the causes. To do so, we define the cost functions for CR and ICR
lucrative. Then CE and CR
do(a)) = -
(x, - -3)
jeL.
and for CE
ca,t(x, x²) = > - - 13.3
The weight Yi ensures that interventions effects more lucrative. It is defined as
where ny is the index of the node when they are (partially) ordered according to the causal graph, that is, causes recieve lower indices than their descendants and therfore more weight in the cost. Further, normalize the cost using each feature's 's variance oz.
32

---

F.2
Experiment setup
Random seeds We conducted each experiment 10 times with seeds 40,..., , 49. In Figure B (left),
we report averages over all runs, that is, the average minimum difference in conditional distribution. the average maximum difference in conditional distribution, and the average expected difference in conditional distribution. In Figure the standard deviation a (the expected value to). In Table [ we report the mean and standard deviation for each metric.
the dots represent averages over 10 runs and the errorbars
We use decision tree with default hyperparamters for the discrete settings and logistic regression with default hyperparamters the real-world setting. We use sk Kllernn to the model. To ensure that the original model is as accurate as possible we sample 10 fresh datapoints as an independent training set.
Decision model
Sampling from the post-recourse distribution Given the SCM and the decision model, ready to compute recourse recommendations. Therefore we sample fresh data and randomly pick 5000 rejected observations in the simulation settings, and 1000 rejected observations for the real- world setting, for which we then generate recourse recommendations. The detailed procedures for generating recourse explained below. Given the recourse recommendation, we compute the true post-recourse outcomes for the individuals. Therefore, we exploit that we sampled the data ourselves, meaning that we have access the unobserved causal influences that determine the observations. In other words, we are able to compute the ground-truth outcomes for the respective recommendation. Specifically, for each individual, we fix the unobserved causal influences the respective values, and then replace the structural equations according to the interventions. Then, we determine the new characteristics of the applicant based on the unobserved causal influences and the structural equations. Based on these ground-truth samples from the post-recourse distribution of rejected applicants we assess Q1 and Q2. We note that sampling from the post- post-recourse distribution expensive, since each post-recourse sample requires solving the recourse optimization problem.
Q1: How did the conditional distribution change? To quantify the change in distribution, we compute the conditional probability of favorable label L both in the original distribution and the part of the distribution that may have changed as a result of recourse. More formally, we compute P(L = 1 X² = x. L = 1) and P(L = 1 X = x). Notably, we do not take X = 7 = 0) into account, since for the subpopulation the conditional distribution must be the same pre and post-recourse, that is P(L
X x. = P(L 1 X xz). To make the estimates as accurate as possible, we sample many observations (5000) and designed the DGPs such that the number of possible values small, and in each bucket enough samples can be found. We aggregate the wisnt-wise differences using the min, max, and expected value. To ensure that buckets with larger sample size get more weight in the expected value, and reflect the perspective of recourse implementing individuals, we weigh the points according to P(XP = x L = 0).
Q2: Does the shift impact acceptance rates? To obtain the updated model and evaluate the impact of the update on acceptance rates, we split the sample of recourse implementing individuals in half. The first half is used to fit the updated model, the second half to evaluate the respective acceptance rate. To fit the updated model, the sample of recourse implementing individuals is matched with their respective pre-recourse observations well as sample of accepted applicants of the same size. As result, the updated model is fitted on one third post-recourse samples and two thirds pre-recourse samples. To compute the acceptance rate, we take the other half of the post-recourse samples and compute the respective decisions with respect to the original and the updated model.
F33
Detailed results
The detailed results are reported in Table
Furthermore, we report model accuracies in Table 2 the model remains optimal post-recourse. The reason that recourse commonly moves data closer to the decision boundary, where model uncertainty is higher, as pointed out in [Fokkema et al. 2024
We note that the model accuracy may drop even
33

---

Table 1: Detailed Experiment Results. We report all detailed results. For each outcome, report the mean and standard deviation over the 10 runs. The outcomes are the expected, maximum, and minimum difference in conditional distribution, well the difference in acceptance rate.
Exp. Dist. DifI. mean
Max. Dist. Diff.
Min. Dist. Diff. mean
Acc. Rate Diff. mean
std
mean
std
std
std
Method
Setting
ind. ICR
LAdd
0.00
#0.01
0.02
#0.05
0.00
#0.00
0.00
10.00
LMult
0.00
#0.00
0.00
#0.00
0.00
#0.00
0.00
#0.00
NLAdd
0.00
#0.00
0.00
#0.00
0.00
#0.00
0.00
#0.00
NLMult
0.00
#0.00
0.00
#0.00
0.00
#0.00
0.00
#0.00
LCubic
0.21
10.00
0.67
#0.00
0.00
10.00
0.00
#0.00
GPA
0.00
+0.00
Credit
0.01
+0.01
sub. ICR
LAdd
0.00
#0.00
0.00
#0.00
0.00
#0.00
0.00
+0.00
LMult
0.00
10.00
0.00
#0.00
0.00
#0.00
0.00
#0.00
NLAdd
0.00
10.00
0.00
#0.00
0.00
10.00
0.00
#0.00
NLMult
0.00
10.00
0.00
10.00
0.00
10.00
0.00
10.00
LCubic
0.00
#0.00
0.00
#0.00
0.00
10.00
0.00
#0.00
GPA
0.00
10.00
Credit
0.01
#0.01
ind. CR
LAdd
0.68
10.13
0046
20.04
0.95
E0.08
0.93
#0.10
LMult
0.65
+0.12
-0.5
+0.08
0.85
+0.18
0.85
+0.18
NLAdd
0..7
+0.03
-0.72
#0.36
-100
#0.00
0.87
+0.02
NLMult
-0.12
+0.00
0.00
#0.00
0.49
#0.02
0.16
#0.01
LCubic
-0.57
+0.01
0.00
10.00
-100
10.00
0.98
10.0
GPA
0.81
+0.05
Credit
0.49
+0.03
sub. CR
LAdd
0.47
10.06
0.13
20.20
0.84
20.06
0.90
#0.08
LMult
0.61
+0.08
-0.54
10.07
0076
#016
0.85
10.18
NLAdd
0.97
10.04
-0.72
10.36
-1.0
10.00
0.87
+0.02
NLMult
0.16
+0.01
0.00
+0.00
-0.47
+0.01
0.01
+0.01
LCubie
-0.88
+0.03
0.00
#0.00
-084
#0.01
0.98
+0.00
GPA
0.79
+0.03
Credit
0.39
+0.04
CE
LAdd
0.71
+0.13
-0.46
10.04
0097
10.07
0.95
+0.09
LMult
0.65
+0.13
-0.55
10.08
-0.82
10.21
-0.85
10.18
NLAdd
-0.97
+0.04
-0.71
+0.38
-1.00
+0.00
0.88
+0.01
NLMult
-0.6
10.01
0.00
10.00
-0.99
+0.02
0.01
10.01
LCubic
-0.41
#0.01
0.00
10.00
-0.83
10.01
-0.98
10.00
GPA
-0.81
10.4
Credit
-0.47
+0.05
34

---

Table 2: Model Accuracies. We report the absolute model accuracies of the original model and the refit both the original data (O and R) and post-recourse data (OP and RP). All accuracies computed on test data and averaged over ten runs.
SCM
Type
R
OP
RP
LAdd
CE
0.91
0.90
0.51
0.89
LAdd
sub. ICR
0.91
0.91
0.96
0.96
LAdd
ind. CR
0.91
0.90
0.51
0.89
LAdd
ind. ICR
0.91
0.91
0.96
0.96
LAdd
sub. CR
0.91
0.90
0.53
0.89
LMult
CE
0.87
0.87
0.50
0.85
LMult
sub. ICR
0.87
0.87
0.94
0.94
LMult
ind. ICR
0.87
0.87
0.94
0.94
LMult
ind. CR
0.87
0.87
0.50
0.85
LMult
sub. CR
0.87
0.87
0.50
0.85
NLAdd
sub. ICR
0.91
0.91
1.00
0.99
NLAdd
ind. ICR
0.91
0.91
0.99
0.99
NLAdd
sub. CR
0.91
0.90
0.58
0.83
NLAdd
ind. CR
0.91
0.90
0.58
0.83
NLAdd
CE
0.91
0.89
0.58
0.83
NLMult
ind. CR
1.00
1.00
0.84
0.92
NLMult
sub. ICR
1.00
1.00
1.00
1.00
NLMult
sub. CR
1.00
1.00
0.84
0,83
NLMult
CE
1.00
1.00
0.84
0.84
NLMult
ind. ICR
1.00
1.00
1.00
1.00
LCubic
sub. CR
0.92
0.86
0.54
0.83
LCubic
CE
0.92
0.86
0.54
0.83
LCubic
ind.
0.92
0.86
0.55
0.83
LCubic
ind. ICR 0.92
0.92
0.95
0.95
LCubic
sub. ICR 0.92
0.92
0.96
0.96
GPA
ind. CR
0.73
0.70
0.51
0.68
GPA
ind. ICR
0.73
0.73
0.81
0.81
GPA
CE
0.73
0.72
0.51
0.70
GPA
sub. CR
0.73
0.72
0.51
0.69
GPA
sub. ICR
0.73
0.73
0.80
0.80
Credit
CE
0.60
0.58
0.57
0.60
Credit
ind. CR
0.61
0.59
0.57
0.59
Credit
sub. CR
0.62
0.59
0.61
0.62
Credit
sub. ICR
0.61
0.60
0.76
0.76
Credit
ind. ICR
0.61
0.59
0.76
0.76
35

---

F.4 Estimation of improvement and acceptance rates
When searching for the optimal recourse recommendation, we have to evaluate the probability of positive outcome for an action given the observation of the individual and given the available causal knowledge. As follows, we explain how this estimation is implemented in our code. The probability estimates are based on samples of size 1000.
Estimating individualized outcomes The individualized versions of CR and ICR are based on individualized effect estimation and assume knowledge of the SCM. To estimate individualized outcomes using a SCM, it the acceptance or improvement probability, we conduct three steps: Abduction, Intervention, and Simulation Pearl 2009]: Specifically, to compute a counterfactual using a SCM, we first use the observation x to abduct the unobserved causal influences, that is, we infer the posterior P(U X z). Then, implement the action in the SCM by replacing the affected structural equations. Last, we sample from the abducted noise P(U X = x) and generate the new outcomes using the updated structural equations. Based the obtained sample from the counterfactual distribution we estimate the probability of a favorable outcome. For ICR, that the probability of the favorable label L and for CR that is the probability of the favorable prediction == 1.
In our experiments we focus on settings with invertible structural equations. That is, given a full observation z, y. we could compute the unique corresponding noise value To perform the abduction in a setting where is not observed, a direct computation of the noise variables is not possible. Specifically, Uy and Ug cannot be determined. However, for each possible uy, there is exactly one possible ug; more formally there exists function ug g(wy). ). And one can show that as result the abducted distribution must be proportional to P(Uy P(UF - g) To sample from this distribution, we employ rejection sampling.
Lemma F.7 (Abduction with invertible structural equations). It holds that
P(UE = -E -E (rs, TC, fy (x)
p(uy xx)
p(øg I0,
Proof.
p(uy 2)) p(uy (cc,
2gg,
p(ac,
Sun Uy, uy uE) dug
((E,, Sun plaglos, IC, uy ug)p(zs, ug) dug
plan se. as pt)cspp((cc.s
Sus p(# Eg, Tc, dy, un Ury duz
p 3c, 2s)
Sug 1[E fø(Øs, ac, fy (#c, uE) dug
p (C,4s)
Sug 1[EE SE (as, TC, fv (IC, #x)) (p(ug) dug
P(Ug = ff p I0,
p(1F #c)
Estimating suhpopulation-based outcomes When no SCM available, the causal recourse methods resort to causal graph. The causal graph does not allow to abduct unobserved causal influences and therefore does not allow the computation of individualized effects. However, causal graphs allow to describe interventional distributions: Therefore, the joint distribution is factorized
36

---

into the conditional distributions of each node given its direct causal parents P(X) -I, X Xpa()). and the conditional distributions for the intervened upon nodes La are replaced with P(X,) 1 ; = 8g]. To obtain the interventional distribution for the subpopulation instead of the whole population, we additionally intervene on the nondescendants (L.) to hold those values fixed [König et al. [2023. In our experiments, instead of learning the conditional distributions, we obtain the ground-
conditional distributions from the SCM: To sample from the interventional distribution, we sample from the intervened-upon SCM. In this sample, we again compute the probabilities of the favorable outcomes.
Translating CEs into actions While CR and ICR recommend actions in the form of an intervention do(X,, = =O,)) on a subset of the variables la. CE only suggests new observation x'. To translate this to a causal action, we intervene on all variables, and set them to the values specified by z. That is, (o(X = z'). In this setting, there is no uncertainty about the post-recourse observation and as a result the acceptance probability can directly be evaluated to zero or one.
F55
Optimization
To solve the optimization problems imposed by each of the recourse methods, we employ evolutionary algorithms (as proposed by Dandl et al. (2020) and rely on the python package deap [Fortin et al.) 2012]. Evolutionary algorithms can naturally deal with both continuous and categorical data. To optimize goal, they randomly draw suggestions (individuals). This sample (population) is then modified and filtered over many rounds (generations).
In our setting, each individual consists of two values per variable: A binary indicator ay that represents whether the variable X₃ shall be changed, and float B3 that represents the respective new value.
When initializing the population, we randomly draw aj ~ Bin 0.5). the variable is continuous, 3j is sampled uniformly from the range Unif (bz, b,), where the lower and upper bound by and b,, determined empirically from a very large sample from the distribution. If the variable is categorical, 3 drawn from the uniform distribution over the empirical support. To cross two individuals, we switch the a and B values for each variable with 0.5 probability. To mutate an individual, we modify each entry with 0.2 probability, where the a, values are simply flipped, and the B3 values are either resampled if they are categorical or modified using Gaussian kernel if they are continuous. We chose the population size 25, the number of generations 25, the crossing probability 0.5, and the mutation probability 0.5.
To select the individuals that make it to the next generation, rely the following fitness function:
cost(x,, a) X(4r // (xr a))
where p puscossss is the probability of a favorable outcome for the given recourse methods (as defined in Section and Appendix B), and 10a.
F.6 Computational resources
We ran the experiments on a MacBook Pro with M3 Pro Chip and a cluster with Intel Xeon Gold processors with 16 cores and 2.9GHz. The main compute is required for sampling from the post- recourse distributions, which takes roughly one hour on one core for one setting and one method, which amounts total of 30 hours per and 300 hours for all experiments. We note that the runs were parallelized over the 16 cores. The post processnccssigg of the results was done on the M3 Pro Chip and requires negligible computational effort.
37


Reviews:
[{"review_id": "4tPPs9jUDS", "reviewer": "Reviewer_rdGh", "summary": "This paper investigates the performative aspect of algorithmic recourse: when we provide an individual with a “recourse action” this very action may shift the model’s future behavior, invalidating the effectiveness of the recourse when the model is retrained. The authors  formalize the performative validity, a criteria to quantify when recourse remains valid under distributional shifts induced by the recourse itself, i.e. when users take a suggestion from a system and improve themselves, then the model retrained on new data points will not reject the users that would originally be accepted. The framework used are Structural Causal Models and they show that two mechanisms can induce such invalidity: actions that are influenced by effect variables and actions that intervene on effect variables, Finally,  the concepts are used to show that Counterfactual Explanations and Causal Recourse can lead to invalidity. While the Improvement-Focused Causal Recourse can satisfy this property.\nThe Formal Condition for Validity is that the recommended action $A$ is conditionally independent of the true post-recourse label $L^p$ given the post-recourse features $X^p$. This reduces the problem to a statistical independence condition: $A\\sqcup L^{p}|X^{p}$. The theorems provide a clear, graph-based diagnostic for when a recourse method is at risk of invalidating itself.\nSome experiments on SCM with various structural model show that ICR preserves the validity, while it is not the case for CE and CR.", "questions": "- can you give more practical / real life examples \n- can we show mathematically that CE and CRa are not valid ? \n- Do we need to know the causal graphs in advance to apply the results proposed in the paper? This seems very difficult in real world. I am also wandering what would be the difficulties if we don't assume assumption 4.4 and 4.6? }", "limitations": "yes", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths and weaknesses. \n\nThe paper is well written with a good balance between necessary and exhaustive information (such as discussion of the state of the art, or proofs) between main paper and supplemental. \n- Various examples are presented and discussed to illustrate the concept and properties of interests. \n- the main theorems are clearly stated and well motivated and explained, and we can apply the methodology.\n\n\nWeaknesses:\n- The impact of retraining the model is not discussed on experiments or theoretically. In the paper, it is not clear what is the real impact, and it seems very useful to link the mixture model (1) (and alpha), with the number of rejected recourses or previously accepted observations. Could you clarify? What is the appropriate concept to anticipate the shift / amplitude of the drop in performance of the recourses.\n- Assumption 4.6 seems to be strong : is it possible de relax it? Do we have equivalence with this assumption in theorem 4.7. \nHow can we use it in practice?\n- I wonder about the significance or interests about this work: although I found the scientific question interesting, I am not fully convinced by the initial examples given in HR, and I feel thai it is a good illustration but not a real problem in practice. Do you have any alternatives?", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the feedback. We are glad that you find the paper well-written, the examples helpful, and the results clear, well-motivated, and applicable.\n\n### Q1/W3: Real-world Examples\n\n*I wonder about the significance or interests about this work: although I found the scientific question interesting, I am not fully convinced by the initial examples given in HR, and I feel thai it is a good illustration but not a real problem in practice. Do you have any alternatives? And: Can you give more practical / real life examples*\n\nAlgorithmic recourse has been suggested in various contexts such as employment services [1], hiring marketplaces [2], health [3,10], academic performance [4,5,7,8], and credit risk [6,9]. Performativity is relevant for any of these examples where ML is applied to individuals, see [10] and references therein. Let us use credit scoring as an additional example to illustrate the performative effects of recourse which is used as a motivating example across the literature on performative prediction and strategic classification. Also, [11] discusses practical examples of performativity in financial markets.\n\nCredit risk scores (such as FICO or SCHUFA) use predictive models to classify the solvency of consumers. They act as gatekeepers in contexts such as loan applications or the housing market. In recent years, an increasing number of companies, including FICO and SCHUFA, have offered so-called credit risk simulators that provide actionable recommendations on how to improve risk scores.\nOne of the most important features for FICO is credit utilization—the ratio of a person's current credit card balance to their total available credit (see e.g., the American Express MyCredit Guide). This feature serves as a proxy for financial discipline and is strongly associated with default risk. However, most credit scoring models only observe credit utilization at a single point in time—typically the time of application. This means that as long as an applicant’s credit utilization is low at that specific moment, it does not matter whether they usually max out their credit limits.\nIf recourse explanations suggest that applicants lower their credit utilization before applying, then the predictive value of this feature diminishes. In such cases, it is conceivable that the feature no longer reliably indicates financial discipline or default risk. As a result, recourse users who follow these suggestions may not be rewarded in future decision rounds, as credit models will be retrained on behavior that no longer correlates with risk—rendering the recourse ineffective or even counterproductive.\n\n\n[1] Raphael Mazzine, Sofie Goethals, Dieter Brughmans, and David Martens. Counterfactual\nExplanations for Employment Services. International workshop on AI for Human Resources728\nand Public Employment Services, 2021.\n\n[2] Nemirovsky, Daniel, et al. \"Providing actionable feedback in hiring marketplaces using generative adversarial networks.\" Proceedings of the 14th ACM International Conference on Web Search and Data Mining. 2021.\n\n[3] Ong, Ming Lun, Anthony Li, and Mehul Motani. \"Explainable and Actionable Machine Learning Models for Electronic Health Record Data.\" International Conference on Biomedical Engineering. Cham: Springer International Publishing, 2019.\n\n[4] Tsiakmaki, Maria, and Omiros Ragos. \"A case study of interpretable counterfactual explanations for the task of predicting student academic performance.\" 2021 25th International Conference on Circuits, Systems, Communications and Computers (CSCC). IEEE, 2021.\n\n[5] Wijekoon, Anjana, et al. \"Counterfactual Explanations for Student Outcome Prediction with Moodle Footprints.\" SICSA XAI. 2021.\n\n[6] Davis, Randall, et al. \"Explainable machine learning models of consumer credit risk.\" Available at SSRN 4006840 (2022).\n\n[7] Ramaswami, Gomathy, Teo Susnjak, and Anuradha Mathrani. \"Supporting students’ academic performance using explainable machine learning with automated prescriptive analytics.\" Big Data and Cognitive Computing 6.4 (2022): 105.\n\n[8] Smith, Bevan I., Charles Chimedza, and Jacoba H. Bührmann. \"Individualized help for at-risk students using model-agnostic and counterfactual explanations.\" Education and Information Technologies 27.2 (2022): 1539-1558.\n\n[9] Wang, Zijie J., et al. \"Gam coach: Towards interactive and user-centered algorithmic recourse.\" Proceedings of the 2023 CHI Conference on Human Factors in Computing Systems. 2023.\n\n[9] Lacerda, Anisio, et al. \"Algorithmic Recourse in Mental Healthcare.\" 2023 International Joint Conference on Neural Networks (IJCNN). IEEE, 2023.\n\n[10] Hardt, Mendler-Duenner. Performative Prediction: Past and Future. Arxiv. 2024. \n\n[11] MacKenzie. An Engine Not a Camera: How Financial Models Shape Markets. Oxford University Press. 2009.\n\n### Q3/W2: Assumptions\n\n*Assumption 4.6 seems to be strong; can it be relaxed? Do we have equivalence with this assumption in theorem 4.7. How can we use it in practice?*\n\nAlthough it is a strong assumption, 4.6 is practically useful since it covers commonly occurring DGPs, such as linear Gaussian data. We prove that the assumption is sufficient for performative validity (Theorem 4.7). It is not necessary for performative validity, as our experiments suggest. Relaxing 4.6 is an interesting avenue of further research.\n\n*How can we use 4.6 in practice?*\n\nFor example, if we know that the data is linear Gaussian and thus that Assumption 4.6 is met, we know that ICR is performatively valid. \n\n*What would be the difficulties if Assumptions 4.4 and 4.6 do not apply?*\n\nIf Assumptions 4.4 and 4.6 do not apply (and we cannot guarantee that only causal variables influence model and recourse generation), we cannot guarantee performative validity, regardless of which specific recourse method is used. We demonstrate this in Example 4.3. As such, we reveal a previously unknown fundamental limitation of any recourse method!\n\n*Do we need to know the causal graphs in advance to apply the results in the paper?*\n\nRecourse methods, such as CR and ICR, require knowledge of at least the causal graph. Causal knowledge is an inevitable requirement for recourse, since it is an inherently causal problem.\n\nIn contrast, for our main results (4.1 & 4.2), you only need to know whether causal sufficiency holds and whether non-causal variables are involved, which is a weaker requirement than knowing the full causal graph. Still, 4.4 and 4.6 can be best assessed given the complete SCM. Extending our work to settings with more limited causal knowledge is an interesting avenue for future work.\n\n### W1: What is the impact of refitting the model? How does alpha play into the results?\n\n*The impact of retraining the model is not discussed on experiments or theoretically.*\n\nOur results concern the impact of retraining a model on a mixture of pre- and post-recourse data. For instance, Theorem 4.2 shows that if no noncausal variables are involved, the optimal refit is equivalent to the optimal pre-recourse model. Furthermore, all experiments investigate how the conditional distribution and acceptance rates change when the model is refitted.\n\n*In the paper, it is not clear what is the real impact, and it seems very useful to link the mixture model (1) (and alpha), with the number of rejected recourses or previously accepted observations. Could you clarify? What is the appropriate concept to anticipate the shift / amplitude of the drop in performance of the recourses.*\n\nOur theoretical results are primarily concerned with whether a shift occurs, rather than its magnitude, and thus they are independent of the choice of $\\alpha$. In Propositions E.1 and F.1 we provide formulas that describe how the distribution shifts when interventions on effects are performed but to predict the strength of the shift in terms of acceptance rates, one would need to make very concrete assumptions about the distribution and decision thresholds. Intuitively, the strength of the shift depends on how strongly the predictor and recourse method rely on non-causal effect variables.\n\n### Q2: Mathematical proof for the invalidity of CE/CR\n*Can we show mathematically that CE and CRa are not valid ?* \n\nWhether CE and CR are performatively valid depends on the concrete setting. In Theorem 4.2, we prove mathematically that any recourse method, including CE and CR, is performatively valid if no effect variables are involved. In Examples 4.3 and 4.8, we prove mathematically that CE and CR may be performatively invalid in other cases. We commit to update the text to clarify that the Examples concern both CE and CR.\n\n**We hope our clarifications were helpful in addressing your questions. If they were, we’d greatly appreciate it if you would consider updating your score.**"}, {"review_id": "BWfIY1bczx", "reviewer": "Reviewer_gDwE", "summary": "The goal of this paper is to address the problem of reliable algorithmic recourse (counterfactual explanations) specifically taking into account how recourses can change the distribution which in turn changes the models. Specifically: when many applicants act according to the recourse recommendations, their collective behavior may shift the data distribution >> shifts the new model trained on the updated data >> changes the model boundary >> makes the recourses invalid (since they are typically the closest points on the decision boundary). The paper investigates this crucial problem through a causal lens, showing that recourses may become invalid if they are influenced by  non-causal variables, and hence, advocates for recourses that specifically focus on causal variables.", "questions": "I have included several questions along with the Weaknesses.\nIt would be great to get clarifications for these questions.\nWith some more clarifications, this can be quite a powerful result.", "limitations": "Some discussion is included.\nGiven the nature of this work, I would encourage more elaborate discussion on the physical meaning of their assumptions, and the role that they would play on the conclusions derived.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\nThis is a very interesting perspective on the challenge of reliable algorithmic recourse. Prior works have looked into this problem, aiming to provide a robust algorithmic recourse (or, robust counterfactual explanations) that finds recourses that would remain valid even under model changes. This work takes a different perspective on this problem, bringing in several novel elements:\n1. noting that data distributions change when people act upon the recourses, potentially leading to certain model changes (rather than a random model change)\n2. bringing in a causal perspective into the problem, showing how recourses that alter non-causal (spurious) features/variables can change the data distribution in a way that new models might no longer depend on that feature, thus making the recourses invalid\n\nProposition 4.1 is quite a surprising result - particularly the if and ONLY if part of the result. Some more clarifications on the assumptions will be needed. After resolving some comments/questions/clarifications, this can be quite a powerful claim.\n\nThey have also included limited empirical results to validate the theoretical findings and included some thoughtful and clever toy examples (more clarifications needed though).\n\n**Weaknesses: (includes questions)**\n\nW1. The results are proven based on certain worldviews/assumptions on HOW the data distribution changes once a recourse is provided, what part of the population receives and acts upon the recourse, consequently leading to the data distribution change. It would be nice to highlight these assumptions/worldviews upfront (not just mathematically, but physically what they mean) for better interpretability of the mathematical results and the conclusions derived from them. This is quite important in interpreting the results because this is quite a surprising claim. \n\n-- Q1: For the post-recourse distribution $P(X_p,Y_p)$, is it assumed that every point in the data distribution acts upon the recourse? Would this be a reasonable assumption in practice? What would be its physical meaning? \nDoes it have provision for:\n (i) only few individuals actually applying and receiving recourse and acting upon it? For instance, not all candidates might actually apply/interview/receive feedback. There might be several candidates who stay the same, and only a handful of them implement the recourse. (I am not sure if low $\\alpha$ alone is sufficient to model this sparsity? Low $\\alpha$ would still place a non-zero mass on *all* altered datapoints, but here only a handful of altered datapoints should have non-zero mass?) \n(ii) entirely new candidates can get added to the pool who have not applied before.\n\n-- Q2: The updated data distribution is assumed to be a mixture of $P(X,Y)$ and $P(X_p,Y_p)$ weighted by $\\alpha$ and $1-\\alpha$. Would this be a reasonable assumption in practice and what is its physical meaning? \n-- Q3: What happens when $\\alpha$ tends towards 0 or 1?\n\nW2. Similarly, there are also implicit assumptions being made on how the new model is being trained once the data changes which need to be made explicit. \n-- Q4: Are you assuming that the new model does not consider the data imbalance in the new data distribution? Depending on the $\\alpha$, there might be severe data imbalance in the new data distribution. What if the new model is trained after fixing the data imbalance, e.g., by reweighting?\n\nW3. A clear example will be very helpful. Though Examples 4.3 and 4.8 is provided, they lack a lot of details and have some notational typos (I also checked the details in Appendix E and F, but still have questions):\n\nIt says L is 1 when Y>0. Is L the true label or the predicted label?  \nIs L_hat/h a deterministic function of the features Xc and XE?\nWhat is beta?\nCould it be clarified what is the initial model h (perhaps provide a concrete model as an example, e.g., a linear classifier)? Then, for this model h and a chosen alpha, what is the hm? Then, for this example, could you show how does this statement mathematically play out: \"Then, the examination score becomes less predictive of the patient’s risk.\"\nAfter acting upon the recourse, are you assuming true label L will change if one acts upon Xc but not if one acts on Xe?\n\nW4. There are typos in the proofs in the Appendix, and also some details are missing in several places. For instance, Y and L have been used interchangeably in places. Some writing errors too.\n\nW5. While the mathematical results seem interesting, I am not fully clear on the final conclusions derived. \n\n-- Q4: What happens if the recourses are about the causal feature, the population updates that causal feature, and why wouldn't those recourses get invalidated?", "was_revised": true, "final_justification": "I liked the core idea of the paper, though there were some technical inconsistencies and missing information/clarity in several places that I pointed out.\nThe authors have responded to my questions to the extent that they could in the rebuttal. \nI maintain my positive rating since I generally agree with their responses, but the rebuttal has little details.\nFor a paper primarily making theory contributions, I would expect more clarity and toy examples in the presentation.", "rebuttal": "We are glad to hear that you find our perspective “interesting” and “novel” in several ways, our claims to be “powerful”, and our examples to be “thoughtful and clever”. Furthermore, we appreciate the reviewer's constructive feedback.\n\n### W1-2/Q1-4: Intuitive Explanations of the Assumptions\n\n*The results are proven based on certain worldviews/assumptions on HOW the data distribution changes once a recourse is provided, what part of the population receives and acts upon the recourse, consequently leading to the data distribution change. It would be nice to highlight these assumptions/worldviews upfront (not just mathematically, but physically what they mean) for better interpretability of the mathematical results and the conclusions derived from them. This is quite important in interpreting the results because this is quite a surprising claim.* \n\nThank you for the helpful comment -- you raise a valid point. We commit to updating the paper to clarify the required assumptions and their underlying intuition.\n\nFor our main results (4.1 and 4.2), we assume that (1) accepted individuals stay the same post-recourse, (2) the underlying causal mechanisms are the same post-recourse, and (3) the post-recourse distribution stems from causal interventions $A(x)$, which are a function of the applicants’ pre-recourse observations. Intuitively, (1) & (2) mean that there is no shift except for the one caused by recourse, and (3) -- although more general -- holds whenever all rejected individuals carry out the recommended recourse action. For results/experiments concerning specific recourse methods, we furthermore assume that $A(x)$ is given by the respective recourse method.\n\nAssumption (1) is captured in (Eq. 1). Apart from that, (Eq. 1) is very general: The applicant distribution $P(L, X)$, the post-recourse (or reapplicant) distribution $P(L^p, X^p|^L(X)=1)$, and $\\alpha$ can be chosen freely. Assumptions (2) and (3) are only introduced later (in Section 3.2) to keep the definition of performative validity (Definition 3.2) as general as possible, such that it can serve as the starting point of follow-up work that relies on a different set of assumptions.\n\nOur assumptions may not hold in practice, for example, additional external distribution shifts may occur, and whether one acts on a recommendation may depend on external factors. However, we highlight that in our analysis, the assumptions serve the purpose of isolating and studying the shifts *caused by specific recourse methods*. We make no claims about the robustness of recourse to other types of distribution shifts!\n\n**Addressing your concrete questions:**  \n\n(Q1)(i) Yes, the applicant distribution $P(X, L)$ can be a subpopulation of some larger population. The recourse action must be expressible with $A(x)$, and $A(x)$ can also map to $do(\\emptyset)$ (do nothing). It is safe to assume that an i.i.d. sample of the rejected applicants implements the recourse recommendation and reapplies. (ii) Yes, as long as the new applicants are an i.i.d. sample from the original applicant distribution $P(X, L)$.\n\n(Q2) $\\alpha$ allows for adjusting the weight given to the original applicant distribution and the post-recourse reapplicant distribution. As discussed above, (Eq. 1) is very general.\n\n(Q3) The larger $\\alpha$, the smaller the impact of post-recourse reapplications in the mixture. The choice of $\\alpha$ does not matter for our main results, as they concern whether a shift occurs, rather than the magnitude of the shift.\n\n(W2/Q4) We assume the model is designed to predict optimally in the mixture (with the respective $\\alpha$), and thus no balancing is employed in our experiments. We note that the choice of alpha does not matter for our main results (see Q3).\n\n### W3 and questions \n\n*A clear example will be very helpful. Though Examples 4.3 and 4.8 is provided, they lack a lot of details and have some notational typos (I also checked the details in Appendix E and F, but still have questions).*\n\nThank you for the valuable reading impression. We commit to updating the description and proof to make the Example easier to parse.\n\n*It says L is 1 when Y>0. Is L the true label or the predicted label?*\n\nIn line with the rest of the paper, $L$ is the true label. In Example 4.3, $L$ is a binary version of $Y$ that reflects whether a patient is qualified to spend the day outside, and $0$ is the qualification threshold.\n\n*Is $\\hat{L}$/$h$ a deterministic function of the features Xc and XE?*\n\nYes, $\\hat{L}$ is the binary predictor, and thus a function of the features. \n\n*What is beta? Could it be clarified what is the initial model h is (perhaps provide a concrete model as an example, e.g., a linear classifier)?*\n\n\nIn Example 4.8 the original model is linear and given by $(X_C + X_E) / \\sqrt(2) >=0$. Why? The model is based on the conditional $h(x) = P(L=1 |X=x)=\\Phi(X_C + X_C) / \\sqrt{2})$, where $\\Phi$ is the cumulative distribution function of a standard normal distribution, and the model's decision threshold $0.5$. \n\n*Then, for this model h and a chosen alpha, what is the hm?*\n\nThe retrained model $h^m$ is a mixture between $h^p$ and $h$, depending on $\\alpha \\in (0,1)$. To derive the post-recourse model, we leverage Proposition F.1 which enables us to describe the post-recourse conditional distribution as a mixture between the original conditional distribution and the conditional distribution where the association between target and effects was broken. Specifically, $h^p(x)= P(L^p=1 |X^p=x) = (1 - \\beta)P(L=1 | X =x) + \\beta * P(L = 1 | X_C =x_C)$. It can again be shown that $P(L=1 |X_C=x_c)  = \\Phi(x_C)$. The parameter $\\beta$ is the proportion of post-recourse individuals that intervened on the effect, and in this Example assumed to be $>0$.\n\n*Then, for this example, could you show how does this statement mathematically play out: \"Then, the examination score becomes less predictive of the patient’s risk.\"*\n\nSince $h^m$ is a mixture between $P(L=1|X=x)$ and $P(L=1|X_C=x_C)$, and since the second term is independent of the effects, the effects, in this case the examination scores, become less predictive post-recourse/in the refit.\n\n*After acting upon the recourse, are you assuming true label L will change if one acts upon Xc but not if one acts on Xe?*\n\nThe reason the true label $L$ does not change when intervening on $X_E$, but $L$ does change when intervening on $X_C$ follows from the fact that $X_C$ is assumed to be a cause of $L$ and $X_E$ an effect of $L$.\n\n### W5/Q5\n\n*While the mathematical results seem interesting, I am not fully clear on the final conclusions derived. What happens if the recourses are about the causal feature, the population updates that causal feature, and why wouldn't those recourses get invalidated?*\n\nAlthough interventions on causes may shift the distribution, they leave the relationship between features and target unaffected. In other words, while $P(X)$ may shift, $P(Y|X)$ stays the same. To make this more intuitive, think of a setting where an applicant’s Covid risk is used to determine whether somebody is allowed to enter a building. Two features are used to predict the covid risk: one cause, the vaccination status, and one effect, the body temperature. Taking antipyretics to get rid of fever (intervening on the symptom/effect) only makes someone appear to have a low covid risk without actually affecting the covid risk, thereby breaking the association between body temperature and covid risk.\nIn contrast, getting a vaccination (intervening on the cause) does indeed decrease the covid risk and the statistical relationship between vaccination and covid risk is preserved.\n\n**We hope our clarifications were helpful in addressing your questions. If they were, we’d greatly appreciate it if you would consider updating your score.**"}, {"review_id": "Mvzxf27ENS", "reviewer": "Reviewer_c6Gq", "summary": "This paper investigates performative effects, a phenomenon where widespread adoption of recourse recommendations shifts the data distribution, potentially rendering the recommendations ineffective after the model is retrained. To address this, the authors introduce the notion of *performative validity* and formally characterize the conditions under which it can be preserved. They show that performative validity fails when recourse actions are either influenced by effect variables or directly intervene on them. Empirical evaluations support the theoretical findings, confirming that avoiding effect variables helps maintain performative validity.", "questions": "Are there any reliable sources that show how often performative effects occur in the real world? I'm ready to raise the Significance score if they're a frequent issue.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "- Strengths\n  - Originality: The paper’s focus on performative effects and its formalization of performative validity is novel. This perspective has received limited attention in prior work, and the study is likely to inspire follow-up research in this emerging area.\n  - Quality: The finding that *improvement-focused causal recourse (ICR)* consistently maintains performative validity across diverse data-generating processes—such as resampling and additive noise—is theoretically and empirically compelling. To the best of my knowledge, this is the first paper to establish the superiority of ICR over CE and CR from the perspective of performative validity.\n  - Clarity: The paper explains related work, prerequisite concepts, and proposed methods with clarity and depth. Examples (e.g., Figure 1) and causal diagrams (e.g., Figure 2) effectively illustrate key ideas. Furthermore, thoughtful presentation choices, including text color and layout, enhance readability. The citations also appropriately cover the foundational work needed to situate this study’s contribution.\n- Weaknesses\n  - Significance: The study's practical impact could be better emphasized. While research on algorithmic recourse has gained substantial attention in academic circles, its societal deployment remains limited. Moreover, it is unclear which real-world domains performative effects occur in and how frequently. Some discussion or empirical justification on this point—possibly based on the experimental results in Q1—would be valuable. Although the GPA prediction task offers a semi-realistic use case, assumptions such as the frequency of model retraining may limit its generalizability.\n  - Practical scope: The reliance on causal knowledge narrows the proposed approach's applicability. As the authors acknowledge in Section 6, extending the framework to settings with incomplete or imperfect causal understanding is an essential and promising direction for future work.", "was_revised": true, "final_justification": "The rebuttal improved my significance and confidence score. The main weaknesses have been addressed, and this paper is worthy of acceptance.", "rebuttal": "We are glad to hear that you find our work to be original, of high quality, compelling, and clear. \n\n### W1: Emphasis on the practical impact\n\n*The study's practical impact could be better emphasized. While research on algorithmic recourse has gained substantial attention in academic circles, its societal deployment remains limited. Moreover, it is unclear which real-world domains performative effects occur in and how frequently. Some discussion or empirical justification on this point—possibly based on the experimental results in Q1—would be valuable. Although the GPA prediction task offers a semi-realistic use case, assumptions such as the frequency of model retraining may limit its generalizability.* \n\nML is increasingly being used for high-stakes decisions [1], and recourse has been identified as one of the central requirements towards restoring the agency of affected individuals [2]. Concretely, their application has been suggested in various contexts such as employment services [3], hiring marketplaces [4], health [5,12], academic performance [6,7,9,10], and credit risk [8,11].\n\nRecourse explanations provide concrete action recommendations to individuals. Thus, as soon as at least some individuals act upon them, they do have a performative effect on the underlying distribution. Our theory concerns the important open question of whether and under which assumptions the performativity of recourse affects the validity of recourse recommendations. Even if we had found that recourse is always performatively valid, that would be a valuable new insight!\n\nOur theoretical and experimental results show that performative validity may be violated when decision model or recourse algorithm rely on noncausal variables. Noncausal variables are parsimonous and often attractive variables for prediction since they can be highly predictive and are often more accessible than the respective causes. Examples include disease symptoms in the context of medical diagnosis/allocation of health resources [5,12], the CV word count in the context of hiring [4], or how long ago the customer last switched their bank in the context of credit approval. As follows, we illustrate the case of credit risk scores in more detail.\n\nCredit risk scores (such as FICO or SCHUFA) use predictive models to classify the solvency of consumers. They act as gatekeepers in important contexts such as loan application or on the housing market. In recent years, more and more companies, including FICO and SCHUFA, offer so-called credit risk simulators that give actionable recommendations on how to improve risk scores. \nOne of the most important features for FICO is credit utilization—the ratio of a person's current credit card balance to their total available credit (see e.g. the American Express MyCredit Guide). This feature serves as a proxy for financial discipline and is strongly associated with default risk. However, most credit scoring models only observe credit utilization at a single point in time—typically the time of application. This means that as long as an applicant’s credit utilization is low at that specific moment, it does not matter whether they usually max out their credit limits.\nIf recourse explanations suggest that applicants lower their credit utilization just before applying, the predictive value of this feature may diminish, since the feature may no longer reliably indicate financial discipline or default risk. As a result, recourse users who follow these suggestions may not be rewarded in future decision rounds, as credit models will be retrained on behavior that no longer correlates with risk -- rendering the recourse ineffective or even counterproductive.\n\n[1] Mehrabi, Ninareh, et al. \"A survey on bias and fairness in machine learning.\" ACM computing surveys (CSUR) 54.6 (2021): 1-35.\n\n[2] Venkatasubramanian, Suresh, and Mark Alfano. \"The philosophical basis of algorithmic recourse.\" Proceedings of the 2020 conference on fairness, accountability, and transparency. 2020.\n\n[3] Raphael Mazzine, Sofie Goethals, Dieter Brughmans, and David Martens. Counterfactual\nExplanations for Employment Services. International workshop on AI for Human Resources728\nand Public Employment Services, 2021.\n\n[4] Nemirovsky, Daniel, et al. \"Providing actionable feedback in hiring marketplaces using generative adversarial networks.\" Proceedings of the 14th ACM International Conference on Web Search and Data Mining. 2021.\n\n[5] Ong, Ming Lun, Anthony Li, and Mehul Motani. \"Explainable and Actionable Machine Learning Models for Electronic Health Record Data.\" International Conference on Biomedical Engineering. Cham: Springer International Publishing, 2019.\n\n[6] Tsiakmaki, Maria, and Omiros Ragos. \"A case study of interpretable counterfactual explanations for the task of predicting student academic performance.\" 2021 25th International Conference on Circuits, Systems, Communications and Computers (CSCC). IEEE, 2021.\n\n[7] Wijekoon, Anjana, et al. \"Counterfactual Explanations for Student Outcome Prediction with Moodle Footprints.\" SICSA XAI. 2021.\n\n[8] Davis, Randall, et al. \"Explainable machine learning models of consumer credit risk.\" Available at SSRN 4006840 (2022).\n\n[9] Ramaswami, Gomathy, Teo Susnjak, and Anuradha Mathrani. \"Supporting students’ academic performance using explainable machine learning with automated prescriptive analytics.\" Big Data and Cognitive Computing 6.4 (2022): 105.\n\n[10] Smith, Bevan I., Charles Chimedza, and Jacoba H. Bührmann. \"Individualized help for at-risk students using model-agnostic and counterfactual explanations.\" Education and Information Technologies 27.2 (2022): 1539-1558.\n\n[11] Wang, Zijie J., et al. \"Gam coach: Towards interactive and user-centered algorithmic recourse.\" Proceedings of the 2023 CHI Conference on Human Factors in Computing Systems. 2023.\n\n[11] Lacerda, Anisio, et al. \"Algorithmic Recourse in Mental Healthcare.\" 2023 International Joint Conference on Neural Networks (IJCNN). IEEE, 2023.\n\n\n\n### W2: Causal knowledge\n\n*The reliance on causal knowledge narrows the proposed approach's applicability. As the authors acknowledge in Section 6, extending the framework to settings with incomplete or imperfect causal understanding is an essential and promising direction for future work.*\n\nIndeed, the requirement of causal knowledge is one of the key factors that limits the application of methods like CR and ICR -- they require at least knowledge of the causal graph. Causal knowledge is an inevitable requirement for recourse, since it is an inherently causal problem.\n\nIn contrast, for our main results (4.1 & 4.2), you only need to know whether causal sufficiency holds and whether non-causal variables are involved, which is a weaker requirement than knowing the full causal graph, making them more practically relevant. 4.4 and 4.6 can best be assessed given the complete SCM; Whether the SCM is always required is an interesting question for future work.\n\n**We hope our clarifications were helpful in addressing your questions. If they were, we’d greatly appreciate it if you would consider updating your score.**"}, {"review_id": "SqemzathoM", "reviewer": "Reviewer_jQ9B", "summary": "The authors introduce the terms performative effect of recourse explanations and performative validity. \nThey then investigate whether recourse explanations remain valid under their performative effects. They theoretically prove that recourse becomes performatively invalid only if the implemented action \ncarries predictive information about the post-recourse outcome.\nLastly, they empirically study the performative validity of three recourse methods: counterfactual explanations, causal recourse, and improvement-focused causal recourse.", "questions": "- While certain features, such as a master's degree, may seem more costly to manipulate compared to others, like GitHub activity, this does not guarantee that acting on them truly reflects genuine qualification.  In the current landscape of generative AI tools, \n(e.g., ChatGPT and Gemini, among others), and cheating rings, among other shortcuts, even ostensibly robust signals (e.g., an academic degree) are manipulable (effect variables).  The feature-level recourse methods (whether CE, CR, or ICR) that suggest how much a user should change each feature (\"Translating CEs into Actions\" G4 in the appendix) are vulnerable to performative invalidity in practice. \nUltimately, decision-makers only see the (new) revealed features, not the underlying process by which users reach there.\n\n- I presume there are cases where the variable \"a\" is an effect variable in model 1 but is no longer an effect variable in model 2. Conversely, the variable \"a\" may not be an effect variable in model 1 but becomes an effect variable in model 2. For example, a model in the year 2012 that used Github activity versus a model in 2025 that used Github activity. How robust are the structural causal model generation processes to these sorts of changes, and how does that affect the performative validity?\n\n- I need some clarification. If \"neither the decision model nor the recourse algorithm is influenced by information about GitHub activity to arrive at their outputs,\" why is GitHub activity considered a feature in the model training? Why is it appearing in the SCM?\n\n- Experimental details are sparse, and lots of key information is missing. \n   - Could the authors include more details on the datasets (size, features, cleaning, etc.) and the generated SCMs?  \n   - From the experimental setting (G.2), the authors say they use logistic regression and decision-tree models. However, it's unclear how accurate they are on the datasets, how the models change after retraining, and so on. It's also unclear to me which model they use on which dataset and when.\n   - How did the authors compute L^p in the ICR case?\n\n\n- From the experimental setup, the authors mention that all settings include one cause and one effect variable. I am unsure how scalable the methods are. \n\n- Since the authors' work is more aligned with the robust CFE generation approaches, I am curious about the empirical comparative analysis between the robust CFE generation methods (https://www.ijcai.org/proceedings/2024/0894.pdf) and the authors' proposed perspective. \n\n- Since theoretical findings show that performative validity depends on the recourse method, would action-based, not feature-level, recourse methods, as those authors use, and would sequence-based recourse methods, and not the one-shot recourse methods authors use, do as well as the ICR at a computationally cheaper rate?", "limitations": "No, there is no limitations section in the main paper or appendix. \nIt would be good to comprehensively address both the theoretical and empirical limitations of their work.\nPlease see the weakness section and questions.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The authors introduce a new perspective on recourse generation and connect the performative prediction to the CFE/recourse generation. \n- The problem is interesting and offers useful insights to the community. \n- The theory is well-written and easy to comprehend. Also, Figure 2 is intuitive.  \n\n**Weaknesses**\n\n- While certain features, such as a master's degree, may seem more costly to manipulate compared to others, like GitHub activity, this does not guarantee that acting on them truly reflects genuine qualification.  In the current landscape of generative AI tools, \n(e.g., ChatGPT and Gemini, among others), and cheating rings, among other shortcuts, even ostensibly robust signals (e.g., an academic degree) are manipulable (effect variables).  The feature-level recourse methods (whether CE, CR, or ICR) that suggest how much a user should change each feature (\"Translating CEs into Actions\" G4 in the appendix) are vulnerable to performative invalidity in practice. \nUltimately, decision-makers only see the (new) revealed features, not the underlying process by which users reach there.\n\n- I presume there are cases where the variable \"a\" is an effect variable in model 1 but is no longer an effect variable in model 2. Conversely, the variable \"a\" may not be an effect variable in model 1 but becomes an effect variable in model 2. For example, a model in the year 2012 that used Github activity versus a model in 2025 that used Github activity. How robust are the structural causal model generation processes to these sorts of changes, and how does that affect the performative validity?\n\n- I need some clarification. If \"neither the decision model nor the recourse algorithm is influenced by information about GitHub activity to arrive at their outputs,\" why is GitHub activity considered a feature in the model training? Why is it appearing in the SCM?\n\n- Experimental details are sparse, and lots of key information is missing. \n   - Could the authors include more details on the datasets (size, features, cleaning, etc.) and the generated SCMs?  \n   - From the experimental setting (G.2), the authors say they use logistic regression and decision-tree models. However, it's unclear how accurate they are on the datasets, how the models change after retraining, and so on. It's also unclear to me which model they use on which dataset and when.\n   - How did the authors compute L^p in the ICR case?\n\n\n- From the experimental setup, the authors mention that all settings include one cause and one effect variable. I am unsure how scalable the methods are. \n\n- Since the authors' work is more aligned with the robust CFE generation approaches, I am curious about the empirical comparative analysis between the robust CFE generation methods (https://www.ijcai.org/proceedings/2024/0894.pdf) and the authors' proposed perspective. \n\n- Since theoretical findings show that performative validity depends on the recourse method, would action-based, not feature-level, recourse methods, as those authors use, and would sequence-based recourse methods, and not the one-shot recourse methods authors use, do as well as the ICR at a computationally cheaper rate?\n\n\n**Miscellaneous**\n\n- When characterizing the invalidity of the recourse, the authors frame the problem from the user's perspective. That is, the issue seems to mainly affect users (unless there is a computed true positive minus false positive rate or true negative minus false negative rate). Therefore, I am curious, what incentives do the decision-makers have to choose an expensive and difficult method like ICR (e.g., gathering causal knowledge is not trivial and is costly) over CE methods? \n- Synthetically generated noise is a feasible way to test the model changes, but it does not sufficiently capture the complexities of change in data distributions and model drift.\n- Authors look at invalidity from the perspective of the recourse not making the user qualified anymore. How about if h(x')=1 but f(x')=0? What are the implications from the casual perspective and long-term effects on both the user and the decision-maker?\n- I am curious about the fairness implications of the suggested setup. \n\n\n**Minor**\n- Typos in the paper, e.g., decison instead of decision", "was_revised": true, "final_justification": "The authors provided detailed responses to the questions I had", "rebuttal": "We are glad to hear that the reviewer finds our paper to be well-written and our insights to be original and useful to the community.\n\n*W1: [Is an academic degree really a robust signal (in contrast to GitHub activity)?]*\n\nIndeed, with the advent of tools like ChatGPT, academic degrees may have lost some of their predictive power. However, people have always attempted to cheat their way into an academic degree, and universities have consistently sought to prevent such practices. As such, we would expect that the recourse recommendation \"get an academic degree\" does not fundamentally alter the relationship between the degree and skill. This is not the case for a comparatively easy-to-manipulate feature such as GitHub activity.\n\n*How robust are the structural causal model generation processes to [changes in the causal structure], and how does that affect the performative validity?*\n\nBy definition, performative validity only concerns shifts resulting from recourse. For this purpose, we assume that all other factors remain constant. Of course, other shifts may occur in practice, and studying them together is an interesting avenue for future research.\n\n*If \"neither the decision model nor the recourse algorithm is influenced by information about GitHub activity to arrive at their outputs,\" why is GitHub activity considered a feature in the model training?*\n\nYou cite from line 213 in the paper, where we argue that performative validity could be guaranteed if model and recourse method were to abstain from noncausal variables like GitHub activity. This may not be the case in practice: Effect variables can be highly predictive of the outcome, and abstaining from them would result in a loss of predictive accuracy. To make this clearer, think of a disease diagnosis setting: Although the symptoms of a disease are not causal for the disease (meaning that treating the symptoms does not make you healthy), they are still valuable for predicting whether somebody is ill or not!\n\n*Experimental details are sparse, and lots of key information is missing.* \n\nDetails on the experiments are provided in Appendix G. The code was made available to the reviewers and will be made publicly available upon publication.\n\n*Could the authors include more details on the datasets (size, features, cleaning, etc.) and the generated SCMs? From the experimental setting (G.2), the authors say they use logistic regression and decision-tree models. However, it's unclear how accurate they are on the datasets, how the models change after retraining, and so on. It's also unclear to me which model they use on which dataset and when. How did the authors compute L^p in the ICR case?*\n\nThe full specification of the SCMs is reported in Appendix G.1. The specification of the models is reported in G.2. Decision trees are used in the synthetic settings, and a logistic regression model for the real-world example (SATGPA). In all settings, $L^p$ is defined as $Y \\geq \\text{median}(Y)$ to ensure balanced classes.\n\nWe updated G.2 to report test set accuracies (averaged over 10 runs) for the Original and Refitted models, both pre- (O and R) and post-recourse (OP and RP).\n\n| SCM      | Type     |    O |    R |   OP |   RP |\n|:---------|:---------|-----:|-----:|-----:|-----:|\n| LinAdd   | CE       | 0.91 | 0.9  | 0.51 | 0.89 |\n| LinAdd   | ind. ICR | 0.91 | 0.91 | 0.96 | 0.96 |\n| LinAdd   | sub. CR  | 0.91 | 0.9  | 0.53 | 0.89 |\n| LinAdd   | ind. CR  | 0.91 | 0.9  | 0.51 | 0.89 |\n| LinAdd   | sub. ICR | 0.91 | 0.91 | 0.96 | 0.96 |\n| LinMult  | sub. CR  | 0.87 | 0.87 | 0.5  | 0.85 |\n| LinMult  | ind. ICR | 0.87 | 0.87 | 0.94 | 0.94 |\n| LinMult  | sub. ICR | 0.87 | 0.87 | 0.94 | 0.94 |\n| LinMult  | CE       | 0.87 | 0.87 | 0.5  | 0.85 |\n| LinMult  | ind. CR  | 0.87 | 0.87 | 0.5  | 0.85 |\n| NlinAdd  | sub. ICR | 0.91 | 0.91 | 1    | 0.99 |\n| NlinAdd  | ind. ICR | 0.91 | 0.91 | 0.99 | 0.99 |\n| NlinAdd  | sub. CR  | 0.91 | 0.9  | 0.58 | 0.83 |\n| NlinAdd  | ind. CR  | 0.91 | 0.9  | 0.58 | 0.83 |\n| NlinAdd  | CE       | 0.91 | 0.89 | 0.58 | 0.83 |\n| NlinMult | sub. CR  | 1    | 1    | 0.84 | 0.83 |\n| NlinMult | sub. ICR | 1    | 1    | 1    | 1    |\n| NlinMult | ind. ICR | 1    | 1    | 1    | 1    |\n| NlinMult | CE       | 1    | 1    | 0.84 | 0.84 |\n| NlinMult | ind. CR  | 1    | 1    | 0.84 | 0.92 |\n| LinCubic | sub. CR  | 0.92 | 0.86 | 0.54 | 0.83 |\n| LinCubic | CE       | 0.92 | 0.86 | 0.54 | 0.83 |\n| LinCubic | ind. ICR | 0.92 | 0.92 | 0.95 | 0.95 |\n| LinCubic | ind. CR  | 0.92 | 0.86 | 0.55 | 0.83 |\n| LinCubic | sub. ICR | 0.92 | 0.92 | 0.96 | 0.96 |\n| GPA      | sub. ICR | 0.73 | 0.73 | 0.8  | 0.8  |\n| GPA      | CE       | 0.73 | 0.72 | 0.51 | 0.7  |\n| GPA      | ind. ICR | 0.73 | 0.73 | 0.81 | 0.81 |\n| GPA      | ind. CR  | 0.73 | 0.7  | 0.51 | 0.68 |\n| GPA      | sub. CR  | 0.73 | 0.72 | 0.51 | 0.69 |\n\n\nWe note that predictive accuracy is not within the scope of the present paper. The model’s accuracy may drop even if the model and conditional distribution $P(Y|X)$ remain the same after recourse. The reason is that recourse may move to regions with higher uncertainty, as demonstrated by [1].\n\n[1] Fokkema, Hidde, Damien Garreau, and Tim van Erven. \"The risks of recourse in binary classification.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2024.\n\n*From the experimental setup, the authors mention that all settings include one cause and one effect variable. I am unsure how scalable the methods are.*\n\nWe chose low-dimensional, discrete settings to enable an accurate empirical comparison of the pre- and post-recourse conditional distributions (Q1). Increasing the dimensionality would require significantly more samples from the post-recourse distribution, which are expensive to obtain since each sample requires solving the recourse optimization problems.\n\nTo address your concern, we added another real-world setting with 11 features, based on the UCI “Default of credit card client” dataset. We chose causes and effects as suggested in [1]. We fit the SCM using random forests, assuming Gaussian noise. The results confirm our theory and previous experiment results: The acceptance rates (Q2) remain the same for ICR, but drop significantly for CR and CE. \n\n| SCM    | Type     |   Acc. Rate Diff. Mean |   Acc. Rate Diff. Std |\n|:-------|:---------|-----------------------:|----------------------:|\n| Credit | CE       |                  -0.47 |                  0.05 |\n| Credit | sub. CR  |                  -0.39 |                  0.04 |\n| Credit | ind. CR  |                  -0.49 |                  0.03 |\n| Credit | sub. ICR |                   0.01 |                  0.01 |\n| Credit | ind. ICR |                   0.01 |                  0.01 |\n\n[1] Chen, Yatong, Jialu Wang, and Yang Liu. \"Learning to incentivize improvements from strategic agents.\" Transactions on Machine Learning Research (2023).\n\n*[...] would action-based, not feature-level, recourse methods, as those authors use, and would sequence-based recourse methods, and not the one-shot recourse methods authors use, do as well as the ICR at a computationally cheaper rate?*\n\nWith CR and ICR, we include two action-based methods. They formalize actions as causal interventions. Only CE does not specify the action to be taken to achieve the desired feature values; therefore, we transform them into an action-based method by intervening on every feature (Karimi, FAccT 2021). Does this answer your question? If not, can you clarify what exactly you mean by \"action-based\" and “sequence-based” recourse?\n\n*When characterizing the invalidity of the recourse, the authors frame the problem from the user's perspective. [...]. Therefore, I am curious, what incentives do the decision-makers have to choose an expensive and difficult method like ICR (e.g., gathering causal knowledge is not trivial and is costly) over CE methods?*\n\nICR is clearly more desirable for the decision maker. CE and CR are allowed to intervene on noncausal features and may thus instruct the applicant on how to deceive the predictor into believing they are qualified without actually making them qualified (acceptance without improvement). This conflicts with the decision maker's objective of making accurate decisions!\n\n*[...]. How about if h(x')=1 but f(x')=0? What are the implications from the casual perspective and long-term effects on both the user and the decision-maker?*\n\nWith h(x’)=1 and f(x’)=0, do you mean that the original model would accept the individual post-recourse but the post-recourse label is negative (false positive)? This is precisely the case that we study! As a result, the post-recourse conditional probability of a positive outcome decreases, and recourse recommendations may be invalidated once the model is updated.\n\n*I am curious about the fairness implications of the suggested setup.*\n\nThis is an interesting question for follow-up work. In particular, it is conceivable that performative invalidity affects different groups disproportionately (i.e., actions on noncausal variables are more likely to be suggested to one group than another), further highlighting the importance of performative validity.\n\n*[Are “robust CE” methods performatively valid?]*\n\nRobust variants of CE and CR are tailored to generic shifts; for instance, they may suggest moving not just onto but further past the decision boundary. However, they do not distinguish between interventions on causes and effects, and thus, in general, do not avoid interventions on noncausal effect variables. As such, we expect them to lead to performative invalidity. We conjecture that combining ICR with ideas from the literature on robust recourse will lead to recourse recommendations that are more expensive but not only performatively valid but also more robust to other types of shifts.\n\n**We hope our clarifications were helpful in addressing your questions. If they were, we’d greatly appreciate it if you would consider updating your score.**"}]

Author rebuttal:

Meta-review: The paper investigates the performative effects of algorithmic recourse, where recommendations can become invalid after a model is retrained on data from users who have followed them. To address this, the authors introduce performative validity, a novel criterion that uses a causal framework to formally characterize the conditions under which recourse remains effective. Theoretical analysis and experiments on Structural Causal Models are used to evaluate existing methods, demonstrating that while Counterfactual Explanations and Causal Recourse can become invalid, Improvement-Focused Causal Recourse is shown to maintain its validity. The reviewers’ concerns were satisfactorily addressed by the authors during the rebuttal period. Both the reviewers and I found the paper to have high potential impact for the machine learning community.
Decision: Accept (poster)