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 more equitable and efficient outcomes. 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.

Counterfactual explanations provide individuals with cost-optimal actions that can alter their labels to desired classes. However, if substantial instances seek state modification, such individual-centric methods can lead to new competitions and unanticipated costs. Furthermore, these recommendations, disregarding the underlying data distribution, may suggest actions that users perceive as outliers. To address these issues, our work proposes a collective approach for formulating counterfactual explanations, with an emphasis on utilizing the current density of the individuals to inform the recommended actions. Our problem naturally casts as an optimal transport problem. Leveraging the extensive literature on optimal transport, we illustrate how this collective method improves upon the desiderata of classical counterfactual explanations. We support our proposal with numerical simulations, illustrating the effectiveness of the proposed approach and its relation to classic methods.

Due to the increasing use of Machine Learning models in high stakes decision making settings, it has become increasingly important to have tools to understand how models arrive at decisions. Assuming a trained Supervised Classification model, explanations can be obtained via counterfactual analysis: a counterfactual explanation of an instance indicates how this instance should be minimally modified so that the perturbed instance is classified in the desired class by the Machine Learning classification model. Most of the Counterfactual Analysis literature focuses on the single-instance single-counterfactual setting, in which the analysis is done for one single instance to provide one single explanation. Taking a stakeholder's perspective, in this paper we introduce the so-called collective counterfactual explanations. By means of novel Mathematical Optimization models, we provide a counterfactual explanation for each instance in a group of interest, so that the total cost of the perturbations is minimized under some linking constraints. Making the process of constructing counterfactuals collective instead of individual enables us to detect the features that are critical to the entire dataset to have the individuals classified in the desired class. Our methodology allows for some instances to be treated individually, performing the collective counterfactual analysis for a fraction of records of the group of interest. This way, outliers are identified and handled appropriately. Under some assumptions on the classifier and the space in which counterfactuals are sought, finding collective counterfactuals is reduced to solving a convex quadratic linearly constrained mixed integer optimization problem, which, for datasets of moderate size, can be solved to optimality using existing solvers. The performance of our approach is illustrated on real-world datasets, demonstrating its usefulness.