Algorithmic Recourse Under Competition

Algorithmic recourse seeks to help individuals reverse unfavorable automated decisions by recommending actionable changes that achieve a desired outcome. As an individual usually has several distinct routes to a favorable decision, and different people can act on different ones, a recourse system should offer multiple realistic alternatives rather than one. Existing approaches formulate recourse as an optimization problem that constructs one or a small set of counterfactuals rather than modeling the underlying space of feasible solutions, and in practice each sacrifices diversity, plausibility, or feasibility to secure the others. We propose Tractable Recourse Distributions, a probabilistic framework that represents the space of feasible alternatives for a given factual instance as a probability distribution over favorable outcomes. For commonly used cost functions based on proximity and the number of feature changes, we show that this distribution admits an exact representation as a probabilistic circuit, obtained by exponentially tilting the circuit; each individual's distribution is therefore available in closed form, without retraining the model. Sampling from these distributions naturally produces diverse and plausible recourses, while the tilting parameters provide explicit control over their proximity and sparsity. Experiments on standard algorithmic recourse benchmark datasets demonstrate that the proposed framework attains diversity, plausibility, and feasibility simultaneously, while retaining sufficient probability mass over feasible counterfactuals for rejection sampling to be practical. A visual study on MNIST illustrates how the tilt strength trades proximity against validity.

Algorithmic recourse promises to help applicants rejected by automated systems by explaining the changes needed to secure acceptance. What incentive do decision-makers, such as banks and employers, have to offer recourse? We study this question in a screening model in which recourse is both productive and selective: completing recourse improves an applicant's value to the decision-maker, but applicants differ in their cost of completion. The optimal policy is a threshold rule: reject applicants with low scores, offer recourse to an intermediate range of scores, and accept applicants with high scores outright. Because the intermediate range spans the cutoff that would separate acceptance from rejection when recourse is not available, some marginal applicants gain a new path to acceptance, while others---who would have been accepted outright---must now clear a costly hurdle.

Algorithmic recourse aims to provide individuals with actionable changes to improve their predicted outcomes in high-stakes classification settings, such as loan and mortgage applications. However, most existing approaches focus only on flipping a model's prediction, without accounting for whether the recommended changes lead to genuine improvement in an individual's true qualifications or merely enable strategic gaming of the classifier. Consequently, deployed recourse policies can induce behavioral responses that degrade predictive accuracy and become ineffective after model retraining. In this work, we formalize this failure mode through a causal performative framework for recourse. We model how recourse actions propagate through a structural causal model, capturing interactions among features as well as their effect on the true label. These causal responses induce a non-convex optimization problem, even under standard convex losses. We characterize conditions under which performatively stable solutions exist and can be efficiently computed via simple iterative dynamics. Our analysis reveals that recourse policies that ignore causal structure can induce large, misaligned behavioral responses, whereas causal recourse leads to stable equilibria that reduce incentives for gaming. Experiments on both semi-synthetic and real credit datasets demonstrate that our approach consistently outperforms standard empirical risk minimization while reducing the need for repeated model retraining to accommodate distribution shifts caused by strategic agent behavior.