Data-driven predictive models are increasingly used to inform decisions that hold important consequences for individuals and society. As a result, decision makers are often obliged, even legally required, to provide explanations about their decisions. In this context, it has been increasingly argued that these explanations should help individuals understand what would have to change for these decisions to be beneficial ones. However, there has been little discussion on the possibility that individuals may use the above counterfactual explanations to invest effort strategically in order to maximize their chances of receiving a beneficial decision. In this paper, our goal is to find policies and counterfactual explanations that are optimal in terms of utility in such a strategic setting. To this end, we first show that, given a pre-defined policy, the problem of finding the optimal set of counterfactual explanations is NP-hard. However, we further show that the corresponding objective is nondecreasing and satisfies submodularity . Therefore, a standard greedy algorithm offers an approximation factor of (1-1/e) at solving the problem. Additionally, we also show that the problem of jointly finding both the optimal policy and set of counterfactual explanations reduces to maximizing a non-monotone submodular function. As a result, we can use a recent randomized algorithm to solve the problem, which offers an approximation factor of 1/e . Finally, we illustrate our theoretical findings by performing experiments on synthetic and real lending data .
As data-driven predictive models are increasingly used to inform decisions , it has been argued that decision makers should provide explanations that help individuals understand what would have to change for these decisions to be beneficial ones. However, there has been little discussion on the possibility that individuals may use the above counterfactual explanations to invest effort strategically and maximize their chances of receiving a beneficial decision. In this paper, our goal is to find policies and counterfactual explanations that are optimal in terms of utility in such a strategic setting. We first show that, given a pre-defined policy, the problem of finding the optimal set of counterfactual explanations is NP-hard. Then, we show that the corresponding objective is nondecreasing and satisfies submodularity and this allows a standard greedy algorithm to enjoy approximation guarantees. In addition, we further show that the problem of jointly finding both the optimal policy and set of counterfactual explanations reduces to maximizing a non-monotone submodular function. As a result, we can use a recent randomized algorithm to solve the problem, which also offers approximation guarantees . Finally, we demonstrate that, by incorporating a matroid constraint into the problem formulation, we can increase the diversity of the optimal set of counterfactual explanations and incentivize individuals across the whole spectrum of the population to self improve. Experiments on synthetic and real lending and credit card data illustrate our theoretical findings and show that the counterfactual explanations and decision policies found by our algorithms achieve higher utility than several competitive baselines .
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