Collective Counterfactual Explanations: Balancing Individual Goals and Collective Dynamics
Authors: Ahmad Reza Ehyaei
Ali Shirali
Samira Samadi
Venue: NeurIPS 2025 poster ()
Keywords: Counterfactual Explanation
Algorithmic Recourse
Population Dynamics
Unbalanced Optimal Transport
Primary area: social_and_economic_aspects_of_machine_learning

Abstract: Counterfactual explanations provide individuals with cost-optimal recommendations to achieve their desired outcomes. However, when a significant number of individuals seek similar state modifications, this individual-centric approach can inadvertently create competition and introduce unforeseen costs. Additionally, disregarding the underlying data distribution may lead to recommendations that individuals perceive as unusual or impractical.
To address these challenges, we propose a novel framework that extends standard counterfactual explanations by incorporating a population dynamics model. This framework penalizes deviations from equilibrium after individuals follow the recommendations, effectively mitigating externalities caused by correlated changes across the population. By balancing individual modification costs with their impact on others, our method ensures a more equitable and efficient outcome.
We show how this approach reframes the counterfactual explanation problem from an individual-centric task to a collective optimization problem. Augmenting our theoretical insights, we design and implement scalable algorithms for computing collective counterfactuals, showcasing their effectiveness and advantages over existing recourse methods, particularly in aligning with collective objectives.

Collective Counterfactual Explanations: Balancing Individual Goals and Collective Dynamics
Ahmad-Reza Ehyaei
Max Planck Institute for Intelligent Systems, Tübingen AI Center, Tübingen, Germany ahmad. ehyaei@ruebingen. .ww.ee..iinnn.....dd.e......
Ali Shirali
University of California, Berkeley, USA
hhirali_ali@berkeleyy
Samira Samadi
Max Planck Institute for Intelligent Systems, Tübingen Al Center, Tübingen, Germany ssamadi @ww.eingeun.nppeeee.e meg.ee...p.deeee..
Abstract
Counterfactual explanations provide individuals with cost-optimal recommenda- tions to achieve their desired outcomes. However, when a significant number of individuals seek similar state modifications, this individual-centric approach can inadvertently create competition and introduce unforeseen costs. Additionally, disregarding the underlying data distribution may lead recommendations that individuals perceive as unusual impractical. To address these challenges, we propose a novel framework that extends standard counterfactual explanations by incorporating population dynamics model. This framework penalizes deviations from equilibrium after individuals follow the recommendations, effectively mitigat- ing externalities caused by correlated changes across the population. By balancing individual modification costs with their impact on others, our method ensures equitable and efficient outcomes. We show how this approach reframes the counterfactual cxplanation problem from an individual-centri task to a collective optimization problem. Augmenting theoretical insights, design and imple- ment scalable algorithms for computing collective counterfactuals, showcasing their effectiveness and advantages existing recourse methods, particularly in aligning with collective objectives.
1 Introduction
Algorithmic decisions are increasingly shaping various aspects of our lives, including access to opportunities and services Karimi et al. [2022). For individuals negatively affected by an algorithmic decision (e.g., loan denial), counterfactual explanations (CE) Wachter al. [2077] provide actionable insights by identifying minimal changes (e.g., increasing savings) needed to achieve a favorable outcome (e.g.. loan approval). Defining CE requires three elements:
feature space X C Rd characterizes individuals and is equipped with probability measure IP E P(Rd), where P(Rd) represents the set of all probability measures on Rdd
cost function e: x X - quantifies the effort to modify features x E to EX
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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3. A binary classifier h X + E {=1} assigns a decision to every 2 E X. This partitions X' into undesirable X' = f2 E X' h(x) = -1} and desirable X far X : h(z)) = +1} subsets. The respective probability distributions, P. and Pt, are induced by restricting PP to X' and X*.
For an individual ar E X receiving an undesirable outcome, the counterfactual explanation CE(x) provides the minimal-cost strategy to achieve the favorable label:
CE(x) = arg min (c(a,x²)) x'Ex*
The standard CE formulation assumes that individuals act in isolation, ignoring interactions between individuals moving to the same destination in X However, real-world dynamics are far more complex. While individual benefits from transitioning from X' to X by receiving positive decision, other factors can influence their overall utility. For instance, individuals moving to an overcrowded region may face increased competition, leading to decline in their utility. By solving CE on a per-individual basis, the standard formulation overlooks these broader societal impacts, failing account for the collective consequences of explanations. This so-called externality resembles the tragedy of the commons Gross and De Dreu [019]] as seen in navigation algorithms optimizing routes independently.
In our work, we model competition by assuming a fixed, yet unknown, amount of resources is available for individuals with a specific feature æc For instance, if x represents job-relevant attributes, the resources at x correspond to the societal demand for the expertise associated with x. While these resources to the population distribution at equilibrium. Assuming the population is in equilibrium before CE generation, the current population density I+, serves as a strong predictor of the available resources.
not directly observable, modeling population dynamics allows us to link them
Leveraging the connection between current population density and available resources, we propose framework called Collective Counterfactual Explanation (CCE). CCE accounts for the limited resources at each œ and generates explanations collectively. It guides individuals in a way that the population, after receiving and partially following these explanations, reaches state close to equilibrium. This ensures that the externalities from increased competition are minimized, making the generated explanations more reliable and beneficial for everyone.
Neglecting the underlying distribution IP raises concerns about robustness to inaccurate cost estimates or feasibility constraints. Social structures and unobserved costs may have already pushed individuals out of low-density areas, rendering recommendations towards those regions ineffective. To address this, recent work highlights data manifold closeness key requirement in Karimi et al. [2022). Guidotti [2022,, Verma et al. 22020). As we discuss in Sec. 3 the CCE formulation naturally aligns with this and related desiderata.
To illustrate these nuances, we use the Moons dataset (Pedregosa et al. [001]] with a non-linear SVM classifier and decision boundary T. in Fig- [ Standard CE methods, such as Wachter et al. (017 guide all applicants to the decision boundary T.. While this approach is cost-effective, neglects ects the feature- -atture-space space distribution P. potentially concentrating individuals in a low- ressucsueee region (left panel). This issue is particularly pronounced for SVM classifiers, which emphasize the margin between L and the nearest data points. In contrast, CCE generates more natural distribution, balancing the costs incurred by individuals with their impact on others (right panel).
Our Contributions. We leverage population dynamics model from mean-field game theory to incorporate competitive interactions between individuals into the CE formulation. Our framework. Collective Counterfactual Explanations (CCE), penalizes deviations from equilibrium to minimize unnecessary competition costs pursuant to the following recommendations. We propose a relaxed version of CCE that reformulates the CE generation problem (OT) problem. This reformulation enables us to draw on extensive techniques and tools from the OT literature to address various challenges in CE. In sum, our main contributions are:
unbalanced optimal transport
Formalize the collective costs of CE using a model of population dynamics (Sec.
Propose the CCE framework that penalizes externalities from individual interactions (Sec. 2.2
Relax CCE unbalanced OT and provide solution existence and consistency guarantees (Sec. 2.3
Design efficient algorithm to solve CCE with the benefit of amortized inference (Sec. 2.4
Demonstrate the advantage of CCE over standard CE along various desiderata (Sec.

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Figure 1: Comparing CE methods with non-linear SVM classifier. (Left) Wachter al. 2017 place all recommendations on the decision boundary L. (Right) In contrast, Collective CE moves individuals to more populated areas, which potentially more resource-rich.
Extend CCE to wide range of new settings and problems, including temporal recourse and recourse based ordered family of classifiers (Sec. 4).
Conduct numerical studies to support the theoretical results and the efficacy of our method (Sec.
2 Collective Counterfactual Explanation (CCE)
In this section, we extend the standard CE framework to address resource scarcity and competition arising from CE. The standard framework assumes individuals seek the minimum-cos action to reach X*. Therefore, it considers the cost of change and the benefit of positive classification as the only factors important for the individual. However, in practice, the benefits of positive classification not uniform; when individuals cluster specific points in X*, congestion can reduce these benefits.
We model congestion and reduced benefits by assigning resource to each X' and denote the resource distribution by P(X). If we ignore resource limitations, following CE may increase demand beyond the resources available at some points. This inefficiency can lower overall utility or force individuals take further actions to establish new equilibrium, which may undermine the value of our recommendations over time.
An ideal collective CE should transport the population from equilibrium another. We assume that individuals are in equilibrium before CE. Our goal is to generate CE that moves individuals to new equilibrium, under the assumption that random subset of individuals fully comply with the recommendations. Suppose there exists a function E (XX) + R that can measure the distance of a distribution from equilibrium. Denote the distribution over X after CE by PPEE- Then, ensure the recommendations lead to new equilibrium, we can include penalty E(PcR) in the objective.
A key property of equilibrium enables us to design function E that penalizes deviations from it. While the resources at each x are not directly observable, mean-field game theory links the resource and equilibrium distributions. Specifically. in Sec. 2.1 show that P S under equilibrium. Consequently, if CE results in an equilibrium, it should satisfy PER x S4. where S represents the resource distribution over X*. To quantify deviations from equilibrium,
measure how far dPxx (known Radon-Nin Radon-Nikodym derivative) is from 1. In particular, we will define E(PCR) is the dP.
x" ddvvergence between Pce and IP, which can capture the extent of deviation effectively.
Under mild assumptions (see Lemma] we can express CE as mapping T from X to X** Let M X²*) denote the space of all such mappings. To describe the distribution of individuals who were initially negatively classified and follow the CE, we use T P., the fosh-forward distribution by the map T. With this terminology, the collective CE problem solves the following problem:
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Proposition 1 (Collective Counterfactual Explanation) Under the population dynamics described in Sec. 2.7 and assuming a Y proportion of individuals X follow the explanations. CCE solves
arg min TEM(X .*))
E (c(2, ((x)) + Dx (TxP..IP.))
(2)
YP
for q E 00) and a competition regularization 1,. Here, À
where p+ and p. are the
Ypp +
proportions of the population in X* and X respectively.
The solution to Eq. 2 may not always exist Royden and Fitzpatrick (0110]. To address this, we
introduce a relaxed version of CCE with existence and consistency guarantees in Sec. 2.3
The rest of this section outlines the tools and details to derive CCE in Prop. I We begin by introducing
population dynamics and equilibrium (Sec. 2.Ip. Then, we provide a step-by-step derivation of CCE
(Sec. 2.2 and its relaxed version (Sec. 2.3 We conclude with algorithms to solve CCE (Sec. 2.4).
2.1 Population Dynamics and Equilibrium
Let t); E P()) be the distribution of individuals over X' time t. analyze U, we use mean- field game theoretic framework Lasry and Lions] [2006). Carmona et al. [2018). In this framework, the utility of an individual x depends on the density of resources S(x) as well as competition which arises from the local population density U(x, t). This interplay between the attraction of resources and the pressure of competition drives how the population redistributes itself over time.
Mathematically, the following PDE explains the evolution of the population:
(U(x t) = V. (U(z, t) V (BU(z, - a8(z)))
(3)
Öt
Attraction: VS captures how individuals move in response to resource availabilities.
Competition: VU captures how individuals spread out depending on the presence of each other. Together, g == V (a(((x) UU(x, ()) is the driving gradient. Individuals move along this gradient. and The term U g is the flux of the population, which is proportional to both U and the driving gradient.
The parameters
determine the significance of resource attraction and competition.
At equilibrium, the population equilibrium density U (z) satisfies (z)) - aS(x)) = 0 which implies (x) = (x)) + C, for some constant C (refer to Appendix Corr additional details).
2.2 Formal Derivation of CCE Formulation
In the first step, we reformulate the standard CE problem in Eq. (T) as an optimization over the space of measurable functions. Let M(X x²) denote the space of all measurable functions mapping from the subspace X to X'* Using Lemma (which we deferred to the appendix for brevity). we
reformulate CE as minimizing the cost function within the space of measurable maps on M(X x²)).
Formally, for any g E [1, oo), the CE in Eq.
(1 is equivalent to
arg min
{ E
(. [c(x, [c, T(=))
TEM(X- (x*)
The second step is to incorporate an additional term in the objective to penalize deviations from equilibrium. As shown in Sec. 2. equilibrium requires the distribution of individual features to align with the resource distribution. Assuming the population starts at cquilibrium and the resource distribution remains unchanged after intervention, by comparing P. and PCE., i.c., the distribution over X before and after CE. A general measure of divee-ivveeeneeeeeee can quantify this difference. We particularly use x=-divergence corresponding to (t)) = (t- 1)a as this will make the connection between CCE and optimal transport theory explicit.
measure deviations from equilibrium
The third and final element needed to derive CCE in Prop. i model of individual responses to CE. We assume that a proportion of individuals in X' follow the to transition to X Let p, and p. represent the fractions of the population initially in 2 and X'= respectively. Under this response model, the distribution of individuals X after receiving CE is PCE P. + A)P.. This completes the preliminaries to prove Prop. I and we leave other details to the appendix.

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2.3 Relaxation of CCE
Generally, there no guarantee for the existence of solution to the CCE problem Eq. 2
common technique to get around this is to search for an optimal plan и E P(X x X²)) instead of an optimal map. We refer the reader to Appendix [ for additional context on optimal transport (OT) theory. Using this technique, we relax Eq. ( with
arg min REP(X xX1))
[c(2, [c(x,y) v) + D (m2 (2,g))a
E
P+)
s.t.
Tı
(5)
Where and 22 are two marginal densities corresponding to the first and second coordinates of π
We formally prove the existence of solution for this relaxed problem:
Proposition 2 (Existence of Plan) When X' C Rd cost function, > 0, and 2 1, the relaxed CCE in Eq. (5) has a solution n. E P(X x x*)).
XxX 10, oo) is lower semicontinuous
To explicitly connect to OT and leverage its extensive tools, we introduce an additional relaxation to Eq. (): We replace the hard constraint
P. with penalty term x D (m P.) define.
arg inf TEP(X- xX*)
E (x.))><
cd,, [c(x,v) v) A (T || IP.) A2 ( P..)
(6)
Here, λ2 == 17 and a are regularization parameters and the choice of and Du are arbitrary. We show that this relaxed version is, in fact, consistent in the following sense:
Proposition 3 Denote the solution to Eq. 55) by n. and the solution to its relaxed problem Eq. 66 by Tx As d1 + oc, we have Xx + Tx..
2.4 Algorithms for CCE
We can build on extensive algorithmic tools from OT to design algorithmic solutions for CCE. Choosing D DKL. and q in Eq. (6). we present a projected-gradiine method solve CCE in Algorithm in the appendix. This algorithm has the following time complexity:
Proposition 4 (Complexity of Algorithm Iy Let m and be the sizes of the discrete sets X and X*, respectively, and let T be the number of iterations in the for-loop (lines 3-8) of Algorithm 7 Then, the overall time complexity of the gradient-batt based unbalanced optimal transport solver is ( m n).
Resembling the well- well-known Sinkhorn algorithm Peyré al. [2017] Séjourné al. 2022 Pham
ct al. (2020], we also present Algorithm 4 fast gradient-based unbalanced OT solver for re relaxed CCE. We refer the reader to the appendix for further details.
3 Collective Counterfactusl Explanation Aligns with Key Desiderata
In this section, we show that the CCE framework not only accounts for limited resources and competition but satisfies key desiderata for a successful recourse highlighted in recent surveys Verma et al. 2020), Karimi et al. 2022
Data Manifold Closeness. To avoid outliers and ensure CE is credible, suggested that CE remain close to the current data distribution Hamer et al. [2023] Movin et al. [2024). This property, known as data manifold closeness, preserves intrinsic feature correlations and leads more realistic, actionable recommendations. In CCE, penalizing deviations from equilibrium naturally enforces similarity to the current distribution. CCE also allows control over the desired level of closeness to the data manifold.
Security & Privacy. Each CE reveals that instances within radius c(æ, CE(z)) around xr belong to the negative class, making the decision boundary easy to identify. Thus, standard CE APIs pose security risks by enabling low-cost construction surrogate models Pawelezyk et al. 2003 Pentyall et al. 2023 Yang al. [2022]. as shown in Fig. 2 (middle). In contrast, CCE issues collective recommendations that account for equilibrium shifts, integrating diverse factors that intuitively complicate boundary manipulation, shown in Fig. (right).
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Robustness & Individual Fairness. In real-world decision crrisiin-mmainn.. robustness and individual fairmess require similar individuals to receive comparable recommendations Ehyae1 al. 2023b]. Guyomard et al. [023]_ Artelt et al. [2021]. Building on OOto et al. [202T]. we can see that under mild conditions on the densities P., P+, and a general class of costs, CCE defines a diffeomorphism. Since the feature space Y' is typically compact, small feature changes result in small changes in CCE
recommendations. In contrast, standard CE may yield sharply different outputs near the decision boundary, leading unequal treatment of similar individuals.
Amortized Inference. The standard CE formulation requires solving an optimization problem for each individual, which can be computationally expensive. Amortized inference addresses this challenge by leveraging patterns learned from prior instances Verma al. [202T], De Toni al. [2023), Majumdar and Valera (2024). Our CCE formulation employs map ' or plan to gen- erate explanations. Leveraging the rich literature of OT, we can efficiently learn this map or plan
using numerical methods. Once learned, we can generate CE for any instance without additional computation.
Actionability. Certain features, such as birthplace (an immutable attribute) or age (which follows naturally increasing trajectory), introduce additional constraints the design of CE. Actionable recourse requires that recommendations respect these constraints Ustun et al. (2019). Joshi et al. Rawal and Lakkaraju 20200. A common approach to handle such constraints is to assign large penalties in the cost function. In CCE, alternatively, we encode these constraints as [inear constraints Zaev (2015) within the OT problem. This feasible because Boolean functions-limits of compact support continuous functions-can generally be formulated as linear constraints. The following proposition formally establishes the existence of valid CCE plan, assuming at least one plan satisfies the specified linear constraints.
Proposition 5 (Actionable CCE Through Linear Constraints) Under conditions of Prop. the relaxed CCE problem with linear constraints has a solution if and only the set I'w == { P(X x X*) wdz 0, W] is empty, where W is the closure of W, subset of continuous functions with compact support on the space X x X
Extensions of Collective Counterfactual Explanation
In Sec. 2 we reformulated CE as the problem finding an optimal coupling between the distributions P. and F., This perspective offers unified framework for addressing key challenges in CE that otherwise difficult to resolve under standard formulations. We outline two such challenges below.
4.1 Path-Guided Counterfactual Explanation
Path-guided CE extends standard CE by offering not just final target point, but sequence of intermediate steps that guide an input toward a desired outcome. To achieve this, we can use displacement interpolation in dynamic OT, where mass moves continuously over time from source distribution to a target distribution.
GRia
Figure 2: (Left) An example CCE recommendation. (Center) The standard CE method poses higher risk of revealing the classifier boundary. (Right) Identification strategies are less effective in uncovering the boundary with the more sophisticated design CCE method.

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10
Ti
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X
Figure 3: (Left) The temporal map showing the flow of each point as it moves toward recourse target. (Right) The back-and-forth method was applied to estimate the CE map. By leveraging displacement interpolation, the optimal flow is depicted across four time steps, showing the transition of the negative region into the positive region using the Moons dataset.
To introduce temporal dimension into CE, define a time-indexed family of maps Tz(z), t [0,1]. where each T; X + X' describes the state of an input at time t. We seek the following:
Initial condition: At i 0, the distribution of transformed points (the push-forward of by To) should match or approximate the source distribution P..
Final condition: At = 1, the distribution induced by Ti should match or approximate the target distribution P.+..
Let (x(x)) be velocity field that describes how each point moves at time t. Then, the path auth-guided CE objective is
(Te,i) (-¹ [/ c(x,,) (T.()) di A [ Dy² ((Tt) ( P
min
Ao Do ((Tb) P.)) + As D2 (TT)4P.1IP
subject to the continuity equation: 8T, + V (T: w2) = O. Here, c(x, (x)x), t) is the instantaneous øt cost of moving point æ with velocity 24 (x) at time
To construct (x)), we assume that each point moves along a constant-speed geodesic in the feature space X' connecting its original position its destination under Ti. That is, (2(x) interpolates between x and T(x) such that To(x) x and Ti (x) T(x). According to results in Villani [2009), such geodesics exist in optimal transport theory, and the time-evolving distribution is then given by (Ti) For numerical implementation, we use the back back-and-forts algorithm from Jacobs and Léger 2020], which efficiently approximates dynamic OT paths. Their implementation is publicly available As demonstration, we apply path-guided CE to the moons dataset using discrete time steps. The results are shown in Fig. which assumes a high competition cost scenario.
4.2 Counterfactual Explanation for Ordered Classifier Families
In conventional CE, eligibility often determined by a single classifier. In practice, however, such as in loan applications, eligibility may vary with the requested loan amount. For example, request for & might be denied, while a lower amount E' could be approved. This necessitates a family of classifiers he, f. E L, where each h X + y = {+1} makes decisions for loan amount é. These classifiers follow natural ordering: & < l2 = he, (x) 2 he, (z), indicating that eligibility becomes less likely as the loan amount increases.

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Figure 4: Comparison of modification and competition costs across 100 experiments with different random seeds. Bar plots show average values, while error bars represent standard deviations. Our method (red bar) achieves lower competition cost but not the lowest modification cost, as it moves points toward higher-density regions. However, when considering the combined metric of modifica- tion cost and competition efficiency, it outperforms baselines, achieving the best trade-off.
Given this structure, CE seeks minimal modification x' such that the individual qualifies for loan of amount least é
CE.(x) arg min (c(x,,x") s.t. he(x') =
x'EX'
This ensures x' satisfies the eligibility criterion for l with minimal cost.
As discussed in Sec. 2 CE can be reframed as an unbalanced OT problem. To extend this to ordered
classifiers, we consider the joint distribution IP over features and loan amounts (x, E) E X x For
simplicity, assume h=)))) sign( [ar e)), where f X' The decision boundary is the set = {(x, E) E (X. L): f(z, f) = 0}. Given that V is non-zero everywhere, by the implicit function theorem [Lang 2012 5], L forms a C' n-dimensional manifold in the (n 1)-dimensional space X' L.
IR. is continuously differentiable
and strictly decreasing in l, i.e.,
0f
DF <
Define P. and P_, the restrictions of TP to the subsets where f (a: E) O / (sr, >> O, respectively. The unbalanced CCE framework constructs recourse map via OT that transports P. P, using the cost e² ((z, f), (x', {')) c(x, x²). To enforce that the resulting loan amount does not decrease, we constrain the OT plan with 1 By leveraging tools to solve OT with linear constraints. we can then find efficient recourse maps tailored to ordered classifiers.
5 Numerical Studies
In this section, we numerically evaluate our proposed collective CE for algorithmic recourse by com- paring against six baseline approaches: Wachter Wachter et [2017] 7. Growing Spheres Laugel
et al. 2017]. CLUE Antorán et al.) [2020). FOCUS Lucic et al. 22022, C-CHVAE Pawelezyk
al. 2021 We selected diverse range of algorithms to ensure broad spectrum of CE methods. We implement baseline methods using the open-source CARLA (awellzall al. (202T) (Counterfactual And Recourse Library) framework in Python, which offers standardized interfaces for generating counterfactual explanations and recourse interventions. Our experimental code will be released after review. conducted experiments on three real-world datasets commonly used in the literature to evaluate recourse methods: Adule and Kohavi [199), COMPASAngwin al. (2016), and HELOCFICO [2018), as well as one synthetic dataset, Moons Pedregosa et al. 2011], which is also frequently utilized in illustrating algorithmic recourse.
2020 and ROAR Upadhyay

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Figure 5: (Left) The blue curve represents the percentage increase in modification of CCE relative to standard CE as X2 varies from 0.01 to 0.3. The red curve illustrates the competition cost obtained by A2 xz (TT I I..). Both curves are supported with confidence intervals. As expected, there is trade-off between modification and competition cost measures. (Right) The result of CCE on the Adult dataset with 22 0.1.
We specifically employed the Moons dataset because it is two-dimensional, allowing us explicitly demonstrate the features and highlight the differences between our method and other baselines.
The real data are derived from the CARLA package and preprocessed for each dataset (for more details, see Appendix F). We use only two continuous actionable features determine the best recourse. Each dataset randomly split into training (80%) and test (20%) sets. Two non-lime lineirn methods, Multilayer Perceptron or Random Forest (corresponding method), are trained on each dataset to serve predictive models for which we seek algorithmic recourse. Baseline hyperparameters tuned according to the guidelines in the CARLA documentation or prior literature (see Table TT. To compute the CCE, employed the algorithm in Sec. 2.4 We also the l Euclidean cost.
To construct P. and I,,, we select or generate 1000 instances for each label, negative or positive, and construct the sample sets D* and D' In each experiment, we run baseline methods and compute the average modification cost, 1 Σ c(x, CE(r:)). Additionally, we compute D² (T. P.). n
to measure the divergence between the transported distribution and the target distribution. In our experiment, we put A = 0.1. To evaluate the competition cost, we discretize the space into grid and determine the proportions p; and q of samples corresponding to D and CE(D)) within each grid cell. Finally, the competition cost is computed as X2 Ei=1 (q==p.) This approach allows for P.
structured comparison of recourse effectiveness and cost across different methods.
We conducted 100 experiments, each with different random seed, and computed both the mod- ification and competition cost metrics for each run. In Fig. 4 the average results across all ex- periments represented by bar plots, while the standard deviation is illustrated using error bars. As expected, our method demonstrates lower competition cost compared to others. However, its modification cost is not the lowest, it tends move points toward higher-density regions. Nev- ertheless, when considering the combined metric of modification cost plus competition efficiency
( Σ c(a,, CE(x,)) + A2 Ei most balanced trade-off.
(a: pp) our method outperforms the baselines, offering the Pi
Another simulation explores the role of X2, the competition cost. We investigate the behavior of the CCE modification and competition as λ2 varies within the range [0.01, 0.3). As expected, and as illustrated by the simulation in Fig. left, there is trade-off between modification and competition adjust the relative weight of modification and
costs. By tuning A in real applications, competition costs to achieve a more realistic recourse.
To find the temporal path for each recourse, explained in Sec. 4.1 employed the back-and-fort method (see Appendix E.3) to determine the optimal temporal curve from the initial state to the modified resource. We demonstrate the construction of path-guided CE for the moons dataset based on discrete temporal steps. The results are illustrated in Fig. 3 In the right figure, the optimal flow that transfers the negative samples into the positive
is depicted over four steps.
Finally, to observe the CCE recourse, after determining the optimal plan, we randomly select the best state using multinomial distribution based on the probabilities derived from the optimal plan matrix. The results are presented for the Moons dataset in Fig. T] the Adult dataset in the right panel of Fig. 5 and the COMPAS and HELOC datasets in Fig.

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Further Related Work
In Poyiadzi et al. [2020], Kanamori et al. [2020), population probability is encoded into the cost function, guiding CE toward denser regions near the boundary L. While this helps mitigate outlier recommendations, these methods remain individual-centric and prone to overcrowding. Moreover, their integration of density lacks principled foundation, appearing somewhat arbitrary.
Group counterfactual explanations improve interpretability and fairess by addressing multiple in- stances simultaneously. Warrenn et al. [2023) developed an algorithm for high-coverage, model-faithful explanations, enhancing user understanding, Wielopolski et al. [2024] introduced gradient- based method linking local, group, and global counterlactuals. [Carrizosa et al. [2024b] proposed optimiza- tion models minimizing perturbation costs with linking constraints. Lodi and Kamirez-Ayerbe [2024) presented a column generation framework for sparse, scalable group explanations. Fragkathoulas al. 2024] developed a graph-based approach ensuring feasible, fair group counterfactuals via subgroup formation.
In Tsittsis and Gomez Rodrigue7 2020)_ utility functions represent decision-makers' objectives, guiding the optimization of policies and CE to maximize desired outcomes in strategic settings. Recently. Carrizosa et al. (2024a] introduced notion of collective CE. focusing on optimizing the modification cost for a group of instances rather than for individuals. This approach aims to harmonize the behavior of CE within a group, thereby mitigating the occurrence of cost-outlier CE. However, by overlooking the underlying density, this method might still result in CEs that are outliers concerning probability measures. Additionally, coupling CE within group could amplify the externalities.
To the best of our knowledge, method is novel in the literature, employing population dynamics to transform the conventional CE problem into a collective version. This approach is more realistic and addresses certain issues inherent in conventional CE.
Our work is also indirectly related to strategic classification Hardt et al. [2016]. Both CE and strategic classification involve individuals seeking positive label. CE recommends cost-minimizing actions under a fixed classifier, which is incentive-compatibe (IC). In contrast, strategic classification accounts for individuals strategically altering features, with the classifier adapting accordingly. Our work adds a new layer of realism to CE: individuals taking similar actions may face competition. This effect has recently been modeled as externality in strategic classification Hossain et al. (2024). Competition challenges the standard IC assumption in CE, as cost-minimizing recommendations may no longer ensure incentive compatibility. While framework does not explicitly model deviations from recommendations, it focuses on maximizing social welfare the presence of externalities. Extending account for strategic deviations would be interesting direction for future work.
7 Discussion and Future works
CCE is designed to address the limitations of standard CE by incorporating the population distribution. By accounting for societal competition for favorable outcomes, CCE mitigates performative effects arising from acaee-ccale behavioral shifts that affect the cost function. Both theory and numerical studies demonstrate CCE's effectiveness in tackling core challenges in counterfactual design.
Importantly, computing the CCE solution does not require estimating P: methods work directly with samples, offering flexibility. No assumption is needed on h, allowing compatibility with black-box classifiers.
Causality is central to algorithmic recourse, as interventions on some features can causally affect others Karimi et al. [2020] Our framework supports causal recourse through the use of causally fair metrics Ehyaei et al. [2024a] 2023a which map from endogenous to exogenous variables, thereby removing direct dependencies before applying CCE. We omit technical details here to maintain focus on collective CE, as full integration into a causal framework warrants separate study.
Choosing 2 is application-speecfic and non-trivial. While this work does not address general models of competition in recourse, we focus on competition over externalities Altmeyer et al. [2023] using a concrete model. Future work should explore broader notions of extermalities in this context.
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1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
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A Supplementary Materials
A.1 Definitions
Definition (Push-forward Measure) Let IP, Q be two probability measures in P(IR) and T R R& is map, the measure Q is called the push-forward of through T is denoted by T if:
Q(B) - P(T
VBCR I
Definition (Weak Topology) Weak topology on a space of probability measures on TR. denoted by P(RR), is defined by convergence in distribution. sequence of probability measures (IP NEN in P(RR)) converges weakly to probability measure P if for every bounded continuous function f: IRd R
lim E [((x)) 10 (X))) X., mu
X-r
Definition 3 (Set of Couplings) The set [(P, Q) represents the couplings of probability distributions P. Q ((TR), comprising distributions Red x Rdd with margins P and Q. measure belongs to [(P, ( if and only if
π (A x R = P(A) and (R x B) = Q(B) VA,B
By extension, random pair (X.Y) where E [(P, Q), will also be called coupling of P and Q
Definition 4 (Linear Constraints) Let (P) and C₁ (Q) be the space of continuous functions on X and X' and Liiitergral respect to IP and measures. Let Ct (IP, Q) be the family of continuous functions on X x X" such that:
(I, Q) {h E C(X x') 3f fi f2 S3. where f1 E (P), f2 (L))).
Definition 5 (The Wasserstein Metric) In the metric space (X.d), d). OT naturally defines metric known as the Wasserstein Earth Mover's distance. define the Wasserstein metric, we consider the space of probability measures with finite p-th moment (Wasserstein space):
(x) {T P(x):
(X,((,,,')) < &0, Va' E X
X-3
For two probability measures IP, Q E P,()), the P- F-Wasserstein distance is defined as:
W2(P, Q
inf
E [d'(XX
(ET((22)
(X.X))~T
This metric is positive-definite, finte, symmetric, and adheres to the triangle inequality (Villani 2009
$ 6].
Definition (Truncated Probability Measure) Let P E P(TR)) be a probability measure and A S Rdd be Borel subset of Rd such that (AA) O, the truncated probability measure IP P(Rd) is defined as:
P(ANB)
PA(B)
P(A)
for all B(Rd).
Definition 7 (lower semi-continuous) A function f Rd + RU is said to be lower semi- continuous (l.s.c) a point To rd for every there exists a neighborhood U of xo such that for all x U, f(x) > f(zo) -e. In economic terms, this concept implies that small perturbations in the input do not lead to a substantial decrease the function value, signifying form of stability or predictability in economic models, such as cost functions in production processes.
Definition 8 (Closure) Given set A in topological space X, the closure of A, denoted by A, is the smallest closed set X that contains A. Equivalently, includes all the points of along with all its limit points (i.e., points that can be approached arbitrarily closely by points in A).
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Definition 9 (Continuous Measure) A measure μ on measurable space (X,F) is said to be continuous if for every A E F, p(A) whenever A is a set of a single point. In other words, [((=))) = for every x E X.
Definition 10 (p-divergence) The --divergence between two probability distributions IP and Q over the same probability space, for a convex function 4, is defined as
(2((1))))= /
dP
dQ
dB is the Radon-Nikodym derivative of with respect to Q. The divergence measures the difference between the two distributions, with different choices of ip leading to different divergence measures. Common examples include the Sellback-Leiblle divergence for ((x) x log x, the Total
where 20
Variation distance for p(x) =//2== 11, and the squared Hellinger distance for u(x) (Vi I 1)2.
Definition (n-dimensional Hausdorff Measure) Let (X,d) be metric space.
The n-
dimensional Hausdorff measure H" of a subset A S X is defined as follows:
(A)
lim inf 5-+0
Σ (diam(U.)) {U.) is 8-cover of A
where a 8-cover of A is countable collection of sets {Ui} with diam(U:) < ó such that A S C U U, and diam(U:) the diameter of the set U4. For N, H. generalizes the notion of n--diensiioal volume, with H. representing length, H2 area, and H. volume.
Definition 12 (6-Confidence Positive Region) 5-confidence positive region is denoted by L con- sists of smallest buffer of L in X i.e. B {xr E X dist(x, L) S r} such that P(B) 2 e. where dist(a, L) inf(c(x, x') E L}.
A.2 Lemmas and Theorems
Lemma (Measurability of the Recourse Map) Suppose X is a standard Borel space, h X
y is measurable function, and c: X' X + R is a jointly measurable cost. Define
R(x) = arg min {c(x,
x'EX"
Then, R is a measurable set-valued map. If R(x) is non-empty and closed for all x E X there exists measurable T X - x with T(x) R(x) for all x.
B Overview of Optimal Transport and Extensions
This section provides an overview of OT and its extensions utilized in this study. OT, initially intro- duced by Monge [781], focused on cost-efficient transportation soil for fortifications. Generally, OT aims transfer probability measures from space X' to X". While X' and X" are typically Polish spaces, this work, they are considered open or closed bounded subsets of Rdd
B.1 Monge Problem
Let PP P(X), Q P(x") be two probability measures, and let e: X X' + R. be ground-cost function representing the cost of transporting a unit mass from x to y. The Monge's OT problem is to find a map T X + 2 that pushes forward IP to @ (i.e., T ? and minimizes transportation effort:
min T=
E X
((X,)(()))
optimal map a T that minimizes this objective, function 7 satisfying TyP Q called push-forward map Ambrosio et al.) 2021 1.2).
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Regardless of the cost function Monge's problem may be ill-posed due to the nonexistence of a mush-forward map and weak sequential closure issues w.r.. the weak topology Ambrosio al.) 2013]. After 150 years, Kantorovich 1942 addressed these limitations by relaxing the problem.
B.2
Kantorovich Problem
Rather than finding an optimal map, Kantorovich proposed minimizing transportation cost for coupling [(P, Q):
min (E
[x X
(8)
(XXX)~~
The solution of the Kantorovich problem, when it exists, is called the optimal plan [Ambrosio et al. 2021 2. The set of push-forward maps, denoted (P, Q), known as Monge couplings, are special cases of couplings characterized by (X,T(X)).
B.3 Kantorovich-Rubine Rubinssein Duality
Kantorovich reformulated OT as convex problem on P(X x X"), with its dual expressed as a
constrained concave maximization problem [Ambrosio et al. 202T $ 3.11]. Kantorovich duality states that the minimum of the Kantorovich problem equals the maximum of the dual problem over bounded and continuous Kantorovich potentials yp X IR and vé X" R:
sup (A.W))
[x() + E (X)), X~P
E
X
subject 4. {(p,,)) (px)) v0x') S c(x,,))).
For example, consider logistics company transporting goods from æ to x'. The company sets a loading fee ()() at a and unloading fee (xx') at x'. Their profit margin p((z) (xx") must not exceed the transportation cost c(x, x'). To maximize profits, the company adjusts the pricing functions oo and v in Eq. (9) (see Galichon 2018 2 for more examples).
B.4 Optimal Transport with Linear Constraints
In practical applications, solutions to the OT problem often need to satisfy constraints. Zaev 2015 incorporated these by adding linear constraints. The constrained OT problem seeks optimal couplings with additional conditions over the family W C CP, Q of continuous and -nttegrabee functions on X' x X²:
min (E.(F
((XXX
s.t.
[w(XXX)) = Vo E W.
(10)
(X.
(x.X))<
Invariant and martingale OT are examples of such constrained problems.
B.5 Unbalanced Optimal Transport (UOT)
Conventional OT assumes total supply equals total demand. UOT extends OT to scenarios where source and target distributions differ in total mass, incorporating terms for creation and annihilation of mass Séjourné et al. (2022). Let μ M_(X), E M+(x") be two positive measures and M. M X"). The UOT problem is:
J dTT D, (H,T1) + (V,
min TEM+(X,
where Dup and Djp, are 0 divergence terms for mass creation and annihilation, with (A1, A2) as pyper-parameters and (T. T2) as marginals of ा.
B.6 Dynamic Optimal Transport (DOT)
DOT extends OT by incorporating temporal dimension to model mass evolution Benamou and Brenier [2000) Let (X) utt, X) represent path of probability measures such that μo IP and M1 and (z) = vtt, X) be a velocity field. DOT is formulated as:
{
/
arg min (14.)))
c(æ, (x t). dul(z) dt

---

subject to the continuity equation 0 /t + V (142)) 0. Here, c(z, 2', represents the transport cost time t, and the integral computes the total transport cost.
B.7 Fundamental Theorem of Optimal Transport
Theorem (Fundamental Theorem of Optimal Transport) Assume e X x X' 7 R is con- tinuous, bounded below, and let 1s E P & P(X') satisfy o(x,,)) I a(x) b(x'), for E L (1), (v). For n E (p,, the following are equivalent:
π is optimal,
The minimum of the Kantorovich problem equals the supremum of the dual problem 9 attained by (pp,4) of the form (p, for some c-concave function ip
C A Mean-Field Population Model: Explanation and Solutions
We consider the PDE
0U (z. v. (U(x,)) V(SUZa) - a(rr))
(11)
where Uzz.t) 2 the population density, S(x) is a static resource function, and a, 3 Equation [II] arises from a continuity equation with velocity
V a a VS V
indicating attraction toward regions of higher S(x) and repulsion from high-density regions U(x, (). Equivalently, one may view 8 U S local potential: a gradient-flom formulation yields the same PDE.
A steady-state (x) satisfies
(U'()) (Z a = 0.
simple family solutions has
a s(x) + C
B (z) = C
(²)
g 8
Boundary conditions normalization (e.g., total mass) determine C. Since suppose U is the density of the distribution, we have C' = 0.
D Proofs
D.1 Proof of Lemma 1
We proved the proposition in more general case. Define the set-valued map A(x) = fa' E h(x') * h(x)). Since h is measurable, the set ((x, M(x') h(x)) is a measurable subset of X X. Consequently, its complement {(x, x') l((x)) + 1(x)) = xz) : : A(x)) is also measurable. Hence, A(-) is measurable set-valued map. Next, define the extended cost function.
(c(x,x') if A(r), g(z,x') = +00
otherwise.
Since e is jointly measurable and A(-) is measurable, the function g is measurable on X x X. Thus, for each fixed x, the minimization problem R(x) arg ming g(z, x') is well-defined.
We assume conditions ensuring the existence of minimizer. For instance, if X is compact (in suitable topological setting) and c lower semicontinuous and coercive, then for each x E XX, the set R(x) is non-empty and closed. These standard conditions often satisfied in algorithmic recourse scenarios, where one restricts attention to compact feasible domains or ensures appropriate behavior of c.
Now, R(x) obtained as the set of minimizers of a measurable function (( x²) over measurable and valued set A(z). Standard results from measurable selection theory (c.g., the Kuratowski-
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Syll-Nardzewski measurable selection theorem Bogachev and Ruas (2007]) ensure that measurable selection from R(x) exists provided R(x) is non-empty and closed.
Specifically, the Kuratowwkii-Ryll-Naddewwwd theorem states that if X' a standard Borel space and T: X = X' is measurable set-valued map with non-empty closed (or compact) values, then there exists a measurable function 1' : X' + N' such that 7 T(x) E R(x) for all и. Applying this theorem to our setting, we obtain such measurable selection T. Therefore, under the stated conditions, R(-) is measurable in the sense that it admits a measurable selection, completing the proof.
D.2 Proof of Prop.
To prove it is sufficient to show that:
Dx² (PEE || IP+) = X² D2² (T# Ip++
(12)
Recall that the x2 -divergence a measure TμP.. W.r.t P. defined as
(I /( dTT.P. ( (v) - 2 dIP (v),
dP,
provided T.P.. < P.. Now consider the mixture XT # P. + (1 A)P.. Its density w.r.t. P, is
x (v)
dT IP. dP,
(1 - A) 1= dT F.
(V)
dP+
dMP
Substituting this expression into the definition of X² -dvverggeeee, we get
D33 XTxxI. + (1- A)I, I / (1 1 +
dTy (v) - -1) - dIP d2)(
dP
dT P. (v) - -1 2
\
dlP(())
dlP
= X² /
dTx
(z) - 1) dP.
dlP
required.
D.3 Proof of Prop.
The optimization is over E P(X' x X') such that its first marginal equals P.,. This implies TΙ P.. Since IP. is fixed, any admissible must have this prescribed marginal. Consider minimizing sequence (Tx) MEEN such that
(Gdrn
1/4 + n (Tn,2|(P.) (TT.2PP)))))
( (d)
(PP) ((cd))"*+*&D +
1/4
inf wimr =P.
If (т₂) attempted to "push mass to infinity," the cost term c would either blow up or, if bounded, the x"-divergence term would penalize deviations of TT.2 from P. significantly. In other words, the >2-penalty encourages T, to remain close to P,, and the cost term controls the large-scale displacement. Together, these terms prevent the mass from escaping, ensuring that (T,, is tight. By Prokhorov's 's theorem Billingsley (003)). there exists a subsequence that converges weakly to some (P('' X)), it means n, m. Weak convergence and the linearity of projection imply that the marginals also converge weakly. Since each Tn satisfies nn, P.,, and F. is fixed, the continuity of the marginalization map ensures π P...
The objective function is:
((T) ( c(z,y) 1/9
The map π + [c is linear in ग,, and a is lower semicontinuous. By the Portmanteau theo- rem Billingsley 20013) for nh - n::
f |ddx S lim inf dddn.
|
n
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Since 2 + 2//9 is continuous and increasing, we have:
(fcuny
1/9
lim inf (f n-
1/9
For the divergnge, Px)+ is lower semicontinuous with respect to weak convergence of measures. Thus, as nn2 T2:
D (T2 P+) s inminf (
Combining these, we have:
J(x) S lim ((T.). 1-+00
Thus, (+) is lower semicontinuous w.r.t weak convergence. Since (TT) is minimizing sequence. we have by definition:
lim J(Tn) inf J(T). n-
m:my P.
By the lower semicontinuity established above:
(TT s lim inf J (TT))
((x)). n':
inf
n=000
Hence, π attains the infimum:
J(x) inf (T).
xom=P.
This shows that a solution n" exists.
D.4 Proof of Prop.
Let n. be the optimal solution of (5). and let ялд be the solution (6) Since T. has nx,1 IP.. plugging T. into the relaxed problem's 's objective shows
(B(ra))=,,(((z, [c(x] v)))
x T2. PP+) 2 (B(a,))aay, [e(z, v) [F.)) (TA,
As À + oc, any deviation of TAIs from P. would make the divergence term x D (TAA))l [IP.) unbounded. Hence, TT1.11 P..
By tightness and lower semicontinuity arguments, any limit point of { has first marginal I.. and cannot exceed the minimal value of (5). Thus, TA + T., completing the proof.
D.5 Proof of Prop.
To establish the time complexity of the Projected-Gradient CCE Solver in Algorithm 1 observe that each iteration from lines 3-9 involves the following steps:
Marginal computations (line 3). Computing क. (1)(1) x0 for all
I,..., m takes (mm) operations. Similarly, computing x2 (j) = -m (i, for all j = 1,..., also takes O(mn) operations. Overall, marginal updates require 0(mn) time.
Gradient computation (line 4). constant number of arithmetic and logarithmic operations. Hence, the gradient calculation for all m x entries is (mm)..
We compute <i(i)) F' for each pair (i, involving only
Update step (line 5). We update ((++)(i.)) max(0, (1. 17 Vn(kn) F), which is a simple arithmetic operation plus comparison, repeated m x n times. Thus, O(m)) operations.
Projection onto feasible set (line 6). For each we normalize (1 (i,5)) by dividing each entry by the sum Σラ= (i++) Computing this sum and the subsequent division also requires (mm) time in total.
Convergence check (line 7). Computing the Frobenius norm (1+11) x x(0)lr requires O(m)) operations.
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Since all five steps above are each (mm) per iteration, the total cost iteration is O(m)). Over T iterations, the overall complexity becomes
(mm) X = O(mnT).
Hence, the time complexity of the algorithm O(mTT).
D.6 Proof of Prop.
This proposition follows directly from the proposition presented in Zaev 2013
Proposition With conditions of Theorem D the OT problem with linear constraints has solution if and only if the set Tw = (x E : wdr O, w E W) is not empty, where W, the closure of W in the Cz topology.
To apply the proposition, we need to verify that its assumptions are satisfied in the C₁ topology. Since the functions continuous and have compact support, they clearly meet the required conditions. Thus, we can use the proposition to complete the proof.
E
Computational Experiments Supplementary Materials
E.1 Projected-Gradies CCE Solver
Algorithm Prooected-Gradien CCE Solver
Require: Feature sets X {z, J N labels Y fy, j=1 probabilities IP, distributions X" {z; V/ -1). X {rr My +1}. cost matrix C = {ct) } with Gy c(X,, Xz). regularization parameters }1. A2. step size threshold e, and max iterations T.
1: Initialize: π ( (i.j) 2 0 for all i (e.g., uniform).
2: for to T 1 do
3: Compute marginals:
(3)
Σ (i.)), 12 4
Σ x (i,)).
4: Compute gradient:
T2 (1) PP(() -P,(i))
V (i(i,) = Cjj + A In (1) P.()
+ 1) + 2
P+(j)
5: Gradient step with positivity projection:
(((+ max ( 0, (1. - [
6: Update plan:
[1+1) (1+1))
7: Check convergence:
if (++1) (1) F < terminate.
8: end for
9: Return (T)
E.2
Unbalanced Sinkhorn's 's algorithm
The unbalanced Sinkhorn algorithm (Peyré al. 2017 & 10) is an extension of the classic Sinkhorn algorithm, adapted for solving optimal transport problems where the mass the distributions does not necessarily match. The Unbalanced Sinkhorn algorithm modifies this problem to allow for
28

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differences in mass between P and Q. The constraints are relaxed using so-called Kullback-Leiller (KL) divergence terms, leading the unbalanced optimal transport problem:
min C e (T) A1 KL(TT [P) A2 KL(T' 1x
T20
Here, A1 and 22 are regularization parameters for the marginal constraints, and the KL divergence terms (L(-II)) measure the discrepancy between the marginals of the transport plan T and the given distributions
In empirical settings, the algorithm deals with discrete distributions often derived from data samples. This involves computing a transport plan between empirical distributions, which are represented sums of Dirac masses. The empirical part of the algorithm refers to its application to empirical distributions, i.c., distributions represented by samples (data points), which is a common scenario in practical applications.
Algorithm 2 Unbalanced Sinkhorn Optimal Transport
Input: probability measures = (pt); E R" and Q = (4)); E R", cost matrix C' (ct))s Raxm regularization parameter c, A1 and X2 regularization parameters and number of iterations
N:
Output: approximated optimal transport matrix T,;
Initialize: Uio ,, U0 1m:
Compute: K -
for n = 0 to N do
Update Un+1 = ( Kun
Update un+1 ( [ un.
r+
end for
Return: T = diag(uN) (
F,3 The back-and-forth method
The Back-and-Fori Jacobs and Léger [2020] method offers robust solution for computing optimal transport maps with strictly convex costs, including p-power costs, for probability densities P and Q on an n-point grid. This method, characterized by its computational efficiency, requires ((n) storage and (n lng(n)) computation per iteration. The iteration count needed to achieve accuracy is proportional to [[maa(||Pll [[]] log(?)) showcasing the method's effectiveness in both storage and computational resource optimization.
In the back-and-forth method, Ω is considered as a convex and compact subset of Rdd and focuses
on specific cost function e: Ω + E defined as c(x, 3) h(y - -)). Here, h: R + R is strictly convex and even function. The dual Kantorovich problem is considered in the two following equivalent forms:
((b) - f th duc I de or J(~) = pd du f du,
where yot, s/ are c-transformation of th. The gradient of [()) in the space of functions from Ω to can be written as Jacobs and Léger.
2020 Lemma 3):
( (-A)
T P),
where the A is Laplacian operator and
(x)
(Vh)
the explicit solution c-transformation (Jacobs and Léger 2020 Lemma 1). We now introduce the gradient descent back-and-forth method.
Theorem establishes that y is the solution to the dual Kantorovich problem, then T. constitutes the optimal map.
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Algorithm 3 Back-and-Forde Method for Optimal Map
Input: probability measures P and Q. cost function and number of iterations N; Output: approximated Kantorovich potential functions;
Initialize: set yu = 0, t 0
for n = N do
Pn+ Pn o (Pn),
$ #n+ = {
on+1 = Un+ + toV n+
Pa++ (
end for Return: PNN
E.4 Entropy-based Algorithm for Collective Counterfactual Explanations
The algorithm provides an entropy-regularized solution for generating collective counterfactual explanations by optimizing transport plan that balances transportation cost, adherence to given probability distributions, and entropy maximization. Starting with initialized potentials, the algorithm iteratively refines using gradient-based updates, ensuring constraints marginals and regularization terms such as Kullback-Leible divergence, x² divierggeec,, and entropy thresholds. Convergence determined by the Frobenius norm of consecutive transport plans, offering robust and interpretable framework for generating collective CE.
Supplementary Simulation
In numerical study, selected two actionable features from each dataset to evaluate the effectiveness our algorithmic recourse method. For the Adult dataset, we used Education Level and Hours per Week, which relevant for assessing socioeconomic mobility. From the COMPAS dataset, we chose Priors Count and Length of Stay, which capture key aspects of criminal history and detention. For the HELOC dataset, Percen! Trades and Trades Number were selected, reflecting financial behavior and creditworthiness. Finally, for the synthetic Moons dataset, we utilized Feature from both dimensions to simulate simple, interpretable feature changes.
F.1 Hyperparameters of Different Methods
This section presents the hyperparameters used for various methods a compact format.
Method
Hyperparameters
Wachter
loss_type tess_type-BCC, t_ min- m/66
Roar
1r=0.01, lambda_=0.0r delta_max-ta mtt.max=00000
t_max_mm mamax_mi=0.55 5, toss_thresholdi1er
1oss_type=BCE, y [0,,
discretize=False, sample=True
CCHVAE
g_gearch__amples=110. p_norm=2, step=1e-2. max_iter=1000.
clamp=True, binary_ cat_ features-True, VAE: layers=[]
limutables 256, 2], train=True, epochs=500, 1r=1e-3, bateh_
lambda 1e
Growing Spheres
No hyperparameterss specified
FOCUS
optimizer-adam, 1r=0.001, n class n_iter=1000, sigma 1.0, temperature=1.00 bastaance_eight=000 distance funcc=11
CLUE
crain_vae=True width=10, depth=5, latent_dim=11 dim=12, batch 20 epochs=5, 1r=0.00: 001, early_stop=20
Table Hyperparameters for different methods used in experiments in the Carla Package.
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Algorithm 4 Entropy-based Solution for Collective Counterfactual Explanations
Require:
Feature sets X = { Ji Y = {
Probability distributions P., F++
Cost matrix C' = {cir}
Regularization parameters Å1, 7,
Step size 7), tolerance max iterations T
1:
Objective Function:
F(T) = Σ 1
Tay Cij + A DKL(Ti + A2 De
+ c.
Σ ij
Tzz log (T)))
where Ti E; Tifh n2(i)= = -i Tzj-
2: Parameterization:
exp d
Tíj
3: Initialize: potentials g( E RN, 2b(0) E R (c.g., zero vectors).
4: for to T 1 do
5: Compute current transport plan:
Φi +g(() Cij
(0) Fij exp
6: Compute marginals:
( (1) =
Σ
iig
22
(1)
(0)
(j)
Σ
Tij
(1)
7:
Compute gradients via chain rule:
F F
Σ
1
(t) Tiz
cij + d (log(
x (1))
+1) 22
T2 (j) P+(1)
(1)
+ (Iog(T!5) + 1)]
P.())
P.(i)
(9) (1) PP(j))
V., F
(1) Tif Cij AA (1og(
11 ( (3) F.(()
+1 + 222
12
ex (10g(= (10) + 1)]
P.()
8: Gradient update:
(++1)
0 1 V F.
(1+1))
w00 V
9: Check convergence (e.g., via plan difference or potential difference): if (++1)
x
then terminate.
10: end for
11: Return final plan n. (T) exp
+ 7
31

---

wacher
compas wooher
of
201
Figure 6: Top left: HELOC dataset with its Wachter counterfactual explanations. Top right: Collective counterfactual explanations. Bottom left and bottom right: Wachter and collective counterfactual explanations, respectively.
32


Reviews:
[{"review_id": "CDwzSmdXtZ", "reviewer": "Reviewer_J14q", "summary": "This paper proposes Collective Counterfactual Explanations (CCE), which reframes algorithmic recourse as a collective optimization problem. By modeling population dynamics using mean-field game theory and optimal transport, CCE accounts for negative externalities when many individuals receive similar recommendations. It balances individual costs with collective impact by penalizing deviations from population equilibrium.", "questions": "- How robust is the CCE method to its core \"equilibrium assumption\"? If the initial population distribution is not in equilibrium before CE generation, how should the framework be adjusted?\n\n- See Weakness. Due to the paper’s lack of completeness, I believe it requires further refinement in both writing and formatting, as well as an additional round of review.", "limitations": "yes", "rating": 2, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 1, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n- The paper addresses a problem of significant value within the field of counterfactual inference.\n- It provides solid theoretical results with proofs and demonstrates originality in its approach.\n\n**Weaknesses:**\n- My primary concern is the paper’s lack of completeness. In addition to not meeting the standard page count and lacking a dedicated conclusion section, the manuscript exhibits several formatting issues—such as a placeholder symbol (a centered dot above line 291), awkward paragraph breaks (e.g., lines 320, 348), and suboptimal paragraph structure in the final section—all of which make the paper feel unfinished.\n- The core \"equilibrium assumption\" is not discussed in sufficient detail regarding its limitations or its feasibility in real-world tasks.\n- There are multiple instances of incorrect citation formatting; the authors do not properly distinguish between \\citep and \\citet where appropriate (e.g., lines 230 and 232).\n- While the paper presents numerous numerical studies, the code is not provided, which prevents the verification or reproduction of the experimental results. The provided GitHub link directs to a repository for a prior work, which is an unnecessary and potentially misleading inclusion.", "was_revised": true, "final_justification": "I appreciate the authors' response. Unfortunately, I think the core concerns I raised were not adequately addressed. Rather than responding directly to the formatting issue I pointed out, the authors shifted the discussion toward the clarity of their writing and motivation, which was not the focus of my critique.", "rebuttal": "We thank the reviewer for reviewing our work. Below we discuss reviewer's concerns in detail.\n\n> Re. My primary concern is the paper’s lack of completeness. ... not meeting the standard page count and lacking a dedicated conclusion section, ... formatting issues\n\nWith all due respect, we disagree with the reviewer’s assessment regarding the completeness of our paper. It is unclear what is meant by not meeting the standard page count or lacking a conclusion section. Our submission uses the full 9 allowed pages, with the last page fully dedicated to further situating our contributions within the broader literature, discussion, and future work, which we believe serve as a conclusion to the paper.\n\nWe have also provided a comprehensive background on optimal transport in Appendices A–C, which Reviewer Aeab found “helpfully organized... for better understanding.” While we welcome concrete suggestions for improvement and thank the reviewer for pointing to the typo and wrong break, we kindly disagree that the paper is incomplete. Other reviewers have also affirmed the completeness and quality of the submission:\n\n* Reviewer kVUx: \"The paper is *very complete and well‑written* with clear motivations, illustrations, assumptions, etc.\"\n* Reviewer fXWP: \"This paper is *well-written and well-organized*.\"\n* Reviewer Aeab: \"The writing quality of this paper is *good*. The logic is clear and rigorous, and the authors have helpfully organized the necessary background material in Appendices A–C for better understanding.\"\n\n\n> Re. While the paper presents numerous numerical studies, the code is not provided, which prevents the verification or reproduction of the experimental results.\n\nWe have not released the code earlier solely to preserve anonymity. We have now addressed this by sharing our project on an anonymous GitHub repository. Since including links is prohibited this year, the code can be found by searching **Collective-Counterfactual-Explanations-CCE** in the GitHub search bar. Even without access to the code, we have aimed to make every aspect of our work as clear and reproducible as possible. Specifically, the appendices include pseudocode for Algorithms 1 and 4, a full listing of hyperparameters (Table 1), and detailed instructions for running each experiment. In terms of baseline, we have also replied on open-source implementation to ensure reproducibility of the comparisons.\n\n> Re. How robust is the CCE method to its core \"equilibrium assumption\"? If the initial population distribution is not in equilibrium before CE generation, how should the framework be adjusted?\n\nThis is an intriguing question. In general, the CCE framework can be extended to non-equilibrium settings under additional assumptions, as we outline below. Recall that unlike standard methods that assume a stationary cost landscape—i.e., that the environment remains unchanged after issuing a recommendation—CCE directly models externalities resulting from recommendations. To retain tractability, we associate the (unobserved) resource landscape \\$S(x)\\$ with the current equilibrium population. At equilibrium, we assume \\$P \\propto S\\$, and deviations from this equilibrium are penalized using the \\$\\chi^2\\$-divergence \\$D\\_{\\chi^2}(P^{\\mathrm{CE}} ,\\Vert, P\\_+)\\$, as simplified in Eq. 2. The framework can accommodate the following deviations from equilibrium:\n\n\n* Small departures from equilibrium: The divergence penalty is continuous, so if the actual population distribution $P_0$ is only approximately at equilibrium, the difference in penalty $D_{\\chi^{2}}(P^{\\mathrm{CE}}\\Vert P_{+})$ remains within $O(\\|P_0 - P_{+}\\|)$. As long as the regularization weight $\\lambda_2$ is appropriately tuned, this discrepancy remains a second-order effect, and the resulting CCE solution stays near-optimal.\n\n* Known transient distribution: If the population distribution $P_t$ evolves over time for $t \\in [0, T]$, and a reliable estimate of the (possibly non-equilibrium) distribution $P_T$ is available at the time an individual requests a recourse—e.g., obtained from recent data or a population dynamics model—then we can directly replace the penalty term as follows:\n$$\nD_{\\chi^{2}}(P^{\\mathrm{CE}} \\Vert P_{+}) \n\\quad\\longrightarrow\\quad \nD_{\\chi^{2}}(P^{\\mathrm{CE}} \\Vert P_T).\n$$\nThe rest of the derivation remains unchanged. Proposition 2 guarantees existence and optimality under the sole condition that the reference distribution has full support, which still holds.\n\n* Fully dynamic settings: If $P_t$ evolves over time, we employ the path-guided extension from Section 4.1, which optimises an entire trajectory $T_t$ and penalises the time-integrated divergence:\n$$\n\\int_0^1 D_{\\chi^{2}}((T_t)_{\\\\#} P \\Vert P_t) \\, dt.\n$$\nThis reformulation turns CCE into a dynamic optimal transport problem, solvable using efficient algorithms such as the back-and-forth method.\n\n* Unknown but estimable equilibrium: If neither $P_t$ nor $S(x)$ is known, we can fit the mean-field PDE described in Appendix C to historical data to estimate a future $\\hat{P}_T$. We then use $D_{\\chi^{2}}(P^{\\mathrm{CE}} \\Vert \\hat{P}_T)$ as the penalty. So Our methods rely only on samples, so no closed-form expression for the density is required.\n\nIn summary, CCE is robust to moderate deviations from equilibrium. Moreover, the framework is flexible: the reference distribution can be adapted, or the full dynamic formulation can be applied when the population is far from equilibrium. These extensions ensure that CCE remains broadly applicable in both static and evolving environments. We thank the reviwer for this question and will clarify this in our updated manuscript.\n\n> Re. citations\n\nWe apologize for the inconsistent use of citation commands. We ensure the use of citep and citet is consistent in our update."}, {"review_id": "kpAqdAPkdZ", "reviewer": "Reviewer_kVUx", "summary": "This work introduces a new formulation of counterfactual explanations generation which targets to evenly distribute the counterfactual points of negatively affected individuals within the data distribution of the favourable region. The motivation behind this approach is to mitigate the potential competition for limited resources after counterfactuals are implemented, especially if the counterfactuals mostly converge to similar spaces in the favourable region. The method is evaluated theoretically and experimentally. The evaluation involves 3 tabular datasets, 2 types of machine learning models (MLP and random forest), 6 baseline methods (2 of which target plausibility of counterfactuals). Results on the cost of competition are promising, but recourse cost (distance between counterfactuals and inputs) for the proposed method is not the best.", "questions": "- Could you comment on how different your targeted problem setting is to the plausibility problem, i.e., requiring the generated explanations to be located well within the data manifold?\n- It seems to me intuitively that existing methods targeting plausibility and diversity, like DiCE or MOC, can naturally address the collective dynamics problem. Do you have any insights on that?\n- Minor point: there's a typo in 28 regarding desirable vs. undesirable; there's a dot above line 291 which takes some spaces.\n\nAlso see comments on weaknesses.", "limitations": "It would be important to address specific differences between the targeted scenario and the plausibility problem.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 4, "strengths_and_weaknesses": "Strengths: \n- The paper is very complete and well-written with clear motivations, illustrations, assumptions, etc. \n- The mathematical formulations of the problem and the methodology are clear and easy to follow. Theoretical analyses are also adequate.\n- While the problem setting targeted in this paper seems very similar to the plausibility of counterfactuals (requiring the generated explanations to be located well within the data manifold), the characterisations regarding resources and equilibrium are novel.\n\nWeaknesses:\n- The main concern I have is the validity and significance of experimental results. Given the similarity between the problem setting and the plausibility of counterfactuals, it would make sense to compare the proposed method with existing methods which also target plausibility. There are 6 baselines involved in the experiment, but only 2 of them (CLUE and CCHVAE) consider the plausibility aspect. Those two methods both require, and are quite sensitive to the quality of, some external generative models. There exist many other methods that might perform better in terms of plausibility, therefore potentially better cost of competition. For example, FACE, DiCE (these two are also in the CARLA library), Guided Prototypes, MOC. All names are taken from Table 1, page 5 of Karimi et al.'s ACM Computing Survey (https://dl.acm.org/doi/pdf/10.1145/3527848). Personally, I've observed much better plausibility results from FACE and DiCE than CCHVAE. The presented good performance on cost of competition also comes with significant trade-offs with recourse cost, especially given that the baselines do not give the best recourse cost in the literature - those by mixed integer programming (e.g., Mohammadi et al. AIES 2021, https://dl.acm.org/doi/10.1145/3461702.3462514) usually perform better. I think addressing this point will make the paper a much stronger one.\n- In addition to the presented evaluation metrics, including quantitative measures for plausibility and computation time would be desirable.\n- Clarity on related works: following the above points, it'd be clearer if the characteristics, especially which desirable properties the baselines consider, are explicitly discussed.", "was_revised": true, "final_justification": "The paper is sound in terms of methodology. The rebuttal with additional baselines addressed my main concerns about the significance of results. The targeted problem is quite artificial for the counterfactual explanations pipeline, which could be a fundamental limitation of this work.", "rebuttal": "We thank the reviewer for the insightful review of our work. We are glad they found the paper complete and well-written, with clear motivations, illustrations, and assumptions, as well as adequate theoretical analysis and a novel characterization of resources and equilibrium. Below, we address the reviewer’s specific concerns and questions.\n\n\n> Re. Could you comment on how different your targeted problem setting is to the plausibility problem, i.e., requiring the generated explanations to be located well within the data manifold?\n\nGenerally, plausibility-focused methods like DiCE, MOC, or Guided Prototypes focus on **individual** feasibility, plausibility, or sparsity but not on the externalities that arise when many individuals receive similar recommendations. In contrast, our framework directly incorporates collective impact of recommendations in addition to individual costs. This collective optimization perspective sets our method apart.\n\nTo illustrate this distinction more clearly: in plausibility-focused methods, it's possible for all individuals to be recommended the same counterfactual point, as long as it lies on the data manifold and the individual cost is low. However, in real-world scenarios, this leads to overcrowding, which traditional methods fail to anticipate. Our approach asserts that if a significant number of individuals are advised to move toward the same point, **externalities will arise** due to natural resource constraints, thus, any responsible recourse method should incorporate such effects.\n\n> Re. ... it would make sense to compare the proposed method with existing methods which also target plausibility. There are 6 baselines involved in the experiment, but only 2 of them (CLUE and CCHVAE) consider the plausibility aspect. ... There exist many other methods that might perform better in terms of plausibility, therefore potentially better cost of competition. For example, FACE, DiCE, Guided Prototypes, MOC. \n\nWe thank the reviewer for this suggestion. To further enrich our comparisons and better support our claims empirically, we have conducted additional experiments based on the reviewer’s suggestions, comparing our method (CCE) with four plausibility-oriented baselines: **FACE, DiCE, Guided Prototypes, and MOC**. Table 1 and 2 below report these new results.\n\nAs expected, since these plausibility-based methods impose only local constraints on individual recourse, congestion arises when a large share of the population seeks recourse, leading to significantly **higher competition costs**. Our method, by design, mitigates this effect.\n\n#### Table 1: Competition Cost Comparison\nBelow is a comparison of the **competition cost** across methods:\n\n| AR       | Moons      | HELOC      | COMPAS     | Adult      |\n| -------- | ---------- | ---------- | ---------- | ---------- |\n| **CCE** | **0.0599** | **0.0633** | **0.0796** | **0.0688** |\n| DICE     | 0.3961     | 0.1760     | 0.3423     | 0.3749     |\n| FACE     | 0.6700     | 0.3560     | 0.2523     | 0.6753     |\n| Guided Prototypes       | 0.8361     | 0.7959     | 0.7336     | 0.6118     |\n| MOC      | 0.4827     | 0.4613     | 0.5135     | 0.2996     |\n\n*Mean values of competition cost (lower is better).*\n\nConsidering the **total cost** (modification + competition), our method consistently outperforms the baselines. This is because **CCE uniquely and significantly addresses the cost of competition**.\n\n#### Table 2: Total Cost Comparison\n\n| AR       | Moons      | HELOC      | COMPAS     | Adult      |\n| -------- | ---------- | ---------- | ---------- | ---------- |\n| **CCE** | **0.7127** | **0.8271** | **0.7629** | **0.5579** |\n| DICE     | 1.7231     | 1.1252     | 1.4965     | 1.5560     |\n| FACE     | 2.1115     | 1.1892     | 0.8661     | 2.0792     |\n| Guided Prototypes       | 2.7834     | 2.6358     | 2.4119     | 1.9578     |\n| MOC      | 1.8392     | 1.7983     | 1.9086     | 1.2055     |\n\n*Mean values of total cost (lower is better).*\n\n\n> Re. It seems to me intuitively that existing methods targeting plausibility and diversity, like DiCE or MOC, can naturally address the collective dynamics problem. Do you have any insights on that?\n\nWhile related, individual plausibility does not necessarily address competition costs, for the reasons discussed above. In prior work, plausibility is often equated with closeness to the data manifold. These approaches focus on individual recourse and typically overlook the *distributional* properties of the manifold, which are crucial when many individuals seek similar recourse and may crowd the same regions. In contrast, our work explicitly addresses *distribution-aware* recourse. By incorporating equilibrium population density into the objective, we offer a principled mechanism for managing collective dynamics and mitigating congestion.\n\n\n> Re. computation time\n\nRegarding computation time, we provide a theoretical analysis in Proposition 4, which characterizes the time complexity of our gradient-based unbalanced optimal transport solver as $O(Tmn)$, where $T$ is the number of iterations, and $m, n$ are the sizes of the source and target distributions, respectively. This offers a practical estimate of scalability and efficiency. We also report the computation time of our algorithm alongside other baselines, as run on a local machine with a single CPU, in our response to Reviewer fXWP below.\n\nLast but not least, we thank the reviewer for their helpful suggestions to explicitly report desirable properties of the baselines and include additional plausibility metrics. We also thank them for spotting a typo. We will incorporate these in our revision."}, {"review_id": "CpLzQS0t47", "reviewer": "Reviewer_Aeab", "summary": "Most previous works on counterfactual explanations (CE) focus only on individual cases and overlook the externalities that occur when many people follow similar recommendations. This work proposes a new framework to address this limitation. Building on a population dynamics model, the authors introduce a penalty term into the standard CE objective to balance private and external costs and develop a scalable algorithm to solve it efficiently. In addition, they analyze how this approach improves over prior methods across different criteria for successful recourse and demonstrate its versatility by extending it to path-guided CE and ordered classifier families.", "questions": "1. Why use KL-divergence for the regularization term corresponding to $\\lambda_1$?\n2. What happens if the two regularization terms in Equation (6) use the same or different divergences? How does this affect choosing $\\lambda_1$ and $\\lambda_2$?\n3. Is using a discretized grid a good way to compute $\\chi^2$-divergence? It seems hard to control the grid resolution, and this might strongly affect the results.", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "I recommend accepting this paper based on the following strengths:\n\n- On the topic of accounting for externalities, prior work [56] was the first to highlight the existence of this problem, but it only proposed some ad-hoc mitigation strategies. To the best of my knowledge, this work is the first to provide a complete and practical framework as a solution. For this reason, it is likely to serve as a solid benchmark for future research in this area. I also look forward to the authors releasing the source code.\n- The writing quality of this paper is good. The logic is clear and rigorous, and the authors have helpfully organized the necessary background material in Appendices A–C for better understanding. I briefly reviewed the proofs in Appendix D and did not notice any major issues. Additionally, the appendices detailedly provide the algorithms and experimental, ensuring the reproducibility of this work.\n- The experimental results strongly support the authors’ arguments. For example, Figures 1, 5b, and 6 demonstrate that the proposed method indeed produces actionable recommendations and that the individuals receiving recourse end up closer to a realistic distribution. Meanwhile, Figure 4 confirms that the approach balances private and external costs as expected.\n\nIn addition to these strengths, I would like to point out a few weaknesses that the authors could improve:\n\n- I would appreciate an explanation of why the regularization term corresponding to $\\lambda_1$ uses KL-divergence. Using different divergence measures for the two regularization terms could potentially introduce challenges, such as making the choice of $\\lambda_2$ more sensitive.\n- The assumption that “individuals are in equilibrium before CE” is an important premise in this paper. I suggest that the authors explicitly state this assumption in a formal way.\n\nNote that I didn't pay much attention to this field in the past, so I gave a relatively low confidence.", "was_revised": true, "final_justification": "Since most of my concerns are addressed, I remain supportive of acceptance.", "rebuttal": "We sincerely thank the reviewer for the insightful and encouraging comments. We are heartwarmed that the reviewer found our approach novel and impactful, and we appreciate their close reading of both the theoretical and experimental components. Below we discuss their thoughtful suggestions and questions in detail.\n\n> Re. Why use KL-divergence for the regularization term corresponding to $\\lambda_1$?\n\nWe use different divergence measures for the two regularization terms in Eq. (6) to take advantage of their respective computational and modeling properties:\n\n* For the divergence between the *second marginal* and the equilibrium distribution $D_{\\chi^2}(\\pi_2 \\,\\|\\, P^+)$, we use the $\\chi^2$-divergence because it naturally aligns with our equilibrium penalty, which quantifies deviations from the target equilibrium distribution (see Proposition 1 and Section 2.2). Specifically,\n  $$\n  D_{\\chi^2}(P^{\\text{CE}} \\,\\|\\, P^+) = \\int \\left( \\frac{dP^{\\text{CE}}}{dP^+} - 1 \\right)^2 dP^+,\n  $$\n  provides a tractable and interpretable penalty that reflects the degree of over- or under-allocation of individuals relative to available resources. This choice is particularly suitable because it penalizes both *overcrowding* and *underutilization* in a simple quadratic form—exactly the kind of behavior we want to discourage in population-level optimization. While we could use KL-divergence instead of $D_{\\chi^2}$, the resulting penalty would be asymmetric: it more strongly penalizes cases where $\\frac{dP}{dQ}$ is large (i.e., overcrowding) than when $\\frac{dP}{dQ} \\to 0$ (i.e., underutilization). We found $\\chi^2$-divergence to better align with our intuition that deviations from equilibrium should be penalized symmetrically. Of course, this is a design choice, and we will clarify our reasoning in the revision.\n\n\n* For the regularization over the *first marginal* $D_{\\text{KL}}(\\pi_1 \\,\\|\\, P^-)$, we adopt KL-divergence due to the popularity of entropy-regularized optimal transport and its well-known properties. This choice enables the use of scalable solvers inspired by Sinkhorn-style algorithms and ensures the relaxed optimization in Eq. (6) remains strongly convex under mild assumptions.\n\n> Re. Using different divergence measures for the two regularization terms could potentially introduce challenges ... What happens if the two regularization terms in Equation (6) use the same or different divergences? How does this affect choosing $\\lambda_1$ and $\\lambda_2$?\n\n* We believe that using different divergences does not compromise the solution’s stability, and our experiments show no indication of such issues. In fact, the KL term $\\lambda_1 D_{\\text{KL}}(\\pi_1 ,|, P^-)$ acts as a soft constraint, and Proposition 3 shows that as $\\lambda_1 \\to \\infty$, the relaxed solution converges to the exact one, i.e., $\\pi^{(\\lambda_1)} \\to \\pi^*$. Thus, we generally expect this divergence term to provide both computational flexibility and theoretical guarantees without introducing any difficulty. We appreciate the reviewer’s careful attention to this detail.\n\n* Regarding the choice of hyperparameters when the divergences are the same: $\\lambda_1$ is primarily an optimization parameter, controlling the trade-off between enforcing a hard constraint and easing optimization. On the other hand, $\\lambda_2$ reflects the policy maker’s trade-off between individual modification cost and collective competition. A higher $\\lambda_2$ prioritizes population-level balance, reducing overcrowding but allowing higher individual effort, while a lower value favors minimal recourse cost at the risk of systemic congestion. Therefore, regardless of the specific divergence used (for which we recommend $\\chi^2$), $\\lambda_2$ should be tuned based on the desired balance between social externalities and individual burden in the given application.\n\n> Re. I suggest that the authors explicitly state [the equilibrium] assumption in a formal way.\n\nThis is a great suggestion, and we will ensure the equilibrium assumption is clearly highlighted in our updated manuscript. That being said, our framework can be extended to non-equilibrium settings as we have discussed in response to Reviewer J14q.\n\n> Re. Is using a discretized grid a good way to compute $\\chi^2$-divergence?\n\nWe appreciate this insightful question. Discretizing the feature space into a grid is a practical approximation for computing $D_{\\chi^2}(P^{\\mathrm{CE}} \\| P^+)$, especially in low-dimensional settings such as the Moons dataset or when interpretability is important. In our simulations, the number of actionable features is fewer than three, making grid-based estimation both feasible and stable. While grid resolution can influence the computed values, we ensure consistency across all methods by using a fixed grid and validating results through convergence checks. For higher-dimensional or more sensitive scenarios, we agree that adaptive approaches—such as kernel density estimation or nearest-neighbor smoothing—could offer improved robustness and are a promising direction for future refinement."}, {"review_id": "FsAZ1uR4gb", "reviewer": "Reviewer_fXWP", "summary": "This paper introduces a new framework for group-wise counterfactual explanation, named collective counterfactual explanation, that aims to maintain the distribution of perturbed instances and positive instances as unchanged as possible while minimizing the total modification costs. \nThe authors show that this task can be reduced to the optimal transport problem, which can be efficiently solved by existing algorithms such as the Sinkhorn algorithm. \nExperimental results demonstrated that the proposed method could achieve significantly lower competition costs than existing standard methods and balance the trade-off between competition cost and modification cost by tuning the hyperparameter.", "questions": "1. In experiments, the authors state, \"We use only two continuous actionable features to determine the best recourse.\" What is the reason for this setting? Can the proposed method be applied to situations with more than two actionable features?\n1. How were the average running times of the proposed method compared to the other baseline methods?\n1. (typos) In line 28, \"desirable\" and \"undesirable\" may be reversed.", "limitations": "The authors discuss the limitations of their work, but there is no discussion of its potential negative societal impacts.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. This paper is well-written and well-organized. \n1. The authors formulate a new, well-motivated problem of group-wise counterfactual explanation. I find it interesting to solve the formulated problem by leveraging techniques from optimal transport. \n1. The efficacy of the proposed method is well-supported by the experimental results. \n\nWeaknesses\n1. One of my concerns is that counterfactuals are limited to a predefined finite set of positive samples $\\mathcal{X}^+$, in contrast to the existing standard methods that optimize counterfactuals over an infinite set of perturbations. I think it is challenging for the proposed method to meet some of the desiderata of counterfactual explanation, such as sparsity. \n1. Another concern is the computational cost of the proposed method, as the experiments are limited to a setting where only two continuous features can be modified, and the average running time of the proposed method is not reported.", "was_revised": true, "final_justification": "Thank you for taking the time to answer all of my questions and concerns. Since the rebuttal addressed my concerns in part, I have decided to keep my score positive.", "rebuttal": "We thank the reviewer for their insightful review. We're pleased that they found the paper well-written and well-organized, the problem well-motivated, our optimal transport framing interesting, and the proposed method efficient and well-supported by experiment. Below, we address the reviewer’s questions and concerns.\n\n> Re. I think it is challenging for the proposed method to meet some of the desiderata of counterfactual explanation, such as sparsity.\n\nWe thank the reviewer for the clarifying question. To clarify, $\\mathcal{X}^+$ is defined by the classifier and can be a continuous set. In this sense, we are not losing generality compared to standard methods. However, we acknowledge that the available positive samples may sometimes be insufficient to define a reliable support for transport. In such cases, it is possible to augment the data—e.g., by generating synthetic points or using bootstrapping—to ensure a more representative distribution of both negative and positive instances.\n\n\n> Re. Another concern is the computational cost of the proposed method, as the experiments are limited to a setting where only two continuous features can be modified ... Can the proposed method be applied to situations with more than two actionable features?\n\nWe thank the reviewer for raising this important point. In many benchmark settings—including those established in the CARLA library, which is widely adopted in the recourse literature—it is common practice to restrict evaluation to 2–3 continuous actionable features. This choice is also made in prior works such as Pawelczyk et al. (2020, C-CHVAE) and Mothilal et al. (2020, DiCE). **We have followed this convention, though our method can be applied to a larger set of features**. Adhering to this setup also allowed us to visually illustrate how CCE distributes individuals under collective dynamics. \n\nRegarding computational cost, **Proposition 4** establishes that our unbalanced optimal transport (UOT) solver has a **theoretical time complexity of $O(n^2)$**, where $n$ is the number of instances. Thus, the method scales reasonably in practice and does not suffer from the curse of dimensionality in its algorithmic formulation. However, we acknowledge that empirical distributions in high-dimensional spaces are inherently sparse, which can affect the stability of the transport plan. In such cases, optimal literature rich literature offers different solutions: For example, one can impose smoothness assumptions on the underlying data distribution like as *smoothed optimal transport* or leverage techniques such as *sliced optimal transport*, which project high-dimensional data into one-dimensional subspaces and significantly reduce computational complexity while preserving key geometric structure\n\n\n> Re. How were the average running times of the proposed method compared to the other baseline methods?\n\n\nWhile we did not report average running times in the current submission, the proposed method is based on unbalanced optimal transport with **$O(n^2)$** complexity (see Proposition 4), and it typically runs in **seconds** for standard benchmark settings. While we cannot make precise complexity claims about the implementations of baseline methods in the CARLA package---since their internal optimizations and dependencies may vary---we provide empirical runtime comparisons for the Moons dataset to offer practical insight.\n\nThe table below shows the time (in seconds) required to generate **1,000 counterfactuals** on the Moons dataset. These experiments were run on a standard personal laptop a single CPU and 16 GB RAM. \n\n| Method           | Time (s) |\n| ---------------- | -------- |\n| wacher           | 6.38   |\n| growing\\_spheres | 10.82  |\n| cce (ours)       | 15.64  |\n| focus            | 39.66  |\n| cchvae           | 90.18  |\n| cchvae           | 147.56 |\n\nNote that runtime can vary across datasets and methods, particularly for approaches like C-CHVAE that rely on generative models and are more sensitive to the number of features. Nevertheless, **CCE demonstrates competitive runtime performance**, even without GPU acceleration, and scales well for practical applications."}]

Author rebuttal:

Meta-review: This paper proposes a framework for collective counterfactual explanations, which generates group-wise counterfactuals while preserving the distribution of both perturbed and positive instances and minimizing modification costs.
The approach addresses resource competition issues that arise when counterfactuals cluster in similar favorable regions.
The reviewers appreciated the clear motivations, illustrative examples, and a novel characterization of resources.
There was a few comments on the clarity of the paper, this could be easily fixed at the time of publication.
Decision: Accept (poster)