Collective Counterfactual Explanations: Balancing Individual Goals and Collective Dynamics

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.

Re. I think it is challenging for the proposed method to meet some of the desiderata of counterfactual explanation, such as sparsity.

We 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.

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?

We 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.

Regarding 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

Re. How were the average running times of the proposed method compared to the other baseline methods?

While 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.

The 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.

Note 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.

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.

Re. Why use KL-divergence for the regularization term corresponding to $\lambda_1$?

We use different divergence measures for the two regularization terms in Eq. (6) to take advantage of their respective computational and modeling properties:

• 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,

  $$D_{\chi^2}(P^{\text{CE}} \,\|\, P^+) = \int \left( \frac{dP^{\text{CE}}}{dP^+} - 1 \right)^2 dP^+,$$

  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.

• 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.

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 $\lambda_2$?

• 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.

• 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.

Re. I suggest that the authors explicitly state [the equilibrium] assumption in a formal way.

This 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.

Re. Is using a discretized grid a good way to compute $\chi^2$-divergence?

We 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.

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.

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?

Generally, 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.

To 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.

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.

We 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.

As 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.

Table 1: Competition Cost Comparison

Below is a comparison of the competition cost across methods:

Mean values of competition cost (lower is better).

Considering the total cost (modification + competition), our method consistently outperforms the baselines. This is because CCE uniquely and significantly addresses the cost of competition.

Table 2: Total Cost Comparison

Mean values of total cost (lower is better).

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?

While 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.

Re. computation time

Regarding 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.

Last 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.

We thank the reviewer for reviewing our work. Below we discuss reviewer's concerns in detail.

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

With 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.

We 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:

• Reviewer kVUx: "The paper is very complete and well‑written with clear motivations, illustrations, assumptions, etc."
• Reviewer fXWP: "This paper is well-written and well-organized."
• 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."

Re. While the paper presents numerous numerical studies, the code is not provided, which prevents the verification or reproduction of the experimental results.

We 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.

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?

This 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:

• 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.

• 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:

$$D_{\chi^{2}}(P^{\mathrm{CE}} \Vert P_{+}) \quad\longrightarrow\quad D_{\chi^{2}}(P^{\mathrm{CE}} \Vert P_T).$$

The 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.

• 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:

\int_0^1 D_{\chi^{2}}((T_t)_{\\#} P \Vert P_t) \, dt.

This reformulation turns CCE into a dynamic optimal transport problem, solvable using efficient algorithms such as the back-and-forth method.

• 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.

In 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.

Re. citations

We apologize for the inconsistent use of citation commands. We ensure the use of citep and citet is consistent in our update.

\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{figs/compate.pdf}
    \includegraphics[width=0.45\textwidth]{figs/adult_cce_10.pdf}
    \caption{(Left) The blue curve represents the percentage increase in modification cost of CCE relative to standard CE as $\lambda_2$ varies from 0.01 to 0.3. The red curve illustrates the competition cost obtained by $\lambda_2 D_{\chi^2}(T_{\#}\bP_\ms \parallel \bP_\ps)$. Both curves are supported with confidence intervals. As expected, there is a trade-off between modification and competition cost measures. (Right) The result of CCE on the Adult dataset with $\lambda_2 = 0.1$.}.
    \label{fig:tradeoff}
\end{figure}